Split (1)

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Context stringlengths 285 157k	file_name stringlengths 21 79	start int64 14 3.67k	end int64 18 3.69k	theorem stringlengths 25 2.71k	proof stringlengths 5 10.6k
/- Copyright (c) 2022 Jujian Zhang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jujian Zhang -/ import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup import Mathlib.GroupTheory.QuotientGroup #align_import algebra.category.Group.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" /-! # Monomorphisms and epimorphisms in `Group` In this file, we prove monomorphisms in the category of groups are injective homomorphisms and epimorphisms are surjective homomorphisms. -/ noncomputable section open scoped Pointwise universe u v namespace MonoidHom open QuotientGroup variable {A : Type u} {B : Type v} section variable [Group A] [Group B] @[to_additive] theorem ker_eq_bot_of_cancel {f : A →* B} (h : ∀ u v : f.ker →* A, f.comp u = f.comp v → u = v) : f.ker = ⊥ := by simpa using _root_.congr_arg range (h f.ker.subtype 1 (by aesop_cat)) #align monoid_hom.ker_eq_bot_of_cancel MonoidHom.ker_eq_bot_of_cancel #align add_monoid_hom.ker_eq_bot_of_cancel AddMonoidHom.ker_eq_bot_of_cancel end section variable [CommGroup A] [CommGroup B] @[to_additive] theorem range_eq_top_of_cancel {f : A →* B} (h : ∀ u v : B →* B ⧸ f.range, u.comp f = v.comp f → u = v) : f.range = ⊤ := by specialize h 1 (QuotientGroup.mk' _) _ · ext1 x simp only [one_apply, coe_comp, coe_mk', Function.comp_apply] rw [show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _, QuotientGroup.eq, inv_one, one_mul] exact ⟨x, rfl⟩ replace h : (QuotientGroup.mk' f.range).ker = (1 : B →* B ⧸ f.range).ker := by rw [h] rwa [ker_one, QuotientGroup.ker_mk'] at h #align monoid_hom.range_eq_top_of_cancel MonoidHom.range_eq_top_of_cancel #align add_monoid_hom.range_eq_top_of_cancel AddMonoidHom.range_eq_top_of_cancel end end MonoidHom section open CategoryTheory namespace GroupCat set_option linter.uppercaseLean3 false -- Porting note: already have Group G but Lean can't use that @[to_additive] instance (G : GroupCat) : Group G.α := G.str variable {A B : GroupCat.{u}} (f : A ⟶ B) @[to_additive] theorem ker_eq_bot_of_mono [Mono f] : f.ker = ⊥ := MonoidHom.ker_eq_bot_of_cancel fun u _ => (@cancel_mono _ _ _ _ _ f _ (show GroupCat.of f.ker ⟶ A from u) _).1 #align Group.ker_eq_bot_of_mono GroupCat.ker_eq_bot_of_mono #align AddGroup.ker_eq_bot_of_mono AddGroupCat.ker_eq_bot_of_mono @[to_additive] theorem mono_iff_ker_eq_bot : Mono f ↔ f.ker = ⊥ := ⟨fun _ => ker_eq_bot_of_mono f, fun h => ConcreteCategory.mono_of_injective _ <\| (MonoidHom.ker_eq_bot_iff f).1 h⟩ #align Group.mono_iff_ker_eq_bot GroupCat.mono_iff_ker_eq_bot #align AddGroup.mono_iff_ker_eq_bot AddGroupCat.mono_iff_ker_eq_bot @[to_additive] theorem mono_iff_injective : Mono f ↔ Function.Injective f := Iff.trans (mono_iff_ker_eq_bot f) <\| MonoidHom.ker_eq_bot_iff f #align Group.mono_iff_injective GroupCat.mono_iff_injective #align AddGroup.mono_iff_injective AddGroupCat.mono_iff_injective namespace SurjectiveOfEpiAuxs set_option quotPrecheck false in local notation "X" => Set.range (· • (f.range : Set B) : B → Set B) /-- Define `X'` to be the set of all left cosets with an extra point at "infinity". -/ inductive XWithInfinity \| fromCoset : Set.range (· • (f.range : Set B) : B → Set B) → XWithInfinity \| infinity : XWithInfinity #align Group.surjective_of_epi_auxs.X_with_infinity GroupCat.SurjectiveOfEpiAuxs.XWithInfinity open XWithInfinity Equiv.Perm local notation "X'" => XWithInfinity f local notation "∞" => XWithInfinity.infinity local notation "SX'" => Equiv.Perm X' instance : SMul B X' where smul b x := match x with \| fromCoset y => fromCoset ⟨b • y, by rw [← y.2.choose_spec, leftCoset_assoc] -- Porting note: should we make `Bundled.α` reducible? let b' : B := y.2.choose use b * b'⟩ \| ∞ => ∞ theorem mul_smul (b b' : B) (x : X') : (b * b') • x = b • b' • x := match x with \| fromCoset y => by change fromCoset _ = fromCoset _ simp only [leftCoset_assoc] \| ∞ => rfl #align Group.surjective_of_epi_auxs.mul_smul GroupCat.SurjectiveOfEpiAuxs.mul_smul theorem one_smul (x : X') : (1 : B) • x = x := match x with \| fromCoset y => by change fromCoset _ = fromCoset _ simp only [one_leftCoset, Subtype.ext_iff_val] \| ∞ => rfl #align Group.surjective_of_epi_auxs.one_smul GroupCat.SurjectiveOfEpiAuxs.one_smul theorem fromCoset_eq_of_mem_range {b : B} (hb : b ∈ f.range) : fromCoset ⟨b • ↑f.range, b, rfl⟩ = fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by congr let b : B.α := b change b • (f.range : Set B) = f.range nth_rw 2 [show (f.range : Set B.α) = (1 : B) • f.range from (one_leftCoset _).symm] rw [leftCoset_eq_iff, mul_one] exact Subgroup.inv_mem _ hb #align Group.surjective_of_epi_auxs.from_coset_eq_of_mem_range GroupCat.SurjectiveOfEpiAuxs.fromCoset_eq_of_mem_range example (G : Type) [Group G] (S : Subgroup G) : Set G := S theorem fromCoset_ne_of_nin_range {b : B} (hb : b ∉ f.range) : fromCoset ⟨b • ↑f.range, b, rfl⟩ ≠ fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by intro r simp only [fromCoset.injEq, Subtype.mk.injEq] at r -- Porting note: annoying dance between types CoeSort.coe B, B.α, and B let b' : B.α := b change b' • (f.range : Set B) = f.range at r nth_rw 2 [show (f.range : Set B.α) = (1 : B) • f.range from (one_leftCoset _).symm] at r rw [leftCoset_eq_iff, mul_one] at r exact hb (inv_inv b ▸ Subgroup.inv_mem _ r) #align Group.surjective_of_epi_auxs.from_coset_ne_of_nin_range GroupCat.SurjectiveOfEpiAuxs.fromCoset_ne_of_nin_range instance : DecidableEq X' := Classical.decEq _ /-- Let `τ` be the permutation on `X'` exchanging `f.range` and the point at infinity. -/ noncomputable def tau : SX' := Equiv.swap (fromCoset ⟨↑f.range, ⟨1, one_leftCoset _⟩⟩) ∞ #align Group.surjective_of_epi_auxs.tau GroupCat.SurjectiveOfEpiAuxs.tau local notation "τ" => tau f theorem τ_apply_infinity : τ ∞ = fromCoset ⟨f.range, 1, one_leftCoset _⟩ := Equiv.swap_apply_right _ _ #align Group.surjective_of_epi_auxs.τ_apply_infinity GroupCat.SurjectiveOfEpiAuxs.τ_apply_infinity theorem τ_apply_fromCoset : τ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = ∞ := Equiv.swap_apply_left _ _ #align Group.surjective_of_epi_auxs.τ_apply_fromCoset GroupCat.SurjectiveOfEpiAuxs.τ_apply_fromCoset theorem τ_apply_fromCoset' (x : B) (hx : x ∈ f.range) : τ (fromCoset ⟨x • ↑f.range, ⟨x, rfl⟩⟩) = ∞ := (fromCoset_eq_of_mem_range _ hx).symm ▸ τ_apply_fromCoset _ #align Group.surjective_of_epi_auxs.τ_apply_fromCoset' GroupCat.SurjectiveOfEpiAuxs.τ_apply_fromCoset' theorem τ_symm_apply_fromCoset : Equiv.symm τ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = ∞ := by rw [tau, Equiv.symm_swap, Equiv.swap_apply_left] #align Group.surjective_of_epi_auxs.τ_symm_apply_fromCoset GroupCat.SurjectiveOfEpiAuxs.τ_symm_apply_fromCoset theorem τ_symm_apply_infinity : Equiv.symm τ ∞ = fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by rw [tau, Equiv.symm_swap, Equiv.swap_apply_right] #align Group.surjective_of_epi_auxs.τ_symm_apply_infinity GroupCat.SurjectiveOfEpiAuxs.τ_symm_apply_infinity /-- Let `g : B ⟶ S(X')` be defined as such that, for any `β : B`, `g(β)` is the function sending point at infinity to point at infinity and sending coset `y` to `β • y`. -/ def g : B →* SX' where toFun β := { toFun := fun x => β • x invFun := fun x => β⁻¹ • x left_inv := fun x => by dsimp only rw [← mul_smul, mul_left_inv, one_smul] right_inv := fun x => by dsimp only rw [← mul_smul, mul_right_inv, one_smul] } map_one' := by ext simp [one_smul] map_mul' b1 b2 := by ext simp [mul_smul] #align Group.surjective_of_epi_auxs.G GroupCat.SurjectiveOfEpiAuxs.g local notation "g" => g f /-- Define `h : B ⟶ S(X')` to be `τ g τ⁻¹` -/ def h : B →* SX' where -- Porting note: mathport removed () from (τ) which are needed toFun β := ((τ).symm.trans (g β)).trans τ map_one' := by ext simp map_mul' b1 b2 := by ext simp #align Group.surjective_of_epi_auxs.H GroupCat.SurjectiveOfEpiAuxs.h local notation "h" => h f /-! The strategy is the following: assuming `epi f` * prove that `f.range = {x \| h x = g x}`; * thus `f ≫ h = f ≫ g` so that `h = g`; * but if `f` is not surjective, then some `x ∉ f.range`, then `h x ≠ g x` at the coset `f.range`. -/ theorem g_apply_fromCoset (x : B) (y : Set.range (· • (f.range : Set B) : B → Set B)) : g x (fromCoset y) = fromCoset ⟨x • ↑y, by obtain ⟨z, hz⟩ := y.2; exact ⟨x * z, by simp [← hz, smul_smul]⟩⟩ := rfl #align Group.surjective_of_epi_auxs.g_apply_fromCoset GroupCat.SurjectiveOfEpiAuxs.g_apply_fromCoset theorem g_apply_infinity (x : B) : (g x) ∞ = ∞ := rfl #align Group.surjective_of_epi_auxs.g_apply_infinity GroupCat.SurjectiveOfEpiAuxs.g_apply_infinity theorem h_apply_infinity (x : B) (hx : x ∈ f.range) : (h x) ∞ = ∞ := by change ((τ).symm.trans (g x)).trans τ _ = _ simp only [MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply] rw [τ_symm_apply_infinity, g_apply_fromCoset] simpa only using τ_apply_fromCoset' f x hx #align Group.surjective_of_epi_auxs.h_apply_infinity GroupCat.SurjectiveOfEpiAuxs.h_apply_infinity theorem h_apply_fromCoset (x : B) : (h x) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by change ((τ).symm.trans (g x)).trans τ _ = _ simp [-MonoidHom.coe_range, τ_symm_apply_fromCoset, g_apply_infinity, τ_apply_infinity] #align Group.surjective_of_epi_auxs.h_apply_fromCoset GroupCat.SurjectiveOfEpiAuxs.h_apply_fromCoset theorem h_apply_fromCoset' (x : B) (b : B) (hb : b ∈ f.range) : h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨b • ↑f.range, b, rfl⟩ := (fromCoset_eq_of_mem_range _ hb).symm ▸ h_apply_fromCoset f x #align Group.surjective_of_epi_auxs.h_apply_fromCoset' GroupCat.SurjectiveOfEpiAuxs.h_apply_fromCoset' theorem h_apply_fromCoset_nin_range (x : B) (hx : x ∈ f.range) (b : B) (hb : b ∉ f.range) : h x (fromCoset ⟨b • f.range, b, rfl⟩) = fromCoset ⟨(x * b) • ↑f.range, x * b, rfl⟩ := by change ((τ).symm.trans (g x)).trans τ _ = _ simp only [tau, MonoidHom.coe_mk, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply] rw [Equiv.symm_swap, @Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.range, 1, one_leftCoset _⟩) ∞ (fromCoset ⟨b • ↑f.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)] simp only [g_apply_fromCoset, leftCoset_assoc] refine Equiv.swap_apply_of_ne_of_ne (fromCoset_ne_of_nin_range _ fun r => hb ?_) (by simp) convert Subgroup.mul_mem _ (Subgroup.inv_mem _ hx) r rw [← mul_assoc, mul_left_inv, one_mul] #align Group.surjective_of_epi_auxs.h_apply_fromCoset_nin_range GroupCat.SurjectiveOfEpiAuxs.h_apply_fromCoset_nin_range theorem agree : f.range = { x \| h x = g x } := by refine Set.ext fun b => ⟨?_, fun hb : h b = g b => by_contradiction fun r => ?_⟩ · rintro ⟨a, rfl⟩ change h (f a) = g (f a) ext ⟨⟨_, ⟨y, rfl⟩⟩⟩ · rw [g_apply_fromCoset] by_cases m : y ∈ f.range · rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m] change fromCoset _ = fromCoset ⟨f a • (y • _), _⟩ simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m), smul_smul] · rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m] simp only [leftCoset_assoc] · rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩] · have eq1 : (h b) (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = fromCoset ⟨f.range, 1, one_leftCoset _⟩ := by change ((τ).symm.trans (g b)).trans τ _ = _ dsimp [tau] simp [g_apply_infinity f] have eq2 : g b (fromCoset ⟨f.range, 1, one_leftCoset _⟩) = fromCoset ⟨b • ↑f.range, b, rfl⟩ := rfl exact (fromCoset_ne_of_nin_range _ r).symm (by rw [← eq1, ← eq2, DFunLike.congr_fun hb]) #align Group.surjective_of_epi_auxs.agree GroupCat.SurjectiveOfEpiAuxs.agree theorem comp_eq : (f ≫ show B ⟶ GroupCat.of SX' from g) = f ≫ show B ⟶ GroupCat.of SX' from h := by ext a change g (f a) = h (f a) have : f a ∈ { b \| h b = g b } := by rw [← agree] use a rw [this] #align Group.surjective_of_epi_auxs.comp_eq GroupCat.SurjectiveOfEpiAuxs.comp_eq theorem g_ne_h (x : B) (hx : x ∉ f.range) : g ≠ h := by intro r replace r := DFunLike.congr_fun (DFunLike.congr_fun r x) (fromCoset ⟨f.range, ⟨1, one_leftCoset _⟩⟩) change _ = ((τ).symm.trans (g x)).trans τ _ at r rw [g_apply_fromCoset, MonoidHom.coe_mk] at r simp only [MonoidHom.coe_range, Subtype.coe_mk, Equiv.symm_swap, Equiv.toFun_as_coe, Equiv.coe_trans, Function.comp_apply] at r erw [Equiv.swap_apply_left, g_apply_infinity, Equiv.swap_apply_right] at r exact fromCoset_ne_of_nin_range _ hx r #align Group.surjective_of_epi_auxs.g_ne_h GroupCat.SurjectiveOfEpiAuxs.g_ne_h end SurjectiveOfEpiAuxs theorem surjective_of_epi [Epi f] : Function.Surjective f := by by_contra r dsimp [Function.Surjective] at r push_neg at r rcases r with ⟨b, hb⟩ exact SurjectiveOfEpiAuxs.g_ne_h f b (fun ⟨c, hc⟩ => hb _ hc) ((cancel_epi f).1 (SurjectiveOfEpiAuxs.comp_eq f)) #align Group.surjective_of_epi GroupCat.surjective_of_epi theorem epi_iff_surjective : Epi f ↔ Function.Surjective f := ⟨fun _ => surjective_of_epi f, ConcreteCategory.epi_of_surjective f⟩ #align Group.epi_iff_surjective GroupCat.epi_iff_surjective theorem epi_iff_range_eq_top : Epi f ↔ f.range = ⊤ := Iff.trans (epi_iff_surjective _) (Subgroup.eq_top_iff' f.range).symm #align Group.epi_iff_range_eq_top GroupCat.epi_iff_range_eq_top end GroupCat namespace AddGroupCat set_option linter.uppercaseLean3 false variable {A B : AddGroupCat.{u}} (f : A ⟶ B)	Mathlib/Algebra/Category/GroupCat/EpiMono.lean	360	365	theorem epi_iff_surjective : Epi f ↔ Function.Surjective f := by	have i1 : Epi f ↔ Epi (groupAddGroupEquivalence.inverse.map f) := by refine ⟨?_, groupAddGroupEquivalence.inverse.epi_of_epi_map⟩ intro e' apply groupAddGroupEquivalence.inverse.map_epi rwa [GroupCat.epi_iff_surjective] at i1
/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Floris van Doorn -/ import Mathlib.Analysis.Calculus.ContDiff.Defs import Mathlib.Analysis.Calculus.FDeriv.Add import Mathlib.Analysis.Calculus.FDeriv.Mul import Mathlib.Analysis.Calculus.Deriv.Inverse #align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # Higher differentiability of usual operations We prove that the usual operations (addition, multiplication, difference, composition, and so on) preserve `C^n` functions. We also expand the API around `C^n` functions. ## Main results * `ContDiff.comp` states that the composition of two `C^n` functions is `C^n`. Similar results are given for `C^n` functions on domains. ## Notations We use the notation `E [×n]→L[𝕜] F` for the space of continuous multilinear maps on `E^n` with values in `F`. This is the space in which the `n`-th derivative of a function from `E` to `F` lives. In this file, we denote `⊤ : ℕ∞` with `∞`. ## Tags derivative, differentiability, higher derivative, `C^n`, multilinear, Taylor series, formal series -/ noncomputable section open scoped Classical NNReal Nat local notation "∞" => (⊤ : ℕ∞) universe u v w uD uE uF uG attribute [local instance 1001] NormedAddCommGroup.toAddCommGroup NormedSpace.toModule' AddCommGroup.toAddCommMonoid open Set Fin Filter Function open scoped Topology variable {𝕜 : Type} [NontriviallyNormedField 𝕜] {D : Type uD} [NormedAddCommGroup D] [NormedSpace 𝕜 D] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {X : Type} [NormedAddCommGroup X] [NormedSpace 𝕜 X] {s s₁ t u : Set E} {f f₁ : E → F} {g : F → G} {x x₀ : E} {c : F} {b : E × F → G} {m n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} /-! ### Constants -/ @[simp] theorem iteratedFDerivWithin_zero_fun (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} : iteratedFDerivWithin 𝕜 i (fun _ : E ↦ (0 : F)) s x = 0 := by induction i generalizing x with \| zero => ext; simp \| succ i IH => ext m rw [iteratedFDerivWithin_succ_apply_left, fderivWithin_congr (fun _ ↦ IH) (IH hx)] rw [fderivWithin_const_apply _ (hs x hx)] rfl @[simp] theorem iteratedFDeriv_zero_fun {n : ℕ} : (iteratedFDeriv 𝕜 n fun _ : E ↦ (0 : F)) = 0 := funext fun x ↦ by simpa [← iteratedFDerivWithin_univ] using iteratedFDerivWithin_zero_fun uniqueDiffOn_univ (mem_univ x) #align iterated_fderiv_zero_fun iteratedFDeriv_zero_fun theorem contDiff_zero_fun : ContDiff 𝕜 n fun _ : E => (0 : F) := contDiff_of_differentiable_iteratedFDeriv fun m _ => by rw [iteratedFDeriv_zero_fun] exact differentiable_const (0 : E[×m]→L[𝕜] F) #align cont_diff_zero_fun contDiff_zero_fun /-- Constants are `C^∞`. -/ theorem contDiff_const {c : F} : ContDiff 𝕜 n fun _ : E => c := by suffices h : ContDiff 𝕜 ∞ fun _ : E => c from h.of_le le_top rw [contDiff_top_iff_fderiv] refine ⟨differentiable_const c, ?_⟩ rw [fderiv_const] exact contDiff_zero_fun #align cont_diff_const contDiff_const theorem contDiffOn_const {c : F} {s : Set E} : ContDiffOn 𝕜 n (fun _ : E => c) s := contDiff_const.contDiffOn #align cont_diff_on_const contDiffOn_const theorem contDiffAt_const {c : F} : ContDiffAt 𝕜 n (fun _ : E => c) x := contDiff_const.contDiffAt #align cont_diff_at_const contDiffAt_const theorem contDiffWithinAt_const {c : F} : ContDiffWithinAt 𝕜 n (fun _ : E => c) s x := contDiffAt_const.contDiffWithinAt #align cont_diff_within_at_const contDiffWithinAt_const @[nontriviality] theorem contDiff_of_subsingleton [Subsingleton F] : ContDiff 𝕜 n f := by rw [Subsingleton.elim f fun _ => 0]; exact contDiff_const #align cont_diff_of_subsingleton contDiff_of_subsingleton @[nontriviality] theorem contDiffAt_of_subsingleton [Subsingleton F] : ContDiffAt 𝕜 n f x := by rw [Subsingleton.elim f fun _ => 0]; exact contDiffAt_const #align cont_diff_at_of_subsingleton contDiffAt_of_subsingleton @[nontriviality] theorem contDiffWithinAt_of_subsingleton [Subsingleton F] : ContDiffWithinAt 𝕜 n f s x := by rw [Subsingleton.elim f fun _ => 0]; exact contDiffWithinAt_const #align cont_diff_within_at_of_subsingleton contDiffWithinAt_of_subsingleton @[nontriviality] theorem contDiffOn_of_subsingleton [Subsingleton F] : ContDiffOn 𝕜 n f s := by rw [Subsingleton.elim f fun _ => 0]; exact contDiffOn_const #align cont_diff_on_of_subsingleton contDiffOn_of_subsingleton theorem iteratedFDerivWithin_succ_const (n : ℕ) (c : F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) : iteratedFDerivWithin 𝕜 (n + 1) (fun _ : E ↦ c) s x = 0 := by ext m rw [iteratedFDerivWithin_succ_apply_right hs hx] rw [iteratedFDerivWithin_congr (fun y hy ↦ fderivWithin_const_apply c (hs y hy)) hx] rw [iteratedFDerivWithin_zero_fun hs hx] simp [ContinuousMultilinearMap.zero_apply (R := 𝕜)] theorem iteratedFDeriv_succ_const (n : ℕ) (c : F) : (iteratedFDeriv 𝕜 (n + 1) fun _ : E ↦ c) = 0 := funext fun x ↦ by simpa [← iteratedFDerivWithin_univ] using iteratedFDerivWithin_succ_const n c uniqueDiffOn_univ (mem_univ x) #align iterated_fderiv_succ_const iteratedFDeriv_succ_const theorem iteratedFDerivWithin_const_of_ne {n : ℕ} (hn : n ≠ 0) (c : F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) : iteratedFDerivWithin 𝕜 n (fun _ : E ↦ c) s x = 0 := by cases n with \| zero => contradiction \| succ n => exact iteratedFDerivWithin_succ_const n c hs hx theorem iteratedFDeriv_const_of_ne {n : ℕ} (hn : n ≠ 0) (c : F) : (iteratedFDeriv 𝕜 n fun _ : E ↦ c) = 0 := funext fun x ↦ by simpa [← iteratedFDerivWithin_univ] using iteratedFDerivWithin_const_of_ne hn c uniqueDiffOn_univ (mem_univ x) #align iterated_fderiv_const_of_ne iteratedFDeriv_const_of_ne /-! ### Smoothness of linear functions -/ /-- Unbundled bounded linear functions are `C^∞`. -/ theorem IsBoundedLinearMap.contDiff (hf : IsBoundedLinearMap 𝕜 f) : ContDiff 𝕜 n f := by suffices h : ContDiff 𝕜 ∞ f from h.of_le le_top rw [contDiff_top_iff_fderiv] refine ⟨hf.differentiable, ?_⟩ simp_rw [hf.fderiv] exact contDiff_const #align is_bounded_linear_map.cont_diff IsBoundedLinearMap.contDiff theorem ContinuousLinearMap.contDiff (f : E →L[𝕜] F) : ContDiff 𝕜 n f := f.isBoundedLinearMap.contDiff #align continuous_linear_map.cont_diff ContinuousLinearMap.contDiff theorem ContinuousLinearEquiv.contDiff (f : E ≃L[𝕜] F) : ContDiff 𝕜 n f := (f : E →L[𝕜] F).contDiff #align continuous_linear_equiv.cont_diff ContinuousLinearEquiv.contDiff theorem LinearIsometry.contDiff (f : E →ₗᵢ[𝕜] F) : ContDiff 𝕜 n f := f.toContinuousLinearMap.contDiff #align linear_isometry.cont_diff LinearIsometry.contDiff theorem LinearIsometryEquiv.contDiff (f : E ≃ₗᵢ[𝕜] F) : ContDiff 𝕜 n f := (f : E →L[𝕜] F).contDiff #align linear_isometry_equiv.cont_diff LinearIsometryEquiv.contDiff /-- The identity is `C^∞`. -/ theorem contDiff_id : ContDiff 𝕜 n (id : E → E) := IsBoundedLinearMap.id.contDiff #align cont_diff_id contDiff_id theorem contDiffWithinAt_id {s x} : ContDiffWithinAt 𝕜 n (id : E → E) s x := contDiff_id.contDiffWithinAt #align cont_diff_within_at_id contDiffWithinAt_id theorem contDiffAt_id {x} : ContDiffAt 𝕜 n (id : E → E) x := contDiff_id.contDiffAt #align cont_diff_at_id contDiffAt_id theorem contDiffOn_id {s} : ContDiffOn 𝕜 n (id : E → E) s := contDiff_id.contDiffOn #align cont_diff_on_id contDiffOn_id /-- Bilinear functions are `C^∞`. -/ theorem IsBoundedBilinearMap.contDiff (hb : IsBoundedBilinearMap 𝕜 b) : ContDiff 𝕜 n b := by suffices h : ContDiff 𝕜 ∞ b from h.of_le le_top rw [contDiff_top_iff_fderiv] refine ⟨hb.differentiable, ?_⟩ simp only [hb.fderiv] exact hb.isBoundedLinearMap_deriv.contDiff #align is_bounded_bilinear_map.cont_diff IsBoundedBilinearMap.contDiff /-- If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `g ∘ f` admits a Taylor series whose `k`-th term is given by `g ∘ (p k)`. -/ theorem HasFTaylorSeriesUpToOn.continuousLinearMap_comp (g : F →L[𝕜] G) (hf : HasFTaylorSeriesUpToOn n f p s) : HasFTaylorSeriesUpToOn n (g ∘ f) (fun x k => g.compContinuousMultilinearMap (p x k)) s where zero_eq x hx := congr_arg g (hf.zero_eq x hx) fderivWithin m hm x hx := (ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 (fun _ : Fin m => E) F G g).hasFDerivAt.comp_hasFDerivWithinAt x (hf.fderivWithin m hm x hx) cont m hm := (ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 (fun _ : Fin m => E) F G g).continuous.comp_continuousOn (hf.cont m hm) #align has_ftaylor_series_up_to_on.continuous_linear_map_comp HasFTaylorSeriesUpToOn.continuousLinearMap_comp /-- Composition by continuous linear maps on the left preserves `C^n` functions in a domain at a point. -/ theorem ContDiffWithinAt.continuousLinearMap_comp (g : F →L[𝕜] G) (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (g ∘ f) s x := fun m hm ↦ by rcases hf m hm with ⟨u, hu, p, hp⟩ exact ⟨u, hu, _, hp.continuousLinearMap_comp g⟩ #align cont_diff_within_at.continuous_linear_map_comp ContDiffWithinAt.continuousLinearMap_comp /-- Composition by continuous linear maps on the left preserves `C^n` functions in a domain at a point. -/ theorem ContDiffAt.continuousLinearMap_comp (g : F →L[𝕜] G) (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (g ∘ f) x := ContDiffWithinAt.continuousLinearMap_comp g hf #align cont_diff_at.continuous_linear_map_comp ContDiffAt.continuousLinearMap_comp /-- Composition by continuous linear maps on the left preserves `C^n` functions on domains. -/ theorem ContDiffOn.continuousLinearMap_comp (g : F →L[𝕜] G) (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (g ∘ f) s := fun x hx => (hf x hx).continuousLinearMap_comp g #align cont_diff_on.continuous_linear_map_comp ContDiffOn.continuousLinearMap_comp /-- Composition by continuous linear maps on the left preserves `C^n` functions. -/ theorem ContDiff.continuousLinearMap_comp {f : E → F} (g : F →L[𝕜] G) (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => g (f x) := contDiffOn_univ.1 <\| ContDiffOn.continuousLinearMap_comp _ (contDiffOn_univ.2 hf) #align cont_diff.continuous_linear_map_comp ContDiff.continuousLinearMap_comp /-- The iterated derivative within a set of the composition with a linear map on the left is obtained by applying the linear map to the iterated derivative. -/ theorem ContinuousLinearMap.iteratedFDerivWithin_comp_left {f : E → F} (g : F →L[𝕜] G) (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) : iteratedFDerivWithin 𝕜 i (g ∘ f) s x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) := (((hf.ftaylorSeriesWithin hs).continuousLinearMap_comp g).eq_iteratedFDerivWithin_of_uniqueDiffOn hi hs hx).symm #align continuous_linear_map.iterated_fderiv_within_comp_left ContinuousLinearMap.iteratedFDerivWithin_comp_left /-- The iterated derivative of the composition with a linear map on the left is obtained by applying the linear map to the iterated derivative. -/ theorem ContinuousLinearMap.iteratedFDeriv_comp_left {f : E → F} (g : F →L[𝕜] G) (hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) : iteratedFDeriv 𝕜 i (g ∘ f) x = g.compContinuousMultilinearMap (iteratedFDeriv 𝕜 i f x) := by simp only [← iteratedFDerivWithin_univ] exact g.iteratedFDerivWithin_comp_left hf.contDiffOn uniqueDiffOn_univ (mem_univ x) hi #align continuous_linear_map.iterated_fderiv_comp_left ContinuousLinearMap.iteratedFDeriv_comp_left /-- The iterated derivative within a set of the composition with a linear equiv on the left is obtained by applying the linear equiv to the iterated derivative. This is true without differentiability assumptions. -/ theorem ContinuousLinearEquiv.iteratedFDerivWithin_comp_left (g : F ≃L[𝕜] G) (f : E → F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ) : iteratedFDerivWithin 𝕜 i (g ∘ f) s x = (g : F →L[𝕜] G).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) := by induction' i with i IH generalizing x · ext1 m simp only [Nat.zero_eq, iteratedFDerivWithin_zero_apply, comp_apply, ContinuousLinearMap.compContinuousMultilinearMap_coe, coe_coe] · ext1 m rw [iteratedFDerivWithin_succ_apply_left] have Z : fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (g ∘ f) s) s x = fderivWithin 𝕜 (g.compContinuousMultilinearMapL (fun _ : Fin i => E) ∘ iteratedFDerivWithin 𝕜 i f s) s x := fderivWithin_congr' (@IH) hx simp_rw [Z] rw [(g.compContinuousMultilinearMapL fun _ : Fin i => E).comp_fderivWithin (hs x hx)] simp only [ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe, comp_apply, ContinuousLinearEquiv.compContinuousMultilinearMapL_apply, ContinuousLinearMap.compContinuousMultilinearMap_coe, EmbeddingLike.apply_eq_iff_eq] rw [iteratedFDerivWithin_succ_apply_left] #align continuous_linear_equiv.iterated_fderiv_within_comp_left ContinuousLinearEquiv.iteratedFDerivWithin_comp_left /-- Composition with a linear isometry on the left preserves the norm of the iterated derivative within a set. -/ theorem LinearIsometry.norm_iteratedFDerivWithin_comp_left {f : E → F} (g : F →ₗᵢ[𝕜] G) (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) : ‖iteratedFDerivWithin 𝕜 i (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖ := by have : iteratedFDerivWithin 𝕜 i (g ∘ f) s x = g.toContinuousLinearMap.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) := g.toContinuousLinearMap.iteratedFDerivWithin_comp_left hf hs hx hi rw [this] apply LinearIsometry.norm_compContinuousMultilinearMap #align linear_isometry.norm_iterated_fderiv_within_comp_left LinearIsometry.norm_iteratedFDerivWithin_comp_left /-- Composition with a linear isometry on the left preserves the norm of the iterated derivative. -/ theorem LinearIsometry.norm_iteratedFDeriv_comp_left {f : E → F} (g : F →ₗᵢ[𝕜] G) (hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) : ‖iteratedFDeriv 𝕜 i (g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i f x‖ := by simp only [← iteratedFDerivWithin_univ] exact g.norm_iteratedFDerivWithin_comp_left hf.contDiffOn uniqueDiffOn_univ (mem_univ x) hi #align linear_isometry.norm_iterated_fderiv_comp_left LinearIsometry.norm_iteratedFDeriv_comp_left /-- Composition with a linear isometry equiv on the left preserves the norm of the iterated derivative within a set. -/ theorem LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left (g : F ≃ₗᵢ[𝕜] G) (f : E → F) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (i : ℕ) : ‖iteratedFDerivWithin 𝕜 i (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖ := by have : iteratedFDerivWithin 𝕜 i (g ∘ f) s x = (g : F →L[𝕜] G).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) := g.toContinuousLinearEquiv.iteratedFDerivWithin_comp_left f hs hx i rw [this] apply LinearIsometry.norm_compContinuousMultilinearMap g.toLinearIsometry #align linear_isometry_equiv.norm_iterated_fderiv_within_comp_left LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left /-- Composition with a linear isometry equiv on the left preserves the norm of the iterated derivative. -/ theorem LinearIsometryEquiv.norm_iteratedFDeriv_comp_left (g : F ≃ₗᵢ[𝕜] G) (f : E → F) (x : E) (i : ℕ) : ‖iteratedFDeriv 𝕜 i (g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i f x‖ := by rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ] apply g.norm_iteratedFDerivWithin_comp_left f uniqueDiffOn_univ (mem_univ x) i #align linear_isometry_equiv.norm_iterated_fderiv_comp_left LinearIsometryEquiv.norm_iteratedFDeriv_comp_left /-- Composition by continuous linear equivs on the left respects higher differentiability at a point in a domain. -/ theorem ContinuousLinearEquiv.comp_contDiffWithinAt_iff (e : F ≃L[𝕜] G) : ContDiffWithinAt 𝕜 n (e ∘ f) s x ↔ ContDiffWithinAt 𝕜 n f s x := ⟨fun H => by simpa only [(· ∘ ·), e.symm.coe_coe, e.symm_apply_apply] using H.continuousLinearMap_comp (e.symm : G →L[𝕜] F), fun H => H.continuousLinearMap_comp (e : F →L[𝕜] G)⟩ #align continuous_linear_equiv.comp_cont_diff_within_at_iff ContinuousLinearEquiv.comp_contDiffWithinAt_iff /-- Composition by continuous linear equivs on the left respects higher differentiability at a point. -/ theorem ContinuousLinearEquiv.comp_contDiffAt_iff (e : F ≃L[𝕜] G) : ContDiffAt 𝕜 n (e ∘ f) x ↔ ContDiffAt 𝕜 n f x := by simp only [← contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff] #align continuous_linear_equiv.comp_cont_diff_at_iff ContinuousLinearEquiv.comp_contDiffAt_iff /-- Composition by continuous linear equivs on the left respects higher differentiability on domains. -/ theorem ContinuousLinearEquiv.comp_contDiffOn_iff (e : F ≃L[𝕜] G) : ContDiffOn 𝕜 n (e ∘ f) s ↔ ContDiffOn 𝕜 n f s := by simp [ContDiffOn, e.comp_contDiffWithinAt_iff] #align continuous_linear_equiv.comp_cont_diff_on_iff ContinuousLinearEquiv.comp_contDiffOn_iff /-- Composition by continuous linear equivs on the left respects higher differentiability. -/ theorem ContinuousLinearEquiv.comp_contDiff_iff (e : F ≃L[𝕜] G) : ContDiff 𝕜 n (e ∘ f) ↔ ContDiff 𝕜 n f := by simp only [← contDiffOn_univ, e.comp_contDiffOn_iff] #align continuous_linear_equiv.comp_cont_diff_iff ContinuousLinearEquiv.comp_contDiff_iff /-- If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `f ∘ g` admits a Taylor series in `g ⁻¹' s`, whose `k`-th term is given by `p k (g v₁, ..., g vₖ)` . -/ theorem HasFTaylorSeriesUpToOn.compContinuousLinearMap (hf : HasFTaylorSeriesUpToOn n f p s) (g : G →L[𝕜] E) : HasFTaylorSeriesUpToOn n (f ∘ g) (fun x k => (p (g x) k).compContinuousLinearMap fun _ => g) (g ⁻¹' s) := by let A : ∀ m : ℕ, (E[×m]→L[𝕜] F) → G[×m]→L[𝕜] F := fun m h => h.compContinuousLinearMap fun _ => g have hA : ∀ m, IsBoundedLinearMap 𝕜 (A m) := fun m => isBoundedLinearMap_continuousMultilinearMap_comp_linear g constructor · intro x hx simp only [(hf.zero_eq (g x) hx).symm, Function.comp_apply] change (p (g x) 0 fun _ : Fin 0 => g 0) = p (g x) 0 0 rw [ContinuousLinearMap.map_zero] rfl · intro m hm x hx convert (hA m).hasFDerivAt.comp_hasFDerivWithinAt x ((hf.fderivWithin m hm (g x) hx).comp x g.hasFDerivWithinAt (Subset.refl _)) ext y v change p (g x) (Nat.succ m) (g ∘ cons y v) = p (g x) m.succ (cons (g y) (g ∘ v)) rw [comp_cons] · intro m hm exact (hA m).continuous.comp_continuousOn <\| (hf.cont m hm).comp g.continuous.continuousOn <\| Subset.refl _ #align has_ftaylor_series_up_to_on.comp_continuous_linear_map HasFTaylorSeriesUpToOn.compContinuousLinearMap /-- Composition by continuous linear maps on the right preserves `C^n` functions at a point on a domain. -/ theorem ContDiffWithinAt.comp_continuousLinearMap {x : G} (g : G →L[𝕜] E) (hf : ContDiffWithinAt 𝕜 n f s (g x)) : ContDiffWithinAt 𝕜 n (f ∘ g) (g ⁻¹' s) x := by intro m hm rcases hf m hm with ⟨u, hu, p, hp⟩ refine ⟨g ⁻¹' u, ?_, _, hp.compContinuousLinearMap g⟩ refine g.continuous.continuousWithinAt.tendsto_nhdsWithin ?_ hu exact (mapsTo_singleton.2 <\| mem_singleton _).union_union (mapsTo_preimage _ _) #align cont_diff_within_at.comp_continuous_linear_map ContDiffWithinAt.comp_continuousLinearMap /-- Composition by continuous linear maps on the right preserves `C^n` functions on domains. -/ theorem ContDiffOn.comp_continuousLinearMap (hf : ContDiffOn 𝕜 n f s) (g : G →L[𝕜] E) : ContDiffOn 𝕜 n (f ∘ g) (g ⁻¹' s) := fun x hx => (hf (g x) hx).comp_continuousLinearMap g #align cont_diff_on.comp_continuous_linear_map ContDiffOn.comp_continuousLinearMap /-- Composition by continuous linear maps on the right preserves `C^n` functions. -/ theorem ContDiff.comp_continuousLinearMap {f : E → F} {g : G →L[𝕜] E} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n (f ∘ g) := contDiffOn_univ.1 <\| ContDiffOn.comp_continuousLinearMap (contDiffOn_univ.2 hf) _ #align cont_diff.comp_continuous_linear_map ContDiff.comp_continuousLinearMap /-- The iterated derivative within a set of the composition with a linear map on the right is obtained by composing the iterated derivative with the linear map. -/ theorem ContinuousLinearMap.iteratedFDerivWithin_comp_right {f : E → F} (g : G →L[𝕜] E) (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (h's : UniqueDiffOn 𝕜 (g ⁻¹' s)) {x : G} (hx : g x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) : iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x = (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g := (((hf.ftaylorSeriesWithin hs).compContinuousLinearMap g).eq_iteratedFDerivWithin_of_uniqueDiffOn hi h's hx).symm #align continuous_linear_map.iterated_fderiv_within_comp_right ContinuousLinearMap.iteratedFDerivWithin_comp_right /-- The iterated derivative within a set of the composition with a linear equiv on the right is obtained by composing the iterated derivative with the linear equiv. -/ theorem ContinuousLinearEquiv.iteratedFDerivWithin_comp_right (g : G ≃L[𝕜] E) (f : E → F) (hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) : iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x = (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g := by induction' i with i IH generalizing x · ext1 simp only [Nat.zero_eq, iteratedFDerivWithin_zero_apply, comp_apply, ContinuousMultilinearMap.compContinuousLinearMap_apply] · ext1 m simp only [ContinuousMultilinearMap.compContinuousLinearMap_apply, ContinuousLinearEquiv.coe_coe, iteratedFDerivWithin_succ_apply_left] have : fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s)) (g ⁻¹' s) x = fderivWithin 𝕜 (ContinuousMultilinearMap.compContinuousLinearMapEquivL _ (fun _x : Fin i => g) ∘ (iteratedFDerivWithin 𝕜 i f s ∘ g)) (g ⁻¹' s) x := fderivWithin_congr' (@IH) hx rw [this, ContinuousLinearEquiv.comp_fderivWithin _ (g.uniqueDiffOn_preimage_iff.2 hs x hx)] simp only [ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe, comp_apply, ContinuousMultilinearMap.compContinuousLinearMapEquivL_apply, ContinuousMultilinearMap.compContinuousLinearMap_apply] rw [ContinuousLinearEquiv.comp_right_fderivWithin _ (g.uniqueDiffOn_preimage_iff.2 hs x hx), ContinuousLinearMap.coe_comp', coe_coe, comp_apply, tail_def, tail_def] #align continuous_linear_equiv.iterated_fderiv_within_comp_right ContinuousLinearEquiv.iteratedFDerivWithin_comp_right /-- The iterated derivative of the composition with a linear map on the right is obtained by composing the iterated derivative with the linear map. -/ theorem ContinuousLinearMap.iteratedFDeriv_comp_right (g : G →L[𝕜] E) {f : E → F} (hf : ContDiff 𝕜 n f) (x : G) {i : ℕ} (hi : (i : ℕ∞) ≤ n) : iteratedFDeriv 𝕜 i (f ∘ g) x = (iteratedFDeriv 𝕜 i f (g x)).compContinuousLinearMap fun _ => g := by simp only [← iteratedFDerivWithin_univ] exact g.iteratedFDerivWithin_comp_right hf.contDiffOn uniqueDiffOn_univ uniqueDiffOn_univ (mem_univ _) hi #align continuous_linear_map.iterated_fderiv_comp_right ContinuousLinearMap.iteratedFDeriv_comp_right /-- Composition with a linear isometry on the right preserves the norm of the iterated derivative within a set. -/ theorem LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right (g : G ≃ₗᵢ[𝕜] E) (f : E → F) (hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) : ‖iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x‖ = ‖iteratedFDerivWithin 𝕜 i f s (g x)‖ := by have : iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x = (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g := g.toContinuousLinearEquiv.iteratedFDerivWithin_comp_right f hs hx i rw [this, ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv] #align linear_isometry_equiv.norm_iterated_fderiv_within_comp_right LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right /-- Composition with a linear isometry on the right preserves the norm of the iterated derivative within a set. -/ theorem LinearIsometryEquiv.norm_iteratedFDeriv_comp_right (g : G ≃ₗᵢ[𝕜] E) (f : E → F) (x : G) (i : ℕ) : ‖iteratedFDeriv 𝕜 i (f ∘ g) x‖ = ‖iteratedFDeriv 𝕜 i f (g x)‖ := by simp only [← iteratedFDerivWithin_univ] apply g.norm_iteratedFDerivWithin_comp_right f uniqueDiffOn_univ (mem_univ (g x)) i #align linear_isometry_equiv.norm_iterated_fderiv_comp_right LinearIsometryEquiv.norm_iteratedFDeriv_comp_right /-- Composition by continuous linear equivs on the right respects higher differentiability at a point in a domain. -/ theorem ContinuousLinearEquiv.contDiffWithinAt_comp_iff (e : G ≃L[𝕜] E) : ContDiffWithinAt 𝕜 n (f ∘ e) (e ⁻¹' s) (e.symm x) ↔ ContDiffWithinAt 𝕜 n f s x := by constructor · intro H simpa [← preimage_comp, (· ∘ ·)] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G) · intro H rw [← e.apply_symm_apply x, ← e.coe_coe] at H exact H.comp_continuousLinearMap _ #align continuous_linear_equiv.cont_diff_within_at_comp_iff ContinuousLinearEquiv.contDiffWithinAt_comp_iff /-- Composition by continuous linear equivs on the right respects higher differentiability at a point. -/ theorem ContinuousLinearEquiv.contDiffAt_comp_iff (e : G ≃L[𝕜] E) : ContDiffAt 𝕜 n (f ∘ e) (e.symm x) ↔ ContDiffAt 𝕜 n f x := by rw [← contDiffWithinAt_univ, ← contDiffWithinAt_univ, ← preimage_univ] exact e.contDiffWithinAt_comp_iff #align continuous_linear_equiv.cont_diff_at_comp_iff ContinuousLinearEquiv.contDiffAt_comp_iff /-- Composition by continuous linear equivs on the right respects higher differentiability on domains. -/ theorem ContinuousLinearEquiv.contDiffOn_comp_iff (e : G ≃L[𝕜] E) : ContDiffOn 𝕜 n (f ∘ e) (e ⁻¹' s) ↔ ContDiffOn 𝕜 n f s := ⟨fun H => by simpa [(· ∘ ·)] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G), fun H => H.comp_continuousLinearMap (e : G →L[𝕜] E)⟩ #align continuous_linear_equiv.cont_diff_on_comp_iff ContinuousLinearEquiv.contDiffOn_comp_iff /-- Composition by continuous linear equivs on the right respects higher differentiability. -/ theorem ContinuousLinearEquiv.contDiff_comp_iff (e : G ≃L[𝕜] E) : ContDiff 𝕜 n (f ∘ e) ↔ ContDiff 𝕜 n f := by rw [← contDiffOn_univ, ← contDiffOn_univ, ← preimage_univ] exact e.contDiffOn_comp_iff #align continuous_linear_equiv.cont_diff_comp_iff ContinuousLinearEquiv.contDiff_comp_iff /-- If two functions `f` and `g` admit Taylor series `p` and `q` in a set `s`, then the cartesian product of `f` and `g` admits the cartesian product of `p` and `q` as a Taylor series. -/ theorem HasFTaylorSeriesUpToOn.prod (hf : HasFTaylorSeriesUpToOn n f p s) {g : E → G} {q : E → FormalMultilinearSeries 𝕜 E G} (hg : HasFTaylorSeriesUpToOn n g q s) : HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => (p y k).prod (q y k)) s := by set L := fun m => ContinuousMultilinearMap.prodL 𝕜 (fun _ : Fin m => E) F G constructor · intro x hx; rw [← hf.zero_eq x hx, ← hg.zero_eq x hx]; rfl · intro m hm x hx convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x ((hf.fderivWithin m hm x hx).prod (hg.fderivWithin m hm x hx)) · intro m hm exact (L m).continuous.comp_continuousOn ((hf.cont m hm).prod (hg.cont m hm)) #align has_ftaylor_series_up_to_on.prod HasFTaylorSeriesUpToOn.prod /-- The cartesian product of `C^n` functions at a point in a domain is `C^n`. -/ theorem ContDiffWithinAt.prod {s : Set E} {f : E → F} {g : E → G} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x : E => (f x, g x)) s x := by intro m hm rcases hf m hm with ⟨u, hu, p, hp⟩ rcases hg m hm with ⟨v, hv, q, hq⟩ exact ⟨u ∩ v, Filter.inter_mem hu hv, _, (hp.mono inter_subset_left).prod (hq.mono inter_subset_right)⟩ #align cont_diff_within_at.prod ContDiffWithinAt.prod /-- The cartesian product of `C^n` functions on domains is `C^n`. -/ theorem ContDiffOn.prod {s : Set E} {f : E → F} {g : E → G} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x : E => (f x, g x)) s := fun x hx => (hf x hx).prod (hg x hx) #align cont_diff_on.prod ContDiffOn.prod /-- The cartesian product of `C^n` functions at a point is `C^n`. -/ theorem ContDiffAt.prod {f : E → F} {g : E → G} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x : E => (f x, g x)) x := contDiffWithinAt_univ.1 <\| ContDiffWithinAt.prod (contDiffWithinAt_univ.2 hf) (contDiffWithinAt_univ.2 hg) #align cont_diff_at.prod ContDiffAt.prod /-- The cartesian product of `C^n` functions is `C^n`. -/ theorem ContDiff.prod {f : E → F} {g : E → G} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x : E => (f x, g x) := contDiffOn_univ.1 <\| ContDiffOn.prod (contDiffOn_univ.2 hf) (contDiffOn_univ.2 hg) #align cont_diff.prod ContDiff.prod /-! ### Composition of `C^n` functions We show that the composition of `C^n` functions is `C^n`. One way to prove it would be to write the `n`-th derivative of the composition (this is Faà di Bruno's formula) and check its continuity, but this is very painful. Instead, we go for a simple inductive proof. Assume it is done for `n`. Then, to check it for `n+1`, one needs to check that the derivative of `g ∘ f` is `C^n`, i.e., that `Dg(f x) ⬝ Df(x)` is `C^n`. The term `Dg (f x)` is the composition of two `C^n` functions, so it is `C^n` by the inductive assumption. The term `Df(x)` is also `C^n`. Then, the matrix multiplication is the application of a bilinear map (which is `C^∞`, and therefore `C^n`) to `x ↦ (Dg(f x), Df x)`. As the composition of two `C^n` maps, it is again `C^n`, and we are done. There is a subtlety in this argument: we apply the inductive assumption to functions on other Banach spaces. In maths, one would say: prove by induction over `n` that, for all `C^n` maps between all pairs of Banach spaces, their composition is `C^n`. In Lean, this is fine as long as the spaces stay in the same universe. This is not the case in the above argument: if `E` lives in universe `u` and `F` lives in universe `v`, then linear maps from `E` to `F` (to which the derivative of `f` belongs) is in universe `max u v`. If one could quantify over finitely many universes, the above proof would work fine, but this is not the case. One could still write the proof considering spaces in any universe in `u, v, w, max u v, max v w, max u v w`, but it would be extremely tedious and lead to a lot of duplication. Instead, we formulate the above proof when all spaces live in the same universe (where everything is fine), and then we deduce the general result by lifting all our spaces to a common universe through `ULift`. This lifting is done through a continuous linear equiv. We have already proved that composing with such a linear equiv does not change the fact of being `C^n`, which concludes the proof. -/ /-- Auxiliary lemma proving that the composition of `C^n` functions on domains is `C^n` when all spaces live in the same universe. Use instead `ContDiffOn.comp` which removes the universe assumption (but is deduced from this one). -/ private theorem ContDiffOn.comp_same_univ {Eu : Type u} [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu] {Fu : Type u} [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] {Gu : Type u} [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {s : Set Eu} {t : Set Fu} {g : Fu → Gu} {f : Eu → Fu} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s) (st : s ⊆ f ⁻¹' t) : ContDiffOn 𝕜 n (g ∘ f) s := by induction' n using ENat.nat_induction with n IH Itop generalizing Eu Fu Gu · rw [contDiffOn_zero] at hf hg ⊢ exact ContinuousOn.comp hg hf st · rw [contDiffOn_succ_iff_hasFDerivWithinAt] at hg ⊢ intro x hx rcases (contDiffOn_succ_iff_hasFDerivWithinAt.1 hf) x hx with ⟨u, hu, f', hf', f'_diff⟩ rcases hg (f x) (st hx) with ⟨v, hv, g', hg', g'_diff⟩ rw [insert_eq_of_mem hx] at hu ⊢ have xu : x ∈ u := mem_of_mem_nhdsWithin hx hu let w := s ∩ (u ∩ f ⁻¹' v) have wv : w ⊆ f ⁻¹' v := fun y hy => hy.2.2 have wu : w ⊆ u := fun y hy => hy.2.1 have ws : w ⊆ s := fun y hy => hy.1 refine ⟨w, ?_, fun y => (g' (f y)).comp (f' y), ?_, ?_⟩ · show w ∈ 𝓝[s] x apply Filter.inter_mem self_mem_nhdsWithin apply Filter.inter_mem hu apply ContinuousWithinAt.preimage_mem_nhdsWithin' · rw [← continuousWithinAt_inter' hu] exact (hf' x xu).differentiableWithinAt.continuousWithinAt.mono inter_subset_right · apply nhdsWithin_mono _ _ hv exact Subset.trans (image_subset_iff.mpr st) (subset_insert (f x) t) · show ∀ y ∈ w, HasFDerivWithinAt (g ∘ f) ((g' (f y)).comp (f' y)) w y rintro y ⟨-, yu, yv⟩ exact (hg' (f y) yv).comp y ((hf' y yu).mono wu) wv · show ContDiffOn 𝕜 n (fun y => (g' (f y)).comp (f' y)) w have A : ContDiffOn 𝕜 n (fun y => g' (f y)) w := IH g'_diff ((hf.of_le (WithTop.coe_le_coe.2 (Nat.le_succ n))).mono ws) wv have B : ContDiffOn 𝕜 n f' w := f'_diff.mono wu have C : ContDiffOn 𝕜 n (fun y => (g' (f y), f' y)) w := A.prod B have D : ContDiffOn 𝕜 n (fun p : (Fu →L[𝕜] Gu) × (Eu →L[𝕜] Fu) => p.1.comp p.2) univ := isBoundedBilinearMap_comp.contDiff.contDiffOn exact IH D C (subset_univ _) · rw [contDiffOn_top] at hf hg ⊢ exact fun n => Itop n (hg n) (hf n) st /-- The composition of `C^n` functions on domains is `C^n`. -/ theorem ContDiffOn.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s) (st : s ⊆ f ⁻¹' t) : ContDiffOn 𝕜 n (g ∘ f) s := by /- we lift all the spaces to a common universe, as we have already proved the result in this situation. -/ let Eu : Type max uE uF uG := ULift.{max uF uG} E let Fu : Type max uE uF uG := ULift.{max uE uG} F let Gu : Type max uE uF uG := ULift.{max uE uF} G -- declare the isomorphisms have isoE : Eu ≃L[𝕜] E := ContinuousLinearEquiv.ulift have isoF : Fu ≃L[𝕜] F := ContinuousLinearEquiv.ulift have isoG : Gu ≃L[𝕜] G := ContinuousLinearEquiv.ulift -- lift the functions to the new spaces, check smoothness there, and then go back. let fu : Eu → Fu := (isoF.symm ∘ f) ∘ isoE have fu_diff : ContDiffOn 𝕜 n fu (isoE ⁻¹' s) := by rwa [isoE.contDiffOn_comp_iff, isoF.symm.comp_contDiffOn_iff] let gu : Fu → Gu := (isoG.symm ∘ g) ∘ isoF have gu_diff : ContDiffOn 𝕜 n gu (isoF ⁻¹' t) := by rwa [isoF.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff] have main : ContDiffOn 𝕜 n (gu ∘ fu) (isoE ⁻¹' s) := by apply ContDiffOn.comp_same_univ gu_diff fu_diff intro y hy simp only [fu, ContinuousLinearEquiv.coe_apply, Function.comp_apply, mem_preimage] rw [isoF.apply_symm_apply (f (isoE y))] exact st hy have : gu ∘ fu = (isoG.symm ∘ g ∘ f) ∘ isoE := by ext y simp only [fu, gu, Function.comp_apply] rw [isoF.apply_symm_apply (f (isoE y))] rwa [this, isoE.contDiffOn_comp_iff, isoG.symm.comp_contDiffOn_iff] at main #align cont_diff_on.comp ContDiffOn.comp /-- The composition of `C^n` functions on domains is `C^n`. -/ theorem ContDiffOn.comp' {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t) := hg.comp (hf.mono inter_subset_left) inter_subset_right #align cont_diff_on.comp' ContDiffOn.comp' /-- The composition of a `C^n` function on a domain with a `C^n` function is `C^n`. -/ theorem ContDiff.comp_contDiffOn {s : Set E} {g : F → G} {f : E → F} (hg : ContDiff 𝕜 n g) (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (g ∘ f) s := (contDiffOn_univ.2 hg).comp hf subset_preimage_univ #align cont_diff.comp_cont_diff_on ContDiff.comp_contDiffOn /-- The composition of `C^n` functions is `C^n`. -/ theorem ContDiff.comp {g : F → G} {f : E → F} (hg : ContDiff 𝕜 n g) (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n (g ∘ f) := contDiffOn_univ.1 <\| ContDiffOn.comp (contDiffOn_univ.2 hg) (contDiffOn_univ.2 hf) (subset_univ _) #align cont_diff.comp ContDiff.comp /-- The composition of `C^n` functions at points in domains is `C^n`. -/ theorem ContDiffWithinAt.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E) (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) (st : s ⊆ f ⁻¹' t) : ContDiffWithinAt 𝕜 n (g ∘ f) s x := by intro m hm rcases hg.contDiffOn hm with ⟨u, u_nhd, _, hu⟩ rcases hf.contDiffOn hm with ⟨v, v_nhd, vs, hv⟩ have xmem : x ∈ f ⁻¹' u ∩ v := ⟨(mem_of_mem_nhdsWithin (mem_insert (f x) _) u_nhd : _), mem_of_mem_nhdsWithin (mem_insert x s) v_nhd⟩ have : f ⁻¹' u ∈ 𝓝[insert x s] x := by apply hf.continuousWithinAt.insert_self.preimage_mem_nhdsWithin' apply nhdsWithin_mono _ _ u_nhd rw [image_insert_eq] exact insert_subset_insert (image_subset_iff.mpr st) have Z := (hu.comp (hv.mono inter_subset_right) inter_subset_left).contDiffWithinAt xmem m le_rfl have : 𝓝[f ⁻¹' u ∩ v] x = 𝓝[insert x s] x := by have A : f ⁻¹' u ∩ v = insert x s ∩ (f ⁻¹' u ∩ v) := by apply Subset.antisymm _ inter_subset_right rintro y ⟨hy1, hy2⟩ simpa only [mem_inter_iff, mem_preimage, hy2, and_true, true_and, vs hy2] using hy1 rw [A, ← nhdsWithin_restrict''] exact Filter.inter_mem this v_nhd rwa [insert_eq_of_mem xmem, this] at Z #align cont_diff_within_at.comp ContDiffWithinAt.comp /-- The composition of `C^n` functions at points in domains is `C^n`, with a weaker condition on `s` and `t`. -/ theorem ContDiffWithinAt.comp_of_mem {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E) (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) (hs : t ∈ 𝓝[f '' s] f x) : ContDiffWithinAt 𝕜 n (g ∘ f) s x := (hg.mono_of_mem hs).comp x hf (subset_preimage_image f s) #align cont_diff_within_at.comp_of_mem ContDiffWithinAt.comp_of_mem /-- The composition of `C^n` functions at points in domains is `C^n`. -/ theorem ContDiffWithinAt.comp' {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E) (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (g ∘ f) (s ∩ f ⁻¹' t) x := hg.comp x (hf.mono inter_subset_left) inter_subset_right #align cont_diff_within_at.comp' ContDiffWithinAt.comp' theorem ContDiffAt.comp_contDiffWithinAt {n} (x : E) (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (g ∘ f) s x := hg.comp x hf (mapsTo_univ _ _) #align cont_diff_at.comp_cont_diff_within_at ContDiffAt.comp_contDiffWithinAt /-- The composition of `C^n` functions at points is `C^n`. -/ nonrec theorem ContDiffAt.comp (x : E) (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (g ∘ f) x := hg.comp x hf subset_preimage_univ #align cont_diff_at.comp ContDiffAt.comp theorem ContDiff.comp_contDiffWithinAt {g : F → G} {f : E → F} (h : ContDiff 𝕜 n g) (hf : ContDiffWithinAt 𝕜 n f t x) : ContDiffWithinAt 𝕜 n (g ∘ f) t x := haveI : ContDiffWithinAt 𝕜 n g univ (f x) := h.contDiffAt.contDiffWithinAt this.comp x hf (subset_univ _) #align cont_diff.comp_cont_diff_within_at ContDiff.comp_contDiffWithinAt theorem ContDiff.comp_contDiffAt {g : F → G} {f : E → F} (x : E) (hg : ContDiff 𝕜 n g) (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (g ∘ f) x := hg.comp_contDiffWithinAt hf #align cont_diff.comp_cont_diff_at ContDiff.comp_contDiffAt /-! ### Smoothness of projections -/ /-- The first projection in a product is `C^∞`. -/ theorem contDiff_fst : ContDiff 𝕜 n (Prod.fst : E × F → E) := IsBoundedLinearMap.contDiff IsBoundedLinearMap.fst #align cont_diff_fst contDiff_fst /-- Postcomposing `f` with `Prod.fst` is `C^n` -/ theorem ContDiff.fst {f : E → F × G} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => (f x).1 := contDiff_fst.comp hf #align cont_diff.fst ContDiff.fst /-- Precomposing `f` with `Prod.fst` is `C^n` -/ theorem ContDiff.fst' {f : E → G} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x : E × F => f x.1 := hf.comp contDiff_fst #align cont_diff.fst' ContDiff.fst' /-- The first projection on a domain in a product is `C^∞`. -/ theorem contDiffOn_fst {s : Set (E × F)} : ContDiffOn 𝕜 n (Prod.fst : E × F → E) s := ContDiff.contDiffOn contDiff_fst #align cont_diff_on_fst contDiffOn_fst theorem ContDiffOn.fst {f : E → F × G} {s : Set E} (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (fun x => (f x).1) s := contDiff_fst.comp_contDiffOn hf #align cont_diff_on.fst ContDiffOn.fst /-- The first projection at a point in a product is `C^∞`. -/ theorem contDiffAt_fst {p : E × F} : ContDiffAt 𝕜 n (Prod.fst : E × F → E) p := contDiff_fst.contDiffAt #align cont_diff_at_fst contDiffAt_fst /-- Postcomposing `f` with `Prod.fst` is `C^n` at `(x, y)` -/ theorem ContDiffAt.fst {f : E → F × G} {x : E} (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun x => (f x).1) x := contDiffAt_fst.comp x hf #align cont_diff_at.fst ContDiffAt.fst /-- Precomposing `f` with `Prod.fst` is `C^n` at `(x, y)` -/ theorem ContDiffAt.fst' {f : E → G} {x : E} {y : F} (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun x : E × F => f x.1) (x, y) := ContDiffAt.comp (x, y) hf contDiffAt_fst #align cont_diff_at.fst' ContDiffAt.fst' /-- Precomposing `f` with `Prod.fst` is `C^n` at `x : E × F` -/ theorem ContDiffAt.fst'' {f : E → G} {x : E × F} (hf : ContDiffAt 𝕜 n f x.1) : ContDiffAt 𝕜 n (fun x : E × F => f x.1) x := hf.comp x contDiffAt_fst #align cont_diff_at.fst'' ContDiffAt.fst'' /-- The first projection within a domain at a point in a product is `C^∞`. -/ theorem contDiffWithinAt_fst {s : Set (E × F)} {p : E × F} : ContDiffWithinAt 𝕜 n (Prod.fst : E × F → E) s p := contDiff_fst.contDiffWithinAt #align cont_diff_within_at_fst contDiffWithinAt_fst /-- The second projection in a product is `C^∞`. -/ theorem contDiff_snd : ContDiff 𝕜 n (Prod.snd : E × F → F) := IsBoundedLinearMap.contDiff IsBoundedLinearMap.snd #align cont_diff_snd contDiff_snd /-- Postcomposing `f` with `Prod.snd` is `C^n` -/ theorem ContDiff.snd {f : E → F × G} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => (f x).2 := contDiff_snd.comp hf #align cont_diff.snd ContDiff.snd /-- Precomposing `f` with `Prod.snd` is `C^n` -/ theorem ContDiff.snd' {f : F → G} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x : E × F => f x.2 := hf.comp contDiff_snd #align cont_diff.snd' ContDiff.snd' /-- The second projection on a domain in a product is `C^∞`. -/ theorem contDiffOn_snd {s : Set (E × F)} : ContDiffOn 𝕜 n (Prod.snd : E × F → F) s := ContDiff.contDiffOn contDiff_snd #align cont_diff_on_snd contDiffOn_snd theorem ContDiffOn.snd {f : E → F × G} {s : Set E} (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (fun x => (f x).2) s := contDiff_snd.comp_contDiffOn hf #align cont_diff_on.snd ContDiffOn.snd /-- The second projection at a point in a product is `C^∞`. -/ theorem contDiffAt_snd {p : E × F} : ContDiffAt 𝕜 n (Prod.snd : E × F → F) p := contDiff_snd.contDiffAt #align cont_diff_at_snd contDiffAt_snd /-- Postcomposing `f` with `Prod.snd` is `C^n` at `x` -/ theorem ContDiffAt.snd {f : E → F × G} {x : E} (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun x => (f x).2) x := contDiffAt_snd.comp x hf #align cont_diff_at.snd ContDiffAt.snd /-- Precomposing `f` with `Prod.snd` is `C^n` at `(x, y)` -/ theorem ContDiffAt.snd' {f : F → G} {x : E} {y : F} (hf : ContDiffAt 𝕜 n f y) : ContDiffAt 𝕜 n (fun x : E × F => f x.2) (x, y) := ContDiffAt.comp (x, y) hf contDiffAt_snd #align cont_diff_at.snd' ContDiffAt.snd' /-- Precomposing `f` with `Prod.snd` is `C^n` at `x : E × F` -/ theorem ContDiffAt.snd'' {f : F → G} {x : E × F} (hf : ContDiffAt 𝕜 n f x.2) : ContDiffAt 𝕜 n (fun x : E × F => f x.2) x := hf.comp x contDiffAt_snd #align cont_diff_at.snd'' ContDiffAt.snd'' /-- The second projection within a domain at a point in a product is `C^∞`. -/ theorem contDiffWithinAt_snd {s : Set (E × F)} {p : E × F} : ContDiffWithinAt 𝕜 n (Prod.snd : E × F → F) s p := contDiff_snd.contDiffWithinAt #align cont_diff_within_at_snd contDiffWithinAt_snd section NAry variable {E₁ E₂ E₃ E₄ : Type} variable [NormedAddCommGroup E₁] [NormedAddCommGroup E₂] [NormedAddCommGroup E₃] [NormedAddCommGroup E₄] [NormedSpace 𝕜 E₁] [NormedSpace 𝕜 E₂] [NormedSpace 𝕜 E₃] [NormedSpace 𝕜 E₄] theorem ContDiff.comp₂ {g : E₁ × E₂ → G} {f₁ : F → E₁} {f₂ : F → E₂} (hg : ContDiff 𝕜 n g) (hf₁ : ContDiff 𝕜 n f₁) (hf₂ : ContDiff 𝕜 n f₂) : ContDiff 𝕜 n fun x => g (f₁ x, f₂ x) := hg.comp <\| hf₁.prod hf₂ #align cont_diff.comp₂ ContDiff.comp₂ theorem ContDiff.comp₃ {g : E₁ × E₂ × E₃ → G} {f₁ : F → E₁} {f₂ : F → E₂} {f₃ : F → E₃} (hg : ContDiff 𝕜 n g) (hf₁ : ContDiff 𝕜 n f₁) (hf₂ : ContDiff 𝕜 n f₂) (hf₃ : ContDiff 𝕜 n f₃) : ContDiff 𝕜 n fun x => g (f₁ x, f₂ x, f₃ x) := hg.comp₂ hf₁ <\| hf₂.prod hf₃ #align cont_diff.comp₃ ContDiff.comp₃ theorem ContDiff.comp_contDiff_on₂ {g : E₁ × E₂ → G} {f₁ : F → E₁} {f₂ : F → E₂} {s : Set F} (hg : ContDiff 𝕜 n g) (hf₁ : ContDiffOn 𝕜 n f₁ s) (hf₂ : ContDiffOn 𝕜 n f₂ s) : ContDiffOn 𝕜 n (fun x => g (f₁ x, f₂ x)) s := hg.comp_contDiffOn <\| hf₁.prod hf₂ #align cont_diff.comp_cont_diff_on₂ ContDiff.comp_contDiff_on₂ theorem ContDiff.comp_contDiff_on₃ {g : E₁ × E₂ × E₃ → G} {f₁ : F → E₁} {f₂ : F → E₂} {f₃ : F → E₃} {s : Set F} (hg : ContDiff 𝕜 n g) (hf₁ : ContDiffOn 𝕜 n f₁ s) (hf₂ : ContDiffOn 𝕜 n f₂ s) (hf₃ : ContDiffOn 𝕜 n f₃ s) : ContDiffOn 𝕜 n (fun x => g (f₁ x, f₂ x, f₃ x)) s := hg.comp_contDiff_on₂ hf₁ <\| hf₂.prod hf₃ #align cont_diff.comp_cont_diff_on₃ ContDiff.comp_contDiff_on₃ end NAry section SpecificBilinearMaps theorem ContDiff.clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F} (hg : ContDiff 𝕜 n g) (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => (g x).comp (f x) := isBoundedBilinearMap_comp.contDiff.comp₂ hg hf #align cont_diff.clm_comp ContDiff.clm_comp theorem ContDiffOn.clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F} {s : Set X} (hg : ContDiffOn 𝕜 n g s) (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (fun x => (g x).comp (f x)) s := isBoundedBilinearMap_comp.contDiff.comp_contDiff_on₂ hg hf #align cont_diff_on.clm_comp ContDiffOn.clm_comp theorem ContDiff.clm_apply {f : E → F →L[𝕜] G} {g : E → F} {n : ℕ∞} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => (f x) (g x) := isBoundedBilinearMap_apply.contDiff.comp₂ hf hg #align cont_diff.clm_apply ContDiff.clm_apply theorem ContDiffOn.clm_apply {f : E → F →L[𝕜] G} {g : E → F} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => (f x) (g x)) s := isBoundedBilinearMap_apply.contDiff.comp_contDiff_on₂ hf hg #align cont_diff_on.clm_apply ContDiffOn.clm_apply -- Porting note: In Lean 3 we had to give implicit arguments in proofs like the following, -- to speed up elaboration. In Lean 4 this isn't necessary anymore. theorem ContDiff.smulRight {f : E → F →L[𝕜] 𝕜} {g : E → G} {n : ℕ∞} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => (f x).smulRight (g x) := isBoundedBilinearMap_smulRight.contDiff.comp₂ hf hg #align cont_diff.smul_right ContDiff.smulRight end SpecificBilinearMaps section ClmApplyConst /-- Application of a `ContinuousLinearMap` to a constant commutes with `iteratedFDerivWithin`. -/ theorem iteratedFDerivWithin_clm_apply_const_apply {s : Set E} (hs : UniqueDiffOn 𝕜 s) {n : ℕ∞} {c : E → F →L[𝕜] G} (hc : ContDiffOn 𝕜 n c s) {i : ℕ} (hi : i ≤ n) {x : E} (hx : x ∈ s) {u : F} {m : Fin i → E} : (iteratedFDerivWithin 𝕜 i (fun y ↦ (c y) u) s x) m = (iteratedFDerivWithin 𝕜 i c s x) m u := by induction i generalizing x with \| zero => simp \| succ i ih => replace hi : i < n := lt_of_lt_of_le (by norm_cast; simp) hi have h_deriv_apply : DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 i (fun y ↦ (c y) u) s) s := (hc.clm_apply contDiffOn_const).differentiableOn_iteratedFDerivWithin hi hs have h_deriv : DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 i c s) s := hc.differentiableOn_iteratedFDerivWithin hi hs simp only [iteratedFDerivWithin_succ_apply_left] rw [← fderivWithin_continuousMultilinear_apply_const_apply (hs x hx) (h_deriv_apply x hx)] rw [fderivWithin_congr' (fun x hx ↦ ih hi.le hx) hx] rw [fderivWithin_clm_apply (hs x hx) (h_deriv.continuousMultilinear_apply_const _ x hx) (differentiableWithinAt_const u)] rw [fderivWithin_const_apply _ (hs x hx)] simp only [ContinuousLinearMap.flip_apply, ContinuousLinearMap.comp_zero, zero_add] rw [fderivWithin_continuousMultilinear_apply_const_apply (hs x hx) (h_deriv x hx)] /-- Application of a `ContinuousLinearMap` to a constant commutes with `iteratedFDeriv`. -/ theorem iteratedFDeriv_clm_apply_const_apply {n : ℕ∞} {c : E → F →L[𝕜] G} (hc : ContDiff 𝕜 n c) {i : ℕ} (hi : i ≤ n) {x : E} {u : F} {m : Fin i → E} : (iteratedFDeriv 𝕜 i (fun y ↦ (c y) u) x) m = (iteratedFDeriv 𝕜 i c x) m u := by simp only [← iteratedFDerivWithin_univ] exact iteratedFDerivWithin_clm_apply_const_apply uniqueDiffOn_univ hc.contDiffOn hi (mem_univ _) end ClmApplyConst /-- The natural equivalence `(E × F) × G ≃ E × (F × G)` is smooth. Warning: if you think you need this lemma, it is likely that you can simplify your proof by reformulating the lemma that you're applying next using the tips in Note [continuity lemma statement] -/ theorem contDiff_prodAssoc : ContDiff 𝕜 ⊤ <\| Equiv.prodAssoc E F G := (LinearIsometryEquiv.prodAssoc 𝕜 E F G).contDiff #align cont_diff_prod_assoc contDiff_prodAssoc /-- The natural equivalence `E × (F × G) ≃ (E × F) × G` is smooth. Warning: see remarks attached to `contDiff_prodAssoc` -/ theorem contDiff_prodAssoc_symm : ContDiff 𝕜 ⊤ <\| (Equiv.prodAssoc E F G).symm := (LinearIsometryEquiv.prodAssoc 𝕜 E F G).symm.contDiff #align cont_diff_prod_assoc_symm contDiff_prodAssoc_symm /-! ### Bundled derivatives are smooth -/ /-- One direction of `contDiffWithinAt_succ_iff_hasFDerivWithinAt`, but where all derivatives taken within the same set. Version for partial derivatives / functions with parameters. `f x` is a `C^n+1` family of functions and `g x` is a `C^n` family of points, then the derivative of `f x` at `g x` depends in a `C^n` way on `x`. We give a general version of this fact relative to sets which may not have unique derivatives, in the following form. If `f : E × F → G` is `C^n+1` at `(x₀, g(x₀))` in `(s ∪ {x₀}) × t ⊆ E × F` and `g : E → F` is `C^n` at `x₀` within some set `s ⊆ E`, then there is a function `f' : E → F →L[𝕜] G` that is `C^n` at `x₀` within `s` such that for all `x` sufficiently close to `x₀` within `s ∪ {x₀}` the function `y ↦ f x y` has derivative `f' x` at `g x` within `t ⊆ F`. For convenience, we return an explicit set of `x`'s where this holds that is a subset of `s ∪ {x₀}`. We need one additional condition, namely that `t` is a neighborhood of `g(x₀)` within `g '' s`. -/ theorem ContDiffWithinAt.hasFDerivWithinAt_nhds {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ} {x₀ : E} (hf : ContDiffWithinAt 𝕜 (n + 1) (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)) (hg : ContDiffWithinAt 𝕜 n g s x₀) (hgt : t ∈ 𝓝[g '' s] g x₀) : ∃ v ∈ 𝓝[insert x₀ s] x₀, v ⊆ insert x₀ s ∧ ∃ f' : E → F →L[𝕜] G, (∀ x ∈ v, HasFDerivWithinAt (f x) (f' x) t (g x)) ∧ ContDiffWithinAt 𝕜 n (fun x => f' x) s x₀ := by have hst : insert x₀ s ×ˢ t ∈ 𝓝[(fun x => (x, g x)) '' s] (x₀, g x₀) := by refine nhdsWithin_mono _ ?_ (nhdsWithin_prod self_mem_nhdsWithin hgt) simp_rw [image_subset_iff, mk_preimage_prod, preimage_id', subset_inter_iff, subset_insert, true_and_iff, subset_preimage_image] obtain ⟨v, hv, hvs, f', hvf', hf'⟩ := contDiffWithinAt_succ_iff_hasFDerivWithinAt'.mp hf refine ⟨(fun z => (z, g z)) ⁻¹' v ∩ insert x₀ s, ?_, inter_subset_right, fun z => (f' (z, g z)).comp (ContinuousLinearMap.inr 𝕜 E F), ?_, ?_⟩ · refine inter_mem ?_ self_mem_nhdsWithin have := mem_of_mem_nhdsWithin (mem_insert _ _) hv refine mem_nhdsWithin_insert.mpr ⟨this, ?_⟩ refine (continuousWithinAt_id.prod hg.continuousWithinAt).preimage_mem_nhdsWithin' ?_ rw [← nhdsWithin_le_iff] at hst hv ⊢ exact (hst.trans <\| nhdsWithin_mono _ <\| subset_insert _ _).trans hv · intro z hz have := hvf' (z, g z) hz.1 refine this.comp _ (hasFDerivAt_prod_mk_right _ _).hasFDerivWithinAt ?_ exact mapsTo'.mpr (image_prod_mk_subset_prod_right hz.2) · exact (hf'.continuousLinearMap_comp <\| (ContinuousLinearMap.compL 𝕜 F (E × F) G).flip (ContinuousLinearMap.inr 𝕜 E F)).comp_of_mem x₀ (contDiffWithinAt_id.prod hg) hst #align cont_diff_within_at.has_fderiv_within_at_nhds ContDiffWithinAt.hasFDerivWithinAt_nhds /-- The most general lemma stating that `x ↦ fderivWithin 𝕜 (f x) t (g x)` is `C^n` at a point within a set. To show that `x ↦ D_yf(x,y)g(x)` (taken within `t`) is `C^m` at `x₀` within `s`, we require that `f` is `C^n` at `(x₀, g(x₀))` within `(s ∪ {x₀}) × t` for `n ≥ m+1`. * `g` is `C^m` at `x₀` within `s`; * Derivatives are unique at `g(x)` within `t` for `x` sufficiently close to `x₀` within `s ∪ {x₀}`; * `t` is a neighborhood of `g(x₀)` within `g '' s`; -/ theorem ContDiffWithinAt.fderivWithin'' {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)) (hg : ContDiffWithinAt 𝕜 m g s x₀) (ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)) (hmn : m + 1 ≤ n) (hgt : t ∈ 𝓝[g '' s] g x₀) : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := by have : ∀ k : ℕ, (k : ℕ∞) ≤ m → ContDiffWithinAt 𝕜 k (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := fun k hkm ↦ by obtain ⟨v, hv, -, f', hvf', hf'⟩ := (hf.of_le <\| (add_le_add_right hkm 1).trans hmn).hasFDerivWithinAt_nhds (hg.of_le hkm) hgt refine hf'.congr_of_eventuallyEq_insert ?_ filter_upwards [hv, ht] exact fun y hy h2y => (hvf' y hy).fderivWithin h2y induction' m with m · obtain rfl := eq_top_iff.mpr hmn rw [contDiffWithinAt_top] exact fun m => this m le_top exact this _ le_rfl #align cont_diff_within_at.fderiv_within'' ContDiffWithinAt.fderivWithin'' /-- A special case of `ContDiffWithinAt.fderivWithin''` where we require that `s ⊆ g⁻¹(t)`. -/ theorem ContDiffWithinAt.fderivWithin' {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)) (hg : ContDiffWithinAt 𝕜 m g s x₀) (ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)) (hmn : m + 1 ≤ n) (hst : s ⊆ g ⁻¹' t) : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := hf.fderivWithin'' hg ht hmn <\| mem_of_superset self_mem_nhdsWithin <\| image_subset_iff.mpr hst #align cont_diff_within_at.fderiv_within' ContDiffWithinAt.fderivWithin' /-- A special case of `ContDiffWithinAt.fderivWithin'` where we require that `x₀ ∈ s` and there are unique derivatives everywhere within `t`. -/ protected theorem ContDiffWithinAt.fderivWithin {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (s ×ˢ t) (x₀, g x₀)) (hg : ContDiffWithinAt 𝕜 m g s x₀) (ht : UniqueDiffOn 𝕜 t) (hmn : m + 1 ≤ n) (hx₀ : x₀ ∈ s) (hst : s ⊆ g ⁻¹' t) : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := by rw [← insert_eq_self.mpr hx₀] at hf refine hf.fderivWithin' hg ?_ hmn hst rw [insert_eq_self.mpr hx₀] exact eventually_of_mem self_mem_nhdsWithin fun x hx => ht _ (hst hx) #align cont_diff_within_at.fderiv_within ContDiffWithinAt.fderivWithin /-- `x ↦ fderivWithin 𝕜 (f x) t (g x) (k x)` is smooth at a point within a set. -/ theorem ContDiffWithinAt.fderivWithin_apply {f : E → F → G} {g k : E → F} {t : Set F} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (s ×ˢ t) (x₀, g x₀)) (hg : ContDiffWithinAt 𝕜 m g s x₀) (hk : ContDiffWithinAt 𝕜 m k s x₀) (ht : UniqueDiffOn 𝕜 t) (hmn : m + 1 ≤ n) (hx₀ : x₀ ∈ s) (hst : s ⊆ g ⁻¹' t) : ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x) (k x)) s x₀ := (contDiff_fst.clm_apply contDiff_snd).contDiffAt.comp_contDiffWithinAt x₀ ((hf.fderivWithin hg ht hmn hx₀ hst).prod hk) #align cont_diff_within_at.fderiv_within_apply ContDiffWithinAt.fderivWithin_apply /-- `fderivWithin 𝕜 f s` is smooth at `x₀` within `s`. -/ theorem ContDiffWithinAt.fderivWithin_right (hf : ContDiffWithinAt 𝕜 n f s x₀) (hs : UniqueDiffOn 𝕜 s) (hmn : (m + 1 : ℕ∞) ≤ n) (hx₀s : x₀ ∈ s) : ContDiffWithinAt 𝕜 m (fderivWithin 𝕜 f s) s x₀ := ContDiffWithinAt.fderivWithin (ContDiffWithinAt.comp (x₀, x₀) hf contDiffWithinAt_snd <\| prod_subset_preimage_snd s s) contDiffWithinAt_id hs hmn hx₀s (by rw [preimage_id']) #align cont_diff_within_at.fderiv_within_right ContDiffWithinAt.fderivWithin_right -- TODO: can we make a version of `ContDiffWithinAt.fderivWithin` for iterated derivatives? theorem ContDiffWithinAt.iteratedFderivWithin_right {i : ℕ} (hf : ContDiffWithinAt 𝕜 n f s x₀) (hs : UniqueDiffOn 𝕜 s) (hmn : (m + i : ℕ∞) ≤ n) (hx₀s : x₀ ∈ s) : ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 i f s) s x₀ := by induction' i with i hi generalizing m · rw [ENat.coe_zero, add_zero] at hmn exact (hf.of_le hmn).continuousLinearMap_comp ((continuousMultilinearCurryFin0 𝕜 E F).symm : _ →L[𝕜] E [×0]→L[𝕜] F) · rw [Nat.cast_succ, add_comm _ 1, ← add_assoc] at hmn exact ((hi hmn).fderivWithin_right hs le_rfl hx₀s).continuousLinearMap_comp (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (i+1) ↦ E) F : _ →L[𝕜] E [×(i+1)]→L[𝕜] F) /-- `x ↦ fderiv 𝕜 (f x) (g x)` is smooth at `x₀`. -/ protected theorem ContDiffAt.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞} (hf : ContDiffAt 𝕜 n (Function.uncurry f) (x₀, g x₀)) (hg : ContDiffAt 𝕜 m g x₀) (hmn : m + 1 ≤ n) : ContDiffAt 𝕜 m (fun x => fderiv 𝕜 (f x) (g x)) x₀ := by simp_rw [← fderivWithin_univ] refine (ContDiffWithinAt.fderivWithin hf.contDiffWithinAt hg.contDiffWithinAt uniqueDiffOn_univ hmn (mem_univ x₀) ?_).contDiffAt univ_mem rw [preimage_univ] #align cont_diff_at.fderiv ContDiffAt.fderiv /-- `fderiv 𝕜 f` is smooth at `x₀`. -/ theorem ContDiffAt.fderiv_right (hf : ContDiffAt 𝕜 n f x₀) (hmn : (m + 1 : ℕ∞) ≤ n) : ContDiffAt 𝕜 m (fderiv 𝕜 f) x₀ := ContDiffAt.fderiv (ContDiffAt.comp (x₀, x₀) hf contDiffAt_snd) contDiffAt_id hmn #align cont_diff_at.fderiv_right ContDiffAt.fderiv_right theorem ContDiffAt.iteratedFDeriv_right {i : ℕ} (hf : ContDiffAt 𝕜 n f x₀) (hmn : (m + i : ℕ∞) ≤ n) : ContDiffAt 𝕜 m (iteratedFDeriv 𝕜 i f) x₀ := by rw [← iteratedFDerivWithin_univ, ← contDiffWithinAt_univ] at * exact hf.iteratedFderivWithin_right uniqueDiffOn_univ hmn trivial /-- `x ↦ fderiv 𝕜 (f x) (g x)` is smooth. -/ protected theorem ContDiff.fderiv {f : E → F → G} {g : E → F} {n m : ℕ∞} (hf : ContDiff 𝕜 m <\| Function.uncurry f) (hg : ContDiff 𝕜 n g) (hnm : n + 1 ≤ m) : ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) (g x) := contDiff_iff_contDiffAt.mpr fun _ => hf.contDiffAt.fderiv hg.contDiffAt hnm #align cont_diff.fderiv ContDiff.fderiv /-- `fderiv 𝕜 f` is smooth. -/ theorem ContDiff.fderiv_right (hf : ContDiff 𝕜 n f) (hmn : (m + 1 : ℕ∞) ≤ n) : ContDiff 𝕜 m (fderiv 𝕜 f) := contDiff_iff_contDiffAt.mpr fun _x => hf.contDiffAt.fderiv_right hmn #align cont_diff.fderiv_right ContDiff.fderiv_right theorem ContDiff.iteratedFDeriv_right {i : ℕ} (hf : ContDiff 𝕜 n f) (hmn : (m + i : ℕ∞) ≤ n) : ContDiff 𝕜 m (iteratedFDeriv 𝕜 i f) := contDiff_iff_contDiffAt.mpr fun _x => hf.contDiffAt.iteratedFDeriv_right hmn /-- `x ↦ fderiv 𝕜 (f x) (g x)` is continuous. -/ theorem Continuous.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞} (hf : ContDiff 𝕜 n <\| Function.uncurry f) (hg : Continuous g) (hn : 1 ≤ n) : Continuous fun x => fderiv 𝕜 (f x) (g x) := (hf.fderiv (contDiff_zero.mpr hg) hn).continuous #align continuous.fderiv Continuous.fderiv /-- `x ↦ fderiv 𝕜 (f x) (g x) (k x)` is smooth. -/ theorem ContDiff.fderiv_apply {f : E → F → G} {g k : E → F} {n m : ℕ∞} (hf : ContDiff 𝕜 m <\| Function.uncurry f) (hg : ContDiff 𝕜 n g) (hk : ContDiff 𝕜 n k) (hnm : n + 1 ≤ m) : ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) (g x) (k x) := (hf.fderiv hg hnm).clm_apply hk #align cont_diff.fderiv_apply ContDiff.fderiv_apply /-- The bundled derivative of a `C^{n+1}` function is `C^n`. -/ theorem contDiffOn_fderivWithin_apply {m n : ℕ∞} {s : Set E} {f : E → F} (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun p : E × E => (fderivWithin 𝕜 f s p.1 : E →L[𝕜] F) p.2) (s ×ˢ univ) := ((hf.fderivWithin hs hmn).comp contDiffOn_fst (prod_subset_preimage_fst _ _)).clm_apply contDiffOn_snd #align cont_diff_on_fderiv_within_apply contDiffOn_fderivWithin_apply /-- If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is continuous. -/ theorem ContDiffOn.continuousOn_fderivWithin_apply (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hn : 1 ≤ n) : ContinuousOn (fun p : E × E => (fderivWithin 𝕜 f s p.1 : E → F) p.2) (s ×ˢ univ) := (contDiffOn_fderivWithin_apply hf hs <\| by rwa [zero_add]).continuousOn #align cont_diff_on.continuous_on_fderiv_within_apply ContDiffOn.continuousOn_fderivWithin_apply /-- The bundled derivative of a `C^{n+1}` function is `C^n`. -/ theorem ContDiff.contDiff_fderiv_apply {f : E → F} (hf : ContDiff 𝕜 n f) (hmn : m + 1 ≤ n) : ContDiff 𝕜 m fun p : E × E => (fderiv 𝕜 f p.1 : E →L[𝕜] F) p.2 := by rw [← contDiffOn_univ] at hf ⊢ rw [← fderivWithin_univ, ← univ_prod_univ] exact contDiffOn_fderivWithin_apply hf uniqueDiffOn_univ hmn #align cont_diff.cont_diff_fderiv_apply ContDiff.contDiff_fderiv_apply /-! ### Smoothness of functions `f : E → Π i, F' i` -/ section Pi variable {ι ι' : Type} [Fintype ι] [Fintype ι'] {F' : ι → Type} [∀ i, NormedAddCommGroup (F' i)] [∀ i, NormedSpace 𝕜 (F' i)] {φ : ∀ i, E → F' i} {p' : ∀ i, E → FormalMultilinearSeries 𝕜 E (F' i)} {Φ : E → ∀ i, F' i} {P' : E → FormalMultilinearSeries 𝕜 E (∀ i, F' i)} theorem hasFTaylorSeriesUpToOn_pi : HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔ ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m => ContinuousMultilinearMap.piₗᵢ _ _ refine ⟨fun h i => ?_, fun h => ⟨fun x hx => ?_, ?_, ?_⟩⟩ · convert h.continuousLinearMap_comp (pr i) · ext1 i exact (h i).zero_eq x hx · intro m hm x hx have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this · intro m hm have := continuousOn_pi.2 fun i => (h i).cont m hm convert (L m).continuous.comp_continuousOn this #align has_ftaylor_series_up_to_on_pi hasFTaylorSeriesUpToOn_pi @[simp] theorem hasFTaylorSeriesUpToOn_pi' : HasFTaylorSeriesUpToOn n Φ P' s ↔ ∀ i, HasFTaylorSeriesUpToOn n (fun x => Φ x i) (fun x m => (@ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ i).compContinuousMultilinearMap (P' x m)) s := by convert hasFTaylorSeriesUpToOn_pi (𝕜 := 𝕜) (φ := fun i x ↦ Φ x i); ext; rfl #align has_ftaylor_series_up_to_on_pi' hasFTaylorSeriesUpToOn_pi' theorem contDiffWithinAt_pi : ContDiffWithinAt 𝕜 n Φ s x ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => Φ x i) s x := by set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ refine ⟨fun h i => h.continuousLinearMap_comp (pr i), fun h m hm => ?_⟩ choose u hux p hp using fun i => h i m hm exact ⟨⋂ i, u i, Filter.iInter_mem.2 hux, _, hasFTaylorSeriesUpToOn_pi.2 fun i => (hp i).mono <\| iInter_subset _ _⟩ #align cont_diff_within_at_pi contDiffWithinAt_pi theorem contDiffOn_pi : ContDiffOn 𝕜 n Φ s ↔ ∀ i, ContDiffOn 𝕜 n (fun x => Φ x i) s := ⟨fun h _ x hx => contDiffWithinAt_pi.1 (h x hx) _, fun h x hx => contDiffWithinAt_pi.2 fun i => h i x hx⟩ #align cont_diff_on_pi contDiffOn_pi theorem contDiffAt_pi : ContDiffAt 𝕜 n Φ x ↔ ∀ i, ContDiffAt 𝕜 n (fun x => Φ x i) x := contDiffWithinAt_pi #align cont_diff_at_pi contDiffAt_pi theorem contDiff_pi : ContDiff 𝕜 n Φ ↔ ∀ i, ContDiff 𝕜 n fun x => Φ x i := by simp only [← contDiffOn_univ, contDiffOn_pi] #align cont_diff_pi contDiff_pi theorem contDiff_update [DecidableEq ι] (k : ℕ∞) (x : ∀ i, F' i) (i : ι) : ContDiff 𝕜 k (update x i) := by rw [contDiff_pi] intro j dsimp [Function.update] split_ifs with h · subst h exact contDiff_id · exact contDiff_const variable (F') in theorem contDiff_single [DecidableEq ι] (k : ℕ∞) (i : ι) : ContDiff 𝕜 k (Pi.single i : F' i → ∀ i, F' i) := contDiff_update k 0 i variable (𝕜 E) theorem contDiff_apply (i : ι) : ContDiff 𝕜 n fun f : ι → E => f i := contDiff_pi.mp contDiff_id i #align cont_diff_apply contDiff_apply theorem contDiff_apply_apply (i : ι) (j : ι') : ContDiff 𝕜 n fun f : ι → ι' → E => f i j := contDiff_pi.mp (contDiff_apply 𝕜 (ι' → E) i) j #align cont_diff_apply_apply contDiff_apply_apply end Pi /-! ### Sum of two functions -/ section Add theorem HasFTaylorSeriesUpToOn.add {q g} (hf : HasFTaylorSeriesUpToOn n f p s) (hg : HasFTaylorSeriesUpToOn n g q s) : HasFTaylorSeriesUpToOn n (f + g) (p + q) s := by convert HasFTaylorSeriesUpToOn.continuousLinearMap_comp (ContinuousLinearMap.fst 𝕜 F F + .snd 𝕜 F F) (hf.prod hg) -- The sum is smooth. theorem contDiff_add : ContDiff 𝕜 n fun p : F × F => p.1 + p.2 := (IsBoundedLinearMap.fst.add IsBoundedLinearMap.snd).contDiff #align cont_diff_add contDiff_add /-- The sum of two `C^n` functions within a set at a point is `C^n` within this set at this point. -/ theorem ContDiffWithinAt.add {s : Set E} {f g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x + g x) s x := contDiff_add.contDiffWithinAt.comp x (hf.prod hg) subset_preimage_univ #align cont_diff_within_at.add ContDiffWithinAt.add /-- The sum of two `C^n` functions at a point is `C^n` at this point. -/ theorem ContDiffAt.add {f g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x => f x + g x) x := by rw [← contDiffWithinAt_univ] at ; exact hf.add hg #align cont_diff_at.add ContDiffAt.add /-- The sum of two `C^n`functions is `C^n`. -/ theorem ContDiff.add {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => f x + g x := contDiff_add.comp (hf.prod hg) #align cont_diff.add ContDiff.add /-- The sum of two `C^n` functions on a domain is `C^n`. -/ theorem ContDiffOn.add {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x + g x) s := fun x hx => (hf x hx).add (hg x hx) #align cont_diff_on.add ContDiffOn.add variable {i : ℕ} /-- The iterated derivative of the sum of two functions is the sum of the iterated derivatives. See also `iteratedFDerivWithin_add_apply'`, which uses the spelling `(fun x ↦ f x + g x)` instead of `f + g`. -/ theorem iteratedFDerivWithin_add_apply {f g : E → F} (hf : ContDiffOn 𝕜 i f s) (hg : ContDiffOn 𝕜 i g s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) : iteratedFDerivWithin 𝕜 i (f + g) s x = iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x := Eq.symm <\| ((hf.ftaylorSeriesWithin hu).add (hg.ftaylorSeriesWithin hu)).eq_iteratedFDerivWithin_of_uniqueDiffOn le_rfl hu hx #align iterated_fderiv_within_add_apply iteratedFDerivWithin_add_apply /-- The iterated derivative of the sum of two functions is the sum of the iterated derivatives. This is the same as `iteratedFDerivWithin_add_apply`, but using the spelling `(fun x ↦ f x + g x)` instead of `f + g`, which can be handy for some rewrites. TODO: use one form consistently. -/ theorem iteratedFDerivWithin_add_apply' {f g : E → F} (hf : ContDiffOn 𝕜 i f s) (hg : ContDiffOn 𝕜 i g s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) : iteratedFDerivWithin 𝕜 i (fun x => f x + g x) s x = iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x := iteratedFDerivWithin_add_apply hf hg hu hx #align iterated_fderiv_within_add_apply' iteratedFDerivWithin_add_apply' theorem iteratedFDeriv_add_apply {i : ℕ} {f g : E → F} (hf : ContDiff 𝕜 i f) (hg : ContDiff 𝕜 i g) : iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x := by simp_rw [← contDiffOn_univ, ← iteratedFDerivWithin_univ] at hf hg ⊢ exact iteratedFDerivWithin_add_apply hf hg uniqueDiffOn_univ (Set.mem_univ _) #align iterated_fderiv_add_apply iteratedFDeriv_add_apply theorem iteratedFDeriv_add_apply' {i : ℕ} {f g : E → F} (hf : ContDiff 𝕜 i f) (hg : ContDiff 𝕜 i g) : iteratedFDeriv 𝕜 i (fun x => f x + g x) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x := iteratedFDeriv_add_apply hf hg #align iterated_fderiv_add_apply' iteratedFDeriv_add_apply' end Add /-! ### Negative -/ section Neg -- The negative is smooth. theorem contDiff_neg : ContDiff 𝕜 n fun p : F => -p := IsBoundedLinearMap.id.neg.contDiff #align cont_diff_neg contDiff_neg /-- The negative of a `C^n` function within a domain at a point is `C^n` within this domain at this point. -/ theorem ContDiffWithinAt.neg {s : Set E} {f : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (fun x => -f x) s x := contDiff_neg.contDiffWithinAt.comp x hf subset_preimage_univ #align cont_diff_within_at.neg ContDiffWithinAt.neg /-- The negative of a `C^n` function at a point is `C^n` at this point. -/ theorem ContDiffAt.neg {f : E → F} (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun x => -f x) x := by rw [← contDiffWithinAt_univ] at ; exact hf.neg #align cont_diff_at.neg ContDiffAt.neg /-- The negative of a `C^n`function is `C^n`. -/ theorem ContDiff.neg {f : E → F} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => -f x := contDiff_neg.comp hf #align cont_diff.neg ContDiff.neg /-- The negative of a `C^n` function on a domain is `C^n`. -/ theorem ContDiffOn.neg {s : Set E} {f : E → F} (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (fun x => -f x) s := fun x hx => (hf x hx).neg #align cont_diff_on.neg ContDiffOn.neg variable {i : ℕ} -- Porting note (#11215): TODO: define `Neg` instance on `ContinuousLinearEquiv`, -- prove it from `ContinuousLinearEquiv.iteratedFDerivWithin_comp_left` theorem iteratedFDerivWithin_neg_apply {f : E → F} (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) : iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x := by induction' i with i hi generalizing x · ext; simp · ext h calc iteratedFDerivWithin 𝕜 (i + 1) (-f) s x h = fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (-f) s) s x (h 0) (Fin.tail h) := rfl _ = fderivWithin 𝕜 (-iteratedFDerivWithin 𝕜 i f s) s x (h 0) (Fin.tail h) := by rw [fderivWithin_congr' (@hi) hx]; rfl _ = -(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s) x (h 0) (Fin.tail h) := by rw [Pi.neg_def, fderivWithin_neg (hu x hx)]; rfl _ = -(iteratedFDerivWithin 𝕜 (i + 1) f s) x h := rfl #align iterated_fderiv_within_neg_apply iteratedFDerivWithin_neg_apply theorem iteratedFDeriv_neg_apply {i : ℕ} {f : E → F} : iteratedFDeriv 𝕜 i (-f) x = -iteratedFDeriv 𝕜 i f x := by simp_rw [← iteratedFDerivWithin_univ] exact iteratedFDerivWithin_neg_apply uniqueDiffOn_univ (Set.mem_univ _) #align iterated_fderiv_neg_apply iteratedFDeriv_neg_apply end Neg /-! ### Subtraction -/ /-- The difference of two `C^n` functions within a set at a point is `C^n` within this set at this point. -/ theorem ContDiffWithinAt.sub {s : Set E} {f g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x - g x) s x := by simpa only [sub_eq_add_neg] using hf.add hg.neg #align cont_diff_within_at.sub ContDiffWithinAt.sub /-- The difference of two `C^n` functions at a point is `C^n` at this point. -/ theorem ContDiffAt.sub {f g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x => f x - g x) x := by simpa only [sub_eq_add_neg] using hf.add hg.neg #align cont_diff_at.sub ContDiffAt.sub /-- The difference of two `C^n` functions on a domain is `C^n`. -/ theorem ContDiffOn.sub {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x - g x) s := by simpa only [sub_eq_add_neg] using hf.add hg.neg #align cont_diff_on.sub ContDiffOn.sub /-- The difference of two `C^n` functions is `C^n`. -/ theorem ContDiff.sub {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => f x - g x := by simpa only [sub_eq_add_neg] using hf.add hg.neg #align cont_diff.sub ContDiff.sub /-! ### Sum of finitely many functions -/ theorem ContDiffWithinAt.sum {ι : Type} {f : ι → E → F} {s : Finset ι} {t : Set E} {x : E} (h : ∀ i ∈ s, ContDiffWithinAt 𝕜 n (fun x => f i x) t x) : ContDiffWithinAt 𝕜 n (fun x => ∑ i ∈ s, f i x) t x := by classical induction' s using Finset.induction_on with i s is IH · simp [contDiffWithinAt_const] · simp only [is, Finset.sum_insert, not_false_iff] exact (h _ (Finset.mem_insert_self i s)).add (IH fun j hj => h _ (Finset.mem_insert_of_mem hj)) #align cont_diff_within_at.sum ContDiffWithinAt.sum theorem ContDiffAt.sum {ι : Type} {f : ι → E → F} {s : Finset ι} {x : E} (h : ∀ i ∈ s, ContDiffAt 𝕜 n (fun x => f i x) x) : ContDiffAt 𝕜 n (fun x => ∑ i ∈ s, f i x) x := by rw [← contDiffWithinAt_univ] at ; exact ContDiffWithinAt.sum h #align cont_diff_at.sum ContDiffAt.sum theorem ContDiffOn.sum {ι : Type} {f : ι → E → F} {s : Finset ι} {t : Set E} (h : ∀ i ∈ s, ContDiffOn 𝕜 n (fun x => f i x) t) : ContDiffOn 𝕜 n (fun x => ∑ i ∈ s, f i x) t := fun x hx => ContDiffWithinAt.sum fun i hi => h i hi x hx #align cont_diff_on.sum ContDiffOn.sum theorem ContDiff.sum {ι : Type} {f : ι → E → F} {s : Finset ι} (h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x) : ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x := by simp only [← contDiffOn_univ] at ; exact ContDiffOn.sum h #align cont_diff.sum ContDiff.sum theorem iteratedFDerivWithin_sum_apply {ι : Type} {f : ι → E → F} {u : Finset ι} {i : ℕ} {x : E} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (h : ∀ j ∈ u, ContDiffOn 𝕜 i (f j) s) : iteratedFDerivWithin 𝕜 i (∑ j ∈ u, f j ·) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x := by induction u using Finset.cons_induction with \| empty => ext; simp [hs, hx] \| cons a u ha IH => simp only [Finset.mem_cons, forall_eq_or_imp] at h simp only [Finset.sum_cons] rw [iteratedFDerivWithin_add_apply' h.1 (ContDiffOn.sum h.2) hs hx, IH h.2] theorem iteratedFDeriv_sum {ι : Type} {f : ι → E → F} {u : Finset ι} {i : ℕ} (h : ∀ j ∈ u, ContDiff 𝕜 i (f j)) : iteratedFDeriv 𝕜 i (∑ j ∈ u, f j ·) = ∑ j ∈ u, iteratedFDeriv 𝕜 i (f j) := funext fun x ↦ by simpa [iteratedFDerivWithin_univ] using iteratedFDerivWithin_sum_apply uniqueDiffOn_univ (mem_univ x) fun j hj ↦ (h j hj).contDiffOn /-! ### Product of two functions -/ section MulProd variable {𝔸 𝔸' ι 𝕜' : Type} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] -- The product is smooth. theorem contDiff_mul : ContDiff 𝕜 n fun p : 𝔸 × 𝔸 => p.1 p.2 := (ContinuousLinearMap.mul 𝕜 𝔸).isBoundedBilinearMap.contDiff #align cont_diff_mul contDiff_mul /-- The product of two `C^n` functions within a set at a point is `C^n` within this set at this point. -/ theorem ContDiffWithinAt.mul {s : Set E} {f g : E → 𝔸} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x * g x) s x := contDiff_mul.comp_contDiffWithinAt (hf.prod hg) #align cont_diff_within_at.mul ContDiffWithinAt.mul /-- The product of two `C^n` functions at a point is `C^n` at this point. -/ nonrec theorem ContDiffAt.mul {f g : E → 𝔸} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x => f x * g x) x := hf.mul hg #align cont_diff_at.mul ContDiffAt.mul /-- The product of two `C^n` functions on a domain is `C^n`. -/ theorem ContDiffOn.mul {f g : E → 𝔸} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x * g x) s := fun x hx => (hf x hx).mul (hg x hx) #align cont_diff_on.mul ContDiffOn.mul /-- The product of two `C^n`functions is `C^n`. -/ theorem ContDiff.mul {f g : E → 𝔸} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => f x * g x := contDiff_mul.comp (hf.prod hg) #align cont_diff.mul ContDiff.mul theorem contDiffWithinAt_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x) : ContDiffWithinAt 𝕜 n (∏ i ∈ t, f i) s x := Finset.prod_induction f (fun f => ContDiffWithinAt 𝕜 n f s x) (fun _ _ => ContDiffWithinAt.mul) (contDiffWithinAt_const (c := 1)) h #align cont_diff_within_at_prod' contDiffWithinAt_prod' theorem contDiffWithinAt_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x) : ContDiffWithinAt 𝕜 n (fun y => ∏ i ∈ t, f i y) s x := by simpa only [← Finset.prod_apply] using contDiffWithinAt_prod' h #align cont_diff_within_at_prod contDiffWithinAt_prod theorem contDiffAt_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffAt 𝕜 n (f i) x) : ContDiffAt 𝕜 n (∏ i ∈ t, f i) x := contDiffWithinAt_prod' h #align cont_diff_at_prod' contDiffAt_prod' theorem contDiffAt_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffAt 𝕜 n (f i) x) : ContDiffAt 𝕜 n (fun y => ∏ i ∈ t, f i y) x := contDiffWithinAt_prod h #align cont_diff_at_prod contDiffAt_prod theorem contDiffOn_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffOn 𝕜 n (f i) s) : ContDiffOn 𝕜 n (∏ i ∈ t, f i) s := fun x hx => contDiffWithinAt_prod' fun i hi => h i hi x hx #align cont_diff_on_prod' contDiffOn_prod' theorem contDiffOn_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffOn 𝕜 n (f i) s) : ContDiffOn 𝕜 n (fun y => ∏ i ∈ t, f i y) s := fun x hx => contDiffWithinAt_prod fun i hi => h i hi x hx #align cont_diff_on_prod contDiffOn_prod theorem contDiff_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiff 𝕜 n (f i)) : ContDiff 𝕜 n (∏ i ∈ t, f i) := contDiff_iff_contDiffAt.mpr fun _ => contDiffAt_prod' fun i hi => (h i hi).contDiffAt #align cont_diff_prod' contDiff_prod' theorem contDiff_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiff 𝕜 n (f i)) : ContDiff 𝕜 n fun y => ∏ i ∈ t, f i y := contDiff_iff_contDiffAt.mpr fun _ => contDiffAt_prod fun i hi => (h i hi).contDiffAt #align cont_diff_prod contDiff_prod theorem ContDiff.pow {f : E → 𝔸} (hf : ContDiff 𝕜 n f) : ∀ m : ℕ, ContDiff 𝕜 n fun x => f x ^ m \| 0 => by simpa using contDiff_const \| m + 1 => by simpa [pow_succ] using (hf.pow m).mul hf #align cont_diff.pow ContDiff.pow theorem ContDiffWithinAt.pow {f : E → 𝔸} (hf : ContDiffWithinAt 𝕜 n f s x) (m : ℕ) : ContDiffWithinAt 𝕜 n (fun y => f y ^ m) s x := (contDiff_id.pow m).comp_contDiffWithinAt hf #align cont_diff_within_at.pow ContDiffWithinAt.pow nonrec theorem ContDiffAt.pow {f : E → 𝔸} (hf : ContDiffAt 𝕜 n f x) (m : ℕ) : ContDiffAt 𝕜 n (fun y => f y ^ m) x := hf.pow m #align cont_diff_at.pow ContDiffAt.pow theorem ContDiffOn.pow {f : E → 𝔸} (hf : ContDiffOn 𝕜 n f s) (m : ℕ) : ContDiffOn 𝕜 n (fun y => f y ^ m) s := fun y hy => (hf y hy).pow m #align cont_diff_on.pow ContDiffOn.pow theorem ContDiffWithinAt.div_const {f : E → 𝕜'} {n} (hf : ContDiffWithinAt 𝕜 n f s x) (c : 𝕜') : ContDiffWithinAt 𝕜 n (fun x => f x / c) s x := by simpa only [div_eq_mul_inv] using hf.mul contDiffWithinAt_const #align cont_diff_within_at.div_const ContDiffWithinAt.div_const nonrec theorem ContDiffAt.div_const {f : E → 𝕜'} {n} (hf : ContDiffAt 𝕜 n f x) (c : 𝕜') : ContDiffAt 𝕜 n (fun x => f x / c) x := hf.div_const c #align cont_diff_at.div_const ContDiffAt.div_const theorem ContDiffOn.div_const {f : E → 𝕜'} {n} (hf : ContDiffOn 𝕜 n f s) (c : 𝕜') : ContDiffOn 𝕜 n (fun x => f x / c) s := fun x hx => (hf x hx).div_const c #align cont_diff_on.div_const ContDiffOn.div_const theorem ContDiff.div_const {f : E → 𝕜'} {n} (hf : ContDiff 𝕜 n f) (c : 𝕜') : ContDiff 𝕜 n fun x => f x / c := by simpa only [div_eq_mul_inv] using hf.mul contDiff_const #align cont_diff.div_const ContDiff.div_const end MulProd /-! ### Scalar multiplication -/ section SMul -- The scalar multiplication is smooth. theorem contDiff_smul : ContDiff 𝕜 n fun p : 𝕜 × F => p.1 • p.2 := isBoundedBilinearMap_smul.contDiff #align cont_diff_smul contDiff_smul /-- The scalar multiplication of two `C^n` functions within a set at a point is `C^n` within this set at this point. -/ theorem ContDiffWithinAt.smul {s : Set E} {f : E → 𝕜} {g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x • g x) s x := contDiff_smul.contDiffWithinAt.comp x (hf.prod hg) subset_preimage_univ #align cont_diff_within_at.smul ContDiffWithinAt.smul /-- The scalar multiplication of two `C^n` functions at a point is `C^n` at this point. -/ theorem ContDiffAt.smul {f : E → 𝕜} {g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x => f x • g x) x := by rw [← contDiffWithinAt_univ] at ; exact hf.smul hg #align cont_diff_at.smul ContDiffAt.smul /-- The scalar multiplication of two `C^n` functions is `C^n`. -/ theorem ContDiff.smul {f : E → 𝕜} {g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => f x • g x := contDiff_smul.comp (hf.prod hg) #align cont_diff.smul ContDiff.smul /-- The scalar multiplication of two `C^n` functions on a domain is `C^n`. -/ theorem ContDiffOn.smul {s : Set E} {f : E → 𝕜} {g : E → F} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x • g x) s := fun x hx => (hf x hx).smul (hg x hx) #align cont_diff_on.smul ContDiffOn.smul end SMul /-! ### Constant scalar multiplication Porting note (#11215): TODO: generalize results in this section. 1. It should be possible to assume `[Monoid R] [DistribMulAction R F] [SMulCommClass 𝕜 R F]`. 2. If `c` is a unit (or `R` is a group), then one can drop `ContDiff` assumptions in some lemmas. -/ section ConstSMul variable {R : Type} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] variable [ContinuousConstSMul R F] -- The scalar multiplication with a constant is smooth. theorem contDiff_const_smul (c : R) : ContDiff 𝕜 n fun p : F => c • p := (c • ContinuousLinearMap.id 𝕜 F).contDiff #align cont_diff_const_smul contDiff_const_smul /-- The scalar multiplication of a constant and a `C^n` function within a set at a point is `C^n` within this set at this point. -/ theorem ContDiffWithinAt.const_smul {s : Set E} {f : E → F} {x : E} (c : R) (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (fun y => c • f y) s x := (contDiff_const_smul c).contDiffAt.comp_contDiffWithinAt x hf #align cont_diff_within_at.const_smul ContDiffWithinAt.const_smul /-- The scalar multiplication of a constant and a `C^n` function at a point is `C^n` at this point. -/ theorem ContDiffAt.const_smul {f : E → F} {x : E} (c : R) (hf : ContDiffAt 𝕜 n f x) : ContDiffAt 𝕜 n (fun y => c • f y) x := by rw [← contDiffWithinAt_univ] at ; exact hf.const_smul c #align cont_diff_at.const_smul ContDiffAt.const_smul /-- The scalar multiplication of a constant and a `C^n` function is `C^n`. -/ theorem ContDiff.const_smul {f : E → F} (c : R) (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun y => c • f y := (contDiff_const_smul c).comp hf #align cont_diff.const_smul ContDiff.const_smul /-- The scalar multiplication of a constant and a `C^n` on a domain is `C^n`. -/ theorem ContDiffOn.const_smul {s : Set E} {f : E → F} (c : R) (hf : ContDiffOn 𝕜 n f s) : ContDiffOn 𝕜 n (fun y => c • f y) s := fun x hx => (hf x hx).const_smul c #align cont_diff_on.const_smul ContDiffOn.const_smul variable {i : ℕ} {a : R} theorem iteratedFDerivWithin_const_smul_apply (hf : ContDiffOn 𝕜 i f s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) : iteratedFDerivWithin 𝕜 i (a • f) s x = a • iteratedFDerivWithin 𝕜 i f s x := (a • (1 : F →L[𝕜] F)).iteratedFDerivWithin_comp_left hf hu hx le_rfl #align iterated_fderiv_within_const_smul_apply iteratedFDerivWithin_const_smul_apply theorem iteratedFDeriv_const_smul_apply {x : E} (hf : ContDiff 𝕜 i f) : iteratedFDeriv 𝕜 i (a • f) x = a • iteratedFDeriv 𝕜 i f x := by simp_rw [← contDiffOn_univ, ← iteratedFDerivWithin_univ] at * exact iteratedFDerivWithin_const_smul_apply hf uniqueDiffOn_univ (Set.mem_univ _) #align iterated_fderiv_const_smul_apply iteratedFDeriv_const_smul_apply theorem iteratedFDeriv_const_smul_apply' {x : E} (hf : ContDiff 𝕜 i f) : iteratedFDeriv 𝕜 i (fun x ↦ a • f x) x = a • iteratedFDeriv 𝕜 i f x := iteratedFDeriv_const_smul_apply hf end ConstSMul /-! ### Cartesian product of two functions -/ section prodMap variable {E' : Type} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] variable {F' : Type} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] /-- The product map of two `C^n` functions within a set at a point is `C^n` within the product set at the product point. -/ theorem ContDiffWithinAt.prod_map' {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} {p : E × E'} (hf : ContDiffWithinAt 𝕜 n f s p.1) (hg : ContDiffWithinAt 𝕜 n g t p.2) : ContDiffWithinAt 𝕜 n (Prod.map f g) (s ×ˢ t) p := (hf.comp p contDiffWithinAt_fst (prod_subset_preimage_fst _ _)).prod (hg.comp p contDiffWithinAt_snd (prod_subset_preimage_snd _ _)) #align cont_diff_within_at.prod_map' ContDiffWithinAt.prod_map' theorem ContDiffWithinAt.prod_map {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} {x : E} {y : E'} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g t y) : ContDiffWithinAt 𝕜 n (Prod.map f g) (s ×ˢ t) (x, y) := ContDiffWithinAt.prod_map' hf hg #align cont_diff_within_at.prod_map ContDiffWithinAt.prod_map /-- The product map of two `C^n` functions on a set is `C^n` on the product set. -/ theorem ContDiffOn.prod_map {E' : Type} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g t) : ContDiffOn 𝕜 n (Prod.map f g) (s ×ˢ t) := (hf.comp contDiffOn_fst (prod_subset_preimage_fst _ _)).prod (hg.comp contDiffOn_snd (prod_subset_preimage_snd _ _)) #align cont_diff_on.prod_map ContDiffOn.prod_map /-- The product map of two `C^n` functions within a set at a point is `C^n` within the product set at the product point. -/ theorem ContDiffAt.prod_map {f : E → F} {g : E' → F'} {x : E} {y : E'} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g y) : ContDiffAt 𝕜 n (Prod.map f g) (x, y) := by rw [ContDiffAt] at * convert hf.prod_map hg simp only [univ_prod_univ] #align cont_diff_at.prod_map ContDiffAt.prod_map /-- The product map of two `C^n` functions within a set at a point is `C^n` within the product set at the product point. -/ theorem ContDiffAt.prod_map' {f : E → F} {g : E' → F'} {p : E × E'} (hf : ContDiffAt 𝕜 n f p.1) (hg : ContDiffAt 𝕜 n g p.2) : ContDiffAt 𝕜 n (Prod.map f g) p := by rcases p with ⟨⟩ exact ContDiffAt.prod_map hf hg #align cont_diff_at.prod_map' ContDiffAt.prod_map' /-- The product map of two `C^n` functions is `C^n`. -/ theorem ContDiff.prod_map {f : E → F} {g : E' → F'} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n (Prod.map f g) := by rw [contDiff_iff_contDiffAt] at * exact fun ⟨x, y⟩ => (hf x).prod_map (hg y) #align cont_diff.prod_map ContDiff.prod_map theorem contDiff_prod_mk_left (f₀ : F) : ContDiff 𝕜 n fun e : E => (e, f₀) := contDiff_id.prod contDiff_const #align cont_diff_prod_mk_left contDiff_prod_mk_left theorem contDiff_prod_mk_right (e₀ : E) : ContDiff 𝕜 n fun f : F => (e₀, f) := contDiff_const.prod contDiff_id #align cont_diff_prod_mk_right contDiff_prod_mk_right end prodMap /-! ### Inversion in a complete normed algebra -/ section AlgebraInverse variable (𝕜) {R : Type*} [NormedRing R] -- Porting note: this couldn't be on the same line as the binder type update of `𝕜` variable [NormedAlgebra 𝕜 R] open NormedRing ContinuousLinearMap Ring /-- In a complete normed algebra, the operation of inversion is `C^n`, for all `n`, at each invertible element. The proof is by induction, bootstrapping using an identity expressing the derivative of inversion as a bilinear map of inversion itself. -/	Mathlib/Analysis/Calculus/ContDiff/Basic.lean	1,758	1,779	theorem contDiffAt_ring_inverse [CompleteSpace R] (x : Rˣ) : ContDiffAt 𝕜 n Ring.inverse (x : R) := by	induction' n using ENat.nat_induction with n IH Itop · intro m hm refine ⟨{ y : R \| IsUnit y }, ?_, ?_⟩ · simp [nhdsWithin_univ] exact x.nhds · use ftaylorSeriesWithin 𝕜 inverse univ rw [le_antisymm hm bot_le, hasFTaylorSeriesUpToOn_zero_iff] constructor · rintro _ ⟨x', rfl⟩ exact (inverse_continuousAt x').continuousWithinAt · simp [ftaylorSeriesWithin] · rw [contDiffAt_succ_iff_hasFDerivAt] refine ⟨fun x : R => -mulLeftRight 𝕜 R (inverse x) (inverse x), ?_, ?_⟩ · refine ⟨{ y : R \| IsUnit y }, x.nhds, ?_⟩ rintro _ ⟨y, rfl⟩ simp_rw [inverse_unit] exact hasFDerivAt_ring_inverse y · convert (mulLeftRight_isBoundedBilinear 𝕜 R).contDiff.neg.comp_contDiffAt (x : R) (IH.prod IH) · exact contDiffAt_top.mpr Itop
/- Copyright (c) 2019 Jan-David Salchow. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo -/ import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm import Mathlib.Analysis.NormedSpace.Span /-! # Operator norm for maps on normed spaces This file contains statements about operator norm for which it really matters that the underlying space has a norm (rather than just a seminorm). -/ suppress_compilation open Bornology open Filter hiding map_smul open scoped Classical NNReal Topology Uniformity -- the `ₗ` subscript variables are for special cases about linear (as opposed to semilinear) maps variable {𝕜 𝕜₂ 𝕜₃ E Eₗ F Fₗ G Gₗ 𝓕 : Type} section Normed variable [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] [NormedAddCommGroup Fₗ] open Metric ContinuousLinearMap section variable [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] [NormedSpace 𝕜 Fₗ] (c : 𝕜) {σ₁₂ : 𝕜 →+ 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} (f g : E →SL[σ₁₂] F) (x y z : E) namespace LinearMap theorem bound_of_shell [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) : ‖f x‖ ≤ C * ‖x‖ := by by_cases hx : x = 0; · simp [hx] exact SemilinearMapClass.bound_of_shell_semi_normed f ε_pos hc hf (norm_ne_zero_iff.2 hx) #align linear_map.bound_of_shell LinearMap.bound_of_shell /-- `LinearMap.bound_of_ball_bound'` is a version of this lemma over a field satisfying `RCLike` that produces a concrete bound. -/ theorem bound_of_ball_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] Fₗ) (h : ∀ z ∈ Metric.ball (0 : E) r, ‖f z‖ ≤ c) : ∃ C, ∀ z : E, ‖f z‖ ≤ C * ‖z‖ := by cases' @NontriviallyNormedField.non_trivial 𝕜 _ with k hk use c * (‖k‖ / r) intro z refine bound_of_shell _ r_pos hk (fun x hko hxo => ?_) _ calc ‖f x‖ ≤ c := h _ (mem_ball_zero_iff.mpr hxo) _ ≤ c * (‖x‖ * ‖k‖ / r) := le_mul_of_one_le_right ?_ ?_ _ = _ := by ring · exact le_trans (norm_nonneg _) (h 0 (by simp [r_pos])) · rw [div_le_iff (zero_lt_one.trans hk)] at hko exact (one_le_div r_pos).mpr hko #align linear_map.bound_of_ball_bound LinearMap.bound_of_ball_bound theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩ simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε lift ε to ℝ≥0 using ε0.le rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩ refine ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => ?_⟩ by_cases hx : f x = 0 · rw [← hx] at hf obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <\| ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf) exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _)) have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc) rw [← h.1] at hc rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩ simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle calc ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖ := by rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)] _ ≤ ‖c ^ n‖⁻¹ * 1 := (mul_le_mul_of_nonneg_left (hε _ hlt).le (inv_nonneg.2 (norm_nonneg _))) _ ≤ ε⁻¹ * ‖c‖ * ‖f x‖ := by rwa [mul_one] #align linear_map.antilipschitz_of_comap_nhds_le LinearMap.antilipschitz_of_comap_nhds_le end LinearMap namespace ContinuousLinearMap section OpNorm open Set Real /-- An operator is zero iff its norm vanishes. -/	Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean	99	107	theorem opNorm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 := Iff.intro (fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1 (calc _ ≤ ‖f‖ * ‖x‖ := le_opNorm _ _ _ = _ := by	rw [hn, zero_mul])) (by rintro rfl exact opNorm_zero)
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import Mathlib.MeasureTheory.OuterMeasure.Caratheodory /-! # Induced Outer Measure We can extend a function defined on a subset of `Set α` to an outer measure. The underlying function is called `extend`, and the measure it induces is called `inducedOuterMeasure`. Some lemmas below are proven twice, once in the general case, and one where the function `m` is only defined on measurable sets (i.e. when `P = MeasurableSet`). In the latter cases, we can remove some hypotheses in the statement. The general version has the same name, but with a prime at the end. ## Tags outer measure -/ #align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55" noncomputable section open Set Function Filter open scoped Classical NNReal Topology ENNReal namespace MeasureTheory open OuterMeasure section Extend variable {α : Type} {P : α → Prop} variable (m : ∀ s : α, P s → ℝ≥0∞) /-- We can trivially extend a function defined on a subclass of objects (with codomain `ℝ≥0∞`) to all objects by defining it to be `∞` on the objects not in the class. -/ def extend (s : α) : ℝ≥0∞ := ⨅ h : P s, m s h #align measure_theory.extend MeasureTheory.extend theorem extend_eq {s : α} (h : P s) : extend m s = m s h := by simp [extend, h] #align measure_theory.extend_eq MeasureTheory.extend_eq theorem extend_eq_top {s : α} (h : ¬P s) : extend m s = ∞ := by simp [extend, h] #align measure_theory.extend_eq_top MeasureTheory.extend_eq_top theorem smul_extend {R} [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [NoZeroSMulDivisors R ℝ≥0∞] {c : R} (hc : c ≠ 0) : c • extend m = extend fun s h => c • m s h := by ext1 s dsimp [extend] by_cases h : P s · simp [h] · simp [h, ENNReal.smul_top, hc] #align measure_theory.smul_extend MeasureTheory.smul_extend theorem le_extend {s : α} (h : P s) : m s h ≤ extend m s := by simp only [extend, le_iInf_iff] intro rfl #align measure_theory.le_extend MeasureTheory.le_extend -- TODO: why this is a bad `congr` lemma? theorem extend_congr {β : Type} {Pb : β → Prop} {mb : ∀ s : β, Pb s → ℝ≥0∞} {sa : α} {sb : β} (hP : P sa ↔ Pb sb) (hm : ∀ (ha : P sa) (hb : Pb sb), m sa ha = mb sb hb) : extend m sa = extend mb sb := iInf_congr_Prop hP fun _h => hm _ _ #align measure_theory.extend_congr MeasureTheory.extend_congr @[simp] theorem extend_top {α : Type} {P : α → Prop} : extend (fun _ _ => ∞ : ∀ s : α, P s → ℝ≥0∞) = ⊤ := funext fun _ => iInf_eq_top.mpr fun _ => rfl #align measure_theory.extend_top MeasureTheory.extend_top end Extend section ExtendSet variable {α : Type} {P : Set α → Prop} variable {m : ∀ s : Set α, P s → ℝ≥0∞} variable (P0 : P ∅) (m0 : m ∅ P0 = 0) variable (PU : ∀ ⦃f : ℕ → Set α⦄ (_hm : ∀ i, P (f i)), P (⋃ i, f i)) variable (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ i, P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (PU hm) = ∑' i, m (f i) (hm i)) variable (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ i, P (f i)), m (⋃ i, f i) (PU hm) ≤ ∑' i, m (f i) (hm i)) variable (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) theorem extend_empty : extend m ∅ = 0 := (extend_eq _ P0).trans m0 #align measure_theory.extend_empty MeasureTheory.extend_empty theorem extend_iUnion_nat {f : ℕ → Set α} (hm : ∀ i, P (f i)) (mU : m (⋃ i, f i) (PU hm) = ∑' i, m (f i) (hm i)) : extend m (⋃ i, f i) = ∑' i, extend m (f i) := (extend_eq _ _).trans <\| mU.trans <\| by congr with i rw [extend_eq] #align measure_theory.extend_Union_nat MeasureTheory.extend_iUnion_nat section Subadditive theorem extend_iUnion_le_tsum_nat' (s : ℕ → Set α) : extend m (⋃ i, s i) ≤ ∑' i, extend m (s i) := by by_cases h : ∀ i, P (s i) · rw [extend_eq _ (PU h), congr_arg tsum _] · apply msU h funext i apply extend_eq _ (h i) · cases' not_forall.1 h with i hi exact le_trans (le_iInf fun h => hi.elim h) (ENNReal.le_tsum i) #align measure_theory.extend_Union_le_tsum_nat' MeasureTheory.extend_iUnion_le_tsum_nat' end Subadditive section Mono theorem extend_mono' ⦃s₁ s₂ : Set α⦄ (h₁ : P s₁) (hs : s₁ ⊆ s₂) : extend m s₁ ≤ extend m s₂ := by refine le_iInf ?_ intro h₂ rw [extend_eq m h₁] exact m_mono h₁ h₂ hs #align measure_theory.extend_mono' MeasureTheory.extend_mono' end Mono section Unions theorem extend_iUnion {β} [Countable β] {f : β → Set α} (hd : Pairwise (Disjoint on f)) (hm : ∀ i, P (f i)) : extend m (⋃ i, f i) = ∑' i, extend m (f i) := by cases nonempty_encodable β rw [← Encodable.iUnion_decode₂, ← tsum_iUnion_decode₂] · exact extend_iUnion_nat PU (fun n => Encodable.iUnion_decode₂_cases P0 hm) (mU _ (Encodable.iUnion_decode₂_disjoint_on hd)) · exact extend_empty P0 m0 #align measure_theory.extend_Union MeasureTheory.extend_iUnion theorem extend_union {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h₁ : P s₁) (h₂ : P s₂) : extend m (s₁ ∪ s₂) = extend m s₁ + extend m s₂ := by rw [union_eq_iUnion, extend_iUnion P0 m0 PU mU (pairwise_disjoint_on_bool.2 hd) (Bool.forall_bool.2 ⟨h₂, h₁⟩), tsum_fintype] simp #align measure_theory.extend_union MeasureTheory.extend_union end Unions variable (m) /-- Given an arbitrary function on a subset of sets, we can define the outer measure corresponding to it (this is the unique maximal outer measure that is at most `m` on the domain of `m`). -/ def inducedOuterMeasure : OuterMeasure α := OuterMeasure.ofFunction (extend m) (extend_empty P0 m0) #align measure_theory.induced_outer_measure MeasureTheory.inducedOuterMeasure variable {m P0 m0} theorem le_inducedOuterMeasure {μ : OuterMeasure α} : μ ≤ inducedOuterMeasure m P0 m0 ↔ ∀ (s) (hs : P s), μ s ≤ m s hs := le_ofFunction.trans <\| forall_congr' fun _s => le_iInf_iff #align measure_theory.le_induced_outer_measure MeasureTheory.le_inducedOuterMeasure /-- If `P u` is `False` for any set `u` that has nonempty intersection both with `s` and `t`, then `μ (s ∪ t) = μ s + μ t`, where `μ = inducedOuterMeasure m P0 m0`. E.g., if `α` is an (e)metric space and `P u = diam u < r`, then this lemma implies that `μ (s ∪ t) = μ s + μ t` on any two sets such that `r ≤ edist x y` for all `x ∈ s` and `y ∈ t`. -/ theorem inducedOuterMeasure_union_of_false_of_nonempty_inter {s t : Set α} (h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → ¬P u) : inducedOuterMeasure m P0 m0 (s ∪ t) = inducedOuterMeasure m P0 m0 s + inducedOuterMeasure m P0 m0 t := ofFunction_union_of_top_of_nonempty_inter fun u hsu htu => @iInf_of_empty _ _ _ ⟨h u hsu htu⟩ _ #align measure_theory.induced_outer_measure_union_of_false_of_nonempty_inter MeasureTheory.inducedOuterMeasure_union_of_false_of_nonempty_inter theorem inducedOuterMeasure_eq_extend' {s : Set α} (hs : P s) : inducedOuterMeasure m P0 m0 s = extend m s := ofFunction_eq s (fun _t => extend_mono' m_mono hs) (extend_iUnion_le_tsum_nat' PU msU) #align measure_theory.induced_outer_measure_eq_extend' MeasureTheory.inducedOuterMeasure_eq_extend' theorem inducedOuterMeasure_eq' {s : Set α} (hs : P s) : inducedOuterMeasure m P0 m0 s = m s hs := (inducedOuterMeasure_eq_extend' PU msU m_mono hs).trans <\| extend_eq _ _ #align measure_theory.induced_outer_measure_eq' MeasureTheory.inducedOuterMeasure_eq' theorem inducedOuterMeasure_eq_iInf (s : Set α) : inducedOuterMeasure m P0 m0 s = ⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht := by apply le_antisymm · simp only [le_iInf_iff] intro t ht hs refine le_trans (measure_mono hs) ?_ exact le_of_eq (inducedOuterMeasure_eq' _ msU m_mono _) · refine le_iInf ?_ intro f refine le_iInf ?_ intro hf refine le_trans ?_ (extend_iUnion_le_tsum_nat' _ msU _) refine le_iInf ?_ intro h2f exact iInf_le_of_le _ (iInf_le_of_le h2f <\| iInf_le _ hf) #align measure_theory.induced_outer_measure_eq_infi MeasureTheory.inducedOuterMeasure_eq_iInf theorem inducedOuterMeasure_preimage (f : α ≃ α) (Pm : ∀ s : Set α, P (f ⁻¹' s) ↔ P s) (mm : ∀ (s : Set α) (hs : P s), m (f ⁻¹' s) ((Pm _).mpr hs) = m s hs) {A : Set α} : inducedOuterMeasure m P0 m0 (f ⁻¹' A) = inducedOuterMeasure m P0 m0 A := by rw [inducedOuterMeasure_eq_iInf _ msU m_mono, inducedOuterMeasure_eq_iInf _ msU m_mono]; symm refine f.injective.preimage_surjective.iInf_congr (preimage f) fun s => ?_ refine iInf_congr_Prop (Pm s) ?_; intro hs refine iInf_congr_Prop f.surjective.preimage_subset_preimage_iff ?_ intro _; exact mm s hs #align measure_theory.induced_outer_measure_preimage MeasureTheory.inducedOuterMeasure_preimage theorem inducedOuterMeasure_exists_set {s : Set α} (hs : inducedOuterMeasure m P0 m0 s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ t : Set α, P t ∧ s ⊆ t ∧ inducedOuterMeasure m P0 m0 t ≤ inducedOuterMeasure m P0 m0 s + ε := by have h := ENNReal.lt_add_right hs hε conv at h => lhs rw [inducedOuterMeasure_eq_iInf _ msU m_mono] simp only [iInf_lt_iff] at h rcases h with ⟨t, h1t, h2t, h3t⟩ exact ⟨t, h1t, h2t, le_trans (le_of_eq <\| inducedOuterMeasure_eq' _ msU m_mono h1t) (le_of_lt h3t)⟩ #align measure_theory.induced_outer_measure_exists_set MeasureTheory.inducedOuterMeasure_exists_set /-- To test whether `s` is Carathéodory-measurable we only need to check the sets `t` for which `P t` holds. See `ofFunction_caratheodory` for another way to show the Carathéodory-measurability of `s`. -/ theorem inducedOuterMeasure_caratheodory (s : Set α) : MeasurableSet[(inducedOuterMeasure m P0 m0).caratheodory] s ↔ ∀ t : Set α, P t → inducedOuterMeasure m P0 m0 (t ∩ s) + inducedOuterMeasure m P0 m0 (t \ s) ≤ inducedOuterMeasure m P0 m0 t := by rw [isCaratheodory_iff_le] constructor · intro h t _ht exact h t · intro h u conv_rhs => rw [inducedOuterMeasure_eq_iInf _ msU m_mono] refine le_iInf ?_ intro t refine le_iInf ?_ intro ht refine le_iInf ?_ intro h2t refine le_trans ?_ ((h t ht).trans_eq <\| inducedOuterMeasure_eq' _ msU m_mono ht) gcongr #align measure_theory.induced_outer_measure_caratheodory MeasureTheory.inducedOuterMeasure_caratheodory end ExtendSet /-! If `P` is `MeasurableSet` for some measurable space, then we can remove some hypotheses of the above lemmas. -/ section MeasurableSpace variable {α : Type} [MeasurableSpace α] variable {m : ∀ s : Set α, MeasurableSet s → ℝ≥0∞} variable (m0 : m ∅ MeasurableSet.empty = 0) variable (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ i, MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (MeasurableSet.iUnion hm) = ∑' i, m (f i) (hm i)) theorem extend_mono {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (hs : s₁ ⊆ s₂) : extend m s₁ ≤ extend m s₂ := by refine le_iInf ?_; intro h₂ have := extend_union MeasurableSet.empty m0 MeasurableSet.iUnion mU disjoint_sdiff_self_right h₁ (h₂.diff h₁) rw [union_diff_cancel hs] at this rw [← extend_eq m] exact le_iff_exists_add.2 ⟨_, this⟩ #align measure_theory.extend_mono MeasureTheory.extend_mono theorem extend_iUnion_le_tsum_nat : ∀ s : ℕ → Set α, extend m (⋃ i, s i) ≤ ∑' i, extend m (s i) := by refine extend_iUnion_le_tsum_nat' MeasurableSet.iUnion ?_; intro f h simp (config := { singlePass := true }) only [iUnion_disjointed.symm] rw [mU (MeasurableSet.disjointed h) (disjoint_disjointed _)] refine ENNReal.tsum_le_tsum fun i => ?_ rw [← extend_eq m, ← extend_eq m] exact extend_mono m0 mU (MeasurableSet.disjointed h _) (disjointed_le f _) #align measure_theory.extend_Union_le_tsum_nat MeasureTheory.extend_iUnion_le_tsum_nat theorem inducedOuterMeasure_eq_extend {s : Set α} (hs : MeasurableSet s) : inducedOuterMeasure m MeasurableSet.empty m0 s = extend m s := ofFunction_eq s (fun _t => extend_mono m0 mU hs) (extend_iUnion_le_tsum_nat m0 mU) #align measure_theory.induced_outer_measure_eq_extend MeasureTheory.inducedOuterMeasure_eq_extend theorem inducedOuterMeasure_eq {s : Set α} (hs : MeasurableSet s) : inducedOuterMeasure m MeasurableSet.empty m0 s = m s hs := (inducedOuterMeasure_eq_extend m0 mU hs).trans <\| extend_eq _ _ #align measure_theory.induced_outer_measure_eq MeasureTheory.inducedOuterMeasure_eq end MeasurableSpace namespace OuterMeasure variable {α : Type} [MeasurableSpace α] (m : OuterMeasure α) /-- Given an outer measure `m` we can forget its value on non-measurable sets, and then consider `m.trim`, the unique maximal outer measure less than that function. -/ def trim : OuterMeasure α := inducedOuterMeasure (fun s _ => m s) MeasurableSet.empty m.empty #align measure_theory.outer_measure.trim MeasureTheory.OuterMeasure.trim theorem le_trim_iff {m₁ m₂ : OuterMeasure α} : m₁ ≤ m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s ≤ m₂ s := le_inducedOuterMeasure #align measure_theory.outer_measure.le_trim_iff MeasureTheory.OuterMeasure.le_trim_iff theorem le_trim : m ≤ m.trim := le_trim_iff.2 fun _ _ ↦ le_rfl #align measure_theory.outer_measure.le_trim MeasureTheory.OuterMeasure.le_trim @[simp] -- Porting note: added `simp` theorem trim_eq {s : Set α} (hs : MeasurableSet s) : m.trim s = m s := inducedOuterMeasure_eq' MeasurableSet.iUnion (fun f _hf => measure_iUnion_le f) (fun _ _ _ _ h => measure_mono h) hs #align measure_theory.outer_measure.trim_eq MeasureTheory.OuterMeasure.trim_eq theorem trim_congr {m₁ m₂ : OuterMeasure α} (H : ∀ {s : Set α}, MeasurableSet s → m₁ s = m₂ s) : m₁.trim = m₂.trim := by simp (config := { contextual := true }) only [trim, H] #align measure_theory.outer_measure.trim_congr MeasureTheory.OuterMeasure.trim_congr @[mono] theorem trim_mono : Monotone (trim : OuterMeasure α → OuterMeasure α) := fun _m₁ _m₂ H _s => iInf₂_mono fun _f _hs => ENNReal.tsum_le_tsum fun _b => iInf_mono fun _hf => H _ #align measure_theory.outer_measure.trim_mono MeasureTheory.OuterMeasure.trim_mono /-- `OuterMeasure.trim` is antitone in the σ-algebra. -/ theorem trim_anti_measurableSpace (m : OuterMeasure α) {m0 m1 : MeasurableSpace α} (h : m0 ≤ m1) : @trim _ m1 m ≤ @trim _ m0 m := by simp only [le_trim_iff] intro s hs rw [trim_eq _ (h s hs)] theorem trim_le_trim_iff {m₁ m₂ : OuterMeasure α} : m₁.trim ≤ m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s ≤ m₂ s := le_trim_iff.trans <\| forall₂_congr fun s hs => by rw [trim_eq _ hs] #align measure_theory.outer_measure.trim_le_trim_iff MeasureTheory.OuterMeasure.trim_le_trim_iff theorem trim_eq_trim_iff {m₁ m₂ : OuterMeasure α} : m₁.trim = m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s = m₂ s := by simp only [le_antisymm_iff, trim_le_trim_iff, forall_and] #align measure_theory.outer_measure.trim_eq_trim_iff MeasureTheory.OuterMeasure.trim_eq_trim_iff	Mathlib/MeasureTheory/OuterMeasure/Induced.lean	363	367	theorem trim_eq_iInf (s : Set α) : m.trim s = ⨅ (t) (_ : s ⊆ t) (_ : MeasurableSet t), m t := by	simp (config := { singlePass := true }) only [iInf_comm] exact inducedOuterMeasure_eq_iInf MeasurableSet.iUnion (fun f _ => measure_iUnion_le f) (fun _ _ _ _ h => measure_mono h) s
/- Copyright (c) 2020 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel -/ import Mathlib.Analysis.Calculus.FDeriv.Measurable import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.Deriv.Add import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.NormedSpace.Dual import Mathlib.MeasureTheory.Integral.DominatedConvergence import Mathlib.MeasureTheory.Integral.VitaliCaratheodory #align_import measure_theory.integral.fund_thm_calculus from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # Fundamental Theorem of Calculus We prove various versions of the [fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) for interval integrals in `ℝ`. Recall that its first version states that the function `(u, v) ↦ ∫ x in u..v, f x` has derivative `(δu, δv) ↦ δv • f b - δu • f a` at `(a, b)` provided that `f` is continuous at `a` and `b`, and its second version states that, if `f` has an integrable derivative on `[a, b]`, then `∫ x in a..b, f' x = f b - f a`. ## Main statements ### FTC-1 for Lebesgue measure We prove several versions of FTC-1, all in the `intervalIntegral` namespace. Many of them follow the naming scheme `integral_has(Strict?)(F?)Deriv(Within?)At(_of_tendsto_ae?)(_right\|_left?)`. They formulate FTC in terms of `Has(Strict?)(F?)Deriv(Within?)At`. Let us explain the meaning of each part of the name: * `Strict` means that the theorem is about strict differentiability, see `HasStrictDerivAt` and `HasStrictFDerivAt`; * `F` means that the theorem is about differentiability in both endpoints; incompatible with `_right\|_left`; * `Within` means that the theorem is about one-sided derivatives, see below for details; * `_of_tendsto_ae` means that instead of continuity the theorem assumes that `f` has a finite limit almost surely as `x` tends to `a` and/or `b`; * `_right` or `_left` mean that the theorem is about differentiability in the right (resp., left) endpoint. We also reformulate these theorems in terms of `(f?)deriv(Within?)`. These theorems are named `(f?)deriv(Within?)_integral(_of_tendsto_ae?)(_right\|_left?)` with the same meaning of parts of the name. ### One-sided derivatives Theorem `intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae` states that `(u, v) ↦ ∫ x in u..v, f x` has a derivative `(δu, δv) ↦ δv • cb - δu • ca` within the set `s × t` at `(a, b)` provided that `f` tends to `ca` (resp., `cb`) almost surely at `la` (resp., `lb`), where possible values of `s`, `t`, and corresponding filters `la`, `lb` are given in the following table. \| `s` \| `la` \| `t` \| `lb` \| \| ------- \| ---- \| --- \| ---- \| \| `Iic a` \| `𝓝[≤] a` \| `Iic b` \| `𝓝[≤] b` \| \| `Ici a` \| `𝓝[>] a` \| `Ici b` \| `𝓝[>] b` \| \| `{a}` \| `⊥` \| `{b}` \| `⊥` \| \| `univ` \| `𝓝 a` \| `univ` \| `𝓝 b` \| We use a typeclass `intervalIntegral.FTCFilter` to make Lean automatically find `la`/`lb` based on `s`/`t`. This way we can formulate one theorem instead of `16` (or `8` if we leave only non-trivial ones not covered by `integral_hasDerivWithinAt_of_tendsto_ae_(left\|right)` and `integral_hasFDerivAt_of_tendsto_ae`). Similarly, `integral_hasDerivWithinAt_of_tendsto_ae_right` works for both one-sided derivatives using the same typeclass to find an appropriate filter. ### FTC for a locally finite measure Before proving FTC for the Lebesgue measure, we prove a few statements that can be seen as FTC for any measure. The most general of them, `measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae`, states the following. Let `(la, la')` be an `intervalIntegral.FTCFilter` pair of filters around `a` (i.e., `intervalIntegral.FTCFilter a la la'`) and let `(lb, lb')` be an `intervalIntegral.FTCFilter` pair of filters around `b`. If `f` has finite limits `ca` and `cb` almost surely at `la'` and `lb'`, respectively, then $$ \int_{va}^{vb} f ∂μ - \int_{ua}^{ub} f ∂μ = \int_{ub}^{vb} cb ∂μ - \int_{ua}^{va} ca ∂μ + o(‖∫_{ua}^{va} 1 ∂μ‖ + ‖∫_{ub}^{vb} (1:ℝ) ∂μ‖) $$ as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`. ### FTC-2 and corollaries We use FTC-1 to prove several versions of FTC-2 for the Lebesgue measure, using a similar naming scheme as for the versions of FTC-1. They include: * `intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le` - most general version, for functions with a right derivative * `intervalIntegral.integral_eq_sub_of_hasDerivAt` - version for functions with a derivative on an open set * `intervalIntegral.integral_deriv_eq_sub'` - version that is easiest to use when computing the integral of a specific function We then derive additional integration techniques from FTC-2: * `intervalIntegral.integral_mul_deriv_eq_deriv_mul` - integration by parts * `intervalIntegral.integral_comp_mul_deriv''` - integration by substitution Many applications of these theorems can be found in the file `Mathlib/Analysis/SpecialFunctions/Integrals.lean`. Note that the assumptions of FTC-2 are formulated in the form that `f'` is integrable. To use it in a context with the stronger assumption that `f'` is continuous, one can use `ContinuousOn.intervalIntegrable` or `ContinuousOn.integrableOn_Icc` or `ContinuousOn.integrableOn_uIcc`. ### `intervalIntegral.FTCFilter` class As explained above, many theorems in this file rely on the typeclass `intervalIntegral.FTCFilter (a : ℝ) (l l' : Filter ℝ)` to avoid code duplication. This typeclass combines four assumptions: - `pure a ≤ l`; - `l' ≤ 𝓝 a`; - `l'` has a basis of measurable sets; - if `u n` and `v n` tend to `l`, then for any `s ∈ l'`, `Ioc (u n) (v n)` is eventually included in `s`. This typeclass has the following “real” instances: `(a, pure a, ⊥)`, `(a, 𝓝[≥] a, 𝓝[>] a)`, `(a, 𝓝[≤] a, 𝓝[≤] a)`, `(a, 𝓝 a, 𝓝 a)`. Furthermore, we have the following instances that are equal to the previously mentioned instances: `(a, 𝓝[{a}] a, ⊥)` and `(a, 𝓝[univ] a, 𝓝[univ] a)`. While the difference between `Ici a` and `Ioi a` doesn't matter for theorems about Lebesgue measure, it becomes important in the versions of FTC about any locally finite measure if this measure has an atom at one of the endpoints. ### Combining one-sided and two-sided derivatives There are some `intervalIntegral.FTCFilter` instances where the fact that it is one-sided or two-sided depends on the point, namely `(x, 𝓝[Set.Icc a b] x, 𝓝[Set.Icc a b] x)` (resp. `(x, 𝓝[Set.uIcc a b] x, 𝓝[Set.uIcc a b] x)`, with `x ∈ Icc a b` (resp. `x ∈ uIcc a b`). This results in a two-sided derivatives for `x ∈ Set.Ioo a b` and one-sided derivatives for `x ∈ {a, b}`. Other instances could be added when needed (in that case, one also needs to add instances for `Filter.IsMeasurablyGenerated` and `Filter.TendstoIxxClass`). ## Tags integral, fundamental theorem of calculus, FTC-1, FTC-2, change of variables in integrals -/ set_option autoImplicit true noncomputable section open scoped Classical open MeasureTheory Set Filter Function open scoped Classical Topology Filter ENNReal Interval NNReal variable {ι 𝕜 E F A : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] namespace intervalIntegral section FTC1 /-! ### Fundamental theorem of calculus, part 1, for any measure In this section we prove a few lemmas that can be seen as versions of FTC-1 for interval integrals w.r.t. any measure. Many theorems are formulated for one or two pairs of filters related by `intervalIntegral.FTCFilter a l l'`. This typeclass has exactly four “real” instances: `(a, pure a, ⊥)`, `(a, 𝓝[≥] a, 𝓝[>] a)`, `(a, 𝓝[≤] a, 𝓝[≤] a)`, `(a, 𝓝 a, 𝓝 a)`, and two instances that are equal to the first and last “real” instances: `(a, 𝓝[{a}] a, ⊥)` and `(a, 𝓝[univ] a, 𝓝[univ] a)`. We use this approach to avoid repeating arguments in many very similar cases. Lean can automatically find both `a` and `l'` based on `l`. The most general theorem `measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae` can be seen as a generalization of lemma `integral_hasStrictFDerivAt` below which states strict differentiability of `∫ x in u..v, f x` in `(u, v)` at `(a, b)` for a measurable function `f` that is integrable on `a..b` and is continuous at `a` and `b`. The lemma is generalized in three directions: first, `measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae` deals with any locally finite measure `μ`; second, it works for one-sided limits/derivatives; third, it assumes only that `f` has finite limits almost surely at `a` and `b`. Namely, let `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `intervalIntegral.FTCFilter`s around `a`; let `(lb, lb')` be a pair of `intervalIntegral.FTCFilter`s around `b`. Suppose that `f` has finite limits `ca` and `cb` at `la' ⊓ ae μ` and `lb' ⊓ ae μ`, respectively. Then `∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ + o(‖∫ x in ua..va, (1:ℝ) ∂μ‖ + ‖∫ x in ub..vb, (1:ℝ) ∂μ‖)` as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`. This theorem is formulated with integral of constants instead of measures in the right hand sides for two reasons: first, this way we avoid `min`/`max` in the statements; second, often it is possible to write better `simp` lemmas for these integrals, see `integral_const` and `integral_const_of_cdf`. In the next subsection we apply this theorem to prove various theorems about differentiability of the integral w.r.t. Lebesgue measure. -/ /-- An auxiliary typeclass for the Fundamental theorem of calculus, part 1. It is used to formulate theorems that work simultaneously for left and right one-sided derivatives of `∫ x in u..v, f x`. -/ class FTCFilter (a : outParam ℝ) (outer : Filter ℝ) (inner : outParam <\| Filter ℝ) extends TendstoIxxClass Ioc outer inner : Prop where pure_le : pure a ≤ outer le_nhds : inner ≤ 𝓝 a [meas_gen : IsMeasurablyGenerated inner] set_option linter.uppercaseLean3 false in #align interval_integral.FTC_filter intervalIntegral.FTCFilter namespace FTCFilter set_option linter.uppercaseLean3 false -- `FTC` in every name instance pure (a : ℝ) : FTCFilter a (pure a) ⊥ where pure_le := le_rfl le_nhds := bot_le #align interval_integral.FTC_filter.pure intervalIntegral.FTCFilter.pure instance nhdsWithinSingleton (a : ℝ) : FTCFilter a (𝓝[{a}] a) ⊥ := by rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)]; infer_instance #align interval_integral.FTC_filter.nhds_within_singleton intervalIntegral.FTCFilter.nhdsWithinSingleton theorem finiteAt_inner {a : ℝ} (l : Filter ℝ) {l'} [h : FTCFilter a l l'] {μ : Measure ℝ} [IsLocallyFiniteMeasure μ] : μ.FiniteAtFilter l' := (μ.finiteAt_nhds a).filter_mono h.le_nhds #align interval_integral.FTC_filter.finite_at_inner intervalIntegral.FTCFilter.finiteAt_inner instance nhds (a : ℝ) : FTCFilter a (𝓝 a) (𝓝 a) where pure_le := pure_le_nhds a le_nhds := le_rfl #align interval_integral.FTC_filter.nhds intervalIntegral.FTCFilter.nhds instance nhdsUniv (a : ℝ) : FTCFilter a (𝓝[univ] a) (𝓝 a) := by rw [nhdsWithin_univ]; infer_instance #align interval_integral.FTC_filter.nhds_univ intervalIntegral.FTCFilter.nhdsUniv instance nhdsLeft (a : ℝ) : FTCFilter a (𝓝[≤] a) (𝓝[≤] a) where pure_le := pure_le_nhdsWithin right_mem_Iic le_nhds := inf_le_left #align interval_integral.FTC_filter.nhds_left intervalIntegral.FTCFilter.nhdsLeft instance nhdsRight (a : ℝ) : FTCFilter a (𝓝[≥] a) (𝓝[>] a) where pure_le := pure_le_nhdsWithin left_mem_Ici le_nhds := inf_le_left #align interval_integral.FTC_filter.nhds_right intervalIntegral.FTCFilter.nhdsRight instance nhdsIcc {x a b : ℝ} [h : Fact (x ∈ Icc a b)] : FTCFilter x (𝓝[Icc a b] x) (𝓝[Icc a b] x) where pure_le := pure_le_nhdsWithin h.out le_nhds := inf_le_left #align interval_integral.FTC_filter.nhds_Icc intervalIntegral.FTCFilter.nhdsIcc instance nhdsUIcc {x a b : ℝ} [h : Fact (x ∈ [[a, b]])] : FTCFilter x (𝓝[[[a, b]]] x) (𝓝[[[a, b]]] x) := .nhdsIcc (h := h) #align interval_integral.FTC_filter.nhds_uIcc intervalIntegral.FTCFilter.nhdsUIcc end FTCFilter open Asymptotics section variable {f : ℝ → E} {a b : ℝ} {c ca cb : E} {l l' la la' lb lb' : Filter ℝ} {lt : Filter ι} {μ : Measure ℝ} {u v ua va ub vb : ι → ℝ} /-- Fundamental theorem of calculus-1, local version for any measure. Let filters `l` and `l'` be related by `TendstoIxxClass Ioc`. If `f` has a finite limit `c` at `l' ⊓ ae μ`, where `μ` is a measure finite at `l'`, then `∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both `u` and `v` tend to `l`. See also `measure_integral_sub_linear_isLittleO_of_tendsto_ae` for a version assuming `[intervalIntegral.FTCFilter a l l']` and `[MeasureTheory.IsLocallyFiniteMeasure μ]`. If `l` is one of `𝓝[≥] a`, `𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version. The primed version also works, e.g., for `l = l' = atTop`. We use integrals of constants instead of measures because this way it is easier to formulate a statement that works in both cases `u ≤ v` and `v ≤ u`. -/ theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae' [IsMeasurablyGenerated l'] [TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l) (hv : Tendsto v lt l) : (fun t => (∫ x in u t..v t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt] fun t => ∫ _ in u t..v t, (1 : ℝ) ∂μ := by by_cases hE : CompleteSpace E; swap · simp [intervalIntegral, integral, hE] have A := hf.integral_sub_linear_isLittleO_ae hfm hl (hu.Ioc hv) have B := hf.integral_sub_linear_isLittleO_ae hfm hl (hv.Ioc hu) simp_rw [integral_const', sub_smul] refine ((A.trans_le fun t ↦ ?_).sub (B.trans_le fun t ↦ ?_)).congr_left fun t ↦ ?_ · cases le_total (u t) (v t) <;> simp [] · cases le_total (u t) (v t) <;> simp [] · simp_rw [intervalIntegral] abel #align interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae' intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae' variable [CompleteSpace E] /-- Fundamental theorem of calculus-1, local version for any measure. Let filters `l` and `l'` be related by `TendstoIxxClass Ioc`. If `f` has a finite limit `c` at `l ⊓ ae μ`, where `μ` is a measure finite at `l`, then `∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both `u` and `v` tend to `l` so that `u ≤ v`. See also `measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le` for a version assuming `[intervalIntegral.FTCFilter a l l']` and `[MeasureTheory.IsLocallyFiniteMeasure μ]`. If `l` is one of `𝓝[≥] a`, `𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version. The primed version also works, e.g., for `l = l' = Filter.atTop`. -/ theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' [IsMeasurablyGenerated l'] [TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l) (hv : Tendsto v lt l) (huv : u ≤ᶠ[lt] v) : (fun t => (∫ x in u t..v t, f x ∂μ) - (μ (Ioc (u t) (v t))).toReal • c) =o[lt] fun t => (μ <\| Ioc (u t) (v t)).toReal := (measure_integral_sub_linear_isLittleO_of_tendsto_ae' hfm hf hl hu hv).congr' (huv.mono fun x hx => by simp [integral_const', hx]) (huv.mono fun x hx => by simp [integral_const', hx]) #align interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_le' intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' /-- Fundamental theorem of calculus-1, local version for any measure. Let filters `l` and `l'` be related by `TendstoIxxClass Ioc`. If `f` has a finite limit `c` at `l ⊓ ae μ`, where `μ` is a measure finite at `l`, then `∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both `u` and `v` tend to `l` so that `v ≤ u`. See also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge` for a version assuming `[intervalIntegral.FTCFilter a l l']` and `[MeasureTheory.IsLocallyFiniteMeasure μ]`. If `l` is one of `𝓝[≥] a`, `𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version. The primed version also works, e.g., for `l = l' = Filter.atTop`. -/ theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' [IsMeasurablyGenerated l'] [TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l) (hv : Tendsto v lt l) (huv : v ≤ᶠ[lt] u) : (fun t => (∫ x in u t..v t, f x ∂μ) + (μ (Ioc (v t) (u t))).toReal • c) =o[lt] fun t => (μ <\| Ioc (v t) (u t)).toReal := (measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' hfm hf hl hv hu huv).neg_left.congr_left fun t => by simp [integral_symm (u t), add_comm] #align interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge' intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' section variable [IsLocallyFiniteMeasure μ] [FTCFilter a l l'] /-- Fundamental theorem of calculus-1, local version for any measure. Let filters `l` and `l'` be related by `[intervalIntegral.FTCFilter a l l']`; let `μ` be a locally finite measure. If `f` has a finite limit `c` at `l' ⊓ ae μ`, then `∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both `u` and `v` tend to `l`. See also `measure_integral_sub_linear_isLittleO_of_tendsto_ae'` for a version that also works, e.g., for `l = l' = Filter.atTop`. We use integrals of constants instead of measures because this way it is easier to formulate a statement that works in both cases `u ≤ v` and `v ≤ u`. -/ theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt l) (hv : Tendsto v lt l) : (fun t => (∫ x in u t..v t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt] fun t => ∫ _ in u t..v t, (1 : ℝ) ∂μ := haveI := FTCFilter.meas_gen l measure_integral_sub_linear_isLittleO_of_tendsto_ae' hfm hf (FTCFilter.finiteAt_inner l) hu hv #align interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae /-- Fundamental theorem of calculus-1, local version for any measure. Let filters `l` and `l'` be related by `[intervalIntegral.FTCFilter a l l']`; let `μ` be a locally finite measure. If `f` has a finite limit `c` at `l' ⊓ ae μ`, then `∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both `u` and `v` tend to `l`. See also `measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le'` for a version that also works, e.g., for `l = l' = Filter.atTop`. -/ theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt l) (hv : Tendsto v lt l) (huv : u ≤ᶠ[lt] v) : (fun t => (∫ x in u t..v t, f x ∂μ) - (μ (Ioc (u t) (v t))).toReal • c) =o[lt] fun t => (μ <\| Ioc (u t) (v t)).toReal := haveI := FTCFilter.meas_gen l measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' hfm hf (FTCFilter.finiteAt_inner l) hu hv huv #align interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_le intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le /-- Fundamental theorem of calculus-1, local version for any measure. Let filters `l` and `l'` be related by `[intervalIntegral.FTCFilter a l l']`; let `μ` be a locally finite measure. If `f` has a finite limit `c` at `l' ⊓ ae μ`, then `∫ x in u..v, f x ∂μ = -μ (Set.Ioc v u) • c + o(μ(Set.Ioc v u))` as both `u` and `v` tend to `l`. See also `measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge'` for a version that also works, e.g., for `l = l' = Filter.atTop`. -/ theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt l) (hv : Tendsto v lt l) (huv : v ≤ᶠ[lt] u) : (fun t => (∫ x in u t..v t, f x ∂μ) + (μ (Ioc (v t) (u t))).toReal • c) =o[lt] fun t => (μ <\| Ioc (v t) (u t)).toReal := haveI := FTCFilter.meas_gen l measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' hfm hf (FTCFilter.finiteAt_inner l) hu hv huv #align interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge end variable [FTCFilter a la la'] [FTCFilter b lb lb'] [IsLocallyFiniteMeasure μ] /-- Fundamental theorem of calculus-1, strict derivative in both limits for a locally finite measure. Let `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `intervalIntegral.FTCFilter`s around `a`; let `(lb, lb')` be a pair of `intervalIntegral.FTCFilter`s around `b`. Suppose that `f` has finite limits `ca` and `cb` at `la' ⊓ ae μ` and `lb' ⊓ ae μ`, respectively. Then `∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ + o(‖∫ x in ua..va, (1:ℝ) ∂μ‖ + ‖∫ x in ub..vb, (1:ℝ) ∂μ‖)` as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`. -/ theorem measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae (hab : IntervalIntegrable f μ a b) (hmeas_a : StronglyMeasurableAtFilter f la' μ) (hmeas_b : StronglyMeasurableAtFilter f lb' μ) (ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)) (hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)) (hua : Tendsto ua lt la) (hva : Tendsto va lt la) (hub : Tendsto ub lt lb) (hvb : Tendsto vb lt lb) : (fun t => ((∫ x in va t..vb t, f x ∂μ) - ∫ x in ua t..ub t, f x ∂μ) - ((∫ _ in ub t..vb t, cb ∂μ) - ∫ _ in ua t..va t, ca ∂μ)) =o[lt] fun t => ‖∫ _ in ua t..va t, (1 : ℝ) ∂μ‖ + ‖∫ _ in ub t..vb t, (1 : ℝ) ∂μ‖ := by haveI := FTCFilter.meas_gen la; haveI := FTCFilter.meas_gen lb refine ((measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add (measure_integral_sub_linear_isLittleO_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr' ?_ EventuallyEq.rfl have A : ∀ᶠ t in lt, IntervalIntegrable f μ (ua t) (va t) := ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la) hua hva have A' : ∀ᶠ t in lt, IntervalIntegrable f μ a (ua t) := ha_lim.eventually_intervalIntegrable_ae hmeas_a (FTCFilter.finiteAt_inner la) (tendsto_const_pure.mono_right FTCFilter.pure_le) hua have B : ∀ᶠ t in lt, IntervalIntegrable f μ (ub t) (vb t) := hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb) hub hvb have B' : ∀ᶠ t in lt, IntervalIntegrable f μ b (ub t) := hb_lim.eventually_intervalIntegrable_ae hmeas_b (FTCFilter.finiteAt_inner lb) (tendsto_const_pure.mono_right FTCFilter.pure_le) hub filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub rw [← integral_interval_sub_interval_comm'] · abel exacts [ub_vb, ua_va, b_ub.symm.trans <\| hab.symm.trans a_ua] #align interval_integral.measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae /-- Fundamental theorem of calculus-1, strict derivative in right endpoint for a locally finite measure. Let `f` be a measurable function integrable on `a..b`. Let `(lb, lb')` be a pair of `intervalIntegral.FTCFilter`s around `b`. Suppose that `f` has a finite limit `c` at `lb' ⊓ ae μ`. Then `∫ x in a..v, f x ∂μ - ∫ x in a..u, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)` as `u` and `v` tend to `lb`. -/ theorem measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right (hab : IntervalIntegrable f μ a b) (hmeas : StronglyMeasurableAtFilter f lb' μ) (hf : Tendsto f (lb' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt lb) (hv : Tendsto v lt lb) : (fun t => ((∫ x in a..v t, f x ∂μ) - ∫ x in a..u t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt] fun t => ∫ _ in u t..v t, (1 : ℝ) ∂μ := by simpa using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab stronglyMeasurableAt_bot hmeas ((tendsto_bot : Tendsto _ ⊥ (𝓝 (0 : E))).mono_left inf_le_left) hf (tendsto_const_pure : Tendsto _ _ (pure a)) tendsto_const_pure hu hv #align interval_integral.measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right /-- Fundamental theorem of calculus-1, strict derivative in left endpoint for a locally finite measure. Let `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `intervalIntegral.FTCFilter`s around `a`. Suppose that `f` has a finite limit `c` at `la' ⊓ ae μ`. Then `∫ x in v..b, f x ∂μ - ∫ x in u..b, f x ∂μ = -∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)` as `u` and `v` tend to `la`. -/ theorem measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left (hab : IntervalIntegrable f μ a b) (hmeas : StronglyMeasurableAtFilter f la' μ) (hf : Tendsto f (la' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt la) (hv : Tendsto v lt la) : (fun t => ((∫ x in v t..b, f x ∂μ) - ∫ x in u t..b, f x ∂μ) + ∫ _ in u t..v t, c ∂μ) =o[lt] fun t => ∫ _ in u t..v t, (1 : ℝ) ∂μ := by simpa using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab hmeas stronglyMeasurableAt_bot hf ((tendsto_bot : Tendsto _ ⊥ (𝓝 (0 : E))).mono_left inf_le_left) hu hv (tendsto_const_pure : Tendsto _ _ (pure b)) tendsto_const_pure #align interval_integral.measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left end /-! ### Fundamental theorem of calculus-1 for Lebesgue measure In this section we restate theorems from the previous section for Lebesgue measure. In particular, we prove that `∫ x in u..v, f x` is strictly differentiable in `(u, v)` at `(a, b)` provided that `f` is integrable on `a..b` and is continuous at `a` and `b`. -/ variable [CompleteSpace E] {f : ℝ → E} {c ca cb : E} {l l' la la' lb lb' : Filter ℝ} {lt : Filter ι} {a b z : ℝ} {u v ua ub va vb : ι → ℝ} [FTCFilter a la la'] [FTCFilter b lb lb'] /-! #### Auxiliary `Asymptotics.IsLittleO` statements In this section we prove several lemmas that can be interpreted as strict differentiability of `(u, v) ↦ ∫ x in u..v, f x ∂μ` in `u` and/or `v` at a filter. The statements use `Asymptotics.isLittleO` because we have no definition of `HasStrict(F)DerivAtFilter` in the library. -/ /-- Fundamental theorem of calculus-1, local version. If `f` has a finite limit `c` almost surely at `l'`, where `(l, l')` is an `intervalIntegral.FTCFilter` pair around `a`, then `∫ x in u..v, f x ∂μ = (v - u) • c + o (v - u)` as both `u` and `v` tend to `l`. -/ theorem integral_sub_linear_isLittleO_of_tendsto_ae [FTCFilter a l l'] (hfm : StronglyMeasurableAtFilter f l') (hf : Tendsto f (l' ⊓ ae volume) (𝓝 c)) {u v : ι → ℝ} (hu : Tendsto u lt l) (hv : Tendsto v lt l) : (fun t => (∫ x in u t..v t, f x) - (v t - u t) • c) =o[lt] (v - u) := by simpa [integral_const] using measure_integral_sub_linear_isLittleO_of_tendsto_ae hfm hf hu hv #align interval_integral.integral_sub_linear_is_o_of_tendsto_ae intervalIntegral.integral_sub_linear_isLittleO_of_tendsto_ae /-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints. If `f` is a measurable function integrable on `a..b`, `(la, la')` is an `intervalIntegral.FTCFilter` pair around `a`, and `(lb, lb')` is an `intervalIntegral.FTCFilter` pair around `b`, and `f` has finite limits `ca` and `cb` almost surely at `la'` and `lb'`, respectively, then `(∫ x in va..vb, f x) - ∫ x in ua..ub, f x = (vb - ub) • cb - (va - ua) • ca + o(‖va - ua‖ + ‖vb - ub‖)` as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`. This lemma could've been formulated using `HasStrictFDerivAtFilter` if we had this definition. -/ theorem integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae (hab : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f la') (hmeas_b : StronglyMeasurableAtFilter f lb') (ha_lim : Tendsto f (la' ⊓ ae volume) (𝓝 ca)) (hb_lim : Tendsto f (lb' ⊓ ae volume) (𝓝 cb)) (hua : Tendsto ua lt la) (hva : Tendsto va lt la) (hub : Tendsto ub lt lb) (hvb : Tendsto vb lt lb) : (fun t => ((∫ x in va t..vb t, f x) - ∫ x in ua t..ub t, f x) - ((vb t - ub t) • cb - (va t - ua t) • ca)) =o[lt] fun t => ‖va t - ua t‖ + ‖vb t - ub t‖ := by simpa [integral_const] using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab hmeas_a hmeas_b ha_lim hb_lim hua hva hub hvb #align interval_integral.integral_sub_integral_sub_linear_is_o_of_tendsto_ae intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae /-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints. If `f` is a measurable function integrable on `a..b`, `(lb, lb')` is an `intervalIntegral.FTCFilter` pair around `b`, and `f` has a finite limit `c` almost surely at `lb'`, then `(∫ x in a..v, f x) - ∫ x in a..u, f x = (v - u) • c + o(‖v - u‖)` as `u` and `v` tend to `lb`. This lemma could've been formulated using `HasStrictDerivAtFilter` if we had this definition. -/ theorem integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right (hab : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f lb') (hf : Tendsto f (lb' ⊓ ae volume) (𝓝 c)) (hu : Tendsto u lt lb) (hv : Tendsto v lt lb) : (fun t => ((∫ x in a..v t, f x) - ∫ x in a..u t, f x) - (v t - u t) • c) =o[lt] (v - u) := by simpa only [integral_const, smul_eq_mul, mul_one] using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right hab hmeas hf hu hv #align interval_integral.integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right /-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints. If `f` is a measurable function integrable on `a..b`, `(la, la')` is an `intervalIntegral.FTCFilter` pair around `a`, and `f` has a finite limit `c` almost surely at `la'`, then `(∫ x in v..b, f x) - ∫ x in u..b, f x = -(v - u) • c + o(‖v - u‖)` as `u` and `v` tend to `la`. This lemma could've been formulated using `HasStrictDerivAtFilter` if we had this definition. -/ theorem integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left (hab : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f la') (hf : Tendsto f (la' ⊓ ae volume) (𝓝 c)) (hu : Tendsto u lt la) (hv : Tendsto v lt la) : (fun t => ((∫ x in v t..b, f x) - ∫ x in u t..b, f x) + (v t - u t) • c) =o[lt] (v - u) := by simpa only [integral_const, smul_eq_mul, mul_one] using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left hab hmeas hf hu hv #align interval_integral.integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left open ContinuousLinearMap (fst snd smulRight sub_apply smulRight_apply coe_fst' coe_snd' map_sub) /-! #### Strict differentiability In this section we prove that for a measurable function `f` integrable on `a..b`, `integral_hasStrictFDerivAt_of_tendsto_ae`: the function `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)` in the sense of strict differentiability provided that `f` tends to `ca` and `cb` almost surely as `x` tendsto to `a` and `b`, respectively; * `integral_hasStrictFDerivAt`: the function `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • f b - u • f a` at `(a, b)` in the sense of strict differentiability provided that `f` is continuous at `a` and `b`; * `integral_hasStrictDerivAt_of_tendsto_ae_right`: the function `u ↦ ∫ x in a..u, f x` has derivative `c` at `b` in the sense of strict differentiability provided that `f` tends to `c` almost surely as `x` tends to `b`; * `integral_hasStrictDerivAt_right`: the function `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b` in the sense of strict differentiability provided that `f` is continuous at `b`; * `integral_hasStrictDerivAt_of_tendsto_ae_left`: the function `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a` in the sense of strict differentiability provided that `f` tends to `c` almost surely as `x` tends to `a`; * `integral_hasStrictDerivAt_left`: the function `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a` in the sense of strict differentiability provided that `f` is continuous at `a`. -/ /-- Fundamental theorem of calculus-1, strict differentiability in both endpoints. If `f : ℝ → E` is integrable on `a..b` and `f x` has finite limits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)` in the sense of strict differentiability. -/ theorem integral_hasStrictFDerivAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)) : HasStrictFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (a, b) := by have := integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb (continuous_snd.fst.tendsto ((a, b), (a, b))) (continuous_fst.fst.tendsto ((a, b), (a, b))) (continuous_snd.snd.tendsto ((a, b), (a, b))) (continuous_fst.snd.tendsto ((a, b), (a, b))) refine (this.congr_left ?_).trans_isBigO ?_ · intro x; simp [sub_smul]; abel · exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left #align interval_integral.integral_has_strict_fderiv_at_of_tendsto_ae intervalIntegral.integral_hasStrictFDerivAt_of_tendsto_ae /-- Fundamental theorem of calculus-1, strict differentiability in both endpoints. If `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)` in the sense of strict differentiability. -/ theorem integral_hasStrictFDerivAt (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b)) (ha : ContinuousAt f a) (hb : ContinuousAt f b) : HasStrictFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a)) (a, b) := integral_hasStrictFDerivAt_of_tendsto_ae hf hmeas_a hmeas_b (ha.mono_left inf_le_left) (hb.mono_left inf_le_left) #align interval_integral.integral_has_strict_fderiv_at intervalIntegral.integral_hasStrictFDerivAt /-- Fundamental theorem of calculus-1, strict differentiability in the right endpoint. If `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b` in the sense of strict differentiability. -/ theorem integral_hasStrictDerivAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 c)) : HasStrictDerivAt (fun u => ∫ x in a..u, f x) c b := integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right hf hmeas hb continuousAt_snd continuousAt_fst #align interval_integral.integral_has_strict_deriv_at_of_tendsto_ae_right intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right /-- Fundamental theorem of calculus-1, strict differentiability in the right endpoint. If `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b` in the sense of strict differentiability. -/ theorem integral_hasStrictDerivAt_right (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : ContinuousAt f b) : HasStrictDerivAt (fun u => ∫ x in a..u, f x) (f b) b := integral_hasStrictDerivAt_of_tendsto_ae_right hf hmeas (hb.mono_left inf_le_left) #align interval_integral.integral_has_strict_deriv_at_right intervalIntegral.integral_hasStrictDerivAt_right /-- Fundamental theorem of calculus-1, strict differentiability in the left endpoint. If `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a` in the sense of strict differentiability. -/ theorem integral_hasStrictDerivAt_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)) : HasStrictDerivAt (fun u => ∫ x in u..b, f x) (-c) a := by simpa only [← integral_symm] using (integral_hasStrictDerivAt_of_tendsto_ae_right hf.symm hmeas ha).neg #align interval_integral.integral_has_strict_deriv_at_of_tendsto_ae_left intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_left /-- Fundamental theorem of calculus-1, strict differentiability in the left endpoint. If `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a` in the sense of strict differentiability. -/ theorem integral_hasStrictDerivAt_left (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : ContinuousAt f a) : HasStrictDerivAt (fun u => ∫ x in u..b, f x) (-f a) a := by simpa only [← integral_symm] using (integral_hasStrictDerivAt_right hf.symm hmeas ha).neg #align interval_integral.integral_has_strict_deriv_at_left intervalIntegral.integral_hasStrictDerivAt_left /-- Fundamental theorem of calculus-1, strict differentiability in the right endpoint. If `f : ℝ → E` is continuous, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b` in the sense of strict differentiability. -/ theorem _root_.Continuous.integral_hasStrictDerivAt {f : ℝ → E} (hf : Continuous f) (a b : ℝ) : HasStrictDerivAt (fun u => ∫ x : ℝ in a..u, f x) (f b) b := integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _) hf.continuousAt #align continuous.integral_has_strict_deriv_at Continuous.integral_hasStrictDerivAt /-- Fundamental theorem of calculus-1, derivative in the right endpoint. If `f : ℝ → E` is continuous, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` is `f b`. -/ theorem _root_.Continuous.deriv_integral (f : ℝ → E) (hf : Continuous f) (a b : ℝ) : deriv (fun u => ∫ x : ℝ in a..u, f x) b = f b := (hf.integral_hasStrictDerivAt a b).hasDerivAt.deriv #align continuous.deriv_integral Continuous.deriv_integral /-! #### Fréchet differentiability In this subsection we restate results from the previous subsection in terms of `HasFDerivAt`, `HasDerivAt`, `fderiv`, and `deriv`. -/ /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite limits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`. -/ theorem integral_hasFDerivAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)) : HasFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (a, b) := (integral_hasStrictFDerivAt_of_tendsto_ae hf hmeas_a hmeas_b ha hb).hasFDerivAt #align interval_integral.integral_has_fderiv_at_of_tendsto_ae intervalIntegral.integral_hasFDerivAt_of_tendsto_ae /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`. -/ theorem integral_hasFDerivAt (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b)) (ha : ContinuousAt f a) (hb : ContinuousAt f b) : HasFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a)) (a, b) := (integral_hasStrictFDerivAt hf hmeas_a hmeas_b ha hb).hasFDerivAt #align interval_integral.integral_has_fderiv_at intervalIntegral.integral_hasFDerivAt /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite limits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `fderiv` derivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦ v • cb - u • ca`. -/ theorem fderiv_integral_of_tendsto_ae (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)) : fderiv ℝ (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) (a, b) = (snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca := (integral_hasFDerivAt_of_tendsto_ae hf hmeas_a hmeas_b ha hb).fderiv #align interval_integral.fderiv_integral_of_tendsto_ae intervalIntegral.fderiv_integral_of_tendsto_ae /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `a` and `b`, then `fderiv` derivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦ v • cb - u • ca`. -/ theorem fderiv_integral (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b)) (ha : ContinuousAt f a) (hb : ContinuousAt f b) : fderiv ℝ (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) (a, b) = (snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a) := (integral_hasFDerivAt hf hmeas_a hmeas_b ha hb).fderiv #align interval_integral.fderiv_integral intervalIntegral.fderiv_integral /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b`. -/ theorem integral_hasDerivAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 c)) : HasDerivAt (fun u => ∫ x in a..u, f x) c b := (integral_hasStrictDerivAt_of_tendsto_ae_right hf hmeas hb).hasDerivAt #align interval_integral.integral_has_deriv_at_of_tendsto_ae_right intervalIntegral.integral_hasDerivAt_of_tendsto_ae_right /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b`. -/ theorem integral_hasDerivAt_right (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : ContinuousAt f b) : HasDerivAt (fun u => ∫ x in a..u, f x) (f b) b := (integral_hasStrictDerivAt_right hf hmeas hb).hasDerivAt #align interval_integral.integral_has_deriv_at_right intervalIntegral.integral_hasDerivAt_right /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite limit `c` almost surely at `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/ theorem deriv_integral_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 c)) : deriv (fun u => ∫ x in a..u, f x) b = c := (integral_hasDerivAt_of_tendsto_ae_right hf hmeas hb).deriv #align interval_integral.deriv_integral_of_tendsto_ae_right intervalIntegral.deriv_integral_of_tendsto_ae_right /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/ theorem deriv_integral_right (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : ContinuousAt f b) : deriv (fun u => ∫ x in a..u, f x) b = f b := (integral_hasDerivAt_right hf hmeas hb).deriv #align interval_integral.deriv_integral_right intervalIntegral.deriv_integral_right /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a`. -/ theorem integral_hasDerivAt_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)) : HasDerivAt (fun u => ∫ x in u..b, f x) (-c) a := (integral_hasStrictDerivAt_of_tendsto_ae_left hf hmeas ha).hasDerivAt #align interval_integral.integral_has_deriv_at_of_tendsto_ae_left intervalIntegral.integral_hasDerivAt_of_tendsto_ae_left /-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a`. -/ theorem integral_hasDerivAt_left (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : ContinuousAt f a) : HasDerivAt (fun u => ∫ x in u..b, f x) (-f a) a := (integral_hasStrictDerivAt_left hf hmeas ha).hasDerivAt #align interval_integral.integral_has_deriv_at_left intervalIntegral.integral_hasDerivAt_left /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite limit `c` almost surely at `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/ theorem deriv_integral_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (hb : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)) : deriv (fun u => ∫ x in u..b, f x) a = -c := (integral_hasDerivAt_of_tendsto_ae_left hf hmeas hb).deriv #align interval_integral.deriv_integral_of_tendsto_ae_left intervalIntegral.deriv_integral_of_tendsto_ae_left /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous at `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/ theorem deriv_integral_left (hf : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (hb : ContinuousAt f a) : deriv (fun u => ∫ x in u..b, f x) a = -f a := (integral_hasDerivAt_left hf hmeas hb).deriv #align interval_integral.deriv_integral_left intervalIntegral.deriv_integral_left /-! #### One-sided derivatives -/ /-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` within `s × t` at `(a, b)`, where `s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to `ca` and `cb` almost surely at the filters `la` and `lb` from the following table. \| `s` \| `la` \| `t` \| `lb` \| \| ------- \| ---- \| --- \| ---- \| \| `Iic a` \| `𝓝[≤] a` \| `Iic b` \| `𝓝[≤] b` \| \| `Ici a` \| `𝓝[>] a` \| `Ici b` \| `𝓝[>] b` \| \| `{a}` \| `⊥` \| `{b}` \| `⊥` \| \| `univ` \| `𝓝 a` \| `univ` \| `𝓝 b` \| -/ theorem integral_hasFDerivWithinAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb] (hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb) (ha : Tendsto f (la ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (lb ⊓ ae volume) (𝓝 cb)) : HasFDerivWithinAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (s ×ˢ t) (a, b) := by rw [HasFDerivWithinAt, nhdsWithin_prod_eq] have := integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb (tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[s] a)) tendsto_fst (tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[t] b)) tendsto_snd refine .of_isLittleO <\| (this.congr_left ?_).trans_isBigO ?_ · intro x; simp [sub_smul]; abel · exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left #align interval_integral.integral_has_fderiv_within_at_of_tendsto_ae intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae /-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • f b - u • f a` within `s × t` at `(a, b)`, where `s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to `f a` and `f b` at the filters `la` and `lb` from the following table. In most cases this assumption is definitionally equal `ContinuousAt f _` or `ContinuousWithinAt f _ _`. \| `s` \| `la` \| `t` \| `lb` \| \| ------- \| ---- \| --- \| ---- \| \| `Iic a` \| `𝓝[≤] a` \| `Iic b` \| `𝓝[≤] b` \| \| `Ici a` \| `𝓝[>] a` \| `Ici b` \| `𝓝[>] b` \| \| `{a}` \| `⊥` \| `{b}` \| `⊥` \| \| `univ` \| `𝓝 a` \| `univ` \| `𝓝 b` \| -/ theorem integral_hasFDerivWithinAt (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb] (ha : Tendsto f la (𝓝 <\| f a)) (hb : Tendsto f lb (𝓝 <\| f b)) : HasFDerivWithinAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) ((snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a)) (s ×ˢ t) (a, b) := integral_hasFDerivWithinAt_of_tendsto_ae hf hmeas_a hmeas_b (ha.mono_left inf_le_left) (hb.mono_left inf_le_left) #align interval_integral.integral_has_fderiv_within_at intervalIntegral.integral_hasFDerivWithinAt /-- An auxiliary tactic closing goals `UniqueDiffWithinAt ℝ s a` where `s ∈ {Iic a, Ici a, univ}`. -/ macro "uniqueDiffWithinAt_Ici_Iic_univ" : tactic => `(tactic\| (first \| exact uniqueDiffOn_Ici _ _ left_mem_Ici \| exact uniqueDiffOn_Iic _ _ right_mem_Iic \| exact uniqueDiffWithinAt_univ (𝕜 := ℝ) (E := ℝ))) #noalign interval_integral.unique_diff_within_at_Ici_Iic_univ /-- Let `f` be a measurable function integrable on `a..b`. Choose `s ∈ {Iic a, Ici a, univ}` and `t ∈ {Iic b, Ici b, univ}`. Suppose that `f` tends to `ca` and `cb` almost surely at the filters `la` and `lb` from the table below. Then `fderivWithin ℝ (fun p ↦ ∫ x in p.1..p.2, f x) (s ×ˢ t)` is equal to `(u, v) ↦ u • cb - v • ca`. \| `s` \| `la` \| `t` \| `lb` \| \| ------- \| ---- \| --- \| ---- \| \| `Iic a` \| `𝓝[≤] a` \| `Iic b` \| `𝓝[≤] b` \| \| `Ici a` \| `𝓝[>] a` \| `Ici b` \| `𝓝[>] b` \| \| `{a}` \| `⊥` \| `{b}` \| `⊥` \| \| `univ` \| `𝓝 a` \| `univ` \| `𝓝 b` \| -/ theorem fderivWithin_integral_of_tendsto_ae (hf : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb] (ha : Tendsto f (la ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (lb ⊓ ae volume) (𝓝 cb)) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) (ht : UniqueDiffWithinAt ℝ t b := by uniqueDiffWithinAt_Ici_Iic_univ) : fderivWithin ℝ (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) (s ×ˢ t) (a, b) = (snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca := (integral_hasFDerivWithinAt_of_tendsto_ae hf hmeas_a hmeas_b ha hb).fderivWithin <\| hs.prod ht #align interval_integral.fderiv_within_integral_of_tendsto_ae intervalIntegral.fderivWithin_integral_of_tendsto_ae /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely as `x` tends to `b` from the right or from the left, then `u ↦ ∫ x in a..u, f x` has right (resp., left) derivative `c` at `b`. -/ theorem integral_hasDerivWithinAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b)) (hb : Tendsto f (𝓝[t] b ⊓ ae volume) (𝓝 c)) : HasDerivWithinAt (fun u => ∫ x in a..u, f x) c s b := .of_isLittleO <\| integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right hf hmeas hb (tendsto_const_pure.mono_right FTCFilter.pure_le) tendsto_id #align interval_integral.integral_has_deriv_within_at_of_tendsto_ae_right intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_right /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous from the left or from the right at `b`, then `u ↦ ∫ x in a..u, f x` has left (resp., right) derivative `f b` at `b`. -/ theorem integral_hasDerivWithinAt_right (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b)) (hb : ContinuousWithinAt f t b) : HasDerivWithinAt (fun u => ∫ x in a..u, f x) (f b) s b := integral_hasDerivWithinAt_of_tendsto_ae_right hf hmeas (hb.mono_left inf_le_left) #align interval_integral.integral_has_deriv_within_at_right intervalIntegral.integral_hasDerivWithinAt_right /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely as `x` tends to `b` from the right or from the left, then the right (resp., left) derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/ theorem derivWithin_integral_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b)) (hb : Tendsto f (𝓝[t] b ⊓ ae volume) (𝓝 c)) (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ) : derivWithin (fun u => ∫ x in a..u, f x) s b = c := (integral_hasDerivWithinAt_of_tendsto_ae_right hf hmeas hb).derivWithin hs #align interval_integral.deriv_within_integral_of_tendsto_ae_right intervalIntegral.derivWithin_integral_of_tendsto_ae_right /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous on the right or on the left at `b`, then the right (resp., left) derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/ theorem derivWithin_integral_right (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b)) (hb : ContinuousWithinAt f t b) (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ) : derivWithin (fun u => ∫ x in a..u, f x) s b = f b := (integral_hasDerivWithinAt_right hf hmeas hb).derivWithin hs #align interval_integral.deriv_within_integral_right intervalIntegral.derivWithin_integral_right /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely as `x` tends to `a` from the right or from the left, then `u ↦ ∫ x in u..b, f x` has right (resp., left) derivative `-c` at `a`. -/ theorem integral_hasDerivWithinAt_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a)) (ha : Tendsto f (𝓝[t] a ⊓ ae volume) (𝓝 c)) : HasDerivWithinAt (fun u => ∫ x in u..b, f x) (-c) s a := by simp only [integral_symm b] exact (integral_hasDerivWithinAt_of_tendsto_ae_right hf.symm hmeas ha).neg #align interval_integral.integral_has_deriv_within_at_of_tendsto_ae_left intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_left /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous from the left or from the right at `a`, then `u ↦ ∫ x in u..b, f x` has left (resp., right) derivative `-f a` at `a`. -/ theorem integral_hasDerivWithinAt_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a)) (ha : ContinuousWithinAt f t a) : HasDerivWithinAt (fun u => ∫ x in u..b, f x) (-f a) s a := integral_hasDerivWithinAt_of_tendsto_ae_left hf hmeas (ha.mono_left inf_le_left) #align interval_integral.integral_has_deriv_within_at_left intervalIntegral.integral_hasDerivWithinAt_left /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite limit `c` almost surely as `x` tends to `a` from the right or from the left, then the right (resp., left) derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/ theorem derivWithin_integral_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a)) (ha : Tendsto f (𝓝[t] a ⊓ ae volume) (𝓝 c)) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) : derivWithin (fun u => ∫ x in u..b, f x) s a = -c := (integral_hasDerivWithinAt_of_tendsto_ae_left hf hmeas ha).derivWithin hs #align interval_integral.deriv_within_integral_of_tendsto_ae_left intervalIntegral.derivWithin_integral_of_tendsto_ae_left /-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous on the right or on the left at `a`, then the right (resp., left) derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/ theorem derivWithin_integral_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ} [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a)) (ha : ContinuousWithinAt f t a) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) : derivWithin (fun u => ∫ x in u..b, f x) s a = -f a := (integral_hasDerivWithinAt_left hf hmeas ha).derivWithin hs #align interval_integral.deriv_within_integral_left intervalIntegral.derivWithin_integral_left /-- The integral of a continuous function is differentiable on a real set `s`. -/ theorem differentiableOn_integral_of_continuous {s : Set ℝ} (hintg : ∀ x ∈ s, IntervalIntegrable f volume a x) (hcont : Continuous f) : DifferentiableOn ℝ (fun u => ∫ x in a..u, f x) s := fun y hy => (integral_hasDerivAt_right (hintg y hy) hcont.aestronglyMeasurable.stronglyMeasurableAtFilter hcont.continuousAt).differentiableAt.differentiableWithinAt #align interval_integral.differentiable_on_integral_of_continuous intervalIntegral.differentiableOn_integral_of_continuous end FTC1 /-! ### Fundamental theorem of calculus, part 2 This section contains theorems pertaining to FTC-2 for interval integrals, i.e., the assertion that `∫ x in a..b, f' x = f b - f a` under suitable assumptions. The most classical version of this theorem assumes that `f'` is continuous. However, this is unnecessarily strong: the result holds if `f'` is just integrable. We prove the strong version, following [Rudin, Real and Complex Analysis (Theorem 7.21)][rudin2006real]. The proof is first given for real-valued functions, and then deduced for functions with a general target space. For a real-valued function `g`, it suffices to show that `g b - g a ≤ (∫ x in a..b, g' x) + ε` for all positive `ε`. To prove this, choose a lower-semicontinuous function `G'` with `g' < G'` and with integral close to that of `g'` (its existence is guaranteed by the Vitali-Carathéodory theorem). It satisfies `g t - g a ≤ ∫ x in a..t, G' x` for all `t ∈ [a, b]`: this inequality holds at `a`, and if it holds at `t` then it holds for `u` close to `t` on its right, as the left hand side increases by `g u - g t ∼ (u -t) g' t`, while the right hand side increases by `∫ x in t..u, G' x` which is roughly at least `∫ x in t..u, G' t = (u - t) G' t`, by lower semicontinuity. As `g' t < G' t`, this gives the conclusion. One can therefore push progressively this inequality to the right until the point `b`, where it gives the desired conclusion. -/ variable {f : ℝ → E} {g' g φ : ℝ → ℝ} /-- Hard part of FTC-2 for integrable derivatives, real-valued functions: one has `g b - g a ≤ ∫ y in a..b, g' y` when `g'` is integrable. Auxiliary lemma in the proof of `integral_eq_sub_of_hasDeriv_right_of_le`. We give the slightly more general version that `g b - g a ≤ ∫ y in a..b, φ y` when `g' ≤ φ` and `φ` is integrable (even if `g'` is not known to be integrable). Version assuming that `g` is differentiable on `[a, b)`. -/ theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (hab : a ≤ b) (hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x) (φint : IntegrableOn φ (Icc a b)) (hφg : ∀ x ∈ Ico a b, g' x ≤ φ x) : g b - g a ≤ ∫ y in a..b, φ y := by refine le_of_forall_pos_le_add fun ε εpos => ?_ -- Bound from above `g'` by a lower-semicontinuous function `G'`. rcases exists_lt_lowerSemicontinuous_integral_lt φ φint εpos with ⟨G', f_lt_G', G'cont, G'int, G'lt_top, hG'⟩ -- we will show by "induction" that `g t - g a ≤ ∫ u in a..t, G' u` for all `t ∈ [a, b]`. set s := {t \| g t - g a ≤ ∫ u in a..t, (G' u).toReal} ∩ Icc a b -- the set `s` of points where this property holds is closed. have s_closed : IsClosed s := by have : ContinuousOn (fun t => (g t - g a, ∫ u in a..t, (G' u).toReal)) (Icc a b) := by rw [← uIcc_of_le hab] at G'int hcont ⊢ exact (hcont.sub continuousOn_const).prod (continuousOn_primitive_interval G'int) simp only [s, inter_comm] exact this.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' have main : Icc a b ⊆ {t \| g t - g a ≤ ∫ u in a..t, (G' u).toReal} := by -- to show that the set `s` is all `[a, b]`, it suffices to show that any point `t` in `s` -- with `t < b` admits another point in `s` slightly to its right -- (this is a sort of real induction). refine s_closed.Icc_subset_of_forall_exists_gt (by simp only [integral_same, mem_setOf_eq, sub_self, le_rfl]) fun t ht v t_lt_v => ?_ obtain ⟨y, g'_lt_y', y_lt_G'⟩ : ∃ y : ℝ, (g' t : EReal) < y ∧ (y : EReal) < G' t := EReal.lt_iff_exists_real_btwn.1 ((EReal.coe_le_coe_iff.2 (hφg t ht.2)).trans_lt (f_lt_G' t)) -- bound from below the increase of `∫ x in a..u, G' x` on the right of `t`, using the lower -- semicontinuity of `G'`. have I1 : ∀ᶠ u in 𝓝[>] t, (u - t) * y ≤ ∫ w in t..u, (G' w).toReal := by have B : ∀ᶠ u in 𝓝 t, (y : EReal) < G' u := G'cont.lowerSemicontinuousAt _ _ y_lt_G' rcases mem_nhds_iff_exists_Ioo_subset.1 B with ⟨m, M, ⟨hm, hM⟩, H⟩ have : Ioo t (min M b) ∈ 𝓝[>] t := Ioo_mem_nhdsWithin_Ioi' (lt_min hM ht.right.right) filter_upwards [this] with u hu have I : Icc t u ⊆ Icc a b := Icc_subset_Icc ht.2.1 (hu.2.le.trans (min_le_right _ _)) calc (u - t) * y = ∫ _ in Icc t u, y := by simp only [hu.left.le, MeasureTheory.integral_const, Algebra.id.smul_eq_mul, sub_nonneg, MeasurableSet.univ, Real.volume_Icc, Measure.restrict_apply, univ_inter, ENNReal.toReal_ofReal] _ ≤ ∫ w in t..u, (G' w).toReal := by rw [intervalIntegral.integral_of_le hu.1.le, ← integral_Icc_eq_integral_Ioc] apply setIntegral_mono_ae_restrict · simp only [integrableOn_const, Real.volume_Icc, ENNReal.ofReal_lt_top, or_true_iff] · exact IntegrableOn.mono_set G'int I · have C1 : ∀ᵐ x : ℝ ∂volume.restrict (Icc t u), G' x < ∞ := ae_mono (Measure.restrict_mono I le_rfl) G'lt_top have C2 : ∀ᵐ x : ℝ ∂volume.restrict (Icc t u), x ∈ Icc t u := ae_restrict_mem measurableSet_Icc filter_upwards [C1, C2] with x G'x hx apply EReal.coe_le_coe_iff.1 have : x ∈ Ioo m M := by simp only [hm.trans_le hx.left, (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self_iff] refine (H this).out.le.trans_eq ?_ exact (EReal.coe_toReal G'x.ne (f_lt_G' x).ne_bot).symm -- bound from above the increase of `g u - g a` on the right of `t`, using the derivative at `t` have I2 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ (u - t) * y := by have g'_lt_y : g' t < y := EReal.coe_lt_coe_iff.1 g'_lt_y' filter_upwards [(hderiv t ⟨ht.2.1, ht.2.2⟩).limsup_slope_le' (not_mem_Ioi.2 le_rfl) g'_lt_y, self_mem_nhdsWithin] with u hu t_lt_u have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u.out).le rwa [← smul_eq_mul, sub_smul_slope] at this -- combine the previous two bounds to show that `g u - g a` increases less quickly than -- `∫ x in a..u, G' x`. have I3 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ ∫ w in t..u, (G' w).toReal := by filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1 have I4 : ∀ᶠ u in 𝓝[>] t, u ∈ Ioc t (min v b) := by refine mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.2 ⟨min v b, ?_, Subset.rfl⟩ simp only [lt_min_iff, mem_Ioi] exact ⟨t_lt_v, ht.2.2⟩ -- choose a point `x` slightly to the right of `t` which satisfies the above bound rcases (I3.and I4).exists with ⟨x, hx, h'x⟩ -- we check that it belongs to `s`, essentially by construction refine ⟨x, ?_, Ioc_subset_Ioc le_rfl (min_le_left _ _) h'x⟩ calc g x - g a = g t - g a + (g x - g t) := by abel _ ≤ (∫ w in a..t, (G' w).toReal) + ∫ w in t..x, (G' w).toReal := add_le_add ht.1 hx _ = ∫ w in a..x, (G' w).toReal := by apply integral_add_adjacent_intervals · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le ht.2.1] exact IntegrableOn.mono_set G'int (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le)) · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x.1.le] apply IntegrableOn.mono_set G'int exact Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _))) -- now that we know that `s` contains `[a, b]`, we get the desired result by applying this to `b`. calc g b - g a ≤ ∫ y in a..b, (G' y).toReal := main (right_mem_Icc.2 hab) _ ≤ (∫ y in a..b, φ y) + ε := by convert hG'.le <;> · rw [intervalIntegral.integral_of_le hab] simp only [integral_Icc_eq_integral_Ioc', Real.volume_singleton] #align interval_integral.sub_le_integral_of_has_deriv_right_of_le_Ico intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico -- Porting note: Lean was adding `lb`/`lb'` to the arguments of this theorem, so I enclosed FTC-1 -- into a `section` /-- Hard part of FTC-2 for integrable derivatives, real-valued functions: one has `g b - g a ≤ ∫ y in a..b, g' y` when `g'` is integrable. Auxiliary lemma in the proof of `integral_eq_sub_of_hasDeriv_right_of_le`. We give the slightly more general version that `g b - g a ≤ ∫ y in a..b, φ y` when `g' ≤ φ` and `φ` is integrable (even if `g'` is not known to be integrable). Version assuming that `g` is differentiable on `(a, b)`. -/ theorem sub_le_integral_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x) (φint : IntegrableOn φ (Icc a b)) (hφg : ∀ x ∈ Ioo a b, g' x ≤ φ x) : g b - g a ≤ ∫ y in a..b, φ y := by -- This follows from the version on a closed-open interval (applied to `[t, b)` for `t` close to -- `a`) and a continuity argument. obtain rfl \| a_lt_b := hab.eq_or_lt · simp set s := {t \| g b - g t ≤ ∫ u in t..b, φ u} ∩ Icc a b have s_closed : IsClosed s := by have : ContinuousOn (fun t => (g b - g t, ∫ u in t..b, φ u)) (Icc a b) := by rw [← uIcc_of_le hab] at hcont φint ⊢ exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left φint) simp only [s, inter_comm] exact this.preimage_isClosed_of_isClosed isClosed_Icc isClosed_le_prod have A : closure (Ioc a b) ⊆ s := by apply s_closed.closure_subset_iff.2 intro t ht refine ⟨?_, ⟨ht.1.le, ht.2⟩⟩ exact sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl)) (fun x hx => hderiv x ⟨ht.1.trans_le hx.1, hx.2⟩) (φint.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => hφg x ⟨ht.1.trans_le hx.1, hx.2⟩ rw [closure_Ioc a_lt_b.ne] at A exact (A (left_mem_Icc.2 hab)).1 #align interval_integral.sub_le_integral_of_has_deriv_right_of_le intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le /-- Auxiliary lemma in the proof of `integral_eq_sub_of_hasDeriv_right_of_le`. -/ theorem integral_le_sub_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x) (φint : IntegrableOn φ (Icc a b)) (hφg : ∀ x ∈ Ioo a b, φ x ≤ g' x) : (∫ y in a..b, φ y) ≤ g b - g a := by rw [← neg_le_neg_iff] convert sub_le_integral_of_hasDeriv_right_of_le hab hcont.neg (fun x hx => (hderiv x hx).neg) φint.neg fun x hx => neg_le_neg (hφg x hx) using 1 · abel · simp only [← integral_neg]; rfl #align interval_integral.integral_le_sub_of_has_deriv_right_of_le intervalIntegral.integral_le_sub_of_hasDeriv_right_of_le /-- Auxiliary lemma in the proof of `integral_eq_sub_of_hasDeriv_right_of_le`: real version -/ theorem integral_eq_sub_of_hasDeriv_right_of_le_real (hab : a ≤ b) (hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x) (g'int : IntegrableOn g' (Icc a b)) : ∫ y in a..b, g' y = g b - g a := le_antisymm (integral_le_sub_of_hasDeriv_right_of_le hab hcont hderiv g'int fun _ _ => le_rfl) (sub_le_integral_of_hasDeriv_right_of_le hab hcont hderiv g'int fun _ _ => le_rfl) #align interval_integral.integral_eq_sub_of_has_deriv_right_of_le_real intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le_real variable [CompleteSpace E] {f f' : ℝ → E} /-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`) and has a right derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`, then `∫ y in a..b, f' y` equals `f b - f a`. -/ theorem integral_eq_sub_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' x) (Ioi x) x) (f'int : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a := by refine (NormedSpace.eq_iff_forall_dual_eq ℝ).2 fun g => ?_ rw [← g.intervalIntegral_comp_comm f'int, g.map_sub] exact integral_eq_sub_of_hasDeriv_right_of_le_real hab (g.continuous.comp_continuousOn hcont) (fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx)) (g.integrable_comp ((intervalIntegrable_iff_integrableOn_Icc_of_le hab).1 f'int)) #align interval_integral.integral_eq_sub_of_has_deriv_right_of_le intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le /-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` and has a right derivative at `f' x` for all `x` in `[a, b)`, and `f'` is integrable on `[a, b]` then `∫ y in a..b, f' y` equals `f b - f a`. -/ theorem integral_eq_sub_of_hasDeriv_right (hcont : ContinuousOn f (uIcc a b)) (hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x) (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a := by rcases le_total a b with hab \| hab · simp only [uIcc_of_le, min_eq_left, max_eq_right, hab] at hcont hderiv hint apply integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint · simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab] at hcont hderiv rw [integral_symm, integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint.symm, neg_sub] #align interval_integral.integral_eq_sub_of_has_deriv_right intervalIntegral.integral_eq_sub_of_hasDeriv_right /-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`) and has a derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`, then `∫ y in a..b, f' y` equals `f b - f a`. -/ theorem integral_eq_sub_of_hasDerivAt_of_le (hab : a ≤ b) (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a := integral_eq_sub_of_hasDeriv_right_of_le hab hcont (fun x hx => (hderiv x hx).hasDerivWithinAt) hint #align interval_integral.integral_eq_sub_of_has_deriv_at_of_le intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le /-- Fundamental theorem of calculus-2: If `f : ℝ → E` has a derivative at `f' x` for all `x` in `[a, b]` and `f'` is integrable on `[a, b]`, then `∫ y in a..b, f' y` equals `f b - f a`. -/ theorem integral_eq_sub_of_hasDerivAt (hderiv : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a := integral_eq_sub_of_hasDeriv_right (HasDerivAt.continuousOn hderiv) (fun _x hx => (hderiv _ (mem_Icc_of_Ioo hx)).hasDerivWithinAt) hint #align interval_integral.integral_eq_sub_of_has_deriv_at intervalIntegral.integral_eq_sub_of_hasDerivAt theorem integral_eq_sub_of_hasDerivAt_of_tendsto (hab : a < b) {fa fb} (hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) (ha : Tendsto f (𝓝[>] a) (𝓝 fa)) (hb : Tendsto f (𝓝[<] b) (𝓝 fb)) : ∫ y in a..b, f' y = fb - fa := by set F : ℝ → E := update (update f a fa) b fb have Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x := by refine fun x hx => (hderiv x hx).congr_of_eventuallyEq ?_ filter_upwards [Ioo_mem_nhds hx.1 hx.2] with _ hy unfold_let F rw [update_noteq hy.2.ne, update_noteq hy.1.ne'] have hcont : ContinuousOn F (Icc a b) := by rw [continuousOn_update_iff, continuousOn_update_iff, Icc_diff_right, Ico_diff_left] refine ⟨⟨fun z hz => (hderiv z hz).continuousAt.continuousWithinAt, ?_⟩, ?_⟩ · exact fun _ => ha.mono_left (nhdsWithin_mono _ Ioo_subset_Ioi_self) · rintro - refine (hb.congr' ?_).mono_left (nhdsWithin_mono _ Ico_subset_Iio_self) filter_upwards [Ioo_mem_nhdsWithin_Iio (right_mem_Ioc.2 hab)] with _ hz using (update_noteq hz.1.ne' _ _).symm simpa [F, hab.ne, hab.ne'] using integral_eq_sub_of_hasDerivAt_of_le hab.le hcont Fderiv hint #align interval_integral.integral_eq_sub_of_has_deriv_at_of_tendsto intervalIntegral.integral_eq_sub_of_hasDerivAt_of_tendsto /-- Fundamental theorem of calculus-2: If `f : ℝ → E` is differentiable at every `x` in `[a, b]` and its derivative is integrable on `[a, b]`, then `∫ y in a..b, deriv f y` equals `f b - f a`. -/ theorem integral_deriv_eq_sub (hderiv : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x) (hint : IntervalIntegrable (deriv f) volume a b) : ∫ y in a..b, deriv f y = f b - f a := integral_eq_sub_of_hasDerivAt (fun x hx => (hderiv x hx).hasDerivAt) hint #align interval_integral.integral_deriv_eq_sub intervalIntegral.integral_deriv_eq_sub	Mathlib/MeasureTheory/Integral/FundThmCalculus.lean	1,244	1,249	theorem integral_deriv_eq_sub' (f) (hderiv : deriv f = f') (hdiff : ∀ x ∈ uIcc a b, DifferentiableAt ℝ f x) (hcont : ContinuousOn f' (uIcc a b)) : ∫ y in a..b, f' y = f b - f a := by	rw [← hderiv, integral_deriv_eq_sub hdiff] rw [hderiv] exact hcont.intervalIntegrable
/- Copyright (c) 2021 Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kyle Miller -/ import Mathlib.CategoryTheory.Filtered.Basic import Mathlib.Topology.Category.TopCat.Limits.Basic #align_import topology.category.Top.limits.konig from "leanprover-community/mathlib"@"dbdf71cee7bb20367cb7e37279c08b0c218cf967" /-! # Topological Kőnig's lemma A topological version of Kőnig's lemma is that the inverse limit of nonempty compact Hausdorff spaces is nonempty. (Note: this can be generalized further to inverse limits of nonempty compact T0 spaces, where all the maps are closed maps; see [Stone1979] --- however there is an erratum for Theorem 4 that the element in the inverse limit can have cofinally many components that are not closed points.) We give this in a more general form, which is that cofiltered limits of nonempty compact Hausdorff spaces are nonempty (`nonempty_limitCone_of_compact_t2_cofiltered_system`). This also applies to inverse limits, where `{J : Type u} [Preorder J] [IsDirected J (≤)]` and `F : Jᵒᵖ ⥤ TopCat`. The theorem is specialized to nonempty finite types (which are compact Hausdorff with the discrete topology) in lemmas `nonempty_sections_of_finite_cofiltered_system` and `nonempty_sections_of_finite_inverse_system` in the file `Mathlib.CategoryTheory.CofilteredSystem`. (See <https://stacks.math.columbia.edu/tag/086J> for the Set version.) -/ -- Porting note: every ML3 decl has an uppercase letter set_option linter.uppercaseLean3 false open CategoryTheory open CategoryTheory.Limits -- Porting note: changed universe order as `v` is usually passed explicitly universe v u w noncomputable section namespace TopCat section TopologicalKonig variable {J : Type u} [SmallCategory J] -- Porting note: generalized `F` to land in `v` not `u` variable (F : J ⥤ TopCat.{v}) private abbrev FiniteDiagramArrow {J : Type u} [SmallCategory J] (G : Finset J) := Σ' (X Y : J) (_ : X ∈ G) (_ : Y ∈ G), X ⟶ Y private abbrev FiniteDiagram (J : Type u) [SmallCategory J] := Σ G : Finset J, Finset (FiniteDiagramArrow G) /-- Partial sections of a cofiltered limit are sections when restricted to a finite subset of objects and morphisms of `J`. -/ -- Porting note: generalized `F` to land in `v` not `u` def partialSections {J : Type u} [SmallCategory J] (F : J ⥤ TopCat.{v}) {G : Finset J} (H : Finset (FiniteDiagramArrow G)) : Set (∀ j, F.obj j) := {u \| ∀ {f : FiniteDiagramArrow G} (_ : f ∈ H), F.map f.2.2.2.2 (u f.1) = u f.2.1} #align Top.partial_sections TopCat.partialSections theorem partialSections.nonempty [IsCofilteredOrEmpty J] [h : ∀ j : J, Nonempty (F.obj j)] {G : Finset J} (H : Finset (FiniteDiagramArrow G)) : (partialSections F H).Nonempty := by classical cases isEmpty_or_nonempty J · exact ⟨isEmptyElim, fun {j} => IsEmpty.elim' inferInstance j.1⟩ haveI : IsCofiltered J := ⟨⟩ use fun j : J => if hj : j ∈ G then F.map (IsCofiltered.infTo G H hj) (h (IsCofiltered.inf G H)).some else (h _).some rintro ⟨X, Y, hX, hY, f⟩ hf dsimp only rwa [dif_pos hX, dif_pos hY, ← comp_app, ← F.map_comp, @IsCofiltered.infTo_commutes _ _ _ G H] #align Top.partial_sections.nonempty TopCat.partialSections.nonempty theorem partialSections.directed : Directed Superset fun G : FiniteDiagram J => partialSections F G.2 := by classical intro A B let ιA : FiniteDiagramArrow A.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f => ⟨f.1, f.2.1, Finset.mem_union_left _ f.2.2.1, Finset.mem_union_left _ f.2.2.2.1, f.2.2.2.2⟩ let ιB : FiniteDiagramArrow B.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f => ⟨f.1, f.2.1, Finset.mem_union_right _ f.2.2.1, Finset.mem_union_right _ f.2.2.2.1, f.2.2.2.2⟩ refine ⟨⟨A.1 ⊔ B.1, A.2.image ιA ⊔ B.2.image ιB⟩, ?_, ?_⟩ · rintro u hu f hf have : ιA f ∈ A.2.image ιA ⊔ B.2.image ιB := by apply Finset.mem_union_left rw [Finset.mem_image] exact ⟨f, hf, rfl⟩ exact hu this · rintro u hu f hf have : ιB f ∈ A.2.image ιA ⊔ B.2.image ιB := by apply Finset.mem_union_right rw [Finset.mem_image] exact ⟨f, hf, rfl⟩ exact hu this #align Top.partial_sections.directed TopCat.partialSections.directed	Mathlib/Topology/Category/TopCat/Limits/Konig.lean	107	124	theorem partialSections.closed [∀ j : J, T2Space (F.obj j)] {G : Finset J} (H : Finset (FiniteDiagramArrow G)) : IsClosed (partialSections F H) := by	have : partialSections F H = ⋂ (f : FiniteDiagramArrow G) (_ : f ∈ H), {u \| F.map f.2.2.2.2 (u f.1) = u f.2.1} := by ext1 simp only [Set.mem_iInter, Set.mem_setOf_eq] rfl rw [this] apply isClosed_biInter intro f _ -- Porting note: can't see through forget have : T2Space ((forget TopCat).obj (F.obj f.snd.fst)) := inferInstanceAs (T2Space (F.obj f.snd.fst)) apply isClosed_eq -- Porting note: used to be a single `continuity` that closed both goals · exact (F.map f.snd.snd.snd.snd).continuous.comp (continuous_apply f.fst) · continuity
/- Copyright (c) 2022 Thomas Browning. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Thomas Browning -/ import Mathlib.Algebra.Polynomial.Mirror import Mathlib.Analysis.Complex.Polynomial #align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836" /-! # Unit Trinomials This file defines irreducible trinomials and proves an irreducibility criterion. ## Main definitions - `Polynomial.IsUnitTrinomial` ## Main results - `Polynomial.IsUnitTrinomial.irreducible_of_coprime`: An irreducibility criterion for unit trinomials. -/ namespace Polynomial open scoped Polynomial open Finset section Semiring variable {R : Type} [Semiring R] (k m n : ℕ) (u v w : R) /-- Shorthand for a trinomial -/ noncomputable def trinomial := C u X ^ k + C v * X ^ m + C w * X ^ n #align polynomial.trinomial Polynomial.trinomial theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n := rfl #align polynomial.trinomial_def Polynomial.trinomial_def variable {k m n u v w} theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) : (trinomial k m n u v w).coeff n = w := by rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow, if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add] #align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff' theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) : (trinomial k m n u v w).coeff m = v := by rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow, if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero] #align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) : (trinomial k m n u v w).coeff k = u := by rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow, if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero] #align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff' theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) : (trinomial k m n u v w).natDegree = n := by refine natDegree_eq_of_degree_eq_some ((Finset.sup_le fun i h => ?_).antisymm <\| le_degree_of_ne_zero <\| by rwa [trinomial_leading_coeff' hkm hmn]) replace h := support_trinomial' k m n u v w h rw [mem_insert, mem_insert, mem_singleton] at h rcases h with (rfl \| rfl \| rfl) · exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le · exact WithBot.coe_le_coe.mpr hmn.le · exact le_rfl #align polynomial.trinomial_nat_degree Polynomial.trinomial_natDegree theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) : (trinomial k m n u v w).natTrailingDegree = k := by refine natTrailingDegree_eq_of_trailingDegree_eq_some ((Finset.le_inf fun i h => ?_).antisymm <\| trailingDegree_le_of_ne_zero <\| by rwa [trinomial_trailing_coeff' hkm hmn]).symm replace h := support_trinomial' k m n u v w h rw [mem_insert, mem_insert, mem_singleton] at h rcases h with (rfl \| rfl \| rfl) · exact le_rfl · exact WithTop.coe_le_coe.mpr hkm.le · exact WithTop.coe_le_coe.mpr (hkm.trans hmn).le #align polynomial.trinomial_nat_trailing_degree Polynomial.trinomial_natTrailingDegree theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) : (trinomial k m n u v w).leadingCoeff = w := by rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn] #align polynomial.trinomial_leading_coeff Polynomial.trinomial_leadingCoeff	Mathlib/Algebra/Polynomial/UnitTrinomial.lean	100	102	theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) : (trinomial k m n u v w).trailingCoeff = u := by	rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
/- Copyright (c) 2019 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Yury Kudryashov -/ import Mathlib.Analysis.Normed.Group.InfiniteSum import Mathlib.Analysis.Normed.MulAction import Mathlib.Topology.Algebra.Order.LiminfLimsup import Mathlib.Topology.PartialHomeomorph #align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Asymptotics We introduce these relations: * `IsBigOWith c l f g` : "f is big O of g along l with constant c"; * `f =O[l] g` : "f is big O of g along l"; * `f =o[l] g` : "f is little o of g along l". Here `l` is any filter on the domain of `f` and `g`, which are assumed to be the same. The codomains of `f` and `g` do not need to be the same; all that is needed that there is a norm associated with these types, and it is the norm that is compared asymptotically. The relation `IsBigOWith c` is introduced to factor out common algebraic arguments in the proofs of similar properties of `IsBigO` and `IsLittleO`. Usually proofs outside of this file should use `IsBigO` instead. Often the ranges of `f` and `g` will be the real numbers, in which case the norm is the absolute value. In general, we have `f =O[l] g ↔ (fun x ↦ ‖f x‖) =O[l] (fun x ↦ ‖g x‖)`, and similarly for `IsLittleO`. But our setup allows us to use the notions e.g. with functions to the integers, rationals, complex numbers, or any normed vector space without mentioning the norm explicitly. If `f` and `g` are functions to a normed field like the reals or complex numbers and `g` is always nonzero, we have `f =o[l] g ↔ Tendsto (fun x ↦ f x / (g x)) l (𝓝 0)`. In fact, the right-to-left direction holds without the hypothesis on `g`, and in the other direction it suffices to assume that `f` is zero wherever `g` is. (This generalization is useful in defining the Fréchet derivative.) -/ open Filter Set open scoped Classical open Topology Filter NNReal namespace Asymptotics set_option linter.uppercaseLean3 false variable {α : Type} {β : Type} {E : Type} {F : Type} {G : Type} {E' : Type} {F' : Type} {G' : Type} {E'' : Type} {F'' : Type} {G'' : Type} {E''' : Type} {R : Type} {R' : Type} {𝕜 : Type} {𝕜' : Type} variable [Norm E] [Norm F] [Norm G] variable [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G'] [NormedAddCommGroup E''] [NormedAddCommGroup F''] [NormedAddCommGroup G''] [SeminormedRing R] [SeminormedAddGroup E'''] [SeminormedRing R'] variable [NormedDivisionRing 𝕜] [NormedDivisionRing 𝕜'] variable {c c' c₁ c₂ : ℝ} {f : α → E} {g : α → F} {k : α → G} variable {f' : α → E'} {g' : α → F'} {k' : α → G'} variable {f'' : α → E''} {g'' : α → F''} {k'' : α → G''} variable {l l' : Filter α} section Defs /-! ### Definitions -/ /-- This version of the Landau notation `IsBigOWith C l f g` where `f` and `g` are two functions on a type `α` and `l` is a filter on `α`, means that eventually for `l`, `‖f‖` is bounded by `C * ‖g‖`. In other words, `‖f‖ / ‖g‖` is eventually bounded by `C`, modulo division by zero issues that are avoided by this definition. Probably you want to use `IsBigO` instead of this relation. -/ irreducible_def IsBigOWith (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : Prop := ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ #align asymptotics.is_O_with Asymptotics.IsBigOWith /-- Definition of `IsBigOWith`. We record it in a lemma as `IsBigOWith` is irreducible. -/ theorem isBigOWith_iff : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by rw [IsBigOWith_def] #align asymptotics.is_O_with_iff Asymptotics.isBigOWith_iff alias ⟨IsBigOWith.bound, IsBigOWith.of_bound⟩ := isBigOWith_iff #align asymptotics.is_O_with.bound Asymptotics.IsBigOWith.bound #align asymptotics.is_O_with.of_bound Asymptotics.IsBigOWith.of_bound /-- The Landau notation `f =O[l] g` where `f` and `g` are two functions on a type `α` and `l` is a filter on `α`, means that eventually for `l`, `‖f‖` is bounded by a constant multiple of `‖g‖`. In other words, `‖f‖ / ‖g‖` is eventually bounded, modulo division by zero issues that are avoided by this definition. -/ irreducible_def IsBigO (l : Filter α) (f : α → E) (g : α → F) : Prop := ∃ c : ℝ, IsBigOWith c l f g #align asymptotics.is_O Asymptotics.IsBigO @[inherit_doc] notation:100 f " =O[" l "] " g:100 => IsBigO l f g /-- Definition of `IsBigO` in terms of `IsBigOWith`. We record it in a lemma as `IsBigO` is irreducible. -/ theorem isBigO_iff_isBigOWith : f =O[l] g ↔ ∃ c : ℝ, IsBigOWith c l f g := by rw [IsBigO_def] #align asymptotics.is_O_iff_is_O_with Asymptotics.isBigO_iff_isBigOWith /-- Definition of `IsBigO` in terms of filters. -/ theorem isBigO_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by simp only [IsBigO_def, IsBigOWith_def] #align asymptotics.is_O_iff Asymptotics.isBigO_iff /-- Definition of `IsBigO` in terms of filters, with a positive constant. -/ theorem isBigO_iff' {g : α → E'''} : f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by refine ⟨fun h => ?mp, fun h => ?mpr⟩ case mp => rw [isBigO_iff] at h obtain ⟨c, hc⟩ := h refine ⟨max c 1, zero_lt_one.trans_le (le_max_right _ _), ?_⟩ filter_upwards [hc] with x hx apply hx.trans gcongr exact le_max_left _ _ case mpr => rw [isBigO_iff] obtain ⟨c, ⟨_, hc⟩⟩ := h exact ⟨c, hc⟩ /-- Definition of `IsBigO` in terms of filters, with the constant in the lower bound. -/ theorem isBigO_iff'' {g : α → E'''} : f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ := by refine ⟨fun h => ?mp, fun h => ?mpr⟩ case mp => rw [isBigO_iff'] at h obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩ filter_upwards [hc] with x hx rwa [inv_mul_le_iff (by positivity)] case mpr => rw [isBigO_iff'] obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩ filter_upwards [hc] with x hx rwa [← inv_inv c, inv_mul_le_iff (by positivity)] at hx theorem IsBigO.of_bound (c : ℝ) (h : ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖) : f =O[l] g := isBigO_iff.2 ⟨c, h⟩ #align asymptotics.is_O.of_bound Asymptotics.IsBigO.of_bound theorem IsBigO.of_bound' (h : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖) : f =O[l] g := IsBigO.of_bound 1 <\| by simp_rw [one_mul] exact h #align asymptotics.is_O.of_bound' Asymptotics.IsBigO.of_bound' theorem IsBigO.bound : f =O[l] g → ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigO_iff.1 #align asymptotics.is_O.bound Asymptotics.IsBigO.bound /-- The Landau notation `f =o[l] g` where `f` and `g` are two functions on a type `α` and `l` is a filter on `α`, means that eventually for `l`, `‖f‖` is bounded by an arbitrarily small constant multiple of `‖g‖`. In other words, `‖f‖ / ‖g‖` tends to `0` along `l`, modulo division by zero issues that are avoided by this definition. -/ irreducible_def IsLittleO (l : Filter α) (f : α → E) (g : α → F) : Prop := ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g #align asymptotics.is_o Asymptotics.IsLittleO @[inherit_doc] notation:100 f " =o[" l "] " g:100 => IsLittleO l f g /-- Definition of `IsLittleO` in terms of `IsBigOWith`. -/ theorem isLittleO_iff_forall_isBigOWith : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g := by rw [IsLittleO_def] #align asymptotics.is_o_iff_forall_is_O_with Asymptotics.isLittleO_iff_forall_isBigOWith alias ⟨IsLittleO.forall_isBigOWith, IsLittleO.of_isBigOWith⟩ := isLittleO_iff_forall_isBigOWith #align asymptotics.is_o.forall_is_O_with Asymptotics.IsLittleO.forall_isBigOWith #align asymptotics.is_o.of_is_O_with Asymptotics.IsLittleO.of_isBigOWith /-- Definition of `IsLittleO` in terms of filters. -/ theorem isLittleO_iff : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by simp only [IsLittleO_def, IsBigOWith_def] #align asymptotics.is_o_iff Asymptotics.isLittleO_iff alias ⟨IsLittleO.bound, IsLittleO.of_bound⟩ := isLittleO_iff #align asymptotics.is_o.bound Asymptotics.IsLittleO.bound #align asymptotics.is_o.of_bound Asymptotics.IsLittleO.of_bound theorem IsLittleO.def (h : f =o[l] g) (hc : 0 < c) : ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isLittleO_iff.1 h hc #align asymptotics.is_o.def Asymptotics.IsLittleO.def theorem IsLittleO.def' (h : f =o[l] g) (hc : 0 < c) : IsBigOWith c l f g := isBigOWith_iff.2 <\| isLittleO_iff.1 h hc #align asymptotics.is_o.def' Asymptotics.IsLittleO.def' theorem IsLittleO.eventuallyLE (h : f =o[l] g) : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖ := by simpa using h.def zero_lt_one end Defs /-! ### Conversions -/ theorem IsBigOWith.isBigO (h : IsBigOWith c l f g) : f =O[l] g := by rw [IsBigO_def]; exact ⟨c, h⟩ #align asymptotics.is_O_with.is_O Asymptotics.IsBigOWith.isBigO theorem IsLittleO.isBigOWith (hgf : f =o[l] g) : IsBigOWith 1 l f g := hgf.def' zero_lt_one #align asymptotics.is_o.is_O_with Asymptotics.IsLittleO.isBigOWith theorem IsLittleO.isBigO (hgf : f =o[l] g) : f =O[l] g := hgf.isBigOWith.isBigO #align asymptotics.is_o.is_O Asymptotics.IsLittleO.isBigO theorem IsBigO.isBigOWith : f =O[l] g → ∃ c : ℝ, IsBigOWith c l f g := isBigO_iff_isBigOWith.1 #align asymptotics.is_O.is_O_with Asymptotics.IsBigO.isBigOWith theorem IsBigOWith.weaken (h : IsBigOWith c l f g') (hc : c ≤ c') : IsBigOWith c' l f g' := IsBigOWith.of_bound <\| mem_of_superset h.bound fun x hx => calc ‖f x‖ ≤ c * ‖g' x‖ := hx _ ≤ _ := by gcongr #align asymptotics.is_O_with.weaken Asymptotics.IsBigOWith.weaken theorem IsBigOWith.exists_pos (h : IsBigOWith c l f g') : ∃ c' > 0, IsBigOWith c' l f g' := ⟨max c 1, lt_of_lt_of_le zero_lt_one (le_max_right c 1), h.weaken <\| le_max_left c 1⟩ #align asymptotics.is_O_with.exists_pos Asymptotics.IsBigOWith.exists_pos theorem IsBigO.exists_pos (h : f =O[l] g') : ∃ c > 0, IsBigOWith c l f g' := let ⟨_c, hc⟩ := h.isBigOWith hc.exists_pos #align asymptotics.is_O.exists_pos Asymptotics.IsBigO.exists_pos theorem IsBigOWith.exists_nonneg (h : IsBigOWith c l f g') : ∃ c' ≥ 0, IsBigOWith c' l f g' := let ⟨c, cpos, hc⟩ := h.exists_pos ⟨c, le_of_lt cpos, hc⟩ #align asymptotics.is_O_with.exists_nonneg Asymptotics.IsBigOWith.exists_nonneg theorem IsBigO.exists_nonneg (h : f =O[l] g') : ∃ c ≥ 0, IsBigOWith c l f g' := let ⟨_c, hc⟩ := h.isBigOWith hc.exists_nonneg #align asymptotics.is_O.exists_nonneg Asymptotics.IsBigO.exists_nonneg /-- `f = O(g)` if and only if `IsBigOWith c f g` for all sufficiently large `c`. -/ theorem isBigO_iff_eventually_isBigOWith : f =O[l] g' ↔ ∀ᶠ c in atTop, IsBigOWith c l f g' := isBigO_iff_isBigOWith.trans ⟨fun ⟨c, hc⟩ => mem_atTop_sets.2 ⟨c, fun _c' hc' => hc.weaken hc'⟩, fun h => h.exists⟩ #align asymptotics.is_O_iff_eventually_is_O_with Asymptotics.isBigO_iff_eventually_isBigOWith /-- `f = O(g)` if and only if `∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖` for all sufficiently large `c`. -/ theorem isBigO_iff_eventually : f =O[l] g' ↔ ∀ᶠ c in atTop, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g' x‖ := isBigO_iff_eventually_isBigOWith.trans <\| by simp only [IsBigOWith_def] #align asymptotics.is_O_iff_eventually Asymptotics.isBigO_iff_eventually theorem IsBigO.exists_mem_basis {ι} {p : ι → Prop} {s : ι → Set α} (h : f =O[l] g') (hb : l.HasBasis p s) : ∃ c > 0, ∃ i : ι, p i ∧ ∀ x ∈ s i, ‖f x‖ ≤ c * ‖g' x‖ := flip Exists.imp h.exists_pos fun c h => by simpa only [isBigOWith_iff, hb.eventually_iff, exists_prop] using h #align asymptotics.is_O.exists_mem_basis Asymptotics.IsBigO.exists_mem_basis theorem isBigOWith_inv (hc : 0 < c) : IsBigOWith c⁻¹ l f g ↔ ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ := by simp only [IsBigOWith_def, ← div_eq_inv_mul, le_div_iff' hc] #align asymptotics.is_O_with_inv Asymptotics.isBigOWith_inv -- We prove this lemma with strange assumptions to get two lemmas below automatically theorem isLittleO_iff_nat_mul_le_aux (h₀ : (∀ x, 0 ≤ ‖f x‖) ∨ ∀ x, 0 ≤ ‖g x‖) : f =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g x‖ := by constructor · rintro H (_ \| n) · refine (H.def one_pos).mono fun x h₀' => ?_ rw [Nat.cast_zero, zero_mul] refine h₀.elim (fun hf => (hf x).trans ?_) fun hg => hg x rwa [one_mul] at h₀' · have : (0 : ℝ) < n.succ := Nat.cast_pos.2 n.succ_pos exact (isBigOWith_inv this).1 (H.def' <\| inv_pos.2 this) · refine fun H => isLittleO_iff.2 fun ε ε0 => ?_ rcases exists_nat_gt ε⁻¹ with ⟨n, hn⟩ have hn₀ : (0 : ℝ) < n := (inv_pos.2 ε0).trans hn refine ((isBigOWith_inv hn₀).2 (H n)).bound.mono fun x hfg => ?_ refine hfg.trans (mul_le_mul_of_nonneg_right (inv_le_of_inv_le ε0 hn.le) ?_) refine h₀.elim (fun hf => nonneg_of_mul_nonneg_right ((hf x).trans hfg) ?_) fun h => h x exact inv_pos.2 hn₀ #align asymptotics.is_o_iff_nat_mul_le_aux Asymptotics.isLittleO_iff_nat_mul_le_aux theorem isLittleO_iff_nat_mul_le : f =o[l] g' ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g' x‖ := isLittleO_iff_nat_mul_le_aux (Or.inr fun _x => norm_nonneg _) #align asymptotics.is_o_iff_nat_mul_le Asymptotics.isLittleO_iff_nat_mul_le theorem isLittleO_iff_nat_mul_le' : f' =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f' x‖ ≤ ‖g x‖ := isLittleO_iff_nat_mul_le_aux (Or.inl fun _x => norm_nonneg _) #align asymptotics.is_o_iff_nat_mul_le' Asymptotics.isLittleO_iff_nat_mul_le' /-! ### Subsingleton -/ @[nontriviality] theorem isLittleO_of_subsingleton [Subsingleton E'] : f' =o[l] g' := IsLittleO.of_bound fun c hc => by simp [Subsingleton.elim (f' _) 0, mul_nonneg hc.le] #align asymptotics.is_o_of_subsingleton Asymptotics.isLittleO_of_subsingleton @[nontriviality] theorem isBigO_of_subsingleton [Subsingleton E'] : f' =O[l] g' := isLittleO_of_subsingleton.isBigO #align asymptotics.is_O_of_subsingleton Asymptotics.isBigO_of_subsingleton section congr variable {f₁ f₂ : α → E} {g₁ g₂ : α → F} /-! ### Congruence -/ theorem isBigOWith_congr (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : IsBigOWith c₁ l f₁ g₁ ↔ IsBigOWith c₂ l f₂ g₂ := by simp only [IsBigOWith_def] subst c₂ apply Filter.eventually_congr filter_upwards [hf, hg] with _ e₁ e₂ rw [e₁, e₂] #align asymptotics.is_O_with_congr Asymptotics.isBigOWith_congr theorem IsBigOWith.congr' (h : IsBigOWith c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : IsBigOWith c₂ l f₂ g₂ := (isBigOWith_congr hc hf hg).mp h #align asymptotics.is_O_with.congr' Asymptotics.IsBigOWith.congr' theorem IsBigOWith.congr (h : IsBigOWith c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : ∀ x, f₁ x = f₂ x) (hg : ∀ x, g₁ x = g₂ x) : IsBigOWith c₂ l f₂ g₂ := h.congr' hc (univ_mem' hf) (univ_mem' hg) #align asymptotics.is_O_with.congr Asymptotics.IsBigOWith.congr theorem IsBigOWith.congr_left (h : IsBigOWith c l f₁ g) (hf : ∀ x, f₁ x = f₂ x) : IsBigOWith c l f₂ g := h.congr rfl hf fun _ => rfl #align asymptotics.is_O_with.congr_left Asymptotics.IsBigOWith.congr_left theorem IsBigOWith.congr_right (h : IsBigOWith c l f g₁) (hg : ∀ x, g₁ x = g₂ x) : IsBigOWith c l f g₂ := h.congr rfl (fun _ => rfl) hg #align asymptotics.is_O_with.congr_right Asymptotics.IsBigOWith.congr_right theorem IsBigOWith.congr_const (h : IsBigOWith c₁ l f g) (hc : c₁ = c₂) : IsBigOWith c₂ l f g := h.congr hc (fun _ => rfl) fun _ => rfl #align asymptotics.is_O_with.congr_const Asymptotics.IsBigOWith.congr_const theorem isBigO_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =O[l] g₁ ↔ f₂ =O[l] g₂ := by simp only [IsBigO_def] exact exists_congr fun c => isBigOWith_congr rfl hf hg #align asymptotics.is_O_congr Asymptotics.isBigO_congr theorem IsBigO.congr' (h : f₁ =O[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₂ =O[l] g₂ := (isBigO_congr hf hg).mp h #align asymptotics.is_O.congr' Asymptotics.IsBigO.congr' theorem IsBigO.congr (h : f₁ =O[l] g₁) (hf : ∀ x, f₁ x = f₂ x) (hg : ∀ x, g₁ x = g₂ x) : f₂ =O[l] g₂ := h.congr' (univ_mem' hf) (univ_mem' hg) #align asymptotics.is_O.congr Asymptotics.IsBigO.congr theorem IsBigO.congr_left (h : f₁ =O[l] g) (hf : ∀ x, f₁ x = f₂ x) : f₂ =O[l] g := h.congr hf fun _ => rfl #align asymptotics.is_O.congr_left Asymptotics.IsBigO.congr_left theorem IsBigO.congr_right (h : f =O[l] g₁) (hg : ∀ x, g₁ x = g₂ x) : f =O[l] g₂ := h.congr (fun _ => rfl) hg #align asymptotics.is_O.congr_right Asymptotics.IsBigO.congr_right theorem isLittleO_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =o[l] g₁ ↔ f₂ =o[l] g₂ := by simp only [IsLittleO_def] exact forall₂_congr fun c _hc => isBigOWith_congr (Eq.refl c) hf hg #align asymptotics.is_o_congr Asymptotics.isLittleO_congr theorem IsLittleO.congr' (h : f₁ =o[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₂ =o[l] g₂ := (isLittleO_congr hf hg).mp h #align asymptotics.is_o.congr' Asymptotics.IsLittleO.congr' theorem IsLittleO.congr (h : f₁ =o[l] g₁) (hf : ∀ x, f₁ x = f₂ x) (hg : ∀ x, g₁ x = g₂ x) : f₂ =o[l] g₂ := h.congr' (univ_mem' hf) (univ_mem' hg) #align asymptotics.is_o.congr Asymptotics.IsLittleO.congr theorem IsLittleO.congr_left (h : f₁ =o[l] g) (hf : ∀ x, f₁ x = f₂ x) : f₂ =o[l] g := h.congr hf fun _ => rfl #align asymptotics.is_o.congr_left Asymptotics.IsLittleO.congr_left theorem IsLittleO.congr_right (h : f =o[l] g₁) (hg : ∀ x, g₁ x = g₂ x) : f =o[l] g₂ := h.congr (fun _ => rfl) hg #align asymptotics.is_o.congr_right Asymptotics.IsLittleO.congr_right @[trans] theorem _root_.Filter.EventuallyEq.trans_isBigO {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂) (h : f₂ =O[l] g) : f₁ =O[l] g := h.congr' hf.symm EventuallyEq.rfl #align filter.eventually_eq.trans_is_O Filter.EventuallyEq.trans_isBigO instance transEventuallyEqIsBigO : @Trans (α → E) (α → E) (α → F) (· =ᶠ[l] ·) (· =O[l] ·) (· =O[l] ·) where trans := Filter.EventuallyEq.trans_isBigO @[trans] theorem _root_.Filter.EventuallyEq.trans_isLittleO {f₁ f₂ : α → E} {g : α → F} (hf : f₁ =ᶠ[l] f₂) (h : f₂ =o[l] g) : f₁ =o[l] g := h.congr' hf.symm EventuallyEq.rfl #align filter.eventually_eq.trans_is_o Filter.EventuallyEq.trans_isLittleO instance transEventuallyEqIsLittleO : @Trans (α → E) (α → E) (α → F) (· =ᶠ[l] ·) (· =o[l] ·) (· =o[l] ·) where trans := Filter.EventuallyEq.trans_isLittleO @[trans] theorem IsBigO.trans_eventuallyEq {f : α → E} {g₁ g₂ : α → F} (h : f =O[l] g₁) (hg : g₁ =ᶠ[l] g₂) : f =O[l] g₂ := h.congr' EventuallyEq.rfl hg #align asymptotics.is_O.trans_eventually_eq Asymptotics.IsBigO.trans_eventuallyEq instance transIsBigOEventuallyEq : @Trans (α → E) (α → F) (α → F) (· =O[l] ·) (· =ᶠ[l] ·) (· =O[l] ·) where trans := IsBigO.trans_eventuallyEq @[trans] theorem IsLittleO.trans_eventuallyEq {f : α → E} {g₁ g₂ : α → F} (h : f =o[l] g₁) (hg : g₁ =ᶠ[l] g₂) : f =o[l] g₂ := h.congr' EventuallyEq.rfl hg #align asymptotics.is_o.trans_eventually_eq Asymptotics.IsLittleO.trans_eventuallyEq instance transIsLittleOEventuallyEq : @Trans (α → E) (α → F) (α → F) (· =o[l] ·) (· =ᶠ[l] ·) (· =o[l] ·) where trans := IsLittleO.trans_eventuallyEq end congr /-! ### Filter operations and transitivity -/ theorem IsBigOWith.comp_tendsto (hcfg : IsBigOWith c l f g) {k : β → α} {l' : Filter β} (hk : Tendsto k l' l) : IsBigOWith c l' (f ∘ k) (g ∘ k) := IsBigOWith.of_bound <\| hk hcfg.bound #align asymptotics.is_O_with.comp_tendsto Asymptotics.IsBigOWith.comp_tendsto theorem IsBigO.comp_tendsto (hfg : f =O[l] g) {k : β → α} {l' : Filter β} (hk : Tendsto k l' l) : (f ∘ k) =O[l'] (g ∘ k) := isBigO_iff_isBigOWith.2 <\| hfg.isBigOWith.imp fun _c h => h.comp_tendsto hk #align asymptotics.is_O.comp_tendsto Asymptotics.IsBigO.comp_tendsto theorem IsLittleO.comp_tendsto (hfg : f =o[l] g) {k : β → α} {l' : Filter β} (hk : Tendsto k l' l) : (f ∘ k) =o[l'] (g ∘ k) := IsLittleO.of_isBigOWith fun _c cpos => (hfg.forall_isBigOWith cpos).comp_tendsto hk #align asymptotics.is_o.comp_tendsto Asymptotics.IsLittleO.comp_tendsto @[simp] theorem isBigOWith_map {k : β → α} {l : Filter β} : IsBigOWith c (map k l) f g ↔ IsBigOWith c l (f ∘ k) (g ∘ k) := by simp only [IsBigOWith_def] exact eventually_map #align asymptotics.is_O_with_map Asymptotics.isBigOWith_map @[simp] theorem isBigO_map {k : β → α} {l : Filter β} : f =O[map k l] g ↔ (f ∘ k) =O[l] (g ∘ k) := by simp only [IsBigO_def, isBigOWith_map] #align asymptotics.is_O_map Asymptotics.isBigO_map @[simp] theorem isLittleO_map {k : β → α} {l : Filter β} : f =o[map k l] g ↔ (f ∘ k) =o[l] (g ∘ k) := by simp only [IsLittleO_def, isBigOWith_map] #align asymptotics.is_o_map Asymptotics.isLittleO_map theorem IsBigOWith.mono (h : IsBigOWith c l' f g) (hl : l ≤ l') : IsBigOWith c l f g := IsBigOWith.of_bound <\| hl h.bound #align asymptotics.is_O_with.mono Asymptotics.IsBigOWith.mono theorem IsBigO.mono (h : f =O[l'] g) (hl : l ≤ l') : f =O[l] g := isBigO_iff_isBigOWith.2 <\| h.isBigOWith.imp fun _c h => h.mono hl #align asymptotics.is_O.mono Asymptotics.IsBigO.mono theorem IsLittleO.mono (h : f =o[l'] g) (hl : l ≤ l') : f =o[l] g := IsLittleO.of_isBigOWith fun _c cpos => (h.forall_isBigOWith cpos).mono hl #align asymptotics.is_o.mono Asymptotics.IsLittleO.mono theorem IsBigOWith.trans (hfg : IsBigOWith c l f g) (hgk : IsBigOWith c' l g k) (hc : 0 ≤ c) : IsBigOWith (c * c') l f k := by simp only [IsBigOWith_def] at * filter_upwards [hfg, hgk] with x hx hx' calc ‖f x‖ ≤ c * ‖g x‖ := hx _ ≤ c * (c' * ‖k x‖) := by gcongr _ = c * c' * ‖k x‖ := (mul_assoc _ _ _).symm #align asymptotics.is_O_with.trans Asymptotics.IsBigOWith.trans @[trans] theorem IsBigO.trans {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g) (hgk : g =O[l] k) : f =O[l] k := let ⟨_c, cnonneg, hc⟩ := hfg.exists_nonneg let ⟨_c', hc'⟩ := hgk.isBigOWith (hc.trans hc' cnonneg).isBigO #align asymptotics.is_O.trans Asymptotics.IsBigO.trans instance transIsBigOIsBigO : @Trans (α → E) (α → F') (α → G) (· =O[l] ·) (· =O[l] ·) (· =O[l] ·) where trans := IsBigO.trans theorem IsLittleO.trans_isBigOWith (hfg : f =o[l] g) (hgk : IsBigOWith c l g k) (hc : 0 < c) : f =o[l] k := by simp only [IsLittleO_def] at * intro c' c'pos have : 0 < c' / c := div_pos c'pos hc exact ((hfg this).trans hgk this.le).congr_const (div_mul_cancel₀ _ hc.ne') #align asymptotics.is_o.trans_is_O_with Asymptotics.IsLittleO.trans_isBigOWith @[trans] theorem IsLittleO.trans_isBigO {f : α → E} {g : α → F} {k : α → G'} (hfg : f =o[l] g) (hgk : g =O[l] k) : f =o[l] k := let ⟨_c, cpos, hc⟩ := hgk.exists_pos hfg.trans_isBigOWith hc cpos #align asymptotics.is_o.trans_is_O Asymptotics.IsLittleO.trans_isBigO instance transIsLittleOIsBigO : @Trans (α → E) (α → F) (α → G') (· =o[l] ·) (· =O[l] ·) (· =o[l] ·) where trans := IsLittleO.trans_isBigO theorem IsBigOWith.trans_isLittleO (hfg : IsBigOWith c l f g) (hgk : g =o[l] k) (hc : 0 < c) : f =o[l] k := by simp only [IsLittleO_def] at * intro c' c'pos have : 0 < c' / c := div_pos c'pos hc exact (hfg.trans (hgk this) hc.le).congr_const (mul_div_cancel₀ _ hc.ne') #align asymptotics.is_O_with.trans_is_o Asymptotics.IsBigOWith.trans_isLittleO @[trans] theorem IsBigO.trans_isLittleO {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g) (hgk : g =o[l] k) : f =o[l] k := let ⟨_c, cpos, hc⟩ := hfg.exists_pos hc.trans_isLittleO hgk cpos #align asymptotics.is_O.trans_is_o Asymptotics.IsBigO.trans_isLittleO instance transIsBigOIsLittleO : @Trans (α → E) (α → F') (α → G) (· =O[l] ·) (· =o[l] ·) (· =o[l] ·) where trans := IsBigO.trans_isLittleO @[trans] theorem IsLittleO.trans {f : α → E} {g : α → F} {k : α → G} (hfg : f =o[l] g) (hgk : g =o[l] k) : f =o[l] k := hfg.trans_isBigOWith hgk.isBigOWith one_pos #align asymptotics.is_o.trans Asymptotics.IsLittleO.trans instance transIsLittleOIsLittleO : @Trans (α → E) (α → F) (α → G) (· =o[l] ·) (· =o[l] ·) (· =o[l] ·) where trans := IsLittleO.trans theorem _root_.Filter.Eventually.trans_isBigO {f : α → E} {g : α → F'} {k : α → G} (hfg : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖) (hgk : g =O[l] k) : f =O[l] k := (IsBigO.of_bound' hfg).trans hgk #align filter.eventually.trans_is_O Filter.Eventually.trans_isBigO theorem _root_.Filter.Eventually.isBigO {f : α → E} {g : α → ℝ} {l : Filter α} (hfg : ∀ᶠ x in l, ‖f x‖ ≤ g x) : f =O[l] g := IsBigO.of_bound' <\| hfg.mono fun _x hx => hx.trans <\| Real.le_norm_self _ #align filter.eventually.is_O Filter.Eventually.isBigO section variable (l) theorem isBigOWith_of_le' (hfg : ∀ x, ‖f x‖ ≤ c * ‖g x‖) : IsBigOWith c l f g := IsBigOWith.of_bound <\| univ_mem' hfg #align asymptotics.is_O_with_of_le' Asymptotics.isBigOWith_of_le' theorem isBigOWith_of_le (hfg : ∀ x, ‖f x‖ ≤ ‖g x‖) : IsBigOWith 1 l f g := isBigOWith_of_le' l fun x => by rw [one_mul] exact hfg x #align asymptotics.is_O_with_of_le Asymptotics.isBigOWith_of_le theorem isBigO_of_le' (hfg : ∀ x, ‖f x‖ ≤ c * ‖g x‖) : f =O[l] g := (isBigOWith_of_le' l hfg).isBigO #align asymptotics.is_O_of_le' Asymptotics.isBigO_of_le' theorem isBigO_of_le (hfg : ∀ x, ‖f x‖ ≤ ‖g x‖) : f =O[l] g := (isBigOWith_of_le l hfg).isBigO #align asymptotics.is_O_of_le Asymptotics.isBigO_of_le end theorem isBigOWith_refl (f : α → E) (l : Filter α) : IsBigOWith 1 l f f := isBigOWith_of_le l fun _ => le_rfl #align asymptotics.is_O_with_refl Asymptotics.isBigOWith_refl theorem isBigO_refl (f : α → E) (l : Filter α) : f =O[l] f := (isBigOWith_refl f l).isBigO #align asymptotics.is_O_refl Asymptotics.isBigO_refl theorem _root_.Filter.EventuallyEq.isBigO {f₁ f₂ : α → E} (hf : f₁ =ᶠ[l] f₂) : f₁ =O[l] f₂ := hf.trans_isBigO (isBigO_refl _ _) theorem IsBigOWith.trans_le (hfg : IsBigOWith c l f g) (hgk : ∀ x, ‖g x‖ ≤ ‖k x‖) (hc : 0 ≤ c) : IsBigOWith c l f k := (hfg.trans (isBigOWith_of_le l hgk) hc).congr_const <\| mul_one c #align asymptotics.is_O_with.trans_le Asymptotics.IsBigOWith.trans_le theorem IsBigO.trans_le (hfg : f =O[l] g') (hgk : ∀ x, ‖g' x‖ ≤ ‖k x‖) : f =O[l] k := hfg.trans (isBigO_of_le l hgk) #align asymptotics.is_O.trans_le Asymptotics.IsBigO.trans_le theorem IsLittleO.trans_le (hfg : f =o[l] g) (hgk : ∀ x, ‖g x‖ ≤ ‖k x‖) : f =o[l] k := hfg.trans_isBigOWith (isBigOWith_of_le _ hgk) zero_lt_one #align asymptotics.is_o.trans_le Asymptotics.IsLittleO.trans_le theorem isLittleO_irrefl' (h : ∃ᶠ x in l, ‖f' x‖ ≠ 0) : ¬f' =o[l] f' := by intro ho rcases ((ho.bound one_half_pos).and_frequently h).exists with ⟨x, hle, hne⟩ rw [one_div, ← div_eq_inv_mul] at hle exact (half_lt_self (lt_of_le_of_ne (norm_nonneg _) hne.symm)).not_le hle #align asymptotics.is_o_irrefl' Asymptotics.isLittleO_irrefl' theorem isLittleO_irrefl (h : ∃ᶠ x in l, f'' x ≠ 0) : ¬f'' =o[l] f'' := isLittleO_irrefl' <\| h.mono fun _x => norm_ne_zero_iff.mpr #align asymptotics.is_o_irrefl Asymptotics.isLittleO_irrefl theorem IsBigO.not_isLittleO (h : f'' =O[l] g') (hf : ∃ᶠ x in l, f'' x ≠ 0) : ¬g' =o[l] f'' := fun h' => isLittleO_irrefl hf (h.trans_isLittleO h') #align asymptotics.is_O.not_is_o Asymptotics.IsBigO.not_isLittleO theorem IsLittleO.not_isBigO (h : f'' =o[l] g') (hf : ∃ᶠ x in l, f'' x ≠ 0) : ¬g' =O[l] f'' := fun h' => isLittleO_irrefl hf (h.trans_isBigO h') #align asymptotics.is_o.not_is_O Asymptotics.IsLittleO.not_isBigO section Bot variable (c f g) @[simp] theorem isBigOWith_bot : IsBigOWith c ⊥ f g := IsBigOWith.of_bound <\| trivial #align asymptotics.is_O_with_bot Asymptotics.isBigOWith_bot @[simp] theorem isBigO_bot : f =O[⊥] g := (isBigOWith_bot 1 f g).isBigO #align asymptotics.is_O_bot Asymptotics.isBigO_bot @[simp] theorem isLittleO_bot : f =o[⊥] g := IsLittleO.of_isBigOWith fun c _ => isBigOWith_bot c f g #align asymptotics.is_o_bot Asymptotics.isLittleO_bot end Bot @[simp] theorem isBigOWith_pure {x} : IsBigOWith c (pure x) f g ↔ ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff #align asymptotics.is_O_with_pure Asymptotics.isBigOWith_pure theorem IsBigOWith.sup (h : IsBigOWith c l f g) (h' : IsBigOWith c l' f g) : IsBigOWith c (l ⊔ l') f g := IsBigOWith.of_bound <\| mem_sup.2 ⟨h.bound, h'.bound⟩ #align asymptotics.is_O_with.sup Asymptotics.IsBigOWith.sup theorem IsBigOWith.sup' (h : IsBigOWith c l f g') (h' : IsBigOWith c' l' f g') : IsBigOWith (max c c') (l ⊔ l') f g' := IsBigOWith.of_bound <\| mem_sup.2 ⟨(h.weaken <\| le_max_left c c').bound, (h'.weaken <\| le_max_right c c').bound⟩ #align asymptotics.is_O_with.sup' Asymptotics.IsBigOWith.sup' theorem IsBigO.sup (h : f =O[l] g') (h' : f =O[l'] g') : f =O[l ⊔ l'] g' := let ⟨_c, hc⟩ := h.isBigOWith let ⟨_c', hc'⟩ := h'.isBigOWith (hc.sup' hc').isBigO #align asymptotics.is_O.sup Asymptotics.IsBigO.sup theorem IsLittleO.sup (h : f =o[l] g) (h' : f =o[l'] g) : f =o[l ⊔ l'] g := IsLittleO.of_isBigOWith fun _c cpos => (h.forall_isBigOWith cpos).sup (h'.forall_isBigOWith cpos) #align asymptotics.is_o.sup Asymptotics.IsLittleO.sup @[simp] theorem isBigO_sup : f =O[l ⊔ l'] g' ↔ f =O[l] g' ∧ f =O[l'] g' := ⟨fun h => ⟨h.mono le_sup_left, h.mono le_sup_right⟩, fun h => h.1.sup h.2⟩ #align asymptotics.is_O_sup Asymptotics.isBigO_sup @[simp] theorem isLittleO_sup : f =o[l ⊔ l'] g ↔ f =o[l] g ∧ f =o[l'] g := ⟨fun h => ⟨h.mono le_sup_left, h.mono le_sup_right⟩, fun h => h.1.sup h.2⟩ #align asymptotics.is_o_sup Asymptotics.isLittleO_sup theorem isBigOWith_insert [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E} {g' : α → F} (h : ‖g x‖ ≤ C * ‖g' x‖) : IsBigOWith C (𝓝[insert x s] x) g g' ↔ IsBigOWith C (𝓝[s] x) g g' := by simp_rw [IsBigOWith_def, nhdsWithin_insert, eventually_sup, eventually_pure, h, true_and_iff] #align asymptotics.is_O_with_insert Asymptotics.isBigOWith_insert protected theorem IsBigOWith.insert [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E} {g' : α → F} (h1 : IsBigOWith C (𝓝[s] x) g g') (h2 : ‖g x‖ ≤ C * ‖g' x‖) : IsBigOWith C (𝓝[insert x s] x) g g' := (isBigOWith_insert h2).mpr h1 #align asymptotics.is_O_with.insert Asymptotics.IsBigOWith.insert theorem isLittleO_insert [TopologicalSpace α] {x : α} {s : Set α} {g : α → E'} {g' : α → F'} (h : g x = 0) : g =o[𝓝[insert x s] x] g' ↔ g =o[𝓝[s] x] g' := by simp_rw [IsLittleO_def] refine forall_congr' fun c => forall_congr' fun hc => ?_ rw [isBigOWith_insert] rw [h, norm_zero] exact mul_nonneg hc.le (norm_nonneg _) #align asymptotics.is_o_insert Asymptotics.isLittleO_insert protected theorem IsLittleO.insert [TopologicalSpace α] {x : α} {s : Set α} {g : α → E'} {g' : α → F'} (h1 : g =o[𝓝[s] x] g') (h2 : g x = 0) : g =o[𝓝[insert x s] x] g' := (isLittleO_insert h2).mpr h1 #align asymptotics.is_o.insert Asymptotics.IsLittleO.insert /-! ### Simplification : norm, abs -/ section NormAbs variable {u v : α → ℝ} @[simp] theorem isBigOWith_norm_right : (IsBigOWith c l f fun x => ‖g' x‖) ↔ IsBigOWith c l f g' := by simp only [IsBigOWith_def, norm_norm] #align asymptotics.is_O_with_norm_right Asymptotics.isBigOWith_norm_right @[simp] theorem isBigOWith_abs_right : (IsBigOWith c l f fun x => \|u x\|) ↔ IsBigOWith c l f u := @isBigOWith_norm_right _ _ _ _ _ _ f u l #align asymptotics.is_O_with_abs_right Asymptotics.isBigOWith_abs_right alias ⟨IsBigOWith.of_norm_right, IsBigOWith.norm_right⟩ := isBigOWith_norm_right #align asymptotics.is_O_with.of_norm_right Asymptotics.IsBigOWith.of_norm_right #align asymptotics.is_O_with.norm_right Asymptotics.IsBigOWith.norm_right alias ⟨IsBigOWith.of_abs_right, IsBigOWith.abs_right⟩ := isBigOWith_abs_right #align asymptotics.is_O_with.of_abs_right Asymptotics.IsBigOWith.of_abs_right #align asymptotics.is_O_with.abs_right Asymptotics.IsBigOWith.abs_right @[simp] theorem isBigO_norm_right : (f =O[l] fun x => ‖g' x‖) ↔ f =O[l] g' := by simp only [IsBigO_def] exact exists_congr fun _ => isBigOWith_norm_right #align asymptotics.is_O_norm_right Asymptotics.isBigO_norm_right @[simp] theorem isBigO_abs_right : (f =O[l] fun x => \|u x\|) ↔ f =O[l] u := @isBigO_norm_right _ _ ℝ _ _ _ _ _ #align asymptotics.is_O_abs_right Asymptotics.isBigO_abs_right alias ⟨IsBigO.of_norm_right, IsBigO.norm_right⟩ := isBigO_norm_right #align asymptotics.is_O.of_norm_right Asymptotics.IsBigO.of_norm_right #align asymptotics.is_O.norm_right Asymptotics.IsBigO.norm_right alias ⟨IsBigO.of_abs_right, IsBigO.abs_right⟩ := isBigO_abs_right #align asymptotics.is_O.of_abs_right Asymptotics.IsBigO.of_abs_right #align asymptotics.is_O.abs_right Asymptotics.IsBigO.abs_right @[simp] theorem isLittleO_norm_right : (f =o[l] fun x => ‖g' x‖) ↔ f =o[l] g' := by simp only [IsLittleO_def] exact forall₂_congr fun _ _ => isBigOWith_norm_right #align asymptotics.is_o_norm_right Asymptotics.isLittleO_norm_right @[simp] theorem isLittleO_abs_right : (f =o[l] fun x => \|u x\|) ↔ f =o[l] u := @isLittleO_norm_right _ _ ℝ _ _ _ _ _ #align asymptotics.is_o_abs_right Asymptotics.isLittleO_abs_right alias ⟨IsLittleO.of_norm_right, IsLittleO.norm_right⟩ := isLittleO_norm_right #align asymptotics.is_o.of_norm_right Asymptotics.IsLittleO.of_norm_right #align asymptotics.is_o.norm_right Asymptotics.IsLittleO.norm_right alias ⟨IsLittleO.of_abs_right, IsLittleO.abs_right⟩ := isLittleO_abs_right #align asymptotics.is_o.of_abs_right Asymptotics.IsLittleO.of_abs_right #align asymptotics.is_o.abs_right Asymptotics.IsLittleO.abs_right @[simp] theorem isBigOWith_norm_left : IsBigOWith c l (fun x => ‖f' x‖) g ↔ IsBigOWith c l f' g := by simp only [IsBigOWith_def, norm_norm] #align asymptotics.is_O_with_norm_left Asymptotics.isBigOWith_norm_left @[simp] theorem isBigOWith_abs_left : IsBigOWith c l (fun x => \|u x\|) g ↔ IsBigOWith c l u g := @isBigOWith_norm_left _ _ _ _ _ _ g u l #align asymptotics.is_O_with_abs_left Asymptotics.isBigOWith_abs_left alias ⟨IsBigOWith.of_norm_left, IsBigOWith.norm_left⟩ := isBigOWith_norm_left #align asymptotics.is_O_with.of_norm_left Asymptotics.IsBigOWith.of_norm_left #align asymptotics.is_O_with.norm_left Asymptotics.IsBigOWith.norm_left alias ⟨IsBigOWith.of_abs_left, IsBigOWith.abs_left⟩ := isBigOWith_abs_left #align asymptotics.is_O_with.of_abs_left Asymptotics.IsBigOWith.of_abs_left #align asymptotics.is_O_with.abs_left Asymptotics.IsBigOWith.abs_left @[simp] theorem isBigO_norm_left : (fun x => ‖f' x‖) =O[l] g ↔ f' =O[l] g := by simp only [IsBigO_def] exact exists_congr fun _ => isBigOWith_norm_left #align asymptotics.is_O_norm_left Asymptotics.isBigO_norm_left @[simp] theorem isBigO_abs_left : (fun x => \|u x\|) =O[l] g ↔ u =O[l] g := @isBigO_norm_left _ _ _ _ _ g u l #align asymptotics.is_O_abs_left Asymptotics.isBigO_abs_left alias ⟨IsBigO.of_norm_left, IsBigO.norm_left⟩ := isBigO_norm_left #align asymptotics.is_O.of_norm_left Asymptotics.IsBigO.of_norm_left #align asymptotics.is_O.norm_left Asymptotics.IsBigO.norm_left alias ⟨IsBigO.of_abs_left, IsBigO.abs_left⟩ := isBigO_abs_left #align asymptotics.is_O.of_abs_left Asymptotics.IsBigO.of_abs_left #align asymptotics.is_O.abs_left Asymptotics.IsBigO.abs_left @[simp] theorem isLittleO_norm_left : (fun x => ‖f' x‖) =o[l] g ↔ f' =o[l] g := by simp only [IsLittleO_def] exact forall₂_congr fun _ _ => isBigOWith_norm_left #align asymptotics.is_o_norm_left Asymptotics.isLittleO_norm_left @[simp] theorem isLittleO_abs_left : (fun x => \|u x\|) =o[l] g ↔ u =o[l] g := @isLittleO_norm_left _ _ _ _ _ g u l #align asymptotics.is_o_abs_left Asymptotics.isLittleO_abs_left alias ⟨IsLittleO.of_norm_left, IsLittleO.norm_left⟩ := isLittleO_norm_left #align asymptotics.is_o.of_norm_left Asymptotics.IsLittleO.of_norm_left #align asymptotics.is_o.norm_left Asymptotics.IsLittleO.norm_left alias ⟨IsLittleO.of_abs_left, IsLittleO.abs_left⟩ := isLittleO_abs_left #align asymptotics.is_o.of_abs_left Asymptotics.IsLittleO.of_abs_left #align asymptotics.is_o.abs_left Asymptotics.IsLittleO.abs_left theorem isBigOWith_norm_norm : (IsBigOWith c l (fun x => ‖f' x‖) fun x => ‖g' x‖) ↔ IsBigOWith c l f' g' := isBigOWith_norm_left.trans isBigOWith_norm_right #align asymptotics.is_O_with_norm_norm Asymptotics.isBigOWith_norm_norm theorem isBigOWith_abs_abs : (IsBigOWith c l (fun x => \|u x\|) fun x => \|v x\|) ↔ IsBigOWith c l u v := isBigOWith_abs_left.trans isBigOWith_abs_right #align asymptotics.is_O_with_abs_abs Asymptotics.isBigOWith_abs_abs alias ⟨IsBigOWith.of_norm_norm, IsBigOWith.norm_norm⟩ := isBigOWith_norm_norm #align asymptotics.is_O_with.of_norm_norm Asymptotics.IsBigOWith.of_norm_norm #align asymptotics.is_O_with.norm_norm Asymptotics.IsBigOWith.norm_norm alias ⟨IsBigOWith.of_abs_abs, IsBigOWith.abs_abs⟩ := isBigOWith_abs_abs #align asymptotics.is_O_with.of_abs_abs Asymptotics.IsBigOWith.of_abs_abs #align asymptotics.is_O_with.abs_abs Asymptotics.IsBigOWith.abs_abs theorem isBigO_norm_norm : ((fun x => ‖f' x‖) =O[l] fun x => ‖g' x‖) ↔ f' =O[l] g' := isBigO_norm_left.trans isBigO_norm_right #align asymptotics.is_O_norm_norm Asymptotics.isBigO_norm_norm theorem isBigO_abs_abs : ((fun x => \|u x\|) =O[l] fun x => \|v x\|) ↔ u =O[l] v := isBigO_abs_left.trans isBigO_abs_right #align asymptotics.is_O_abs_abs Asymptotics.isBigO_abs_abs alias ⟨IsBigO.of_norm_norm, IsBigO.norm_norm⟩ := isBigO_norm_norm #align asymptotics.is_O.of_norm_norm Asymptotics.IsBigO.of_norm_norm #align asymptotics.is_O.norm_norm Asymptotics.IsBigO.norm_norm alias ⟨IsBigO.of_abs_abs, IsBigO.abs_abs⟩ := isBigO_abs_abs #align asymptotics.is_O.of_abs_abs Asymptotics.IsBigO.of_abs_abs #align asymptotics.is_O.abs_abs Asymptotics.IsBigO.abs_abs theorem isLittleO_norm_norm : ((fun x => ‖f' x‖) =o[l] fun x => ‖g' x‖) ↔ f' =o[l] g' := isLittleO_norm_left.trans isLittleO_norm_right #align asymptotics.is_o_norm_norm Asymptotics.isLittleO_norm_norm theorem isLittleO_abs_abs : ((fun x => \|u x\|) =o[l] fun x => \|v x\|) ↔ u =o[l] v := isLittleO_abs_left.trans isLittleO_abs_right #align asymptotics.is_o_abs_abs Asymptotics.isLittleO_abs_abs alias ⟨IsLittleO.of_norm_norm, IsLittleO.norm_norm⟩ := isLittleO_norm_norm #align asymptotics.is_o.of_norm_norm Asymptotics.IsLittleO.of_norm_norm #align asymptotics.is_o.norm_norm Asymptotics.IsLittleO.norm_norm alias ⟨IsLittleO.of_abs_abs, IsLittleO.abs_abs⟩ := isLittleO_abs_abs #align asymptotics.is_o.of_abs_abs Asymptotics.IsLittleO.of_abs_abs #align asymptotics.is_o.abs_abs Asymptotics.IsLittleO.abs_abs end NormAbs /-! ### Simplification: negate -/ @[simp] theorem isBigOWith_neg_right : (IsBigOWith c l f fun x => -g' x) ↔ IsBigOWith c l f g' := by simp only [IsBigOWith_def, norm_neg] #align asymptotics.is_O_with_neg_right Asymptotics.isBigOWith_neg_right alias ⟨IsBigOWith.of_neg_right, IsBigOWith.neg_right⟩ := isBigOWith_neg_right #align asymptotics.is_O_with.of_neg_right Asymptotics.IsBigOWith.of_neg_right #align asymptotics.is_O_with.neg_right Asymptotics.IsBigOWith.neg_right @[simp] theorem isBigO_neg_right : (f =O[l] fun x => -g' x) ↔ f =O[l] g' := by simp only [IsBigO_def] exact exists_congr fun _ => isBigOWith_neg_right #align asymptotics.is_O_neg_right Asymptotics.isBigO_neg_right alias ⟨IsBigO.of_neg_right, IsBigO.neg_right⟩ := isBigO_neg_right #align asymptotics.is_O.of_neg_right Asymptotics.IsBigO.of_neg_right #align asymptotics.is_O.neg_right Asymptotics.IsBigO.neg_right @[simp] theorem isLittleO_neg_right : (f =o[l] fun x => -g' x) ↔ f =o[l] g' := by simp only [IsLittleO_def] exact forall₂_congr fun _ _ => isBigOWith_neg_right #align asymptotics.is_o_neg_right Asymptotics.isLittleO_neg_right alias ⟨IsLittleO.of_neg_right, IsLittleO.neg_right⟩ := isLittleO_neg_right #align asymptotics.is_o.of_neg_right Asymptotics.IsLittleO.of_neg_right #align asymptotics.is_o.neg_right Asymptotics.IsLittleO.neg_right @[simp] theorem isBigOWith_neg_left : IsBigOWith c l (fun x => -f' x) g ↔ IsBigOWith c l f' g := by simp only [IsBigOWith_def, norm_neg] #align asymptotics.is_O_with_neg_left Asymptotics.isBigOWith_neg_left alias ⟨IsBigOWith.of_neg_left, IsBigOWith.neg_left⟩ := isBigOWith_neg_left #align asymptotics.is_O_with.of_neg_left Asymptotics.IsBigOWith.of_neg_left #align asymptotics.is_O_with.neg_left Asymptotics.IsBigOWith.neg_left @[simp] theorem isBigO_neg_left : (fun x => -f' x) =O[l] g ↔ f' =O[l] g := by simp only [IsBigO_def] exact exists_congr fun _ => isBigOWith_neg_left #align asymptotics.is_O_neg_left Asymptotics.isBigO_neg_left alias ⟨IsBigO.of_neg_left, IsBigO.neg_left⟩ := isBigO_neg_left #align asymptotics.is_O.of_neg_left Asymptotics.IsBigO.of_neg_left #align asymptotics.is_O.neg_left Asymptotics.IsBigO.neg_left @[simp] theorem isLittleO_neg_left : (fun x => -f' x) =o[l] g ↔ f' =o[l] g := by simp only [IsLittleO_def] exact forall₂_congr fun _ _ => isBigOWith_neg_left #align asymptotics.is_o_neg_left Asymptotics.isLittleO_neg_left alias ⟨IsLittleO.of_neg_left, IsLittleO.neg_left⟩ := isLittleO_neg_left #align asymptotics.is_o.of_neg_left Asymptotics.IsLittleO.of_neg_left #align asymptotics.is_o.neg_left Asymptotics.IsLittleO.neg_left /-! ### Product of functions (right) -/ theorem isBigOWith_fst_prod : IsBigOWith 1 l f' fun x => (f' x, g' x) := isBigOWith_of_le l fun _x => le_max_left _ _ #align asymptotics.is_O_with_fst_prod Asymptotics.isBigOWith_fst_prod theorem isBigOWith_snd_prod : IsBigOWith 1 l g' fun x => (f' x, g' x) := isBigOWith_of_le l fun _x => le_max_right _ _ #align asymptotics.is_O_with_snd_prod Asymptotics.isBigOWith_snd_prod theorem isBigO_fst_prod : f' =O[l] fun x => (f' x, g' x) := isBigOWith_fst_prod.isBigO #align asymptotics.is_O_fst_prod Asymptotics.isBigO_fst_prod theorem isBigO_snd_prod : g' =O[l] fun x => (f' x, g' x) := isBigOWith_snd_prod.isBigO #align asymptotics.is_O_snd_prod Asymptotics.isBigO_snd_prod theorem isBigO_fst_prod' {f' : α → E' × F'} : (fun x => (f' x).1) =O[l] f' := by simpa [IsBigO_def, IsBigOWith_def] using isBigO_fst_prod (E' := E') (F' := F') #align asymptotics.is_O_fst_prod' Asymptotics.isBigO_fst_prod' theorem isBigO_snd_prod' {f' : α → E' × F'} : (fun x => (f' x).2) =O[l] f' := by simpa [IsBigO_def, IsBigOWith_def] using isBigO_snd_prod (E' := E') (F' := F') #align asymptotics.is_O_snd_prod' Asymptotics.isBigO_snd_prod' section variable (f' k') theorem IsBigOWith.prod_rightl (h : IsBigOWith c l f g') (hc : 0 ≤ c) : IsBigOWith c l f fun x => (g' x, k' x) := (h.trans isBigOWith_fst_prod hc).congr_const (mul_one c) #align asymptotics.is_O_with.prod_rightl Asymptotics.IsBigOWith.prod_rightl theorem IsBigO.prod_rightl (h : f =O[l] g') : f =O[l] fun x => (g' x, k' x) := let ⟨_c, cnonneg, hc⟩ := h.exists_nonneg (hc.prod_rightl k' cnonneg).isBigO #align asymptotics.is_O.prod_rightl Asymptotics.IsBigO.prod_rightl theorem IsLittleO.prod_rightl (h : f =o[l] g') : f =o[l] fun x => (g' x, k' x) := IsLittleO.of_isBigOWith fun _c cpos => (h.forall_isBigOWith cpos).prod_rightl k' cpos.le #align asymptotics.is_o.prod_rightl Asymptotics.IsLittleO.prod_rightl theorem IsBigOWith.prod_rightr (h : IsBigOWith c l f g') (hc : 0 ≤ c) : IsBigOWith c l f fun x => (f' x, g' x) := (h.trans isBigOWith_snd_prod hc).congr_const (mul_one c) #align asymptotics.is_O_with.prod_rightr Asymptotics.IsBigOWith.prod_rightr theorem IsBigO.prod_rightr (h : f =O[l] g') : f =O[l] fun x => (f' x, g' x) := let ⟨_c, cnonneg, hc⟩ := h.exists_nonneg (hc.prod_rightr f' cnonneg).isBigO #align asymptotics.is_O.prod_rightr Asymptotics.IsBigO.prod_rightr theorem IsLittleO.prod_rightr (h : f =o[l] g') : f =o[l] fun x => (f' x, g' x) := IsLittleO.of_isBigOWith fun _c cpos => (h.forall_isBigOWith cpos).prod_rightr f' cpos.le #align asymptotics.is_o.prod_rightr Asymptotics.IsLittleO.prod_rightr end theorem IsBigOWith.prod_left_same (hf : IsBigOWith c l f' k') (hg : IsBigOWith c l g' k') : IsBigOWith c l (fun x => (f' x, g' x)) k' := by rw [isBigOWith_iff] at ; filter_upwards [hf, hg] with x using max_le #align asymptotics.is_O_with.prod_left_same Asymptotics.IsBigOWith.prod_left_same theorem IsBigOWith.prod_left (hf : IsBigOWith c l f' k') (hg : IsBigOWith c' l g' k') : IsBigOWith (max c c') l (fun x => (f' x, g' x)) k' := (hf.weaken <\| le_max_left c c').prod_left_same (hg.weaken <\| le_max_right c c') #align asymptotics.is_O_with.prod_left Asymptotics.IsBigOWith.prod_left theorem IsBigOWith.prod_left_fst (h : IsBigOWith c l (fun x => (f' x, g' x)) k') : IsBigOWith c l f' k' := (isBigOWith_fst_prod.trans h zero_le_one).congr_const <\| one_mul c #align asymptotics.is_O_with.prod_left_fst Asymptotics.IsBigOWith.prod_left_fst theorem IsBigOWith.prod_left_snd (h : IsBigOWith c l (fun x => (f' x, g' x)) k') : IsBigOWith c l g' k' := (isBigOWith_snd_prod.trans h zero_le_one).congr_const <\| one_mul c #align asymptotics.is_O_with.prod_left_snd Asymptotics.IsBigOWith.prod_left_snd theorem isBigOWith_prod_left : IsBigOWith c l (fun x => (f' x, g' x)) k' ↔ IsBigOWith c l f' k' ∧ IsBigOWith c l g' k' := ⟨fun h => ⟨h.prod_left_fst, h.prod_left_snd⟩, fun h => h.1.prod_left_same h.2⟩ #align asymptotics.is_O_with_prod_left Asymptotics.isBigOWith_prod_left theorem IsBigO.prod_left (hf : f' =O[l] k') (hg : g' =O[l] k') : (fun x => (f' x, g' x)) =O[l] k' := let ⟨_c, hf⟩ := hf.isBigOWith let ⟨_c', hg⟩ := hg.isBigOWith (hf.prod_left hg).isBigO #align asymptotics.is_O.prod_left Asymptotics.IsBigO.prod_left theorem IsBigO.prod_left_fst : (fun x => (f' x, g' x)) =O[l] k' → f' =O[l] k' := IsBigO.trans isBigO_fst_prod #align asymptotics.is_O.prod_left_fst Asymptotics.IsBigO.prod_left_fst theorem IsBigO.prod_left_snd : (fun x => (f' x, g' x)) =O[l] k' → g' =O[l] k' := IsBigO.trans isBigO_snd_prod #align asymptotics.is_O.prod_left_snd Asymptotics.IsBigO.prod_left_snd @[simp] theorem isBigO_prod_left : (fun x => (f' x, g' x)) =O[l] k' ↔ f' =O[l] k' ∧ g' =O[l] k' := ⟨fun h => ⟨h.prod_left_fst, h.prod_left_snd⟩, fun h => h.1.prod_left h.2⟩ #align asymptotics.is_O_prod_left Asymptotics.isBigO_prod_left theorem IsLittleO.prod_left (hf : f' =o[l] k') (hg : g' =o[l] k') : (fun x => (f' x, g' x)) =o[l] k' := IsLittleO.of_isBigOWith fun _c hc => (hf.forall_isBigOWith hc).prod_left_same (hg.forall_isBigOWith hc) #align asymptotics.is_o.prod_left Asymptotics.IsLittleO.prod_left theorem IsLittleO.prod_left_fst : (fun x => (f' x, g' x)) =o[l] k' → f' =o[l] k' := IsBigO.trans_isLittleO isBigO_fst_prod #align asymptotics.is_o.prod_left_fst Asymptotics.IsLittleO.prod_left_fst theorem IsLittleO.prod_left_snd : (fun x => (f' x, g' x)) =o[l] k' → g' =o[l] k' := IsBigO.trans_isLittleO isBigO_snd_prod #align asymptotics.is_o.prod_left_snd Asymptotics.IsLittleO.prod_left_snd @[simp] theorem isLittleO_prod_left : (fun x => (f' x, g' x)) =o[l] k' ↔ f' =o[l] k' ∧ g' =o[l] k' := ⟨fun h => ⟨h.prod_left_fst, h.prod_left_snd⟩, fun h => h.1.prod_left h.2⟩ #align asymptotics.is_o_prod_left Asymptotics.isLittleO_prod_left theorem IsBigOWith.eq_zero_imp (h : IsBigOWith c l f'' g'') : ∀ᶠ x in l, g'' x = 0 → f'' x = 0 := Eventually.mono h.bound fun x hx hg => norm_le_zero_iff.1 <\| by simpa [hg] using hx #align asymptotics.is_O_with.eq_zero_imp Asymptotics.IsBigOWith.eq_zero_imp theorem IsBigO.eq_zero_imp (h : f'' =O[l] g'') : ∀ᶠ x in l, g'' x = 0 → f'' x = 0 := let ⟨_C, hC⟩ := h.isBigOWith hC.eq_zero_imp #align asymptotics.is_O.eq_zero_imp Asymptotics.IsBigO.eq_zero_imp /-! ### Addition and subtraction -/ section add_sub variable {f₁ f₂ : α → E'} {g₁ g₂ : α → F'} theorem IsBigOWith.add (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : IsBigOWith c₂ l f₂ g) : IsBigOWith (c₁ + c₂) l (fun x => f₁ x + f₂ x) g := by rw [IsBigOWith_def] at filter_upwards [h₁, h₂] with x hx₁ hx₂ using calc ‖f₁ x + f₂ x‖ ≤ c₁ * ‖g x‖ + c₂ * ‖g x‖ := norm_add_le_of_le hx₁ hx₂ _ = (c₁ + c₂) * ‖g x‖ := (add_mul _ _ _).symm #align asymptotics.is_O_with.add Asymptotics.IsBigOWith.add theorem IsBigO.add (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g) : (fun x => f₁ x + f₂ x) =O[l] g := let ⟨_c₁, hc₁⟩ := h₁.isBigOWith let ⟨_c₂, hc₂⟩ := h₂.isBigOWith (hc₁.add hc₂).isBigO #align asymptotics.is_O.add Asymptotics.IsBigO.add theorem IsLittleO.add (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g) : (fun x => f₁ x + f₂ x) =o[l] g := IsLittleO.of_isBigOWith fun c cpos => ((h₁.forall_isBigOWith <\| half_pos cpos).add (h₂.forall_isBigOWith <\| half_pos cpos)).congr_const (add_halves c) #align asymptotics.is_o.add Asymptotics.IsLittleO.add theorem IsLittleO.add_add (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =o[l] g₂) : (fun x => f₁ x + f₂ x) =o[l] fun x => ‖g₁ x‖ + ‖g₂ x‖ := by refine (h₁.trans_le fun x => ?_).add (h₂.trans_le ?_) <;> simp [abs_of_nonneg, add_nonneg] #align asymptotics.is_o.add_add Asymptotics.IsLittleO.add_add theorem IsBigO.add_isLittleO (h₁ : f₁ =O[l] g) (h₂ : f₂ =o[l] g) : (fun x => f₁ x + f₂ x) =O[l] g := h₁.add h₂.isBigO #align asymptotics.is_O.add_is_o Asymptotics.IsBigO.add_isLittleO theorem IsLittleO.add_isBigO (h₁ : f₁ =o[l] g) (h₂ : f₂ =O[l] g) : (fun x => f₁ x + f₂ x) =O[l] g := h₁.isBigO.add h₂ #align asymptotics.is_o.add_is_O Asymptotics.IsLittleO.add_isBigO theorem IsBigOWith.add_isLittleO (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂) : IsBigOWith c₂ l (fun x => f₁ x + f₂ x) g := (h₁.add (h₂.forall_isBigOWith (sub_pos.2 hc))).congr_const (add_sub_cancel _ _) #align asymptotics.is_O_with.add_is_o Asymptotics.IsBigOWith.add_isLittleO theorem IsLittleO.add_isBigOWith (h₁ : f₁ =o[l] g) (h₂ : IsBigOWith c₁ l f₂ g) (hc : c₁ < c₂) : IsBigOWith c₂ l (fun x => f₁ x + f₂ x) g := (h₂.add_isLittleO h₁ hc).congr_left fun _ => add_comm _ _ #align asymptotics.is_o.add_is_O_with Asymptotics.IsLittleO.add_isBigOWith theorem IsBigOWith.sub (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : IsBigOWith c₂ l f₂ g) : IsBigOWith (c₁ + c₂) l (fun x => f₁ x - f₂ x) g := by simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left #align asymptotics.is_O_with.sub Asymptotics.IsBigOWith.sub theorem IsBigOWith.sub_isLittleO (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂) : IsBigOWith c₂ l (fun x => f₁ x - f₂ x) g := by simpa only [sub_eq_add_neg] using h₁.add_isLittleO h₂.neg_left hc #align asymptotics.is_O_with.sub_is_o Asymptotics.IsBigOWith.sub_isLittleO theorem IsBigO.sub (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g) : (fun x => f₁ x - f₂ x) =O[l] g := by simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left #align asymptotics.is_O.sub Asymptotics.IsBigO.sub theorem IsLittleO.sub (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g) : (fun x => f₁ x - f₂ x) =o[l] g := by simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left #align asymptotics.is_o.sub Asymptotics.IsLittleO.sub end add_sub /-! ### Lemmas about `IsBigO (f₁ - f₂) g l` / `IsLittleO (f₁ - f₂) g l` treated as a binary relation -/ section IsBigOOAsRel variable {f₁ f₂ f₃ : α → E'} theorem IsBigOWith.symm (h : IsBigOWith c l (fun x => f₁ x - f₂ x) g) : IsBigOWith c l (fun x => f₂ x - f₁ x) g := h.neg_left.congr_left fun _x => neg_sub _ _ #align asymptotics.is_O_with.symm Asymptotics.IsBigOWith.symm theorem isBigOWith_comm : IsBigOWith c l (fun x => f₁ x - f₂ x) g ↔ IsBigOWith c l (fun x => f₂ x - f₁ x) g := ⟨IsBigOWith.symm, IsBigOWith.symm⟩ #align asymptotics.is_O_with_comm Asymptotics.isBigOWith_comm theorem IsBigO.symm (h : (fun x => f₁ x - f₂ x) =O[l] g) : (fun x => f₂ x - f₁ x) =O[l] g := h.neg_left.congr_left fun _x => neg_sub _ _ #align asymptotics.is_O.symm Asymptotics.IsBigO.symm theorem isBigO_comm : (fun x => f₁ x - f₂ x) =O[l] g ↔ (fun x => f₂ x - f₁ x) =O[l] g := ⟨IsBigO.symm, IsBigO.symm⟩ #align asymptotics.is_O_comm Asymptotics.isBigO_comm theorem IsLittleO.symm (h : (fun x => f₁ x - f₂ x) =o[l] g) : (fun x => f₂ x - f₁ x) =o[l] g := by simpa only [neg_sub] using h.neg_left #align asymptotics.is_o.symm Asymptotics.IsLittleO.symm theorem isLittleO_comm : (fun x => f₁ x - f₂ x) =o[l] g ↔ (fun x => f₂ x - f₁ x) =o[l] g := ⟨IsLittleO.symm, IsLittleO.symm⟩ #align asymptotics.is_o_comm Asymptotics.isLittleO_comm theorem IsBigOWith.triangle (h₁ : IsBigOWith c l (fun x => f₁ x - f₂ x) g) (h₂ : IsBigOWith c' l (fun x => f₂ x - f₃ x) g) : IsBigOWith (c + c') l (fun x => f₁ x - f₃ x) g := (h₁.add h₂).congr_left fun _x => sub_add_sub_cancel _ _ _ #align asymptotics.is_O_with.triangle Asymptotics.IsBigOWith.triangle theorem IsBigO.triangle (h₁ : (fun x => f₁ x - f₂ x) =O[l] g) (h₂ : (fun x => f₂ x - f₃ x) =O[l] g) : (fun x => f₁ x - f₃ x) =O[l] g := (h₁.add h₂).congr_left fun _x => sub_add_sub_cancel _ _ _ #align asymptotics.is_O.triangle Asymptotics.IsBigO.triangle theorem IsLittleO.triangle (h₁ : (fun x => f₁ x - f₂ x) =o[l] g) (h₂ : (fun x => f₂ x - f₃ x) =o[l] g) : (fun x => f₁ x - f₃ x) =o[l] g := (h₁.add h₂).congr_left fun _x => sub_add_sub_cancel _ _ _ #align asymptotics.is_o.triangle Asymptotics.IsLittleO.triangle theorem IsBigO.congr_of_sub (h : (fun x => f₁ x - f₂ x) =O[l] g) : f₁ =O[l] g ↔ f₂ =O[l] g := ⟨fun h' => (h'.sub h).congr_left fun _x => sub_sub_cancel _ _, fun h' => (h.add h').congr_left fun _x => sub_add_cancel _ _⟩ #align asymptotics.is_O.congr_of_sub Asymptotics.IsBigO.congr_of_sub theorem IsLittleO.congr_of_sub (h : (fun x => f₁ x - f₂ x) =o[l] g) : f₁ =o[l] g ↔ f₂ =o[l] g := ⟨fun h' => (h'.sub h).congr_left fun _x => sub_sub_cancel _ _, fun h' => (h.add h').congr_left fun _x => sub_add_cancel _ _⟩ #align asymptotics.is_o.congr_of_sub Asymptotics.IsLittleO.congr_of_sub end IsBigOOAsRel /-! ### Zero, one, and other constants -/ section ZeroConst variable (g g' l) theorem isLittleO_zero : (fun _x => (0 : E')) =o[l] g' := IsLittleO.of_bound fun c hc => univ_mem' fun x => by simpa using mul_nonneg hc.le (norm_nonneg <\| g' x) #align asymptotics.is_o_zero Asymptotics.isLittleO_zero theorem isBigOWith_zero (hc : 0 ≤ c) : IsBigOWith c l (fun _x => (0 : E')) g' := IsBigOWith.of_bound <\| univ_mem' fun x => by simpa using mul_nonneg hc (norm_nonneg <\| g' x) #align asymptotics.is_O_with_zero Asymptotics.isBigOWith_zero theorem isBigOWith_zero' : IsBigOWith 0 l (fun _x => (0 : E')) g := IsBigOWith.of_bound <\| univ_mem' fun x => by simp #align asymptotics.is_O_with_zero' Asymptotics.isBigOWith_zero' theorem isBigO_zero : (fun _x => (0 : E')) =O[l] g := isBigO_iff_isBigOWith.2 ⟨0, isBigOWith_zero' _ _⟩ #align asymptotics.is_O_zero Asymptotics.isBigO_zero theorem isBigO_refl_left : (fun x => f' x - f' x) =O[l] g' := (isBigO_zero g' l).congr_left fun _x => (sub_self _).symm #align asymptotics.is_O_refl_left Asymptotics.isBigO_refl_left theorem isLittleO_refl_left : (fun x => f' x - f' x) =o[l] g' := (isLittleO_zero g' l).congr_left fun _x => (sub_self _).symm #align asymptotics.is_o_refl_left Asymptotics.isLittleO_refl_left variable {g g' l} @[simp] theorem isBigOWith_zero_right_iff : (IsBigOWith c l f'' fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 := by simp only [IsBigOWith_def, exists_prop, true_and_iff, norm_zero, mul_zero, norm_le_zero_iff, EventuallyEq, Pi.zero_apply] #align asymptotics.is_O_with_zero_right_iff Asymptotics.isBigOWith_zero_right_iff @[simp] theorem isBigO_zero_right_iff : (f'' =O[l] fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 := ⟨fun h => let ⟨_c, hc⟩ := h.isBigOWith isBigOWith_zero_right_iff.1 hc, fun h => (isBigOWith_zero_right_iff.2 h : IsBigOWith 1 _ _ _).isBigO⟩ #align asymptotics.is_O_zero_right_iff Asymptotics.isBigO_zero_right_iff @[simp] theorem isLittleO_zero_right_iff : (f'' =o[l] fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 := ⟨fun h => isBigO_zero_right_iff.1 h.isBigO, fun h => IsLittleO.of_isBigOWith fun _c _hc => isBigOWith_zero_right_iff.2 h⟩ #align asymptotics.is_o_zero_right_iff Asymptotics.isLittleO_zero_right_iff theorem isBigOWith_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α) : IsBigOWith (‖c‖ / ‖c'‖) l (fun _x : α => c) fun _x => c' := by simp only [IsBigOWith_def] apply univ_mem' intro x rw [mem_setOf, div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hc')] #align asymptotics.is_O_with_const_const Asymptotics.isBigOWith_const_const theorem isBigO_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α) : (fun _x : α => c) =O[l] fun _x => c' := (isBigOWith_const_const c hc' l).isBigO #align asymptotics.is_O_const_const Asymptotics.isBigO_const_const @[simp] theorem isBigO_const_const_iff {c : E''} {c' : F''} (l : Filter α) [l.NeBot] : ((fun _x : α => c) =O[l] fun _x => c') ↔ c' = 0 → c = 0 := by rcases eq_or_ne c' 0 with (rfl \| hc') · simp [EventuallyEq] · simp [hc', isBigO_const_const _ hc'] #align asymptotics.is_O_const_const_iff Asymptotics.isBigO_const_const_iff @[simp] theorem isBigO_pure {x} : f'' =O[pure x] g'' ↔ g'' x = 0 → f'' x = 0 := calc f'' =O[pure x] g'' ↔ (fun _y : α => f'' x) =O[pure x] fun _ => g'' x := isBigO_congr rfl rfl _ ↔ g'' x = 0 → f'' x = 0 := isBigO_const_const_iff _ #align asymptotics.is_O_pure Asymptotics.isBigO_pure end ZeroConst @[simp] theorem isBigOWith_principal {s : Set α} : IsBigOWith c (𝓟 s) f g ↔ ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖ := by rw [IsBigOWith_def, eventually_principal] #align asymptotics.is_O_with_principal Asymptotics.isBigOWith_principal theorem isBigO_principal {s : Set α} : f =O[𝓟 s] g ↔ ∃ c, ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖ := by simp_rw [isBigO_iff, eventually_principal] #align asymptotics.is_O_principal Asymptotics.isBigO_principal @[simp] theorem isLittleO_principal {s : Set α} : f'' =o[𝓟 s] g' ↔ ∀ x ∈ s, f'' x = 0 := by refine ⟨fun h x hx ↦ norm_le_zero_iff.1 ?_, fun h ↦ ?_⟩ · simp only [isLittleO_iff, isBigOWith_principal] at h have : Tendsto (fun c : ℝ => c * ‖g' x‖) (𝓝[>] 0) (𝓝 0) := ((continuous_id.mul continuous_const).tendsto' _ _ (zero_mul _)).mono_left inf_le_left apply le_of_tendsto_of_tendsto tendsto_const_nhds this apply eventually_nhdsWithin_iff.2 (eventually_of_forall (fun c hc ↦ ?_)) exact eventually_principal.1 (h hc) x hx · apply (isLittleO_zero g' _).congr' ?_ EventuallyEq.rfl exact fun x hx ↦ (h x hx).symm @[simp] theorem isBigOWith_top : IsBigOWith c ⊤ f g ↔ ∀ x, ‖f x‖ ≤ c * ‖g x‖ := by rw [IsBigOWith_def, eventually_top] #align asymptotics.is_O_with_top Asymptotics.isBigOWith_top @[simp] theorem isBigO_top : f =O[⊤] g ↔ ∃ C, ∀ x, ‖f x‖ ≤ C * ‖g x‖ := by simp_rw [isBigO_iff, eventually_top] #align asymptotics.is_O_top Asymptotics.isBigO_top @[simp]	Mathlib/Analysis/Asymptotics/Asymptotics.lean	1,338	1,339	theorem isLittleO_top : f'' =o[⊤] g' ↔ ∀ x, f'' x = 0 := by	simp only [← principal_univ, isLittleO_principal, mem_univ, forall_true_left]
/- Copyright (c) 2021 Eric Wieser. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Wieser -/ import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv #align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101" /-! # Quadratic form on product and pi types ## Main definitions * `QuadraticForm.prod Q₁ Q₂`: the quadratic form constructed elementwise on a product * `QuadraticForm.pi Q`: the quadratic form constructed elementwise on a pi type ## Main results * `QuadraticForm.Equivalent.prod`, `QuadraticForm.Equivalent.pi`: quadratic forms are equivalent if their components are equivalent * `QuadraticForm.nonneg_prod_iff`, `QuadraticForm.nonneg_pi_iff`: quadratic forms are positive- semidefinite if and only if their components are positive-semidefinite. * `QuadraticForm.posDef_prod_iff`, `QuadraticForm.posDef_pi_iff`: quadratic forms are positive- definite if and only if their components are positive-definite. ## Implementation notes Many of the lemmas in this file could be generalized into results about sums of positive and non-negative elements, and would generalize to any map `Q` where `Q 0 = 0`, not just quadratic forms specifically. -/ universe u v w variable {ι : Type} {R : Type} {M₁ M₂ N₁ N₂ : Type} {Mᵢ Nᵢ : ι → Type} namespace QuadraticForm section Prod section Semiring variable [CommSemiring R] variable [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid N₁] [AddCommMonoid N₂] variable [Module R M₁] [Module R M₂] [Module R N₁] [Module R N₂] /-- Construct a quadratic form on a product of two modules from the quadratic form on each module. -/ @[simps!] def prod (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : QuadraticForm R (M₁ × M₂) := Q₁.comp (LinearMap.fst _ _ _) + Q₂.comp (LinearMap.snd _ _ _) #align quadratic_form.prod QuadraticForm.prod /-- An isometry between quadratic forms generated by `QuadraticForm.prod` can be constructed from a pair of isometries between the left and right parts. -/ @[simps toLinearEquiv] def IsometryEquiv.prod {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₁' : QuadraticForm R N₁} {Q₂' : QuadraticForm R N₂} (e₁ : Q₁.IsometryEquiv Q₁') (e₂ : Q₂.IsometryEquiv Q₂') : (Q₁.prod Q₂).IsometryEquiv (Q₁'.prod Q₂') where map_app' x := congr_arg₂ (· + ·) (e₁.map_app x.1) (e₂.map_app x.2) toLinearEquiv := LinearEquiv.prod e₁.toLinearEquiv e₂.toLinearEquiv #align quadratic_form.isometry.prod QuadraticForm.IsometryEquiv.prod /-- `LinearMap.inl` as an isometry. -/ @[simps!] def Isometry.inl (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : Q₁ →qᵢ (Q₁.prod Q₂) where toLinearMap := LinearMap.inl R _ _ map_app' m₁ := by simp /-- `LinearMap.inr` as an isometry. -/ @[simps!] def Isometry.inr (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : Q₂ →qᵢ (Q₁.prod Q₂) where toLinearMap := LinearMap.inr R _ _ map_app' m₁ := by simp variable (M₂) in /-- `LinearMap.fst` as an isometry, when the second space has the zero quadratic form. -/ @[simps!] def Isometry.fst (Q₁ : QuadraticForm R M₁) : (Q₁.prod (0 : QuadraticForm R M₂)) →qᵢ Q₁ where toLinearMap := LinearMap.fst R _ _ map_app' m₁ := by simp variable (M₁) in /-- `LinearMap.snd` as an isometry, when the first space has the zero quadratic form. -/ @[simps!] def Isometry.snd (Q₂ : QuadraticForm R M₂) : ((0 : QuadraticForm R M₁).prod Q₂) →qᵢ Q₂ where toLinearMap := LinearMap.snd R _ _ map_app' m₁ := by simp @[simp] lemma Isometry.fst_comp_inl (Q₁ : QuadraticForm R M₁) : (fst M₂ Q₁).comp (inl Q₁ (0 : QuadraticForm R M₂)) = .id _ := ext fun _ => rfl @[simp] lemma Isometry.snd_comp_inr (Q₂ : QuadraticForm R M₂) : (snd M₁ Q₂).comp (inr (0 : QuadraticForm R M₁) Q₂) = .id _ := ext fun _ => rfl @[simp] lemma Isometry.snd_comp_inl (Q₂ : QuadraticForm R M₂) : (snd M₁ Q₂).comp (inl (0 : QuadraticForm R M₁) Q₂) = 0 := ext fun _ => rfl @[simp] lemma Isometry.fst_comp_inr (Q₁ : QuadraticForm R M₁) : (fst M₂ Q₁).comp (inr Q₁ (0 : QuadraticForm R M₂)) = 0 := ext fun _ => rfl theorem Equivalent.prod {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₁' : QuadraticForm R N₁} {Q₂' : QuadraticForm R N₂} (e₁ : Q₁.Equivalent Q₁') (e₂ : Q₂.Equivalent Q₂') : (Q₁.prod Q₂).Equivalent (Q₁'.prod Q₂') := Nonempty.map2 IsometryEquiv.prod e₁ e₂ #align quadratic_form.equivalent.prod QuadraticForm.Equivalent.prod /-- `LinearEquiv.prodComm` is isometric. -/ @[simps!] def IsometryEquiv.prodComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : (Q₁.prod Q₂).IsometryEquiv (Q₂.prod Q₁) where toLinearEquiv := LinearEquiv.prodComm _ _ _ map_app' _ := add_comm _ _ /-- `LinearEquiv.prodProdProdComm` is isometric. -/ @[simps!] def IsometryEquiv.prodProdProdComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R N₁) (Q₄ : QuadraticForm R N₂) : ((Q₁.prod Q₂).prod (Q₃.prod Q₄)).IsometryEquiv ((Q₁.prod Q₃).prod (Q₂.prod Q₄)) where toLinearEquiv := LinearEquiv.prodProdProdComm _ _ _ _ _ map_app' _ := add_add_add_comm _ _ _ _ /-- If a product is anisotropic then its components must be. The converse is not true. -/ theorem anisotropic_of_prod {R} [OrderedCommRing R] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (h : (Q₁.prod Q₂).Anisotropic) : Q₁.Anisotropic ∧ Q₂.Anisotropic := by simp_rw [Anisotropic, prod_apply, Prod.forall, Prod.mk_eq_zero] at h constructor · intro x hx refine (h x 0 ?_).1 rw [hx, zero_add, map_zero] · intro x hx refine (h 0 x ?_).2 rw [hx, add_zero, map_zero] #align quadratic_form.anisotropic_of_prod QuadraticForm.anisotropic_of_prod theorem nonneg_prod_iff {R} [OrderedCommRing R] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} : (∀ x, 0 ≤ (Q₁.prod Q₂) x) ↔ (∀ x, 0 ≤ Q₁ x) ∧ ∀ x, 0 ≤ Q₂ x := by simp_rw [Prod.forall, prod_apply] constructor · intro h constructor · intro x; simpa only [add_zero, map_zero] using h x 0 · intro x; simpa only [zero_add, map_zero] using h 0 x · rintro ⟨h₁, h₂⟩ x₁ x₂ exact add_nonneg (h₁ x₁) (h₂ x₂) #align quadratic_form.nonneg_prod_iff QuadraticForm.nonneg_prod_iff theorem posDef_prod_iff {R} [OrderedCommRing R] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} : (Q₁.prod Q₂).PosDef ↔ Q₁.PosDef ∧ Q₂.PosDef := by simp_rw [posDef_iff_nonneg, nonneg_prod_iff] constructor · rintro ⟨⟨hle₁, hle₂⟩, ha⟩ obtain ⟨ha₁, ha₂⟩ := anisotropic_of_prod ha exact ⟨⟨hle₁, ha₁⟩, ⟨hle₂, ha₂⟩⟩ · rintro ⟨⟨hle₁, ha₁⟩, ⟨hle₂, ha₂⟩⟩ refine ⟨⟨hle₁, hle₂⟩, ?_⟩ rintro ⟨x₁, x₂⟩ (hx : Q₁ x₁ + Q₂ x₂ = 0) rw [add_eq_zero_iff' (hle₁ x₁) (hle₂ x₂), ha₁.eq_zero_iff, ha₂.eq_zero_iff] at hx rwa [Prod.mk_eq_zero] #align quadratic_form.pos_def_prod_iff QuadraticForm.posDef_prod_iff theorem PosDef.prod {R} [OrderedCommRing R] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (h₁ : Q₁.PosDef) (h₂ : Q₂.PosDef) : (Q₁.prod Q₂).PosDef := posDef_prod_iff.mpr ⟨h₁, h₂⟩ #align quadratic_form.pos_def.prod QuadraticForm.PosDef.prod theorem IsOrtho.prod {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {v w : M₁ × M₂} (h₁ : Q₁.IsOrtho v.1 w.1) (h₂ : Q₂.IsOrtho v.2 w.2) : (Q₁.prod Q₂).IsOrtho v w := (congr_arg₂ HAdd.hAdd h₁ h₂).trans <\| add_add_add_comm _ _ _ _ @[simp] theorem IsOrtho.inl_inr {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (m₁ : M₁) (m₂ : M₂) : (Q₁.prod Q₂).IsOrtho (m₁, 0) (0, m₂) := .prod (.zero_right _) (.zero_left _) @[simp] theorem IsOrtho.inr_inl {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (m₁ : M₁) (m₂ : M₂) : (Q₁.prod Q₂).IsOrtho (0, m₂) (m₁, 0) := (IsOrtho.inl_inr _ _).symm @[simp] theorem isOrtho_inl_inl_iff {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (m₁ m₁' : M₁) : (Q₁.prod Q₂).IsOrtho (m₁, 0) (m₁', 0) ↔ Q₁.IsOrtho m₁ m₁' := by simp [isOrtho_def] @[simp] theorem isOrtho_inr_inr_iff {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (m₂ m₂' : M₂) : (Q₁.prod Q₂).IsOrtho (0, m₂) (0, m₂') ↔ Q₂.IsOrtho m₂ m₂' := by simp [isOrtho_def] end Semiring section Ring variable [CommRing R] variable [AddCommGroup M₁] [AddCommGroup M₂] variable [Module R M₁] [Module R M₂] @[simp] theorem polar_prod (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (x y : M₁ × M₂) : polar (Q₁.prod Q₂) x y = polar Q₁ x.1 y.1 + polar Q₂ x.2 y.2 := by dsimp [polar] abel @[simp] theorem polarBilin_prod (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : (Q₁.prod Q₂).polarBilin = Q₁.polarBilin.compl₁₂ (.fst R M₁ M₂) (.fst R M₁ M₂) + Q₂.polarBilin.compl₁₂ (.snd R M₁ M₂) (.snd R M₁ M₂) := LinearMap.ext₂ <\| polar_prod _ _ @[simp] theorem associated_prod [Invertible (2 : R)] (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) : associated (Q₁.prod Q₂) = (associated Q₁).compl₁₂ (.fst R M₁ M₂) (.fst R M₁ M₂) + (associated Q₂).compl₁₂ (.snd R M₁ M₂) (.snd R M₁ M₂) := by dsimp [associated, associatedHom] rw [polarBilin_prod, smul_add] rfl end Ring end Prod section Pi section Semiring variable [CommSemiring R] variable [∀ i, AddCommMonoid (Mᵢ i)] [∀ i, AddCommMonoid (Nᵢ i)] variable [∀ i, Module R (Mᵢ i)] [∀ i, Module R (Nᵢ i)] /-- Construct a quadratic form on a family of modules from the quadratic form on each module. -/ def pi [Fintype ι] (Q : ∀ i, QuadraticForm R (Mᵢ i)) : QuadraticForm R (∀ i, Mᵢ i) := ∑ i, (Q i).comp (LinearMap.proj i : _ →ₗ[R] Mᵢ i) #align quadratic_form.pi QuadraticForm.pi @[simp] theorem pi_apply [Fintype ι] (Q : ∀ i, QuadraticForm R (Mᵢ i)) (x : ∀ i, Mᵢ i) : pi Q x = ∑ i, Q i (x i) := sum_apply _ _ _ #align quadratic_form.pi_apply QuadraticForm.pi_apply theorem pi_apply_single [Fintype ι] [DecidableEq ι] (Q : ∀ i, QuadraticForm R (Mᵢ i)) (i : ι) (m : Mᵢ i) : pi Q (Pi.single i m) = Q i m := by rw [pi_apply, Fintype.sum_eq_single i fun j hj => ?_, Pi.single_eq_same] rw [Pi.single_eq_of_ne hj, map_zero] /-- An isometry between quadratic forms generated by `QuadraticForm.pi` can be constructed from a pair of isometries between the left and right parts. -/ @[simps toLinearEquiv] def IsometryEquiv.pi [Fintype ι] {Q : ∀ i, QuadraticForm R (Mᵢ i)} {Q' : ∀ i, QuadraticForm R (Nᵢ i)} (e : ∀ i, (Q i).IsometryEquiv (Q' i)) : (pi Q).IsometryEquiv (pi Q') where map_app' x := by simp only [pi_apply, LinearEquiv.piCongrRight, LinearEquiv.toFun_eq_coe, IsometryEquiv.coe_toLinearEquiv, IsometryEquiv.map_app] toLinearEquiv := LinearEquiv.piCongrRight fun i => (e i : Mᵢ i ≃ₗ[R] Nᵢ i) #align quadratic_form.isometry.pi QuadraticForm.IsometryEquiv.pi /-- `LinearMap.single` as an isometry. -/ @[simps!] def Isometry.single [Fintype ι] [DecidableEq ι] (Q : ∀ i, QuadraticForm R (Mᵢ i)) (i : ι) : Q i →qᵢ pi Q where toLinearMap := LinearMap.single i map_app' := pi_apply_single _ _ /-- `LinearMap.proj` as an isometry, when all but one quadratic form is zero. -/ @[simps!] def Isometry.proj [Fintype ι] [DecidableEq ι] (i : ι) (Q : QuadraticForm R (Mᵢ i)) : pi (Pi.single i Q) →qᵢ Q where toLinearMap := LinearMap.proj i map_app' m := by dsimp rw [pi_apply, Fintype.sum_eq_single i (fun j hij => ?_), Pi.single_eq_same] rw [Pi.single_eq_of_ne hij, zero_apply] /-- Note that `QuadraticForm.Isometry.id` would not be well-typed as the RHS. -/ @[simp, nolint simpNF] -- ignore the bogus "Left-hand side does not simplify" lint error theorem Isometry.proj_comp_single_of_same [Fintype ι] [DecidableEq ι] (i : ι) (Q : QuadraticForm R (Mᵢ i)) : (proj i Q).comp (single _ i) = .ofEq (Pi.single_eq_same _ _) := ext fun _ => Pi.single_eq_same _ _ /-- Note that `0 : 0 →qᵢ Q` alone would not be well-typed as the RHS. -/ @[simp] theorem Isometry.proj_comp_single_of_ne [Fintype ι] [DecidableEq ι] {i j : ι} (h : i ≠ j) (Q : QuadraticForm R (Mᵢ i)) : (proj i Q).comp (single _ j) = (0 : 0 →qᵢ Q).comp (ofEq (Pi.single_eq_of_ne h.symm _)) := ext fun _ => Pi.single_eq_of_ne h _ theorem Equivalent.pi [Fintype ι] {Q : ∀ i, QuadraticForm R (Mᵢ i)} {Q' : ∀ i, QuadraticForm R (Nᵢ i)} (e : ∀ i, (Q i).Equivalent (Q' i)) : (pi Q).Equivalent (pi Q') := ⟨IsometryEquiv.pi fun i => Classical.choice (e i)⟩ #align quadratic_form.equivalent.pi QuadraticForm.Equivalent.pi /-- If a family is anisotropic then its components must be. The converse is not true. -/ theorem anisotropic_of_pi [Fintype ι] {R} [OrderedCommRing R] [∀ i, Module R (Mᵢ i)] {Q : ∀ i, QuadraticForm R (Mᵢ i)} (h : (pi Q).Anisotropic) : ∀ i, (Q i).Anisotropic := by simp_rw [Anisotropic, pi_apply, Function.funext_iff, Pi.zero_apply] at h intro i x hx classical have := h (Pi.single i x) ?_ i · rw [Pi.single_eq_same] at this exact this apply Finset.sum_eq_zero intro j _ by_cases hji : j = i · subst hji; rw [Pi.single_eq_same, hx] · rw [Pi.single_eq_of_ne hji, map_zero] #align quadratic_form.anisotropic_of_pi QuadraticForm.anisotropic_of_pi	Mathlib/LinearAlgebra/QuadraticForm/Prod.lean	328	339	theorem nonneg_pi_iff [Fintype ι] {R} [OrderedCommRing R] [∀ i, Module R (Mᵢ i)] {Q : ∀ i, QuadraticForm R (Mᵢ i)} : (∀ x, 0 ≤ pi Q x) ↔ ∀ i x, 0 ≤ Q i x := by	simp_rw [pi, sum_apply, comp_apply, LinearMap.proj_apply] constructor -- TODO: does this generalize to a useful lemma independent of `QuadraticForm`? · intro h i x classical convert h (Pi.single i x) using 1 rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same] rw [Pi.single_eq_of_ne hji, map_zero] · rintro h x exact Finset.sum_nonneg fun i _ => h i (x i)
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Mathlib.Algebra.Ring.Divisibility.Basic import Mathlib.Init.Data.Ordering.Lemmas import Mathlib.SetTheory.Ordinal.Principal import Mathlib.Tactic.NormNum #align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d" /-! # Ordinal notation Constructive ordinal arithmetic for ordinals below `ε₀`. We define a type `ONote`, with constructors `0 : ONote` and `ONote.oadd e n a` representing `ω ^ e * n + a`. We say that `o` is in Cantor normal form - `ONote.NF o` - if either `o = 0` or `o = ω ^ e * n + a` with `a < ω ^ e` and `a` in Cantor normal form. The type `NONote` is the type of ordinals below `ε₀` in Cantor normal form. Various operations (addition, subtraction, multiplication, power function) are defined on `ONote` and `NONote`. -/ set_option linter.uppercaseLean3 false open Ordinal Order -- Porting note: the generated theorem is warned by `simpNF`. set_option genSizeOfSpec false in /-- Recursive definition of an ordinal notation. `zero` denotes the ordinal 0, and `oadd e n a` is intended to refer to `ω^e * n + a`. For this to be valid Cantor normal form, we must have the exponents decrease to the right, but we can't state this condition until we've defined `repr`, so it is a separate definition `NF`. -/ inductive ONote : Type \| zero : ONote \| oadd : ONote → ℕ+ → ONote → ONote deriving DecidableEq #align onote ONote compile_inductive% ONote namespace ONote /-- Notation for 0 -/ instance : Zero ONote := ⟨zero⟩ @[simp] theorem zero_def : zero = 0 := rfl #align onote.zero_def ONote.zero_def instance : Inhabited ONote := ⟨0⟩ /-- Notation for 1 -/ instance : One ONote := ⟨oadd 0 1 0⟩ /-- Notation for ω -/ def omega : ONote := oadd 1 1 0 #align onote.omega ONote.omega /-- The ordinal denoted by a notation -/ @[simp] noncomputable def repr : ONote → Ordinal.{0} \| 0 => 0 \| oadd e n a => ω ^ repr e * n + repr a #align onote.repr ONote.repr /-- Auxiliary definition to print an ordinal notation -/ def toStringAux1 (e : ONote) (n : ℕ) (s : String) : String := if e = 0 then toString n else (if e = 1 then "ω" else "ω^(" ++ s ++ ")") ++ if n = 1 then "" else "" ++ toString n #align onote.to_string_aux1 ONote.toStringAux1 /-- Print an ordinal notation -/ def toString : ONote → String \| zero => "0" \| oadd e n 0 => toStringAux1 e n (toString e) \| oadd e n a => toStringAux1 e n (toString e) ++ " + " ++ toString a #align onote.to_string ONote.toString open Lean in /-- Print an ordinal notation -/ def repr' (prec : ℕ) : ONote → Format \| zero => "0" \| oadd e n a => Repr.addAppParen ("oadd " ++ (repr' max_prec e) ++ " " ++ Nat.repr (n : ℕ) ++ " " ++ (repr' max_prec a)) prec #align onote.repr' ONote.repr instance : ToString ONote := ⟨toString⟩ instance : Repr ONote where reprPrec o prec := repr' prec o instance : Preorder ONote where le x y := repr x ≤ repr y lt x y := repr x < repr y le_refl _ := @le_refl Ordinal _ _ le_trans _ _ _ := @le_trans Ordinal _ _ _ _ lt_iff_le_not_le _ _ := @lt_iff_le_not_le Ordinal _ _ _ theorem lt_def {x y : ONote} : x < y ↔ repr x < repr y := Iff.rfl #align onote.lt_def ONote.lt_def theorem le_def {x y : ONote} : x ≤ y ↔ repr x ≤ repr y := Iff.rfl #align onote.le_def ONote.le_def instance : WellFoundedRelation ONote := ⟨(· < ·), InvImage.wf repr Ordinal.lt_wf⟩ /-- Convert a `Nat` into an ordinal -/ @[coe] def ofNat : ℕ → ONote \| 0 => 0 \| Nat.succ n => oadd 0 n.succPNat 0 #align onote.of_nat ONote.ofNat -- Porting note (#11467): during the port we marked these lemmas with `@[eqns]` -- to emulate the old Lean 3 behaviour. @[simp] theorem ofNat_zero : ofNat 0 = 0 := rfl @[simp] theorem ofNat_succ (n) : ofNat (Nat.succ n) = oadd 0 n.succPNat 0 := rfl instance nat (n : ℕ) : OfNat ONote n where ofNat := ofNat n @[simp 1200] theorem ofNat_one : ofNat 1 = 1 := rfl #align onote.of_nat_one ONote.ofNat_one @[simp] theorem repr_ofNat (n : ℕ) : repr (ofNat n) = n := by cases n <;> simp #align onote.repr_of_nat ONote.repr_ofNat -- @[simp] -- Porting note (#10618): simp can prove this theorem repr_one : repr (ofNat 1) = (1 : ℕ) := repr_ofNat 1 #align onote.repr_one ONote.repr_one theorem omega_le_oadd (e n a) : ω ^ repr e ≤ repr (oadd e n a) := by refine le_trans ?_ (le_add_right _ _) simpa using (Ordinal.mul_le_mul_iff_left <\| opow_pos (repr e) omega_pos).2 (natCast_le.2 n.2) #align onote.omega_le_oadd ONote.omega_le_oadd theorem oadd_pos (e n a) : 0 < oadd e n a := @lt_of_lt_of_le _ _ _ (ω ^ repr e) _ (opow_pos (repr e) omega_pos) (omega_le_oadd e n a) #align onote.oadd_pos ONote.oadd_pos /-- Compare ordinal notations -/ def cmp : ONote → ONote → Ordering \| 0, 0 => Ordering.eq \| _, 0 => Ordering.gt \| 0, _ => Ordering.lt \| _o₁@(oadd e₁ n₁ a₁), _o₂@(oadd e₂ n₂ a₂) => (cmp e₁ e₂).orElse <\| (_root_.cmp (n₁ : ℕ) n₂).orElse (cmp a₁ a₂) #align onote.cmp ONote.cmp theorem eq_of_cmp_eq : ∀ {o₁ o₂}, cmp o₁ o₂ = Ordering.eq → o₁ = o₂ \| 0, 0, _ => rfl \| oadd e n a, 0, h => by injection h \| 0, oadd e n a, h => by injection h \| oadd e₁ n₁ a₁, oadd e₂ n₂ a₂, h => by revert h; simp only [cmp] cases h₁ : cmp e₁ e₂ <;> intro h <;> try cases h obtain rfl := eq_of_cmp_eq h₁ revert h; cases h₂ : _root_.cmp (n₁ : ℕ) n₂ <;> intro h <;> try cases h obtain rfl := eq_of_cmp_eq h rw [_root_.cmp, cmpUsing_eq_eq] at h₂ obtain rfl := Subtype.eq (eq_of_incomp h₂) simp #align onote.eq_of_cmp_eq ONote.eq_of_cmp_eq protected theorem zero_lt_one : (0 : ONote) < 1 := by simp only [lt_def, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero, zero_lt_one] #align onote.zero_lt_one ONote.zero_lt_one /-- `NFBelow o b` says that `o` is a normal form ordinal notation satisfying `repr o < ω ^ b`. -/ inductive NFBelow : ONote → Ordinal.{0} → Prop \| zero {b} : NFBelow 0 b \| oadd' {e n a eb b} : NFBelow e eb → NFBelow a (repr e) → repr e < b → NFBelow (oadd e n a) b #align onote.NF_below ONote.NFBelow /-- A normal form ordinal notation has the form ω ^ a₁ n₁ + ω ^ a₂ * n₂ + ... ω ^ aₖ * nₖ where `a₁ > a₂ > ... > aₖ` and all the `aᵢ` are also in normal form. We will essentially only be interested in normal form ordinal notations, but to avoid complicating the algorithms we define everything over general ordinal notations and only prove correctness with normal form as an invariant. -/ class NF (o : ONote) : Prop where out : Exists (NFBelow o) #align onote.NF ONote.NF instance NF.zero : NF 0 := ⟨⟨0, NFBelow.zero⟩⟩ #align onote.NF.zero ONote.NF.zero theorem NFBelow.oadd {e n a b} : NF e → NFBelow a (repr e) → repr e < b → NFBelow (oadd e n a) b \| ⟨⟨_, h⟩⟩ => NFBelow.oadd' h #align onote.NF_below.oadd ONote.NFBelow.oadd theorem NFBelow.fst {e n a b} (h : NFBelow (ONote.oadd e n a) b) : NF e := by cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact ⟨⟨_, h₁⟩⟩ #align onote.NF_below.fst ONote.NFBelow.fst theorem NF.fst {e n a} : NF (oadd e n a) → NF e \| ⟨⟨_, h⟩⟩ => h.fst #align onote.NF.fst ONote.NF.fst theorem NFBelow.snd {e n a b} (h : NFBelow (ONote.oadd e n a) b) : NFBelow a (repr e) := by cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact h₂ #align onote.NF_below.snd ONote.NFBelow.snd theorem NF.snd' {e n a} : NF (oadd e n a) → NFBelow a (repr e) \| ⟨⟨_, h⟩⟩ => h.snd #align onote.NF.snd' ONote.NF.snd' theorem NF.snd {e n a} (h : NF (oadd e n a)) : NF a := ⟨⟨_, h.snd'⟩⟩ #align onote.NF.snd ONote.NF.snd theorem NF.oadd {e a} (h₁ : NF e) (n) (h₂ : NFBelow a (repr e)) : NF (oadd e n a) := ⟨⟨_, NFBelow.oadd h₁ h₂ (lt_succ _)⟩⟩ #align onote.NF.oadd ONote.NF.oadd instance NF.oadd_zero (e n) [h : NF e] : NF (ONote.oadd e n 0) := h.oadd _ NFBelow.zero #align onote.NF.oadd_zero ONote.NF.oadd_zero theorem NFBelow.lt {e n a b} (h : NFBelow (ONote.oadd e n a) b) : repr e < b := by cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact h₃ #align onote.NF_below.lt ONote.NFBelow.lt theorem NFBelow_zero : ∀ {o}, NFBelow o 0 ↔ o = 0 \| 0 => ⟨fun _ => rfl, fun _ => NFBelow.zero⟩ \| oadd _ _ _ => ⟨fun h => (not_le_of_lt h.lt).elim (Ordinal.zero_le _), fun e => e.symm ▸ NFBelow.zero⟩ #align onote.NF_below_zero ONote.NFBelow_zero theorem NF.zero_of_zero {e n a} (h : NF (ONote.oadd e n a)) (e0 : e = 0) : a = 0 := by simpa [e0, NFBelow_zero] using h.snd' #align onote.NF.zero_of_zero ONote.NF.zero_of_zero theorem NFBelow.repr_lt {o b} (h : NFBelow o b) : repr o < ω ^ b := by induction' h with _ e n a eb b h₁ h₂ h₃ _ IH · exact opow_pos _ omega_pos · rw [repr] apply ((add_lt_add_iff_left _).2 IH).trans_le rw [← mul_succ] apply (mul_le_mul_left' (succ_le_of_lt (nat_lt_omega _)) _).trans rw [← opow_succ] exact opow_le_opow_right omega_pos (succ_le_of_lt h₃) #align onote.NF_below.repr_lt ONote.NFBelow.repr_lt theorem NFBelow.mono {o b₁ b₂} (bb : b₁ ≤ b₂) (h : NFBelow o b₁) : NFBelow o b₂ := by induction' h with _ e n a eb b h₁ h₂ h₃ _ _ <;> constructor exacts [h₁, h₂, lt_of_lt_of_le h₃ bb] #align onote.NF_below.mono ONote.NFBelow.mono theorem NF.below_of_lt {e n a b} (H : repr e < b) : NF (ONote.oadd e n a) → NFBelow (ONote.oadd e n a) b \| ⟨⟨b', h⟩⟩ => by (cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact NFBelow.oadd' h₁ h₂ H) #align onote.NF.below_of_lt ONote.NF.below_of_lt theorem NF.below_of_lt' : ∀ {o b}, repr o < ω ^ b → NF o → NFBelow o b \| 0, _, _, _ => NFBelow.zero \| ONote.oadd _ _ _, _, H, h => h.below_of_lt <\| (opow_lt_opow_iff_right one_lt_omega).1 <\| lt_of_le_of_lt (omega_le_oadd _ _ _) H #align onote.NF.below_of_lt' ONote.NF.below_of_lt' theorem nfBelow_ofNat : ∀ n, NFBelow (ofNat n) 1 \| 0 => NFBelow.zero \| Nat.succ _ => NFBelow.oadd NF.zero NFBelow.zero zero_lt_one #align onote.NF_below_of_nat ONote.nfBelow_ofNat instance nf_ofNat (n) : NF (ofNat n) := ⟨⟨_, nfBelow_ofNat n⟩⟩ #align onote.NF_of_nat ONote.nf_ofNat instance nf_one : NF 1 := by rw [← ofNat_one]; infer_instance #align onote.NF_one ONote.nf_one theorem oadd_lt_oadd_1 {e₁ n₁ o₁ e₂ n₂ o₂} (h₁ : NF (oadd e₁ n₁ o₁)) (h : e₁ < e₂) : oadd e₁ n₁ o₁ < oadd e₂ n₂ o₂ := @lt_of_lt_of_le _ _ (repr (oadd e₁ n₁ o₁)) _ _ (NF.below_of_lt h h₁).repr_lt (omega_le_oadd e₂ n₂ o₂) #align onote.oadd_lt_oadd_1 ONote.oadd_lt_oadd_1 theorem oadd_lt_oadd_2 {e o₁ o₂ : ONote} {n₁ n₂ : ℕ+} (h₁ : NF (oadd e n₁ o₁)) (h : (n₁ : ℕ) < n₂) : oadd e n₁ o₁ < oadd e n₂ o₂ := by simp only [lt_def, repr] refine lt_of_lt_of_le ((add_lt_add_iff_left _).2 h₁.snd'.repr_lt) (le_trans ?_ (le_add_right _ _)) rwa [← mul_succ,Ordinal.mul_le_mul_iff_left (opow_pos _ omega_pos), succ_le_iff, natCast_lt] #align onote.oadd_lt_oadd_2 ONote.oadd_lt_oadd_2 theorem oadd_lt_oadd_3 {e n a₁ a₂} (h : a₁ < a₂) : oadd e n a₁ < oadd e n a₂ := by rw [lt_def]; unfold repr exact @add_lt_add_left _ _ _ _ (repr a₁) _ h _ #align onote.oadd_lt_oadd_3 ONote.oadd_lt_oadd_3 theorem cmp_compares : ∀ (a b : ONote) [NF a] [NF b], (cmp a b).Compares a b \| 0, 0, _, _ => rfl \| oadd e n a, 0, _, _ => oadd_pos _ _ _ \| 0, oadd e n a, _, _ => oadd_pos _ _ _ \| o₁@(oadd e₁ n₁ a₁), o₂@(oadd e₂ n₂ a₂), h₁, h₂ => by -- TODO: golf rw [cmp] have IHe := @cmp_compares _ _ h₁.fst h₂.fst simp only [Ordering.Compares, gt_iff_lt] at IHe; revert IHe cases cmp e₁ e₂ case lt => intro IHe; exact oadd_lt_oadd_1 h₁ IHe case gt => intro IHe; exact oadd_lt_oadd_1 h₂ IHe case eq => intro IHe; dsimp at IHe; subst IHe unfold _root_.cmp; cases nh : cmpUsing (· < ·) (n₁ : ℕ) n₂ <;> rw [cmpUsing, ite_eq_iff, not_lt] at nh case lt => cases' nh with nh nh · exact oadd_lt_oadd_2 h₁ nh.left · rw [ite_eq_iff] at nh; cases' nh.right with nh nh <;> cases nh <;> contradiction case gt => cases' nh with nh nh · cases nh; contradiction · cases' nh with _ nh rw [ite_eq_iff] at nh; cases' nh with nh nh · exact oadd_lt_oadd_2 h₂ nh.left · cases nh; contradiction cases' nh with nh nh · cases nh; contradiction cases' nh with nhl nhr rw [ite_eq_iff] at nhr cases' nhr with nhr nhr · cases nhr; contradiction obtain rfl := Subtype.eq (eq_of_incomp ⟨(not_lt_of_ge nhl), nhr.left⟩) have IHa := @cmp_compares _ _ h₁.snd h₂.snd revert IHa; cases cmp a₁ a₂ <;> intro IHa <;> dsimp at IHa case lt => exact oadd_lt_oadd_3 IHa case gt => exact oadd_lt_oadd_3 IHa subst IHa; exact rfl #align onote.cmp_compares ONote.cmp_compares theorem repr_inj {a b} [NF a] [NF b] : repr a = repr b ↔ a = b := ⟨fun e => match cmp a b, cmp_compares a b with \| Ordering.lt, (h : repr a < repr b) => (ne_of_lt h e).elim \| Ordering.gt, (h : repr a > repr b)=> (ne_of_gt h e).elim \| Ordering.eq, h => h, congr_arg _⟩ #align onote.repr_inj ONote.repr_inj theorem NF.of_dvd_omega_opow {b e n a} (h : NF (ONote.oadd e n a)) (d : ω ^ b ∣ repr (ONote.oadd e n a)) : b ≤ repr e ∧ ω ^ b ∣ repr a := by have := mt repr_inj.1 (fun h => by injection h : ONote.oadd e n a ≠ 0) have L := le_of_not_lt fun l => not_le_of_lt (h.below_of_lt l).repr_lt (le_of_dvd this d) simp only [repr] at d exact ⟨L, (dvd_add_iff <\| (opow_dvd_opow _ L).mul_right _).1 d⟩ #align onote.NF.of_dvd_omega_opow ONote.NF.of_dvd_omega_opow theorem NF.of_dvd_omega {e n a} (h : NF (ONote.oadd e n a)) : ω ∣ repr (ONote.oadd e n a) → repr e ≠ 0 ∧ ω ∣ repr a := by (rw [← opow_one ω, ← one_le_iff_ne_zero]; exact h.of_dvd_omega_opow) #align onote.NF.of_dvd_omega ONote.NF.of_dvd_omega /-- `TopBelow b o` asserts that the largest exponent in `o`, if it exists, is less than `b`. This is an auxiliary definition for decidability of `NF`. -/ def TopBelow (b : ONote) : ONote → Prop \| 0 => True \| oadd e _ _ => cmp e b = Ordering.lt #align onote.top_below ONote.TopBelow instance decidableTopBelow : DecidableRel TopBelow := by intro b o cases o <;> delta TopBelow <;> infer_instance #align onote.decidable_top_below ONote.decidableTopBelow theorem nfBelow_iff_topBelow {b} [NF b] : ∀ {o}, NFBelow o (repr b) ↔ NF o ∧ TopBelow b o \| 0 => ⟨fun h => ⟨⟨⟨_, h⟩⟩, trivial⟩, fun _ => NFBelow.zero⟩ \| oadd _ _ _ => ⟨fun h => ⟨⟨⟨_, h⟩⟩, (@cmp_compares _ b h.fst _).eq_lt.2 h.lt⟩, fun ⟨h₁, h₂⟩ => h₁.below_of_lt <\| (@cmp_compares _ b h₁.fst _).eq_lt.1 h₂⟩ #align onote.NF_below_iff_top_below ONote.nfBelow_iff_topBelow instance decidableNF : DecidablePred NF \| 0 => isTrue NF.zero \| oadd e n a => by have := decidableNF e have := decidableNF a apply decidable_of_iff (NF e ∧ NF a ∧ TopBelow e a) rw [← and_congr_right fun h => @nfBelow_iff_topBelow _ h _] exact ⟨fun ⟨h₁, h₂⟩ => NF.oadd h₁ n h₂, fun h => ⟨h.fst, h.snd'⟩⟩ #align onote.decidable_NF ONote.decidableNF /-- Auxiliary definition for `add` -/ def addAux (e : ONote) (n : ℕ+) (o : ONote) : ONote := match o with \| 0 => oadd e n 0 \| o'@(oadd e' n' a') => match cmp e e' with \| Ordering.lt => o' \| Ordering.eq => oadd e (n + n') a' \| Ordering.gt => oadd e n o' /-- Addition of ordinal notations (correct only for normal input) -/ def add : ONote → ONote → ONote \| 0, o => o \| oadd e n a, o => addAux e n (add a o) #align onote.add ONote.add instance : Add ONote := ⟨add⟩ @[simp] theorem zero_add (o : ONote) : 0 + o = o := rfl #align onote.zero_add ONote.zero_add theorem oadd_add (e n a o) : oadd e n a + o = addAux e n (a + o) := rfl #align onote.oadd_add ONote.oadd_add /-- Subtraction of ordinal notations (correct only for normal input) -/ def sub : ONote → ONote → ONote \| 0, _ => 0 \| o, 0 => o \| o₁@(oadd e₁ n₁ a₁), oadd e₂ n₂ a₂ => match cmp e₁ e₂ with \| Ordering.lt => 0 \| Ordering.gt => o₁ \| Ordering.eq => match (n₁ : ℕ) - n₂ with \| 0 => if n₁ = n₂ then sub a₁ a₂ else 0 \| Nat.succ k => oadd e₁ k.succPNat a₁ #align onote.sub ONote.sub instance : Sub ONote := ⟨sub⟩ theorem add_nfBelow {b} : ∀ {o₁ o₂}, NFBelow o₁ b → NFBelow o₂ b → NFBelow (o₁ + o₂) b \| 0, _, _, h₂ => h₂ \| oadd e n a, o, h₁, h₂ => by have h' := add_nfBelow (h₁.snd.mono <\| le_of_lt h₁.lt) h₂ simp [oadd_add]; revert h'; cases' a + o with e' n' a' <;> intro h' · exact NFBelow.oadd h₁.fst NFBelow.zero h₁.lt have : ((e.cmp e').Compares e e') := @cmp_compares _ _ h₁.fst h'.fst cases h: cmp e e' <;> dsimp [addAux] <;> simp [h] · exact h' · simp [h] at this subst e' exact NFBelow.oadd h'.fst h'.snd h'.lt · simp [h] at this exact NFBelow.oadd h₁.fst (NF.below_of_lt this ⟨⟨_, h'⟩⟩) h₁.lt #align onote.add_NF_below ONote.add_nfBelow instance add_nf (o₁ o₂) : ∀ [NF o₁] [NF o₂], NF (o₁ + o₂) \| ⟨⟨b₁, h₁⟩⟩, ⟨⟨b₂, h₂⟩⟩ => ⟨(le_total b₁ b₂).elim (fun h => ⟨b₂, add_nfBelow (h₁.mono h) h₂⟩) fun h => ⟨b₁, add_nfBelow h₁ (h₂.mono h)⟩⟩ #align onote.add_NF ONote.add_nf @[simp] theorem repr_add : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ + o₂) = repr o₁ + repr o₂ \| 0, o, _, _ => by simp \| oadd e n a, o, h₁, h₂ => by haveI := h₁.snd; have h' := repr_add a o conv_lhs at h' => simp [HAdd.hAdd, Add.add] have nf := ONote.add_nf a o conv at nf => simp [HAdd.hAdd, Add.add] conv in _ + o => simp [HAdd.hAdd, Add.add] cases' h : add a o with e' n' a' <;> simp only [Add.add, add, addAux, h'.symm, h, add_assoc, repr] at nf h₁ ⊢ have := h₁.fst; haveI := nf.fst; have ee := cmp_compares e e' cases he: cmp e e' <;> simp only [he, Ordering.compares_gt, Ordering.compares_lt, Ordering.compares_eq, repr, gt_iff_lt, PNat.add_coe, Nat.cast_add] at ee ⊢ · rw [← add_assoc, @add_absorp _ (repr e') (ω ^ repr e' * (n' : ℕ))] · have := (h₁.below_of_lt ee).repr_lt unfold repr at this cases he': e' <;> simp only [he', zero_def, opow_zero, repr, gt_iff_lt] at this ⊢ <;> exact lt_of_le_of_lt (le_add_right _ _) this · simpa using (Ordinal.mul_le_mul_iff_left <\| opow_pos (repr e') omega_pos).2 (natCast_le.2 n'.pos) · rw [ee, ← add_assoc, ← mul_add] #align onote.repr_add ONote.repr_add theorem sub_nfBelow : ∀ {o₁ o₂ b}, NFBelow o₁ b → NF o₂ → NFBelow (o₁ - o₂) b \| 0, o, b, _, h₂ => by cases o <;> exact NFBelow.zero \| oadd _ _ _, 0, _, h₁, _ => h₁ \| oadd e₁ n₁ a₁, oadd e₂ n₂ a₂, b, h₁, h₂ => by have h' := sub_nfBelow h₁.snd h₂.snd simp only [HSub.hSub, Sub.sub, sub] at h' ⊢ have := @cmp_compares _ _ h₁.fst h₂.fst cases h : cmp e₁ e₂ <;> simp [sub] · apply NFBelow.zero · simp only [h, Ordering.compares_eq] at this subst e₂ cases (n₁ : ℕ) - n₂ <;> simp [sub] · by_cases en : n₁ = n₂ <;> simp [en] · exact h'.mono (le_of_lt h₁.lt) · exact NFBelow.zero · exact NFBelow.oadd h₁.fst h₁.snd h₁.lt · exact h₁ #align onote.sub_NF_below ONote.sub_nfBelow instance sub_nf (o₁ o₂) : ∀ [NF o₁] [NF o₂], NF (o₁ - o₂) \| ⟨⟨b₁, h₁⟩⟩, h₂ => ⟨⟨b₁, sub_nfBelow h₁ h₂⟩⟩ #align onote.sub_NF ONote.sub_nf @[simp] theorem repr_sub : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ - o₂) = repr o₁ - repr o₂ \| 0, o, _, h₂ => by cases o <;> exact (Ordinal.zero_sub _).symm \| oadd e n a, 0, _, _ => (Ordinal.sub_zero _).symm \| oadd e₁ n₁ a₁, oadd e₂ n₂ a₂, h₁, h₂ => by haveI := h₁.snd; haveI := h₂.snd; have h' := repr_sub a₁ a₂ conv_lhs at h' => dsimp [HSub.hSub, Sub.sub, sub] conv_lhs => dsimp only [HSub.hSub, Sub.sub]; dsimp only [sub] have ee := @cmp_compares _ _ h₁.fst h₂.fst cases h : cmp e₁ e₂ <;> simp only [h] at ee · rw [Ordinal.sub_eq_zero_iff_le.2] · rfl exact le_of_lt (oadd_lt_oadd_1 h₁ ee) · change e₁ = e₂ at ee subst e₂ dsimp only cases mn : (n₁ : ℕ) - n₂ <;> dsimp only · by_cases en : n₁ = n₂ · simpa [en] · simp only [en, ite_false] exact (Ordinal.sub_eq_zero_iff_le.2 <\| le_of_lt <\| oadd_lt_oadd_2 h₁ <\| lt_of_le_of_ne (tsub_eq_zero_iff_le.1 mn) (mt PNat.eq en)).symm · simp [Nat.succPNat] rw [(tsub_eq_iff_eq_add_of_le <\| le_of_lt <\| Nat.lt_of_sub_eq_succ mn).1 mn, add_comm, Nat.cast_add, mul_add, add_assoc, add_sub_add_cancel] refine (Ordinal.sub_eq_of_add_eq <\| add_absorp h₂.snd'.repr_lt <\| le_trans ?_ (le_add_right _ _)).symm simpa using mul_le_mul_left' (natCast_le.2 <\| Nat.succ_pos _) _ · exact (Ordinal.sub_eq_of_add_eq <\| add_absorp (h₂.below_of_lt ee).repr_lt <\| omega_le_oadd _ _ _).symm #align onote.repr_sub ONote.repr_sub /-- Multiplication of ordinal notations (correct only for normal input) -/ def mul : ONote → ONote → ONote \| 0, _ => 0 \| _, 0 => 0 \| o₁@(oadd e₁ n₁ a₁), oadd e₂ n₂ a₂ => if e₂ = 0 then oadd e₁ (n₁ * n₂) a₁ else oadd (e₁ + e₂) n₂ (mul o₁ a₂) #align onote.mul ONote.mul instance : Mul ONote := ⟨mul⟩ instance : MulZeroClass ONote where mul := (· * ·) zero := 0 zero_mul o := by cases o <;> rfl mul_zero o := by cases o <;> rfl theorem oadd_mul (e₁ n₁ a₁ e₂ n₂ a₂) : oadd e₁ n₁ a₁ * oadd e₂ n₂ a₂ = if e₂ = 0 then oadd e₁ (n₁ * n₂) a₁ else oadd (e₁ + e₂) n₂ (oadd e₁ n₁ a₁ * a₂) := rfl #align onote.oadd_mul ONote.oadd_mul theorem oadd_mul_nfBelow {e₁ n₁ a₁ b₁} (h₁ : NFBelow (oadd e₁ n₁ a₁) b₁) : ∀ {o₂ b₂}, NFBelow o₂ b₂ → NFBelow (oadd e₁ n₁ a₁ * o₂) (repr e₁ + b₂) \| 0, b₂, _ => NFBelow.zero \| oadd e₂ n₂ a₂, b₂, h₂ => by have IH := oadd_mul_nfBelow h₁ h₂.snd by_cases e0 : e₂ = 0 <;> simp [e0, oadd_mul] · apply NFBelow.oadd h₁.fst h₁.snd simpa using (add_lt_add_iff_left (repr e₁)).2 (lt_of_le_of_lt (Ordinal.zero_le _) h₂.lt) · haveI := h₁.fst haveI := h₂.fst apply NFBelow.oadd · infer_instance · rwa [repr_add] · rw [repr_add, add_lt_add_iff_left] exact h₂.lt #align onote.oadd_mul_NF_below ONote.oadd_mul_nfBelow instance mul_nf : ∀ (o₁ o₂) [NF o₁] [NF o₂], NF (o₁ * o₂) \| 0, o, _, h₂ => by cases o <;> exact NF.zero \| oadd e n a, o, ⟨⟨b₁, hb₁⟩⟩, ⟨⟨b₂, hb₂⟩⟩ => ⟨⟨_, oadd_mul_nfBelow hb₁ hb₂⟩⟩ #align onote.mul_NF ONote.mul_nf @[simp] theorem repr_mul : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ * o₂) = repr o₁ * repr o₂ \| 0, o, _, h₂ => by cases o <;> exact (zero_mul _).symm \| oadd e₁ n₁ a₁, 0, _, _ => (mul_zero _).symm \| oadd e₁ n₁ a₁, oadd e₂ n₂ a₂, h₁, h₂ => by have IH : repr (mul _ _) = _ := @repr_mul _ _ h₁ h₂.snd conv => lhs simp [(· * ·)] have ao : repr a₁ + ω ^ repr e₁ * (n₁ : ℕ) = ω ^ repr e₁ * (n₁ : ℕ) := by apply add_absorp h₁.snd'.repr_lt simpa using (Ordinal.mul_le_mul_iff_left <\| opow_pos _ omega_pos).2 (natCast_le.2 n₁.2) by_cases e0 : e₂ = 0 <;> simp [e0, mul] · cases' Nat.exists_eq_succ_of_ne_zero n₂.ne_zero with x xe simp only [xe, h₂.zero_of_zero e0, repr, add_zero] rw [natCast_succ x, add_mul_succ _ ao, mul_assoc] · haveI := h₁.fst haveI := h₂.fst simp only [Mul.mul, mul, e0, ite_false, repr.eq_2, repr_add, opow_add, IH, repr, mul_add] rw [← mul_assoc] congr 2 have := mt repr_inj.1 e0 rw [add_mul_limit ao (opow_isLimit_left omega_isLimit this), mul_assoc, mul_omega_dvd (natCast_pos.2 n₁.pos) (nat_lt_omega _)] simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 this) #align onote.repr_mul ONote.repr_mul /-- Calculate division and remainder of `o` mod ω. `split' o = (a, n)` means `o = ω * a + n`. -/ def split' : ONote → ONote × ℕ \| 0 => (0, 0) \| oadd e n a => if e = 0 then (0, n) else let (a', m) := split' a (oadd (e - 1) n a', m) #align onote.split' ONote.split' /-- Calculate division and remainder of `o` mod ω. `split o = (a, n)` means `o = a + n`, where `ω ∣ a`. -/ def split : ONote → ONote × ℕ \| 0 => (0, 0) \| oadd e n a => if e = 0 then (0, n) else let (a', m) := split a (oadd e n a', m) #align onote.split ONote.split /-- `scale x o` is the ordinal notation for `ω ^ x * o`. -/ def scale (x : ONote) : ONote → ONote \| 0 => 0 \| oadd e n a => oadd (x + e) n (scale x a) #align onote.scale ONote.scale /-- `mulNat o n` is the ordinal notation for `o * n`. -/ def mulNat : ONote → ℕ → ONote \| 0, _ => 0 \| _, 0 => 0 \| oadd e n a, m + 1 => oadd e (n * m.succPNat) a #align onote.mul_nat ONote.mulNat /-- Auxiliary definition to compute the ordinal notation for the ordinal exponentiation in `opow` -/ def opowAux (e a0 a : ONote) : ℕ → ℕ → ONote \| _, 0 => 0 \| 0, m + 1 => oadd e m.succPNat 0 \| k + 1, m => scale (e + mulNat a0 k) a + (opowAux e a0 a k m) #align onote.opow_aux ONote.opowAux /-- Auxiliary definition to compute the ordinal notation for the ordinal exponentiation in `opow` -/ def opowAux2 (o₂ : ONote) (o₁ : ONote × ℕ) : ONote := match o₁ with \| (0, 0) => if o₂ = 0 then 1 else 0 \| (0, 1) => 1 \| (0, m + 1) => let (b', k) := split' o₂ oadd b' (m.succPNat ^ k) 0 \| (a@(oadd a0 _ _), m) => match split o₂ with \| (b, 0) => oadd (a0 * b) 1 0 \| (b, k + 1) => let eb := a0 * b scale (eb + mulNat a0 k) a + opowAux eb a0 (mulNat a m) k m /-- `opow o₁ o₂` calculates the ordinal notation for the ordinal exponential `o₁ ^ o₂`. -/ def opow (o₁ o₂ : ONote) : ONote := opowAux2 o₂ (split o₁) #align onote.opow ONote.opow instance : Pow ONote ONote := ⟨opow⟩ theorem opow_def (o₁ o₂ : ONote) : o₁ ^ o₂ = opowAux2 o₂ (split o₁) := rfl #align onote.opow_def ONote.opow_def theorem split_eq_scale_split' : ∀ {o o' m} [NF o], split' o = (o', m) → split o = (scale 1 o', m) \| 0, o', m, _, p => by injection p; substs o' m; rfl \| oadd e n a, o', m, h, p => by by_cases e0 : e = 0 <;> simp [e0, split, split'] at p ⊢ · rcases p with ⟨rfl, rfl⟩ exact ⟨rfl, rfl⟩ · revert p cases' h' : split' a with a' m' haveI := h.fst haveI := h.snd simp only [split_eq_scale_split' h', and_imp] have : 1 + (e - 1) = e := by refine repr_inj.1 ?_ simp only [repr_add, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero, repr_sub] have := mt repr_inj.1 e0 refine Ordinal.add_sub_cancel_of_le ?_ have := one_le_iff_ne_zero.2 this exact this intros substs o' m simp [scale, this] #align onote.split_eq_scale_split' ONote.split_eq_scale_split' theorem nf_repr_split' : ∀ {o o' m} [NF o], split' o = (o', m) → NF o' ∧ repr o = ω * repr o' + m \| 0, o', m, _, p => by injection p; substs o' m; simp [NF.zero] \| oadd e n a, o', m, h, p => by by_cases e0 : e = 0 <;> simp [e0, split, split'] at p ⊢ · rcases p with ⟨rfl, rfl⟩ simp [h.zero_of_zero e0, NF.zero] · revert p cases' h' : split' a with a' m' haveI := h.fst haveI := h.snd cases' nf_repr_split' h' with IH₁ IH₂ simp only [IH₂, and_imp] intros substs o' m have : (ω : Ordinal.{0}) ^ repr e = ω ^ (1 : Ordinal.{0}) * ω ^ (repr e - 1) := by have := mt repr_inj.1 e0 rw [← opow_add, Ordinal.add_sub_cancel_of_le (one_le_iff_ne_zero.2 this)] refine ⟨NF.oadd (by infer_instance) _ ?_, ?_⟩ · simp at this ⊢ refine IH₁.below_of_lt' ((Ordinal.mul_lt_mul_iff_left omega_pos).1 <\| lt_of_le_of_lt (le_add_right _ m') ?_) rw [← this, ← IH₂] exact h.snd'.repr_lt · rw [this] simp [mul_add, mul_assoc, add_assoc] #align onote.NF_repr_split' ONote.nf_repr_split' theorem scale_eq_mul (x) [NF x] : ∀ (o) [NF o], scale x o = oadd x 1 0 * o \| 0, _ => rfl \| oadd e n a, h => by simp only [HMul.hMul]; simp only [scale] haveI := h.snd by_cases e0 : e = 0 · simp_rw [scale_eq_mul] simp [Mul.mul, mul, scale_eq_mul, e0, h.zero_of_zero, show x + 0 = x from repr_inj.1 (by simp)] · simp [e0, Mul.mul, mul, scale_eq_mul, (· * ·)] #align onote.scale_eq_mul ONote.scale_eq_mul instance nf_scale (x) [NF x] (o) [NF o] : NF (scale x o) := by rw [scale_eq_mul] infer_instance #align onote.NF_scale ONote.nf_scale @[simp] theorem repr_scale (x) [NF x] (o) [NF o] : repr (scale x o) = ω ^ repr x * repr o := by simp only [scale_eq_mul, repr_mul, repr, PNat.one_coe, Nat.cast_one, mul_one, add_zero] #align onote.repr_scale ONote.repr_scale theorem nf_repr_split {o o' m} [NF o] (h : split o = (o', m)) : NF o' ∧ repr o = repr o' + m := by cases' e : split' o with a n cases' nf_repr_split' e with s₁ s₂ rw [split_eq_scale_split' e] at h injection h; substs o' n simp only [repr_scale, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero, opow_one, s₂.symm, and_true] infer_instance #align onote.NF_repr_split ONote.nf_repr_split theorem split_dvd {o o' m} [NF o] (h : split o = (o', m)) : ω ∣ repr o' := by cases' e : split' o with a n rw [split_eq_scale_split' e] at h injection h; subst o' cases nf_repr_split' e; simp #align onote.split_dvd ONote.split_dvd theorem split_add_lt {o e n a m} [NF o] (h : split o = (oadd e n a, m)) : repr a + m < ω ^ repr e := by cases' nf_repr_split h with h₁ h₂ cases' h₁.of_dvd_omega (split_dvd h) with e0 d apply principal_add_omega_opow _ h₁.snd'.repr_lt (lt_of_lt_of_le (nat_lt_omega _) _) simpa using opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0) #align onote.split_add_lt ONote.split_add_lt @[simp] theorem mulNat_eq_mul (n o) : mulNat o n = o * ofNat n := by cases o <;> cases n <;> rfl #align onote.mul_nat_eq_mul ONote.mulNat_eq_mul instance nf_mulNat (o) [NF o] (n) : NF (mulNat o n) := by simp; exact ONote.mul_nf o (ofNat n) #align onote.NF_mul_nat ONote.nf_mulNat instance nf_opowAux (e a0 a) [NF e] [NF a0] [NF a] : ∀ k m, NF (opowAux e a0 a k m) := by intro k m unfold opowAux cases' m with m m · cases k <;> exact NF.zero cases' k with k k · exact NF.oadd_zero _ _ · haveI := nf_opowAux e a0 a k simp only [Nat.succ_ne_zero m, IsEmpty.forall_iff, mulNat_eq_mul]; infer_instance #align onote.NF_opow_aux ONote.nf_opowAux instance nf_opow (o₁ o₂) [NF o₁] [NF o₂] : NF (o₁ ^ o₂) := by cases' e₁ : split o₁ with a m have na := (nf_repr_split e₁).1 cases' e₂ : split' o₂ with b' k haveI := (nf_repr_split' e₂).1 cases' a with a0 n a' · cases' m with m · by_cases o₂ = 0 <;> simp only [(· ^ ·), Pow.pow, pow, opow, opowAux2, ] <;> decide · by_cases m = 0 · simp only [(· ^ ·), Pow.pow, pow, opow, opowAux2, , zero_def] decide · simp only [(· ^ ·), Pow.pow, pow, opow, opowAux2, mulNat_eq_mul, ofNat, ] infer_instance · simp [(· ^ ·),Pow.pow,pow, opow, opowAux2, e₁, e₂, split_eq_scale_split' e₂] have := na.fst cases' k with k <;> simp · infer_instance · cases k <;> cases m <;> infer_instance #align onote.NF_opow ONote.nf_opow theorem scale_opowAux (e a0 a : ONote) [NF e] [NF a0] [NF a] : ∀ k m, repr (opowAux e a0 a k m) = ω ^ repr e repr (opowAux 0 a0 a k m) \| 0, m => by cases m <;> simp [opowAux] \| k + 1, m => by by_cases h : m = 0 · simp [h, opowAux, mul_add, opow_add, mul_assoc, scale_opowAux _ _ _ k] · -- Porting note: rewrote proof rw [opowAux]; swap · assumption rw [opowAux]; swap · assumption rw [repr_add, repr_scale, scale_opowAux _ _ _ k] simp only [repr_add, repr_scale, opow_add, mul_assoc, zero_add, mul_add] #align onote.scale_opow_aux ONote.scale_opowAux theorem repr_opow_aux₁ {e a} [Ne : NF e] [Na : NF a] {a' : Ordinal} (e0 : repr e ≠ 0) (h : a' < (ω : Ordinal.{0}) ^ repr e) (aa : repr a = a') (n : ℕ+) : ((ω : Ordinal.{0}) ^ repr e * (n : ℕ) + a') ^ (ω : Ordinal.{0}) = (ω ^ repr e) ^ (ω : Ordinal.{0}) := by subst aa have No := Ne.oadd n (Na.below_of_lt' h) have := omega_le_oadd e n a rw [repr] at this refine le_antisymm ?_ (opow_le_opow_left _ this) apply (opow_le_of_limit ((opow_pos _ omega_pos).trans_le this).ne' omega_isLimit).2 intro b l have := (No.below_of_lt (lt_succ _)).repr_lt rw [repr] at this apply (opow_le_opow_left b <\| this.le).trans rw [← opow_mul, ← opow_mul] apply opow_le_opow_right omega_pos rcases le_or_lt ω (repr e) with h \| h · apply (mul_le_mul_left' (le_succ b) _).trans rw [← add_one_eq_succ, add_mul_succ _ (one_add_of_omega_le h), add_one_eq_succ, succ_le_iff, Ordinal.mul_lt_mul_iff_left (Ordinal.pos_iff_ne_zero.2 e0)] exact omega_isLimit.2 _ l · apply (principal_mul_omega (omega_isLimit.2 _ h) l).le.trans simpa using mul_le_mul_right' (one_le_iff_ne_zero.2 e0) ω #align onote.repr_opow_aux₁ ONote.repr_opow_aux₁ section -- Porting note: `R'` is used in the proof but marked as an unused variable. set_option linter.unusedVariables false in theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω ∣ repr a') (e0 : repr a0 ≠ 0) (h : repr a' + m < (ω ^ repr a0)) (n : ℕ+) (k : ℕ) : let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m) (k ≠ 0 → R < ((ω ^ repr a0) ^ succ (k : Ordinal))) ∧ ((ω ^ repr a0) ^ (k : Ordinal)) * ((ω ^ repr a0) * (n : ℕ) + repr a') + R = ((ω ^ repr a0) * (n : ℕ) + repr a' + m) ^ succ (k : Ordinal) := by intro R' haveI No : NF (oadd a0 n a') := N0.oadd n (Na'.below_of_lt' <\| lt_of_le_of_lt (le_add_right _ _) h) induction' k with k IH · cases m <;> simp [R', opowAux] -- rename R => R' let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m) let ω0 := ω ^ repr a0 let α' := ω0 * n + repr a' change (k ≠ 0 → R < (ω0 ^ succ (k : Ordinal))) ∧ (ω0 ^ (k : Ordinal)) * α' + R = (α' + m) ^ (succ ↑k : Ordinal) at IH have RR : R' = ω0 ^ (k : Ordinal) * (α' * m) + R := by by_cases h : m = 0 · simp only [R, R', h, ONote.ofNat, Nat.cast_zero, zero_add, ONote.repr, mul_zero, ONote.opowAux, add_zero] · simp only [R', ONote.repr_scale, ONote.repr, ONote.mulNat_eq_mul, ONote.opowAux, ONote.repr_ofNat, ONote.repr_mul, ONote.repr_add, Ordinal.opow_mul, ONote.zero_add] have α0 : 0 < α' := by simpa [lt_def, repr] using oadd_pos a0 n a' have ω00 : 0 < ω0 ^ (k : Ordinal) := opow_pos _ (opow_pos _ omega_pos) have Rl : R < ω ^ (repr a0 * succ ↑k) := by by_cases k0 : k = 0 · simp [R, k0] refine lt_of_lt_of_le ?_ (opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0)) cases' m with m <;> simp [opowAux, omega_pos] rw [← add_one_eq_succ, ← Nat.cast_succ] apply nat_lt_omega · rw [opow_mul] exact IH.1 k0 refine ⟨fun _ => ?_, ?_⟩ · rw [RR, ← opow_mul _ _ (succ k.succ)] have e0 := Ordinal.pos_iff_ne_zero.2 e0 have rr0 : 0 < repr a0 + repr a0 := lt_of_lt_of_le e0 (le_add_left _ _) apply principal_add_omega_opow · simp [opow_mul, opow_add, mul_assoc] rw [Ordinal.mul_lt_mul_iff_left ω00, ← Ordinal.opow_add] have : _ < ω ^ (repr a0 + repr a0) := (No.below_of_lt ?_).repr_lt · exact mul_lt_omega_opow rr0 this (nat_lt_omega _) · simpa using (add_lt_add_iff_left (repr a0)).2 e0 · exact lt_of_lt_of_le Rl (opow_le_opow_right omega_pos <\| mul_le_mul_left' (succ_le_succ_iff.2 (natCast_le.2 (le_of_lt k.lt_succ_self))) _) calc (ω0 ^ (k.succ : Ordinal)) * α' + R' _ = (ω0 ^ succ (k : Ordinal)) * α' + ((ω0 ^ (k : Ordinal)) * α' * m + R) := by rw [natCast_succ, RR, ← mul_assoc] _ = ((ω0 ^ (k : Ordinal)) * α' + R) * α' + ((ω0 ^ (k : Ordinal)) * α' + R) * m := ?_ _ = (α' + m) ^ succ (k.succ : Ordinal) := by rw [← mul_add, natCast_succ, opow_succ, IH.2] congr 1 · have αd : ω ∣ α' := dvd_add (dvd_mul_of_dvd_left (by simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 e0)) _) d rw [mul_add (ω0 ^ (k : Ordinal)), add_assoc, ← mul_assoc, ← opow_succ, add_mul_limit _ (isLimit_iff_omega_dvd.2 ⟨ne_of_gt α0, αd⟩), mul_assoc, @mul_omega_dvd n (natCast_pos.2 n.pos) (nat_lt_omega _) _ αd] apply @add_absorp _ (repr a0 * succ ↑k) · refine principal_add_omega_opow _ ?_ Rl rw [opow_mul, opow_succ, Ordinal.mul_lt_mul_iff_left ω00] exact No.snd'.repr_lt · have := mul_le_mul_left' (one_le_iff_pos.2 <\| natCast_pos.2 n.pos) (ω0 ^ succ (k : Ordinal)) rw [opow_mul] simpa [-opow_succ] · cases m · have : R = 0 := by cases k <;> simp [R, opowAux] simp [this] · rw [natCast_succ, add_mul_succ] apply add_absorp Rl rw [opow_mul, opow_succ] apply mul_le_mul_left' simpa [repr] using omega_le_oadd a0 n a' #align onote.repr_opow_aux₂ ONote.repr_opow_aux₂ end	Mathlib/SetTheory/Ordinal/Notation.lean	970	1,014	theorem repr_opow (o₁ o₂) [NF o₁] [NF o₂] : repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ := by	cases' e₁ : split o₁ with a m cases' nf_repr_split e₁ with N₁ r₁ cases' a with a0 n a' · cases' m with m · by_cases h : o₂ = 0 <;> simp [opow_def, opowAux2, opow, e₁, h, r₁] have := mt repr_inj.1 h rw [zero_opow this] · cases' e₂ : split' o₂ with b' k cases' nf_repr_split' e₂ with _ r₂ by_cases h : m = 0 · simp [opow_def, opow, e₁, h, r₁, e₂, r₂, ← Nat.one_eq_succ_zero] simp only [opow_def, opowAux2, opow, e₁, h, r₁, e₂, r₂, repr, opow_zero, Nat.succPNat_coe, Nat.cast_succ, Nat.cast_zero, _root_.zero_add, mul_one, add_zero, one_opow, npow_eq_pow] rw [opow_add, opow_mul, opow_omega, add_one_eq_succ] · congr conv_lhs => dsimp [(· ^ ·)] simp [Pow.pow, opow, Ordinal.succ_ne_zero] · simpa [Nat.one_le_iff_ne_zero] · rw [← Nat.cast_succ, lt_omega] exact ⟨_, rfl⟩ · haveI := N₁.fst haveI := N₁.snd cases' N₁.of_dvd_omega (split_dvd e₁) with a00 ad have al := split_add_lt e₁ have aa : repr (a' + ofNat m) = repr a' + m := by simp only [eq_self_iff_true, ONote.repr_ofNat, ONote.repr_add] cases' e₂ : split' o₂ with b' k cases' nf_repr_split' e₂ with _ r₂ simp only [opow_def, opow, e₁, r₁, split_eq_scale_split' e₂, opowAux2, repr] cases' k with k · simp [r₂, opow_mul, repr_opow_aux₁ a00 al aa, add_assoc] · simp? [r₂, opow_add, opow_mul, mul_assoc, add_assoc, -repr] says simp only [mulNat_eq_mul, repr_add, repr_scale, repr_mul, repr_ofNat, opow_add, opow_mul, mul_assoc, add_assoc, r₂, Nat.cast_add, Nat.cast_one, add_one_eq_succ, opow_succ] simp only [repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero, opow_one] rw [repr_opow_aux₁ a00 al aa, scale_opowAux] simp only [repr_mul, repr_scale, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero, opow_one, opow_mul] rw [← mul_add, ← add_assoc ((ω : Ordinal.{0}) ^ repr a0 * (n : ℕ))] congr 1 rw [← opow_succ] exact (repr_opow_aux₂ _ ad a00 al _ _).2
/- Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies, Bhavik Mehta -/ import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.Combinatorics.Additive.AP.Three.Defs import Mathlib.Combinatorics.Pigeonhole import Mathlib.Data.Complex.ExponentialBounds #align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8" /-! # Behrend's bound on Roth numbers This file proves Behrend's lower bound on Roth numbers. This says that we can find a subset of `{1, ..., n}` of size `n / exp (O (sqrt (log n)))` which does not contain arithmetic progressions of length `3`. The idea is that the sphere (in the `n` dimensional Euclidean space) doesn't contain arithmetic progressions (literally) because the corresponding ball is strictly convex. Thus we can take integer points on that sphere and map them onto `ℕ` in a way that preserves arithmetic progressions (`Behrend.map`). ## Main declarations * `Behrend.sphere`: The intersection of the Euclidean sphere with the positive integer quadrant. This is the set that we will map on `ℕ`. * `Behrend.map`: Given a natural number `d`, `Behrend.map d : ℕⁿ → ℕ` reads off the coordinates as digits in base `d`. * `Behrend.card_sphere_le_rothNumberNat`: Implicit lower bound on Roth numbers in terms of `Behrend.sphere`. * `Behrend.roth_lower_bound`: Behrend's explicit lower bound on Roth numbers. ## References * [Bryan Gillespie, Behrend’s Construction] (http://www.epsilonsmall.com/resources/behrends-construction/behrend.pdf) * Behrend, F. A., "On sets of integers which contain no three terms in arithmetical progression" * [Wikipedia, Salem-Spencer set](https://en.wikipedia.org/wiki/Salem–Spencer_set) ## Tags 3AP-free, Salem-Spencer, Behrend construction, arithmetic progression, sphere, strictly convex -/ open Nat hiding log open Finset Metric Real open scoped Pointwise /-- The frontier of a closed strictly convex set only contains trivial arithmetic progressions. The idea is that an arithmetic progression is contained on a line and the frontier of a strictly convex set does not contain lines. -/ lemma threeAPFree_frontier {𝕜 E : Type} [LinearOrderedField 𝕜] [TopologicalSpace E] [AddCommMonoid E] [Module 𝕜 E] {s : Set E} (hs₀ : IsClosed s) (hs₁ : StrictConvex 𝕜 s) : ThreeAPFree (frontier s) := by intro a ha b hb c hc habc obtain rfl : (1 / 2 : 𝕜) • a + (1 / 2 : 𝕜) • c = b := by rwa [← smul_add, one_div, inv_smul_eq_iff₀ (show (2 : 𝕜) ≠ 0 by norm_num), two_smul] have := hs₁.eq (hs₀.frontier_subset ha) (hs₀.frontier_subset hc) one_half_pos one_half_pos (add_halves _) hb.2 simp [this, ← add_smul] ring_nf simp #align add_salem_spencer_frontier threeAPFree_frontier lemma threeAPFree_sphere {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] (x : E) (r : ℝ) : ThreeAPFree (sphere x r) := by obtain rfl \| hr := eq_or_ne r 0 · rw [sphere_zero] exact threeAPFree_singleton _ · convert threeAPFree_frontier isClosed_ball (strictConvex_closedBall ℝ x r) exact (frontier_closedBall _ hr).symm #align add_salem_spencer_sphere threeAPFree_sphere namespace Behrend variable {α β : Type} {n d k N : ℕ} {x : Fin n → ℕ} /-! ### Turning the sphere into 3AP-free set We define `Behrend.sphere`, the intersection of the $L^2$ sphere with the positive quadrant of integer points. Because the $L^2$ closed ball is strictly convex, the $L^2$ sphere and `Behrend.sphere` are 3AP-free (`threeAPFree_sphere`). Then we can turn this set in `Fin n → ℕ` into a set in `ℕ` using `Behrend.map`, which preserves `ThreeAPFree` because it is an additive monoid homomorphism. -/ /-- The box `{0, ..., d - 1}^n` as a `Finset`. -/ def box (n d : ℕ) : Finset (Fin n → ℕ) := Fintype.piFinset fun _ => range d #align behrend.box Behrend.box theorem mem_box : x ∈ box n d ↔ ∀ i, x i < d := by simp only [box, Fintype.mem_piFinset, mem_range] #align behrend.mem_box Behrend.mem_box @[simp] theorem card_box : (box n d).card = d ^ n := by simp [box] #align behrend.card_box Behrend.card_box @[simp] theorem box_zero : box (n + 1) 0 = ∅ := by simp [box] #align behrend.box_zero Behrend.box_zero /-- The intersection of the sphere of radius `√k` with the integer points in the positive quadrant. -/ def sphere (n d k : ℕ) : Finset (Fin n → ℕ) := (box n d).filter fun x => ∑ i, x i ^ 2 = k #align behrend.sphere Behrend.sphere theorem sphere_zero_subset : sphere n d 0 ⊆ 0 := fun x => by simp [sphere, Function.funext_iff] #align behrend.sphere_zero_subset Behrend.sphere_zero_subset @[simp] theorem sphere_zero_right (n k : ℕ) : sphere (n + 1) 0 k = ∅ := by simp [sphere] #align behrend.sphere_zero_right Behrend.sphere_zero_right theorem sphere_subset_box : sphere n d k ⊆ box n d := filter_subset _ _ #align behrend.sphere_subset_box Behrend.sphere_subset_box theorem norm_of_mem_sphere {x : Fin n → ℕ} (hx : x ∈ sphere n d k) : ‖(WithLp.equiv 2 _).symm ((↑) ∘ x : Fin n → ℝ)‖ = √↑k := by rw [EuclideanSpace.norm_eq] dsimp simp_rw [abs_cast, ← cast_pow, ← cast_sum, (mem_filter.1 hx).2] #align behrend.norm_of_mem_sphere Behrend.norm_of_mem_sphere theorem sphere_subset_preimage_metric_sphere : (sphere n d k : Set (Fin n → ℕ)) ⊆ (fun x : Fin n → ℕ => (WithLp.equiv 2 _).symm ((↑) ∘ x : Fin n → ℝ)) ⁻¹' Metric.sphere (0 : PiLp 2 fun _ : Fin n => ℝ) (√↑k) := fun x hx => by rw [Set.mem_preimage, mem_sphere_zero_iff_norm, norm_of_mem_sphere hx] #align behrend.sphere_subset_preimage_metric_sphere Behrend.sphere_subset_preimage_metric_sphere /-- The map that appears in Behrend's bound on Roth numbers. -/ @[simps] def map (d : ℕ) : (Fin n → ℕ) →+ ℕ where toFun a := ∑ i, a i d ^ (i : ℕ) map_zero' := by simp_rw [Pi.zero_apply, zero_mul, sum_const_zero] map_add' a b := by simp_rw [Pi.add_apply, add_mul, sum_add_distrib] #align behrend.map Behrend.map -- @[simp] -- Porting note (#10618): simp can prove this theorem map_zero (d : ℕ) (a : Fin 0 → ℕ) : map d a = 0 := by simp [map] #align behrend.map_zero Behrend.map_zero theorem map_succ (a : Fin (n + 1) → ℕ) : map d a = a 0 + (∑ x : Fin n, a x.succ * d ^ (x : ℕ)) * d := by simp [map, Fin.sum_univ_succ, _root_.pow_succ, ← mul_assoc, ← sum_mul] #align behrend.map_succ Behrend.map_succ theorem map_succ' (a : Fin (n + 1) → ℕ) : map d a = a 0 + map d (a ∘ Fin.succ) * d := map_succ _ #align behrend.map_succ' Behrend.map_succ' theorem map_monotone (d : ℕ) : Monotone (map d : (Fin n → ℕ) → ℕ) := fun x y h => by dsimp; exact sum_le_sum fun i _ => Nat.mul_le_mul_right _ <\| h i #align behrend.map_monotone Behrend.map_monotone theorem map_mod (a : Fin n.succ → ℕ) : map d a % d = a 0 % d := by rw [map_succ, Nat.add_mul_mod_self_right] #align behrend.map_mod Behrend.map_mod theorem map_eq_iff {x₁ x₂ : Fin n.succ → ℕ} (hx₁ : ∀ i, x₁ i < d) (hx₂ : ∀ i, x₂ i < d) : map d x₁ = map d x₂ ↔ x₁ 0 = x₂ 0 ∧ map d (x₁ ∘ Fin.succ) = map d (x₂ ∘ Fin.succ) := by refine ⟨fun h => ?_, fun h => by rw [map_succ', map_succ', h.1, h.2]⟩ have : x₁ 0 = x₂ 0 := by rw [← mod_eq_of_lt (hx₁ _), ← map_mod, ← mod_eq_of_lt (hx₂ _), ← map_mod, h] rw [map_succ, map_succ, this, add_right_inj, mul_eq_mul_right_iff] at h exact ⟨this, h.resolve_right (pos_of_gt (hx₁ 0)).ne'⟩ #align behrend.map_eq_iff Behrend.map_eq_iff theorem map_injOn : {x : Fin n → ℕ \| ∀ i, x i < d}.InjOn (map d) := by intro x₁ hx₁ x₂ hx₂ h induction' n with n ih · simp [eq_iff_true_of_subsingleton] rw [forall_const] at ih ext i have x := (map_eq_iff hx₁ hx₂).1 h refine Fin.cases x.1 (congr_fun <\| ih (fun _ => ?_) (fun _ => ?_) x.2) i · exact hx₁ _ · exact hx₂ _ #align behrend.map_inj_on Behrend.map_injOn theorem map_le_of_mem_box (hx : x ∈ box n d) : map (2 * d - 1) x ≤ ∑ i : Fin n, (d - 1) * (2 * d - 1) ^ (i : ℕ) := map_monotone (2 * d - 1) fun _ => Nat.le_sub_one_of_lt <\| mem_box.1 hx _ #align behrend.map_le_of_mem_box Behrend.map_le_of_mem_box nonrec theorem threeAPFree_sphere : ThreeAPFree (sphere n d k : Set (Fin n → ℕ)) := by set f : (Fin n → ℕ) →+ EuclideanSpace ℝ (Fin n) := { toFun := fun f => ((↑) : ℕ → ℝ) ∘ f map_zero' := funext fun _ => cast_zero map_add' := fun _ _ => funext fun _ => cast_add _ _ } refine ThreeAPFree.of_image (AddMonoidHomClass.isAddFreimanHom f (Set.mapsTo_image _ _)) cast_injective.comp_left.injOn (Set.subset_univ _) ?_ refine (threeAPFree_sphere 0 (√↑k)).mono (Set.image_subset_iff.2 fun x => ?_) rw [Set.mem_preimage, mem_sphere_zero_iff_norm] exact norm_of_mem_sphere #align behrend.add_salem_spencer_sphere Behrend.threeAPFree_sphere theorem threeAPFree_image_sphere : ThreeAPFree ((sphere n d k).image (map (2 * d - 1)) : Set ℕ) := by rw [coe_image] apply ThreeAPFree.image' (α := Fin n → ℕ) (β := ℕ) (s := sphere n d k) (map (2 * d - 1)) (map_injOn.mono _) threeAPFree_sphere · rw [Set.add_subset_iff] rintro a ha b hb i have hai := mem_box.1 (sphere_subset_box ha) i have hbi := mem_box.1 (sphere_subset_box hb) i rw [lt_tsub_iff_right, ← succ_le_iff, two_mul] exact (add_add_add_comm _ _ 1 1).trans_le (_root_.add_le_add hai hbi) · exact x #align behrend.add_salem_spencer_image_sphere Behrend.threeAPFree_image_sphere theorem sum_sq_le_of_mem_box (hx : x ∈ box n d) : ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2 := by rw [mem_box] at hx have : ∀ i, x i ^ 2 ≤ (d - 1) ^ 2 := fun i => Nat.pow_le_pow_left (Nat.le_sub_one_of_lt (hx i)) _ exact (sum_le_card_nsmul univ _ _ fun i _ => this i).trans (by rw [card_fin, smul_eq_mul]) #align behrend.sum_sq_le_of_mem_box Behrend.sum_sq_le_of_mem_box theorem sum_eq : (∑ i : Fin n, d * (2 * d + 1) ^ (i : ℕ)) = ((2 * d + 1) ^ n - 1) / 2 := by refine (Nat.div_eq_of_eq_mul_left zero_lt_two ?_).symm rw [← sum_range fun i => d * (2 * d + 1) ^ (i : ℕ), ← mul_sum, mul_right_comm, mul_comm d, ← geom_sum_mul_add, add_tsub_cancel_right, mul_comm] #align behrend.sum_eq Behrend.sum_eq theorem sum_lt : (∑ i : Fin n, d * (2 * d + 1) ^ (i : ℕ)) < (2 * d + 1) ^ n := sum_eq.trans_lt <\| (Nat.div_le_self _ 2).trans_lt <\| pred_lt (pow_pos (succ_pos _) _).ne' #align behrend.sum_lt Behrend.sum_lt theorem card_sphere_le_rothNumberNat (n d k : ℕ) : (sphere n d k).card ≤ rothNumberNat ((2 * d - 1) ^ n) := by cases n · dsimp; refine (card_le_univ _).trans_eq ?_; rfl cases d · simp apply threeAPFree_image_sphere.le_rothNumberNat _ _ (card_image_of_injOn _) · intro; assumption · simp only [subset_iff, mem_image, and_imp, forall_exists_index, mem_range, forall_apply_eq_imp_iff₂, sphere, mem_filter] rintro _ x hx _ rfl exact (map_le_of_mem_box hx).trans_lt sum_lt apply map_injOn.mono fun x => ?_ · intro; assumption simp only [mem_coe, sphere, mem_filter, mem_box, and_imp, two_mul] exact fun h _ i => (h i).trans_le le_self_add #align behrend.card_sphere_le_roth_number_nat Behrend.card_sphere_le_rothNumberNat /-! ### Optimization Now that we know how to turn the integer points of any sphere into a 3AP-free set, we find a sphere containing many integer points by the pigeonhole principle. This gives us an implicit bound that we then optimize by tweaking the parameters. The (almost) optimal parameters are `Behrend.nValue` and `Behrend.dValue`. -/ theorem exists_large_sphere_aux (n d : ℕ) : ∃ k ∈ range (n * (d - 1) ^ 2 + 1), (↑(d ^ n) / ((n * (d - 1) ^ 2 :) + 1) : ℝ) ≤ (sphere n d k).card := by refine exists_le_card_fiber_of_nsmul_le_card_of_maps_to (fun x hx => ?_) nonempty_range_succ ?_ · rw [mem_range, Nat.lt_succ_iff] exact sum_sq_le_of_mem_box hx · rw [card_range, _root_.nsmul_eq_mul, mul_div_assoc', cast_add_one, mul_div_cancel_left₀, card_box] exact (cast_add_one_pos _).ne' #align behrend.exists_large_sphere_aux Behrend.exists_large_sphere_aux theorem exists_large_sphere (n d : ℕ) : ∃ k, ((d ^ n :) / (n * d ^ 2 :) : ℝ) ≤ (sphere n d k).card := by obtain ⟨k, -, hk⟩ := exists_large_sphere_aux n d refine ⟨k, ?_⟩ obtain rfl \| hn := n.eq_zero_or_pos · simp obtain rfl \| hd := d.eq_zero_or_pos · simp refine (div_le_div_of_nonneg_left ?_ ?_ ?_).trans hk · exact cast_nonneg _ · exact cast_add_one_pos _ simp only [← le_sub_iff_add_le', cast_mul, ← mul_sub, cast_pow, cast_sub hd, sub_sq, one_pow, cast_one, mul_one, sub_add, sub_sub_self] apply one_le_mul_of_one_le_of_one_le · rwa [one_le_cast] rw [_root_.le_sub_iff_add_le] set_option tactic.skipAssignedInstances false in norm_num exact one_le_cast.2 hd #align behrend.exists_large_sphere Behrend.exists_large_sphere theorem bound_aux' (n d : ℕ) : ((d ^ n :) / (n * d ^ 2 :) : ℝ) ≤ rothNumberNat ((2 * d - 1) ^ n) := let ⟨_, h⟩ := exists_large_sphere n d h.trans <\| cast_le.2 <\| card_sphere_le_rothNumberNat _ _ _ #align behrend.bound_aux' Behrend.bound_aux' theorem bound_aux (hd : d ≠ 0) (hn : 2 ≤ n) : (d ^ (n - 2 :) / n : ℝ) ≤ rothNumberNat ((2 * d - 1) ^ n) := by convert bound_aux' n d using 1 rw [cast_mul, cast_pow, mul_comm, ← div_div, pow_sub₀ _ _ hn, ← div_eq_mul_inv, cast_pow] rwa [cast_ne_zero] #align behrend.bound_aux Behrend.bound_aux open scoped Filter Topology open Real section NumericalBounds theorem log_two_mul_two_le_sqrt_log_eight : log 2 * 2 ≤ √(log 8) := by have : (8 : ℝ) = 2 ^ ((3 : ℕ) : ℝ) := by rw [rpow_natCast]; norm_num rw [this, log_rpow zero_lt_two (3 : ℕ)] apply le_sqrt_of_sq_le rw [mul_pow, sq (log 2), mul_assoc, mul_comm] refine mul_le_mul_of_nonneg_right ?_ (log_nonneg one_le_two) rw [← le_div_iff] on_goal 1 => apply log_two_lt_d9.le.trans all_goals norm_num1 #align behrend.log_two_mul_two_le_sqrt_log_eight Behrend.log_two_mul_two_le_sqrt_log_eight theorem two_div_one_sub_two_div_e_le_eight : 2 / (1 - 2 / exp 1) ≤ 8 := by rw [div_le_iff, mul_sub, mul_one, mul_div_assoc', le_sub_comm, div_le_iff (exp_pos _)] · have : 16 < 6 * (2.7182818283 : ℝ) := by norm_num linarith [exp_one_gt_d9] rw [sub_pos, div_lt_one] <;> exact exp_one_gt_d9.trans' (by norm_num) #align behrend.two_div_one_sub_two_div_e_le_eight Behrend.two_div_one_sub_two_div_e_le_eight theorem le_sqrt_log (hN : 4096 ≤ N) : log (2 / (1 - 2 / exp 1)) * (69 / 50) ≤ √(log ↑N) := by have : (12 : ℕ) * log 2 ≤ log N := by rw [← log_rpow zero_lt_two, rpow_natCast] exact log_le_log (by positivity) (mod_cast hN) refine (mul_le_mul_of_nonneg_right (log_le_log ?_ two_div_one_sub_two_div_e_le_eight) <\| by norm_num1).trans ?_ · refine div_pos zero_lt_two ?_ rw [sub_pos, div_lt_one (exp_pos _)] exact exp_one_gt_d9.trans_le' (by norm_num1) have l8 : log 8 = (3 : ℕ) * log 2 := by rw [← log_rpow zero_lt_two, rpow_natCast] norm_num rw [l8] apply le_sqrt_of_sq_le (le_trans _ this) rw [mul_right_comm, mul_pow, sq (log 2), ← mul_assoc] apply mul_le_mul_of_nonneg_right _ (log_nonneg one_le_two) rw [← le_div_iff'] · exact log_two_lt_d9.le.trans (by norm_num1) exact sq_pos_of_ne_zero (by norm_num1) #align behrend.le_sqrt_log Behrend.le_sqrt_log theorem exp_neg_two_mul_le {x : ℝ} (hx : 0 < x) : exp (-2 * x) < exp (2 - ⌈x⌉₊) / ⌈x⌉₊ := by have h₁ := ceil_lt_add_one hx.le have h₂ : 1 - x ≤ 2 - ⌈x⌉₊ := by linarith calc _ ≤ exp (1 - x) / (x + 1) := ?_ _ ≤ exp (2 - ⌈x⌉₊) / (x + 1) := by gcongr _ < _ := by gcongr rw [le_div_iff (add_pos hx zero_lt_one), ← le_div_iff' (exp_pos _), ← exp_sub, neg_mul, sub_neg_eq_add, two_mul, sub_add_add_cancel, add_comm _ x] exact le_trans (le_add_of_nonneg_right zero_le_one) (add_one_le_exp _) #align behrend.exp_neg_two_mul_le Behrend.exp_neg_two_mul_le theorem div_lt_floor {x : ℝ} (hx : 2 / (1 - 2 / exp 1) ≤ x) : x / exp 1 < (⌊x / 2⌋₊ : ℝ) := by apply lt_of_le_of_lt _ (sub_one_lt_floor _) have : 0 < 1 - 2 / exp 1 := by rw [sub_pos, div_lt_one (exp_pos _)] exact lt_of_le_of_lt (by norm_num) exp_one_gt_d9 rwa [le_sub_comm, div_eq_mul_one_div x, div_eq_mul_one_div x, ← mul_sub, div_sub', ← div_eq_mul_one_div, mul_div_assoc', one_le_div, ← div_le_iff this] · exact zero_lt_two · exact two_ne_zero #align behrend.div_lt_floor Behrend.div_lt_floor theorem ceil_lt_mul {x : ℝ} (hx : 50 / 19 ≤ x) : (⌈x⌉₊ : ℝ) < 1.38 * x := by refine (ceil_lt_add_one <\| hx.trans' <\| by norm_num).trans_le ?_ rw [← le_sub_iff_add_le', ← sub_one_mul] have : (1.38 : ℝ) = 69 / 50 := by norm_num rwa [this, show (69 / 50 - 1 : ℝ) = (50 / 19)⁻¹ by norm_num1, ← div_eq_inv_mul, one_le_div] norm_num1 #align behrend.ceil_lt_mul Behrend.ceil_lt_mul end NumericalBounds /-- The (almost) optimal value of `n` in `Behrend.bound_aux`. -/ noncomputable def nValue (N : ℕ) : ℕ := ⌈√(log N)⌉₊ #align behrend.n_value Behrend.nValue /-- The (almost) optimal value of `d` in `Behrend.bound_aux`. -/ noncomputable def dValue (N : ℕ) : ℕ := ⌊(N : ℝ) ^ (nValue N : ℝ)⁻¹ / 2⌋₊ #align behrend.d_value Behrend.dValue theorem nValue_pos (hN : 2 ≤ N) : 0 < nValue N := ceil_pos.2 <\| Real.sqrt_pos.2 <\| log_pos <\| one_lt_cast.2 <\| hN #align behrend.n_value_pos Behrend.nValue_pos #noalign behrend.two_le_n_value theorem three_le_nValue (hN : 64 ≤ N) : 3 ≤ nValue N := by rw [nValue, ← lt_iff_add_one_le, lt_ceil, cast_two] apply lt_sqrt_of_sq_lt have : (2 : ℝ) ^ ((6 : ℕ) : ℝ) ≤ N := by rw [rpow_natCast] exact (cast_le.2 hN).trans' (by norm_num1) apply lt_of_lt_of_le _ (log_le_log (rpow_pos_of_pos zero_lt_two _) this) rw [log_rpow zero_lt_two, ← div_lt_iff'] · exact log_two_gt_d9.trans_le' (by norm_num1) · norm_num1 #align behrend.three_le_n_value Behrend.three_le_nValue theorem dValue_pos (hN₃ : 8 ≤ N) : 0 < dValue N := by have hN₀ : 0 < (N : ℝ) := cast_pos.2 (succ_pos'.trans_le hN₃) rw [dValue, floor_pos, ← log_le_log_iff zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN₀, inv_mul_eq_div, sub_nonneg, le_div_iff] · have : (nValue N : ℝ) ≤ 2 * √(log N) := by apply (ceil_lt_add_one <\| sqrt_nonneg _).le.trans rw [two_mul, add_le_add_iff_left] apply le_sqrt_of_sq_le rw [one_pow, le_log_iff_exp_le hN₀] exact (exp_one_lt_d9.le.trans <\| by norm_num).trans (cast_le.2 hN₃) apply (mul_le_mul_of_nonneg_left this <\| log_nonneg one_le_two).trans _ rw [← mul_assoc, ← le_div_iff (Real.sqrt_pos.2 <\| log_pos <\| one_lt_cast.2 _), div_sqrt] · apply log_two_mul_two_le_sqrt_log_eight.trans apply Real.sqrt_le_sqrt exact log_le_log (by norm_num) (mod_cast hN₃) exact hN₃.trans_lt' (by norm_num) · exact cast_pos.2 (nValue_pos <\| hN₃.trans' <\| by norm_num) · exact (rpow_pos_of_pos hN₀ _).ne' · exact div_pos (rpow_pos_of_pos hN₀ _) zero_lt_two #align behrend.d_value_pos Behrend.dValue_pos theorem le_N (hN : 2 ≤ N) : (2 * dValue N - 1) ^ nValue N ≤ N := by have : (2 * dValue N - 1) ^ nValue N ≤ (2 * dValue N) ^ nValue N := Nat.pow_le_pow_left (Nat.sub_le _ _) _ apply this.trans suffices ((2 * dValue N) ^ nValue N : ℝ) ≤ N from mod_cast this suffices i : (2 * dValue N : ℝ) ≤ (N : ℝ) ^ (nValue N : ℝ)⁻¹ by rw [← rpow_natCast] apply (rpow_le_rpow (mul_nonneg zero_le_two (cast_nonneg _)) i (cast_nonneg _)).trans rw [← rpow_mul (cast_nonneg _), inv_mul_cancel, rpow_one] rw [cast_ne_zero] apply (nValue_pos hN).ne' rw [← le_div_iff'] · exact floor_le (div_nonneg (rpow_nonneg (cast_nonneg _) _) zero_le_two) apply zero_lt_two set_option linter.uppercaseLean3 false in #align behrend.le_N Behrend.le_N theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by apply div_lt_floor _ rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv] · apply le_trans _ (div_le_div_of_nonneg_left _ _ (ceil_lt_mul _).le) · rw [mul_comm, ← div_div, div_sqrt, le_div_iff] · set_option tactic.skipAssignedInstances false in norm_num; exact le_sqrt_log hN · norm_num1 · apply log_nonneg rw [one_le_cast] exact hN.trans' (by norm_num1) · rw [cast_pos, lt_ceil, cast_zero, Real.sqrt_pos] refine log_pos ?_ rw [one_lt_cast] exact hN.trans_lt' (by norm_num1) apply le_sqrt_of_sq_le have : (12 : ℕ) * log 2 ≤ log N := by rw [← log_rpow zero_lt_two, rpow_natCast] exact log_le_log (by positivity) (mod_cast hN) refine le_trans ?_ this rw [← div_le_iff'] · exact log_two_gt_d9.le.trans' (by norm_num1) · norm_num1 · rw [cast_pos] exact hN.trans_lt' (by norm_num1) · refine div_pos zero_lt_two ?_ rw [sub_pos, div_lt_one (exp_pos _)] exact lt_of_le_of_lt (by norm_num1) exp_one_gt_d9 positivity #align behrend.bound Behrend.bound theorem roth_lower_bound_explicit (hN : 4096 ≤ N) : (N : ℝ) * exp (-4 * √(log N)) < rothNumberNat N := by let n := nValue N have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <\| hN.trans' <\| by norm_num1) have hd : 0 < dValue N := dValue_pos (hN.trans' <\| by norm_num1) have hN₀ : 0 < (N : ℝ) := cast_pos.2 (hN.trans' <\| by norm_num1) have hn₂ : 2 < n := three_le_nValue <\| hN.trans' <\| by norm_num1 have : (2 * dValue N - 1) ^ n ≤ N := le_N (hN.trans' <\| by norm_num1) calc _ ≤ (N ^ (nValue N : ℝ)⁻¹ / rexp 1 : ℝ) ^ (n - 2) / n := ?_ _ < _ := by gcongr; exacts [(tsub_pos_of_lt hn₂).ne', bound hN] _ ≤ rothNumberNat ((2 * dValue N - 1) ^ n) := bound_aux hd.ne' hn₂.le _ ≤ rothNumberNat N := mod_cast rothNumberNat.mono this rw [← rpow_natCast, div_rpow (rpow_nonneg hN₀.le _) (exp_pos _).le, ← rpow_mul hN₀.le, inv_mul_eq_div, cast_sub hn₂.le, cast_two, same_sub_div hn.ne', exp_one_rpow, div_div, rpow_sub hN₀, rpow_one, div_div, div_eq_mul_inv] refine mul_le_mul_of_nonneg_left ?_ (cast_nonneg _) rw [mul_inv, mul_inv, ← exp_neg, ← rpow_neg (cast_nonneg _), neg_sub, ← div_eq_mul_inv] have : exp (-4 * √(log N)) = exp (-2 * √(log N)) * exp (-2 * √(log N)) := by rw [← exp_add, ← add_mul] norm_num rw [this] refine mul_le_mul ?_ (exp_neg_two_mul_le <\| Real.sqrt_pos.2 <\| log_pos ?_).le (exp_pos _).le <\| rpow_nonneg (cast_nonneg _) _ · rw [← le_log_iff_exp_le (rpow_pos_of_pos hN₀ _), log_rpow hN₀, ← le_div_iff, mul_div_assoc, div_sqrt, neg_mul, neg_le_neg_iff, div_mul_eq_mul_div, div_le_iff hn] · exact mul_le_mul_of_nonneg_left (le_ceil _) zero_le_two refine Real.sqrt_pos.2 (log_pos ?_) rw [one_lt_cast] exact hN.trans_lt' (by norm_num1) · rw [one_lt_cast] exact hN.trans_lt' (by norm_num1) #align behrend.roth_lower_bound_explicit Behrend.roth_lower_bound_explicit theorem exp_four_lt : exp 4 < 64 := by rw [show (64 : ℝ) = 2 ^ ((6 : ℕ) : ℝ) by rw [rpow_natCast]; norm_num1, ← lt_log_iff_exp_lt (rpow_pos_of_pos zero_lt_two _), log_rpow zero_lt_two, ← div_lt_iff'] · exact log_two_gt_d9.trans_le' (by norm_num1) · norm_num #align behrend.exp_four_lt Behrend.exp_four_lt	Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean	521	525	theorem four_zero_nine_six_lt_exp_sixteen : 4096 < exp 16 := by	rw [← log_lt_iff_lt_exp (show (0 : ℝ) < 4096 by norm_num), show (4096 : ℝ) = 2 ^ 12 by norm_cast, ← rpow_natCast, log_rpow zero_lt_two, cast_ofNat] have : 12 * (0.6931471808 : ℝ) < 16 := by norm_num linarith [log_two_lt_d9]
/- Copyright (c) 2019 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.FDeriv.Linear import Mathlib.Analysis.Calculus.FDeriv.Comp #align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee" /-! # Additive operations on derivatives For detailed documentation of the Fréchet derivative, see the module docstring of `Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of * sum of finitely many functions * multiplication of a function by a scalar constant * negative of a function * subtraction of two functions -/ open Filter Asymptotics ContinuousLinearMap Set Metric open scoped Classical open Topology NNReal Filter Asymptotics ENNReal noncomputable section section variable {𝕜 : Type} [NontriviallyNormedField 𝕜] variable {E : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E] variable {F : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F] variable {G : Type} [NormedAddCommGroup G] [NormedSpace 𝕜 G] variable {G' : Type} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] variable {f f₀ f₁ g : E → F} variable {f' f₀' f₁' g' : E →L[𝕜] F} variable (e : E →L[𝕜] F) variable {x : E} variable {s t : Set E} variable {L L₁ L₂ : Filter E} section ConstSMul variable {R : Type} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] /-! ### Derivative of a function multiplied by a constant -/ @[fun_prop] theorem HasStrictFDerivAt.const_smul (h : HasStrictFDerivAt f f' x) (c : R) : HasStrictFDerivAt (fun x => c • f x) (c • f') x := (c • (1 : F →L[𝕜] F)).hasStrictFDerivAt.comp x h #align has_strict_fderiv_at.const_smul HasStrictFDerivAt.const_smul theorem HasFDerivAtFilter.const_smul (h : HasFDerivAtFilter f f' x L) (c : R) : HasFDerivAtFilter (fun x => c • f x) (c • f') x L := (c • (1 : F →L[𝕜] F)).hasFDerivAtFilter.comp x h tendsto_map #align has_fderiv_at_filter.const_smul HasFDerivAtFilter.const_smul @[fun_prop] nonrec theorem HasFDerivWithinAt.const_smul (h : HasFDerivWithinAt f f' s x) (c : R) : HasFDerivWithinAt (fun x => c • f x) (c • f') s x := h.const_smul c #align has_fderiv_within_at.const_smul HasFDerivWithinAt.const_smul @[fun_prop] nonrec theorem HasFDerivAt.const_smul (h : HasFDerivAt f f' x) (c : R) : HasFDerivAt (fun x => c • f x) (c • f') x := h.const_smul c #align has_fderiv_at.const_smul HasFDerivAt.const_smul @[fun_prop] theorem DifferentiableWithinAt.const_smul (h : DifferentiableWithinAt 𝕜 f s x) (c : R) : DifferentiableWithinAt 𝕜 (fun y => c • f y) s x := (h.hasFDerivWithinAt.const_smul c).differentiableWithinAt #align differentiable_within_at.const_smul DifferentiableWithinAt.const_smul @[fun_prop] theorem DifferentiableAt.const_smul (h : DifferentiableAt 𝕜 f x) (c : R) : DifferentiableAt 𝕜 (fun y => c • f y) x := (h.hasFDerivAt.const_smul c).differentiableAt #align differentiable_at.const_smul DifferentiableAt.const_smul @[fun_prop] theorem DifferentiableOn.const_smul (h : DifferentiableOn 𝕜 f s) (c : R) : DifferentiableOn 𝕜 (fun y => c • f y) s := fun x hx => (h x hx).const_smul c #align differentiable_on.const_smul DifferentiableOn.const_smul @[fun_prop] theorem Differentiable.const_smul (h : Differentiable 𝕜 f) (c : R) : Differentiable 𝕜 fun y => c • f y := fun x => (h x).const_smul c #align differentiable.const_smul Differentiable.const_smul theorem fderivWithin_const_smul (hxs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f s x) (c : R) : fderivWithin 𝕜 (fun y => c • f y) s x = c • fderivWithin 𝕜 f s x := (h.hasFDerivWithinAt.const_smul c).fderivWithin hxs #align fderiv_within_const_smul fderivWithin_const_smul theorem fderiv_const_smul (h : DifferentiableAt 𝕜 f x) (c : R) : fderiv 𝕜 (fun y => c • f y) x = c • fderiv 𝕜 f x := (h.hasFDerivAt.const_smul c).fderiv #align fderiv_const_smul fderiv_const_smul end ConstSMul section Add /-! ### Derivative of the sum of two functions -/ @[fun_prop] nonrec theorem HasStrictFDerivAt.add (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) : HasStrictFDerivAt (fun y => f y + g y) (f' + g') x := (hf.add hg).congr_left fun y => by simp only [LinearMap.sub_apply, LinearMap.add_apply, map_sub, map_add, add_apply] abel #align has_strict_fderiv_at.add HasStrictFDerivAt.add theorem HasFDerivAtFilter.add (hf : HasFDerivAtFilter f f' x L) (hg : HasFDerivAtFilter g g' x L) : HasFDerivAtFilter (fun y => f y + g y) (f' + g') x L := .of_isLittleO <\| (hf.isLittleO.add hg.isLittleO).congr_left fun _ => by simp only [LinearMap.sub_apply, LinearMap.add_apply, map_sub, map_add, add_apply] abel #align has_fderiv_at_filter.add HasFDerivAtFilter.add @[fun_prop] nonrec theorem HasFDerivWithinAt.add (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x) : HasFDerivWithinAt (fun y => f y + g y) (f' + g') s x := hf.add hg #align has_fderiv_within_at.add HasFDerivWithinAt.add @[fun_prop] nonrec theorem HasFDerivAt.add (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) : HasFDerivAt (fun x => f x + g x) (f' + g') x := hf.add hg #align has_fderiv_at.add HasFDerivAt.add @[fun_prop] theorem DifferentiableWithinAt.add (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) : DifferentiableWithinAt 𝕜 (fun y => f y + g y) s x := (hf.hasFDerivWithinAt.add hg.hasFDerivWithinAt).differentiableWithinAt #align differentiable_within_at.add DifferentiableWithinAt.add @[simp, fun_prop] theorem DifferentiableAt.add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : DifferentiableAt 𝕜 (fun y => f y + g y) x := (hf.hasFDerivAt.add hg.hasFDerivAt).differentiableAt #align differentiable_at.add DifferentiableAt.add @[fun_prop] theorem DifferentiableOn.add (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) : DifferentiableOn 𝕜 (fun y => f y + g y) s := fun x hx => (hf x hx).add (hg x hx) #align differentiable_on.add DifferentiableOn.add @[simp, fun_prop] theorem Differentiable.add (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) : Differentiable 𝕜 fun y => f y + g y := fun x => (hf x).add (hg x) #align differentiable.add Differentiable.add theorem fderivWithin_add (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) : fderivWithin 𝕜 (fun y => f y + g y) s x = fderivWithin 𝕜 f s x + fderivWithin 𝕜 g s x := (hf.hasFDerivWithinAt.add hg.hasFDerivWithinAt).fderivWithin hxs #align fderiv_within_add fderivWithin_add theorem fderiv_add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : fderiv 𝕜 (fun y => f y + g y) x = fderiv 𝕜 f x + fderiv 𝕜 g x := (hf.hasFDerivAt.add hg.hasFDerivAt).fderiv #align fderiv_add fderiv_add @[fun_prop] theorem HasStrictFDerivAt.add_const (hf : HasStrictFDerivAt f f' x) (c : F) : HasStrictFDerivAt (fun y => f y + c) f' x := add_zero f' ▸ hf.add (hasStrictFDerivAt_const _ _) #align has_strict_fderiv_at.add_const HasStrictFDerivAt.add_const theorem HasFDerivAtFilter.add_const (hf : HasFDerivAtFilter f f' x L) (c : F) : HasFDerivAtFilter (fun y => f y + c) f' x L := add_zero f' ▸ hf.add (hasFDerivAtFilter_const _ _ _) #align has_fderiv_at_filter.add_const HasFDerivAtFilter.add_const @[fun_prop] nonrec theorem HasFDerivWithinAt.add_const (hf : HasFDerivWithinAt f f' s x) (c : F) : HasFDerivWithinAt (fun y => f y + c) f' s x := hf.add_const c #align has_fderiv_within_at.add_const HasFDerivWithinAt.add_const @[fun_prop] nonrec theorem HasFDerivAt.add_const (hf : HasFDerivAt f f' x) (c : F) : HasFDerivAt (fun x => f x + c) f' x := hf.add_const c #align has_fderiv_at.add_const HasFDerivAt.add_const @[fun_prop] theorem DifferentiableWithinAt.add_const (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) : DifferentiableWithinAt 𝕜 (fun y => f y + c) s x := (hf.hasFDerivWithinAt.add_const c).differentiableWithinAt #align differentiable_within_at.add_const DifferentiableWithinAt.add_const @[simp] theorem differentiableWithinAt_add_const_iff (c : F) : DifferentiableWithinAt 𝕜 (fun y => f y + c) s x ↔ DifferentiableWithinAt 𝕜 f s x := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_within_at_add_const_iff differentiableWithinAt_add_const_iff @[fun_prop] theorem DifferentiableAt.add_const (hf : DifferentiableAt 𝕜 f x) (c : F) : DifferentiableAt 𝕜 (fun y => f y + c) x := (hf.hasFDerivAt.add_const c).differentiableAt #align differentiable_at.add_const DifferentiableAt.add_const @[simp] theorem differentiableAt_add_const_iff (c : F) : DifferentiableAt 𝕜 (fun y => f y + c) x ↔ DifferentiableAt 𝕜 f x := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_at_add_const_iff differentiableAt_add_const_iff @[fun_prop] theorem DifferentiableOn.add_const (hf : DifferentiableOn 𝕜 f s) (c : F) : DifferentiableOn 𝕜 (fun y => f y + c) s := fun x hx => (hf x hx).add_const c #align differentiable_on.add_const DifferentiableOn.add_const @[simp] theorem differentiableOn_add_const_iff (c : F) : DifferentiableOn 𝕜 (fun y => f y + c) s ↔ DifferentiableOn 𝕜 f s := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_on_add_const_iff differentiableOn_add_const_iff @[fun_prop] theorem Differentiable.add_const (hf : Differentiable 𝕜 f) (c : F) : Differentiable 𝕜 fun y => f y + c := fun x => (hf x).add_const c #align differentiable.add_const Differentiable.add_const @[simp] theorem differentiable_add_const_iff (c : F) : (Differentiable 𝕜 fun y => f y + c) ↔ Differentiable 𝕜 f := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_add_const_iff differentiable_add_const_iff theorem fderivWithin_add_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) : fderivWithin 𝕜 (fun y => f y + c) s x = fderivWithin 𝕜 f s x := if hf : DifferentiableWithinAt 𝕜 f s x then (hf.hasFDerivWithinAt.add_const c).fderivWithin hxs else by rw [fderivWithin_zero_of_not_differentiableWithinAt hf, fderivWithin_zero_of_not_differentiableWithinAt] simpa #align fderiv_within_add_const fderivWithin_add_const theorem fderiv_add_const (c : F) : fderiv 𝕜 (fun y => f y + c) x = fderiv 𝕜 f x := by simp only [← fderivWithin_univ, fderivWithin_add_const uniqueDiffWithinAt_univ] #align fderiv_add_const fderiv_add_const @[fun_prop] theorem HasStrictFDerivAt.const_add (hf : HasStrictFDerivAt f f' x) (c : F) : HasStrictFDerivAt (fun y => c + f y) f' x := zero_add f' ▸ (hasStrictFDerivAt_const _ _).add hf #align has_strict_fderiv_at.const_add HasStrictFDerivAt.const_add theorem HasFDerivAtFilter.const_add (hf : HasFDerivAtFilter f f' x L) (c : F) : HasFDerivAtFilter (fun y => c + f y) f' x L := zero_add f' ▸ (hasFDerivAtFilter_const _ _ _).add hf #align has_fderiv_at_filter.const_add HasFDerivAtFilter.const_add @[fun_prop] nonrec theorem HasFDerivWithinAt.const_add (hf : HasFDerivWithinAt f f' s x) (c : F) : HasFDerivWithinAt (fun y => c + f y) f' s x := hf.const_add c #align has_fderiv_within_at.const_add HasFDerivWithinAt.const_add @[fun_prop] nonrec theorem HasFDerivAt.const_add (hf : HasFDerivAt f f' x) (c : F) : HasFDerivAt (fun x => c + f x) f' x := hf.const_add c #align has_fderiv_at.const_add HasFDerivAt.const_add @[fun_prop] theorem DifferentiableWithinAt.const_add (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) : DifferentiableWithinAt 𝕜 (fun y => c + f y) s x := (hf.hasFDerivWithinAt.const_add c).differentiableWithinAt #align differentiable_within_at.const_add DifferentiableWithinAt.const_add @[simp] theorem differentiableWithinAt_const_add_iff (c : F) : DifferentiableWithinAt 𝕜 (fun y => c + f y) s x ↔ DifferentiableWithinAt 𝕜 f s x := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_within_at_const_add_iff differentiableWithinAt_const_add_iff @[fun_prop] theorem DifferentiableAt.const_add (hf : DifferentiableAt 𝕜 f x) (c : F) : DifferentiableAt 𝕜 (fun y => c + f y) x := (hf.hasFDerivAt.const_add c).differentiableAt #align differentiable_at.const_add DifferentiableAt.const_add @[simp] theorem differentiableAt_const_add_iff (c : F) : DifferentiableAt 𝕜 (fun y => c + f y) x ↔ DifferentiableAt 𝕜 f x := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_at_const_add_iff differentiableAt_const_add_iff @[fun_prop] theorem DifferentiableOn.const_add (hf : DifferentiableOn 𝕜 f s) (c : F) : DifferentiableOn 𝕜 (fun y => c + f y) s := fun x hx => (hf x hx).const_add c #align differentiable_on.const_add DifferentiableOn.const_add @[simp] theorem differentiableOn_const_add_iff (c : F) : DifferentiableOn 𝕜 (fun y => c + f y) s ↔ DifferentiableOn 𝕜 f s := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_on_const_add_iff differentiableOn_const_add_iff @[fun_prop] theorem Differentiable.const_add (hf : Differentiable 𝕜 f) (c : F) : Differentiable 𝕜 fun y => c + f y := fun x => (hf x).const_add c #align differentiable.const_add Differentiable.const_add @[simp] theorem differentiable_const_add_iff (c : F) : (Differentiable 𝕜 fun y => c + f y) ↔ Differentiable 𝕜 f := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_const_add_iff differentiable_const_add_iff theorem fderivWithin_const_add (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) : fderivWithin 𝕜 (fun y => c + f y) s x = fderivWithin 𝕜 f s x := by simpa only [add_comm] using fderivWithin_add_const hxs c #align fderiv_within_const_add fderivWithin_const_add theorem fderiv_const_add (c : F) : fderiv 𝕜 (fun y => c + f y) x = fderiv 𝕜 f x := by simp only [add_comm c, fderiv_add_const] #align fderiv_const_add fderiv_const_add end Add section Sum /-! ### Derivative of a finite sum of functions -/ variable {ι : Type} {u : Finset ι} {A : ι → E → F} {A' : ι → E →L[𝕜] F} @[fun_prop] theorem HasStrictFDerivAt.sum (h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x) : HasStrictFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x := by dsimp [HasStrictFDerivAt] at convert IsLittleO.sum h simp [Finset.sum_sub_distrib, ContinuousLinearMap.sum_apply] #align has_strict_fderiv_at.sum HasStrictFDerivAt.sum	Mathlib/Analysis/Calculus/FDeriv/Add.lean	353	357	theorem HasFDerivAtFilter.sum (h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) x L) : HasFDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L := by	simp only [hasFDerivAtFilter_iff_isLittleO] at * convert IsLittleO.sum h simp [ContinuousLinearMap.sum_apply]
/- Copyright (c) 2019 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison, Bhavik Mehta -/ import Mathlib.CategoryTheory.Comma.Over import Mathlib.CategoryTheory.DiscreteCategory import Mathlib.CategoryTheory.EpiMono import Mathlib.CategoryTheory.Limits.Shapes.Terminal #align_import category_theory.limits.shapes.binary_products from "leanprover-community/mathlib"@"fec1d95fc61c750c1ddbb5b1f7f48b8e811a80d7" /-! # Binary (co)products We define a category `WalkingPair`, which is the index category for a binary (co)product diagram. A convenience method `pair X Y` constructs the functor from the walking pair, hitting the given objects. We define `prod X Y` and `coprod X Y` as limits and colimits of such functors. Typeclasses `HasBinaryProducts` and `HasBinaryCoproducts` assert the existence of (co)limits shaped as walking pairs. We include lemmas for simplifying equations involving projections and coprojections, and define braiding and associating isomorphisms, and the product comparison morphism. ## References * [Stacks: Products of pairs](https://stacks.math.columbia.edu/tag/001R) * [Stacks: coproducts of pairs](https://stacks.math.columbia.edu/tag/04AN) -/ noncomputable section universe v u u₂ open CategoryTheory namespace CategoryTheory.Limits /-- The type of objects for the diagram indexing a binary (co)product. -/ inductive WalkingPair : Type \| left \| right deriving DecidableEq, Inhabited #align category_theory.limits.walking_pair CategoryTheory.Limits.WalkingPair open WalkingPair /-- The equivalence swapping left and right. -/ def WalkingPair.swap : WalkingPair ≃ WalkingPair where toFun j := WalkingPair.recOn j right left invFun j := WalkingPair.recOn j right left left_inv j := by cases j; repeat rfl right_inv j := by cases j; repeat rfl #align category_theory.limits.walking_pair.swap CategoryTheory.Limits.WalkingPair.swap @[simp] theorem WalkingPair.swap_apply_left : WalkingPair.swap left = right := rfl #align category_theory.limits.walking_pair.swap_apply_left CategoryTheory.Limits.WalkingPair.swap_apply_left @[simp] theorem WalkingPair.swap_apply_right : WalkingPair.swap right = left := rfl #align category_theory.limits.walking_pair.swap_apply_right CategoryTheory.Limits.WalkingPair.swap_apply_right @[simp] theorem WalkingPair.swap_symm_apply_tt : WalkingPair.swap.symm left = right := rfl #align category_theory.limits.walking_pair.swap_symm_apply_tt CategoryTheory.Limits.WalkingPair.swap_symm_apply_tt @[simp] theorem WalkingPair.swap_symm_apply_ff : WalkingPair.swap.symm right = left := rfl #align category_theory.limits.walking_pair.swap_symm_apply_ff CategoryTheory.Limits.WalkingPair.swap_symm_apply_ff /-- An equivalence from `WalkingPair` to `Bool`, sometimes useful when reindexing limits. -/ def WalkingPair.equivBool : WalkingPair ≃ Bool where toFun j := WalkingPair.recOn j true false -- to match equiv.sum_equiv_sigma_bool invFun b := Bool.recOn b right left left_inv j := by cases j; repeat rfl right_inv b := by cases b; repeat rfl #align category_theory.limits.walking_pair.equiv_bool CategoryTheory.Limits.WalkingPair.equivBool @[simp] theorem WalkingPair.equivBool_apply_left : WalkingPair.equivBool left = true := rfl #align category_theory.limits.walking_pair.equiv_bool_apply_left CategoryTheory.Limits.WalkingPair.equivBool_apply_left @[simp] theorem WalkingPair.equivBool_apply_right : WalkingPair.equivBool right = false := rfl #align category_theory.limits.walking_pair.equiv_bool_apply_right CategoryTheory.Limits.WalkingPair.equivBool_apply_right @[simp] theorem WalkingPair.equivBool_symm_apply_true : WalkingPair.equivBool.symm true = left := rfl #align category_theory.limits.walking_pair.equiv_bool_symm_apply_tt CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_true @[simp] theorem WalkingPair.equivBool_symm_apply_false : WalkingPair.equivBool.symm false = right := rfl #align category_theory.limits.walking_pair.equiv_bool_symm_apply_ff CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_false variable {C : Type u} /-- The function on the walking pair, sending the two points to `X` and `Y`. -/ def pairFunction (X Y : C) : WalkingPair → C := fun j => WalkingPair.casesOn j X Y #align category_theory.limits.pair_function CategoryTheory.Limits.pairFunction @[simp] theorem pairFunction_left (X Y : C) : pairFunction X Y left = X := rfl #align category_theory.limits.pair_function_left CategoryTheory.Limits.pairFunction_left @[simp] theorem pairFunction_right (X Y : C) : pairFunction X Y right = Y := rfl #align category_theory.limits.pair_function_right CategoryTheory.Limits.pairFunction_right variable [Category.{v} C] /-- The diagram on the walking pair, sending the two points to `X` and `Y`. -/ def pair (X Y : C) : Discrete WalkingPair ⥤ C := Discrete.functor fun j => WalkingPair.casesOn j X Y #align category_theory.limits.pair CategoryTheory.Limits.pair @[simp] theorem pair_obj_left (X Y : C) : (pair X Y).obj ⟨left⟩ = X := rfl #align category_theory.limits.pair_obj_left CategoryTheory.Limits.pair_obj_left @[simp] theorem pair_obj_right (X Y : C) : (pair X Y).obj ⟨right⟩ = Y := rfl #align category_theory.limits.pair_obj_right CategoryTheory.Limits.pair_obj_right section variable {F G : Discrete WalkingPair ⥤ C} (f : F.obj ⟨left⟩ ⟶ G.obj ⟨left⟩) (g : F.obj ⟨right⟩ ⟶ G.obj ⟨right⟩) attribute [local aesop safe tactic (rule_sets := [CategoryTheory])] CategoryTheory.Discrete.discreteCases /-- The natural transformation between two functors out of the walking pair, specified by its components. -/ def mapPair : F ⟶ G where app j := Discrete.recOn j fun j => WalkingPair.casesOn j f g naturality := fun ⟨X⟩ ⟨Y⟩ ⟨⟨u⟩⟩ => by aesop_cat #align category_theory.limits.map_pair CategoryTheory.Limits.mapPair @[simp] theorem mapPair_left : (mapPair f g).app ⟨left⟩ = f := rfl #align category_theory.limits.map_pair_left CategoryTheory.Limits.mapPair_left @[simp] theorem mapPair_right : (mapPair f g).app ⟨right⟩ = g := rfl #align category_theory.limits.map_pair_right CategoryTheory.Limits.mapPair_right /-- The natural isomorphism between two functors out of the walking pair, specified by its components. -/ @[simps!] def mapPairIso (f : F.obj ⟨left⟩ ≅ G.obj ⟨left⟩) (g : F.obj ⟨right⟩ ≅ G.obj ⟨right⟩) : F ≅ G := NatIso.ofComponents (fun j => Discrete.recOn j fun j => WalkingPair.casesOn j f g) (fun ⟨⟨u⟩⟩ => by aesop_cat) #align category_theory.limits.map_pair_iso CategoryTheory.Limits.mapPairIso end /-- Every functor out of the walking pair is naturally isomorphic (actually, equal) to a `pair` -/ @[simps!] def diagramIsoPair (F : Discrete WalkingPair ⥤ C) : F ≅ pair (F.obj ⟨WalkingPair.left⟩) (F.obj ⟨WalkingPair.right⟩) := mapPairIso (Iso.refl _) (Iso.refl _) #align category_theory.limits.diagram_iso_pair CategoryTheory.Limits.diagramIsoPair section variable {D : Type u} [Category.{v} D] /-- The natural isomorphism between `pair X Y ⋙ F` and `pair (F.obj X) (F.obj Y)`. -/ def pairComp (X Y : C) (F : C ⥤ D) : pair X Y ⋙ F ≅ pair (F.obj X) (F.obj Y) := diagramIsoPair _ #align category_theory.limits.pair_comp CategoryTheory.Limits.pairComp end /-- A binary fan is just a cone on a diagram indexing a product. -/ abbrev BinaryFan (X Y : C) := Cone (pair X Y) #align category_theory.limits.binary_fan CategoryTheory.Limits.BinaryFan /-- The first projection of a binary fan. -/ abbrev BinaryFan.fst {X Y : C} (s : BinaryFan X Y) := s.π.app ⟨WalkingPair.left⟩ #align category_theory.limits.binary_fan.fst CategoryTheory.Limits.BinaryFan.fst /-- The second projection of a binary fan. -/ abbrev BinaryFan.snd {X Y : C} (s : BinaryFan X Y) := s.π.app ⟨WalkingPair.right⟩ #align category_theory.limits.binary_fan.snd CategoryTheory.Limits.BinaryFan.snd @[simp] theorem BinaryFan.π_app_left {X Y : C} (s : BinaryFan X Y) : s.π.app ⟨WalkingPair.left⟩ = s.fst := rfl #align category_theory.limits.binary_fan.π_app_left CategoryTheory.Limits.BinaryFan.π_app_left @[simp] theorem BinaryFan.π_app_right {X Y : C} (s : BinaryFan X Y) : s.π.app ⟨WalkingPair.right⟩ = s.snd := rfl #align category_theory.limits.binary_fan.π_app_right CategoryTheory.Limits.BinaryFan.π_app_right /-- A convenient way to show that a binary fan is a limit. -/ def BinaryFan.IsLimit.mk {X Y : C} (s : BinaryFan X Y) (lift : ∀ {T : C} (_ : T ⟶ X) (_ : T ⟶ Y), T ⟶ s.pt) (hl₁ : ∀ {T : C} (f : T ⟶ X) (g : T ⟶ Y), lift f g ≫ s.fst = f) (hl₂ : ∀ {T : C} (f : T ⟶ X) (g : T ⟶ Y), lift f g ≫ s.snd = g) (uniq : ∀ {T : C} (f : T ⟶ X) (g : T ⟶ Y) (m : T ⟶ s.pt) (_ : m ≫ s.fst = f) (_ : m ≫ s.snd = g), m = lift f g) : IsLimit s := Limits.IsLimit.mk (fun t => lift (BinaryFan.fst t) (BinaryFan.snd t)) (by rintro t (rfl \| rfl) · exact hl₁ _ _ · exact hl₂ _ _) fun t m h => uniq _ _ _ (h ⟨WalkingPair.left⟩) (h ⟨WalkingPair.right⟩) #align category_theory.limits.binary_fan.is_limit.mk CategoryTheory.Limits.BinaryFan.IsLimit.mk theorem BinaryFan.IsLimit.hom_ext {W X Y : C} {s : BinaryFan X Y} (h : IsLimit s) {f g : W ⟶ s.pt} (h₁ : f ≫ s.fst = g ≫ s.fst) (h₂ : f ≫ s.snd = g ≫ s.snd) : f = g := h.hom_ext fun j => Discrete.recOn j fun j => WalkingPair.casesOn j h₁ h₂ #align category_theory.limits.binary_fan.is_limit.hom_ext CategoryTheory.Limits.BinaryFan.IsLimit.hom_ext /-- A binary cofan is just a cocone on a diagram indexing a coproduct. -/ abbrev BinaryCofan (X Y : C) := Cocone (pair X Y) #align category_theory.limits.binary_cofan CategoryTheory.Limits.BinaryCofan /-- The first inclusion of a binary cofan. -/ abbrev BinaryCofan.inl {X Y : C} (s : BinaryCofan X Y) := s.ι.app ⟨WalkingPair.left⟩ #align category_theory.limits.binary_cofan.inl CategoryTheory.Limits.BinaryCofan.inl /-- The second inclusion of a binary cofan. -/ abbrev BinaryCofan.inr {X Y : C} (s : BinaryCofan X Y) := s.ι.app ⟨WalkingPair.right⟩ #align category_theory.limits.binary_cofan.inr CategoryTheory.Limits.BinaryCofan.inr @[simp] theorem BinaryCofan.ι_app_left {X Y : C} (s : BinaryCofan X Y) : s.ι.app ⟨WalkingPair.left⟩ = s.inl := rfl #align category_theory.limits.binary_cofan.ι_app_left CategoryTheory.Limits.BinaryCofan.ι_app_left @[simp] theorem BinaryCofan.ι_app_right {X Y : C} (s : BinaryCofan X Y) : s.ι.app ⟨WalkingPair.right⟩ = s.inr := rfl #align category_theory.limits.binary_cofan.ι_app_right CategoryTheory.Limits.BinaryCofan.ι_app_right /-- A convenient way to show that a binary cofan is a colimit. -/ def BinaryCofan.IsColimit.mk {X Y : C} (s : BinaryCofan X Y) (desc : ∀ {T : C} (_ : X ⟶ T) (_ : Y ⟶ T), s.pt ⟶ T) (hd₁ : ∀ {T : C} (f : X ⟶ T) (g : Y ⟶ T), s.inl ≫ desc f g = f) (hd₂ : ∀ {T : C} (f : X ⟶ T) (g : Y ⟶ T), s.inr ≫ desc f g = g) (uniq : ∀ {T : C} (f : X ⟶ T) (g : Y ⟶ T) (m : s.pt ⟶ T) (_ : s.inl ≫ m = f) (_ : s.inr ≫ m = g), m = desc f g) : IsColimit s := Limits.IsColimit.mk (fun t => desc (BinaryCofan.inl t) (BinaryCofan.inr t)) (by rintro t (rfl \| rfl) · exact hd₁ _ _ · exact hd₂ _ _) fun t m h => uniq _ _ _ (h ⟨WalkingPair.left⟩) (h ⟨WalkingPair.right⟩) #align category_theory.limits.binary_cofan.is_colimit.mk CategoryTheory.Limits.BinaryCofan.IsColimit.mk theorem BinaryCofan.IsColimit.hom_ext {W X Y : C} {s : BinaryCofan X Y} (h : IsColimit s) {f g : s.pt ⟶ W} (h₁ : s.inl ≫ f = s.inl ≫ g) (h₂ : s.inr ≫ f = s.inr ≫ g) : f = g := h.hom_ext fun j => Discrete.recOn j fun j => WalkingPair.casesOn j h₁ h₂ #align category_theory.limits.binary_cofan.is_colimit.hom_ext CategoryTheory.Limits.BinaryCofan.IsColimit.hom_ext variable {X Y : C} section attribute [local aesop safe tactic (rule_sets := [CategoryTheory])] CategoryTheory.Discrete.discreteCases -- Porting note: would it be okay to use this more generally? attribute [local aesop safe cases (rule_sets := [CategoryTheory])] Eq /-- A binary fan with vertex `P` consists of the two projections `π₁ : P ⟶ X` and `π₂ : P ⟶ Y`. -/ @[simps pt] def BinaryFan.mk {P : C} (π₁ : P ⟶ X) (π₂ : P ⟶ Y) : BinaryFan X Y where pt := P π := { app := fun ⟨j⟩ => by cases j <;> simpa } #align category_theory.limits.binary_fan.mk CategoryTheory.Limits.BinaryFan.mk /-- A binary cofan with vertex `P` consists of the two inclusions `ι₁ : X ⟶ P` and `ι₂ : Y ⟶ P`. -/ @[simps pt] def BinaryCofan.mk {P : C} (ι₁ : X ⟶ P) (ι₂ : Y ⟶ P) : BinaryCofan X Y where pt := P ι := { app := fun ⟨j⟩ => by cases j <;> simpa } #align category_theory.limits.binary_cofan.mk CategoryTheory.Limits.BinaryCofan.mk end @[simp] theorem BinaryFan.mk_fst {P : C} (π₁ : P ⟶ X) (π₂ : P ⟶ Y) : (BinaryFan.mk π₁ π₂).fst = π₁ := rfl #align category_theory.limits.binary_fan.mk_fst CategoryTheory.Limits.BinaryFan.mk_fst @[simp] theorem BinaryFan.mk_snd {P : C} (π₁ : P ⟶ X) (π₂ : P ⟶ Y) : (BinaryFan.mk π₁ π₂).snd = π₂ := rfl #align category_theory.limits.binary_fan.mk_snd CategoryTheory.Limits.BinaryFan.mk_snd @[simp] theorem BinaryCofan.mk_inl {P : C} (ι₁ : X ⟶ P) (ι₂ : Y ⟶ P) : (BinaryCofan.mk ι₁ ι₂).inl = ι₁ := rfl #align category_theory.limits.binary_cofan.mk_inl CategoryTheory.Limits.BinaryCofan.mk_inl @[simp] theorem BinaryCofan.mk_inr {P : C} (ι₁ : X ⟶ P) (ι₂ : Y ⟶ P) : (BinaryCofan.mk ι₁ ι₂).inr = ι₂ := rfl #align category_theory.limits.binary_cofan.mk_inr CategoryTheory.Limits.BinaryCofan.mk_inr /-- Every `BinaryFan` is isomorphic to an application of `BinaryFan.mk`. -/ def isoBinaryFanMk {X Y : C} (c : BinaryFan X Y) : c ≅ BinaryFan.mk c.fst c.snd := Cones.ext (Iso.refl _) fun j => by cases' j with l; cases l; repeat simp #align category_theory.limits.iso_binary_fan_mk CategoryTheory.Limits.isoBinaryFanMk /-- Every `BinaryFan` is isomorphic to an application of `BinaryFan.mk`. -/ def isoBinaryCofanMk {X Y : C} (c : BinaryCofan X Y) : c ≅ BinaryCofan.mk c.inl c.inr := Cocones.ext (Iso.refl _) fun j => by cases' j with l; cases l; repeat simp #align category_theory.limits.iso_binary_cofan_mk CategoryTheory.Limits.isoBinaryCofanMk /-- This is a more convenient formulation to show that a `BinaryFan` constructed using `BinaryFan.mk` is a limit cone. -/ def BinaryFan.isLimitMk {W : C} {fst : W ⟶ X} {snd : W ⟶ Y} (lift : ∀ s : BinaryFan X Y, s.pt ⟶ W) (fac_left : ∀ s : BinaryFan X Y, lift s ≫ fst = s.fst) (fac_right : ∀ s : BinaryFan X Y, lift s ≫ snd = s.snd) (uniq : ∀ (s : BinaryFan X Y) (m : s.pt ⟶ W) (_ : m ≫ fst = s.fst) (_ : m ≫ snd = s.snd), m = lift s) : IsLimit (BinaryFan.mk fst snd) := { lift := lift fac := fun s j => by rcases j with ⟨⟨⟩⟩ exacts [fac_left s, fac_right s] uniq := fun s m w => uniq s m (w ⟨WalkingPair.left⟩) (w ⟨WalkingPair.right⟩) } #align category_theory.limits.binary_fan.is_limit_mk CategoryTheory.Limits.BinaryFan.isLimitMk /-- This is a more convenient formulation to show that a `BinaryCofan` constructed using `BinaryCofan.mk` is a colimit cocone. -/ def BinaryCofan.isColimitMk {W : C} {inl : X ⟶ W} {inr : Y ⟶ W} (desc : ∀ s : BinaryCofan X Y, W ⟶ s.pt) (fac_left : ∀ s : BinaryCofan X Y, inl ≫ desc s = s.inl) (fac_right : ∀ s : BinaryCofan X Y, inr ≫ desc s = s.inr) (uniq : ∀ (s : BinaryCofan X Y) (m : W ⟶ s.pt) (_ : inl ≫ m = s.inl) (_ : inr ≫ m = s.inr), m = desc s) : IsColimit (BinaryCofan.mk inl inr) := { desc := desc fac := fun s j => by rcases j with ⟨⟨⟩⟩ exacts [fac_left s, fac_right s] uniq := fun s m w => uniq s m (w ⟨WalkingPair.left⟩) (w ⟨WalkingPair.right⟩) } #align category_theory.limits.binary_cofan.is_colimit_mk CategoryTheory.Limits.BinaryCofan.isColimitMk /-- If `s` is a limit binary fan over `X` and `Y`, then every pair of morphisms `f : W ⟶ X` and `g : W ⟶ Y` induces a morphism `l : W ⟶ s.pt` satisfying `l ≫ s.fst = f` and `l ≫ s.snd = g`. -/ @[simps] def BinaryFan.IsLimit.lift' {W X Y : C} {s : BinaryFan X Y} (h : IsLimit s) (f : W ⟶ X) (g : W ⟶ Y) : { l : W ⟶ s.pt // l ≫ s.fst = f ∧ l ≫ s.snd = g } := ⟨h.lift <\| BinaryFan.mk f g, h.fac _ _, h.fac _ _⟩ #align category_theory.limits.binary_fan.is_limit.lift' CategoryTheory.Limits.BinaryFan.IsLimit.lift' /-- If `s` is a colimit binary cofan over `X` and `Y`,, then every pair of morphisms `f : X ⟶ W` and `g : Y ⟶ W` induces a morphism `l : s.pt ⟶ W` satisfying `s.inl ≫ l = f` and `s.inr ≫ l = g`. -/ @[simps] def BinaryCofan.IsColimit.desc' {W X Y : C} {s : BinaryCofan X Y} (h : IsColimit s) (f : X ⟶ W) (g : Y ⟶ W) : { l : s.pt ⟶ W // s.inl ≫ l = f ∧ s.inr ≫ l = g } := ⟨h.desc <\| BinaryCofan.mk f g, h.fac _ _, h.fac _ _⟩ #align category_theory.limits.binary_cofan.is_colimit.desc' CategoryTheory.Limits.BinaryCofan.IsColimit.desc' /-- Binary products are symmetric. -/ def BinaryFan.isLimitFlip {X Y : C} {c : BinaryFan X Y} (hc : IsLimit c) : IsLimit (BinaryFan.mk c.snd c.fst) := BinaryFan.isLimitMk (fun s => hc.lift (BinaryFan.mk s.snd s.fst)) (fun _ => hc.fac _ _) (fun _ => hc.fac _ _) fun s _ e₁ e₂ => BinaryFan.IsLimit.hom_ext hc (e₂.trans (hc.fac (BinaryFan.mk s.snd s.fst) ⟨WalkingPair.left⟩).symm) (e₁.trans (hc.fac (BinaryFan.mk s.snd s.fst) ⟨WalkingPair.right⟩).symm) #align category_theory.limits.binary_fan.is_limit_flip CategoryTheory.Limits.BinaryFan.isLimitFlip theorem BinaryFan.isLimit_iff_isIso_fst {X Y : C} (h : IsTerminal Y) (c : BinaryFan X Y) : Nonempty (IsLimit c) ↔ IsIso c.fst := by constructor · rintro ⟨H⟩ obtain ⟨l, hl, -⟩ := BinaryFan.IsLimit.lift' H (𝟙 X) (h.from X) exact ⟨⟨l, BinaryFan.IsLimit.hom_ext H (by simpa [hl, -Category.comp_id] using Category.comp_id _) (h.hom_ext _ _), hl⟩⟩ · intro exact ⟨BinaryFan.IsLimit.mk _ (fun f _ => f ≫ inv c.fst) (fun _ _ => by simp) (fun _ _ => h.hom_ext _ _) fun _ _ _ e _ => by simp [← e]⟩ #align category_theory.limits.binary_fan.is_limit_iff_is_iso_fst CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_fst theorem BinaryFan.isLimit_iff_isIso_snd {X Y : C} (h : IsTerminal X) (c : BinaryFan X Y) : Nonempty (IsLimit c) ↔ IsIso c.snd := by refine Iff.trans ?_ (BinaryFan.isLimit_iff_isIso_fst h (BinaryFan.mk c.snd c.fst)) exact ⟨fun h => ⟨BinaryFan.isLimitFlip h.some⟩, fun h => ⟨(BinaryFan.isLimitFlip h.some).ofIsoLimit (isoBinaryFanMk c).symm⟩⟩ #align category_theory.limits.binary_fan.is_limit_iff_is_iso_snd CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_snd /-- If `X' ≅ X`, then `X × Y` also is the product of `X'` and `Y`. -/ noncomputable def BinaryFan.isLimitCompLeftIso {X Y X' : C} (c : BinaryFan X Y) (f : X ⟶ X') [IsIso f] (h : IsLimit c) : IsLimit (BinaryFan.mk (c.fst ≫ f) c.snd) := by fapply BinaryFan.isLimitMk · exact fun s => h.lift (BinaryFan.mk (s.fst ≫ inv f) s.snd) · intro s -- Porting note: simp timed out here simp only [Category.comp_id,BinaryFan.π_app_left,IsIso.inv_hom_id, BinaryFan.mk_fst,IsLimit.fac_assoc,eq_self_iff_true,Category.assoc] · intro s -- Porting note: simp timed out here simp only [BinaryFan.π_app_right,BinaryFan.mk_snd,eq_self_iff_true,IsLimit.fac] · intro s m e₁ e₂ -- Porting note: simpa timed out here also apply BinaryFan.IsLimit.hom_ext h · simpa only [BinaryFan.π_app_left,BinaryFan.mk_fst,Category.assoc,IsLimit.fac,IsIso.eq_comp_inv] · simpa only [BinaryFan.π_app_right,BinaryFan.mk_snd,IsLimit.fac] #align category_theory.limits.binary_fan.is_limit_comp_left_iso CategoryTheory.Limits.BinaryFan.isLimitCompLeftIso /-- If `Y' ≅ Y`, then `X x Y` also is the product of `X` and `Y'`. -/ noncomputable def BinaryFan.isLimitCompRightIso {X Y Y' : C} (c : BinaryFan X Y) (f : Y ⟶ Y') [IsIso f] (h : IsLimit c) : IsLimit (BinaryFan.mk c.fst (c.snd ≫ f)) := BinaryFan.isLimitFlip <\| BinaryFan.isLimitCompLeftIso _ f (BinaryFan.isLimitFlip h) #align category_theory.limits.binary_fan.is_limit_comp_right_iso CategoryTheory.Limits.BinaryFan.isLimitCompRightIso /-- Binary coproducts are symmetric. -/ def BinaryCofan.isColimitFlip {X Y : C} {c : BinaryCofan X Y} (hc : IsColimit c) : IsColimit (BinaryCofan.mk c.inr c.inl) := BinaryCofan.isColimitMk (fun s => hc.desc (BinaryCofan.mk s.inr s.inl)) (fun _ => hc.fac _ _) (fun _ => hc.fac _ _) fun s _ e₁ e₂ => BinaryCofan.IsColimit.hom_ext hc (e₂.trans (hc.fac (BinaryCofan.mk s.inr s.inl) ⟨WalkingPair.left⟩).symm) (e₁.trans (hc.fac (BinaryCofan.mk s.inr s.inl) ⟨WalkingPair.right⟩).symm) #align category_theory.limits.binary_cofan.is_colimit_flip CategoryTheory.Limits.BinaryCofan.isColimitFlip theorem BinaryCofan.isColimit_iff_isIso_inl {X Y : C} (h : IsInitial Y) (c : BinaryCofan X Y) : Nonempty (IsColimit c) ↔ IsIso c.inl := by constructor · rintro ⟨H⟩ obtain ⟨l, hl, -⟩ := BinaryCofan.IsColimit.desc' H (𝟙 X) (h.to X) refine ⟨⟨l, hl, BinaryCofan.IsColimit.hom_ext H (?_) (h.hom_ext _ _)⟩⟩ rw [Category.comp_id] have e : (inl c ≫ l) ≫ inl c = 𝟙 X ≫ inl c := congrArg (·≫inl c) hl rwa [Category.assoc,Category.id_comp] at e · intro exact ⟨BinaryCofan.IsColimit.mk _ (fun f _ => inv c.inl ≫ f) (fun _ _ => IsIso.hom_inv_id_assoc _ _) (fun _ _ => h.hom_ext _ _) fun _ _ _ e _ => (IsIso.eq_inv_comp _).mpr e⟩ #align category_theory.limits.binary_cofan.is_colimit_iff_is_iso_inl CategoryTheory.Limits.BinaryCofan.isColimit_iff_isIso_inl theorem BinaryCofan.isColimit_iff_isIso_inr {X Y : C} (h : IsInitial X) (c : BinaryCofan X Y) : Nonempty (IsColimit c) ↔ IsIso c.inr := by refine Iff.trans ?_ (BinaryCofan.isColimit_iff_isIso_inl h (BinaryCofan.mk c.inr c.inl)) exact ⟨fun h => ⟨BinaryCofan.isColimitFlip h.some⟩, fun h => ⟨(BinaryCofan.isColimitFlip h.some).ofIsoColimit (isoBinaryCofanMk c).symm⟩⟩ #align category_theory.limits.binary_cofan.is_colimit_iff_is_iso_inr CategoryTheory.Limits.BinaryCofan.isColimit_iff_isIso_inr /-- If `X' ≅ X`, then `X ⨿ Y` also is the coproduct of `X'` and `Y`. -/ noncomputable def BinaryCofan.isColimitCompLeftIso {X Y X' : C} (c : BinaryCofan X Y) (f : X' ⟶ X) [IsIso f] (h : IsColimit c) : IsColimit (BinaryCofan.mk (f ≫ c.inl) c.inr) := by fapply BinaryCofan.isColimitMk · exact fun s => h.desc (BinaryCofan.mk (inv f ≫ s.inl) s.inr) · intro s -- Porting note: simp timed out here too simp only [IsColimit.fac,BinaryCofan.ι_app_left,eq_self_iff_true, Category.assoc,BinaryCofan.mk_inl,IsIso.hom_inv_id_assoc] · intro s -- Porting note: simp timed out here too simp only [IsColimit.fac,BinaryCofan.ι_app_right,eq_self_iff_true,BinaryCofan.mk_inr] · intro s m e₁ e₂ apply BinaryCofan.IsColimit.hom_ext h · rw [← cancel_epi f] -- Porting note: simp timed out here too simpa only [IsColimit.fac,BinaryCofan.ι_app_left,eq_self_iff_true, Category.assoc,BinaryCofan.mk_inl,IsIso.hom_inv_id_assoc] using e₁ -- Porting note: simp timed out here too · simpa only [IsColimit.fac,BinaryCofan.ι_app_right,eq_self_iff_true,BinaryCofan.mk_inr] #align category_theory.limits.binary_cofan.is_colimit_comp_left_iso CategoryTheory.Limits.BinaryCofan.isColimitCompLeftIso /-- If `Y' ≅ Y`, then `X ⨿ Y` also is the coproduct of `X` and `Y'`. -/ noncomputable def BinaryCofan.isColimitCompRightIso {X Y Y' : C} (c : BinaryCofan X Y) (f : Y' ⟶ Y) [IsIso f] (h : IsColimit c) : IsColimit (BinaryCofan.mk c.inl (f ≫ c.inr)) := BinaryCofan.isColimitFlip <\| BinaryCofan.isColimitCompLeftIso _ f (BinaryCofan.isColimitFlip h) #align category_theory.limits.binary_cofan.is_colimit_comp_right_iso CategoryTheory.Limits.BinaryCofan.isColimitCompRightIso /-- An abbreviation for `HasLimit (pair X Y)`. -/ abbrev HasBinaryProduct (X Y : C) := HasLimit (pair X Y) #align category_theory.limits.has_binary_product CategoryTheory.Limits.HasBinaryProduct /-- An abbreviation for `HasColimit (pair X Y)`. -/ abbrev HasBinaryCoproduct (X Y : C) := HasColimit (pair X Y) #align category_theory.limits.has_binary_coproduct CategoryTheory.Limits.HasBinaryCoproduct /-- If we have a product of `X` and `Y`, we can access it using `prod X Y` or `X ⨯ Y`. -/ abbrev prod (X Y : C) [HasBinaryProduct X Y] := limit (pair X Y) #align category_theory.limits.prod CategoryTheory.Limits.prod /-- If we have a coproduct of `X` and `Y`, we can access it using `coprod X Y` or `X ⨿ Y`. -/ abbrev coprod (X Y : C) [HasBinaryCoproduct X Y] := colimit (pair X Y) #align category_theory.limits.coprod CategoryTheory.Limits.coprod /-- Notation for the product -/ notation:20 X " ⨯ " Y:20 => prod X Y /-- Notation for the coproduct -/ notation:20 X " ⨿ " Y:20 => coprod X Y /-- The projection map to the first component of the product. -/ abbrev prod.fst {X Y : C} [HasBinaryProduct X Y] : X ⨯ Y ⟶ X := limit.π (pair X Y) ⟨WalkingPair.left⟩ #align category_theory.limits.prod.fst CategoryTheory.Limits.prod.fst /-- The projection map to the second component of the product. -/ abbrev prod.snd {X Y : C} [HasBinaryProduct X Y] : X ⨯ Y ⟶ Y := limit.π (pair X Y) ⟨WalkingPair.right⟩ #align category_theory.limits.prod.snd CategoryTheory.Limits.prod.snd /-- The inclusion map from the first component of the coproduct. -/ abbrev coprod.inl {X Y : C} [HasBinaryCoproduct X Y] : X ⟶ X ⨿ Y := colimit.ι (pair X Y) ⟨WalkingPair.left⟩ #align category_theory.limits.coprod.inl CategoryTheory.Limits.coprod.inl /-- The inclusion map from the second component of the coproduct. -/ abbrev coprod.inr {X Y : C} [HasBinaryCoproduct X Y] : Y ⟶ X ⨿ Y := colimit.ι (pair X Y) ⟨WalkingPair.right⟩ #align category_theory.limits.coprod.inr CategoryTheory.Limits.coprod.inr /-- The binary fan constructed from the projection maps is a limit. -/ def prodIsProd (X Y : C) [HasBinaryProduct X Y] : IsLimit (BinaryFan.mk (prod.fst : X ⨯ Y ⟶ X) prod.snd) := (limit.isLimit _).ofIsoLimit (Cones.ext (Iso.refl _) (fun ⟨u⟩ => by cases u · dsimp; simp only [Category.id_comp]; rfl · dsimp; simp only [Category.id_comp]; rfl )) #align category_theory.limits.prod_is_prod CategoryTheory.Limits.prodIsProd /-- The binary cofan constructed from the coprojection maps is a colimit. -/ def coprodIsCoprod (X Y : C) [HasBinaryCoproduct X Y] : IsColimit (BinaryCofan.mk (coprod.inl : X ⟶ X ⨿ Y) coprod.inr) := (colimit.isColimit _).ofIsoColimit (Cocones.ext (Iso.refl _) (fun ⟨u⟩ => by cases u · dsimp; simp only [Category.comp_id] · dsimp; simp only [Category.comp_id] )) #align category_theory.limits.coprod_is_coprod CategoryTheory.Limits.coprodIsCoprod @[ext 1100] theorem prod.hom_ext {W X Y : C} [HasBinaryProduct X Y] {f g : W ⟶ X ⨯ Y} (h₁ : f ≫ prod.fst = g ≫ prod.fst) (h₂ : f ≫ prod.snd = g ≫ prod.snd) : f = g := BinaryFan.IsLimit.hom_ext (limit.isLimit _) h₁ h₂ #align category_theory.limits.prod.hom_ext CategoryTheory.Limits.prod.hom_ext @[ext 1100] theorem coprod.hom_ext {W X Y : C} [HasBinaryCoproduct X Y] {f g : X ⨿ Y ⟶ W} (h₁ : coprod.inl ≫ f = coprod.inl ≫ g) (h₂ : coprod.inr ≫ f = coprod.inr ≫ g) : f = g := BinaryCofan.IsColimit.hom_ext (colimit.isColimit _) h₁ h₂ #align category_theory.limits.coprod.hom_ext CategoryTheory.Limits.coprod.hom_ext /-- If the product of `X` and `Y` exists, then every pair of morphisms `f : W ⟶ X` and `g : W ⟶ Y` induces a morphism `prod.lift f g : W ⟶ X ⨯ Y`. -/ abbrev prod.lift {W X Y : C} [HasBinaryProduct X Y] (f : W ⟶ X) (g : W ⟶ Y) : W ⟶ X ⨯ Y := limit.lift _ (BinaryFan.mk f g) #align category_theory.limits.prod.lift CategoryTheory.Limits.prod.lift /-- diagonal arrow of the binary product in the category `fam I` -/ abbrev diag (X : C) [HasBinaryProduct X X] : X ⟶ X ⨯ X := prod.lift (𝟙 _) (𝟙 _) #align category_theory.limits.diag CategoryTheory.Limits.diag /-- If the coproduct of `X` and `Y` exists, then every pair of morphisms `f : X ⟶ W` and `g : Y ⟶ W` induces a morphism `coprod.desc f g : X ⨿ Y ⟶ W`. -/ abbrev coprod.desc {W X Y : C} [HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : X ⨿ Y ⟶ W := colimit.desc _ (BinaryCofan.mk f g) #align category_theory.limits.coprod.desc CategoryTheory.Limits.coprod.desc /-- codiagonal arrow of the binary coproduct -/ abbrev codiag (X : C) [HasBinaryCoproduct X X] : X ⨿ X ⟶ X := coprod.desc (𝟙 _) (𝟙 _) #align category_theory.limits.codiag CategoryTheory.Limits.codiag -- Porting note (#10618): simp removes as simp can prove this @[reassoc] theorem prod.lift_fst {W X Y : C} [HasBinaryProduct X Y] (f : W ⟶ X) (g : W ⟶ Y) : prod.lift f g ≫ prod.fst = f := limit.lift_π _ _ #align category_theory.limits.prod.lift_fst CategoryTheory.Limits.prod.lift_fst #align category_theory.limits.prod.lift_fst_assoc CategoryTheory.Limits.prod.lift_fst_assoc -- Porting note (#10618): simp removes as simp can prove this @[reassoc] theorem prod.lift_snd {W X Y : C} [HasBinaryProduct X Y] (f : W ⟶ X) (g : W ⟶ Y) : prod.lift f g ≫ prod.snd = g := limit.lift_π _ _ #align category_theory.limits.prod.lift_snd CategoryTheory.Limits.prod.lift_snd #align category_theory.limits.prod.lift_snd_assoc CategoryTheory.Limits.prod.lift_snd_assoc -- The simp linter says simp can prove the reassoc version of this lemma. -- Porting note: it can also prove the og version @[reassoc] theorem coprod.inl_desc {W X Y : C} [HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : coprod.inl ≫ coprod.desc f g = f := colimit.ι_desc _ _ #align category_theory.limits.coprod.inl_desc CategoryTheory.Limits.coprod.inl_desc #align category_theory.limits.coprod.inl_desc_assoc CategoryTheory.Limits.coprod.inl_desc_assoc -- The simp linter says simp can prove the reassoc version of this lemma. -- Porting note: it can also prove the og version @[reassoc] theorem coprod.inr_desc {W X Y : C} [HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : coprod.inr ≫ coprod.desc f g = g := colimit.ι_desc _ _ #align category_theory.limits.coprod.inr_desc CategoryTheory.Limits.coprod.inr_desc #align category_theory.limits.coprod.inr_desc_assoc CategoryTheory.Limits.coprod.inr_desc_assoc instance prod.mono_lift_of_mono_left {W X Y : C} [HasBinaryProduct X Y] (f : W ⟶ X) (g : W ⟶ Y) [Mono f] : Mono (prod.lift f g) := mono_of_mono_fac <\| prod.lift_fst _ _ #align category_theory.limits.prod.mono_lift_of_mono_left CategoryTheory.Limits.prod.mono_lift_of_mono_left instance prod.mono_lift_of_mono_right {W X Y : C} [HasBinaryProduct X Y] (f : W ⟶ X) (g : W ⟶ Y) [Mono g] : Mono (prod.lift f g) := mono_of_mono_fac <\| prod.lift_snd _ _ #align category_theory.limits.prod.mono_lift_of_mono_right CategoryTheory.Limits.prod.mono_lift_of_mono_right instance coprod.epi_desc_of_epi_left {W X Y : C} [HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) [Epi f] : Epi (coprod.desc f g) := epi_of_epi_fac <\| coprod.inl_desc _ _ #align category_theory.limits.coprod.epi_desc_of_epi_left CategoryTheory.Limits.coprod.epi_desc_of_epi_left instance coprod.epi_desc_of_epi_right {W X Y : C} [HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) [Epi g] : Epi (coprod.desc f g) := epi_of_epi_fac <\| coprod.inr_desc _ _ #align category_theory.limits.coprod.epi_desc_of_epi_right CategoryTheory.Limits.coprod.epi_desc_of_epi_right /-- If the product of `X` and `Y` exists, then every pair of morphisms `f : W ⟶ X` and `g : W ⟶ Y` induces a morphism `l : W ⟶ X ⨯ Y` satisfying `l ≫ Prod.fst = f` and `l ≫ Prod.snd = g`. -/ def prod.lift' {W X Y : C} [HasBinaryProduct X Y] (f : W ⟶ X) (g : W ⟶ Y) : { l : W ⟶ X ⨯ Y // l ≫ prod.fst = f ∧ l ≫ prod.snd = g } := ⟨prod.lift f g, prod.lift_fst _ _, prod.lift_snd _ _⟩ #align category_theory.limits.prod.lift' CategoryTheory.Limits.prod.lift' /-- If the coproduct of `X` and `Y` exists, then every pair of morphisms `f : X ⟶ W` and `g : Y ⟶ W` induces a morphism `l : X ⨿ Y ⟶ W` satisfying `coprod.inl ≫ l = f` and `coprod.inr ≫ l = g`. -/ def coprod.desc' {W X Y : C} [HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : { l : X ⨿ Y ⟶ W // coprod.inl ≫ l = f ∧ coprod.inr ≫ l = g } := ⟨coprod.desc f g, coprod.inl_desc _ _, coprod.inr_desc _ _⟩ #align category_theory.limits.coprod.desc' CategoryTheory.Limits.coprod.desc' /-- If the products `W ⨯ X` and `Y ⨯ Z` exist, then every pair of morphisms `f : W ⟶ Y` and `g : X ⟶ Z` induces a morphism `prod.map f g : W ⨯ X ⟶ Y ⨯ Z`. -/ def prod.map {W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) : W ⨯ X ⟶ Y ⨯ Z := limMap (mapPair f g) #align category_theory.limits.prod.map CategoryTheory.Limits.prod.map /-- If the coproducts `W ⨿ X` and `Y ⨿ Z` exist, then every pair of morphisms `f : W ⟶ Y` and `g : W ⟶ Z` induces a morphism `coprod.map f g : W ⨿ X ⟶ Y ⨿ Z`. -/ def coprod.map {W X Y Z : C} [HasBinaryCoproduct W X] [HasBinaryCoproduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) : W ⨿ X ⟶ Y ⨿ Z := colimMap (mapPair f g) #align category_theory.limits.coprod.map CategoryTheory.Limits.coprod.map section ProdLemmas -- Making the reassoc version of this a simp lemma seems to be more harmful than helpful. @[reassoc, simp] theorem prod.comp_lift {V W X Y : C} [HasBinaryProduct X Y] (f : V ⟶ W) (g : W ⟶ X) (h : W ⟶ Y) : f ≫ prod.lift g h = prod.lift (f ≫ g) (f ≫ h) := by ext <;> simp #align category_theory.limits.prod.comp_lift CategoryTheory.Limits.prod.comp_lift #align category_theory.limits.prod.comp_lift_assoc CategoryTheory.Limits.prod.comp_lift_assoc theorem prod.comp_diag {X Y : C} [HasBinaryProduct Y Y] (f : X ⟶ Y) : f ≫ diag Y = prod.lift f f := by simp #align category_theory.limits.prod.comp_diag CategoryTheory.Limits.prod.comp_diag @[reassoc (attr := simp)] theorem prod.map_fst {W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) : prod.map f g ≫ prod.fst = prod.fst ≫ f := limMap_π _ _ #align category_theory.limits.prod.map_fst CategoryTheory.Limits.prod.map_fst #align category_theory.limits.prod.map_fst_assoc CategoryTheory.Limits.prod.map_fst_assoc @[reassoc (attr := simp)] theorem prod.map_snd {W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) : prod.map f g ≫ prod.snd = prod.snd ≫ g := limMap_π _ _ #align category_theory.limits.prod.map_snd CategoryTheory.Limits.prod.map_snd #align category_theory.limits.prod.map_snd_assoc CategoryTheory.Limits.prod.map_snd_assoc @[simp] theorem prod.map_id_id {X Y : C} [HasBinaryProduct X Y] : prod.map (𝟙 X) (𝟙 Y) = 𝟙 _ := by ext <;> simp #align category_theory.limits.prod.map_id_id CategoryTheory.Limits.prod.map_id_id @[simp] theorem prod.lift_fst_snd {X Y : C} [HasBinaryProduct X Y] : prod.lift prod.fst prod.snd = 𝟙 (X ⨯ Y) := by ext <;> simp #align category_theory.limits.prod.lift_fst_snd CategoryTheory.Limits.prod.lift_fst_snd @[reassoc (attr := simp)] theorem prod.lift_map {V W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z] (f : V ⟶ W) (g : V ⟶ X) (h : W ⟶ Y) (k : X ⟶ Z) : prod.lift f g ≫ prod.map h k = prod.lift (f ≫ h) (g ≫ k) := by ext <;> simp #align category_theory.limits.prod.lift_map CategoryTheory.Limits.prod.lift_map #align category_theory.limits.prod.lift_map_assoc CategoryTheory.Limits.prod.lift_map_assoc @[simp] theorem prod.lift_fst_comp_snd_comp {W X Y Z : C} [HasBinaryProduct W Y] [HasBinaryProduct X Z] (g : W ⟶ X) (g' : Y ⟶ Z) : prod.lift (prod.fst ≫ g) (prod.snd ≫ g') = prod.map g g' := by rw [← prod.lift_map] simp #align category_theory.limits.prod.lift_fst_comp_snd_comp CategoryTheory.Limits.prod.lift_fst_comp_snd_comp -- We take the right hand side here to be simp normal form, as this way composition lemmas for -- `f ≫ h` and `g ≫ k` can fire (eg `id_comp`) , while `map_fst` and `map_snd` can still work just -- as well. @[reassoc (attr := simp)] theorem prod.map_map {A₁ A₂ A₃ B₁ B₂ B₃ : C} [HasBinaryProduct A₁ B₁] [HasBinaryProduct A₂ B₂] [HasBinaryProduct A₃ B₃] (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) (h : A₂ ⟶ A₃) (k : B₂ ⟶ B₃) : prod.map f g ≫ prod.map h k = prod.map (f ≫ h) (g ≫ k) := by ext <;> simp #align category_theory.limits.prod.map_map CategoryTheory.Limits.prod.map_map #align category_theory.limits.prod.map_map_assoc CategoryTheory.Limits.prod.map_map_assoc -- TODO: is it necessary to weaken the assumption here? @[reassoc] theorem prod.map_swap {A B X Y : C} (f : A ⟶ B) (g : X ⟶ Y) [HasLimitsOfShape (Discrete WalkingPair) C] : prod.map (𝟙 X) f ≫ prod.map g (𝟙 B) = prod.map g (𝟙 A) ≫ prod.map (𝟙 Y) f := by simp #align category_theory.limits.prod.map_swap CategoryTheory.Limits.prod.map_swap #align category_theory.limits.prod.map_swap_assoc CategoryTheory.Limits.prod.map_swap_assoc @[reassoc] theorem prod.map_comp_id {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasBinaryProduct X W] [HasBinaryProduct Z W] [HasBinaryProduct Y W] : prod.map (f ≫ g) (𝟙 W) = prod.map f (𝟙 W) ≫ prod.map g (𝟙 W) := by simp #align category_theory.limits.prod.map_comp_id CategoryTheory.Limits.prod.map_comp_id #align category_theory.limits.prod.map_comp_id_assoc CategoryTheory.Limits.prod.map_comp_id_assoc @[reassoc] theorem prod.map_id_comp {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasBinaryProduct W X] [HasBinaryProduct W Y] [HasBinaryProduct W Z] : prod.map (𝟙 W) (f ≫ g) = prod.map (𝟙 W) f ≫ prod.map (𝟙 W) g := by simp #align category_theory.limits.prod.map_id_comp CategoryTheory.Limits.prod.map_id_comp #align category_theory.limits.prod.map_id_comp_assoc CategoryTheory.Limits.prod.map_id_comp_assoc /-- If the products `W ⨯ X` and `Y ⨯ Z` exist, then every pair of isomorphisms `f : W ≅ Y` and `g : X ≅ Z` induces an isomorphism `prod.mapIso f g : W ⨯ X ≅ Y ⨯ Z`. -/ @[simps] def prod.mapIso {W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z] (f : W ≅ Y) (g : X ≅ Z) : W ⨯ X ≅ Y ⨯ Z where hom := prod.map f.hom g.hom inv := prod.map f.inv g.inv #align category_theory.limits.prod.map_iso CategoryTheory.Limits.prod.mapIso instance isIso_prod {W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) [IsIso f] [IsIso g] : IsIso (prod.map f g) := (prod.mapIso (asIso f) (asIso g)).isIso_hom #align category_theory.limits.is_iso_prod CategoryTheory.Limits.isIso_prod instance prod.map_mono {C : Type*} [Category C] {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [Mono f] [Mono g] [HasBinaryProduct W X] [HasBinaryProduct Y Z] : Mono (prod.map f g) := ⟨fun i₁ i₂ h => by ext · rw [← cancel_mono f] simpa using congr_arg (fun f => f ≫ prod.fst) h · rw [← cancel_mono g] simpa using congr_arg (fun f => f ≫ prod.snd) h⟩ #align category_theory.limits.prod.map_mono CategoryTheory.Limits.prod.map_mono @[reassoc] -- Porting note (#10618): simp can prove these	Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean	810	811	theorem prod.diag_map {X Y : C} (f : X ⟶ Y) [HasBinaryProduct X X] [HasBinaryProduct Y Y] : diag X ≫ prod.map f f = f ≫ diag Y := by	simp
/- Copyright (c) 2023 Heather Macbeth. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Heather Macbeth -/ import Mathlib.Geometry.Manifold.Sheaf.Smooth import Mathlib.Geometry.RingedSpace.LocallyRingedSpace /-! # Smooth manifolds as locally ringed spaces This file equips a smooth manifold-with-corners with the structure of a locally ringed space. ## Main results * `smoothSheafCommRing.isUnit_stalk_iff`: The units of the stalk at `x` of the sheaf of smooth functions from a smooth manifold `M` to its scalar field `𝕜`, considered as a sheaf of commutative rings, are the functions whose values at `x` are nonzero. ## Main definitions * `SmoothManifoldWithCorners.locallyRingedSpace`: A smooth manifold-with-corners can be considered as a locally ringed space. ## TODO Characterize morphisms-of-locally-ringed-spaces (`AlgebraicGeometry.LocallyRingedSpace.Hom`) between smooth manifolds. -/ noncomputable section universe u variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {EM : Type} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] open AlgebraicGeometry Manifold TopologicalSpace Topology /-- The units of the stalk at `x` of the sheaf of smooth functions from `M` to `𝕜`, considered as a sheaf of commutative rings, are the functions whose values at `x` are nonzero. -/	Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean	43	98	theorem smoothSheafCommRing.isUnit_stalk_iff {x : M} (f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) : IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) := by	constructor · rintro ⟨⟨f, g, hf, hg⟩, rfl⟩ (h' : smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x f = 0) simpa [h'] using congr_arg (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) hf · let S := (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf -- Suppose that `f`, in the stalk at `x`, is nonzero at `x` rintro (hf : _ ≠ 0) -- Represent `f` as the germ of some function (also called `f`) on an open neighbourhood `U` of -- `x`, which is nonzero at `x` obtain ⟨U : Opens M, hxU, f : C^∞⟮IM, U; 𝓘(𝕜), 𝕜⟯, rfl⟩ := S.germ_exist x f have hf' : f ⟨x, hxU⟩ ≠ 0 := by convert hf exact (smoothSheafCommRing.eval_germ U ⟨x, hxU⟩ f).symm -- In fact, by continuity, `f` is nonzero on a neighbourhood `V` of `x` have H : ∀ᶠ (z : U) in 𝓝 ⟨x, hxU⟩, f z ≠ 0 := f.2.continuous.continuousAt.eventually_ne hf' rw [eventually_nhds_iff] at H obtain ⟨V₀, hV₀f, hV₀, hxV₀⟩ := H let V : Opens M := ⟨Subtype.val '' V₀, U.2.isOpenMap_subtype_val V₀ hV₀⟩ have hUV : V ≤ U := Subtype.coe_image_subset (U : Set M) V₀ have hV : V₀ = Set.range (Set.inclusion hUV) := by convert (Set.range_inclusion hUV).symm ext y show _ ↔ y ∈ Subtype.val ⁻¹' (Subtype.val '' V₀) rw [Set.preimage_image_eq _ Subtype.coe_injective] clear_value V subst hV have hxV : x ∈ (V : Set M) := by obtain ⟨x₀, hxx₀⟩ := hxV₀ convert x₀.2 exact congr_arg Subtype.val hxx₀.symm have hVf : ∀ y : V, f (Set.inclusion hUV y) ≠ 0 := fun y ↦ hV₀f (Set.inclusion hUV y) (Set.mem_range_self y) -- Let `g` be the pointwise inverse of `f` on `V`, which is smooth since `f` is nonzero there let g : C^∞⟮IM, V; 𝓘(𝕜), 𝕜⟯ := ⟨(f ∘ Set.inclusion hUV)⁻¹, ?_⟩ -- The germ of `g` is inverse to the germ of `f`, so `f` is a unit · refine ⟨⟨S.germ ⟨x, hxV⟩ (SmoothMap.restrictRingHom IM 𝓘(𝕜) 𝕜 hUV f), S.germ ⟨x, hxV⟩ g, ?_, ?_⟩, S.germ_res_apply hUV.hom ⟨x, hxV⟩ f⟩ · rw [← map_mul] -- Qualified the name to avoid Lean not finding a `OneHomClass` #8386 convert RingHom.map_one _ apply Subtype.ext ext y apply mul_inv_cancel exact hVf y · rw [← map_mul] -- Qualified the name to avoid Lean not finding a `OneHomClass` #8386 convert RingHom.map_one _ apply Subtype.ext ext y apply inv_mul_cancel exact hVf y · intro y exact ((contDiffAt_inv _ (hVf y)).contMDiffAt).comp y (f.smooth.comp (smooth_inclusion hUV)).smoothAt
/- Copyright (c) 2020 Patrick Massot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Massot, Scott Morrison -/ import Mathlib.Algebra.Order.Interval.Set.Instances import Mathlib.Order.Interval.Set.ProjIcc import Mathlib.Topology.Instances.Real #align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # The unit interval, as a topological space Use `open unitInterval` to turn on the notation `I := Set.Icc (0 : ℝ) (1 : ℝ)`. We provide basic instances, as well as a custom tactic for discharging `0 ≤ ↑x`, `0 ≤ 1 - ↑x`, `↑x ≤ 1`, and `1 - ↑x ≤ 1` when `x : I`. -/ noncomputable section open scoped Classical open Topology Filter open Set Int Set.Icc /-! ### The unit interval -/ /-- The unit interval `[0,1]` in ℝ. -/ abbrev unitInterval : Set ℝ := Set.Icc 0 1 #align unit_interval unitInterval @[inherit_doc] scoped[unitInterval] notation "I" => unitInterval namespace unitInterval theorem zero_mem : (0 : ℝ) ∈ I := ⟨le_rfl, zero_le_one⟩ #align unit_interval.zero_mem unitInterval.zero_mem theorem one_mem : (1 : ℝ) ∈ I := ⟨zero_le_one, le_rfl⟩ #align unit_interval.one_mem unitInterval.one_mem theorem mul_mem {x y : ℝ} (hx : x ∈ I) (hy : y ∈ I) : x * y ∈ I := ⟨mul_nonneg hx.1 hy.1, mul_le_one hx.2 hy.1 hy.2⟩ #align unit_interval.mul_mem unitInterval.mul_mem theorem div_mem {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y) : x / y ∈ I := ⟨div_nonneg hx hy, div_le_one_of_le hxy hy⟩ #align unit_interval.div_mem unitInterval.div_mem theorem fract_mem (x : ℝ) : fract x ∈ I := ⟨fract_nonneg _, (fract_lt_one _).le⟩ #align unit_interval.fract_mem unitInterval.fract_mem theorem mem_iff_one_sub_mem {t : ℝ} : t ∈ I ↔ 1 - t ∈ I := by rw [mem_Icc, mem_Icc] constructor <;> intro <;> constructor <;> linarith #align unit_interval.mem_iff_one_sub_mem unitInterval.mem_iff_one_sub_mem instance hasZero : Zero I := ⟨⟨0, zero_mem⟩⟩ #align unit_interval.has_zero unitInterval.hasZero instance hasOne : One I := ⟨⟨1, by constructor <;> norm_num⟩⟩ #align unit_interval.has_one unitInterval.hasOne instance : ZeroLEOneClass I := ⟨zero_le_one (α := ℝ)⟩ instance : BoundedOrder I := Set.Icc.boundedOrder zero_le_one lemma univ_eq_Icc : (univ : Set I) = Icc (0 : I) (1 : I) := Icc_bot_top.symm theorem coe_ne_zero {x : I} : (x : ℝ) ≠ 0 ↔ x ≠ 0 := not_iff_not.mpr coe_eq_zero #align unit_interval.coe_ne_zero unitInterval.coe_ne_zero theorem coe_ne_one {x : I} : (x : ℝ) ≠ 1 ↔ x ≠ 1 := not_iff_not.mpr coe_eq_one #align unit_interval.coe_ne_one unitInterval.coe_ne_one instance : Nonempty I := ⟨0⟩ instance : Mul I := ⟨fun x y => ⟨x * y, mul_mem x.2 y.2⟩⟩ -- todo: we could set up a `LinearOrderedCommMonoidWithZero I` instance theorem mul_le_left {x y : I} : x * y ≤ x := Subtype.coe_le_coe.mp <\| mul_le_of_le_one_right x.2.1 y.2.2 #align unit_interval.mul_le_left unitInterval.mul_le_left theorem mul_le_right {x y : I} : x * y ≤ y := Subtype.coe_le_coe.mp <\| mul_le_of_le_one_left y.2.1 x.2.2 #align unit_interval.mul_le_right unitInterval.mul_le_right /-- Unit interval central symmetry. -/ def symm : I → I := fun t => ⟨1 - t, mem_iff_one_sub_mem.mp t.prop⟩ #align unit_interval.symm unitInterval.symm @[inherit_doc] scoped notation "σ" => unitInterval.symm @[simp] theorem symm_zero : σ 0 = 1 := Subtype.ext <\| by simp [symm] #align unit_interval.symm_zero unitInterval.symm_zero @[simp] theorem symm_one : σ 1 = 0 := Subtype.ext <\| by simp [symm] #align unit_interval.symm_one unitInterval.symm_one @[simp] theorem symm_symm (x : I) : σ (σ x) = x := Subtype.ext <\| by simp [symm] #align unit_interval.symm_symm unitInterval.symm_symm theorem symm_involutive : Function.Involutive (symm : I → I) := symm_symm theorem symm_bijective : Function.Bijective (symm : I → I) := symm_involutive.bijective @[simp] theorem coe_symm_eq (x : I) : (σ x : ℝ) = 1 - x := rfl #align unit_interval.coe_symm_eq unitInterval.coe_symm_eq -- Porting note: Proof used to be `by continuity!` @[continuity] theorem continuous_symm : Continuous σ := (continuous_const.add continuous_induced_dom.neg).subtype_mk _ #align unit_interval.continuous_symm unitInterval.continuous_symm /-- `unitInterval.symm` as a `Homeomorph`. -/ @[simps] def symmHomeomorph : I ≃ₜ I where toFun := symm invFun := symm left_inv := symm_symm right_inv := symm_symm theorem strictAnti_symm : StrictAnti σ := fun _ _ h ↦ sub_lt_sub_left (α := ℝ) h _ @[deprecated (since := "2024-02-27")] alias involutive_symm := symm_involutive @[deprecated (since := "2024-02-27")] alias bijective_symm := symm_bijective theorem half_le_symm_iff (t : I) : 1 / 2 ≤ (σ t : ℝ) ↔ (t : ℝ) ≤ 1 / 2 := by rw [coe_symm_eq, le_sub_iff_add_le, add_comm, ← le_sub_iff_add_le, sub_half] instance : ConnectedSpace I := Subtype.connectedSpace ⟨nonempty_Icc.mpr zero_le_one, isPreconnected_Icc⟩ /-- Verify there is an instance for `CompactSpace I`. -/ example : CompactSpace I := by infer_instance theorem nonneg (x : I) : 0 ≤ (x : ℝ) := x.2.1 #align unit_interval.nonneg unitInterval.nonneg theorem one_minus_nonneg (x : I) : 0 ≤ 1 - (x : ℝ) := by simpa using x.2.2 #align unit_interval.one_minus_nonneg unitInterval.one_minus_nonneg theorem le_one (x : I) : (x : ℝ) ≤ 1 := x.2.2 #align unit_interval.le_one unitInterval.le_one theorem one_minus_le_one (x : I) : 1 - (x : ℝ) ≤ 1 := by simpa using x.2.1 #align unit_interval.one_minus_le_one unitInterval.one_minus_le_one theorem add_pos {t : I} {x : ℝ} (hx : 0 < x) : 0 < (x + t : ℝ) := add_pos_of_pos_of_nonneg hx <\| nonneg _ #align unit_interval.add_pos unitInterval.add_pos /-- like `unitInterval.nonneg`, but with the inequality in `I`. -/ theorem nonneg' {t : I} : 0 ≤ t := t.2.1 #align unit_interval.nonneg' unitInterval.nonneg' /-- like `unitInterval.le_one`, but with the inequality in `I`. -/ theorem le_one' {t : I} : t ≤ 1 := t.2.2 #align unit_interval.le_one' unitInterval.le_one' instance : Nontrivial I := ⟨⟨1, 0, (one_ne_zero <\| congrArg Subtype.val ·)⟩⟩	Mathlib/Topology/UnitInterval.lean	193	198	theorem mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ Set.Icc (0 : ℝ) (1 / a) := by	constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor · exact nonneg_of_mul_nonneg_right h₁ ha · rwa [le_div_iff ha, mul_comm] · exact mul_nonneg ha.le h₁ · rwa [le_div_iff ha, mul_comm] at h₂
/- Copyright (c) 2022 Oliver Nash. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Oliver Nash -/ import Mathlib.Algebra.Order.ToIntervalMod import Mathlib.Algebra.Ring.AddAut import Mathlib.Data.Nat.Totient import Mathlib.GroupTheory.Divisible import Mathlib.Topology.Connected.PathConnected import Mathlib.Topology.IsLocalHomeomorph #align_import topology.instances.add_circle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829ce42efec" /-! # The additive circle We define the additive circle `AddCircle p` as the quotient `𝕜 ⧸ (ℤ ∙ p)` for some period `p : 𝕜`. See also `Circle` and `Real.angle`. For the normed group structure on `AddCircle`, see `AddCircle.NormedAddCommGroup` in a later file. ## Main definitions and results: * `AddCircle`: the additive circle `𝕜 ⧸ (ℤ ∙ p)` for some period `p : 𝕜` * `UnitAddCircle`: the special case `ℝ ⧸ ℤ` * `AddCircle.equivAddCircle`: the rescaling equivalence `AddCircle p ≃+ AddCircle q` * `AddCircle.equivIco`: the natural equivalence `AddCircle p ≃ Ico a (a + p)` * `AddCircle.addOrderOf_div_of_gcd_eq_one`: rational points have finite order * `AddCircle.exists_gcd_eq_one_of_isOfFinAddOrder`: finite-order points are rational * `AddCircle.homeoIccQuot`: the natural topological equivalence between `AddCircle p` and `Icc a (a + p)` with its endpoints identified. * `AddCircle.liftIco_continuous`: if `f : ℝ → B` is continuous, and `f a = f (a + p)` for some `a`, then there is a continuous function `AddCircle p → B` which agrees with `f` on `Icc a (a + p)`. ## Implementation notes: Although the most important case is `𝕜 = ℝ` we wish to support other types of scalars, such as the rational circle `AddCircle (1 : ℚ)`, and so we set things up more generally. ## TODO * Link with periodicity * Lie group structure * Exponential equivalence to `Circle` -/ noncomputable section open AddCommGroup Set Function AddSubgroup TopologicalSpace open Topology variable {𝕜 B : Type*} section Continuity variable [LinearOrderedAddCommGroup 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p : 𝕜} (hp : 0 < p) (a x : 𝕜)	Mathlib/Topology/Instances/AddCircle.lean	64	79	theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x := by	intro s h rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter] haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩ simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢ obtain ⟨l, u, hxI, hIs⟩ := h let d := toIcoDiv hp a x • p have hd := toIcoMod_mem_Ico hp a x simp_rw [subset_def, mem_inter_iff] refine ⟨_, ⟨l + d, min (a + p) u + d, ?_, fun x => id⟩, fun y => ?_⟩ <;> simp_rw [← sub_mem_Ioo_iff_left, mem_Ioo, lt_min_iff] · exact ⟨hxI.1, hd.2, hxI.2⟩ · rintro ⟨h, h'⟩ apply hIs rw [← toIcoMod_sub_zsmul, (toIcoMod_eq_self _).2] exacts [⟨h.1, h.2.2⟩, ⟨hd.1.trans (sub_le_sub_right h' _), h.2.1⟩]
/- Copyright (c) 2024 Peter Nelson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Peter Nelson -/ import Mathlib.Data.Matroid.Restrict /-! # Some constructions of matroids This file defines some very elementary examples of matroids, namely those with at most one base. ## Main definitions * `emptyOn α` is the matroid on `α` with empty ground set. For `E : Set α`, ... * `loopyOn E` is the matroid on `E` whose elements are all loops, or equivalently in which `∅` is the only base. * `freeOn E` is the 'free matroid' whose ground set `E` is the only base. * For `I ⊆ E`, `uniqueBaseOn I E` is the matroid with ground set `E` in which `I` is the only base. ## Implementation details To avoid the tedious process of certifying the matroid axioms for each of these easy examples, we bootstrap the definitions starting with `emptyOn α` (which `simp` can prove is a matroid) and then construct the other examples using duality and restriction. -/ variable {α : Type} {M : Matroid α} {E B I X R J : Set α} namespace Matroid open Set section EmptyOn /-- The `Matroid α` with empty ground set. -/ def emptyOn (α : Type) : Matroid α where E := ∅ Base := (· = ∅) Indep := (· = ∅) indep_iff' := by simp [subset_empty_iff] exists_base := ⟨∅, rfl⟩ base_exchange := by rintro _ _ rfl; simp maximality := by rintro _ _ _ rfl -; exact ⟨∅, by simp [mem_maximals_iff]⟩ subset_ground := by simp @[simp] theorem emptyOn_ground : (emptyOn α).E = ∅ := rfl @[simp] theorem emptyOn_base_iff : (emptyOn α).Base B ↔ B = ∅ := Iff.rfl @[simp] theorem emptyOn_indep_iff : (emptyOn α).Indep I ↔ I = ∅ := Iff.rfl theorem ground_eq_empty_iff : (M.E = ∅) ↔ M = emptyOn α := by simp only [emptyOn, eq_iff_indep_iff_indep_forall, iff_self_and] exact fun h ↦ by simp [h, subset_empty_iff] @[simp] theorem emptyOn_dual_eq : (emptyOn α)✶ = emptyOn α := by rw [← ground_eq_empty_iff]; rfl @[simp] theorem restrict_empty (M : Matroid α) : M ↾ (∅ : Set α) = emptyOn α := by simp [← ground_eq_empty_iff] theorem eq_emptyOn_or_nonempty (M : Matroid α) : M = emptyOn α ∨ Matroid.Nonempty M := by rw [← ground_eq_empty_iff] exact M.E.eq_empty_or_nonempty.elim Or.inl (fun h ↦ Or.inr ⟨h⟩) theorem eq_emptyOn [IsEmpty α] (M : Matroid α) : M = emptyOn α := by rw [← ground_eq_empty_iff] exact M.E.eq_empty_of_isEmpty instance finite_emptyOn (α : Type) : (emptyOn α).Finite := ⟨finite_empty⟩ end EmptyOn section LoopyOn /-- The `Matroid α` with ground set `E` whose only base is `∅` -/ def loopyOn (E : Set α) : Matroid α := emptyOn α ↾ E @[simp] theorem loopyOn_ground (E : Set α) : (loopyOn E).E = E := rfl @[simp] theorem loopyOn_empty (α : Type) : loopyOn (∅ : Set α) = emptyOn α := by rw [← ground_eq_empty_iff, loopyOn_ground] @[simp] theorem loopyOn_indep_iff : (loopyOn E).Indep I ↔ I = ∅ := by simp only [loopyOn, restrict_indep_iff, emptyOn_indep_iff, and_iff_left_iff_imp] rintro rfl; apply empty_subset theorem eq_loopyOn_iff : M = loopyOn E ↔ M.E = E ∧ ∀ X ⊆ M.E, M.Indep X → X = ∅ := by simp only [eq_iff_indep_iff_indep_forall, loopyOn_ground, loopyOn_indep_iff, and_congr_right_iff] rintro rfl refine ⟨fun h I hI ↦ (h I hI).1, fun h I hIE ↦ ⟨h I hIE, by rintro rfl; simp⟩⟩ @[simp] theorem loopyOn_base_iff : (loopyOn E).Base B ↔ B = ∅ := by simp only [base_iff_maximal_indep, loopyOn_indep_iff, forall_eq, and_iff_left_iff_imp] exact fun h _ ↦ h @[simp] theorem loopyOn_basis_iff : (loopyOn E).Basis I X ↔ I = ∅ ∧ X ⊆ E := ⟨fun h ↦ ⟨loopyOn_indep_iff.mp h.indep, h.subset_ground⟩, by rintro ⟨rfl, hX⟩; rw [basis_iff]; simp⟩ instance : FiniteRk (loopyOn E) := ⟨⟨∅, loopyOn_base_iff.2 rfl, finite_empty⟩⟩ theorem Finite.loopyOn_finite (hE : E.Finite) : Matroid.Finite (loopyOn E) := ⟨hE⟩ @[simp] theorem loopyOn_restrict (E R : Set α) : (loopyOn E) ↾ R = loopyOn R := by refine eq_of_indep_iff_indep_forall rfl ?_ simp only [restrict_ground_eq, restrict_indep_iff, loopyOn_indep_iff, and_iff_left_iff_imp] exact fun _ h _ ↦ h theorem empty_base_iff : M.Base ∅ ↔ M = loopyOn M.E := by simp only [base_iff_maximal_indep, empty_indep, empty_subset, eq_comm (a := ∅), true_implies, true_and, eq_iff_indep_iff_indep_forall, loopyOn_ground, loopyOn_indep_iff] exact ⟨fun h I _ ↦ ⟨h I, by rintro rfl; simp⟩, fun h I hI ↦ (h I hI.subset_ground).1 hI⟩ theorem eq_loopyOn_or_rkPos (M : Matroid α) : M = loopyOn M.E ∨ RkPos M := by rw [← empty_base_iff, rkPos_iff_empty_not_base]; apply em theorem not_rkPos_iff : ¬RkPos M ↔ M = loopyOn M.E := by rw [rkPos_iff_empty_not_base, not_iff_comm, empty_base_iff] end LoopyOn section FreeOn /-- The `Matroid α` with ground set `E` whose only base is `E`. -/ def freeOn (E : Set α) : Matroid α := (loopyOn E)✶ @[simp] theorem freeOn_ground : (freeOn E).E = E := rfl @[simp] theorem freeOn_dual_eq : (freeOn E)✶ = loopyOn E := by rw [freeOn, dual_dual] @[simp] theorem loopyOn_dual_eq : (loopyOn E)✶ = freeOn E := rfl @[simp] theorem freeOn_empty (α : Type*) : freeOn (∅ : Set α) = emptyOn α := by simp [freeOn] @[simp] theorem freeOn_base_iff : (freeOn E).Base B ↔ B = E := by simp only [freeOn, loopyOn_ground, dual_base_iff', loopyOn_base_iff, diff_eq_empty, ← subset_antisymm_iff, eq_comm (a := E)] @[simp] theorem freeOn_indep_iff : (freeOn E).Indep I ↔ I ⊆ E := by simp [indep_iff] theorem freeOn_indep (hIE : I ⊆ E) : (freeOn E).Indep I := freeOn_indep_iff.2 hIE @[simp] theorem freeOn_basis_iff : (freeOn E).Basis I X ↔ I = X ∧ X ⊆ E := by use fun h ↦ ⟨(freeOn_indep h.subset_ground).eq_of_basis h ,h.subset_ground⟩ rintro ⟨rfl, hIE⟩ exact (freeOn_indep hIE).basis_self @[simp] theorem freeOn_basis'_iff : (freeOn E).Basis' I X ↔ I = X ∩ E := by rw [basis'_iff_basis_inter_ground, freeOn_basis_iff, freeOn_ground, and_iff_left inter_subset_right] theorem eq_freeOn_iff : M = freeOn E ↔ M.E = E ∧ M.Indep E := by refine ⟨?_, fun h ↦ ?_⟩ · rintro rfl; simp [Subset.rfl] simp only [eq_iff_indep_iff_indep_forall, freeOn_ground, freeOn_indep_iff, h.1, true_and] exact fun I hIX ↦ iff_of_true (h.2.subset hIX) hIX theorem ground_indep_iff_eq_freeOn : M.Indep M.E ↔ M = freeOn M.E := by simp [eq_freeOn_iff]	Mathlib/Data/Matroid/Constructions.lean	174	175	theorem freeOn_restrict (h : R ⊆ E) : (freeOn E) ↾ R = freeOn R := by	simp [h, eq_freeOn_iff, Subset.rfl]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne -/ import Mathlib.Order.MinMax import Mathlib.Data.Set.Subsingleton import Mathlib.Tactic.Says #align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c" /-! # Intervals In any preorder `α`, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. This file contains these definitions, and basic facts on inclusion, intersection, difference of intervals (where the precise statements may depend on the properties of the order, in particular for some statements it should be `LinearOrder` or `DenselyOrdered`). TODO: This is just the beginning; a lot of rules are missing -/ open Function open OrderDual (toDual ofDual) variable {α β : Type} namespace Set section Preorder variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α} /-- Left-open right-open interval -/ def Ioo (a b : α) := { x \| a < x ∧ x < b } #align set.Ioo Set.Ioo /-- Left-closed right-open interval -/ def Ico (a b : α) := { x \| a ≤ x ∧ x < b } #align set.Ico Set.Ico /-- Left-infinite right-open interval -/ def Iio (a : α) := { x \| x < a } #align set.Iio Set.Iio /-- Left-closed right-closed interval -/ def Icc (a b : α) := { x \| a ≤ x ∧ x ≤ b } #align set.Icc Set.Icc /-- Left-infinite right-closed interval -/ def Iic (b : α) := { x \| x ≤ b } #align set.Iic Set.Iic /-- Left-open right-closed interval -/ def Ioc (a b : α) := { x \| a < x ∧ x ≤ b } #align set.Ioc Set.Ioc /-- Left-closed right-infinite interval -/ def Ici (a : α) := { x \| a ≤ x } #align set.Ici Set.Ici /-- Left-open right-infinite interval -/ def Ioi (a : α) := { x \| a < x } #align set.Ioi Set.Ioi theorem Ioo_def (a b : α) : { x \| a < x ∧ x < b } = Ioo a b := rfl #align set.Ioo_def Set.Ioo_def theorem Ico_def (a b : α) : { x \| a ≤ x ∧ x < b } = Ico a b := rfl #align set.Ico_def Set.Ico_def theorem Iio_def (a : α) : { x \| x < a } = Iio a := rfl #align set.Iio_def Set.Iio_def theorem Icc_def (a b : α) : { x \| a ≤ x ∧ x ≤ b } = Icc a b := rfl #align set.Icc_def Set.Icc_def theorem Iic_def (b : α) : { x \| x ≤ b } = Iic b := rfl #align set.Iic_def Set.Iic_def theorem Ioc_def (a b : α) : { x \| a < x ∧ x ≤ b } = Ioc a b := rfl #align set.Ioc_def Set.Ioc_def theorem Ici_def (a : α) : { x \| a ≤ x } = Ici a := rfl #align set.Ici_def Set.Ici_def theorem Ioi_def (a : α) : { x \| a < x } = Ioi a := rfl #align set.Ioi_def Set.Ioi_def @[simp] theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b := Iff.rfl #align set.mem_Ioo Set.mem_Ioo @[simp] theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b := Iff.rfl #align set.mem_Ico Set.mem_Ico @[simp] theorem mem_Iio : x ∈ Iio b ↔ x < b := Iff.rfl #align set.mem_Iio Set.mem_Iio @[simp] theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b := Iff.rfl #align set.mem_Icc Set.mem_Icc @[simp] theorem mem_Iic : x ∈ Iic b ↔ x ≤ b := Iff.rfl #align set.mem_Iic Set.mem_Iic @[simp] theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b := Iff.rfl #align set.mem_Ioc Set.mem_Ioc @[simp] theorem mem_Ici : x ∈ Ici a ↔ a ≤ x := Iff.rfl #align set.mem_Ici Set.mem_Ici @[simp] theorem mem_Ioi : x ∈ Ioi a ↔ a < x := Iff.rfl #align set.mem_Ioi Set.mem_Ioi instance decidableMemIoo [Decidable (a < x ∧ x < b)] : Decidable (x ∈ Ioo a b) := by assumption #align set.decidable_mem_Ioo Set.decidableMemIoo instance decidableMemIco [Decidable (a ≤ x ∧ x < b)] : Decidable (x ∈ Ico a b) := by assumption #align set.decidable_mem_Ico Set.decidableMemIco instance decidableMemIio [Decidable (x < b)] : Decidable (x ∈ Iio b) := by assumption #align set.decidable_mem_Iio Set.decidableMemIio instance decidableMemIcc [Decidable (a ≤ x ∧ x ≤ b)] : Decidable (x ∈ Icc a b) := by assumption #align set.decidable_mem_Icc Set.decidableMemIcc instance decidableMemIic [Decidable (x ≤ b)] : Decidable (x ∈ Iic b) := by assumption #align set.decidable_mem_Iic Set.decidableMemIic instance decidableMemIoc [Decidable (a < x ∧ x ≤ b)] : Decidable (x ∈ Ioc a b) := by assumption #align set.decidable_mem_Ioc Set.decidableMemIoc instance decidableMemIci [Decidable (a ≤ x)] : Decidable (x ∈ Ici a) := by assumption #align set.decidable_mem_Ici Set.decidableMemIci instance decidableMemIoi [Decidable (a < x)] : Decidable (x ∈ Ioi a) := by assumption #align set.decidable_mem_Ioi Set.decidableMemIoi -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioo Set.left_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl] #align set.left_mem_Ico Set.left_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.left_mem_Icc Set.left_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioc Set.left_mem_Ioc theorem left_mem_Ici : a ∈ Ici a := by simp #align set.left_mem_Ici Set.left_mem_Ici -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioo : b ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ioo Set.right_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ico : b ∈ Ico a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ico Set.right_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.right_mem_Icc Set.right_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl] #align set.right_mem_Ioc Set.right_mem_Ioc theorem right_mem_Iic : a ∈ Iic a := by simp #align set.right_mem_Iic Set.right_mem_Iic @[simp] theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a := rfl #align set.dual_Ici Set.dual_Ici @[simp] theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a := rfl #align set.dual_Iic Set.dual_Iic @[simp] theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a := rfl #align set.dual_Ioi Set.dual_Ioi @[simp] theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a := rfl #align set.dual_Iio Set.dual_Iio @[simp] theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a := Set.ext fun _ => and_comm #align set.dual_Icc Set.dual_Icc @[simp] theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a := Set.ext fun _ => and_comm #align set.dual_Ioc Set.dual_Ioc @[simp] theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a := Set.ext fun _ => and_comm #align set.dual_Ico Set.dual_Ico @[simp] theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a := Set.ext fun _ => and_comm #align set.dual_Ioo Set.dual_Ioo @[simp] theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := ⟨fun ⟨_, hx⟩ => hx.1.trans hx.2, fun h => ⟨a, left_mem_Icc.2 h⟩⟩ #align set.nonempty_Icc Set.nonempty_Icc @[simp] theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_lt hx.2, fun h => ⟨a, left_mem_Ico.2 h⟩⟩ #align set.nonempty_Ico Set.nonempty_Ico @[simp] theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_le hx.2, fun h => ⟨b, right_mem_Ioc.2 h⟩⟩ #align set.nonempty_Ioc Set.nonempty_Ioc @[simp] theorem nonempty_Ici : (Ici a).Nonempty := ⟨a, left_mem_Ici⟩ #align set.nonempty_Ici Set.nonempty_Ici @[simp] theorem nonempty_Iic : (Iic a).Nonempty := ⟨a, right_mem_Iic⟩ #align set.nonempty_Iic Set.nonempty_Iic @[simp] theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := ⟨fun ⟨_, ha, hb⟩ => ha.trans hb, exists_between⟩ #align set.nonempty_Ioo Set.nonempty_Ioo @[simp] theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty := exists_gt a #align set.nonempty_Ioi Set.nonempty_Ioi @[simp] theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty := exists_lt a #align set.nonempty_Iio Set.nonempty_Iio theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) := Nonempty.to_subtype (nonempty_Icc.mpr h) #align set.nonempty_Icc_subtype Set.nonempty_Icc_subtype theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) := Nonempty.to_subtype (nonempty_Ico.mpr h) #align set.nonempty_Ico_subtype Set.nonempty_Ico_subtype theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) := Nonempty.to_subtype (nonempty_Ioc.mpr h) #align set.nonempty_Ioc_subtype Set.nonempty_Ioc_subtype /-- An interval `Ici a` is nonempty. -/ instance nonempty_Ici_subtype : Nonempty (Ici a) := Nonempty.to_subtype nonempty_Ici #align set.nonempty_Ici_subtype Set.nonempty_Ici_subtype /-- An interval `Iic a` is nonempty. -/ instance nonempty_Iic_subtype : Nonempty (Iic a) := Nonempty.to_subtype nonempty_Iic #align set.nonempty_Iic_subtype Set.nonempty_Iic_subtype theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) := Nonempty.to_subtype (nonempty_Ioo.mpr h) #align set.nonempty_Ioo_subtype Set.nonempty_Ioo_subtype /-- In an order without maximal elements, the intervals `Ioi` are nonempty. -/ instance nonempty_Ioi_subtype [NoMaxOrder α] : Nonempty (Ioi a) := Nonempty.to_subtype nonempty_Ioi #align set.nonempty_Ioi_subtype Set.nonempty_Ioi_subtype /-- In an order without minimal elements, the intervals `Iio` are nonempty. -/ instance nonempty_Iio_subtype [NoMinOrder α] : Nonempty (Iio a) := Nonempty.to_subtype nonempty_Iio #align set.nonempty_Iio_subtype Set.nonempty_Iio_subtype instance [NoMinOrder α] : NoMinOrder (Iio a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, lt_trans hb a.2⟩, hb⟩⟩ instance [NoMinOrder α] : NoMinOrder (Iic a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, hb.le.trans a.2⟩, hb⟩⟩ instance [NoMaxOrder α] : NoMaxOrder (Ioi a) := OrderDual.noMaxOrder (α := Iio (toDual a)) instance [NoMaxOrder α] : NoMaxOrder (Ici a) := OrderDual.noMaxOrder (α := Iic (toDual a)) @[simp] theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Icc_eq_empty Set.Icc_eq_empty @[simp] theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_lt hb) #align set.Ico_eq_empty Set.Ico_eq_empty @[simp] theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_le hb) #align set.Ioc_eq_empty Set.Ioc_eq_empty @[simp] theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Ioo_eq_empty Set.Ioo_eq_empty @[simp] theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ := Icc_eq_empty h.not_le #align set.Icc_eq_empty_of_lt Set.Icc_eq_empty_of_lt @[simp] theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ := Ico_eq_empty h.not_lt #align set.Ico_eq_empty_of_le Set.Ico_eq_empty_of_le @[simp] theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ := Ioc_eq_empty h.not_lt #align set.Ioc_eq_empty_of_le Set.Ioc_eq_empty_of_le @[simp] theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ := Ioo_eq_empty h.not_lt #align set.Ioo_eq_empty_of_le Set.Ioo_eq_empty_of_le -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ico_self (a : α) : Ico a a = ∅ := Ico_eq_empty <\| lt_irrefl _ #align set.Ico_self Set.Ico_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioc_self (a : α) : Ioc a a = ∅ := Ioc_eq_empty <\| lt_irrefl _ #align set.Ioc_self Set.Ioc_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioo_self (a : α) : Ioo a a = ∅ := Ioo_eq_empty <\| lt_irrefl _ #align set.Ioo_self Set.Ioo_self theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a := ⟨fun h => h <\| left_mem_Ici, fun h _ hx => h.trans hx⟩ #align set.Ici_subset_Ici Set.Ici_subset_Ici @[gcongr] alias ⟨_, _root_.GCongr.Ici_subset_Ici_of_le⟩ := Ici_subset_Ici theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b := @Ici_subset_Ici αᵒᵈ _ _ _ #align set.Iic_subset_Iic Set.Iic_subset_Iic @[gcongr] alias ⟨_, _root_.GCongr.Iic_subset_Iic_of_le⟩ := Iic_subset_Iic theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a := ⟨fun h => h left_mem_Ici, fun h _ hx => h.trans_le hx⟩ #align set.Ici_subset_Ioi Set.Ici_subset_Ioi theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b := ⟨fun h => h right_mem_Iic, fun h _ hx => lt_of_le_of_lt hx h⟩ #align set.Iic_subset_Iio Set.Iic_subset_Iio @[gcongr] theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans_le h₂⟩ #align set.Ioo_subset_Ioo Set.Ioo_subset_Ioo @[gcongr] theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b := Ioo_subset_Ioo h le_rfl #align set.Ioo_subset_Ioo_left Set.Ioo_subset_Ioo_left @[gcongr] theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ := Ioo_subset_Ioo le_rfl h #align set.Ioo_subset_Ioo_right Set.Ioo_subset_Ioo_right @[gcongr] theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, hx₂.trans_le h₂⟩ #align set.Ico_subset_Ico Set.Ico_subset_Ico @[gcongr] theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b := Ico_subset_Ico h le_rfl #align set.Ico_subset_Ico_left Set.Ico_subset_Ico_left @[gcongr] theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ := Ico_subset_Ico le_rfl h #align set.Ico_subset_Ico_right Set.Ico_subset_Ico_right @[gcongr] theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, le_trans hx₂ h₂⟩ #align set.Icc_subset_Icc Set.Icc_subset_Icc @[gcongr] theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b := Icc_subset_Icc h le_rfl #align set.Icc_subset_Icc_left Set.Icc_subset_Icc_left @[gcongr] theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ := Icc_subset_Icc le_rfl h #align set.Icc_subset_Icc_right Set.Icc_subset_Icc_right theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ hx => ⟨ha.trans_le hx.1, hx.2.trans_lt hb⟩ #align set.Icc_subset_Ioo Set.Icc_subset_Ioo theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := fun _ => And.left #align set.Icc_subset_Ici_self Set.Icc_subset_Ici_self theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := fun _ => And.right #align set.Icc_subset_Iic_self Set.Icc_subset_Iic_self theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := fun _ => And.right #align set.Ioc_subset_Iic_self Set.Ioc_subset_Iic_self @[gcongr] theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans h₂⟩ #align set.Ioc_subset_Ioc Set.Ioc_subset_Ioc @[gcongr] theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b := Ioc_subset_Ioc h le_rfl #align set.Ioc_subset_Ioc_left Set.Ioc_subset_Ioc_left @[gcongr] theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ := Ioc_subset_Ioc le_rfl h #align set.Ioc_subset_Ioc_right Set.Ioc_subset_Ioc_right theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := fun _ => And.imp_left h₁.trans_le #align set.Ico_subset_Ioo_left Set.Ico_subset_Ioo_left theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := fun _ => And.imp_right fun h' => h'.trans_lt h #align set.Ioc_subset_Ioo_right Set.Ioc_subset_Ioo_right theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := fun _ => And.imp_right fun h₂ => h₂.trans_lt h₁ #align set.Icc_subset_Ico_right Set.Icc_subset_Ico_right theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := fun _ => And.imp_left le_of_lt #align set.Ioo_subset_Ico_self Set.Ioo_subset_Ico_self theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := fun _ => And.imp_right le_of_lt #align set.Ioo_subset_Ioc_self Set.Ioo_subset_Ioc_self theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := fun _ => And.imp_right le_of_lt #align set.Ico_subset_Icc_self Set.Ico_subset_Icc_self theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := fun _ => And.imp_left le_of_lt #align set.Ioc_subset_Icc_self Set.Ioc_subset_Icc_self theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b := Subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self #align set.Ioo_subset_Icc_self Set.Ioo_subset_Icc_self theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := fun _ => And.right #align set.Ico_subset_Iio_self Set.Ico_subset_Iio_self theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := fun _ => And.right #align set.Ioo_subset_Iio_self Set.Ioo_subset_Iio_self theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := fun _ => And.left #align set.Ioc_subset_Ioi_self Set.Ioc_subset_Ioi_self theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := fun _ => And.left #align set.Ioo_subset_Ioi_self Set.Ioo_subset_Ioi_self theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a := fun _ hx => le_of_lt hx #align set.Ioi_subset_Ici_self Set.Ioi_subset_Ici_self theorem Iio_subset_Iic_self : Iio a ⊆ Iic a := fun _ hx => le_of_lt hx #align set.Iio_subset_Iic_self Set.Iio_subset_Iic_self theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := fun _ => And.left #align set.Ico_subset_Ici_self Set.Ico_subset_Ici_self theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a := ⟨Ioi_subset_Ici_self, fun h => lt_irrefl a (h le_rfl)⟩ #align set.Ioi_ssubset_Ici_self Set.Ioi_ssubset_Ici_self theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a := @Ioi_ssubset_Ici_self αᵒᵈ _ _ #align set.Iio_ssubset_Iic_self Set.Iio_ssubset_Iic_self theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Icc_iff Set.Icc_subset_Icc_iff theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ioo_iff Set.Icc_subset_Ioo_iff theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ico_iff Set.Icc_subset_Ico_iff theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Ioc_iff Set.Icc_subset_Ioc_iff theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans_lt h⟩ #align set.Icc_subset_Iio_iff Set.Icc_subset_Iio_iff theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans_le hx⟩ #align set.Icc_subset_Ioi_iff Set.Icc_subset_Ioi_iff theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans h⟩ #align set.Icc_subset_Iic_iff Set.Icc_subset_Iic_iff theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans hx⟩ #align set.Icc_subset_Ici_iff Set.Icc_subset_Ici_iff theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc (le_of_lt ha) hb)).mpr ⟨a₂, left_mem_Icc.mpr hI, not_and.mpr fun f _ => lt_irrefl a₂ (ha.trans_le f)⟩ #align set.Icc_ssubset_Icc_left Set.Icc_ssubset_Icc_left theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc ha (le_of_lt hb))).mpr ⟨b₂, right_mem_Icc.mpr hI, fun f => lt_irrefl b₁ (hb.trans_le f.2)⟩ #align set.Icc_ssubset_Icc_right Set.Icc_ssubset_Icc_right /-- If `a ≤ b`, then `(b, +∞) ⊆ (a, +∞)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Ioi_subset_Ioi_iff`. -/ @[gcongr] theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := fun _ hx => h.trans_lt hx #align set.Ioi_subset_Ioi Set.Ioi_subset_Ioi /-- If `a ≤ b`, then `(b, +∞) ⊆ [a, +∞)`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Ioi_subset_Ici_iff`. -/ theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a := Subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self #align set.Ioi_subset_Ici Set.Ioi_subset_Ici /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Iio_subset_Iio_iff`. -/ @[gcongr] theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := fun _ hx => lt_of_lt_of_le hx h #align set.Iio_subset_Iio Set.Iio_subset_Iio /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b]`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Iio_subset_Iic_iff`. -/ theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b := Subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self #align set.Iio_subset_Iic Set.Iio_subset_Iic theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b := rfl #align set.Ici_inter_Iic Set.Ici_inter_Iic theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b := rfl #align set.Ici_inter_Iio Set.Ici_inter_Iio theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b := rfl #align set.Ioi_inter_Iic Set.Ioi_inter_Iic theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b := rfl #align set.Ioi_inter_Iio Set.Ioi_inter_Iio theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a := inter_comm _ _ #align set.Iic_inter_Ici Set.Iic_inter_Ici theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a := inter_comm _ _ #align set.Iio_inter_Ici Set.Iio_inter_Ici theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a := inter_comm _ _ #align set.Iic_inter_Ioi Set.Iic_inter_Ioi theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a := inter_comm _ _ #align set.Iio_inter_Ioi Set.Iio_inter_Ioi theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b := Ioo_subset_Icc_self h #align set.mem_Icc_of_Ioo Set.mem_Icc_of_Ioo theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b := Ioo_subset_Ico_self h #align set.mem_Ico_of_Ioo Set.mem_Ico_of_Ioo theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b := Ioo_subset_Ioc_self h #align set.mem_Ioc_of_Ioo Set.mem_Ioc_of_Ioo theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b := Ico_subset_Icc_self h #align set.mem_Icc_of_Ico Set.mem_Icc_of_Ico theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b := Ioc_subset_Icc_self h #align set.mem_Icc_of_Ioc Set.mem_Icc_of_Ioc theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a := Ioi_subset_Ici_self h #align set.mem_Ici_of_Ioi Set.mem_Ici_of_Ioi theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a := Iio_subset_Iic_self h #align set.mem_Iic_of_Iio Set.mem_Iic_of_Iio theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc] #align set.Icc_eq_empty_iff Set.Icc_eq_empty_iff theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico] #align set.Ico_eq_empty_iff Set.Ico_eq_empty_iff theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioc] #align set.Ioc_eq_empty_iff Set.Ioc_eq_empty_iff theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo] #align set.Ioo_eq_empty_iff Set.Ioo_eq_empty_iff theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ := eq_univ_of_forall h #align is_top.Iic_eq IsTop.Iic_eq theorem _root_.IsBot.Ici_eq (h : IsBot a) : Ici a = univ := eq_univ_of_forall h #align is_bot.Ici_eq IsBot.Ici_eq theorem _root_.IsMax.Ioi_eq (h : IsMax a) : Ioi a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_max.Ioi_eq IsMax.Ioi_eq theorem _root_.IsMin.Iio_eq (h : IsMin a) : Iio a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_min.Iio_eq IsMin.Iio_eq theorem Iic_inter_Ioc_of_le (h : a ≤ c) : Iic a ∩ Ioc b c = Ioc b a := ext fun _ => ⟨fun H => ⟨H.2.1, H.1⟩, fun H => ⟨H.2, H.1, H.2.trans h⟩⟩ #align set.Iic_inter_Ioc_of_le Set.Iic_inter_Ioc_of_le theorem not_mem_Icc_of_lt (ha : c < a) : c ∉ Icc a b := fun h => ha.not_le h.1 #align set.not_mem_Icc_of_lt Set.not_mem_Icc_of_lt theorem not_mem_Icc_of_gt (hb : b < c) : c ∉ Icc a b := fun h => hb.not_le h.2 #align set.not_mem_Icc_of_gt Set.not_mem_Icc_of_gt theorem not_mem_Ico_of_lt (ha : c < a) : c ∉ Ico a b := fun h => ha.not_le h.1 #align set.not_mem_Ico_of_lt Set.not_mem_Ico_of_lt theorem not_mem_Ioc_of_gt (hb : b < c) : c ∉ Ioc a b := fun h => hb.not_le h.2 #align set.not_mem_Ioc_of_gt Set.not_mem_Ioc_of_gt -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Ioi_self : a ∉ Ioi a := lt_irrefl _ #align set.not_mem_Ioi_self Set.not_mem_Ioi_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Iio_self : b ∉ Iio b := lt_irrefl _ #align set.not_mem_Iio_self Set.not_mem_Iio_self theorem not_mem_Ioc_of_le (ha : c ≤ a) : c ∉ Ioc a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioc_of_le Set.not_mem_Ioc_of_le theorem not_mem_Ico_of_ge (hb : b ≤ c) : c ∉ Ico a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ico_of_ge Set.not_mem_Ico_of_ge theorem not_mem_Ioo_of_le (ha : c ≤ a) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioo_of_le Set.not_mem_Ioo_of_le theorem not_mem_Ioo_of_ge (hb : b ≤ c) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ioo_of_ge Set.not_mem_Ioo_of_ge end Preorder section PartialOrder variable [PartialOrder α] {a b c : α} @[simp] theorem Icc_self (a : α) : Icc a a = {a} := Set.ext <\| by simp [Icc, le_antisymm_iff, and_comm] #align set.Icc_self Set.Icc_self instance instIccUnique : Unique (Set.Icc a a) where default := ⟨a, by simp⟩ uniq y := Subtype.ext <\| by simpa using y.2 @[simp] theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by refine ⟨fun h => ?_, ?_⟩ · have hab : a ≤ b := nonempty_Icc.1 (h.symm.subst <\| singleton_nonempty c) exact ⟨eq_of_mem_singleton <\| h.subst <\| left_mem_Icc.2 hab, eq_of_mem_singleton <\| h.subst <\| right_mem_Icc.2 hab⟩ · rintro ⟨rfl, rfl⟩ exact Icc_self _ #align set.Icc_eq_singleton_iff Set.Icc_eq_singleton_iff lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) := fun _x ⟨hax, hxb⟩ _y ⟨hay, hyb⟩ ↦ le_antisymm (le_implies_le_of_le_of_le hxb hay hba) (le_implies_le_of_le_of_le hyb hax hba) #align set.subsingleton_Icc_of_ge Set.subsingleton_Icc_of_ge @[simp] lemma subsingleton_Icc_iff {α : Type} [LinearOrder α] {a b : α} : Set.Subsingleton (Icc a b) ↔ b ≤ a := by refine ⟨fun h ↦ ?_, subsingleton_Icc_of_ge⟩ contrapose! h simp only [ge_iff_le, gt_iff_lt, not_subsingleton_iff] exact ⟨a, ⟨le_refl _, h.le⟩, b, ⟨h.le, le_refl _⟩, h.ne⟩ @[simp] theorem Icc_diff_left : Icc a b \ {a} = Ioc a b := ext fun x => by simp [lt_iff_le_and_ne, eq_comm, and_right_comm] #align set.Icc_diff_left Set.Icc_diff_left @[simp] theorem Icc_diff_right : Icc a b \ {b} = Ico a b := ext fun x => by simp [lt_iff_le_and_ne, and_assoc] #align set.Icc_diff_right Set.Icc_diff_right @[simp] theorem Ico_diff_left : Ico a b \ {a} = Ioo a b := ext fun x => by simp [and_right_comm, ← lt_iff_le_and_ne, eq_comm] #align set.Ico_diff_left Set.Ico_diff_left @[simp] theorem Ioc_diff_right : Ioc a b \ {b} = Ioo a b := ext fun x => by simp [and_assoc, ← lt_iff_le_and_ne] #align set.Ioc_diff_right Set.Ioc_diff_right @[simp] theorem Icc_diff_both : Icc a b \ {a, b} = Ioo a b := by rw [insert_eq, ← diff_diff, Icc_diff_left, Ioc_diff_right] #align set.Icc_diff_both Set.Icc_diff_both @[simp] theorem Ici_diff_left : Ici a \ {a} = Ioi a := ext fun x => by simp [lt_iff_le_and_ne, eq_comm] #align set.Ici_diff_left Set.Ici_diff_left @[simp] theorem Iic_diff_right : Iic a \ {a} = Iio a := ext fun x => by simp [lt_iff_le_and_ne] #align set.Iic_diff_right Set.Iic_diff_right @[simp] theorem Ico_diff_Ioo_same (h : a < b) : Ico a b \ Ioo a b = {a} := by rw [← Ico_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Ico.2 h)] #align set.Ico_diff_Ioo_same Set.Ico_diff_Ioo_same @[simp] theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} := by rw [← Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Ioc.2 h)] #align set.Ioc_diff_Ioo_same Set.Ioc_diff_Ioo_same @[simp] theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Icc.2 h)] #align set.Icc_diff_Ico_same Set.Icc_diff_Ico_same @[simp] theorem Icc_diff_Ioc_same (h : a ≤ b) : Icc a b \ Ioc a b = {a} := by rw [← Icc_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Icc.2 h)] #align set.Icc_diff_Ioc_same Set.Icc_diff_Ioc_same @[simp] theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} := by rw [← Icc_diff_both, diff_diff_cancel_left] simp [insert_subset_iff, h] #align set.Icc_diff_Ioo_same Set.Icc_diff_Ioo_same @[simp] theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} := by rw [← Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)] #align set.Ici_diff_Ioi_same Set.Ici_diff_Ioi_same @[simp] theorem Iic_diff_Iio_same : Iic a \ Iio a = {a} := by rw [← Iic_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 right_mem_Iic)] #align set.Iic_diff_Iio_same Set.Iic_diff_Iio_same -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioi_union_left : Ioi a ∪ {a} = Ici a := ext fun x => by simp [eq_comm, le_iff_eq_or_lt] #align set.Ioi_union_left Set.Ioi_union_left -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Iio_union_right : Iio a ∪ {a} = Iic a := ext fun _ => le_iff_lt_or_eq.symm #align set.Iio_union_right Set.Iio_union_right theorem Ioo_union_left (hab : a < b) : Ioo a b ∪ {a} = Ico a b := by rw [← Ico_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Ico.2 hab)] #align set.Ioo_union_left Set.Ioo_union_left theorem Ioo_union_right (hab : a < b) : Ioo a b ∪ {b} = Ioc a b := by simpa only [dual_Ioo, dual_Ico] using Ioo_union_left hab.dual #align set.Ioo_union_right Set.Ioo_union_right theorem Ioo_union_both (h : a ≤ b) : Ioo a b ∪ {a, b} = Icc a b := by have : (Icc a b \ {a, b}) ∪ {a, b} = Icc a b := diff_union_of_subset fun \| x, .inl rfl => left_mem_Icc.mpr h \| x, .inr rfl => right_mem_Icc.mpr h rw [← this, Icc_diff_both] theorem Ioc_union_left (hab : a ≤ b) : Ioc a b ∪ {a} = Icc a b := by rw [← Icc_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Icc.2 hab)] #align set.Ioc_union_left Set.Ioc_union_left theorem Ico_union_right (hab : a ≤ b) : Ico a b ∪ {b} = Icc a b := by simpa only [dual_Ioc, dual_Icc] using Ioc_union_left hab.dual #align set.Ico_union_right Set.Ico_union_right @[simp] theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b := by rw [insert_eq, union_comm, Ico_union_right h] #align set.Ico_insert_right Set.Ico_insert_right @[simp] theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b := by rw [insert_eq, union_comm, Ioc_union_left h] #align set.Ioc_insert_left Set.Ioc_insert_left @[simp] theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b := by rw [insert_eq, union_comm, Ioo_union_left h] #align set.Ioo_insert_left Set.Ioo_insert_left @[simp] theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b := by rw [insert_eq, union_comm, Ioo_union_right h] #align set.Ioo_insert_right Set.Ioo_insert_right @[simp] theorem Iio_insert : insert a (Iio a) = Iic a := ext fun _ => le_iff_eq_or_lt.symm #align set.Iio_insert Set.Iio_insert @[simp] theorem Ioi_insert : insert a (Ioi a) = Ici a := ext fun _ => (or_congr_left eq_comm).trans le_iff_eq_or_lt.symm #align set.Ioi_insert Set.Ioi_insert theorem mem_Ici_Ioi_of_subset_of_subset {s : Set α} (ho : Ioi a ⊆ s) (hc : s ⊆ Ici a) : s ∈ ({Ici a, Ioi a} : Set (Set α)) := by_cases (fun h : a ∈ s => Or.inl <\| Subset.antisymm hc <\| by rw [← Ioi_union_left, union_subset_iff]; simp []) fun h => Or.inr <\| Subset.antisymm (fun x hx => lt_of_le_of_ne (hc hx) fun heq => h <\| heq.symm ▸ hx) ho #align set.mem_Ici_Ioi_of_subset_of_subset Set.mem_Ici_Ioi_of_subset_of_subset theorem mem_Iic_Iio_of_subset_of_subset {s : Set α} (ho : Iio a ⊆ s) (hc : s ⊆ Iic a) : s ∈ ({Iic a, Iio a} : Set (Set α)) := @mem_Ici_Ioi_of_subset_of_subset αᵒᵈ _ a s ho hc #align set.mem_Iic_Iio_of_subset_of_subset Set.mem_Iic_Iio_of_subset_of_subset theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) : s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α)) := by classical by_cases ha : a ∈ s <;> by_cases hb : b ∈ s · refine Or.inl (Subset.antisymm hc ?_) rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, ← Icc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_right] exact subset_diff_singleton hc hb · rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho · refine Or.inr <\| Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_left] exact subset_diff_singleton hc ha · rwa [← Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inr <\| Or.inr <\| Subset.antisymm ?_ ho rw [← Ico_diff_left, ← Icc_diff_right] apply_rules [subset_diff_singleton] #align set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset theorem eq_left_or_mem_Ioo_of_mem_Ico {x : α} (hmem : x ∈ Ico a b) : x = a ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => ⟨h, hmem.2⟩ #align set.eq_left_or_mem_Ioo_of_mem_Ico Set.eq_left_or_mem_Ioo_of_mem_Ico theorem eq_right_or_mem_Ioo_of_mem_Ioc {x : α} (hmem : x ∈ Ioc a b) : x = b ∨ x ∈ Ioo a b := hmem.2.eq_or_lt.imp_right <\| And.intro hmem.1 #align set.eq_right_or_mem_Ioo_of_mem_Ioc Set.eq_right_or_mem_Ioo_of_mem_Ioc theorem eq_endpoints_or_mem_Ioo_of_mem_Icc {x : α} (hmem : x ∈ Icc a b) : x = a ∨ x = b ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => eq_right_or_mem_Ioo_of_mem_Ioc ⟨h, hmem.2⟩ #align set.eq_endpoints_or_mem_Ioo_of_mem_Icc Set.eq_endpoints_or_mem_Ioo_of_mem_Icc theorem _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} := eq_singleton_iff_unique_mem.2 ⟨left_mem_Ici, fun _ => h.eq_of_ge⟩ #align is_max.Ici_eq IsMax.Ici_eq theorem _root_.IsMin.Iic_eq (h : IsMin a) : Iic a = {a} := h.toDual.Ici_eq #align is_min.Iic_eq IsMin.Iic_eq theorem Ici_injective : Injective (Ici : α → Set α) := fun _ _ => eq_of_forall_ge_iff ∘ Set.ext_iff.1 #align set.Ici_injective Set.Ici_injective theorem Iic_injective : Injective (Iic : α → Set α) := fun _ _ => eq_of_forall_le_iff ∘ Set.ext_iff.1 #align set.Iic_injective Set.Iic_injective theorem Ici_inj : Ici a = Ici b ↔ a = b := Ici_injective.eq_iff #align set.Ici_inj Set.Ici_inj theorem Iic_inj : Iic a = Iic b ↔ a = b := Iic_injective.eq_iff #align set.Iic_inj Set.Iic_inj end PartialOrder section OrderTop @[simp] theorem Ici_top [PartialOrder α] [OrderTop α] : Ici (⊤ : α) = {⊤} := isMax_top.Ici_eq #align set.Ici_top Set.Ici_top variable [Preorder α] [OrderTop α] {a : α} @[simp] theorem Ioi_top : Ioi (⊤ : α) = ∅ := isMax_top.Ioi_eq #align set.Ioi_top Set.Ioi_top @[simp] theorem Iic_top : Iic (⊤ : α) = univ := isTop_top.Iic_eq #align set.Iic_top Set.Iic_top @[simp] theorem Icc_top : Icc a ⊤ = Ici a := by simp [← Ici_inter_Iic] #align set.Icc_top Set.Icc_top @[simp] theorem Ioc_top : Ioc a ⊤ = Ioi a := by simp [← Ioi_inter_Iic] #align set.Ioc_top Set.Ioc_top end OrderTop section OrderBot @[simp] theorem Iic_bot [PartialOrder α] [OrderBot α] : Iic (⊥ : α) = {⊥} := isMin_bot.Iic_eq #align set.Iic_bot Set.Iic_bot variable [Preorder α] [OrderBot α] {a : α} @[simp] theorem Iio_bot : Iio (⊥ : α) = ∅ := isMin_bot.Iio_eq #align set.Iio_bot Set.Iio_bot @[simp] theorem Ici_bot : Ici (⊥ : α) = univ := isBot_bot.Ici_eq #align set.Ici_bot Set.Ici_bot @[simp] theorem Icc_bot : Icc ⊥ a = Iic a := by simp [← Ici_inter_Iic] #align set.Icc_bot Set.Icc_bot @[simp] theorem Ico_bot : Ico ⊥ a = Iio a := by simp [← Ici_inter_Iio] #align set.Ico_bot Set.Ico_bot end OrderBot theorem Icc_bot_top [PartialOrder α] [BoundedOrder α] : Icc (⊥ : α) ⊤ = univ := by simp #align set.Icc_bot_top Set.Icc_bot_top section LinearOrder variable [LinearOrder α] {a a₁ a₂ b b₁ b₂ c d : α} theorem not_mem_Ici : c ∉ Ici a ↔ c < a := not_le #align set.not_mem_Ici Set.not_mem_Ici theorem not_mem_Iic : c ∉ Iic b ↔ b < c := not_le #align set.not_mem_Iic Set.not_mem_Iic theorem not_mem_Ioi : c ∉ Ioi a ↔ c ≤ a := not_lt #align set.not_mem_Ioi Set.not_mem_Ioi theorem not_mem_Iio : c ∉ Iio b ↔ b ≤ c := not_lt #align set.not_mem_Iio Set.not_mem_Iio @[simp] theorem compl_Iic : (Iic a)ᶜ = Ioi a := ext fun _ => not_le #align set.compl_Iic Set.compl_Iic @[simp] theorem compl_Ici : (Ici a)ᶜ = Iio a := ext fun _ => not_le #align set.compl_Ici Set.compl_Ici @[simp] theorem compl_Iio : (Iio a)ᶜ = Ici a := ext fun _ => not_lt #align set.compl_Iio Set.compl_Iio @[simp] theorem compl_Ioi : (Ioi a)ᶜ = Iic a := ext fun _ => not_lt #align set.compl_Ioi Set.compl_Ioi @[simp] theorem Ici_diff_Ici : Ici a \ Ici b = Ico a b := by rw [diff_eq, compl_Ici, Ici_inter_Iio] #align set.Ici_diff_Ici Set.Ici_diff_Ici @[simp] theorem Ici_diff_Ioi : Ici a \ Ioi b = Icc a b := by rw [diff_eq, compl_Ioi, Ici_inter_Iic] #align set.Ici_diff_Ioi Set.Ici_diff_Ioi @[simp] theorem Ioi_diff_Ioi : Ioi a \ Ioi b = Ioc a b := by rw [diff_eq, compl_Ioi, Ioi_inter_Iic] #align set.Ioi_diff_Ioi Set.Ioi_diff_Ioi @[simp] theorem Ioi_diff_Ici : Ioi a \ Ici b = Ioo a b := by rw [diff_eq, compl_Ici, Ioi_inter_Iio] #align set.Ioi_diff_Ici Set.Ioi_diff_Ici @[simp] theorem Iic_diff_Iic : Iic b \ Iic a = Ioc a b := by rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iic] #align set.Iic_diff_Iic Set.Iic_diff_Iic @[simp] theorem Iio_diff_Iic : Iio b \ Iic a = Ioo a b := by rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iio] #align set.Iio_diff_Iic Set.Iio_diff_Iic @[simp] theorem Iic_diff_Iio : Iic b \ Iio a = Icc a b := by rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iic] #align set.Iic_diff_Iio Set.Iic_diff_Iio @[simp] theorem Iio_diff_Iio : Iio b \ Iio a = Ico a b := by rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iio] #align set.Iio_diff_Iio Set.Iio_diff_Iio theorem Ioi_injective : Injective (Ioi : α → Set α) := fun _ _ => eq_of_forall_gt_iff ∘ Set.ext_iff.1 #align set.Ioi_injective Set.Ioi_injective theorem Iio_injective : Injective (Iio : α → Set α) := fun _ _ => eq_of_forall_lt_iff ∘ Set.ext_iff.1 #align set.Iio_injective Set.Iio_injective theorem Ioi_inj : Ioi a = Ioi b ↔ a = b := Ioi_injective.eq_iff #align set.Ioi_inj Set.Ioi_inj theorem Iio_inj : Iio a = Iio b ↔ a = b := Iio_injective.eq_iff #align set.Iio_inj Set.Iio_inj theorem Ico_subset_Ico_iff (h₁ : a₁ < b₁) : Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => have : a₂ ≤ a₁ ∧ a₁ < b₂ := h ⟨le_rfl, h₁⟩ ⟨this.1, le_of_not_lt fun h' => lt_irrefl b₂ (h ⟨this.2.le, h'⟩).2⟩, fun ⟨h₁, h₂⟩ => Ico_subset_Ico h₁ h₂⟩ #align set.Ico_subset_Ico_iff Set.Ico_subset_Ico_iff theorem Ioc_subset_Ioc_iff (h₁ : a₁ < b₁) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ b₁ ≤ b₂ ∧ a₂ ≤ a₁ := by convert @Ico_subset_Ico_iff αᵒᵈ _ b₁ b₂ a₁ a₂ h₁ using 2 <;> exact (@dual_Ico α _ _ _).symm #align set.Ioc_subset_Ioc_iff Set.Ioc_subset_Ioc_iff theorem Ioo_subset_Ioo_iff [DenselyOrdered α] (h₁ : a₁ < b₁) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => by rcases exists_between h₁ with ⟨x, xa, xb⟩ constructor <;> refine le_of_not_lt fun h' => ?_ · have ab := (h ⟨xa, xb⟩).1.trans xb exact lt_irrefl _ (h ⟨h', ab⟩).1 · have ab := xa.trans (h ⟨xa, xb⟩).2 exact lt_irrefl _ (h ⟨ab, h'⟩).2, fun ⟨h₁, h₂⟩ => Ioo_subset_Ioo h₁ h₂⟩ #align set.Ioo_subset_Ioo_iff Set.Ioo_subset_Ioo_iff theorem Ico_eq_Ico_iff (h : a₁ < b₁ ∨ a₂ < b₂) : Ico a₁ b₁ = Ico a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ := ⟨fun e => by simp only [Subset.antisymm_iff] at e simp only [le_antisymm_iff] cases' h with h h <;> simp only [gt_iff_lt, not_lt, ge_iff_le, Ico_subset_Ico_iff h] at e <;> [ rcases e with ⟨⟨h₁, h₂⟩, e'⟩; rcases e with ⟨e', ⟨h₁, h₂⟩⟩ ] <;> -- Porting note: restore `tauto` have hab := (Ico_subset_Ico_iff <\| h₁.trans_lt <\| h.trans_le h₂).1 e' <;> [ exact ⟨⟨hab.left, h₁⟩, ⟨h₂, hab.right⟩⟩; exact ⟨⟨h₁, hab.left⟩, ⟨hab.right, h₂⟩⟩ ], fun ⟨h₁, h₂⟩ => by rw [h₁, h₂]⟩ #align set.Ico_eq_Ico_iff Set.Ico_eq_Ico_iff lemma Ici_eq_singleton_iff_isTop {x : α} : (Ici x = {x}) ↔ IsTop x := by refine ⟨fun h y ↦ ?_, fun h ↦ by ext y; simp [(h y).ge_iff_eq]⟩ by_contra! H have : y ∈ Ici x := H.le rw [h, mem_singleton_iff] at this exact lt_irrefl y (this.le.trans_lt H) open scoped Classical @[simp] theorem Ioi_subset_Ioi_iff : Ioi b ⊆ Ioi a ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Ioi_subset_Ioi h⟩ by_contra ba exact lt_irrefl _ (h (not_le.mp ba)) #align set.Ioi_subset_Ioi_iff Set.Ioi_subset_Ioi_iff @[simp] theorem Ioi_subset_Ici_iff [DenselyOrdered α] : Ioi b ⊆ Ici a ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Ioi_subset_Ici h⟩ by_contra ba obtain ⟨c, bc, ca⟩ : ∃ c, b < c ∧ c < a := exists_between (not_le.mp ba) exact lt_irrefl _ (ca.trans_le (h bc)) #align set.Ioi_subset_Ici_iff Set.Ioi_subset_Ici_iff @[simp] theorem Iio_subset_Iio_iff : Iio a ⊆ Iio b ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Iio_subset_Iio h⟩ by_contra ab exact lt_irrefl _ (h (not_le.mp ab)) #align set.Iio_subset_Iio_iff Set.Iio_subset_Iio_iff @[simp] theorem Iio_subset_Iic_iff [DenselyOrdered α] : Iio a ⊆ Iic b ↔ a ≤ b := by rw [← diff_eq_empty, Iio_diff_Iic, Ioo_eq_empty_iff, not_lt] #align set.Iio_subset_Iic_iff Set.Iio_subset_Iic_iff /-! ### Unions of adjacent intervals -/ /-! #### Two infinite intervals -/ theorem Iic_union_Ioi_of_le (h : a ≤ b) : Iic b ∪ Ioi a = univ := eq_univ_of_forall fun x => (h.lt_or_le x).symm #align set.Iic_union_Ioi_of_le Set.Iic_union_Ioi_of_le theorem Iio_union_Ici_of_le (h : a ≤ b) : Iio b ∪ Ici a = univ := eq_univ_of_forall fun x => (h.le_or_lt x).symm #align set.Iio_union_Ici_of_le Set.Iio_union_Ici_of_le theorem Iic_union_Ici_of_le (h : a ≤ b) : Iic b ∪ Ici a = univ := eq_univ_of_forall fun x => (h.le_or_le x).symm #align set.Iic_union_Ici_of_le Set.Iic_union_Ici_of_le theorem Iio_union_Ioi_of_lt (h : a < b) : Iio b ∪ Ioi a = univ := eq_univ_of_forall fun x => (h.lt_or_lt x).symm #align set.Iio_union_Ioi_of_lt Set.Iio_union_Ioi_of_lt @[simp] theorem Iic_union_Ici : Iic a ∪ Ici a = univ := Iic_union_Ici_of_le le_rfl #align set.Iic_union_Ici Set.Iic_union_Ici @[simp] theorem Iio_union_Ici : Iio a ∪ Ici a = univ := Iio_union_Ici_of_le le_rfl #align set.Iio_union_Ici Set.Iio_union_Ici @[simp] theorem Iic_union_Ioi : Iic a ∪ Ioi a = univ := Iic_union_Ioi_of_le le_rfl #align set.Iic_union_Ioi Set.Iic_union_Ioi @[simp] theorem Iio_union_Ioi : Iio a ∪ Ioi a = {a}ᶜ := ext fun _ => lt_or_lt_iff_ne #align set.Iio_union_Ioi Set.Iio_union_Ioi /-! #### A finite and an infinite interval -/ theorem Ioo_union_Ioi' (h₁ : c < b) : Ioo a b ∪ Ioi c = Ioi (min a c) := by ext1 x simp_rw [mem_union, mem_Ioo, mem_Ioi, min_lt_iff] by_cases hc : c < x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x < b := (le_of_not_gt hc).trans_lt h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ioo_union_Ioi' Set.Ioo_union_Ioi' theorem Ioo_union_Ioi (h : c < max a b) : Ioo a b ∪ Ioi c = Ioi (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ioo_union_Ioi' h · rw [min_comm] simp [, min_eq_left_of_lt] #align set.Ioo_union_Ioi Set.Ioo_union_Ioi theorem Ioi_subset_Ioo_union_Ici : Ioi a ⊆ Ioo a b ∪ Ici b := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ioi_subset_Ioo_union_Ici Set.Ioi_subset_Ioo_union_Ici @[simp] theorem Ioo_union_Ici_eq_Ioi (h : a < b) : Ioo a b ∪ Ici b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_le) Ioi_subset_Ioo_union_Ici #align set.Ioo_union_Ici_eq_Ioi Set.Ioo_union_Ici_eq_Ioi theorem Ici_subset_Ico_union_Ici : Ici a ⊆ Ico a b ∪ Ici b := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ici_subset_Ico_union_Ici Set.Ici_subset_Ico_union_Ici @[simp] theorem Ico_union_Ici_eq_Ici (h : a ≤ b) : Ico a b ∪ Ici b = Ici a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans) Ici_subset_Ico_union_Ici #align set.Ico_union_Ici_eq_Ici Set.Ico_union_Ici_eq_Ici theorem Ico_union_Ici' (h₁ : c ≤ b) : Ico a b ∪ Ici c = Ici (min a c) := by ext1 x simp_rw [mem_union, mem_Ico, mem_Ici, min_le_iff] by_cases hc : c ≤ x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ico_union_Ici' Set.Ico_union_Ici' theorem Ico_union_Ici (h : c ≤ max a b) : Ico a b ∪ Ici c = Ici (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ico_union_Ici' h · simp [] #align set.Ico_union_Ici Set.Ico_union_Ici theorem Ioi_subset_Ioc_union_Ioi : Ioi a ⊆ Ioc a b ∪ Ioi b := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ioi_subset_Ioc_union_Ioi Set.Ioi_subset_Ioc_union_Ioi @[simp] theorem Ioc_union_Ioi_eq_Ioi (h : a ≤ b) : Ioc a b ∪ Ioi b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_lt) Ioi_subset_Ioc_union_Ioi #align set.Ioc_union_Ioi_eq_Ioi Set.Ioc_union_Ioi_eq_Ioi theorem Ioc_union_Ioi' (h₁ : c ≤ b) : Ioc a b ∪ Ioi c = Ioi (min a c) := by ext1 x simp_rw [mem_union, mem_Ioc, mem_Ioi, min_lt_iff] by_cases hc : c < x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ioc_union_Ioi' Set.Ioc_union_Ioi' theorem Ioc_union_Ioi (h : c ≤ max a b) : Ioc a b ∪ Ioi c = Ioi (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ioc_union_Ioi' h · simp [] #align set.Ioc_union_Ioi Set.Ioc_union_Ioi theorem Ici_subset_Icc_union_Ioi : Ici a ⊆ Icc a b ∪ Ioi b := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ici_subset_Icc_union_Ioi Set.Ici_subset_Icc_union_Ioi @[simp] theorem Icc_union_Ioi_eq_Ici (h : a ≤ b) : Icc a b ∪ Ioi b = Ici a := Subset.antisymm (fun _ hx => (hx.elim And.left) fun hx' => h.trans <\| le_of_lt hx') Ici_subset_Icc_union_Ioi #align set.Icc_union_Ioi_eq_Ici Set.Icc_union_Ioi_eq_Ici theorem Ioi_subset_Ioc_union_Ici : Ioi a ⊆ Ioc a b ∪ Ici b := Subset.trans Ioi_subset_Ioo_union_Ici (union_subset_union_left _ Ioo_subset_Ioc_self) #align set.Ioi_subset_Ioc_union_Ici Set.Ioi_subset_Ioc_union_Ici @[simp] theorem Ioc_union_Ici_eq_Ioi (h : a < b) : Ioc a b ∪ Ici b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_le) Ioi_subset_Ioc_union_Ici #align set.Ioc_union_Ici_eq_Ioi Set.Ioc_union_Ici_eq_Ioi theorem Ici_subset_Icc_union_Ici : Ici a ⊆ Icc a b ∪ Ici b := Subset.trans Ici_subset_Ico_union_Ici (union_subset_union_left _ Ico_subset_Icc_self) #align set.Ici_subset_Icc_union_Ici Set.Ici_subset_Icc_union_Ici @[simp] theorem Icc_union_Ici_eq_Ici (h : a ≤ b) : Icc a b ∪ Ici b = Ici a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans) Ici_subset_Icc_union_Ici #align set.Icc_union_Ici_eq_Ici Set.Icc_union_Ici_eq_Ici theorem Icc_union_Ici' (h₁ : c ≤ b) : Icc a b ∪ Ici c = Ici (min a c) := by ext1 x simp_rw [mem_union, mem_Icc, mem_Ici, min_le_iff] by_cases hc : c ≤ x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Icc_union_Ici' Set.Icc_union_Ici' theorem Icc_union_Ici (h : c ≤ max a b) : Icc a b ∪ Ici c = Ici (min a c) := by rcases le_or_lt a b with hab \| hab <;> simp [hab] at h · exact Icc_union_Ici' h · cases' h with h h · simp [] · have hca : c ≤ a := h.trans hab.le simp [] #align set.Icc_union_Ici Set.Icc_union_Ici /-! #### An infinite and a finite interval -/ theorem Iic_subset_Iio_union_Icc : Iic b ⊆ Iio a ∪ Icc a b := fun x hx => (lt_or_le x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iic_subset_Iio_union_Icc Set.Iic_subset_Iio_union_Icc @[simp] theorem Iio_union_Icc_eq_Iic (h : a ≤ b) : Iio a ∪ Icc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx => (le_of_lt hx).trans h) And.right) Iic_subset_Iio_union_Icc #align set.Iio_union_Icc_eq_Iic Set.Iio_union_Icc_eq_Iic theorem Iio_subset_Iio_union_Ico : Iio b ⊆ Iio a ∪ Ico a b := fun x hx => (lt_or_le x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iio_subset_Iio_union_Ico Set.Iio_subset_Iio_union_Ico @[simp] theorem Iio_union_Ico_eq_Iio (h : a ≤ b) : Iio a ∪ Ico a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_lt_of_le hx' h) And.right) Iio_subset_Iio_union_Ico #align set.Iio_union_Ico_eq_Iio Set.Iio_union_Ico_eq_Iio theorem Iio_union_Ico' (h₁ : c ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by ext1 x simp_rw [mem_union, mem_Iio, mem_Ico, lt_max_iff] by_cases hc : c ≤ x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iio_union_Ico' Set.Iio_union_Ico' theorem Iio_union_Ico (h : min c d ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iio_union_Ico' h · simp [] #align set.Iio_union_Ico Set.Iio_union_Ico theorem Iic_subset_Iic_union_Ioc : Iic b ⊆ Iic a ∪ Ioc a b := fun x hx => (le_or_lt x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iic_subset_Iic_union_Ioc Set.Iic_subset_Iic_union_Ioc @[simp] theorem Iic_union_Ioc_eq_Iic (h : a ≤ b) : Iic a ∪ Ioc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => le_trans hx' h) And.right) Iic_subset_Iic_union_Ioc #align set.Iic_union_Ioc_eq_Iic Set.Iic_union_Ioc_eq_Iic theorem Iic_union_Ioc' (h₁ : c < b) : Iic b ∪ Ioc c d = Iic (max b d) := by ext1 x simp_rw [mem_union, mem_Iic, mem_Ioc, le_max_iff] by_cases hc : c < x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁.le simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iic_union_Ioc' Set.Iic_union_Ioc' theorem Iic_union_Ioc (h : min c d < b) : Iic b ∪ Ioc c d = Iic (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iic_union_Ioc' h · rw [max_comm] simp [, max_eq_right_of_lt h] #align set.Iic_union_Ioc Set.Iic_union_Ioc theorem Iio_subset_Iic_union_Ioo : Iio b ⊆ Iic a ∪ Ioo a b := fun x hx => (le_or_lt x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iio_subset_Iic_union_Ioo Set.Iio_subset_Iic_union_Ioo @[simp] theorem Iic_union_Ioo_eq_Iio (h : a < b) : Iic a ∪ Ioo a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_le_of_lt hx' h) And.right) Iio_subset_Iic_union_Ioo #align set.Iic_union_Ioo_eq_Iio Set.Iic_union_Ioo_eq_Iio theorem Iio_union_Ioo' (h₁ : c < b) : Iio b ∪ Ioo c d = Iio (max b d) := by ext x cases' lt_or_le x b with hba hba · simp [hba, h₁] · simp only [mem_Iio, mem_union, mem_Ioo, lt_max_iff] refine or_congr Iff.rfl ⟨And.right, ?_⟩ exact fun h₂ => ⟨h₁.trans_le hba, h₂⟩ #align set.Iio_union_Ioo' Set.Iio_union_Ioo' theorem Iio_union_Ioo (h : min c d < b) : Iio b ∪ Ioo c d = Iio (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iio_union_Ioo' h · rw [max_comm] simp [, max_eq_right_of_lt h] #align set.Iio_union_Ioo Set.Iio_union_Ioo theorem Iic_subset_Iic_union_Icc : Iic b ⊆ Iic a ∪ Icc a b := Subset.trans Iic_subset_Iic_union_Ioc (union_subset_union_right _ Ioc_subset_Icc_self) #align set.Iic_subset_Iic_union_Icc Set.Iic_subset_Iic_union_Icc @[simp] theorem Iic_union_Icc_eq_Iic (h : a ≤ b) : Iic a ∪ Icc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => le_trans hx' h) And.right) Iic_subset_Iic_union_Icc #align set.Iic_union_Icc_eq_Iic Set.Iic_union_Icc_eq_Iic theorem Iic_union_Icc' (h₁ : c ≤ b) : Iic b ∪ Icc c d = Iic (max b d) := by ext1 x simp_rw [mem_union, mem_Iic, mem_Icc, le_max_iff] by_cases hc : c ≤ x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iic_union_Icc' Set.Iic_union_Icc' theorem Iic_union_Icc (h : min c d ≤ b) : Iic b ∪ Icc c d = Iic (max b d) := by rcases le_or_lt c d with hcd \| hcd <;> simp [hcd] at h · exact Iic_union_Icc' h · cases' h with h h · have hdb : d ≤ b := hcd.le.trans h simp [] · simp [] #align set.Iic_union_Icc Set.Iic_union_Icc theorem Iio_subset_Iic_union_Ico : Iio b ⊆ Iic a ∪ Ico a b := Subset.trans Iio_subset_Iic_union_Ioo (union_subset_union_right _ Ioo_subset_Ico_self) #align set.Iio_subset_Iic_union_Ico Set.Iio_subset_Iic_union_Ico @[simp] theorem Iic_union_Ico_eq_Iio (h : a < b) : Iic a ∪ Ico a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_le_of_lt hx' h) And.right) Iio_subset_Iic_union_Ico #align set.Iic_union_Ico_eq_Iio Set.Iic_union_Ico_eq_Iio /-! #### Two finite intervals, `I?o` and `Ic?` -/ theorem Ioo_subset_Ioo_union_Ico : Ioo a c ⊆ Ioo a b ∪ Ico b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioo_subset_Ioo_union_Ico Set.Ioo_subset_Ioo_union_Ico @[simp] theorem Ioo_union_Ico_eq_Ioo (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Ico b c = Ioo a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_le h₂⟩) fun hx => ⟨h₁.trans_le hx.1, hx.2⟩) Ioo_subset_Ioo_union_Ico #align set.Ioo_union_Ico_eq_Ioo Set.Ioo_union_Ico_eq_Ioo theorem Ico_subset_Ico_union_Ico : Ico a c ⊆ Ico a b ∪ Ico b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ico_subset_Ico_union_Ico Set.Ico_subset_Ico_union_Ico @[simp] theorem Ico_union_Ico_eq_Ico (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Ico b c = Ico a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_le h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Ico_subset_Ico_union_Ico #align set.Ico_union_Ico_eq_Ico Set.Ico_union_Ico_eq_Ico theorem Ico_union_Ico' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ico a b ∪ Ico c d = Ico (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Ico, min_le_iff, lt_max_iff] by_cases hc : c ≤ x <;> by_cases hd : x < d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a ≤ x := h₂.trans (le_of_not_gt hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Ico_union_Ico' Set.Ico_union_Ico' theorem Ico_union_Ico (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b) : Ico a b ∪ Ico c d = Ico (min a c) (max b d) := by rcases le_total a b with hab \| hab <;> rcases le_total c d with hcd \| hcd <;> simp [] at h₁ h₂ · exact Ico_union_Ico' h₂ h₁ all_goals simp [] #align set.Ico_union_Ico Set.Ico_union_Ico theorem Icc_subset_Ico_union_Icc : Icc a c ⊆ Ico a b ∪ Icc b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Icc_subset_Ico_union_Icc Set.Icc_subset_Ico_union_Icc @[simp] theorem Ico_union_Icc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Icc b c = Icc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.le.trans h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Icc_subset_Ico_union_Icc #align set.Ico_union_Icc_eq_Icc Set.Ico_union_Icc_eq_Icc theorem Ioc_subset_Ioo_union_Icc : Ioc a c ⊆ Ioo a b ∪ Icc b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioc_subset_Ioo_union_Icc Set.Ioc_subset_Ioo_union_Icc @[simp] theorem Ioo_union_Icc_eq_Ioc (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Icc b c = Ioc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.le.trans h₂⟩) fun hx => ⟨h₁.trans_le hx.1, hx.2⟩) Ioc_subset_Ioo_union_Icc #align set.Ioo_union_Icc_eq_Ioc Set.Ioo_union_Icc_eq_Ioc /-! #### Two finite intervals, `I?c` and `Io?` -/ theorem Ioo_subset_Ioc_union_Ioo : Ioo a c ⊆ Ioc a b ∪ Ioo b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioo_subset_Ioc_union_Ioo Set.Ioo_subset_Ioc_union_Ioo @[simp] theorem Ioc_union_Ioo_eq_Ioo (h₁ : a ≤ b) (h₂ : b < c) : Ioc a b ∪ Ioo b c = Ioo a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans_lt hx.1, hx.2⟩) Ioo_subset_Ioc_union_Ioo #align set.Ioc_union_Ioo_eq_Ioo Set.Ioc_union_Ioo_eq_Ioo theorem Ico_subset_Icc_union_Ioo : Ico a c ⊆ Icc a b ∪ Ioo b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ico_subset_Icc_union_Ioo Set.Ico_subset_Icc_union_Ioo @[simp] theorem Icc_union_Ioo_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ioo b c = Ico a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans hx.1.le, hx.2⟩) Ico_subset_Icc_union_Ioo #align set.Icc_union_Ioo_eq_Ico Set.Icc_union_Ioo_eq_Ico theorem Icc_subset_Icc_union_Ioc : Icc a c ⊆ Icc a b ∪ Ioc b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Icc_subset_Icc_union_Ioc Set.Icc_subset_Icc_union_Ioc @[simp] theorem Icc_union_Ioc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Ioc b c = Icc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans hx.1.le, hx.2⟩) Icc_subset_Icc_union_Ioc #align set.Icc_union_Ioc_eq_Icc Set.Icc_union_Ioc_eq_Icc theorem Ioc_subset_Ioc_union_Ioc : Ioc a c ⊆ Ioc a b ∪ Ioc b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioc_subset_Ioc_union_Ioc Set.Ioc_subset_Ioc_union_Ioc @[simp] theorem Ioc_union_Ioc_eq_Ioc (h₁ : a ≤ b) (h₂ : b ≤ c) : Ioc a b ∪ Ioc b c = Ioc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans_lt hx.1, hx.2⟩) Ioc_subset_Ioc_union_Ioc #align set.Ioc_union_Ioc_eq_Ioc Set.Ioc_union_Ioc_eq_Ioc theorem Ioc_union_Ioc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Ioc, min_lt_iff, le_max_iff] by_cases hc : c < x <;> by_cases hd : x ≤ d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a < x := h₂.trans_lt (lt_of_not_ge hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Ioc_union_Ioc' Set.Ioc_union_Ioc' theorem Ioc_union_Ioc (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b) : Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) := by rcases le_total a b with hab \| hab <;> rcases le_total c d with hcd \| hcd <;> simp [] at h₁ h₂ · exact Ioc_union_Ioc' h₂ h₁ all_goals simp [*] #align set.Ioc_union_Ioc Set.Ioc_union_Ioc /-! #### Two finite intervals with a common point -/ theorem Ioo_subset_Ioc_union_Ico : Ioo a c ⊆ Ioc a b ∪ Ico b c := Subset.trans Ioo_subset_Ioc_union_Ioo (union_subset_union_right _ Ioo_subset_Ico_self) #align set.Ioo_subset_Ioc_union_Ico Set.Ioo_subset_Ioc_union_Ico @[simp] theorem Ioc_union_Ico_eq_Ioo (h₁ : a < b) (h₂ : b < c) : Ioc a b ∪ Ico b c = Ioo a c := Subset.antisymm (fun _ hx => hx.elim (fun hx' => ⟨hx'.1, hx'.2.trans_lt h₂⟩) fun hx' => ⟨h₁.trans_le hx'.1, hx'.2⟩) Ioo_subset_Ioc_union_Ico #align set.Ioc_union_Ico_eq_Ioo Set.Ioc_union_Ico_eq_Ioo theorem Ico_subset_Icc_union_Ico : Ico a c ⊆ Icc a b ∪ Ico b c := Subset.trans Ico_subset_Icc_union_Ioo (union_subset_union_right _ Ioo_subset_Ico_self) #align set.Ico_subset_Icc_union_Ico Set.Ico_subset_Icc_union_Ico @[simp] theorem Icc_union_Ico_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ico b c = Ico a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Ico_subset_Icc_union_Ico #align set.Icc_union_Ico_eq_Ico Set.Icc_union_Ico_eq_Ico theorem Icc_subset_Icc_union_Icc : Icc a c ⊆ Icc a b ∪ Icc b c := Subset.trans Icc_subset_Icc_union_Ioc (union_subset_union_right _ Ioc_subset_Icc_self) #align set.Icc_subset_Icc_union_Icc Set.Icc_subset_Icc_union_Icc @[simp] theorem Icc_union_Icc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Icc b c = Icc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Icc_subset_Icc_union_Icc #align set.Icc_union_Icc_eq_Icc Set.Icc_union_Icc_eq_Icc theorem Icc_union_Icc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Icc a b ∪ Icc c d = Icc (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Icc, min_le_iff, le_max_iff] by_cases hc : c ≤ x <;> by_cases hd : x ≤ d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a ≤ x := h₂.trans (le_of_not_ge hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Icc_union_Icc' Set.Icc_union_Icc' /-- We cannot replace `<` by `≤` in the hypotheses. Otherwise for `b < a = d < c` the l.h.s. is `∅` and the r.h.s. is `{a}`. -/	Mathlib/Order/Interval/Set/Basic.lean	1,715	1,721	theorem Icc_union_Icc (h₁ : min a b < max c d) (h₂ : min c d < max a b) : Icc a b ∪ Icc c d = Icc (min a c) (max b d) := by	rcases le_or_lt a b with hab \| hab <;> rcases le_or_lt c d with hcd \| hcd <;> simp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, hab, hcd] at h₁ h₂ · exact Icc_union_Icc' h₂.le h₁.le all_goals simp [*, min_eq_left_of_lt, max_eq_left_of_lt, min_eq_right_of_lt, max_eq_right_of_lt]
/- Copyright (c) 2022 Eric Rodriguez. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Rodriguez -/ import Mathlib.Algebra.GroupWithZero.Units.Lemmas import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.Data.Fintype.BigOperators #align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c" /-! # Sign function This file defines the sign function for types with zero and a decidable less-than relation, and proves some basic theorems about it. -/ -- Porting note (#11081): cannot automatically derive Fintype, added manually /-- The type of signs. -/ inductive SignType \| zero \| neg \| pos deriving DecidableEq, Inhabited #align sign_type SignType -- Porting note: these lemmas are autogenerated by the inductive definition and are not -- in simple form due to the below `x_eq_x` lemmas attribute [nolint simpNF] SignType.zero.sizeOf_spec attribute [nolint simpNF] SignType.neg.sizeOf_spec attribute [nolint simpNF] SignType.pos.sizeOf_spec namespace SignType -- Porting note: Added Fintype SignType manually instance : Fintype SignType := Fintype.ofMultiset (zero :: neg :: pos :: List.nil) (fun x ↦ by cases x <;> simp) instance : Zero SignType := ⟨zero⟩ instance : One SignType := ⟨pos⟩ instance : Neg SignType := ⟨fun s => match s with \| neg => pos \| zero => zero \| pos => neg⟩ @[simp] theorem zero_eq_zero : zero = 0 := rfl #align sign_type.zero_eq_zero SignType.zero_eq_zero @[simp] theorem neg_eq_neg_one : neg = -1 := rfl #align sign_type.neg_eq_neg_one SignType.neg_eq_neg_one @[simp] theorem pos_eq_one : pos = 1 := rfl #align sign_type.pos_eq_one SignType.pos_eq_one instance : Mul SignType := ⟨fun x y => match x with \| neg => -y \| zero => zero \| pos => y⟩ /-- The less-than-or-equal relation on signs. -/ protected inductive LE : SignType → SignType → Prop \| of_neg (a) : SignType.LE neg a \| zero : SignType.LE zero zero \| of_pos (a) : SignType.LE a pos #align sign_type.le SignType.LE instance : LE SignType := ⟨SignType.LE⟩ instance LE.decidableRel : DecidableRel SignType.LE := fun a b => by cases a <;> cases b <;> first \| exact isTrue (by constructor)\| exact isFalse (by rintro ⟨_⟩) instance decidableEq : DecidableEq SignType := fun a b => by cases a <;> cases b <;> first \| exact isTrue (by constructor)\| exact isFalse (by rintro ⟨_⟩) private lemma mul_comm : ∀ (a b : SignType), a * b = b * a := by rintro ⟨⟩ ⟨⟩ <;> rfl private lemma mul_assoc : ∀ (a b c : SignType), (a * b) * c = a * (b * c) := by rintro ⟨⟩ ⟨⟩ ⟨⟩ <;> rfl /- We can define a `Field` instance on `SignType`, but it's not mathematically sensible, so we only define the `CommGroupWithZero`. -/ instance : CommGroupWithZero SignType where zero := 0 one := 1 mul := (· * ·) inv := id mul_zero a := by cases a <;> rfl zero_mul a := by cases a <;> rfl mul_one a := by cases a <;> rfl one_mul a := by cases a <;> rfl mul_inv_cancel a ha := by cases a <;> trivial mul_comm := mul_comm mul_assoc := mul_assoc exists_pair_ne := ⟨0, 1, by rintro ⟨_⟩⟩ inv_zero := rfl private lemma le_antisymm (a b : SignType) (_ : a ≤ b) (_: b ≤ a) : a = b := by cases a <;> cases b <;> trivial private lemma le_trans (a b c : SignType) (_ : a ≤ b) (_: b ≤ c) : a ≤ c := by cases a <;> cases b <;> cases c <;> tauto instance : LinearOrder SignType where le := (· ≤ ·) le_refl a := by cases a <;> constructor le_total a b := by cases a <;> cases b <;> first \| left; constructor \| right; constructor le_antisymm := le_antisymm le_trans := le_trans decidableLE := LE.decidableRel decidableEq := SignType.decidableEq instance : BoundedOrder SignType where top := 1 le_top := LE.of_pos bot := -1 bot_le := LE.of_neg instance : HasDistribNeg SignType := { neg_neg := fun x => by cases x <;> rfl neg_mul := fun x y => by cases x <;> cases y <;> rfl mul_neg := fun x y => by cases x <;> cases y <;> rfl } /-- `SignType` is equivalent to `Fin 3`. -/ def fin3Equiv : SignType ≃* Fin 3 where toFun a := match a with \| 0 => ⟨0, by simp⟩ \| 1 => ⟨1, by simp⟩ \| -1 => ⟨2, by simp⟩ invFun a := match a with \| ⟨0, _⟩ => 0 \| ⟨1, _⟩ => 1 \| ⟨2, _⟩ => -1 left_inv a := by cases a <;> rfl right_inv a := match a with \| ⟨0, _⟩ => by simp \| ⟨1, _⟩ => by simp \| ⟨2, _⟩ => by simp map_mul' a b := by cases a <;> cases b <;> rfl #align sign_type.fin3_equiv SignType.fin3Equiv section CaseBashing -- Porting note: a lot of these thms used to use decide! which is not implemented yet theorem nonneg_iff {a : SignType} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by cases a <;> decide #align sign_type.nonneg_iff SignType.nonneg_iff	Mathlib/Data/Sign.lean	165	165	theorem nonneg_iff_ne_neg_one {a : SignType} : 0 ≤ a ↔ a ≠ -1 := by	cases a <;> decide
/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou -/ import Mathlib.MeasureTheory.Function.LpOrder #align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f" /-! # Integrable functions and `L¹` space In the first part of this file, the predicate `Integrable` is defined and basic properties of integrable functions are proved. Such a predicate is already available under the name `Memℒp 1`. We give a direct definition which is easier to use, and show that it is equivalent to `Memℒp 1` In the second part, we establish an API between `Integrable` and the space `L¹` of equivalence classes of integrable functions, already defined as a special case of `L^p` spaces for `p = 1`. ## Notation * `α →₁[μ] β` is the type of `L¹` space, where `α` is a `MeasureSpace` and `β` is a `NormedAddCommGroup` with a `SecondCountableTopology`. `f : α →ₘ β` is a "function" in `L¹`. In comments, `[f]` is also used to denote an `L¹` function. `₁` can be typed as `\1`. ## Main definitions * Let `f : α → β` be a function, where `α` is a `MeasureSpace` and `β` a `NormedAddCommGroup`. Then `HasFiniteIntegral f` means `(∫⁻ a, ‖f a‖₊) < ∞`. * If `β` is moreover a `MeasurableSpace` then `f` is called `Integrable` if `f` is `Measurable` and `HasFiniteIntegral f` holds. ## Implementation notes To prove something for an arbitrary integrable function, a useful theorem is `Integrable.induction` in the file `SetIntegral`. ## Tags integrable, function space, l1 -/ noncomputable section open scoped Classical open Topology ENNReal MeasureTheory NNReal open Set Filter TopologicalSpace ENNReal EMetric MeasureTheory variable {α β γ δ : Type} {m : MeasurableSpace α} {μ ν : Measure α} [MeasurableSpace δ] variable [NormedAddCommGroup β] variable [NormedAddCommGroup γ] namespace MeasureTheory /-! ### Some results about the Lebesgue integral involving a normed group -/ theorem lintegral_nnnorm_eq_lintegral_edist (f : α → β) : ∫⁻ a, ‖f a‖₊ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by simp only [edist_eq_coe_nnnorm] #align measure_theory.lintegral_nnnorm_eq_lintegral_edist MeasureTheory.lintegral_nnnorm_eq_lintegral_edist theorem lintegral_norm_eq_lintegral_edist (f : α → β) : ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by simp only [ofReal_norm_eq_coe_nnnorm, edist_eq_coe_nnnorm] #align measure_theory.lintegral_norm_eq_lintegral_edist MeasureTheory.lintegral_norm_eq_lintegral_edist theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ) (hh : AEStronglyMeasurable h μ) : (∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ := by rw [← lintegral_add_left' (hf.edist hh)] refine lintegral_mono fun a => ?_ apply edist_triangle_right #align measure_theory.lintegral_edist_triangle MeasureTheory.lintegral_edist_triangle theorem lintegral_nnnorm_zero : (∫⁻ _ : α, ‖(0 : β)‖₊ ∂μ) = 0 := by simp #align measure_theory.lintegral_nnnorm_zero MeasureTheory.lintegral_nnnorm_zero theorem lintegral_nnnorm_add_left {f : α → β} (hf : AEStronglyMeasurable f μ) (g : α → γ) : ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂μ := lintegral_add_left' hf.ennnorm _ #align measure_theory.lintegral_nnnorm_add_left MeasureTheory.lintegral_nnnorm_add_left theorem lintegral_nnnorm_add_right (f : α → β) {g : α → γ} (hg : AEStronglyMeasurable g μ) : ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂μ := lintegral_add_right' _ hg.ennnorm #align measure_theory.lintegral_nnnorm_add_right MeasureTheory.lintegral_nnnorm_add_right theorem lintegral_nnnorm_neg {f : α → β} : (∫⁻ a, ‖(-f) a‖₊ ∂μ) = ∫⁻ a, ‖f a‖₊ ∂μ := by simp only [Pi.neg_apply, nnnorm_neg] #align measure_theory.lintegral_nnnorm_neg MeasureTheory.lintegral_nnnorm_neg /-! ### The predicate `HasFiniteIntegral` -/ /-- `HasFiniteIntegral f μ` means that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite. `HasFiniteIntegral f` means `HasFiniteIntegral f volume`. -/ def HasFiniteIntegral {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop := (∫⁻ a, ‖f a‖₊ ∂μ) < ∞ #align measure_theory.has_finite_integral MeasureTheory.HasFiniteIntegral theorem hasFiniteIntegral_def {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : HasFiniteIntegral f μ ↔ ((∫⁻ a, ‖f a‖₊ ∂μ) < ∞) := Iff.rfl theorem hasFiniteIntegral_iff_norm (f : α → β) : HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) < ∞ := by simp only [HasFiniteIntegral, ofReal_norm_eq_coe_nnnorm] #align measure_theory.has_finite_integral_iff_norm MeasureTheory.hasFiniteIntegral_iff_norm theorem hasFiniteIntegral_iff_edist (f : α → β) : HasFiniteIntegral f μ ↔ (∫⁻ a, edist (f a) 0 ∂μ) < ∞ := by simp only [hasFiniteIntegral_iff_norm, edist_dist, dist_zero_right] #align measure_theory.has_finite_integral_iff_edist MeasureTheory.hasFiniteIntegral_iff_edist theorem hasFiniteIntegral_iff_ofReal {f : α → ℝ} (h : 0 ≤ᵐ[μ] f) : HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal (f a) ∂μ) < ∞ := by rw [HasFiniteIntegral, lintegral_nnnorm_eq_of_ae_nonneg h] #align measure_theory.has_finite_integral_iff_of_real MeasureTheory.hasFiniteIntegral_iff_ofReal theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} : HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by simp [hasFiniteIntegral_iff_norm] #align measure_theory.has_finite_integral_iff_of_nnreal MeasureTheory.hasFiniteIntegral_iff_ofNNReal theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral f μ := by simp only [hasFiniteIntegral_iff_norm] at calc (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ := lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h) _ < ∞ := hg #align measure_theory.has_finite_integral.mono MeasureTheory.HasFiniteIntegral.mono theorem HasFiniteIntegral.mono' {f : α → β} {g : α → ℝ} (hg : HasFiniteIntegral g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : HasFiniteIntegral f μ := hg.mono <\| h.mono fun _x hx => le_trans hx (le_abs_self _) #align measure_theory.has_finite_integral.mono' MeasureTheory.HasFiniteIntegral.mono' theorem HasFiniteIntegral.congr' {f : α → β} {g : α → γ} (hf : HasFiniteIntegral f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral g μ := hf.mono <\| EventuallyEq.le <\| EventuallyEq.symm h #align measure_theory.has_finite_integral.congr' MeasureTheory.HasFiniteIntegral.congr' theorem hasFiniteIntegral_congr' {f : α → β} {g : α → γ} (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ := ⟨fun hf => hf.congr' h, fun hg => hg.congr' <\| EventuallyEq.symm h⟩ #align measure_theory.has_finite_integral_congr' MeasureTheory.hasFiniteIntegral_congr' theorem HasFiniteIntegral.congr {f g : α → β} (hf : HasFiniteIntegral f μ) (h : f =ᵐ[μ] g) : HasFiniteIntegral g μ := hf.congr' <\| h.fun_comp norm #align measure_theory.has_finite_integral.congr MeasureTheory.HasFiniteIntegral.congr theorem hasFiniteIntegral_congr {f g : α → β} (h : f =ᵐ[μ] g) : HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ := hasFiniteIntegral_congr' <\| h.fun_comp norm #align measure_theory.has_finite_integral_congr MeasureTheory.hasFiniteIntegral_congr theorem hasFiniteIntegral_const_iff {c : β} : HasFiniteIntegral (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ := by simp [HasFiniteIntegral, lintegral_const, lt_top_iff_ne_top, ENNReal.mul_eq_top, or_iff_not_imp_left] #align measure_theory.has_finite_integral_const_iff MeasureTheory.hasFiniteIntegral_const_iff theorem hasFiniteIntegral_const [IsFiniteMeasure μ] (c : β) : HasFiniteIntegral (fun _ : α => c) μ := hasFiniteIntegral_const_iff.2 (Or.inr <\| measure_lt_top _ _) #align measure_theory.has_finite_integral_const MeasureTheory.hasFiniteIntegral_const theorem hasFiniteIntegral_of_bounded [IsFiniteMeasure μ] {f : α → β} {C : ℝ} (hC : ∀ᵐ a ∂μ, ‖f a‖ ≤ C) : HasFiniteIntegral f μ := (hasFiniteIntegral_const C).mono' hC #align measure_theory.has_finite_integral_of_bounded MeasureTheory.hasFiniteIntegral_of_bounded theorem HasFiniteIntegral.of_finite [Finite α] [IsFiniteMeasure μ] {f : α → β} : HasFiniteIntegral f μ := let ⟨_⟩ := nonempty_fintype α hasFiniteIntegral_of_bounded <\| ae_of_all μ <\| norm_le_pi_norm f @[deprecated (since := "2024-02-05")] alias hasFiniteIntegral_of_fintype := HasFiniteIntegral.of_finite theorem HasFiniteIntegral.mono_measure {f : α → β} (h : HasFiniteIntegral f ν) (hμ : μ ≤ ν) : HasFiniteIntegral f μ := lt_of_le_of_lt (lintegral_mono' hμ le_rfl) h #align measure_theory.has_finite_integral.mono_measure MeasureTheory.HasFiniteIntegral.mono_measure theorem HasFiniteIntegral.add_measure {f : α → β} (hμ : HasFiniteIntegral f μ) (hν : HasFiniteIntegral f ν) : HasFiniteIntegral f (μ + ν) := by simp only [HasFiniteIntegral, lintegral_add_measure] at * exact add_lt_top.2 ⟨hμ, hν⟩ #align measure_theory.has_finite_integral.add_measure MeasureTheory.HasFiniteIntegral.add_measure theorem HasFiniteIntegral.left_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) : HasFiniteIntegral f μ := h.mono_measure <\| Measure.le_add_right <\| le_rfl #align measure_theory.has_finite_integral.left_of_add_measure MeasureTheory.HasFiniteIntegral.left_of_add_measure theorem HasFiniteIntegral.right_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) : HasFiniteIntegral f ν := h.mono_measure <\| Measure.le_add_left <\| le_rfl #align measure_theory.has_finite_integral.right_of_add_measure MeasureTheory.HasFiniteIntegral.right_of_add_measure @[simp] theorem hasFiniteIntegral_add_measure {f : α → β} : HasFiniteIntegral f (μ + ν) ↔ HasFiniteIntegral f μ ∧ HasFiniteIntegral f ν := ⟨fun h => ⟨h.left_of_add_measure, h.right_of_add_measure⟩, fun h => h.1.add_measure h.2⟩ #align measure_theory.has_finite_integral_add_measure MeasureTheory.hasFiniteIntegral_add_measure theorem HasFiniteIntegral.smul_measure {f : α → β} (h : HasFiniteIntegral f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) : HasFiniteIntegral f (c • μ) := by simp only [HasFiniteIntegral, lintegral_smul_measure] at * exact mul_lt_top hc h.ne #align measure_theory.has_finite_integral.smul_measure MeasureTheory.HasFiniteIntegral.smul_measure @[simp] theorem hasFiniteIntegral_zero_measure {m : MeasurableSpace α} (f : α → β) : HasFiniteIntegral f (0 : Measure α) := by simp only [HasFiniteIntegral, lintegral_zero_measure, zero_lt_top] #align measure_theory.has_finite_integral_zero_measure MeasureTheory.hasFiniteIntegral_zero_measure variable (α β μ) @[simp] theorem hasFiniteIntegral_zero : HasFiniteIntegral (fun _ : α => (0 : β)) μ := by simp [HasFiniteIntegral] #align measure_theory.has_finite_integral_zero MeasureTheory.hasFiniteIntegral_zero variable {α β μ} theorem HasFiniteIntegral.neg {f : α → β} (hfi : HasFiniteIntegral f μ) : HasFiniteIntegral (-f) μ := by simpa [HasFiniteIntegral] using hfi #align measure_theory.has_finite_integral.neg MeasureTheory.HasFiniteIntegral.neg @[simp] theorem hasFiniteIntegral_neg_iff {f : α → β} : HasFiniteIntegral (-f) μ ↔ HasFiniteIntegral f μ := ⟨fun h => neg_neg f ▸ h.neg, HasFiniteIntegral.neg⟩ #align measure_theory.has_finite_integral_neg_iff MeasureTheory.hasFiniteIntegral_neg_iff theorem HasFiniteIntegral.norm {f : α → β} (hfi : HasFiniteIntegral f μ) : HasFiniteIntegral (fun a => ‖f a‖) μ := by have eq : (fun a => (nnnorm ‖f a‖ : ℝ≥0∞)) = fun a => (‖f a‖₊ : ℝ≥0∞) := by funext rw [nnnorm_norm] rwa [HasFiniteIntegral, eq] #align measure_theory.has_finite_integral.norm MeasureTheory.HasFiniteIntegral.norm theorem hasFiniteIntegral_norm_iff (f : α → β) : HasFiniteIntegral (fun a => ‖f a‖) μ ↔ HasFiniteIntegral f μ := hasFiniteIntegral_congr' <\| eventually_of_forall fun x => norm_norm (f x) #align measure_theory.has_finite_integral_norm_iff MeasureTheory.hasFiniteIntegral_norm_iff theorem hasFiniteIntegral_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hf : (∫⁻ x, f x ∂μ) ≠ ∞) : HasFiniteIntegral (fun x => (f x).toReal) μ := by have : ∀ x, (‖(f x).toReal‖₊ : ℝ≥0∞) = ENNReal.ofNNReal ⟨(f x).toReal, ENNReal.toReal_nonneg⟩ := by intro x rw [Real.nnnorm_of_nonneg] simp_rw [HasFiniteIntegral, this] refine lt_of_le_of_lt (lintegral_mono fun x => ?_) (lt_top_iff_ne_top.2 hf) by_cases hfx : f x = ∞ · simp [hfx] · lift f x to ℝ≥0 using hfx with fx h simp [← h, ← NNReal.coe_le_coe] #align measure_theory.has_finite_integral_to_real_of_lintegral_ne_top MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top theorem isFiniteMeasure_withDensity_ofReal {f : α → ℝ} (hfi : HasFiniteIntegral f μ) : IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal <\| f x) := by refine isFiniteMeasure_withDensity ((lintegral_mono fun x => ?_).trans_lt hfi).ne exact Real.ofReal_le_ennnorm (f x) #align measure_theory.is_finite_measure_with_density_of_real MeasureTheory.isFiniteMeasure_withDensity_ofReal section DominatedConvergence variable {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} theorem all_ae_ofReal_F_le_bound (h : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a) := fun n => (h n).mono fun _ h => ENNReal.ofReal_le_ofReal h set_option linter.uppercaseLean3 false in #align measure_theory.all_ae_of_real_F_le_bound MeasureTheory.all_ae_ofReal_F_le_bound theorem all_ae_tendsto_ofReal_norm (h : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop <\| 𝓝 <\| f a) : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop <\| 𝓝 <\| ENNReal.ofReal ‖f a‖ := h.mono fun _ h => tendsto_ofReal <\| Tendsto.comp (Continuous.tendsto continuous_norm _) h #align measure_theory.all_ae_tendsto_of_real_norm MeasureTheory.all_ae_tendsto_ofReal_norm theorem all_ae_ofReal_f_le_bound (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : ∀ᵐ a ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a) := by have F_le_bound := all_ae_ofReal_F_le_bound h_bound rw [← ae_all_iff] at F_le_bound apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _) intro a tendsto_norm F_le_bound exact le_of_tendsto' tendsto_norm F_le_bound #align measure_theory.all_ae_of_real_f_le_bound MeasureTheory.all_ae_ofReal_f_le_bound theorem hasFiniteIntegral_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (bound_hasFiniteIntegral : HasFiniteIntegral bound μ) (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : HasFiniteIntegral f μ := by /- `‖F n a‖ ≤ bound a` and `‖F n a‖ --> ‖f a‖` implies `‖f a‖ ≤ bound a`, and so `∫ ‖f‖ ≤ ∫ bound < ∞` since `bound` is has_finite_integral -/ rw [hasFiniteIntegral_iff_norm] calc (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := lintegral_mono_ae <\| all_ae_ofReal_f_le_bound h_bound h_lim _ < ∞ := by rw [← hasFiniteIntegral_iff_ofReal] · exact bound_hasFiniteIntegral exact (h_bound 0).mono fun a h => le_trans (norm_nonneg _) h #align measure_theory.has_finite_integral_of_dominated_convergence MeasureTheory.hasFiniteIntegral_of_dominated_convergence theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_hasFiniteIntegral : HasFiniteIntegral bound μ) (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0) := by have f_measurable : AEStronglyMeasurable f μ := aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim let b a := 2 * ENNReal.ofReal (bound a) /- `‖F n a‖ ≤ bound a` and `F n a --> f a` implies `‖f a‖ ≤ bound a`, and thus by the triangle inequality, have `‖F n a - f a‖ ≤ 2 * (bound a)`. -/ have hb : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a := by intro n filter_upwards [all_ae_ofReal_F_le_bound h_bound n, all_ae_ofReal_f_le_bound h_bound h_lim] with a h₁ h₂ calc ENNReal.ofReal ‖F n a - f a‖ ≤ ENNReal.ofReal ‖F n a‖ + ENNReal.ofReal ‖f a‖ := by rw [← ENNReal.ofReal_add] · apply ofReal_le_ofReal apply norm_sub_le · exact norm_nonneg _ · exact norm_nonneg _ _ ≤ ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) := add_le_add h₁ h₂ _ = b a := by rw [← two_mul] -- On the other hand, `F n a --> f a` implies that `‖F n a - f a‖ --> 0` have h : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0) := by rw [← ENNReal.ofReal_zero] refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_ rwa [← tendsto_iff_norm_sub_tendsto_zero] /- Therefore, by the dominated convergence theorem for nonnegative integration, have ` ∫ ‖f a - F n a‖ --> 0 ` -/ suffices Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 (∫⁻ _ : α, 0 ∂μ)) by rwa [lintegral_zero] at this -- Using the dominated convergence theorem. refine tendsto_lintegral_of_dominated_convergence' _ ?_ hb ?_ ?_ -- Show `fun a => ‖f a - F n a‖` is almost everywhere measurable for all `n` · exact fun n => measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable -- Show `2 * bound` `HasFiniteIntegral` · rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral · calc ∫⁻ a, b a ∂μ = 2 * ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := by rw [lintegral_const_mul'] exact coe_ne_top _ ≠ ∞ := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne filter_upwards [h_bound 0] with _ h using le_trans (norm_nonneg _) h -- Show `‖f a - F n a‖ --> 0` · exact h #align measure_theory.tendsto_lintegral_norm_of_dominated_convergence MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence end DominatedConvergence section PosPart /-! Lemmas used for defining the positive part of an `L¹` function -/ theorem HasFiniteIntegral.max_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) : HasFiniteIntegral (fun a => max (f a) 0) μ := hf.mono <\| eventually_of_forall fun x => by simp [abs_le, le_abs_self] #align measure_theory.has_finite_integral.max_zero MeasureTheory.HasFiniteIntegral.max_zero theorem HasFiniteIntegral.min_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) : HasFiniteIntegral (fun a => min (f a) 0) μ := hf.mono <\| eventually_of_forall fun x => by simpa [abs_le] using neg_abs_le _ #align measure_theory.has_finite_integral.min_zero MeasureTheory.HasFiniteIntegral.min_zero end PosPart section NormedSpace variable {𝕜 : Type} theorem HasFiniteIntegral.smul [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) {f : α → β} : HasFiniteIntegral f μ → HasFiniteIntegral (c • f) μ := by simp only [HasFiniteIntegral]; intro hfi calc (∫⁻ a : α, ‖c • f a‖₊ ∂μ) ≤ ∫⁻ a : α, ‖c‖₊ ‖f a‖₊ ∂μ := by refine lintegral_mono ?_ intro i -- After leanprover/lean4#2734, we need to do beta reduction `exact mod_cast` beta_reduce exact mod_cast (nnnorm_smul_le c (f i)) _ < ∞ := by rw [lintegral_const_mul'] exacts [mul_lt_top coe_ne_top hfi.ne, coe_ne_top] #align measure_theory.has_finite_integral.smul MeasureTheory.HasFiniteIntegral.smul theorem hasFiniteIntegral_smul_iff [NormedRing 𝕜] [MulActionWithZero 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜} (hc : IsUnit c) (f : α → β) : HasFiniteIntegral (c • f) μ ↔ HasFiniteIntegral f μ := by obtain ⟨c, rfl⟩ := hc constructor · intro h simpa only [smul_smul, Units.inv_mul, one_smul] using h.smul ((c⁻¹ : 𝕜ˣ) : 𝕜) exact HasFiniteIntegral.smul _ #align measure_theory.has_finite_integral_smul_iff MeasureTheory.hasFiniteIntegral_smul_iff theorem HasFiniteIntegral.const_mul [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) : HasFiniteIntegral (fun x => c * f x) μ := h.smul c #align measure_theory.has_finite_integral.const_mul MeasureTheory.HasFiniteIntegral.const_mul theorem HasFiniteIntegral.mul_const [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) : HasFiniteIntegral (fun x => f x * c) μ := h.smul (MulOpposite.op c) #align measure_theory.has_finite_integral.mul_const MeasureTheory.HasFiniteIntegral.mul_const end NormedSpace /-! ### The predicate `Integrable` -/ -- variable [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] /-- `Integrable f μ` means that `f` is measurable and that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite. `Integrable f` means `Integrable f volume`. -/ def Integrable {α} {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop := AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ #align measure_theory.integrable MeasureTheory.Integrable theorem memℒp_one_iff_integrable {f : α → β} : Memℒp f 1 μ ↔ Integrable f μ := by simp_rw [Integrable, HasFiniteIntegral, Memℒp, snorm_one_eq_lintegral_nnnorm] #align measure_theory.mem_ℒp_one_iff_integrable MeasureTheory.memℒp_one_iff_integrable theorem Integrable.aestronglyMeasurable {f : α → β} (hf : Integrable f μ) : AEStronglyMeasurable f μ := hf.1 #align measure_theory.integrable.ae_strongly_measurable MeasureTheory.Integrable.aestronglyMeasurable theorem Integrable.aemeasurable [MeasurableSpace β] [BorelSpace β] {f : α → β} (hf : Integrable f μ) : AEMeasurable f μ := hf.aestronglyMeasurable.aemeasurable #align measure_theory.integrable.ae_measurable MeasureTheory.Integrable.aemeasurable theorem Integrable.hasFiniteIntegral {f : α → β} (hf : Integrable f μ) : HasFiniteIntegral f μ := hf.2 #align measure_theory.integrable.has_finite_integral MeasureTheory.Integrable.hasFiniteIntegral theorem Integrable.mono {f : α → β} {g : α → γ} (hg : Integrable g μ) (hf : AEStronglyMeasurable f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : Integrable f μ := ⟨hf, hg.hasFiniteIntegral.mono h⟩ #align measure_theory.integrable.mono MeasureTheory.Integrable.mono theorem Integrable.mono' {f : α → β} {g : α → ℝ} (hg : Integrable g μ) (hf : AEStronglyMeasurable f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : Integrable f μ := ⟨hf, hg.hasFiniteIntegral.mono' h⟩ #align measure_theory.integrable.mono' MeasureTheory.Integrable.mono' theorem Integrable.congr' {f : α → β} {g : α → γ} (hf : Integrable f μ) (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Integrable g μ := ⟨hg, hf.hasFiniteIntegral.congr' h⟩ #align measure_theory.integrable.congr' MeasureTheory.Integrable.congr' theorem integrable_congr' {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Integrable f μ ↔ Integrable g μ := ⟨fun h2f => h2f.congr' hg h, fun h2g => h2g.congr' hf <\| EventuallyEq.symm h⟩ #align measure_theory.integrable_congr' MeasureTheory.integrable_congr' theorem Integrable.congr {f g : α → β} (hf : Integrable f μ) (h : f =ᵐ[μ] g) : Integrable g μ := ⟨hf.1.congr h, hf.2.congr h⟩ #align measure_theory.integrable.congr MeasureTheory.Integrable.congr theorem integrable_congr {f g : α → β} (h : f =ᵐ[μ] g) : Integrable f μ ↔ Integrable g μ := ⟨fun hf => hf.congr h, fun hg => hg.congr h.symm⟩ #align measure_theory.integrable_congr MeasureTheory.integrable_congr theorem integrable_const_iff {c : β} : Integrable (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ := by have : AEStronglyMeasurable (fun _ : α => c) μ := aestronglyMeasurable_const rw [Integrable, and_iff_right this, hasFiniteIntegral_const_iff] #align measure_theory.integrable_const_iff MeasureTheory.integrable_const_iff @[simp] theorem integrable_const [IsFiniteMeasure μ] (c : β) : Integrable (fun _ : α => c) μ := integrable_const_iff.2 <\| Or.inr <\| measure_lt_top _ _ #align measure_theory.integrable_const MeasureTheory.integrable_const @[simp] theorem Integrable.of_finite [Finite α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : Measure α) [IsFiniteMeasure μ] (f : α → β) : Integrable (fun a ↦ f a) μ := ⟨(StronglyMeasurable.of_finite f).aestronglyMeasurable, .of_finite⟩ @[deprecated (since := "2024-02-05")] alias integrable_of_fintype := Integrable.of_finite theorem Memℒp.integrable_norm_rpow {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by rw [← memℒp_one_iff_integrable] exact hf.norm_rpow hp_ne_zero hp_ne_top #align measure_theory.mem_ℒp.integrable_norm_rpow MeasureTheory.Memℒp.integrable_norm_rpow theorem Memℒp.integrable_norm_rpow' [IsFiniteMeasure μ] {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by by_cases h_zero : p = 0 · simp [h_zero, integrable_const] by_cases h_top : p = ∞ · simp [h_top, integrable_const] exact hf.integrable_norm_rpow h_zero h_top #align measure_theory.mem_ℒp.integrable_norm_rpow' MeasureTheory.Memℒp.integrable_norm_rpow' theorem Integrable.mono_measure {f : α → β} (h : Integrable f ν) (hμ : μ ≤ ν) : Integrable f μ := ⟨h.aestronglyMeasurable.mono_measure hμ, h.hasFiniteIntegral.mono_measure hμ⟩ #align measure_theory.integrable.mono_measure MeasureTheory.Integrable.mono_measure theorem Integrable.of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (hμ'_le : μ' ≤ c • μ) {f : α → β} (hf : Integrable f μ) : Integrable f μ' := by rw [← memℒp_one_iff_integrable] at hf ⊢ exact hf.of_measure_le_smul c hc hμ'_le #align measure_theory.integrable.of_measure_le_smul MeasureTheory.Integrable.of_measure_le_smul theorem Integrable.add_measure {f : α → β} (hμ : Integrable f μ) (hν : Integrable f ν) : Integrable f (μ + ν) := by simp_rw [← memℒp_one_iff_integrable] at hμ hν ⊢ refine ⟨hμ.aestronglyMeasurable.add_measure hν.aestronglyMeasurable, ?_⟩ rw [snorm_one_add_measure, ENNReal.add_lt_top] exact ⟨hμ.snorm_lt_top, hν.snorm_lt_top⟩ #align measure_theory.integrable.add_measure MeasureTheory.Integrable.add_measure theorem Integrable.left_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) : Integrable f μ := by rw [← memℒp_one_iff_integrable] at h ⊢ exact h.left_of_add_measure #align measure_theory.integrable.left_of_add_measure MeasureTheory.Integrable.left_of_add_measure theorem Integrable.right_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) : Integrable f ν := by rw [← memℒp_one_iff_integrable] at h ⊢ exact h.right_of_add_measure #align measure_theory.integrable.right_of_add_measure MeasureTheory.Integrable.right_of_add_measure @[simp] theorem integrable_add_measure {f : α → β} : Integrable f (μ + ν) ↔ Integrable f μ ∧ Integrable f ν := ⟨fun h => ⟨h.left_of_add_measure, h.right_of_add_measure⟩, fun h => h.1.add_measure h.2⟩ #align measure_theory.integrable_add_measure MeasureTheory.integrable_add_measure @[simp] theorem integrable_zero_measure {_ : MeasurableSpace α} {f : α → β} : Integrable f (0 : Measure α) := ⟨aestronglyMeasurable_zero_measure f, hasFiniteIntegral_zero_measure f⟩ #align measure_theory.integrable_zero_measure MeasureTheory.integrable_zero_measure theorem integrable_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → β} {μ : ι → Measure α} {s : Finset ι} : Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i) := by induction s using Finset.induction_on <;> simp [*] #align measure_theory.integrable_finset_sum_measure MeasureTheory.integrable_finset_sum_measure	Mathlib/MeasureTheory/Function/L1Space.lean	566	569	theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) : Integrable f (c • μ) := by	rw [← memℒp_one_iff_integrable] at h ⊢ exact h.smul_measure hc
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Johannes Hölzl -/ import Mathlib.Dynamics.Ergodic.MeasurePreserving import Mathlib.MeasureTheory.Function.SimpleFunc import Mathlib.MeasureTheory.Measure.MutuallySingular import Mathlib.MeasureTheory.Measure.Count import Mathlib.Topology.IndicatorConstPointwise import Mathlib.MeasureTheory.Constructions.BorelSpace.Real #align_import measure_theory.integral.lebesgue from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" /-! # Lower Lebesgue integral for `ℝ≥0∞`-valued functions We define the lower Lebesgue integral of an `ℝ≥0∞`-valued function. ## Notation We introduce the following notation for the lower Lebesgue integral of a function `f : α → ℝ≥0∞`. * `∫⁻ x, f x ∂μ`: integral of a function `f : α → ℝ≥0∞` with respect to a measure `μ`; * `∫⁻ x, f x`: integral of a function `f : α → ℝ≥0∞` with respect to the canonical measure `volume` on `α`; * `∫⁻ x in s, f x ∂μ`: integral of a function `f : α → ℝ≥0∞` over a set `s` with respect to a measure `μ`, defined as `∫⁻ x, f x ∂(μ.restrict s)`; * `∫⁻ x in s, f x`: integral of a function `f : α → ℝ≥0∞` over a set `s` with respect to the canonical measure `volume`, defined as `∫⁻ x, f x ∂(volume.restrict s)`. -/ assert_not_exists NormedSpace set_option autoImplicit true noncomputable section open Set hiding restrict restrict_apply open Filter ENNReal open Function (support) open scoped Classical open Topology NNReal ENNReal MeasureTheory namespace MeasureTheory local infixr:25 " →ₛ " => SimpleFunc variable {α β γ δ : Type} section Lintegral open SimpleFunc variable {m : MeasurableSpace α} {μ ν : Measure α} /-- The lower Lebesgue integral* of a function `f` with respect to a measure `μ`. -/ irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ := ⨆ (g : α →ₛ ℝ≥0∞) (_ : ⇑g ≤ f), g.lintegral μ #align measure_theory.lintegral MeasureTheory.lintegral /-! In the notation for integrals, an expression like `∫⁻ x, g ‖x‖ ∂μ` will not be parsed correctly, and needs parentheses. We do not set the binding power of `r` to `0`, because then `∫⁻ x, f x = 0` will be parsed incorrectly. -/ @[inherit_doc MeasureTheory.lintegral] notation3 "∫⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => lintegral μ r @[inherit_doc MeasureTheory.lintegral] notation3 "∫⁻ "(...)", "r:60:(scoped f => lintegral volume f) => r @[inherit_doc MeasureTheory.lintegral] notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => lintegral (Measure.restrict μ s) r @[inherit_doc MeasureTheory.lintegral] notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => lintegral (Measure.restrict volume s) f) => r theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) : ∫⁻ a, f a ∂μ = f.lintegral μ := by rw [MeasureTheory.lintegral] exact le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <\| le_rfl) (le_iSup₂_of_le f le_rfl le_rfl) #align measure_theory.simple_func.lintegral_eq_lintegral MeasureTheory.SimpleFunc.lintegral_eq_lintegral @[mono] theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂ν := by rw [lintegral, lintegral] exact iSup_mono fun φ => iSup_mono' fun hφ => ⟨le_trans hφ hfg, lintegral_mono (le_refl φ) hμν⟩ #align measure_theory.lintegral_mono' MeasureTheory.lintegral_mono' -- workaround for the known eta-reduction issue with `@[gcongr]` @[gcongr] theorem lintegral_mono_fn' ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) (h2 : μ ≤ ν) : lintegral μ f ≤ lintegral ν g := lintegral_mono' h2 hfg theorem lintegral_mono ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ := lintegral_mono' (le_refl μ) hfg #align measure_theory.lintegral_mono MeasureTheory.lintegral_mono -- workaround for the known eta-reduction issue with `@[gcongr]` @[gcongr] theorem lintegral_mono_fn ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) : lintegral μ f ≤ lintegral μ g := lintegral_mono hfg theorem lintegral_mono_nnreal {f g : α → ℝ≥0} (h : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ := lintegral_mono fun a => ENNReal.coe_le_coe.2 (h a) #align measure_theory.lintegral_mono_nnreal MeasureTheory.lintegral_mono_nnreal theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) : ⨆ (g : α → ℝ≥0∞) (_ : Measurable g) (_ : g ≤ f), ∫⁻ a, g a ∂μ = ∫⁻ a, f a ∂μ := by apply le_antisymm · exact iSup_le fun i => iSup_le fun _ => iSup_le fun h'i => lintegral_mono h'i · rw [lintegral] refine iSup₂_le fun i hi => le_iSup₂_of_le i i.measurable <\| le_iSup_of_le hi ?_ exact le_of_eq (i.lintegral_eq_lintegral _).symm #align measure_theory.supr_lintegral_measurable_le_eq_lintegral MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral theorem lintegral_mono_set {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞} (hst : s ⊆ t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ := lintegral_mono' (Measure.restrict_mono hst (le_refl μ)) (le_refl f) #align measure_theory.lintegral_mono_set MeasureTheory.lintegral_mono_set theorem lintegral_mono_set' {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞} (hst : s ≤ᵐ[μ] t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ := lintegral_mono' (Measure.restrict_mono' hst (le_refl μ)) (le_refl f) #align measure_theory.lintegral_mono_set' MeasureTheory.lintegral_mono_set' theorem monotone_lintegral {_ : MeasurableSpace α} (μ : Measure α) : Monotone (lintegral μ) := lintegral_mono #align measure_theory.monotone_lintegral MeasureTheory.monotone_lintegral @[simp] theorem lintegral_const (c : ℝ≥0∞) : ∫⁻ _, c ∂μ = c * μ univ := by rw [← SimpleFunc.const_lintegral, ← SimpleFunc.lintegral_eq_lintegral, SimpleFunc.coe_const] rfl #align measure_theory.lintegral_const MeasureTheory.lintegral_const theorem lintegral_zero : ∫⁻ _ : α, 0 ∂μ = 0 := by simp #align measure_theory.lintegral_zero MeasureTheory.lintegral_zero theorem lintegral_zero_fun : lintegral μ (0 : α → ℝ≥0∞) = 0 := lintegral_zero #align measure_theory.lintegral_zero_fun MeasureTheory.lintegral_zero_fun -- @[simp] -- Porting note (#10618): simp can prove this theorem lintegral_one : ∫⁻ _, (1 : ℝ≥0∞) ∂μ = μ univ := by rw [lintegral_const, one_mul] #align measure_theory.lintegral_one MeasureTheory.lintegral_one theorem set_lintegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s := by rw [lintegral_const, Measure.restrict_apply_univ] #align measure_theory.set_lintegral_const MeasureTheory.set_lintegral_const theorem set_lintegral_one (s) : ∫⁻ _ in s, 1 ∂μ = μ s := by rw [set_lintegral_const, one_mul] #align measure_theory.set_lintegral_one MeasureTheory.set_lintegral_one theorem set_lintegral_const_lt_top [IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) : ∫⁻ _ in s, c ∂μ < ∞ := by rw [lintegral_const] exact ENNReal.mul_lt_top hc (measure_ne_top (μ.restrict s) univ) #align measure_theory.set_lintegral_const_lt_top MeasureTheory.set_lintegral_const_lt_top theorem lintegral_const_lt_top [IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : ∫⁻ _, c ∂μ < ∞ := by simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set α) hc #align measure_theory.lintegral_const_lt_top MeasureTheory.lintegral_const_lt_top section variable (μ) /-- For any function `f : α → ℝ≥0∞`, there exists a measurable function `g ≤ f` with the same integral. -/ theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) : ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by rcases eq_or_ne (∫⁻ a, f a ∂μ) 0 with h₀ \| h₀ · exact ⟨0, measurable_zero, zero_le f, h₀.trans lintegral_zero.symm⟩ rcases exists_seq_strictMono_tendsto' h₀.bot_lt with ⟨L, _, hLf, hL_tendsto⟩ have : ∀ n, ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ a, g a ∂μ := by intro n simpa only [← iSup_lintegral_measurable_le_eq_lintegral f, lt_iSup_iff, exists_prop] using (hLf n).2 choose g hgm hgf hLg using this refine ⟨fun x => ⨆ n, g n x, measurable_iSup hgm, fun x => iSup_le fun n => hgf n x, le_antisymm ?_ ?_⟩ · refine le_of_tendsto' hL_tendsto fun n => (hLg n).le.trans <\| lintegral_mono fun x => ?_ exact le_iSup (fun n => g n x) n · exact lintegral_mono fun x => iSup_le fun n => hgf n x #align measure_theory.exists_measurable_le_lintegral_eq MeasureTheory.exists_measurable_le_lintegral_eq end /-- `∫⁻ a in s, f a ∂μ` is defined as the supremum of integrals of simple functions `φ : α →ₛ ℝ≥0∞` such that `φ ≤ f`. This lemma says that it suffices to take functions `φ : α →ₛ ℝ≥0`. -/ theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) : ∫⁻ a, f a ∂μ = ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ x, ↑(φ x) ≤ f x), (φ.map ((↑) : ℝ≥0 → ℝ≥0∞)).lintegral μ := by rw [lintegral] refine le_antisymm (iSup₂_le fun φ hφ => ?_) (iSup_mono' fun φ => ⟨φ.map ((↑) : ℝ≥0 → ℝ≥0∞), le_rfl⟩) by_cases h : ∀ᵐ a ∂μ, φ a ≠ ∞ · let ψ := φ.map ENNReal.toNNReal replace h : ψ.map ((↑) : ℝ≥0 → ℝ≥0∞) =ᵐ[μ] φ := h.mono fun a => ENNReal.coe_toNNReal have : ∀ x, ↑(ψ x) ≤ f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (hφ x) exact le_iSup_of_le (φ.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <\| lintegral_congr h)) · have h_meas : μ (φ ⁻¹' {∞}) ≠ 0 := mt measure_zero_iff_ae_nmem.1 h refine le_trans le_top (ge_of_eq <\| (iSup_eq_top _).2 fun b hb => ?_) obtain ⟨n, hn⟩ : ∃ n : ℕ, b < n * μ (φ ⁻¹' {∞}) := exists_nat_mul_gt h_meas (ne_of_lt hb) use (const α (n : ℝ≥0)).restrict (φ ⁻¹' {∞}) simp only [lt_iSup_iff, exists_prop, coe_restrict, φ.measurableSet_preimage, coe_const, ENNReal.coe_indicator, map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_natCast, restrict_const_lintegral] refine ⟨indicator_le fun x hx => le_trans ?_ (hφ _), hn⟩ simp only [mem_preimage, mem_singleton_iff] at hx simp only [hx, le_top] #align measure_theory.lintegral_eq_nnreal MeasureTheory.lintegral_eq_nnreal theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ φ : α →ₛ ℝ≥0, (∀ x, ↑(φ x) ≤ f x) ∧ ∀ ψ : α →ₛ ℝ≥0, (∀ x, ↑(ψ x) ≤ f x) → (map (↑) (ψ - φ)).lintegral μ < ε := by rw [lintegral_eq_nnreal] at h have := ENNReal.lt_add_right h hε erw [ENNReal.biSup_add] at this <;> [skip; exact ⟨0, fun x => zero_le _⟩] simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this rcases this with ⟨φ, hle : ∀ x, ↑(φ x) ≤ f x, b, hbφ, hb⟩ refine ⟨φ, hle, fun ψ hψ => ?_⟩ have : (map (↑) φ).lintegral μ ≠ ∞ := ne_top_of_le_ne_top h (by exact le_iSup₂ (α := ℝ≥0∞) φ hle) rw [← ENNReal.add_lt_add_iff_left this, ← add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add] refine (hb _ fun x => le_trans ?_ (max_le (hle x) (hψ x))).trans_lt hbφ norm_cast simp only [add_apply, sub_apply, add_tsub_eq_max] rfl #align measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos theorem iSup_lintegral_le {ι : Sort} (f : ι → α → ℝ≥0∞) : ⨆ i, ∫⁻ a, f i a ∂μ ≤ ∫⁻ a, ⨆ i, f i a ∂μ := by simp only [← iSup_apply] exact (monotone_lintegral μ).le_map_iSup #align measure_theory.supr_lintegral_le MeasureTheory.iSup_lintegral_le theorem iSup₂_lintegral_le {ι : Sort} {ι' : ι → Sort} (f : ∀ i, ι' i → α → ℝ≥0∞) : ⨆ (i) (j), ∫⁻ a, f i j a ∂μ ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ := by convert (monotone_lintegral μ).le_map_iSup₂ f with a simp only [iSup_apply] #align measure_theory.supr₂_lintegral_le MeasureTheory.iSup₂_lintegral_le theorem le_iInf_lintegral {ι : Sort} (f : ι → α → ℝ≥0∞) : ∫⁻ a, ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ a, f i a ∂μ := by simp only [← iInf_apply] exact (monotone_lintegral μ).map_iInf_le #align measure_theory.le_infi_lintegral MeasureTheory.le_iInf_lintegral theorem le_iInf₂_lintegral {ι : Sort} {ι' : ι → Sort} (f : ∀ i, ι' i → α → ℝ≥0∞) : ∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ := by convert (monotone_lintegral μ).map_iInf₂_le f with a simp only [iInf_apply] #align measure_theory.le_infi₂_lintegral MeasureTheory.le_iInf₂_lintegral theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ := by rcases exists_measurable_superset_of_null h with ⟨t, hts, ht, ht0⟩ have : ∀ᵐ x ∂μ, x ∉ t := measure_zero_iff_ae_nmem.1 ht0 rw [lintegral, lintegral] refine iSup_le fun s => iSup_le fun hfs => le_iSup_of_le (s.restrict tᶜ) <\| le_iSup_of_le ?_ ?_ · intro a by_cases h : a ∈ t <;> simp only [restrict_apply s ht.compl, mem_compl_iff, h, not_true, not_false_eq_true, indicator_of_not_mem, zero_le, not_false_eq_true, indicator_of_mem] exact le_trans (hfs a) (_root_.by_contradiction fun hnfg => h (hts hnfg)) · refine le_of_eq (SimpleFunc.lintegral_congr <\| this.mono fun a hnt => ?_) by_cases hat : a ∈ t <;> simp only [restrict_apply s ht.compl, mem_compl_iff, hat, not_true, not_false_eq_true, indicator_of_not_mem, not_false_eq_true, indicator_of_mem] exact (hnt hat).elim #align measure_theory.lintegral_mono_ae MeasureTheory.lintegral_mono_ae theorem set_lintegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ := lintegral_mono_ae <\| (ae_restrict_iff <\| measurableSet_le hf hg).2 hfg #align measure_theory.set_lintegral_mono_ae MeasureTheory.set_lintegral_mono_ae theorem set_lintegral_mono {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ := set_lintegral_mono_ae hf hg (ae_of_all _ hfg) #align measure_theory.set_lintegral_mono MeasureTheory.set_lintegral_mono theorem set_lintegral_mono_ae' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s) (hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ := lintegral_mono_ae <\| (ae_restrict_iff' hs).2 hfg theorem set_lintegral_mono' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s) (hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ := set_lintegral_mono_ae' hs (ae_of_all _ hfg) theorem set_lintegral_le_lintegral (s : Set α) (f : α → ℝ≥0∞) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x, f x ∂μ := lintegral_mono' Measure.restrict_le_self le_rfl theorem lintegral_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := le_antisymm (lintegral_mono_ae <\| h.le) (lintegral_mono_ae <\| h.symm.le) #align measure_theory.lintegral_congr_ae MeasureTheory.lintegral_congr_ae theorem lintegral_congr {f g : α → ℝ≥0∞} (h : ∀ a, f a = g a) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by simp only [h] #align measure_theory.lintegral_congr MeasureTheory.lintegral_congr theorem set_lintegral_congr {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t) : ∫⁻ x in s, f x ∂μ = ∫⁻ x in t, f x ∂μ := by rw [Measure.restrict_congr_set h] #align measure_theory.set_lintegral_congr MeasureTheory.set_lintegral_congr theorem set_lintegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s) (hfg : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ := by rw [lintegral_congr_ae] rw [EventuallyEq] rwa [ae_restrict_iff' hs] #align measure_theory.set_lintegral_congr_fun MeasureTheory.set_lintegral_congr_fun theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) : ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖₊ ∂μ := by simp_rw [← ofReal_norm_eq_coe_nnnorm] refine lintegral_mono fun x => ENNReal.ofReal_le_ofReal ?_ rw [Real.norm_eq_abs] exact le_abs_self (f x) #align measure_theory.lintegral_of_real_le_lintegral_nnnorm MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) : ∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by apply lintegral_congr_ae filter_upwards [h_nonneg] with x hx rw [Real.nnnorm_of_nonneg hx, ENNReal.ofReal_eq_coe_nnreal hx] #align measure_theory.lintegral_nnnorm_eq_of_ae_nonneg MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) : ∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := lintegral_nnnorm_eq_of_ae_nonneg (Filter.eventually_of_forall h_nonneg) #align measure_theory.lintegral_nnnorm_eq_of_nonneg MeasureTheory.lintegral_nnnorm_eq_of_nonneg /-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. See `lintegral_iSup_directed` for a more general form. -/ theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by set c : ℝ≥0 → ℝ≥0∞ := (↑) set F := fun a : α => ⨆ n, f n a refine le_antisymm ?_ (iSup_lintegral_le _) rw [lintegral_eq_nnreal] refine iSup_le fun s => iSup_le fun hsf => ?_ refine ENNReal.le_of_forall_lt_one_mul_le fun a ha => ?_ rcases ENNReal.lt_iff_exists_coe.1 ha with ⟨r, rfl, _⟩ have ha : r < 1 := ENNReal.coe_lt_coe.1 ha let rs := s.map fun a => r * a have eq_rs : rs.map c = (const α r : α →ₛ ℝ≥0∞) * map c s := rfl have eq : ∀ p, rs.map c ⁻¹' {p} = ⋃ n, rs.map c ⁻¹' {p} ∩ { a \| p ≤ f n a } := by intro p rw [← inter_iUnion]; nth_rw 1 [← inter_univ (map c rs ⁻¹' {p})] refine Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 ?_ by_cases p_eq : p = 0 · simp [p_eq] simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx subst hx have : r * s x ≠ 0 := by rwa [Ne, ← ENNReal.coe_eq_zero] have : s x ≠ 0 := right_ne_zero_of_mul this have : (rs.map c) x < ⨆ n : ℕ, f n x := by refine lt_of_lt_of_le (ENNReal.coe_lt_coe.2 ?_) (hsf x) suffices r * s x < 1 * s x by simpa exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this) rcases lt_iSup_iff.1 this with ⟨i, hi⟩ exact mem_iUnion.2 ⟨i, le_of_lt hi⟩ have mono : ∀ r : ℝ≥0∞, Monotone fun n => rs.map c ⁻¹' {r} ∩ { a \| r ≤ f n a } := by intro r i j h refine inter_subset_inter_right _ ?_ simp_rw [subset_def, mem_setOf] intro x hx exact le_trans hx (h_mono h x) have h_meas : ∀ n, MeasurableSet {a : α \| map c rs a ≤ f n a} := fun n => measurableSet_le (SimpleFunc.measurable _) (hf n) calc (r : ℝ≥0∞) * (s.map c).lintegral μ = ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r}) := by rw [← const_mul_lintegral, eq_rs, SimpleFunc.lintegral] _ = ∑ r ∈ (rs.map c).range, r * μ (⋃ n, rs.map c ⁻¹' {r} ∩ { a \| r ≤ f n a }) := by simp only [(eq _).symm] _ = ∑ r ∈ (rs.map c).range, ⨆ n, r * μ (rs.map c ⁻¹' {r} ∩ { a \| r ≤ f n a }) := (Finset.sum_congr rfl fun x _ => by rw [measure_iUnion_eq_iSup (mono x).directed_le, ENNReal.mul_iSup]) _ = ⨆ n, ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r} ∩ { a \| r ≤ f n a }) := by refine ENNReal.finset_sum_iSup_nat fun p i j h ↦ ?_ gcongr _ * μ ?_ exact mono p h _ ≤ ⨆ n : ℕ, ((rs.map c).restrict { a \| (rs.map c) a ≤ f n a }).lintegral μ := by gcongr with n rw [restrict_lintegral _ (h_meas n)] refine le_of_eq (Finset.sum_congr rfl fun r _ => ?_) congr 2 with a refine and_congr_right ?_ simp (config := { contextual := true }) _ ≤ ⨆ n, ∫⁻ a, f n a ∂μ := by simp only [← SimpleFunc.lintegral_eq_lintegral] gcongr with n a simp only [map_apply] at h_meas simp only [coe_map, restrict_apply _ (h_meas _), (· ∘ ·)] exact indicator_apply_le id #align measure_theory.lintegral_supr MeasureTheory.lintegral_iSup /-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with ae_measurable functions. -/ theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by simp_rw [← iSup_apply] let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Monotone f' have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_mono have h_ae_seq_mono : Monotone (aeSeq hf p) := by intro n m hnm x by_cases hx : x ∈ aeSeqSet hf p · exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm · simp only [aeSeq, hx, if_false, le_rfl] rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm] simp_rw [iSup_apply] rw [lintegral_iSup (aeSeq.measurable hf p) h_ae_seq_mono] congr with n exact lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n) #align measure_theory.lintegral_supr' MeasureTheory.lintegral_iSup' /-- Monotone convergence theorem expressed with limits -/ theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 <\| F x)) : Tendsto (fun n => ∫⁻ x, f n x ∂μ) atTop (𝓝 <\| ∫⁻ x, F x ∂μ) := by have : Monotone fun n => ∫⁻ x, f n x ∂μ := fun i j hij => lintegral_mono_ae (h_mono.mono fun x hx => hx hij) suffices key : ∫⁻ x, F x ∂μ = ⨆ n, ∫⁻ x, f n x ∂μ by rw [key] exact tendsto_atTop_iSup this rw [← lintegral_iSup' hf h_mono] refine lintegral_congr_ae ?_ filter_upwards [h_mono, h_tendsto] with _ hx_mono hx_tendsto using tendsto_nhds_unique hx_tendsto (tendsto_atTop_iSup hx_mono) #align measure_theory.lintegral_tendsto_of_tendsto_of_monotone MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = ⨆ n, (eapprox f n).lintegral μ := calc ∫⁻ a, f a ∂μ = ∫⁻ a, ⨆ n, (eapprox f n : α → ℝ≥0∞) a ∂μ := by congr; ext a; rw [iSup_eapprox_apply f hf] _ = ⨆ n, ∫⁻ a, (eapprox f n : α → ℝ≥0∞) a ∂μ := by apply lintegral_iSup · measurability · intro i j h exact monotone_eapprox f h _ = ⨆ n, (eapprox f n).lintegral μ := by congr; ext n; rw [(eapprox f n).lintegral_eq_lintegral] #align measure_theory.lintegral_eq_supr_eapprox_lintegral MeasureTheory.lintegral_eq_iSup_eapprox_lintegral /-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends to zero as `μ s` tends to zero. This lemma states this fact in terms of `ε` and `δ`. -/ theorem exists_pos_set_lintegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε := by rcases exists_between (pos_iff_ne_zero.mpr hε) with ⟨ε₂, hε₂0, hε₂ε⟩ rcases exists_between hε₂0 with ⟨ε₁, hε₁0, hε₁₂⟩ rcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h hε₁0.ne' with ⟨φ, _, hφ⟩ rcases φ.exists_forall_le with ⟨C, hC⟩ use (ε₂ - ε₁) / C, ENNReal.div_pos_iff.2 ⟨(tsub_pos_iff_lt.2 hε₁₂).ne', ENNReal.coe_ne_top⟩ refine fun s hs => lt_of_le_of_lt ?_ hε₂ε simp only [lintegral_eq_nnreal, iSup_le_iff] intro ψ hψ calc (map (↑) ψ).lintegral (μ.restrict s) ≤ (map (↑) φ).lintegral (μ.restrict s) + (map (↑) (ψ - φ)).lintegral (μ.restrict s) := by rw [← SimpleFunc.add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add] refine SimpleFunc.lintegral_mono (fun x => ?_) le_rfl simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add, SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, ENNReal.coe_max (φ x) (ψ x)] _ ≤ (map (↑) φ).lintegral (μ.restrict s) + ε₁ := by gcongr refine le_trans ?_ (hφ _ hψ).le exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self _ ≤ (SimpleFunc.const α (C : ℝ≥0∞)).lintegral (μ.restrict s) + ε₁ := by gcongr exact SimpleFunc.lintegral_mono (fun x ↦ ENNReal.coe_le_coe.2 (hC x)) le_rfl _ = C * μ s + ε₁ := by simp only [← SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const] _ ≤ C * ((ε₂ - ε₁) / C) + ε₁ := by gcongr _ ≤ ε₂ - ε₁ + ε₁ := by gcongr; apply mul_div_le _ = ε₂ := tsub_add_cancel_of_le hε₁₂.le #align measure_theory.exists_pos_set_lintegral_lt_of_measure_lt MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt /-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends to zero as `μ s` tends to zero. -/ theorem tendsto_set_lintegral_zero {ι} {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {l : Filter ι} {s : ι → Set α} (hl : Tendsto (μ ∘ s) l (𝓝 0)) : Tendsto (fun i => ∫⁻ x in s i, f x ∂μ) l (𝓝 0) := by simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio, ← pos_iff_ne_zero] at hl ⊢ intro ε ε0 rcases exists_pos_set_lintegral_lt_of_measure_lt h ε0.ne' with ⟨δ, δ0, hδ⟩ exact (hl δ δ0).mono fun i => hδ _ #align measure_theory.tendsto_set_lintegral_zero MeasureTheory.tendsto_set_lintegral_zero /-- The sum of the lower Lebesgue integrals of two functions is less than or equal to the integral of their sum. The other inequality needs one of these functions to be (a.e.-)measurable. -/ theorem le_lintegral_add (f g : α → ℝ≥0∞) : ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ ≤ ∫⁻ a, f a + g a ∂μ := by simp only [lintegral] refine ENNReal.biSup_add_biSup_le' (p := fun h : α →ₛ ℝ≥0∞ => h ≤ f) (q := fun h : α →ₛ ℝ≥0∞ => h ≤ g) ⟨0, zero_le f⟩ ⟨0, zero_le g⟩ fun f' hf' g' hg' => ?_ exact le_iSup₂_of_le (f' + g') (add_le_add hf' hg') (add_lintegral _ _).ge #align measure_theory.le_lintegral_add MeasureTheory.le_lintegral_add -- Use stronger lemmas `lintegral_add_left`/`lintegral_add_right` instead theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) : ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := calc ∫⁻ a, f a + g a ∂μ = ∫⁻ a, (⨆ n, (eapprox f n : α → ℝ≥0∞) a) + ⨆ n, (eapprox g n : α → ℝ≥0∞) a ∂μ := by simp only [iSup_eapprox_apply, hf, hg] _ = ∫⁻ a, ⨆ n, (eapprox f n + eapprox g n : α → ℝ≥0∞) a ∂μ := by congr; funext a rw [ENNReal.iSup_add_iSup_of_monotone] · simp only [Pi.add_apply] · intro i j h exact monotone_eapprox _ h a · intro i j h exact monotone_eapprox _ h a _ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ := by rw [lintegral_iSup] · congr funext n rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral] simp only [Pi.add_apply, SimpleFunc.coe_add] · measurability · intro i j h a dsimp gcongr <;> exact monotone_eapprox _ h _ _ = (⨆ n, (eapprox f n).lintegral μ) + ⨆ n, (eapprox g n).lintegral μ := by refine (ENNReal.iSup_add_iSup_of_monotone ?_ ?_).symm <;> · intro i j h exact SimpleFunc.lintegral_mono (monotone_eapprox _ h) le_rfl _ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by rw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral hg] #align measure_theory.lintegral_add_aux MeasureTheory.lintegral_add_aux /-- If `f g : α → ℝ≥0∞` are two functions and one of them is (a.e.) measurable, then the Lebesgue integral of `f + g` equals the sum of integrals. This lemma assumes that `f` is integrable, see also `MeasureTheory.lintegral_add_right` and primed versions of these lemmas. -/ @[simp] theorem lintegral_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) : ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by refine le_antisymm ?_ (le_lintegral_add _ _) rcases exists_measurable_le_lintegral_eq μ fun a => f a + g a with ⟨φ, hφm, hφ_le, hφ_eq⟩ calc ∫⁻ a, f a + g a ∂μ = ∫⁻ a, φ a ∂μ := hφ_eq _ ≤ ∫⁻ a, f a + (φ a - f a) ∂μ := lintegral_mono fun a => le_add_tsub _ = ∫⁻ a, f a ∂μ + ∫⁻ a, φ a - f a ∂μ := lintegral_add_aux hf (hφm.sub hf) _ ≤ ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := add_le_add_left (lintegral_mono fun a => tsub_le_iff_left.2 <\| hφ_le a) _ #align measure_theory.lintegral_add_left MeasureTheory.lintegral_add_left theorem lintegral_add_left' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞) : ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by rw [lintegral_congr_ae hf.ae_eq_mk, ← lintegral_add_left hf.measurable_mk, lintegral_congr_ae (hf.ae_eq_mk.add (ae_eq_refl g))] #align measure_theory.lintegral_add_left' MeasureTheory.lintegral_add_left' theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) : ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by simpa only [add_comm] using lintegral_add_left' hg f #align measure_theory.lintegral_add_right' MeasureTheory.lintegral_add_right' /-- If `f g : α → ℝ≥0∞` are two functions and one of them is (a.e.) measurable, then the Lebesgue integral of `f + g` equals the sum of integrals. This lemma assumes that `g` is integrable, see also `MeasureTheory.lintegral_add_left` and primed versions of these lemmas. -/ @[simp] theorem lintegral_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) : ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := lintegral_add_right' f hg.aemeasurable #align measure_theory.lintegral_add_right MeasureTheory.lintegral_add_right @[simp] theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂c • μ = c * ∫⁻ a, f a ∂μ := by simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul] #align measure_theory.lintegral_smul_measure MeasureTheory.lintegral_smul_measure lemma set_lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α) : ∫⁻ a in s, f a ∂(c • μ) = c * ∫⁻ a in s, f a ∂μ := by rw [Measure.restrict_smul, lintegral_smul_measure] @[simp] theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) : ∫⁻ a, f a ∂Measure.sum μ = ∑' i, ∫⁻ a, f a ∂μ i := by simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum] rw [iSup_comm] congr; funext s induction' s using Finset.induction_on with i s hi hs · simp simp only [Finset.sum_insert hi, ← hs] refine (ENNReal.iSup_add_iSup ?_).symm intro φ ψ exact ⟨⟨φ ⊔ ψ, fun x => sup_le (φ.2 x) (ψ.2 x)⟩, add_le_add (SimpleFunc.lintegral_mono le_sup_left le_rfl) (Finset.sum_le_sum fun j _ => SimpleFunc.lintegral_mono le_sup_right le_rfl)⟩ #align measure_theory.lintegral_sum_measure MeasureTheory.lintegral_sum_measure theorem hasSum_lintegral_measure {ι} {_ : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → Measure α) : HasSum (fun i => ∫⁻ a, f a ∂μ i) (∫⁻ a, f a ∂Measure.sum μ) := (lintegral_sum_measure f μ).symm ▸ ENNReal.summable.hasSum #align measure_theory.has_sum_lintegral_measure MeasureTheory.hasSum_lintegral_measure @[simp] theorem lintegral_add_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : Measure α) : ∫⁻ a, f a ∂(μ + ν) = ∫⁻ a, f a ∂μ + ∫⁻ a, f a ∂ν := by simpa [tsum_fintype] using lintegral_sum_measure f fun b => cond b μ ν #align measure_theory.lintegral_add_measure MeasureTheory.lintegral_add_measure @[simp] theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞) (μ : ι → Measure α) : ∫⁻ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫⁻ a, f a ∂μ i := by rw [← Measure.sum_coe_finset, lintegral_sum_measure, ← Finset.tsum_subtype'] simp only [Finset.coe_sort_coe] #align measure_theory.lintegral_finset_sum_measure MeasureTheory.lintegral_finset_sum_measure @[simp] theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂(0 : Measure α) = 0 := by simp [lintegral] #align measure_theory.lintegral_zero_measure MeasureTheory.lintegral_zero_measure @[simp] theorem lintegral_of_isEmpty {α} [MeasurableSpace α] [IsEmpty α] (μ : Measure α) (f : α → ℝ≥0∞) : ∫⁻ x, f x ∂μ = 0 := by have : Subsingleton (Measure α) := inferInstance convert lintegral_zero_measure f theorem set_lintegral_empty (f : α → ℝ≥0∞) : ∫⁻ x in ∅, f x ∂μ = 0 := by rw [Measure.restrict_empty, lintegral_zero_measure] #align measure_theory.set_lintegral_empty MeasureTheory.set_lintegral_empty theorem set_lintegral_univ (f : α → ℝ≥0∞) : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by rw [Measure.restrict_univ] #align measure_theory.set_lintegral_univ MeasureTheory.set_lintegral_univ theorem set_lintegral_measure_zero (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0) : ∫⁻ x in s, f x ∂μ = 0 := by convert lintegral_zero_measure _ exact Measure.restrict_eq_zero.2 hs' #align measure_theory.set_lintegral_measure_zero MeasureTheory.set_lintegral_measure_zero theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, AEMeasurable (f b) μ) : ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ := by induction' s using Finset.induction_on with a s has ih · simp · simp only [Finset.sum_insert has] rw [Finset.forall_mem_insert] at hf rw [lintegral_add_left' hf.1, ih hf.2] #align measure_theory.lintegral_finset_sum' MeasureTheory.lintegral_finset_sum' theorem lintegral_finset_sum (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b)) : ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ := lintegral_finset_sum' s fun b hb => (hf b hb).aemeasurable #align measure_theory.lintegral_finset_sum MeasureTheory.lintegral_finset_sum @[simp] theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := calc ∫⁻ a, r * f a ∂μ = ∫⁻ a, ⨆ n, (const α r * eapprox f n) a ∂μ := by congr funext a rw [← iSup_eapprox_apply f hf, ENNReal.mul_iSup] simp _ = ⨆ n, r * (eapprox f n).lintegral μ := by rw [lintegral_iSup] · congr funext n rw [← SimpleFunc.const_mul_lintegral, ← SimpleFunc.lintegral_eq_lintegral] · intro n exact SimpleFunc.measurable _ · intro i j h a exact mul_le_mul_left' (monotone_eapprox _ h _) _ _ = r * ∫⁻ a, f a ∂μ := by rw [← ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf] #align measure_theory.lintegral_const_mul MeasureTheory.lintegral_const_mul theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) : ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by have A : ∫⁻ a, f a ∂μ = ∫⁻ a, hf.mk f a ∂μ := lintegral_congr_ae hf.ae_eq_mk have B : ∫⁻ a, r * f a ∂μ = ∫⁻ a, r * hf.mk f a ∂μ := lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _) rw [A, B, lintegral_const_mul _ hf.measurable_mk] #align measure_theory.lintegral_const_mul'' MeasureTheory.lintegral_const_mul'' theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) : r * ∫⁻ a, f a ∂μ ≤ ∫⁻ a, r * f a ∂μ := by rw [lintegral, ENNReal.mul_iSup] refine iSup_le fun s => ?_ rw [ENNReal.mul_iSup, iSup_le_iff] intro hs rw [← SimpleFunc.const_mul_lintegral, lintegral] refine le_iSup_of_le (const α r * s) (le_iSup_of_le (fun x => ?_) le_rfl) exact mul_le_mul_left' (hs x) _ #align measure_theory.lintegral_const_mul_le MeasureTheory.lintegral_const_mul_le theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) : ∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by by_cases h : r = 0 · simp [h] apply le_antisymm _ (lintegral_const_mul_le r f) have rinv : r * r⁻¹ = 1 := ENNReal.mul_inv_cancel h hr have rinv' : r⁻¹ * r = 1 := by rw [mul_comm] exact rinv have := lintegral_const_mul_le (μ := μ) r⁻¹ fun x => r * f x simp? [(mul_assoc _ _ _).symm, rinv'] at this says simp only [(mul_assoc _ _ _).symm, rinv', one_mul] at this simpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r #align measure_theory.lintegral_const_mul' MeasureTheory.lintegral_const_mul' theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul r hf] #align measure_theory.lintegral_mul_const MeasureTheory.lintegral_mul_const theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) : ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul'' r hf] #align measure_theory.lintegral_mul_const'' MeasureTheory.lintegral_mul_const'' theorem lintegral_mul_const_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) : (∫⁻ a, f a ∂μ) * r ≤ ∫⁻ a, f a * r ∂μ := by simp_rw [mul_comm, lintegral_const_mul_le r f] #align measure_theory.lintegral_mul_const_le MeasureTheory.lintegral_mul_const_le theorem lintegral_mul_const' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) : ∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul' r f hr] #align measure_theory.lintegral_mul_const' MeasureTheory.lintegral_mul_const' /- A double integral of a product where each factor contains only one variable is a product of integrals -/ theorem lintegral_lintegral_mul {β} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g ν) : ∫⁻ x, ∫⁻ y, f x * g y ∂ν ∂μ = (∫⁻ x, f x ∂μ) * ∫⁻ y, g y ∂ν := by simp [lintegral_const_mul'' _ hg, lintegral_mul_const'' _ hf] #align measure_theory.lintegral_lintegral_mul MeasureTheory.lintegral_lintegral_mul -- TODO: Need a better way of rewriting inside of an integral theorem lintegral_rw₁ {f f' : α → β} (h : f =ᵐ[μ] f') (g : β → ℝ≥0∞) : ∫⁻ a, g (f a) ∂μ = ∫⁻ a, g (f' a) ∂μ := lintegral_congr_ae <\| h.mono fun a h => by dsimp only; rw [h] #align measure_theory.lintegral_rw₁ MeasureTheory.lintegral_rw₁ -- TODO: Need a better way of rewriting inside of an integral theorem lintegral_rw₂ {f₁ f₁' : α → β} {f₂ f₂' : α → γ} (h₁ : f₁ =ᵐ[μ] f₁') (h₂ : f₂ =ᵐ[μ] f₂') (g : β → γ → ℝ≥0∞) : ∫⁻ a, g (f₁ a) (f₂ a) ∂μ = ∫⁻ a, g (f₁' a) (f₂' a) ∂μ := lintegral_congr_ae <\| h₁.mp <\| h₂.mono fun _ h₂ h₁ => by dsimp only; rw [h₁, h₂] #align measure_theory.lintegral_rw₂ MeasureTheory.lintegral_rw₂ theorem lintegral_indicator_le (f : α → ℝ≥0∞) (s : Set α) : ∫⁻ a, s.indicator f a ∂μ ≤ ∫⁻ a in s, f a ∂μ := by simp only [lintegral] apply iSup_le (fun g ↦ (iSup_le (fun hg ↦ ?_))) have : g ≤ f := hg.trans (indicator_le_self s f) refine le_iSup_of_le g (le_iSup_of_le this (le_of_eq ?_)) rw [lintegral_restrict, SimpleFunc.lintegral] congr with t by_cases H : t = 0 · simp [H] congr with x simp only [mem_preimage, mem_singleton_iff, mem_inter_iff, iff_self_and] rintro rfl contrapose! H simpa [H] using hg x @[simp] theorem lintegral_indicator (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) : ∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ := by apply le_antisymm (lintegral_indicator_le f s) simp only [lintegral, ← restrict_lintegral_eq_lintegral_restrict _ hs, iSup_subtype'] refine iSup_mono' (Subtype.forall.2 fun φ hφ => ?_) refine ⟨⟨φ.restrict s, fun x => ?_⟩, le_rfl⟩ simp [hφ x, hs, indicator_le_indicator] #align measure_theory.lintegral_indicator MeasureTheory.lintegral_indicator theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMeasurableSet s μ) : ∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ := by rw [← lintegral_congr_ae (indicator_ae_eq_of_ae_eq_set hs.toMeasurable_ae_eq), lintegral_indicator _ (measurableSet_toMeasurable _ _), Measure.restrict_congr_set hs.toMeasurable_ae_eq] #align measure_theory.lintegral_indicator₀ MeasureTheory.lintegral_indicator₀ theorem lintegral_indicator_const_le (s : Set α) (c : ℝ≥0∞) : ∫⁻ a, s.indicator (fun _ => c) a ∂μ ≤ c * μ s := (lintegral_indicator_le _ _).trans (set_lintegral_const s c).le theorem lintegral_indicator_const₀ {s : Set α} (hs : NullMeasurableSet s μ) (c : ℝ≥0∞) : ∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s := by rw [lintegral_indicator₀ _ hs, set_lintegral_const] theorem lintegral_indicator_const {s : Set α} (hs : MeasurableSet s) (c : ℝ≥0∞) : ∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s := lintegral_indicator_const₀ hs.nullMeasurableSet c #align measure_theory.lintegral_indicator_const MeasureTheory.lintegral_indicator_const theorem set_lintegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞) : ∫⁻ x in { x \| f x = r }, f x ∂μ = r * μ { x \| f x = r } := by have : ∀ᵐ x ∂μ, x ∈ { x \| f x = r } → f x = r := ae_of_all μ fun _ hx => hx rw [set_lintegral_congr_fun _ this] · rw [lintegral_const, Measure.restrict_apply MeasurableSet.univ, Set.univ_inter] · exact hf (measurableSet_singleton r) #align measure_theory.set_lintegral_eq_const MeasureTheory.set_lintegral_eq_const theorem lintegral_indicator_one_le (s : Set α) : ∫⁻ a, s.indicator 1 a ∂μ ≤ μ s := (lintegral_indicator_const_le _ _).trans <\| (one_mul _).le @[simp] theorem lintegral_indicator_one₀ (hs : NullMeasurableSet s μ) : ∫⁻ a, s.indicator 1 a ∂μ = μ s := (lintegral_indicator_const₀ hs _).trans <\| one_mul _ @[simp] theorem lintegral_indicator_one (hs : MeasurableSet s) : ∫⁻ a, s.indicator 1 a ∂μ = μ s := (lintegral_indicator_const hs _).trans <\| one_mul _ #align measure_theory.lintegral_indicator_one MeasureTheory.lintegral_indicator_one /-- A version of Markov's inequality for two functions. It doesn't follow from the standard Markov's inequality because we only assume measurability of `g`, not `f`. -/ theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g) (hg : AEMeasurable g μ) (ε : ℝ≥0∞) : ∫⁻ a, f a ∂μ + ε * μ { x \| f x + ε ≤ g x } ≤ ∫⁻ a, g a ∂μ := by rcases exists_measurable_le_lintegral_eq μ f with ⟨φ, hφm, hφ_le, hφ_eq⟩ calc ∫⁻ x, f x ∂μ + ε * μ { x \| f x + ε ≤ g x } = ∫⁻ x, φ x ∂μ + ε * μ { x \| f x + ε ≤ g x } := by rw [hφ_eq] _ ≤ ∫⁻ x, φ x ∂μ + ε * μ { x \| φ x + ε ≤ g x } := by gcongr exact fun x => (add_le_add_right (hφ_le _) _).trans _ = ∫⁻ x, φ x + indicator { x \| φ x + ε ≤ g x } (fun _ => ε) x ∂μ := by rw [lintegral_add_left hφm, lintegral_indicator₀, set_lintegral_const] exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable _ ≤ ∫⁻ x, g x ∂μ := lintegral_mono_ae (hle.mono fun x hx₁ => ?_) simp only [indicator_apply]; split_ifs with hx₂ exacts [hx₂, (add_zero _).trans_le <\| (hφ_le x).trans hx₁] #align measure_theory.lintegral_add_mul_meas_add_le_le_lintegral MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral /-- Markov's inequality also known as Chebyshev's first inequality. -/ theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞) : ε * μ { x \| ε ≤ f x } ≤ ∫⁻ a, f a ∂μ := by simpa only [lintegral_zero, zero_add] using lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le (f x)) hf ε #align measure_theory.mul_meas_ge_le_lintegral₀ MeasureTheory.mul_meas_ge_le_lintegral₀ /-- Markov's inequality also known as Chebyshev's first inequality. For a version assuming `AEMeasurable`, see `mul_meas_ge_le_lintegral₀`. -/ theorem mul_meas_ge_le_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) (ε : ℝ≥0∞) : ε * μ { x \| ε ≤ f x } ≤ ∫⁻ a, f a ∂μ := mul_meas_ge_le_lintegral₀ hf.aemeasurable ε #align measure_theory.mul_meas_ge_le_lintegral MeasureTheory.mul_meas_ge_le_lintegral lemma meas_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {s : Set α} (hs : ∀ x ∈ s, 1 ≤ f x) : μ s ≤ ∫⁻ a, f a ∂μ := by apply le_trans _ (mul_meas_ge_le_lintegral₀ hf 1) rw [one_mul] exact measure_mono hs lemma lintegral_le_meas {s : Set α} {f : α → ℝ≥0∞} (hf : ∀ a, f a ≤ 1) (h'f : ∀ a ∈ sᶜ, f a = 0) : ∫⁻ a, f a ∂μ ≤ μ s := by apply (lintegral_mono (fun x ↦ ?_)).trans (lintegral_indicator_one_le s) by_cases hx : x ∈ s · simpa [hx] using hf x · simpa [hx] using h'f x hx theorem lintegral_eq_top_of_measure_eq_top_ne_zero {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hμf : μ {x \| f x = ∞} ≠ 0) : ∫⁻ x, f x ∂μ = ∞ := eq_top_iff.mpr <\| calc ∞ = ∞ * μ { x \| ∞ ≤ f x } := by simp [mul_eq_top, hμf] _ ≤ ∫⁻ x, f x ∂μ := mul_meas_ge_le_lintegral₀ hf ∞ #align measure_theory.lintegral_eq_top_of_measure_eq_top_ne_zero MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero theorem setLintegral_eq_top_of_measure_eq_top_ne_zero (hf : AEMeasurable f (μ.restrict s)) (hμf : μ ({x ∈ s \| f x = ∞}) ≠ 0) : ∫⁻ x in s, f x ∂μ = ∞ := lintegral_eq_top_of_measure_eq_top_ne_zero hf <\| mt (eq_bot_mono <\| by rw [← setOf_inter_eq_sep]; exact Measure.le_restrict_apply _ _) hμf #align measure_theory.set_lintegral_eq_top_of_measure_eq_top_ne_zero MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero theorem measure_eq_top_of_lintegral_ne_top (hf : AEMeasurable f μ) (hμf : ∫⁻ x, f x ∂μ ≠ ∞) : μ {x \| f x = ∞} = 0 := of_not_not fun h => hμf <\| lintegral_eq_top_of_measure_eq_top_ne_zero hf h #align measure_theory.measure_eq_top_of_lintegral_ne_top MeasureTheory.measure_eq_top_of_lintegral_ne_top theorem measure_eq_top_of_setLintegral_ne_top (hf : AEMeasurable f (μ.restrict s)) (hμf : ∫⁻ x in s, f x ∂μ ≠ ∞) : μ ({x ∈ s \| f x = ∞}) = 0 := of_not_not fun h => hμf <\| setLintegral_eq_top_of_measure_eq_top_ne_zero hf h #align measure_theory.measure_eq_top_of_set_lintegral_ne_top MeasureTheory.measure_eq_top_of_setLintegral_ne_top /-- Markov's inequality also known as Chebyshev's first inequality. -/ theorem meas_ge_le_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) (hε' : ε ≠ ∞) : μ { x \| ε ≤ f x } ≤ (∫⁻ a, f a ∂μ) / ε := (ENNReal.le_div_iff_mul_le (Or.inl hε) (Or.inl hε')).2 <\| by rw [mul_comm] exact mul_meas_ge_le_lintegral₀ hf ε #align measure_theory.meas_ge_le_lintegral_div MeasureTheory.meas_ge_le_lintegral_div theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : ∫⁻ x, f x ∂μ ≠ ∞) (hg : AEMeasurable g μ) (hgf : ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ) : f =ᵐ[μ] g := by have : ∀ n : ℕ, ∀ᵐ x ∂μ, g x < f x + (n : ℝ≥0∞)⁻¹ := by intro n simp only [ae_iff, not_lt] have : ∫⁻ x, f x ∂μ + (↑n)⁻¹ * μ { x : α \| f x + (n : ℝ≥0∞)⁻¹ ≤ g x } ≤ ∫⁻ x, f x ∂μ := (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n⁻¹).trans hgf rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.natCast_ne_top _)) refine hfg.mp ((ae_all_iff.2 this).mono fun x hlt hle => hle.antisymm ?_) suffices Tendsto (fun n : ℕ => f x + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (f x)) from ge_of_tendsto' this fun i => (hlt i).le simpa only [inv_top, add_zero] using tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top) #align measure_theory.ae_eq_of_ae_le_of_lintegral_le MeasureTheory.ae_eq_of_ae_le_of_lintegral_le @[simp] theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 := have : ∫⁻ _ : α, 0 ∂μ ≠ ∞ := by simp [lintegral_zero, zero_ne_top] ⟨fun h => (ae_eq_of_ae_le_of_lintegral_le (ae_of_all _ <\| zero_le f) this hf (h.trans lintegral_zero.symm).le).symm, fun h => (lintegral_congr_ae h).trans lintegral_zero⟩ #align measure_theory.lintegral_eq_zero_iff' MeasureTheory.lintegral_eq_zero_iff' @[simp] theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 := lintegral_eq_zero_iff' hf.aemeasurable #align measure_theory.lintegral_eq_zero_iff MeasureTheory.lintegral_eq_zero_iff theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) : (0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) := by simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support] #align measure_theory.lintegral_pos_iff_support MeasureTheory.lintegral_pos_iff_support theorem setLintegral_pos_iff {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} : 0 < ∫⁻ a in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s) := by rw [lintegral_pos_iff_support hf, Measure.restrict_apply (measurableSet_support hf)] /-- Weaker version of the monotone convergence theorem-/ theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by let ⟨s, hs⟩ := exists_measurable_superset_of_null (ae_iff.1 (ae_all_iff.2 h_mono)) let g n a := if a ∈ s then 0 else f n a have g_eq_f : ∀ᵐ a ∂μ, ∀ n, g n a = f n a := (measure_zero_iff_ae_nmem.1 hs.2.2).mono fun a ha n => if_neg ha calc ∫⁻ a, ⨆ n, f n a ∂μ = ∫⁻ a, ⨆ n, g n a ∂μ := lintegral_congr_ae <\| g_eq_f.mono fun a ha => by simp only [ha] _ = ⨆ n, ∫⁻ a, g n a ∂μ := (lintegral_iSup (fun n => measurable_const.piecewise hs.2.1 (hf n)) (monotone_nat_of_le_succ fun n a => ?_)) _ = ⨆ n, ∫⁻ a, f n a ∂μ := by simp only [lintegral_congr_ae (g_eq_f.mono fun _a ha => ha _)] simp only [g] split_ifs with h · rfl · have := Set.not_mem_subset hs.1 h simp only [not_forall, not_le, mem_setOf_eq, not_exists, not_lt] at this exact this n #align measure_theory.lintegral_supr_ae MeasureTheory.lintegral_iSup_ae theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞) (h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ := by refine ENNReal.eq_sub_of_add_eq hg_fin ?_ rw [← lintegral_add_right' _ hg] exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx) #align measure_theory.lintegral_sub' MeasureTheory.lintegral_sub' theorem lintegral_sub {f g : α → ℝ≥0∞} (hg : Measurable g) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞) (h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ := lintegral_sub' hg.aemeasurable hg_fin h_le #align measure_theory.lintegral_sub MeasureTheory.lintegral_sub theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) : ∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ := by rw [tsub_le_iff_right] by_cases hfi : ∫⁻ x, f x ∂μ = ∞ · rw [hfi, add_top] exact le_top · rw [← lintegral_add_right' _ hf] gcongr exact le_tsub_add #align measure_theory.lintegral_sub_le' MeasureTheory.lintegral_sub_le' theorem lintegral_sub_le (f g : α → ℝ≥0∞) (hf : Measurable f) : ∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ := lintegral_sub_le' f g hf.aemeasurable #align measure_theory.lintegral_sub_le MeasureTheory.lintegral_sub_le theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ x ∂μ, f x ≠ g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := by contrapose! h simp only [not_frequently, Ne, Classical.not_not] exact ae_eq_of_ae_le_of_lintegral_le h_le hfi hg h #align measure_theory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt theorem lintegral_strict_mono_of_ae_le_of_ae_lt_on {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) {s : Set α} (hμs : μ s ≠ 0) (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := lintegral_strict_mono_of_ae_le_of_frequently_ae_lt hg hfi h_le <\| ((frequently_ae_mem_iff.2 hμs).and_eventually h).mono fun _x hx => (hx.2 hx.1).ne #align measure_theory.lintegral_strict_mono_of_ae_le_of_ae_lt_on MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h : ∀ᵐ x ∂μ, f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := by rw [Ne, ← Measure.measure_univ_eq_zero] at hμ refine lintegral_strict_mono_of_ae_le_of_ae_lt_on hg hfi (ae_le_of_ae_lt h) hμ ?_ simpa using h #align measure_theory.lintegral_strict_mono MeasureTheory.lintegral_strict_mono theorem set_lintegral_strict_mono {f g : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s) (hs : μ s ≠ 0) (hg : Measurable g) (hfi : ∫⁻ x in s, f x ∂μ ≠ ∞) (h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x in s, f x ∂μ < ∫⁻ x in s, g x ∂μ := lintegral_strict_mono (by simp [hs]) hg.aemeasurable hfi ((ae_restrict_iff' hsm).mpr h) #align measure_theory.set_lintegral_strict_mono MeasureTheory.set_lintegral_strict_mono /-- Monotone convergence theorem for nonincreasing sequences of functions -/ theorem lintegral_iInf_ae {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_mono : ∀ n : ℕ, f n.succ ≤ᵐ[μ] f n) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) : ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ := have fn_le_f0 : ∫⁻ a, ⨅ n, f n a ∂μ ≤ ∫⁻ a, f 0 a ∂μ := lintegral_mono fun a => iInf_le_of_le 0 le_rfl have fn_le_f0' : ⨅ n, ∫⁻ a, f n a ∂μ ≤ ∫⁻ a, f 0 a ∂μ := iInf_le_of_le 0 le_rfl (ENNReal.sub_right_inj h_fin fn_le_f0 fn_le_f0').1 <\| show ∫⁻ a, f 0 a ∂μ - ∫⁻ a, ⨅ n, f n a ∂μ = ∫⁻ a, f 0 a ∂μ - ⨅ n, ∫⁻ a, f n a ∂μ from calc ∫⁻ a, f 0 a ∂μ - ∫⁻ a, ⨅ n, f n a ∂μ = ∫⁻ a, f 0 a - ⨅ n, f n a ∂μ := (lintegral_sub (measurable_iInf h_meas) (ne_top_of_le_ne_top h_fin <\| lintegral_mono fun a => iInf_le _ _) (ae_of_all _ fun a => iInf_le _ _)).symm _ = ∫⁻ a, ⨆ n, f 0 a - f n a ∂μ := congr rfl (funext fun a => ENNReal.sub_iInf) _ = ⨆ n, ∫⁻ a, f 0 a - f n a ∂μ := (lintegral_iSup_ae (fun n => (h_meas 0).sub (h_meas n)) fun n => (h_mono n).mono fun a ha => tsub_le_tsub le_rfl ha) _ = ⨆ n, ∫⁻ a, f 0 a ∂μ - ∫⁻ a, f n a ∂μ := (have h_mono : ∀ᵐ a ∂μ, ∀ n : ℕ, f n.succ a ≤ f n a := ae_all_iff.2 h_mono have h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f 0 a := fun n => h_mono.mono fun a h => by induction' n with n ih · exact le_rfl · exact le_trans (h n) ih congr_arg iSup <\| funext fun n => lintegral_sub (h_meas _) (ne_top_of_le_ne_top h_fin <\| lintegral_mono_ae <\| h_mono n) (h_mono n)) _ = ∫⁻ a, f 0 a ∂μ - ⨅ n, ∫⁻ a, f n a ∂μ := ENNReal.sub_iInf.symm #align measure_theory.lintegral_infi_ae MeasureTheory.lintegral_iInf_ae /-- Monotone convergence theorem for nonincreasing sequences of functions -/ theorem lintegral_iInf {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_anti : Antitone f) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) : ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ := lintegral_iInf_ae h_meas (fun n => ae_of_all _ <\| h_anti n.le_succ) h_fin #align measure_theory.lintegral_infi MeasureTheory.lintegral_iInf theorem lintegral_iInf' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ) (h_anti : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a)) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) : ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ := by simp_rw [← iInf_apply] let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Antitone f' have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_anti have h_ae_seq_mono : Antitone (aeSeq h_meas p) := by intro n m hnm x by_cases hx : x ∈ aeSeqSet h_meas p · exact aeSeq.prop_of_mem_aeSeqSet h_meas hx hnm · simp only [aeSeq, hx, if_false] exact le_rfl rw [lintegral_congr_ae (aeSeq.iInf h_meas hp).symm] simp_rw [iInf_apply] rw [lintegral_iInf (aeSeq.measurable h_meas p) h_ae_seq_mono] · congr exact funext fun n ↦ lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae h_meas hp n) · rwa [lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae h_meas hp 0)] /-- Monotone convergence for an infimum over a directed family and indexed by a countable type -/ theorem lintegral_iInf_directed_of_measurable {mα : MeasurableSpace α} [Countable β] {f : β → α → ℝ≥0∞} {μ : Measure α} (hμ : μ ≠ 0) (hf : ∀ b, Measurable (f b)) (hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : Directed (· ≥ ·) f) : ∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ := by cases nonempty_encodable β cases isEmpty_or_nonempty β · simp only [iInf_of_empty, lintegral_const, ENNReal.top_mul (Measure.measure_univ_ne_zero.mpr hμ)] inhabit β have : ∀ a, ⨅ b, f b a = ⨅ n, f (h_directed.sequence f n) a := by refine fun a => le_antisymm (le_iInf fun n => iInf_le _ _) (le_iInf fun b => iInf_le_of_le (Encodable.encode b + 1) ?_) exact h_directed.sequence_le b a -- Porting note: used `∘` below to deal with its reduced reducibility calc ∫⁻ a, ⨅ b, f b a ∂μ _ = ∫⁻ a, ⨅ n, (f ∘ h_directed.sequence f) n a ∂μ := by simp only [this, Function.comp_apply] _ = ⨅ n, ∫⁻ a, (f ∘ h_directed.sequence f) n a ∂μ := by rw [lintegral_iInf ?_ h_directed.sequence_anti] · exact hf_int _ · exact fun n => hf _ _ = ⨅ b, ∫⁻ a, f b a ∂μ := by refine le_antisymm (le_iInf fun b => ?_) (le_iInf fun n => ?_) · exact iInf_le_of_le (Encodable.encode b + 1) (lintegral_mono <\| h_directed.sequence_le b) · exact iInf_le (fun b => ∫⁻ a, f b a ∂μ) _ #align lintegral_infi_directed_of_measurable MeasureTheory.lintegral_iInf_directed_of_measurable /-- Known as Fatou's lemma, version with `AEMeasurable` functions -/ theorem lintegral_liminf_le' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ) : ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop := calc ∫⁻ a, liminf (fun n => f n a) atTop ∂μ = ∫⁻ a, ⨆ n : ℕ, ⨅ i ≥ n, f i a ∂μ := by simp only [liminf_eq_iSup_iInf_of_nat] _ = ⨆ n : ℕ, ∫⁻ a, ⨅ i ≥ n, f i a ∂μ := (lintegral_iSup' (fun n => aemeasurable_biInf _ (to_countable _) (fun i _ ↦ h_meas i)) (ae_of_all μ fun a n m hnm => iInf_le_iInf_of_subset fun i hi => le_trans hnm hi)) _ ≤ ⨆ n : ℕ, ⨅ i ≥ n, ∫⁻ a, f i a ∂μ := iSup_mono fun n => le_iInf₂_lintegral _ _ = atTop.liminf fun n => ∫⁻ a, f n a ∂μ := Filter.liminf_eq_iSup_iInf_of_nat.symm #align measure_theory.lintegral_liminf_le' MeasureTheory.lintegral_liminf_le' /-- Known as Fatou's lemma -/ theorem lintegral_liminf_le {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) : ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop := lintegral_liminf_le' fun n => (h_meas n).aemeasurable #align measure_theory.lintegral_liminf_le MeasureTheory.lintegral_liminf_le theorem limsup_lintegral_le {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} (hf_meas : ∀ n, Measurable (f n)) (h_bound : ∀ n, f n ≤ᵐ[μ] g) (h_fin : ∫⁻ a, g a ∂μ ≠ ∞) : limsup (fun n => ∫⁻ a, f n a ∂μ) atTop ≤ ∫⁻ a, limsup (fun n => f n a) atTop ∂μ := calc limsup (fun n => ∫⁻ a, f n a ∂μ) atTop = ⨅ n : ℕ, ⨆ i ≥ n, ∫⁻ a, f i a ∂μ := limsup_eq_iInf_iSup_of_nat _ ≤ ⨅ n : ℕ, ∫⁻ a, ⨆ i ≥ n, f i a ∂μ := iInf_mono fun n => iSup₂_lintegral_le _ _ = ∫⁻ a, ⨅ n : ℕ, ⨆ i ≥ n, f i a ∂μ := by refine (lintegral_iInf ?_ ?_ ?_).symm · intro n exact measurable_biSup _ (to_countable _) (fun i _ ↦ hf_meas i) · intro n m hnm a exact iSup_le_iSup_of_subset fun i hi => le_trans hnm hi · refine ne_top_of_le_ne_top h_fin (lintegral_mono_ae ?_) refine (ae_all_iff.2 h_bound).mono fun n hn => ?_ exact iSup_le fun i => iSup_le fun _ => hn i _ = ∫⁻ a, limsup (fun n => f n a) atTop ∂μ := by simp only [limsup_eq_iInf_iSup_of_nat] #align measure_theory.limsup_lintegral_le MeasureTheory.limsup_lintegral_le /-- Dominated convergence theorem for nonnegative functions -/ theorem tendsto_lintegral_of_dominated_convergence {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ n, Measurable (F n)) (h_bound : ∀ n, F n ≤ᵐ[μ] bound) (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : Tendsto (fun n => ∫⁻ a, F n a ∂μ) atTop (𝓝 (∫⁻ a, f a ∂μ)) := tendsto_of_le_liminf_of_limsup_le (calc ∫⁻ a, f a ∂μ = ∫⁻ a, liminf (fun n : ℕ => F n a) atTop ∂μ := lintegral_congr_ae <\| h_lim.mono fun a h => h.liminf_eq.symm _ ≤ liminf (fun n => ∫⁻ a, F n a ∂μ) atTop := lintegral_liminf_le hF_meas ) (calc limsup (fun n : ℕ => ∫⁻ a, F n a ∂μ) atTop ≤ ∫⁻ a, limsup (fun n => F n a) atTop ∂μ := limsup_lintegral_le hF_meas h_bound h_fin _ = ∫⁻ a, f a ∂μ := lintegral_congr_ae <\| h_lim.mono fun a h => h.limsup_eq ) #align measure_theory.tendsto_lintegral_of_dominated_convergence MeasureTheory.tendsto_lintegral_of_dominated_convergence /-- Dominated convergence theorem for nonnegative functions which are just almost everywhere measurable. -/ theorem tendsto_lintegral_of_dominated_convergence' {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ n, AEMeasurable (F n) μ) (h_bound : ∀ n, F n ≤ᵐ[μ] bound) (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : Tendsto (fun n => ∫⁻ a, F n a ∂μ) atTop (𝓝 (∫⁻ a, f a ∂μ)) := by have : ∀ n, ∫⁻ a, F n a ∂μ = ∫⁻ a, (hF_meas n).mk (F n) a ∂μ := fun n => lintegral_congr_ae (hF_meas n).ae_eq_mk simp_rw [this] apply tendsto_lintegral_of_dominated_convergence bound (fun n => (hF_meas n).measurable_mk) _ h_fin · have : ∀ n, ∀ᵐ a ∂μ, (hF_meas n).mk (F n) a = F n a := fun n => (hF_meas n).ae_eq_mk.symm have : ∀ᵐ a ∂μ, ∀ n, (hF_meas n).mk (F n) a = F n a := ae_all_iff.mpr this filter_upwards [this, h_lim] with a H H' simp_rw [H] exact H' · intro n filter_upwards [h_bound n, (hF_meas n).ae_eq_mk] with a H H' rwa [H'] at H #align measure_theory.tendsto_lintegral_of_dominated_convergence' MeasureTheory.tendsto_lintegral_of_dominated_convergence' /-- Dominated convergence theorem for filters with a countable basis -/ theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ᶠ n in l, Measurable (F n)) (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, F n a ≤ bound a) (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) : Tendsto (fun n => ∫⁻ a, F n a ∂μ) l (𝓝 <\| ∫⁻ a, f a ∂μ) := by rw [tendsto_iff_seq_tendsto] intro x xl have hxl := by rw [tendsto_atTop'] at xl exact xl have h := inter_mem hF_meas h_bound replace h := hxl _ h rcases h with ⟨k, h⟩ rw [← tendsto_add_atTop_iff_nat k] refine tendsto_lintegral_of_dominated_convergence ?_ ?_ ?_ ?_ ?_ · exact bound · intro refine (h _ ?_).1 exact Nat.le_add_left _ _ · intro refine (h _ ?_).2 exact Nat.le_add_left _ _ · assumption · refine h_lim.mono fun a h_lim => ?_ apply @Tendsto.comp _ _ _ (fun n => x (n + k)) fun n => F n a · assumption rw [tendsto_add_atTop_iff_nat] assumption #align measure_theory.tendsto_lintegral_filter_of_dominated_convergence MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence theorem lintegral_tendsto_of_tendsto_of_antitone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ) (h_anti : ∀ᵐ x ∂μ, Antitone fun n ↦ f n x) (h0 : ∫⁻ a, f 0 a ∂μ ≠ ∞) (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))) : Tendsto (fun n ↦ ∫⁻ x, f n x ∂μ) atTop (𝓝 (∫⁻ x, F x ∂μ)) := by have : Antitone fun n ↦ ∫⁻ x, f n x ∂μ := fun i j hij ↦ lintegral_mono_ae (h_anti.mono fun x hx ↦ hx hij) suffices key : ∫⁻ x, F x ∂μ = ⨅ n, ∫⁻ x, f n x ∂μ by rw [key] exact tendsto_atTop_iInf this rw [← lintegral_iInf' hf h_anti h0] refine lintegral_congr_ae ?_ filter_upwards [h_anti, h_tendsto] with _ hx_anti hx_tendsto using tendsto_nhds_unique hx_tendsto (tendsto_atTop_iInf hx_anti) section open Encodable /-- Monotone convergence for a supremum over a directed family and indexed by a countable type -/ theorem lintegral_iSup_directed_of_measurable [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, Measurable (f b)) (h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ := by cases nonempty_encodable β cases isEmpty_or_nonempty β · simp [iSup_of_empty] inhabit β have : ∀ a, ⨆ b, f b a = ⨆ n, f (h_directed.sequence f n) a := by intro a refine le_antisymm (iSup_le fun b => ?_) (iSup_le fun n => le_iSup (fun n => f n a) _) exact le_iSup_of_le (encode b + 1) (h_directed.le_sequence b a) calc ∫⁻ a, ⨆ b, f b a ∂μ = ∫⁻ a, ⨆ n, f (h_directed.sequence f n) a ∂μ := by simp only [this] _ = ⨆ n, ∫⁻ a, f (h_directed.sequence f n) a ∂μ := (lintegral_iSup (fun n => hf _) h_directed.sequence_mono) _ = ⨆ b, ∫⁻ a, f b a ∂μ := by refine le_antisymm (iSup_le fun n => ?_) (iSup_le fun b => ?_) · exact le_iSup (fun b => ∫⁻ a, f b a ∂μ) _ · exact le_iSup_of_le (encode b + 1) (lintegral_mono <\| h_directed.le_sequence b) #align measure_theory.lintegral_supr_directed_of_measurable MeasureTheory.lintegral_iSup_directed_of_measurable /-- Monotone convergence for a supremum over a directed family and indexed by a countable type. -/ theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ) (h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ := by simp_rw [← iSup_apply] let p : α → (β → ENNReal) → Prop := fun x f' => Directed LE.le f' have hp : ∀ᵐ x ∂μ, p x fun i => f i x := by filter_upwards [] with x i j obtain ⟨z, hz₁, hz₂⟩ := h_directed i j exact ⟨z, hz₁ x, hz₂ x⟩ have h_ae_seq_directed : Directed LE.le (aeSeq hf p) := by intro b₁ b₂ obtain ⟨z, hz₁, hz₂⟩ := h_directed b₁ b₂ refine ⟨z, ?_, ?_⟩ <;> · intro x by_cases hx : x ∈ aeSeqSet hf p · repeat rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx] apply_rules [hz₁, hz₂] · simp only [aeSeq, hx, if_false] exact le_rfl convert lintegral_iSup_directed_of_measurable (aeSeq.measurable hf p) h_ae_seq_directed using 1 · simp_rw [← iSup_apply] rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm] · congr 1 ext1 b rw [lintegral_congr_ae] apply EventuallyEq.symm exact aeSeq.aeSeq_n_eq_fun_n_ae hf hp _ #align measure_theory.lintegral_supr_directed MeasureTheory.lintegral_iSup_directed end theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) : ∫⁻ a, ∑' i, f i a ∂μ = ∑' i, ∫⁻ a, f i a ∂μ := by simp only [ENNReal.tsum_eq_iSup_sum] rw [lintegral_iSup_directed] · simp [lintegral_finset_sum' _ fun i _ => hf i] · intro b exact Finset.aemeasurable_sum _ fun i _ => hf i · intro s t use s ∪ t constructor · exact fun a => Finset.sum_le_sum_of_subset Finset.subset_union_left · exact fun a => Finset.sum_le_sum_of_subset Finset.subset_union_right #align measure_theory.lintegral_tsum MeasureTheory.lintegral_tsum open Measure theorem lintegral_iUnion₀ [Countable β] {s : β → Set α} (hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ i, s i, f a ∂μ = ∑' i, ∫⁻ a in s i, f a ∂μ := by simp only [Measure.restrict_iUnion_ae hd hm, lintegral_sum_measure] #align measure_theory.lintegral_Union₀ MeasureTheory.lintegral_iUnion₀ theorem lintegral_iUnion [Countable β] {s : β → Set α} (hm : ∀ i, MeasurableSet (s i)) (hd : Pairwise (Disjoint on s)) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ i, s i, f a ∂μ = ∑' i, ∫⁻ a in s i, f a ∂μ := lintegral_iUnion₀ (fun i => (hm i).nullMeasurableSet) hd.aedisjoint f #align measure_theory.lintegral_Union MeasureTheory.lintegral_iUnion theorem lintegral_biUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, NullMeasurableSet (s i) μ) (hd : t.Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ := by haveI := ht.toEncodable rw [biUnion_eq_iUnion, lintegral_iUnion₀ (SetCoe.forall'.1 hm) (hd.subtype _ _)] #align measure_theory.lintegral_bUnion₀ MeasureTheory.lintegral_biUnion₀ theorem lintegral_biUnion {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, MeasurableSet (s i)) (hd : t.PairwiseDisjoint s) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ := lintegral_biUnion₀ ht (fun i hi => (hm i hi).nullMeasurableSet) hd.aedisjoint f #align measure_theory.lintegral_bUnion MeasureTheory.lintegral_biUnion theorem lintegral_biUnion_finset₀ {s : Finset β} {t : β → Set α} (hd : Set.Pairwise (↑s) (AEDisjoint μ on t)) (hm : ∀ b ∈ s, NullMeasurableSet (t b) μ) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ a in t b, f a ∂μ := by simp only [← Finset.mem_coe, lintegral_biUnion₀ s.countable_toSet hm hd, ← Finset.tsum_subtype'] #align measure_theory.lintegral_bUnion_finset₀ MeasureTheory.lintegral_biUnion_finset₀ theorem lintegral_biUnion_finset {s : Finset β} {t : β → Set α} (hd : Set.PairwiseDisjoint (↑s) t) (hm : ∀ b ∈ s, MeasurableSet (t b)) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ a in t b, f a ∂μ := lintegral_biUnion_finset₀ hd.aedisjoint (fun b hb => (hm b hb).nullMeasurableSet) f #align measure_theory.lintegral_bUnion_finset MeasureTheory.lintegral_biUnion_finset theorem lintegral_iUnion_le [Countable β] (s : β → Set α) (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ i, s i, f a ∂μ ≤ ∑' i, ∫⁻ a in s i, f a ∂μ := by rw [← lintegral_sum_measure] exact lintegral_mono' restrict_iUnion_le le_rfl #align measure_theory.lintegral_Union_le MeasureTheory.lintegral_iUnion_le theorem lintegral_union {f : α → ℝ≥0∞} {A B : Set α} (hB : MeasurableSet B) (hAB : Disjoint A B) : ∫⁻ a in A ∪ B, f a ∂μ = ∫⁻ a in A, f a ∂μ + ∫⁻ a in B, f a ∂μ := by rw [restrict_union hAB hB, lintegral_add_measure] #align measure_theory.lintegral_union MeasureTheory.lintegral_union theorem lintegral_union_le (f : α → ℝ≥0∞) (s t : Set α) : ∫⁻ a in s ∪ t, f a ∂μ ≤ ∫⁻ a in s, f a ∂μ + ∫⁻ a in t, f a ∂μ := by rw [← lintegral_add_measure] exact lintegral_mono' (restrict_union_le _ _) le_rfl theorem lintegral_inter_add_diff {B : Set α} (f : α → ℝ≥0∞) (A : Set α) (hB : MeasurableSet B) : ∫⁻ x in A ∩ B, f x ∂μ + ∫⁻ x in A \ B, f x ∂μ = ∫⁻ x in A, f x ∂μ := by rw [← lintegral_add_measure, restrict_inter_add_diff _ hB] #align measure_theory.lintegral_inter_add_diff MeasureTheory.lintegral_inter_add_diff theorem lintegral_add_compl (f : α → ℝ≥0∞) {A : Set α} (hA : MeasurableSet A) : ∫⁻ x in A, f x ∂μ + ∫⁻ x in Aᶜ, f x ∂μ = ∫⁻ x, f x ∂μ := by rw [← lintegral_add_measure, Measure.restrict_add_restrict_compl hA] #align measure_theory.lintegral_add_compl MeasureTheory.lintegral_add_compl theorem lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) : ∫⁻ x, max (f x) (g x) ∂μ = ∫⁻ x in { x \| f x ≤ g x }, g x ∂μ + ∫⁻ x in { x \| g x < f x }, f x ∂μ := by have hm : MeasurableSet { x \| f x ≤ g x } := measurableSet_le hf hg rw [← lintegral_add_compl (fun x => max (f x) (g x)) hm] simp only [← compl_setOf, ← not_le] refine congr_arg₂ (· + ·) (set_lintegral_congr_fun hm ?_) (set_lintegral_congr_fun hm.compl ?_) exacts [ae_of_all _ fun x => max_eq_right (a := f x) (b := g x), ae_of_all _ fun x (hx : ¬ f x ≤ g x) => max_eq_left (not_le.1 hx).le] #align measure_theory.lintegral_max MeasureTheory.lintegral_max theorem set_lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (s : Set α) : ∫⁻ x in s, max (f x) (g x) ∂μ = ∫⁻ x in s ∩ { x \| f x ≤ g x }, g x ∂μ + ∫⁻ x in s ∩ { x \| g x < f x }, f x ∂μ := by rw [lintegral_max hf hg, restrict_restrict, restrict_restrict, inter_comm s, inter_comm s] exacts [measurableSet_lt hg hf, measurableSet_le hf hg] #align measure_theory.set_lintegral_max MeasureTheory.set_lintegral_max theorem lintegral_map {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f) (hg : Measurable g) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ := by erw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral (hf.comp hg)] congr with n : 1 convert SimpleFunc.lintegral_map _ hg ext1 x; simp only [eapprox_comp hf hg, coe_comp] #align measure_theory.lintegral_map MeasureTheory.lintegral_map theorem lintegral_map' {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : AEMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) : ∫⁻ a, f a ∂Measure.map g μ = ∫⁻ a, f (g a) ∂μ := calc ∫⁻ a, f a ∂Measure.map g μ = ∫⁻ a, hf.mk f a ∂Measure.map g μ := lintegral_congr_ae hf.ae_eq_mk _ = ∫⁻ a, hf.mk f a ∂Measure.map (hg.mk g) μ := by congr 1 exact Measure.map_congr hg.ae_eq_mk _ = ∫⁻ a, hf.mk f (hg.mk g a) ∂μ := lintegral_map hf.measurable_mk hg.measurable_mk _ = ∫⁻ a, hf.mk f (g a) ∂μ := lintegral_congr_ae <\| hg.ae_eq_mk.symm.fun_comp _ _ = ∫⁻ a, f (g a) ∂μ := lintegral_congr_ae (ae_eq_comp hg hf.ae_eq_mk.symm) #align measure_theory.lintegral_map' MeasureTheory.lintegral_map' theorem lintegral_map_le {mβ : MeasurableSpace β} (f : β → ℝ≥0∞) {g : α → β} (hg : Measurable g) : ∫⁻ a, f a ∂Measure.map g μ ≤ ∫⁻ a, f (g a) ∂μ := by rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral] refine iSup₂_le fun i hi => iSup_le fun h'i => ?_ refine le_iSup₂_of_le (i ∘ g) (hi.comp hg) ?_ exact le_iSup_of_le (fun x => h'i (g x)) (le_of_eq (lintegral_map hi hg)) #align measure_theory.lintegral_map_le MeasureTheory.lintegral_map_le theorem lintegral_comp [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f) (hg : Measurable g) : lintegral μ (f ∘ g) = ∫⁻ a, f a ∂map g μ := (lintegral_map hf hg).symm #align measure_theory.lintegral_comp MeasureTheory.lintegral_comp theorem set_lintegral_map [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} {s : Set β} (hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g) : ∫⁻ y in s, f y ∂map g μ = ∫⁻ x in g ⁻¹' s, f (g x) ∂μ := by rw [restrict_map hg hs, lintegral_map hf hg] #align measure_theory.set_lintegral_map MeasureTheory.set_lintegral_map theorem lintegral_indicator_const_comp {mβ : MeasurableSpace β} {f : α → β} {s : Set β} (hf : Measurable f) (hs : MeasurableSet s) (c : ℝ≥0∞) : ∫⁻ a, s.indicator (fun _ => c) (f a) ∂μ = c * μ (f ⁻¹' s) := by erw [lintegral_comp (measurable_const.indicator hs) hf, lintegral_indicator_const hs, Measure.map_apply hf hs] #align measure_theory.lintegral_indicator_const_comp MeasureTheory.lintegral_indicator_const_comp /-- If `g : α → β` is a measurable embedding and `f : β → ℝ≥0∞` is any function (not necessarily measurable), then `∫⁻ a, f a ∂(map g μ) = ∫⁻ a, f (g a) ∂μ`. Compare with `lintegral_map` which applies to any measurable `g : α → β` but requires that `f` is measurable as well. -/ theorem _root_.MeasurableEmbedding.lintegral_map [MeasurableSpace β] {g : α → β} (hg : MeasurableEmbedding g) (f : β → ℝ≥0∞) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ := by rw [lintegral, lintegral] refine le_antisymm (iSup₂_le fun f₀ hf₀ => ?_) (iSup₂_le fun f₀ hf₀ => ?_) · rw [SimpleFunc.lintegral_map _ hg.measurable] have : (f₀.comp g hg.measurable : α → ℝ≥0∞) ≤ f ∘ g := fun x => hf₀ (g x) exact le_iSup_of_le (comp f₀ g hg.measurable) (by exact le_iSup (α := ℝ≥0∞) _ this) · rw [← f₀.extend_comp_eq hg (const _ 0), ← SimpleFunc.lintegral_map, ← SimpleFunc.lintegral_eq_lintegral, ← lintegral] refine lintegral_mono_ae (hg.ae_map_iff.2 <\| eventually_of_forall fun x => ?_) exact (extend_apply _ _ _ _).trans_le (hf₀ _) #align measurable_embedding.lintegral_map MeasurableEmbedding.lintegral_map /-- The `lintegral` transforms appropriately under a measurable equivalence `g : α ≃ᵐ β`. (Compare `lintegral_map`, which applies to a wider class of functions `g : α → β`, but requires measurability of the function being integrated.) -/ theorem lintegral_map_equiv [MeasurableSpace β] (f : β → ℝ≥0∞) (g : α ≃ᵐ β) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ := g.measurableEmbedding.lintegral_map f #align measure_theory.lintegral_map_equiv MeasureTheory.lintegral_map_equiv protected theorem MeasurePreserving.lintegral_map_equiv [MeasurableSpace β] {ν : Measure β} (f : β → ℝ≥0∞) (g : α ≃ᵐ β) (hg : MeasurePreserving g μ ν) : ∫⁻ a, f a ∂ν = ∫⁻ a, f (g a) ∂μ := by rw [← MeasureTheory.lintegral_map_equiv f g, hg.map_eq] theorem MeasurePreserving.lintegral_comp {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) {f : β → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f (g a) ∂μ = ∫⁻ b, f b ∂ν := by rw [← hg.map_eq, lintegral_map hf hg.measurable] #align measure_theory.measure_preserving.lintegral_comp MeasureTheory.MeasurePreserving.lintegral_comp theorem MeasurePreserving.lintegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) : ∫⁻ a, f (g a) ∂μ = ∫⁻ b, f b ∂ν := by rw [← hg.map_eq, hge.lintegral_map] #align measure_theory.measure_preserving.lintegral_comp_emb MeasureTheory.MeasurePreserving.lintegral_comp_emb	Mathlib/MeasureTheory/Integral/Lebesgue.lean	1,473	1,476	theorem MeasurePreserving.set_lintegral_comp_preimage {mb : MeasurableSpace β} {ν : Measure β} {g : α → β} (hg : MeasurePreserving g μ ν) {s : Set β} (hs : MeasurableSet s) {f : β → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν := by	rw [← hg.map_eq, set_lintegral_map hs hf hg.measurable]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne -/ import Mathlib.Order.MinMax import Mathlib.Data.Set.Subsingleton import Mathlib.Tactic.Says #align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c" /-! # Intervals In any preorder `α`, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. This file contains these definitions, and basic facts on inclusion, intersection, difference of intervals (where the precise statements may depend on the properties of the order, in particular for some statements it should be `LinearOrder` or `DenselyOrdered`). TODO: This is just the beginning; a lot of rules are missing -/ open Function open OrderDual (toDual ofDual) variable {α β : Type} namespace Set section Preorder variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α} /-- Left-open right-open interval -/ def Ioo (a b : α) := { x \| a < x ∧ x < b } #align set.Ioo Set.Ioo /-- Left-closed right-open interval -/ def Ico (a b : α) := { x \| a ≤ x ∧ x < b } #align set.Ico Set.Ico /-- Left-infinite right-open interval -/ def Iio (a : α) := { x \| x < a } #align set.Iio Set.Iio /-- Left-closed right-closed interval -/ def Icc (a b : α) := { x \| a ≤ x ∧ x ≤ b } #align set.Icc Set.Icc /-- Left-infinite right-closed interval -/ def Iic (b : α) := { x \| x ≤ b } #align set.Iic Set.Iic /-- Left-open right-closed interval -/ def Ioc (a b : α) := { x \| a < x ∧ x ≤ b } #align set.Ioc Set.Ioc /-- Left-closed right-infinite interval -/ def Ici (a : α) := { x \| a ≤ x } #align set.Ici Set.Ici /-- Left-open right-infinite interval -/ def Ioi (a : α) := { x \| a < x } #align set.Ioi Set.Ioi theorem Ioo_def (a b : α) : { x \| a < x ∧ x < b } = Ioo a b := rfl #align set.Ioo_def Set.Ioo_def theorem Ico_def (a b : α) : { x \| a ≤ x ∧ x < b } = Ico a b := rfl #align set.Ico_def Set.Ico_def theorem Iio_def (a : α) : { x \| x < a } = Iio a := rfl #align set.Iio_def Set.Iio_def theorem Icc_def (a b : α) : { x \| a ≤ x ∧ x ≤ b } = Icc a b := rfl #align set.Icc_def Set.Icc_def theorem Iic_def (b : α) : { x \| x ≤ b } = Iic b := rfl #align set.Iic_def Set.Iic_def theorem Ioc_def (a b : α) : { x \| a < x ∧ x ≤ b } = Ioc a b := rfl #align set.Ioc_def Set.Ioc_def theorem Ici_def (a : α) : { x \| a ≤ x } = Ici a := rfl #align set.Ici_def Set.Ici_def theorem Ioi_def (a : α) : { x \| a < x } = Ioi a := rfl #align set.Ioi_def Set.Ioi_def @[simp] theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b := Iff.rfl #align set.mem_Ioo Set.mem_Ioo @[simp] theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b := Iff.rfl #align set.mem_Ico Set.mem_Ico @[simp] theorem mem_Iio : x ∈ Iio b ↔ x < b := Iff.rfl #align set.mem_Iio Set.mem_Iio @[simp] theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b := Iff.rfl #align set.mem_Icc Set.mem_Icc @[simp] theorem mem_Iic : x ∈ Iic b ↔ x ≤ b := Iff.rfl #align set.mem_Iic Set.mem_Iic @[simp] theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b := Iff.rfl #align set.mem_Ioc Set.mem_Ioc @[simp] theorem mem_Ici : x ∈ Ici a ↔ a ≤ x := Iff.rfl #align set.mem_Ici Set.mem_Ici @[simp] theorem mem_Ioi : x ∈ Ioi a ↔ a < x := Iff.rfl #align set.mem_Ioi Set.mem_Ioi instance decidableMemIoo [Decidable (a < x ∧ x < b)] : Decidable (x ∈ Ioo a b) := by assumption #align set.decidable_mem_Ioo Set.decidableMemIoo instance decidableMemIco [Decidable (a ≤ x ∧ x < b)] : Decidable (x ∈ Ico a b) := by assumption #align set.decidable_mem_Ico Set.decidableMemIco instance decidableMemIio [Decidable (x < b)] : Decidable (x ∈ Iio b) := by assumption #align set.decidable_mem_Iio Set.decidableMemIio instance decidableMemIcc [Decidable (a ≤ x ∧ x ≤ b)] : Decidable (x ∈ Icc a b) := by assumption #align set.decidable_mem_Icc Set.decidableMemIcc instance decidableMemIic [Decidable (x ≤ b)] : Decidable (x ∈ Iic b) := by assumption #align set.decidable_mem_Iic Set.decidableMemIic instance decidableMemIoc [Decidable (a < x ∧ x ≤ b)] : Decidable (x ∈ Ioc a b) := by assumption #align set.decidable_mem_Ioc Set.decidableMemIoc instance decidableMemIci [Decidable (a ≤ x)] : Decidable (x ∈ Ici a) := by assumption #align set.decidable_mem_Ici Set.decidableMemIci instance decidableMemIoi [Decidable (a < x)] : Decidable (x ∈ Ioi a) := by assumption #align set.decidable_mem_Ioi Set.decidableMemIoi -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioo Set.left_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl] #align set.left_mem_Ico Set.left_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.left_mem_Icc Set.left_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioc Set.left_mem_Ioc theorem left_mem_Ici : a ∈ Ici a := by simp #align set.left_mem_Ici Set.left_mem_Ici -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioo : b ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ioo Set.right_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ico : b ∈ Ico a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ico Set.right_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.right_mem_Icc Set.right_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl] #align set.right_mem_Ioc Set.right_mem_Ioc theorem right_mem_Iic : a ∈ Iic a := by simp #align set.right_mem_Iic Set.right_mem_Iic @[simp] theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a := rfl #align set.dual_Ici Set.dual_Ici @[simp] theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a := rfl #align set.dual_Iic Set.dual_Iic @[simp] theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a := rfl #align set.dual_Ioi Set.dual_Ioi @[simp] theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a := rfl #align set.dual_Iio Set.dual_Iio @[simp] theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a := Set.ext fun _ => and_comm #align set.dual_Icc Set.dual_Icc @[simp] theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a := Set.ext fun _ => and_comm #align set.dual_Ioc Set.dual_Ioc @[simp] theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a := Set.ext fun _ => and_comm #align set.dual_Ico Set.dual_Ico @[simp] theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a := Set.ext fun _ => and_comm #align set.dual_Ioo Set.dual_Ioo @[simp] theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := ⟨fun ⟨_, hx⟩ => hx.1.trans hx.2, fun h => ⟨a, left_mem_Icc.2 h⟩⟩ #align set.nonempty_Icc Set.nonempty_Icc @[simp] theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_lt hx.2, fun h => ⟨a, left_mem_Ico.2 h⟩⟩ #align set.nonempty_Ico Set.nonempty_Ico @[simp] theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_le hx.2, fun h => ⟨b, right_mem_Ioc.2 h⟩⟩ #align set.nonempty_Ioc Set.nonempty_Ioc @[simp] theorem nonempty_Ici : (Ici a).Nonempty := ⟨a, left_mem_Ici⟩ #align set.nonempty_Ici Set.nonempty_Ici @[simp] theorem nonempty_Iic : (Iic a).Nonempty := ⟨a, right_mem_Iic⟩ #align set.nonempty_Iic Set.nonempty_Iic @[simp] theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := ⟨fun ⟨_, ha, hb⟩ => ha.trans hb, exists_between⟩ #align set.nonempty_Ioo Set.nonempty_Ioo @[simp] theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty := exists_gt a #align set.nonempty_Ioi Set.nonempty_Ioi @[simp] theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty := exists_lt a #align set.nonempty_Iio Set.nonempty_Iio theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) := Nonempty.to_subtype (nonempty_Icc.mpr h) #align set.nonempty_Icc_subtype Set.nonempty_Icc_subtype theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) := Nonempty.to_subtype (nonempty_Ico.mpr h) #align set.nonempty_Ico_subtype Set.nonempty_Ico_subtype theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) := Nonempty.to_subtype (nonempty_Ioc.mpr h) #align set.nonempty_Ioc_subtype Set.nonempty_Ioc_subtype /-- An interval `Ici a` is nonempty. -/ instance nonempty_Ici_subtype : Nonempty (Ici a) := Nonempty.to_subtype nonempty_Ici #align set.nonempty_Ici_subtype Set.nonempty_Ici_subtype /-- An interval `Iic a` is nonempty. -/ instance nonempty_Iic_subtype : Nonempty (Iic a) := Nonempty.to_subtype nonempty_Iic #align set.nonempty_Iic_subtype Set.nonempty_Iic_subtype theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) := Nonempty.to_subtype (nonempty_Ioo.mpr h) #align set.nonempty_Ioo_subtype Set.nonempty_Ioo_subtype /-- In an order without maximal elements, the intervals `Ioi` are nonempty. -/ instance nonempty_Ioi_subtype [NoMaxOrder α] : Nonempty (Ioi a) := Nonempty.to_subtype nonempty_Ioi #align set.nonempty_Ioi_subtype Set.nonempty_Ioi_subtype /-- In an order without minimal elements, the intervals `Iio` are nonempty. -/ instance nonempty_Iio_subtype [NoMinOrder α] : Nonempty (Iio a) := Nonempty.to_subtype nonempty_Iio #align set.nonempty_Iio_subtype Set.nonempty_Iio_subtype instance [NoMinOrder α] : NoMinOrder (Iio a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, lt_trans hb a.2⟩, hb⟩⟩ instance [NoMinOrder α] : NoMinOrder (Iic a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, hb.le.trans a.2⟩, hb⟩⟩ instance [NoMaxOrder α] : NoMaxOrder (Ioi a) := OrderDual.noMaxOrder (α := Iio (toDual a)) instance [NoMaxOrder α] : NoMaxOrder (Ici a) := OrderDual.noMaxOrder (α := Iic (toDual a)) @[simp] theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Icc_eq_empty Set.Icc_eq_empty @[simp] theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_lt hb) #align set.Ico_eq_empty Set.Ico_eq_empty @[simp] theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_le hb) #align set.Ioc_eq_empty Set.Ioc_eq_empty @[simp] theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Ioo_eq_empty Set.Ioo_eq_empty @[simp] theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ := Icc_eq_empty h.not_le #align set.Icc_eq_empty_of_lt Set.Icc_eq_empty_of_lt @[simp] theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ := Ico_eq_empty h.not_lt #align set.Ico_eq_empty_of_le Set.Ico_eq_empty_of_le @[simp] theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ := Ioc_eq_empty h.not_lt #align set.Ioc_eq_empty_of_le Set.Ioc_eq_empty_of_le @[simp] theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ := Ioo_eq_empty h.not_lt #align set.Ioo_eq_empty_of_le Set.Ioo_eq_empty_of_le -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ico_self (a : α) : Ico a a = ∅ := Ico_eq_empty <\| lt_irrefl _ #align set.Ico_self Set.Ico_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioc_self (a : α) : Ioc a a = ∅ := Ioc_eq_empty <\| lt_irrefl _ #align set.Ioc_self Set.Ioc_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioo_self (a : α) : Ioo a a = ∅ := Ioo_eq_empty <\| lt_irrefl _ #align set.Ioo_self Set.Ioo_self theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a := ⟨fun h => h <\| left_mem_Ici, fun h _ hx => h.trans hx⟩ #align set.Ici_subset_Ici Set.Ici_subset_Ici @[gcongr] alias ⟨_, _root_.GCongr.Ici_subset_Ici_of_le⟩ := Ici_subset_Ici theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b := @Ici_subset_Ici αᵒᵈ _ _ _ #align set.Iic_subset_Iic Set.Iic_subset_Iic @[gcongr] alias ⟨_, _root_.GCongr.Iic_subset_Iic_of_le⟩ := Iic_subset_Iic theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a := ⟨fun h => h left_mem_Ici, fun h _ hx => h.trans_le hx⟩ #align set.Ici_subset_Ioi Set.Ici_subset_Ioi theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b := ⟨fun h => h right_mem_Iic, fun h _ hx => lt_of_le_of_lt hx h⟩ #align set.Iic_subset_Iio Set.Iic_subset_Iio @[gcongr] theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans_le h₂⟩ #align set.Ioo_subset_Ioo Set.Ioo_subset_Ioo @[gcongr] theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b := Ioo_subset_Ioo h le_rfl #align set.Ioo_subset_Ioo_left Set.Ioo_subset_Ioo_left @[gcongr] theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ := Ioo_subset_Ioo le_rfl h #align set.Ioo_subset_Ioo_right Set.Ioo_subset_Ioo_right @[gcongr] theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, hx₂.trans_le h₂⟩ #align set.Ico_subset_Ico Set.Ico_subset_Ico @[gcongr] theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b := Ico_subset_Ico h le_rfl #align set.Ico_subset_Ico_left Set.Ico_subset_Ico_left @[gcongr] theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ := Ico_subset_Ico le_rfl h #align set.Ico_subset_Ico_right Set.Ico_subset_Ico_right @[gcongr] theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, le_trans hx₂ h₂⟩ #align set.Icc_subset_Icc Set.Icc_subset_Icc @[gcongr] theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b := Icc_subset_Icc h le_rfl #align set.Icc_subset_Icc_left Set.Icc_subset_Icc_left @[gcongr] theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ := Icc_subset_Icc le_rfl h #align set.Icc_subset_Icc_right Set.Icc_subset_Icc_right theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ hx => ⟨ha.trans_le hx.1, hx.2.trans_lt hb⟩ #align set.Icc_subset_Ioo Set.Icc_subset_Ioo theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := fun _ => And.left #align set.Icc_subset_Ici_self Set.Icc_subset_Ici_self theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := fun _ => And.right #align set.Icc_subset_Iic_self Set.Icc_subset_Iic_self theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := fun _ => And.right #align set.Ioc_subset_Iic_self Set.Ioc_subset_Iic_self @[gcongr] theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans h₂⟩ #align set.Ioc_subset_Ioc Set.Ioc_subset_Ioc @[gcongr] theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b := Ioc_subset_Ioc h le_rfl #align set.Ioc_subset_Ioc_left Set.Ioc_subset_Ioc_left @[gcongr] theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ := Ioc_subset_Ioc le_rfl h #align set.Ioc_subset_Ioc_right Set.Ioc_subset_Ioc_right theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := fun _ => And.imp_left h₁.trans_le #align set.Ico_subset_Ioo_left Set.Ico_subset_Ioo_left theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := fun _ => And.imp_right fun h' => h'.trans_lt h #align set.Ioc_subset_Ioo_right Set.Ioc_subset_Ioo_right theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := fun _ => And.imp_right fun h₂ => h₂.trans_lt h₁ #align set.Icc_subset_Ico_right Set.Icc_subset_Ico_right theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := fun _ => And.imp_left le_of_lt #align set.Ioo_subset_Ico_self Set.Ioo_subset_Ico_self theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := fun _ => And.imp_right le_of_lt #align set.Ioo_subset_Ioc_self Set.Ioo_subset_Ioc_self theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := fun _ => And.imp_right le_of_lt #align set.Ico_subset_Icc_self Set.Ico_subset_Icc_self theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := fun _ => And.imp_left le_of_lt #align set.Ioc_subset_Icc_self Set.Ioc_subset_Icc_self theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b := Subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self #align set.Ioo_subset_Icc_self Set.Ioo_subset_Icc_self theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := fun _ => And.right #align set.Ico_subset_Iio_self Set.Ico_subset_Iio_self theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := fun _ => And.right #align set.Ioo_subset_Iio_self Set.Ioo_subset_Iio_self theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := fun _ => And.left #align set.Ioc_subset_Ioi_self Set.Ioc_subset_Ioi_self theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := fun _ => And.left #align set.Ioo_subset_Ioi_self Set.Ioo_subset_Ioi_self theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a := fun _ hx => le_of_lt hx #align set.Ioi_subset_Ici_self Set.Ioi_subset_Ici_self theorem Iio_subset_Iic_self : Iio a ⊆ Iic a := fun _ hx => le_of_lt hx #align set.Iio_subset_Iic_self Set.Iio_subset_Iic_self theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := fun _ => And.left #align set.Ico_subset_Ici_self Set.Ico_subset_Ici_self theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a := ⟨Ioi_subset_Ici_self, fun h => lt_irrefl a (h le_rfl)⟩ #align set.Ioi_ssubset_Ici_self Set.Ioi_ssubset_Ici_self theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a := @Ioi_ssubset_Ici_self αᵒᵈ _ _ #align set.Iio_ssubset_Iic_self Set.Iio_ssubset_Iic_self theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Icc_iff Set.Icc_subset_Icc_iff theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ioo_iff Set.Icc_subset_Ioo_iff theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ico_iff Set.Icc_subset_Ico_iff theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Ioc_iff Set.Icc_subset_Ioc_iff theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans_lt h⟩ #align set.Icc_subset_Iio_iff Set.Icc_subset_Iio_iff theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans_le hx⟩ #align set.Icc_subset_Ioi_iff Set.Icc_subset_Ioi_iff theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans h⟩ #align set.Icc_subset_Iic_iff Set.Icc_subset_Iic_iff theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans hx⟩ #align set.Icc_subset_Ici_iff Set.Icc_subset_Ici_iff theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc (le_of_lt ha) hb)).mpr ⟨a₂, left_mem_Icc.mpr hI, not_and.mpr fun f _ => lt_irrefl a₂ (ha.trans_le f)⟩ #align set.Icc_ssubset_Icc_left Set.Icc_ssubset_Icc_left theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc ha (le_of_lt hb))).mpr ⟨b₂, right_mem_Icc.mpr hI, fun f => lt_irrefl b₁ (hb.trans_le f.2)⟩ #align set.Icc_ssubset_Icc_right Set.Icc_ssubset_Icc_right /-- If `a ≤ b`, then `(b, +∞) ⊆ (a, +∞)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Ioi_subset_Ioi_iff`. -/ @[gcongr] theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := fun _ hx => h.trans_lt hx #align set.Ioi_subset_Ioi Set.Ioi_subset_Ioi /-- If `a ≤ b`, then `(b, +∞) ⊆ [a, +∞)`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Ioi_subset_Ici_iff`. -/ theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a := Subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self #align set.Ioi_subset_Ici Set.Ioi_subset_Ici /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Iio_subset_Iio_iff`. -/ @[gcongr] theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := fun _ hx => lt_of_lt_of_le hx h #align set.Iio_subset_Iio Set.Iio_subset_Iio /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b]`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Iio_subset_Iic_iff`. -/ theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b := Subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self #align set.Iio_subset_Iic Set.Iio_subset_Iic theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b := rfl #align set.Ici_inter_Iic Set.Ici_inter_Iic theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b := rfl #align set.Ici_inter_Iio Set.Ici_inter_Iio theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b := rfl #align set.Ioi_inter_Iic Set.Ioi_inter_Iic theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b := rfl #align set.Ioi_inter_Iio Set.Ioi_inter_Iio theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a := inter_comm _ _ #align set.Iic_inter_Ici Set.Iic_inter_Ici theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a := inter_comm _ _ #align set.Iio_inter_Ici Set.Iio_inter_Ici theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a := inter_comm _ _ #align set.Iic_inter_Ioi Set.Iic_inter_Ioi theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a := inter_comm _ _ #align set.Iio_inter_Ioi Set.Iio_inter_Ioi theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b := Ioo_subset_Icc_self h #align set.mem_Icc_of_Ioo Set.mem_Icc_of_Ioo theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b := Ioo_subset_Ico_self h #align set.mem_Ico_of_Ioo Set.mem_Ico_of_Ioo theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b := Ioo_subset_Ioc_self h #align set.mem_Ioc_of_Ioo Set.mem_Ioc_of_Ioo theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b := Ico_subset_Icc_self h #align set.mem_Icc_of_Ico Set.mem_Icc_of_Ico theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b := Ioc_subset_Icc_self h #align set.mem_Icc_of_Ioc Set.mem_Icc_of_Ioc theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a := Ioi_subset_Ici_self h #align set.mem_Ici_of_Ioi Set.mem_Ici_of_Ioi theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a := Iio_subset_Iic_self h #align set.mem_Iic_of_Iio Set.mem_Iic_of_Iio theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc] #align set.Icc_eq_empty_iff Set.Icc_eq_empty_iff theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico] #align set.Ico_eq_empty_iff Set.Ico_eq_empty_iff theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioc] #align set.Ioc_eq_empty_iff Set.Ioc_eq_empty_iff theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo] #align set.Ioo_eq_empty_iff Set.Ioo_eq_empty_iff theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ := eq_univ_of_forall h #align is_top.Iic_eq IsTop.Iic_eq theorem _root_.IsBot.Ici_eq (h : IsBot a) : Ici a = univ := eq_univ_of_forall h #align is_bot.Ici_eq IsBot.Ici_eq theorem _root_.IsMax.Ioi_eq (h : IsMax a) : Ioi a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_max.Ioi_eq IsMax.Ioi_eq theorem _root_.IsMin.Iio_eq (h : IsMin a) : Iio a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_min.Iio_eq IsMin.Iio_eq theorem Iic_inter_Ioc_of_le (h : a ≤ c) : Iic a ∩ Ioc b c = Ioc b a := ext fun _ => ⟨fun H => ⟨H.2.1, H.1⟩, fun H => ⟨H.2, H.1, H.2.trans h⟩⟩ #align set.Iic_inter_Ioc_of_le Set.Iic_inter_Ioc_of_le theorem not_mem_Icc_of_lt (ha : c < a) : c ∉ Icc a b := fun h => ha.not_le h.1 #align set.not_mem_Icc_of_lt Set.not_mem_Icc_of_lt theorem not_mem_Icc_of_gt (hb : b < c) : c ∉ Icc a b := fun h => hb.not_le h.2 #align set.not_mem_Icc_of_gt Set.not_mem_Icc_of_gt theorem not_mem_Ico_of_lt (ha : c < a) : c ∉ Ico a b := fun h => ha.not_le h.1 #align set.not_mem_Ico_of_lt Set.not_mem_Ico_of_lt theorem not_mem_Ioc_of_gt (hb : b < c) : c ∉ Ioc a b := fun h => hb.not_le h.2 #align set.not_mem_Ioc_of_gt Set.not_mem_Ioc_of_gt -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Ioi_self : a ∉ Ioi a := lt_irrefl _ #align set.not_mem_Ioi_self Set.not_mem_Ioi_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Iio_self : b ∉ Iio b := lt_irrefl _ #align set.not_mem_Iio_self Set.not_mem_Iio_self theorem not_mem_Ioc_of_le (ha : c ≤ a) : c ∉ Ioc a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioc_of_le Set.not_mem_Ioc_of_le theorem not_mem_Ico_of_ge (hb : b ≤ c) : c ∉ Ico a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ico_of_ge Set.not_mem_Ico_of_ge theorem not_mem_Ioo_of_le (ha : c ≤ a) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioo_of_le Set.not_mem_Ioo_of_le theorem not_mem_Ioo_of_ge (hb : b ≤ c) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ioo_of_ge Set.not_mem_Ioo_of_ge end Preorder section PartialOrder variable [PartialOrder α] {a b c : α} @[simp] theorem Icc_self (a : α) : Icc a a = {a} := Set.ext <\| by simp [Icc, le_antisymm_iff, and_comm] #align set.Icc_self Set.Icc_self instance instIccUnique : Unique (Set.Icc a a) where default := ⟨a, by simp⟩ uniq y := Subtype.ext <\| by simpa using y.2 @[simp] theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by refine ⟨fun h => ?_, ?_⟩ · have hab : a ≤ b := nonempty_Icc.1 (h.symm.subst <\| singleton_nonempty c) exact ⟨eq_of_mem_singleton <\| h.subst <\| left_mem_Icc.2 hab, eq_of_mem_singleton <\| h.subst <\| right_mem_Icc.2 hab⟩ · rintro ⟨rfl, rfl⟩ exact Icc_self _ #align set.Icc_eq_singleton_iff Set.Icc_eq_singleton_iff lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) := fun _x ⟨hax, hxb⟩ _y ⟨hay, hyb⟩ ↦ le_antisymm (le_implies_le_of_le_of_le hxb hay hba) (le_implies_le_of_le_of_le hyb hax hba) #align set.subsingleton_Icc_of_ge Set.subsingleton_Icc_of_ge @[simp] lemma subsingleton_Icc_iff {α : Type} [LinearOrder α] {a b : α} : Set.Subsingleton (Icc a b) ↔ b ≤ a := by refine ⟨fun h ↦ ?_, subsingleton_Icc_of_ge⟩ contrapose! h simp only [ge_iff_le, gt_iff_lt, not_subsingleton_iff] exact ⟨a, ⟨le_refl _, h.le⟩, b, ⟨h.le, le_refl _⟩, h.ne⟩ @[simp] theorem Icc_diff_left : Icc a b \ {a} = Ioc a b := ext fun x => by simp [lt_iff_le_and_ne, eq_comm, and_right_comm] #align set.Icc_diff_left Set.Icc_diff_left @[simp] theorem Icc_diff_right : Icc a b \ {b} = Ico a b := ext fun x => by simp [lt_iff_le_and_ne, and_assoc] #align set.Icc_diff_right Set.Icc_diff_right @[simp] theorem Ico_diff_left : Ico a b \ {a} = Ioo a b := ext fun x => by simp [and_right_comm, ← lt_iff_le_and_ne, eq_comm] #align set.Ico_diff_left Set.Ico_diff_left @[simp] theorem Ioc_diff_right : Ioc a b \ {b} = Ioo a b := ext fun x => by simp [and_assoc, ← lt_iff_le_and_ne] #align set.Ioc_diff_right Set.Ioc_diff_right @[simp] theorem Icc_diff_both : Icc a b \ {a, b} = Ioo a b := by rw [insert_eq, ← diff_diff, Icc_diff_left, Ioc_diff_right] #align set.Icc_diff_both Set.Icc_diff_both @[simp] theorem Ici_diff_left : Ici a \ {a} = Ioi a := ext fun x => by simp [lt_iff_le_and_ne, eq_comm] #align set.Ici_diff_left Set.Ici_diff_left @[simp] theorem Iic_diff_right : Iic a \ {a} = Iio a := ext fun x => by simp [lt_iff_le_and_ne] #align set.Iic_diff_right Set.Iic_diff_right @[simp] theorem Ico_diff_Ioo_same (h : a < b) : Ico a b \ Ioo a b = {a} := by rw [← Ico_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Ico.2 h)] #align set.Ico_diff_Ioo_same Set.Ico_diff_Ioo_same @[simp] theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} := by rw [← Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Ioc.2 h)] #align set.Ioc_diff_Ioo_same Set.Ioc_diff_Ioo_same @[simp] theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Icc.2 h)] #align set.Icc_diff_Ico_same Set.Icc_diff_Ico_same @[simp] theorem Icc_diff_Ioc_same (h : a ≤ b) : Icc a b \ Ioc a b = {a} := by rw [← Icc_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Icc.2 h)] #align set.Icc_diff_Ioc_same Set.Icc_diff_Ioc_same @[simp] theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} := by rw [← Icc_diff_both, diff_diff_cancel_left] simp [insert_subset_iff, h] #align set.Icc_diff_Ioo_same Set.Icc_diff_Ioo_same @[simp] theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} := by rw [← Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)] #align set.Ici_diff_Ioi_same Set.Ici_diff_Ioi_same @[simp] theorem Iic_diff_Iio_same : Iic a \ Iio a = {a} := by rw [← Iic_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 right_mem_Iic)] #align set.Iic_diff_Iio_same Set.Iic_diff_Iio_same -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioi_union_left : Ioi a ∪ {a} = Ici a := ext fun x => by simp [eq_comm, le_iff_eq_or_lt] #align set.Ioi_union_left Set.Ioi_union_left -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Iio_union_right : Iio a ∪ {a} = Iic a := ext fun _ => le_iff_lt_or_eq.symm #align set.Iio_union_right Set.Iio_union_right theorem Ioo_union_left (hab : a < b) : Ioo a b ∪ {a} = Ico a b := by rw [← Ico_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Ico.2 hab)] #align set.Ioo_union_left Set.Ioo_union_left theorem Ioo_union_right (hab : a < b) : Ioo a b ∪ {b} = Ioc a b := by simpa only [dual_Ioo, dual_Ico] using Ioo_union_left hab.dual #align set.Ioo_union_right Set.Ioo_union_right theorem Ioo_union_both (h : a ≤ b) : Ioo a b ∪ {a, b} = Icc a b := by have : (Icc a b \ {a, b}) ∪ {a, b} = Icc a b := diff_union_of_subset fun \| x, .inl rfl => left_mem_Icc.mpr h \| x, .inr rfl => right_mem_Icc.mpr h rw [← this, Icc_diff_both] theorem Ioc_union_left (hab : a ≤ b) : Ioc a b ∪ {a} = Icc a b := by rw [← Icc_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Icc.2 hab)] #align set.Ioc_union_left Set.Ioc_union_left theorem Ico_union_right (hab : a ≤ b) : Ico a b ∪ {b} = Icc a b := by simpa only [dual_Ioc, dual_Icc] using Ioc_union_left hab.dual #align set.Ico_union_right Set.Ico_union_right @[simp] theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b := by rw [insert_eq, union_comm, Ico_union_right h] #align set.Ico_insert_right Set.Ico_insert_right @[simp] theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b := by rw [insert_eq, union_comm, Ioc_union_left h] #align set.Ioc_insert_left Set.Ioc_insert_left @[simp] theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b := by rw [insert_eq, union_comm, Ioo_union_left h] #align set.Ioo_insert_left Set.Ioo_insert_left @[simp] theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b := by rw [insert_eq, union_comm, Ioo_union_right h] #align set.Ioo_insert_right Set.Ioo_insert_right @[simp] theorem Iio_insert : insert a (Iio a) = Iic a := ext fun _ => le_iff_eq_or_lt.symm #align set.Iio_insert Set.Iio_insert @[simp] theorem Ioi_insert : insert a (Ioi a) = Ici a := ext fun _ => (or_congr_left eq_comm).trans le_iff_eq_or_lt.symm #align set.Ioi_insert Set.Ioi_insert theorem mem_Ici_Ioi_of_subset_of_subset {s : Set α} (ho : Ioi a ⊆ s) (hc : s ⊆ Ici a) : s ∈ ({Ici a, Ioi a} : Set (Set α)) := by_cases (fun h : a ∈ s => Or.inl <\| Subset.antisymm hc <\| by rw [← Ioi_union_left, union_subset_iff]; simp [*]) fun h => Or.inr <\| Subset.antisymm (fun x hx => lt_of_le_of_ne (hc hx) fun heq => h <\| heq.symm ▸ hx) ho #align set.mem_Ici_Ioi_of_subset_of_subset Set.mem_Ici_Ioi_of_subset_of_subset theorem mem_Iic_Iio_of_subset_of_subset {s : Set α} (ho : Iio a ⊆ s) (hc : s ⊆ Iic a) : s ∈ ({Iic a, Iio a} : Set (Set α)) := @mem_Ici_Ioi_of_subset_of_subset αᵒᵈ _ a s ho hc #align set.mem_Iic_Iio_of_subset_of_subset Set.mem_Iic_Iio_of_subset_of_subset theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) : s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α)) := by classical by_cases ha : a ∈ s <;> by_cases hb : b ∈ s · refine Or.inl (Subset.antisymm hc ?_) rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, ← Icc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_right] exact subset_diff_singleton hc hb · rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho · refine Or.inr <\| Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_left] exact subset_diff_singleton hc ha · rwa [← Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inr <\| Or.inr <\| Subset.antisymm ?_ ho rw [← Ico_diff_left, ← Icc_diff_right] apply_rules [subset_diff_singleton] #align set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset theorem eq_left_or_mem_Ioo_of_mem_Ico {x : α} (hmem : x ∈ Ico a b) : x = a ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => ⟨h, hmem.2⟩ #align set.eq_left_or_mem_Ioo_of_mem_Ico Set.eq_left_or_mem_Ioo_of_mem_Ico theorem eq_right_or_mem_Ioo_of_mem_Ioc {x : α} (hmem : x ∈ Ioc a b) : x = b ∨ x ∈ Ioo a b := hmem.2.eq_or_lt.imp_right <\| And.intro hmem.1 #align set.eq_right_or_mem_Ioo_of_mem_Ioc Set.eq_right_or_mem_Ioo_of_mem_Ioc theorem eq_endpoints_or_mem_Ioo_of_mem_Icc {x : α} (hmem : x ∈ Icc a b) : x = a ∨ x = b ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => eq_right_or_mem_Ioo_of_mem_Ioc ⟨h, hmem.2⟩ #align set.eq_endpoints_or_mem_Ioo_of_mem_Icc Set.eq_endpoints_or_mem_Ioo_of_mem_Icc theorem _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} := eq_singleton_iff_unique_mem.2 ⟨left_mem_Ici, fun _ => h.eq_of_ge⟩ #align is_max.Ici_eq IsMax.Ici_eq theorem _root_.IsMin.Iic_eq (h : IsMin a) : Iic a = {a} := h.toDual.Ici_eq #align is_min.Iic_eq IsMin.Iic_eq theorem Ici_injective : Injective (Ici : α → Set α) := fun _ _ => eq_of_forall_ge_iff ∘ Set.ext_iff.1 #align set.Ici_injective Set.Ici_injective theorem Iic_injective : Injective (Iic : α → Set α) := fun _ _ => eq_of_forall_le_iff ∘ Set.ext_iff.1 #align set.Iic_injective Set.Iic_injective theorem Ici_inj : Ici a = Ici b ↔ a = b := Ici_injective.eq_iff #align set.Ici_inj Set.Ici_inj theorem Iic_inj : Iic a = Iic b ↔ a = b := Iic_injective.eq_iff #align set.Iic_inj Set.Iic_inj end PartialOrder section OrderTop @[simp] theorem Ici_top [PartialOrder α] [OrderTop α] : Ici (⊤ : α) = {⊤} := isMax_top.Ici_eq #align set.Ici_top Set.Ici_top variable [Preorder α] [OrderTop α] {a : α} @[simp] theorem Ioi_top : Ioi (⊤ : α) = ∅ := isMax_top.Ioi_eq #align set.Ioi_top Set.Ioi_top @[simp] theorem Iic_top : Iic (⊤ : α) = univ := isTop_top.Iic_eq #align set.Iic_top Set.Iic_top @[simp] theorem Icc_top : Icc a ⊤ = Ici a := by simp [← Ici_inter_Iic] #align set.Icc_top Set.Icc_top @[simp] theorem Ioc_top : Ioc a ⊤ = Ioi a := by simp [← Ioi_inter_Iic] #align set.Ioc_top Set.Ioc_top end OrderTop section OrderBot @[simp] theorem Iic_bot [PartialOrder α] [OrderBot α] : Iic (⊥ : α) = {⊥} := isMin_bot.Iic_eq #align set.Iic_bot Set.Iic_bot variable [Preorder α] [OrderBot α] {a : α} @[simp] theorem Iio_bot : Iio (⊥ : α) = ∅ := isMin_bot.Iio_eq #align set.Iio_bot Set.Iio_bot @[simp] theorem Ici_bot : Ici (⊥ : α) = univ := isBot_bot.Ici_eq #align set.Ici_bot Set.Ici_bot @[simp] theorem Icc_bot : Icc ⊥ a = Iic a := by simp [← Ici_inter_Iic] #align set.Icc_bot Set.Icc_bot @[simp] theorem Ico_bot : Ico ⊥ a = Iio a := by simp [← Ici_inter_Iio] #align set.Ico_bot Set.Ico_bot end OrderBot theorem Icc_bot_top [PartialOrder α] [BoundedOrder α] : Icc (⊥ : α) ⊤ = univ := by simp #align set.Icc_bot_top Set.Icc_bot_top section LinearOrder variable [LinearOrder α] {a a₁ a₂ b b₁ b₂ c d : α} theorem not_mem_Ici : c ∉ Ici a ↔ c < a := not_le #align set.not_mem_Ici Set.not_mem_Ici theorem not_mem_Iic : c ∉ Iic b ↔ b < c := not_le #align set.not_mem_Iic Set.not_mem_Iic theorem not_mem_Ioi : c ∉ Ioi a ↔ c ≤ a := not_lt #align set.not_mem_Ioi Set.not_mem_Ioi theorem not_mem_Iio : c ∉ Iio b ↔ b ≤ c := not_lt #align set.not_mem_Iio Set.not_mem_Iio @[simp] theorem compl_Iic : (Iic a)ᶜ = Ioi a := ext fun _ => not_le #align set.compl_Iic Set.compl_Iic @[simp] theorem compl_Ici : (Ici a)ᶜ = Iio a := ext fun _ => not_le #align set.compl_Ici Set.compl_Ici @[simp] theorem compl_Iio : (Iio a)ᶜ = Ici a := ext fun _ => not_lt #align set.compl_Iio Set.compl_Iio @[simp] theorem compl_Ioi : (Ioi a)ᶜ = Iic a := ext fun _ => not_lt #align set.compl_Ioi Set.compl_Ioi @[simp] theorem Ici_diff_Ici : Ici a \ Ici b = Ico a b := by rw [diff_eq, compl_Ici, Ici_inter_Iio] #align set.Ici_diff_Ici Set.Ici_diff_Ici @[simp]	Mathlib/Order/Interval/Set/Basic.lean	1,120	1,120	theorem Ici_diff_Ioi : Ici a \ Ioi b = Icc a b := by	rw [diff_eq, compl_Ioi, Ici_inter_Iic]
/- Copyright (c) 2019 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes -/ import Mathlib.Algebra.Group.Basic import Mathlib.Algebra.Group.Pi.Basic import Mathlib.Order.Fin import Mathlib.Order.PiLex import Mathlib.Order.Interval.Set.Basic #align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b" /-! # Operation on tuples We interpret maps `∀ i : Fin n, α i` as `n`-tuples of elements of possibly varying type `α i`, `(α 0, …, α (n-1))`. A particular case is `Fin n → α` of elements with all the same type. In this case when `α i` is a constant map, then tuples are isomorphic (but not definitionally equal) to `Vector`s. We define the following operations: * `Fin.tail` : the tail of an `n+1` tuple, i.e., its last `n` entries; * `Fin.cons` : adding an element at the beginning of an `n`-tuple, to get an `n+1`-tuple; * `Fin.init` : the beginning of an `n+1` tuple, i.e., its first `n` entries; * `Fin.snoc` : adding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc` comes from `cons` (i.e., adding an element to the left of a tuple) read in reverse order. * `Fin.insertNth` : insert an element to a tuple at a given position. * `Fin.find p` : returns the first index `n` where `p n` is satisfied, and `none` if it is never satisfied. * `Fin.append a b` : append two tuples. * `Fin.repeat n a` : repeat a tuple `n` times. -/ assert_not_exists MonoidWithZero universe u v namespace Fin variable {m n : ℕ} open Function section Tuple /-- There is exactly one tuple of size zero. -/ example (α : Fin 0 → Sort u) : Unique (∀ i : Fin 0, α i) := by infer_instance theorem tuple0_le {α : Fin 0 → Type} [∀ i, Preorder (α i)] (f g : ∀ i, α i) : f ≤ g := finZeroElim #align fin.tuple0_le Fin.tuple0_le variable {α : Fin (n + 1) → Type u} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ) (i : Fin n) (y : α i.succ) (z : α 0) /-- The tail of an `n+1` tuple, i.e., its last `n` entries. -/ def tail (q : ∀ i, α i) : ∀ i : Fin n, α i.succ := fun i ↦ q i.succ #align fin.tail Fin.tail theorem tail_def {n : ℕ} {α : Fin (n + 1) → Type} {q : ∀ i, α i} : (tail fun k : Fin (n + 1) ↦ q k) = fun k : Fin n ↦ q k.succ := rfl #align fin.tail_def Fin.tail_def /-- Adding an element at the beginning of an `n`-tuple, to get an `n+1`-tuple. -/ def cons (x : α 0) (p : ∀ i : Fin n, α i.succ) : ∀ i, α i := fun j ↦ Fin.cases x p j #align fin.cons Fin.cons @[simp] theorem tail_cons : tail (cons x p) = p := by simp (config := { unfoldPartialApp := true }) [tail, cons] #align fin.tail_cons Fin.tail_cons @[simp] theorem cons_succ : cons x p i.succ = p i := by simp [cons] #align fin.cons_succ Fin.cons_succ @[simp] theorem cons_zero : cons x p 0 = x := by simp [cons] #align fin.cons_zero Fin.cons_zero @[simp] theorem cons_one {α : Fin (n + 2) → Type} (x : α 0) (p : ∀ i : Fin n.succ, α i.succ) : cons x p 1 = p 0 := by rw [← cons_succ x p]; rfl /-- Updating a tuple and adding an element at the beginning commute. -/ @[simp] theorem cons_update : cons x (update p i y) = update (cons x p) i.succ y := by ext j by_cases h : j = 0 · rw [h] simp [Ne.symm (succ_ne_zero i)] · let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this, cons_succ] by_cases h' : j' = i · rw [h'] simp · have : j'.succ ≠ i.succ := by rwa [Ne, succ_inj] rw [update_noteq h', update_noteq this, cons_succ] #align fin.cons_update Fin.cons_update /-- As a binary function, `Fin.cons` is injective. -/ theorem cons_injective2 : Function.Injective2 (@cons n α) := fun x₀ y₀ x y h ↦ ⟨congr_fun h 0, funext fun i ↦ by simpa using congr_fun h (Fin.succ i)⟩ #align fin.cons_injective2 Fin.cons_injective2 @[simp] theorem cons_eq_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} : cons x₀ x = cons y₀ y ↔ x₀ = y₀ ∧ x = y := cons_injective2.eq_iff #align fin.cons_eq_cons Fin.cons_eq_cons theorem cons_left_injective (x : ∀ i : Fin n, α i.succ) : Function.Injective fun x₀ ↦ cons x₀ x := cons_injective2.left _ #align fin.cons_left_injective Fin.cons_left_injective theorem cons_right_injective (x₀ : α 0) : Function.Injective (cons x₀) := cons_injective2.right _ #align fin.cons_right_injective Fin.cons_right_injective /-- Adding an element at the beginning of a tuple and then updating it amounts to adding it directly. -/ theorem update_cons_zero : update (cons x p) 0 z = cons z p := by ext j by_cases h : j = 0 · rw [h] simp · simp only [h, update_noteq, Ne, not_false_iff] let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this, cons_succ, cons_succ] #align fin.update_cons_zero Fin.update_cons_zero /-- Concatenating the first element of a tuple with its tail gives back the original tuple -/ @[simp, nolint simpNF] -- Porting note: linter claims LHS doesn't simplify theorem cons_self_tail : cons (q 0) (tail q) = q := by ext j by_cases h : j = 0 · rw [h] simp · let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this] unfold tail rw [cons_succ] #align fin.cons_self_tail Fin.cons_self_tail -- Porting note: Mathport removes `_root_`? /-- Recurse on an `n+1`-tuple by splitting it into a single element and an `n`-tuple. -/ @[elab_as_elim] def consCases {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x)) (x : ∀ i : Fin n.succ, α i) : P x := _root_.cast (by rw [cons_self_tail]) <\| h (x 0) (tail x) #align fin.cons_cases Fin.consCases @[simp] theorem consCases_cons {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x)) (x₀ : α 0) (x : ∀ i : Fin n, α i.succ) : @consCases _ _ _ h (cons x₀ x) = h x₀ x := by rw [consCases, cast_eq] congr #align fin.cons_cases_cons Fin.consCases_cons /-- Recurse on a tuple by splitting into `Fin.elim0` and `Fin.cons`. -/ @[elab_as_elim] def consInduction {α : Type} {P : ∀ {n : ℕ}, (Fin n → α) → Sort v} (h0 : P Fin.elim0) (h : ∀ {n} (x₀) (x : Fin n → α), P x → P (Fin.cons x₀ x)) : ∀ {n : ℕ} (x : Fin n → α), P x \| 0, x => by convert h0 \| n + 1, x => consCases (fun x₀ x ↦ h _ _ <\| consInduction h0 h _) x #align fin.cons_induction Fin.consInductionₓ -- Porting note: universes theorem cons_injective_of_injective {α} {x₀ : α} {x : Fin n → α} (hx₀ : x₀ ∉ Set.range x) (hx : Function.Injective x) : Function.Injective (cons x₀ x : Fin n.succ → α) := by refine Fin.cases ?_ ?_ · refine Fin.cases ?_ ?_ · intro rfl · intro j h rw [cons_zero, cons_succ] at h exact hx₀.elim ⟨_, h.symm⟩ · intro i refine Fin.cases ?_ ?_ · intro h rw [cons_zero, cons_succ] at h exact hx₀.elim ⟨_, h⟩ · intro j h rw [cons_succ, cons_succ] at h exact congr_arg _ (hx h) #align fin.cons_injective_of_injective Fin.cons_injective_of_injective theorem cons_injective_iff {α} {x₀ : α} {x : Fin n → α} : Function.Injective (cons x₀ x : Fin n.succ → α) ↔ x₀ ∉ Set.range x ∧ Function.Injective x := by refine ⟨fun h ↦ ⟨?_, ?_⟩, fun h ↦ cons_injective_of_injective h.1 h.2⟩ · rintro ⟨i, hi⟩ replace h := @h i.succ 0 simp [hi, succ_ne_zero] at h · simpa [Function.comp] using h.comp (Fin.succ_injective _) #align fin.cons_injective_iff Fin.cons_injective_iff @[simp] theorem forall_fin_zero_pi {α : Fin 0 → Sort} {P : (∀ i, α i) → Prop} : (∀ x, P x) ↔ P finZeroElim := ⟨fun h ↦ h _, fun h x ↦ Subsingleton.elim finZeroElim x ▸ h⟩ #align fin.forall_fin_zero_pi Fin.forall_fin_zero_pi @[simp] theorem exists_fin_zero_pi {α : Fin 0 → Sort} {P : (∀ i, α i) → Prop} : (∃ x, P x) ↔ P finZeroElim := ⟨fun ⟨x, h⟩ ↦ Subsingleton.elim x finZeroElim ▸ h, fun h ↦ ⟨_, h⟩⟩ #align fin.exists_fin_zero_pi Fin.exists_fin_zero_pi theorem forall_fin_succ_pi {P : (∀ i, α i) → Prop} : (∀ x, P x) ↔ ∀ a v, P (Fin.cons a v) := ⟨fun h a v ↦ h (Fin.cons a v), consCases⟩ #align fin.forall_fin_succ_pi Fin.forall_fin_succ_pi theorem exists_fin_succ_pi {P : (∀ i, α i) → Prop} : (∃ x, P x) ↔ ∃ a v, P (Fin.cons a v) := ⟨fun ⟨x, h⟩ ↦ ⟨x 0, tail x, (cons_self_tail x).symm ▸ h⟩, fun ⟨_, _, h⟩ ↦ ⟨_, h⟩⟩ #align fin.exists_fin_succ_pi Fin.exists_fin_succ_pi /-- Updating the first element of a tuple does not change the tail. -/ @[simp] theorem tail_update_zero : tail (update q 0 z) = tail q := by ext j simp [tail, Fin.succ_ne_zero] #align fin.tail_update_zero Fin.tail_update_zero /-- Updating a nonzero element and taking the tail commute. -/ @[simp] theorem tail_update_succ : tail (update q i.succ y) = update (tail q) i y := by ext j by_cases h : j = i · rw [h] simp [tail] · simp [tail, (Fin.succ_injective n).ne h, h] #align fin.tail_update_succ Fin.tail_update_succ theorem comp_cons {α : Type} {β : Type} (g : α → β) (y : α) (q : Fin n → α) : g ∘ cons y q = cons (g y) (g ∘ q) := by ext j by_cases h : j = 0 · rw [h] rfl · let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this, cons_succ, comp_apply, comp_apply, cons_succ] #align fin.comp_cons Fin.comp_cons theorem comp_tail {α : Type} {β : Type} (g : α → β) (q : Fin n.succ → α) : g ∘ tail q = tail (g ∘ q) := by ext j simp [tail] #align fin.comp_tail Fin.comp_tail theorem le_cons [∀ i, Preorder (α i)] {x : α 0} {q : ∀ i, α i} {p : ∀ i : Fin n, α i.succ} : q ≤ cons x p ↔ q 0 ≤ x ∧ tail q ≤ p := forall_fin_succ.trans <\| and_congr Iff.rfl <\| forall_congr' fun j ↦ by simp [tail] #align fin.le_cons Fin.le_cons theorem cons_le [∀ i, Preorder (α i)] {x : α 0} {q : ∀ i, α i} {p : ∀ i : Fin n, α i.succ} : cons x p ≤ q ↔ x ≤ q 0 ∧ p ≤ tail q := @le_cons _ (fun i ↦ (α i)ᵒᵈ) _ x q p #align fin.cons_le Fin.cons_le theorem cons_le_cons [∀ i, Preorder (α i)] {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} : cons x₀ x ≤ cons y₀ y ↔ x₀ ≤ y₀ ∧ x ≤ y := forall_fin_succ.trans <\| and_congr_right' <\| by simp only [cons_succ, Pi.le_def] #align fin.cons_le_cons Fin.cons_le_cons theorem pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} (s : ∀ {i : Fin n.succ}, α i → α i → Prop) : Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔ s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_fin_succ] simp [and_assoc, exists_and_left] #align fin.pi_lex_lt_cons_cons Fin.pi_lex_lt_cons_cons theorem range_fin_succ {α} (f : Fin (n + 1) → α) : Set.range f = insert (f 0) (Set.range (Fin.tail f)) := Set.ext fun _ ↦ exists_fin_succ.trans <\| eq_comm.or Iff.rfl #align fin.range_fin_succ Fin.range_fin_succ @[simp] theorem range_cons {α : Type} {n : ℕ} (x : α) (b : Fin n → α) : Set.range (Fin.cons x b : Fin n.succ → α) = insert x (Set.range b) := by rw [range_fin_succ, cons_zero, tail_cons] #align fin.range_cons Fin.range_cons section Append /-- Append a tuple of length `m` to a tuple of length `n` to get a tuple of length `m + n`. This is a non-dependent version of `Fin.add_cases`. -/ def append {α : Type} (a : Fin m → α) (b : Fin n → α) : Fin (m + n) → α := @Fin.addCases _ _ (fun _ => α) a b #align fin.append Fin.append @[simp] theorem append_left {α : Type} (u : Fin m → α) (v : Fin n → α) (i : Fin m) : append u v (Fin.castAdd n i) = u i := addCases_left _ #align fin.append_left Fin.append_left @[simp] theorem append_right {α : Type} (u : Fin m → α) (v : Fin n → α) (i : Fin n) : append u v (natAdd m i) = v i := addCases_right _ #align fin.append_right Fin.append_right theorem append_right_nil {α : Type} (u : Fin m → α) (v : Fin n → α) (hv : n = 0) : append u v = u ∘ Fin.cast (by rw [hv, Nat.add_zero]) := by refine funext (Fin.addCases (fun l => ?_) fun r => ?_) · rw [append_left, Function.comp_apply] refine congr_arg u (Fin.ext ?_) simp · exact (Fin.cast hv r).elim0 #align fin.append_right_nil Fin.append_right_nil @[simp] theorem append_elim0 {α : Type} (u : Fin m → α) : append u Fin.elim0 = u ∘ Fin.cast (Nat.add_zero _) := append_right_nil _ _ rfl #align fin.append_elim0 Fin.append_elim0 theorem append_left_nil {α : Type} (u : Fin m → α) (v : Fin n → α) (hu : m = 0) : append u v = v ∘ Fin.cast (by rw [hu, Nat.zero_add]) := by refine funext (Fin.addCases (fun l => ?_) fun r => ?_) · exact (Fin.cast hu l).elim0 · rw [append_right, Function.comp_apply] refine congr_arg v (Fin.ext ?_) simp [hu] #align fin.append_left_nil Fin.append_left_nil @[simp] theorem elim0_append {α : Type} (v : Fin n → α) : append Fin.elim0 v = v ∘ Fin.cast (Nat.zero_add _) := append_left_nil _ _ rfl #align fin.elim0_append Fin.elim0_append theorem append_assoc {p : ℕ} {α : Type} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) : append (append a b) c = append a (append b c) ∘ Fin.cast (Nat.add_assoc ..) := by ext i rw [Function.comp_apply] refine Fin.addCases (fun l => ?_) (fun r => ?_) i · rw [append_left] refine Fin.addCases (fun ll => ?_) (fun lr => ?_) l · rw [append_left] simp [castAdd_castAdd] · rw [append_right] simp [castAdd_natAdd] · rw [append_right] simp [← natAdd_natAdd] #align fin.append_assoc Fin.append_assoc /-- Appending a one-tuple to the left is the same as `Fin.cons`. -/ theorem append_left_eq_cons {α : Type} {n : ℕ} (x₀ : Fin 1 → α) (x : Fin n → α) : Fin.append x₀ x = Fin.cons (x₀ 0) x ∘ Fin.cast (Nat.add_comm ..) := by ext i refine Fin.addCases ?_ ?_ i <;> clear i · intro i rw [Subsingleton.elim i 0, Fin.append_left, Function.comp_apply, eq_comm] exact Fin.cons_zero _ _ · intro i rw [Fin.append_right, Function.comp_apply, Fin.cast_natAdd, eq_comm, Fin.addNat_one] exact Fin.cons_succ _ _ _ #align fin.append_left_eq_cons Fin.append_left_eq_cons /-- `Fin.cons` is the same as appending a one-tuple to the left. -/ theorem cons_eq_append {α : Type} (x : α) (xs : Fin n → α) : cons x xs = append (cons x Fin.elim0) xs ∘ Fin.cast (Nat.add_comm ..) := by funext i; simp [append_left_eq_cons] @[simp] lemma append_cast_left {n m} {α : Type} (xs : Fin n → α) (ys : Fin m → α) (n' : ℕ) (h : n' = n) : Fin.append (xs ∘ Fin.cast h) ys = Fin.append xs ys ∘ (Fin.cast <\| by rw [h]) := by subst h; simp @[simp] lemma append_cast_right {n m} {α : Type} (xs : Fin n → α) (ys : Fin m → α) (m' : ℕ) (h : m' = m) : Fin.append xs (ys ∘ Fin.cast h) = Fin.append xs ys ∘ (Fin.cast <\| by rw [h]) := by subst h; simp lemma append_rev {m n} {α : Type} (xs : Fin m → α) (ys : Fin n → α) (i : Fin (m + n)) : append xs ys (rev i) = append (ys ∘ rev) (xs ∘ rev) (cast (Nat.add_comm ..) i) := by rcases rev_surjective i with ⟨i, rfl⟩ rw [rev_rev] induction i using Fin.addCases · simp [rev_castAdd] · simp [cast_rev, rev_addNat] lemma append_comp_rev {m n} {α : Type} (xs : Fin m → α) (ys : Fin n → α) : append xs ys ∘ rev = append (ys ∘ rev) (xs ∘ rev) ∘ cast (Nat.add_comm ..) := funext <\| append_rev xs ys end Append section Repeat /-- Repeat `a` `m` times. For example `Fin.repeat 2 ![0, 3, 7] = ![0, 3, 7, 0, 3, 7]`. -/ -- Porting note: removed @[simp] def «repeat» {α : Type} (m : ℕ) (a : Fin n → α) : Fin (m * n) → α \| i => a i.modNat #align fin.repeat Fin.repeat -- Porting note: added (leanprover/lean4#2042) @[simp] theorem repeat_apply {α : Type} (a : Fin n → α) (i : Fin (m n)) : Fin.repeat m a i = a i.modNat := rfl @[simp] theorem repeat_zero {α : Type} (a : Fin n → α) : Fin.repeat 0 a = Fin.elim0 ∘ cast (Nat.zero_mul _) := funext fun x => (cast (Nat.zero_mul _) x).elim0 #align fin.repeat_zero Fin.repeat_zero @[simp] theorem repeat_one {α : Type} (a : Fin n → α) : Fin.repeat 1 a = a ∘ cast (Nat.one_mul _) := by generalize_proofs h apply funext rw [(Fin.rightInverse_cast h.symm).surjective.forall] intro i simp [modNat, Nat.mod_eq_of_lt i.is_lt] #align fin.repeat_one Fin.repeat_one theorem repeat_succ {α : Type} (a : Fin n → α) (m : ℕ) : Fin.repeat m.succ a = append a (Fin.repeat m a) ∘ cast ((Nat.succ_mul _ _).trans (Nat.add_comm ..)) := by generalize_proofs h apply funext rw [(Fin.rightInverse_cast h.symm).surjective.forall] refine Fin.addCases (fun l => ?_) fun r => ?_ · simp [modNat, Nat.mod_eq_of_lt l.is_lt] · simp [modNat] #align fin.repeat_succ Fin.repeat_succ @[simp] theorem repeat_add {α : Type} (a : Fin n → α) (m₁ m₂ : ℕ) : Fin.repeat (m₁ + m₂) a = append (Fin.repeat m₁ a) (Fin.repeat m₂ a) ∘ cast (Nat.add_mul ..) := by generalize_proofs h apply funext rw [(Fin.rightInverse_cast h.symm).surjective.forall] refine Fin.addCases (fun l => ?_) fun r => ?_ · simp [modNat, Nat.mod_eq_of_lt l.is_lt] · simp [modNat, Nat.add_mod] #align fin.repeat_add Fin.repeat_add theorem repeat_rev {α : Type} (a : Fin n → α) (k : Fin (m n)) : Fin.repeat m a k.rev = Fin.repeat m (a ∘ Fin.rev) k := congr_arg a k.modNat_rev theorem repeat_comp_rev {α} (a : Fin n → α) : Fin.repeat m a ∘ Fin.rev = Fin.repeat m (a ∘ Fin.rev) := funext <\| repeat_rev a end Repeat end Tuple section TupleRight /-! In the previous section, we have discussed inserting or removing elements on the left of a tuple. In this section, we do the same on the right. A difference is that `Fin (n+1)` is constructed inductively from `Fin n` starting from the left, not from the right. This implies that Lean needs more help to realize that elements belong to the right types, i.e., we need to insert casts at several places. -/ -- Porting note: `i.castSucc` does not work like it did in Lean 3; -- `(castSucc i)` must be used. variable {α : Fin (n + 1) → Type u} (x : α (last n)) (q : ∀ i, α i) (p : ∀ i : Fin n, α (castSucc i)) (i : Fin n) (y : α (castSucc i)) (z : α (last n)) /-- The beginning of an `n+1` tuple, i.e., its first `n` entries -/ def init (q : ∀ i, α i) (i : Fin n) : α (castSucc i) := q (castSucc i) #align fin.init Fin.init theorem init_def {n : ℕ} {α : Fin (n + 1) → Type} {q : ∀ i, α i} : (init fun k : Fin (n + 1) ↦ q k) = fun k : Fin n ↦ q (castSucc k) := rfl #align fin.init_def Fin.init_def /-- Adding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc` comes from `cons` (i.e., adding an element to the left of a tuple) read in reverse order. -/ def snoc (p : ∀ i : Fin n, α (castSucc i)) (x : α (last n)) (i : Fin (n + 1)) : α i := if h : i.val < n then _root_.cast (by rw [Fin.castSucc_castLT i h]) (p (castLT i h)) else _root_.cast (by rw [eq_last_of_not_lt h]) x #align fin.snoc Fin.snoc @[simp] theorem init_snoc : init (snoc p x) = p := by ext i simp only [init, snoc, coe_castSucc, is_lt, cast_eq, dite_true] convert cast_eq rfl (p i) #align fin.init_snoc Fin.init_snoc @[simp] theorem snoc_castSucc : snoc p x (castSucc i) = p i := by simp only [snoc, coe_castSucc, is_lt, cast_eq, dite_true] convert cast_eq rfl (p i) #align fin.snoc_cast_succ Fin.snoc_castSucc @[simp] theorem snoc_comp_castSucc {n : ℕ} {α : Sort _} {a : α} {f : Fin n → α} : (snoc f a : Fin (n + 1) → α) ∘ castSucc = f := funext fun i ↦ by rw [Function.comp_apply, snoc_castSucc] #align fin.snoc_comp_cast_succ Fin.snoc_comp_castSucc @[simp] theorem snoc_last : snoc p x (last n) = x := by simp [snoc] #align fin.snoc_last Fin.snoc_last lemma snoc_zero {α : Type} (p : Fin 0 → α) (x : α) : Fin.snoc p x = fun _ ↦ x := by ext y have : Subsingleton (Fin (0 + 1)) := Fin.subsingleton_one simp only [Subsingleton.elim y (Fin.last 0), snoc_last] @[simp] theorem snoc_comp_nat_add {n m : ℕ} {α : Sort _} (f : Fin (m + n) → α) (a : α) : (snoc f a : Fin _ → α) ∘ (natAdd m : Fin (n + 1) → Fin (m + n + 1)) = snoc (f ∘ natAdd m) a := by ext i refine Fin.lastCases ?_ (fun i ↦ ?_) i · simp only [Function.comp_apply] rw [snoc_last, natAdd_last, snoc_last] · simp only [comp_apply, snoc_castSucc] rw [natAdd_castSucc, snoc_castSucc] #align fin.snoc_comp_nat_add Fin.snoc_comp_nat_add @[simp] theorem snoc_cast_add {α : Fin (n + m + 1) → Type} (f : ∀ i : Fin (n + m), α (castSucc i)) (a : α (last (n + m))) (i : Fin n) : (snoc f a) (castAdd (m + 1) i) = f (castAdd m i) := dif_pos _ #align fin.snoc_cast_add Fin.snoc_cast_add -- Porting note: Had to `unfold comp` @[simp] theorem snoc_comp_cast_add {n m : ℕ} {α : Sort _} (f : Fin (n + m) → α) (a : α) : (snoc f a : Fin _ → α) ∘ castAdd (m + 1) = f ∘ castAdd m := funext (by unfold comp; exact snoc_cast_add _ _) #align fin.snoc_comp_cast_add Fin.snoc_comp_cast_add /-- Updating a tuple and adding an element at the end commute. -/ @[simp] theorem snoc_update : snoc (update p i y) x = update (snoc p x) (castSucc i) y := by ext j by_cases h : j.val < n · rw [snoc] simp only [h] simp only [dif_pos] by_cases h' : j = castSucc i · have C1 : α (castSucc i) = α j := by rw [h'] have E1 : update (snoc p x) (castSucc i) y j = _root_.cast C1 y := by have : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y := by simp convert this · exact h'.symm · exact heq_of_cast_eq (congr_arg α (Eq.symm h')) rfl have C2 : α (castSucc i) = α (castSucc (castLT j h)) := by rw [castSucc_castLT, h'] have E2 : update p i y (castLT j h) = _root_.cast C2 y := by have : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y := by simp convert this · simp [h, h'] · exact heq_of_cast_eq C2 rfl rw [E1, E2] exact eq_rec_compose (Eq.trans C2.symm C1) C2 y · have : ¬castLT j h = i := by intro E apply h' rw [← E, castSucc_castLT] simp [h', this, snoc, h] · rw [eq_last_of_not_lt h] simp [Ne.symm (ne_of_lt (castSucc_lt_last i))] #align fin.snoc_update Fin.snoc_update /-- Adding an element at the beginning of a tuple and then updating it amounts to adding it directly. -/ theorem update_snoc_last : update (snoc p x) (last n) z = snoc p z := by ext j by_cases h : j.val < n · have : j ≠ last n := ne_of_lt h simp [h, update_noteq, this, snoc] · rw [eq_last_of_not_lt h] simp #align fin.update_snoc_last Fin.update_snoc_last /-- Concatenating the first element of a tuple with its tail gives back the original tuple -/ @[simp] theorem snoc_init_self : snoc (init q) (q (last n)) = q := by ext j by_cases h : j.val < n · simp only [init, snoc, h, cast_eq, dite_true, castSucc_castLT] · rw [eq_last_of_not_lt h] simp #align fin.snoc_init_self Fin.snoc_init_self /-- Updating the last element of a tuple does not change the beginning. -/ @[simp] theorem init_update_last : init (update q (last n) z) = init q := by ext j simp [init, ne_of_lt, castSucc_lt_last] #align fin.init_update_last Fin.init_update_last /-- Updating an element and taking the beginning commute. -/ @[simp] theorem init_update_castSucc : init (update q (castSucc i) y) = update (init q) i y := by ext j by_cases h : j = i · rw [h] simp [init] · simp [init, h, castSucc_inj] #align fin.init_update_cast_succ Fin.init_update_castSucc /-- `tail` and `init` commute. We state this lemma in a non-dependent setting, as otherwise it would involve a cast to convince Lean that the two types are equal, making it harder to use. -/ theorem tail_init_eq_init_tail {β : Type} (q : Fin (n + 2) → β) : tail (init q) = init (tail q) := by ext i simp [tail, init, castSucc_fin_succ] #align fin.tail_init_eq_init_tail Fin.tail_init_eq_init_tail /-- `cons` and `snoc` commute. We state this lemma in a non-dependent setting, as otherwise it would involve a cast to convince Lean that the two types are equal, making it harder to use. -/ theorem cons_snoc_eq_snoc_cons {β : Type} (a : β) (q : Fin n → β) (b : β) : @cons n.succ (fun _ ↦ β) a (snoc q b) = snoc (cons a q) b := by ext i by_cases h : i = 0 · rw [h] -- Porting note: `refl` finished it here in Lean 3, but I had to add more. simp [snoc, castLT] set j := pred i h with ji have : i = j.succ := by rw [ji, succ_pred] rw [this, cons_succ] by_cases h' : j.val < n · set k := castLT j h' with jk have : j = castSucc k := by rw [jk, castSucc_castLT] rw [this, ← castSucc_fin_succ, snoc] simp [pred, snoc, cons] rw [eq_last_of_not_lt h', succ_last] simp #align fin.cons_snoc_eq_snoc_cons Fin.cons_snoc_eq_snoc_cons theorem comp_snoc {α : Type} {β : Type} (g : α → β) (q : Fin n → α) (y : α) : g ∘ snoc q y = snoc (g ∘ q) (g y) := by ext j by_cases h : j.val < n · simp [h, snoc, castSucc_castLT] · rw [eq_last_of_not_lt h] simp #align fin.comp_snoc Fin.comp_snoc /-- Appending a one-tuple to the right is the same as `Fin.snoc`. -/ theorem append_right_eq_snoc {α : Type} {n : ℕ} (x : Fin n → α) (x₀ : Fin 1 → α) : Fin.append x x₀ = Fin.snoc x (x₀ 0) := by ext i refine Fin.addCases ?_ ?_ i <;> clear i · intro i rw [Fin.append_left] exact (@snoc_castSucc _ (fun _ => α) _ _ i).symm · intro i rw [Subsingleton.elim i 0, Fin.append_right] exact (@snoc_last _ (fun _ => α) _ _).symm #align fin.append_right_eq_snoc Fin.append_right_eq_snoc /-- `Fin.snoc` is the same as appending a one-tuple -/ theorem snoc_eq_append {α : Type} (xs : Fin n → α) (x : α) : snoc xs x = append xs (cons x Fin.elim0) := (append_right_eq_snoc xs (cons x Fin.elim0)).symm theorem append_left_snoc {n m} {α : Type} (xs : Fin n → α) (x : α) (ys : Fin m → α) : Fin.append (Fin.snoc xs x) ys = Fin.append xs (Fin.cons x ys) ∘ Fin.cast (Nat.succ_add_eq_add_succ ..) := by rw [snoc_eq_append, append_assoc, append_left_eq_cons, append_cast_right]; rfl theorem append_right_cons {n m} {α : Type} (xs : Fin n → α) (y : α) (ys : Fin m → α) : Fin.append xs (Fin.cons y ys) = Fin.append (Fin.snoc xs y) ys ∘ Fin.cast (Nat.succ_add_eq_add_succ ..).symm := by rw [append_left_snoc]; rfl theorem append_cons {α} (a : α) (as : Fin n → α) (bs : Fin m → α) : Fin.append (cons a as) bs = cons a (Fin.append as bs) ∘ (Fin.cast <\| Nat.add_right_comm n 1 m) := by funext i rcases i with ⟨i, -⟩ simp only [append, addCases, cons, castLT, cast, comp_apply] cases' i with i · simp · split_ifs with h · have : i < n := Nat.lt_of_succ_lt_succ h simp [addCases, this] · have : ¬i < n := Nat.not_le.mpr <\| Nat.lt_succ.mp <\| Nat.not_le.mp h simp [addCases, this] theorem append_snoc {α} (as : Fin n → α) (bs : Fin m → α) (b : α) : Fin.append as (snoc bs b) = snoc (Fin.append as bs) b := by funext i rcases i with ⟨i, isLt⟩ simp only [append, addCases, castLT, cast_mk, subNat_mk, natAdd_mk, cast, ge_iff_le, snoc.eq_1, cast_eq, eq_rec_constant, Nat.add_eq, Nat.add_zero, castLT_mk] split_ifs with lt_n lt_add sub_lt nlt_add lt_add <;> (try rfl) · have := Nat.lt_add_right m lt_n contradiction · obtain rfl := Nat.eq_of_le_of_lt_succ (Nat.not_lt.mp nlt_add) isLt simp [Nat.add_comm n m] at sub_lt · have := Nat.sub_lt_left_of_lt_add (Nat.not_lt.mp lt_n) lt_add contradiction theorem comp_init {α : Type} {β : Type} (g : α → β) (q : Fin n.succ → α) : g ∘ init q = init (g ∘ q) := by ext j simp [init] #align fin.comp_init Fin.comp_init /-- Recurse on an `n+1`-tuple by splitting it its initial `n`-tuple and its last element. -/ @[elab_as_elim, inline] def snocCases {P : (∀ i : Fin n.succ, α i) → Sort} (h : ∀ xs x, P (Fin.snoc xs x)) (x : ∀ i : Fin n.succ, α i) : P x := _root_.cast (by rw [Fin.snoc_init_self]) <\| h (Fin.init x) (x <\| Fin.last _) @[simp] lemma snocCases_snoc {P : (∀ i : Fin (n+1), α i) → Sort} (h : ∀ x x₀, P (Fin.snoc x x₀)) (x : ∀ i : Fin n, (Fin.init α) i) (x₀ : α (Fin.last _)) : snocCases h (Fin.snoc x x₀) = h x x₀ := by rw [snocCases, cast_eq_iff_heq, Fin.init_snoc, Fin.snoc_last] /-- Recurse on a tuple by splitting into `Fin.elim0` and `Fin.snoc`. -/ @[elab_as_elim] def snocInduction {α : Type} {P : ∀ {n : ℕ}, (Fin n → α) → Sort*} (h0 : P Fin.elim0) (h : ∀ {n} (x : Fin n → α) (x₀), P x → P (Fin.snoc x x₀)) : ∀ {n : ℕ} (x : Fin n → α), P x \| 0, x => by convert h0 \| n + 1, x => snocCases (fun x₀ x ↦ h _ _ <\| snocInduction h0 h _) x end TupleRight section InsertNth variable {α : Fin (n + 1) → Type u} {β : Type v} /- Porting note: Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling automatic insertion and specifying that motive seems to work. -/ /-- Define a function on `Fin (n + 1)` from a value on `i : Fin (n + 1)` and values on each `Fin.succAbove i j`, `j : Fin n`. This version is elaborated as eliminator and works for propositions, see also `Fin.insertNth` for a version without an `@[elab_as_elim]` attribute. -/ @[elab_as_elim] def succAboveCases {α : Fin (n + 1) → Sort u} (i : Fin (n + 1)) (x : α i) (p : ∀ j : Fin n, α (i.succAbove j)) (j : Fin (n + 1)) : α j := if hj : j = i then Eq.rec x hj.symm else if hlt : j < i then @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_castPred_of_lt _ _ hlt) (p _) else @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_pred_of_lt _ _ <\| (Ne.lt_or_lt hj).resolve_left hlt) (p _) #align fin.succ_above_cases Fin.succAboveCases theorem forall_iff_succAbove {p : Fin (n + 1) → Prop} (i : Fin (n + 1)) : (∀ j, p j) ↔ p i ∧ ∀ j, p (i.succAbove j) := ⟨fun h ↦ ⟨h _, fun _ ↦ h _⟩, fun h ↦ succAboveCases i h.1 h.2⟩ #align fin.forall_iff_succ_above Fin.forall_iff_succAbove /-- Insert an element into a tuple at a given position. For `i = 0` see `Fin.cons`, for `i = Fin.last n` see `Fin.snoc`. See also `Fin.succAboveCases` for a version elaborated as an eliminator. -/ def insertNth (i : Fin (n + 1)) (x : α i) (p : ∀ j : Fin n, α (i.succAbove j)) (j : Fin (n + 1)) : α j := succAboveCases i x p j #align fin.insert_nth Fin.insertNth @[simp] theorem insertNth_apply_same (i : Fin (n + 1)) (x : α i) (p : ∀ j, α (i.succAbove j)) : insertNth i x p i = x := by simp [insertNth, succAboveCases] #align fin.insert_nth_apply_same Fin.insertNth_apply_same @[simp] theorem insertNth_apply_succAbove (i : Fin (n + 1)) (x : α i) (p : ∀ j, α (i.succAbove j)) (j : Fin n) : insertNth i x p (i.succAbove j) = p j := by simp only [insertNth, succAboveCases, dif_neg (succAbove_ne _ _), succAbove_lt_iff_castSucc_lt] split_ifs with hlt · generalize_proofs H₁ H₂; revert H₂ generalize hk : castPred ((succAbove i) j) H₁ = k rw [castPred_succAbove _ _ hlt] at hk; cases hk intro; rfl · generalize_proofs H₁ H₂; revert H₂ generalize hk : pred (succAbove i j) H₁ = k erw [pred_succAbove _ _ (le_of_not_lt hlt)] at hk; cases hk intro; rfl #align fin.insert_nth_apply_succ_above Fin.insertNth_apply_succAbove @[simp] theorem succAbove_cases_eq_insertNth : @succAboveCases.{u + 1} = @insertNth.{u} := rfl #align fin.succ_above_cases_eq_insert_nth Fin.succAbove_cases_eq_insertNth /- Porting note: Had to `unfold comp`. Sometimes, when I use a placeholder, if I try to insert what Lean says it synthesized, it gives me a type error anyway. In this case, it's `x` and `p`. -/ @[simp] theorem insertNth_comp_succAbove (i : Fin (n + 1)) (x : β) (p : Fin n → β) : insertNth i x p ∘ i.succAbove = p := funext (by unfold comp; exact insertNth_apply_succAbove i _ _) #align fin.insert_nth_comp_succ_above Fin.insertNth_comp_succAbove theorem insertNth_eq_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} : i.insertNth x p = q ↔ q i = x ∧ p = fun j ↦ q (i.succAbove j) := by simp [funext_iff, forall_iff_succAbove i, eq_comm] #align fin.insert_nth_eq_iff Fin.insertNth_eq_iff theorem eq_insertNth_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} : q = i.insertNth x p ↔ q i = x ∧ p = fun j ↦ q (i.succAbove j) := eq_comm.trans insertNth_eq_iff #align fin.eq_insert_nth_iff Fin.eq_insertNth_iff /- Porting note: Once again, Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling automatic insertion and specifying that motive seems to work. -/ theorem insertNth_apply_below {i j : Fin (n + 1)} (h : j < i) (x : α i) (p : ∀ k, α (i.succAbove k)) : i.insertNth x p j = @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_castPred_of_lt _ _ h) (p <\| j.castPred _) := by rw [insertNth, succAboveCases, dif_neg h.ne, dif_pos h] #align fin.insert_nth_apply_below Fin.insertNth_apply_below /- Porting note: Once again, Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling automatic insertion and specifying that motive seems to work. -/ theorem insertNth_apply_above {i j : Fin (n + 1)} (h : i < j) (x : α i) (p : ∀ k, α (i.succAbove k)) : i.insertNth x p j = @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_pred_of_lt _ _ h) (p <\| j.pred _) := by rw [insertNth, succAboveCases, dif_neg h.ne', dif_neg h.not_lt] #align fin.insert_nth_apply_above Fin.insertNth_apply_above theorem insertNth_zero (x : α 0) (p : ∀ j : Fin n, α (succAbove 0 j)) : insertNth 0 x p = cons x fun j ↦ _root_.cast (congr_arg α (congr_fun succAbove_zero j)) (p j) := by refine insertNth_eq_iff.2 ⟨by simp, ?_⟩ ext j convert (cons_succ x p j).symm #align fin.insert_nth_zero Fin.insertNth_zero @[simp] theorem insertNth_zero' (x : β) (p : Fin n → β) : @insertNth _ (fun _ ↦ β) 0 x p = cons x p := by simp [insertNth_zero] #align fin.insert_nth_zero' Fin.insertNth_zero' theorem insertNth_last (x : α (last n)) (p : ∀ j : Fin n, α ((last n).succAbove j)) : insertNth (last n) x p = snoc (fun j ↦ _root_.cast (congr_arg α (succAbove_last_apply j)) (p j)) x := by refine insertNth_eq_iff.2 ⟨by simp, ?_⟩ ext j apply eq_of_heq trans snoc (fun j ↦ _root_.cast (congr_arg α (succAbove_last_apply j)) (p j)) x (castSucc j) · rw [snoc_castSucc] exact (cast_heq _ _).symm · apply congr_arg_heq rw [succAbove_last] #align fin.insert_nth_last Fin.insertNth_last @[simp]	Mathlib/Data/Fin/Tuple/Basic.lean	861	862	theorem insertNth_last' (x : β) (p : Fin n → β) : @insertNth _ (fun _ ↦ β) (last n) x p = snoc p x := by	simp [insertNth_last]
/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Kenny Lau -/ import Mathlib.Algebra.BigOperators.GroupWithZero.Finset import Mathlib.Algebra.Group.Submonoid.Membership import Mathlib.Algebra.Module.LinearMap.Basic import Mathlib.Data.Finset.Preimage import Mathlib.Data.Set.Finite import Mathlib.GroupTheory.GroupAction.BigOperators #align_import data.dfinsupp.basic from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1" /-! # Dependent functions with finite support For a non-dependent version see `data/finsupp.lean`. ## Notation This file introduces the notation `Π₀ a, β a` as notation for `DFinsupp β`, mirroring the `α →₀ β` notation used for `Finsupp`. This works for nested binders too, with `Π₀ a b, γ a b` as notation for `DFinsupp (fun a ↦ DFinsupp (γ a))`. ## Implementation notes The support is internally represented (in the primed `DFinsupp.support'`) as a `Multiset` that represents a superset of the true support of the function, quotiented by the always-true relation so that this does not impact equality. This approach has computational benefits over storing a `Finset`; it allows us to add together two finitely-supported functions without having to evaluate the resulting function to recompute its support (which would required decidability of `b = 0` for `b : β i`). The true support of the function can still be recovered with `DFinsupp.support`; but these decidability obligations are now postponed to when the support is actually needed. As a consequence, there are two ways to sum a `DFinsupp`: with `DFinsupp.sum` which works over an arbitrary function but requires recomputation of the support and therefore a `Decidable` argument; and with `DFinsupp.sumAddHom` which requires an additive morphism, using its properties to show that summing over a superset of the support is sufficient. `Finsupp` takes an altogether different approach here; it uses `Classical.Decidable` and declares the `Add` instance as noncomputable. This design difference is independent of the fact that `DFinsupp` is dependently-typed and `Finsupp` is not; in future, we may want to align these two definitions, or introduce two more definitions for the other combinations of decisions. -/ universe u u₁ u₂ v v₁ v₂ v₃ w x y l variable {ι : Type u} {γ : Type w} {β : ι → Type v} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} variable (β) /-- A dependent function `Π i, β i` with finite support, with notation `Π₀ i, β i`. Note that `DFinsupp.support` is the preferred API for accessing the support of the function, `DFinsupp.support'` is an implementation detail that aids computability; see the implementation notes in this file for more information. -/ structure DFinsupp [∀ i, Zero (β i)] : Type max u v where mk' :: /-- The underlying function of a dependent function with finite support (aka `DFinsupp`). -/ toFun : ∀ i, β i /-- The support of a dependent function with finite support (aka `DFinsupp`). -/ support' : Trunc { s : Multiset ι // ∀ i, i ∈ s ∨ toFun i = 0 } #align dfinsupp DFinsupp variable {β} /-- `Π₀ i, β i` denotes the type of dependent functions with finite support `DFinsupp β`. -/ notation3 "Π₀ "(...)", "r:(scoped f => DFinsupp f) => r namespace DFinsupp section Basic variable [∀ i, Zero (β i)] [∀ i, Zero (β₁ i)] [∀ i, Zero (β₂ i)] instance instDFunLike : DFunLike (Π₀ i, β i) ι β := ⟨fun f => f.toFun, fun ⟨f₁, s₁⟩ ⟨f₂, s₁⟩ ↦ fun (h : f₁ = f₂) ↦ by subst h congr apply Subsingleton.elim ⟩ #align dfinsupp.fun_like DFinsupp.instDFunLike /-- Helper instance for when there are too many metavariables to apply `DFunLike.coeFunForall` directly. -/ instance : CoeFun (Π₀ i, β i) fun _ => ∀ i, β i := inferInstance @[simp] theorem toFun_eq_coe (f : Π₀ i, β i) : f.toFun = f := rfl #align dfinsupp.to_fun_eq_coe DFinsupp.toFun_eq_coe @[ext] theorem ext {f g : Π₀ i, β i} (h : ∀ i, f i = g i) : f = g := DFunLike.ext _ _ h #align dfinsupp.ext DFinsupp.ext #align dfinsupp.ext_iff DFunLike.ext_iff #align dfinsupp.coe_fn_injective DFunLike.coe_injective lemma ne_iff {f g : Π₀ i, β i} : f ≠ g ↔ ∃ i, f i ≠ g i := DFunLike.ne_iff instance : Zero (Π₀ i, β i) := ⟨⟨0, Trunc.mk <\| ⟨∅, fun _ => Or.inr rfl⟩⟩⟩ instance : Inhabited (Π₀ i, β i) := ⟨0⟩ @[simp, norm_cast] lemma coe_mk' (f : ∀ i, β i) (s) : ⇑(⟨f, s⟩ : Π₀ i, β i) = f := rfl #align dfinsupp.coe_mk' DFinsupp.coe_mk' @[simp, norm_cast] lemma coe_zero : ⇑(0 : Π₀ i, β i) = 0 := rfl #align dfinsupp.coe_zero DFinsupp.coe_zero theorem zero_apply (i : ι) : (0 : Π₀ i, β i) i = 0 := rfl #align dfinsupp.zero_apply DFinsupp.zero_apply /-- The composition of `f : β₁ → β₂` and `g : Π₀ i, β₁ i` is `mapRange f hf g : Π₀ i, β₂ i`, well defined when `f 0 = 0`. This preserves the structure on `f`, and exists in various bundled forms for when `f` is itself bundled: * `DFinsupp.mapRange.addMonoidHom` * `DFinsupp.mapRange.addEquiv` * `dfinsupp.mapRange.linearMap` * `dfinsupp.mapRange.linearEquiv` -/ def mapRange (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0) (x : Π₀ i, β₁ i) : Π₀ i, β₂ i := ⟨fun i => f i (x i), x.support'.map fun s => ⟨s.1, fun i => (s.2 i).imp_right fun h : x i = 0 => by rw [← hf i, ← h]⟩⟩ #align dfinsupp.map_range DFinsupp.mapRange @[simp] theorem mapRange_apply (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0) (g : Π₀ i, β₁ i) (i : ι) : mapRange f hf g i = f i (g i) := rfl #align dfinsupp.map_range_apply DFinsupp.mapRange_apply @[simp] theorem mapRange_id (h : ∀ i, id (0 : β₁ i) = 0 := fun i => rfl) (g : Π₀ i : ι, β₁ i) : mapRange (fun i => (id : β₁ i → β₁ i)) h g = g := by ext rfl #align dfinsupp.map_range_id DFinsupp.mapRange_id theorem mapRange_comp (f : ∀ i, β₁ i → β₂ i) (f₂ : ∀ i, β i → β₁ i) (hf : ∀ i, f i 0 = 0) (hf₂ : ∀ i, f₂ i 0 = 0) (h : ∀ i, (f i ∘ f₂ i) 0 = 0) (g : Π₀ i : ι, β i) : mapRange (fun i => f i ∘ f₂ i) h g = mapRange f hf (mapRange f₂ hf₂ g) := by ext simp only [mapRange_apply]; rfl #align dfinsupp.map_range_comp DFinsupp.mapRange_comp @[simp] theorem mapRange_zero (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0) : mapRange f hf (0 : Π₀ i, β₁ i) = 0 := by ext simp only [mapRange_apply, coe_zero, Pi.zero_apply, hf] #align dfinsupp.map_range_zero DFinsupp.mapRange_zero /-- Let `f i` be a binary operation `β₁ i → β₂ i → β i` such that `f i 0 0 = 0`. Then `zipWith f hf` is a binary operation `Π₀ i, β₁ i → Π₀ i, β₂ i → Π₀ i, β i`. -/ def zipWith (f : ∀ i, β₁ i → β₂ i → β i) (hf : ∀ i, f i 0 0 = 0) (x : Π₀ i, β₁ i) (y : Π₀ i, β₂ i) : Π₀ i, β i := ⟨fun i => f i (x i) (y i), by refine x.support'.bind fun xs => ?_ refine y.support'.map fun ys => ?_ refine ⟨xs + ys, fun i => ?_⟩ obtain h1 \| (h1 : x i = 0) := xs.prop i · left rw [Multiset.mem_add] left exact h1 obtain h2 \| (h2 : y i = 0) := ys.prop i · left rw [Multiset.mem_add] right exact h2 right; rw [← hf, ← h1, ← h2]⟩ #align dfinsupp.zip_with DFinsupp.zipWith @[simp] theorem zipWith_apply (f : ∀ i, β₁ i → β₂ i → β i) (hf : ∀ i, f i 0 0 = 0) (g₁ : Π₀ i, β₁ i) (g₂ : Π₀ i, β₂ i) (i : ι) : zipWith f hf g₁ g₂ i = f i (g₁ i) (g₂ i) := rfl #align dfinsupp.zip_with_apply DFinsupp.zipWith_apply section Piecewise variable (x y : Π₀ i, β i) (s : Set ι) [∀ i, Decidable (i ∈ s)] /-- `x.piecewise y s` is the finitely supported function equal to `x` on the set `s`, and to `y` on its complement. -/ def piecewise : Π₀ i, β i := zipWith (fun i x y => if i ∈ s then x else y) (fun _ => ite_self 0) x y #align dfinsupp.piecewise DFinsupp.piecewise theorem piecewise_apply (i : ι) : x.piecewise y s i = if i ∈ s then x i else y i := zipWith_apply _ _ x y i #align dfinsupp.piecewise_apply DFinsupp.piecewise_apply @[simp, norm_cast] theorem coe_piecewise : ⇑(x.piecewise y s) = s.piecewise x y := by ext apply piecewise_apply #align dfinsupp.coe_piecewise DFinsupp.coe_piecewise end Piecewise end Basic section Algebra instance [∀ i, AddZeroClass (β i)] : Add (Π₀ i, β i) := ⟨zipWith (fun _ => (· + ·)) fun _ => add_zero 0⟩ theorem add_apply [∀ i, AddZeroClass (β i)] (g₁ g₂ : Π₀ i, β i) (i : ι) : (g₁ + g₂) i = g₁ i + g₂ i := rfl #align dfinsupp.add_apply DFinsupp.add_apply @[simp, norm_cast] theorem coe_add [∀ i, AddZeroClass (β i)] (g₁ g₂ : Π₀ i, β i) : ⇑(g₁ + g₂) = g₁ + g₂ := rfl #align dfinsupp.coe_add DFinsupp.coe_add instance addZeroClass [∀ i, AddZeroClass (β i)] : AddZeroClass (Π₀ i, β i) := DFunLike.coe_injective.addZeroClass _ coe_zero coe_add instance instIsLeftCancelAdd [∀ i, AddZeroClass (β i)] [∀ i, IsLeftCancelAdd (β i)] : IsLeftCancelAdd (Π₀ i, β i) where add_left_cancel _ _ _ h := ext fun x => add_left_cancel <\| DFunLike.congr_fun h x instance instIsRightCancelAdd [∀ i, AddZeroClass (β i)] [∀ i, IsRightCancelAdd (β i)] : IsRightCancelAdd (Π₀ i, β i) where add_right_cancel _ _ _ h := ext fun x => add_right_cancel <\| DFunLike.congr_fun h x instance instIsCancelAdd [∀ i, AddZeroClass (β i)] [∀ i, IsCancelAdd (β i)] : IsCancelAdd (Π₀ i, β i) where /-- Note the general `SMul` instance doesn't apply as `ℕ` is not distributive unless `β i`'s addition is commutative. -/ instance hasNatScalar [∀ i, AddMonoid (β i)] : SMul ℕ (Π₀ i, β i) := ⟨fun c v => v.mapRange (fun _ => (c • ·)) fun _ => nsmul_zero _⟩ #align dfinsupp.has_nat_scalar DFinsupp.hasNatScalar theorem nsmul_apply [∀ i, AddMonoid (β i)] (b : ℕ) (v : Π₀ i, β i) (i : ι) : (b • v) i = b • v i := rfl #align dfinsupp.nsmul_apply DFinsupp.nsmul_apply @[simp, norm_cast] theorem coe_nsmul [∀ i, AddMonoid (β i)] (b : ℕ) (v : Π₀ i, β i) : ⇑(b • v) = b • ⇑v := rfl #align dfinsupp.coe_nsmul DFinsupp.coe_nsmul instance [∀ i, AddMonoid (β i)] : AddMonoid (Π₀ i, β i) := DFunLike.coe_injective.addMonoid _ coe_zero coe_add fun _ _ => coe_nsmul _ _ /-- Coercion from a `DFinsupp` to a pi type is an `AddMonoidHom`. -/ def coeFnAddMonoidHom [∀ i, AddZeroClass (β i)] : (Π₀ i, β i) →+ ∀ i, β i where toFun := (⇑) map_zero' := coe_zero map_add' := coe_add #align dfinsupp.coe_fn_add_monoid_hom DFinsupp.coeFnAddMonoidHom /-- Evaluation at a point is an `AddMonoidHom`. This is the finitely-supported version of `Pi.evalAddMonoidHom`. -/ def evalAddMonoidHom [∀ i, AddZeroClass (β i)] (i : ι) : (Π₀ i, β i) →+ β i := (Pi.evalAddMonoidHom β i).comp coeFnAddMonoidHom #align dfinsupp.eval_add_monoid_hom DFinsupp.evalAddMonoidHom instance addCommMonoid [∀ i, AddCommMonoid (β i)] : AddCommMonoid (Π₀ i, β i) := DFunLike.coe_injective.addCommMonoid _ coe_zero coe_add fun _ _ => coe_nsmul _ _ @[simp, norm_cast] theorem coe_finset_sum {α} [∀ i, AddCommMonoid (β i)] (s : Finset α) (g : α → Π₀ i, β i) : ⇑(∑ a ∈ s, g a) = ∑ a ∈ s, ⇑(g a) := map_sum coeFnAddMonoidHom g s #align dfinsupp.coe_finset_sum DFinsupp.coe_finset_sum @[simp] theorem finset_sum_apply {α} [∀ i, AddCommMonoid (β i)] (s : Finset α) (g : α → Π₀ i, β i) (i : ι) : (∑ a ∈ s, g a) i = ∑ a ∈ s, g a i := map_sum (evalAddMonoidHom i) g s #align dfinsupp.finset_sum_apply DFinsupp.finset_sum_apply instance [∀ i, AddGroup (β i)] : Neg (Π₀ i, β i) := ⟨fun f => f.mapRange (fun _ => Neg.neg) fun _ => neg_zero⟩ theorem neg_apply [∀ i, AddGroup (β i)] (g : Π₀ i, β i) (i : ι) : (-g) i = -g i := rfl #align dfinsupp.neg_apply DFinsupp.neg_apply @[simp, norm_cast] lemma coe_neg [∀ i, AddGroup (β i)] (g : Π₀ i, β i) : ⇑(-g) = -g := rfl #align dfinsupp.coe_neg DFinsupp.coe_neg instance [∀ i, AddGroup (β i)] : Sub (Π₀ i, β i) := ⟨zipWith (fun _ => Sub.sub) fun _ => sub_zero 0⟩ theorem sub_apply [∀ i, AddGroup (β i)] (g₁ g₂ : Π₀ i, β i) (i : ι) : (g₁ - g₂) i = g₁ i - g₂ i := rfl #align dfinsupp.sub_apply DFinsupp.sub_apply @[simp, norm_cast] theorem coe_sub [∀ i, AddGroup (β i)] (g₁ g₂ : Π₀ i, β i) : ⇑(g₁ - g₂) = g₁ - g₂ := rfl #align dfinsupp.coe_sub DFinsupp.coe_sub /-- Note the general `SMul` instance doesn't apply as `ℤ` is not distributive unless `β i`'s addition is commutative. -/ instance hasIntScalar [∀ i, AddGroup (β i)] : SMul ℤ (Π₀ i, β i) := ⟨fun c v => v.mapRange (fun _ => (c • ·)) fun _ => zsmul_zero _⟩ #align dfinsupp.has_int_scalar DFinsupp.hasIntScalar theorem zsmul_apply [∀ i, AddGroup (β i)] (b : ℤ) (v : Π₀ i, β i) (i : ι) : (b • v) i = b • v i := rfl #align dfinsupp.zsmul_apply DFinsupp.zsmul_apply @[simp, norm_cast] theorem coe_zsmul [∀ i, AddGroup (β i)] (b : ℤ) (v : Π₀ i, β i) : ⇑(b • v) = b • ⇑v := rfl #align dfinsupp.coe_zsmul DFinsupp.coe_zsmul instance [∀ i, AddGroup (β i)] : AddGroup (Π₀ i, β i) := DFunLike.coe_injective.addGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => coe_nsmul _ _) fun _ _ => coe_zsmul _ _ instance addCommGroup [∀ i, AddCommGroup (β i)] : AddCommGroup (Π₀ i, β i) := DFunLike.coe_injective.addCommGroup _ coe_zero coe_add coe_neg coe_sub (fun _ _ => coe_nsmul _ _) fun _ _ => coe_zsmul _ _ /-- Dependent functions with finite support inherit a semiring action from an action on each coordinate. -/ instance [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] : SMul γ (Π₀ i, β i) := ⟨fun c v => v.mapRange (fun _ => (c • ·)) fun _ => smul_zero _⟩ theorem smul_apply [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] (b : γ) (v : Π₀ i, β i) (i : ι) : (b • v) i = b • v i := rfl #align dfinsupp.smul_apply DFinsupp.smul_apply @[simp, norm_cast] theorem coe_smul [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] (b : γ) (v : Π₀ i, β i) : ⇑(b • v) = b • ⇑v := rfl #align dfinsupp.coe_smul DFinsupp.coe_smul instance smulCommClass {δ : Type} [Monoid γ] [Monoid δ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] [∀ i, DistribMulAction δ (β i)] [∀ i, SMulCommClass γ δ (β i)] : SMulCommClass γ δ (Π₀ i, β i) where smul_comm r s m := ext fun i => by simp only [smul_apply, smul_comm r s (m i)] instance isScalarTower {δ : Type} [Monoid γ] [Monoid δ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] [∀ i, DistribMulAction δ (β i)] [SMul γ δ] [∀ i, IsScalarTower γ δ (β i)] : IsScalarTower γ δ (Π₀ i, β i) where smul_assoc r s m := ext fun i => by simp only [smul_apply, smul_assoc r s (m i)] instance isCentralScalar [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] [∀ i, DistribMulAction γᵐᵒᵖ (β i)] [∀ i, IsCentralScalar γ (β i)] : IsCentralScalar γ (Π₀ i, β i) where op_smul_eq_smul r m := ext fun i => by simp only [smul_apply, op_smul_eq_smul r (m i)] /-- Dependent functions with finite support inherit a `DistribMulAction` structure from such a structure on each coordinate. -/ instance distribMulAction [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] : DistribMulAction γ (Π₀ i, β i) := Function.Injective.distribMulAction coeFnAddMonoidHom DFunLike.coe_injective coe_smul /-- Dependent functions with finite support inherit a module structure from such a structure on each coordinate. -/ instance module [Semiring γ] [∀ i, AddCommMonoid (β i)] [∀ i, Module γ (β i)] : Module γ (Π₀ i, β i) := { inferInstanceAs (DistribMulAction γ (Π₀ i, β i)) with zero_smul := fun c => ext fun i => by simp only [smul_apply, zero_smul, zero_apply] add_smul := fun c x y => ext fun i => by simp only [add_apply, smul_apply, add_smul] } #align dfinsupp.module DFinsupp.module end Algebra section FilterAndSubtypeDomain /-- `Filter p f` is the function which is `f i` if `p i` is true and 0 otherwise. -/ def filter [∀ i, Zero (β i)] (p : ι → Prop) [DecidablePred p] (x : Π₀ i, β i) : Π₀ i, β i := ⟨fun i => if p i then x i else 0, x.support'.map fun xs => ⟨xs.1, fun i => (xs.prop i).imp_right fun H : x i = 0 => by simp only [H, ite_self]⟩⟩ #align dfinsupp.filter DFinsupp.filter @[simp] theorem filter_apply [∀ i, Zero (β i)] (p : ι → Prop) [DecidablePred p] (i : ι) (f : Π₀ i, β i) : f.filter p i = if p i then f i else 0 := rfl #align dfinsupp.filter_apply DFinsupp.filter_apply theorem filter_apply_pos [∀ i, Zero (β i)] {p : ι → Prop} [DecidablePred p] (f : Π₀ i, β i) {i : ι} (h : p i) : f.filter p i = f i := by simp only [filter_apply, if_pos h] #align dfinsupp.filter_apply_pos DFinsupp.filter_apply_pos theorem filter_apply_neg [∀ i, Zero (β i)] {p : ι → Prop} [DecidablePred p] (f : Π₀ i, β i) {i : ι} (h : ¬p i) : f.filter p i = 0 := by simp only [filter_apply, if_neg h] #align dfinsupp.filter_apply_neg DFinsupp.filter_apply_neg theorem filter_pos_add_filter_neg [∀ i, AddZeroClass (β i)] (f : Π₀ i, β i) (p : ι → Prop) [DecidablePred p] : (f.filter p + f.filter fun i => ¬p i) = f := ext fun i => by simp only [add_apply, filter_apply]; split_ifs <;> simp only [add_zero, zero_add] #align dfinsupp.filter_pos_add_filter_neg DFinsupp.filter_pos_add_filter_neg @[simp] theorem filter_zero [∀ i, Zero (β i)] (p : ι → Prop) [DecidablePred p] : (0 : Π₀ i, β i).filter p = 0 := by ext simp #align dfinsupp.filter_zero DFinsupp.filter_zero @[simp] theorem filter_add [∀ i, AddZeroClass (β i)] (p : ι → Prop) [DecidablePred p] (f g : Π₀ i, β i) : (f + g).filter p = f.filter p + g.filter p := by ext simp [ite_add_zero] #align dfinsupp.filter_add DFinsupp.filter_add @[simp] theorem filter_smul [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] (p : ι → Prop) [DecidablePred p] (r : γ) (f : Π₀ i, β i) : (r • f).filter p = r • f.filter p := by ext simp [smul_apply, smul_ite] #align dfinsupp.filter_smul DFinsupp.filter_smul variable (γ β) /-- `DFinsupp.filter` as an `AddMonoidHom`. -/ @[simps] def filterAddMonoidHom [∀ i, AddZeroClass (β i)] (p : ι → Prop) [DecidablePred p] : (Π₀ i, β i) →+ Π₀ i, β i where toFun := filter p map_zero' := filter_zero p map_add' := filter_add p #align dfinsupp.filter_add_monoid_hom DFinsupp.filterAddMonoidHom #align dfinsupp.filter_add_monoid_hom_apply DFinsupp.filterAddMonoidHom_apply /-- `DFinsupp.filter` as a `LinearMap`. -/ @[simps] def filterLinearMap [Semiring γ] [∀ i, AddCommMonoid (β i)] [∀ i, Module γ (β i)] (p : ι → Prop) [DecidablePred p] : (Π₀ i, β i) →ₗ[γ] Π₀ i, β i where toFun := filter p map_add' := filter_add p map_smul' := filter_smul p #align dfinsupp.filter_linear_map DFinsupp.filterLinearMap #align dfinsupp.filter_linear_map_apply DFinsupp.filterLinearMap_apply variable {γ β} @[simp] theorem filter_neg [∀ i, AddGroup (β i)] (p : ι → Prop) [DecidablePred p] (f : Π₀ i, β i) : (-f).filter p = -f.filter p := (filterAddMonoidHom β p).map_neg f #align dfinsupp.filter_neg DFinsupp.filter_neg @[simp] theorem filter_sub [∀ i, AddGroup (β i)] (p : ι → Prop) [DecidablePred p] (f g : Π₀ i, β i) : (f - g).filter p = f.filter p - g.filter p := (filterAddMonoidHom β p).map_sub f g #align dfinsupp.filter_sub DFinsupp.filter_sub /-- `subtypeDomain p f` is the restriction of the finitely supported function `f` to the subtype `p`. -/ def subtypeDomain [∀ i, Zero (β i)] (p : ι → Prop) [DecidablePred p] (x : Π₀ i, β i) : Π₀ i : Subtype p, β i := ⟨fun i => x (i : ι), x.support'.map fun xs => ⟨(Multiset.filter p xs.1).attach.map fun j => ⟨j.1, (Multiset.mem_filter.1 j.2).2⟩, fun i => (xs.prop i).imp_left fun H => Multiset.mem_map.2 ⟨⟨i, Multiset.mem_filter.2 ⟨H, i.2⟩⟩, Multiset.mem_attach _ _, Subtype.eta _ _⟩⟩⟩ #align dfinsupp.subtype_domain DFinsupp.subtypeDomain @[simp] theorem subtypeDomain_zero [∀ i, Zero (β i)] {p : ι → Prop} [DecidablePred p] : subtypeDomain p (0 : Π₀ i, β i) = 0 := rfl #align dfinsupp.subtype_domain_zero DFinsupp.subtypeDomain_zero @[simp] theorem subtypeDomain_apply [∀ i, Zero (β i)] {p : ι → Prop} [DecidablePred p] {i : Subtype p} {v : Π₀ i, β i} : (subtypeDomain p v) i = v i := rfl #align dfinsupp.subtype_domain_apply DFinsupp.subtypeDomain_apply @[simp] theorem subtypeDomain_add [∀ i, AddZeroClass (β i)] {p : ι → Prop} [DecidablePred p] (v v' : Π₀ i, β i) : (v + v').subtypeDomain p = v.subtypeDomain p + v'.subtypeDomain p := DFunLike.coe_injective rfl #align dfinsupp.subtype_domain_add DFinsupp.subtypeDomain_add @[simp] theorem subtypeDomain_smul [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] {p : ι → Prop} [DecidablePred p] (r : γ) (f : Π₀ i, β i) : (r • f).subtypeDomain p = r • f.subtypeDomain p := DFunLike.coe_injective rfl #align dfinsupp.subtype_domain_smul DFinsupp.subtypeDomain_smul variable (γ β) /-- `subtypeDomain` but as an `AddMonoidHom`. -/ @[simps] def subtypeDomainAddMonoidHom [∀ i, AddZeroClass (β i)] (p : ι → Prop) [DecidablePred p] : (Π₀ i : ι, β i) →+ Π₀ i : Subtype p, β i where toFun := subtypeDomain p map_zero' := subtypeDomain_zero map_add' := subtypeDomain_add #align dfinsupp.subtype_domain_add_monoid_hom DFinsupp.subtypeDomainAddMonoidHom #align dfinsupp.subtype_domain_add_monoid_hom_apply DFinsupp.subtypeDomainAddMonoidHom_apply /-- `DFinsupp.subtypeDomain` as a `LinearMap`. -/ @[simps] def subtypeDomainLinearMap [Semiring γ] [∀ i, AddCommMonoid (β i)] [∀ i, Module γ (β i)] (p : ι → Prop) [DecidablePred p] : (Π₀ i, β i) →ₗ[γ] Π₀ i : Subtype p, β i where toFun := subtypeDomain p map_add' := subtypeDomain_add map_smul' := subtypeDomain_smul #align dfinsupp.subtype_domain_linear_map DFinsupp.subtypeDomainLinearMap #align dfinsupp.subtype_domain_linear_map_apply DFinsupp.subtypeDomainLinearMap_apply variable {γ β} @[simp] theorem subtypeDomain_neg [∀ i, AddGroup (β i)] {p : ι → Prop} [DecidablePred p] {v : Π₀ i, β i} : (-v).subtypeDomain p = -v.subtypeDomain p := DFunLike.coe_injective rfl #align dfinsupp.subtype_domain_neg DFinsupp.subtypeDomain_neg @[simp] theorem subtypeDomain_sub [∀ i, AddGroup (β i)] {p : ι → Prop} [DecidablePred p] {v v' : Π₀ i, β i} : (v - v').subtypeDomain p = v.subtypeDomain p - v'.subtypeDomain p := DFunLike.coe_injective rfl #align dfinsupp.subtype_domain_sub DFinsupp.subtypeDomain_sub end FilterAndSubtypeDomain variable [DecidableEq ι] section Basic variable [∀ i, Zero (β i)] theorem finite_support (f : Π₀ i, β i) : Set.Finite { i \| f i ≠ 0 } := Trunc.induction_on f.support' fun xs ↦ xs.1.finite_toSet.subset fun i H ↦ ((xs.prop i).resolve_right H) #align dfinsupp.finite_support DFinsupp.finite_support /-- Create an element of `Π₀ i, β i` from a finset `s` and a function `x` defined on this `Finset`. -/ def mk (s : Finset ι) (x : ∀ i : (↑s : Set ι), β (i : ι)) : Π₀ i, β i := ⟨fun i => if H : i ∈ s then x ⟨i, H⟩ else 0, Trunc.mk ⟨s.1, fun i => if H : i ∈ s then Or.inl H else Or.inr <\| dif_neg H⟩⟩ #align dfinsupp.mk DFinsupp.mk variable {s : Finset ι} {x : ∀ i : (↑s : Set ι), β i} {i : ι} @[simp] theorem mk_apply : (mk s x : ∀ i, β i) i = if H : i ∈ s then x ⟨i, H⟩ else 0 := rfl #align dfinsupp.mk_apply DFinsupp.mk_apply theorem mk_of_mem (hi : i ∈ s) : (mk s x : ∀ i, β i) i = x ⟨i, hi⟩ := dif_pos hi #align dfinsupp.mk_of_mem DFinsupp.mk_of_mem theorem mk_of_not_mem (hi : i ∉ s) : (mk s x : ∀ i, β i) i = 0 := dif_neg hi #align dfinsupp.mk_of_not_mem DFinsupp.mk_of_not_mem theorem mk_injective (s : Finset ι) : Function.Injective (@mk ι β _ _ s) := by intro x y H ext i have h1 : (mk s x : ∀ i, β i) i = (mk s y : ∀ i, β i) i := by rw [H] obtain ⟨i, hi : i ∈ s⟩ := i dsimp only [mk_apply, Subtype.coe_mk] at h1 simpa only [dif_pos hi] using h1 #align dfinsupp.mk_injective DFinsupp.mk_injective instance unique [∀ i, Subsingleton (β i)] : Unique (Π₀ i, β i) := DFunLike.coe_injective.unique #align dfinsupp.unique DFinsupp.unique instance uniqueOfIsEmpty [IsEmpty ι] : Unique (Π₀ i, β i) := DFunLike.coe_injective.unique #align dfinsupp.unique_of_is_empty DFinsupp.uniqueOfIsEmpty /-- Given `Fintype ι`, `equivFunOnFintype` is the `Equiv` between `Π₀ i, β i` and `Π i, β i`. (All dependent functions on a finite type are finitely supported.) -/ @[simps apply] def equivFunOnFintype [Fintype ι] : (Π₀ i, β i) ≃ ∀ i, β i where toFun := (⇑) invFun f := ⟨f, Trunc.mk ⟨Finset.univ.1, fun _ => Or.inl <\| Finset.mem_univ_val _⟩⟩ left_inv _ := DFunLike.coe_injective rfl right_inv _ := rfl #align dfinsupp.equiv_fun_on_fintype DFinsupp.equivFunOnFintype #align dfinsupp.equiv_fun_on_fintype_apply DFinsupp.equivFunOnFintype_apply @[simp] theorem equivFunOnFintype_symm_coe [Fintype ι] (f : Π₀ i, β i) : equivFunOnFintype.symm f = f := Equiv.symm_apply_apply _ _ #align dfinsupp.equiv_fun_on_fintype_symm_coe DFinsupp.equivFunOnFintype_symm_coe /-- The function `single i b : Π₀ i, β i` sends `i` to `b` and all other points to `0`. -/ def single (i : ι) (b : β i) : Π₀ i, β i := ⟨Pi.single i b, Trunc.mk ⟨{i}, fun j => (Decidable.eq_or_ne j i).imp (by simp) fun h => Pi.single_eq_of_ne h _⟩⟩ #align dfinsupp.single DFinsupp.single theorem single_eq_pi_single {i b} : ⇑(single i b : Π₀ i, β i) = Pi.single i b := rfl #align dfinsupp.single_eq_pi_single DFinsupp.single_eq_pi_single @[simp] theorem single_apply {i i' b} : (single i b : Π₀ i, β i) i' = if h : i = i' then Eq.recOn h b else 0 := by rw [single_eq_pi_single, Pi.single, Function.update] simp [@eq_comm _ i i'] #align dfinsupp.single_apply DFinsupp.single_apply @[simp] theorem single_zero (i) : (single i 0 : Π₀ i, β i) = 0 := DFunLike.coe_injective <\| Pi.single_zero _ #align dfinsupp.single_zero DFinsupp.single_zero -- @[simp] -- Porting note (#10618): simp can prove this theorem single_eq_same {i b} : (single i b : Π₀ i, β i) i = b := by simp only [single_apply, dite_eq_ite, ite_true] #align dfinsupp.single_eq_same DFinsupp.single_eq_same theorem single_eq_of_ne {i i' b} (h : i ≠ i') : (single i b : Π₀ i, β i) i' = 0 := by simp only [single_apply, dif_neg h] #align dfinsupp.single_eq_of_ne DFinsupp.single_eq_of_ne theorem single_injective {i} : Function.Injective (single i : β i → Π₀ i, β i) := fun _ _ H => Pi.single_injective β i <\| DFunLike.coe_injective.eq_iff.mpr H #align dfinsupp.single_injective DFinsupp.single_injective /-- Like `Finsupp.single_eq_single_iff`, but with a `HEq` due to dependent types -/ theorem single_eq_single_iff (i j : ι) (xi : β i) (xj : β j) : DFinsupp.single i xi = DFinsupp.single j xj ↔ i = j ∧ HEq xi xj ∨ xi = 0 ∧ xj = 0 := by constructor · intro h by_cases hij : i = j · subst hij exact Or.inl ⟨rfl, heq_of_eq (DFinsupp.single_injective h)⟩ · have h_coe : ⇑(DFinsupp.single i xi) = DFinsupp.single j xj := congr_arg (⇑) h have hci := congr_fun h_coe i have hcj := congr_fun h_coe j rw [DFinsupp.single_eq_same] at hci hcj rw [DFinsupp.single_eq_of_ne (Ne.symm hij)] at hci rw [DFinsupp.single_eq_of_ne hij] at hcj exact Or.inr ⟨hci, hcj.symm⟩ · rintro (⟨rfl, hxi⟩ \| ⟨hi, hj⟩) · rw [eq_of_heq hxi] · rw [hi, hj, DFinsupp.single_zero, DFinsupp.single_zero] #align dfinsupp.single_eq_single_iff DFinsupp.single_eq_single_iff /-- `DFinsupp.single a b` is injective in `a`. For the statement that it is injective in `b`, see `DFinsupp.single_injective` -/ theorem single_left_injective {b : ∀ i : ι, β i} (h : ∀ i, b i ≠ 0) : Function.Injective (fun i => single i (b i) : ι → Π₀ i, β i) := fun _ _ H => (((single_eq_single_iff _ _ _ _).mp H).resolve_right fun hb => h _ hb.1).left #align dfinsupp.single_left_injective DFinsupp.single_left_injective @[simp] theorem single_eq_zero {i : ι} {xi : β i} : single i xi = 0 ↔ xi = 0 := by rw [← single_zero i, single_eq_single_iff] simp #align dfinsupp.single_eq_zero DFinsupp.single_eq_zero theorem filter_single (p : ι → Prop) [DecidablePred p] (i : ι) (x : β i) : (single i x).filter p = if p i then single i x else 0 := by ext j have := apply_ite (fun x : Π₀ i, β i => x j) (p i) (single i x) 0 dsimp at this rw [filter_apply, this] obtain rfl \| hij := Decidable.eq_or_ne i j · rfl · rw [single_eq_of_ne hij, ite_self, ite_self] #align dfinsupp.filter_single DFinsupp.filter_single @[simp] theorem filter_single_pos {p : ι → Prop} [DecidablePred p] (i : ι) (x : β i) (h : p i) : (single i x).filter p = single i x := by rw [filter_single, if_pos h] #align dfinsupp.filter_single_pos DFinsupp.filter_single_pos @[simp] theorem filter_single_neg {p : ι → Prop} [DecidablePred p] (i : ι) (x : β i) (h : ¬p i) : (single i x).filter p = 0 := by rw [filter_single, if_neg h] #align dfinsupp.filter_single_neg DFinsupp.filter_single_neg /-- Equality of sigma types is sufficient (but not necessary) to show equality of `DFinsupp`s. -/ theorem single_eq_of_sigma_eq {i j} {xi : β i} {xj : β j} (h : (⟨i, xi⟩ : Sigma β) = ⟨j, xj⟩) : DFinsupp.single i xi = DFinsupp.single j xj := by cases h rfl #align dfinsupp.single_eq_of_sigma_eq DFinsupp.single_eq_of_sigma_eq @[simp] theorem equivFunOnFintype_single [Fintype ι] (i : ι) (m : β i) : (@DFinsupp.equivFunOnFintype ι β _ _) (DFinsupp.single i m) = Pi.single i m := by ext x dsimp [Pi.single, Function.update] simp [DFinsupp.single_eq_pi_single, @eq_comm _ i] #align dfinsupp.equiv_fun_on_fintype_single DFinsupp.equivFunOnFintype_single @[simp] theorem equivFunOnFintype_symm_single [Fintype ι] (i : ι) (m : β i) : (@DFinsupp.equivFunOnFintype ι β _ _).symm (Pi.single i m) = DFinsupp.single i m := by ext i' simp only [← single_eq_pi_single, equivFunOnFintype_symm_coe] #align dfinsupp.equiv_fun_on_fintype_symm_single DFinsupp.equivFunOnFintype_symm_single section SingleAndZipWith variable [∀ i, Zero (β₁ i)] [∀ i, Zero (β₂ i)] @[simp] theorem zipWith_single_single (f : ∀ i, β₁ i → β₂ i → β i) (hf : ∀ i, f i 0 0 = 0) {i} (b₁ : β₁ i) (b₂ : β₂ i) : zipWith f hf (single i b₁) (single i b₂) = single i (f i b₁ b₂) := by ext j rw [zipWith_apply] obtain rfl \| hij := Decidable.eq_or_ne i j · rw [single_eq_same, single_eq_same, single_eq_same] · rw [single_eq_of_ne hij, single_eq_of_ne hij, single_eq_of_ne hij, hf] end SingleAndZipWith /-- Redefine `f i` to be `0`. -/ def erase (i : ι) (x : Π₀ i, β i) : Π₀ i, β i := ⟨fun j ↦ if j = i then 0 else x.1 j, x.support'.map fun xs ↦ ⟨xs.1, fun j ↦ (xs.prop j).imp_right (by simp only [·, ite_self])⟩⟩ #align dfinsupp.erase DFinsupp.erase @[simp] theorem erase_apply {i j : ι} {f : Π₀ i, β i} : (f.erase i) j = if j = i then 0 else f j := rfl #align dfinsupp.erase_apply DFinsupp.erase_apply -- @[simp] -- Porting note (#10618): simp can prove this theorem erase_same {i : ι} {f : Π₀ i, β i} : (f.erase i) i = 0 := by simp #align dfinsupp.erase_same DFinsupp.erase_same theorem erase_ne {i i' : ι} {f : Π₀ i, β i} (h : i' ≠ i) : (f.erase i) i' = f i' := by simp [h] #align dfinsupp.erase_ne DFinsupp.erase_ne theorem piecewise_single_erase (x : Π₀ i, β i) (i : ι) [∀ i' : ι, Decidable <\| (i' ∈ ({i} : Set ι))] : -- Porting note: added Decidable hypothesis (single i (x i)).piecewise (x.erase i) {i} = x := by ext j; rw [piecewise_apply]; split_ifs with h · rw [(id h : j = i), single_eq_same] · exact erase_ne h #align dfinsupp.piecewise_single_erase DFinsupp.piecewise_single_erase theorem erase_eq_sub_single {β : ι → Type} [∀ i, AddGroup (β i)] (f : Π₀ i, β i) (i : ι) : f.erase i = f - single i (f i) := by ext j rcases eq_or_ne i j with (rfl \| h) · simp · simp [erase_ne h.symm, single_eq_of_ne h, @eq_comm _ j, h] #align dfinsupp.erase_eq_sub_single DFinsupp.erase_eq_sub_single @[simp] theorem erase_zero (i : ι) : erase i (0 : Π₀ i, β i) = 0 := ext fun _ => ite_self _ #align dfinsupp.erase_zero DFinsupp.erase_zero @[simp] theorem filter_ne_eq_erase (f : Π₀ i, β i) (i : ι) : f.filter (· ≠ i) = f.erase i := by ext1 j simp only [DFinsupp.filter_apply, DFinsupp.erase_apply, ite_not] #align dfinsupp.filter_ne_eq_erase DFinsupp.filter_ne_eq_erase @[simp] theorem filter_ne_eq_erase' (f : Π₀ i, β i) (i : ι) : f.filter (i ≠ ·) = f.erase i := by rw [← filter_ne_eq_erase f i] congr with j exact ne_comm #align dfinsupp.filter_ne_eq_erase' DFinsupp.filter_ne_eq_erase' theorem erase_single (j : ι) (i : ι) (x : β i) : (single i x).erase j = if i = j then 0 else single i x := by rw [← filter_ne_eq_erase, filter_single, ite_not] #align dfinsupp.erase_single DFinsupp.erase_single @[simp] theorem erase_single_same (i : ι) (x : β i) : (single i x).erase i = 0 := by rw [erase_single, if_pos rfl] #align dfinsupp.erase_single_same DFinsupp.erase_single_same @[simp] theorem erase_single_ne {i j : ι} (x : β i) (h : i ≠ j) : (single i x).erase j = single i x := by rw [erase_single, if_neg h] #align dfinsupp.erase_single_ne DFinsupp.erase_single_ne section Update variable (f : Π₀ i, β i) (i) (b : β i) /-- Replace the value of a `Π₀ i, β i` at a given point `i : ι` by a given value `b : β i`. If `b = 0`, this amounts to removing `i` from the support. Otherwise, `i` is added to it. This is the (dependent) finitely-supported version of `Function.update`. -/ def update : Π₀ i, β i := ⟨Function.update f i b, f.support'.map fun s => ⟨i ::ₘ s.1, fun j => by rcases eq_or_ne i j with (rfl \| hi) · simp · obtain hj \| (hj : f j = 0) := s.prop j · exact Or.inl (Multiset.mem_cons_of_mem hj) · exact Or.inr ((Function.update_noteq hi.symm b _).trans hj)⟩⟩ #align dfinsupp.update DFinsupp.update variable (j : ι) @[simp, norm_cast] lemma coe_update : (f.update i b : ∀ i : ι, β i) = Function.update f i b := rfl #align dfinsupp.coe_update DFinsupp.coe_update @[simp] theorem update_self : f.update i (f i) = f := by ext simp #align dfinsupp.update_self DFinsupp.update_self @[simp] theorem update_eq_erase : f.update i 0 = f.erase i := by ext j rcases eq_or_ne i j with (rfl \| hi) · simp · simp [hi.symm] #align dfinsupp.update_eq_erase DFinsupp.update_eq_erase theorem update_eq_single_add_erase {β : ι → Type} [∀ i, AddZeroClass (β i)] (f : Π₀ i, β i) (i : ι) (b : β i) : f.update i b = single i b + f.erase i := by ext j rcases eq_or_ne i j with (rfl \| h) · simp · simp [Function.update_noteq h.symm, h, erase_ne, h.symm] #align dfinsupp.update_eq_single_add_erase DFinsupp.update_eq_single_add_erase theorem update_eq_erase_add_single {β : ι → Type} [∀ i, AddZeroClass (β i)] (f : Π₀ i, β i) (i : ι) (b : β i) : f.update i b = f.erase i + single i b := by ext j rcases eq_or_ne i j with (rfl \| h) · simp · simp [Function.update_noteq h.symm, h, erase_ne, h.symm] #align dfinsupp.update_eq_erase_add_single DFinsupp.update_eq_erase_add_single theorem update_eq_sub_add_single {β : ι → Type} [∀ i, AddGroup (β i)] (f : Π₀ i, β i) (i : ι) (b : β i) : f.update i b = f - single i (f i) + single i b := by rw [update_eq_erase_add_single f i b, erase_eq_sub_single f i] #align dfinsupp.update_eq_sub_add_single DFinsupp.update_eq_sub_add_single end Update end Basic section AddMonoid variable [∀ i, AddZeroClass (β i)] @[simp] theorem single_add (i : ι) (b₁ b₂ : β i) : single i (b₁ + b₂) = single i b₁ + single i b₂ := (zipWith_single_single (fun _ => (· + ·)) _ b₁ b₂).symm #align dfinsupp.single_add DFinsupp.single_add @[simp] theorem erase_add (i : ι) (f₁ f₂ : Π₀ i, β i) : erase i (f₁ + f₂) = erase i f₁ + erase i f₂ := ext fun _ => by simp [ite_zero_add] #align dfinsupp.erase_add DFinsupp.erase_add variable (β) /-- `DFinsupp.single` as an `AddMonoidHom`. -/ @[simps] def singleAddHom (i : ι) : β i →+ Π₀ i, β i where toFun := single i map_zero' := single_zero i map_add' := single_add i #align dfinsupp.single_add_hom DFinsupp.singleAddHom #align dfinsupp.single_add_hom_apply DFinsupp.singleAddHom_apply /-- `DFinsupp.erase` as an `AddMonoidHom`. -/ @[simps] def eraseAddHom (i : ι) : (Π₀ i, β i) →+ Π₀ i, β i where toFun := erase i map_zero' := erase_zero i map_add' := erase_add i #align dfinsupp.erase_add_hom DFinsupp.eraseAddHom #align dfinsupp.erase_add_hom_apply DFinsupp.eraseAddHom_apply variable {β} @[simp] theorem single_neg {β : ι → Type v} [∀ i, AddGroup (β i)] (i : ι) (x : β i) : single i (-x) = -single i x := (singleAddHom β i).map_neg x #align dfinsupp.single_neg DFinsupp.single_neg @[simp] theorem single_sub {β : ι → Type v} [∀ i, AddGroup (β i)] (i : ι) (x y : β i) : single i (x - y) = single i x - single i y := (singleAddHom β i).map_sub x y #align dfinsupp.single_sub DFinsupp.single_sub @[simp] theorem erase_neg {β : ι → Type v} [∀ i, AddGroup (β i)] (i : ι) (f : Π₀ i, β i) : (-f).erase i = -f.erase i := (eraseAddHom β i).map_neg f #align dfinsupp.erase_neg DFinsupp.erase_neg @[simp] theorem erase_sub {β : ι → Type v} [∀ i, AddGroup (β i)] (i : ι) (f g : Π₀ i, β i) : (f - g).erase i = f.erase i - g.erase i := (eraseAddHom β i).map_sub f g #align dfinsupp.erase_sub DFinsupp.erase_sub theorem single_add_erase (i : ι) (f : Π₀ i, β i) : single i (f i) + f.erase i = f := ext fun i' => if h : i = i' then by subst h; simp only [add_apply, single_apply, erase_apply, add_zero, dite_eq_ite, if_true] else by simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (Ne.symm h), zero_add] #align dfinsupp.single_add_erase DFinsupp.single_add_erase theorem erase_add_single (i : ι) (f : Π₀ i, β i) : f.erase i + single i (f i) = f := ext fun i' => if h : i = i' then by subst h; simp only [add_apply, single_apply, erase_apply, zero_add, dite_eq_ite, if_true] else by simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (Ne.symm h), add_zero] #align dfinsupp.erase_add_single DFinsupp.erase_add_single protected theorem induction {p : (Π₀ i, β i) → Prop} (f : Π₀ i, β i) (h0 : p 0) (ha : ∀ (i b) (f : Π₀ i, β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)) : p f := by cases' f with f s induction' s using Trunc.induction_on with s cases' s with s H induction' s using Multiset.induction_on with i s ih generalizing f · have : f = 0 := funext fun i => (H i).resolve_left (Multiset.not_mem_zero _) subst this exact h0 have H2 : p (erase i ⟨f, Trunc.mk ⟨i ::ₘ s, H⟩⟩) := by dsimp only [erase, Trunc.map, Trunc.bind, Trunc.liftOn, Trunc.lift_mk, Function.comp, Subtype.coe_mk] have H2 : ∀ j, j ∈ s ∨ ite (j = i) 0 (f j) = 0 := by intro j cases' H j with H2 H2 · cases' Multiset.mem_cons.1 H2 with H3 H3 · right; exact if_pos H3 · left; exact H3 right split_ifs <;> [rfl; exact H2] have H3 : ∀ aux, (⟨fun j : ι => ite (j = i) 0 (f j), Trunc.mk ⟨i ::ₘ s, aux⟩⟩ : Π₀ i, β i) = ⟨fun j : ι => ite (j = i) 0 (f j), Trunc.mk ⟨s, H2⟩⟩ := fun _ ↦ ext fun _ => rfl rw [H3] apply ih have H3 : single i _ + _ = (⟨f, Trunc.mk ⟨i ::ₘ s, H⟩⟩ : Π₀ i, β i) := single_add_erase _ _ rw [← H3] change p (single i (f i) + _) cases' Classical.em (f i = 0) with h h · rw [h, single_zero, zero_add] exact H2 refine ha _ _ _ ?_ h H2 rw [erase_same] #align dfinsupp.induction DFinsupp.induction theorem induction₂ {p : (Π₀ i, β i) → Prop} (f : Π₀ i, β i) (h0 : p 0) (ha : ∀ (i b) (f : Π₀ i, β i), f i = 0 → b ≠ 0 → p f → p (f + single i b)) : p f := DFinsupp.induction f h0 fun i b f h1 h2 h3 => have h4 : f + single i b = single i b + f := by ext j; by_cases H : i = j · subst H simp [h1] · simp [H] Eq.recOn h4 <\| ha i b f h1 h2 h3 #align dfinsupp.induction₂ DFinsupp.induction₂ @[simp] theorem add_closure_iUnion_range_single : AddSubmonoid.closure (⋃ i : ι, Set.range (single i : β i → Π₀ i, β i)) = ⊤ := top_unique fun x _ => by apply DFinsupp.induction x · exact AddSubmonoid.zero_mem _ exact fun a b f _ _ hf => AddSubmonoid.add_mem _ (AddSubmonoid.subset_closure <\| Set.mem_iUnion.2 ⟨a, Set.mem_range_self _⟩) hf #align dfinsupp.add_closure_Union_range_single DFinsupp.add_closure_iUnion_range_single /-- If two additive homomorphisms from `Π₀ i, β i` are equal on each `single a b`, then they are equal. -/ theorem addHom_ext {γ : Type w} [AddZeroClass γ] ⦃f g : (Π₀ i, β i) →+ γ⦄ (H : ∀ (i : ι) (y : β i), f (single i y) = g (single i y)) : f = g := by refine AddMonoidHom.eq_of_eqOn_denseM add_closure_iUnion_range_single fun f hf => ?_ simp only [Set.mem_iUnion, Set.mem_range] at hf rcases hf with ⟨x, y, rfl⟩ apply H #align dfinsupp.add_hom_ext DFinsupp.addHom_ext /-- If two additive homomorphisms from `Π₀ i, β i` are equal on each `single a b`, then they are equal. See note [partially-applied ext lemmas]. -/ @[ext] theorem addHom_ext' {γ : Type w} [AddZeroClass γ] ⦃f g : (Π₀ i, β i) →+ γ⦄ (H : ∀ x, f.comp (singleAddHom β x) = g.comp (singleAddHom β x)) : f = g := addHom_ext fun x => DFunLike.congr_fun (H x) #align dfinsupp.add_hom_ext' DFinsupp.addHom_ext' end AddMonoid @[simp] theorem mk_add [∀ i, AddZeroClass (β i)] {s : Finset ι} {x y : ∀ i : (↑s : Set ι), β i} : mk s (x + y) = mk s x + mk s y := ext fun i => by simp only [add_apply, mk_apply]; split_ifs <;> [rfl; rw [zero_add]] #align dfinsupp.mk_add DFinsupp.mk_add @[simp] theorem mk_zero [∀ i, Zero (β i)] {s : Finset ι} : mk s (0 : ∀ i : (↑s : Set ι), β i.1) = 0 := ext fun i => by simp only [mk_apply]; split_ifs <;> rfl #align dfinsupp.mk_zero DFinsupp.mk_zero @[simp] theorem mk_neg [∀ i, AddGroup (β i)] {s : Finset ι} {x : ∀ i : (↑s : Set ι), β i.1} : mk s (-x) = -mk s x := ext fun i => by simp only [neg_apply, mk_apply]; split_ifs <;> [rfl; rw [neg_zero]] #align dfinsupp.mk_neg DFinsupp.mk_neg @[simp] theorem mk_sub [∀ i, AddGroup (β i)] {s : Finset ι} {x y : ∀ i : (↑s : Set ι), β i.1} : mk s (x - y) = mk s x - mk s y := ext fun i => by simp only [sub_apply, mk_apply]; split_ifs <;> [rfl; rw [sub_zero]] #align dfinsupp.mk_sub DFinsupp.mk_sub /-- If `s` is a subset of `ι` then `mk_addGroupHom s` is the canonical additive group homomorphism from $\prod_{i\in s}\beta_i$ to $\prod_{\mathtt{i : \iota}}\beta_i.$-/ def mkAddGroupHom [∀ i, AddGroup (β i)] (s : Finset ι) : (∀ i : (s : Set ι), β ↑i) →+ Π₀ i : ι, β i where toFun := mk s map_zero' := mk_zero map_add' _ _ := mk_add #align dfinsupp.mk_add_group_hom DFinsupp.mkAddGroupHom section variable [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] @[simp] theorem mk_smul {s : Finset ι} (c : γ) (x : ∀ i : (↑s : Set ι), β (i : ι)) : mk s (c • x) = c • mk s x := ext fun i => by simp only [smul_apply, mk_apply]; split_ifs <;> [rfl; rw [smul_zero]] #align dfinsupp.mk_smul DFinsupp.mk_smul @[simp] theorem single_smul {i : ι} (c : γ) (x : β i) : single i (c • x) = c • single i x := ext fun i => by simp only [smul_apply, single_apply] split_ifs with h · cases h; rfl · rw [smul_zero] #align dfinsupp.single_smul DFinsupp.single_smul end section SupportBasic variable [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] /-- Set `{i \| f x ≠ 0}` as a `Finset`. -/ def support (f : Π₀ i, β i) : Finset ι := (f.support'.lift fun xs => (Multiset.toFinset xs.1).filter fun i => f i ≠ 0) <\| by rintro ⟨sx, hx⟩ ⟨sy, hy⟩ dsimp only [Subtype.coe_mk, toFun_eq_coe] at * ext i; constructor · intro H rcases Finset.mem_filter.1 H with ⟨_, h⟩ exact Finset.mem_filter.2 ⟨Multiset.mem_toFinset.2 <\| (hy i).resolve_right h, h⟩ · intro H rcases Finset.mem_filter.1 H with ⟨_, h⟩ exact Finset.mem_filter.2 ⟨Multiset.mem_toFinset.2 <\| (hx i).resolve_right h, h⟩ #align dfinsupp.support DFinsupp.support @[simp] theorem support_mk_subset {s : Finset ι} {x : ∀ i : (↑s : Set ι), β i.1} : (mk s x).support ⊆ s := fun _ H => Multiset.mem_toFinset.1 (Finset.mem_filter.1 H).1 #align dfinsupp.support_mk_subset DFinsupp.support_mk_subset @[simp] theorem support_mk'_subset {f : ∀ i, β i} {s : Multiset ι} {h} : (mk' f <\| Trunc.mk ⟨s, h⟩).support ⊆ s.toFinset := fun i H => Multiset.mem_toFinset.1 <\| by simpa using (Finset.mem_filter.1 H).1 #align dfinsupp.support_mk'_subset DFinsupp.support_mk'_subset @[simp] theorem mem_support_toFun (f : Π₀ i, β i) (i) : i ∈ f.support ↔ f i ≠ 0 := by cases' f with f s induction' s using Trunc.induction_on with s dsimp only [support, Trunc.lift_mk] rw [Finset.mem_filter, Multiset.mem_toFinset, coe_mk'] exact and_iff_right_of_imp (s.prop i).resolve_right #align dfinsupp.mem_support_to_fun DFinsupp.mem_support_toFun theorem eq_mk_support (f : Π₀ i, β i) : f = mk f.support fun i => f i := by aesop #align dfinsupp.eq_mk_support DFinsupp.eq_mk_support /-- Equivalence between dependent functions with finite support `s : Finset ι` and functions `∀ i, {x : β i // x ≠ 0}`. -/ @[simps] def subtypeSupportEqEquiv (s : Finset ι) : {f : Π₀ i, β i // f.support = s} ≃ ∀ i : s, {x : β i // x ≠ 0} where toFun \| ⟨f, hf⟩ => fun ⟨i, hi⟩ ↦ ⟨f i, (f.mem_support_toFun i).1 <\| hf.symm ▸ hi⟩ invFun f := ⟨mk s fun i ↦ (f i).1, Finset.ext fun i ↦ by -- TODO: `simp` fails to use `(f _).2` inside `∃ _, _` calc i ∈ support (mk s fun i ↦ (f i).1) ↔ ∃ h : i ∈ s, (f ⟨i, h⟩).1 ≠ 0 := by simp _ ↔ ∃ _ : i ∈ s, True := exists_congr fun h ↦ (iff_true _).mpr (f _).2 _ ↔ i ∈ s := by simp⟩ left_inv := by rintro ⟨f, rfl⟩ ext i simpa using Eq.symm right_inv f := by ext1 simp [Subtype.eta]; rfl /-- Equivalence between all dependent finitely supported functions `f : Π₀ i, β i` and type of pairs `⟨s : Finset ι, f : ∀ i : s, {x : β i // x ≠ 0}⟩`. -/ @[simps! apply_fst apply_snd_coe] def sigmaFinsetFunEquiv : (Π₀ i, β i) ≃ Σ s : Finset ι, ∀ i : s, {x : β i // x ≠ 0} := (Equiv.sigmaFiberEquiv DFinsupp.support).symm.trans (.sigmaCongrRight subtypeSupportEqEquiv) @[simp] theorem support_zero : (0 : Π₀ i, β i).support = ∅ := rfl #align dfinsupp.support_zero DFinsupp.support_zero theorem mem_support_iff {f : Π₀ i, β i} {i : ι} : i ∈ f.support ↔ f i ≠ 0 := f.mem_support_toFun _ #align dfinsupp.mem_support_iff DFinsupp.mem_support_iff theorem not_mem_support_iff {f : Π₀ i, β i} {i : ι} : i ∉ f.support ↔ f i = 0 := not_iff_comm.1 mem_support_iff.symm #align dfinsupp.not_mem_support_iff DFinsupp.not_mem_support_iff @[simp] theorem support_eq_empty {f : Π₀ i, β i} : f.support = ∅ ↔ f = 0 := ⟨fun H => ext <\| by simpa [Finset.ext_iff] using H, by simp (config := { contextual := true })⟩ #align dfinsupp.support_eq_empty DFinsupp.support_eq_empty instance decidableZero : DecidablePred (Eq (0 : Π₀ i, β i)) := fun _ => decidable_of_iff _ <\| support_eq_empty.trans eq_comm #align dfinsupp.decidable_zero DFinsupp.decidableZero theorem support_subset_iff {s : Set ι} {f : Π₀ i, β i} : ↑f.support ⊆ s ↔ ∀ i ∉ s, f i = 0 := by simp [Set.subset_def]; exact forall_congr' fun i => not_imp_comm #align dfinsupp.support_subset_iff DFinsupp.support_subset_iff theorem support_single_ne_zero {i : ι} {b : β i} (hb : b ≠ 0) : (single i b).support = {i} := by ext j; by_cases h : i = j · subst h simp [hb] simp [Ne.symm h, h] #align dfinsupp.support_single_ne_zero DFinsupp.support_single_ne_zero theorem support_single_subset {i : ι} {b : β i} : (single i b).support ⊆ {i} := support_mk'_subset #align dfinsupp.support_single_subset DFinsupp.support_single_subset section MapRangeAndZipWith variable [∀ i, Zero (β₁ i)] [∀ i, Zero (β₂ i)] theorem mapRange_def [∀ (i) (x : β₁ i), Decidable (x ≠ 0)] {f : ∀ i, β₁ i → β₂ i} {hf : ∀ i, f i 0 = 0} {g : Π₀ i, β₁ i} : mapRange f hf g = mk g.support fun i => f i.1 (g i.1) := by ext i by_cases h : g i ≠ 0 <;> simp at h <;> simp [h, hf] #align dfinsupp.map_range_def DFinsupp.mapRange_def @[simp] theorem mapRange_single {f : ∀ i, β₁ i → β₂ i} {hf : ∀ i, f i 0 = 0} {i : ι} {b : β₁ i} : mapRange f hf (single i b) = single i (f i b) := DFinsupp.ext fun i' => by by_cases h : i = i' · subst i' simp · simp [h, hf] #align dfinsupp.map_range_single DFinsupp.mapRange_single variable [∀ (i) (x : β₁ i), Decidable (x ≠ 0)] [∀ (i) (x : β₂ i), Decidable (x ≠ 0)] theorem support_mapRange {f : ∀ i, β₁ i → β₂ i} {hf : ∀ i, f i 0 = 0} {g : Π₀ i, β₁ i} : (mapRange f hf g).support ⊆ g.support := by simp [mapRange_def] #align dfinsupp.support_map_range DFinsupp.support_mapRange theorem zipWith_def {ι : Type u} {β : ι → Type v} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [dec : DecidableEq ι] [∀ i : ι, Zero (β i)] [∀ i : ι, Zero (β₁ i)] [∀ i : ι, Zero (β₂ i)] [∀ (i : ι) (x : β₁ i), Decidable (x ≠ 0)] [∀ (i : ι) (x : β₂ i), Decidable (x ≠ 0)] {f : ∀ i, β₁ i → β₂ i → β i} {hf : ∀ i, f i 0 0 = 0} {g₁ : Π₀ i, β₁ i} {g₂ : Π₀ i, β₂ i} : zipWith f hf g₁ g₂ = mk (g₁.support ∪ g₂.support) fun i => f i.1 (g₁ i.1) (g₂ i.1) := by ext i by_cases h1 : g₁ i ≠ 0 <;> by_cases h2 : g₂ i ≠ 0 <;> simp only [not_not, Ne] at h1 h2 <;> simp [h1, h2, hf] #align dfinsupp.zip_with_def DFinsupp.zipWith_def theorem support_zipWith {f : ∀ i, β₁ i → β₂ i → β i} {hf : ∀ i, f i 0 0 = 0} {g₁ : Π₀ i, β₁ i} {g₂ : Π₀ i, β₂ i} : (zipWith f hf g₁ g₂).support ⊆ g₁.support ∪ g₂.support := by simp [zipWith_def] #align dfinsupp.support_zip_with DFinsupp.support_zipWith end MapRangeAndZipWith theorem erase_def (i : ι) (f : Π₀ i, β i) : f.erase i = mk (f.support.erase i) fun j => f j.1 := by ext j by_cases h1 : j = i <;> by_cases h2 : f j ≠ 0 <;> simp at h2 <;> simp [h1, h2] #align dfinsupp.erase_def DFinsupp.erase_def @[simp] theorem support_erase (i : ι) (f : Π₀ i, β i) : (f.erase i).support = f.support.erase i := by ext j by_cases h1 : j = i · simp only [h1, mem_support_toFun, erase_apply, ite_true, ne_eq, not_true, not_not, Finset.mem_erase, false_and] by_cases h2 : f j ≠ 0 <;> simp at h2 <;> simp [h1, h2] #align dfinsupp.support_erase DFinsupp.support_erase theorem support_update_ne_zero (f : Π₀ i, β i) (i : ι) {b : β i} (h : b ≠ 0) : support (f.update i b) = insert i f.support := by ext j rcases eq_or_ne i j with (rfl \| hi) · simp [h] · simp [hi.symm] #align dfinsupp.support_update_ne_zero DFinsupp.support_update_ne_zero theorem support_update (f : Π₀ i, β i) (i : ι) (b : β i) [Decidable (b = 0)] : support (f.update i b) = if b = 0 then support (f.erase i) else insert i f.support := by ext j split_ifs with hb · subst hb simp [update_eq_erase, support_erase] · rw [support_update_ne_zero f _ hb] #align dfinsupp.support_update DFinsupp.support_update section FilterAndSubtypeDomain variable {p : ι → Prop} [DecidablePred p] theorem filter_def (f : Π₀ i, β i) : f.filter p = mk (f.support.filter p) fun i => f i.1 := by ext i; by_cases h1 : p i <;> by_cases h2 : f i ≠ 0 <;> simp at h2 <;> simp [h1, h2] #align dfinsupp.filter_def DFinsupp.filter_def @[simp] theorem support_filter (f : Π₀ i, β i) : (f.filter p).support = f.support.filter p := by ext i; by_cases h : p i <;> simp [h] #align dfinsupp.support_filter DFinsupp.support_filter theorem subtypeDomain_def (f : Π₀ i, β i) : f.subtypeDomain p = mk (f.support.subtype p) fun i => f i := by ext i; by_cases h2 : f i ≠ 0 <;> try simp at h2; dsimp; simp [h2] #align dfinsupp.subtype_domain_def DFinsupp.subtypeDomain_def @[simp, nolint simpNF] -- Porting note: simpNF claims that LHS does not simplify, but it does theorem support_subtypeDomain {f : Π₀ i, β i} : (subtypeDomain p f).support = f.support.subtype p := by ext i simp #align dfinsupp.support_subtype_domain DFinsupp.support_subtypeDomain end FilterAndSubtypeDomain end SupportBasic theorem support_add [∀ i, AddZeroClass (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] {g₁ g₂ : Π₀ i, β i} : (g₁ + g₂).support ⊆ g₁.support ∪ g₂.support := support_zipWith #align dfinsupp.support_add DFinsupp.support_add @[simp] theorem support_neg [∀ i, AddGroup (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] {f : Π₀ i, β i} : support (-f) = support f := by ext i; simp #align dfinsupp.support_neg DFinsupp.support_neg theorem support_smul {γ : Type w} [Semiring γ] [∀ i, AddCommMonoid (β i)] [∀ i, Module γ (β i)] [∀ (i : ι) (x : β i), Decidable (x ≠ 0)] (b : γ) (v : Π₀ i, β i) : (b • v).support ⊆ v.support := support_mapRange #align dfinsupp.support_smul DFinsupp.support_smul instance [∀ i, Zero (β i)] [∀ i, DecidableEq (β i)] : DecidableEq (Π₀ i, β i) := fun f g => decidable_of_iff (f.support = g.support ∧ ∀ i ∈ f.support, f i = g i) ⟨fun ⟨h₁, h₂⟩ => ext fun i => if h : i ∈ f.support then h₂ i h else by have hf : f i = 0 := by rwa [mem_support_iff, not_not] at h have hg : g i = 0 := by rwa [h₁, mem_support_iff, not_not] at h rw [hf, hg], by rintro rfl; simp⟩ section Equiv open Finset variable {κ : Type*} /-- Reindexing (and possibly removing) terms of a dfinsupp. -/ noncomputable def comapDomain [∀ i, Zero (β i)] (h : κ → ι) (hh : Function.Injective h) (f : Π₀ i, β i) : Π₀ k, β (h k) where toFun x := f (h x) support' := f.support'.map fun s => ⟨((Multiset.toFinset s.1).preimage h hh.injOn).val, fun x => (s.prop (h x)).imp_left fun hx => mem_preimage.mpr <\| Multiset.mem_toFinset.mpr hx⟩ #align dfinsupp.comap_domain DFinsupp.comapDomain @[simp] theorem comapDomain_apply [∀ i, Zero (β i)] (h : κ → ι) (hh : Function.Injective h) (f : Π₀ i, β i) (k : κ) : comapDomain h hh f k = f (h k) := rfl #align dfinsupp.comap_domain_apply DFinsupp.comapDomain_apply @[simp] theorem comapDomain_zero [∀ i, Zero (β i)] (h : κ → ι) (hh : Function.Injective h) : comapDomain h hh (0 : Π₀ i, β i) = 0 := by ext rw [zero_apply, comapDomain_apply, zero_apply] #align dfinsupp.comap_domain_zero DFinsupp.comapDomain_zero @[simp] theorem comapDomain_add [∀ i, AddZeroClass (β i)] (h : κ → ι) (hh : Function.Injective h) (f g : Π₀ i, β i) : comapDomain h hh (f + g) = comapDomain h hh f + comapDomain h hh g := by ext rw [add_apply, comapDomain_apply, comapDomain_apply, comapDomain_apply, add_apply] #align dfinsupp.comap_domain_add DFinsupp.comapDomain_add @[simp]	Mathlib/Data/DFinsupp/Basic.lean	1,346	1,350	theorem comapDomain_smul [Monoid γ] [∀ i, AddMonoid (β i)] [∀ i, DistribMulAction γ (β i)] (h : κ → ι) (hh : Function.Injective h) (r : γ) (f : Π₀ i, β i) : comapDomain h hh (r • f) = r • comapDomain h hh f := by	ext rw [smul_apply, comapDomain_apply, smul_apply, comapDomain_apply]
/- Copyright (c) 2018 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import Mathlib.MeasureTheory.Measure.Dirac /-! # Counting measure In this file we define the counting measure `MeasurTheory.Measure.count` as `MeasureTheory.Measure.sum MeasureTheory.Measure.dirac` and prove basic properties of this measure. -/ set_option autoImplicit true open Set open scoped ENNReal Classical variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α} noncomputable section namespace MeasureTheory.Measure /-- Counting measure on any measurable space. -/ def count : Measure α := sum dirac #align measure_theory.measure.count MeasureTheory.Measure.count theorem le_count_apply : ∑' _ : s, (1 : ℝ≥0∞) ≤ count s := calc (∑' _ : s, 1 : ℝ≥0∞) = ∑' i, indicator s 1 i := tsum_subtype s 1 _ ≤ ∑' i, dirac i s := ENNReal.tsum_le_tsum fun _ => le_dirac_apply _ ≤ count s := le_sum_apply _ _ #align measure_theory.measure.le_count_apply MeasureTheory.Measure.le_count_apply theorem count_apply (hs : MeasurableSet s) : count s = ∑' i : s, 1 := by simp only [count, sum_apply, hs, dirac_apply', ← tsum_subtype s (1 : α → ℝ≥0∞), Pi.one_apply] #align measure_theory.measure.count_apply MeasureTheory.Measure.count_apply -- @[simp] -- Porting note (#10618): simp can prove this theorem count_empty : count (∅ : Set α) = 0 := by rw [count_apply MeasurableSet.empty, tsum_empty] #align measure_theory.measure.count_empty MeasureTheory.Measure.count_empty @[simp] theorem count_apply_finset' {s : Finset α} (s_mble : MeasurableSet (s : Set α)) : count (↑s : Set α) = s.card := calc count (↑s : Set α) = ∑' i : (↑s : Set α), 1 := count_apply s_mble _ = ∑ i ∈ s, 1 := s.tsum_subtype 1 _ = s.card := by simp #align measure_theory.measure.count_apply_finset' MeasureTheory.Measure.count_apply_finset' @[simp] theorem count_apply_finset [MeasurableSingletonClass α] (s : Finset α) : count (↑s : Set α) = s.card := count_apply_finset' s.measurableSet #align measure_theory.measure.count_apply_finset MeasureTheory.Measure.count_apply_finset theorem count_apply_finite' {s : Set α} (s_fin : s.Finite) (s_mble : MeasurableSet s) : count s = s_fin.toFinset.card := by simp [← @count_apply_finset' _ _ s_fin.toFinset (by simpa only [Finite.coe_toFinset] using s_mble)] #align measure_theory.measure.count_apply_finite' MeasureTheory.Measure.count_apply_finite' theorem count_apply_finite [MeasurableSingletonClass α] (s : Set α) (hs : s.Finite) : count s = hs.toFinset.card := by rw [← count_apply_finset, Finite.coe_toFinset] #align measure_theory.measure.count_apply_finite MeasureTheory.Measure.count_apply_finite /-- `count` measure evaluates to infinity at infinite sets. -/ theorem count_apply_infinite (hs : s.Infinite) : count s = ∞ := by refine top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => ?_) rcases hs.exists_subset_card_eq n with ⟨t, ht, rfl⟩ calc (t.card : ℝ≥0∞) = ∑ i ∈ t, 1 := by simp _ = ∑' i : (t : Set α), 1 := (t.tsum_subtype 1).symm _ ≤ count (t : Set α) := le_count_apply _ ≤ count s := measure_mono ht #align measure_theory.measure.count_apply_infinite MeasureTheory.Measure.count_apply_infinite @[simp] theorem count_apply_eq_top' (s_mble : MeasurableSet s) : count s = ∞ ↔ s.Infinite := by by_cases hs : s.Finite · simp [Set.Infinite, hs, count_apply_finite' hs s_mble] · change s.Infinite at hs simp [hs, count_apply_infinite] #align measure_theory.measure.count_apply_eq_top' MeasureTheory.Measure.count_apply_eq_top' @[simp] theorem count_apply_eq_top [MeasurableSingletonClass α] : count s = ∞ ↔ s.Infinite := by by_cases hs : s.Finite · exact count_apply_eq_top' hs.measurableSet · change s.Infinite at hs simp [hs, count_apply_infinite] #align measure_theory.measure.count_apply_eq_top MeasureTheory.Measure.count_apply_eq_top @[simp] theorem count_apply_lt_top' (s_mble : MeasurableSet s) : count s < ∞ ↔ s.Finite := calc count s < ∞ ↔ count s ≠ ∞ := lt_top_iff_ne_top _ ↔ ¬s.Infinite := not_congr (count_apply_eq_top' s_mble) _ ↔ s.Finite := Classical.not_not #align measure_theory.measure.count_apply_lt_top' MeasureTheory.Measure.count_apply_lt_top' @[simp] theorem count_apply_lt_top [MeasurableSingletonClass α] : count s < ∞ ↔ s.Finite := calc count s < ∞ ↔ count s ≠ ∞ := lt_top_iff_ne_top _ ↔ ¬s.Infinite := not_congr count_apply_eq_top _ ↔ s.Finite := Classical.not_not #align measure_theory.measure.count_apply_lt_top MeasureTheory.Measure.count_apply_lt_top theorem empty_of_count_eq_zero' (s_mble : MeasurableSet s) (hsc : count s = 0) : s = ∅ := by have hs : s.Finite := by rw [← count_apply_lt_top' s_mble, hsc] exact WithTop.zero_lt_top simpa [count_apply_finite' hs s_mble] using hsc #align measure_theory.measure.empty_of_count_eq_zero' MeasureTheory.Measure.empty_of_count_eq_zero' theorem empty_of_count_eq_zero [MeasurableSingletonClass α] (hsc : count s = 0) : s = ∅ := by have hs : s.Finite := by rw [← count_apply_lt_top, hsc] exact WithTop.zero_lt_top simpa [count_apply_finite _ hs] using hsc #align measure_theory.measure.empty_of_count_eq_zero MeasureTheory.Measure.empty_of_count_eq_zero @[simp] theorem count_eq_zero_iff' (s_mble : MeasurableSet s) : count s = 0 ↔ s = ∅ := ⟨empty_of_count_eq_zero' s_mble, fun h => h.symm ▸ count_empty⟩ #align measure_theory.measure.count_eq_zero_iff' MeasureTheory.Measure.count_eq_zero_iff' @[simp] theorem count_eq_zero_iff [MeasurableSingletonClass α] : count s = 0 ↔ s = ∅ := ⟨empty_of_count_eq_zero, fun h => h.symm ▸ count_empty⟩ #align measure_theory.measure.count_eq_zero_iff MeasureTheory.Measure.count_eq_zero_iff	Mathlib/MeasureTheory/Measure/Count.lean	139	141	theorem count_ne_zero' (hs' : s.Nonempty) (s_mble : MeasurableSet s) : count s ≠ 0 := by	rw [Ne, count_eq_zero_iff' s_mble] exact hs'.ne_empty
/- Copyright (c) 2021 Gabriel Moise. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Gabriel Moise, Yaël Dillies, Kyle Miller -/ import Mathlib.Combinatorics.SimpleGraph.Finite import Mathlib.Data.Finset.Sym import Mathlib.Data.Matrix.Basic #align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496" /-! # Incidence matrix of a simple graph This file defines the unoriented incidence matrix of a simple graph. ## Main definitions * `SimpleGraph.incMatrix`: `G.incMatrix R` is the incidence matrix of `G` over the ring `R`. ## Main results * `SimpleGraph.incMatrix_mul_transpose_diag`: The diagonal entries of the product of `G.incMatrix R` and its transpose are the degrees of the vertices. * `SimpleGraph.incMatrix_mul_transpose`: Gives a complete description of the product of `G.incMatrix R` and its transpose; the diagonal is the degrees of each vertex, and the off-diagonals are 1 or 0 depending on whether or not the vertices are adjacent. * `SimpleGraph.incMatrix_transpose_mul_diag`: The diagonal entries of the product of the transpose of `G.incMatrix R` and `G.inc_matrix R` are `2` or `0` depending on whether or not the unordered pair is an edge of `G`. ## Implementation notes The usual definition of an incidence matrix has one row per vertex and one column per edge. However, this definition has columns indexed by all of `Sym2 α`, where `α` is the vertex type. This appears not to change the theory, and for simple graphs it has the nice effect that every incidence matrix for each `SimpleGraph α` has the same type. ## TODO * Define the oriented incidence matrices for oriented graphs. * Define the graph Laplacian of a simple graph using the oriented incidence matrix from an arbitrary orientation of a simple graph. -/ open Finset Matrix SimpleGraph Sym2 open Matrix namespace SimpleGraph variable (R : Type) {α : Type} (G : SimpleGraph α) /-- `G.incMatrix R` is the `α × Sym2 α` matrix whose `(a, e)`-entry is `1` if `e` is incident to `a` and `0` otherwise. -/ noncomputable def incMatrix [Zero R] [One R] : Matrix α (Sym2 α) R := fun a => (G.incidenceSet a).indicator 1 #align simple_graph.inc_matrix SimpleGraph.incMatrix variable {R} theorem incMatrix_apply [Zero R] [One R] {a : α} {e : Sym2 α} : G.incMatrix R a e = (G.incidenceSet a).indicator 1 e := rfl #align simple_graph.inc_matrix_apply SimpleGraph.incMatrix_apply /-- Entries of the incidence matrix can be computed given additional decidable instances. -/ theorem incMatrix_apply' [Zero R] [One R] [DecidableEq α] [DecidableRel G.Adj] {a : α} {e : Sym2 α} : G.incMatrix R a e = if e ∈ G.incidenceSet a then 1 else 0 := by unfold incMatrix Set.indicator convert rfl #align simple_graph.inc_matrix_apply' SimpleGraph.incMatrix_apply' section MulZeroOneClass variable [MulZeroOneClass R] {a b : α} {e : Sym2 α} theorem incMatrix_apply_mul_incMatrix_apply : G.incMatrix R a e * G.incMatrix R b e = (G.incidenceSet a ∩ G.incidenceSet b).indicator 1 e := by classical simp only [incMatrix, Set.indicator_apply, ite_zero_mul_ite_zero, Pi.one_apply, mul_one, Set.mem_inter_iff] #align simple_graph.inc_matrix_apply_mul_inc_matrix_apply SimpleGraph.incMatrix_apply_mul_incMatrix_apply theorem incMatrix_apply_mul_incMatrix_apply_of_not_adj (hab : a ≠ b) (h : ¬G.Adj a b) : G.incMatrix R a e * G.incMatrix R b e = 0 := by rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem] rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab] exact Set.not_mem_empty e #align simple_graph.inc_matrix_apply_mul_inc_matrix_apply_of_not_adj SimpleGraph.incMatrix_apply_mul_incMatrix_apply_of_not_adj theorem incMatrix_of_not_mem_incidenceSet (h : e ∉ G.incidenceSet a) : G.incMatrix R a e = 0 := by rw [incMatrix_apply, Set.indicator_of_not_mem h] #align simple_graph.inc_matrix_of_not_mem_incidence_set SimpleGraph.incMatrix_of_not_mem_incidenceSet theorem incMatrix_of_mem_incidenceSet (h : e ∈ G.incidenceSet a) : G.incMatrix R a e = 1 := by rw [incMatrix_apply, Set.indicator_of_mem h, Pi.one_apply] #align simple_graph.inc_matrix_of_mem_incidence_set SimpleGraph.incMatrix_of_mem_incidenceSet variable [Nontrivial R] theorem incMatrix_apply_eq_zero_iff : G.incMatrix R a e = 0 ↔ e ∉ G.incidenceSet a := by simp only [incMatrix_apply, Set.indicator_apply_eq_zero, Pi.one_apply, one_ne_zero] #align simple_graph.inc_matrix_apply_eq_zero_iff SimpleGraph.incMatrix_apply_eq_zero_iff theorem incMatrix_apply_eq_one_iff : G.incMatrix R a e = 1 ↔ e ∈ G.incidenceSet a := by -- Porting note: was `convert one_ne_zero.ite_eq_left_iff; infer_instance` unfold incMatrix Set.indicator simp only [Pi.one_apply] apply Iff.intro <;> intro h · split at h <;> simp_all only [zero_ne_one] · simp_all only [ite_true] #align simple_graph.inc_matrix_apply_eq_one_iff SimpleGraph.incMatrix_apply_eq_one_iff end MulZeroOneClass section NonAssocSemiring variable [Fintype (Sym2 α)] [NonAssocSemiring R] {a b : α} {e : Sym2 α} theorem sum_incMatrix_apply [Fintype (neighborSet G a)] : ∑ e, G.incMatrix R a e = G.degree a := by classical simp [incMatrix_apply', sum_boole, Set.filter_mem_univ_eq_toFinset] #align simple_graph.sum_inc_matrix_apply SimpleGraph.sum_incMatrix_apply theorem incMatrix_mul_transpose_diag [Fintype (neighborSet G a)] : (G.incMatrix R * (G.incMatrix R)ᵀ) a a = G.degree a := by classical rw [← sum_incMatrix_apply] simp only [mul_apply, incMatrix_apply', transpose_apply, mul_ite, mul_one, mul_zero] simp_all only [ite_true, sum_boole] #align simple_graph.inc_matrix_mul_transpose_diag SimpleGraph.incMatrix_mul_transpose_diag theorem sum_incMatrix_apply_of_mem_edgeSet [Fintype α] : e ∈ G.edgeSet → ∑ a, G.incMatrix R a e = 2 := by classical refine e.ind ?_ intro a b h rw [mem_edgeSet] at h rw [← Nat.cast_two, ← card_pair h.ne] simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff] congr 2 ext e simp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton] #align simple_graph.sum_inc_matrix_apply_of_mem_edge_set SimpleGraph.sum_incMatrix_apply_of_mem_edgeSet theorem sum_incMatrix_apply_of_not_mem_edgeSet [Fintype α] (h : e ∉ G.edgeSet) : ∑ a, G.incMatrix R a e = 0 := sum_eq_zero fun _ _ => G.incMatrix_of_not_mem_incidenceSet fun he => h he.1 #align simple_graph.sum_inc_matrix_apply_of_not_mem_edge_set SimpleGraph.sum_incMatrix_apply_of_not_mem_edgeSet theorem incMatrix_transpose_mul_diag [Fintype α] [Decidable (e ∈ G.edgeSet)] : ((G.incMatrix R)ᵀ * G.incMatrix R) e e = if e ∈ G.edgeSet then 2 else 0 := by classical simp only [Matrix.mul_apply, incMatrix_apply', transpose_apply, ite_zero_mul_ite_zero, one_mul, sum_boole, and_self_iff] split_ifs with h · revert h refine e.ind ?_ intro v w h rw [← Nat.cast_two, ← card_pair (G.ne_of_adj h)] simp only [mk'_mem_incidenceSet_iff, G.mem_edgeSet.mp h, true_and, mem_univ, forall_true_left, forall_eq_or_imp, forall_eq, and_self, mem_singleton, ne_eq] congr 2 ext u simp · revert h refine e.ind ?_ intro v w h simp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.not.mp h] #align simple_graph.inc_matrix_transpose_mul_diag SimpleGraph.incMatrix_transpose_mul_diag end NonAssocSemiring section Semiring variable [Fintype (Sym2 α)] [Semiring R] {a b : α} {e : Sym2 α}	Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean	179	187	theorem incMatrix_mul_transpose_apply_of_adj (h : G.Adj a b) : (G.incMatrix R * (G.incMatrix R)ᵀ) a b = (1 : R) := by	classical simp_rw [Matrix.mul_apply, Matrix.transpose_apply, incMatrix_apply_mul_incMatrix_apply, Set.indicator_apply, Pi.one_apply, sum_boole] convert @Nat.cast_one R _ convert card_singleton s(a, b) rw [← coe_eq_singleton, coe_filter_univ] exact G.incidenceSet_inter_incidenceSet_of_adj h
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp -/ import Mathlib.Algebra.BigOperators.Finsupp import Mathlib.Algebra.BigOperators.Finprod import Mathlib.Data.Fintype.BigOperators import Mathlib.LinearAlgebra.Finsupp import Mathlib.LinearAlgebra.LinearIndependent import Mathlib.SetTheory.Cardinal.Cofinality #align_import linear_algebra.basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395" /-! # Bases This file defines bases in a module or vector space. It is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light. ## Main definitions All definitions are given for families of vectors, i.e. `v : ι → M` where `M` is the module or vector space and `ι : Type` is an arbitrary indexing type. `Basis ι R M` is the type of `ι`-indexed `R`-bases for a module `M`, represented by a linear equiv `M ≃ₗ[R] ι →₀ R`. * the basis vectors of a basis `b : Basis ι R M` are available as `b i`, where `i : ι` * `Basis.repr` is the isomorphism sending `x : M` to its coordinates `Basis.repr x : ι →₀ R`. The converse, turning this isomorphism into a basis, is called `Basis.ofRepr`. * If `ι` is finite, there is a variant of `repr` called `Basis.equivFun b : M ≃ₗ[R] ι → R` (saving you from having to work with `Finsupp`). The converse, turning this isomorphism into a basis, is called `Basis.ofEquivFun`. * `Basis.constr b R f` constructs a linear map `M₁ →ₗ[R] M₂` given the values `f : ι → M₂` at the basis elements `⇑b : ι → M₁`. * `Basis.reindex` uses an equiv to map a basis to a different indexing set. * `Basis.map` uses a linear equiv to map a basis to a different module. ## Main statements * `Basis.mk`: a linear independent set of vectors spanning the whole module determines a basis * `Basis.ext` states that two linear maps are equal if they coincide on a basis. Similar results are available for linear equivs (if they coincide on the basis vectors), elements (if their coordinates coincide) and the functions `b.repr` and `⇑b`. ## Implementation notes We use families instead of sets because it allows us to say that two identical vectors are linearly dependent. For bases, this is useful as well because we can easily derive ordered bases by using an ordered index type `ι`. ## Tags basis, bases -/ noncomputable section universe u open Function Set Submodule variable {ι : Type} {ι' : Type} {R : Type} {R₂ : Type} {K : Type} variable {M : Type} {M' M'' : Type} {V : Type u} {V' : Type} section Module variable [Semiring R] variable [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] section variable (ι R M) /-- A `Basis ι R M` for a module `M` is the type of `ι`-indexed `R`-bases of `M`. The basis vectors are available as `DFunLike.coe (b : Basis ι R M) : ι → M`. To turn a linear independent family of vectors spanning `M` into a basis, use `Basis.mk`. They are internally represented as linear equivs `M ≃ₗ[R] (ι →₀ R)`, available as `Basis.repr`. -/ structure Basis where /-- `Basis.ofRepr` constructs a basis given an assignment of coordinates to each vector. -/ ofRepr :: /-- `repr` is the linear equivalence sending a vector `x` to its coordinates: the `c`s such that `x = ∑ i, c i`. -/ repr : M ≃ₗ[R] ι →₀ R #align basis Basis #align basis.repr Basis.repr #align basis.of_repr Basis.ofRepr end instance uniqueBasis [Subsingleton R] : Unique (Basis ι R M) := ⟨⟨⟨default⟩⟩, fun ⟨b⟩ => by rw [Subsingleton.elim b]⟩ #align unique_basis uniqueBasis namespace Basis instance : Inhabited (Basis ι R (ι →₀ R)) := ⟨.ofRepr (LinearEquiv.refl _ _)⟩ variable (b b₁ : Basis ι R M) (i : ι) (c : R) (x : M) section repr theorem repr_injective : Injective (repr : Basis ι R M → M ≃ₗ[R] ι →₀ R) := fun f g h => by cases f; cases g; congr #align basis.repr_injective Basis.repr_injective /-- `b i` is the `i`th basis vector. -/ instance instFunLike : FunLike (Basis ι R M) ι M where coe b i := b.repr.symm (Finsupp.single i 1) coe_injective' f g h := repr_injective <\| LinearEquiv.symm_bijective.injective <\| LinearEquiv.toLinearMap_injective <\| by ext; exact congr_fun h _ #align basis.fun_like Basis.instFunLike @[simp] theorem coe_ofRepr (e : M ≃ₗ[R] ι →₀ R) : ⇑(ofRepr e) = fun i => e.symm (Finsupp.single i 1) := rfl #align basis.coe_of_repr Basis.coe_ofRepr protected theorem injective [Nontrivial R] : Injective b := b.repr.symm.injective.comp fun _ _ => (Finsupp.single_left_inj (one_ne_zero : (1 : R) ≠ 0)).mp #align basis.injective Basis.injective theorem repr_symm_single_one : b.repr.symm (Finsupp.single i 1) = b i := rfl #align basis.repr_symm_single_one Basis.repr_symm_single_one theorem repr_symm_single : b.repr.symm (Finsupp.single i c) = c • b i := calc b.repr.symm (Finsupp.single i c) = b.repr.symm (c • Finsupp.single i (1 : R)) := by { rw [Finsupp.smul_single', mul_one] } _ = c • b i := by rw [LinearEquiv.map_smul, repr_symm_single_one] #align basis.repr_symm_single Basis.repr_symm_single @[simp] theorem repr_self : b.repr (b i) = Finsupp.single i 1 := LinearEquiv.apply_symm_apply _ _ #align basis.repr_self Basis.repr_self theorem repr_self_apply (j) [Decidable (i = j)] : b.repr (b i) j = if i = j then 1 else 0 := by rw [repr_self, Finsupp.single_apply] #align basis.repr_self_apply Basis.repr_self_apply @[simp] theorem repr_symm_apply (v) : b.repr.symm v = Finsupp.total ι M R b v := calc b.repr.symm v = b.repr.symm (v.sum Finsupp.single) := by simp _ = v.sum fun i vi => b.repr.symm (Finsupp.single i vi) := map_finsupp_sum .. _ = Finsupp.total ι M R b v := by simp only [repr_symm_single, Finsupp.total_apply] #align basis.repr_symm_apply Basis.repr_symm_apply @[simp] theorem coe_repr_symm : ↑b.repr.symm = Finsupp.total ι M R b := LinearMap.ext fun v => b.repr_symm_apply v #align basis.coe_repr_symm Basis.coe_repr_symm @[simp] theorem repr_total (v) : b.repr (Finsupp.total _ _ _ b v) = v := by rw [← b.coe_repr_symm] exact b.repr.apply_symm_apply v #align basis.repr_total Basis.repr_total @[simp] theorem total_repr : Finsupp.total _ _ _ b (b.repr x) = x := by rw [← b.coe_repr_symm] exact b.repr.symm_apply_apply x #align basis.total_repr Basis.total_repr theorem repr_range : LinearMap.range (b.repr : M →ₗ[R] ι →₀ R) = Finsupp.supported R R univ := by rw [LinearEquiv.range, Finsupp.supported_univ] #align basis.repr_range Basis.repr_range theorem mem_span_repr_support (m : M) : m ∈ span R (b '' (b.repr m).support) := (Finsupp.mem_span_image_iff_total _).2 ⟨b.repr m, by simp [Finsupp.mem_supported_support]⟩ #align basis.mem_span_repr_support Basis.mem_span_repr_support theorem repr_support_subset_of_mem_span (s : Set ι) {m : M} (hm : m ∈ span R (b '' s)) : ↑(b.repr m).support ⊆ s := by rcases (Finsupp.mem_span_image_iff_total _).1 hm with ⟨l, hl, rfl⟩ rwa [repr_total, ← Finsupp.mem_supported R l] #align basis.repr_support_subset_of_mem_span Basis.repr_support_subset_of_mem_span theorem mem_span_image {m : M} {s : Set ι} : m ∈ span R (b '' s) ↔ ↑(b.repr m).support ⊆ s := ⟨repr_support_subset_of_mem_span _ _, fun h ↦ span_mono (image_subset _ h) (mem_span_repr_support b _)⟩ @[simp] theorem self_mem_span_image [Nontrivial R] {i : ι} {s : Set ι} : b i ∈ span R (b '' s) ↔ i ∈ s := by simp [mem_span_image, Finsupp.support_single_ne_zero] end repr section Coord /-- `b.coord i` is the linear function giving the `i`'th coordinate of a vector with respect to the basis `b`. `b.coord i` is an element of the dual space. In particular, for finite-dimensional spaces it is the `ι`th basis vector of the dual space. -/ @[simps!] def coord : M →ₗ[R] R := Finsupp.lapply i ∘ₗ ↑b.repr #align basis.coord Basis.coord theorem forall_coord_eq_zero_iff {x : M} : (∀ i, b.coord i x = 0) ↔ x = 0 := Iff.trans (by simp only [b.coord_apply, DFunLike.ext_iff, Finsupp.zero_apply]) b.repr.map_eq_zero_iff #align basis.forall_coord_eq_zero_iff Basis.forall_coord_eq_zero_iff /-- The sum of the coordinates of an element `m : M` with respect to a basis. -/ noncomputable def sumCoords : M →ₗ[R] R := (Finsupp.lsum ℕ fun _ => LinearMap.id) ∘ₗ (b.repr : M →ₗ[R] ι →₀ R) #align basis.sum_coords Basis.sumCoords @[simp] theorem coe_sumCoords : (b.sumCoords : M → R) = fun m => (b.repr m).sum fun _ => id := rfl #align basis.coe_sum_coords Basis.coe_sumCoords theorem coe_sumCoords_eq_finsum : (b.sumCoords : M → R) = fun m => ∑ᶠ i, b.coord i m := by ext m simp only [Basis.sumCoords, Basis.coord, Finsupp.lapply_apply, LinearMap.id_coe, LinearEquiv.coe_coe, Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, finsum_eq_sum _ (b.repr m).finite_support, Finsupp.sum, Finset.finite_toSet_toFinset, id, Finsupp.fun_support_eq] #align basis.coe_sum_coords_eq_finsum Basis.coe_sumCoords_eq_finsum @[simp high] theorem coe_sumCoords_of_fintype [Fintype ι] : (b.sumCoords : M → R) = ∑ i, b.coord i := by ext m -- Porting note: - `eq_self_iff_true` -- + `comp_apply` `LinearMap.coeFn_sum` simp only [sumCoords, Finsupp.sum_fintype, LinearMap.id_coe, LinearEquiv.coe_coe, coord_apply, id, Fintype.sum_apply, imp_true_iff, Finsupp.coe_lsum, LinearMap.coe_comp, comp_apply, LinearMap.coeFn_sum] #align basis.coe_sum_coords_of_fintype Basis.coe_sumCoords_of_fintype @[simp] theorem sumCoords_self_apply : b.sumCoords (b i) = 1 := by simp only [Basis.sumCoords, LinearMap.id_coe, LinearEquiv.coe_coe, id, Basis.repr_self, Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, Finsupp.sum_single_index] #align basis.sum_coords_self_apply Basis.sumCoords_self_apply theorem dvd_coord_smul (i : ι) (m : M) (r : R) : r ∣ b.coord i (r • m) := ⟨b.coord i m, by simp⟩ #align basis.dvd_coord_smul Basis.dvd_coord_smul theorem coord_repr_symm (b : Basis ι R M) (i : ι) (f : ι →₀ R) : b.coord i (b.repr.symm f) = f i := by simp only [repr_symm_apply, coord_apply, repr_total] #align basis.coord_repr_symm Basis.coord_repr_symm end Coord section Ext variable {R₁ : Type} [Semiring R₁] {σ : R →+ R₁} {σ' : R₁ →+* R} variable [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] variable {M₁ : Type} [AddCommMonoid M₁] [Module R₁ M₁] /-- Two linear maps are equal if they are equal on basis vectors. -/ theorem ext {f₁ f₂ : M →ₛₗ[σ] M₁} (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ := by ext x rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum] simp only [map_sum, LinearMap.map_smulₛₗ, h] #align basis.ext Basis.ext /-- Two linear equivs are equal if they are equal on basis vectors. -/ theorem ext' {f₁ f₂ : M ≃ₛₗ[σ] M₁} (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ := by ext x rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum] simp only [map_sum, LinearEquiv.map_smulₛₗ, h] #align basis.ext' Basis.ext' /-- Two elements are equal iff their coordinates are equal. -/ theorem ext_elem_iff {x y : M} : x = y ↔ ∀ i, b.repr x i = b.repr y i := by simp only [← DFunLike.ext_iff, EmbeddingLike.apply_eq_iff_eq] #align basis.ext_elem_iff Basis.ext_elem_iff alias ⟨_, _root_.Basis.ext_elem⟩ := ext_elem_iff #align basis.ext_elem Basis.ext_elem theorem repr_eq_iff {b : Basis ι R M} {f : M →ₗ[R] ι →₀ R} : ↑b.repr = f ↔ ∀ i, f (b i) = Finsupp.single i 1 := ⟨fun h i => h ▸ b.repr_self i, fun h => b.ext fun i => (b.repr_self i).trans (h i).symm⟩ #align basis.repr_eq_iff Basis.repr_eq_iff theorem repr_eq_iff' {b : Basis ι R M} {f : M ≃ₗ[R] ι →₀ R} : b.repr = f ↔ ∀ i, f (b i) = Finsupp.single i 1 := ⟨fun h i => h ▸ b.repr_self i, fun h => b.ext' fun i => (b.repr_self i).trans (h i).symm⟩ #align basis.repr_eq_iff' Basis.repr_eq_iff' theorem apply_eq_iff {b : Basis ι R M} {x : M} {i : ι} : b i = x ↔ b.repr x = Finsupp.single i 1 := ⟨fun h => h ▸ b.repr_self i, fun h => b.repr.injective ((b.repr_self i).trans h.symm)⟩ #align basis.apply_eq_iff Basis.apply_eq_iff /-- An unbundled version of `repr_eq_iff` -/ theorem repr_apply_eq (f : M → ι → R) (hadd : ∀ x y, f (x + y) = f x + f y) (hsmul : ∀ (c : R) (x : M), f (c • x) = c • f x) (f_eq : ∀ i, f (b i) = Finsupp.single i 1) (x : M) (i : ι) : b.repr x i = f x i := by let f_i : M →ₗ[R] R := { toFun := fun x => f x i -- Porting note(#12129): additional beta reduction needed map_add' := fun _ _ => by beta_reduce; rw [hadd, Pi.add_apply] map_smul' := fun _ _ => by simp [hsmul, Pi.smul_apply] } have : Finsupp.lapply i ∘ₗ ↑b.repr = f_i := by refine b.ext fun j => ?_ show b.repr (b j) i = f (b j) i rw [b.repr_self, f_eq] calc b.repr x i = f_i x := by { rw [← this] rfl } _ = f x i := rfl #align basis.repr_apply_eq Basis.repr_apply_eq /-- Two bases are equal if they assign the same coordinates. -/ theorem eq_ofRepr_eq_repr {b₁ b₂ : Basis ι R M} (h : ∀ x i, b₁.repr x i = b₂.repr x i) : b₁ = b₂ := repr_injective <\| by ext; apply h #align basis.eq_of_repr_eq_repr Basis.eq_ofRepr_eq_repr /-- Two bases are equal if their basis vectors are the same. -/ @[ext] theorem eq_of_apply_eq {b₁ b₂ : Basis ι R M} : (∀ i, b₁ i = b₂ i) → b₁ = b₂ := DFunLike.ext _ _ #align basis.eq_of_apply_eq Basis.eq_of_apply_eq end Ext section Map variable (f : M ≃ₗ[R] M') /-- Apply the linear equivalence `f` to the basis vectors. -/ @[simps] protected def map : Basis ι R M' := ofRepr (f.symm.trans b.repr) #align basis.map Basis.map @[simp] theorem map_apply (i) : b.map f i = f (b i) := rfl #align basis.map_apply Basis.map_apply theorem coe_map : (b.map f : ι → M') = f ∘ b := rfl end Map section MapCoeffs variable {R' : Type} [Semiring R'] [Module R' M] (f : R ≃+* R') (h : ∀ (c) (x : M), f c • x = c • x) attribute [local instance] SMul.comp.isScalarTower /-- If `R` and `R'` are isomorphic rings that act identically on a module `M`, then a basis for `M` as `R`-module is also a basis for `M` as `R'`-module. See also `Basis.algebraMapCoeffs` for the case where `f` is equal to `algebraMap`. -/ @[simps (config := { simpRhs := true })] def mapCoeffs : Basis ι R' M := by letI : Module R' R := Module.compHom R (↑f.symm : R' →+* R) haveI : IsScalarTower R' R M := { smul_assoc := fun x y z => by -- Porting note: `dsimp [(· • ·)]` is unavailable because -- `HSMul.hsmul` becomes `SMul.smul`. change (f.symm x * y) • z = x • (y • z) rw [mul_smul, ← h, f.apply_symm_apply] } exact ofRepr <\| (b.repr.restrictScalars R').trans <\| Finsupp.mapRange.linearEquiv (Module.compHom.toLinearEquiv f.symm).symm #align basis.map_coeffs Basis.mapCoeffs theorem mapCoeffs_apply (i : ι) : b.mapCoeffs f h i = b i := apply_eq_iff.mpr <\| by -- Porting note: in Lean 3, these were automatically inferred from the definition of -- `mapCoeffs`. letI : Module R' R := Module.compHom R (↑f.symm : R' →+* R) haveI : IsScalarTower R' R M := { smul_assoc := fun x y z => by -- Porting note: `dsimp [(· • ·)]` is unavailable because -- `HSMul.hsmul` becomes `SMul.smul`. change (f.symm x * y) • z = x • (y • z) rw [mul_smul, ← h, f.apply_symm_apply] } simp #align basis.map_coeffs_apply Basis.mapCoeffs_apply @[simp] theorem coe_mapCoeffs : (b.mapCoeffs f h : ι → M) = b := funext <\| b.mapCoeffs_apply f h #align basis.coe_map_coeffs Basis.coe_mapCoeffs end MapCoeffs section Reindex variable (b' : Basis ι' R M') variable (e : ι ≃ ι') /-- `b.reindex (e : ι ≃ ι')` is a basis indexed by `ι'` -/ def reindex : Basis ι' R M := .ofRepr (b.repr.trans (Finsupp.domLCongr e)) #align basis.reindex Basis.reindex theorem reindex_apply (i' : ι') : b.reindex e i' = b (e.symm i') := show (b.repr.trans (Finsupp.domLCongr e)).symm (Finsupp.single i' 1) = b.repr.symm (Finsupp.single (e.symm i') 1) by rw [LinearEquiv.symm_trans_apply, Finsupp.domLCongr_symm, Finsupp.domLCongr_single] #align basis.reindex_apply Basis.reindex_apply @[simp] theorem coe_reindex : (b.reindex e : ι' → M) = b ∘ e.symm := funext (b.reindex_apply e) #align basis.coe_reindex Basis.coe_reindex theorem repr_reindex_apply (i' : ι') : (b.reindex e).repr x i' = b.repr x (e.symm i') := show (Finsupp.domLCongr e : _ ≃ₗ[R] _) (b.repr x) i' = _ by simp #align basis.repr_reindex_apply Basis.repr_reindex_apply @[simp] theorem repr_reindex : (b.reindex e).repr x = (b.repr x).mapDomain e := DFunLike.ext _ _ <\| by simp [repr_reindex_apply] #align basis.repr_reindex Basis.repr_reindex @[simp] theorem reindex_refl : b.reindex (Equiv.refl ι) = b := eq_of_apply_eq fun i => by simp #align basis.reindex_refl Basis.reindex_refl /-- `simp` can prove this as `Basis.coe_reindex` + `EquivLike.range_comp` -/ theorem range_reindex : Set.range (b.reindex e) = Set.range b := by simp [coe_reindex, range_comp] #align basis.range_reindex Basis.range_reindex @[simp] theorem sumCoords_reindex : (b.reindex e).sumCoords = b.sumCoords := by ext x simp only [coe_sumCoords, repr_reindex] exact Finsupp.sum_mapDomain_index (fun _ => rfl) fun _ _ _ => rfl #align basis.sum_coords_reindex Basis.sumCoords_reindex /-- `b.reindex_range` is a basis indexed by `range b`, the basis vectors themselves. -/ def reindexRange : Basis (range b) R M := haveI := Classical.dec (Nontrivial R) if h : Nontrivial R then letI := h b.reindex (Equiv.ofInjective b (Basis.injective b)) else letI : Subsingleton R := not_nontrivial_iff_subsingleton.mp h .ofRepr (Module.subsingletonEquiv R M (range b)) #align basis.reindex_range Basis.reindexRange theorem reindexRange_self (i : ι) (h := Set.mem_range_self i) : b.reindexRange ⟨b i, h⟩ = b i := by by_cases htr : Nontrivial R · letI := htr simp [htr, reindexRange, reindex_apply, Equiv.apply_ofInjective_symm b.injective, Subtype.coe_mk] · letI : Subsingleton R := not_nontrivial_iff_subsingleton.mp htr letI := Module.subsingleton R M simp [reindexRange, eq_iff_true_of_subsingleton] #align basis.reindex_range_self Basis.reindexRange_self theorem reindexRange_repr_self (i : ι) : b.reindexRange.repr (b i) = Finsupp.single ⟨b i, mem_range_self i⟩ 1 := calc b.reindexRange.repr (b i) = b.reindexRange.repr (b.reindexRange ⟨b i, mem_range_self i⟩) := congr_arg _ (b.reindexRange_self _ _).symm _ = Finsupp.single ⟨b i, mem_range_self i⟩ 1 := b.reindexRange.repr_self _ #align basis.reindex_range_repr_self Basis.reindexRange_repr_self @[simp] theorem reindexRange_apply (x : range b) : b.reindexRange x = x := by rcases x with ⟨bi, ⟨i, rfl⟩⟩ exact b.reindexRange_self i #align basis.reindex_range_apply Basis.reindexRange_apply theorem reindexRange_repr' (x : M) {bi : M} {i : ι} (h : b i = bi) : b.reindexRange.repr x ⟨bi, ⟨i, h⟩⟩ = b.repr x i := by nontriviality subst h apply (b.repr_apply_eq (fun x i => b.reindexRange.repr x ⟨b i, _⟩) _ _ _ x i).symm · intro x y ext i simp only [Pi.add_apply, LinearEquiv.map_add, Finsupp.coe_add] · intro c x ext i simp only [Pi.smul_apply, LinearEquiv.map_smul, Finsupp.coe_smul] · intro i ext j simp only [reindexRange_repr_self] apply Finsupp.single_apply_left (f := fun i => (⟨b i, _⟩ : Set.range b)) exact fun i j h => b.injective (Subtype.mk.inj h) #align basis.reindex_range_repr' Basis.reindexRange_repr' @[simp] theorem reindexRange_repr (x : M) (i : ι) (h := Set.mem_range_self i) : b.reindexRange.repr x ⟨b i, h⟩ = b.repr x i := b.reindexRange_repr' _ rfl #align basis.reindex_range_repr Basis.reindexRange_repr section Fintype variable [Fintype ι] [DecidableEq M] /-- `b.reindexFinsetRange` is a basis indexed by `Finset.univ.image b`, the finite set of basis vectors themselves. -/ def reindexFinsetRange : Basis (Finset.univ.image b) R M := b.reindexRange.reindex ((Equiv.refl M).subtypeEquiv (by simp)) #align basis.reindex_finset_range Basis.reindexFinsetRange theorem reindexFinsetRange_self (i : ι) (h := Finset.mem_image_of_mem b (Finset.mem_univ i)) : b.reindexFinsetRange ⟨b i, h⟩ = b i := by rw [reindexFinsetRange, reindex_apply, reindexRange_apply] rfl #align basis.reindex_finset_range_self Basis.reindexFinsetRange_self @[simp] theorem reindexFinsetRange_apply (x : Finset.univ.image b) : b.reindexFinsetRange x = x := by rcases x with ⟨bi, hbi⟩ rcases Finset.mem_image.mp hbi with ⟨i, -, rfl⟩ exact b.reindexFinsetRange_self i #align basis.reindex_finset_range_apply Basis.reindexFinsetRange_apply theorem reindexFinsetRange_repr_self (i : ι) : b.reindexFinsetRange.repr (b i) = Finsupp.single ⟨b i, Finset.mem_image_of_mem b (Finset.mem_univ i)⟩ 1 := by ext ⟨bi, hbi⟩ rw [reindexFinsetRange, repr_reindex, Finsupp.mapDomain_equiv_apply, reindexRange_repr_self] -- Porting note: replaced a `convert; refl` with `simp` simp [Finsupp.single_apply] #align basis.reindex_finset_range_repr_self Basis.reindexFinsetRange_repr_self @[simp] theorem reindexFinsetRange_repr (x : M) (i : ι) (h := Finset.mem_image_of_mem b (Finset.mem_univ i)) : b.reindexFinsetRange.repr x ⟨b i, h⟩ = b.repr x i := by simp [reindexFinsetRange] #align basis.reindex_finset_range_repr Basis.reindexFinsetRange_repr end Fintype end Reindex protected theorem linearIndependent : LinearIndependent R b := linearIndependent_iff.mpr fun l hl => calc l = b.repr (Finsupp.total _ _ _ b l) := (b.repr_total l).symm _ = 0 := by rw [hl, LinearEquiv.map_zero] #align basis.linear_independent Basis.linearIndependent protected theorem ne_zero [Nontrivial R] (i) : b i ≠ 0 := b.linearIndependent.ne_zero i #align basis.ne_zero Basis.ne_zero protected theorem mem_span (x : M) : x ∈ span R (range b) := span_mono (image_subset_range _ _) (mem_span_repr_support b x) #align basis.mem_span Basis.mem_span @[simp] protected theorem span_eq : span R (range b) = ⊤ := eq_top_iff.mpr fun x _ => b.mem_span x #align basis.span_eq Basis.span_eq theorem index_nonempty (b : Basis ι R M) [Nontrivial M] : Nonempty ι := by obtain ⟨x, y, ne⟩ : ∃ x y : M, x ≠ y := Nontrivial.exists_pair_ne obtain ⟨i, _⟩ := not_forall.mp (mt b.ext_elem_iff.2 ne) exact ⟨i⟩ #align basis.index_nonempty Basis.index_nonempty /-- If the submodule `P` has a basis, `x ∈ P` iff it is a linear combination of basis vectors. -/ theorem mem_submodule_iff {P : Submodule R M} (b : Basis ι R P) {x : M} : x ∈ P ↔ ∃ c : ι →₀ R, x = Finsupp.sum c fun i x => x • (b i : M) := by conv_lhs => rw [← P.range_subtype, ← Submodule.map_top, ← b.span_eq, Submodule.map_span, ← Set.range_comp, ← Finsupp.range_total] simp [@eq_comm _ x, Function.comp, Finsupp.total_apply] #align basis.mem_submodule_iff Basis.mem_submodule_iff section Constr variable (S : Type*) [Semiring S] [Module S M'] variable [SMulCommClass R S M'] /-- Construct a linear map given the value at the basis, called `Basis.constr b S f` where `b` is a basis, `f` is the value of the linear map over the elements of the basis, and `S` is an extra semiring (typically `S = R` or `S = ℕ`). This definition is parameterized over an extra `Semiring S`, such that `SMulCommClass R S M'` holds. If `R` is commutative, you can set `S := R`; if `R` is not commutative, you can recover an `AddEquiv` by setting `S := ℕ`. See library note [bundled maps over different rings]. -/ def constr : (ι → M') ≃ₗ[S] M →ₗ[R] M' where toFun f := (Finsupp.total M' M' R id).comp <\| Finsupp.lmapDomain R R f ∘ₗ ↑b.repr invFun f i := f (b i) left_inv f := by ext simp right_inv f := by refine b.ext fun i => ?_ simp map_add' f g := by refine b.ext fun i => ?_ simp map_smul' c f := by refine b.ext fun i => ?_ simp #align basis.constr Basis.constr theorem constr_def (f : ι → M') : constr (M' := M') b S f = Finsupp.total M' M' R id ∘ₗ Finsupp.lmapDomain R R f ∘ₗ ↑b.repr := rfl #align basis.constr_def Basis.constr_def theorem constr_apply (f : ι → M') (x : M) : constr (M' := M') b S f x = (b.repr x).sum fun b a => a • f b := by simp only [constr_def, LinearMap.comp_apply, Finsupp.lmapDomain_apply, Finsupp.total_apply] rw [Finsupp.sum_mapDomain_index] <;> simp [add_smul] #align basis.constr_apply Basis.constr_apply @[simp] theorem constr_basis (f : ι → M') (i : ι) : (constr (M' := M') b S f : M → M') (b i) = f i := by simp [Basis.constr_apply, b.repr_self] #align basis.constr_basis Basis.constr_basis theorem constr_eq {g : ι → M'} {f : M →ₗ[R] M'} (h : ∀ i, g i = f (b i)) : constr (M' := M') b S g = f := b.ext fun i => (b.constr_basis S g i).trans (h i) #align basis.constr_eq Basis.constr_eq theorem constr_self (f : M →ₗ[R] M') : (constr (M' := M') b S fun i => f (b i)) = f := b.constr_eq S fun _ => rfl #align basis.constr_self Basis.constr_self	Mathlib/LinearAlgebra/Basis.lean	647	651	theorem constr_range {f : ι → M'} : LinearMap.range (constr (M' := M') b S f) = span R (range f) := by	rw [b.constr_def S f, LinearMap.range_comp, LinearMap.range_comp, LinearEquiv.range, ← Finsupp.supported_univ, Finsupp.lmapDomain_supported, ← Set.image_univ, ← Finsupp.span_image_eq_map_total, Set.image_id]
/- Copyright (c) 2020 Anatole Dedecker. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anatole Dedecker -/ import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Analysis.Calculus.Deriv.Inv #align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # L'Hôpital's rule for 0/0 indeterminate forms In this file, we prove several forms of "L'Hôpital's rule" for computing 0/0 indeterminate forms. The proof of `HasDerivAt.lhopital_zero_right_on_Ioo` is based on the one given in the corresponding [Wikibooks](https://en.wikibooks.org/wiki/Calculus/L%27H%C3%B4pital%27s_Rule) chapter, and all other statements are derived from this one by composing by carefully chosen functions. Note that the filter `f'/g'` tends to isn't required to be one of `𝓝 a`, `atTop` or `atBot`. In fact, we give a slightly stronger statement by allowing it to be any filter on `ℝ`. Each statement is available in a `HasDerivAt` form and a `deriv` form, which is denoted by each statement being in either the `HasDerivAt` or the `deriv` namespace. ## Tags L'Hôpital's rule, L'Hopital's rule -/ open Filter Set open scoped Filter Topology Pointwise variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ} /-! ## Interval-based versions We start by proving statements where all conditions (derivability, `g' ≠ 0`) have to be satisfied on an explicitly-provided interval. -/ namespace HasDerivAt theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0)) (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) : Tendsto (fun x => f x / g x) (𝓝[>] a) l := by have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := fun x hx => Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2) have hg : ∀ x ∈ Ioo a b, g x ≠ 0 := by intro x hx h have : Tendsto g (𝓝[<] x) (𝓝 0) := by rw [← h, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1] exact ((hgg' x hx).continuousAt.continuousWithinAt.mono <\| sub x hx).tendsto obtain ⟨y, hyx, hy⟩ : ∃ c ∈ Ioo a x, g' c = 0 := exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <\| sub x hx hy exact hg' y (sub x hx hyx) hy have : ∀ x ∈ Ioo a b, ∃ c ∈ Ioo a x, f x * g' c = g x * f' c := by intro x hx rw [← sub_zero (f x), ← sub_zero (g x)] exact exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <\| sub x hx hy) (fun y hy => hff' y <\| sub x hx hy) hga hfa (tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).continuousAt.tendsto) (tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).continuousAt.tendsto) choose! c hc using this have : ∀ x ∈ Ioo a b, ((fun x' => f' x' / g' x') ∘ c) x = f x / g x := by intro x hx rcases hc x hx with ⟨h₁, h₂⟩ field_simp [hg x hx, hg' (c x) ((sub x hx) h₁)] simp only [h₂] rw [mul_comm] have cmp : ∀ x ∈ Ioo a b, a < c x ∧ c x < x := fun x hx => (hc x hx).1 rw [← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab] apply tendsto_nhdsWithin_congr this apply hdiv.comp refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ (tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) ?_ ?_) ?_ all_goals apply eventually_nhdsWithin_of_forall intro x hx have := cmp x hx try simp linarith [this] #align has_deriv_at.lhopital_zero_right_on_Ioo HasDerivAt.lhopital_zero_right_on_Ioo theorem lhopital_zero_right_on_Ico (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ico a b)) (hcg : ContinuousOn g (Ico a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfa : f a = 0) (hga : g a = 0) (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) : Tendsto (fun x => f x / g x) (𝓝[>] a) l := by refine lhopital_zero_right_on_Ioo hab hff' hgg' hg' ?_ ?_ hdiv · rw [← hfa, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab] exact ((hcf a <\| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto · rw [← hga, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab] exact ((hcg a <\| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto #align has_deriv_at.lhopital_zero_right_on_Ico HasDerivAt.lhopital_zero_right_on_Ico theorem lhopital_zero_left_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfb : Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Tendsto g (𝓝[<] b) (𝓝 0)) (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) : Tendsto (fun x => f x / g x) (𝓝[<] b) l := by -- Here, we essentially compose by `Neg.neg`. The following is mostly technical details. have hdnf : ∀ x ∈ -Ioo a b, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx => comp x (hff' (-x) hx) (hasDerivAt_neg x) have hdng : ∀ x ∈ -Ioo a b, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx => comp x (hgg' (-x) hx) (hasDerivAt_neg x) rw [preimage_neg_Ioo] at hdnf rw [preimage_neg_Ioo] at hdng have := lhopital_zero_right_on_Ioo (neg_lt_neg hab) hdnf hdng (by intro x hx h apply hg' _ (by rw [← preimage_neg_Ioo] at hx; exact hx) rwa [mul_comm, ← neg_eq_neg_one_mul, neg_eq_zero] at h) (hfb.comp tendsto_neg_nhdsWithin_Ioi_neg) (hgb.comp tendsto_neg_nhdsWithin_Ioi_neg) (by simp only [neg_div_neg_eq, mul_one, mul_neg] exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_nhdsWithin_Ioi_neg)) have := this.comp tendsto_neg_nhdsWithin_Iio unfold Function.comp at this simpa only [neg_neg] #align has_deriv_at.lhopital_zero_left_on_Ioo HasDerivAt.lhopital_zero_left_on_Ioo theorem lhopital_zero_left_on_Ioc (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ioc a b)) (hcg : ContinuousOn g (Ioc a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfb : f b = 0) (hgb : g b = 0) (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) : Tendsto (fun x => f x / g x) (𝓝[<] b) l := by refine lhopital_zero_left_on_Ioo hab hff' hgg' hg' ?_ ?_ hdiv · rw [← hfb, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hab] exact ((hcf b <\| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto · rw [← hgb, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hab] exact ((hcg b <\| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto #align has_deriv_at.lhopital_zero_left_on_Ioc HasDerivAt.lhopital_zero_left_on_Ioc	Mathlib/Analysis/Calculus/LHopital.lean	144	174	theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0) (hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l := by	obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0, ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _), lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩ have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2) have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx => comp x (hff' x⁻¹ <\| fact2 x hx) (hasDerivAt_inv <\| fact1 x hx) have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx => comp x (hgg' x⁻¹ <\| fact2 x hx) (hasDerivAt_inv <\| fact1 x hx) have := lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng (by intro x hx refine mul_ne_zero ?_ (neg_ne_zero.mpr <\| inv_ne_zero <\| pow_ne_zero _ <\| fact1 x hx) exact hg' _ (fact2 x hx)) (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop) (by refine (tendsto_congr' ?_).mp (hdiv.comp tendsto_inv_zero_atTop) rw [eventuallyEq_iff_exists_mem] use Ioi 0, self_mem_nhdsWithin intro x hx unfold Function.comp simp only erw [mul_div_mul_right] exact neg_ne_zero.mpr (inv_ne_zero <\| pow_ne_zero _ <\| ne_of_gt hx)) have := this.comp tendsto_inv_atTop_zero' unfold Function.comp at this simpa only [inv_inv]
/- Copyright (c) 2022 Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies -/ import Mathlib.Order.PropInstances #align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4" /-! # Heyting algebras This file defines Heyting, co-Heyting and bi-Heyting algebras. A Heyting algebra is a bounded distributive lattice with an implication operation `⇨` such that `a ≤ b ⇨ c ↔ a ⊓ b ≤ c`. It also comes with a pseudo-complement `ᶜ`, such that `aᶜ = a ⇨ ⊥`. Co-Heyting algebras are dual to Heyting algebras. They have a difference `\` and a negation `￢` such that `a \ b ≤ c ↔ a ≤ b ⊔ c` and `￢a = ⊤ \ a`. Bi-Heyting algebras are Heyting algebras that are also co-Heyting algebras. From a logic standpoint, Heyting algebras precisely model intuitionistic logic, whereas boolean algebras model classical logic. Heyting algebras are the order theoretic equivalent of cartesian-closed categories. ## Main declarations * `GeneralizedHeytingAlgebra`: Heyting algebra without a top element (nor negation). * `GeneralizedCoheytingAlgebra`: Co-Heyting algebra without a bottom element (nor complement). * `HeytingAlgebra`: Heyting algebra. * `CoheytingAlgebra`: Co-Heyting algebra. * `BiheytingAlgebra`: bi-Heyting algebra. ## References * [Francis Borceux, Handbook of Categorical Algebra III][borceux-vol3] ## Tags Heyting, Brouwer, algebra, implication, negation, intuitionistic -/ open Function OrderDual universe u variable {ι α β : Type} /-! ### Notation -/ section variable (α β) instance Prod.instHImp [HImp α] [HImp β] : HImp (α × β) := ⟨fun a b => (a.1 ⇨ b.1, a.2 ⇨ b.2)⟩ instance Prod.instHNot [HNot α] [HNot β] : HNot (α × β) := ⟨fun a => (￢a.1, ￢a.2)⟩ instance Prod.instSDiff [SDiff α] [SDiff β] : SDiff (α × β) := ⟨fun a b => (a.1 \ b.1, a.2 \ b.2)⟩ instance Prod.instHasCompl [HasCompl α] [HasCompl β] : HasCompl (α × β) := ⟨fun a => (a.1ᶜ, a.2ᶜ)⟩ end @[simp] theorem fst_himp [HImp α] [HImp β] (a b : α × β) : (a ⇨ b).1 = a.1 ⇨ b.1 := rfl #align fst_himp fst_himp @[simp] theorem snd_himp [HImp α] [HImp β] (a b : α × β) : (a ⇨ b).2 = a.2 ⇨ b.2 := rfl #align snd_himp snd_himp @[simp] theorem fst_hnot [HNot α] [HNot β] (a : α × β) : (￢a).1 = ￢a.1 := rfl #align fst_hnot fst_hnot @[simp] theorem snd_hnot [HNot α] [HNot β] (a : α × β) : (￢a).2 = ￢a.2 := rfl #align snd_hnot snd_hnot @[simp] theorem fst_sdiff [SDiff α] [SDiff β] (a b : α × β) : (a \ b).1 = a.1 \ b.1 := rfl #align fst_sdiff fst_sdiff @[simp] theorem snd_sdiff [SDiff α] [SDiff β] (a b : α × β) : (a \ b).2 = a.2 \ b.2 := rfl #align snd_sdiff snd_sdiff @[simp] theorem fst_compl [HasCompl α] [HasCompl β] (a : α × β) : aᶜ.1 = a.1ᶜ := rfl #align fst_compl fst_compl @[simp] theorem snd_compl [HasCompl α] [HasCompl β] (a : α × β) : aᶜ.2 = a.2ᶜ := rfl #align snd_compl snd_compl namespace Pi variable {π : ι → Type} instance [∀ i, HImp (π i)] : HImp (∀ i, π i) := ⟨fun a b i => a i ⇨ b i⟩ instance [∀ i, HNot (π i)] : HNot (∀ i, π i) := ⟨fun a i => ￢a i⟩ theorem himp_def [∀ i, HImp (π i)] (a b : ∀ i, π i) : a ⇨ b = fun i => a i ⇨ b i := rfl #align pi.himp_def Pi.himp_def theorem hnot_def [∀ i, HNot (π i)] (a : ∀ i, π i) : ￢a = fun i => ￢a i := rfl #align pi.hnot_def Pi.hnot_def @[simp] theorem himp_apply [∀ i, HImp (π i)] (a b : ∀ i, π i) (i : ι) : (a ⇨ b) i = a i ⇨ b i := rfl #align pi.himp_apply Pi.himp_apply @[simp] theorem hnot_apply [∀ i, HNot (π i)] (a : ∀ i, π i) (i : ι) : (￢a) i = ￢a i := rfl #align pi.hnot_apply Pi.hnot_apply end Pi /-- A generalized Heyting algebra is a lattice with an additional binary operation `⇨` called Heyting implication such that `a ⇨` is right adjoint to `a ⊓`. This generalizes `HeytingAlgebra` by not requiring a bottom element. -/ class GeneralizedHeytingAlgebra (α : Type) extends Lattice α, OrderTop α, HImp α where /-- `a ⇨` is right adjoint to `a ⊓` -/ le_himp_iff (a b c : α) : a ≤ b ⇨ c ↔ a ⊓ b ≤ c #align generalized_heyting_algebra GeneralizedHeytingAlgebra #align generalized_heyting_algebra.to_order_top GeneralizedHeytingAlgebra.toOrderTop /-- A generalized co-Heyting algebra is a lattice with an additional binary difference operation `\` such that `\ a` is right adjoint to `⊔ a`. This generalizes `CoheytingAlgebra` by not requiring a top element. -/ class GeneralizedCoheytingAlgebra (α : Type) extends Lattice α, OrderBot α, SDiff α where /-- `\ a` is right adjoint to `⊔ a` -/ sdiff_le_iff (a b c : α) : a \ b ≤ c ↔ a ≤ b ⊔ c #align generalized_coheyting_algebra GeneralizedCoheytingAlgebra #align generalized_coheyting_algebra.to_order_bot GeneralizedCoheytingAlgebra.toOrderBot /-- A Heyting algebra is a bounded lattice with an additional binary operation `⇨` called Heyting implication such that `a ⇨` is right adjoint to `a ⊓`. -/ class HeytingAlgebra (α : Type) extends GeneralizedHeytingAlgebra α, OrderBot α, HasCompl α where /-- `a ⇨` is right adjoint to `a ⊓` -/ himp_bot (a : α) : a ⇨ ⊥ = aᶜ #align heyting_algebra HeytingAlgebra /-- A co-Heyting algebra is a bounded lattice with an additional binary difference operation `\` such that `\ a` is right adjoint to `⊔ a`. -/ class CoheytingAlgebra (α : Type) extends GeneralizedCoheytingAlgebra α, OrderTop α, HNot α where /-- `⊤ \ a` is `￢a` -/ top_sdiff (a : α) : ⊤ \ a = ￢a #align coheyting_algebra CoheytingAlgebra /-- A bi-Heyting algebra is a Heyting algebra that is also a co-Heyting algebra. -/ class BiheytingAlgebra (α : Type) extends HeytingAlgebra α, SDiff α, HNot α where /-- `\ a` is right adjoint to `⊔ a` -/ sdiff_le_iff (a b c : α) : a \ b ≤ c ↔ a ≤ b ⊔ c /-- `⊤ \ a` is `￢a` -/ top_sdiff (a : α) : ⊤ \ a = ￢a #align biheyting_algebra BiheytingAlgebra -- See note [lower instance priority] attribute [instance 100] GeneralizedHeytingAlgebra.toOrderTop attribute [instance 100] GeneralizedCoheytingAlgebra.toOrderBot -- See note [lower instance priority] instance (priority := 100) HeytingAlgebra.toBoundedOrder [HeytingAlgebra α] : BoundedOrder α := { bot_le := ‹HeytingAlgebra α›.bot_le } --#align heyting_algebra.to_bounded_order HeytingAlgebra.toBoundedOrder -- See note [lower instance priority] instance (priority := 100) CoheytingAlgebra.toBoundedOrder [CoheytingAlgebra α] : BoundedOrder α := { ‹CoheytingAlgebra α› with } #align coheyting_algebra.to_bounded_order CoheytingAlgebra.toBoundedOrder -- See note [lower instance priority] instance (priority := 100) BiheytingAlgebra.toCoheytingAlgebra [BiheytingAlgebra α] : CoheytingAlgebra α := { ‹BiheytingAlgebra α› with } #align biheyting_algebra.to_coheyting_algebra BiheytingAlgebra.toCoheytingAlgebra -- See note [reducible non-instances] /-- Construct a Heyting algebra from the lattice structure and Heyting implication alone. -/ abbrev HeytingAlgebra.ofHImp [DistribLattice α] [BoundedOrder α] (himp : α → α → α) (le_himp_iff : ∀ a b c, a ≤ himp b c ↔ a ⊓ b ≤ c) : HeytingAlgebra α := { ‹DistribLattice α›, ‹BoundedOrder α› with himp, compl := fun a => himp a ⊥, le_himp_iff, himp_bot := fun a => rfl } #align heyting_algebra.of_himp HeytingAlgebra.ofHImp -- See note [reducible non-instances] /-- Construct a Heyting algebra from the lattice structure and complement operator alone. -/ abbrev HeytingAlgebra.ofCompl [DistribLattice α] [BoundedOrder α] (compl : α → α) (le_himp_iff : ∀ a b c, a ≤ compl b ⊔ c ↔ a ⊓ b ≤ c) : HeytingAlgebra α where himp := (compl · ⊔ ·) compl := compl le_himp_iff := le_himp_iff himp_bot _ := sup_bot_eq _ #align heyting_algebra.of_compl HeytingAlgebra.ofCompl -- See note [reducible non-instances] /-- Construct a co-Heyting algebra from the lattice structure and the difference alone. -/ abbrev CoheytingAlgebra.ofSDiff [DistribLattice α] [BoundedOrder α] (sdiff : α → α → α) (sdiff_le_iff : ∀ a b c, sdiff a b ≤ c ↔ a ≤ b ⊔ c) : CoheytingAlgebra α := { ‹DistribLattice α›, ‹BoundedOrder α› with sdiff, hnot := fun a => sdiff ⊤ a, sdiff_le_iff, top_sdiff := fun a => rfl } #align coheyting_algebra.of_sdiff CoheytingAlgebra.ofSDiff -- See note [reducible non-instances] /-- Construct a co-Heyting algebra from the difference and Heyting negation alone. -/ abbrev CoheytingAlgebra.ofHNot [DistribLattice α] [BoundedOrder α] (hnot : α → α) (sdiff_le_iff : ∀ a b c, a ⊓ hnot b ≤ c ↔ a ≤ b ⊔ c) : CoheytingAlgebra α where sdiff a b := a ⊓ hnot b hnot := hnot sdiff_le_iff := sdiff_le_iff top_sdiff _ := top_inf_eq _ #align coheyting_algebra.of_hnot CoheytingAlgebra.ofHNot /-! In this section, we'll give interpretations of these results in the Heyting algebra model of intuitionistic logic,- where `≤` can be interpreted as "validates", `⇨` as "implies", `⊓` as "and", `⊔` as "or", `⊥` as "false" and `⊤` as "true". Note that we confuse `→` and `⊢` because those are the same in this logic. See also `Prop.heytingAlgebra`. -/ section GeneralizedHeytingAlgebra variable [GeneralizedHeytingAlgebra α] {a b c d : α} /-- `p → q → r ↔ p ∧ q → r` -/ @[simp] theorem le_himp_iff : a ≤ b ⇨ c ↔ a ⊓ b ≤ c := GeneralizedHeytingAlgebra.le_himp_iff _ _ _ #align le_himp_iff le_himp_iff /-- `p → q → r ↔ q ∧ p → r` -/ theorem le_himp_iff' : a ≤ b ⇨ c ↔ b ⊓ a ≤ c := by rw [le_himp_iff, inf_comm] #align le_himp_iff' le_himp_iff' /-- `p → q → r ↔ q → p → r` -/ theorem le_himp_comm : a ≤ b ⇨ c ↔ b ≤ a ⇨ c := by rw [le_himp_iff, le_himp_iff'] #align le_himp_comm le_himp_comm /-- `p → q → p` -/ theorem le_himp : a ≤ b ⇨ a := le_himp_iff.2 inf_le_left #align le_himp le_himp /-- `p → p → q ↔ p → q` -/ theorem le_himp_iff_left : a ≤ a ⇨ b ↔ a ≤ b := by rw [le_himp_iff, inf_idem] #align le_himp_iff_left le_himp_iff_left /-- `p → p` -/ @[simp] theorem himp_self : a ⇨ a = ⊤ := top_le_iff.1 <\| le_himp_iff.2 inf_le_right #align himp_self himp_self /-- `(p → q) ∧ p → q` -/ theorem himp_inf_le : (a ⇨ b) ⊓ a ≤ b := le_himp_iff.1 le_rfl #align himp_inf_le himp_inf_le /-- `p ∧ (p → q) → q` -/ theorem inf_himp_le : a ⊓ (a ⇨ b) ≤ b := by rw [inf_comm, ← le_himp_iff] #align inf_himp_le inf_himp_le /-- `p ∧ (p → q) ↔ p ∧ q` -/ @[simp] theorem inf_himp (a b : α) : a ⊓ (a ⇨ b) = a ⊓ b := le_antisymm (le_inf inf_le_left <\| by rw [inf_comm, ← le_himp_iff]) <\| inf_le_inf_left _ le_himp #align inf_himp inf_himp /-- `(p → q) ∧ p ↔ q ∧ p` -/ @[simp] theorem himp_inf_self (a b : α) : (a ⇨ b) ⊓ a = b ⊓ a := by rw [inf_comm, inf_himp, inf_comm] #align himp_inf_self himp_inf_self /-- The deduction theorem* in the Heyting algebra model of intuitionistic logic: an implication holds iff the conclusion follows from the hypothesis. -/ @[simp] theorem himp_eq_top_iff : a ⇨ b = ⊤ ↔ a ≤ b := by rw [← top_le_iff, le_himp_iff, top_inf_eq] #align himp_eq_top_iff himp_eq_top_iff /-- `p → true`, `true → p ↔ p` -/ @[simp] theorem himp_top : a ⇨ ⊤ = ⊤ := himp_eq_top_iff.2 le_top #align himp_top himp_top @[simp] theorem top_himp : ⊤ ⇨ a = a := eq_of_forall_le_iff fun b => by rw [le_himp_iff, inf_top_eq] #align top_himp top_himp /-- `p → q → r ↔ p ∧ q → r` -/ theorem himp_himp (a b c : α) : a ⇨ b ⇨ c = a ⊓ b ⇨ c := eq_of_forall_le_iff fun d => by simp_rw [le_himp_iff, inf_assoc] #align himp_himp himp_himp /-- `(q → r) → (p → q) → q → r` -/ theorem himp_le_himp_himp_himp : b ⇨ c ≤ (a ⇨ b) ⇨ a ⇨ c := by rw [le_himp_iff, le_himp_iff, inf_assoc, himp_inf_self, ← inf_assoc, himp_inf_self, inf_assoc] exact inf_le_left #align himp_le_himp_himp_himp himp_le_himp_himp_himp @[simp] theorem himp_inf_himp_inf_le : (b ⇨ c) ⊓ (a ⇨ b) ⊓ a ≤ c := by simpa using @himp_le_himp_himp_himp /-- `p → q → r ↔ q → p → r` -/ theorem himp_left_comm (a b c : α) : a ⇨ b ⇨ c = b ⇨ a ⇨ c := by simp_rw [himp_himp, inf_comm] #align himp_left_comm himp_left_comm @[simp] theorem himp_idem : b ⇨ b ⇨ a = b ⇨ a := by rw [himp_himp, inf_idem] #align himp_idem himp_idem theorem himp_inf_distrib (a b c : α) : a ⇨ b ⊓ c = (a ⇨ b) ⊓ (a ⇨ c) := eq_of_forall_le_iff fun d => by simp_rw [le_himp_iff, le_inf_iff, le_himp_iff] #align himp_inf_distrib himp_inf_distrib theorem sup_himp_distrib (a b c : α) : a ⊔ b ⇨ c = (a ⇨ c) ⊓ (b ⇨ c) := eq_of_forall_le_iff fun d => by rw [le_inf_iff, le_himp_comm, sup_le_iff] simp_rw [le_himp_comm] #align sup_himp_distrib sup_himp_distrib theorem himp_le_himp_left (h : a ≤ b) : c ⇨ a ≤ c ⇨ b := le_himp_iff.2 <\| himp_inf_le.trans h #align himp_le_himp_left himp_le_himp_left theorem himp_le_himp_right (h : a ≤ b) : b ⇨ c ≤ a ⇨ c := le_himp_iff.2 <\| (inf_le_inf_left _ h).trans himp_inf_le #align himp_le_himp_right himp_le_himp_right theorem himp_le_himp (hab : a ≤ b) (hcd : c ≤ d) : b ⇨ c ≤ a ⇨ d := (himp_le_himp_right hab).trans <\| himp_le_himp_left hcd #align himp_le_himp himp_le_himp @[simp] theorem sup_himp_self_left (a b : α) : a ⊔ b ⇨ a = b ⇨ a := by rw [sup_himp_distrib, himp_self, top_inf_eq] #align sup_himp_self_left sup_himp_self_left @[simp] theorem sup_himp_self_right (a b : α) : a ⊔ b ⇨ b = a ⇨ b := by rw [sup_himp_distrib, himp_self, inf_top_eq] #align sup_himp_self_right sup_himp_self_right theorem Codisjoint.himp_eq_right (h : Codisjoint a b) : b ⇨ a = a := by conv_rhs => rw [← @top_himp _ _ a] rw [← h.eq_top, sup_himp_self_left] #align codisjoint.himp_eq_right Codisjoint.himp_eq_right theorem Codisjoint.himp_eq_left (h : Codisjoint a b) : a ⇨ b = b := h.symm.himp_eq_right #align codisjoint.himp_eq_left Codisjoint.himp_eq_left theorem Codisjoint.himp_inf_cancel_right (h : Codisjoint a b) : a ⇨ a ⊓ b = b := by rw [himp_inf_distrib, himp_self, top_inf_eq, h.himp_eq_left] #align codisjoint.himp_inf_cancel_right Codisjoint.himp_inf_cancel_right theorem Codisjoint.himp_inf_cancel_left (h : Codisjoint a b) : b ⇨ a ⊓ b = a := by rw [himp_inf_distrib, himp_self, inf_top_eq, h.himp_eq_right] #align codisjoint.himp_inf_cancel_left Codisjoint.himp_inf_cancel_left /-- See `himp_le` for a stronger version in Boolean algebras. -/ theorem Codisjoint.himp_le_of_right_le (hac : Codisjoint a c) (hba : b ≤ a) : c ⇨ b ≤ a := (himp_le_himp_left hba).trans_eq hac.himp_eq_right #align codisjoint.himp_le_of_right_le Codisjoint.himp_le_of_right_le theorem le_himp_himp : a ≤ (a ⇨ b) ⇨ b := le_himp_iff.2 inf_himp_le #align le_himp_himp le_himp_himp @[simp] lemma himp_eq_himp_iff : b ⇨ a = a ⇨ b ↔ a = b := by simp [le_antisymm_iff] lemma himp_ne_himp_iff : b ⇨ a ≠ a ⇨ b ↔ a ≠ b := himp_eq_himp_iff.not theorem himp_triangle (a b c : α) : (a ⇨ b) ⊓ (b ⇨ c) ≤ a ⇨ c := by rw [le_himp_iff, inf_right_comm, ← le_himp_iff] exact himp_inf_le.trans le_himp_himp #align himp_triangle himp_triangle theorem himp_inf_himp_cancel (hba : b ≤ a) (hcb : c ≤ b) : (a ⇨ b) ⊓ (b ⇨ c) = a ⇨ c := (himp_triangle _ _ _).antisymm <\| le_inf (himp_le_himp_left hcb) (himp_le_himp_right hba) #align himp_inf_himp_cancel himp_inf_himp_cancel -- See note [lower instance priority] instance (priority := 100) GeneralizedHeytingAlgebra.toDistribLattice : DistribLattice α := DistribLattice.ofInfSupLe fun a b c => by simp_rw [inf_comm a, ← le_himp_iff, sup_le_iff, le_himp_iff, ← sup_le_iff]; rfl #align generalized_heyting_algebra.to_distrib_lattice GeneralizedHeytingAlgebra.toDistribLattice instance OrderDual.instGeneralizedCoheytingAlgebra : GeneralizedCoheytingAlgebra αᵒᵈ where sdiff a b := toDual (ofDual b ⇨ ofDual a) sdiff_le_iff a b c := by rw [sup_comm]; exact le_himp_iff instance Prod.instGeneralizedHeytingAlgebra [GeneralizedHeytingAlgebra β] : GeneralizedHeytingAlgebra (α × β) where le_himp_iff _ _ _ := and_congr le_himp_iff le_himp_iff #align prod.generalized_heyting_algebra Prod.instGeneralizedHeytingAlgebra instance Pi.instGeneralizedHeytingAlgebra {α : ι → Type*} [∀ i, GeneralizedHeytingAlgebra (α i)] : GeneralizedHeytingAlgebra (∀ i, α i) where le_himp_iff i := by simp [le_def] #align pi.generalized_heyting_algebra Pi.instGeneralizedHeytingAlgebra end GeneralizedHeytingAlgebra section GeneralizedCoheytingAlgebra variable [GeneralizedCoheytingAlgebra α] {a b c d : α} @[simp] theorem sdiff_le_iff : a \ b ≤ c ↔ a ≤ b ⊔ c := GeneralizedCoheytingAlgebra.sdiff_le_iff _ _ _ #align sdiff_le_iff sdiff_le_iff theorem sdiff_le_iff' : a \ b ≤ c ↔ a ≤ c ⊔ b := by rw [sdiff_le_iff, sup_comm] #align sdiff_le_iff' sdiff_le_iff' theorem sdiff_le_comm : a \ b ≤ c ↔ a \ c ≤ b := by rw [sdiff_le_iff, sdiff_le_iff'] #align sdiff_le_comm sdiff_le_comm theorem sdiff_le : a \ b ≤ a := sdiff_le_iff.2 le_sup_right #align sdiff_le sdiff_le theorem Disjoint.disjoint_sdiff_left (h : Disjoint a b) : Disjoint (a \ c) b := h.mono_left sdiff_le #align disjoint.disjoint_sdiff_left Disjoint.disjoint_sdiff_left theorem Disjoint.disjoint_sdiff_right (h : Disjoint a b) : Disjoint a (b \ c) := h.mono_right sdiff_le #align disjoint.disjoint_sdiff_right Disjoint.disjoint_sdiff_right theorem sdiff_le_iff_left : a \ b ≤ b ↔ a ≤ b := by rw [sdiff_le_iff, sup_idem] #align sdiff_le_iff_left sdiff_le_iff_left @[simp] theorem sdiff_self : a \ a = ⊥ := le_bot_iff.1 <\| sdiff_le_iff.2 le_sup_left #align sdiff_self sdiff_self theorem le_sup_sdiff : a ≤ b ⊔ a \ b := sdiff_le_iff.1 le_rfl #align le_sup_sdiff le_sup_sdiff theorem le_sdiff_sup : a ≤ a \ b ⊔ b := by rw [sup_comm, ← sdiff_le_iff] #align le_sdiff_sup le_sdiff_sup theorem sup_sdiff_left : a ⊔ a \ b = a := sup_of_le_left sdiff_le #align sup_sdiff_left sup_sdiff_left theorem sup_sdiff_right : a \ b ⊔ a = a := sup_of_le_right sdiff_le #align sup_sdiff_right sup_sdiff_right theorem inf_sdiff_left : a \ b ⊓ a = a \ b := inf_of_le_left sdiff_le #align inf_sdiff_left inf_sdiff_left theorem inf_sdiff_right : a ⊓ a \ b = a \ b := inf_of_le_right sdiff_le #align inf_sdiff_right inf_sdiff_right @[simp] theorem sup_sdiff_self (a b : α) : a ⊔ b \ a = a ⊔ b := le_antisymm (sup_le_sup_left sdiff_le _) (sup_le le_sup_left le_sup_sdiff) #align sup_sdiff_self sup_sdiff_self @[simp] theorem sdiff_sup_self (a b : α) : b \ a ⊔ a = b ⊔ a := by rw [sup_comm, sup_sdiff_self, sup_comm] #align sdiff_sup_self sdiff_sup_self alias sup_sdiff_self_left := sdiff_sup_self #align sup_sdiff_self_left sup_sdiff_self_left alias sup_sdiff_self_right := sup_sdiff_self #align sup_sdiff_self_right sup_sdiff_self_right theorem sup_sdiff_eq_sup (h : c ≤ a) : a ⊔ b \ c = a ⊔ b := sup_congr_left (sdiff_le.trans le_sup_right) <\| le_sup_sdiff.trans <\| sup_le_sup_right h _ #align sup_sdiff_eq_sup sup_sdiff_eq_sup -- cf. `Set.union_diff_cancel'` theorem sup_sdiff_cancel' (hab : a ≤ b) (hbc : b ≤ c) : b ⊔ c \ a = c := by rw [sup_sdiff_eq_sup hab, sup_of_le_right hbc] #align sup_sdiff_cancel' sup_sdiff_cancel' theorem sup_sdiff_cancel_right (h : a ≤ b) : a ⊔ b \ a = b := sup_sdiff_cancel' le_rfl h #align sup_sdiff_cancel_right sup_sdiff_cancel_right	Mathlib/Order/Heyting/Basic.lean	521	521	theorem sdiff_sup_cancel (h : b ≤ a) : a \ b ⊔ b = a := by	rw [sup_comm, sup_sdiff_cancel_right h]
/- Copyright (c) 2021 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Batteries.Data.UnionFind.Basic namespace Batteries.UnionFind @[simp] theorem arr_empty : empty.arr = #[] := rfl @[simp] theorem parent_empty : empty.parent a = a := rfl @[simp] theorem rank_empty : empty.rank a = 0 := rfl @[simp] theorem rootD_empty : empty.rootD a = a := rfl @[simp] theorem arr_push {m : UnionFind} : m.push.arr = m.arr.push ⟨m.arr.size, 0⟩ := rfl @[simp] theorem parentD_push {arr : Array UFNode} : parentD (arr.push ⟨arr.size, 0⟩) a = parentD arr a := by simp [parentD]; split <;> split <;> try simp [Array.get_push, ] · next h1 h2 => simp [Nat.lt_succ] at h1 h2 exact Nat.le_antisymm h2 h1 · next h1 h2 => cases h1 (Nat.lt_succ_of_lt h2) @[simp] theorem parent_push {m : UnionFind} : m.push.parent a = m.parent a := by simp [parent] @[simp] theorem rankD_push {arr : Array UFNode} : rankD (arr.push ⟨arr.size, 0⟩) a = rankD arr a := by simp [rankD]; split <;> split <;> try simp [Array.get_push, ] next h1 h2 => cases h1 (Nat.lt_succ_of_lt h2) @[simp] theorem rank_push {m : UnionFind} : m.push.rank a = m.rank a := by simp [rank] @[simp] theorem rankMax_push {m : UnionFind} : m.push.rankMax = m.rankMax := by simp [rankMax] @[simp] theorem root_push {self : UnionFind} : self.push.rootD x = self.rootD x := rootD_ext fun _ => parent_push @[simp] theorem arr_link : (link self x y yroot).arr = linkAux self.arr x y := rfl theorem parentD_linkAux {self} {x y : Fin self.size} : parentD (linkAux self x y) i = if x.1 = y then parentD self i else if (self.get y).rank < (self.get x).rank then if y = i then x else parentD self i else if x = i then y else parentD self i := by dsimp only [linkAux]; split <;> [rfl; split] <;> [rw [parentD_set]; split] <;> rw [parentD_set] split <;> [(subst i; rwa [if_neg, parentD_eq]); rw [parentD_set]] theorem parent_link {self} {x y : Fin self.size} (yroot) {i} : (link self x y yroot).parent i = if x.1 = y then self.parent i else if self.rank y < self.rank x then if y = i then x else self.parent i else if x = i then y else self.parent i := by simp [rankD_eq]; exact parentD_linkAux theorem root_link {self : UnionFind} {x y : Fin self.size} (xroot : self.parent x = x) (yroot : self.parent y = y) : ∃ r, (r = x ∨ r = y) ∧ ∀ i, (link self x y yroot).rootD i = if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i := by if h : x.1 = y then refine ⟨x, .inl rfl, fun i => ?_⟩ rw [rootD_ext (m2 := self) (fun _ => by rw [parent_link, if_pos h])] split <;> [obtain _ \| _ := ‹_› <;> simp []; rfl] else have {x y : Fin self.size} (xroot : self.parent x = x) (yroot : self.parent y = y) {m : UnionFind} (hm : ∀ i, m.parent i = if y = i then x.1 else self.parent i) : ∃ r, (r = x ∨ r = y) ∧ ∀ i, m.rootD i = if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i := by let rec go (i) : m.rootD i = if self.rootD i = x ∨ self.rootD i = y then x.1 else self.rootD i := by if h : m.parent i = i then rw [rootD_eq_self.2 h]; rw [hm i] at h; split at h · rw [if_pos, h]; simp [← h, rootD_eq_self, xroot] · rw [rootD_eq_self.2 ‹_›]; split <;> [skip; rfl] next h' => exact h'.resolve_right (Ne.symm ‹_›) else have _ := Nat.sub_lt_sub_left (m.lt_rankMax i) (m.rank_lt h) rw [← rootD_parent, go (m.parent i)] rw [hm i]; split <;> [subst i; rw [rootD_parent]] simp [rootD_eq_self.2 xroot, rootD_eq_self.2 yroot] termination_by m.rankMax - m.rank i exact ⟨x, .inl rfl, go⟩ if hr : self.rank y < self.rank x then exact this xroot yroot fun i => by simp [parent_link, h, hr] else simpa (config := {singlePass := true}) [or_comm] using this yroot xroot fun i => by simp [parent_link, h, hr] nonrec theorem Equiv.rfl : Equiv self a a := rfl theorem Equiv.symm : Equiv self a b → Equiv self b a := .symm theorem Equiv.trans : Equiv self a b → Equiv self b c → Equiv self a c := .trans @[simp] theorem equiv_empty : Equiv empty a b ↔ a = b := by simp [Equiv] @[simp] theorem equiv_push : Equiv self.push a b ↔ Equiv self a b := by simp [Equiv] @[simp] theorem equiv_rootD : Equiv self (self.rootD a) a := by simp [Equiv, rootD_rootD] @[simp] theorem equiv_rootD_l : Equiv self (self.rootD a) b ↔ Equiv self a b := by simp [Equiv, rootD_rootD] @[simp] theorem equiv_rootD_r : Equiv self a (self.rootD b) ↔ Equiv self a b := by simp [Equiv, rootD_rootD] theorem equiv_find : Equiv (self.find x).1 a b ↔ Equiv self a b := by simp [Equiv, find_root_1] theorem equiv_link {self : UnionFind} {x y : Fin self.size} (xroot : self.parent x = x) (yroot : self.parent y = y) : Equiv (link self x y yroot) a b ↔ Equiv self a b ∨ Equiv self a x ∧ Equiv self y b ∨ Equiv self a y ∧ Equiv self x b := by have {m : UnionFind} {x y : Fin self.size} (xroot : self.rootD x = x) (yroot : self.rootD y = y) (hm : ∀ i, m.rootD i = if self.rootD i = x ∨ self.rootD i = y then x.1 else self.rootD i) : Equiv m a b ↔ Equiv self a b ∨ Equiv self a x ∧ Equiv self y b ∨ Equiv self a y ∧ Equiv self x b := by simp [Equiv, hm, xroot, yroot] by_cases h1 : rootD self a = x <;> by_cases h2 : rootD self b = x <;> simp [h1, h2, imp_false, Decidable.not_not] · simp [h2, Ne.symm h2]; split <;> simp [@eq_comm _ _ (rootD self b), ] · by_cases h1 : rootD self a = y <;> by_cases h2 : rootD self b = y <;> simp [h1, h2, @eq_comm _ _ (rootD self b), *] obtain ⟨r, ha, hr⟩ := root_link xroot yroot; revert hr rw [← rootD_eq_self] at xroot yroot obtain rfl \| rfl := ha · exact this xroot yroot · simpa [or_comm, and_comm] using this yroot xroot	.lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean	136	139	theorem equiv_union {self : UnionFind} {x y : Fin self.size} : Equiv (union self x y) a b ↔ Equiv self a b ∨ Equiv self a x ∧ Equiv self y b ∨ Equiv self a y ∧ Equiv self x b := by	simp [union]; rw [equiv_link (by simp [← rootD_eq_self, rootD_rootD])]; simp [equiv_find]
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker -/ import Mathlib.Algebra.MonoidAlgebra.Degree import Mathlib.Algebra.Polynomial.Coeff import Mathlib.Algebra.Polynomial.Monomial import Mathlib.Data.Fintype.BigOperators import Mathlib.Data.Nat.WithBot import Mathlib.Data.Nat.Cast.WithTop import Mathlib.Data.Nat.SuccPred #align_import data.polynomial.degree.definitions from "leanprover-community/mathlib"@"808ea4ebfabeb599f21ec4ae87d6dc969597887f" /-! # Theory of univariate polynomials The definitions include `degree`, `Monic`, `leadingCoeff` Results include - `degree_mul` : The degree of the product is the sum of degrees - `leadingCoeff_add_of_degree_eq` and `leadingCoeff_add_of_degree_lt` : The leading_coefficient of a sum is determined by the leading coefficients and degrees -/ -- Porting note: `Mathlib.Data.Nat.Cast.WithTop` should be imported for `Nat.cast_withBot`. set_option linter.uppercaseLean3 false noncomputable section open Finsupp Finset open Polynomial namespace Polynomial universe u v variable {R : Type u} {S : Type v} {a b c d : R} {n m : ℕ} section Semiring variable [Semiring R] {p q r : R[X]} /-- `degree p` is the degree of the polynomial `p`, i.e. the largest `X`-exponent in `p`. `degree p = some n` when `p ≠ 0` and `n` is the highest power of `X` that appears in `p`, otherwise `degree 0 = ⊥`. -/ def degree (p : R[X]) : WithBot ℕ := p.support.max #align polynomial.degree Polynomial.degree theorem supDegree_eq_degree (p : R[X]) : p.toFinsupp.supDegree WithBot.some = p.degree := max_eq_sup_coe theorem degree_lt_wf : WellFounded fun p q : R[X] => degree p < degree q := InvImage.wf degree wellFounded_lt #align polynomial.degree_lt_wf Polynomial.degree_lt_wf instance : WellFoundedRelation R[X] := ⟨_, degree_lt_wf⟩ /-- `natDegree p` forces `degree p` to ℕ, by defining `natDegree 0 = 0`. -/ def natDegree (p : R[X]) : ℕ := (degree p).unbot' 0 #align polynomial.nat_degree Polynomial.natDegree /-- `leadingCoeff p` gives the coefficient of the highest power of `X` in `p`-/ def leadingCoeff (p : R[X]) : R := coeff p (natDegree p) #align polynomial.leading_coeff Polynomial.leadingCoeff /-- a polynomial is `Monic` if its leading coefficient is 1 -/ def Monic (p : R[X]) := leadingCoeff p = (1 : R) #align polynomial.monic Polynomial.Monic @[nontriviality] theorem monic_of_subsingleton [Subsingleton R] (p : R[X]) : Monic p := Subsingleton.elim _ _ #align polynomial.monic_of_subsingleton Polynomial.monic_of_subsingleton theorem Monic.def : Monic p ↔ leadingCoeff p = 1 := Iff.rfl #align polynomial.monic.def Polynomial.Monic.def instance Monic.decidable [DecidableEq R] : Decidable (Monic p) := by unfold Monic; infer_instance #align polynomial.monic.decidable Polynomial.Monic.decidable @[simp] theorem Monic.leadingCoeff {p : R[X]} (hp : p.Monic) : leadingCoeff p = 1 := hp #align polynomial.monic.leading_coeff Polynomial.Monic.leadingCoeff theorem Monic.coeff_natDegree {p : R[X]} (hp : p.Monic) : p.coeff p.natDegree = 1 := hp #align polynomial.monic.coeff_nat_degree Polynomial.Monic.coeff_natDegree @[simp] theorem degree_zero : degree (0 : R[X]) = ⊥ := rfl #align polynomial.degree_zero Polynomial.degree_zero @[simp] theorem natDegree_zero : natDegree (0 : R[X]) = 0 := rfl #align polynomial.nat_degree_zero Polynomial.natDegree_zero @[simp] theorem coeff_natDegree : coeff p (natDegree p) = leadingCoeff p := rfl #align polynomial.coeff_nat_degree Polynomial.coeff_natDegree @[simp] theorem degree_eq_bot : degree p = ⊥ ↔ p = 0 := ⟨fun h => support_eq_empty.1 (Finset.max_eq_bot.1 h), fun h => h.symm ▸ rfl⟩ #align polynomial.degree_eq_bot Polynomial.degree_eq_bot @[nontriviality] theorem degree_of_subsingleton [Subsingleton R] : degree p = ⊥ := by rw [Subsingleton.elim p 0, degree_zero] #align polynomial.degree_of_subsingleton Polynomial.degree_of_subsingleton @[nontriviality] theorem natDegree_of_subsingleton [Subsingleton R] : natDegree p = 0 := by rw [Subsingleton.elim p 0, natDegree_zero] #align polynomial.nat_degree_of_subsingleton Polynomial.natDegree_of_subsingleton theorem degree_eq_natDegree (hp : p ≠ 0) : degree p = (natDegree p : WithBot ℕ) := by let ⟨n, hn⟩ := not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt degree_eq_bot.1 hp)) have hn : degree p = some n := Classical.not_not.1 hn rw [natDegree, hn]; rfl #align polynomial.degree_eq_nat_degree Polynomial.degree_eq_natDegree theorem supDegree_eq_natDegree (p : R[X]) : p.toFinsupp.supDegree id = p.natDegree := by obtain rfl\|h := eq_or_ne p 0 · simp apply WithBot.coe_injective rw [← AddMonoidAlgebra.supDegree_withBot_some_comp, Function.comp_id, supDegree_eq_degree, degree_eq_natDegree h, Nat.cast_withBot] rwa [support_toFinsupp, nonempty_iff_ne_empty, Ne, support_eq_empty] theorem degree_eq_iff_natDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) : p.degree = n ↔ p.natDegree = n := by rw [degree_eq_natDegree hp]; exact WithBot.coe_eq_coe #align polynomial.degree_eq_iff_nat_degree_eq Polynomial.degree_eq_iff_natDegree_eq theorem degree_eq_iff_natDegree_eq_of_pos {p : R[X]} {n : ℕ} (hn : 0 < n) : p.degree = n ↔ p.natDegree = n := by obtain rfl\|h := eq_or_ne p 0 · simp [hn.ne] · exact degree_eq_iff_natDegree_eq h #align polynomial.degree_eq_iff_nat_degree_eq_of_pos Polynomial.degree_eq_iff_natDegree_eq_of_pos theorem natDegree_eq_of_degree_eq_some {p : R[X]} {n : ℕ} (h : degree p = n) : natDegree p = n := by -- Porting note: `Nat.cast_withBot` is required. rw [natDegree, h, Nat.cast_withBot, WithBot.unbot'_coe] #align polynomial.nat_degree_eq_of_degree_eq_some Polynomial.natDegree_eq_of_degree_eq_some theorem degree_ne_of_natDegree_ne {n : ℕ} : p.natDegree ≠ n → degree p ≠ n := mt natDegree_eq_of_degree_eq_some #align polynomial.degree_ne_of_nat_degree_ne Polynomial.degree_ne_of_natDegree_ne @[simp] theorem degree_le_natDegree : degree p ≤ natDegree p := WithBot.giUnbot'Bot.gc.le_u_l _ #align polynomial.degree_le_nat_degree Polynomial.degree_le_natDegree theorem natDegree_eq_of_degree_eq [Semiring S] {q : S[X]} (h : degree p = degree q) : natDegree p = natDegree q := by unfold natDegree; rw [h] #align polynomial.nat_degree_eq_of_degree_eq Polynomial.natDegree_eq_of_degree_eq theorem le_degree_of_ne_zero (h : coeff p n ≠ 0) : (n : WithBot ℕ) ≤ degree p := by rw [Nat.cast_withBot] exact Finset.le_sup (mem_support_iff.2 h) #align polynomial.le_degree_of_ne_zero Polynomial.le_degree_of_ne_zero theorem le_natDegree_of_ne_zero (h : coeff p n ≠ 0) : n ≤ natDegree p := by rw [← Nat.cast_le (α := WithBot ℕ), ← degree_eq_natDegree] · exact le_degree_of_ne_zero h · rintro rfl exact h rfl #align polynomial.le_nat_degree_of_ne_zero Polynomial.le_natDegree_of_ne_zero theorem le_natDegree_of_mem_supp (a : ℕ) : a ∈ p.support → a ≤ natDegree p := le_natDegree_of_ne_zero ∘ mem_support_iff.mp #align polynomial.le_nat_degree_of_mem_supp Polynomial.le_natDegree_of_mem_supp theorem degree_eq_of_le_of_coeff_ne_zero (pn : p.degree ≤ n) (p1 : p.coeff n ≠ 0) : p.degree = n := pn.antisymm (le_degree_of_ne_zero p1) #align polynomial.degree_eq_of_le_of_coeff_ne_zero Polynomial.degree_eq_of_le_of_coeff_ne_zero theorem natDegree_eq_of_le_of_coeff_ne_zero (pn : p.natDegree ≤ n) (p1 : p.coeff n ≠ 0) : p.natDegree = n := pn.antisymm (le_natDegree_of_ne_zero p1) #align polynomial.nat_degree_eq_of_le_of_coeff_ne_zero Polynomial.natDegree_eq_of_le_of_coeff_ne_zero theorem degree_mono [Semiring S] {f : R[X]} {g : S[X]} (h : f.support ⊆ g.support) : f.degree ≤ g.degree := Finset.sup_mono h #align polynomial.degree_mono Polynomial.degree_mono theorem supp_subset_range (h : natDegree p < m) : p.support ⊆ Finset.range m := fun _n hn => mem_range.2 <\| (le_natDegree_of_mem_supp _ hn).trans_lt h #align polynomial.supp_subset_range Polynomial.supp_subset_range theorem supp_subset_range_natDegree_succ : p.support ⊆ Finset.range (natDegree p + 1) := supp_subset_range (Nat.lt_succ_self _) #align polynomial.supp_subset_range_nat_degree_succ Polynomial.supp_subset_range_natDegree_succ theorem degree_le_degree (h : coeff q (natDegree p) ≠ 0) : degree p ≤ degree q := by by_cases hp : p = 0 · rw [hp, degree_zero] exact bot_le · rw [degree_eq_natDegree hp] exact le_degree_of_ne_zero h #align polynomial.degree_le_degree Polynomial.degree_le_degree theorem natDegree_le_iff_degree_le {n : ℕ} : natDegree p ≤ n ↔ degree p ≤ n := WithBot.unbot'_le_iff (fun _ ↦ bot_le) #align polynomial.nat_degree_le_iff_degree_le Polynomial.natDegree_le_iff_degree_le theorem natDegree_lt_iff_degree_lt (hp : p ≠ 0) : p.natDegree < n ↔ p.degree < ↑n := WithBot.unbot'_lt_iff (absurd · (degree_eq_bot.not.mpr hp)) #align polynomial.nat_degree_lt_iff_degree_lt Polynomial.natDegree_lt_iff_degree_lt alias ⟨degree_le_of_natDegree_le, natDegree_le_of_degree_le⟩ := natDegree_le_iff_degree_le #align polynomial.degree_le_of_nat_degree_le Polynomial.degree_le_of_natDegree_le #align polynomial.nat_degree_le_of_degree_le Polynomial.natDegree_le_of_degree_le theorem natDegree_le_natDegree [Semiring S] {q : S[X]} (hpq : p.degree ≤ q.degree) : p.natDegree ≤ q.natDegree := WithBot.giUnbot'Bot.gc.monotone_l hpq #align polynomial.nat_degree_le_nat_degree Polynomial.natDegree_le_natDegree theorem natDegree_lt_natDegree {p q : R[X]} (hp : p ≠ 0) (hpq : p.degree < q.degree) : p.natDegree < q.natDegree := by by_cases hq : q = 0 · exact (not_lt_bot <\| hq ▸ hpq).elim rwa [degree_eq_natDegree hp, degree_eq_natDegree hq, Nat.cast_lt] at hpq #align polynomial.nat_degree_lt_nat_degree Polynomial.natDegree_lt_natDegree @[simp] theorem degree_C (ha : a ≠ 0) : degree (C a) = (0 : WithBot ℕ) := by rw [degree, ← monomial_zero_left, support_monomial 0 ha, max_eq_sup_coe, sup_singleton, WithBot.coe_zero] #align polynomial.degree_C Polynomial.degree_C theorem degree_C_le : degree (C a) ≤ 0 := by by_cases h : a = 0 · rw [h, C_0] exact bot_le · rw [degree_C h] #align polynomial.degree_C_le Polynomial.degree_C_le theorem degree_C_lt : degree (C a) < 1 := degree_C_le.trans_lt <\| WithBot.coe_lt_coe.mpr zero_lt_one #align polynomial.degree_C_lt Polynomial.degree_C_lt theorem degree_one_le : degree (1 : R[X]) ≤ (0 : WithBot ℕ) := by rw [← C_1]; exact degree_C_le #align polynomial.degree_one_le Polynomial.degree_one_le @[simp] theorem natDegree_C (a : R) : natDegree (C a) = 0 := by by_cases ha : a = 0 · have : C a = 0 := by rw [ha, C_0] rw [natDegree, degree_eq_bot.2 this, WithBot.unbot'_bot] · rw [natDegree, degree_C ha, WithBot.unbot_zero'] #align polynomial.nat_degree_C Polynomial.natDegree_C @[simp] theorem natDegree_one : natDegree (1 : R[X]) = 0 := natDegree_C 1 #align polynomial.nat_degree_one Polynomial.natDegree_one @[simp] theorem natDegree_natCast (n : ℕ) : natDegree (n : R[X]) = 0 := by simp only [← C_eq_natCast, natDegree_C] #align polynomial.nat_degree_nat_cast Polynomial.natDegree_natCast @[deprecated (since := "2024-04-17")] alias natDegree_nat_cast := natDegree_natCast theorem degree_natCast_le (n : ℕ) : degree (n : R[X]) ≤ 0 := degree_le_of_natDegree_le (by simp) @[deprecated (since := "2024-04-17")] alias degree_nat_cast_le := degree_natCast_le @[simp] theorem degree_monomial (n : ℕ) (ha : a ≠ 0) : degree (monomial n a) = n := by rw [degree, support_monomial n ha, max_singleton, Nat.cast_withBot] #align polynomial.degree_monomial Polynomial.degree_monomial @[simp] theorem degree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : degree (C a * X ^ n) = n := by rw [C_mul_X_pow_eq_monomial, degree_monomial n ha] #align polynomial.degree_C_mul_X_pow Polynomial.degree_C_mul_X_pow theorem degree_C_mul_X (ha : a ≠ 0) : degree (C a * X) = 1 := by simpa only [pow_one] using degree_C_mul_X_pow 1 ha #align polynomial.degree_C_mul_X Polynomial.degree_C_mul_X theorem degree_monomial_le (n : ℕ) (a : R) : degree (monomial n a) ≤ n := letI := Classical.decEq R if h : a = 0 then by rw [h, (monomial n).map_zero, degree_zero]; exact bot_le else le_of_eq (degree_monomial n h) #align polynomial.degree_monomial_le Polynomial.degree_monomial_le theorem degree_C_mul_X_pow_le (n : ℕ) (a : R) : degree (C a * X ^ n) ≤ n := by rw [C_mul_X_pow_eq_monomial] apply degree_monomial_le #align polynomial.degree_C_mul_X_pow_le Polynomial.degree_C_mul_X_pow_le theorem degree_C_mul_X_le (a : R) : degree (C a * X) ≤ 1 := by simpa only [pow_one] using degree_C_mul_X_pow_le 1 a #align polynomial.degree_C_mul_X_le Polynomial.degree_C_mul_X_le @[simp] theorem natDegree_C_mul_X_pow (n : ℕ) (a : R) (ha : a ≠ 0) : natDegree (C a * X ^ n) = n := natDegree_eq_of_degree_eq_some (degree_C_mul_X_pow n ha) #align polynomial.nat_degree_C_mul_X_pow Polynomial.natDegree_C_mul_X_pow @[simp] theorem natDegree_C_mul_X (a : R) (ha : a ≠ 0) : natDegree (C a * X) = 1 := by simpa only [pow_one] using natDegree_C_mul_X_pow 1 a ha #align polynomial.nat_degree_C_mul_X Polynomial.natDegree_C_mul_X @[simp] theorem natDegree_monomial [DecidableEq R] (i : ℕ) (r : R) : natDegree (monomial i r) = if r = 0 then 0 else i := by split_ifs with hr · simp [hr] · rw [← C_mul_X_pow_eq_monomial, natDegree_C_mul_X_pow i r hr] #align polynomial.nat_degree_monomial Polynomial.natDegree_monomial theorem natDegree_monomial_le (a : R) {m : ℕ} : (monomial m a).natDegree ≤ m := by classical rw [Polynomial.natDegree_monomial] split_ifs exacts [Nat.zero_le _, le_rfl] #align polynomial.nat_degree_monomial_le Polynomial.natDegree_monomial_le theorem natDegree_monomial_eq (i : ℕ) {r : R} (r0 : r ≠ 0) : (monomial i r).natDegree = i := letI := Classical.decEq R Eq.trans (natDegree_monomial _ _) (if_neg r0) #align polynomial.nat_degree_monomial_eq Polynomial.natDegree_monomial_eq theorem coeff_eq_zero_of_degree_lt (h : degree p < n) : coeff p n = 0 := Classical.not_not.1 (mt le_degree_of_ne_zero (not_le_of_gt h)) #align polynomial.coeff_eq_zero_of_degree_lt Polynomial.coeff_eq_zero_of_degree_lt theorem coeff_eq_zero_of_natDegree_lt {p : R[X]} {n : ℕ} (h : p.natDegree < n) : p.coeff n = 0 := by apply coeff_eq_zero_of_degree_lt by_cases hp : p = 0 · subst hp exact WithBot.bot_lt_coe n · rwa [degree_eq_natDegree hp, Nat.cast_lt] #align polynomial.coeff_eq_zero_of_nat_degree_lt Polynomial.coeff_eq_zero_of_natDegree_lt theorem ext_iff_natDegree_le {p q : R[X]} {n : ℕ} (hp : p.natDegree ≤ n) (hq : q.natDegree ≤ n) : p = q ↔ ∀ i ≤ n, p.coeff i = q.coeff i := by refine Iff.trans Polynomial.ext_iff ?_ refine forall_congr' fun i => ⟨fun h _ => h, fun h => ?_⟩ refine (le_or_lt i n).elim h fun k => ?_ exact (coeff_eq_zero_of_natDegree_lt (hp.trans_lt k)).trans (coeff_eq_zero_of_natDegree_lt (hq.trans_lt k)).symm #align polynomial.ext_iff_nat_degree_le Polynomial.ext_iff_natDegree_le theorem ext_iff_degree_le {p q : R[X]} {n : ℕ} (hp : p.degree ≤ n) (hq : q.degree ≤ n) : p = q ↔ ∀ i ≤ n, p.coeff i = q.coeff i := ext_iff_natDegree_le (natDegree_le_of_degree_le hp) (natDegree_le_of_degree_le hq) #align polynomial.ext_iff_degree_le Polynomial.ext_iff_degree_le @[simp] theorem coeff_natDegree_succ_eq_zero {p : R[X]} : p.coeff (p.natDegree + 1) = 0 := coeff_eq_zero_of_natDegree_lt (lt_add_one _) #align polynomial.coeff_nat_degree_succ_eq_zero Polynomial.coeff_natDegree_succ_eq_zero -- We need the explicit `Decidable` argument here because an exotic one shows up in a moment! theorem ite_le_natDegree_coeff (p : R[X]) (n : ℕ) (I : Decidable (n < 1 + natDegree p)) : @ite _ (n < 1 + natDegree p) I (coeff p n) 0 = coeff p n := by split_ifs with h · rfl · exact (coeff_eq_zero_of_natDegree_lt (not_le.1 fun w => h (Nat.lt_one_add_iff.2 w))).symm #align polynomial.ite_le_nat_degree_coeff Polynomial.ite_le_natDegree_coeff theorem as_sum_support (p : R[X]) : p = ∑ i ∈ p.support, monomial i (p.coeff i) := (sum_monomial_eq p).symm #align polynomial.as_sum_support Polynomial.as_sum_support theorem as_sum_support_C_mul_X_pow (p : R[X]) : p = ∑ i ∈ p.support, C (p.coeff i) * X ^ i := _root_.trans p.as_sum_support <\| by simp only [C_mul_X_pow_eq_monomial] #align polynomial.as_sum_support_C_mul_X_pow Polynomial.as_sum_support_C_mul_X_pow /-- We can reexpress a sum over `p.support` as a sum over `range n`, for any `n` satisfying `p.natDegree < n`. -/ theorem sum_over_range' [AddCommMonoid S] (p : R[X]) {f : ℕ → R → S} (h : ∀ n, f n 0 = 0) (n : ℕ) (w : p.natDegree < n) : p.sum f = ∑ a ∈ range n, f a (coeff p a) := by rcases p with ⟨⟩ have := supp_subset_range w simp only [Polynomial.sum, support, coeff, natDegree, degree] at this ⊢ exact Finsupp.sum_of_support_subset _ this _ fun n _hn => h n #align polynomial.sum_over_range' Polynomial.sum_over_range' /-- We can reexpress a sum over `p.support` as a sum over `range (p.natDegree + 1)`. -/ theorem sum_over_range [AddCommMonoid S] (p : R[X]) {f : ℕ → R → S} (h : ∀ n, f n 0 = 0) : p.sum f = ∑ a ∈ range (p.natDegree + 1), f a (coeff p a) := sum_over_range' p h (p.natDegree + 1) (lt_add_one _) #align polynomial.sum_over_range Polynomial.sum_over_range -- TODO this is essentially a duplicate of `sum_over_range`, and should be removed. theorem sum_fin [AddCommMonoid S] (f : ℕ → R → S) (hf : ∀ i, f i 0 = 0) {n : ℕ} {p : R[X]} (hn : p.degree < n) : (∑ i : Fin n, f i (p.coeff i)) = p.sum f := by by_cases hp : p = 0 · rw [hp, sum_zero_index, Finset.sum_eq_zero] intro i _ exact hf i rw [sum_over_range' _ hf n ((natDegree_lt_iff_degree_lt hp).mpr hn), Fin.sum_univ_eq_sum_range fun i => f i (p.coeff i)] #align polynomial.sum_fin Polynomial.sum_fin theorem as_sum_range' (p : R[X]) (n : ℕ) (w : p.natDegree < n) : p = ∑ i ∈ range n, monomial i (coeff p i) := p.sum_monomial_eq.symm.trans <\| p.sum_over_range' monomial_zero_right _ w #align polynomial.as_sum_range' Polynomial.as_sum_range' theorem as_sum_range (p : R[X]) : p = ∑ i ∈ range (p.natDegree + 1), monomial i (coeff p i) := p.sum_monomial_eq.symm.trans <\| p.sum_over_range <\| monomial_zero_right #align polynomial.as_sum_range Polynomial.as_sum_range theorem as_sum_range_C_mul_X_pow (p : R[X]) : p = ∑ i ∈ range (p.natDegree + 1), C (coeff p i) * X ^ i := p.as_sum_range.trans <\| by simp only [C_mul_X_pow_eq_monomial] #align polynomial.as_sum_range_C_mul_X_pow Polynomial.as_sum_range_C_mul_X_pow theorem coeff_ne_zero_of_eq_degree (hn : degree p = n) : coeff p n ≠ 0 := fun h => mem_support_iff.mp (mem_of_max hn) h #align polynomial.coeff_ne_zero_of_eq_degree Polynomial.coeff_ne_zero_of_eq_degree theorem eq_X_add_C_of_degree_le_one (h : degree p ≤ 1) : p = C (p.coeff 1) * X + C (p.coeff 0) := ext fun n => Nat.casesOn n (by simp) fun n => Nat.casesOn n (by simp [coeff_C]) fun m => by -- Porting note: `by decide` → `Iff.mpr ..` have : degree p < m.succ.succ := lt_of_le_of_lt h (Iff.mpr WithBot.coe_lt_coe <\| Nat.succ_lt_succ <\| Nat.zero_lt_succ m) simp [coeff_eq_zero_of_degree_lt this, coeff_C, Nat.succ_ne_zero, coeff_X, Nat.succ_inj', @eq_comm ℕ 0] #align polynomial.eq_X_add_C_of_degree_le_one Polynomial.eq_X_add_C_of_degree_le_one theorem eq_X_add_C_of_degree_eq_one (h : degree p = 1) : p = C p.leadingCoeff * X + C (p.coeff 0) := (eq_X_add_C_of_degree_le_one h.le).trans (by rw [← Nat.cast_one] at h; rw [leadingCoeff, natDegree_eq_of_degree_eq_some h]) #align polynomial.eq_X_add_C_of_degree_eq_one Polynomial.eq_X_add_C_of_degree_eq_one theorem eq_X_add_C_of_natDegree_le_one (h : natDegree p ≤ 1) : p = C (p.coeff 1) * X + C (p.coeff 0) := eq_X_add_C_of_degree_le_one <\| degree_le_of_natDegree_le h #align polynomial.eq_X_add_C_of_nat_degree_le_one Polynomial.eq_X_add_C_of_natDegree_le_one theorem Monic.eq_X_add_C (hm : p.Monic) (hnd : p.natDegree = 1) : p = X + C (p.coeff 0) := by rw [← one_mul X, ← C_1, ← hm.coeff_natDegree, hnd, ← eq_X_add_C_of_natDegree_le_one hnd.le] #align polynomial.monic.eq_X_add_C Polynomial.Monic.eq_X_add_C theorem exists_eq_X_add_C_of_natDegree_le_one (h : natDegree p ≤ 1) : ∃ a b, p = C a * X + C b := ⟨p.coeff 1, p.coeff 0, eq_X_add_C_of_natDegree_le_one h⟩ #align polynomial.exists_eq_X_add_C_of_natDegree_le_one Polynomial.exists_eq_X_add_C_of_natDegree_le_one theorem degree_X_pow_le (n : ℕ) : degree (X ^ n : R[X]) ≤ n := by simpa only [C_1, one_mul] using degree_C_mul_X_pow_le n (1 : R) #align polynomial.degree_X_pow_le Polynomial.degree_X_pow_le theorem degree_X_le : degree (X : R[X]) ≤ 1 := degree_monomial_le _ _ #align polynomial.degree_X_le Polynomial.degree_X_le theorem natDegree_X_le : (X : R[X]).natDegree ≤ 1 := natDegree_le_of_degree_le degree_X_le #align polynomial.nat_degree_X_le Polynomial.natDegree_X_le theorem mem_support_C_mul_X_pow {n a : ℕ} {c : R} (h : a ∈ support (C c * X ^ n)) : a = n := mem_singleton.1 <\| support_C_mul_X_pow' n c h #align polynomial.mem_support_C_mul_X_pow Polynomial.mem_support_C_mul_X_pow theorem card_support_C_mul_X_pow_le_one {c : R} {n : ℕ} : card (support (C c * X ^ n)) ≤ 1 := by rw [← card_singleton n] apply card_le_card (support_C_mul_X_pow' n c) #align polynomial.card_support_C_mul_X_pow_le_one Polynomial.card_support_C_mul_X_pow_le_one theorem card_supp_le_succ_natDegree (p : R[X]) : p.support.card ≤ p.natDegree + 1 := by rw [← Finset.card_range (p.natDegree + 1)] exact Finset.card_le_card supp_subset_range_natDegree_succ #align polynomial.card_supp_le_succ_nat_degree Polynomial.card_supp_le_succ_natDegree theorem le_degree_of_mem_supp (a : ℕ) : a ∈ p.support → ↑a ≤ degree p := le_degree_of_ne_zero ∘ mem_support_iff.mp #align polynomial.le_degree_of_mem_supp Polynomial.le_degree_of_mem_supp theorem nonempty_support_iff : p.support.Nonempty ↔ p ≠ 0 := by rw [Ne, nonempty_iff_ne_empty, Ne, ← support_eq_empty] #align polynomial.nonempty_support_iff Polynomial.nonempty_support_iff end Semiring section NonzeroSemiring variable [Semiring R] [Nontrivial R] {p q : R[X]} @[simp] theorem degree_one : degree (1 : R[X]) = (0 : WithBot ℕ) := degree_C one_ne_zero #align polynomial.degree_one Polynomial.degree_one @[simp] theorem degree_X : degree (X : R[X]) = 1 := degree_monomial _ one_ne_zero #align polynomial.degree_X Polynomial.degree_X @[simp] theorem natDegree_X : (X : R[X]).natDegree = 1 := natDegree_eq_of_degree_eq_some degree_X #align polynomial.nat_degree_X Polynomial.natDegree_X end NonzeroSemiring section Ring variable [Ring R] theorem coeff_mul_X_sub_C {p : R[X]} {r : R} {a : ℕ} : coeff (p * (X - C r)) (a + 1) = coeff p a - coeff p (a + 1) * r := by simp [mul_sub] #align polynomial.coeff_mul_X_sub_C Polynomial.coeff_mul_X_sub_C @[simp] theorem degree_neg (p : R[X]) : degree (-p) = degree p := by unfold degree; rw [support_neg] #align polynomial.degree_neg Polynomial.degree_neg theorem degree_neg_le_of_le {a : WithBot ℕ} {p : R[X]} (hp : degree p ≤ a) : degree (-p) ≤ a := p.degree_neg.le.trans hp @[simp] theorem natDegree_neg (p : R[X]) : natDegree (-p) = natDegree p := by simp [natDegree] #align polynomial.nat_degree_neg Polynomial.natDegree_neg theorem natDegree_neg_le_of_le {p : R[X]} (hp : natDegree p ≤ m) : natDegree (-p) ≤ m := (natDegree_neg p).le.trans hp @[simp] theorem natDegree_intCast (n : ℤ) : natDegree (n : R[X]) = 0 := by rw [← C_eq_intCast, natDegree_C] #align polynomial.nat_degree_intCast Polynomial.natDegree_intCast @[deprecated (since := "2024-04-17")] alias natDegree_int_cast := natDegree_intCast theorem degree_intCast_le (n : ℤ) : degree (n : R[X]) ≤ 0 := degree_le_of_natDegree_le (by simp) @[deprecated (since := "2024-04-17")] alias degree_int_cast_le := degree_intCast_le @[simp] theorem leadingCoeff_neg (p : R[X]) : (-p).leadingCoeff = -p.leadingCoeff := by rw [leadingCoeff, leadingCoeff, natDegree_neg, coeff_neg] #align polynomial.leading_coeff_neg Polynomial.leadingCoeff_neg end Ring section Semiring variable [Semiring R] {p : R[X]} /-- The second-highest coefficient, or 0 for constants -/ def nextCoeff (p : R[X]) : R := if p.natDegree = 0 then 0 else p.coeff (p.natDegree - 1) #align polynomial.next_coeff Polynomial.nextCoeff lemma nextCoeff_eq_zero : p.nextCoeff = 0 ↔ p.natDegree = 0 ∨ 0 < p.natDegree ∧ p.coeff (p.natDegree - 1) = 0 := by simp [nextCoeff, or_iff_not_imp_left, pos_iff_ne_zero]; aesop lemma nextCoeff_ne_zero : p.nextCoeff ≠ 0 ↔ p.natDegree ≠ 0 ∧ p.coeff (p.natDegree - 1) ≠ 0 := by simp [nextCoeff] @[simp] theorem nextCoeff_C_eq_zero (c : R) : nextCoeff (C c) = 0 := by rw [nextCoeff] simp #align polynomial.next_coeff_C_eq_zero Polynomial.nextCoeff_C_eq_zero theorem nextCoeff_of_natDegree_pos (hp : 0 < p.natDegree) : nextCoeff p = p.coeff (p.natDegree - 1) := by rw [nextCoeff, if_neg] contrapose! hp simpa #align polynomial.next_coeff_of_pos_nat_degree Polynomial.nextCoeff_of_natDegree_pos variable {p q : R[X]} {ι : Type*} theorem coeff_natDegree_eq_zero_of_degree_lt (h : degree p < degree q) : coeff p (natDegree q) = 0 := coeff_eq_zero_of_degree_lt (lt_of_lt_of_le h degree_le_natDegree) #align polynomial.coeff_nat_degree_eq_zero_of_degree_lt Polynomial.coeff_natDegree_eq_zero_of_degree_lt theorem ne_zero_of_degree_gt {n : WithBot ℕ} (h : n < degree p) : p ≠ 0 := mt degree_eq_bot.2 h.ne_bot #align polynomial.ne_zero_of_degree_gt Polynomial.ne_zero_of_degree_gt theorem ne_zero_of_degree_ge_degree (hpq : p.degree ≤ q.degree) (hp : p ≠ 0) : q ≠ 0 := Polynomial.ne_zero_of_degree_gt (lt_of_lt_of_le (bot_lt_iff_ne_bot.mpr (by rwa [Ne, Polynomial.degree_eq_bot])) hpq : q.degree > ⊥) #align polynomial.ne_zero_of_degree_ge_degree Polynomial.ne_zero_of_degree_ge_degree theorem ne_zero_of_natDegree_gt {n : ℕ} (h : n < natDegree p) : p ≠ 0 := fun H => by simp [H, Nat.not_lt_zero] at h #align polynomial.ne_zero_of_nat_degree_gt Polynomial.ne_zero_of_natDegree_gt theorem degree_lt_degree (h : natDegree p < natDegree q) : degree p < degree q := by by_cases hp : p = 0 · simp [hp] rw [bot_lt_iff_ne_bot] intro hq simp [hp, degree_eq_bot.mp hq, lt_irrefl] at h · rwa [degree_eq_natDegree hp, degree_eq_natDegree <\| ne_zero_of_natDegree_gt h, Nat.cast_lt] #align polynomial.degree_lt_degree Polynomial.degree_lt_degree theorem natDegree_lt_natDegree_iff (hp : p ≠ 0) : natDegree p < natDegree q ↔ degree p < degree q := ⟨degree_lt_degree, fun h ↦ by have hq : q ≠ 0 := ne_zero_of_degree_gt h rwa [degree_eq_natDegree hp, degree_eq_natDegree hq, Nat.cast_lt] at h⟩ #align polynomial.nat_degree_lt_nat_degree_iff Polynomial.natDegree_lt_natDegree_iff theorem eq_C_of_degree_le_zero (h : degree p ≤ 0) : p = C (coeff p 0) := by ext (_ \| n) · simp rw [coeff_C, if_neg (Nat.succ_ne_zero _), coeff_eq_zero_of_degree_lt] exact h.trans_lt (WithBot.coe_lt_coe.2 n.succ_pos) #align polynomial.eq_C_of_degree_le_zero Polynomial.eq_C_of_degree_le_zero theorem eq_C_of_degree_eq_zero (h : degree p = 0) : p = C (coeff p 0) := eq_C_of_degree_le_zero h.le #align polynomial.eq_C_of_degree_eq_zero Polynomial.eq_C_of_degree_eq_zero theorem degree_le_zero_iff : degree p ≤ 0 ↔ p = C (coeff p 0) := ⟨eq_C_of_degree_le_zero, fun h => h.symm ▸ degree_C_le⟩ #align polynomial.degree_le_zero_iff Polynomial.degree_le_zero_iff theorem degree_add_le (p q : R[X]) : degree (p + q) ≤ max (degree p) (degree q) := by simpa only [degree, ← support_toFinsupp, toFinsupp_add] using AddMonoidAlgebra.sup_support_add_le _ _ _ #align polynomial.degree_add_le Polynomial.degree_add_le theorem degree_add_le_of_degree_le {p q : R[X]} {n : ℕ} (hp : degree p ≤ n) (hq : degree q ≤ n) : degree (p + q) ≤ n := (degree_add_le p q).trans <\| max_le hp hq #align polynomial.degree_add_le_of_degree_le Polynomial.degree_add_le_of_degree_le theorem degree_add_le_of_le {a b : WithBot ℕ} (hp : degree p ≤ a) (hq : degree q ≤ b) : degree (p + q) ≤ max a b := (p.degree_add_le q).trans <\| max_le_max ‹_› ‹_› theorem natDegree_add_le (p q : R[X]) : natDegree (p + q) ≤ max (natDegree p) (natDegree q) := by cases' le_max_iff.1 (degree_add_le p q) with h h <;> simp [natDegree_le_natDegree h] #align polynomial.nat_degree_add_le Polynomial.natDegree_add_le theorem natDegree_add_le_of_degree_le {p q : R[X]} {n : ℕ} (hp : natDegree p ≤ n) (hq : natDegree q ≤ n) : natDegree (p + q) ≤ n := (natDegree_add_le p q).trans <\| max_le hp hq #align polynomial.nat_degree_add_le_of_degree_le Polynomial.natDegree_add_le_of_degree_le theorem natDegree_add_le_of_le (hp : natDegree p ≤ m) (hq : natDegree q ≤ n) : natDegree (p + q) ≤ max m n := (p.natDegree_add_le q).trans <\| max_le_max ‹_› ‹_› @[simp] theorem leadingCoeff_zero : leadingCoeff (0 : R[X]) = 0 := rfl #align polynomial.leading_coeff_zero Polynomial.leadingCoeff_zero @[simp] theorem leadingCoeff_eq_zero : leadingCoeff p = 0 ↔ p = 0 := ⟨fun h => Classical.by_contradiction fun hp => mt mem_support_iff.1 (Classical.not_not.2 h) (mem_of_max (degree_eq_natDegree hp)), fun h => h.symm ▸ leadingCoeff_zero⟩ #align polynomial.leading_coeff_eq_zero Polynomial.leadingCoeff_eq_zero theorem leadingCoeff_ne_zero : leadingCoeff p ≠ 0 ↔ p ≠ 0 := by rw [Ne, leadingCoeff_eq_zero] #align polynomial.leading_coeff_ne_zero Polynomial.leadingCoeff_ne_zero theorem leadingCoeff_eq_zero_iff_deg_eq_bot : leadingCoeff p = 0 ↔ degree p = ⊥ := by rw [leadingCoeff_eq_zero, degree_eq_bot] #align polynomial.leading_coeff_eq_zero_iff_deg_eq_bot Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot lemma natDegree_le_pred (hf : p.natDegree ≤ n) (hn : p.coeff n = 0) : p.natDegree ≤ n - 1 := by obtain _ \| n := n · exact hf · refine (Nat.le_succ_iff_eq_or_le.1 hf).resolve_left fun h ↦ ?_ rw [← Nat.succ_eq_add_one, ← h, coeff_natDegree, leadingCoeff_eq_zero] at hn aesop	Mathlib/Algebra/Polynomial/Degree/Definitions.lean	709	711	theorem natDegree_mem_support_of_nonzero (H : p ≠ 0) : p.natDegree ∈ p.support := by	rw [mem_support_iff] exact (not_congr leadingCoeff_eq_zero).mpr H
/- Copyright (c) 2020 Oliver Nash. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Oliver Nash, Antoine Labelle -/ import Mathlib.LinearAlgebra.Dual import Mathlib.LinearAlgebra.Matrix.ToLin #align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec" /-! # Contractions Given modules $M, N$ over a commutative ring $R$, this file defines the natural linear maps: $M^* \otimes M \to R$, $M \otimes M^* \to R$, and $M^* \otimes N → Hom(M, N)$, as well as proving some basic properties of these maps. ## Tags contraction, dual module, tensor product -/ suppress_compilation -- Porting note: universe metavariables behave oddly universe w u v₁ v₂ v₃ v₄ variable {ι : Type w} (R : Type u) (M : Type v₁) (N : Type v₂) (P : Type v₃) (Q : Type v₄) -- Porting note: we need high priority for this to fire first; not the case in ML3 attribute [local ext high] TensorProduct.ext section Contraction open TensorProduct LinearMap Matrix Module open TensorProduct section CommSemiring variable [CommSemiring R] variable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] variable [Module R M] [Module R N] [Module R P] [Module R Q] variable [DecidableEq ι] [Fintype ι] (b : Basis ι R M) -- Porting note: doesn't like implicit ring in the tensor product /-- The natural left-handed pairing between a module and its dual. -/ def contractLeft : Module.Dual R M ⊗[R] M →ₗ[R] R := (uncurry _ _ _ _).toFun LinearMap.id #align contract_left contractLeft -- Porting note: doesn't like implicit ring in the tensor product /-- The natural right-handed pairing between a module and its dual. -/ def contractRight : M ⊗[R] Module.Dual R M →ₗ[R] R := (uncurry _ _ _ _).toFun (LinearMap.flip LinearMap.id) #align contract_right contractRight -- Porting note: doesn't like implicit ring in the tensor product /-- The natural map associating a linear map to the tensor product of two modules. -/ def dualTensorHom : Module.Dual R M ⊗[R] N →ₗ[R] M →ₗ[R] N := let M' := Module.Dual R M (uncurry R M' N (M →ₗ[R] N) : _ → M' ⊗ N →ₗ[R] M →ₗ[R] N) LinearMap.smulRightₗ #align dual_tensor_hom dualTensorHom variable {R M N P Q} @[simp] theorem contractLeft_apply (f : Module.Dual R M) (m : M) : contractLeft R M (f ⊗ₜ m) = f m := rfl #align contract_left_apply contractLeft_apply @[simp] theorem contractRight_apply (f : Module.Dual R M) (m : M) : contractRight R M (m ⊗ₜ f) = f m := rfl #align contract_right_apply contractRight_apply @[simp] theorem dualTensorHom_apply (f : Module.Dual R M) (m : M) (n : N) : dualTensorHom R M N (f ⊗ₜ n) m = f m • n := rfl #align dual_tensor_hom_apply dualTensorHom_apply @[simp] theorem transpose_dualTensorHom (f : Module.Dual R M) (m : M) : Dual.transpose (R := R) (dualTensorHom R M M (f ⊗ₜ m)) = dualTensorHom R _ _ (Dual.eval R M m ⊗ₜ f) := by ext f' m' simp only [Dual.transpose_apply, coe_comp, Function.comp_apply, dualTensorHom_apply, LinearMap.map_smulₛₗ, RingHom.id_apply, Algebra.id.smul_eq_mul, Dual.eval_apply, LinearMap.smul_apply] exact mul_comm _ _ #align transpose_dual_tensor_hom transpose_dualTensorHom @[simp] theorem dualTensorHom_prodMap_zero (f : Module.Dual R M) (p : P) : ((dualTensorHom R M P) (f ⊗ₜ[R] p)).prodMap (0 : N →ₗ[R] Q) = dualTensorHom R (M × N) (P × Q) ((f ∘ₗ fst R M N) ⊗ₜ inl R P Q p) := by ext <;> simp only [coe_comp, coe_inl, Function.comp_apply, prodMap_apply, dualTensorHom_apply, fst_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero] #align dual_tensor_hom_prod_map_zero dualTensorHom_prodMap_zero @[simp] theorem zero_prodMap_dualTensorHom (g : Module.Dual R N) (q : Q) : (0 : M →ₗ[R] P).prodMap ((dualTensorHom R N Q) (g ⊗ₜ[R] q)) = dualTensorHom R (M × N) (P × Q) ((g ∘ₗ snd R M N) ⊗ₜ inr R P Q q) := by ext <;> simp only [coe_comp, coe_inr, Function.comp_apply, prodMap_apply, dualTensorHom_apply, snd_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero] #align zero_prod_map_dual_tensor_hom zero_prodMap_dualTensorHom theorem map_dualTensorHom (f : Module.Dual R M) (p : P) (g : Module.Dual R N) (q : Q) : TensorProduct.map (dualTensorHom R M P (f ⊗ₜ[R] p)) (dualTensorHom R N Q (g ⊗ₜ[R] q)) = dualTensorHom R (M ⊗[R] N) (P ⊗[R] Q) (dualDistrib R M N (f ⊗ₜ g) ⊗ₜ[R] p ⊗ₜ[R] q) := by ext m n simp only [compr₂_apply, mk_apply, map_tmul, dualTensorHom_apply, dualDistrib_apply, ← smul_tmul_smul] #align map_dual_tensor_hom map_dualTensorHom @[simp] theorem comp_dualTensorHom (f : Module.Dual R M) (n : N) (g : Module.Dual R N) (p : P) : dualTensorHom R N P (g ⊗ₜ[R] p) ∘ₗ dualTensorHom R M N (f ⊗ₜ[R] n) = g n • dualTensorHom R M P (f ⊗ₜ p) := by ext m simp only [coe_comp, Function.comp_apply, dualTensorHom_apply, LinearMap.map_smul, RingHom.id_apply, LinearMap.smul_apply] rw [smul_comm] #align comp_dual_tensor_hom comp_dualTensorHom /-- As a matrix, `dualTensorHom` evaluated on a basis element of `M* ⊗ N` is a matrix with a single one and zeros elsewhere -/ theorem toMatrix_dualTensorHom {m : Type} {n : Type} [Fintype m] [Finite n] [DecidableEq m] [DecidableEq n] (bM : Basis m R M) (bN : Basis n R N) (j : m) (i : n) : toMatrix bM bN (dualTensorHom R M N (bM.coord j ⊗ₜ bN i)) = stdBasisMatrix i j 1 := by ext i' j' by_cases hij : i = i' ∧ j = j' <;> simp [LinearMap.toMatrix_apply, Finsupp.single_eq_pi_single, hij] rw [and_iff_not_or_not, Classical.not_not] at hij cases' hij with hij hij <;> simp [hij] #align to_matrix_dual_tensor_hom toMatrix_dualTensorHom end CommSemiring section CommRing variable [CommRing R] variable [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [AddCommGroup Q] variable [Module R M] [Module R N] [Module R P] [Module R Q] variable [DecidableEq ι] [Fintype ι] (b : Basis ι R M) variable {R M N P Q} /-- If `M` is free, the natural linear map $M^* ⊗ N → Hom(M, N)$ is an equivalence. This function provides this equivalence in return for a basis of `M`. -/ -- @[simps! apply] -- Porting note: removed and created manually; malformed noncomputable def dualTensorHomEquivOfBasis : Module.Dual R M ⊗[R] N ≃ₗ[R] M →ₗ[R] N := LinearEquiv.ofLinear (dualTensorHom R M N) (∑ i, TensorProduct.mk R _ N (b.dualBasis i) ∘ₗ (LinearMap.applyₗ (R := R) (b i))) (by ext f m simp only [applyₗ_apply_apply, coeFn_sum, dualTensorHom_apply, mk_apply, id_coe, _root_.id, Fintype.sum_apply, Function.comp_apply, Basis.coe_dualBasis, coe_comp, Basis.coord_apply, ← f.map_smul, _root_.map_sum (dualTensorHom R M N), ← _root_.map_sum f, b.sum_repr]) (by ext f m simp only [applyₗ_apply_apply, coeFn_sum, dualTensorHom_apply, mk_apply, id_coe, _root_.id, Fintype.sum_apply, Function.comp_apply, Basis.coe_dualBasis, coe_comp, compr₂_apply, tmul_smul, smul_tmul', ← sum_tmul, Basis.sum_dual_apply_smul_coord]) #align dual_tensor_hom_equiv_of_basis dualTensorHomEquivOfBasis @[simp] theorem dualTensorHomEquivOfBasis_apply (x : Module.Dual R M ⊗[R] N) : (dualTensorHomEquivOfBasis (N := N) b : Module.Dual R M ⊗[R] N → (M →ₗ[R] N)) x = (dualTensorHom R M N) x := by ext; rfl @[simp] theorem dualTensorHomEquivOfBasis_toLinearMap : (dualTensorHomEquivOfBasis b : Module.Dual R M ⊗[R] N ≃ₗ[R] M →ₗ[R] N).toLinearMap = dualTensorHom R M N := rfl #align dual_tensor_hom_equiv_of_basis_to_linear_map dualTensorHomEquivOfBasis_toLinearMap -- Porting note: should N be explicit in dualTensorHomEquivOfBasis? @[simp] theorem dualTensorHomEquivOfBasis_symm_cancel_left (x : Module.Dual R M ⊗[R] N) : (dualTensorHomEquivOfBasis (N := N) b).symm (dualTensorHom R M N x) = x := by rw [← dualTensorHomEquivOfBasis_apply b, LinearEquiv.symm_apply_apply <\| dualTensorHomEquivOfBasis (N := N) b] #align dual_tensor_hom_equiv_of_basis_symm_cancel_left dualTensorHomEquivOfBasis_symm_cancel_left @[simp]	Mathlib/LinearAlgebra/Contraction.lean	193	195	theorem dualTensorHomEquivOfBasis_symm_cancel_right (x : M →ₗ[R] N) : dualTensorHom R M N ((dualTensorHomEquivOfBasis (N := N) b).symm x) = x := by	rw [← dualTensorHomEquivOfBasis_apply b, LinearEquiv.apply_symm_apply]
/- Copyright (c) 2018 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson -/ import Mathlib.Algebra.BigOperators.Associated import Mathlib.Algebra.GCDMonoid.Basic import Mathlib.Data.Finsupp.Multiset import Mathlib.Data.Nat.Factors import Mathlib.RingTheory.Noetherian import Mathlib.RingTheory.Multiplicity #align_import ring_theory.unique_factorization_domain from "leanprover-community/mathlib"@"570e9f4877079b3a923135b3027ac3be8695ab8c" /-! # Unique factorization ## Main Definitions * `WfDvdMonoid` holds for `Monoid`s for which a strict divisibility relation is well-founded. * `UniqueFactorizationMonoid` holds for `WfDvdMonoid`s where `Irreducible` is equivalent to `Prime` ## To do * set up the complete lattice structure on `FactorSet`. -/ variable {α : Type} local infixl:50 " ~ᵤ " => Associated /-- Well-foundedness of the strict version of \|, which is equivalent to the descending chain condition on divisibility and to the ascending chain condition on principal ideals in an integral domain. -/ class WfDvdMonoid (α : Type) [CommMonoidWithZero α] : Prop where wellFounded_dvdNotUnit : WellFounded (@DvdNotUnit α _) #align wf_dvd_monoid WfDvdMonoid export WfDvdMonoid (wellFounded_dvdNotUnit) -- see Note [lower instance priority] instance (priority := 100) IsNoetherianRing.wfDvdMonoid [CommRing α] [IsDomain α] [IsNoetherianRing α] : WfDvdMonoid α := ⟨by convert InvImage.wf (fun a => Ideal.span ({a} : Set α)) (wellFounded_submodule_gt _ _) ext exact Ideal.span_singleton_lt_span_singleton.symm⟩ #align is_noetherian_ring.wf_dvd_monoid IsNoetherianRing.wfDvdMonoid namespace WfDvdMonoid variable [CommMonoidWithZero α] open Associates Nat theorem of_wfDvdMonoid_associates (_ : WfDvdMonoid (Associates α)) : WfDvdMonoid α := ⟨(mk_surjective.wellFounded_iff mk_dvdNotUnit_mk_iff.symm).2 wellFounded_dvdNotUnit⟩ #align wf_dvd_monoid.of_wf_dvd_monoid_associates WfDvdMonoid.of_wfDvdMonoid_associates variable [WfDvdMonoid α] instance wfDvdMonoid_associates : WfDvdMonoid (Associates α) := ⟨(mk_surjective.wellFounded_iff mk_dvdNotUnit_mk_iff.symm).1 wellFounded_dvdNotUnit⟩ #align wf_dvd_monoid.wf_dvd_monoid_associates WfDvdMonoid.wfDvdMonoid_associates theorem wellFounded_associates : WellFounded ((· < ·) : Associates α → Associates α → Prop) := Subrelation.wf dvdNotUnit_of_lt wellFounded_dvdNotUnit #align wf_dvd_monoid.well_founded_associates WfDvdMonoid.wellFounded_associates -- Porting note: elab_as_elim can only be global and cannot be changed on an imported decl -- attribute [local elab_as_elim] WellFounded.fix theorem exists_irreducible_factor {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : ∃ i, Irreducible i ∧ i ∣ a := let ⟨b, hs, hr⟩ := wellFounded_dvdNotUnit.has_min { b \| b ∣ a ∧ ¬IsUnit b } ⟨a, dvd_rfl, ha⟩ ⟨b, ⟨hs.2, fun c d he => let h := dvd_trans ⟨d, he⟩ hs.1 or_iff_not_imp_left.2 fun hc => of_not_not fun hd => hr c ⟨h, hc⟩ ⟨ne_zero_of_dvd_ne_zero ha0 h, d, hd, he⟩⟩, hs.1⟩ #align wf_dvd_monoid.exists_irreducible_factor WfDvdMonoid.exists_irreducible_factor @[elab_as_elim] theorem induction_on_irreducible {P : α → Prop} (a : α) (h0 : P 0) (hu : ∀ u : α, IsUnit u → P u) (hi : ∀ a i : α, a ≠ 0 → Irreducible i → P a → P (i * a)) : P a := haveI := Classical.dec wellFounded_dvdNotUnit.fix (fun a ih => if ha0 : a = 0 then ha0.substr h0 else if hau : IsUnit a then hu a hau else let ⟨i, hii, b, hb⟩ := exists_irreducible_factor hau ha0 let hb0 : b ≠ 0 := ne_zero_of_dvd_ne_zero ha0 ⟨i, mul_comm i b ▸ hb⟩ hb.symm ▸ hi b i hb0 hii <\| ih b ⟨hb0, i, hii.1, mul_comm i b ▸ hb⟩) a #align wf_dvd_monoid.induction_on_irreducible WfDvdMonoid.induction_on_irreducible theorem exists_factors (a : α) : a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ Associated f.prod a := induction_on_irreducible a (fun h => (h rfl).elim) (fun u hu _ => ⟨0, fun _ h => False.elim (Multiset.not_mem_zero _ h), hu.unit, one_mul _⟩) fun a i ha0 hi ih _ => let ⟨s, hs⟩ := ih ha0 ⟨i ::ₘ s, fun b H => (Multiset.mem_cons.1 H).elim (fun h => h.symm ▸ hi) (hs.1 b), by rw [s.prod_cons i] exact hs.2.mul_left i⟩ #align wf_dvd_monoid.exists_factors WfDvdMonoid.exists_factors theorem not_unit_iff_exists_factors_eq (a : α) (hn0 : a ≠ 0) : ¬IsUnit a ↔ ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod = a ∧ f ≠ ∅ := ⟨fun hnu => by obtain ⟨f, hi, u, rfl⟩ := exists_factors a hn0 obtain ⟨b, h⟩ := Multiset.exists_mem_of_ne_zero fun h : f = 0 => hnu <\| by simp [h] classical refine ⟨(f.erase b).cons (b * u), fun a ha => ?_, ?_, Multiset.cons_ne_zero⟩ · obtain rfl \| ha := Multiset.mem_cons.1 ha exacts [Associated.irreducible ⟨u, rfl⟩ (hi b h), hi a (Multiset.mem_of_mem_erase ha)] · rw [Multiset.prod_cons, mul_comm b, mul_assoc, Multiset.prod_erase h, mul_comm], fun ⟨f, hi, he, hne⟩ => let ⟨b, h⟩ := Multiset.exists_mem_of_ne_zero hne not_isUnit_of_not_isUnit_dvd (hi b h).not_unit <\| he ▸ Multiset.dvd_prod h⟩ #align wf_dvd_monoid.not_unit_iff_exists_factors_eq WfDvdMonoid.not_unit_iff_exists_factors_eq theorem isRelPrime_of_no_irreducible_factors {x y : α} (nonzero : ¬(x = 0 ∧ y = 0)) (H : ∀ z : α, Irreducible z → z ∣ x → ¬z ∣ y) : IsRelPrime x y := isRelPrime_of_no_nonunits_factors nonzero fun _z znu znz zx zy ↦ have ⟨i, h1, h2⟩ := exists_irreducible_factor znu znz H i h1 (h2.trans zx) (h2.trans zy) end WfDvdMonoid theorem WfDvdMonoid.of_wellFounded_associates [CancelCommMonoidWithZero α] (h : WellFounded ((· < ·) : Associates α → Associates α → Prop)) : WfDvdMonoid α := WfDvdMonoid.of_wfDvdMonoid_associates ⟨by convert h ext exact Associates.dvdNotUnit_iff_lt⟩ #align wf_dvd_monoid.of_well_founded_associates WfDvdMonoid.of_wellFounded_associates theorem WfDvdMonoid.iff_wellFounded_associates [CancelCommMonoidWithZero α] : WfDvdMonoid α ↔ WellFounded ((· < ·) : Associates α → Associates α → Prop) := ⟨by apply WfDvdMonoid.wellFounded_associates, WfDvdMonoid.of_wellFounded_associates⟩ #align wf_dvd_monoid.iff_well_founded_associates WfDvdMonoid.iff_wellFounded_associates theorem WfDvdMonoid.max_power_factor' [CommMonoidWithZero α] [WfDvdMonoid α] {a₀ x : α} (h : a₀ ≠ 0) (hx : ¬IsUnit x) : ∃ (n : ℕ) (a : α), ¬x ∣ a ∧ a₀ = x ^ n * a := by obtain ⟨a, ⟨n, rfl⟩, hm⟩ := wellFounded_dvdNotUnit.has_min {a \| ∃ n, x ^ n * a = a₀} ⟨a₀, 0, by rw [pow_zero, one_mul]⟩ refine ⟨n, a, ?_, rfl⟩; rintro ⟨d, rfl⟩ exact hm d ⟨n + 1, by rw [pow_succ, mul_assoc]⟩ ⟨(right_ne_zero_of_mul <\| right_ne_zero_of_mul h), x, hx, mul_comm _ _⟩ theorem WfDvdMonoid.max_power_factor [CommMonoidWithZero α] [WfDvdMonoid α] {a₀ x : α} (h : a₀ ≠ 0) (hx : Irreducible x) : ∃ (n : ℕ) (a : α), ¬x ∣ a ∧ a₀ = x ^ n * a := max_power_factor' h hx.not_unit theorem multiplicity.finite_of_not_isUnit [CancelCommMonoidWithZero α] [WfDvdMonoid α] {a b : α} (ha : ¬IsUnit a) (hb : b ≠ 0) : multiplicity.Finite a b := by obtain ⟨n, c, ndvd, rfl⟩ := WfDvdMonoid.max_power_factor' hb ha exact ⟨n, by rwa [pow_succ, mul_dvd_mul_iff_left (left_ne_zero_of_mul hb)]⟩ section Prio -- set_option default_priority 100 -- see Note [default priority] /-- unique factorization monoids. These are defined as `CancelCommMonoidWithZero`s with well-founded strict divisibility relations, but this is equivalent to more familiar definitions: Each element (except zero) is uniquely represented as a multiset of irreducible factors. Uniqueness is only up to associated elements. Each element (except zero) is non-uniquely represented as a multiset of prime factors. To define a UFD using the definition in terms of multisets of irreducible factors, use the definition `of_exists_unique_irreducible_factors` To define a UFD using the definition in terms of multisets of prime factors, use the definition `of_exists_prime_factors` -/ class UniqueFactorizationMonoid (α : Type) [CancelCommMonoidWithZero α] extends WfDvdMonoid α : Prop where protected irreducible_iff_prime : ∀ {a : α}, Irreducible a ↔ Prime a #align unique_factorization_monoid UniqueFactorizationMonoid /-- Can't be an instance because it would cause a loop `ufm → WfDvdMonoid → ufm → ...`. -/ theorem ufm_of_decomposition_of_wfDvdMonoid [CancelCommMonoidWithZero α] [WfDvdMonoid α] [DecompositionMonoid α] : UniqueFactorizationMonoid α := { ‹WfDvdMonoid α› with irreducible_iff_prime := irreducible_iff_prime } #align ufm_of_gcd_of_wf_dvd_monoid ufm_of_decomposition_of_wfDvdMonoid @[deprecated] alias ufm_of_gcd_of_wfDvdMonoid := ufm_of_decomposition_of_wfDvdMonoid instance Associates.ufm [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] : UniqueFactorizationMonoid (Associates α) := { (WfDvdMonoid.wfDvdMonoid_associates : WfDvdMonoid (Associates α)) with irreducible_iff_prime := by rw [← Associates.irreducible_iff_prime_iff] apply UniqueFactorizationMonoid.irreducible_iff_prime } #align associates.ufm Associates.ufm end Prio namespace UniqueFactorizationMonoid variable [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] theorem exists_prime_factors (a : α) : a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a := by simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime] apply WfDvdMonoid.exists_factors a #align unique_factorization_monoid.exists_prime_factors UniqueFactorizationMonoid.exists_prime_factors instance : DecompositionMonoid α where primal a := by obtain rfl \| ha := eq_or_ne a 0; · exact isPrimal_zero obtain ⟨f, hf, u, rfl⟩ := exists_prime_factors a ha exact ((Submonoid.isPrimal α).multiset_prod_mem f (hf · ·\|>.isPrimal)).mul u.isUnit.isPrimal lemma exists_prime_iff : (∃ (p : α), Prime p) ↔ ∃ (x : α), x ≠ 0 ∧ ¬ IsUnit x := by refine ⟨fun ⟨p, hp⟩ ↦ ⟨p, hp.ne_zero, hp.not_unit⟩, fun ⟨x, hx₀, hxu⟩ ↦ ?_⟩ obtain ⟨f, hf, -⟩ := WfDvdMonoid.exists_irreducible_factor hxu hx₀ exact ⟨f, UniqueFactorizationMonoid.irreducible_iff_prime.mp hf⟩ @[elab_as_elim] theorem induction_on_prime {P : α → Prop} (a : α) (h₁ : P 0) (h₂ : ∀ x : α, IsUnit x → P x) (h₃ : ∀ a p : α, a ≠ 0 → Prime p → P a → P (p a)) : P a := by simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime] at h₃ exact WfDvdMonoid.induction_on_irreducible a h₁ h₂ h₃ #align unique_factorization_monoid.induction_on_prime UniqueFactorizationMonoid.induction_on_prime end UniqueFactorizationMonoid theorem prime_factors_unique [CancelCommMonoidWithZero α] : ∀ {f g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by classical intro f induction' f using Multiset.induction_on with p f ih · intros g _ hg h exact Multiset.rel_zero_left.2 <\| Multiset.eq_zero_of_forall_not_mem fun x hx => have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm (hg x hx).not_unit <\| isUnit_iff_dvd_one.2 <\| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this) · intros g hf hg hfg let ⟨b, hbg, hb⟩ := (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <\| hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp) haveI := Classical.decEq α rw [← Multiset.cons_erase hbg] exact Multiset.Rel.cons hb (ih (fun q hq => hf _ (by simp [hq])) (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq)) (Associated.of_mul_left (by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb (hf p (by simp)).ne_zero)) #align prime_factors_unique prime_factors_unique namespace UniqueFactorizationMonoid variable [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] theorem factors_unique {f g : Multiset α} (hf : ∀ x ∈ f, Irreducible x) (hg : ∀ x ∈ g, Irreducible x) (h : f.prod ~ᵤ g.prod) : Multiset.Rel Associated f g := prime_factors_unique (fun x hx => UniqueFactorizationMonoid.irreducible_iff_prime.mp (hf x hx)) (fun x hx => UniqueFactorizationMonoid.irreducible_iff_prime.mp (hg x hx)) h #align unique_factorization_monoid.factors_unique UniqueFactorizationMonoid.factors_unique end UniqueFactorizationMonoid /-- If an irreducible has a prime factorization, then it is an associate of one of its prime factors. -/ theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α} (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by haveI := Classical.decEq α refine @Multiset.induction_on _ (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1 · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim · rintro p s _ ⟨u, hu⟩ hs use p have hs0 : s = 0 := by by_contra hs0 obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0 apply (hs q (by simp [hq])).2.1 refine (ha.isUnit_or_isUnit (?_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left ?_ · rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu, mul_comm, mul_comm p _, mul_assoc] simp apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _) apply (hs p (Multiset.mem_cons_self _ _)).2.1 simp only [mul_one, Multiset.prod_cons, Multiset.prod_zero, hs0] at * exact ⟨Associated.symm ⟨u, hu⟩, rfl⟩ #align prime_factors_irreducible prime_factors_irreducible section ExistsPrimeFactors variable [CancelCommMonoidWithZero α] variable (pf : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α := ⟨by classical refine RelHomClass.wellFounded (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt · intro a by_cases h : a = 0 · exact ⊤ exact ↑(Multiset.card (Classical.choose (pf a h))) rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩ rw [dif_neg ane0] by_cases h : b = 0 · simp [h, lt_top_iff_ne_top] · rw [dif_neg h] erw [WithTop.coe_lt_coe] have cne0 : c ≠ 0 := by refine mt (fun con => ?_) h rw [b_eq, con, mul_zero] calc Multiset.card (Classical.choose (pf a ane0)) < _ + Multiset.card (Classical.choose (pf c cne0)) := lt_add_of_pos_right _ (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_)) _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) := (Multiset.card_add _ _).symm _ = Multiset.card (Classical.choose (pf b h)) := Multiset.card_eq_card_of_rel (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_) · convert (Classical.choose_spec (pf c cne0)).2.symm rw [con, Multiset.prod_zero] · intro x hadd rw [Multiset.mem_add] at hadd cases' hadd with h h <;> apply (Classical.choose_spec (pf _ _)).1 _ h <;> assumption · rw [Multiset.prod_add] trans a * c · apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption · rw [← b_eq] apply (Classical.choose_spec (pf _ _)).2.symm; assumption⟩ #align wf_dvd_monoid.of_exists_prime_factors WfDvdMonoid.of_exists_prime_factors theorem irreducible_iff_prime_of_exists_prime_factors {p : α} : Irreducible p ↔ Prime p := by by_cases hp0 : p = 0 · simp [hp0] refine ⟨fun h => ?_, Prime.irreducible⟩ obtain ⟨f, hf⟩ := pf p hp0 obtain ⟨q, hq, rfl⟩ := prime_factors_irreducible h hf rw [hq.prime_iff] exact hf.1 q (Multiset.mem_singleton_self _) #align irreducible_iff_prime_of_exists_prime_factors irreducible_iff_prime_of_exists_prime_factors theorem UniqueFactorizationMonoid.of_exists_prime_factors : UniqueFactorizationMonoid α := { WfDvdMonoid.of_exists_prime_factors pf with irreducible_iff_prime := irreducible_iff_prime_of_exists_prime_factors pf } #align unique_factorization_monoid.of_exists_prime_factors UniqueFactorizationMonoid.of_exists_prime_factors end ExistsPrimeFactors theorem UniqueFactorizationMonoid.iff_exists_prime_factors [CancelCommMonoidWithZero α] : UniqueFactorizationMonoid α ↔ ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a := ⟨fun h => @UniqueFactorizationMonoid.exists_prime_factors _ _ h, UniqueFactorizationMonoid.of_exists_prime_factors⟩ #align unique_factorization_monoid.iff_exists_prime_factors UniqueFactorizationMonoid.iff_exists_prime_factors section variable {β : Type} [CancelCommMonoidWithZero α] [CancelCommMonoidWithZero β] theorem MulEquiv.uniqueFactorizationMonoid (e : α ≃ β) (hα : UniqueFactorizationMonoid α) : UniqueFactorizationMonoid β := by rw [UniqueFactorizationMonoid.iff_exists_prime_factors] at hα ⊢ intro a ha obtain ⟨w, hp, u, h⟩ := hα (e.symm a) fun h => ha <\| by convert← map_zero e simp [← h] exact ⟨w.map e, fun b hb => let ⟨c, hc, he⟩ := Multiset.mem_map.1 hb he ▸ e.prime_iff.1 (hp c hc), Units.map e.toMonoidHom u, by erw [Multiset.prod_hom, ← e.map_mul, h] simp⟩ #align mul_equiv.unique_factorization_monoid MulEquiv.uniqueFactorizationMonoid theorem MulEquiv.uniqueFactorizationMonoid_iff (e : α ≃* β) : UniqueFactorizationMonoid α ↔ UniqueFactorizationMonoid β := ⟨e.uniqueFactorizationMonoid, e.symm.uniqueFactorizationMonoid⟩ #align mul_equiv.unique_factorization_monoid_iff MulEquiv.uniqueFactorizationMonoid_iff end theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α] (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a) (uif : ∀ f g : Multiset α, (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g) (p : α) : Irreducible p ↔ Prime p := letI := Classical.decEq α ⟨ fun hpi => ⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ => if hab0 : a * b = 0 then (eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by simp [hb0] else by have hx0 : x ≠ 0 := fun hx0 => by simp_all have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0 have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0 cases' eif x hx0 with fx hfx cases' eif a ha0 with fa hfa cases' eif b hb0 with fb hfb have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by apply uif · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _) · exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _) calc Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _ _ ~ᵤ fa.prod * fb.prod := hfa.2.symm.mul_mul hfb.2.symm _ = _ := by rw [Multiset.prod_add] exact let ⟨q, hqf, hq⟩ := Multiset.exists_mem_of_rel_of_mem h (Multiset.mem_cons_self p _) (Multiset.mem_add.1 hqf).elim (fun hqa => Or.inl <\| hq.dvd_iff_dvd_left.2 <\| hfa.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqa)) fun hqb => Or.inr <\| hq.dvd_iff_dvd_left.2 <\| hfb.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqb)⟩, Prime.irreducible⟩ #align irreducible_iff_prime_of_exists_unique_irreducible_factors irreducible_iff_prime_of_exists_unique_irreducible_factors theorem UniqueFactorizationMonoid.of_exists_unique_irreducible_factors [CancelCommMonoidWithZero α] (eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a) (uif : ∀ f g : Multiset α, (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g) : UniqueFactorizationMonoid α := UniqueFactorizationMonoid.of_exists_prime_factors (by convert eif using 7 simp_rw [irreducible_iff_prime_of_exists_unique_irreducible_factors eif uif]) #align unique_factorization_monoid.of_exists_unique_irreducible_factors UniqueFactorizationMonoid.of_exists_unique_irreducible_factors namespace UniqueFactorizationMonoid variable [CancelCommMonoidWithZero α] variable [UniqueFactorizationMonoid α] open Classical in /-- Noncomputably determines the multiset of prime factors. -/ noncomputable def factors (a : α) : Multiset α := if h : a = 0 then 0 else Classical.choose (UniqueFactorizationMonoid.exists_prime_factors a h) #align unique_factorization_monoid.factors UniqueFactorizationMonoid.factors theorem factors_prod {a : α} (ane0 : a ≠ 0) : Associated (factors a).prod a := by rw [factors, dif_neg ane0] exact (Classical.choose_spec (exists_prime_factors a ane0)).2 #align unique_factorization_monoid.factors_prod UniqueFactorizationMonoid.factors_prod @[simp] theorem factors_zero : factors (0 : α) = 0 := by simp [factors] #align unique_factorization_monoid.factors_zero UniqueFactorizationMonoid.factors_zero theorem ne_zero_of_mem_factors {p a : α} (h : p ∈ factors a) : a ≠ 0 := by rintro rfl simp at h #align unique_factorization_monoid.ne_zero_of_mem_factors UniqueFactorizationMonoid.ne_zero_of_mem_factors theorem dvd_of_mem_factors {p a : α} (h : p ∈ factors a) : p ∣ a := dvd_trans (Multiset.dvd_prod h) (Associated.dvd (factors_prod (ne_zero_of_mem_factors h))) #align unique_factorization_monoid.dvd_of_mem_factors UniqueFactorizationMonoid.dvd_of_mem_factors theorem prime_of_factor {a : α} (x : α) (hx : x ∈ factors a) : Prime x := by have ane0 := ne_zero_of_mem_factors hx rw [factors, dif_neg ane0] at hx exact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 x hx #align unique_factorization_monoid.prime_of_factor UniqueFactorizationMonoid.prime_of_factor theorem irreducible_of_factor {a : α} : ∀ x : α, x ∈ factors a → Irreducible x := fun x h => (prime_of_factor x h).irreducible #align unique_factorization_monoid.irreducible_of_factor UniqueFactorizationMonoid.irreducible_of_factor @[simp] theorem factors_one : factors (1 : α) = 0 := by nontriviality α using factors rw [← Multiset.rel_zero_right] refine factors_unique irreducible_of_factor (fun x hx => (Multiset.not_mem_zero x hx).elim) ?_ rw [Multiset.prod_zero] exact factors_prod one_ne_zero #align unique_factorization_monoid.factors_one UniqueFactorizationMonoid.factors_one theorem exists_mem_factors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) : p ∣ a → ∃ q ∈ factors a, p ~ᵤ q := fun ⟨b, hb⟩ => have hb0 : b ≠ 0 := fun hb0 => by simp_all have : Multiset.Rel Associated (p ::ₘ factors b) (factors a) := factors_unique (fun x hx => (Multiset.mem_cons.1 hx).elim (fun h => h.symm ▸ hp) (irreducible_of_factor _)) irreducible_of_factor (Associated.symm <\| calc Multiset.prod (factors a) ~ᵤ a := factors_prod ha0 _ = p * b := hb _ ~ᵤ Multiset.prod (p ::ₘ factors b) := by rw [Multiset.prod_cons]; exact (factors_prod hb0).symm.mul_left _ ) Multiset.exists_mem_of_rel_of_mem this (by simp) #align unique_factorization_monoid.exists_mem_factors_of_dvd UniqueFactorizationMonoid.exists_mem_factors_of_dvd theorem exists_mem_factors {x : α} (hx : x ≠ 0) (h : ¬IsUnit x) : ∃ p, p ∈ factors x := by obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx obtain ⟨p, hp, _⟩ := exists_mem_factors_of_dvd hx hp' hp'x exact ⟨p, hp⟩ #align unique_factorization_monoid.exists_mem_factors UniqueFactorizationMonoid.exists_mem_factors open Classical in theorem factors_mul {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) : Multiset.Rel Associated (factors (x * y)) (factors x + factors y) := by refine factors_unique irreducible_of_factor (fun a ha => (Multiset.mem_add.mp ha).by_cases (irreducible_of_factor _) (irreducible_of_factor _)) ((factors_prod (mul_ne_zero hx hy)).trans ?_) rw [Multiset.prod_add] exact (Associated.mul_mul (factors_prod hx) (factors_prod hy)).symm #align unique_factorization_monoid.factors_mul UniqueFactorizationMonoid.factors_mul theorem factors_pow {x : α} (n : ℕ) : Multiset.Rel Associated (factors (x ^ n)) (n • factors x) := by match n with \| 0 => rw [zero_smul, pow_zero, factors_one, Multiset.rel_zero_right] \| n+1 => by_cases h0 : x = 0 · simp [h0, zero_pow n.succ_ne_zero, smul_zero] · rw [pow_succ', succ_nsmul'] refine Multiset.Rel.trans _ (factors_mul h0 (pow_ne_zero n h0)) ?_ refine Multiset.Rel.add ?_ <\| factors_pow n exact Multiset.rel_refl_of_refl_on fun y _ => Associated.refl _ #align unique_factorization_monoid.factors_pow UniqueFactorizationMonoid.factors_pow @[simp] theorem factors_pos (x : α) (hx : x ≠ 0) : 0 < factors x ↔ ¬IsUnit x := by constructor · intro h hx obtain ⟨p, hp⟩ := Multiset.exists_mem_of_ne_zero h.ne' exact (prime_of_factor _ hp).not_unit (isUnit_of_dvd_unit (dvd_of_mem_factors hp) hx) · intro h obtain ⟨p, hp⟩ := exists_mem_factors hx h exact bot_lt_iff_ne_bot.mpr (mt Multiset.eq_zero_iff_forall_not_mem.mp (not_forall.mpr ⟨p, not_not.mpr hp⟩)) #align unique_factorization_monoid.factors_pos UniqueFactorizationMonoid.factors_pos open Multiset in theorem factors_pow_count_prod [DecidableEq α] {x : α} (hx : x ≠ 0) : (∏ p ∈ (factors x).toFinset, p ^ (factors x).count p) ~ᵤ x := calc _ = prod (∑ a ∈ toFinset (factors x), count a (factors x) • {a}) := by simp only [prod_sum, prod_nsmul, prod_singleton] _ = prod (factors x) := by rw [toFinset_sum_count_nsmul_eq (factors x)] _ ~ᵤ x := factors_prod hx end UniqueFactorizationMonoid namespace UniqueFactorizationMonoid variable [CancelCommMonoidWithZero α] [NormalizationMonoid α] variable [UniqueFactorizationMonoid α] /-- Noncomputably determines the multiset of prime factors. -/ noncomputable def normalizedFactors (a : α) : Multiset α := Multiset.map normalize <\| factors a #align unique_factorization_monoid.normalized_factors UniqueFactorizationMonoid.normalizedFactors /-- An arbitrary choice of factors of `x : M` is exactly the (unique) normalized set of factors, if `M` has a trivial group of units. -/ @[simp] theorem factors_eq_normalizedFactors {M : Type} [CancelCommMonoidWithZero M] [UniqueFactorizationMonoid M] [Unique Mˣ] (x : M) : factors x = normalizedFactors x := by unfold normalizedFactors convert (Multiset.map_id (factors x)).symm ext p exact normalize_eq p #align unique_factorization_monoid.factors_eq_normalized_factors UniqueFactorizationMonoid.factors_eq_normalizedFactors theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) : Associated (normalizedFactors a).prod a := by rw [normalizedFactors, factors, dif_neg ane0] refine Associated.trans ?_ (Classical.choose_spec (exists_prime_factors a ane0)).2 rw [← Associates.mk_eq_mk_iff_associated, ← Associates.prod_mk, ← Associates.prod_mk, Multiset.map_map] congr 2 ext rw [Function.comp_apply, Associates.mk_normalize] #align unique_factorization_monoid.normalized_factors_prod UniqueFactorizationMonoid.normalizedFactors_prod theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by rw [normalizedFactors, factors] split_ifs with ane0; · simp intro x hx; rcases Multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩ rw [(normalize_associated _).prime_iff] exact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 y hy #align unique_factorization_monoid.prime_of_normalized_factor UniqueFactorizationMonoid.prime_of_normalized_factor theorem irreducible_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Irreducible x := fun x h => (prime_of_normalized_factor x h).irreducible #align unique_factorization_monoid.irreducible_of_normalized_factor UniqueFactorizationMonoid.irreducible_of_normalized_factor theorem normalize_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by rw [normalizedFactors, factors] split_ifs with h; · simp intro x hx obtain ⟨y, _, rfl⟩ := Multiset.mem_map.1 hx apply normalize_idem #align unique_factorization_monoid.normalize_normalized_factor UniqueFactorizationMonoid.normalize_normalized_factor theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) : normalizedFactors a = {normalize a} := by obtain ⟨p, a_assoc, hp⟩ := prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩ have p_mem : p ∈ normalizedFactors a := by rw [hp] exact Multiset.mem_singleton_self _ convert hp rwa [← normalize_normalized_factor p p_mem, normalize_eq_normalize_iff, dvd_dvd_iff_associated] #align unique_factorization_monoid.normalized_factors_irreducible UniqueFactorizationMonoid.normalizedFactors_irreducible theorem normalizedFactors_eq_of_dvd (a : α) : ∀ᵉ (p ∈ normalizedFactors a) (q ∈ normalizedFactors a), p ∣ q → p = q := by intro p hp q hq hdvd convert normalize_eq_normalize hdvd ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd) <;> apply (normalize_normalized_factor _ ‹_›).symm #align unique_factorization_monoid.normalized_factors_eq_of_dvd UniqueFactorizationMonoid.normalizedFactors_eq_of_dvd theorem exists_mem_normalizedFactors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) : p ∣ a → ∃ q ∈ normalizedFactors a, p ~ᵤ q := fun ⟨b, hb⟩ => have hb0 : b ≠ 0 := fun hb0 => by simp_all have : Multiset.Rel Associated (p ::ₘ normalizedFactors b) (normalizedFactors a) := factors_unique (fun x hx => (Multiset.mem_cons.1 hx).elim (fun h => h.symm ▸ hp) (irreducible_of_normalized_factor _)) irreducible_of_normalized_factor (Associated.symm <\| calc Multiset.prod (normalizedFactors a) ~ᵤ a := normalizedFactors_prod ha0 _ = p b := hb _ ~ᵤ Multiset.prod (p ::ₘ normalizedFactors b) := by rw [Multiset.prod_cons] exact (normalizedFactors_prod hb0).symm.mul_left _ ) Multiset.exists_mem_of_rel_of_mem this (by simp) #align unique_factorization_monoid.exists_mem_normalized_factors_of_dvd UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd theorem exists_mem_normalizedFactors {x : α} (hx : x ≠ 0) (h : ¬IsUnit x) : ∃ p, p ∈ normalizedFactors x := by obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx obtain ⟨p, hp, _⟩ := exists_mem_normalizedFactors_of_dvd hx hp' hp'x exact ⟨p, hp⟩ #align unique_factorization_monoid.exists_mem_normalized_factors UniqueFactorizationMonoid.exists_mem_normalizedFactors @[simp] theorem normalizedFactors_zero : normalizedFactors (0 : α) = 0 := by simp [normalizedFactors, factors] #align unique_factorization_monoid.normalized_factors_zero UniqueFactorizationMonoid.normalizedFactors_zero @[simp] theorem normalizedFactors_one : normalizedFactors (1 : α) = 0 := by cases' subsingleton_or_nontrivial α with h h · dsimp [normalizedFactors, factors] simp [Subsingleton.elim (1:α) 0] · rw [← Multiset.rel_zero_right] apply factors_unique irreducible_of_normalized_factor · intro x hx exfalso apply Multiset.not_mem_zero x hx · apply normalizedFactors_prod one_ne_zero #align unique_factorization_monoid.normalized_factors_one UniqueFactorizationMonoid.normalizedFactors_one @[simp] theorem normalizedFactors_mul {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) : normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y := by have h : (normalize : α → α) = Associates.out ∘ Associates.mk := by ext rw [Function.comp_apply, Associates.out_mk] rw [← Multiset.map_id' (normalizedFactors (x * y)), ← Multiset.map_id' (normalizedFactors x), ← Multiset.map_id' (normalizedFactors y), ← Multiset.map_congr rfl normalize_normalized_factor, ← Multiset.map_congr rfl normalize_normalized_factor, ← Multiset.map_congr rfl normalize_normalized_factor, ← Multiset.map_add, h, ← Multiset.map_map Associates.out, eq_comm, ← Multiset.map_map Associates.out] refine congr rfl ?_ apply Multiset.map_mk_eq_map_mk_of_rel apply factors_unique · intro x hx rcases Multiset.mem_add.1 hx with (hx \| hx) <;> exact irreducible_of_normalized_factor x hx · exact irreducible_of_normalized_factor · rw [Multiset.prod_add] exact ((normalizedFactors_prod hx).mul_mul (normalizedFactors_prod hy)).trans (normalizedFactors_prod (mul_ne_zero hx hy)).symm #align unique_factorization_monoid.normalized_factors_mul UniqueFactorizationMonoid.normalizedFactors_mul @[simp] theorem normalizedFactors_pow {x : α} (n : ℕ) : normalizedFactors (x ^ n) = n • normalizedFactors x := by induction' n with n ih · simp by_cases h0 : x = 0 · simp [h0, zero_pow n.succ_ne_zero, smul_zero] rw [pow_succ', succ_nsmul', normalizedFactors_mul h0 (pow_ne_zero _ h0), ih] #align unique_factorization_monoid.normalized_factors_pow UniqueFactorizationMonoid.normalizedFactors_pow theorem _root_.Irreducible.normalizedFactors_pow {p : α} (hp : Irreducible p) (k : ℕ) : normalizedFactors (p ^ k) = Multiset.replicate k (normalize p) := by rw [UniqueFactorizationMonoid.normalizedFactors_pow, normalizedFactors_irreducible hp, Multiset.nsmul_singleton] #align irreducible.normalized_factors_pow Irreducible.normalizedFactors_pow theorem normalizedFactors_prod_eq (s : Multiset α) (hs : ∀ a ∈ s, Irreducible a) : normalizedFactors s.prod = s.map normalize := by induction' s using Multiset.induction with a s ih · rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero] · have ia := hs a (Multiset.mem_cons_self a _) have ib := fun b h => hs b (Multiset.mem_cons_of_mem h) obtain rfl \| ⟨b, hb⟩ := s.empty_or_exists_mem · rw [Multiset.cons_zero, Multiset.prod_singleton, Multiset.map_singleton, normalizedFactors_irreducible ia] haveI := nontrivial_of_ne b 0 (ib b hb).ne_zero rw [Multiset.prod_cons, Multiset.map_cons, normalizedFactors_mul ia.ne_zero (Multiset.prod_ne_zero fun h => (ib 0 h).ne_zero rfl), normalizedFactors_irreducible ia, ih ib, Multiset.singleton_add] #align unique_factorization_monoid.normalized_factors_prod_eq UniqueFactorizationMonoid.normalizedFactors_prod_eq theorem dvd_iff_normalizedFactors_le_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) : x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y := by constructor · rintro ⟨c, rfl⟩ simp [hx, right_ne_zero_of_mul hy] · rw [← (normalizedFactors_prod hx).dvd_iff_dvd_left, ← (normalizedFactors_prod hy).dvd_iff_dvd_right] apply Multiset.prod_dvd_prod_of_le #align unique_factorization_monoid.dvd_iff_normalized_factors_le_normalized_factors UniqueFactorizationMonoid.dvd_iff_normalizedFactors_le_normalizedFactors theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) : x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by refine ⟨fun h => ?_, fun h => (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩ apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors] all_goals simp [, h.dvd, h.symm.dvd] #align unique_factorization_monoid.associated_iff_normalized_factors_eq_normalized_factors UniqueFactorizationMonoid.associated_iff_normalizedFactors_eq_normalizedFactors theorem normalizedFactors_of_irreducible_pow {p : α} (hp : Irreducible p) (k : ℕ) : normalizedFactors (p ^ k) = Multiset.replicate k (normalize p) := by rw [normalizedFactors_pow, normalizedFactors_irreducible hp, Multiset.nsmul_singleton] #align unique_factorization_monoid.normalized_factors_of_irreducible_pow UniqueFactorizationMonoid.normalizedFactors_of_irreducible_pow theorem zero_not_mem_normalizedFactors (x : α) : (0 : α) ∉ normalizedFactors x := fun h => Prime.ne_zero (prime_of_normalized_factor _ h) rfl #align unique_factorization_monoid.zero_not_mem_normalized_factors UniqueFactorizationMonoid.zero_not_mem_normalizedFactors theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a := by by_cases hcases : a = 0 · rw [hcases] exact dvd_zero p · exact dvd_trans (Multiset.dvd_prod H) (Associated.dvd (normalizedFactors_prod hcases)) #align unique_factorization_monoid.dvd_of_mem_normalized_factors UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) : p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by constructor · intro h exact ⟨prime_of_normalized_factor p h, dvd_of_mem_normalizedFactors h⟩ · rintro ⟨hprime, hdvd⟩ obtain ⟨q, hqmem, hqeq⟩ := exists_mem_normalizedFactors_of_dvd hx hprime.irreducible hdvd rw [associated_iff_eq] at hqeq exact hqeq ▸ hqmem theorem exists_associated_prime_pow_of_unique_normalized_factor {p r : α} (h : ∀ {m}, m ∈ normalizedFactors r → m = p) (hr : r ≠ 0) : ∃ i : ℕ, Associated (p ^ i) r := by use Multiset.card.toFun (normalizedFactors r) have := UniqueFactorizationMonoid.normalizedFactors_prod hr rwa [Multiset.eq_replicate_of_mem fun b => h, Multiset.prod_replicate] at this #align unique_factorization_monoid.exists_associated_prime_pow_of_unique_normalized_factor UniqueFactorizationMonoid.exists_associated_prime_pow_of_unique_normalized_factor theorem normalizedFactors_prod_of_prime [Nontrivial α] [Unique αˣ] {m : Multiset α} (h : ∀ p ∈ m, Prime p) : normalizedFactors m.prod = m := by simpa only [← Multiset.rel_eq, ← associated_eq_eq] using prime_factors_unique prime_of_normalized_factor h (normalizedFactors_prod (m.prod_ne_zero_of_prime h)) #align unique_factorization_monoid.normalized_factors_prod_of_prime UniqueFactorizationMonoid.normalizedFactors_prod_of_prime theorem mem_normalizedFactors_eq_of_associated {a b c : α} (ha : a ∈ normalizedFactors c) (hb : b ∈ normalizedFactors c) (h : Associated a b) : a = b := by rw [← normalize_normalized_factor a ha, ← normalize_normalized_factor b hb, normalize_eq_normalize_iff] exact Associated.dvd_dvd h #align unique_factorization_monoid.mem_normalized_factors_eq_of_associated UniqueFactorizationMonoid.mem_normalizedFactors_eq_of_associated @[simp] theorem normalizedFactors_pos (x : α) (hx : x ≠ 0) : 0 < normalizedFactors x ↔ ¬IsUnit x := by constructor · intro h hx obtain ⟨p, hp⟩ := Multiset.exists_mem_of_ne_zero h.ne' exact (prime_of_normalized_factor _ hp).not_unit (isUnit_of_dvd_unit (dvd_of_mem_normalizedFactors hp) hx) · intro h obtain ⟨p, hp⟩ := exists_mem_normalizedFactors hx h exact bot_lt_iff_ne_bot.mpr (mt Multiset.eq_zero_iff_forall_not_mem.mp (not_forall.mpr ⟨p, not_not.mpr hp⟩)) #align unique_factorization_monoid.normalized_factors_pos UniqueFactorizationMonoid.normalizedFactors_pos theorem dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) : DvdNotUnit x y ↔ normalizedFactors x < normalizedFactors y := by constructor · rintro ⟨_, c, hc, rfl⟩ simp only [hx, right_ne_zero_of_mul hy, normalizedFactors_mul, Ne, not_false_iff, lt_add_iff_pos_right, normalizedFactors_pos, hc] · intro h exact dvdNotUnit_of_dvd_of_not_dvd ((dvd_iff_normalizedFactors_le_normalizedFactors hx hy).mpr h.le) (mt (dvd_iff_normalizedFactors_le_normalizedFactors hy hx).mp h.not_le) #align unique_factorization_monoid.dvd_not_unit_iff_normalized_factors_lt_normalized_factors UniqueFactorizationMonoid.dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors theorem normalizedFactors_multiset_prod (s : Multiset α) (hs : 0 ∉ s) : normalizedFactors (s.prod) = (s.map normalizedFactors).sum := by cases subsingleton_or_nontrivial α · obtain rfl : s = 0 := by apply Multiset.eq_zero_of_forall_not_mem intro _ convert hs simp induction s using Multiset.induction with \| empty => simp \| cons _ _ IH => rw [Multiset.prod_cons, Multiset.map_cons, Multiset.sum_cons, normalizedFactors_mul, IH] · exact fun h ↦ hs (Multiset.mem_cons_of_mem h) · exact fun h ↦ hs (h ▸ Multiset.mem_cons_self _ _) · apply Multiset.prod_ne_zero exact fun h ↦ hs (Multiset.mem_cons_of_mem h) end UniqueFactorizationMonoid namespace UniqueFactorizationMonoid open scoped Classical open Multiset Associates variable [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] /-- Noncomputably defines a `normalizationMonoid` structure on a `UniqueFactorizationMonoid`. -/ protected noncomputable def normalizationMonoid : NormalizationMonoid α := normalizationMonoidOfMonoidHomRightInverse { toFun := fun a : Associates α => if a = 0 then 0 else ((normalizedFactors a).map (Classical.choose mk_surjective.hasRightInverse : Associates α → α)).prod map_one' := by nontriviality α; simp map_mul' := fun x y => by by_cases hx : x = 0 · simp [hx] by_cases hy : y = 0 · simp [hy] simp [hx, hy] } (by intro x dsimp by_cases hx : x = 0 · simp [hx] have h : Associates.mkMonoidHom ∘ Classical.choose mk_surjective.hasRightInverse = (id : Associates α → Associates α) := by ext x rw [Function.comp_apply, mkMonoidHom_apply, Classical.choose_spec mk_surjective.hasRightInverse x] rfl rw [if_neg hx, ← mkMonoidHom_apply, MonoidHom.map_multiset_prod, map_map, h, map_id, ← associated_iff_eq] apply normalizedFactors_prod hx) #align unique_factorization_monoid.normalization_monoid UniqueFactorizationMonoid.normalizationMonoid end UniqueFactorizationMonoid namespace UniqueFactorizationMonoid variable {R : Type} [CancelCommMonoidWithZero R] [UniqueFactorizationMonoid R] theorem isRelPrime_iff_no_prime_factors {a b : R} (ha : a ≠ 0) : IsRelPrime a b ↔ ∀ ⦃d⦄, d ∣ a → d ∣ b → ¬Prime d := ⟨fun h _ ha hb ↦ (·.not_unit <\| h ha hb), fun h ↦ WfDvdMonoid.isRelPrime_of_no_irreducible_factors (ha ·.1) fun _ irr ha hb ↦ h ha hb (UniqueFactorizationMonoid.irreducible_iff_prime.mp irr)⟩ #align unique_factorization_monoid.no_factors_of_no_prime_factors UniqueFactorizationMonoid.isRelPrime_iff_no_prime_factors /-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`. Compare `IsCoprime.dvd_of_dvd_mul_left`. -/ theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) (h : ∀ ⦃d⦄, d ∣ a → d ∣ c → ¬Prime d) : a ∣ b * c → a ∣ b := ((isRelPrime_iff_no_prime_factors ha).mpr h).dvd_of_dvd_mul_right #align unique_factorization_monoid.dvd_of_dvd_mul_left_of_no_prime_factors UniqueFactorizationMonoid.dvd_of_dvd_mul_left_of_no_prime_factors /-- Euclid's lemma: if `a ∣ b * c` and `a` and `b` have no common prime factors, `a ∣ c`. Compare `IsCoprime.dvd_of_dvd_mul_right`. -/ theorem dvd_of_dvd_mul_right_of_no_prime_factors {a b c : R} (ha : a ≠ 0) (no_factors : ∀ {d}, d ∣ a → d ∣ b → ¬Prime d) : a ∣ b * c → a ∣ c := by simpa [mul_comm b c] using dvd_of_dvd_mul_left_of_no_prime_factors ha @no_factors #align unique_factorization_monoid.dvd_of_dvd_mul_right_of_no_prime_factors UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors /-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing out their common factor `c'` gives `a'` and `b'` with no factors in common. -/ theorem exists_reduced_factors : ∀ a ≠ (0 : R), ∀ b, ∃ a' b' c', IsRelPrime a' b' ∧ c' * a' = a ∧ c' * b' = b := by intro a refine induction_on_prime a ?_ ?_ ?_ · intros contradiction · intro a a_unit _ b use a, b, 1 constructor · intro p p_dvd_a _ exact isUnit_of_dvd_unit p_dvd_a a_unit · simp · intro a p a_ne_zero p_prime ih_a pa_ne_zero b by_cases h : p ∣ b · rcases h with ⟨b, rfl⟩ obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b refine ⟨a', b', p * c', @no_factor, ?_, ?_⟩ · rw [mul_assoc, ha'] · rw [mul_assoc, hb'] · obtain ⟨a', b', c', coprime, rfl, rfl⟩ := ih_a a_ne_zero b refine ⟨p * a', b', c', ?_, mul_left_comm _ _ _, rfl⟩ intro q q_dvd_pa' q_dvd_b' cases' p_prime.left_dvd_or_dvd_right_of_dvd_mul q_dvd_pa' with p_dvd_q q_dvd_a' · have : p ∣ c' * b' := dvd_mul_of_dvd_right (p_dvd_q.trans q_dvd_b') _ contradiction exact coprime q_dvd_a' q_dvd_b' #align unique_factorization_monoid.exists_reduced_factors UniqueFactorizationMonoid.exists_reduced_factors theorem exists_reduced_factors' (a b : R) (hb : b ≠ 0) : ∃ a' b' c', IsRelPrime a' b' ∧ c' * a' = a ∧ c' * b' = b := let ⟨b', a', c', no_factor, hb, ha⟩ := exists_reduced_factors b hb a ⟨a', b', c', fun _ hpb hpa => no_factor hpa hpb, ha, hb⟩ #align unique_factorization_monoid.exists_reduced_factors' UniqueFactorizationMonoid.exists_reduced_factors' theorem pow_right_injective {a : R} (ha0 : a ≠ 0) (ha1 : ¬IsUnit a) : Function.Injective (a ^ · : ℕ → R) := by letI := Classical.decEq R intro i j hij letI : Nontrivial R := ⟨⟨a, 0, ha0⟩⟩ letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid obtain ⟨p', hp', dvd'⟩ := WfDvdMonoid.exists_irreducible_factor ha1 ha0 obtain ⟨p, mem, _⟩ := exists_mem_normalizedFactors_of_dvd ha0 hp' dvd' have := congr_arg (fun x => Multiset.count p (normalizedFactors x)) hij simp only [normalizedFactors_pow, Multiset.count_nsmul] at this exact mul_right_cancel₀ (Multiset.count_ne_zero.mpr mem) this #align unique_factorization_monoid.pow_right_injective UniqueFactorizationMonoid.pow_right_injective theorem pow_eq_pow_iff {a : R} (ha0 : a ≠ 0) (ha1 : ¬IsUnit a) {i j : ℕ} : a ^ i = a ^ j ↔ i = j := (pow_right_injective ha0 ha1).eq_iff #align unique_factorization_monoid.pow_eq_pow_iff UniqueFactorizationMonoid.pow_eq_pow_iff section multiplicity variable [NormalizationMonoid R] variable [DecidableRel (Dvd.dvd : R → R → Prop)] open multiplicity Multiset theorem le_multiplicity_iff_replicate_le_normalizedFactors {a b : R} {n : ℕ} (ha : Irreducible a) (hb : b ≠ 0) : ↑n ≤ multiplicity a b ↔ replicate n (normalize a) ≤ normalizedFactors b := by rw [← pow_dvd_iff_le_multiplicity] revert b induction' n with n ih; · simp intro b hb constructor · rintro ⟨c, rfl⟩ rw [Ne, pow_succ', mul_assoc, mul_eq_zero, not_or] at hb rw [pow_succ', mul_assoc, normalizedFactors_mul hb.1 hb.2, replicate_succ, normalizedFactors_irreducible ha, singleton_add, cons_le_cons_iff, ← ih hb.2] apply Dvd.intro _ rfl · rw [Multiset.le_iff_exists_add] rintro ⟨u, hu⟩ rw [← (normalizedFactors_prod hb).dvd_iff_dvd_right, hu, prod_add, prod_replicate] exact (Associated.pow_pow <\| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl) #align unique_factorization_monoid.le_multiplicity_iff_replicate_le_normalized_factors UniqueFactorizationMonoid.le_multiplicity_iff_replicate_le_normalizedFactors /-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times the normalized factor occurs in the `normalizedFactors`. See also `count_normalizedFactors_eq` which expands the definition of `multiplicity` to produce a specification for `count (normalizedFactors _) _`.. -/ theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a) (hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by apply le_antisymm · apply PartENat.le_of_lt_add_one rw [← Nat.cast_one, ← Nat.cast_add, lt_iff_not_ge, ge_iff_le, le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le] simp rw [le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le] #align unique_factorization_monoid.multiplicity_eq_count_normalized_factors UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors /-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by the number of times it divides `x`. See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`. -/ theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p) (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) : (normalizedFactors x).count p = n := by letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _ by_cases hx0 : x = 0 · simp [hx0] at hlt rw [← PartENat.natCast_inj] convert (multiplicity_eq_count_normalizedFactors hp hx0).symm · exact hnorm.symm exact (multiplicity.eq_coe_iff.mpr ⟨hle, hlt⟩).symm #align unique_factorization_monoid.count_normalized_factors_eq UniqueFactorizationMonoid.count_normalizedFactors_eq /-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by the number of times it divides `x`. This is a slightly more general version of `UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`. See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`. -/ theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p) (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) : (normalizedFactors x).count p = n := by rcases hp with (rfl \| hp) · cases n · exact count_eq_zero.2 (zero_not_mem_normalizedFactors _) · rw [zero_pow (Nat.succ_ne_zero _)] at hle hlt exact absurd hle hlt · exact count_normalizedFactors_eq hp hnorm hle hlt #align unique_factorization_monoid.count_normalized_factors_eq' UniqueFactorizationMonoid.count_normalizedFactors_eq' /-- Deprecated. Use `WfDvdMonoid.max_power_factor` instead. -/ @[deprecated WfDvdMonoid.max_power_factor] theorem max_power_factor {a₀ x : R} (h : a₀ ≠ 0) (hx : Irreducible x) : ∃ n : ℕ, ∃ a : R, ¬x ∣ a ∧ a₀ = x ^ n * a := WfDvdMonoid.max_power_factor h hx #align unique_factorization_monoid.max_power_factor UniqueFactorizationMonoid.max_power_factor end multiplicity section Multiplicative variable [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] variable {β : Type} [CancelCommMonoidWithZero β] theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α) (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q) (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') : IsRelPrime (p ^ i p) (∏ p' ∈ s, p' ^ i p') := by have hp := is_prime _ (Finset.mem_insert_self _ _) refine (isRelPrime_iff_no_prime_factors <\| pow_ne_zero _ hp.ne_zero).mpr ?_ intro d hdp hdprod hd apply hps replace hdp := hd.dvd_of_dvd_pow hdp obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem' replace hdq := hd.dvd_of_dvd_pow hdq have : p ∣ q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq convert q_mem rw [Finset.mem_val, is_coprime _ (Finset.mem_insert_self p s) _ (Finset.mem_insert_of_mem q_mem) this] #align unique_factorization_monoid.prime_pow_coprime_prod_of_coprime_insert UniqueFactorizationMonoid.prime_pow_coprime_prod_of_coprime_insert /-- If `P` holds for units and powers of primes, and `P x ∧ P y` for coprime `x, y` implies `P (x y)`, then `P` holds on a product of powers of distinct primes. -/ -- @[elab_as_elim] Porting note: commented out theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ) (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q) (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i)) (hcp : ∀ {x y}, IsRelPrime x y → P x → P y → P (x * y)) : P (∏ p ∈ s, p ^ i p) := by letI := Classical.decEq α induction' s using Finset.induction_on with p f' hpf' ih · simpa using h1 isUnit_one rw [Finset.prod_insert hpf'] exact hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime) (hpr (i p) (is_prime _ (Finset.mem_insert_self _ _))) (ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' => is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq')) #align unique_factorization_monoid.induction_on_prime_power UniqueFactorizationMonoid.induction_on_prime_power /-- If `P` holds for `0`, units and powers of primes, and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`, then `P` holds on all `a : α`. -/ @[elab_as_elim] theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i)) (hcp : ∀ {x y}, IsRelPrime x y → P x → P y → P (x * y)) : P a := by letI := Classical.decEq α have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by rintro x y ⟨u, rfl⟩ hx exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit) by_cases ha0 : a = 0 · rwa [ha0] haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩ letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid refine P_of_associated (normalizedFactors_prod ha0) ?_ rw [← (normalizedFactors a).map_id, Finset.prod_multiset_map_count] refine induction_on_prime_power _ _ ?_ ?_ @h1 @hpr @hcp <;> simp only [Multiset.mem_toFinset] · apply prime_of_normalized_factor · apply normalizedFactors_eq_of_dvd #align unique_factorization_monoid.induction_on_coprime UniqueFactorizationMonoid.induction_on_coprime /-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f` is multiplicative on coprime elements, then `f` is multiplicative on all products of primes. -/ -- @[elab_as_elim] Porting note: commented out theorem multiplicative_prime_power {f : α → β} (s : Finset α) (i j : α → ℕ) (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q) (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y) (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i) (hcp : ∀ {x y}, IsRelPrime x y → f (x * y) = f x * f y) : f (∏ p ∈ s, p ^ (i p + j p)) = f (∏ p ∈ s, p ^ i p) * f (∏ p ∈ s, p ^ j p) := by letI := Classical.decEq α induction' s using Finset.induction_on with p s hps ih · simpa using h1 isUnit_one have hpr_p := is_prime _ (Finset.mem_insert_self _ _) have hpr_s : ∀ p ∈ s, Prime p := fun p hp => is_prime _ (Finset.mem_insert_of_mem hp) have hcp_p := fun i => prime_pow_coprime_prod_of_coprime_insert i p hps is_prime is_coprime have hcp_s : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q := fun p hp q hq => is_coprime p (Finset.mem_insert_of_mem hp) q (Finset.mem_insert_of_mem hq) rw [Finset.prod_insert hps, Finset.prod_insert hps, Finset.prod_insert hps, hcp (hcp_p _), hpr _ hpr_p, hcp (hcp_p _), hpr _ hpr_p, hcp (hcp_p (fun p => i p + j p)), hpr _ hpr_p, ih hpr_s hcp_s, pow_add, mul_assoc, mul_left_comm (f p ^ j p), mul_assoc] #align unique_factorization_monoid.multiplicative_prime_power UniqueFactorizationMonoid.multiplicative_prime_power /-- If `f` maps `p ^ i` to `(f p) ^ i` for primes `p`, and `f` is multiplicative on coprime elements, then `f` is multiplicative everywhere. -/ theorem multiplicative_of_coprime (f : α → β) (a b : α) (h0 : f 0 = 0) (h1 : ∀ {x y}, IsUnit y → f (x * y) = f x * f y) (hpr : ∀ {p} (i : ℕ), Prime p → f (p ^ i) = f p ^ i) (hcp : ∀ {x y}, IsRelPrime x y → f (x * y) = f x * f y) : f (a * b) = f a * f b := by letI := Classical.decEq α by_cases ha0 : a = 0 · rw [ha0, zero_mul, h0, zero_mul] by_cases hb0 : b = 0 · rw [hb0, mul_zero, h0, mul_zero] by_cases hf1 : f 1 = 0 · calc f (a * b) = f (a * b * 1) := by rw [mul_one] _ = 0 := by simp only [h1 isUnit_one, hf1, mul_zero] _ = f a * f (b * 1) := by simp only [h1 isUnit_one, hf1, mul_zero] _ = f a * f b := by rw [mul_one] haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩ letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid suffices f (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset, p ^ ((normalizedFactors a).count p + (normalizedFactors b).count p)) = f (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset, p ^ (normalizedFactors a).count p) * f (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset, p ^ (normalizedFactors b).count p) by obtain ⟨ua, a_eq⟩ := normalizedFactors_prod ha0 obtain ⟨ub, b_eq⟩ := normalizedFactors_prod hb0 rw [← a_eq, ← b_eq, mul_right_comm (Multiset.prod (normalizedFactors a)) ua (Multiset.prod (normalizedFactors b) * ub), h1 ua.isUnit, h1 ub.isUnit, h1 ua.isUnit, ← mul_assoc, h1 ub.isUnit, mul_right_comm _ (f ua), ← mul_assoc] congr rw [← (normalizedFactors a).map_id, ← (normalizedFactors b).map_id, Finset.prod_multiset_map_count, Finset.prod_multiset_map_count, Finset.prod_subset (Finset.subset_union_left (s₂:=(normalizedFactors b).toFinset)), Finset.prod_subset (Finset.subset_union_right (s₂:=(normalizedFactors b).toFinset)), ← Finset.prod_mul_distrib] · simp_rw [id, ← pow_add, this] all_goals simp only [Multiset.mem_toFinset] · intro p _ hpb simp [hpb] · intro p _ hpa simp [hpa] refine multiplicative_prime_power _ _ _ ?_ ?_ @h1 @hpr @hcp all_goals simp only [Multiset.mem_toFinset, Finset.mem_union] · rintro p (hpa \| hpb) <;> apply prime_of_normalized_factor <;> assumption · rintro p (hp \| hp) q (hq \| hq) hdvd <;> rw [← normalize_normalized_factor _ hp, ← normalize_normalized_factor _ hq] <;> exact normalize_eq_normalize hdvd ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd) #align unique_factorization_monoid.multiplicative_of_coprime UniqueFactorizationMonoid.multiplicative_of_coprime end Multiplicative end UniqueFactorizationMonoid namespace Associates open UniqueFactorizationMonoid Associated Multiset variable [CancelCommMonoidWithZero α] /-- `FactorSet α` representation elements of unique factorization domain as multisets. `Multiset α` produced by `normalizedFactors` are only unique up to associated elements, while the multisets in `FactorSet α` are unique by equality and restricted to irreducible elements. This gives us a representation of each element as a unique multisets (or the added ⊤ for 0), which has a complete lattice structure. Infimum is the greatest common divisor and supremum is the least common multiple. -/ abbrev FactorSet.{u} (α : Type u) [CancelCommMonoidWithZero α] : Type u := WithTop (Multiset { a : Associates α // Irreducible a }) #align associates.factor_set Associates.FactorSet attribute [local instance] Associated.setoid theorem FactorSet.coe_add {a b : Multiset { a : Associates α // Irreducible a }} : (↑(a + b) : FactorSet α) = a + b := by norm_cast #align associates.factor_set.coe_add Associates.FactorSet.coe_add theorem FactorSet.sup_add_inf_eq_add [DecidableEq (Associates α)] : ∀ a b : FactorSet α, a ⊔ b + a ⊓ b = a + b \| ⊤, b => show ⊤ ⊔ b + ⊤ ⊓ b = ⊤ + b by simp \| a, ⊤ => show a ⊔ ⊤ + a ⊓ ⊤ = a + ⊤ by simp \| WithTop.some a, WithTop.some b => show (a : FactorSet α) ⊔ b + (a : FactorSet α) ⊓ b = a + b by rw [← WithTop.coe_sup, ← WithTop.coe_inf, ← WithTop.coe_add, ← WithTop.coe_add, WithTop.coe_eq_coe] exact Multiset.union_add_inter _ _ #align associates.factor_set.sup_add_inf_eq_add Associates.FactorSet.sup_add_inf_eq_add /-- Evaluates the product of a `FactorSet` to be the product of the corresponding multiset, or `0` if there is none. -/ def FactorSet.prod : FactorSet α → Associates α \| ⊤ => 0 \| WithTop.some s => (s.map (↑)).prod #align associates.factor_set.prod Associates.FactorSet.prod @[simp] theorem prod_top : (⊤ : FactorSet α).prod = 0 := rfl #align associates.prod_top Associates.prod_top @[simp] theorem prod_coe {s : Multiset { a : Associates α // Irreducible a }} : FactorSet.prod (s : FactorSet α) = (s.map (↑)).prod := rfl #align associates.prod_coe Associates.prod_coe @[simp] theorem prod_add : ∀ a b : FactorSet α, (a + b).prod = a.prod * b.prod \| ⊤, b => show (⊤ + b).prod = (⊤ : FactorSet α).prod * b.prod by simp \| a, ⊤ => show (a + ⊤).prod = a.prod * (⊤ : FactorSet α).prod by simp \| WithTop.some a, WithTop.some b => by rw [← FactorSet.coe_add, prod_coe, prod_coe, prod_coe, Multiset.map_add, Multiset.prod_add] #align associates.prod_add Associates.prod_add @[gcongr] theorem prod_mono : ∀ {a b : FactorSet α}, a ≤ b → a.prod ≤ b.prod \| ⊤, b, h => by have : b = ⊤ := top_unique h rw [this, prod_top] \| a, ⊤, _ => show a.prod ≤ (⊤ : FactorSet α).prod by simp \| WithTop.some a, WithTop.some b, h => prod_le_prod <\| Multiset.map_le_map <\| WithTop.coe_le_coe.1 <\| h #align associates.prod_mono Associates.prod_mono theorem FactorSet.prod_eq_zero_iff [Nontrivial α] (p : FactorSet α) : p.prod = 0 ↔ p = ⊤ := by unfold FactorSet at p induction p -- TODO: `induction_eliminator` doesn't work with `abbrev` · simp only [iff_self_iff, eq_self_iff_true, Associates.prod_top] · rw [prod_coe, Multiset.prod_eq_zero_iff, Multiset.mem_map, eq_false WithTop.coe_ne_top, iff_false_iff, not_exists] exact fun a => not_and_of_not_right _ a.prop.ne_zero #align associates.factor_set.prod_eq_zero_iff Associates.FactorSet.prod_eq_zero_iff section count variable [DecidableEq (Associates α)] /-- `bcount p s` is the multiplicity of `p` in the FactorSet `s` (with bundled `p`)-/ def bcount (p : { a : Associates α // Irreducible a }) : FactorSet α → ℕ \| ⊤ => 0 \| WithTop.some s => s.count p #align associates.bcount Associates.bcount variable [∀ p : Associates α, Decidable (Irreducible p)] {p : Associates α} /-- `count p s` is the multiplicity of the irreducible `p` in the FactorSet `s`. If `p` is not irreducible, `count p s` is defined to be `0`. -/ def count (p : Associates α) : FactorSet α → ℕ := if hp : Irreducible p then bcount ⟨p, hp⟩ else 0 #align associates.count Associates.count @[simp] theorem count_some (hp : Irreducible p) (s : Multiset _) : count p (WithTop.some s) = s.count ⟨p, hp⟩ := by simp only [count, dif_pos hp, bcount] #align associates.count_some Associates.count_some @[simp] theorem count_zero (hp : Irreducible p) : count p (0 : FactorSet α) = 0 := by simp only [count, dif_pos hp, bcount, Multiset.count_zero] #align associates.count_zero Associates.count_zero theorem count_reducible (hp : ¬Irreducible p) : count p = 0 := dif_neg hp #align associates.count_reducible Associates.count_reducible end count section Mem /-- membership in a FactorSet (bundled version) -/ def BfactorSetMem : { a : Associates α // Irreducible a } → FactorSet α → Prop \| _, ⊤ => True \| p, some l => p ∈ l #align associates.bfactor_set_mem Associates.BfactorSetMem /-- `FactorSetMem p s` is the predicate that the irreducible `p` is a member of `s : FactorSet α`. If `p` is not irreducible, `p` is not a member of any `FactorSet`. -/ def FactorSetMem (p : Associates α) (s : FactorSet α) : Prop := letI : Decidable (Irreducible p) := Classical.dec _ if hp : Irreducible p then BfactorSetMem ⟨p, hp⟩ s else False #align associates.factor_set_mem Associates.FactorSetMem instance : Membership (Associates α) (FactorSet α) := ⟨FactorSetMem⟩ @[simp] theorem factorSetMem_eq_mem (p : Associates α) (s : FactorSet α) : FactorSetMem p s = (p ∈ s) := rfl #align associates.factor_set_mem_eq_mem Associates.factorSetMem_eq_mem theorem mem_factorSet_top {p : Associates α} {hp : Irreducible p} : p ∈ (⊤ : FactorSet α) := by dsimp only [Membership.mem]; dsimp only [FactorSetMem]; split_ifs; exact trivial #align associates.mem_factor_set_top Associates.mem_factorSet_top theorem mem_factorSet_some {p : Associates α} {hp : Irreducible p} {l : Multiset { a : Associates α // Irreducible a }} : p ∈ (l : FactorSet α) ↔ Subtype.mk p hp ∈ l := by dsimp only [Membership.mem]; dsimp only [FactorSetMem]; split_ifs; rfl #align associates.mem_factor_set_some Associates.mem_factorSet_some theorem reducible_not_mem_factorSet {p : Associates α} (hp : ¬Irreducible p) (s : FactorSet α) : ¬p ∈ s := fun h ↦ by rwa [← factorSetMem_eq_mem, FactorSetMem, dif_neg hp] at h #align associates.reducible_not_mem_factor_set Associates.reducible_not_mem_factorSet theorem irreducible_of_mem_factorSet {p : Associates α} {s : FactorSet α} (h : p ∈ s) : Irreducible p := by_contra fun hp ↦ reducible_not_mem_factorSet hp s h end Mem variable [UniqueFactorizationMonoid α] theorem unique' {p q : Multiset (Associates α)} : (∀ a ∈ p, Irreducible a) → (∀ a ∈ q, Irreducible a) → p.prod = q.prod → p = q := by apply Multiset.induction_on_multiset_quot p apply Multiset.induction_on_multiset_quot q intro s t hs ht eq refine Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ ?_) · exact fun a ha => irreducible_mk.1 <\| hs _ <\| Multiset.mem_map_of_mem _ ha · exact fun a ha => irreducible_mk.1 <\| ht _ <\| Multiset.mem_map_of_mem _ ha have eq' : (Quot.mk Setoid.r : α → Associates α) = Associates.mk := funext quot_mk_eq_mk rwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq #align associates.unique' Associates.unique' theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by -- TODO: `induction_eliminator` doesn't work with `abbrev` unfold FactorSet at p q induction p <;> induction q · rfl · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top] · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top] · congr 1 rw [← Multiset.map_eq_map Subtype.coe_injective] apply unique' _ _ h <;> · intro a ha obtain ⟨⟨a', irred⟩, -, rfl⟩ := Multiset.mem_map.mp ha rwa [Subtype.coe_mk] #align associates.factor_set.unique Associates.FactorSet.unique theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)} (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q := by refine ⟨?_, prod_le_prod⟩ rintro ⟨c, eqc⟩ refine Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx ↦ ?_) ?_⟩ · obtain h \| h := Multiset.mem_add.1 hx · exact hp x h · exact irreducible_of_factor _ h · rw [eqc, Multiset.prod_add] congr refine associated_iff_eq.mp (factors_prod fun hc => ?_).symm refine not_irreducible_zero (hq _ ?_) rw [← prod_eq_zero_iff, eqc, hc, mul_zero] #align associates.prod_le_prod_iff_le Associates.prod_le_prod_iff_le /-- This returns the multiset of irreducible factors as a `FactorSet`, a multiset of irreducible associates `WithTop`. -/ noncomputable def factors' (a : α) : Multiset { a : Associates α // Irreducible a } := (factors a).pmap (fun a ha => ⟨Associates.mk a, irreducible_mk.2 ha⟩) irreducible_of_factor #align associates.factors' Associates.factors' @[simp] theorem map_subtype_coe_factors' {a : α} : (factors' a).map (↑) = (factors a).map Associates.mk := by simp [factors', Multiset.map_pmap, Multiset.pmap_eq_map] #align associates.map_subtype_coe_factors' Associates.map_subtype_coe_factors' theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by obtain rfl \| hb := eq_or_ne b 0 · rw [associated_zero_iff_eq_zero] at h rw [h] have ha : a ≠ 0 := by contrapose! hb with ha rw [← associated_zero_iff_eq_zero, ← ha] exact h.symm rw [← Multiset.map_eq_map Subtype.coe_injective, map_subtype_coe_factors', map_subtype_coe_factors', ← rel_associated_iff_map_eq_map] exact factors_unique irreducible_of_factor irreducible_of_factor ((factors_prod ha).trans <\| h.trans <\| (factors_prod hb).symm) #align associates.factors'_cong Associates.factors'_cong /-- This returns the multiset of irreducible factors of an associate as a `FactorSet`, a multiset of irreducible associates `WithTop`. -/ noncomputable def factors (a : Associates α) : FactorSet α := by classical refine if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => factors' x) ?_ h intro a b hab apply Function.hfunext · have : a ~ᵤ 0 ↔ b ~ᵤ 0 := Iff.intro (fun ha0 => hab.symm.trans ha0) fun hb0 => hab.trans hb0 simp only [associated_zero_iff_eq_zero] at this simp only [quotient_mk_eq_mk, this, mk_eq_zero] exact fun ha hb _ => heq_of_eq <\| congr_arg some <\| factors'_cong hab #align associates.factors Associates.factors @[simp] theorem factors_zero : (0 : Associates α).factors = ⊤ := dif_pos rfl #align associates.factors_0 Associates.factors_zero @[deprecated (since := "2024-03-16")] alias factors_0 := factors_zero @[simp] theorem factors_mk (a : α) (h : a ≠ 0) : (Associates.mk a).factors = factors' a := by classical apply dif_neg apply mt mk_eq_zero.1 h #align associates.factors_mk Associates.factors_mk @[simp] theorem factors_prod (a : Associates α) : a.factors.prod = a := by rcases Associates.mk_surjective a with ⟨a, rfl⟩ rcases eq_or_ne a 0 with rfl \| ha · simp · simp [ha, prod_mk, mk_eq_mk_iff_associated, UniqueFactorizationMonoid.factors_prod, -Quotient.eq] #align associates.factors_prod Associates.factors_prod @[simp] theorem prod_factors [Nontrivial α] (s : FactorSet α) : s.prod.factors = s := FactorSet.unique <\| factors_prod _ #align associates.prod_factors Associates.prod_factors @[nontriviality] theorem factors_subsingleton [Subsingleton α] {a : Associates α} : a.factors = ⊤ := by have : Subsingleton (Associates α) := inferInstance convert factors_zero #align associates.factors_subsingleton Associates.factors_subsingleton theorem factors_eq_top_iff_zero {a : Associates α} : a.factors = ⊤ ↔ a = 0 := by nontriviality α exact ⟨fun h ↦ by rwa [← factors_prod a, FactorSet.prod_eq_zero_iff], fun h ↦ h ▸ factors_zero⟩ #align associates.factors_eq_none_iff_zero Associates.factors_eq_top_iff_zero @[deprecated] alias factors_eq_none_iff_zero := factors_eq_top_iff_zero theorem factors_eq_some_iff_ne_zero {a : Associates α} : (∃ s : Multiset { p : Associates α // Irreducible p }, a.factors = s) ↔ a ≠ 0 := by simp_rw [@eq_comm _ a.factors, ← WithTop.ne_top_iff_exists] exact factors_eq_top_iff_zero.not #align associates.factors_eq_some_iff_ne_zero Associates.factors_eq_some_iff_ne_zero theorem eq_of_factors_eq_factors {a b : Associates α} (h : a.factors = b.factors) : a = b := by have : a.factors.prod = b.factors.prod := by rw [h] rwa [factors_prod, factors_prod] at this #align associates.eq_of_factors_eq_factors Associates.eq_of_factors_eq_factors theorem eq_of_prod_eq_prod [Nontrivial α] {a b : FactorSet α} (h : a.prod = b.prod) : a = b := by have : a.prod.factors = b.prod.factors := by rw [h] rwa [prod_factors, prod_factors] at this #align associates.eq_of_prod_eq_prod Associates.eq_of_prod_eq_prod @[simp] theorem factors_mul (a b : Associates α) : (a * b).factors = a.factors + b.factors := by nontriviality α refine eq_of_prod_eq_prod <\| eq_of_factors_eq_factors ?_ rw [prod_add, factors_prod, factors_prod, factors_prod] #align associates.factors_mul Associates.factors_mul @[gcongr] theorem factors_mono : ∀ {a b : Associates α}, a ≤ b → a.factors ≤ b.factors \| s, t, ⟨d, eq⟩ => by rw [eq, factors_mul]; exact le_add_of_nonneg_right bot_le #align associates.factors_mono Associates.factors_mono @[simp] theorem factors_le {a b : Associates α} : a.factors ≤ b.factors ↔ a ≤ b := by refine ⟨fun h ↦ ?_, factors_mono⟩ have : a.factors.prod ≤ b.factors.prod := prod_mono h rwa [factors_prod, factors_prod] at this #align associates.factors_le Associates.factors_le section count variable [DecidableEq (Associates α)] [∀ p : Associates α, Decidable (Irreducible p)] theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0) (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) : a.factors = b.factors := by obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb rw [h_sa, h_sb] at h ⊢ rw [WithTop.coe_eq_coe] have h_count : ∀ (p : Associates α) (hp : Irreducible p), sa.count ⟨p, hp⟩ = sb.count ⟨p, hp⟩ := by intro p hp rw [← count_some, ← count_some, h p hp] apply Multiset.toFinsupp.injective ext ⟨p, hp⟩ rw [Multiset.toFinsupp_apply, Multiset.toFinsupp_apply, h_count p hp] #align associates.eq_factors_of_eq_counts Associates.eq_factors_of_eq_counts theorem eq_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0) (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) : a = b := eq_of_factors_eq_factors (eq_factors_of_eq_counts ha hb h) #align associates.eq_of_eq_counts Associates.eq_of_eq_counts	Mathlib/RingTheory/UniqueFactorizationDomain.lean	1,561	1,569	theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p) (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by	by_cases ha : a = 0 · simp_all obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb rw [h_sa, h_sb] at h ⊢ rw [count_some hp, count_some hp]; rw [WithTop.coe_le_coe] at h exact Multiset.count_le_of_le _ h
/- Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser -/ import Mathlib.Data.Finset.Sigma import Mathlib.Data.Finset.Pairwise import Mathlib.Data.Finset.Powerset import Mathlib.Data.Fintype.Basic import Mathlib.Order.CompleteLatticeIntervals #align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d" /-! # Supremum independence In this file, we define supremum independence of indexed sets. An indexed family `f : ι → α` is sup-independent if, for all `a`, `f a` and the supremum of the rest are disjoint. ## Main definitions * `Finset.SupIndep s f`: a family of elements `f` are supremum independent on the finite set `s`. * `CompleteLattice.SetIndependent s`: a set of elements are supremum independent. * `CompleteLattice.Independent f`: a family of elements are supremum independent. ## Main statements * In a distributive lattice, supremum independence is equivalent to pairwise disjointness: * `Finset.supIndep_iff_pairwiseDisjoint` * `CompleteLattice.setIndependent_iff_pairwiseDisjoint` * `CompleteLattice.independent_iff_pairwiseDisjoint` * Otherwise, supremum independence is stronger than pairwise disjointness: * `Finset.SupIndep.pairwiseDisjoint` * `CompleteLattice.SetIndependent.pairwiseDisjoint` * `CompleteLattice.Independent.pairwiseDisjoint` ## Implementation notes For the finite version, we avoid the "obvious" definition `∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f)` because `erase` would require decidable equality on `ι`. -/ variable {α β ι ι' : Type} /-! ### On lattices with a bottom element, via `Finset.sup` -/ namespace Finset section Lattice variable [Lattice α] [OrderBot α] /-- Supremum independence of finite sets. We avoid the "obvious" definition using `s.erase i` because `erase` would require decidable equality on `ι`. -/ def SupIndep (s : Finset ι) (f : ι → α) : Prop := ∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f) #align finset.sup_indep Finset.SupIndep variable {s t : Finset ι} {f : ι → α} {i : ι} instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_) rintro t - refine @Finset.decidableDforallFinset _ _ _ (?_) rintro i - have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff infer_instance theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi => ht (hu.trans h) (h hi) #align finset.sup_indep.subset Finset.SupIndep.subset @[simp] theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha => (not_mem_empty a ha).elim #align finset.sup_indep_empty Finset.supIndep_empty theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f := fun s hs j hji hj => by rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty] exact disjoint_bot_right #align finset.sup_indep_singleton Finset.supIndep_singleton theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f := fun _ ha _ hb hab => sup_singleton.subst <\| hs (singleton_subset_iff.2 hb) ha <\| not_mem_singleton.2 hab #align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) : f i ≤ t.sup f ↔ i ∈ t := by refine ⟨fun h => ?_, le_sup⟩ by_contra hit exact hf i (disjoint_self.1 <\| (hs hts hi hit).mono_right h) #align finset.sup_indep.le_sup_iff Finset.SupIndep.le_sup_iff /-- The RHS looks like the definition of `CompleteLattice.Independent`. -/ theorem supIndep_iff_disjoint_erase [DecidableEq ι] : s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) := ⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit => (hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩ #align finset.sup_indep_iff_disjoint_erase Finset.supIndep_iff_disjoint_erase theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) : (s.image g).SupIndep f := by intro t ht i hi hit rw [mem_image] at hi obtain ⟨i, hi, rfl⟩ := hi haveI : DecidableEq ι' := Classical.decEq _ suffices hts : t ⊆ (s.erase i).image g by refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_) rw [sup_image] rintro j hjt obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt) exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩) #align finset.sup_indep.image Finset.SupIndep.image theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩ · rw [← sup_map] exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map']) · classical rw [map_eq_image] exact hs.image #align finset.sup_indep_map Finset.supIndep_map @[simp] theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) : ({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) := ⟨fun h => h.pairwiseDisjoint (by simp) (by simp) hij, fun h => by rw [supIndep_iff_disjoint_erase] intro k hk rw [Finset.mem_insert, Finset.mem_singleton] at hk obtain rfl \| rfl := hk · convert h using 1 rw [Finset.erase_insert, Finset.sup_singleton] simpa using hij · convert h.symm using 1 have : ({i, k} : Finset ι).erase k = {i} := by ext rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne, not_and_self_iff, or_false_iff, and_iff_right_of_imp] rintro rfl exact hij rw [this, Finset.sup_singleton]⟩ #align finset.sup_indep_pair Finset.supIndep_pair theorem supIndep_univ_bool (f : Bool → α) : (Finset.univ : Finset Bool).SupIndep f ↔ Disjoint (f false) (f true) := haveI : true ≠ false := by simp only [Ne, not_false_iff] (supIndep_pair this).trans disjoint_comm #align finset.sup_indep_univ_bool Finset.supIndep_univ_bool @[simp] theorem supIndep_univ_fin_two (f : Fin 2 → α) : (Finset.univ : Finset (Fin 2)).SupIndep f ↔ Disjoint (f 0) (f 1) := haveI : (0 : Fin 2) ≠ 1 := by simp supIndep_pair this #align finset.sup_indep_univ_fin_two Finset.supIndep_univ_fin_two theorem SupIndep.attach (hs : s.SupIndep f) : s.attach.SupIndep fun a => f a := by intro t _ i _ hi classical have : (fun (a : { x // x ∈ s }) => f ↑a) = f ∘ (fun a : { x // x ∈ s } => ↑a) := rfl rw [this, ← Finset.sup_image] refine hs (image_subset_iff.2 fun (j : { x // x ∈ s }) _ => j.2) i.2 fun hi' => hi ?_ rw [mem_image] at hi' obtain ⟨j, hj, hji⟩ := hi' rwa [Subtype.ext hji] at hj #align finset.sup_indep.attach Finset.SupIndep.attach /- Porting note: simpNF linter returns "Left-hand side does not simplify, when using the simp lemma on itself." However, simp does indeed solve the following. leanprover/std4#71 is related. example {α ι} [Lattice α] [OrderBot α] (s : Finset ι) (f : ι → α) : (s.attach.SupIndep fun a => f a) ↔ s.SupIndep f := by simp -/ @[simp, nolint simpNF] theorem supIndep_attach : (s.attach.SupIndep fun a => f a) ↔ s.SupIndep f := by refine ⟨fun h t ht i his hit => ?_, SupIndep.attach⟩ classical convert h (filter_subset (fun (i : { x // x ∈ s }) => (i : ι) ∈ t) _) (mem_attach _ ⟨i, ‹_›⟩) fun hi => hit <\| by simpa using hi using 1 refine eq_of_forall_ge_iff ?_ simp only [Finset.sup_le_iff, mem_filter, mem_attach, true_and_iff, Function.comp_apply, Subtype.forall, Subtype.coe_mk] exact fun a => forall_congr' fun j => ⟨fun h _ => h, fun h hj => h (ht hj) hj⟩ #align finset.sup_indep_attach Finset.supIndep_attach end Lattice section DistribLattice variable [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α} theorem supIndep_iff_pairwiseDisjoint : s.SupIndep f ↔ (s : Set ι).PairwiseDisjoint f := ⟨SupIndep.pairwiseDisjoint, fun hs _ ht _ hi hit => Finset.disjoint_sup_right.2 fun _ hj => hs hi (ht hj) (ne_of_mem_of_not_mem hj hit).symm⟩ #align finset.sup_indep_iff_pairwise_disjoint Finset.supIndep_iff_pairwiseDisjoint alias ⟨sup_indep.pairwise_disjoint, _root_.Set.PairwiseDisjoint.supIndep⟩ := supIndep_iff_pairwiseDisjoint #align set.pairwise_disjoint.sup_indep Set.PairwiseDisjoint.supIndep /-- Bind operation for `SupIndep`. -/ theorem SupIndep.sup [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) : (s.sup g).SupIndep f := by simp_rw [supIndep_iff_pairwiseDisjoint] at hs hg ⊢ rw [sup_eq_biUnion, coe_biUnion] exact hs.biUnion_finset hg #align finset.sup_indep.sup Finset.SupIndep.sup /-- Bind operation for `SupIndep`. -/ theorem SupIndep.biUnion [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) : (s.biUnion g).SupIndep f := by rw [← sup_eq_biUnion] exact hs.sup hg #align finset.sup_indep.bUnion Finset.SupIndep.biUnion /-- Bind operation for `SupIndep`. -/ theorem SupIndep.sigma {β : ι → Type} {s : Finset ι} {g : ∀ i, Finset (β i)} {f : Sigma β → α} (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩) (hg : ∀ i ∈ s, (g i).SupIndep fun b => f ⟨i, b⟩) : (s.sigma g).SupIndep f := by rintro t ht ⟨i, b⟩ hi hit rw [Finset.disjoint_sup_right] rintro ⟨j, c⟩ hj have hbc := (ne_of_mem_of_not_mem hj hit).symm replace hj := ht hj rw [mem_sigma] at hi hj obtain rfl \| hij := eq_or_ne i j · exact (hg _ hj.1).pairwiseDisjoint hi.2 hj.2 (sigma_mk_injective.ne_iff.1 hbc) · refine (hs.pairwiseDisjoint hi.1 hj.1 hij).mono ?_ ?_ · convert le_sup (α := α) hi.2; simp · convert le_sup (α := α) hj.2; simp #align finset.sup_indep.sigma Finset.SupIndep.sigma theorem SupIndep.product {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} (hs : s.SupIndep fun i => t.sup fun i' => f (i, i')) (ht : t.SupIndep fun i' => s.sup fun i => f (i, i')) : (s ×ˢ t).SupIndep f := by rintro u hu ⟨i, i'⟩ hi hiu rw [Finset.disjoint_sup_right] rintro ⟨j, j'⟩ hj have hij := (ne_of_mem_of_not_mem hj hiu).symm replace hj := hu hj rw [mem_product] at hi hj obtain rfl \| hij := eq_or_ne i j · refine (ht.pairwiseDisjoint hi.2 hj.2 <\| (Prod.mk.inj_left _).ne_iff.1 hij).mono ?_ ?_ · convert le_sup (α := α) hi.1; simp · convert le_sup (α := α) hj.1; simp · refine (hs.pairwiseDisjoint hi.1 hj.1 hij).mono ?_ ?_ · convert le_sup (α := α) hi.2; simp · convert le_sup (α := α) hj.2; simp #align finset.sup_indep.product Finset.SupIndep.product theorem supIndep_product_iff {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} : (s.product t).SupIndep f ↔ (s.SupIndep fun i => t.sup fun i' => f (i, i')) ∧ t.SupIndep fun i' => s.sup fun i => f (i, i') := by refine ⟨?_, fun h => h.1.product h.2⟩ simp_rw [supIndep_iff_pairwiseDisjoint] refine fun h => ⟨fun i hi j hj hij => ?_, fun i hi j hj hij => ?_⟩ <;> simp_rw [Finset.disjoint_sup_left, Finset.disjoint_sup_right] <;> intro i' hi' j' hj' · exact h (mk_mem_product hi hi') (mk_mem_product hj hj') (ne_of_apply_ne Prod.fst hij) · exact h (mk_mem_product hi' hi) (mk_mem_product hj' hj) (ne_of_apply_ne Prod.snd hij) #align finset.sup_indep_product_iff Finset.supIndep_product_iff end DistribLattice end Finset /-! ### On complete lattices via `sSup` -/ namespace CompleteLattice variable [CompleteLattice α] open Set Function /-- An independent set of elements in a complete lattice is one in which every element is disjoint from the `Sup` of the rest. -/ def SetIndependent (s : Set α) : Prop := ∀ ⦃a⦄, a ∈ s → Disjoint a (sSup (s \ {a})) #align complete_lattice.set_independent CompleteLattice.SetIndependent variable {s : Set α} (hs : SetIndependent s) @[simp] theorem setIndependent_empty : SetIndependent (∅ : Set α) := fun x hx => (Set.not_mem_empty x hx).elim #align complete_lattice.set_independent_empty CompleteLattice.setIndependent_empty theorem SetIndependent.mono {t : Set α} (hst : t ⊆ s) : SetIndependent t := fun _ ha => (hs (hst ha)).mono_right (sSup_le_sSup (diff_subset_diff_left hst)) #align complete_lattice.set_independent.mono CompleteLattice.SetIndependent.mono /-- If the elements of a set are independent, then any pair within that set is disjoint. -/ theorem SetIndependent.pairwiseDisjoint : s.PairwiseDisjoint id := fun _ hx y hy h => disjoint_sSup_right (hs hx) ((mem_diff y).mpr ⟨hy, h.symm⟩) #align complete_lattice.set_independent.pairwise_disjoint CompleteLattice.SetIndependent.pairwiseDisjoint theorem setIndependent_singleton (a : α) : SetIndependent ({a} : Set α) := fun i hi ↦ by simp_all theorem setIndependent_pair {a b : α} (hab : a ≠ b) : SetIndependent ({a, b} : Set α) ↔ Disjoint a b := by constructor · intro h exact h.pairwiseDisjoint (mem_insert _ _) (mem_insert_of_mem _ (mem_singleton _)) hab · rintro h c ((rfl : c = a) \| (rfl : c = b)) · convert h using 1 simp [hab, sSup_singleton] · convert h.symm using 1 simp [hab, sSup_singleton] #align complete_lattice.set_independent_pair CompleteLattice.setIndependent_pair /-- If the elements of a set are independent, then any element is disjoint from the `sSup` of some subset of the rest. -/ theorem SetIndependent.disjoint_sSup {x : α} {y : Set α} (hx : x ∈ s) (hy : y ⊆ s) (hxy : x ∉ y) : Disjoint x (sSup y) := by have := (hs.mono <\| insert_subset_iff.mpr ⟨hx, hy⟩) (mem_insert x _) rw [insert_diff_of_mem _ (mem_singleton _), diff_singleton_eq_self hxy] at this exact this #align complete_lattice.set_independent.disjoint_Sup CompleteLattice.SetIndependent.disjoint_sSup /-- An independent indexed family of elements in a complete lattice is one in which every element is disjoint from the `iSup` of the rest. Example: an indexed family of non-zero elements in a vector space is linearly independent iff the indexed family of subspaces they generate is independent in this sense. Example: an indexed family of submodules of a module is independent in this sense if and only the natural map from the direct sum of the submodules to the module is injective. -/ def Independent {ι : Sort} {α : Type} [CompleteLattice α] (t : ι → α) : Prop := ∀ i : ι, Disjoint (t i) (⨆ (j) (_ : j ≠ i), t j) #align complete_lattice.independent CompleteLattice.Independent theorem setIndependent_iff {α : Type} [CompleteLattice α] (s : Set α) : SetIndependent s ↔ Independent ((↑) : s → α) := by simp_rw [Independent, SetIndependent, SetCoe.forall, sSup_eq_iSup] refine forall₂_congr fun a ha => ?_ simp [iSup_subtype, iSup_and] #align complete_lattice.set_independent_iff CompleteLattice.setIndependent_iff variable {t : ι → α} (ht : Independent t) theorem independent_def : Independent t ↔ ∀ i : ι, Disjoint (t i) (⨆ (j) (_ : j ≠ i), t j) := Iff.rfl #align complete_lattice.independent_def CompleteLattice.independent_def theorem independent_def' : Independent t ↔ ∀ i, Disjoint (t i) (sSup (t '' { j \| j ≠ i })) := by simp_rw [sSup_image] rfl #align complete_lattice.independent_def' CompleteLattice.independent_def' theorem independent_def'' : Independent t ↔ ∀ i, Disjoint (t i) (sSup { a \| ∃ j ≠ i, t j = a }) := by rw [independent_def'] aesop #align complete_lattice.independent_def'' CompleteLattice.independent_def'' @[simp] theorem independent_empty (t : Empty → α) : Independent t := nofun #align complete_lattice.independent_empty CompleteLattice.independent_empty @[simp] theorem independent_pempty (t : PEmpty → α) : Independent t := nofun #align complete_lattice.independent_pempty CompleteLattice.independent_pempty /-- If the elements of a set are independent, then any pair within that set is disjoint. -/ theorem Independent.pairwiseDisjoint : Pairwise (Disjoint on t) := fun x y h => disjoint_sSup_right (ht x) ⟨y, iSup_pos h.symm⟩ #align complete_lattice.independent.pairwise_disjoint CompleteLattice.Independent.pairwiseDisjoint theorem Independent.mono {s t : ι → α} (hs : Independent s) (hst : t ≤ s) : Independent t := fun i => (hs i).mono (hst i) <\| iSup₂_mono fun j _ => hst j #align complete_lattice.independent.mono CompleteLattice.Independent.mono /-- Composing an independent indexed family with an injective function on the index results in another indepedendent indexed family. -/ theorem Independent.comp {ι ι' : Sort} {t : ι → α} {f : ι' → ι} (ht : Independent t) (hf : Injective f) : Independent (t ∘ f) := fun i => (ht (f i)).mono_right <\| by refine (iSup_mono fun i => ?_).trans (iSup_comp_le _ f) exact iSup_const_mono hf.ne #align complete_lattice.independent.comp CompleteLattice.Independent.comp theorem Independent.comp' {ι ι' : Sort*} {t : ι → α} {f : ι' → ι} (ht : Independent <\| t ∘ f) (hf : Surjective f) : Independent t := by intro i obtain ⟨i', rfl⟩ := hf i rw [← hf.iSup_comp] exact (ht i').mono_right (biSup_mono fun j' hij => mt (congr_arg f) hij) #align complete_lattice.independent.comp' CompleteLattice.Independent.comp' theorem Independent.setIndependent_range (ht : Independent t) : SetIndependent <\| range t := by rw [setIndependent_iff] rw [← coe_comp_rangeFactorization t] at ht exact ht.comp' surjective_onto_range #align complete_lattice.independent.set_independent_range CompleteLattice.Independent.setIndependent_range @[simp] theorem independent_ne_bot_iff_independent : Independent (fun i : {i // t i ≠ ⊥} ↦ t i) ↔ Independent t := by refine ⟨fun h ↦ ?_, fun h ↦ h.comp Subtype.val_injective⟩ simp only [independent_def] at h ⊢ intro i cases eq_or_ne (t i) ⊥ with \| inl hi => simp [hi] \| inr hi => ?_ convert h ⟨i, hi⟩ have : ∀ j, ⨆ (_ : t j = ⊥), t j = ⊥ := fun j ↦ by simp only [iSup_eq_bot, imp_self] rw [iSup_split _ (fun j ↦ t j = ⊥), iSup_subtype] simp only [iSup_comm (ι' := _ ≠ i), this, ne_eq, sup_of_le_right, Subtype.mk.injEq, iSup_bot, bot_le] theorem Independent.injOn (ht : Independent t) : InjOn t {i \| t i ≠ ⊥} := by rintro i _ j (hj : t j ≠ ⊥) h by_contra! contra apply hj suffices t j ≤ ⨆ (k) (_ : k ≠ i), t k by replace ht := (ht i).mono_right this rwa [h, disjoint_self] at ht replace contra : j ≠ i := Ne.symm contra -- Porting note: needs explicit `f` exact le_iSup₂ (f := fun x _ ↦ t x) j contra theorem Independent.injective (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Injective t := by suffices univ = {i \| t i ≠ ⊥} by rw [injective_iff_injOn_univ, this]; exact ht.injOn aesop #align complete_lattice.independent.injective CompleteLattice.Independent.injective	Mathlib/Order/SupIndep.lean	445	454	theorem independent_pair {i j : ι} (hij : i ≠ j) (huniv : ∀ k, k = i ∨ k = j) : Independent t ↔ Disjoint (t i) (t j) := by	constructor · exact fun h => h.pairwiseDisjoint hij · rintro h k obtain rfl \| rfl := huniv k · refine h.mono_right (iSup_le fun i => iSup_le fun hi => Eq.le ?_) rw [(huniv i).resolve_left hi] · refine h.symm.mono_right (iSup_le fun j => iSup_le fun hj => Eq.le ?_) rw [(huniv j).resolve_right hj]
/- Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies, Bhavik Mehta -/ import Mathlib.Algebra.BigOperators.Group.Finset import Mathlib.Order.SupIndep import Mathlib.Order.Atoms #align_import order.partition.finpartition from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce" /-! # Finite partitions In this file, we define finite partitions. A finpartition of `a : α` is a finite set of pairwise disjoint parts `parts : Finset α` which does not contain `⊥` and whose supremum is `a`. Finpartitions of a finset are at the heart of Szemerédi's regularity lemma. They are also studied purely order theoretically in Sperner theory. ## Constructions We provide many ways to build finpartitions: * `Finpartition.ofErase`: Builds a finpartition by erasing `⊥` for you. * `Finpartition.ofSubset`: Builds a finpartition from a subset of the parts of a previous finpartition. * `Finpartition.empty`: The empty finpartition of `⊥`. * `Finpartition.indiscrete`: The indiscrete, aka trivial, aka pure, finpartition made of a single part. * `Finpartition.discrete`: The discrete finpartition of `s : Finset α` made of singletons. * `Finpartition.bind`: Puts together the finpartitions of the parts of a finpartition into a new finpartition. * `Finpartition.ofSetoid`: With `Fintype α`, constructs the finpartition of `univ : Finset α` induced by the equivalence classes of `s : Setoid α`. * `Finpartition.atomise`: Makes a finpartition of `s : Finset α` by breaking `s` along all finsets in `F : Finset (Finset α)`. Two elements of `s` belong to the same part iff they belong to the same elements of `F`. `Finpartition.indiscrete` and `Finpartition.bind` together form the monadic structure of `Finpartition`. ## Implementation notes Forbidding `⊥` as a part follows mathematical tradition and is a pragmatic choice concerning operations on `Finpartition`. Not caring about `⊥` being a part or not breaks extensionality (it's not because the parts of `P` and the parts of `Q` have the same elements that `P = Q`). Enforcing `⊥` to be a part makes `Finpartition.bind` uglier and doesn't rid us of the need of `Finpartition.ofErase`. ## TODO The order is the wrong way around to make `Finpartition a` a graded order. Is it bad to depart from the literature and turn the order around? -/ open Finset Function variable {α : Type} /-- A finite partition of `a : α` is a pairwise disjoint finite set of elements whose supremum is `a`. We forbid `⊥` as a part. -/ @[ext] structure Finpartition [Lattice α] [OrderBot α] (a : α) where -- Porting note: Docstrings added /-- The elements of the finite partition of `a` -/ parts : Finset α /-- The partition is supremum-independent -/ supIndep : parts.SupIndep id /-- The supremum of the partition is `a` -/ sup_parts : parts.sup id = a /-- No element of the partition is bottom-/ not_bot_mem : ⊥ ∉ parts deriving DecidableEq #align finpartition Finpartition #align finpartition.parts Finpartition.parts #align finpartition.sup_indep Finpartition.supIndep #align finpartition.sup_parts Finpartition.sup_parts #align finpartition.not_bot_mem Finpartition.not_bot_mem -- Porting note: attribute [protected] doesn't work -- attribute [protected] Finpartition.supIndep namespace Finpartition section Lattice variable [Lattice α] [OrderBot α] /-- A `Finpartition` constructor which does not insist on `⊥` not being a part. -/ @[simps] def ofErase [DecidableEq α] {a : α} (parts : Finset α) (sup_indep : parts.SupIndep id) (sup_parts : parts.sup id = a) : Finpartition a where parts := parts.erase ⊥ supIndep := sup_indep.subset (erase_subset _ _) sup_parts := (sup_erase_bot _).trans sup_parts not_bot_mem := not_mem_erase _ _ #align finpartition.of_erase Finpartition.ofErase /-- A `Finpartition` constructor from a bigger existing finpartition. -/ @[simps] def ofSubset {a b : α} (P : Finpartition a) {parts : Finset α} (subset : parts ⊆ P.parts) (sup_parts : parts.sup id = b) : Finpartition b := { parts := parts supIndep := P.supIndep.subset subset sup_parts := sup_parts not_bot_mem := fun h ↦ P.not_bot_mem (subset h) } #align finpartition.of_subset Finpartition.ofSubset /-- Changes the type of a finpartition to an equal one. -/ @[simps] def copy {a b : α} (P : Finpartition a) (h : a = b) : Finpartition b where parts := P.parts supIndep := P.supIndep sup_parts := h ▸ P.sup_parts not_bot_mem := P.not_bot_mem #align finpartition.copy Finpartition.copy /-- Transfer a finpartition over an order isomorphism. -/ def map {β : Type} [Lattice β] [OrderBot β] {a : α} (e : α ≃o β) (P : Finpartition a) : Finpartition (e a) where parts := P.parts.map e supIndep u hu _ hb hbu _ hx hxu := by rw [← map_symm_subset] at hu simp only [mem_map_equiv] at hb have := P.supIndep hu hb (by simp [hbu]) (map_rel e.symm hx) ?_ · rw [← e.symm.map_bot] at this exact e.symm.map_rel_iff.mp this · convert e.symm.map_rel_iff.mpr hxu rw [map_finset_sup, sup_map] rfl sup_parts := by simp [← P.sup_parts] not_bot_mem := by rw [mem_map_equiv] convert P.not_bot_mem exact e.symm.map_bot @[simp] theorem parts_map {β : Type*} [Lattice β] [OrderBot β] {a : α} {e : α ≃o β} {P : Finpartition a} : (P.map e).parts = P.parts.map e := rfl variable (α) /-- The empty finpartition. -/ @[simps] protected def empty : Finpartition (⊥ : α) where parts := ∅ supIndep := supIndep_empty _ sup_parts := Finset.sup_empty not_bot_mem := not_mem_empty ⊥ #align finpartition.empty Finpartition.empty instance : Inhabited (Finpartition (⊥ : α)) := ⟨Finpartition.empty α⟩ @[simp] theorem default_eq_empty : (default : Finpartition (⊥ : α)) = Finpartition.empty α := rfl #align finpartition.default_eq_empty Finpartition.default_eq_empty variable {α} {a : α} /-- The finpartition in one part, aka indiscrete finpartition. -/ @[simps] def indiscrete (ha : a ≠ ⊥) : Finpartition a where parts := {a} supIndep := supIndep_singleton _ _ sup_parts := Finset.sup_singleton not_bot_mem h := ha (mem_singleton.1 h).symm #align finpartition.indiscrete Finpartition.indiscrete variable (P : Finpartition a) protected theorem le {b : α} (hb : b ∈ P.parts) : b ≤ a := (le_sup hb).trans P.sup_parts.le #align finpartition.le Finpartition.le theorem ne_bot {b : α} (hb : b ∈ P.parts) : b ≠ ⊥ := by intro h refine P.not_bot_mem (?_) rw [h] at hb exact hb #align finpartition.ne_bot Finpartition.ne_bot protected theorem disjoint : (P.parts : Set α).PairwiseDisjoint id := P.supIndep.pairwiseDisjoint #align finpartition.disjoint Finpartition.disjoint variable {P} theorem parts_eq_empty_iff : P.parts = ∅ ↔ a = ⊥ := by simp_rw [← P.sup_parts] refine ⟨fun h ↦ ?_, fun h ↦ eq_empty_iff_forall_not_mem.2 fun b hb ↦ P.not_bot_mem ?_⟩ · rw [h] exact Finset.sup_empty · rwa [← le_bot_iff.1 ((le_sup hb).trans h.le)] #align finpartition.parts_eq_empty_iff Finpartition.parts_eq_empty_iff theorem parts_nonempty_iff : P.parts.Nonempty ↔ a ≠ ⊥ := by rw [nonempty_iff_ne_empty, not_iff_not, parts_eq_empty_iff] #align finpartition.parts_nonempty_iff Finpartition.parts_nonempty_iff theorem parts_nonempty (P : Finpartition a) (ha : a ≠ ⊥) : P.parts.Nonempty := parts_nonempty_iff.2 ha #align finpartition.parts_nonempty Finpartition.parts_nonempty instance : Unique (Finpartition (⊥ : α)) := { (inferInstance : Inhabited (Finpartition (⊥ : α))) with uniq := fun P ↦ by ext a exact iff_of_false (fun h ↦ P.ne_bot h <\| le_bot_iff.1 <\| P.le h) (not_mem_empty a) } -- See note [reducible non instances] /-- There's a unique partition of an atom. -/ abbrev _root_.IsAtom.uniqueFinpartition (ha : IsAtom a) : Unique (Finpartition a) where default := indiscrete ha.1 uniq P := by have h : ∀ b ∈ P.parts, b = a := fun _ hb ↦ (ha.le_iff.mp <\| P.le hb).resolve_left (P.ne_bot hb) ext b refine Iff.trans ⟨h b, ?_⟩ mem_singleton.symm rintro rfl obtain ⟨c, hc⟩ := P.parts_nonempty ha.1 simp_rw [← h c hc] exact hc #align is_atom.unique_finpartition IsAtom.uniqueFinpartition instance [Fintype α] [DecidableEq α] (a : α) : Fintype (Finpartition a) := @Fintype.ofSurjective { p : Finset α // p.SupIndep id ∧ p.sup id = a ∧ ⊥ ∉ p } (Finpartition a) _ (Subtype.fintype _) (fun i ↦ ⟨i.1, i.2.1, i.2.2.1, i.2.2.2⟩) fun ⟨_, y, z, w⟩ ↦ ⟨⟨_, y, z, w⟩, rfl⟩ /-! ### Refinement order -/ section Order /-- We say that `P ≤ Q` if `P` refines `Q`: each part of `P` is less than some part of `Q`. -/ instance : LE (Finpartition a) := ⟨fun P Q ↦ ∀ ⦃b⦄, b ∈ P.parts → ∃ c ∈ Q.parts, b ≤ c⟩ instance : PartialOrder (Finpartition a) := { (inferInstance : LE (Finpartition a)) with le_refl := fun P b hb ↦ ⟨b, hb, le_rfl⟩ le_trans := fun P Q R hPQ hQR b hb ↦ by obtain ⟨c, hc, hbc⟩ := hPQ hb obtain ⟨d, hd, hcd⟩ := hQR hc exact ⟨d, hd, hbc.trans hcd⟩ le_antisymm := fun P Q hPQ hQP ↦ by ext b refine ⟨fun hb ↦ ?_, fun hb ↦ ?_⟩ · obtain ⟨c, hc, hbc⟩ := hPQ hb obtain ⟨d, hd, hcd⟩ := hQP hc rwa [hbc.antisymm] rwa [P.disjoint.eq_of_le hb hd (P.ne_bot hb) (hbc.trans hcd)] · obtain ⟨c, hc, hbc⟩ := hQP hb obtain ⟨d, hd, hcd⟩ := hPQ hc rwa [hbc.antisymm] rwa [Q.disjoint.eq_of_le hb hd (Q.ne_bot hb) (hbc.trans hcd)] } instance [Decidable (a = ⊥)] : OrderTop (Finpartition a) where top := if ha : a = ⊥ then (Finpartition.empty α).copy ha.symm else indiscrete ha le_top P := by split_ifs with h · intro x hx simpa [h, P.ne_bot hx] using P.le hx · exact fun b hb ↦ ⟨a, mem_singleton_self _, P.le hb⟩ theorem parts_top_subset (a : α) [Decidable (a = ⊥)] : (⊤ : Finpartition a).parts ⊆ {a} := by intro b hb have hb : b ∈ Finpartition.parts (dite _ _ _) := hb split_ifs at hb · simp only [copy_parts, empty_parts, not_mem_empty] at hb · exact hb #align finpartition.parts_top_subset Finpartition.parts_top_subset theorem parts_top_subsingleton (a : α) [Decidable (a = ⊥)] : ((⊤ : Finpartition a).parts : Set α).Subsingleton := Set.subsingleton_of_subset_singleton fun _ hb ↦ mem_singleton.1 <\| parts_top_subset _ hb #align finpartition.parts_top_subsingleton Finpartition.parts_top_subsingleton end Order end Lattice section DistribLattice variable [DistribLattice α] [OrderBot α] section Inf variable [DecidableEq α] {a b c : α} instance : Inf (Finpartition a) := ⟨fun P Q ↦ ofErase ((P.parts ×ˢ Q.parts).image fun bc ↦ bc.1 ⊓ bc.2) (by rw [supIndep_iff_disjoint_erase] simp only [mem_image, and_imp, exists_prop, forall_exists_index, id, Prod.exists, mem_product, Finset.disjoint_sup_right, mem_erase, Ne] rintro _ x₁ y₁ hx₁ hy₁ rfl _ h x₂ y₂ hx₂ hy₂ rfl rcases eq_or_ne x₁ x₂ with (rfl \| xdiff) · refine Disjoint.mono inf_le_right inf_le_right (Q.disjoint hy₁ hy₂ ?_) intro t simp [t] at h exact Disjoint.mono inf_le_left inf_le_left (P.disjoint hx₁ hx₂ xdiff)) (by rw [sup_image, id_comp, sup_product_left] trans P.parts.sup id ⊓ Q.parts.sup id · simp_rw [Finset.sup_inf_distrib_right, Finset.sup_inf_distrib_left] rfl · rw [P.sup_parts, Q.sup_parts, inf_idem])⟩ @[simp] theorem parts_inf (P Q : Finpartition a) : (P ⊓ Q).parts = ((P.parts ×ˢ Q.parts).image fun bc : α × α ↦ bc.1 ⊓ bc.2).erase ⊥ := rfl #align finpartition.parts_inf Finpartition.parts_inf instance : SemilatticeInf (Finpartition a) := { (inferInstance : PartialOrder (Finpartition a)), (inferInstance : Inf (Finpartition a)) with inf_le_left := fun P Q b hb ↦ by obtain ⟨c, hc, rfl⟩ := mem_image.1 (mem_of_mem_erase hb) rw [mem_product] at hc exact ⟨c.1, hc.1, inf_le_left⟩ inf_le_right := fun P Q b hb ↦ by obtain ⟨c, hc, rfl⟩ := mem_image.1 (mem_of_mem_erase hb) rw [mem_product] at hc exact ⟨c.2, hc.2, inf_le_right⟩ le_inf := fun P Q R hPQ hPR b hb ↦ by obtain ⟨c, hc, hbc⟩ := hPQ hb obtain ⟨d, hd, hbd⟩ := hPR hb have h := _root_.le_inf hbc hbd refine ⟨c ⊓ d, mem_erase_of_ne_of_mem (ne_bot_of_le_ne_bot (P.ne_bot hb) h) (mem_image.2 ⟨(c, d), mem_product.2 ⟨hc, hd⟩, rfl⟩), h⟩ } end Inf theorem exists_le_of_le {a b : α} {P Q : Finpartition a} (h : P ≤ Q) (hb : b ∈ Q.parts) : ∃ c ∈ P.parts, c ≤ b := by by_contra H refine Q.ne_bot hb (disjoint_self.1 <\| Disjoint.mono_right (Q.le hb) ?_) rw [← P.sup_parts, Finset.disjoint_sup_right] rintro c hc obtain ⟨d, hd, hcd⟩ := h hc refine (Q.disjoint hb hd ?_).mono_right hcd rintro rfl simp only [not_exists, not_and] at H exact H _ hc hcd #align finpartition.exists_le_of_le Finpartition.exists_le_of_le theorem card_mono {a : α} {P Q : Finpartition a} (h : P ≤ Q) : Q.parts.card ≤ P.parts.card := by classical have : ∀ b ∈ Q.parts, ∃ c ∈ P.parts, c ≤ b := fun b ↦ exists_le_of_le h choose f hP hf using this rw [← card_attach] refine card_le_card_of_inj_on (fun b ↦ f _ b.2) (fun b _ ↦ hP _ b.2) fun b _ c _ h ↦ ?_ exact Subtype.coe_injective (Q.disjoint.elim b.2 c.2 fun H ↦ P.ne_bot (hP _ b.2) <\| disjoint_self.1 <\| H.mono (hf _ b.2) <\| h.le.trans <\| hf _ c.2) #align finpartition.card_mono Finpartition.card_mono variable [DecidableEq α] {a b c : α} section Bind variable {P : Finpartition a} {Q : ∀ i ∈ P.parts, Finpartition i} /-- Given a finpartition `P` of `a` and finpartitions of each part of `P`, this yields the finpartition of `a` obtained by juxtaposing all the subpartitions. -/ @[simps] def bind (P : Finpartition a) (Q : ∀ i ∈ P.parts, Finpartition i) : Finpartition a where parts := P.parts.attach.biUnion fun i ↦ (Q i.1 i.2).parts supIndep := by rw [supIndep_iff_pairwiseDisjoint] rintro a ha b hb h rw [Finset.mem_coe, Finset.mem_biUnion] at ha hb obtain ⟨⟨A, hA⟩, -, ha⟩ := ha obtain ⟨⟨B, hB⟩, -, hb⟩ := hb obtain rfl \| hAB := eq_or_ne A B · exact (Q A hA).disjoint ha hb h · exact (P.disjoint hA hB hAB).mono ((Q A hA).le ha) ((Q B hB).le hb) sup_parts := by simp_rw [sup_biUnion] trans (sup P.parts id) · rw [eq_comm, ← Finset.sup_attach] exact sup_congr rfl fun b _hb ↦ (Q b.1 b.2).sup_parts.symm · exact P.sup_parts not_bot_mem h := by rw [Finset.mem_biUnion] at h obtain ⟨⟨A, hA⟩, -, h⟩ := h exact (Q A hA).not_bot_mem h #align finpartition.bind Finpartition.bind theorem mem_bind : b ∈ (P.bind Q).parts ↔ ∃ A hA, b ∈ (Q A hA).parts := by rw [bind, mem_biUnion] constructor · rintro ⟨⟨A, hA⟩, -, h⟩ exact ⟨A, hA, h⟩ · rintro ⟨A, hA, h⟩ exact ⟨⟨A, hA⟩, mem_attach _ ⟨A, hA⟩, h⟩ #align finpartition.mem_bind Finpartition.mem_bind theorem card_bind (Q : ∀ i ∈ P.parts, Finpartition i) : (P.bind Q).parts.card = ∑ A ∈ P.parts.attach, (Q _ A.2).parts.card := by apply card_biUnion rintro ⟨b, hb⟩ - ⟨c, hc⟩ - hbc rw [Finset.disjoint_left] rintro d hdb hdc rw [Ne, Subtype.mk_eq_mk] at hbc exact (Q b hb).ne_bot hdb (eq_bot_iff.2 <\| (le_inf ((Q b hb).le hdb) <\| (Q c hc).le hdc).trans <\| (P.disjoint hb hc hbc).le_bot) #align finpartition.card_bind Finpartition.card_bind end Bind /-- Adds `b` to a finpartition of `a` to make a finpartition of `a ⊔ b`. -/ @[simps] def extend (P : Finpartition a) (hb : b ≠ ⊥) (hab : Disjoint a b) (hc : a ⊔ b = c) : Finpartition c where parts := insert b P.parts supIndep := by rw [supIndep_iff_pairwiseDisjoint, coe_insert] exact P.disjoint.insert fun d hd _ ↦ hab.symm.mono_right <\| P.le hd sup_parts := by rwa [sup_insert, P.sup_parts, id, _root_.sup_comm] not_bot_mem h := (mem_insert.1 h).elim hb.symm P.not_bot_mem #align finpartition.extend Finpartition.extend theorem card_extend (P : Finpartition a) (b c : α) {hb : b ≠ ⊥} {hab : Disjoint a b} {hc : a ⊔ b = c} : (P.extend hb hab hc).parts.card = P.parts.card + 1 := card_insert_of_not_mem fun h ↦ hb <\| hab.symm.eq_bot_of_le <\| P.le h #align finpartition.card_extend Finpartition.card_extend end DistribLattice section GeneralizedBooleanAlgebra variable [GeneralizedBooleanAlgebra α] [DecidableEq α] {a b c : α} (P : Finpartition a) /-- Restricts a finpartition to avoid a given element. -/ @[simps!] def avoid (b : α) : Finpartition (a \ b) := ofErase (P.parts.image (· \ b)) (P.disjoint.image_finset_of_le fun a ↦ sdiff_le).supIndep (by rw [sup_image, id_comp, Finset.sup_sdiff_right, ← Function.id_def, P.sup_parts]) #align finpartition.avoid Finpartition.avoid @[simp] theorem mem_avoid : c ∈ (P.avoid b).parts ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \ b = c := by simp only [avoid, ofErase, mem_erase, Ne, mem_image, exists_prop, ← exists_and_left, @and_left_comm (c ≠ ⊥)] refine exists_congr fun d ↦ and_congr_right' <\| and_congr_left ?_ rintro rfl rw [sdiff_eq_bot_iff] #align finpartition.mem_avoid Finpartition.mem_avoid end GeneralizedBooleanAlgebra end Finpartition /-! ### Finite partitions of finsets -/ namespace Finpartition variable [DecidableEq α] {s t u : Finset α} (P : Finpartition s) {a : α} theorem nonempty_of_mem_parts {a : Finset α} (ha : a ∈ P.parts) : a.Nonempty := nonempty_iff_ne_empty.2 <\| P.ne_bot ha #align finpartition.nonempty_of_mem_parts Finpartition.nonempty_of_mem_parts lemma eq_of_mem_parts (ht : t ∈ P.parts) (hu : u ∈ P.parts) (hat : a ∈ t) (hau : a ∈ u) : t = u := P.disjoint.elim ht hu <\| not_disjoint_iff.2 ⟨a, hat, hau⟩ theorem exists_mem (ha : a ∈ s) : ∃ t ∈ P.parts, a ∈ t := by simp_rw [← P.sup_parts] at ha exact mem_sup.1 ha #align finpartition.exists_mem Finpartition.exists_mem theorem biUnion_parts : P.parts.biUnion id = s := (sup_eq_biUnion _ _).symm.trans P.sup_parts #align finpartition.bUnion_parts Finpartition.biUnion_parts theorem existsUnique_mem (ha : a ∈ s) : ∃! t, t ∈ P.parts ∧ a ∈ t := by obtain ⟨t, ht, ht'⟩ := P.exists_mem ha refine ⟨t, ⟨ht, ht'⟩, ?_⟩ rintro u ⟨hu, hu'⟩ exact P.eq_of_mem_parts hu ht hu' ht' /-- The part of the finpartition that `a` lies in. -/ def part (a : α) : Finset α := if ha : a ∈ s then choose (hp := P.existsUnique_mem ha) else ∅	Mathlib/Order/Partition/Finpartition.lean	501	501	theorem part_mem (ha : a ∈ s) : P.part a ∈ P.parts := by	simp [part, ha, choose_mem]
/- Copyright (c) 2022 Damiano Testa. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Damiano Testa -/ import Mathlib.Algebra.Polynomial.Degree.Definitions #align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0" /-! # Interactions between `R[X]` and `Rᵐᵒᵖ[X]` This file contains the basic API for "pushing through" the isomorphism `opRingEquiv : R[X]ᵐᵒᵖ ≃+* Rᵐᵒᵖ[X]`. It allows going back and forth between a polynomial ring over a semiring and the polynomial ring over the opposite semiring. -/ open Polynomial open Polynomial MulOpposite variable {R : Type} [Semiring R] noncomputable section namespace Polynomial /-- Ring isomorphism between `R[X]ᵐᵒᵖ` and `Rᵐᵒᵖ[X]` sending each coefficient of a polynomial to the corresponding element of the opposite ring. -/ def opRingEquiv (R : Type) [Semiring R] : R[X]ᵐᵒᵖ ≃+* Rᵐᵒᵖ[X] := ((toFinsuppIso R).op.trans AddMonoidAlgebra.opRingEquiv).trans (toFinsuppIso _).symm #align polynomial.op_ring_equiv Polynomial.opRingEquiv /-! Lemmas to get started, using `opRingEquiv R` on the various expressions of `Finsupp.single`: `monomial`, `C a`, `X`, `C a * X ^ n`. -/ @[simp] theorem opRingEquiv_op_monomial (n : ℕ) (r : R) : opRingEquiv R (op (monomial n r : R[X])) = monomial n (op r) := by simp only [opRingEquiv, RingEquiv.coe_trans, Function.comp_apply, AddMonoidAlgebra.opRingEquiv_apply, RingEquiv.op_apply_apply, toFinsuppIso_apply, unop_op, toFinsupp_monomial, Finsupp.mapRange_single, toFinsuppIso_symm_apply, ofFinsupp_single] #align polynomial.op_ring_equiv_op_monomial Polynomial.opRingEquiv_op_monomial @[simp] theorem opRingEquiv_op_C (a : R) : opRingEquiv R (op (C a)) = C (op a) := opRingEquiv_op_monomial 0 a set_option linter.uppercaseLean3 false in #align polynomial.op_ring_equiv_op_C Polynomial.opRingEquiv_op_C @[simp] theorem opRingEquiv_op_X : opRingEquiv R (op (X : R[X])) = X := opRingEquiv_op_monomial 1 1 set_option linter.uppercaseLean3 false in #align polynomial.op_ring_equiv_op_X Polynomial.opRingEquiv_op_X theorem opRingEquiv_op_C_mul_X_pow (r : R) (n : ℕ) : opRingEquiv R (op (C r * X ^ n : R[X])) = C (op r) * X ^ n := by simp only [X_pow_mul, op_mul, op_pow, map_mul, map_pow, opRingEquiv_op_X, opRingEquiv_op_C] set_option linter.uppercaseLean3 false in #align polynomial.op_ring_equiv_op_C_mul_X_pow Polynomial.opRingEquiv_op_C_mul_X_pow /-! Lemmas to get started, using `(opRingEquiv R).symm` on the various expressions of `Finsupp.single`: `monomial`, `C a`, `X`, `C a * X ^ n`. -/ @[simp] theorem opRingEquiv_symm_monomial (n : ℕ) (r : Rᵐᵒᵖ) : (opRingEquiv R).symm (monomial n r) = op (monomial n (unop r)) := (opRingEquiv R).injective (by simp) #align polynomial.op_ring_equiv_symm_monomial Polynomial.opRingEquiv_symm_monomial @[simp] theorem opRingEquiv_symm_C (a : Rᵐᵒᵖ) : (opRingEquiv R).symm (C a) = op (C (unop a)) := opRingEquiv_symm_monomial 0 a set_option linter.uppercaseLean3 false in #align polynomial.op_ring_equiv_symm_C Polynomial.opRingEquiv_symm_C @[simp] theorem opRingEquiv_symm_X : (opRingEquiv R).symm (X : Rᵐᵒᵖ[X]) = op X := opRingEquiv_symm_monomial 1 1 set_option linter.uppercaseLean3 false in #align polynomial.op_ring_equiv_symm_X Polynomial.opRingEquiv_symm_X theorem opRingEquiv_symm_C_mul_X_pow (r : Rᵐᵒᵖ) (n : ℕ) : (opRingEquiv R).symm (C r * X ^ n : Rᵐᵒᵖ[X]) = op (C (unop r) * X ^ n) := by rw [C_mul_X_pow_eq_monomial, opRingEquiv_symm_monomial, C_mul_X_pow_eq_monomial] set_option linter.uppercaseLean3 false in #align polynomial.op_ring_equiv_symm_C_mul_X_pow Polynomial.opRingEquiv_symm_C_mul_X_pow /-! Lemmas about more global properties of polynomials and opposites. -/ @[simp] theorem coeff_opRingEquiv (p : R[X]ᵐᵒᵖ) (n : ℕ) : (opRingEquiv R p).coeff n = op ((unop p).coeff n) := by induction' p using MulOpposite.rec' with p cases p rfl #align polynomial.coeff_op_ring_equiv Polynomial.coeff_opRingEquiv @[simp] theorem support_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).support = (unop p).support := by induction' p using MulOpposite.rec' with p cases p exact Finsupp.support_mapRange_of_injective (map_zero _) _ op_injective #align polynomial.support_op_ring_equiv Polynomial.support_opRingEquiv @[simp]	Mathlib/RingTheory/Polynomial/Opposites.lean	110	114	theorem natDegree_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).natDegree = (unop p).natDegree := by	by_cases p0 : p = 0 · simp only [p0, _root_.map_zero, natDegree_zero, unop_zero] · simp only [p0, natDegree_eq_support_max', Ne, AddEquivClass.map_eq_zero_iff, not_false_iff, support_opRingEquiv, unop_eq_zero_iff]
/- Copyright (c) 2021 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension import Mathlib.Geometry.Manifold.ContMDiff.Atlas import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace #align_import geometry.manifold.bump_function from "leanprover-community/mathlib"@"b018406ad2f2a73223a3a9e198ccae61e6f05318" /-! # Smooth bump functions on a smooth manifold In this file we define `SmoothBumpFunction I c` to be a bundled smooth "bump" function centered at `c`. It is a structure that consists of two real numbers `0 < rIn < rOut` with small enough `rOut`. We define a coercion to function for this type, and for `f : SmoothBumpFunction I c`, the function `⇑f` written in the extended chart at `c` has the following properties: * `f x = 1` in the closed ball of radius `f.rIn` centered at `c`; * `f x = 0` outside of the ball of radius `f.rOut` centered at `c`; * `0 ≤ f x ≤ 1` for all `x`. The actual statements involve (pre)images under `extChartAt I f` and are given as lemmas in the `SmoothBumpFunction` namespace. ## Tags manifold, smooth bump function -/ universe uE uF uH uM variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] open Function Filter FiniteDimensional Set Metric open scoped Topology Manifold Classical Filter noncomputable section /-! ### Smooth bump function In this section we define a structure for a bundled smooth bump function and prove its properties. -/ /-- Given a smooth manifold modelled on a finite dimensional space `E`, `f : SmoothBumpFunction I M` is a smooth function on `M` such that in the extended chart `e` at `f.c`: * `f x = 1` in the closed ball of radius `f.rIn` centered at `f.c`; * `f x = 0` outside of the ball of radius `f.rOut` centered at `f.c`; * `0 ≤ f x ≤ 1` for all `x`. The structure contains data required to construct a function with these properties. The function is available as `⇑f` or `f x`. Formal statements of the properties listed above involve some (pre)images under `extChartAt I f.c` and are given as lemmas in the `SmoothBumpFunction` namespace. -/ structure SmoothBumpFunction (c : M) extends ContDiffBump (extChartAt I c c) where closedBall_subset : closedBall (extChartAt I c c) rOut ∩ range I ⊆ (extChartAt I c).target #align smooth_bump_function SmoothBumpFunction namespace SmoothBumpFunction variable {c : M} (f : SmoothBumpFunction I c) {x : M} {I} /-- The function defined by `f : SmoothBumpFunction c`. Use automatic coercion to function instead. -/ @[coe] def toFun : M → ℝ := indicator (chartAt H c).source (f.toContDiffBump ∘ extChartAt I c) #align smooth_bump_function.to_fun SmoothBumpFunction.toFun instance : CoeFun (SmoothBumpFunction I c) fun _ => M → ℝ := ⟨toFun⟩ theorem coe_def : ⇑f = indicator (chartAt H c).source (f.toContDiffBump ∘ extChartAt I c) := rfl #align smooth_bump_function.coe_def SmoothBumpFunction.coe_def theorem rOut_pos : 0 < f.rOut := f.toContDiffBump.rOut_pos set_option linter.uppercaseLean3 false in #align smooth_bump_function.R_pos SmoothBumpFunction.rOut_pos theorem ball_subset : ball (extChartAt I c c) f.rOut ∩ range I ⊆ (extChartAt I c).target := Subset.trans (inter_subset_inter_left _ ball_subset_closedBall) f.closedBall_subset #align smooth_bump_function.ball_subset SmoothBumpFunction.ball_subset theorem ball_inter_range_eq_ball_inter_target : ball (extChartAt I c c) f.rOut ∩ range I = ball (extChartAt I c c) f.rOut ∩ (extChartAt I c).target := (subset_inter inter_subset_left f.ball_subset).antisymm <\| inter_subset_inter_right _ <\| extChartAt_target_subset_range _ _ theorem eqOn_source : EqOn f (f.toContDiffBump ∘ extChartAt I c) (chartAt H c).source := eqOn_indicator #align smooth_bump_function.eq_on_source SmoothBumpFunction.eqOn_source theorem eventuallyEq_of_mem_source (hx : x ∈ (chartAt H c).source) : f =ᶠ[𝓝 x] f.toContDiffBump ∘ extChartAt I c := f.eqOn_source.eventuallyEq_of_mem <\| (chartAt H c).open_source.mem_nhds hx #align smooth_bump_function.eventually_eq_of_mem_source SmoothBumpFunction.eventuallyEq_of_mem_source theorem one_of_dist_le (hs : x ∈ (chartAt H c).source) (hd : dist (extChartAt I c x) (extChartAt I c c) ≤ f.rIn) : f x = 1 := by simp only [f.eqOn_source hs, (· ∘ ·), f.one_of_mem_closedBall hd] #align smooth_bump_function.one_of_dist_le SmoothBumpFunction.one_of_dist_le theorem support_eq_inter_preimage : support f = (chartAt H c).source ∩ extChartAt I c ⁻¹' ball (extChartAt I c c) f.rOut := by rw [coe_def, support_indicator, support_comp_eq_preimage, ← extChartAt_source I, ← (extChartAt I c).symm_image_target_inter_eq', ← (extChartAt I c).symm_image_target_inter_eq', f.support_eq] #align smooth_bump_function.support_eq_inter_preimage SmoothBumpFunction.support_eq_inter_preimage theorem isOpen_support : IsOpen (support f) := by rw [support_eq_inter_preimage] exact isOpen_extChartAt_preimage I c isOpen_ball #align smooth_bump_function.is_open_support SmoothBumpFunction.isOpen_support	Mathlib/Geometry/Manifold/BumpFunction.lean	124	128	theorem support_eq_symm_image : support f = (extChartAt I c).symm '' (ball (extChartAt I c c) f.rOut ∩ range I) := by	rw [f.support_eq_inter_preimage, ← extChartAt_source I, ← (extChartAt I c).symm_image_target_inter_eq', inter_comm, ball_inter_range_eq_ball_inter_target]
/- Copyright (c) 2021 Jakob Scholbach. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jakob Scholbach -/ import Mathlib.Algebra.CharP.Basic import Mathlib.Algebra.CharP.Algebra import Mathlib.Data.Nat.Prime #align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" /-! # Exponential characteristic This file defines the exponential characteristic, which is defined to be 1 for a ring with characteristic 0 and the same as the ordinary characteristic, if the ordinary characteristic is prime. This concept is useful to simplify some theorem statements. This file establishes a few basic results relating it to the (ordinary characteristic). The definition is stated for a semiring, but the actual results are for nontrivial rings (as far as exponential characteristic one is concerned), respectively a ring without zero-divisors (for prime characteristic). ## Main results - `ExpChar`: the definition of exponential characteristic - `expChar_is_prime_or_one`: the exponential characteristic is a prime or one - `char_eq_expChar_iff`: the characteristic equals the exponential characteristic iff the characteristic is prime ## Tags exponential characteristic, characteristic -/ universe u variable (R : Type u) section Semiring variable [Semiring R] /-- The definition of the exponential characteristic of a semiring. -/ class inductive ExpChar (R : Type u) [Semiring R] : ℕ → Prop \| zero [CharZero R] : ExpChar R 1 \| prime {q : ℕ} (hprime : q.Prime) [hchar : CharP R q] : ExpChar R q #align exp_char ExpChar #align exp_char.prime ExpChar.prime instance expChar_prime (p) [CharP R p] [Fact p.Prime] : ExpChar R p := ExpChar.prime Fact.out instance expChar_zero [CharZero R] : ExpChar R 1 := ExpChar.zero instance (S : Type) [Semiring S] (p) [ExpChar R p] [ExpChar S p] : ExpChar (R × S) p := by obtain hp \| ⟨hp⟩ := ‹ExpChar R p› · have := Prod.charZero_of_left R S; exact .zero obtain _ \| _ := ‹ExpChar S p› · exact (Nat.not_prime_one hp).elim · have := Prod.charP R S p; exact .prime hp variable {R} in /-- The exponential characteristic is unique. -/ theorem ExpChar.eq {p q : ℕ} (hp : ExpChar R p) (hq : ExpChar R q) : p = q := by cases' hp with hp _ hp' hp · cases' hq with hq _ hq' hq exacts [rfl, False.elim (Nat.not_prime_zero (CharP.eq R hq (CharP.ofCharZero R) ▸ hq'))] · cases' hq with hq _ hq' hq exacts [False.elim (Nat.not_prime_zero (CharP.eq R hp (CharP.ofCharZero R) ▸ hp')), CharP.eq R hp hq] theorem ExpChar.congr {p : ℕ} (q : ℕ) [hq : ExpChar R q] (h : q = p) : ExpChar R p := h ▸ hq /-- Noncomputable function that outputs the unique exponential characteristic of a semiring. -/ noncomputable def ringExpChar (R : Type) [NonAssocSemiring R] : ℕ := max (ringChar R) 1 theorem ringExpChar.eq (q : ℕ) [h : ExpChar R q] : ringExpChar R = q := by cases' h with _ _ h _ · haveI := CharP.ofCharZero R rw [ringExpChar, ringChar.eq R 0]; rfl rw [ringExpChar, ringChar.eq R q] exact Nat.max_eq_left h.one_lt.le @[simp] theorem ringExpChar.eq_one (R : Type*) [NonAssocSemiring R] [CharZero R] : ringExpChar R = 1 := by rw [ringExpChar, ringChar.eq_zero, max_eq_right zero_le_one] /-- The exponential characteristic is one if the characteristic is zero. -/ theorem expChar_one_of_char_zero (q : ℕ) [hp : CharP R 0] [hq : ExpChar R q] : q = 1 := by cases' hq with q hq_one hq_prime hq_hchar · rfl · exact False.elim <\| hq_prime.ne_zero <\| hq_hchar.eq R hp #align exp_char_one_of_char_zero expChar_one_of_char_zero /-- The characteristic equals the exponential characteristic iff the former is prime. -/ theorem char_eq_expChar_iff (p q : ℕ) [hp : CharP R p] [hq : ExpChar R q] : p = q ↔ p.Prime := by cases' hq with q hq_one hq_prime hq_hchar · rw [(CharP.eq R hp inferInstance : p = 0)] decide · exact ⟨fun hpq => hpq.symm ▸ hq_prime, fun _ => CharP.eq R hp hq_hchar⟩ #align char_eq_exp_char_iff char_eq_expChar_iff section Nontrivial variable [Nontrivial R] /-- The exponential characteristic is one if the characteristic is zero. -/	Mathlib/Algebra/CharP/ExpChar.lean	105	108	theorem char_zero_of_expChar_one (p : ℕ) [hp : CharP R p] [hq : ExpChar R 1] : p = 0 := by	cases hq · exact CharP.eq R hp inferInstance · exact False.elim (CharP.char_ne_one R 1 rfl)
/- Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel -/ import Mathlib.Topology.EMetricSpace.Basic import Mathlib.Topology.Bornology.Constructions import Mathlib.Data.Set.Pointwise.Interval import Mathlib.Topology.Order.DenselyOrdered /-! ## Pseudo-metric spaces This file defines pseudo-metric spaces: these differ from metric spaces by not imposing the condition `dist x y = 0 → x = y`. Many definitions and theorems expected on (pseudo-)metric spaces are already introduced on uniform spaces and topological spaces. For example: open and closed sets, compactness, completeness, continuity and uniform continuity. ## Main definitions * `Dist α`: Endows a space `α` with a function `dist a b`. * `PseudoMetricSpace α`: A space endowed with a distance function, which can be zero even if the two elements are non-equal. * `Metric.ball x ε`: The set of all points `y` with `dist y x < ε`. * `Metric.Bounded s`: Whether a subset of a `PseudoMetricSpace` is bounded. * `MetricSpace α`: A `PseudoMetricSpace` with the guarantee `dist x y = 0 → x = y`. Additional useful definitions: * `nndist a b`: `dist` as a function to the non-negative reals. * `Metric.closedBall x ε`: The set of all points `y` with `dist y x ≤ ε`. * `Metric.sphere x ε`: The set of all points `y` with `dist y x = ε`. TODO (anyone): Add "Main results" section. ## Tags pseudo_metric, dist -/ open Set Filter TopologicalSpace Bornology open scoped ENNReal NNReal Uniformity Topology universe u v w variable {α : Type u} {β : Type v} {X ι : Type} theorem UniformSpace.ofDist_aux (ε : ℝ) (hε : 0 < ε) : ∃ δ > (0 : ℝ), ∀ x < δ, ∀ y < δ, x + y < ε := ⟨ε / 2, half_pos hε, fun _x hx _y hy => add_halves ε ▸ add_lt_add hx hy⟩ /-- Construct a uniform structure from a distance function and metric space axioms -/ def UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x) (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace α := .ofFun dist dist_self dist_comm dist_triangle ofDist_aux #align uniform_space_of_dist UniformSpace.ofDist -- Porting note: dropped the `dist_self` argument /-- Construct a bornology from a distance function and metric space axioms. -/ abbrev Bornology.ofDist {α : Type} (dist : α → α → ℝ) (dist_comm : ∀ x y, dist x y = dist y x) (dist_triangle : ∀ x y z, dist x z ≤ dist x y + dist y z) : Bornology α := Bornology.ofBounded { s : Set α \| ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C } ⟨0, fun x hx y => hx.elim⟩ (fun s ⟨c, hc⟩ t h => ⟨c, fun x hx y hy => hc (h hx) (h hy)⟩) (fun s hs t ht => by rcases s.eq_empty_or_nonempty with rfl \| ⟨x, hx⟩ · rwa [empty_union] rcases t.eq_empty_or_nonempty with rfl \| ⟨y, hy⟩ · rwa [union_empty] rsuffices ⟨C, hC⟩ : ∃ C, ∀ z ∈ s ∪ t, dist x z ≤ C · refine ⟨C + C, fun a ha b hb => (dist_triangle a x b).trans ?_⟩ simpa only [dist_comm] using add_le_add (hC _ ha) (hC _ hb) rcases hs with ⟨Cs, hs⟩; rcases ht with ⟨Ct, ht⟩ refine ⟨max Cs (dist x y + Ct), fun z hz => hz.elim (fun hz => (hs hx hz).trans (le_max_left _ _)) (fun hz => (dist_triangle x y z).trans <\| (add_le_add le_rfl (ht hy hz)).trans (le_max_right _ _))⟩) fun z => ⟨dist z z, forall_eq.2 <\| forall_eq.2 le_rfl⟩ #align bornology.of_dist Bornology.ofDistₓ /-- The distance function (given an ambient metric space on `α`), which returns a nonnegative real number `dist x y` given `x y : α`. -/ @[ext] class Dist (α : Type) where dist : α → α → ℝ #align has_dist Dist export Dist (dist) -- the uniform structure and the emetric space structure are embedded in the metric space structure -- to avoid instance diamond issues. See Note [forgetful inheritance]. /-- This is an internal lemma used inside the default of `PseudoMetricSpace.edist`. -/ private theorem dist_nonneg' {α} {x y : α} (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x) (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : 0 ≤ dist x y := have : 0 ≤ 2 dist x y := calc 0 = dist x x := (dist_self _).symm _ ≤ dist x y + dist y x := dist_triangle _ _ _ _ = 2 * dist x y := by rw [two_mul, dist_comm] nonneg_of_mul_nonneg_right this two_pos #noalign pseudo_metric_space.edist_dist_tac -- Porting note (#11215): TODO: restore /-- Pseudo metric and Metric spaces A pseudo metric space is endowed with a distance for which the requirement `d(x,y)=0 → x = y` might not hold. A metric space is a pseudo metric space such that `d(x,y)=0 → x = y`. Each pseudo metric space induces a canonical `UniformSpace` and hence a canonical `TopologicalSpace` This is enforced in the type class definition, by extending the `UniformSpace` structure. When instantiating a `PseudoMetricSpace` structure, the uniformity fields are not necessary, they will be filled in by default. In the same way, each (pseudo) metric space induces a (pseudo) emetric space structure. It is included in the structure, but filled in by default. -/ class PseudoMetricSpace (α : Type u) extends Dist α : Type u where dist_self : ∀ x : α, dist x x = 0 dist_comm : ∀ x y : α, dist x y = dist y x dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z edist : α → α → ℝ≥0∞ := fun x y => ENNReal.ofNNReal ⟨dist x y, dist_nonneg' _ ‹_› ‹_› ‹_›⟩ edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) -- Porting note (#11215): TODO: add := by _ toUniformSpace : UniformSpace α := .ofDist dist dist_self dist_comm dist_triangle uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α \| dist p.1 p.2 < ε } := by intros; rfl toBornology : Bornology α := Bornology.ofDist dist dist_comm dist_triangle cobounded_sets : (Bornology.cobounded α).sets = { s \| ∃ C : ℝ, ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C } := by intros; rfl #align pseudo_metric_space PseudoMetricSpace /-- Two pseudo metric space structures with the same distance function coincide. -/ @[ext] theorem PseudoMetricSpace.ext {α : Type} {m m' : PseudoMetricSpace α} (h : m.toDist = m'.toDist) : m = m' := by cases' m with d _ _ _ ed hed U hU B hB cases' m' with d' _ _ _ ed' hed' U' hU' B' hB' obtain rfl : d = d' := h congr · ext x y : 2 rw [hed, hed'] · exact UniformSpace.ext (hU.trans hU'.symm) · ext : 2 rw [← Filter.mem_sets, ← Filter.mem_sets, hB, hB'] #align pseudo_metric_space.ext PseudoMetricSpace.ext variable [PseudoMetricSpace α] attribute [instance] PseudoMetricSpace.toUniformSpace PseudoMetricSpace.toBornology -- see Note [lower instance priority] instance (priority := 200) PseudoMetricSpace.toEDist : EDist α := ⟨PseudoMetricSpace.edist⟩ #align pseudo_metric_space.to_has_edist PseudoMetricSpace.toEDist /-- Construct a pseudo-metric space structure whose underlying topological space structure (definitionally) agrees which a pre-existing topology which is compatible with a given distance function. -/ def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x) (dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) (H : ∀ s : Set α, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) : PseudoMetricSpace α := { dist := dist dist_self := dist_self dist_comm := dist_comm dist_triangle := dist_triangle edist_dist := fun x y => by exact ENNReal.coe_nnreal_eq _ toUniformSpace := (UniformSpace.ofDist dist dist_self dist_comm dist_triangle).replaceTopology <\| TopologicalSpace.ext_iff.2 fun s ↦ (H s).trans <\| forall₂_congr fun x _ ↦ ((UniformSpace.hasBasis_ofFun (exists_gt (0 : ℝ)) dist dist_self dist_comm dist_triangle UniformSpace.ofDist_aux).comap (Prod.mk x)).mem_iff.symm uniformity_dist := rfl toBornology := Bornology.ofDist dist dist_comm dist_triangle cobounded_sets := rfl } #align pseudo_metric_space.of_dist_topology PseudoMetricSpace.ofDistTopology @[simp] theorem dist_self (x : α) : dist x x = 0 := PseudoMetricSpace.dist_self x #align dist_self dist_self theorem dist_comm (x y : α) : dist x y = dist y x := PseudoMetricSpace.dist_comm x y #align dist_comm dist_comm theorem edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) := PseudoMetricSpace.edist_dist x y #align edist_dist edist_dist theorem dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z := PseudoMetricSpace.dist_triangle x y z #align dist_triangle dist_triangle theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by rw [dist_comm z]; apply dist_triangle #align dist_triangle_left dist_triangle_left theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by rw [dist_comm y]; apply dist_triangle #align dist_triangle_right dist_triangle_right theorem dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w := calc dist x w ≤ dist x z + dist z w := dist_triangle x z w _ ≤ dist x y + dist y z + dist z w := add_le_add_right (dist_triangle x y z) _ #align dist_triangle4 dist_triangle4 theorem dist_triangle4_left (x₁ y₁ x₂ y₂ : α) : dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by rw [add_left_comm, dist_comm x₁, ← add_assoc] apply dist_triangle4 #align dist_triangle4_left dist_triangle4_left theorem dist_triangle4_right (x₁ y₁ x₂ y₂ : α) : dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ := by rw [add_right_comm, dist_comm y₁] apply dist_triangle4 #align dist_triangle4_right dist_triangle4_right /-- The triangle (polygon) inequality for sequences of points; `Finset.Ico` version. -/ theorem dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1)) := by induction n, h using Nat.le_induction with \| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self] \| succ n hle ihn => calc dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _ _ ≤ (∑ i ∈ Finset.Ico m n, _) + _ := add_le_add ihn le_rfl _ = ∑ i ∈ Finset.Ico m (n + 1), _ := by { rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp } #align dist_le_Ico_sum_dist dist_le_Ico_sum_dist /-- The triangle (polygon) inequality for sequences of points; `Finset.range` version. -/ theorem dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) : dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, dist (f i) (f (i + 1)) := Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_dist f (Nat.zero_le n) #align dist_le_range_sum_dist dist_le_range_sum_dist /-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced with an upper estimate. -/ theorem dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ} (hd : ∀ {k}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i := le_trans (dist_le_Ico_sum_dist f hmn) <\| Finset.sum_le_sum fun _k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2 #align dist_le_Ico_sum_of_dist_le dist_le_Ico_sum_of_dist_le /-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced with an upper estimate. -/ theorem dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ} (hd : ∀ {k}, k < n → dist (f k) (f (k + 1)) ≤ d k) : dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, d i := Nat.Ico_zero_eq_range ▸ dist_le_Ico_sum_of_dist_le (zero_le n) fun _ => hd #align dist_le_range_sum_of_dist_le dist_le_range_sum_of_dist_le theorem swap_dist : Function.swap (@dist α _) = dist := by funext x y; exact dist_comm _ _ #align swap_dist swap_dist theorem abs_dist_sub_le (x y z : α) : \|dist x z - dist y z\| ≤ dist x y := abs_sub_le_iff.2 ⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩ #align abs_dist_sub_le abs_dist_sub_le theorem dist_nonneg {x y : α} : 0 ≤ dist x y := dist_nonneg' dist dist_self dist_comm dist_triangle #align dist_nonneg dist_nonneg namespace Mathlib.Meta.Positivity open Lean Meta Qq Function /-- Extension for the `positivity` tactic: distances are nonnegative. -/ @[positivity Dist.dist _ _] def evalDist : PositivityExt where eval {u α} _zα _pα e := do match u, α, e with \| 0, ~q(ℝ), ~q(@Dist.dist $β $inst $a $b) => let _inst ← synthInstanceQ q(PseudoMetricSpace $β) assertInstancesCommute pure (.nonnegative q(dist_nonneg)) \| _, _, _ => throwError "not dist" end Mathlib.Meta.Positivity example {x y : α} : 0 ≤ dist x y := by positivity @[simp] theorem abs_dist {a b : α} : \|dist a b\| = dist a b := abs_of_nonneg dist_nonneg #align abs_dist abs_dist /-- A version of `Dist` that takes value in `ℝ≥0`. -/ class NNDist (α : Type) where nndist : α → α → ℝ≥0 #align has_nndist NNDist export NNDist (nndist) -- see Note [lower instance priority] /-- Distance as a nonnegative real number. -/ instance (priority := 100) PseudoMetricSpace.toNNDist : NNDist α := ⟨fun a b => ⟨dist a b, dist_nonneg⟩⟩ #align pseudo_metric_space.to_has_nndist PseudoMetricSpace.toNNDist /-- Express `dist` in terms of `nndist`-/ theorem dist_nndist (x y : α) : dist x y = nndist x y := rfl #align dist_nndist dist_nndist @[simp, norm_cast] theorem coe_nndist (x y : α) : ↑(nndist x y) = dist x y := rfl #align coe_nndist coe_nndist /-- Express `edist` in terms of `nndist`-/ theorem edist_nndist (x y : α) : edist x y = nndist x y := by rw [edist_dist, dist_nndist, ENNReal.ofReal_coe_nnreal] #align edist_nndist edist_nndist /-- Express `nndist` in terms of `edist`-/ theorem nndist_edist (x y : α) : nndist x y = (edist x y).toNNReal := by simp [edist_nndist] #align nndist_edist nndist_edist @[simp, norm_cast] theorem coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y := (edist_nndist x y).symm #align coe_nnreal_ennreal_nndist coe_nnreal_ennreal_nndist @[simp, norm_cast] theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by rw [edist_nndist, ENNReal.coe_lt_coe] #align edist_lt_coe edist_lt_coe @[simp, norm_cast] theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by rw [edist_nndist, ENNReal.coe_le_coe] #align edist_le_coe edist_le_coe /-- In a pseudometric space, the extended distance is always finite-/ theorem edist_lt_top {α : Type} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ := (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top #align edist_lt_top edist_lt_top /-- In a pseudometric space, the extended distance is always finite-/ theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ := (edist_lt_top x y).ne #align edist_ne_top edist_ne_top /-- `nndist x x` vanishes-/ @[simp] theorem nndist_self (a : α) : nndist a a = 0 := NNReal.coe_eq_zero.1 (dist_self a) #align nndist_self nndist_self -- Porting note: `dist_nndist` and `coe_nndist` moved up @[simp, norm_cast] theorem dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c := Iff.rfl #align dist_lt_coe dist_lt_coe @[simp, norm_cast] theorem dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c := Iff.rfl #align dist_le_coe dist_le_coe @[simp] theorem edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg] #align edist_lt_of_real edist_lt_ofReal @[simp] theorem edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) : edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr] #align edist_le_of_real edist_le_ofReal /-- Express `nndist` in terms of `dist`-/ theorem nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) := by rw [dist_nndist, Real.toNNReal_coe] #align nndist_dist nndist_dist theorem nndist_comm (x y : α) : nndist x y = nndist y x := NNReal.eq <\| dist_comm x y #align nndist_comm nndist_comm /-- Triangle inequality for the nonnegative distance-/ theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z := dist_triangle _ _ _ #align nndist_triangle nndist_triangle theorem nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y := dist_triangle_left _ _ _ #align nndist_triangle_left nndist_triangle_left theorem nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z := dist_triangle_right _ _ _ #align nndist_triangle_right nndist_triangle_right /-- Express `dist` in terms of `edist`-/ theorem dist_edist (x y : α) : dist x y = (edist x y).toReal := by rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg] #align dist_edist dist_edist namespace Metric -- instantiate pseudometric space as a topology variable {x y z : α} {δ ε ε₁ ε₂ : ℝ} {s : Set α} /-- `ball x ε` is the set of all points `y` with `dist y x < ε` -/ def ball (x : α) (ε : ℝ) : Set α := { y \| dist y x < ε } #align metric.ball Metric.ball @[simp] theorem mem_ball : y ∈ ball x ε ↔ dist y x < ε := Iff.rfl #align metric.mem_ball Metric.mem_ball theorem mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw [dist_comm, mem_ball] #align metric.mem_ball' Metric.mem_ball' theorem pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε := dist_nonneg.trans_lt hy #align metric.pos_of_mem_ball Metric.pos_of_mem_ball theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by rwa [mem_ball, dist_self] #align metric.mem_ball_self Metric.mem_ball_self @[simp] theorem nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε := ⟨fun ⟨_x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩ #align metric.nonempty_ball Metric.nonempty_ball @[simp] theorem ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 := by rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt] #align metric.ball_eq_empty Metric.ball_eq_empty @[simp] theorem ball_zero : ball x 0 = ∅ := by rw [ball_eq_empty] #align metric.ball_zero Metric.ball_zero /-- If a point belongs to an open ball, then there is a strictly smaller radius whose ball also contains it. See also `exists_lt_subset_ball`. -/ theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' := by simp only [mem_ball] at h ⊢ exact ⟨(dist x y + ε) / 2, by linarith, by linarith⟩ #align metric.exists_lt_mem_ball_of_mem_ball Metric.exists_lt_mem_ball_of_mem_ball theorem ball_eq_ball (ε : ℝ) (x : α) : UniformSpace.ball x { p \| dist p.2 p.1 < ε } = Metric.ball x ε := rfl #align metric.ball_eq_ball Metric.ball_eq_ball theorem ball_eq_ball' (ε : ℝ) (x : α) : UniformSpace.ball x { p \| dist p.1 p.2 < ε } = Metric.ball x ε := by ext simp [dist_comm, UniformSpace.ball] #align metric.ball_eq_ball' Metric.ball_eq_ball' @[simp] theorem iUnion_ball_nat (x : α) : ⋃ n : ℕ, ball x n = univ := iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x) #align metric.Union_ball_nat Metric.iUnion_ball_nat @[simp] theorem iUnion_ball_nat_succ (x : α) : ⋃ n : ℕ, ball x (n + 1) = univ := iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _) #align metric.Union_ball_nat_succ Metric.iUnion_ball_nat_succ /-- `closedBall x ε` is the set of all points `y` with `dist y x ≤ ε` -/ def closedBall (x : α) (ε : ℝ) := { y \| dist y x ≤ ε } #align metric.closed_ball Metric.closedBall @[simp] theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε := Iff.rfl #align metric.mem_closed_ball Metric.mem_closedBall theorem mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closedBall] #align metric.mem_closed_ball' Metric.mem_closedBall' /-- `sphere x ε` is the set of all points `y` with `dist y x = ε` -/ def sphere (x : α) (ε : ℝ) := { y \| dist y x = ε } #align metric.sphere Metric.sphere @[simp] theorem mem_sphere : y ∈ sphere x ε ↔ dist y x = ε := Iff.rfl #align metric.mem_sphere Metric.mem_sphere theorem mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε := by rw [dist_comm, mem_sphere] #align metric.mem_sphere' Metric.mem_sphere' theorem ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x := ne_of_mem_of_not_mem h <\| by simpa using hε.symm #align metric.ne_of_mem_sphere Metric.ne_of_mem_sphere theorem nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε := dist_nonneg.trans_eq hy #align metric.nonneg_of_mem_sphere Metric.nonneg_of_mem_sphere @[simp] theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ := Set.eq_empty_iff_forall_not_mem.mpr fun _y hy => (nonneg_of_mem_sphere hy).not_lt hε #align metric.sphere_eq_empty_of_neg Metric.sphere_eq_empty_of_neg theorem sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ := Set.eq_empty_iff_forall_not_mem.mpr fun _ h => ne_of_mem_sphere h hε (Subsingleton.elim _ _) #align metric.sphere_eq_empty_of_subsingleton Metric.sphere_eq_empty_of_subsingleton instance sphere_isEmpty_of_subsingleton [Subsingleton α] [NeZero ε] : IsEmpty (sphere x ε) := by rw [sphere_eq_empty_of_subsingleton (NeZero.ne ε)]; infer_instance #align metric.sphere_is_empty_of_subsingleton Metric.sphere_isEmpty_of_subsingleton theorem mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε := by rwa [mem_closedBall, dist_self] #align metric.mem_closed_ball_self Metric.mem_closedBall_self @[simp] theorem nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε := ⟨fun ⟨_x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩ #align metric.nonempty_closed_ball Metric.nonempty_closedBall @[simp] theorem closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le] #align metric.closed_ball_eq_empty Metric.closedBall_eq_empty /-- Closed balls and spheres coincide when the radius is non-positive -/ theorem closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε := Set.ext fun _ => (hε.trans dist_nonneg).le_iff_eq #align metric.closed_ball_eq_sphere_of_nonpos Metric.closedBall_eq_sphere_of_nonpos theorem ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun _y hy => mem_closedBall.2 (le_of_lt hy) #align metric.ball_subset_closed_ball Metric.ball_subset_closedBall theorem sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun _ => le_of_eq #align metric.sphere_subset_closed_ball Metric.sphere_subset_closedBall lemma sphere_subset_ball {r R : ℝ} (h : r < R) : sphere x r ⊆ ball x R := fun _x hx ↦ (mem_sphere.1 hx).trans_lt h theorem closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) := Set.disjoint_left.mpr fun _a ha1 ha2 => (h.trans <\| dist_triangle_left _ _ _).not_lt <\| add_lt_add_of_le_of_lt ha1 ha2 #align metric.closed_ball_disjoint_ball Metric.closedBall_disjoint_ball theorem ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) := (closedBall_disjoint_ball <\| by rwa [add_comm, dist_comm]).symm #align metric.ball_disjoint_closed_ball Metric.ball_disjoint_closedBall theorem ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) := (closedBall_disjoint_ball h).mono_left ball_subset_closedBall #align metric.ball_disjoint_ball Metric.ball_disjoint_ball theorem closedBall_disjoint_closedBall (h : δ + ε < dist x y) : Disjoint (closedBall x δ) (closedBall y ε) := Set.disjoint_left.mpr fun _a ha1 ha2 => h.not_le <\| (dist_triangle_left _ _ _).trans <\| add_le_add ha1 ha2 #align metric.closed_ball_disjoint_closed_ball Metric.closedBall_disjoint_closedBall theorem sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) := Set.disjoint_left.mpr fun _y hy₁ hy₂ => absurd hy₁ <\| ne_of_lt hy₂ #align metric.sphere_disjoint_ball Metric.sphere_disjoint_ball @[simp] theorem ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε := Set.ext fun _y => (@le_iff_lt_or_eq ℝ _ _ _).symm #align metric.ball_union_sphere Metric.ball_union_sphere @[simp] theorem sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε := by rw [union_comm, ball_union_sphere] #align metric.sphere_union_ball Metric.sphere_union_ball @[simp] theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot] #align metric.closed_ball_diff_sphere Metric.closedBall_diff_sphere @[simp] theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot] #align metric.closed_ball_diff_ball Metric.closedBall_diff_ball theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by rw [mem_ball', mem_ball] #align metric.mem_ball_comm Metric.mem_ball_comm theorem mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by rw [mem_closedBall', mem_closedBall] #align metric.mem_closed_ball_comm Metric.mem_closedBall_comm theorem mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε := by rw [mem_sphere', mem_sphere] #align metric.mem_sphere_comm Metric.mem_sphere_comm @[gcongr] theorem ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun _y yx => lt_of_lt_of_le (mem_ball.1 yx) h #align metric.ball_subset_ball Metric.ball_subset_ball theorem closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by ext y; rw [mem_closedBall, ← forall_lt_iff_le', mem_iInter₂]; rfl #align metric.closed_ball_eq_bInter_ball Metric.closedBall_eq_bInter_ball theorem ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz => calc dist z y ≤ dist z x + dist x y := dist_triangle _ _ _ _ < ε₁ + dist x y := add_lt_add_right (mem_ball.1 hz) _ _ ≤ ε₂ := h #align metric.ball_subset_ball' Metric.ball_subset_ball' @[gcongr] theorem closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ := fun _y (yx : _ ≤ ε₁) => le_trans yx h #align metric.closed_ball_subset_closed_ball Metric.closedBall_subset_closedBall theorem closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) : closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz => calc dist z y ≤ dist z x + dist x y := dist_triangle _ _ _ _ ≤ ε₁ + dist x y := add_le_add_right (mem_closedBall.1 hz) _ _ ≤ ε₂ := h #align metric.closed_ball_subset_closed_ball' Metric.closedBall_subset_closedBall' theorem closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ := fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h #align metric.closed_ball_subset_ball Metric.closedBall_subset_ball theorem closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) : closedBall x ε₁ ⊆ ball y ε₂ := fun z hz => calc dist z y ≤ dist z x + dist x y := dist_triangle _ _ _ _ ≤ ε₁ + dist x y := add_le_add_right (mem_closedBall.1 hz) _ _ < ε₂ := h #align metric.closed_ball_subset_ball' Metric.closedBall_subset_ball' theorem dist_le_add_of_nonempty_closedBall_inter_closedBall (h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ := let ⟨z, hz⟩ := h calc dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _ _ ≤ ε₁ + ε₂ := add_le_add hz.1 hz.2 #align metric.dist_le_add_of_nonempty_closed_ball_inter_closed_ball Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) : dist x y < ε₁ + ε₂ := let ⟨z, hz⟩ := h calc dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _ _ < ε₁ + ε₂ := add_lt_add_of_le_of_lt hz.1 hz.2 #align metric.dist_lt_add_of_nonempty_closed_ball_inter_ball Metric.dist_lt_add_of_nonempty_closedBall_inter_ball theorem dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y < ε₁ + ε₂ := by rw [inter_comm] at h rw [add_comm, dist_comm] exact dist_lt_add_of_nonempty_closedBall_inter_ball h #align metric.dist_lt_add_of_nonempty_ball_inter_closed_ball Metric.dist_lt_add_of_nonempty_ball_inter_closedBall theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) : dist x y < ε₁ + ε₂ := dist_lt_add_of_nonempty_closedBall_inter_ball <\| h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl) #align metric.dist_lt_add_of_nonempty_ball_inter_ball Metric.dist_lt_add_of_nonempty_ball_inter_ball @[simp] theorem iUnion_closedBall_nat (x : α) : ⋃ n : ℕ, closedBall x n = univ := iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x) #align metric.Union_closed_ball_nat Metric.iUnion_closedBall_nat theorem iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s := by rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ] #align metric.Union_inter_closed_ball_nat Metric.iUnion_inter_closedBall_nat theorem ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by rw [← add_sub_cancel ε₁ ε₂] exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h) #align metric.ball_subset Metric.ball_subset theorem ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε := ball_subset <\| by rw [sub_self_div_two]; exact le_of_lt h #align metric.ball_half_subset Metric.ball_half_subset theorem exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε := ⟨_, sub_pos.2 h, ball_subset <\| by rw [sub_sub_self]⟩ #align metric.exists_ball_subset_ball Metric.exists_ball_subset_ball /-- If a property holds for all points in closed balls of arbitrarily large radii, then it holds for all points. -/ theorem forall_of_forall_mem_closedBall (p : α → Prop) (x : α) (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ closedBall x R, p y) (y : α) : p y := by obtain ⟨R, hR, h⟩ : ∃ R ≥ dist y x, ∀ z : α, z ∈ closedBall x R → p z := frequently_iff.1 H (Ici_mem_atTop (dist y x)) exact h _ hR #align metric.forall_of_forall_mem_closed_ball Metric.forall_of_forall_mem_closedBall /-- If a property holds for all points in balls of arbitrarily large radii, then it holds for all points. -/ theorem forall_of_forall_mem_ball (p : α → Prop) (x : α) (H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y := by obtain ⟨R, hR, h⟩ : ∃ R > dist y x, ∀ z : α, z ∈ ball x R → p z := frequently_iff.1 H (Ioi_mem_atTop (dist y x)) exact h _ hR #align metric.forall_of_forall_mem_ball Metric.forall_of_forall_mem_ball theorem isBounded_iff {s : Set α} : IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := by rw [isBounded_def, ← Filter.mem_sets, @PseudoMetricSpace.cobounded_sets α, mem_setOf_eq, compl_compl] #align metric.is_bounded_iff Metric.isBounded_iff theorem isBounded_iff_eventually {s : Set α} : IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := isBounded_iff.trans ⟨fun ⟨C, h⟩ => eventually_atTop.2 ⟨C, fun _C' hC' _x hx _y hy => (h hx hy).trans hC'⟩, Eventually.exists⟩ #align metric.is_bounded_iff_eventually Metric.isBounded_iff_eventually theorem isBounded_iff_exists_ge {s : Set α} (c : ℝ) : IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := ⟨fun h => ((eventually_ge_atTop c).and (isBounded_iff_eventually.1 h)).exists, fun h => isBounded_iff.2 <\| h.imp fun _ => And.right⟩ #align metric.is_bounded_iff_exists_ge Metric.isBounded_iff_exists_ge theorem isBounded_iff_nndist {s : Set α} : IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by simp only [isBounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist, NNReal.coe_mk, exists_prop] #align metric.is_bounded_iff_nndist Metric.isBounded_iff_nndist theorem toUniformSpace_eq : ‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle := UniformSpace.ext PseudoMetricSpace.uniformity_dist #align metric.to_uniform_space_eq Metric.toUniformSpace_eq theorem uniformity_basis_dist : (𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α \| dist p.1 p.2 < ε } := by rw [toUniformSpace_eq] exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _ #align metric.uniformity_basis_dist Metric.uniformity_basis_dist /-- Given `f : β → ℝ`, if `f` sends `{i \| p i}` to a set of positive numbers accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`. For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`, and `uniformity_basis_dist_inv_nat_pos`. -/ protected theorem mk_uniformity_basis {β : Type} {p : β → Prop} {f : β → ℝ} (hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε) : (𝓤 α).HasBasis p fun i => { p : α × α \| dist p.1 p.2 < f i } := by refine ⟨fun s => uniformity_basis_dist.mem_iff.trans ?_⟩ constructor · rintro ⟨ε, ε₀, hε⟩ rcases hf ε₀ with ⟨i, hi, H⟩ exact ⟨i, hi, fun x (hx : _ < _) => hε <\| lt_of_lt_of_le hx H⟩ · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩ #align metric.mk_uniformity_basis Metric.mk_uniformity_basis theorem uniformity_basis_dist_rat : (𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α \| dist p.1 p.2 < r } := Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun _ε hε => let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε ⟨r, Rat.cast_pos.1 hr0, hrε.le⟩ #align metric.uniformity_basis_dist_rat Metric.uniformity_basis_dist_rat theorem uniformity_basis_dist_inv_nat_succ : (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α \| dist p.1 p.2 < 1 / (↑n + 1) } := Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <\| Nat.cast_add_one_pos n) fun _ε ε0 => (exists_nat_one_div_lt ε0).imp fun _n hn => ⟨trivial, le_of_lt hn⟩ #align metric.uniformity_basis_dist_inv_nat_succ Metric.uniformity_basis_dist_inv_nat_succ theorem uniformity_basis_dist_inv_nat_pos : (𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α \| dist p.1 p.2 < 1 / ↑n } := Metric.mk_uniformity_basis (fun _ hn => div_pos zero_lt_one <\| Nat.cast_pos.2 hn) fun _ ε0 => let ⟨n, hn⟩ := exists_nat_one_div_lt ε0 ⟨n + 1, Nat.succ_pos n, mod_cast hn.le⟩ #align metric.uniformity_basis_dist_inv_nat_pos Metric.uniformity_basis_dist_inv_nat_pos theorem uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α \| dist p.1 p.2 < r ^ n } := Metric.mk_uniformity_basis (fun _ _ => pow_pos h0 _) fun _ε ε0 => let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1 ⟨n, trivial, hn.le⟩ #align metric.uniformity_basis_dist_pow Metric.uniformity_basis_dist_pow theorem uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) : (𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α \| dist p.1 p.2 < r } := Metric.mk_uniformity_basis (fun _ => And.left) fun r hr => ⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 <\| Or.inr (half_lt_self hR)⟩, min_le_left _ _⟩ #align metric.uniformity_basis_dist_lt Metric.uniformity_basis_dist_lt /-- Given `f : β → ℝ`, if `f` sends `{i \| p i}` to a set of positive numbers accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i \| p i}` form a basis of `𝓤 α`. Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor. More can be easily added if needed in the future. -/ protected theorem mk_uniformity_basis_le {β : Type} {p : β → Prop} {f : β → ℝ} (hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) : (𝓤 α).HasBasis p fun x => { p : α × α \| dist p.1 p.2 ≤ f x } := by refine ⟨fun s => uniformity_basis_dist.mem_iff.trans ?_⟩ constructor · rintro ⟨ε, ε₀, hε⟩ rcases exists_between ε₀ with ⟨ε', hε'⟩ rcases hf ε' hε'.1 with ⟨i, hi, H⟩ exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <\| lt_of_le_of_lt (le_trans hx H) hε'.2⟩ · exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)⟩ #align metric.mk_uniformity_basis_le Metric.mk_uniformity_basis_le /-- Constant size closed neighborhoods of the diagonal form a basis of the uniformity filter. -/ theorem uniformity_basis_dist_le : (𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α \| dist p.1 p.2 ≤ ε } := Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩ #align metric.uniformity_basis_dist_le Metric.uniformity_basis_dist_le theorem uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α \| dist p.1 p.2 ≤ r ^ n } := Metric.mk_uniformity_basis_le (fun _ _ => pow_pos h0 _) fun _ε ε0 => let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1 ⟨n, trivial, hn.le⟩ #align metric.uniformity_basis_dist_le_pow Metric.uniformity_basis_dist_le_pow theorem mem_uniformity_dist {s : Set (α × α)} : s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s := uniformity_basis_dist.mem_uniformity_iff #align metric.mem_uniformity_dist Metric.mem_uniformity_dist /-- A constant size neighborhood of the diagonal is an entourage. -/ theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α \| dist p.1 p.2 < ε } ∈ 𝓤 α := mem_uniformity_dist.2 ⟨ε, ε0, id⟩ #align metric.dist_mem_uniformity Metric.dist_mem_uniformity theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε := uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist #align metric.uniform_continuous_iff Metric.uniformContinuous_iff theorem uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε := Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist #align metric.uniform_continuous_on_iff Metric.uniformContinuousOn_iff theorem uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε := Metric.uniformity_basis_dist_le.uniformContinuousOn_iff Metric.uniformity_basis_dist_le #align metric.uniform_continuous_on_iff_le Metric.uniformContinuousOn_iff_le nonrec theorem uniformInducing_iff [PseudoMetricSpace β] {f : α → β} : UniformInducing f ↔ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ := uniformInducing_iff'.trans <\| Iff.rfl.and <\| ((uniformity_basis_dist.comap _).le_basis_iff uniformity_basis_dist).trans <\| by simp only [subset_def, Prod.forall, gt_iff_lt, preimage_setOf_eq, Prod.map_apply, mem_setOf] nonrec theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} : UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ := by rw [uniformEmbedding_iff, and_comm, uniformInducing_iff] #align metric.uniform_embedding_iff Metric.uniformEmbedding_iff /-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x` and `f y` is controlled in terms of the distance between `x` and `y`. -/ theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} (h : UniformEmbedding f) : (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ := ⟨uniformContinuous_iff.1 h.uniformContinuous, (uniformEmbedding_iff.1 h).2.2⟩ #align metric.controlled_of_uniform_embedding Metric.controlled_of_uniformEmbedding theorem totallyBounded_iff {s : Set α} : TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε := uniformity_basis_dist.totallyBounded_iff #align metric.totally_bounded_iff Metric.totallyBounded_iff /-- A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the space from finitely many data. -/ theorem totallyBounded_of_finite_discretization {s : Set α} (H : ∀ ε > (0 : ℝ), ∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) : TotallyBounded s := by rcases s.eq_empty_or_nonempty with hs \| hs · rw [hs] exact totallyBounded_empty rcases hs with ⟨x0, hx0⟩ haveI : Inhabited s := ⟨⟨x0, hx0⟩⟩ refine totallyBounded_iff.2 fun ε ε0 => ?_ rcases H ε ε0 with ⟨β, fβ, F, hF⟩ let Finv := Function.invFun F refine ⟨range (Subtype.val ∘ Finv), finite_range _, fun x xs => ?_⟩ let x' := Finv (F ⟨x, xs⟩) have : F x' = F ⟨x, xs⟩ := Function.invFun_eq ⟨⟨x, xs⟩, rfl⟩ simp only [Set.mem_iUnion, Set.mem_range] exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩ #align metric.totally_bounded_of_finite_discretization Metric.totallyBounded_of_finite_discretization theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) : ∀ ε > 0, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε := by intro ε ε_pos rw [totallyBounded_iff_subset] at hs exact hs _ (dist_mem_uniformity ε_pos) #align metric.finite_approx_of_totally_bounded Metric.finite_approx_of_totallyBounded /-- Expressing uniform convergence using `dist` -/ theorem tendstoUniformlyOnFilter_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} : TendstoUniformlyOnFilter F f p p' ↔ ∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε := by refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩ rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩ exact (H ε εpos).mono fun n hn => hε hn #align metric.tendsto_uniformly_on_filter_iff Metric.tendstoUniformlyOnFilter_iff /-- Expressing locally uniform convergence on a set using `dist`. -/ theorem tendstoLocallyUniformlyOn_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} : TendstoLocallyUniformlyOn F f p s ↔ ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu x hx => ?_⟩ rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩ rcases H ε εpos x hx with ⟨t, ht, Ht⟩ exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩ #align metric.tendsto_locally_uniformly_on_iff Metric.tendstoLocallyUniformlyOn_iff /-- Expressing uniform convergence on a set using `dist`. -/ theorem tendstoUniformlyOn_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} : TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε := by refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩ rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩ exact (H ε εpos).mono fun n hs x hx => hε (hs x hx) #align metric.tendsto_uniformly_on_iff Metric.tendstoUniformlyOn_iff /-- Expressing locally uniform convergence using `dist`. -/ theorem tendstoLocallyUniformly_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α} {p : Filter ι} : TendstoLocallyUniformly F f p ↔ ∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε := by simp only [← tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, nhdsWithin_univ, mem_univ, forall_const, exists_prop] #align metric.tendsto_locally_uniformly_iff Metric.tendstoLocallyUniformly_iff /-- Expressing uniform convergence using `dist`. -/ theorem tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} : TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff] simp #align metric.tendsto_uniformly_iff Metric.tendstoUniformly_iff protected theorem cauchy_iff {f : Filter α} : Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε := uniformity_basis_dist.cauchy_iff #align metric.cauchy_iff Metric.cauchy_iff theorem nhds_basis_ball : (𝓝 x).HasBasis (0 < ·) (ball x) := nhds_basis_uniformity uniformity_basis_dist #align metric.nhds_basis_ball Metric.nhds_basis_ball theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s := nhds_basis_ball.mem_iff #align metric.mem_nhds_iff Metric.mem_nhds_iff theorem eventually_nhds_iff {p : α → Prop} : (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y := mem_nhds_iff #align metric.eventually_nhds_iff Metric.eventually_nhds_iff theorem eventually_nhds_iff_ball {p : α → Prop} : (∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y := mem_nhds_iff #align metric.eventually_nhds_iff_ball Metric.eventually_nhds_iff_ball /-- A version of `Filter.eventually_prod_iff` where the first filter consists of neighborhoods in a pseudo-metric space. -/ theorem eventually_nhds_prod_iff {f : Filter ι} {x₀ : α} {p : α × ι → Prop} : (∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧ ∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) := by refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_ simp only [Prod.exists, forall_prod_set, id, mem_ball, and_assoc, exists_and_left, and_imp] rfl #align metric.eventually_nhds_prod_iff Metric.eventually_nhds_prod_iff /-- A version of `Filter.eventually_prod_iff` where the second filter consists of neighborhoods in a pseudo-metric space. -/ theorem eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} : (∀ᶠ x in f ×ˢ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧ ∃ ε > 0, ∀ {i}, pa i → ∀ {x}, dist x x₀ < ε → p (i, x) := by rw [eventually_swap_iff, Metric.eventually_nhds_prod_iff] constructor <;> · rintro ⟨a1, a2, a3, a4, a5⟩ exact ⟨a3, a4, a1, a2, fun b1 b2 b3 => a5 b3 b1⟩ #align metric.eventually_prod_nhds_iff Metric.eventually_prod_nhds_iff theorem nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) := nhds_basis_uniformity uniformity_basis_dist_le #align metric.nhds_basis_closed_ball Metric.nhds_basis_closedBall theorem nhds_basis_ball_inv_nat_succ : (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) := nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ #align metric.nhds_basis_ball_inv_nat_succ Metric.nhds_basis_ball_inv_nat_succ theorem nhds_basis_ball_inv_nat_pos : (𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) := nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos #align metric.nhds_basis_ball_inv_nat_pos Metric.nhds_basis_ball_inv_nat_pos theorem nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (r ^ n) := nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1) #align metric.nhds_basis_ball_pow Metric.nhds_basis_ball_pow theorem nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (𝓝 x).HasBasis (fun _ => True) fun n : ℕ => closedBall x (r ^ n) := nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1) #align metric.nhds_basis_closed_ball_pow Metric.nhds_basis_closedBall_pow theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by simp only [isOpen_iff_mem_nhds, mem_nhds_iff] #align metric.is_open_iff Metric.isOpen_iff theorem isOpen_ball : IsOpen (ball x ε) := isOpen_iff.2 fun _ => exists_ball_subset_ball #align metric.is_open_ball Metric.isOpen_ball theorem ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x := isOpen_ball.mem_nhds (mem_ball_self ε0) #align metric.ball_mem_nhds Metric.ball_mem_nhds theorem closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x := mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall #align metric.closed_ball_mem_nhds Metric.closedBall_mem_nhds theorem closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x := mem_of_superset (isOpen_ball.mem_nhds h) ball_subset_closedBall #align metric.closed_ball_mem_nhds_of_mem Metric.closedBall_mem_nhds_of_mem theorem nhdsWithin_basis_ball {s : Set α} : (𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s := nhdsWithin_hasBasis nhds_basis_ball s #align metric.nhds_within_basis_ball Metric.nhdsWithin_basis_ball theorem mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s := nhdsWithin_basis_ball.mem_iff #align metric.mem_nhds_within_iff Metric.mem_nhdsWithin_iff theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} : Tendsto f (𝓝[s] a) (𝓝[t] b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε := (nhdsWithin_basis_ball.tendsto_iff nhdsWithin_basis_ball).trans <\| by simp only [inter_comm _ s, inter_comm _ t, mem_inter_iff, and_imp, gt_iff_lt, mem_ball] #align metric.tendsto_nhds_within_nhds_within Metric.tendsto_nhdsWithin_nhdsWithin theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} : Tendsto f (𝓝[s] a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε := by rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin] simp only [mem_univ, true_and_iff] #align metric.tendsto_nhds_within_nhds Metric.tendsto_nhdsWithin_nhds theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} : Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε := nhds_basis_ball.tendsto_iff nhds_basis_ball #align metric.tendsto_nhds_nhds Metric.tendsto_nhds_nhds theorem continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} : ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε := by rw [ContinuousAt, tendsto_nhds_nhds] #align metric.continuous_at_iff Metric.continuousAt_iff theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} : ContinuousWithinAt f s a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε := by rw [ContinuousWithinAt, tendsto_nhdsWithin_nhds] #align metric.continuous_within_at_iff Metric.continuousWithinAt_iff theorem continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} : ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by simp [ContinuousOn, continuousWithinAt_iff] #align metric.continuous_on_iff Metric.continuousOn_iff theorem continuous_iff [PseudoMetricSpace β] {f : α → β} : Continuous f ↔ ∀ b, ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε := continuous_iff_continuousAt.trans <\| forall_congr' fun _ => tendsto_nhds_nhds #align metric.continuous_iff Metric.continuous_iff theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} : Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε := nhds_basis_ball.tendsto_right_iff #align metric.tendsto_nhds Metric.tendsto_nhds theorem continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε := by rw [ContinuousAt, tendsto_nhds] #align metric.continuous_at_iff' Metric.continuousAt_iff' theorem continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} : ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by rw [ContinuousWithinAt, tendsto_nhds] #align metric.continuous_within_at_iff' Metric.continuousWithinAt_iff' theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} : ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by simp [ContinuousOn, continuousWithinAt_iff'] #align metric.continuous_on_iff' Metric.continuousOn_iff' theorem continuous_iff' [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε := continuous_iff_continuousAt.trans <\| forall_congr' fun _ => tendsto_nhds #align metric.continuous_iff' Metric.continuous_iff' theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := (atTop_basis.tendsto_iff nhds_basis_ball).trans <\| by simp only [true_and, mem_ball, mem_Ici] #align metric.tendsto_at_top Metric.tendsto_atTop /-- A variant of `tendsto_atTop` that uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...` -/ theorem tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε := (atTop_basis_Ioi.tendsto_iff nhds_basis_ball).trans <\| by simp only [true_and, gt_iff_lt, mem_Ioi, mem_ball] #align metric.tendsto_at_top' Metric.tendsto_atTop' theorem isOpen_singleton_iff {α : Type} [PseudoMetricSpace α] {x : α} : IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by simp [isOpen_iff, subset_singleton_iff, mem_ball] #align metric.is_open_singleton_iff Metric.isOpen_singleton_iff /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball centered at `x` and intersecting `s` only at `x`. -/ theorem exists_ball_inter_eq_singleton_of_mem_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) : ∃ ε > 0, Metric.ball x ε ∩ s = {x} := nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx #align metric.exists_ball_inter_eq_singleton_of_mem_discrete Metric.exists_ball_inter_eq_singleton_of_mem_discrete /-- Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball of positive radius centered at `x` and intersecting `s` only at `x`. -/ theorem exists_closedBall_inter_eq_singleton_of_discrete [DiscreteTopology s] {x : α} (hx : x ∈ s) : ∃ ε > 0, Metric.closedBall x ε ∩ s = {x} := nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hx #align metric.exists_closed_ball_inter_eq_singleton_of_discrete Metric.exists_closedBall_inter_eq_singleton_of_discrete theorem _root_.Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) : ∃ y ∈ s, dist x y < ε := by have : (ball x ε).Nonempty := by simp [hε] simpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this #align dense.exists_dist_lt Dense.exists_dist_lt nonrec theorem _root_.DenseRange.exists_dist_lt {β : Type} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε := exists_range_iff.1 (hf.exists_dist_lt x hε) #align dense_range.exists_dist_lt DenseRange.exists_dist_lt end Metric open Metric /- Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance, we need to show that the uniform structure coming from the edistance and the distance coincide. -/ -- Porting note (#10756): new theorem theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) : ⨅ ε > (0 : ℝ), 𝓟 { p : α × α \| ↑(d p.1 p.2) < ε } = ⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α \| ↑(d p.1 p.2) < ε } := by simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff] refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩ · rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hε with ⟨ε', ε'0, ε'ε⟩ refine mem_iInf_of_mem (ε' : ℝ) (mem_iInf_of_mem (ENNReal.coe_pos.1 ε'0) ?_) exact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) ε'ε · lift ε to ℝ≥0 using le_of_lt hε refine mem_iInf_of_mem (ε : ℝ≥0∞) (mem_iInf_of_mem (ENNReal.coe_pos.2 hε) ?_) exact fun _ => ENNReal.coe_lt_coe.1 theorem Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α \| edist p.1 p.2 < ε } := by simp only [PseudoMetricSpace.uniformity_dist, dist_nndist, edist_nndist, Metric.uniformity_edist_aux] #align metric.uniformity_edist Metric.uniformity_edist -- see Note [lower instance priority] /-- A pseudometric space induces a pseudoemetric space -/ instance (priority := 100) PseudoMetricSpace.toPseudoEMetricSpace : PseudoEMetricSpace α := { ‹PseudoMetricSpace α› with edist_self := by simp [edist_dist] edist_comm := fun _ _ => by simp only [edist_dist, dist_comm] edist_triangle := fun x y z => by simp only [edist_dist, ← ENNReal.ofReal_add, dist_nonneg] rw [ENNReal.ofReal_le_ofReal_iff _] · exact dist_triangle _ _ _ · simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg uniformity_edist := Metric.uniformity_edist } #align pseudo_metric_space.to_pseudo_emetric_space PseudoMetricSpace.toPseudoEMetricSpace /-- Expressing the uniformity in terms of `edist` -/ @[deprecated _root_.uniformity_basis_edist] protected theorem Metric.uniformity_basis_edist : (𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => { p \| edist p.1 p.2 < ε } := uniformity_basis_edist #align pseudo_metric.uniformity_basis_edist Metric.uniformity_basis_edist /-- In a pseudometric space, an open ball of infinite radius is the whole space -/ theorem Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ := Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x #align metric.eball_top_eq_univ Metric.eball_top_eq_univ /-- Balls defined using the distance or the edistance coincide -/ @[simp] theorem Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = ball x ε := by ext y simp only [EMetric.mem_ball, mem_ball, edist_dist] exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg #align metric.emetric_ball Metric.emetric_ball /-- Balls defined using the distance or the edistance coincide -/ @[simp] theorem Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε := by rw [← Metric.emetric_ball] simp #align metric.emetric_ball_nnreal Metric.emetric_ball_nnreal /-- Closed balls defined using the distance or the edistance coincide -/ theorem Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) : EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by ext y; simp [edist_le_ofReal h] #align metric.emetric_closed_ball Metric.emetric_closedBall /-- Closed balls defined using the distance or the edistance coincide -/ @[simp] theorem Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} : EMetric.closedBall x ε = closedBall x ε := by rw [← Metric.emetric_closedBall ε.coe_nonneg, ENNReal.ofReal_coe_nnreal] #align metric.emetric_closed_ball_nnreal Metric.emetric_closedBall_nnreal @[simp] theorem Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ := eq_univ_of_forall fun _ => edist_lt_top _ _ #align metric.emetric_ball_top Metric.emetric_ball_top theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero] #align metric.inseparable_iff Metric.inseparable_iff /-- Build a new pseudometric space from an old one where the bundled uniform structure is provably (but typically non-definitionaly) equal to some given uniform structure. See Note [forgetful inheritance]. -/ abbrev PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMetricSpace α) (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α := { m with toUniformSpace := U uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist } #align pseudo_metric_space.replace_uniformity PseudoMetricSpace.replaceUniformity theorem PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α) (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by ext rfl #align pseudo_metric_space.replace_uniformity_eq PseudoMetricSpace.replaceUniformity_eq -- ensure that the bornology is unchanged when replacing the uniformity. example {α} [U : UniformSpace α] (m : PseudoMetricSpace α) (H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : (PseudoMetricSpace.replaceUniformity m H).toBornology = m.toBornology := rfl /-- Build a new pseudo metric space from an old one where the bundled topological structure is provably (but typically non-definitionaly) equal to some given topological structure. See Note [forgetful inheritance]. -/ abbrev PseudoMetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ) (H : U = m.toUniformSpace.toTopologicalSpace) : PseudoMetricSpace γ := @PseudoMetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl #align pseudo_metric_space.replace_topology PseudoMetricSpace.replaceTopology theorem PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ) (H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m := by ext rfl #align pseudo_metric_space.replace_topology_eq PseudoMetricSpace.replaceTopology_eq /-- One gets a pseudometric space from an emetric space if the edistance is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the distance is given separately, to be able to prescribe some expression which is not defeq to the push-forward of the edistance to reals. See note [reducible non-instances]. -/ abbrev PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetricSpace α] (dist : α → α → ℝ) (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤) (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) : PseudoMetricSpace α where dist := dist dist_self x := by simp [h] dist_comm x y := by simp [h, edist_comm] dist_triangle x y z := by simp only [h] exact ENNReal.toReal_le_add (edist_triangle _ _ _) (edist_ne_top _ _) (edist_ne_top _ _) edist := edist edist_dist _ _ := by simp only [h, ENNReal.ofReal_toReal (edist_ne_top _ _)] toUniformSpace := e.toUniformSpace uniformity_dist := e.uniformity_edist.trans <\| by simpa only [ENNReal.coe_toNNReal (edist_ne_top _ _), h] using (Metric.uniformity_edist_aux fun x y : α => (edist x y).toNNReal).symm #align pseudo_emetric_space.to_pseudo_metric_space_of_dist PseudoEMetricSpace.toPseudoMetricSpaceOfDist /-- One gets a pseudometric space from an emetric space if the edistance is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the uniformity are defeq in the pseudometric space and the emetric space. -/ abbrev PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [PseudoEMetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun _ _ => rfl #align pseudo_emetric_space.to_pseudo_metric_space PseudoEMetricSpace.toPseudoMetricSpace /-- Build a new pseudometric space from an old one where the bundled bornology structure is provably (but typically non-definitionaly) equal to some given bornology structure. See Note [forgetful inheritance]. -/ abbrev PseudoMetricSpace.replaceBornology {α} [B : Bornology α] (m : PseudoMetricSpace α) (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) : PseudoMetricSpace α := { m with toBornology := B cobounded_sets := Set.ext <\| compl_surjective.forall.2 fun s => (H s).trans <\| by rw [isBounded_iff, mem_setOf_eq, compl_compl] } #align pseudo_metric_space.replace_bornology PseudoMetricSpace.replaceBornology theorem PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α] (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) : PseudoMetricSpace.replaceBornology _ H = m := by ext rfl #align pseudo_metric_space.replace_bornology_eq PseudoMetricSpace.replaceBornology_eq -- ensure that the uniformity is unchanged when replacing the bornology. example {α} [B : Bornology α] (m : PseudoMetricSpace α) (H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) : (PseudoMetricSpace.replaceBornology m H).toUniformSpace = m.toUniformSpace := rfl section Real /-- Instantiate the reals as a pseudometric space. -/ instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ where dist x y := \|x - y\| dist_self := by simp [abs_zero] dist_comm x y := abs_sub_comm _ _ dist_triangle x y z := abs_sub_le _ _ _ edist_dist := fun x y => by exact ENNReal.coe_nnreal_eq _ #align real.pseudo_metric_space Real.pseudoMetricSpace theorem Real.dist_eq (x y : ℝ) : dist x y = \|x - y\| := rfl #align real.dist_eq Real.dist_eq theorem Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) := rfl #align real.nndist_eq Real.nndist_eq theorem Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) := nndist_comm _ _ #align real.nndist_eq' Real.nndist_eq' theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = \|x\| := by simp [Real.dist_eq] #align real.dist_0_eq_abs Real.dist_0_eq_abs theorem Real.sub_le_dist (x y : ℝ) : x - y ≤ dist x y := by rw [Real.dist_eq, le_abs] exact Or.inl (le_refl _) theorem Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h #align real.dist_left_le_of_mem_uIcc Real.dist_left_le_of_mem_uIcc theorem Real.dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z := by simpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h #align real.dist_right_le_of_mem_uIcc Real.dist_right_le_of_mem_uIcc theorem Real.dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') : dist x y ≤ dist x' y' := abs_sub_le_of_uIcc_subset_uIcc <\| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm]) #align real.dist_le_of_mem_uIcc Real.dist_le_of_mem_uIcc theorem Real.dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') : dist x y ≤ y' - x' := by simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <\| hx.1.trans hx.2), neg_sub] using Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy) #align real.dist_le_of_mem_Icc Real.dist_le_of_mem_Icc theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) : dist x y ≤ 1 := by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy #align real.dist_le_of_mem_Icc_01 Real.dist_le_of_mem_Icc_01 instance : OrderTopology ℝ := orderTopology_of_nhds_abs fun x => by simp only [nhds_basis_ball.eq_biInf, ball, Real.dist_eq, abs_sub_comm] theorem Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) := Set.ext fun y => by rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add', sub_lt_comm] #align real.ball_eq_Ioo Real.ball_eq_Ioo theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by ext y rw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add', sub_le_comm] #align real.closed_ball_eq_Icc Real.closedBall_eq_Icc theorem Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two, ← add_div, add_assoc, add_sub_cancel, add_self_div_two] #align real.Ioo_eq_ball Real.Ioo_eq_ball theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two, ← add_div, add_assoc, add_sub_cancel, add_self_div_two] #align real.Icc_eq_closed_ball Real.Icc_eq_closedBall /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/ theorem squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t) (hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) := tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft #align squeeze_zero' squeeze_zero' /-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le` and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/ theorem squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t) (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) := squeeze_zero' (eventually_of_forall hf) (eventually_of_forall hft) g0 #align squeeze_zero squeeze_zero theorem Metric.uniformity_eq_comap_nhds_zero : 𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) := by ext s simp only [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff] simp [subset_def, Real.dist_0_eq_abs] #align metric.uniformity_eq_comap_nhds_zero Metric.uniformity_eq_comap_nhds_zero theorem cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} : CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) := by rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Function.comp_def] simp_rw [Prod.map_apply] #align cauchy_seq_iff_tendsto_dist_at_top_0 cauchySeq_iff_tendsto_dist_atTop_0 theorem tendsto_uniformity_iff_dist_tendsto_zero {f : ι → α × α} {p : Filter ι} : Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Function.comp_def] #align tendsto_uniformity_iff_dist_tendsto_zero tendsto_uniformity_iff_dist_tendsto_zero theorem Filter.Tendsto.congr_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α} (h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₂ p (𝓝 a) := h₁.congr_uniformity <\| tendsto_uniformity_iff_dist_tendsto_zero.2 h #align filter.tendsto.congr_dist Filter.Tendsto.congr_dist alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist #align tendsto_of_tendsto_of_dist tendsto_of_tendsto_of_dist theorem tendsto_iff_of_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α} (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) := Uniform.tendsto_congr <\| tendsto_uniformity_iff_dist_tendsto_zero.2 h #align tendsto_iff_of_dist tendsto_iff_of_dist /-- If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball `Metric.closedBall x r` is contained in `u`. -/ theorem eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) : ∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u := by obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ closedBall x ε ⊆ u := nhds_basis_closedBall.mem_iff.1 hu have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos filter_upwards [this] with _ hr using Subset.trans (closedBall_subset_closedBall hr) hε #align eventually_closed_ball_subset eventually_closedBall_subset theorem tendsto_closedBall_smallSets (x : α) : Tendsto (closedBall x) (𝓝 0) (𝓝 x).smallSets := tendsto_smallSets_iff.2 fun _ ↦ eventually_closedBall_subset end Real /-- Pseudometric space structure pulled back by a function. -/ abbrev PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β) : PseudoMetricSpace α where dist x y := dist (f x) (f y) dist_self x := dist_self _ dist_comm x y := dist_comm _ _ dist_triangle x y z := dist_triangle _ _ _ edist x y := edist (f x) (f y) edist_dist x y := edist_dist _ _ toUniformSpace := UniformSpace.comap f m.toUniformSpace uniformity_dist := (uniformity_basis_dist.comap _).eq_biInf toBornology := Bornology.induced f cobounded_sets := Set.ext fun s => mem_comap_iff_compl.trans <\| by simp only [← isBounded_def, isBounded_iff, forall_mem_image, mem_setOf] #align pseudo_metric_space.induced PseudoMetricSpace.induced /-- Pull back a pseudometric space structure by an inducing map. This is a version of `PseudoMetricSpace.induced` useful in case if the domain already has a `TopologicalSpace` structure. -/ def Inducing.comapPseudoMetricSpace {α β} [TopologicalSpace α] [m : PseudoMetricSpace β] {f : α → β} (hf : Inducing f) : PseudoMetricSpace α := .replaceTopology (.induced f m) hf.induced #align inducing.comap_pseudo_metric_space Inducing.comapPseudoMetricSpace /-- Pull back a pseudometric space structure by a uniform inducing map. This is a version of `PseudoMetricSpace.induced` useful in case if the domain already has a `UniformSpace` structure. -/ def UniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [m : PseudoMetricSpace β] (f : α → β) (h : UniformInducing f) : PseudoMetricSpace α := .replaceUniformity (.induced f m) h.comap_uniformity.symm #align uniform_inducing.comap_pseudo_metric_space UniformInducing.comapPseudoMetricSpace instance Subtype.pseudoMetricSpace {p : α → Prop} : PseudoMetricSpace (Subtype p) := PseudoMetricSpace.induced Subtype.val ‹_› #align subtype.pseudo_metric_space Subtype.pseudoMetricSpace theorem Subtype.dist_eq {p : α → Prop} (x y : Subtype p) : dist x y = dist (x : α) y := rfl #align subtype.dist_eq Subtype.dist_eq theorem Subtype.nndist_eq {p : α → Prop} (x y : Subtype p) : nndist x y = nndist (x : α) y := rfl #align subtype.nndist_eq Subtype.nndist_eq namespace MulOpposite @[to_additive] instance instPseudoMetricSpace : PseudoMetricSpace αᵐᵒᵖ := PseudoMetricSpace.induced MulOpposite.unop ‹_› @[to_additive (attr := simp)] theorem dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y := rfl #align mul_opposite.dist_unop MulOpposite.dist_unop #align add_opposite.dist_unop AddOpposite.dist_unop @[to_additive (attr := simp)] theorem dist_op (x y : α) : dist (op x) (op y) = dist x y := rfl #align mul_opposite.dist_op MulOpposite.dist_op #align add_opposite.dist_op AddOpposite.dist_op @[to_additive (attr := simp)] theorem nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y := rfl #align mul_opposite.nndist_unop MulOpposite.nndist_unop #align add_opposite.nndist_unop AddOpposite.nndist_unop @[to_additive (attr := simp)] theorem nndist_op (x y : α) : nndist (op x) (op y) = nndist x y := rfl #align mul_opposite.nndist_op MulOpposite.nndist_op #align add_opposite.nndist_op AddOpposite.nndist_op end MulOpposite section NNReal instance : PseudoMetricSpace ℝ≥0 := Subtype.pseudoMetricSpace theorem NNReal.dist_eq (a b : ℝ≥0) : dist a b = \|(a : ℝ) - b\| := rfl #align nnreal.dist_eq NNReal.dist_eq theorem NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) := eq_of_forall_ge_iff fun _ => by simp only [max_le_iff, tsub_le_iff_right (α := ℝ≥0)] simp only [← NNReal.coe_le_coe, coe_nndist, dist_eq, abs_sub_le_iff, tsub_le_iff_right, NNReal.coe_add] #align nnreal.nndist_eq NNReal.nndist_eq @[simp] theorem NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le'] #align nnreal.nndist_zero_eq_val NNReal.nndist_zero_eq_val @[simp] theorem NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by rw [nndist_comm] exact NNReal.nndist_zero_eq_val z #align nnreal.nndist_zero_eq_val' NNReal.nndist_zero_eq_val' theorem NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b := by suffices (a : ℝ) ≤ (b : ℝ) + dist a b by rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] rw [← sub_le_iff_le_add'] exact le_of_abs_le (dist_eq a b).ge #align nnreal.le_add_nndist NNReal.le_add_nndist lemma NNReal.ball_zero_eq_Ico' (c : ℝ≥0) : Metric.ball (0 : ℝ≥0) c.toReal = Set.Ico 0 c := by ext x; simp lemma NNReal.ball_zero_eq_Ico (c : ℝ) : Metric.ball (0 : ℝ≥0) c = Set.Ico 0 c.toNNReal := by by_cases c_pos : 0 < c · convert NNReal.ball_zero_eq_Ico' ⟨c, c_pos.le⟩ simp [Real.toNNReal, c_pos.le] simp [not_lt.mp c_pos] lemma NNReal.closedBall_zero_eq_Icc' (c : ℝ≥0) : Metric.closedBall (0 : ℝ≥0) c.toReal = Set.Icc 0 c := by ext x; simp lemma NNReal.closedBall_zero_eq_Icc {c : ℝ} (c_nn : 0 ≤ c) : Metric.closedBall (0 : ℝ≥0) c = Set.Icc 0 c.toNNReal := by convert NNReal.closedBall_zero_eq_Icc' ⟨c, c_nn⟩ simp [Real.toNNReal, c_nn] end NNReal section ULift variable [PseudoMetricSpace β] instance : PseudoMetricSpace (ULift β) := PseudoMetricSpace.induced ULift.down ‹_› theorem ULift.dist_eq (x y : ULift β) : dist x y = dist x.down y.down := rfl #align ulift.dist_eq ULift.dist_eq theorem ULift.nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down := rfl #align ulift.nndist_eq ULift.nndist_eq @[simp] theorem ULift.dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y := rfl #align ulift.dist_up_up ULift.dist_up_up @[simp] theorem ULift.nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y := rfl #align ulift.nndist_up_up ULift.nndist_up_up end ULift section Prod variable [PseudoMetricSpace β] -- Porting note: added `let`, otherwise `simp` failed instance Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) := let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2) (fun x y => (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).ne) fun x y => by simp only [sup_eq_max, dist_edist, ← ENNReal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _), Prod.edist_eq] i.replaceBornology fun s => by simp only [← isBounded_image_fst_and_snd, isBounded_iff_eventually, forall_mem_image, ← eventually_and, ← forall_and, ← max_le_iff] rfl #align prod.pseudo_metric_space_max Prod.pseudoMetricSpaceMax theorem Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) := rfl #align prod.dist_eq Prod.dist_eq @[simp] theorem dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ := by simp [Prod.dist_eq, dist_nonneg] #align dist_prod_same_left dist_prod_same_left @[simp] theorem dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ := by simp [Prod.dist_eq, dist_nonneg] #align dist_prod_same_right dist_prod_same_right theorem ball_prod_same (x : α) (y : β) (r : ℝ) : ball x r ×ˢ ball y r = ball (x, y) r := ext fun z => by simp [Prod.dist_eq] #align ball_prod_same ball_prod_same theorem closedBall_prod_same (x : α) (y : β) (r : ℝ) : closedBall x r ×ˢ closedBall y r = closedBall (x, y) r := ext fun z => by simp [Prod.dist_eq] #align closed_ball_prod_same closedBall_prod_same theorem sphere_prod (x : α × β) (r : ℝ) : sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r := by obtain hr \| rfl \| hr := lt_trichotomy r 0 · simp [hr] · cases x simp_rw [← closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same] · ext ⟨x', y'⟩ simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq, max_eq_iff] refine or_congr (and_congr_right ?_) (and_comm.trans (and_congr_left ?_)) all_goals rintro rfl; rfl #align sphere_prod sphere_prod end Prod -- Porting note: 3 new lemmas theorem dist_dist_dist_le_left (x y z : α) : dist (dist x z) (dist y z) ≤ dist x y := abs_dist_sub_le .. theorem dist_dist_dist_le_right (x y z : α) : dist (dist x y) (dist x z) ≤ dist y z := by simpa only [dist_comm x] using dist_dist_dist_le_left y z x theorem dist_dist_dist_le (x y x' y' : α) : dist (dist x y) (dist x' y') ≤ dist x x' + dist y y' := (dist_triangle _ _ _).trans <\| add_le_add (dist_dist_dist_le_left _ _ _) (dist_dist_dist_le_right _ _ _) theorem uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 := Metric.uniformContinuous_iff.2 fun ε ε0 => ⟨ε / 2, half_pos ε0, fun {a b} h => calc dist (dist a.1 a.2) (dist b.1 b.2) ≤ dist a.1 b.1 + dist a.2 b.2 := dist_dist_dist_le _ _ _ _ _ ≤ dist a b + dist a b := add_le_add (le_max_left _ _) (le_max_right _ _) _ < ε / 2 + ε / 2 := add_lt_add h h _ = ε := add_halves ε⟩ #align uniform_continuous_dist uniformContinuous_dist protected theorem UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun b => dist (f b) (g b) := uniformContinuous_dist.comp (hf.prod_mk hg) #align uniform_continuous.dist UniformContinuous.dist @[continuity] theorem continuous_dist : Continuous fun p : α × α => dist p.1 p.2 := uniformContinuous_dist.continuous #align continuous_dist continuous_dist @[continuity, fun_prop] protected theorem Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : Continuous fun b => dist (f b) (g b) := continuous_dist.comp (hf.prod_mk hg : _) #align continuous.dist Continuous.dist protected theorem Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b)) := (continuous_dist.tendsto (a, b)).comp (hf.prod_mk_nhds hg) #align filter.tendsto.dist Filter.Tendsto.dist theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a := by simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap, (· ∘ ·), dist_comm] #align nhds_comap_dist nhds_comap_dist theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} : Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by rw [← nhds_comap_dist a, tendsto_comap_iff, Function.comp_def] #align tendsto_iff_dist_tendsto_zero tendsto_iff_dist_tendsto_zero theorem continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} : Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) := ⟨fun h => h.fst'.dist h.snd', fun h => continuous_iff_continuousAt.2 fun _ => tendsto_iff_dist_tendsto_zero.2 <\| (h.comp (continuous_id.prod_mk continuous_const)).tendsto' _ _ <\| dist_self _⟩ #align continuous_iff_continuous_dist continuous_iff_continuous_dist theorem uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 := uniformContinuous_dist.subtype_mk _ #align uniform_continuous_nndist uniformContinuous_nndist protected theorem UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) := uniformContinuous_nndist.comp (hf.prod_mk hg) #align uniform_continuous.nndist UniformContinuous.nndist theorem continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 := uniformContinuous_nndist.continuous #align continuous_nndist continuous_nndist @[fun_prop] protected theorem Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : Continuous fun b => nndist (f b) (g b) := continuous_nndist.comp (hf.prod_mk hg : _) #align continuous.nndist Continuous.nndist protected theorem Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) := (continuous_nndist.tendsto (a, b)).comp (hf.prod_mk_nhds hg) #align filter.tendsto.nndist Filter.Tendsto.nndist namespace Metric variable {x y z : α} {ε ε₁ ε₂ : ℝ} {s : Set α} theorem isClosed_ball : IsClosed (closedBall x ε) := isClosed_le (continuous_id.dist continuous_const) continuous_const #align metric.is_closed_ball Metric.isClosed_ball theorem isClosed_sphere : IsClosed (sphere x ε) := isClosed_eq (continuous_id.dist continuous_const) continuous_const #align metric.is_closed_sphere Metric.isClosed_sphere @[simp] theorem closure_closedBall : closure (closedBall x ε) = closedBall x ε := isClosed_ball.closure_eq #align metric.closure_closed_ball Metric.closure_closedBall @[simp] theorem closure_sphere : closure (sphere x ε) = sphere x ε := isClosed_sphere.closure_eq #align metric.closure_sphere Metric.closure_sphere theorem closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε := closure_minimal ball_subset_closedBall isClosed_ball #align metric.closure_ball_subset_closed_ball Metric.closure_ball_subset_closedBall theorem frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε := frontier_lt_subset_eq (continuous_id.dist continuous_const) continuous_const #align metric.frontier_ball_subset_sphere Metric.frontier_ball_subset_sphere theorem frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε := frontier_le_subset_eq (continuous_id.dist continuous_const) continuous_const #align metric.frontier_closed_ball_subset_sphere Metric.frontier_closedBall_subset_sphere theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) := interior_maximal ball_subset_closedBall isOpen_ball #align metric.ball_subset_interior_closed_ball Metric.ball_subset_interior_closedBall /-- ε-characterization of the closure in pseudometric spaces-/ theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := (mem_closure_iff_nhds_basis nhds_basis_ball).trans <\| by simp only [mem_ball, dist_comm] #align metric.mem_closure_iff Metric.mem_closure_iff theorem mem_closure_range_iff {e : β → α} {a : α} : a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by simp only [mem_closure_iff, exists_range_iff] #align metric.mem_closure_range_iff Metric.mem_closure_range_iff theorem mem_closure_range_iff_nat {e : β → α} {a : α} : a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) := (mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <\| by simp only [mem_ball, dist_comm, exists_range_iff, forall_const] #align metric.mem_closure_range_iff_nat Metric.mem_closure_range_iff_nat theorem mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} : a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := by simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a #align metric.mem_of_closed' Metric.mem_of_closed' theorem closedBall_zero' (x : α) : closedBall x 0 = closure {x} := Subset.antisymm (fun _y hy => mem_closure_iff.2 fun _ε ε0 => ⟨x, mem_singleton x, (mem_closedBall.1 hy).trans_lt ε0⟩) (closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_ball) #align metric.closed_ball_zero' Metric.closedBall_zero' lemma eventually_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) : ∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) := by rcases exists_compact_mem_nhds x with ⟨s, s_compact, hs⟩ filter_upwards [eventually_closedBall_subset hs] with r hr exact IsCompact.of_isClosed_subset s_compact isClosed_ball hr lemma exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) : ∃ r, 0 < r ∧ IsCompact (closedBall x r) := by have : ∀ᶠ r in 𝓝[>] 0, IsCompact (closedBall x r) := eventually_nhdsWithin_of_eventually_nhds (eventually_isCompact_closedBall x) simpa only [and_comm] using (this.and self_mem_nhdsWithin).exists theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty := forall_congr' fun x => by simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm] #align metric.dense_iff Metric.dense_iff theorem denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r := forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff] #align metric.dense_range_iff Metric.denseRange_iff -- Porting note: `TopologicalSpace.IsSeparable.separableSpace` moved to `EMetricSpace` /-- The preimage of a separable set by an inducing map is separable. -/ protected theorem _root_.Inducing.isSeparable_preimage {f : β → α} [TopologicalSpace β] (hf : Inducing f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) := by have : SeparableSpace s := hs.separableSpace have : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _ have : Inducing ((mapsTo_preimage f s).restrict _ _ _) := (hf.comp inducing_subtype_val).codRestrict _ have := this.secondCountableTopology exact .of_subtype _ #align inducing.is_separable_preimage Inducing.isSeparable_preimage protected theorem _root_.Embedding.isSeparable_preimage {f : β → α} [TopologicalSpace β] (hf : Embedding f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) := hf.toInducing.isSeparable_preimage hs #align embedding.is_separable_preimage Embedding.isSeparable_preimage /-- If a map is continuous on a separable set `s`, then the image of `s` is also separable. -/ theorem _root_.ContinuousOn.isSeparable_image [TopologicalSpace β] {f : α → β} {s : Set α} (hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s) := by rw [image_eq_range, ← image_univ] exact (isSeparable_univ_iff.2 hs.separableSpace).image hf.restrict #align continuous_on.is_separable_image ContinuousOn.isSeparable_image end Metric /-- A compact set is separable. -/ theorem IsCompact.isSeparable {s : Set α} (hs : IsCompact s) : IsSeparable s := haveI : CompactSpace s := isCompact_iff_compactSpace.mp hs .of_subtype s #align is_compact.is_separable IsCompact.isSeparable section Pi open Finset variable {π : β → Type} [Fintype β] [∀ b, PseudoMetricSpace (π b)] /-- A finite product of pseudometric spaces is a pseudometric space, with the sup distance. -/ instance pseudoMetricSpacePi : PseudoMetricSpace (∀ b, π b) := by /- we construct the instance from the pseudoemetric space instance to avoid checking again that the uniformity is the same as the product uniformity, but we register nevertheless a nice formula for the distance -/ let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g : ∀ b, π b => ((sup univ fun b => nndist (f b) (g b) : ℝ≥0) : ℝ)) (fun f g => ((Finset.sup_lt_iff bot_lt_top).2 fun b _ => edist_lt_top _ _).ne) (fun f g => by simp only [edist_pi_def, edist_nndist, ← ENNReal.coe_finset_sup, ENNReal.coe_toReal]) refine i.replaceBornology fun s => ?_ simp only [← isBounded_def, isBounded_iff_eventually, ← forall_isBounded_image_eval_iff, forall_mem_image, ← Filter.eventually_all, Function.eval_apply, @dist_nndist (π _)] refine eventually_congr ((eventually_ge_atTop 0).mono fun C hC ↦ ?_) lift C to ℝ≥0 using hC refine ⟨fun H x hx y hy ↦ NNReal.coe_le_coe.2 <\| Finset.sup_le fun b _ ↦ H b hx hy, fun H b x hx y hy ↦ NNReal.coe_le_coe.2 ?_⟩ simpa only using Finset.sup_le_iff.1 (NNReal.coe_le_coe.1 <\| H hx hy) b (Finset.mem_univ b) #align pseudo_metric_space_pi pseudoMetricSpacePi theorem nndist_pi_def (f g : ∀ b, π b) : nndist f g = sup univ fun b => nndist (f b) (g b) := NNReal.eq rfl #align nndist_pi_def nndist_pi_def theorem dist_pi_def (f g : ∀ b, π b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) := rfl #align dist_pi_def dist_pi_def theorem nndist_pi_le_iff {f g : ∀ b, π b} {r : ℝ≥0} : nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def] #align nndist_pi_le_iff nndist_pi_le_iff theorem nndist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) : nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r := by rw [← bot_eq_zero'] at hr simp [nndist_pi_def, Finset.sup_lt_iff hr] #align nndist_pi_lt_iff nndist_pi_lt_iff theorem nndist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ≥0} (hr : 0 < r) : nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r := by rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm] simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff] intro h refine exists_congr fun b => ?_ apply (and_iff_right <\| h _).symm #align nndist_pi_eq_iff nndist_pi_eq_iff theorem dist_pi_lt_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) : dist f g < r ↔ ∀ b, dist (f b) (g b) < r := by lift r to ℝ≥0 using hr.le exact nndist_pi_lt_iff hr #align dist_pi_lt_iff dist_pi_lt_iff theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) : dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by lift r to ℝ≥0 using hr exact nndist_pi_le_iff #align dist_pi_le_iff dist_pi_le_iff theorem dist_pi_eq_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 < r) : dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r := by lift r to ℝ≥0 using hr.le simp_rw [← coe_nndist, NNReal.coe_inj, nndist_pi_eq_iff hr, NNReal.coe_le_coe] #align dist_pi_eq_iff dist_pi_eq_iff theorem dist_pi_le_iff' [Nonempty β] {f g : ∀ b, π b} {r : ℝ} : dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by by_cases hr : 0 ≤ r · exact dist_pi_le_iff hr · exact iff_of_false (fun h => hr <\| dist_nonneg.trans h) fun h => hr <\| dist_nonneg.trans <\| h <\| Classical.arbitrary _ #align dist_pi_le_iff' dist_pi_le_iff' theorem dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b := (dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl #align dist_pi_const_le dist_pi_const_le theorem nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b := nndist_pi_le_iff.2 fun _ => le_rfl #align nndist_pi_const_le nndist_pi_const_le @[simp] theorem dist_pi_const [Nonempty β] (a b : α) : (dist (fun _ : β => a) fun _ => b) = dist a b := by simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b) #align dist_pi_const dist_pi_const @[simp] theorem nndist_pi_const [Nonempty β] (a b : α) : (nndist (fun _ : β => a) fun _ => b) = nndist a b := NNReal.eq <\| dist_pi_const a b #align nndist_pi_const nndist_pi_const theorem nndist_le_pi_nndist (f g : ∀ b, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g := by rw [← ENNReal.coe_le_coe, ← edist_nndist, ← edist_nndist] exact edist_le_pi_edist f g b #align nndist_le_pi_nndist nndist_le_pi_nndist theorem dist_le_pi_dist (f g : ∀ b, π b) (b : β) : dist (f b) (g b) ≤ dist f g := by simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b] #align dist_le_pi_dist dist_le_pi_dist /-- An open ball in a product space is a product of open balls. See also `ball_pi'` for a version assuming `Nonempty β` instead of `0 < r`. -/ theorem ball_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 < r) : ball x r = Set.pi univ fun b => ball (x b) r := by ext p simp [dist_pi_lt_iff hr] #align ball_pi ball_pi /-- An open ball in a product space is a product of open balls. See also `ball_pi` for a version assuming `0 < r` instead of `Nonempty β`. -/ theorem ball_pi' [Nonempty β] (x : ∀ b, π b) (r : ℝ) : ball x r = Set.pi univ fun b => ball (x b) r := (lt_or_le 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr] #align ball_pi' ball_pi' /-- A closed ball in a product space is a product of closed balls. See also `closedBall_pi'` for a version assuming `Nonempty β` instead of `0 ≤ r`. -/	Mathlib/Topology/MetricSpace/PseudoMetric.lean	2,000	2,003	theorem closedBall_pi (x : ∀ b, π b) {r : ℝ} (hr : 0 ≤ r) : closedBall x r = Set.pi univ fun b => closedBall (x b) r := by	ext p simp [dist_pi_le_iff hr]
/- Copyright (c) 2021 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import Mathlib.Analysis.SpecialFunctions.Log.Deriv import Mathlib.MeasureTheory.Integral.FundThmCalculus #align_import analysis.special_functions.non_integrable from "leanprover-community/mathlib"@"55ec6e9af7d3e0043f57e394cb06a72f6275273e" /-! # Non integrable functions In this file we prove that the derivative of a function that tends to infinity is not interval integrable, see `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter` and `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured`. Then we apply the latter lemma to prove that the function `fun x => x⁻¹` is integrable on `a..b` if and only if `a = b` or `0 ∉ [a, b]`. ## Main results * `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured`: if `f` tends to infinity along `𝓝[≠] c` and `f' = O(g)` along the same filter, then `g` is not interval integrable on any nontrivial integral `a..b`, `c ∈ [a, b]`. * `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter`: a version of `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured` that works for one-sided neighborhoods; * `not_intervalIntegrable_of_sub_inv_isBigO_punctured`: if `1 / (x - c) = O(f)` as `x → c`, `x ≠ c`, then `f` is not interval integrable on any nontrivial interval `a..b`, `c ∈ [a, b]`; * `intervalIntegrable_sub_inv_iff`, `intervalIntegrable_inv_iff`: integrability conditions for `(x - c)⁻¹` and `x⁻¹`. ## Tags integrable function -/ open scoped MeasureTheory Topology Interval NNReal ENNReal open MeasureTheory TopologicalSpace Set Filter Asymptotics intervalIntegral variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] /-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'` is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`. Auxiliary version assuming that `E` is complete. -/	Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean	52	96	theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux [CompleteSpace E] {f : ℝ → E} {g : ℝ → F} {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l] (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop) (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by	intro hgi obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ : ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧ (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧ ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩ have h : ∀ᶠ x : ℝ × ℝ in l.prod l, ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k := (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩ simp only [prod_subset_iff, mem_setOf_eq] at hs exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz => (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩ replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩ have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ := hf.eventually (eventually_gt_atTop _) exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩ specialize hsub c hc d hd; specialize hfd c hc d hd replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ := fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩ have hg_ae : ∀ᵐ x ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖ := (ae_restrict_mem measurableSet_uIoc).mono hg have hsub' : Ι c d ⊆ k := Subset.trans Ioc_subset_Icc_self hsub have hfi : IntervalIntegrable (deriv f) volume c d := by rw [intervalIntegrable_iff] have : IntegrableOn (fun x ↦ C * ‖g x‖) (Ι c d) := IntegrableOn.mono hgi hsub' le_rfl exact Integrable.mono' this (aestronglyMeasurable_deriv _ _) hg_ae refine hlt.not_le (sub_le_iff_le_add'.1 ?_) calc ‖f d‖ - ‖f c‖ ≤ ‖f d - f c‖ := norm_sub_norm_le _ _ _ = ‖∫ x in c..d, deriv f x‖ := congr_arg _ (integral_deriv_eq_sub hfd hfi).symm _ = ‖∫ x in Ι c d, deriv f x‖ := norm_integral_eq_norm_integral_Ioc _ _ ≤ ∫ x in Ι c d, ‖deriv f x‖ := norm_integral_le_integral_norm _ _ ≤ ∫ x in Ι c d, C * ‖g x‖ := setIntegral_mono_on hfi.norm.def' (hgi.mono_set hsub') measurableSet_uIoc hg _ ≤ ∫ x in k, C * ‖g x‖ := by apply setIntegral_mono_set hgi (ae_of_all _ fun x => mul_nonneg hC₀ (norm_nonneg _)) hsub'.eventuallyLE
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Thomas Browning -/ import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.SetLike.Fintype import Mathlib.GroupTheory.GroupAction.ConjAct import Mathlib.GroupTheory.PGroup import Mathlib.GroupTheory.NoncommPiCoprod import Mathlib.Order.Atoms.Finite import Mathlib.Data.Set.Lattice #align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef" /-! # Sylow theorems The Sylow theorems are the following results for every finite group `G` and every prime number `p`. * There exists a Sylow `p`-subgroup of `G`. * All Sylow `p`-subgroups of `G` are conjugate to each other. * Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow `p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow `p`-subgroup in `G`. ## Main definitions * `Sylow p G` : The type of Sylow `p`-subgroups of `G`. ## Main statements * `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem: For every prime power `pⁿ` dividing the cardinality of `G`, there exists a subgroup of `G` of order `pⁿ`. * `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem: Every `p`-subgroup is contained in a Sylow `p`-subgroup. * `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n` where `n` is the multiplicity of `p` in the group order. * `sylow_conjugate`: A generalization of Sylow's second theorem: If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. * `card_sylow_modEq_one`: A generalization of Sylow's third theorem: If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/ open Fintype MulAction Subgroup section InfiniteSylow variable (p : ℕ) (G : Type) [Group G] /-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/ structure Sylow extends Subgroup G where isPGroup' : IsPGroup p toSubgroup is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup #align sylow Sylow variable {p} {G} namespace Sylow attribute [coe] Sylow.toSubgroup -- Porting note: Changed to `CoeOut` instance : CoeOut (Sylow p G) (Subgroup G) := ⟨Sylow.toSubgroup⟩ -- Porting note: syntactic tautology -- @[simp] -- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P := -- rfl #noalign sylow.to_subgroup_eq_coe @[ext] theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr #align sylow.ext Sylow.ext theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q := ⟨congr_arg _, ext⟩ #align sylow.ext_iff Sylow.ext_iff instance : SetLike (Sylow p G) G where coe := (↑) coe_injective' _ _ h := ext (SetLike.coe_injective h) instance : SubgroupClass (Sylow p G) G where mul_mem := Subgroup.mul_mem _ one_mem _ := Subgroup.one_mem _ inv_mem := Subgroup.inv_mem _ variable (P : Sylow p G) /-- The action by a Sylow subgroup is the action by the underlying group. -/ instance mulActionLeft {α : Type} [MulAction G α] : MulAction P α := inferInstanceAs (MulAction (P : Subgroup G) α) #align sylow.mul_action_left Sylow.mulActionLeft variable {K : Type} [Group K] (ϕ : K → G) {N : Subgroup G} /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/ def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K := { P.1.comap ϕ with isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ is_maximal' := fun {Q} hQ hle => by show Q = P.1.comap ϕ rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))] exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm } #align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup @[simp] theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : (P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P := rfl #align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup /-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/ def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K := P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h #align sylow.comap_of_injective Sylow.comapOfInjective @[simp] theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : ↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P := rfl #align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective /-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/ protected def subtype (h : ↑P ≤ N) : Sylow p N := P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range]) #align sylow.subtype Sylow.subtype @[simp] theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N := rfl #align sylow.coe_subtype Sylow.coe_subtype theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N} (h : P.subtype hP = Q.subtype hQ) : P = Q := by rw [SetLike.ext_iff] at h ⊢ exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩ #align sylow.subtype_injective Sylow.subtype_injective end Sylow /-- A generalization of Sylow's first theorem. Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/ theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q := Exists.elim (zorn_nonempty_partialOrder₀ { Q : Subgroup G \| IsPGroup p Q } (fun c hc1 hc2 Q hQ => ⟨{ carrier := ⋃ R : c, R one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩ inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩ mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ => (hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T => ⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ }, fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by refine Exists.imp (fun k hk => ?_) (hc1 S.2 ⟨g, hg⟩) rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩) P hP) fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩ #align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow instance Sylow.nonempty : Nonempty (Sylow p G) := nonempty_of_exists IsPGroup.of_bot.exists_le_sylow #align sylow.nonempty Sylow.nonempty noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) := Classical.inhabited_of_nonempty Sylow.nonempty #align sylow.inhabited Sylow.inhabited theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type} [Group H] (P : Sylow p H) {f : H → G} (hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P := Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ)) (P.2.map f).exists_le_sylow #align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup theorem Sylow.exists_comap_eq_of_injective {H : Type} [Group H] (P : Sylow p H) {f : H → G} (hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P := P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf) #align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) : ∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P := P.exists_comap_eq_of_injective Subtype.coe_injective #align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq /-- If the kernel of `f : H →* G` is a `p`-group, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/ noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type} [Group H] {f : H → G} (hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) := let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P) have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P) Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec) #align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup /-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/ noncomputable def Sylow.fintypeOfInjective {H : Type} [Group H] {f : H → G} (hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) := Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf) #align sylow.fintype_of_injective Sylow.fintypeOfInjective /-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/ noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) := Sylow.fintypeOfInjective H.subtype_injective /-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/ instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by cases nonempty_fintype (Sylow p G) infer_instance open Pointwise /-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/ instance Sylow.pointwiseMulAction {α : Type} [Group α] [MulDistribMulAction α G] : MulAction α (Sylow p G) where smul g P := ⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS => inv_smul_eq_iff.mp (P.3 (hQ.map _) fun s hs => (congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp (smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩ one_smul P := Sylow.ext (one_smul α P.toSubgroup) mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup) #align sylow.pointwise_mul_action Sylow.pointwiseMulAction theorem Sylow.pointwise_smul_def {α : Type} [Group α] [MulDistribMulAction α G] {g : α} {P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) := rfl #align sylow.pointwise_smul_def Sylow.pointwise_smul_def instance Sylow.mulAction : MulAction G (Sylow p G) := compHom _ MulAut.conj #align sylow.mul_action Sylow.mulAction theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P := rfl #align sylow.smul_def Sylow.smul_def theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Subgroup G) := rfl #align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) := rfl #align sylow.coe_smul Sylow.coe_smul theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H := Subgroup.conj_smul_le_of_le hP h #align sylow.smul_le Sylow.smul_le theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) := Sylow.ext (Subgroup.conj_smul_subgroupOf hP h) #align sylow.smul_subtype Sylow.smul_subtype theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} : g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup), mem_normalizer_iff, inv_inv] exact forall_congr' fun h => iff_congr Iff.rfl ⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b, fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩ #align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P := by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top] #align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} : P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall #align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) : P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q := le_antisymm (le_inf inf_le_left (sup_eq_right.mp (Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right))) (inf_le_inf_left P le_normalizer) #align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} : Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left] #align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff /-- A generalization of Sylow's second theorem. If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/ instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) := ⟨fun P Q => by classical cases nonempty_fintype (Sylow p G) have H := fun {R : Sylow p G} {S : orbit G P} => calc S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) := forall_congr' fun a => Subtype.ext_iff _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩ suffices Set.Nonempty (fixedPoints Q (orbit G P)) by exact Exists.elim this fun R hR => by rw [← Sylow.ext (H.mp hR)] exact R.2 apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_) calc 1 = card (fixedPoints P (orbit G P)) := ?_ _ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm _ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)] refine card_congr' (congr_arg _ (Eq.symm ?_)) rw [Set.eq_singleton_iff_unique_mem] exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩ variable (p) (G) /-- A generalization of Sylow's third theorem. If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/ theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] : card (Sylow p G) ≡ 1 [MOD p] := by refine Sylow.nonempty.elim fun P : Sylow p G => ?_ have : fixedPoints P.1 (Sylow p G) = {P} := Set.ext fun Q : Sylow p G => calc Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩ _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by rw [this] infer_instance have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this] exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this]) #align card_sylow_modeq_one card_sylow_modEq_one theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) := fun h => hp.1.ne_one (Nat.dvd_one.mp ((Nat.modEq_iff_dvd' zero_le_one).mp ((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G)))) #align not_dvd_card_sylow not_dvd_card_sylow variable {p} {G} /-- Sylow subgroups are isomorphic -/ nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) := equivSMul (MulAut.conj g) P.toSubgroup #align sylow.equiv_smul Sylow.equivSMul /-- Sylow subgroups are isomorphic -/ noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by rw [← Classical.choose_spec (exists_smul_eq G P Q)] exact P.equivSMul (Classical.choose (exists_smul_eq G P Q)) #align sylow.equiv Sylow.equiv @[simp] theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ := top_le_iff.mp fun Q _ => exists_smul_eq G P Q #align sylow.orbit_eq_top Sylow.orbit_eq_top theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) : stabilizer G P = (P : Subgroup G).normalizer := by ext; simp [Sylow.smul_eq_iff_mem_normalizer] #align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G)) (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) : ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le] have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by rw [le_centralizer_iff, zpowers_le] rintro - ⟨z, hz, rfl⟩ specialize hy z hz rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc, eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy obtain ⟨h, hh⟩ := exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1) simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh refine ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), ?_⟩ rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right] #align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) [_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) : ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy) #align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/ noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer := calc Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm _ ≃ orbit G P := Equiv.setCongr P.orbit_eq_top.symm _ ≃ G ⧸ stabilizer G P := orbitEquivQuotientStabilizer G P _ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer] #align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) : Fintype (G ⧸ (P : Subgroup G).normalizer) := ofEquiv (Sylow p G) P.equivQuotientNormalizer theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) := card_congr P.equivQuotientNormalizer #align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) : card (Sylow p G) = (P : Subgroup G).normalizer.index := (card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm #align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) : card (Sylow p G) ∣ (P : Subgroup G).index := ((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans (index_dvd_of_le le_normalizer) #align card_sylow_dvd_index card_sylow_dvd_index theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal] [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by intro h letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex rw [index_eq_card (P : Subgroup G)] at h obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm) let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G)) have hQ : IsPGroup p Q := by apply h.comap_of_ker_isPGroup rw [QuotientGroup.ker_mk'] exact P.2 replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one) rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ← comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot, QuotientGroup.ker_mk'] at hp exact hp.ne' (P.3 hQ hp.le) #align not_dvd_index_sylow' not_dvd_index_sylow'	Mathlib/GroupTheory/Sylow.lean	447	455	theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) (hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by	cases nonempty_fintype (Sylow p G) rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer] haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal := Subgroup.normal_in_normalizer haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩ replace hP := not_dvd_index_sylow' (P.subtype le_normalizer) exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
/- Copyright (c) 2023 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov, Patrick Massot -/ import Mathlib.LinearAlgebra.Basis import Mathlib.LinearAlgebra.Dual import Mathlib.Data.Fin.FlagRange /-! # Flag of submodules defined by a basis In this file we define `Basis.flag b k`, where `b : Basis (Fin n) R M`, `k : Fin (n + 1)`, to be the subspace spanned by the first `k` vectors of the basis `b`. We also prove some lemmas about this definition. -/ open Set Submodule namespace Basis section Semiring variable {R M : Type} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ} /-- The subspace spanned by the first `k` vectors of the basis `b`. -/ def flag (b : Basis (Fin n) R M) (k : Fin (n + 1)) : Submodule R M := .span R <\| b '' {i \| i.castSucc < k} @[simp] theorem flag_zero (b : Basis (Fin n) R M) : b.flag 0 = ⊥ := by simp [flag] @[simp] theorem flag_last (b : Basis (Fin n) R M) : b.flag (.last n) = ⊤ := by simp [flag, Fin.castSucc_lt_last] theorem flag_le_iff (b : Basis (Fin n) R M) {k p} : b.flag k ≤ p ↔ ∀ i : Fin n, i.castSucc < k → b i ∈ p := span_le.trans forall_mem_image theorem flag_succ (b : Basis (Fin n) R M) (k : Fin n) : b.flag k.succ = (R ∙ b k) ⊔ b.flag k.castSucc := by simp only [flag, Fin.castSucc_lt_castSucc_iff] simp [Fin.castSucc_lt_iff_succ_le, le_iff_eq_or_lt, setOf_or, image_insert_eq, span_insert] theorem self_mem_flag (b : Basis (Fin n) R M) {i : Fin n} {k : Fin (n + 1)} (h : i.castSucc < k) : b i ∈ b.flag k := subset_span <\| mem_image_of_mem _ h @[simp] theorem self_mem_flag_iff [Nontrivial R] (b : Basis (Fin n) R M) {i : Fin n} {k : Fin (n + 1)} : b i ∈ b.flag k ↔ i.castSucc < k := b.self_mem_span_image @[mono] theorem flag_mono (b : Basis (Fin n) R M) : Monotone b.flag := Fin.monotone_iff_le_succ.2 fun k ↦ by rw [flag_succ]; exact le_sup_right theorem isChain_range_flag (b : Basis (Fin n) R M) : IsChain (· ≤ ·) (range b.flag) := b.flag_mono.isChain_range @[mono] theorem flag_strictMono [Nontrivial R] (b : Basis (Fin n) R M) : StrictMono b.flag := Fin.strictMono_iff_lt_succ.2 fun _ ↦ by simp [flag_succ] end Semiring section CommRing variable {R M : Type} [CommRing R] [AddCommGroup M] [Module R M] {n : ℕ} @[simp]	Mathlib/LinearAlgebra/Basis/Flag.lean	74	76	theorem flag_le_ker_coord_iff [Nontrivial R] (b : Basis (Fin n) R M) {k : Fin (n + 1)} {l : Fin n} : b.flag k ≤ LinearMap.ker (b.coord l) ↔ k ≤ l.castSucc := by	simp [flag_le_iff, Finsupp.single_apply_eq_zero, imp_false, imp_not_comm]
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Thomas Browning -/ import Mathlib.Algebra.Group.ConjFinite import Mathlib.Data.Fintype.BigOperators import Mathlib.Dynamics.PeriodicPts import Mathlib.GroupTheory.Commutator import Mathlib.GroupTheory.Coset import Mathlib.GroupTheory.GroupAction.ConjAct import Mathlib.GroupTheory.GroupAction.Hom #align_import group_theory.group_action.quotient from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef" /-! # Properties of group actions involving quotient groups This file proves properties of group actions which use the quotient group construction, notably * the orbit-stabilizer theorem `card_orbit_mul_card_stabilizer_eq_card_group` * the class formula `card_eq_sum_card_group_div_card_stabilizer'` * Burnside's lemma `sum_card_fixedBy_eq_card_orbits_mul_card_group` -/ universe u v w variable {α : Type u} {β : Type v} {γ : Type w} open Function namespace MulAction variable [Group α] section QuotientAction open Subgroup MulOpposite QuotientGroup variable (β) [Monoid β] [MulAction β α] (H : Subgroup α) /-- A typeclass for when a `MulAction β α` descends to the quotient `α ⧸ H`. -/ class QuotientAction : Prop where /-- The action fulfils a normality condition on products that lie in `H`. This ensures that the action descends to an action on the quotient `α ⧸ H`. -/ inv_mul_mem : ∀ (b : β) {a a' : α}, a⁻¹ * a' ∈ H → (b • a)⁻¹ * b • a' ∈ H #align mul_action.quotient_action MulAction.QuotientAction /-- A typeclass for when an `AddAction β α` descends to the quotient `α ⧸ H`. -/ class _root_.AddAction.QuotientAction {α : Type u} (β : Type v) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) : Prop where /-- The action fulfils a normality condition on summands that lie in `H`. This ensures that the action descends to an action on the quotient `α ⧸ H`. -/ inv_mul_mem : ∀ (b : β) {a a' : α}, -a + a' ∈ H → -(b +ᵥ a) + (b +ᵥ a') ∈ H #align add_action.quotient_action AddAction.QuotientAction attribute [to_additive] MulAction.QuotientAction @[to_additive] instance left_quotientAction : QuotientAction α H := ⟨fun _ _ _ _ => by rwa [smul_eq_mul, smul_eq_mul, mul_inv_rev, mul_assoc, inv_mul_cancel_left]⟩ #align mul_action.left_quotient_action MulAction.left_quotientAction #align add_action.left_quotient_action AddAction.left_quotientAction @[to_additive] instance right_quotientAction : QuotientAction (normalizer H).op H := ⟨fun b c _ _ => by rwa [smul_def, smul_def, smul_eq_mul_unop, smul_eq_mul_unop, mul_inv_rev, ← mul_assoc, mem_normalizer_iff'.mp b.prop, mul_assoc, mul_inv_cancel_left]⟩ #align mul_action.right_quotient_action MulAction.right_quotientAction #align add_action.right_quotient_action AddAction.right_quotientAction @[to_additive] instance right_quotientAction' [hH : H.Normal] : QuotientAction αᵐᵒᵖ H := ⟨fun _ _ _ _ => by rwa [smul_eq_mul_unop, smul_eq_mul_unop, mul_inv_rev, mul_assoc, hH.mem_comm_iff, mul_assoc, mul_inv_cancel_right]⟩ #align mul_action.right_quotient_action' MulAction.right_quotientAction' #align add_action.right_quotient_action' AddAction.right_quotientAction' @[to_additive] instance quotient [QuotientAction β H] : MulAction β (α ⧸ H) where smul b := Quotient.map' (b • ·) fun _ _ h => leftRel_apply.mpr <\| QuotientAction.inv_mul_mem b <\| leftRel_apply.mp h one_smul q := Quotient.inductionOn' q fun a => congr_arg Quotient.mk'' (one_smul β a) mul_smul b b' q := Quotient.inductionOn' q fun a => congr_arg Quotient.mk'' (mul_smul b b' a) #align mul_action.quotient MulAction.quotient #align add_action.quotient AddAction.quotient variable {β} @[to_additive (attr := simp)] theorem Quotient.smul_mk [QuotientAction β H] (b : β) (a : α) : (b • QuotientGroup.mk a : α ⧸ H) = QuotientGroup.mk (b • a) := rfl #align mul_action.quotient.smul_mk MulAction.Quotient.smul_mk #align add_action.quotient.vadd_mk AddAction.Quotient.vadd_mk @[to_additive (attr := simp)] theorem Quotient.smul_coe [QuotientAction β H] (b : β) (a : α) : b • (a : α ⧸ H) = (↑(b • a) : α ⧸ H) := rfl #align mul_action.quotient.smul_coe MulAction.Quotient.smul_coe #align add_action.quotient.vadd_coe AddAction.Quotient.vadd_coe @[to_additive (attr := simp)] theorem Quotient.mk_smul_out' [QuotientAction β H] (b : β) (q : α ⧸ H) : QuotientGroup.mk (b • q.out') = b • q := by rw [← Quotient.smul_mk, QuotientGroup.out_eq'] #align mul_action.quotient.mk_smul_out' MulAction.Quotient.mk_smul_out' #align add_action.quotient.mk_vadd_out' AddAction.Quotient.mk_vadd_out' -- Porting note: removed simp attribute, simp can prove this @[to_additive] theorem Quotient.coe_smul_out' [QuotientAction β H] (b : β) (q : α ⧸ H) : ↑(b • q.out') = b • q := Quotient.mk_smul_out' H b q #align mul_action.quotient.coe_smul_out' MulAction.Quotient.coe_smul_out' #align add_action.quotient.coe_vadd_out' AddAction.Quotient.coe_vadd_out'	Mathlib/GroupTheory/GroupAction/Quotient.lean	120	123	theorem _root_.QuotientGroup.out'_conj_pow_minimalPeriod_mem (a : α) (q : α ⧸ H) : q.out'⁻¹ * a ^ Function.minimalPeriod (a • ·) q * q.out' ∈ H := by	rw [mul_assoc, ← QuotientGroup.eq', QuotientGroup.out_eq', ← smul_eq_mul, Quotient.mk_smul_out', eq_comm, pow_smul_eq_iff_minimalPeriod_dvd]
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker -/ import Mathlib.Algebra.Polynomial.Degree.Definitions import Mathlib.Algebra.Polynomial.Induction #align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f" /-! # Theory of univariate polynomials The main defs here are `eval₂`, `eval`, and `map`. We give several lemmas about their interaction with each other and with module operations. -/ set_option linter.uppercaseLean3 false noncomputable section open Finset AddMonoidAlgebra open Polynomial namespace Polynomial universe u v w y variable {R : Type u} {S : Type v} {T : Type w} {ι : Type y} {a b : R} {m n : ℕ} section Semiring variable [Semiring R] {p q r : R[X]} section variable [Semiring S] variable (f : R →+* S) (x : S) /-- Evaluate a polynomial `p` given a ring hom `f` from the scalar ring to the target and a value `x` for the variable in the target -/ irreducible_def eval₂ (p : R[X]) : S := p.sum fun e a => f a * x ^ e #align polynomial.eval₂ Polynomial.eval₂ theorem eval₂_eq_sum {f : R →+* S} {x : S} : p.eval₂ f x = p.sum fun e a => f a * x ^ e := by rw [eval₂_def] #align polynomial.eval₂_eq_sum Polynomial.eval₂_eq_sum theorem eval₂_congr {R S : Type} [Semiring R] [Semiring S] {f g : R →+ S} {s t : S} {φ ψ : R[X]} : f = g → s = t → φ = ψ → eval₂ f s φ = eval₂ g t ψ := by rintro rfl rfl rfl; rfl #align polynomial.eval₂_congr Polynomial.eval₂_congr @[simp] theorem eval₂_at_zero : p.eval₂ f 0 = f (coeff p 0) := by simp (config := { contextual := true }) only [eval₂_eq_sum, zero_pow_eq, mul_ite, mul_zero, mul_one, sum, Classical.not_not, mem_support_iff, sum_ite_eq', ite_eq_left_iff, RingHom.map_zero, imp_true_iff, eq_self_iff_true] #align polynomial.eval₂_at_zero Polynomial.eval₂_at_zero @[simp] theorem eval₂_zero : (0 : R[X]).eval₂ f x = 0 := by simp [eval₂_eq_sum] #align polynomial.eval₂_zero Polynomial.eval₂_zero @[simp] theorem eval₂_C : (C a).eval₂ f x = f a := by simp [eval₂_eq_sum] #align polynomial.eval₂_C Polynomial.eval₂_C @[simp] theorem eval₂_X : X.eval₂ f x = x := by simp [eval₂_eq_sum] #align polynomial.eval₂_X Polynomial.eval₂_X @[simp] theorem eval₂_monomial {n : ℕ} {r : R} : (monomial n r).eval₂ f x = f r * x ^ n := by simp [eval₂_eq_sum] #align polynomial.eval₂_monomial Polynomial.eval₂_monomial @[simp] theorem eval₂_X_pow {n : ℕ} : (X ^ n).eval₂ f x = x ^ n := by rw [X_pow_eq_monomial] convert eval₂_monomial f x (n := n) (r := 1) simp #align polynomial.eval₂_X_pow Polynomial.eval₂_X_pow @[simp] theorem eval₂_add : (p + q).eval₂ f x = p.eval₂ f x + q.eval₂ f x := by simp only [eval₂_eq_sum] apply sum_add_index <;> simp [add_mul] #align polynomial.eval₂_add Polynomial.eval₂_add @[simp] theorem eval₂_one : (1 : R[X]).eval₂ f x = 1 := by rw [← C_1, eval₂_C, f.map_one] #align polynomial.eval₂_one Polynomial.eval₂_one set_option linter.deprecated false in @[simp] theorem eval₂_bit0 : (bit0 p).eval₂ f x = bit0 (p.eval₂ f x) := by rw [bit0, eval₂_add, bit0] #align polynomial.eval₂_bit0 Polynomial.eval₂_bit0 set_option linter.deprecated false in @[simp] theorem eval₂_bit1 : (bit1 p).eval₂ f x = bit1 (p.eval₂ f x) := by rw [bit1, eval₂_add, eval₂_bit0, eval₂_one, bit1] #align polynomial.eval₂_bit1 Polynomial.eval₂_bit1 @[simp] theorem eval₂_smul (g : R →+* S) (p : R[X]) (x : S) {s : R} : eval₂ g x (s • p) = g s * eval₂ g x p := by have A : p.natDegree < p.natDegree.succ := Nat.lt_succ_self _ have B : (s • p).natDegree < p.natDegree.succ := (natDegree_smul_le _ _).trans_lt A rw [eval₂_eq_sum, eval₂_eq_sum, sum_over_range' _ _ _ A, sum_over_range' _ _ _ B] <;> simp [mul_sum, mul_assoc] #align polynomial.eval₂_smul Polynomial.eval₂_smul @[simp] theorem eval₂_C_X : eval₂ C X p = p := Polynomial.induction_on' p (fun p q hp hq => by simp [hp, hq]) fun n x => by rw [eval₂_monomial, ← smul_X_eq_monomial, C_mul'] #align polynomial.eval₂_C_X Polynomial.eval₂_C_X /-- `eval₂AddMonoidHom (f : R →+* S) (x : S)` is the `AddMonoidHom` from `R[X]` to `S` obtained by evaluating the pushforward of `p` along `f` at `x`. -/ @[simps] def eval₂AddMonoidHom : R[X] →+ S where toFun := eval₂ f x map_zero' := eval₂_zero _ _ map_add' _ _ := eval₂_add _ _ #align polynomial.eval₂_add_monoid_hom Polynomial.eval₂AddMonoidHom #align polynomial.eval₂_add_monoid_hom_apply Polynomial.eval₂AddMonoidHom_apply @[simp] theorem eval₂_natCast (n : ℕ) : (n : R[X]).eval₂ f x = n := by induction' n with n ih -- Porting note: `Nat.zero_eq` is required. · simp only [eval₂_zero, Nat.cast_zero, Nat.zero_eq] · rw [n.cast_succ, eval₂_add, ih, eval₂_one, n.cast_succ] #align polynomial.eval₂_nat_cast Polynomial.eval₂_natCast @[deprecated (since := "2024-04-17")] alias eval₂_nat_cast := eval₂_natCast -- See note [no_index around OfNat.ofNat] @[simp] lemma eval₂_ofNat {S : Type} [Semiring S] (n : ℕ) [n.AtLeastTwo] (f : R →+ S) (a : S) : (no_index (OfNat.ofNat n : R[X])).eval₂ f a = OfNat.ofNat n := by simp [OfNat.ofNat] variable [Semiring T] theorem eval₂_sum (p : T[X]) (g : ℕ → T → R[X]) (x : S) : (p.sum g).eval₂ f x = p.sum fun n a => (g n a).eval₂ f x := by let T : R[X] →+ S := { toFun := eval₂ f x map_zero' := eval₂_zero _ _ map_add' := fun p q => eval₂_add _ _ } have A : ∀ y, eval₂ f x y = T y := fun y => rfl simp only [A] rw [sum, map_sum, sum] #align polynomial.eval₂_sum Polynomial.eval₂_sum theorem eval₂_list_sum (l : List R[X]) (x : S) : eval₂ f x l.sum = (l.map (eval₂ f x)).sum := map_list_sum (eval₂AddMonoidHom f x) l #align polynomial.eval₂_list_sum Polynomial.eval₂_list_sum theorem eval₂_multiset_sum (s : Multiset R[X]) (x : S) : eval₂ f x s.sum = (s.map (eval₂ f x)).sum := map_multiset_sum (eval₂AddMonoidHom f x) s #align polynomial.eval₂_multiset_sum Polynomial.eval₂_multiset_sum theorem eval₂_finset_sum (s : Finset ι) (g : ι → R[X]) (x : S) : (∑ i ∈ s, g i).eval₂ f x = ∑ i ∈ s, (g i).eval₂ f x := map_sum (eval₂AddMonoidHom f x) _ _ #align polynomial.eval₂_finset_sum Polynomial.eval₂_finset_sum theorem eval₂_ofFinsupp {f : R →+* S} {x : S} {p : R[ℕ]} : eval₂ f x (⟨p⟩ : R[X]) = liftNC (↑f) (powersHom S x) p := by simp only [eval₂_eq_sum, sum, toFinsupp_sum, support, coeff] rfl #align polynomial.eval₂_of_finsupp Polynomial.eval₂_ofFinsupp theorem eval₂_mul_noncomm (hf : ∀ k, Commute (f <\| q.coeff k) x) : eval₂ f x (p * q) = eval₂ f x p * eval₂ f x q := by rcases p with ⟨p⟩; rcases q with ⟨q⟩ simp only [coeff] at hf simp only [← ofFinsupp_mul, eval₂_ofFinsupp] exact liftNC_mul _ _ p q fun {k n} _hn => (hf k).pow_right n #align polynomial.eval₂_mul_noncomm Polynomial.eval₂_mul_noncomm @[simp] theorem eval₂_mul_X : eval₂ f x (p * X) = eval₂ f x p * x := by refine _root_.trans (eval₂_mul_noncomm _ _ fun k => ?_) (by rw [eval₂_X]) rcases em (k = 1) with (rfl \| hk) · simp · simp [coeff_X_of_ne_one hk] #align polynomial.eval₂_mul_X Polynomial.eval₂_mul_X @[simp] theorem eval₂_X_mul : eval₂ f x (X * p) = eval₂ f x p * x := by rw [X_mul, eval₂_mul_X] #align polynomial.eval₂_X_mul Polynomial.eval₂_X_mul theorem eval₂_mul_C' (h : Commute (f a) x) : eval₂ f x (p * C a) = eval₂ f x p * f a := by rw [eval₂_mul_noncomm, eval₂_C] intro k by_cases hk : k = 0 · simp only [hk, h, coeff_C_zero, coeff_C_ne_zero] · simp only [coeff_C_ne_zero hk, RingHom.map_zero, Commute.zero_left] #align polynomial.eval₂_mul_C' Polynomial.eval₂_mul_C' theorem eval₂_list_prod_noncomm (ps : List R[X]) (hf : ∀ p ∈ ps, ∀ (k), Commute (f <\| coeff p k) x) : eval₂ f x ps.prod = (ps.map (Polynomial.eval₂ f x)).prod := by induction' ps using List.reverseRecOn with ps p ihp · simp · simp only [List.forall_mem_append, List.forall_mem_singleton] at hf simp [eval₂_mul_noncomm _ _ hf.2, ihp hf.1] #align polynomial.eval₂_list_prod_noncomm Polynomial.eval₂_list_prod_noncomm /-- `eval₂` as a `RingHom` for noncommutative rings -/ @[simps] def eval₂RingHom' (f : R →+* S) (x : S) (hf : ∀ a, Commute (f a) x) : R[X] →+* S where toFun := eval₂ f x map_add' _ _ := eval₂_add _ _ map_zero' := eval₂_zero _ _ map_mul' _p q := eval₂_mul_noncomm f x fun k => hf <\| coeff q k map_one' := eval₂_one _ _ #align polynomial.eval₂_ring_hom' Polynomial.eval₂RingHom' end /-! We next prove that eval₂ is multiplicative as long as target ring is commutative (even if the source ring is not). -/ section Eval₂ section variable [Semiring S] (f : R →+* S) (x : S) theorem eval₂_eq_sum_range : p.eval₂ f x = ∑ i ∈ Finset.range (p.natDegree + 1), f (p.coeff i) * x ^ i := _root_.trans (congr_arg _ p.as_sum_range) (_root_.trans (eval₂_finset_sum f _ _ x) (congr_arg _ (by simp))) #align polynomial.eval₂_eq_sum_range Polynomial.eval₂_eq_sum_range theorem eval₂_eq_sum_range' (f : R →+* S) {p : R[X]} {n : ℕ} (hn : p.natDegree < n) (x : S) : eval₂ f x p = ∑ i ∈ Finset.range n, f (p.coeff i) * x ^ i := by rw [eval₂_eq_sum, p.sum_over_range' _ _ hn] intro i rw [f.map_zero, zero_mul] #align polynomial.eval₂_eq_sum_range' Polynomial.eval₂_eq_sum_range' end section variable [CommSemiring S] (f : R →+* S) (x : S) @[simp] theorem eval₂_mul : (p * q).eval₂ f x = p.eval₂ f x * q.eval₂ f x := eval₂_mul_noncomm _ _ fun _k => Commute.all _ _ #align polynomial.eval₂_mul Polynomial.eval₂_mul theorem eval₂_mul_eq_zero_of_left (q : R[X]) (hp : p.eval₂ f x = 0) : (p * q).eval₂ f x = 0 := by rw [eval₂_mul f x] exact mul_eq_zero_of_left hp (q.eval₂ f x) #align polynomial.eval₂_mul_eq_zero_of_left Polynomial.eval₂_mul_eq_zero_of_left theorem eval₂_mul_eq_zero_of_right (p : R[X]) (hq : q.eval₂ f x = 0) : (p * q).eval₂ f x = 0 := by rw [eval₂_mul f x] exact mul_eq_zero_of_right (p.eval₂ f x) hq #align polynomial.eval₂_mul_eq_zero_of_right Polynomial.eval₂_mul_eq_zero_of_right /-- `eval₂` as a `RingHom` -/ def eval₂RingHom (f : R →+* S) (x : S) : R[X] →+* S := { eval₂AddMonoidHom f x with map_one' := eval₂_one _ _ map_mul' := fun _ _ => eval₂_mul _ _ } #align polynomial.eval₂_ring_hom Polynomial.eval₂RingHom @[simp] theorem coe_eval₂RingHom (f : R →+* S) (x) : ⇑(eval₂RingHom f x) = eval₂ f x := rfl #align polynomial.coe_eval₂_ring_hom Polynomial.coe_eval₂RingHom theorem eval₂_pow (n : ℕ) : (p ^ n).eval₂ f x = p.eval₂ f x ^ n := (eval₂RingHom _ _).map_pow _ _ #align polynomial.eval₂_pow Polynomial.eval₂_pow theorem eval₂_dvd : p ∣ q → eval₂ f x p ∣ eval₂ f x q := (eval₂RingHom f x).map_dvd #align polynomial.eval₂_dvd Polynomial.eval₂_dvd theorem eval₂_eq_zero_of_dvd_of_eval₂_eq_zero (h : p ∣ q) (h0 : eval₂ f x p = 0) : eval₂ f x q = 0 := zero_dvd_iff.mp (h0 ▸ eval₂_dvd f x h) #align polynomial.eval₂_eq_zero_of_dvd_of_eval₂_eq_zero Polynomial.eval₂_eq_zero_of_dvd_of_eval₂_eq_zero theorem eval₂_list_prod (l : List R[X]) (x : S) : eval₂ f x l.prod = (l.map (eval₂ f x)).prod := map_list_prod (eval₂RingHom f x) l #align polynomial.eval₂_list_prod Polynomial.eval₂_list_prod end end Eval₂ section Eval variable {x : R} /-- `eval x p` is the evaluation of the polynomial `p` at `x` -/ def eval : R → R[X] → R := eval₂ (RingHom.id _) #align polynomial.eval Polynomial.eval theorem eval_eq_sum : p.eval x = p.sum fun e a => a * x ^ e := by rw [eval, eval₂_eq_sum] rfl #align polynomial.eval_eq_sum Polynomial.eval_eq_sum theorem eval_eq_sum_range {p : R[X]} (x : R) : p.eval x = ∑ i ∈ Finset.range (p.natDegree + 1), p.coeff i * x ^ i := by rw [eval_eq_sum, sum_over_range]; simp #align polynomial.eval_eq_sum_range Polynomial.eval_eq_sum_range theorem eval_eq_sum_range' {p : R[X]} {n : ℕ} (hn : p.natDegree < n) (x : R) : p.eval x = ∑ i ∈ Finset.range n, p.coeff i * x ^ i := by rw [eval_eq_sum, p.sum_over_range' _ _ hn]; simp #align polynomial.eval_eq_sum_range' Polynomial.eval_eq_sum_range' @[simp] theorem eval₂_at_apply {S : Type} [Semiring S] (f : R →+ S) (r : R) : p.eval₂ f (f r) = f (p.eval r) := by rw [eval₂_eq_sum, eval_eq_sum, sum, sum, map_sum f] simp only [f.map_mul, f.map_pow] #align polynomial.eval₂_at_apply Polynomial.eval₂_at_apply @[simp] theorem eval₂_at_one {S : Type} [Semiring S] (f : R →+ S) : p.eval₂ f 1 = f (p.eval 1) := by convert eval₂_at_apply (p := p) f 1 simp #align polynomial.eval₂_at_one Polynomial.eval₂_at_one @[simp] theorem eval₂_at_natCast {S : Type} [Semiring S] (f : R →+ S) (n : ℕ) : p.eval₂ f n = f (p.eval n) := by convert eval₂_at_apply (p := p) f n simp #align polynomial.eval₂_at_nat_cast Polynomial.eval₂_at_natCast @[deprecated (since := "2024-04-17")] alias eval₂_at_nat_cast := eval₂_at_natCast -- See note [no_index around OfNat.ofNat] @[simp] theorem eval₂_at_ofNat {S : Type} [Semiring S] (f : R →+ S) (n : ℕ) [n.AtLeastTwo] : p.eval₂ f (no_index (OfNat.ofNat n)) = f (p.eval (OfNat.ofNat n)) := by simp [OfNat.ofNat] @[simp] theorem eval_C : (C a).eval x = a := eval₂_C _ _ #align polynomial.eval_C Polynomial.eval_C @[simp] theorem eval_natCast {n : ℕ} : (n : R[X]).eval x = n := by simp only [← C_eq_natCast, eval_C] #align polynomial.eval_nat_cast Polynomial.eval_natCast @[deprecated (since := "2024-04-17")] alias eval_nat_cast := eval_natCast -- See note [no_index around OfNat.ofNat] @[simp] lemma eval_ofNat (n : ℕ) [n.AtLeastTwo] (a : R) : (no_index (OfNat.ofNat n : R[X])).eval a = OfNat.ofNat n := by simp only [OfNat.ofNat, eval_natCast] @[simp] theorem eval_X : X.eval x = x := eval₂_X _ _ #align polynomial.eval_X Polynomial.eval_X @[simp] theorem eval_monomial {n a} : (monomial n a).eval x = a * x ^ n := eval₂_monomial _ _ #align polynomial.eval_monomial Polynomial.eval_monomial @[simp] theorem eval_zero : (0 : R[X]).eval x = 0 := eval₂_zero _ _ #align polynomial.eval_zero Polynomial.eval_zero @[simp] theorem eval_add : (p + q).eval x = p.eval x + q.eval x := eval₂_add _ _ #align polynomial.eval_add Polynomial.eval_add @[simp] theorem eval_one : (1 : R[X]).eval x = 1 := eval₂_one _ _ #align polynomial.eval_one Polynomial.eval_one set_option linter.deprecated false in @[simp] theorem eval_bit0 : (bit0 p).eval x = bit0 (p.eval x) := eval₂_bit0 _ _ #align polynomial.eval_bit0 Polynomial.eval_bit0 set_option linter.deprecated false in @[simp] theorem eval_bit1 : (bit1 p).eval x = bit1 (p.eval x) := eval₂_bit1 _ _ #align polynomial.eval_bit1 Polynomial.eval_bit1 @[simp] theorem eval_smul [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p : R[X]) (x : R) : (s • p).eval x = s • p.eval x := by rw [← smul_one_smul R s p, eval, eval₂_smul, RingHom.id_apply, smul_one_mul] #align polynomial.eval_smul Polynomial.eval_smul @[simp] theorem eval_C_mul : (C a * p).eval x = a * p.eval x := by induction p using Polynomial.induction_on' with \| h_add p q ph qh => simp only [mul_add, eval_add, ph, qh] \| h_monomial n b => simp only [mul_assoc, C_mul_monomial, eval_monomial] #align polynomial.eval_C_mul Polynomial.eval_C_mul /-- A reformulation of the expansion of (1 + y)^d: $$(d + 1) (1 + y)^d - (d + 1)y^d = \sum_{i = 0}^d {d + 1 \choose i} \cdot i \cdot y^{i - 1}.$$ -/ theorem eval_monomial_one_add_sub [CommRing S] (d : ℕ) (y : S) : eval (1 + y) (monomial d (d + 1 : S)) - eval y (monomial d (d + 1 : S)) = ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * y ^ (x_1 - 1)) := by have cast_succ : (d + 1 : S) = ((d.succ : ℕ) : S) := by simp only [Nat.cast_succ] rw [cast_succ, eval_monomial, eval_monomial, add_comm, add_pow] -- Porting note: `apply_congr` hadn't been ported yet, so `congr` & `ext` is used. conv_lhs => congr · congr · skip · congr · skip · ext rw [one_pow, mul_one, mul_comm] rw [sum_range_succ, mul_add, Nat.choose_self, Nat.cast_one, one_mul, add_sub_cancel_right, mul_sum, sum_range_succ', Nat.cast_zero, zero_mul, mul_zero, add_zero] refine sum_congr rfl fun y _hy => ?_ rw [← mul_assoc, ← mul_assoc, ← Nat.cast_mul, Nat.succ_mul_choose_eq, Nat.cast_mul, Nat.add_sub_cancel] #align polynomial.eval_monomial_one_add_sub Polynomial.eval_monomial_one_add_sub /-- `Polynomial.eval` as linear map -/ @[simps] def leval {R : Type} [Semiring R] (r : R) : R[X] →ₗ[R] R where toFun f := f.eval r map_add' _f _g := eval_add map_smul' c f := eval_smul c f r #align polynomial.leval Polynomial.leval #align polynomial.leval_apply Polynomial.leval_apply @[simp] theorem eval_natCast_mul {n : ℕ} : ((n : R[X]) p).eval x = n * p.eval x := by rw [← C_eq_natCast, eval_C_mul] #align polynomial.eval_nat_cast_mul Polynomial.eval_natCast_mul @[deprecated (since := "2024-04-17")] alias eval_nat_cast_mul := eval_natCast_mul @[simp] theorem eval_mul_X : (p * X).eval x = p.eval x * x := by induction p using Polynomial.induction_on' with \| h_add p q ph qh => simp only [add_mul, eval_add, ph, qh] \| h_monomial n a => simp only [← monomial_one_one_eq_X, monomial_mul_monomial, eval_monomial, mul_one, pow_succ, mul_assoc] #align polynomial.eval_mul_X Polynomial.eval_mul_X @[simp] theorem eval_mul_X_pow {k : ℕ} : (p * X ^ k).eval x = p.eval x * x ^ k := by induction' k with k ih · simp · simp [pow_succ, ← mul_assoc, ih] #align polynomial.eval_mul_X_pow Polynomial.eval_mul_X_pow theorem eval_sum (p : R[X]) (f : ℕ → R → R[X]) (x : R) : (p.sum f).eval x = p.sum fun n a => (f n a).eval x := eval₂_sum _ _ _ _ #align polynomial.eval_sum Polynomial.eval_sum theorem eval_finset_sum (s : Finset ι) (g : ι → R[X]) (x : R) : (∑ i ∈ s, g i).eval x = ∑ i ∈ s, (g i).eval x := eval₂_finset_sum _ _ _ _ #align polynomial.eval_finset_sum Polynomial.eval_finset_sum /-- `IsRoot p x` implies `x` is a root of `p`. The evaluation of `p` at `x` is zero -/ def IsRoot (p : R[X]) (a : R) : Prop := p.eval a = 0 #align polynomial.is_root Polynomial.IsRoot instance IsRoot.decidable [DecidableEq R] : Decidable (IsRoot p a) := by unfold IsRoot; infer_instance #align polynomial.is_root.decidable Polynomial.IsRoot.decidable @[simp] theorem IsRoot.def : IsRoot p a ↔ p.eval a = 0 := Iff.rfl #align polynomial.is_root.def Polynomial.IsRoot.def theorem IsRoot.eq_zero (h : IsRoot p x) : eval x p = 0 := h #align polynomial.is_root.eq_zero Polynomial.IsRoot.eq_zero theorem coeff_zero_eq_eval_zero (p : R[X]) : coeff p 0 = p.eval 0 := calc coeff p 0 = coeff p 0 * 0 ^ 0 := by simp _ = p.eval 0 := by symm rw [eval_eq_sum] exact Finset.sum_eq_single _ (fun b _ hb => by simp [zero_pow hb]) (by simp) #align polynomial.coeff_zero_eq_eval_zero Polynomial.coeff_zero_eq_eval_zero theorem zero_isRoot_of_coeff_zero_eq_zero {p : R[X]} (hp : p.coeff 0 = 0) : IsRoot p 0 := by rwa [coeff_zero_eq_eval_zero] at hp #align polynomial.zero_is_root_of_coeff_zero_eq_zero Polynomial.zero_isRoot_of_coeff_zero_eq_zero theorem IsRoot.dvd {R : Type} [CommSemiring R] {p q : R[X]} {x : R} (h : p.IsRoot x) (hpq : p ∣ q) : q.IsRoot x := by rwa [IsRoot, eval, eval₂_eq_zero_of_dvd_of_eval₂_eq_zero _ _ hpq] #align polynomial.is_root.dvd Polynomial.IsRoot.dvd theorem not_isRoot_C (r a : R) (hr : r ≠ 0) : ¬IsRoot (C r) a := by simpa using hr #align polynomial.not_is_root_C Polynomial.not_isRoot_C theorem eval_surjective (x : R) : Function.Surjective <\| eval x := fun y => ⟨C y, eval_C⟩ #align polynomial.eval_surjective Polynomial.eval_surjective end Eval section Comp /-- The composition of polynomials as a polynomial. -/ def comp (p q : R[X]) : R[X] := p.eval₂ C q #align polynomial.comp Polynomial.comp theorem comp_eq_sum_left : p.comp q = p.sum fun e a => C a q ^ e := by rw [comp, eval₂_eq_sum] #align polynomial.comp_eq_sum_left Polynomial.comp_eq_sum_left @[simp] theorem comp_X : p.comp X = p := by simp only [comp, eval₂_def, C_mul_X_pow_eq_monomial] exact sum_monomial_eq _ #align polynomial.comp_X Polynomial.comp_X @[simp] theorem X_comp : X.comp p = p := eval₂_X _ _ #align polynomial.X_comp Polynomial.X_comp @[simp] theorem comp_C : p.comp (C a) = C (p.eval a) := by simp [comp, map_sum (C : R →+* _)] #align polynomial.comp_C Polynomial.comp_C @[simp] theorem C_comp : (C a).comp p = C a := eval₂_C _ _ #align polynomial.C_comp Polynomial.C_comp @[simp] theorem natCast_comp {n : ℕ} : (n : R[X]).comp p = n := by rw [← C_eq_natCast, C_comp] #align polynomial.nat_cast_comp Polynomial.natCast_comp @[deprecated (since := "2024-04-17")] alias nat_cast_comp := natCast_comp -- Porting note (#10756): new theorem @[simp] theorem ofNat_comp (n : ℕ) [n.AtLeastTwo] : (no_index (OfNat.ofNat n) : R[X]).comp p = n := natCast_comp @[simp] theorem comp_zero : p.comp (0 : R[X]) = C (p.eval 0) := by rw [← C_0, comp_C] #align polynomial.comp_zero Polynomial.comp_zero @[simp] theorem zero_comp : comp (0 : R[X]) p = 0 := by rw [← C_0, C_comp] #align polynomial.zero_comp Polynomial.zero_comp @[simp] theorem comp_one : p.comp 1 = C (p.eval 1) := by rw [← C_1, comp_C] #align polynomial.comp_one Polynomial.comp_one @[simp] theorem one_comp : comp (1 : R[X]) p = 1 := by rw [← C_1, C_comp] #align polynomial.one_comp Polynomial.one_comp @[simp] theorem add_comp : (p + q).comp r = p.comp r + q.comp r := eval₂_add _ _ #align polynomial.add_comp Polynomial.add_comp @[simp] theorem monomial_comp (n : ℕ) : (monomial n a).comp p = C a * p ^ n := eval₂_monomial _ _ #align polynomial.monomial_comp Polynomial.monomial_comp @[simp] theorem mul_X_comp : (p * X).comp r = p.comp r * r := by induction p using Polynomial.induction_on' with \| h_add p q hp hq => simp only [hp, hq, add_mul, add_comp] \| h_monomial n b => simp only [pow_succ, mul_assoc, monomial_mul_X, monomial_comp] #align polynomial.mul_X_comp Polynomial.mul_X_comp @[simp] theorem X_pow_comp {k : ℕ} : (X ^ k).comp p = p ^ k := by induction' k with k ih · simp · simp [pow_succ, mul_X_comp, ih] #align polynomial.X_pow_comp Polynomial.X_pow_comp @[simp] theorem mul_X_pow_comp {k : ℕ} : (p * X ^ k).comp r = p.comp r * r ^ k := by induction' k with k ih · simp · simp [ih, pow_succ, ← mul_assoc, mul_X_comp] #align polynomial.mul_X_pow_comp Polynomial.mul_X_pow_comp @[simp] theorem C_mul_comp : (C a * p).comp r = C a * p.comp r := by induction p using Polynomial.induction_on' with \| h_add p q hp hq => simp [hp, hq, mul_add] \| h_monomial n b => simp [mul_assoc] #align polynomial.C_mul_comp Polynomial.C_mul_comp @[simp] theorem natCast_mul_comp {n : ℕ} : ((n : R[X]) * p).comp r = n * p.comp r := by rw [← C_eq_natCast, C_mul_comp] #align polynomial.nat_cast_mul_comp Polynomial.natCast_mul_comp @[deprecated (since := "2024-04-17")] alias nat_cast_mul_comp := natCast_mul_comp theorem mul_X_add_natCast_comp {n : ℕ} : (p * (X + (n : R[X]))).comp q = p.comp q * (q + n) := by rw [mul_add, add_comp, mul_X_comp, ← Nat.cast_comm, natCast_mul_comp, Nat.cast_comm, mul_add] set_option linter.uppercaseLean3 false in #align polynomial.mul_X_add_nat_cast_comp Polynomial.mul_X_add_natCast_comp @[deprecated (since := "2024-04-17")] alias mul_X_add_nat_cast_comp := mul_X_add_natCast_comp @[simp] theorem mul_comp {R : Type} [CommSemiring R] (p q r : R[X]) : (p q).comp r = p.comp r * q.comp r := eval₂_mul _ _ #align polynomial.mul_comp Polynomial.mul_comp @[simp] theorem pow_comp {R : Type} [CommSemiring R] (p q : R[X]) (n : ℕ) : (p ^ n).comp q = p.comp q ^ n := (MonoidHom.mk (OneHom.mk (fun r : R[X] => r.comp q) one_comp) fun r s => mul_comp r s q).map_pow p n #align polynomial.pow_comp Polynomial.pow_comp set_option linter.deprecated false in @[simp] theorem bit0_comp : comp (bit0 p : R[X]) q = bit0 (p.comp q) := by simp only [bit0, add_comp] #align polynomial.bit0_comp Polynomial.bit0_comp set_option linter.deprecated false in @[simp] theorem bit1_comp : comp (bit1 p : R[X]) q = bit1 (p.comp q) := by simp only [bit1, add_comp, bit0_comp, one_comp] #align polynomial.bit1_comp Polynomial.bit1_comp @[simp] theorem smul_comp [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p q : R[X]) : (s • p).comp q = s • p.comp q := by rw [← smul_one_smul R s p, comp, comp, eval₂_smul, ← smul_eq_C_mul, smul_assoc, one_smul] #align polynomial.smul_comp Polynomial.smul_comp theorem comp_assoc {R : Type} [CommSemiring R] (φ ψ χ : R[X]) : (φ.comp ψ).comp χ = φ.comp (ψ.comp χ) := by refine Polynomial.induction_on φ ?_ ?_ ?_ <;> · intros simp_all only [add_comp, mul_comp, C_comp, X_comp, pow_succ, ← mul_assoc] #align polynomial.comp_assoc Polynomial.comp_assoc theorem coeff_comp_degree_mul_degree (hqd0 : natDegree q ≠ 0) : coeff (p.comp q) (natDegree p * natDegree q) = leadingCoeff p * leadingCoeff q ^ natDegree p := by rw [comp, eval₂_def, coeff_sum] -- Porting note: `convert` → `refine` refine Eq.trans (Finset.sum_eq_single p.natDegree ?h₀ ?h₁) ?h₂ case h₂ => simp only [coeff_natDegree, coeff_C_mul, coeff_pow_mul_natDegree] case h₀ => intro b hbs hbp refine coeff_eq_zero_of_natDegree_lt (natDegree_mul_le.trans_lt ?_) rw [natDegree_C, zero_add] refine natDegree_pow_le.trans_lt ((mul_lt_mul_right (pos_iff_ne_zero.mpr hqd0)).mpr ?_) exact lt_of_le_of_ne (le_natDegree_of_mem_supp _ hbs) hbp case h₁ => simp (config := { contextual := true }) #align polynomial.coeff_comp_degree_mul_degree Polynomial.coeff_comp_degree_mul_degree @[simp] lemma sum_comp (s : Finset ι) (p : ι → R[X]) (q : R[X]) : (∑ i ∈ s, p i).comp q = ∑ i ∈ s, (p i).comp q := Polynomial.eval₂_finset_sum _ _ _ _ end Comp section Map variable [Semiring S] variable (f : R →+* S) /-- `map f p` maps a polynomial `p` across a ring hom `f` -/ def map : R[X] → S[X] := eval₂ (C.comp f) X #align polynomial.map Polynomial.map @[simp] theorem map_C : (C a).map f = C (f a) := eval₂_C _ _ #align polynomial.map_C Polynomial.map_C @[simp] theorem map_X : X.map f = X := eval₂_X _ _ #align polynomial.map_X Polynomial.map_X @[simp] theorem map_monomial {n a} : (monomial n a).map f = monomial n (f a) := by dsimp only [map] rw [eval₂_monomial, ← C_mul_X_pow_eq_monomial]; rfl #align polynomial.map_monomial Polynomial.map_monomial @[simp] protected theorem map_zero : (0 : R[X]).map f = 0 := eval₂_zero _ _ #align polynomial.map_zero Polynomial.map_zero @[simp] protected theorem map_add : (p + q).map f = p.map f + q.map f := eval₂_add _ _ #align polynomial.map_add Polynomial.map_add @[simp] protected theorem map_one : (1 : R[X]).map f = 1 := eval₂_one _ _ #align polynomial.map_one Polynomial.map_one @[simp] protected theorem map_mul : (p * q).map f = p.map f * q.map f := by rw [map, eval₂_mul_noncomm] exact fun k => (commute_X _).symm #align polynomial.map_mul Polynomial.map_mul @[simp] protected theorem map_smul (r : R) : (r • p).map f = f r • p.map f := by rw [map, eval₂_smul, RingHom.comp_apply, C_mul'] #align polynomial.map_smul Polynomial.map_smul -- `map` is a ring-hom unconditionally, and theoretically the definition could be replaced, -- but this turns out not to be easy because `p.map f` does not resolve to `Polynomial.map` -- if `map` is a `RingHom` instead of a plain function; the elaborator does not try to coerce -- to a function before trying field (dot) notation (this may be technically infeasible); -- the relevant code is (both lines): https://github.com/leanprover-community/ -- lean/blob/487ac5d7e9b34800502e1ddf3c7c806c01cf9d51/src/frontends/lean/elaborator.cpp#L1876-L1913 /-- `Polynomial.map` as a `RingHom`. -/ def mapRingHom (f : R →+* S) : R[X] →+* S[X] where toFun := Polynomial.map f map_add' _ _ := Polynomial.map_add f map_zero' := Polynomial.map_zero f map_mul' _ _ := Polynomial.map_mul f map_one' := Polynomial.map_one f #align polynomial.map_ring_hom Polynomial.mapRingHom @[simp] theorem coe_mapRingHom (f : R →+* S) : ⇑(mapRingHom f) = map f := rfl #align polynomial.coe_map_ring_hom Polynomial.coe_mapRingHom -- This is protected to not clash with the global `map_natCast`. @[simp] protected theorem map_natCast (n : ℕ) : (n : R[X]).map f = n := map_natCast (mapRingHom f) n #align polynomial.map_nat_cast Polynomial.map_natCast @[deprecated (since := "2024-04-17")] alias map_nat_cast := map_natCast -- Porting note (#10756): new theorem -- See note [no_index around OfNat.ofNat] @[simp] protected theorem map_ofNat (n : ℕ) [n.AtLeastTwo] : (no_index (OfNat.ofNat n) : R[X]).map f = OfNat.ofNat n := show (n : R[X]).map f = n by rw [Polynomial.map_natCast] #noalign polynomial.map_bit0 #noalign polynomial.map_bit1 --TODO rename to `map_dvd_map` theorem map_dvd (f : R →+* S) {x y : R[X]} : x ∣ y → x.map f ∣ y.map f := (mapRingHom f).map_dvd #align polynomial.map_dvd Polynomial.map_dvd @[simp] theorem coeff_map (n : ℕ) : coeff (p.map f) n = f (coeff p n) := by rw [map, eval₂_def, coeff_sum, sum] conv_rhs => rw [← sum_C_mul_X_pow_eq p, coeff_sum, sum, map_sum] refine Finset.sum_congr rfl fun x _hx => ?_ simp only [RingHom.coe_comp, Function.comp, coeff_C_mul_X_pow] split_ifs <;> simp [f.map_zero] #align polynomial.coeff_map Polynomial.coeff_map /-- If `R` and `S` are isomorphic, then so are their polynomial rings. -/ @[simps!] def mapEquiv (e : R ≃+* S) : R[X] ≃+* S[X] := RingEquiv.ofHomInv (mapRingHom (e : R →+* S)) (mapRingHom (e.symm : S →+* R)) (by ext <;> simp) (by ext <;> simp) #align polynomial.map_equiv Polynomial.mapEquiv #align polynomial.map_equiv_apply Polynomial.mapEquiv_apply #align polynomial.map_equiv_symm_apply Polynomial.mapEquiv_symm_apply theorem map_map [Semiring T] (g : S →+* T) (p : R[X]) : (p.map f).map g = p.map (g.comp f) := ext (by simp [coeff_map]) #align polynomial.map_map Polynomial.map_map @[simp] theorem map_id : p.map (RingHom.id _) = p := by simp [Polynomial.ext_iff, coeff_map] #align polynomial.map_id Polynomial.map_id /-- The polynomial ring over a finite product of rings is isomorphic to the product of polynomial rings over individual rings. -/ def piEquiv {ι} [Finite ι] (R : ι → Type) [∀ i, Semiring (R i)] : (∀ i, R i)[X] ≃+ ∀ i, (R i)[X] := .ofBijective (Pi.ringHom fun i ↦ mapRingHom (Pi.evalRingHom R i)) ⟨fun p q h ↦ by ext n i; simpa using congr_arg (fun p ↦ coeff (p i) n) h, fun p ↦ ⟨.ofFinsupp (.ofSupportFinite (fun n i ↦ coeff (p i) n) <\| (Set.finite_iUnion fun i ↦ (p i).support.finite_toSet).subset fun n hn ↦ by simp only [Set.mem_iUnion, Finset.mem_coe, mem_support_iff, Function.mem_support] at hn ⊢ contrapose! hn; exact funext hn), by ext i n; exact coeff_map _ _⟩⟩ theorem eval₂_eq_eval_map {x : S} : p.eval₂ f x = (p.map f).eval x := by induction p using Polynomial.induction_on' with \| h_add p q hp hq => simp [hp, hq] \| h_monomial n r => simp #align polynomial.eval₂_eq_eval_map Polynomial.eval₂_eq_eval_map theorem map_injective (hf : Function.Injective f) : Function.Injective (map f) := fun p q h => ext fun m => hf <\| by rw [← coeff_map f, ← coeff_map f, h] #align polynomial.map_injective Polynomial.map_injective theorem map_surjective (hf : Function.Surjective f) : Function.Surjective (map f) := fun p => Polynomial.induction_on' p (fun p q hp hq => let ⟨p', hp'⟩ := hp let ⟨q', hq'⟩ := hq ⟨p' + q', by rw [Polynomial.map_add f, hp', hq']⟩) fun n s => let ⟨r, hr⟩ := hf s ⟨monomial n r, by rw [map_monomial f, hr]⟩ #align polynomial.map_surjective Polynomial.map_surjective theorem degree_map_le (p : R[X]) : degree (p.map f) ≤ degree p := by refine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_ rw [degree_lt_iff_coeff_zero] at hm simp [hm m le_rfl] #align polynomial.degree_map_le Polynomial.degree_map_le theorem natDegree_map_le (p : R[X]) : natDegree (p.map f) ≤ natDegree p := natDegree_le_natDegree (degree_map_le f p) #align polynomial.nat_degree_map_le Polynomial.natDegree_map_le variable {f} protected theorem map_eq_zero_iff (hf : Function.Injective f) : p.map f = 0 ↔ p = 0 := map_eq_zero_iff (mapRingHom f) (map_injective f hf) #align polynomial.map_eq_zero_iff Polynomial.map_eq_zero_iff protected theorem map_ne_zero_iff (hf : Function.Injective f) : p.map f ≠ 0 ↔ p ≠ 0 := (Polynomial.map_eq_zero_iff hf).not #align polynomial.map_ne_zero_iff Polynomial.map_ne_zero_iff theorem map_monic_eq_zero_iff (hp : p.Monic) : p.map f = 0 ↔ ∀ x, f x = 0 := ⟨fun hfp x => calc f x = f x * f p.leadingCoeff := by simp only [mul_one, hp.leadingCoeff, f.map_one] _ = f x * (p.map f).coeff p.natDegree := congr_arg _ (coeff_map _ _).symm _ = 0 := by simp only [hfp, mul_zero, coeff_zero] , fun h => ext fun n => by simp only [h, coeff_map, coeff_zero]⟩ #align polynomial.map_monic_eq_zero_iff Polynomial.map_monic_eq_zero_iff theorem map_monic_ne_zero (hp : p.Monic) [Nontrivial S] : p.map f ≠ 0 := fun h => f.map_one_ne_zero ((map_monic_eq_zero_iff hp).mp h _) #align polynomial.map_monic_ne_zero Polynomial.map_monic_ne_zero theorem degree_map_eq_of_leadingCoeff_ne_zero (f : R →+* S) (hf : f (leadingCoeff p) ≠ 0) : degree (p.map f) = degree p := le_antisymm (degree_map_le f _) <\| by have hp0 : p ≠ 0 := leadingCoeff_ne_zero.mp fun hp0 => hf (_root_.trans (congr_arg _ hp0) f.map_zero) rw [degree_eq_natDegree hp0] refine le_degree_of_ne_zero ?_ rw [coeff_map] exact hf #align polynomial.degree_map_eq_of_leading_coeff_ne_zero Polynomial.degree_map_eq_of_leadingCoeff_ne_zero theorem natDegree_map_of_leadingCoeff_ne_zero (f : R →+* S) (hf : f (leadingCoeff p) ≠ 0) : natDegree (p.map f) = natDegree p := natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f hf) #align polynomial.nat_degree_map_of_leading_coeff_ne_zero Polynomial.natDegree_map_of_leadingCoeff_ne_zero theorem leadingCoeff_map_of_leadingCoeff_ne_zero (f : R →+* S) (hf : f (leadingCoeff p) ≠ 0) : leadingCoeff (p.map f) = f (leadingCoeff p) := by unfold leadingCoeff rw [coeff_map, natDegree_map_of_leadingCoeff_ne_zero f hf] #align polynomial.leading_coeff_map_of_leading_coeff_ne_zero Polynomial.leadingCoeff_map_of_leadingCoeff_ne_zero variable (f) @[simp] theorem mapRingHom_id : mapRingHom (RingHom.id R) = RingHom.id R[X] := RingHom.ext fun _x => map_id #align polynomial.map_ring_hom_id Polynomial.mapRingHom_id @[simp] theorem mapRingHom_comp [Semiring T] (f : S →+* T) (g : R →+* S) : (mapRingHom f).comp (mapRingHom g) = mapRingHom (f.comp g) := RingHom.ext <\| Polynomial.map_map g f #align polynomial.map_ring_hom_comp Polynomial.mapRingHom_comp protected theorem map_list_prod (L : List R[X]) : L.prod.map f = (L.map <\| map f).prod := Eq.symm <\| List.prod_hom _ (mapRingHom f).toMonoidHom #align polynomial.map_list_prod Polynomial.map_list_prod @[simp] protected theorem map_pow (n : ℕ) : (p ^ n).map f = p.map f ^ n := (mapRingHom f).map_pow _ _ #align polynomial.map_pow Polynomial.map_pow theorem mem_map_rangeS {p : S[X]} : p ∈ (mapRingHom f).rangeS ↔ ∀ n, p.coeff n ∈ f.rangeS := by constructor · rintro ⟨p, rfl⟩ n rw [coe_mapRingHom, coeff_map] exact Set.mem_range_self _ · intro h rw [p.as_sum_range_C_mul_X_pow] refine (mapRingHom f).rangeS.sum_mem ?_ intro i _hi rcases h i with ⟨c, hc⟩ use C c * X ^ i rw [coe_mapRingHom, Polynomial.map_mul, map_C, hc, Polynomial.map_pow, map_X] #align polynomial.mem_map_srange Polynomial.mem_map_rangeS theorem mem_map_range {R S : Type} [Ring R] [Ring S] (f : R →+ S) {p : S[X]} : p ∈ (mapRingHom f).range ↔ ∀ n, p.coeff n ∈ f.range := mem_map_rangeS f #align polynomial.mem_map_range Polynomial.mem_map_range theorem eval₂_map [Semiring T] (g : S →+* T) (x : T) : (p.map f).eval₂ g x = p.eval₂ (g.comp f) x := by rw [eval₂_eq_eval_map, eval₂_eq_eval_map, map_map] #align polynomial.eval₂_map Polynomial.eval₂_map theorem eval_map (x : S) : (p.map f).eval x = p.eval₂ f x := (eval₂_eq_eval_map f).symm #align polynomial.eval_map Polynomial.eval_map protected theorem map_sum {ι : Type*} (g : ι → R[X]) (s : Finset ι) : (∑ i ∈ s, g i).map f = ∑ i ∈ s, (g i).map f := map_sum (mapRingHom f) _ _ #align polynomial.map_sum Polynomial.map_sum theorem map_comp (p q : R[X]) : map f (p.comp q) = (map f p).comp (map f q) := Polynomial.induction_on p (by simp only [map_C, forall_const, C_comp, eq_self_iff_true]) (by simp (config := { contextual := true }) only [Polynomial.map_add, add_comp, forall_const, imp_true_iff, eq_self_iff_true]) (by simp (config := { contextual := true }) only [pow_succ, ← mul_assoc, comp, forall_const, eval₂_mul_X, imp_true_iff, eq_self_iff_true, map_X, Polynomial.map_mul]) #align polynomial.map_comp Polynomial.map_comp @[simp]	Mathlib/Algebra/Polynomial/Eval.lean	1,003	1,004	theorem eval_zero_map (f : R →+* S) (p : R[X]) : (p.map f).eval 0 = f (p.eval 0) := by	simp [← coeff_zero_eq_eval_zero]
/- Copyright (c) 2019 Reid Barton. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Topology.Constructions #align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494" /-! # Neighborhoods and continuity relative to a subset This file defines relative versions * `nhdsWithin` of `nhds` * `ContinuousOn` of `Continuous` * `ContinuousWithinAt` of `ContinuousAt` and proves their basic properties, including the relationships between these restricted notions and the corresponding notions for the subtype equipped with the subspace topology. ## Notation * `𝓝 x`: the filter of neighborhoods of a point `x`; * `𝓟 s`: the principal filter of a set `s`; * `𝓝[s] x`: the filter `nhdsWithin x s` of neighborhoods of a point `x` within a set `s`. -/ open Set Filter Function Topology Filter variable {α : Type} {β : Type} {γ : Type} {δ : Type} variable [TopologicalSpace α] @[simp] theorem nhds_bind_nhdsWithin {a : α} {s : Set α} : ((𝓝 a).bind fun x => 𝓝[s] x) = 𝓝[s] a := bind_inf_principal.trans <\| congr_arg₂ _ nhds_bind_nhds rfl #align nhds_bind_nhds_within nhds_bind_nhdsWithin @[simp] theorem eventually_nhds_nhdsWithin {a : α} {s : Set α} {p : α → Prop} : (∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := Filter.ext_iff.1 nhds_bind_nhdsWithin { x \| p x } #align eventually_nhds_nhds_within eventually_nhds_nhdsWithin theorem eventually_nhdsWithin_iff {a : α} {s : Set α} {p : α → Prop} : (∀ᶠ x in 𝓝[s] a, p x) ↔ ∀ᶠ x in 𝓝 a, x ∈ s → p x := eventually_inf_principal #align eventually_nhds_within_iff eventually_nhdsWithin_iff theorem frequently_nhdsWithin_iff {z : α} {s : Set α} {p : α → Prop} : (∃ᶠ x in 𝓝[s] z, p x) ↔ ∃ᶠ x in 𝓝 z, p x ∧ x ∈ s := frequently_inf_principal.trans <\| by simp only [and_comm] #align frequently_nhds_within_iff frequently_nhdsWithin_iff theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} : z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s := by simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff] #align mem_closure_ne_iff_frequently_within mem_closure_ne_iff_frequently_within @[simp] theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} : (∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩ simp only [eventually_nhdsWithin_iff] at h ⊢ exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs #align eventually_nhds_within_nhds_within eventually_nhdsWithin_nhdsWithin theorem nhdsWithin_eq (a : α) (s : Set α) : 𝓝[s] a = ⨅ t ∈ { t : Set α \| a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) := ((nhds_basis_opens a).inf_principal s).eq_biInf #align nhds_within_eq nhdsWithin_eq theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by rw [nhdsWithin, principal_univ, inf_top_eq] #align nhds_within_univ nhdsWithin_univ theorem nhdsWithin_hasBasis {p : β → Prop} {s : β → Set α} {a : α} (h : (𝓝 a).HasBasis p s) (t : Set α) : (𝓝[t] a).HasBasis p fun i => s i ∩ t := h.inf_principal t #align nhds_within_has_basis nhdsWithin_hasBasis theorem nhdsWithin_basis_open (a : α) (t : Set α) : (𝓝[t] a).HasBasis (fun u => a ∈ u ∧ IsOpen u) fun u => u ∩ t := nhdsWithin_hasBasis (nhds_basis_opens a) t #align nhds_within_basis_open nhdsWithin_basis_open theorem mem_nhdsWithin {t : Set α} {a : α} {s : Set α} : t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t := by simpa only [and_assoc, and_left_comm] using (nhdsWithin_basis_open a s).mem_iff #align mem_nhds_within mem_nhdsWithin theorem mem_nhdsWithin_iff_exists_mem_nhds_inter {t : Set α} {a : α} {s : Set α} : t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t := (nhdsWithin_hasBasis (𝓝 a).basis_sets s).mem_iff #align mem_nhds_within_iff_exists_mem_nhds_inter mem_nhdsWithin_iff_exists_mem_nhds_inter theorem diff_mem_nhdsWithin_compl {x : α} {s : Set α} (hs : s ∈ 𝓝 x) (t : Set α) : s \ t ∈ 𝓝[tᶜ] x := diff_mem_inf_principal_compl hs t #align diff_mem_nhds_within_compl diff_mem_nhdsWithin_compl theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α) : s \ t' ∈ 𝓝[t \ t'] x := by rw [nhdsWithin, diff_eq, diff_eq, ← inf_principal, ← inf_assoc] exact inter_mem_inf hs (mem_principal_self _) #align diff_mem_nhds_within_diff diff_mem_nhdsWithin_diff theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) : t ∈ 𝓝 a := by rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp h2 with ⟨_, Hw, hw⟩ exact (𝓝 a).sets_of_superset ((𝓝 a).inter_sets Hw h1) hw #align nhds_of_nhds_within_of_nhds nhds_of_nhdsWithin_of_nhds theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} : t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t := eventually_inf_principal #align mem_nhds_within_iff_eventually mem_nhdsWithin_iff_eventually theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} : t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : Set α) := by simp_rw [mem_nhdsWithin_iff_eventually, eventuallyEq_set, mem_inter_iff, iff_self_and] #align mem_nhds_within_iff_eventually_eq mem_nhdsWithin_iff_eventuallyEq theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t := set_eventuallyEq_iff_inf_principal.symm #align nhds_within_eq_iff_eventually_eq nhdsWithin_eq_iff_eventuallyEq theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x := set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal #align nhds_within_le_iff nhdsWithin_le_iff -- Porting note: golfed, dropped an unneeded assumption theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t) (hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) : π ⁻¹' s ∈ 𝓝[t] a := by lift a to t using h replace hs : (fun x : t => π x) ⁻¹' s ∈ 𝓝 a := preimage_nhds_coinduced hs rwa [← map_nhds_subtype_val, mem_map] #align preimage_nhds_within_coinduced' preimage_nhdsWithin_coinduced'ₓ theorem mem_nhdsWithin_of_mem_nhds {s t : Set α} {a : α} (h : s ∈ 𝓝 a) : s ∈ 𝓝[t] a := mem_inf_of_left h #align mem_nhds_within_of_mem_nhds mem_nhdsWithin_of_mem_nhds theorem self_mem_nhdsWithin {a : α} {s : Set α} : s ∈ 𝓝[s] a := mem_inf_of_right (mem_principal_self s) #align self_mem_nhds_within self_mem_nhdsWithin theorem eventually_mem_nhdsWithin {a : α} {s : Set α} : ∀ᶠ x in 𝓝[s] a, x ∈ s := self_mem_nhdsWithin #align eventually_mem_nhds_within eventually_mem_nhdsWithin theorem inter_mem_nhdsWithin (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝 a) : s ∩ t ∈ 𝓝[s] a := inter_mem self_mem_nhdsWithin (mem_inf_of_left h) #align inter_mem_nhds_within inter_mem_nhdsWithin theorem nhdsWithin_mono (a : α) {s t : Set α} (h : s ⊆ t) : 𝓝[s] a ≤ 𝓝[t] a := inf_le_inf_left _ (principal_mono.mpr h) #align nhds_within_mono nhdsWithin_mono theorem pure_le_nhdsWithin {a : α} {s : Set α} (ha : a ∈ s) : pure a ≤ 𝓝[s] a := le_inf (pure_le_nhds a) (le_principal_iff.2 ha) #align pure_le_nhds_within pure_le_nhdsWithin theorem mem_of_mem_nhdsWithin {a : α} {s t : Set α} (ha : a ∈ s) (ht : t ∈ 𝓝[s] a) : a ∈ t := pure_le_nhdsWithin ha ht #align mem_of_mem_nhds_within mem_of_mem_nhdsWithin theorem Filter.Eventually.self_of_nhdsWithin {p : α → Prop} {s : Set α} {x : α} (h : ∀ᶠ y in 𝓝[s] x, p y) (hx : x ∈ s) : p x := mem_of_mem_nhdsWithin hx h #align filter.eventually.self_of_nhds_within Filter.Eventually.self_of_nhdsWithin theorem tendsto_const_nhdsWithin {l : Filter β} {s : Set α} {a : α} (ha : a ∈ s) : Tendsto (fun _ : β => a) l (𝓝[s] a) := tendsto_const_pure.mono_right <\| pure_le_nhdsWithin ha #align tendsto_const_nhds_within tendsto_const_nhdsWithin theorem nhdsWithin_restrict'' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝[s] a) : 𝓝[s] a = 𝓝[s ∩ t] a := le_antisymm (le_inf inf_le_left (le_principal_iff.mpr (inter_mem self_mem_nhdsWithin h))) (inf_le_inf_left _ (principal_mono.mpr Set.inter_subset_left)) #align nhds_within_restrict'' nhdsWithin_restrict'' theorem nhdsWithin_restrict' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝 a) : 𝓝[s] a = 𝓝[s ∩ t] a := nhdsWithin_restrict'' s <\| mem_inf_of_left h #align nhds_within_restrict' nhdsWithin_restrict' theorem nhdsWithin_restrict {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t) (h₁ : IsOpen t) : 𝓝[s] a = 𝓝[s ∩ t] a := nhdsWithin_restrict' s (IsOpen.mem_nhds h₁ h₀) #align nhds_within_restrict nhdsWithin_restrict theorem nhdsWithin_le_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t] a ≤ 𝓝[s] a := nhdsWithin_le_iff.mpr h #align nhds_within_le_of_mem nhdsWithin_le_of_mem	Mathlib/Topology/ContinuousOn.lean	200	203	theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by	rw [← nhdsWithin_univ] apply nhdsWithin_le_of_mem exact univ_mem
/- Copyright (c) 2020 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import Mathlib.Algebra.Group.Ext import Mathlib.CategoryTheory.Limits.Shapes.Biproducts import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products import Mathlib.CategoryTheory.Preadditive.Basic import Mathlib.Tactic.Abel #align_import category_theory.preadditive.biproducts from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af" /-! # Basic facts about biproducts in preadditive categories. In (or between) preadditive categories, * Any biproduct satisfies the equality `total : ∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f)`, or, in the binary case, `total : fst ≫ inl + snd ≫ inr = 𝟙 X`. * Any (binary) `product` or (binary) `coproduct` is a (binary) `biproduct`. * In any category (with zero morphisms), if `biprod.map f g` is an isomorphism, then both `f` and `g` are isomorphisms. * If `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism, then we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂` so that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`), via Gaussian elimination. * As a corollary of the previous two facts, if we have an isomorphism `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism, we can construct an isomorphism `X₂ ≅ Y₂`. * If `f : W ⊞ X ⟶ Y ⊞ Z` is an isomorphism, either `𝟙 W = 0`, or at least one of the component maps `W ⟶ Y` and `W ⟶ Z` is nonzero. * If `f : ⨁ S ⟶ ⨁ T` is an isomorphism, then every column (corresponding to a nonzero summand in the domain) has some nonzero matrix entry. * A functor preserves a biproduct if and only if it preserves the corresponding product if and only if it preserves the corresponding coproduct. There are connections between this material and the special case of the category whose morphisms are matrices over a ring, in particular the Schur complement (see `Mathlib.LinearAlgebra.Matrix.SchurComplement`). In particular, the declarations `CategoryTheory.Biprod.isoElim`, `CategoryTheory.Biprod.gaussian` and `Matrix.invertibleOfFromBlocks₁₁Invertible` are all closely related. -/ open CategoryTheory open CategoryTheory.Preadditive open CategoryTheory.Limits open CategoryTheory.Functor open CategoryTheory.Preadditive open scoped Classical universe v v' u u' noncomputable section namespace CategoryTheory variable {C : Type u} [Category.{v} C] [Preadditive C] namespace Limits section Fintype variable {J : Type} [Fintype J] /-- In a preadditive category, we can construct a biproduct for `f : J → C` from any bicone `b` for `f` satisfying `total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X`. (That is, such a bicone is a limit cone and a colimit cocone.) -/ def isBilimitOfTotal {f : J → C} (b : Bicone f) (total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.pt) : b.IsBilimit where isLimit := { lift := fun s => ∑ j : J, s.π.app ⟨j⟩ ≫ b.ι j uniq := fun s m h => by erw [← Category.comp_id m, ← total, comp_sum] apply Finset.sum_congr rfl intro j _ have reassoced : m ≫ Bicone.π b j ≫ Bicone.ι b j = s.π.app ⟨j⟩ ≫ Bicone.ι b j := by erw [← Category.assoc, eq_whisker (h ⟨j⟩)] rw [reassoced] fac := fun s j => by cases j simp only [sum_comp, Category.assoc, Bicone.toCone_π_app, b.ι_π, comp_dite] -- See note [dsimp, simp]. dsimp; simp } isColimit := { desc := fun s => ∑ j : J, b.π j ≫ s.ι.app ⟨j⟩ uniq := fun s m h => by erw [← Category.id_comp m, ← total, sum_comp] apply Finset.sum_congr rfl intro j _ erw [Category.assoc, h ⟨j⟩] fac := fun s j => by cases j simp only [comp_sum, ← Category.assoc, Bicone.toCocone_ι_app, b.ι_π, dite_comp] dsimp; simp } #align category_theory.limits.is_bilimit_of_total CategoryTheory.Limits.isBilimitOfTotal theorem IsBilimit.total {f : J → C} {b : Bicone f} (i : b.IsBilimit) : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.pt := i.isLimit.hom_ext fun j => by cases j simp [sum_comp, b.ι_π, comp_dite] #align category_theory.limits.is_bilimit.total CategoryTheory.Limits.IsBilimit.total /-- In a preadditive category, we can construct a biproduct for `f : J → C` from any bicone `b` for `f` satisfying `total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X`. (That is, such a bicone is a limit cone and a colimit cocone.) -/ theorem hasBiproduct_of_total {f : J → C} (b : Bicone f) (total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.pt) : HasBiproduct f := HasBiproduct.mk { bicone := b isBilimit := isBilimitOfTotal b total } #align category_theory.limits.has_biproduct_of_total CategoryTheory.Limits.hasBiproduct_of_total /-- In a preadditive category, any finite bicone which is a limit cone is in fact a bilimit bicone. -/ def isBilimitOfIsLimit {f : J → C} (t : Bicone f) (ht : IsLimit t.toCone) : t.IsBilimit := isBilimitOfTotal _ <\| ht.hom_ext fun j => by cases j simp [sum_comp, t.ι_π, dite_comp, comp_dite] #align category_theory.limits.is_bilimit_of_is_limit CategoryTheory.Limits.isBilimitOfIsLimit /-- We can turn any limit cone over a pair into a bilimit bicone. -/ def biconeIsBilimitOfLimitConeOfIsLimit {f : J → C} {t : Cone (Discrete.functor f)} (ht : IsLimit t) : (Bicone.ofLimitCone ht).IsBilimit := isBilimitOfIsLimit _ <\| IsLimit.ofIsoLimit ht <\| Cones.ext (Iso.refl _) (by rintro ⟨j⟩ aesop_cat) #align category_theory.limits.bicone_is_bilimit_of_limit_cone_of_is_limit CategoryTheory.Limits.biconeIsBilimitOfLimitConeOfIsLimit /-- In a preadditive category, any finite bicone which is a colimit cocone is in fact a bilimit bicone. -/ def isBilimitOfIsColimit {f : J → C} (t : Bicone f) (ht : IsColimit t.toCocone) : t.IsBilimit := isBilimitOfTotal _ <\| ht.hom_ext fun j => by cases j simp_rw [Bicone.toCocone_ι_app, comp_sum, ← Category.assoc, t.ι_π, dite_comp] simp #align category_theory.limits.is_bilimit_of_is_colimit CategoryTheory.Limits.isBilimitOfIsColimit /-- We can turn any limit cone over a pair into a bilimit bicone. -/ def biconeIsBilimitOfColimitCoconeOfIsColimit {f : J → C} {t : Cocone (Discrete.functor f)} (ht : IsColimit t) : (Bicone.ofColimitCocone ht).IsBilimit := isBilimitOfIsColimit _ <\| IsColimit.ofIsoColimit ht <\| Cocones.ext (Iso.refl _) <\| by rintro ⟨j⟩; simp #align category_theory.limits.bicone_is_bilimit_of_colimit_cocone_of_is_colimit CategoryTheory.Limits.biconeIsBilimitOfColimitCoconeOfIsColimit end Fintype section Finite variable {J : Type} [Finite J] /-- In a preadditive category, if the product over `f : J → C` exists, then the biproduct over `f` exists. -/ theorem HasBiproduct.of_hasProduct (f : J → C) [HasProduct f] : HasBiproduct f := by cases nonempty_fintype J exact HasBiproduct.mk { bicone := _ isBilimit := biconeIsBilimitOfLimitConeOfIsLimit (limit.isLimit _) } #align category_theory.limits.has_biproduct.of_has_product CategoryTheory.Limits.HasBiproduct.of_hasProduct /-- In a preadditive category, if the coproduct over `f : J → C` exists, then the biproduct over `f` exists. -/ theorem HasBiproduct.of_hasCoproduct (f : J → C) [HasCoproduct f] : HasBiproduct f := by cases nonempty_fintype J exact HasBiproduct.mk { bicone := _ isBilimit := biconeIsBilimitOfColimitCoconeOfIsColimit (colimit.isColimit _) } #align category_theory.limits.has_biproduct.of_has_coproduct CategoryTheory.Limits.HasBiproduct.of_hasCoproduct end Finite /-- A preadditive category with finite products has finite biproducts. -/ theorem HasFiniteBiproducts.of_hasFiniteProducts [HasFiniteProducts C] : HasFiniteBiproducts C := ⟨fun _ => { has_biproduct := fun _ => HasBiproduct.of_hasProduct _ }⟩ #align category_theory.limits.has_finite_biproducts.of_has_finite_products CategoryTheory.Limits.HasFiniteBiproducts.of_hasFiniteProducts /-- A preadditive category with finite coproducts has finite biproducts. -/ theorem HasFiniteBiproducts.of_hasFiniteCoproducts [HasFiniteCoproducts C] : HasFiniteBiproducts C := ⟨fun _ => { has_biproduct := fun _ => HasBiproduct.of_hasCoproduct _ }⟩ #align category_theory.limits.has_finite_biproducts.of_has_finite_coproducts CategoryTheory.Limits.HasFiniteBiproducts.of_hasFiniteCoproducts section HasBiproduct variable {J : Type} [Fintype J] {f : J → C} [HasBiproduct f] /-- In any preadditive category, any biproduct satsifies `∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f)` -/ @[simp] theorem biproduct.total : ∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f) := IsBilimit.total (biproduct.isBilimit _) #align category_theory.limits.biproduct.total CategoryTheory.Limits.biproduct.total theorem biproduct.lift_eq {T : C} {g : ∀ j, T ⟶ f j} : biproduct.lift g = ∑ j, g j ≫ biproduct.ι f j := by ext j simp only [sum_comp, biproduct.ι_π, comp_dite, biproduct.lift_π, Category.assoc, comp_zero, Finset.sum_dite_eq', Finset.mem_univ, eqToHom_refl, Category.comp_id, if_true] #align category_theory.limits.biproduct.lift_eq CategoryTheory.Limits.biproduct.lift_eq theorem biproduct.desc_eq {T : C} {g : ∀ j, f j ⟶ T} : biproduct.desc g = ∑ j, biproduct.π f j ≫ g j := by ext j simp [comp_sum, biproduct.ι_π_assoc, dite_comp] #align category_theory.limits.biproduct.desc_eq CategoryTheory.Limits.biproduct.desc_eq @[reassoc] theorem biproduct.lift_desc {T U : C} {g : ∀ j, T ⟶ f j} {h : ∀ j, f j ⟶ U} : biproduct.lift g ≫ biproduct.desc h = ∑ j : J, g j ≫ h j := by simp [biproduct.lift_eq, biproduct.desc_eq, comp_sum, sum_comp, biproduct.ι_π_assoc, comp_dite, dite_comp] #align category_theory.limits.biproduct.lift_desc CategoryTheory.Limits.biproduct.lift_desc	Mathlib/CategoryTheory/Preadditive/Biproducts.lean	244	247	theorem biproduct.map_eq [HasFiniteBiproducts C] {f g : J → C} {h : ∀ j, f j ⟶ g j} : biproduct.map h = ∑ j : J, biproduct.π f j ≫ h j ≫ biproduct.ι g j := by	ext simp [biproduct.ι_π, biproduct.ι_π_assoc, comp_sum, sum_comp, comp_dite, dite_comp]
/- Copyright (c) 2019 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison, Yaël Dillies -/ import Mathlib.Order.Cover import Mathlib.Order.Interval.Finset.Defs #align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d" /-! # Intervals as finsets This file provides basic results about all the `Finset.Ixx`, which are defined in `Order.Interval.Finset.Defs`. In addition, it shows that in a locally finite order `≤` and `<` are the transitive closures of, respectively, `⩿` and `⋖`, which then leads to a characterization of monotone and strictly functions whose domain is a locally finite order. In particular, this file proves: * `le_iff_transGen_wcovBy`: `≤` is the transitive closure of `⩿` * `lt_iff_transGen_covBy`: `≤` is the transitive closure of `⩿` * `monotone_iff_forall_wcovBy`: Characterization of monotone functions * `strictMono_iff_forall_covBy`: Characterization of strictly monotone functions ## TODO This file was originally only about `Finset.Ico a b` where `a b : ℕ`. No care has yet been taken to generalize these lemmas properly and many lemmas about `Icc`, `Ioc`, `Ioo` are missing. In general, what's to do is taking the lemmas in `Data.X.Intervals` and abstract away the concrete structure. Complete the API. See https://github.com/leanprover-community/mathlib/pull/14448#discussion_r906109235 for some ideas. -/ assert_not_exists MonoidWithZero assert_not_exists Finset.sum open Function OrderDual open FinsetInterval variable {ι α : Type*} namespace Finset section Preorder variable [Preorder α] section LocallyFiniteOrder variable [LocallyFiniteOrder α] {a a₁ a₂ b b₁ b₂ c x : α} @[simp, aesop safe apply (rule_sets := [finsetNonempty])] theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := by rw [← coe_nonempty, coe_Icc, Set.nonempty_Icc] #align finset.nonempty_Icc Finset.nonempty_Icc @[simp, aesop safe apply (rule_sets := [finsetNonempty])] theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := by rw [← coe_nonempty, coe_Ico, Set.nonempty_Ico] #align finset.nonempty_Ico Finset.nonempty_Ico @[simp, aesop safe apply (rule_sets := [finsetNonempty])] theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := by rw [← coe_nonempty, coe_Ioc, Set.nonempty_Ioc] #align finset.nonempty_Ioc Finset.nonempty_Ioc -- TODO: This is nonsense. A locally finite order is never densely ordered @[simp] theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := by rw [← coe_nonempty, coe_Ioo, Set.nonempty_Ioo] #align finset.nonempty_Ioo Finset.nonempty_Ioo @[simp] theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by rw [← coe_eq_empty, coe_Icc, Set.Icc_eq_empty_iff] #align finset.Icc_eq_empty_iff Finset.Icc_eq_empty_iff @[simp] theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by rw [← coe_eq_empty, coe_Ico, Set.Ico_eq_empty_iff] #align finset.Ico_eq_empty_iff Finset.Ico_eq_empty_iff @[simp] theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by rw [← coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff] #align finset.Ioc_eq_empty_iff Finset.Ioc_eq_empty_iff -- TODO: This is nonsense. A locally finite order is never densely ordered @[simp] theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by rw [← coe_eq_empty, coe_Ioo, Set.Ioo_eq_empty_iff] #align finset.Ioo_eq_empty_iff Finset.Ioo_eq_empty_iff alias ⟨_, Icc_eq_empty⟩ := Icc_eq_empty_iff #align finset.Icc_eq_empty Finset.Icc_eq_empty alias ⟨_, Ico_eq_empty⟩ := Ico_eq_empty_iff #align finset.Ico_eq_empty Finset.Ico_eq_empty alias ⟨_, Ioc_eq_empty⟩ := Ioc_eq_empty_iff #align finset.Ioc_eq_empty Finset.Ioc_eq_empty @[simp] theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ hx => h ((mem_Ioo.1 hx).1.trans (mem_Ioo.1 hx).2) #align finset.Ioo_eq_empty Finset.Ioo_eq_empty @[simp] theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ := Icc_eq_empty h.not_le #align finset.Icc_eq_empty_of_lt Finset.Icc_eq_empty_of_lt @[simp] theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ := Ico_eq_empty h.not_lt #align finset.Ico_eq_empty_of_le Finset.Ico_eq_empty_of_le @[simp] theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ := Ioc_eq_empty h.not_lt #align finset.Ioc_eq_empty_of_le Finset.Ioc_eq_empty_of_le @[simp] theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ := Ioo_eq_empty h.not_lt #align finset.Ioo_eq_empty_of_le Finset.Ioo_eq_empty_of_le -- porting note (#10618): simp can prove this -- @[simp] theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp only [mem_Icc, true_and_iff, le_rfl] #align finset.left_mem_Icc Finset.left_mem_Icc -- porting note (#10618): simp can prove this -- @[simp] theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp only [mem_Ico, true_and_iff, le_refl] #align finset.left_mem_Ico Finset.left_mem_Ico -- porting note (#10618): simp can prove this -- @[simp] theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp only [mem_Icc, and_true_iff, le_rfl] #align finset.right_mem_Icc Finset.right_mem_Icc -- porting note (#10618): simp can prove this -- @[simp] theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp only [mem_Ioc, and_true_iff, le_rfl] #align finset.right_mem_Ioc Finset.right_mem_Ioc -- porting note (#10618): simp can prove this -- @[simp] theorem left_not_mem_Ioc : a ∉ Ioc a b := fun h => lt_irrefl _ (mem_Ioc.1 h).1 #align finset.left_not_mem_Ioc Finset.left_not_mem_Ioc -- porting note (#10618): simp can prove this -- @[simp] theorem left_not_mem_Ioo : a ∉ Ioo a b := fun h => lt_irrefl _ (mem_Ioo.1 h).1 #align finset.left_not_mem_Ioo Finset.left_not_mem_Ioo -- porting note (#10618): simp can prove this -- @[simp] theorem right_not_mem_Ico : b ∉ Ico a b := fun h => lt_irrefl _ (mem_Ico.1 h).2 #align finset.right_not_mem_Ico Finset.right_not_mem_Ico -- porting note (#10618): simp can prove this -- @[simp] theorem right_not_mem_Ioo : b ∉ Ioo a b := fun h => lt_irrefl _ (mem_Ioo.1 h).2 #align finset.right_not_mem_Ioo Finset.right_not_mem_Ioo theorem Icc_subset_Icc (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := by simpa [← coe_subset] using Set.Icc_subset_Icc ha hb #align finset.Icc_subset_Icc Finset.Icc_subset_Icc theorem Ico_subset_Ico (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := by simpa [← coe_subset] using Set.Ico_subset_Ico ha hb #align finset.Ico_subset_Ico Finset.Ico_subset_Ico theorem Ioc_subset_Ioc (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := by simpa [← coe_subset] using Set.Ioc_subset_Ioc ha hb #align finset.Ioc_subset_Ioc Finset.Ioc_subset_Ioc theorem Ioo_subset_Ioo (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := by simpa [← coe_subset] using Set.Ioo_subset_Ioo ha hb #align finset.Ioo_subset_Ioo Finset.Ioo_subset_Ioo theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b := Icc_subset_Icc h le_rfl #align finset.Icc_subset_Icc_left Finset.Icc_subset_Icc_left theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b := Ico_subset_Ico h le_rfl #align finset.Ico_subset_Ico_left Finset.Ico_subset_Ico_left theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b := Ioc_subset_Ioc h le_rfl #align finset.Ioc_subset_Ioc_left Finset.Ioc_subset_Ioc_left theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b := Ioo_subset_Ioo h le_rfl #align finset.Ioo_subset_Ioo_left Finset.Ioo_subset_Ioo_left theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ := Icc_subset_Icc le_rfl h #align finset.Icc_subset_Icc_right Finset.Icc_subset_Icc_right theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ := Ico_subset_Ico le_rfl h #align finset.Ico_subset_Ico_right Finset.Ico_subset_Ico_right theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ := Ioc_subset_Ioc le_rfl h #align finset.Ioc_subset_Ioc_right Finset.Ioc_subset_Ioc_right theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ := Ioo_subset_Ioo le_rfl h #align finset.Ioo_subset_Ioo_right Finset.Ioo_subset_Ioo_right theorem Ico_subset_Ioo_left (h : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := by rw [← coe_subset, coe_Ico, coe_Ioo] exact Set.Ico_subset_Ioo_left h #align finset.Ico_subset_Ioo_left Finset.Ico_subset_Ioo_left theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := by rw [← coe_subset, coe_Ioc, coe_Ioo] exact Set.Ioc_subset_Ioo_right h #align finset.Ioc_subset_Ioo_right Finset.Ioc_subset_Ioo_right theorem Icc_subset_Ico_right (h : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := by rw [← coe_subset, coe_Icc, coe_Ico] exact Set.Icc_subset_Ico_right h #align finset.Icc_subset_Ico_right Finset.Icc_subset_Ico_right theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := by rw [← coe_subset, coe_Ioo, coe_Ico] exact Set.Ioo_subset_Ico_self #align finset.Ioo_subset_Ico_self Finset.Ioo_subset_Ico_self theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := by rw [← coe_subset, coe_Ioo, coe_Ioc] exact Set.Ioo_subset_Ioc_self #align finset.Ioo_subset_Ioc_self Finset.Ioo_subset_Ioc_self theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := by rw [← coe_subset, coe_Ico, coe_Icc] exact Set.Ico_subset_Icc_self #align finset.Ico_subset_Icc_self Finset.Ico_subset_Icc_self theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := by rw [← coe_subset, coe_Ioc, coe_Icc] exact Set.Ioc_subset_Icc_self #align finset.Ioc_subset_Icc_self Finset.Ioc_subset_Icc_self theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b := Ioo_subset_Ico_self.trans Ico_subset_Icc_self #align finset.Ioo_subset_Icc_self Finset.Ioo_subset_Icc_self theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := by rw [← coe_subset, coe_Icc, coe_Icc, Set.Icc_subset_Icc_iff h₁] #align finset.Icc_subset_Icc_iff Finset.Icc_subset_Icc_iff theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := by rw [← coe_subset, coe_Icc, coe_Ioo, Set.Icc_subset_Ioo_iff h₁] #align finset.Icc_subset_Ioo_iff Finset.Icc_subset_Ioo_iff theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ := by rw [← coe_subset, coe_Icc, coe_Ico, Set.Icc_subset_Ico_iff h₁] #align finset.Icc_subset_Ico_iff Finset.Icc_subset_Ico_iff theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ := (Icc_subset_Ico_iff h₁.dual).trans and_comm #align finset.Icc_subset_Ioc_iff Finset.Icc_subset_Ioc_iff --TODO: `Ico_subset_Ioo_iff`, `Ioc_subset_Ioo_iff` theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := by rw [← coe_ssubset, coe_Icc, coe_Icc] exact Set.Icc_ssubset_Icc_left hI ha hb #align finset.Icc_ssubset_Icc_left Finset.Icc_ssubset_Icc_left theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := by rw [← coe_ssubset, coe_Icc, coe_Icc] exact Set.Icc_ssubset_Icc_right hI ha hb #align finset.Icc_ssubset_Icc_right Finset.Icc_ssubset_Icc_right variable (a) -- porting note (#10618): simp can prove this -- @[simp] theorem Ico_self : Ico a a = ∅ := Ico_eq_empty <\| lt_irrefl _ #align finset.Ico_self Finset.Ico_self -- porting note (#10618): simp can prove this -- @[simp] theorem Ioc_self : Ioc a a = ∅ := Ioc_eq_empty <\| lt_irrefl _ #align finset.Ioc_self Finset.Ioc_self -- porting note (#10618): simp can prove this -- @[simp] theorem Ioo_self : Ioo a a = ∅ := Ioo_eq_empty <\| lt_irrefl _ #align finset.Ioo_self Finset.Ioo_self variable {a} /-- A set with upper and lower bounds in a locally finite order is a fintype -/ def _root_.Set.fintypeOfMemBounds {s : Set α} [DecidablePred (· ∈ s)] (ha : a ∈ lowerBounds s) (hb : b ∈ upperBounds s) : Fintype s := Set.fintypeSubset (Set.Icc a b) fun _ hx => ⟨ha hx, hb hx⟩ #align set.fintype_of_mem_bounds Set.fintypeOfMemBounds section Filter theorem Ico_filter_lt_of_le_left [DecidablePred (· < c)] (hca : c ≤ a) : (Ico a b).filter (· < c) = ∅ := filter_false_of_mem fun _ hx => (hca.trans (mem_Ico.1 hx).1).not_lt #align finset.Ico_filter_lt_of_le_left Finset.Ico_filter_lt_of_le_left theorem Ico_filter_lt_of_right_le [DecidablePred (· < c)] (hbc : b ≤ c) : (Ico a b).filter (· < c) = Ico a b := filter_true_of_mem fun _ hx => (mem_Ico.1 hx).2.trans_le hbc #align finset.Ico_filter_lt_of_right_le Finset.Ico_filter_lt_of_right_le theorem Ico_filter_lt_of_le_right [DecidablePred (· < c)] (hcb : c ≤ b) : (Ico a b).filter (· < c) = Ico a c := by ext x rw [mem_filter, mem_Ico, mem_Ico, and_right_comm] exact and_iff_left_of_imp fun h => h.2.trans_le hcb #align finset.Ico_filter_lt_of_le_right Finset.Ico_filter_lt_of_le_right theorem Ico_filter_le_of_le_left {a b c : α} [DecidablePred (c ≤ ·)] (hca : c ≤ a) : (Ico a b).filter (c ≤ ·) = Ico a b := filter_true_of_mem fun _ hx => hca.trans (mem_Ico.1 hx).1 #align finset.Ico_filter_le_of_le_left Finset.Ico_filter_le_of_le_left theorem Ico_filter_le_of_right_le {a b : α} [DecidablePred (b ≤ ·)] : (Ico a b).filter (b ≤ ·) = ∅ := filter_false_of_mem fun _ hx => (mem_Ico.1 hx).2.not_le #align finset.Ico_filter_le_of_right_le Finset.Ico_filter_le_of_right_le theorem Ico_filter_le_of_left_le {a b c : α} [DecidablePred (c ≤ ·)] (hac : a ≤ c) : (Ico a b).filter (c ≤ ·) = Ico c b := by ext x rw [mem_filter, mem_Ico, mem_Ico, and_comm, and_left_comm] exact and_iff_right_of_imp fun h => hac.trans h.1 #align finset.Ico_filter_le_of_left_le Finset.Ico_filter_le_of_left_le theorem Icc_filter_lt_of_lt_right {a b c : α} [DecidablePred (· < c)] (h : b < c) : (Icc a b).filter (· < c) = Icc a b := filter_true_of_mem fun _ hx => lt_of_le_of_lt (mem_Icc.1 hx).2 h #align finset.Icc_filter_lt_of_lt_right Finset.Icc_filter_lt_of_lt_right theorem Ioc_filter_lt_of_lt_right {a b c : α} [DecidablePred (· < c)] (h : b < c) : (Ioc a b).filter (· < c) = Ioc a b := filter_true_of_mem fun _ hx => lt_of_le_of_lt (mem_Ioc.1 hx).2 h #align finset.Ioc_filter_lt_of_lt_right Finset.Ioc_filter_lt_of_lt_right theorem Iic_filter_lt_of_lt_right {α} [Preorder α] [LocallyFiniteOrderBot α] {a c : α} [DecidablePred (· < c)] (h : a < c) : (Iic a).filter (· < c) = Iic a := filter_true_of_mem fun _ hx => lt_of_le_of_lt (mem_Iic.1 hx) h #align finset.Iic_filter_lt_of_lt_right Finset.Iic_filter_lt_of_lt_right variable (a b) [Fintype α] theorem filter_lt_lt_eq_Ioo [DecidablePred fun j => a < j ∧ j < b] : (univ.filter fun j => a < j ∧ j < b) = Ioo a b := by ext simp #align finset.filter_lt_lt_eq_Ioo Finset.filter_lt_lt_eq_Ioo theorem filter_lt_le_eq_Ioc [DecidablePred fun j => a < j ∧ j ≤ b] : (univ.filter fun j => a < j ∧ j ≤ b) = Ioc a b := by ext simp #align finset.filter_lt_le_eq_Ioc Finset.filter_lt_le_eq_Ioc theorem filter_le_lt_eq_Ico [DecidablePred fun j => a ≤ j ∧ j < b] : (univ.filter fun j => a ≤ j ∧ j < b) = Ico a b := by ext simp #align finset.filter_le_lt_eq_Ico Finset.filter_le_lt_eq_Ico theorem filter_le_le_eq_Icc [DecidablePred fun j => a ≤ j ∧ j ≤ b] : (univ.filter fun j => a ≤ j ∧ j ≤ b) = Icc a b := by ext simp #align finset.filter_le_le_eq_Icc Finset.filter_le_le_eq_Icc end Filter section LocallyFiniteOrderTop variable [LocallyFiniteOrderTop α] @[simp, aesop safe apply (rule_sets := [finsetNonempty])] lemma nonempty_Ici : (Ici a).Nonempty := ⟨a, mem_Ici.2 le_rfl⟩ @[simp, aesop safe apply (rule_sets := [finsetNonempty])] lemma nonempty_Ioi : (Ioi a).Nonempty ↔ ¬ IsMax a := by simp [Finset.Nonempty]	Mathlib/Order/Interval/Finset/Basic.lean	404	405	theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := by	simpa [← coe_subset] using Set.Icc_subset_Ici_self
/- Copyright (c) 2021 Shing Tak Lam. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Shing Tak Lam -/ import Mathlib.Topology.Order.ProjIcc import Mathlib.Topology.ContinuousFunction.Ordered import Mathlib.Topology.CompactOpen import Mathlib.Topology.UnitInterval #align_import topology.homotopy.basic from "leanprover-community/mathlib"@"11c53f174270aa43140c0b26dabce5fc4a253e80" /-! # Homotopy between functions In this file, we define a homotopy between two functions `f₀` and `f₁`. First we define `ContinuousMap.Homotopy` between the two functions, with no restrictions on the intermediate maps. Then, as in the formalisation in HOL-Analysis, we define `ContinuousMap.HomotopyWith f₀ f₁ P`, for homotopies between `f₀` and `f₁`, where the intermediate maps satisfy the predicate `P`. Finally, we define `ContinuousMap.HomotopyRel f₀ f₁ S`, for homotopies between `f₀` and `f₁` which are fixed on `S`. ## Definitions * `ContinuousMap.Homotopy f₀ f₁` is the type of homotopies between `f₀` and `f₁`. * `ContinuousMap.HomotopyWith f₀ f₁ P` is the type of homotopies between `f₀` and `f₁`, where the intermediate maps satisfy the predicate `P`. * `ContinuousMap.HomotopyRel f₀ f₁ S` is the type of homotopies between `f₀` and `f₁` which are fixed on `S`. For each of the above, we have * `refl f`, which is the constant homotopy from `f` to `f`. * `symm F`, which reverses the homotopy `F`. For example, if `F : ContinuousMap.Homotopy f₀ f₁`, then `F.symm : ContinuousMap.Homotopy f₁ f₀`. * `trans F G`, which concatenates the homotopies `F` and `G`. For example, if `F : ContinuousMap.Homotopy f₀ f₁` and `G : ContinuousMap.Homotopy f₁ f₂`, then `F.trans G : ContinuousMap.Homotopy f₀ f₂`. We also define the relations * `ContinuousMap.Homotopic f₀ f₁` is defined to be `Nonempty (ContinuousMap.Homotopy f₀ f₁)` * `ContinuousMap.HomotopicWith f₀ f₁ P` is defined to be `Nonempty (ContinuousMap.HomotopyWith f₀ f₁ P)` * `ContinuousMap.HomotopicRel f₀ f₁ P` is defined to be `Nonempty (ContinuousMap.HomotopyRel f₀ f₁ P)` and for `ContinuousMap.homotopic` and `ContinuousMap.homotopic_rel`, we also define the `setoid` and `quotient` in `C(X, Y)` by these relations. ## References - [HOL-Analysis formalisation](https://isabelle.in.tum.de/library/HOL/HOL-Analysis/Homotopy.html) -/ noncomputable section universe u v w x variable {F : Type} {X : Type u} {Y : Type v} {Z : Type w} {Z' : Type x} {ι : Type} variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace Z'] open unitInterval namespace ContinuousMap /-- `ContinuousMap.Homotopy f₀ f₁` is the type of homotopies from `f₀` to `f₁`. When possible, instead of parametrizing results over `(f : Homotopy f₀ f₁)`, you should parametrize over `{F : Type} [HomotopyLike F f₀ f₁] (f : F)`. When you extend this structure, make sure to extend `ContinuousMap.HomotopyLike`. -/ structure Homotopy (f₀ f₁ : C(X, Y)) extends C(I × X, Y) where /-- value of the homotopy at 0 -/ map_zero_left : ∀ x, toFun (0, x) = f₀ x /-- value of the homotopy at 1 -/ map_one_left : ∀ x, toFun (1, x) = f₁ x #align continuous_map.homotopy ContinuousMap.Homotopy section /-- `ContinuousMap.HomotopyLike F f₀ f₁` states that `F` is a type of homotopies between `f₀` and `f₁`. You should extend this class when you extend `ContinuousMap.Homotopy`. -/ class HomotopyLike {X Y : outParam Type} [TopologicalSpace X] [TopologicalSpace Y] (F : Type*) (f₀ f₁ : outParam <\| C(X, Y)) [FunLike F (I × X) Y] extends ContinuousMapClass F (I × X) Y : Prop where /-- value of the homotopy at 0 -/ map_zero_left (f : F) : ∀ x, f (0, x) = f₀ x /-- value of the homotopy at 1 -/ map_one_left (f : F) : ∀ x, f (1, x) = f₁ x #align continuous_map.homotopy_like ContinuousMap.HomotopyLike end namespace Homotopy section variable {f₀ f₁ : C(X, Y)} instance instFunLike : FunLike (Homotopy f₀ f₁) (I × X) Y where coe f := f.toFun coe_injective' f g h := by obtain ⟨⟨_, _⟩, _⟩ := f obtain ⟨⟨_, _⟩, _⟩ := g congr instance : HomotopyLike (Homotopy f₀ f₁) f₀ f₁ where map_continuous f := f.continuous_toFun map_zero_left f := f.map_zero_left map_one_left f := f.map_one_left @[ext] theorem ext {F G : Homotopy f₀ f₁} (h : ∀ x, F x = G x) : F = G := DFunLike.ext _ _ h #align continuous_map.homotopy.ext ContinuousMap.Homotopy.ext /-- See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections. -/ def Simps.apply (F : Homotopy f₀ f₁) : I × X → Y := F #align continuous_map.homotopy.simps.apply ContinuousMap.Homotopy.Simps.apply initialize_simps_projections Homotopy (toFun → apply, -toContinuousMap) /-- Deprecated. Use `map_continuous` instead. -/ protected theorem continuous (F : Homotopy f₀ f₁) : Continuous F := F.continuous_toFun #align continuous_map.homotopy.continuous ContinuousMap.Homotopy.continuous @[simp] theorem apply_zero (F : Homotopy f₀ f₁) (x : X) : F (0, x) = f₀ x := F.map_zero_left x #align continuous_map.homotopy.apply_zero ContinuousMap.Homotopy.apply_zero @[simp] theorem apply_one (F : Homotopy f₀ f₁) (x : X) : F (1, x) = f₁ x := F.map_one_left x #align continuous_map.homotopy.apply_one ContinuousMap.Homotopy.apply_one @[simp] theorem coe_toContinuousMap (F : Homotopy f₀ f₁) : ⇑F.toContinuousMap = F := rfl #align continuous_map.homotopy.coe_to_continuous_map ContinuousMap.Homotopy.coe_toContinuousMap /-- Currying a homotopy to a continuous function from `I` to `C(X, Y)`. -/ def curry (F : Homotopy f₀ f₁) : C(I, C(X, Y)) := F.toContinuousMap.curry #align continuous_map.homotopy.curry ContinuousMap.Homotopy.curry @[simp] theorem curry_apply (F : Homotopy f₀ f₁) (t : I) (x : X) : F.curry t x = F (t, x) := rfl #align continuous_map.homotopy.curry_apply ContinuousMap.Homotopy.curry_apply /-- Continuously extending a curried homotopy to a function from `ℝ` to `C(X, Y)`. -/ def extend (F : Homotopy f₀ f₁) : C(ℝ, C(X, Y)) := F.curry.IccExtend zero_le_one #align continuous_map.homotopy.extend ContinuousMap.Homotopy.extend theorem extend_apply_of_le_zero (F : Homotopy f₀ f₁) {t : ℝ} (ht : t ≤ 0) (x : X) : F.extend t x = f₀ x := by rw [← F.apply_zero] exact ContinuousMap.congr_fun (Set.IccExtend_of_le_left (zero_le_one' ℝ) F.curry ht) x #align continuous_map.homotopy.extend_apply_of_le_zero ContinuousMap.Homotopy.extend_apply_of_le_zero theorem extend_apply_of_one_le (F : Homotopy f₀ f₁) {t : ℝ} (ht : 1 ≤ t) (x : X) : F.extend t x = f₁ x := by rw [← F.apply_one] exact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' ℝ) F.curry ht) x #align continuous_map.homotopy.extend_apply_of_one_le ContinuousMap.Homotopy.extend_apply_of_one_le @[simp] theorem extend_apply_coe (F : Homotopy f₀ f₁) (t : I) (x : X) : F.extend t x = F (t, x) := ContinuousMap.congr_fun (Set.IccExtend_val (zero_le_one' ℝ) F.curry t) x #align continuous_map.homotopy.extend_apply_coe ContinuousMap.Homotopy.extend_apply_coe @[simp] theorem extend_apply_of_mem_I (F : Homotopy f₀ f₁) {t : ℝ} (ht : t ∈ I) (x : X) : F.extend t x = F (⟨t, ht⟩, x) := ContinuousMap.congr_fun (Set.IccExtend_of_mem (zero_le_one' ℝ) F.curry ht) x set_option linter.uppercaseLean3 false in #align continuous_map.homotopy.extend_apply_of_mem_I ContinuousMap.Homotopy.extend_apply_of_mem_I theorem congr_fun {F G : Homotopy f₀ f₁} (h : F = G) (x : I × X) : F x = G x := ContinuousMap.congr_fun (congr_arg _ h) x #align continuous_map.homotopy.congr_fun ContinuousMap.Homotopy.congr_fun theorem congr_arg (F : Homotopy f₀ f₁) {x y : I × X} (h : x = y) : F x = F y := F.toContinuousMap.congr_arg h #align continuous_map.homotopy.congr_arg ContinuousMap.Homotopy.congr_arg end /-- Given a continuous function `f`, we can define a `Homotopy f f` by `F (t, x) = f x` -/ @[simps] def refl (f : C(X, Y)) : Homotopy f f where toFun x := f x.2 map_zero_left _ := rfl map_one_left _ := rfl #align continuous_map.homotopy.refl ContinuousMap.Homotopy.refl instance : Inhabited (Homotopy (ContinuousMap.id X) (ContinuousMap.id X)) := ⟨Homotopy.refl _⟩ /-- Given a `Homotopy f₀ f₁`, we can define a `Homotopy f₁ f₀` by reversing the homotopy. -/ @[simps] def symm {f₀ f₁ : C(X, Y)} (F : Homotopy f₀ f₁) : Homotopy f₁ f₀ where toFun x := F (σ x.1, x.2) map_zero_left := by norm_num map_one_left := by norm_num #align continuous_map.homotopy.symm ContinuousMap.Homotopy.symm @[simp]	Mathlib/Topology/Homotopy/Basic.lean	222	224	theorem symm_symm {f₀ f₁ : C(X, Y)} (F : Homotopy f₀ f₁) : F.symm.symm = F := by	ext simp
/- Copyright (c) 2020 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.ContDiff.Basic import Mathlib.Analysis.NormedSpace.FiniteDimension #align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # Infinitely smooth "bump" functions A smooth bump function is an infinitely smooth function `f : E → ℝ` supported on a ball that is equal to `1` on a ball of smaller radius. These functions have many uses in real analysis. E.g., - they can be used to construct a smooth partition of unity which is a very useful tool; - they can be used to approximate a continuous function by infinitely smooth functions. There are two classes of spaces where bump functions are guaranteed to exist: inner product spaces and finite dimensional spaces. In this file we define a typeclass `HasContDiffBump` saying that a normed space has a family of smooth bump functions with certain properties. We also define a structure `ContDiffBump` that holds the center and radii of the balls from above. An element `f : ContDiffBump c` can be coerced to a function which is an infinitely smooth function such that - `f` is equal to `1` in `Metric.closedBall c f.rIn`; - `support f = Metric.ball c f.rOut`; - `0 ≤ f x ≤ 1` for all `x`. ## Main Definitions - `ContDiffBump (c : E)`: a structure holding data needed to construct an infinitely smooth bump function. - `ContDiffBumpBase (E : Type)`: a family of infinitely smooth bump functions that can be used to construct coercion of a `ContDiffBump (c : E)` to a function. - `HasContDiffBump (E : Type)`: a typeclass saying that `E` has a `ContDiffBumpBase`. Two instances of this typeclass (for inner product spaces and for finite dimensional spaces) are provided elsewhere. ## Keywords smooth function, smooth bump function -/ noncomputable section open Function Set Filter open scoped Topology Filter variable {E X : Type} /-- `f : ContDiffBump c`, where `c` is a point in a normed vector space, is a bundled smooth function such that - `f` is equal to `1` in `Metric.closedBall c f.rIn`; - `support f = Metric.ball c f.rOut`; - `0 ≤ f x ≤ 1` for all `x`. The structure `ContDiffBump` contains the data required to construct the function: real numbers `rIn`, `rOut`, and proofs of `0 < rIn < rOut`. The function itself is available through `CoeFun` when the space is nice enough, i.e., satisfies the `HasContDiffBump` typeclass. -/ structure ContDiffBump (c : E) where /-- real numbers `0 < rIn < rOut` -/ (rIn rOut : ℝ) rIn_pos : 0 < rIn rIn_lt_rOut : rIn < rOut #align cont_diff_bump ContDiffBump #align cont_diff_bump.r ContDiffBump.rIn set_option linter.uppercaseLean3 false in #align cont_diff_bump.R ContDiffBump.rOut #align cont_diff_bump.r_pos ContDiffBump.rIn_pos set_option linter.uppercaseLean3 false in #align cont_diff_bump.r_lt_R ContDiffBump.rIn_lt_rOut /-- The base function from which one will construct a family of bump functions. One could add more properties if they are useful and satisfied in the examples of inner product spaces and finite dimensional vector spaces, notably derivative norm control in terms of `R - 1`. TODO: do we ever need `f x = 1 ↔ ‖x‖ ≤ 1`? -/ -- Porting note(#5171): linter not yet ported; was @[nolint has_nonempty_instance] structure ContDiffBumpBase (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] where /-- The function underlying this family of bump functions -/ toFun : ℝ → E → ℝ mem_Icc : ∀ (R : ℝ) (x : E), toFun R x ∈ Icc (0 : ℝ) 1 symmetric : ∀ (R : ℝ) (x : E), toFun R (-x) = toFun R x smooth : ContDiffOn ℝ ⊤ (uncurry toFun) (Ioi (1 : ℝ) ×ˢ (univ : Set E)) eq_one : ∀ R : ℝ, 1 < R → ∀ x : E, ‖x‖ ≤ 1 → toFun R x = 1 support : ∀ R : ℝ, 1 < R → Function.support (toFun R) = Metric.ball (0 : E) R #align cont_diff_bump_base ContDiffBumpBase /-- A class registering that a real vector space admits bump functions. This will be instantiated first for inner product spaces, and then for finite-dimensional normed spaces. We use a specific class instead of `Nonempty (ContDiffBumpBase E)` for performance reasons. -/ class HasContDiffBump (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop where out : Nonempty (ContDiffBumpBase E) #align has_cont_diff_bump HasContDiffBump /-- In a space with `C^∞` bump functions, register some function that will be used as a basis to construct bump functions of arbitrary size around any point. -/ def someContDiffBumpBase (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [hb : HasContDiffBump E] : ContDiffBumpBase E := Nonempty.some hb.out #align some_cont_diff_bump_base someContDiffBumpBase namespace ContDiffBump theorem rOut_pos {c : E} (f : ContDiffBump c) : 0 < f.rOut := f.rIn_pos.trans f.rIn_lt_rOut set_option linter.uppercaseLean3 false in #align cont_diff_bump.R_pos ContDiffBump.rOut_pos theorem one_lt_rOut_div_rIn {c : E} (f : ContDiffBump c) : 1 < f.rOut / f.rIn := by rw [one_lt_div f.rIn_pos] exact f.rIn_lt_rOut set_option linter.uppercaseLean3 false in #align cont_diff_bump.one_lt_R_div_r ContDiffBump.one_lt_rOut_div_rIn instance (c : E) : Inhabited (ContDiffBump c) := ⟨⟨1, 2, zero_lt_one, one_lt_two⟩⟩ variable [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup X] [NormedSpace ℝ X] [HasContDiffBump E] {c : E} (f : ContDiffBump c) {x : E} {n : ℕ∞} /-- The function defined by `f : ContDiffBump c`. Use automatic coercion to function instead. -/ @[coe] def toFun {c : E} (f : ContDiffBump c) : E → ℝ := (someContDiffBumpBase E).toFun (f.rOut / f.rIn) ∘ fun x ↦ (f.rIn⁻¹ • (x - c)) #align cont_diff_bump.to_fun ContDiffBump.toFun instance : CoeFun (ContDiffBump c) fun _ => E → ℝ := ⟨toFun⟩ protected theorem apply (x : E) : f x = (someContDiffBumpBase E).toFun (f.rOut / f.rIn) (f.rIn⁻¹ • (x - c)) := rfl #align cont_diff_bump.def ContDiffBump.apply protected theorem sub (x : E) : f (c - x) = f (c + x) := by simp [f.apply, ContDiffBumpBase.symmetric] #align cont_diff_bump.sub ContDiffBump.sub protected theorem neg (f : ContDiffBump (0 : E)) (x : E) : f (-x) = f x := by simp_rw [← zero_sub, f.sub, zero_add] #align cont_diff_bump.neg ContDiffBump.neg open Metric theorem one_of_mem_closedBall (hx : x ∈ closedBall c f.rIn) : f x = 1 := by apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn simpa only [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg f.rIn_pos.le, ← div_eq_inv_mul, div_le_one f.rIn_pos] using mem_closedBall_iff_norm.1 hx #align cont_diff_bump.one_of_mem_closed_ball ContDiffBump.one_of_mem_closedBall theorem nonneg : 0 ≤ f x := (ContDiffBumpBase.mem_Icc (someContDiffBumpBase E) _ _).1 #align cont_diff_bump.nonneg ContDiffBump.nonneg /-- A version of `ContDiffBump.nonneg` with `x` explicit -/ theorem nonneg' (x : E) : 0 ≤ f x := f.nonneg #align cont_diff_bump.nonneg' ContDiffBump.nonneg' theorem le_one : f x ≤ 1 := (ContDiffBumpBase.mem_Icc (someContDiffBumpBase E) _ _).2 #align cont_diff_bump.le_one ContDiffBump.le_one	Mathlib/Analysis/Calculus/BumpFunction/Basic.lean	172	176	theorem support_eq : Function.support f = Metric.ball c f.rOut := by	simp only [toFun, support_comp_eq_preimage, ContDiffBumpBase.support _ _ f.one_lt_rOut_div_rIn] ext x simp only [mem_ball_iff_norm, sub_zero, norm_smul, mem_preimage, Real.norm_eq_abs, abs_inv, abs_of_pos f.rIn_pos, ← div_eq_inv_mul, div_lt_div_right f.rIn_pos]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import Mathlib.Data.Set.Finite import Mathlib.Data.Countable.Basic import Mathlib.Logic.Equiv.List import Mathlib.Data.Set.Subsingleton #align_import data.set.countable from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58" /-! # Countable sets In this file we define `Set.Countable s` as `Countable s` and prove basic properties of this definition. Note that this definition does not provide a computable encoding. For a noncomputable conversion to `Encodable s`, use `Set.Countable.nonempty_encodable`. ## Keywords sets, countable set -/ noncomputable section open scoped Classical open Function Set Encodable universe u v w x variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x} namespace Set /-- A set `s` is countable if the corresponding subtype is countable, i.e., there exists an injective map `f : s → ℕ`. Note that this is an abbreviation, so `hs : Set.Countable s` in the proof context is the same as an instance `Countable s`. For a constructive version, see `Encodable`. -/ protected def Countable (s : Set α) : Prop := Countable s #align set.countable Set.Countable @[simp] theorem countable_coe_iff {s : Set α} : Countable s ↔ s.Countable := .rfl #align set.countable_coe_iff Set.countable_coe_iff /-- Prove `Set.Countable` from a `Countable` instance on the subtype. -/ theorem to_countable (s : Set α) [Countable s] : s.Countable := ‹_› #align set.to_countable Set.to_countable /-- Restate `Set.Countable` as a `Countable` instance. -/ alias ⟨_root_.Countable.to_set, Countable.to_subtype⟩ := countable_coe_iff #align countable.to_set Countable.to_set #align set.countable.to_subtype Set.Countable.to_subtype protected theorem countable_iff_exists_injective {s : Set α} : s.Countable ↔ ∃ f : s → ℕ, Injective f := countable_iff_exists_injective s #align set.countable_iff_exists_injective Set.countable_iff_exists_injective /-- A set `s : Set α` is countable if and only if there exists a function `α → ℕ` injective on `s`. -/ theorem countable_iff_exists_injOn {s : Set α} : s.Countable ↔ ∃ f : α → ℕ, InjOn f s := Set.countable_iff_exists_injective.trans exists_injOn_iff_injective.symm #align set.countable_iff_exists_inj_on Set.countable_iff_exists_injOn theorem countable_iff_nonempty_encodable {s : Set α} : s.Countable ↔ Nonempty (Encodable s) := Encodable.nonempty_encodable.symm alias ⟨Countable.nonempty_encodable, _⟩ := countable_iff_nonempty_encodable /-- Convert `Set.Countable s` to `Encodable s` (noncomputable). -/ protected def Countable.toEncodable {s : Set α} (hs : s.Countable) : Encodable s := Classical.choice hs.nonempty_encodable #align set.countable.to_encodable Set.Countable.toEncodable section Enumerate /-- Noncomputably enumerate elements in a set. The `default` value is used to extend the domain to all of `ℕ`. -/ def enumerateCountable {s : Set α} (h : s.Countable) (default : α) : ℕ → α := fun n => match @Encodable.decode s h.toEncodable n with \| some y => y \| none => default #align set.enumerate_countable Set.enumerateCountable theorem subset_range_enumerate {s : Set α} (h : s.Countable) (default : α) : s ⊆ range (enumerateCountable h default) := fun x hx => ⟨@Encodable.encode s h.toEncodable ⟨x, hx⟩, by letI := h.toEncodable simp [enumerateCountable, Encodable.encodek]⟩ #align set.subset_range_enumerate Set.subset_range_enumerate lemma range_enumerateCountable_subset {s : Set α} (h : s.Countable) (default : α) : range (enumerateCountable h default) ⊆ insert default s := by refine range_subset_iff.mpr (fun n ↦ ?_) rw [enumerateCountable] match @decode s (Countable.toEncodable h) n with \| none => exact mem_insert _ _ \| some val => simp lemma range_enumerateCountable_of_mem {s : Set α} (h : s.Countable) {default : α} (h_mem : default ∈ s) : range (enumerateCountable h default) = s := subset_antisymm ((range_enumerateCountable_subset h _).trans_eq (insert_eq_of_mem h_mem)) (subset_range_enumerate h default) lemma enumerateCountable_mem {s : Set α} (h : s.Countable) {default : α} (h_mem : default ∈ s) (n : ℕ) : enumerateCountable h default n ∈ s := by conv_rhs => rw [← range_enumerateCountable_of_mem h h_mem] exact mem_range_self n end Enumerate theorem Countable.mono {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (hs : s₂.Countable) : s₁.Countable := have := hs.to_subtype; (inclusion_injective h).countable #align set.countable.mono Set.Countable.mono theorem countable_range [Countable ι] (f : ι → β) : (range f).Countable := surjective_onto_range.countable.to_set #align set.countable_range Set.countable_range theorem countable_iff_exists_subset_range [Nonempty α] {s : Set α} : s.Countable ↔ ∃ f : ℕ → α, s ⊆ range f := ⟨fun h => by inhabit α exact ⟨enumerateCountable h default, subset_range_enumerate _ _⟩, fun ⟨f, hsf⟩ => (countable_range f).mono hsf⟩ #align set.countable_iff_exists_subset_range Set.countable_iff_exists_subset_range /-- A non-empty set is countable iff there exists a surjection from the natural numbers onto the subtype induced by the set. -/ protected theorem countable_iff_exists_surjective {s : Set α} (hs : s.Nonempty) : s.Countable ↔ ∃ f : ℕ → s, Surjective f := @countable_iff_exists_surjective s hs.to_subtype #align set.countable_iff_exists_surjective Set.countable_iff_exists_surjective alias ⟨Countable.exists_surjective, _⟩ := Set.countable_iff_exists_surjective #align set.countable.exists_surjective Set.Countable.exists_surjective theorem countable_univ [Countable α] : (univ : Set α).Countable := to_countable univ #align set.countable_univ Set.countable_univ theorem countable_univ_iff : (univ : Set α).Countable ↔ Countable α := countable_coe_iff.symm.trans (Equiv.Set.univ _).countable_iff /-- If `s : Set α` is a nonempty countable set, then there exists a map `f : ℕ → α` such that `s = range f`. -/ theorem Countable.exists_eq_range {s : Set α} (hc : s.Countable) (hs : s.Nonempty) : ∃ f : ℕ → α, s = range f := by rcases hc.exists_surjective hs with ⟨f, hf⟩ refine ⟨(↑) ∘ f, ?_⟩ rw [hf.range_comp, Subtype.range_coe] #align set.countable.exists_eq_range Set.Countable.exists_eq_range @[simp] theorem countable_empty : (∅ : Set α).Countable := to_countable _ #align set.countable_empty Set.countable_empty @[simp] theorem countable_singleton (a : α) : ({a} : Set α).Countable := to_countable _ #align set.countable_singleton Set.countable_singleton theorem Countable.image {s : Set α} (hs : s.Countable) (f : α → β) : (f '' s).Countable := by rw [image_eq_range] have := hs.to_subtype apply countable_range #align set.countable.image Set.Countable.image theorem MapsTo.countable_of_injOn {s : Set α} {t : Set β} {f : α → β} (hf : MapsTo f s t) (hf' : InjOn f s) (ht : t.Countable) : s.Countable := have := ht.to_subtype have : Injective (hf.restrict f s t) := (injOn_iff_injective.1 hf').codRestrict _ this.countable #align set.maps_to.countable_of_inj_on Set.MapsTo.countable_of_injOn theorem Countable.preimage_of_injOn {s : Set β} (hs : s.Countable) {f : α → β} (hf : InjOn f (f ⁻¹' s)) : (f ⁻¹' s).Countable := (mapsTo_preimage f s).countable_of_injOn hf hs #align set.countable.preimage_of_inj_on Set.Countable.preimage_of_injOn protected theorem Countable.preimage {s : Set β} (hs : s.Countable) {f : α → β} (hf : Injective f) : (f ⁻¹' s).Countable := hs.preimage_of_injOn hf.injOn #align set.countable.preimage Set.Countable.preimage theorem exists_seq_iSup_eq_top_iff_countable [CompleteLattice α] {p : α → Prop} (h : ∃ x, p x) : (∃ s : ℕ → α, (∀ n, p (s n)) ∧ ⨆ n, s n = ⊤) ↔ ∃ S : Set α, S.Countable ∧ (∀ s ∈ S, p s) ∧ sSup S = ⊤ := by constructor · rintro ⟨s, hps, hs⟩ refine ⟨range s, countable_range s, forall_mem_range.2 hps, ?_⟩ rwa [sSup_range] · rintro ⟨S, hSc, hps, hS⟩ rcases eq_empty_or_nonempty S with (rfl \| hne) · rw [sSup_empty] at hS haveI := subsingleton_of_bot_eq_top hS rcases h with ⟨x, hx⟩ exact ⟨fun _ => x, fun _ => hx, Subsingleton.elim _ _⟩ · rcases (Set.countable_iff_exists_surjective hne).1 hSc with ⟨s, hs⟩ refine ⟨fun n => s n, fun n => hps _ (s n).coe_prop, ?_⟩ rwa [hs.iSup_comp, ← sSup_eq_iSup'] #align set.exists_seq_supr_eq_top_iff_countable Set.exists_seq_iSup_eq_top_iff_countable theorem exists_seq_cover_iff_countable {p : Set α → Prop} (h : ∃ s, p s) : (∃ s : ℕ → Set α, (∀ n, p (s n)) ∧ ⋃ n, s n = univ) ↔ ∃ S : Set (Set α), S.Countable ∧ (∀ s ∈ S, p s) ∧ ⋃₀ S = univ := exists_seq_iSup_eq_top_iff_countable h #align set.exists_seq_cover_iff_countable Set.exists_seq_cover_iff_countable theorem countable_of_injective_of_countable_image {s : Set α} {f : α → β} (hf : InjOn f s) (hs : (f '' s).Countable) : s.Countable := (mapsTo_image _ _).countable_of_injOn hf hs #align set.countable_of_injective_of_countable_image Set.countable_of_injective_of_countable_image theorem countable_iUnion {t : ι → Set α} [Countable ι] (ht : ∀ i, (t i).Countable) : (⋃ i, t i).Countable := by have := fun i ↦ (ht i).to_subtype rw [iUnion_eq_range_psigma] apply countable_range #align set.countable_Union Set.countable_iUnion @[simp] theorem countable_iUnion_iff [Countable ι] {t : ι → Set α} : (⋃ i, t i).Countable ↔ ∀ i, (t i).Countable := ⟨fun h _ => h.mono <\| subset_iUnion _ _, countable_iUnion⟩ #align set.countable_Union_iff Set.countable_iUnion_iff theorem Countable.biUnion_iff {s : Set α} {t : ∀ a ∈ s, Set β} (hs : s.Countable) : (⋃ a ∈ s, t a ‹_›).Countable ↔ ∀ a (ha : a ∈ s), (t a ha).Countable := by have := hs.to_subtype rw [biUnion_eq_iUnion, countable_iUnion_iff, SetCoe.forall'] #align set.countable.bUnion_iff Set.Countable.biUnion_iff theorem Countable.sUnion_iff {s : Set (Set α)} (hs : s.Countable) : (⋃₀ s).Countable ↔ ∀ a ∈ s, a.Countable := by rw [sUnion_eq_biUnion, hs.biUnion_iff] #align set.countable.sUnion_iff Set.Countable.sUnion_iff alias ⟨_, Countable.biUnion⟩ := Countable.biUnion_iff #align set.countable.bUnion Set.Countable.biUnion alias ⟨_, Countable.sUnion⟩ := Countable.sUnion_iff #align set.countable.sUnion Set.Countable.sUnion @[simp] theorem countable_union {s t : Set α} : (s ∪ t).Countable ↔ s.Countable ∧ t.Countable := by simp [union_eq_iUnion, and_comm] #align set.countable_union Set.countable_union theorem Countable.union {s t : Set α} (hs : s.Countable) (ht : t.Countable) : (s ∪ t).Countable := countable_union.2 ⟨hs, ht⟩ #align set.countable.union Set.Countable.union theorem Countable.of_diff {s t : Set α} (h : (s \ t).Countable) (ht : t.Countable) : s.Countable := (h.union ht).mono (subset_diff_union _ _) @[simp] theorem countable_insert {s : Set α} {a : α} : (insert a s).Countable ↔ s.Countable := by simp only [insert_eq, countable_union, countable_singleton, true_and_iff] #align set.countable_insert Set.countable_insert protected theorem Countable.insert {s : Set α} (a : α) (h : s.Countable) : (insert a s).Countable := countable_insert.2 h #align set.countable.insert Set.Countable.insert theorem Finite.countable {s : Set α} (hs : s.Finite) : s.Countable := have := hs.to_subtype; s.to_countable #align set.finite.countable Set.Finite.countable @[nontriviality] theorem Countable.of_subsingleton [Subsingleton α] (s : Set α) : s.Countable := (Finite.of_subsingleton s).countable #align set.countable.of_subsingleton Set.Countable.of_subsingleton theorem Subsingleton.countable {s : Set α} (hs : s.Subsingleton) : s.Countable := hs.finite.countable #align set.subsingleton.countable Set.Subsingleton.countable theorem countable_isTop (α : Type) [PartialOrder α] : { x : α \| IsTop x }.Countable := (finite_isTop α).countable #align set.countable_is_top Set.countable_isTop theorem countable_isBot (α : Type) [PartialOrder α] : { x : α \| IsBot x }.Countable := (finite_isBot α).countable #align set.countable_is_bot Set.countable_isBot /-- The set of finite subsets of a countable set is countable. -/ theorem countable_setOf_finite_subset {s : Set α} (hs : s.Countable) : { t \| Set.Finite t ∧ t ⊆ s }.Countable := by have := hs.to_subtype refine (countable_range fun t : Finset s => Subtype.val '' (t : Set s)).mono ?_ rintro t ⟨ht, hts⟩ lift t to Set s using hts lift t to Finset s using ht.of_finite_image Subtype.val_injective.injOn exact mem_range_self _ #align set.countable_set_of_finite_subset Set.countable_setOf_finite_subset theorem countable_univ_pi {π : α → Type} [Finite α] {s : ∀ a, Set (π a)} (hs : ∀ a, (s a).Countable) : (pi univ s).Countable := have := fun a ↦ (hs a).to_subtype; .of_equiv _ (Equiv.Set.univPi s).symm #align set.countable_univ_pi Set.countable_univ_pi theorem countable_pi {π : α → Type} [Finite α] {s : ∀ a, Set (π a)} (hs : ∀ a, (s a).Countable) : { f : ∀ a, π a \| ∀ a, f a ∈ s a }.Countable := by simpa only [← mem_univ_pi] using countable_univ_pi hs #align set.countable_pi Set.countable_pi protected theorem Countable.prod {s : Set α} {t : Set β} (hs : s.Countable) (ht : t.Countable) : Set.Countable (s ×ˢ t) := have := hs.to_subtype; have := ht.to_subtype; .of_equiv _ <\| (Equiv.Set.prod _ _).symm #align set.countable.prod Set.Countable.prod	Mathlib/Data/Set/Countable.lean	319	322	theorem Countable.image2 {s : Set α} {t : Set β} (hs : s.Countable) (ht : t.Countable) (f : α → β → γ) : (image2 f s t).Countable := by	rw [← image_prod] exact (hs.prod ht).image _
/- Copyright (c) 2022 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anne Baanen -/ import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed import Mathlib.RingTheory.PowerBasis #align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c" /-! # A predicate on adjoining roots of polynomial This file defines a predicate `IsAdjoinRoot S f`, which states that the ring `S` can be constructed by adjoining a specified root of the polynomial `f : R[X]` to `R`. This predicate is useful when the same ring can be generated by adjoining the root of different polynomials, and you want to vary which polynomial you're considering. The results in this file are intended to mirror those in `RingTheory.AdjoinRoot`, in order to provide an easier way to translate results from one to the other. ## Motivation `AdjoinRoot` presents one construction of a ring `R[α]`. However, it is possible to obtain rings of this form in many ways, such as `NumberField.ringOfIntegers ℚ(√-5)`, or `Algebra.adjoin R {α, α^2}`, or `IntermediateField.adjoin R {α, 2 - α}`, or even if we want to view `ℂ` as adjoining a root of `X^2 + 1` to `ℝ`. ## Main definitions The two main predicates in this file are: * `IsAdjoinRoot S f`: `S` is generated by adjoining a specified root of `f : R[X]` to `R` * `IsAdjoinRootMonic S f`: `S` is generated by adjoining a root of the monic polynomial `f : R[X]` to `R` Using `IsAdjoinRoot` to map into `S`: * `IsAdjoinRoot.map`: inclusion from `R[X]` to `S` * `IsAdjoinRoot.root`: the specific root adjoined to `R` to give `S` Using `IsAdjoinRoot` to map out of `S`: * `IsAdjoinRoot.repr`: choose a non-unique representative in `R[X]` * `IsAdjoinRoot.lift`, `IsAdjoinRoot.liftHom`: lift a morphism `R →+* T` to `S →+* T` * `IsAdjoinRootMonic.modByMonicHom`: a unique representative in `R[X]` if `f` is monic ## Main results * `AdjoinRoot.isAdjoinRoot` and `AdjoinRoot.isAdjoinRootMonic`: `AdjoinRoot` satisfies the conditions on `IsAdjoinRoot`(`_monic`) * `IsAdjoinRootMonic.powerBasis`: the `root` generates a power basis on `S` over `R` * `IsAdjoinRoot.aequiv`: algebra isomorphism showing adjoining a root gives a unique ring up to isomorphism * `IsAdjoinRoot.ofEquiv`: transfer `IsAdjoinRoot` across an algebra isomorphism * `IsAdjoinRootMonic.minpoly_eq`: the minimal polynomial of the adjoined root of `f` is equal to `f`, if `f` is irreducible and monic, and `R` is a GCD domain -/ open scoped Polynomial open Polynomial noncomputable section universe u v -- Porting note: this looks like something that should not be here -- section MoveMe -- -- end MoveMe -- This class doesn't really make sense on a predicate /-- `IsAdjoinRoot S f` states that the ring `S` can be constructed by adjoining a specified root of the polynomial `f : R[X]` to `R`. Compare `PowerBasis R S`, which does not explicitly specify which polynomial we adjoin a root of (in particular `f` does not need to be the minimal polynomial of the root we adjoin), and `AdjoinRoot` which constructs a new type. This is not a typeclass because the choice of root given `S` and `f` is not unique. -/ -- Porting note(#5171): this linter isn't ported yet. -- @[nolint has_nonempty_instance] structure IsAdjoinRoot {R : Type u} (S : Type v) [CommSemiring R] [Semiring S] [Algebra R S] (f : R[X]) : Type max u v where map : R[X] →+* S map_surjective : Function.Surjective map ker_map : RingHom.ker map = Ideal.span {f} algebraMap_eq : algebraMap R S = map.comp Polynomial.C #align is_adjoin_root IsAdjoinRoot -- This class doesn't really make sense on a predicate /-- `IsAdjoinRootMonic S f` states that the ring `S` can be constructed by adjoining a specified root of the monic polynomial `f : R[X]` to `R`. As long as `f` is monic, there is a well-defined representation of elements of `S` as polynomials in `R[X]` of degree lower than `deg f` (see `modByMonicHom` and `coeff`). In particular, we have `IsAdjoinRootMonic.powerBasis`. Bundling `Monic` into this structure is very useful when working with explicit `f`s such as `X^2 - C a * X - C b` since it saves you carrying around the proofs of monicity. -/ -- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet. structure IsAdjoinRootMonic {R : Type u} (S : Type v) [CommSemiring R] [Semiring S] [Algebra R S] (f : R[X]) extends IsAdjoinRoot S f where Monic : Monic f #align is_adjoin_root_monic IsAdjoinRootMonic section Ring variable {R : Type u} {S : Type v} [CommRing R] [Ring S] {f : R[X]} [Algebra R S] namespace IsAdjoinRoot /-- `(h : IsAdjoinRoot S f).root` is the root of `f` that can be adjoined to generate `S`. -/ def root (h : IsAdjoinRoot S f) : S := h.map X #align is_adjoin_root.root IsAdjoinRoot.root theorem subsingleton (h : IsAdjoinRoot S f) [Subsingleton R] : Subsingleton S := h.map_surjective.subsingleton #align is_adjoin_root.subsingleton IsAdjoinRoot.subsingleton theorem algebraMap_apply (h : IsAdjoinRoot S f) (x : R) : algebraMap R S x = h.map (Polynomial.C x) := by rw [h.algebraMap_eq, RingHom.comp_apply] #align is_adjoin_root.algebra_map_apply IsAdjoinRoot.algebraMap_apply @[simp] theorem mem_ker_map (h : IsAdjoinRoot S f) {p} : p ∈ RingHom.ker h.map ↔ f ∣ p := by rw [h.ker_map, Ideal.mem_span_singleton] #align is_adjoin_root.mem_ker_map IsAdjoinRoot.mem_ker_map theorem map_eq_zero_iff (h : IsAdjoinRoot S f) {p} : h.map p = 0 ↔ f ∣ p := by rw [← h.mem_ker_map, RingHom.mem_ker] #align is_adjoin_root.map_eq_zero_iff IsAdjoinRoot.map_eq_zero_iff @[simp] theorem map_X (h : IsAdjoinRoot S f) : h.map X = h.root := rfl set_option linter.uppercaseLean3 false in #align is_adjoin_root.map_X IsAdjoinRoot.map_X @[simp] theorem map_self (h : IsAdjoinRoot S f) : h.map f = 0 := h.map_eq_zero_iff.mpr dvd_rfl #align is_adjoin_root.map_self IsAdjoinRoot.map_self @[simp] theorem aeval_eq (h : IsAdjoinRoot S f) (p : R[X]) : aeval h.root p = h.map p := Polynomial.induction_on p (fun x => by rw [aeval_C, h.algebraMap_apply]) (fun p q ihp ihq => by rw [AlgHom.map_add, RingHom.map_add, ihp, ihq]) fun n x _ => by rw [AlgHom.map_mul, aeval_C, AlgHom.map_pow, aeval_X, RingHom.map_mul, ← h.algebraMap_apply, RingHom.map_pow, map_X] #align is_adjoin_root.aeval_eq IsAdjoinRoot.aeval_eq -- @[simp] -- Porting note (#10618): simp can prove this theorem aeval_root (h : IsAdjoinRoot S f) : aeval h.root f = 0 := by rw [aeval_eq, map_self] #align is_adjoin_root.aeval_root IsAdjoinRoot.aeval_root /-- Choose an arbitrary representative so that `h.map (h.repr x) = x`. If `f` is monic, use `IsAdjoinRootMonic.modByMonicHom` for a unique choice of representative. -/ def repr (h : IsAdjoinRoot S f) (x : S) : R[X] := (h.map_surjective x).choose #align is_adjoin_root.repr IsAdjoinRoot.repr theorem map_repr (h : IsAdjoinRoot S f) (x : S) : h.map (h.repr x) = x := (h.map_surjective x).choose_spec #align is_adjoin_root.map_repr IsAdjoinRoot.map_repr /-- `repr` preserves zero, up to multiples of `f` -/ theorem repr_zero_mem_span (h : IsAdjoinRoot S f) : h.repr 0 ∈ Ideal.span ({f} : Set R[X]) := by rw [← h.ker_map, RingHom.mem_ker, h.map_repr] #align is_adjoin_root.repr_zero_mem_span IsAdjoinRoot.repr_zero_mem_span /-- `repr` preserves addition, up to multiples of `f` -/ theorem repr_add_sub_repr_add_repr_mem_span (h : IsAdjoinRoot S f) (x y : S) : h.repr (x + y) - (h.repr x + h.repr y) ∈ Ideal.span ({f} : Set R[X]) := by rw [← h.ker_map, RingHom.mem_ker, map_sub, h.map_repr, map_add, h.map_repr, h.map_repr, sub_self] #align is_adjoin_root.repr_add_sub_repr_add_repr_mem_span IsAdjoinRoot.repr_add_sub_repr_add_repr_mem_span /-- Extensionality of the `IsAdjoinRoot` structure itself. See `IsAdjoinRootMonic.ext_elem` for extensionality of the ring elements. -/ theorem ext_map (h h' : IsAdjoinRoot S f) (eq : ∀ x, h.map x = h'.map x) : h = h' := by cases h; cases h'; congr exact RingHom.ext eq #align is_adjoin_root.ext_map IsAdjoinRoot.ext_map /-- Extensionality of the `IsAdjoinRoot` structure itself. See `IsAdjoinRootMonic.ext_elem` for extensionality of the ring elements. -/ @[ext] theorem ext (h h' : IsAdjoinRoot S f) (eq : h.root = h'.root) : h = h' := h.ext_map h' fun x => by rw [← h.aeval_eq, ← h'.aeval_eq, eq] #align is_adjoin_root.ext IsAdjoinRoot.ext section lift variable {T : Type} [CommRing T] {i : R →+ T} {x : T} (hx : f.eval₂ i x = 0) /-- Auxiliary lemma for `IsAdjoinRoot.lift` -/ theorem eval₂_repr_eq_eval₂_of_map_eq (h : IsAdjoinRoot S f) (z : S) (w : R[X]) (hzw : h.map w = z) : (h.repr z).eval₂ i x = w.eval₂ i x := by rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw obtain ⟨y, hy⟩ := hzw rw [← sub_eq_zero, ← eval₂_sub, hy, eval₂_mul, hx, zero_mul] #align is_adjoin_root.eval₂_repr_eq_eval₂_of_map_eq IsAdjoinRoot.eval₂_repr_eq_eval₂_of_map_eq variable (i x) -- To match `AdjoinRoot.lift` /-- Lift a ring homomorphism `R →+* T` to `S →+* T` by specifying a root `x` of `f` in `T`, where `S` is given by adjoining a root of `f` to `R`. -/ def lift (h : IsAdjoinRoot S f) : S →+* T where toFun z := (h.repr z).eval₂ i x map_zero' := by dsimp only -- Porting note (#10752): added `dsimp only` rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ _ (map_zero _), eval₂_zero] map_add' z w := by dsimp only -- Porting note (#10752): added `dsimp only` rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ (h.repr z + h.repr w), eval₂_add] rw [map_add, map_repr, map_repr] map_one' := by beta_reduce -- Porting note (#12129): additional beta reduction needed rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ _ (map_one _), eval₂_one] map_mul' z w := by dsimp only -- Porting note (#10752): added `dsimp only` rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ (h.repr z * h.repr w), eval₂_mul] rw [map_mul, map_repr, map_repr] #align is_adjoin_root.lift IsAdjoinRoot.lift variable {i x} @[simp] theorem lift_map (h : IsAdjoinRoot S f) (z : R[X]) : h.lift i x hx (h.map z) = z.eval₂ i x := by rw [lift, RingHom.coe_mk] dsimp -- Porting note (#11227):added a `dsimp` rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ _ rfl] #align is_adjoin_root.lift_map IsAdjoinRoot.lift_map @[simp] theorem lift_root (h : IsAdjoinRoot S f) : h.lift i x hx h.root = x := by rw [← h.map_X, lift_map, eval₂_X] #align is_adjoin_root.lift_root IsAdjoinRoot.lift_root @[simp] theorem lift_algebraMap (h : IsAdjoinRoot S f) (a : R) : h.lift i x hx (algebraMap R S a) = i a := by rw [h.algebraMap_apply, lift_map, eval₂_C] #align is_adjoin_root.lift_algebra_map IsAdjoinRoot.lift_algebraMap /-- Auxiliary lemma for `apply_eq_lift` -/ theorem apply_eq_lift (h : IsAdjoinRoot S f) (g : S →+* T) (hmap : ∀ a, g (algebraMap R S a) = i a) (hroot : g h.root = x) (a : S) : g a = h.lift i x hx a := by rw [← h.map_repr a, Polynomial.as_sum_range_C_mul_X_pow (h.repr a)] simp only [map_sum, map_mul, map_pow, h.map_X, hroot, ← h.algebraMap_apply, hmap, lift_root, lift_algebraMap] #align is_adjoin_root.apply_eq_lift IsAdjoinRoot.apply_eq_lift /-- Unicity of `lift`: a map that agrees on `R` and `h.root` agrees with `lift` everywhere. -/ theorem eq_lift (h : IsAdjoinRoot S f) (g : S →+* T) (hmap : ∀ a, g (algebraMap R S a) = i a) (hroot : g h.root = x) : g = h.lift i x hx := RingHom.ext (h.apply_eq_lift hx g hmap hroot) #align is_adjoin_root.eq_lift IsAdjoinRoot.eq_lift variable [Algebra R T] (hx' : aeval x f = 0) variable (x) -- To match `AdjoinRoot.liftHom` /-- Lift the algebra map `R → T` to `S →ₐ[R] T` by specifying a root `x` of `f` in `T`, where `S` is given by adjoining a root of `f` to `R`. -/ def liftHom (h : IsAdjoinRoot S f) : S →ₐ[R] T := { h.lift (algebraMap R T) x hx' with commutes' := fun a => h.lift_algebraMap hx' a } #align is_adjoin_root.lift_hom IsAdjoinRoot.liftHom variable {x} @[simp] theorem coe_liftHom (h : IsAdjoinRoot S f) : (h.liftHom x hx' : S →+* T) = h.lift (algebraMap R T) x hx' := rfl #align is_adjoin_root.coe_lift_hom IsAdjoinRoot.coe_liftHom theorem lift_algebraMap_apply (h : IsAdjoinRoot S f) (z : S) : h.lift (algebraMap R T) x hx' z = h.liftHom x hx' z := rfl #align is_adjoin_root.lift_algebra_map_apply IsAdjoinRoot.lift_algebraMap_apply @[simp] theorem liftHom_map (h : IsAdjoinRoot S f) (z : R[X]) : h.liftHom x hx' (h.map z) = aeval x z := by rw [← lift_algebraMap_apply, lift_map, aeval_def] #align is_adjoin_root.lift_hom_map IsAdjoinRoot.liftHom_map @[simp] theorem liftHom_root (h : IsAdjoinRoot S f) : h.liftHom x hx' h.root = x := by rw [← lift_algebraMap_apply, lift_root] #align is_adjoin_root.lift_hom_root IsAdjoinRoot.liftHom_root /-- Unicity of `liftHom`: a map that agrees on `h.root` agrees with `liftHom` everywhere. -/ theorem eq_liftHom (h : IsAdjoinRoot S f) (g : S →ₐ[R] T) (hroot : g h.root = x) : g = h.liftHom x hx' := AlgHom.ext (h.apply_eq_lift hx' g g.commutes hroot) #align is_adjoin_root.eq_lift_hom IsAdjoinRoot.eq_liftHom end lift end IsAdjoinRoot namespace AdjoinRoot variable (f) /-- `AdjoinRoot f` is indeed given by adjoining a root of `f`. -/ protected def isAdjoinRoot : IsAdjoinRoot (AdjoinRoot f) f where map := AdjoinRoot.mk f map_surjective := Ideal.Quotient.mk_surjective ker_map := by ext rw [RingHom.mem_ker, ← @AdjoinRoot.mk_self _ _ f, AdjoinRoot.mk_eq_mk, Ideal.mem_span_singleton, ← dvd_add_left (dvd_refl f), sub_add_cancel] algebraMap_eq := AdjoinRoot.algebraMap_eq f #align adjoin_root.is_adjoin_root AdjoinRoot.isAdjoinRoot /-- `AdjoinRoot f` is indeed given by adjoining a root of `f`. If `f` is monic this is more powerful than `AdjoinRoot.isAdjoinRoot`. -/ protected def isAdjoinRootMonic (hf : Monic f) : IsAdjoinRootMonic (AdjoinRoot f) f := { AdjoinRoot.isAdjoinRoot f with Monic := hf } #align adjoin_root.is_adjoin_root_monic AdjoinRoot.isAdjoinRootMonic @[simp] theorem isAdjoinRoot_map_eq_mk : (AdjoinRoot.isAdjoinRoot f).map = AdjoinRoot.mk f := rfl #align adjoin_root.is_adjoin_root_map_eq_mk AdjoinRoot.isAdjoinRoot_map_eq_mk @[simp] theorem isAdjoinRootMonic_map_eq_mk (hf : f.Monic) : (AdjoinRoot.isAdjoinRootMonic f hf).map = AdjoinRoot.mk f := rfl #align adjoin_root.is_adjoin_root_monic_map_eq_mk AdjoinRoot.isAdjoinRootMonic_map_eq_mk @[simp] theorem isAdjoinRoot_root_eq_root : (AdjoinRoot.isAdjoinRoot f).root = AdjoinRoot.root f := by simp only [IsAdjoinRoot.root, AdjoinRoot.root, AdjoinRoot.isAdjoinRoot_map_eq_mk] #align adjoin_root.is_adjoin_root_root_eq_root AdjoinRoot.isAdjoinRoot_root_eq_root @[simp] theorem isAdjoinRootMonic_root_eq_root (hf : Monic f) : (AdjoinRoot.isAdjoinRootMonic f hf).root = AdjoinRoot.root f := by simp only [IsAdjoinRoot.root, AdjoinRoot.root, AdjoinRoot.isAdjoinRootMonic_map_eq_mk] #align adjoin_root.is_adjoin_root_monic_root_eq_root AdjoinRoot.isAdjoinRootMonic_root_eq_root end AdjoinRoot namespace IsAdjoinRootMonic open IsAdjoinRoot theorem map_modByMonic (h : IsAdjoinRootMonic S f) (g : R[X]) : h.map (g %ₘ f) = h.map g := by rw [← RingHom.sub_mem_ker_iff, mem_ker_map, modByMonic_eq_sub_mul_div _ h.Monic, sub_right_comm, sub_self, zero_sub, dvd_neg] exact ⟨_, rfl⟩ #align is_adjoin_root_monic.map_mod_by_monic IsAdjoinRootMonic.map_modByMonic theorem modByMonic_repr_map (h : IsAdjoinRootMonic S f) (g : R[X]) : h.repr (h.map g) %ₘ f = g %ₘ f := modByMonic_eq_of_dvd_sub h.Monic <\| by rw [← h.mem_ker_map, RingHom.sub_mem_ker_iff, map_repr] #align is_adjoin_root_monic.mod_by_monic_repr_map IsAdjoinRootMonic.modByMonic_repr_map /-- `IsAdjoinRoot.modByMonicHom` sends the equivalence class of `f` mod `g` to `f %ₘ g`. -/ def modByMonicHom (h : IsAdjoinRootMonic S f) : S →ₗ[R] R[X] where toFun x := h.repr x %ₘ f map_add' x y := by conv_lhs => rw [← h.map_repr x, ← h.map_repr y, ← map_add] beta_reduce -- Porting note (#12129): additional beta reduction needed rw [h.modByMonic_repr_map, add_modByMonic] map_smul' c x := by rw [RingHom.id_apply, ← h.map_repr x, Algebra.smul_def, h.algebraMap_apply, ← map_mul] dsimp only -- Porting note (#10752): added `dsimp only` rw [h.modByMonic_repr_map, ← smul_eq_C_mul, smul_modByMonic, h.map_repr] #align is_adjoin_root_monic.mod_by_monic_hom IsAdjoinRootMonic.modByMonicHom @[simp] theorem modByMonicHom_map (h : IsAdjoinRootMonic S f) (g : R[X]) : h.modByMonicHom (h.map g) = g %ₘ f := h.modByMonic_repr_map g #align is_adjoin_root_monic.mod_by_monic_hom_map IsAdjoinRootMonic.modByMonicHom_map @[simp] theorem map_modByMonicHom (h : IsAdjoinRootMonic S f) (x : S) : h.map (h.modByMonicHom x) = x := by rw [modByMonicHom, LinearMap.coe_mk] dsimp -- Porting note (#11227):added a `dsimp` rw [map_modByMonic, map_repr] #align is_adjoin_root_monic.map_mod_by_monic_hom IsAdjoinRootMonic.map_modByMonicHom @[simp] theorem modByMonicHom_root_pow (h : IsAdjoinRootMonic S f) {n : ℕ} (hdeg : n < natDegree f) : h.modByMonicHom (h.root ^ n) = X ^ n := by nontriviality R rw [← h.map_X, ← map_pow, modByMonicHom_map, modByMonic_eq_self_iff h.Monic, degree_X_pow] contrapose! hdeg simpa [natDegree_le_iff_degree_le] using hdeg #align is_adjoin_root_monic.mod_by_monic_hom_root_pow IsAdjoinRootMonic.modByMonicHom_root_pow @[simp] theorem modByMonicHom_root (h : IsAdjoinRootMonic S f) (hdeg : 1 < natDegree f) : h.modByMonicHom h.root = X := by simpa using modByMonicHom_root_pow h hdeg #align is_adjoin_root_monic.mod_by_monic_hom_root IsAdjoinRootMonic.modByMonicHom_root /-- The basis on `S` generated by powers of `h.root`. Auxiliary definition for `IsAdjoinRootMonic.powerBasis`. -/ def basis (h : IsAdjoinRootMonic S f) : Basis (Fin (natDegree f)) R S := Basis.ofRepr { toFun := fun x => (h.modByMonicHom x).toFinsupp.comapDomain _ Fin.val_injective.injOn invFun := fun g => h.map (ofFinsupp (g.mapDomain _)) left_inv := fun x => by cases subsingleton_or_nontrivial R · haveI := h.subsingleton exact Subsingleton.elim _ _ simp only rw [Finsupp.mapDomain_comapDomain, Polynomial.eta, h.map_modByMonicHom x] · exact Fin.val_injective intro i hi refine Set.mem_range.mpr ⟨⟨i, ?_⟩, rfl⟩ contrapose! hi simp only [Polynomial.toFinsupp_apply, Classical.not_not, Finsupp.mem_support_iff, Ne, modByMonicHom, LinearMap.coe_mk, Finset.mem_coe] by_cases hx : h.toIsAdjoinRoot.repr x %ₘ f = 0 · simp [hx] refine coeff_eq_zero_of_natDegree_lt (lt_of_lt_of_le ?_ hi) dsimp -- Porting note (#11227):added a `dsimp` rw [natDegree_lt_natDegree_iff hx] exact degree_modByMonic_lt _ h.Monic right_inv := fun g => by nontriviality R ext i simp only [h.modByMonicHom_map, Finsupp.comapDomain_apply, Polynomial.toFinsupp_apply] rw [(Polynomial.modByMonic_eq_self_iff h.Monic).mpr, Polynomial.coeff] · dsimp only -- Porting note (#10752): added `dsimp only` rw [Finsupp.mapDomain_apply Fin.val_injective] rw [degree_eq_natDegree h.Monic.ne_zero, degree_lt_iff_coeff_zero] intro m hm rw [Polynomial.coeff] dsimp only -- Porting note (#10752): added `dsimp only` rw [Finsupp.mapDomain_notin_range] rw [Set.mem_range, not_exists] rintro i rfl exact i.prop.not_le hm map_add' := fun x y => by beta_reduce -- Porting note (#12129): additional beta reduction needed rw [map_add, toFinsupp_add, Finsupp.comapDomain_add_of_injective Fin.val_injective] -- Porting note: the original simp proof with the same lemmas does not work -- See https://github.com/leanprover-community/mathlib4/issues/5026 -- simp only [map_add, Finsupp.comapDomain_add_of_injective Fin.val_injective, toFinsupp_add] map_smul' := fun c x => by dsimp only -- Porting note (#10752): added `dsimp only` rw [map_smul, toFinsupp_smul, Finsupp.comapDomain_smul_of_injective Fin.val_injective, RingHom.id_apply] } -- Porting note: the original simp proof with the same lemmas does not work -- See https://github.com/leanprover-community/mathlib4/issues/5026 -- simp only [map_smul, Finsupp.comapDomain_smul_of_injective Fin.val_injective, -- RingHom.id_apply, toFinsupp_smul] } #align is_adjoin_root_monic.basis IsAdjoinRootMonic.basis @[simp] theorem basis_apply (h : IsAdjoinRootMonic S f) (i) : h.basis i = h.root ^ (i : ℕ) := Basis.apply_eq_iff.mpr <\| show (h.modByMonicHom (h.toIsAdjoinRoot.root ^ (i : ℕ))).toFinsupp.comapDomain _ Fin.val_injective.injOn = Finsupp.single _ _ by ext j rw [Finsupp.comapDomain_apply, modByMonicHom_root_pow] · rw [X_pow_eq_monomial, toFinsupp_monomial, Finsupp.single_apply_left Fin.val_injective] · exact i.is_lt #align is_adjoin_root_monic.basis_apply IsAdjoinRootMonic.basis_apply theorem deg_pos [Nontrivial S] (h : IsAdjoinRootMonic S f) : 0 < natDegree f := by rcases h.basis.index_nonempty with ⟨⟨i, hi⟩⟩ exact (Nat.zero_le _).trans_lt hi #align is_adjoin_root_monic.deg_pos IsAdjoinRootMonic.deg_pos theorem deg_ne_zero [Nontrivial S] (h : IsAdjoinRootMonic S f) : natDegree f ≠ 0 := h.deg_pos.ne' #align is_adjoin_root_monic.deg_ne_zero IsAdjoinRootMonic.deg_ne_zero /-- If `f` is monic, the powers of `h.root` form a basis. -/ @[simps! gen dim basis] def powerBasis (h : IsAdjoinRootMonic S f) : PowerBasis R S where gen := h.root dim := natDegree f basis := h.basis basis_eq_pow := h.basis_apply #align is_adjoin_root_monic.power_basis IsAdjoinRootMonic.powerBasis @[simp] theorem basis_repr (h : IsAdjoinRootMonic S f) (x : S) (i : Fin (natDegree f)) : h.basis.repr x i = (h.modByMonicHom x).coeff (i : ℕ) := by change (h.modByMonicHom x).toFinsupp.comapDomain _ Fin.val_injective.injOn i = _ rw [Finsupp.comapDomain_apply, Polynomial.toFinsupp_apply] #align is_adjoin_root_monic.basis_repr IsAdjoinRootMonic.basis_repr theorem basis_one (h : IsAdjoinRootMonic S f) (hdeg : 1 < natDegree f) : h.basis ⟨1, hdeg⟩ = h.root := by rw [h.basis_apply, Fin.val_mk, pow_one] #align is_adjoin_root_monic.basis_one IsAdjoinRootMonic.basis_one /-- `IsAdjoinRootMonic.liftPolyₗ` lifts a linear map on polynomials to a linear map on `S`. -/ @[simps!] def liftPolyₗ {T : Type*} [AddCommGroup T] [Module R T] (h : IsAdjoinRootMonic S f) (g : R[X] →ₗ[R] T) : S →ₗ[R] T := g.comp h.modByMonicHom #align is_adjoin_root_monic.lift_polyₗ IsAdjoinRootMonic.liftPolyₗ /-- `IsAdjoinRootMonic.coeff h x i` is the `i`th coefficient of the representative of `x : S`. -/ def coeff (h : IsAdjoinRootMonic S f) : S →ₗ[R] ℕ → R := h.liftPolyₗ { toFun := Polynomial.coeff map_add' := fun p q => funext (Polynomial.coeff_add p q) map_smul' := fun c p => funext (Polynomial.coeff_smul c p) } #align is_adjoin_root_monic.coeff IsAdjoinRootMonic.coeff theorem coeff_apply_lt (h : IsAdjoinRootMonic S f) (z : S) (i : ℕ) (hi : i < natDegree f) : h.coeff z i = h.basis.repr z ⟨i, hi⟩ := by simp only [coeff, LinearMap.comp_apply, Finsupp.lcoeFun_apply, Finsupp.lmapDomain_apply, LinearEquiv.coe_coe, liftPolyₗ_apply, LinearMap.coe_mk, h.basis_repr] rfl #align is_adjoin_root_monic.coeff_apply_lt IsAdjoinRootMonic.coeff_apply_lt theorem coeff_apply_coe (h : IsAdjoinRootMonic S f) (z : S) (i : Fin (natDegree f)) : h.coeff z i = h.basis.repr z i := h.coeff_apply_lt z i i.prop #align is_adjoin_root_monic.coeff_apply_coe IsAdjoinRootMonic.coeff_apply_coe theorem coeff_apply_le (h : IsAdjoinRootMonic S f) (z : S) (i : ℕ) (hi : natDegree f ≤ i) : h.coeff z i = 0 := by simp only [coeff, LinearMap.comp_apply, Finsupp.lcoeFun_apply, Finsupp.lmapDomain_apply, LinearEquiv.coe_coe, liftPolyₗ_apply, LinearMap.coe_mk, h.basis_repr] nontriviality R exact Polynomial.coeff_eq_zero_of_degree_lt ((degree_modByMonic_lt _ h.Monic).trans_le (Polynomial.degree_le_of_natDegree_le hi)) #align is_adjoin_root_monic.coeff_apply_le IsAdjoinRootMonic.coeff_apply_le	Mathlib/RingTheory/IsAdjoinRoot.lean	540	544	theorem coeff_apply (h : IsAdjoinRootMonic S f) (z : S) (i : ℕ) : h.coeff z i = if hi : i < natDegree f then h.basis.repr z ⟨i, hi⟩ else 0 := by	split_ifs with hi · exact h.coeff_apply_lt z i hi · exact h.coeff_apply_le z i (le_of_not_lt hi)
/- Copyright (c) 2017 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis, Keeley Hoek -/ import Mathlib.Algebra.NeZero import Mathlib.Data.Nat.Defs import Mathlib.Logic.Embedding.Basic import Mathlib.Logic.Equiv.Set import Mathlib.Tactic.Common #align_import data.fin.basic from "leanprover-community/mathlib"@"3a2b5524a138b5d0b818b858b516d4ac8a484b03" /-! # The finite type with `n` elements `Fin n` is the type whose elements are natural numbers smaller than `n`. This file expands on the development in the core library. ## Main definitions ### Induction principles * `finZeroElim` : Elimination principle for the empty set `Fin 0`, generalizes `Fin.elim0`. * `Fin.succRec` : Define `C n i` by induction on `i : Fin n` interpreted as `(0 : Fin (n - i)).succ.succ…`. This function has two arguments: `H0 n` defines `0`-th element `C (n+1) 0` of an `(n+1)`-tuple, and `Hs n i` defines `(i+1)`-st element of `(n+1)`-tuple based on `n`, `i`, and `i`-th element of `n`-tuple. * `Fin.succRecOn` : same as `Fin.succRec` but `i : Fin n` is the first argument; * `Fin.induction` : Define `C i` by induction on `i : Fin (n + 1)`, separating into the `Nat`-like base cases of `C 0` and `C (i.succ)`. * `Fin.inductionOn` : same as `Fin.induction` but with `i : Fin (n + 1)` as the first argument. * `Fin.cases` : define `f : Π i : Fin n.succ, C i` by separately handling the cases `i = 0` and `i = Fin.succ j`, `j : Fin n`, defined using `Fin.induction`. * `Fin.reverseInduction`: reverse induction on `i : Fin (n + 1)`; given `C (Fin.last n)` and `∀ i : Fin n, C (Fin.succ i) → C (Fin.castSucc i)`, constructs all values `C i` by going down; * `Fin.lastCases`: define `f : Π i, Fin (n + 1), C i` by separately handling the cases `i = Fin.last n` and `i = Fin.castSucc j`, a special case of `Fin.reverseInduction`; * `Fin.addCases`: define a function on `Fin (m + n)` by separately handling the cases `Fin.castAdd n i` and `Fin.natAdd m i`; * `Fin.succAboveCases`: given `i : Fin (n + 1)`, define a function on `Fin (n + 1)` by separately handling the cases `j = i` and `j = Fin.succAbove i k`, same as `Fin.insertNth` but marked as eliminator and works for `Sort`. -- Porting note: this is in another file ### Embeddings and isomorphisms `Fin.valEmbedding` : coercion to natural numbers as an `Embedding`; * `Fin.succEmb` : `Fin.succ` as an `Embedding`; * `Fin.castLEEmb h` : `Fin.castLE` as an `Embedding`, embed `Fin n` into `Fin m`, `h : n ≤ m`; * `finCongr` : `Fin.cast` as an `Equiv`, equivalence between `Fin n` and `Fin m` when `n = m`; * `Fin.castAddEmb m` : `Fin.castAdd` as an `Embedding`, embed `Fin n` into `Fin (n+m)`; * `Fin.castSuccEmb` : `Fin.castSucc` as an `Embedding`, embed `Fin n` into `Fin (n+1)`; * `Fin.addNatEmb m i` : `Fin.addNat` as an `Embedding`, add `m` on `i` on the right, generalizes `Fin.succ`; * `Fin.natAddEmb n i` : `Fin.natAdd` as an `Embedding`, adds `n` on `i` on the left; ### Other casts * `Fin.ofNat'`: given a positive number `n` (deduced from `[NeZero n]`), `Fin.ofNat' i` is `i % n` interpreted as an element of `Fin n`; * `Fin.divNat i` : divides `i : Fin (m * n)` by `n`; * `Fin.modNat i` : takes the mod of `i : Fin (m * n)` by `n`; ### Misc definitions * `Fin.revPerm : Equiv.Perm (Fin n)` : `Fin.rev` as an `Equiv.Perm`, the antitone involution given by `i ↦ n-(i+1)` -/ assert_not_exists Monoid universe u v open Fin Nat Function /-- Elimination principle for the empty set `Fin 0`, dependent version. -/ def finZeroElim {α : Fin 0 → Sort} (x : Fin 0) : α x := x.elim0 #align fin_zero_elim finZeroElim namespace Fin instance {n : ℕ} : CanLift ℕ (Fin n) Fin.val (· < n) where prf k hk := ⟨⟨k, hk⟩, rfl⟩ /-- A dependent variant of `Fin.elim0`. -/ def rec0 {α : Fin 0 → Sort} (i : Fin 0) : α i := absurd i.2 (Nat.not_lt_zero _) #align fin.elim0' Fin.elim0 variable {n m : ℕ} --variable {a b : Fin n} -- this really breaks stuff #align fin.fin_to_nat Fin.coeToNat theorem val_injective : Function.Injective (@Fin.val n) := @Fin.eq_of_val_eq n #align fin.val_injective Fin.val_injective /-- If you actually have an element of `Fin n`, then the `n` is always positive -/ lemma size_positive : Fin n → 0 < n := Fin.pos lemma size_positive' [Nonempty (Fin n)] : 0 < n := ‹Nonempty (Fin n)›.elim Fin.pos protected theorem prop (a : Fin n) : a.val < n := a.2 #align fin.prop Fin.prop #align fin.is_lt Fin.is_lt #align fin.pos Fin.pos #align fin.pos_iff_nonempty Fin.pos_iff_nonempty section Order variable {a b c : Fin n} protected lemma lt_of_le_of_lt : a ≤ b → b < c → a < c := Nat.lt_of_le_of_lt protected lemma lt_of_lt_of_le : a < b → b ≤ c → a < c := Nat.lt_of_lt_of_le protected lemma le_rfl : a ≤ a := Nat.le_refl _ protected lemma lt_iff_le_and_ne : a < b ↔ a ≤ b ∧ a ≠ b := by rw [← val_ne_iff]; exact Nat.lt_iff_le_and_ne protected lemma lt_or_lt_of_ne (h : a ≠ b) : a < b ∨ b < a := Nat.lt_or_lt_of_ne $ val_ne_iff.2 h protected lemma lt_or_le (a b : Fin n) : a < b ∨ b ≤ a := Nat.lt_or_ge _ _ protected lemma le_or_lt (a b : Fin n) : a ≤ b ∨ b < a := (b.lt_or_le a).symm protected lemma le_of_eq (hab : a = b) : a ≤ b := Nat.le_of_eq $ congr_arg val hab protected lemma ge_of_eq (hab : a = b) : b ≤ a := Fin.le_of_eq hab.symm protected lemma eq_or_lt_of_le : a ≤ b → a = b ∨ a < b := by rw [ext_iff]; exact Nat.eq_or_lt_of_le protected lemma lt_or_eq_of_le : a ≤ b → a < b ∨ a = b := by rw [ext_iff]; exact Nat.lt_or_eq_of_le end Order lemma lt_last_iff_ne_last {a : Fin (n + 1)} : a < last n ↔ a ≠ last n := by simp [Fin.lt_iff_le_and_ne, le_last] lemma ne_zero_of_lt {a b : Fin (n + 1)} (hab : a < b) : b ≠ 0 := Fin.ne_of_gt $ Fin.lt_of_le_of_lt a.zero_le hab lemma ne_last_of_lt {a b : Fin (n + 1)} (hab : a < b) : a ≠ last n := Fin.ne_of_lt $ Fin.lt_of_lt_of_le hab b.le_last /-- Equivalence between `Fin n` and `{ i // i < n }`. -/ @[simps apply symm_apply] def equivSubtype : Fin n ≃ { i // i < n } where toFun a := ⟨a.1, a.2⟩ invFun a := ⟨a.1, a.2⟩ left_inv := fun ⟨_, _⟩ => rfl right_inv := fun ⟨_, _⟩ => rfl #align fin.equiv_subtype Fin.equivSubtype #align fin.equiv_subtype_symm_apply Fin.equivSubtype_symm_apply #align fin.equiv_subtype_apply Fin.equivSubtype_apply section coe /-! ### coercions and constructions -/ #align fin.eta Fin.eta #align fin.ext Fin.ext #align fin.ext_iff Fin.ext_iff #align fin.coe_injective Fin.val_injective theorem val_eq_val (a b : Fin n) : (a : ℕ) = b ↔ a = b := ext_iff.symm #align fin.coe_eq_coe Fin.val_eq_val @[deprecated ext_iff (since := "2024-02-20")] theorem eq_iff_veq (a b : Fin n) : a = b ↔ a.1 = b.1 := ext_iff #align fin.eq_iff_veq Fin.eq_iff_veq theorem ne_iff_vne (a b : Fin n) : a ≠ b ↔ a.1 ≠ b.1 := ext_iff.not #align fin.ne_iff_vne Fin.ne_iff_vne -- Porting note: I'm not sure if this comment still applies. -- built-in reduction doesn't always work @[simp, nolint simpNF] theorem mk_eq_mk {a h a' h'} : @mk n a h = @mk n a' h' ↔ a = a' := ext_iff #align fin.mk_eq_mk Fin.mk_eq_mk #align fin.mk.inj_iff Fin.mk.inj_iff #align fin.mk_val Fin.val_mk #align fin.eq_mk_iff_coe_eq Fin.eq_mk_iff_val_eq #align fin.coe_mk Fin.val_mk #align fin.mk_coe Fin.mk_val -- syntactic tautologies now #noalign fin.coe_eq_val #noalign fin.val_eq_coe /-- Assume `k = l`. If two functions defined on `Fin k` and `Fin l` are equal on each element, then they coincide (in the heq sense). -/ protected theorem heq_fun_iff {α : Sort} {k l : ℕ} (h : k = l) {f : Fin k → α} {g : Fin l → α} : HEq f g ↔ ∀ i : Fin k, f i = g ⟨(i : ℕ), h ▸ i.2⟩ := by subst h simp [Function.funext_iff] #align fin.heq_fun_iff Fin.heq_fun_iff /-- Assume `k = l` and `k' = l'`. If two functions `Fin k → Fin k' → α` and `Fin l → Fin l' → α` are equal on each pair, then they coincide (in the heq sense). -/ protected theorem heq_fun₂_iff {α : Sort} {k l k' l' : ℕ} (h : k = l) (h' : k' = l') {f : Fin k → Fin k' → α} {g : Fin l → Fin l' → α} : HEq f g ↔ ∀ (i : Fin k) (j : Fin k'), f i j = g ⟨(i : ℕ), h ▸ i.2⟩ ⟨(j : ℕ), h' ▸ j.2⟩ := by subst h subst h' simp [Function.funext_iff] protected theorem heq_ext_iff {k l : ℕ} (h : k = l) {i : Fin k} {j : Fin l} : HEq i j ↔ (i : ℕ) = (j : ℕ) := by subst h simp [val_eq_val] #align fin.heq_ext_iff Fin.heq_ext_iff #align fin.exists_iff Fin.exists_iff #align fin.forall_iff Fin.forall_iff end coe section Order /-! ### order -/ #align fin.is_le Fin.is_le #align fin.is_le' Fin.is_le' #align fin.lt_iff_coe_lt_coe Fin.lt_iff_val_lt_val theorem le_iff_val_le_val {a b : Fin n} : a ≤ b ↔ (a : ℕ) ≤ b := Iff.rfl #align fin.le_iff_coe_le_coe Fin.le_iff_val_le_val #align fin.mk_lt_of_lt_coe Fin.mk_lt_of_lt_val #align fin.mk_le_of_le_coe Fin.mk_le_of_le_val /-- `a < b` as natural numbers if and only if `a < b` in `Fin n`. -/ @[norm_cast, simp] theorem val_fin_lt {n : ℕ} {a b : Fin n} : (a : ℕ) < (b : ℕ) ↔ a < b := Iff.rfl #align fin.coe_fin_lt Fin.val_fin_lt /-- `a ≤ b` as natural numbers if and only if `a ≤ b` in `Fin n`. -/ @[norm_cast, simp] theorem val_fin_le {n : ℕ} {a b : Fin n} : (a : ℕ) ≤ (b : ℕ) ↔ a ≤ b := Iff.rfl #align fin.coe_fin_le Fin.val_fin_le #align fin.mk_le_mk Fin.mk_le_mk #align fin.mk_lt_mk Fin.mk_lt_mk -- @[simp] -- Porting note (#10618): simp can prove this theorem min_val {a : Fin n} : min (a : ℕ) n = a := by simp #align fin.min_coe Fin.min_val -- @[simp] -- Porting note (#10618): simp can prove this theorem max_val {a : Fin n} : max (a : ℕ) n = n := by simp #align fin.max_coe Fin.max_val /-- The inclusion map `Fin n → ℕ` is an embedding. -/ @[simps apply] def valEmbedding : Fin n ↪ ℕ := ⟨val, val_injective⟩ #align fin.coe_embedding Fin.valEmbedding @[simp] theorem equivSubtype_symm_trans_valEmbedding : equivSubtype.symm.toEmbedding.trans valEmbedding = Embedding.subtype (· < n) := rfl #align fin.equiv_subtype_symm_trans_val_embedding Fin.equivSubtype_symm_trans_valEmbedding /-- Use the ordering on `Fin n` for checking recursive definitions. For example, the following definition is not accepted by the termination checker, unless we declare the `WellFoundedRelation` instance: ```lean def factorial {n : ℕ} : Fin n → ℕ \| ⟨0, _⟩ := 1 \| ⟨i + 1, hi⟩ := (i + 1) * factorial ⟨i, i.lt_succ_self.trans hi⟩ ``` -/ instance {n : ℕ} : WellFoundedRelation (Fin n) := measure (val : Fin n → ℕ) /-- Given a positive `n`, `Fin.ofNat' i` is `i % n` as an element of `Fin n`. -/ def ofNat'' [NeZero n] (i : ℕ) : Fin n := ⟨i % n, mod_lt _ n.pos_of_neZero⟩ #align fin.of_nat' Fin.ofNat''ₓ -- Porting note: `Fin.ofNat'` conflicts with something in core (there the hypothesis is `n > 0`), -- so for now we make this double-prime `''`. This is also the reason for the dubious translation. instance {n : ℕ} [NeZero n] : Zero (Fin n) := ⟨ofNat'' 0⟩ instance {n : ℕ} [NeZero n] : One (Fin n) := ⟨ofNat'' 1⟩ #align fin.coe_zero Fin.val_zero /-- The `Fin.val_zero` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ @[simp] theorem val_zero' (n : ℕ) [NeZero n] : ((0 : Fin n) : ℕ) = 0 := rfl #align fin.val_zero' Fin.val_zero' #align fin.mk_zero Fin.mk_zero /-- The `Fin.zero_le` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ @[simp] protected theorem zero_le' [NeZero n] (a : Fin n) : 0 ≤ a := Nat.zero_le a.val #align fin.zero_le Fin.zero_le' #align fin.zero_lt_one Fin.zero_lt_one #align fin.not_lt_zero Fin.not_lt_zero /-- The `Fin.pos_iff_ne_zero` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ theorem pos_iff_ne_zero' [NeZero n] (a : Fin n) : 0 < a ↔ a ≠ 0 := by rw [← val_fin_lt, val_zero', Nat.pos_iff_ne_zero, Ne, Ne, ext_iff, val_zero'] #align fin.pos_iff_ne_zero Fin.pos_iff_ne_zero' #align fin.eq_zero_or_eq_succ Fin.eq_zero_or_eq_succ #align fin.eq_succ_of_ne_zero Fin.eq_succ_of_ne_zero @[simp] lemma cast_eq_self (a : Fin n) : cast rfl a = a := rfl theorem rev_involutive : Involutive (rev : Fin n → Fin n) := fun i => ext <\| by dsimp only [rev] rw [← Nat.sub_sub, Nat.sub_sub_self (Nat.add_one_le_iff.2 i.is_lt), Nat.add_sub_cancel_right] #align fin.rev_involutive Fin.rev_involutive /-- `Fin.rev` as an `Equiv.Perm`, the antitone involution `Fin n → Fin n` given by `i ↦ n-(i+1)`. -/ @[simps! apply symm_apply] def revPerm : Equiv.Perm (Fin n) := Involutive.toPerm rev rev_involutive #align fin.rev Fin.revPerm #align fin.coe_rev Fin.val_revₓ theorem rev_injective : Injective (@rev n) := rev_involutive.injective #align fin.rev_injective Fin.rev_injective theorem rev_surjective : Surjective (@rev n) := rev_involutive.surjective #align fin.rev_surjective Fin.rev_surjective theorem rev_bijective : Bijective (@rev n) := rev_involutive.bijective #align fin.rev_bijective Fin.rev_bijective #align fin.rev_inj Fin.rev_injₓ #align fin.rev_rev Fin.rev_revₓ @[simp] theorem revPerm_symm : (@revPerm n).symm = revPerm := rfl #align fin.rev_symm Fin.revPerm_symm #align fin.rev_eq Fin.rev_eqₓ #align fin.rev_le_rev Fin.rev_le_revₓ #align fin.rev_lt_rev Fin.rev_lt_revₓ theorem cast_rev (i : Fin n) (h : n = m) : cast h i.rev = (i.cast h).rev := by subst h; simp theorem rev_eq_iff {i j : Fin n} : rev i = j ↔ i = rev j := by rw [← rev_inj, rev_rev] theorem rev_ne_iff {i j : Fin n} : rev i ≠ j ↔ i ≠ rev j := rev_eq_iff.not theorem rev_lt_iff {i j : Fin n} : rev i < j ↔ rev j < i := by rw [← rev_lt_rev, rev_rev] theorem rev_le_iff {i j : Fin n} : rev i ≤ j ↔ rev j ≤ i := by rw [← rev_le_rev, rev_rev] theorem lt_rev_iff {i j : Fin n} : i < rev j ↔ j < rev i := by rw [← rev_lt_rev, rev_rev] theorem le_rev_iff {i j : Fin n} : i ≤ rev j ↔ j ≤ rev i := by rw [← rev_le_rev, rev_rev] #align fin.last Fin.last #align fin.coe_last Fin.val_last -- Porting note: this is now syntactically equal to `val_last` #align fin.last_val Fin.val_last #align fin.le_last Fin.le_last #align fin.last_pos Fin.last_pos #align fin.eq_last_of_not_lt Fin.eq_last_of_not_lt theorem last_pos' [NeZero n] : 0 < last n := n.pos_of_neZero theorem one_lt_last [NeZero n] : 1 < last (n + 1) := Nat.lt_add_left_iff_pos.2 n.pos_of_neZero end Order section Add /-! ### addition, numerals, and coercion from Nat -/ #align fin.val_one Fin.val_one #align fin.coe_one Fin.val_one @[simp] theorem val_one' (n : ℕ) [NeZero n] : ((1 : Fin n) : ℕ) = 1 % n := rfl #align fin.coe_one' Fin.val_one' -- Porting note: Delete this lemma after porting theorem val_one'' {n : ℕ} : ((1 : Fin (n + 1)) : ℕ) = 1 % (n + 1) := rfl #align fin.one_val Fin.val_one'' #align fin.mk_one Fin.mk_one instance nontrivial {n : ℕ} : Nontrivial (Fin (n + 2)) where exists_pair_ne := ⟨0, 1, (ne_iff_vne 0 1).mpr (by simp [val_one, val_zero])⟩ theorem nontrivial_iff_two_le : Nontrivial (Fin n) ↔ 2 ≤ n := by rcases n with (_ \| _ \| n) <;> simp [← Nat.one_eq_succ_zero, Fin.nontrivial, not_nontrivial, Nat.succ_le_iff] -- Porting note: here and in the next lemma, had to use `← Nat.one_eq_succ_zero`. #align fin.nontrivial_iff_two_le Fin.nontrivial_iff_two_le #align fin.subsingleton_iff_le_one Fin.subsingleton_iff_le_one section Monoid -- Porting note (#10618): removing `simp`, `simp` can prove it with AddCommMonoid instance protected theorem add_zero [NeZero n] (k : Fin n) : k + 0 = k := by simp only [add_def, val_zero', Nat.add_zero, mod_eq_of_lt (is_lt k)] #align fin.add_zero Fin.add_zero -- Porting note (#10618): removing `simp`, `simp` can prove it with AddCommMonoid instance protected theorem zero_add [NeZero n] (k : Fin n) : 0 + k = k := by simp [ext_iff, add_def, mod_eq_of_lt (is_lt k)] #align fin.zero_add Fin.zero_add instance {a : ℕ} [NeZero n] : OfNat (Fin n) a where ofNat := Fin.ofNat' a n.pos_of_neZero instance inhabited (n : ℕ) [NeZero n] : Inhabited (Fin n) := ⟨0⟩ instance inhabitedFinOneAdd (n : ℕ) : Inhabited (Fin (1 + n)) := haveI : NeZero (1 + n) := by rw [Nat.add_comm]; infer_instance inferInstance @[simp] theorem default_eq_zero (n : ℕ) [NeZero n] : (default : Fin n) = 0 := rfl #align fin.default_eq_zero Fin.default_eq_zero section from_ad_hoc @[simp] lemma ofNat'_zero {h : 0 < n} [NeZero n] : (Fin.ofNat' 0 h : Fin n) = 0 := rfl @[simp] lemma ofNat'_one {h : 0 < n} [NeZero n] : (Fin.ofNat' 1 h : Fin n) = 1 := rfl end from_ad_hoc instance instNatCast [NeZero n] : NatCast (Fin n) where natCast n := Fin.ofNat'' n lemma natCast_def [NeZero n] (a : ℕ) : (a : Fin n) = ⟨a % n, mod_lt _ n.pos_of_neZero⟩ := rfl end Monoid #align fin.val_add Fin.val_add #align fin.coe_add Fin.val_add theorem val_add_eq_ite {n : ℕ} (a b : Fin n) : (↑(a + b) : ℕ) = if n ≤ a + b then a + b - n else a + b := by rw [Fin.val_add, Nat.add_mod_eq_ite, Nat.mod_eq_of_lt (show ↑a < n from a.2), Nat.mod_eq_of_lt (show ↑b < n from b.2)] #align fin.coe_add_eq_ite Fin.val_add_eq_ite section deprecated set_option linter.deprecated false @[deprecated] theorem val_bit0 {n : ℕ} (k : Fin n) : ((bit0 k : Fin n) : ℕ) = bit0 (k : ℕ) % n := by cases k rfl #align fin.coe_bit0 Fin.val_bit0 @[deprecated] theorem val_bit1 {n : ℕ} [NeZero n] (k : Fin n) : ((bit1 k : Fin n) : ℕ) = bit1 (k : ℕ) % n := by cases n; · cases' k with k h cases k · show _ % _ = _ simp at h cases' h with _ h simp [bit1, Fin.val_bit0, Fin.val_add, Fin.val_one] #align fin.coe_bit1 Fin.val_bit1 end deprecated #align fin.coe_add_one_of_lt Fin.val_add_one_of_lt #align fin.last_add_one Fin.last_add_one #align fin.coe_add_one Fin.val_add_one section Bit set_option linter.deprecated false @[simp, deprecated] theorem mk_bit0 {m n : ℕ} (h : bit0 m < n) : (⟨bit0 m, h⟩ : Fin n) = (bit0 ⟨m, (Nat.le_add_right m m).trans_lt h⟩ : Fin _) := eq_of_val_eq (Nat.mod_eq_of_lt h).symm #align fin.mk_bit0 Fin.mk_bit0 @[simp, deprecated] theorem mk_bit1 {m n : ℕ} [NeZero n] (h : bit1 m < n) : (⟨bit1 m, h⟩ : Fin n) = (bit1 ⟨m, (Nat.le_add_right m m).trans_lt ((m + m).lt_succ_self.trans h)⟩ : Fin _) := by ext simp only [bit1, bit0] at h simp only [bit1, bit0, val_add, val_one', ← Nat.add_mod, Nat.mod_eq_of_lt h] #align fin.mk_bit1 Fin.mk_bit1 end Bit #align fin.val_two Fin.val_two --- Porting note: syntactically the same as the above #align fin.coe_two Fin.val_two section OfNatCoe @[simp] theorem ofNat''_eq_cast (n : ℕ) [NeZero n] (a : ℕ) : (Fin.ofNat'' a : Fin n) = a := rfl #align fin.of_nat_eq_coe Fin.ofNat''_eq_cast @[simp] lemma val_natCast (a n : ℕ) [NeZero n] : (a : Fin n).val = a % n := rfl @[deprecated (since := "2024-04-17")] alias val_nat_cast := val_natCast -- Porting note: is this the right name for things involving `Nat.cast`? /-- Converting an in-range number to `Fin (n + 1)` produces a result whose value is the original number. -/ theorem val_cast_of_lt {n : ℕ} [NeZero n] {a : ℕ} (h : a < n) : (a : Fin n).val = a := Nat.mod_eq_of_lt h #align fin.coe_val_of_lt Fin.val_cast_of_lt /-- If `n` is non-zero, converting the value of a `Fin n` to `Fin n` results in the same value. -/ @[simp] theorem cast_val_eq_self {n : ℕ} [NeZero n] (a : Fin n) : (a.val : Fin n) = a := ext <\| val_cast_of_lt a.isLt #align fin.coe_val_eq_self Fin.cast_val_eq_self -- Porting note: this is syntactically the same as `val_cast_of_lt` #align fin.coe_coe_of_lt Fin.val_cast_of_lt -- Porting note: this is syntactically the same as `cast_val_of_lt` #align fin.coe_coe_eq_self Fin.cast_val_eq_self @[simp] lemma natCast_self (n : ℕ) [NeZero n] : (n : Fin n) = 0 := by ext; simp @[deprecated (since := "2024-04-17")] alias nat_cast_self := natCast_self @[simp] lemma natCast_eq_zero {a n : ℕ} [NeZero n] : (a : Fin n) = 0 ↔ n ∣ a := by simp [ext_iff, Nat.dvd_iff_mod_eq_zero] @[deprecated (since := "2024-04-17")] alias nat_cast_eq_zero := natCast_eq_zero @[simp] theorem natCast_eq_last (n) : (n : Fin (n + 1)) = Fin.last n := by ext; simp #align fin.coe_nat_eq_last Fin.natCast_eq_last @[deprecated (since := "2024-05-04")] alias cast_nat_eq_last := natCast_eq_last theorem le_val_last (i : Fin (n + 1)) : i ≤ n := by rw [Fin.natCast_eq_last] exact Fin.le_last i #align fin.le_coe_last Fin.le_val_last variable {a b : ℕ} lemma natCast_le_natCast (han : a ≤ n) (hbn : b ≤ n) : (a : Fin (n + 1)) ≤ b ↔ a ≤ b := by rw [← Nat.lt_succ_iff] at han hbn simp [le_iff_val_le_val, -val_fin_le, Nat.mod_eq_of_lt, han, hbn] lemma natCast_lt_natCast (han : a ≤ n) (hbn : b ≤ n) : (a : Fin (n + 1)) < b ↔ a < b := by rw [← Nat.lt_succ_iff] at han hbn; simp [lt_iff_val_lt_val, Nat.mod_eq_of_lt, han, hbn] lemma natCast_mono (hbn : b ≤ n) (hab : a ≤ b) : (a : Fin (n + 1)) ≤ b := (natCast_le_natCast (hab.trans hbn) hbn).2 hab lemma natCast_strictMono (hbn : b ≤ n) (hab : a < b) : (a : Fin (n + 1)) < b := (natCast_lt_natCast (hab.le.trans hbn) hbn).2 hab end OfNatCoe #align fin.add_one_pos Fin.add_one_pos #align fin.one_pos Fin.one_pos #align fin.zero_ne_one Fin.zero_ne_one @[simp] theorem one_eq_zero_iff [NeZero n] : (1 : Fin n) = 0 ↔ n = 1 := by obtain _ \| _ \| n := n <;> simp [Fin.ext_iff] #align fin.one_eq_zero_iff Fin.one_eq_zero_iff @[simp] theorem zero_eq_one_iff [NeZero n] : (0 : Fin n) = 1 ↔ n = 1 := by rw [eq_comm, one_eq_zero_iff] #align fin.zero_eq_one_iff Fin.zero_eq_one_iff end Add section Succ /-! ### succ and casts into larger Fin types -/ #align fin.coe_succ Fin.val_succ #align fin.succ_pos Fin.succ_pos lemma succ_injective (n : ℕ) : Injective (@Fin.succ n) := fun a b ↦ by simp [ext_iff] #align fin.succ_injective Fin.succ_injective /-- `Fin.succ` as an `Embedding` -/ def succEmb (n : ℕ) : Fin n ↪ Fin (n + 1) where toFun := succ inj' := succ_injective _ @[simp] theorem val_succEmb : ⇑(succEmb n) = Fin.succ := rfl #align fin.succ_le_succ_iff Fin.succ_le_succ_iff #align fin.succ_lt_succ_iff Fin.succ_lt_succ_iff @[simp] theorem exists_succ_eq {x : Fin (n + 1)} : (∃ y, Fin.succ y = x) ↔ x ≠ 0 := ⟨fun ⟨_, hy⟩ => hy ▸ succ_ne_zero _, x.cases (fun h => h.irrefl.elim) (fun _ _ => ⟨_, rfl⟩)⟩ #align fin.exists_succ_eq_iff Fin.exists_succ_eq theorem exists_succ_eq_of_ne_zero {x : Fin (n + 1)} (h : x ≠ 0) : ∃ y, Fin.succ y = x := exists_succ_eq.mpr h #align fin.succ_inj Fin.succ_inj #align fin.succ_ne_zero Fin.succ_ne_zero @[simp] theorem succ_zero_eq_one' [NeZero n] : Fin.succ (0 : Fin n) = 1 := by cases n · exact (NeZero.ne 0 rfl).elim · rfl #align fin.succ_zero_eq_one Fin.succ_zero_eq_one' theorem one_pos' [NeZero n] : (0 : Fin (n + 1)) < 1 := succ_zero_eq_one' (n := n) ▸ succ_pos _ theorem zero_ne_one' [NeZero n] : (0 : Fin (n + 1)) ≠ 1 := Fin.ne_of_lt one_pos' #align fin.succ_zero_eq_one' Fin.succ_zero_eq_one /-- The `Fin.succ_one_eq_two` in `Lean` only applies in `Fin (n+2)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ @[simp] theorem succ_one_eq_two' [NeZero n] : Fin.succ (1 : Fin (n + 1)) = 2 := by cases n · exact (NeZero.ne 0 rfl).elim · rfl #align fin.succ_one_eq_two Fin.succ_one_eq_two' -- Version of `succ_one_eq_two` to be used by `dsimp`. -- Note the `'` swapped around due to a move to std4. #align fin.succ_one_eq_two' Fin.succ_one_eq_two #align fin.succ_mk Fin.succ_mk #align fin.mk_succ_pos Fin.mk_succ_pos #align fin.one_lt_succ_succ Fin.one_lt_succ_succ #align fin.add_one_lt_iff Fin.add_one_lt_iff #align fin.add_one_le_iff Fin.add_one_le_iff #align fin.last_le_iff Fin.last_le_iff #align fin.lt_add_one_iff Fin.lt_add_one_iff /-- The `Fin.le_zero_iff` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ @[simp] theorem le_zero_iff' {n : ℕ} [NeZero n] {k : Fin n} : k ≤ 0 ↔ k = 0 := ⟨fun h => Fin.ext <\| by rw [Nat.eq_zero_of_le_zero h]; rfl, by rintro rfl; exact Nat.le_refl _⟩ #align fin.le_zero_iff Fin.le_zero_iff' #align fin.succ_succ_ne_one Fin.succ_succ_ne_one #align fin.cast_lt Fin.castLT #align fin.coe_cast_lt Fin.coe_castLT #align fin.cast_lt_mk Fin.castLT_mk -- Move to Batteries? @[simp] theorem cast_refl {n : Nat} (h : n = n) : Fin.cast h = id := rfl -- TODO: Move to Batteries @[simp] lemma castLE_inj {hmn : m ≤ n} {a b : Fin m} : castLE hmn a = castLE hmn b ↔ a = b := by simp [ext_iff] @[simp] lemma castAdd_inj {a b : Fin m} : castAdd n a = castAdd n b ↔ a = b := by simp [ext_iff] attribute [simp] castSucc_inj lemma castLE_injective (hmn : m ≤ n) : Injective (castLE hmn) := fun a b hab ↦ ext (by have := congr_arg val hab; exact this) lemma castAdd_injective (m n : ℕ) : Injective (@Fin.castAdd m n) := castLE_injective _ lemma castSucc_injective (n : ℕ) : Injective (@Fin.castSucc n) := castAdd_injective _ _ #align fin.cast_succ_injective Fin.castSucc_injective /-- `Fin.castLE` as an `Embedding`, `castLEEmb h i` embeds `i` into a larger `Fin` type. -/ @[simps! apply] def castLEEmb (h : n ≤ m) : Fin n ↪ Fin m where toFun := castLE h inj' := castLE_injective _ @[simp, norm_cast] lemma coe_castLEEmb {m n} (hmn : m ≤ n) : castLEEmb hmn = castLE hmn := rfl #align fin.coe_cast_le Fin.coe_castLE #align fin.cast_le_mk Fin.castLE_mk #align fin.cast_le_zero Fin.castLE_zero /- The next proof can be golfed a lot using `Fintype.card`. It is written this way to define `ENat.card` and `Nat.card` without a `Fintype` dependency (not done yet). -/ assert_not_exists Fintype lemma nonempty_embedding_iff : Nonempty (Fin n ↪ Fin m) ↔ n ≤ m := by refine ⟨fun h ↦ ?_, fun h ↦ ⟨castLEEmb h⟩⟩ induction n generalizing m with \| zero => exact m.zero_le \| succ n ihn => cases' h with e rcases exists_eq_succ_of_ne_zero (pos_iff_nonempty.2 (Nonempty.map e inferInstance)).ne' with ⟨m, rfl⟩ refine Nat.succ_le_succ <\| ihn ⟨?_⟩ refine ⟨fun i ↦ (e.setValue 0 0 i.succ).pred (mt e.setValue_eq_iff.1 i.succ_ne_zero), fun i j h ↦ ?_⟩ simpa only [pred_inj, EmbeddingLike.apply_eq_iff_eq, succ_inj] using h lemma equiv_iff_eq : Nonempty (Fin m ≃ Fin n) ↔ m = n := ⟨fun ⟨e⟩ ↦ le_antisymm (nonempty_embedding_iff.1 ⟨e⟩) (nonempty_embedding_iff.1 ⟨e.symm⟩), fun h ↦ h ▸ ⟨.refl _⟩⟩ #align fin.equiv_iff_eq Fin.equiv_iff_eq @[simp] lemma castLE_castSucc {n m} (i : Fin n) (h : n + 1 ≤ m) : i.castSucc.castLE h = i.castLE (Nat.le_of_succ_le h) := rfl @[simp] lemma castLE_comp_castSucc {n m} (h : n + 1 ≤ m) : Fin.castLE h ∘ Fin.castSucc = Fin.castLE (Nat.le_of_succ_le h) := rfl @[simp] lemma castLE_rfl (n : ℕ) : Fin.castLE (le_refl n) = id := rfl @[simp] theorem range_castLE {n k : ℕ} (h : n ≤ k) : Set.range (castLE h) = { i : Fin k \| (i : ℕ) < n } := Set.ext fun x => ⟨fun ⟨y, hy⟩ => hy ▸ y.2, fun hx => ⟨⟨x, hx⟩, Fin.ext rfl⟩⟩ #align fin.range_cast_le Fin.range_castLE @[simp] theorem coe_of_injective_castLE_symm {n k : ℕ} (h : n ≤ k) (i : Fin k) (hi) : ((Equiv.ofInjective _ (castLE_injective h)).symm ⟨i, hi⟩ : ℕ) = i := by rw [← coe_castLE h] exact congr_arg Fin.val (Equiv.apply_ofInjective_symm _ _) #align fin.coe_of_injective_cast_le_symm Fin.coe_of_injective_castLE_symm #align fin.cast_le_succ Fin.castLE_succ #align fin.cast_le_cast_le Fin.castLE_castLE #align fin.cast_le_comp_cast_le Fin.castLE_comp_castLE theorem leftInverse_cast (eq : n = m) : LeftInverse (cast eq.symm) (cast eq) := fun _ => rfl theorem rightInverse_cast (eq : n = m) : RightInverse (cast eq.symm) (cast eq) := fun _ => rfl theorem cast_le_cast (eq : n = m) {a b : Fin n} : cast eq a ≤ cast eq b ↔ a ≤ b := Iff.rfl /-- The 'identity' equivalence between `Fin m` and `Fin n` when `m = n`. -/ @[simps] def _root_.finCongr (eq : n = m) : Fin n ≃ Fin m where toFun := cast eq invFun := cast eq.symm left_inv := leftInverse_cast eq right_inv := rightInverse_cast eq #align fin_congr finCongr @[simp] lemma _root_.finCongr_apply_mk (h : m = n) (k : ℕ) (hk : k < m) : finCongr h ⟨k, hk⟩ = ⟨k, h ▸ hk⟩ := rfl #align fin_congr_apply_mk finCongr_apply_mk @[simp] lemma _root_.finCongr_refl (h : n = n := rfl) : finCongr h = Equiv.refl (Fin n) := by ext; simp @[simp] lemma _root_.finCongr_symm (h : m = n) : (finCongr h).symm = finCongr h.symm := rfl #align fin_congr_symm finCongr_symm @[simp] lemma _root_.finCongr_apply_coe (h : m = n) (k : Fin m) : (finCongr h k : ℕ) = k := rfl #align fin_congr_apply_coe finCongr_apply_coe lemma _root_.finCongr_symm_apply_coe (h : m = n) (k : Fin n) : ((finCongr h).symm k : ℕ) = k := rfl #align fin_congr_symm_apply_coe finCongr_symm_apply_coe /-- While in many cases `finCongr` is better than `Equiv.cast`/`cast`, sometimes we want to apply a generic theorem about `cast`. -/ lemma _root_.finCongr_eq_equivCast (h : n = m) : finCongr h = .cast (h ▸ rfl) := by subst h; simp #align fin.coe_cast Fin.coe_castₓ @[simp] theorem cast_zero {n' : ℕ} [NeZero n] {h : n = n'} : cast h (0 : Fin n) = by { haveI : NeZero n' := by {rw [← h]; infer_instance}; exact 0} := ext rfl #align fin.cast_zero Fin.cast_zero #align fin.cast_last Fin.cast_lastₓ #align fin.cast_mk Fin.cast_mkₓ #align fin.cast_trans Fin.cast_transₓ #align fin.cast_le_of_eq Fin.castLE_of_eq /-- While in many cases `Fin.cast` is better than `Equiv.cast`/`cast`, sometimes we want to apply a generic theorem about `cast`. -/ theorem cast_eq_cast (h : n = m) : (cast h : Fin n → Fin m) = _root_.cast (h ▸ rfl) := by subst h ext rfl #align fin.cast_eq_cast Fin.cast_eq_cast /-- `Fin.castAdd` as an `Embedding`, `castAddEmb m i` embeds `i : Fin n` in `Fin (n+m)`. See also `Fin.natAddEmb` and `Fin.addNatEmb`. -/ @[simps! apply] def castAddEmb (m) : Fin n ↪ Fin (n + m) := castLEEmb (le_add_right n m) #align fin.coe_cast_add Fin.coe_castAdd #align fin.cast_add_zero Fin.castAdd_zeroₓ #align fin.cast_add_lt Fin.castAdd_lt #align fin.cast_add_mk Fin.castAdd_mk #align fin.cast_add_cast_lt Fin.castAdd_castLT #align fin.cast_lt_cast_add Fin.castLT_castAdd #align fin.cast_add_cast Fin.castAdd_castₓ #align fin.cast_cast_add_left Fin.cast_castAdd_leftₓ #align fin.cast_cast_add_right Fin.cast_castAdd_rightₓ #align fin.cast_add_cast_add Fin.castAdd_castAdd #align fin.cast_succ_eq Fin.cast_succ_eqₓ #align fin.succ_cast_eq Fin.succ_cast_eqₓ /-- `Fin.castSucc` as an `Embedding`, `castSuccEmb i` embeds `i : Fin n` in `Fin (n+1)`. -/ @[simps! apply] def castSuccEmb : Fin n ↪ Fin (n + 1) := castAddEmb _ @[simp, norm_cast] lemma coe_castSuccEmb : (castSuccEmb : Fin n → Fin (n + 1)) = Fin.castSucc := rfl #align fin.coe_cast_succ Fin.coe_castSucc #align fin.cast_succ_mk Fin.castSucc_mk #align fin.cast_cast_succ Fin.cast_castSuccₓ #align fin.cast_succ_lt_succ Fin.castSucc_lt_succ #align fin.le_cast_succ_iff Fin.le_castSucc_iff #align fin.cast_succ_lt_iff_succ_le Fin.castSucc_lt_iff_succ_le #align fin.succ_last Fin.succ_last #align fin.succ_eq_last_succ Fin.succ_eq_last_succ #align fin.cast_succ_cast_lt Fin.castSucc_castLT #align fin.cast_lt_cast_succ Fin.castLT_castSucc #align fin.cast_succ_lt_cast_succ_iff Fin.castSucc_lt_castSucc_iff @[simp] theorem castSucc_le_castSucc_iff {a b : Fin n} : castSucc a ≤ castSucc b ↔ a ≤ b := Iff.rfl @[simp] theorem succ_le_castSucc_iff {a b : Fin n} : succ a ≤ castSucc b ↔ a < b := by rw [le_castSucc_iff, succ_lt_succ_iff] @[simp] theorem castSucc_lt_succ_iff {a b : Fin n} : castSucc a < succ b ↔ a ≤ b := by rw [castSucc_lt_iff_succ_le, succ_le_succ_iff] theorem le_of_castSucc_lt_of_succ_lt {a b : Fin (n + 1)} {i : Fin n} (hl : castSucc i < a) (hu : b < succ i) : b < a := by simp [Fin.lt_def, -val_fin_lt] at *; omega theorem castSucc_lt_or_lt_succ (p : Fin (n + 1)) (i : Fin n) : castSucc i < p ∨ p < i.succ := by simp [Fin.lt_def, -val_fin_lt]; omega #align fin.succ_above_lt_gt Fin.castSucc_lt_or_lt_succ @[deprecated] alias succAbove_lt_gt := castSucc_lt_or_lt_succ theorem succ_le_or_le_castSucc (p : Fin (n + 1)) (i : Fin n) : succ i ≤ p ∨ p ≤ i.castSucc := by rw [le_castSucc_iff, ← castSucc_lt_iff_succ_le] exact p.castSucc_lt_or_lt_succ i theorem exists_castSucc_eq_of_ne_last {x : Fin (n + 1)} (h : x ≠ (last _)) : ∃ y, Fin.castSucc y = x := exists_castSucc_eq.mpr h #align fin.cast_succ_inj Fin.castSucc_inj #align fin.cast_succ_lt_last Fin.castSucc_lt_last theorem forall_fin_succ' {P : Fin (n + 1) → Prop} : (∀ i, P i) ↔ (∀ i : Fin n, P i.castSucc) ∧ P (.last _) := ⟨fun H => ⟨fun _ => H _, H _⟩, fun ⟨H0, H1⟩ i => Fin.lastCases H1 H0 i⟩ -- to match `Fin.eq_zero_or_eq_succ` theorem eq_castSucc_or_eq_last {n : Nat} (i : Fin (n + 1)) : (∃ j : Fin n, i = j.castSucc) ∨ i = last n := i.lastCases (Or.inr rfl) (Or.inl ⟨·, rfl⟩) theorem exists_fin_succ' {P : Fin (n + 1) → Prop} : (∃ i, P i) ↔ (∃ i : Fin n, P i.castSucc) ∨ P (.last _) := ⟨fun ⟨i, h⟩ => Fin.lastCases Or.inr (fun i hi => Or.inl ⟨i, hi⟩) i h, fun h => h.elim (fun ⟨i, hi⟩ => ⟨i.castSucc, hi⟩) (fun h => ⟨.last _, h⟩)⟩ /-- The `Fin.castSucc_zero` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ @[simp] theorem castSucc_zero' [NeZero n] : castSucc (0 : Fin n) = 0 := ext rfl #align fin.cast_succ_zero Fin.castSucc_zero' #align fin.cast_succ_one Fin.castSucc_one /-- `castSucc i` is positive when `i` is positive. The `Fin.castSucc_pos` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ theorem castSucc_pos' [NeZero n] {i : Fin n} (h : 0 < i) : 0 < castSucc i := by simpa [lt_iff_val_lt_val] using h #align fin.cast_succ_pos Fin.castSucc_pos' /-- The `Fin.castSucc_eq_zero_iff` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ @[simp] theorem castSucc_eq_zero_iff' [NeZero n] (a : Fin n) : castSucc a = 0 ↔ a = 0 := Fin.ext_iff.trans <\| (Fin.ext_iff.trans <\| by simp).symm #align fin.cast_succ_eq_zero_iff Fin.castSucc_eq_zero_iff' /-- The `Fin.castSucc_ne_zero_iff` in `Lean` only applies in `Fin (n+1)`. This one instead uses a `NeZero n` typeclass hypothesis. -/ theorem castSucc_ne_zero_iff' [NeZero n] (a : Fin n) : castSucc a ≠ 0 ↔ a ≠ 0 := not_iff_not.mpr <\| castSucc_eq_zero_iff' a #align fin.cast_succ_ne_zero_iff Fin.castSucc_ne_zero_iff theorem castSucc_ne_zero_of_lt {p i : Fin n} (h : p < i) : castSucc i ≠ 0 := by cases n · exact i.elim0 · rw [castSucc_ne_zero_iff', Ne, ext_iff] exact ((zero_le _).trans_lt h).ne' theorem succ_ne_last_iff (a : Fin (n + 1)) : succ a ≠ last (n + 1) ↔ a ≠ last n := not_iff_not.mpr <\| succ_eq_last_succ a theorem succ_ne_last_of_lt {p i : Fin n} (h : i < p) : succ i ≠ last n := by cases n · exact i.elim0 · rw [succ_ne_last_iff, Ne, ext_iff] exact ((le_last _).trans_lt' h).ne #align fin.cast_succ_fin_succ Fin.castSucc_fin_succ @[norm_cast, simp] theorem coe_eq_castSucc {a : Fin n} : (a : Fin (n + 1)) = castSucc a := by ext exact val_cast_of_lt (Nat.lt.step a.is_lt) #align fin.coe_eq_cast_succ Fin.coe_eq_castSucc theorem coe_succ_lt_iff_lt {n : ℕ} {j k : Fin n} : (j : Fin <\| n + 1) < k ↔ j < k := by simp only [coe_eq_castSucc, castSucc_lt_castSucc_iff] #align fin.coe_succ_eq_succ Fin.coeSucc_eq_succ #align fin.lt_succ Fin.lt_succ @[simp] theorem range_castSucc {n : ℕ} : Set.range (castSucc : Fin n → Fin n.succ) = ({ i \| (i : ℕ) < n } : Set (Fin n.succ)) := range_castLE (by omega) #align fin.range_cast_succ Fin.range_castSucc @[simp] theorem coe_of_injective_castSucc_symm {n : ℕ} (i : Fin n.succ) (hi) : ((Equiv.ofInjective castSucc (castSucc_injective _)).symm ⟨i, hi⟩ : ℕ) = i := by rw [← coe_castSucc] exact congr_arg val (Equiv.apply_ofInjective_symm _ _) #align fin.coe_of_injective_cast_succ_symm Fin.coe_of_injective_castSucc_symm #align fin.succ_cast_succ Fin.succ_castSucc /-- `Fin.addNat` as an `Embedding`, `addNatEmb m i` adds `m` to `i`, generalizes `Fin.succ`. -/ @[simps! apply] def addNatEmb (m) : Fin n ↪ Fin (n + m) where toFun := (addNat · m) inj' a b := by simp [ext_iff] #align fin.coe_add_nat Fin.coe_addNat #align fin.add_nat_one Fin.addNat_one #align fin.le_coe_add_nat Fin.le_coe_addNat #align fin.add_nat_mk Fin.addNat_mk #align fin.cast_add_nat_zero Fin.cast_addNat_zeroₓ #align fin.add_nat_cast Fin.addNat_castₓ #align fin.cast_add_nat_left Fin.cast_addNat_leftₓ #align fin.cast_add_nat_right Fin.cast_addNat_rightₓ /-- `Fin.natAdd` as an `Embedding`, `natAddEmb n i` adds `n` to `i` "on the left". -/ @[simps! apply] def natAddEmb (n) {m} : Fin m ↪ Fin (n + m) where toFun := natAdd n inj' a b := by simp [ext_iff] #align fin.coe_nat_add Fin.coe_natAdd #align fin.nat_add_mk Fin.natAdd_mk #align fin.le_coe_nat_add Fin.le_coe_natAdd #align fin.nat_add_zero Fin.natAdd_zeroₓ #align fin.nat_add_cast Fin.natAdd_castₓ #align fin.cast_nat_add_right Fin.cast_natAdd_rightₓ #align fin.cast_nat_add_left Fin.cast_natAdd_leftₓ #align fin.cast_add_nat_add Fin.castAdd_natAddₓ #align fin.nat_add_cast_add Fin.natAdd_castAddₓ #align fin.nat_add_nat_add Fin.natAdd_natAddₓ #align fin.cast_nat_add_zero Fin.cast_natAdd_zeroₓ #align fin.cast_nat_add Fin.cast_natAddₓ #align fin.cast_add_nat Fin.cast_addNatₓ #align fin.nat_add_last Fin.natAdd_last #align fin.nat_add_cast_succ Fin.natAdd_castSucc end Succ section Pred /-! ### pred -/ #align fin.pred Fin.pred #align fin.coe_pred Fin.coe_pred #align fin.succ_pred Fin.succ_pred #align fin.pred_succ Fin.pred_succ #align fin.pred_eq_iff_eq_succ Fin.pred_eq_iff_eq_succ #align fin.pred_mk_succ Fin.pred_mk_succ #align fin.pred_mk Fin.pred_mk #align fin.pred_le_pred_iff Fin.pred_le_pred_iff #align fin.pred_lt_pred_iff Fin.pred_lt_pred_iff #align fin.pred_inj Fin.pred_inj #align fin.pred_one Fin.pred_one #align fin.pred_add_one Fin.pred_add_one #align fin.sub_nat Fin.subNat #align fin.coe_sub_nat Fin.coe_subNat #align fin.sub_nat_mk Fin.subNat_mk #align fin.pred_cast_succ_succ Fin.pred_castSucc_succ #align fin.add_nat_sub_nat Fin.addNat_subNat #align fin.sub_nat_add_nat Fin.subNat_addNat #align fin.nat_add_sub_nat_cast Fin.natAdd_subNat_castₓ theorem pred_one' [NeZero n] (h := (zero_ne_one' (n := n)).symm) : Fin.pred (1 : Fin (n + 1)) h = 0 := by simp_rw [Fin.ext_iff, coe_pred, val_one', val_zero', Nat.sub_eq_zero_iff_le, Nat.mod_le] theorem pred_last (h := ext_iff.not.2 last_pos'.ne') : pred (last (n + 1)) h = last n := by simp_rw [← succ_last, pred_succ] theorem pred_lt_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ 0) : pred i hi < j ↔ i < succ j := by rw [← succ_lt_succ_iff, succ_pred] theorem lt_pred_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ 0) : j < pred i hi ↔ succ j < i := by rw [← succ_lt_succ_iff, succ_pred] theorem pred_le_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ 0) : pred i hi ≤ j ↔ i ≤ succ j := by rw [← succ_le_succ_iff, succ_pred] theorem le_pred_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ 0) : j ≤ pred i hi ↔ succ j ≤ i := by rw [← succ_le_succ_iff, succ_pred] theorem castSucc_pred_eq_pred_castSucc {a : Fin (n + 1)} (ha : a ≠ 0) (ha' := a.castSucc_ne_zero_iff.mpr ha) : (a.pred ha).castSucc = (castSucc a).pred ha' := rfl #align fin.cast_succ_pred_eq_pred_cast_succ Fin.castSucc_pred_eq_pred_castSucc theorem castSucc_pred_add_one_eq {a : Fin (n + 1)} (ha : a ≠ 0) : (a.pred ha).castSucc + 1 = a := by cases' a using cases with a · exact (ha rfl).elim · rw [pred_succ, coeSucc_eq_succ]	Mathlib/Data/Fin/Basic.lean	1,137	1,139	theorem le_pred_castSucc_iff {a b : Fin (n + 1)} (ha : castSucc a ≠ 0) : b ≤ (castSucc a).pred ha ↔ b < a := by	rw [le_pred_iff, succ_le_castSucc_iff]
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel, Rémy Degenne, David Loeffler -/ import Mathlib.Analysis.SpecialFunctions.Pow.Complex import Qq #align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8" /-! # Power function on `ℝ` We construct the power functions `x ^ y`, where `x` and `y` are real numbers. -/ noncomputable section open scoped Classical open Real ComplexConjugate open Finset Set /- ## Definitions -/ namespace Real variable {x y z : ℝ} /-- The real power function `x ^ y`, defined as the real part of the complex power function. For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`, one sets `0 ^ 0=1` and `0 ^ y=0` for `y ≠ 0`. For `x < 0`, the definition is somewhat arbitrary as it depends on the choice of a complex determination of the logarithm. With our conventions, it is equal to `exp (y log x) cos (π y)`. -/ noncomputable def rpow (x y : ℝ) := ((x : ℂ) ^ (y : ℂ)).re #align real.rpow Real.rpow noncomputable instance : Pow ℝ ℝ := ⟨rpow⟩ @[simp] theorem rpow_eq_pow (x y : ℝ) : rpow x y = x ^ y := rfl #align real.rpow_eq_pow Real.rpow_eq_pow theorem rpow_def (x y : ℝ) : x ^ y = ((x : ℂ) ^ (y : ℂ)).re := rfl #align real.rpow_def Real.rpow_def theorem rpow_def_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) := by simp only [rpow_def, Complex.cpow_def]; split_ifs <;> simp_all [(Complex.ofReal_log hx).symm, -Complex.ofReal_mul, -RCLike.ofReal_mul, (Complex.ofReal_mul _ _).symm, Complex.exp_ofReal_re, Complex.ofReal_eq_zero] #align real.rpow_def_of_nonneg Real.rpow_def_of_nonneg theorem rpow_def_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : x ^ y = exp (log x * y) := by rw [rpow_def_of_nonneg (le_of_lt hx), if_neg (ne_of_gt hx)] #align real.rpow_def_of_pos Real.rpow_def_of_pos theorem exp_mul (x y : ℝ) : exp (x * y) = exp x ^ y := by rw [rpow_def_of_pos (exp_pos _), log_exp] #align real.exp_mul Real.exp_mul @[simp, norm_cast] theorem rpow_intCast (x : ℝ) (n : ℤ) : x ^ (n : ℝ) = x ^ n := by simp only [rpow_def, ← Complex.ofReal_zpow, Complex.cpow_intCast, Complex.ofReal_intCast, Complex.ofReal_re] #align real.rpow_int_cast Real.rpow_intCast @[deprecated (since := "2024-04-17")] alias rpow_int_cast := rpow_intCast @[simp, norm_cast] theorem rpow_natCast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n := by simpa using rpow_intCast x n #align real.rpow_nat_cast Real.rpow_natCast @[deprecated (since := "2024-04-17")] alias rpow_nat_cast := rpow_natCast @[simp] theorem exp_one_rpow (x : ℝ) : exp 1 ^ x = exp x := by rw [← exp_mul, one_mul] #align real.exp_one_rpow Real.exp_one_rpow @[simp] lemma exp_one_pow (n : ℕ) : exp 1 ^ n = exp n := by rw [← rpow_natCast, exp_one_rpow] theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by simp only [rpow_def_of_nonneg hx] split_ifs <;> simp [, exp_ne_zero] #align real.rpow_eq_zero_iff_of_nonneg Real.rpow_eq_zero_iff_of_nonneg @[simp] lemma rpow_eq_zero (hx : 0 ≤ x) (hy : y ≠ 0) : x ^ y = 0 ↔ x = 0 := by simp [rpow_eq_zero_iff_of_nonneg, ] @[simp] lemma rpow_ne_zero (hx : 0 ≤ x) (hy : y ≠ 0) : x ^ y ≠ 0 ↔ x ≠ 0 := Real.rpow_eq_zero hx hy \|>.not open Real theorem rpow_def_of_neg {x : ℝ} (hx : x < 0) (y : ℝ) : x ^ y = exp (log x * y) * cos (y * π) := by rw [rpow_def, Complex.cpow_def, if_neg] · have : Complex.log x * y = ↑(log (-x) * y) + ↑(y * π) * Complex.I := by simp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal, Complex.ofReal_mul] ring rw [this, Complex.exp_add_mul_I, ← Complex.ofReal_exp, ← Complex.ofReal_cos, ← Complex.ofReal_sin, mul_add, ← Complex.ofReal_mul, ← mul_assoc, ← Complex.ofReal_mul, Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.ofReal_im, Real.log_neg_eq_log] ring · rw [Complex.ofReal_eq_zero] exact ne_of_lt hx #align real.rpow_def_of_neg Real.rpow_def_of_neg theorem rpow_def_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℝ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * π) := by split_ifs with h <;> simp [rpow_def, ]; exact rpow_def_of_neg (lt_of_le_of_ne hx h) _ #align real.rpow_def_of_nonpos Real.rpow_def_of_nonpos theorem rpow_pos_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : 0 < x ^ y := by rw [rpow_def_of_pos hx]; apply exp_pos #align real.rpow_pos_of_pos Real.rpow_pos_of_pos @[simp] theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by simp [rpow_def] #align real.rpow_zero Real.rpow_zero theorem rpow_zero_pos (x : ℝ) : 0 < x ^ (0 : ℝ) := by simp @[simp] theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 := by simp [rpow_def, ] #align real.zero_rpow Real.zero_rpow theorem zero_rpow_eq_iff {x : ℝ} {a : ℝ} : 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by constructor · intro hyp simp only [rpow_def, Complex.ofReal_zero] at hyp by_cases h : x = 0 · subst h simp only [Complex.one_re, Complex.ofReal_zero, Complex.cpow_zero] at hyp exact Or.inr ⟨rfl, hyp.symm⟩ · rw [Complex.zero_cpow (Complex.ofReal_ne_zero.mpr h)] at hyp exact Or.inl ⟨h, hyp.symm⟩ · rintro (⟨h, rfl⟩ \| ⟨rfl, rfl⟩) · exact zero_rpow h · exact rpow_zero _ #align real.zero_rpow_eq_iff Real.zero_rpow_eq_iff theorem eq_zero_rpow_iff {x : ℝ} {a : ℝ} : a = 0 ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by rw [← zero_rpow_eq_iff, eq_comm] #align real.eq_zero_rpow_iff Real.eq_zero_rpow_iff @[simp] theorem rpow_one (x : ℝ) : x ^ (1 : ℝ) = x := by simp [rpow_def] #align real.rpow_one Real.rpow_one @[simp] theorem one_rpow (x : ℝ) : (1 : ℝ) ^ x = 1 := by simp [rpow_def] #align real.one_rpow Real.one_rpow theorem zero_rpow_le_one (x : ℝ) : (0 : ℝ) ^ x ≤ 1 := by by_cases h : x = 0 <;> simp [h, zero_le_one] #align real.zero_rpow_le_one Real.zero_rpow_le_one theorem zero_rpow_nonneg (x : ℝ) : 0 ≤ (0 : ℝ) ^ x := by by_cases h : x = 0 <;> simp [h, zero_le_one] #align real.zero_rpow_nonneg Real.zero_rpow_nonneg theorem rpow_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : 0 ≤ x ^ y := by rw [rpow_def_of_nonneg hx]; split_ifs <;> simp only [zero_le_one, le_refl, le_of_lt (exp_pos _)] #align real.rpow_nonneg_of_nonneg Real.rpow_nonneg theorem abs_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : \|x ^ y\| = \|x\| ^ y := by have h_rpow_nonneg : 0 ≤ x ^ y := Real.rpow_nonneg hx_nonneg _ rw [abs_eq_self.mpr hx_nonneg, abs_eq_self.mpr h_rpow_nonneg] #align real.abs_rpow_of_nonneg Real.abs_rpow_of_nonneg theorem abs_rpow_le_abs_rpow (x y : ℝ) : \|x ^ y\| ≤ \|x\| ^ y := by rcases le_or_lt 0 x with hx \| hx · rw [abs_rpow_of_nonneg hx] · rw [abs_of_neg hx, rpow_def_of_neg hx, rpow_def_of_pos (neg_pos.2 hx), log_neg_eq_log, abs_mul, abs_of_pos (exp_pos _)] exact mul_le_of_le_one_right (exp_pos _).le (abs_cos_le_one _) #align real.abs_rpow_le_abs_rpow Real.abs_rpow_le_abs_rpow theorem abs_rpow_le_exp_log_mul (x y : ℝ) : \|x ^ y\| ≤ exp (log x * y) := by refine (abs_rpow_le_abs_rpow x y).trans ?_ by_cases hx : x = 0 · by_cases hy : y = 0 <;> simp [hx, hy, zero_le_one] · rw [rpow_def_of_pos (abs_pos.2 hx), log_abs] #align real.abs_rpow_le_exp_log_mul Real.abs_rpow_le_exp_log_mul theorem norm_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : ‖x ^ y‖ = ‖x‖ ^ y := by simp_rw [Real.norm_eq_abs] exact abs_rpow_of_nonneg hx_nonneg #align real.norm_rpow_of_nonneg Real.norm_rpow_of_nonneg variable {w x y z : ℝ} theorem rpow_add (hx : 0 < x) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z := by simp only [rpow_def_of_pos hx, mul_add, exp_add] #align real.rpow_add Real.rpow_add theorem rpow_add' (hx : 0 ≤ x) (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := by rcases hx.eq_or_lt with (rfl \| pos) · rw [zero_rpow h, zero_eq_mul] have : y ≠ 0 ∨ z ≠ 0 := not_and_or.1 fun ⟨hy, hz⟩ => h <\| hy.symm ▸ hz.symm ▸ zero_add 0 exact this.imp zero_rpow zero_rpow · exact rpow_add pos _ _ #align real.rpow_add' Real.rpow_add' /-- Variant of `Real.rpow_add'` that avoids having to prove `y + z = w` twice. -/ lemma rpow_of_add_eq (hx : 0 ≤ x) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z := by rw [← h, rpow_add' hx]; rwa [h] theorem rpow_add_of_nonneg (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : x ^ (y + z) = x ^ y * x ^ z := by rcases hy.eq_or_lt with (rfl \| hy) · rw [zero_add, rpow_zero, one_mul] exact rpow_add' hx (ne_of_gt <\| add_pos_of_pos_of_nonneg hy hz) #align real.rpow_add_of_nonneg Real.rpow_add_of_nonneg /-- For `0 ≤ x`, the only problematic case in the equality `x ^ y * x ^ z = x ^ (y + z)` is for `x = 0` and `y + z = 0`, where the right hand side is `1` while the left hand side can vanish. The inequality is always true, though, and given in this lemma. -/ theorem le_rpow_add {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ y * x ^ z ≤ x ^ (y + z) := by rcases le_iff_eq_or_lt.1 hx with (H \| pos) · by_cases h : y + z = 0 · simp only [H.symm, h, rpow_zero] calc (0 : ℝ) ^ y * 0 ^ z ≤ 1 * 1 := mul_le_mul (zero_rpow_le_one y) (zero_rpow_le_one z) (zero_rpow_nonneg z) zero_le_one _ = 1 := by simp · simp [rpow_add', ← H, h] · simp [rpow_add pos] #align real.le_rpow_add Real.le_rpow_add theorem rpow_sum_of_pos {ι : Type} {a : ℝ} (ha : 0 < a) (f : ι → ℝ) (s : Finset ι) : (a ^ ∑ x ∈ s, f x) = ∏ x ∈ s, a ^ f x := map_sum (⟨⟨fun (x : ℝ) => (a ^ x : ℝ), rpow_zero a⟩, rpow_add ha⟩ : ℝ →+ (Additive ℝ)) f s #align real.rpow_sum_of_pos Real.rpow_sum_of_pos theorem rpow_sum_of_nonneg {ι : Type} {a : ℝ} (ha : 0 ≤ a) {s : Finset ι} {f : ι → ℝ} (h : ∀ x ∈ s, 0 ≤ f x) : (a ^ ∑ x ∈ s, f x) = ∏ x ∈ s, a ^ f x := by induction' s using Finset.cons_induction with i s hi ihs · rw [sum_empty, Finset.prod_empty, rpow_zero] · rw [forall_mem_cons] at h rw [sum_cons, prod_cons, ← ihs h.2, rpow_add_of_nonneg ha h.1 (sum_nonneg h.2)] #align real.rpow_sum_of_nonneg Real.rpow_sum_of_nonneg theorem rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ := by simp only [rpow_def_of_nonneg hx]; split_ifs <;> simp_all [exp_neg] #align real.rpow_neg Real.rpow_neg theorem rpow_sub {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z := by simp only [sub_eq_add_neg, rpow_add hx, rpow_neg (le_of_lt hx), div_eq_mul_inv] #align real.rpow_sub Real.rpow_sub theorem rpow_sub' {x : ℝ} (hx : 0 ≤ x) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := by simp only [sub_eq_add_neg] at h ⊢ simp only [rpow_add' hx h, rpow_neg hx, div_eq_mul_inv] #align real.rpow_sub' Real.rpow_sub' end Real /-! ## Comparing real and complex powers -/ namespace Complex theorem ofReal_cpow {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : ((x ^ y : ℝ) : ℂ) = (x : ℂ) ^ (y : ℂ) := by simp only [Real.rpow_def_of_nonneg hx, Complex.cpow_def, ofReal_eq_zero]; split_ifs <;> simp [Complex.ofReal_log hx] #align complex.of_real_cpow Complex.ofReal_cpow theorem ofReal_cpow_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℂ) : (x : ℂ) ^ y = (-x : ℂ) ^ y * exp (π * I * y) := by rcases hx.eq_or_lt with (rfl \| hlt) · rcases eq_or_ne y 0 with (rfl \| hy) <;> simp [] have hne : (x : ℂ) ≠ 0 := ofReal_ne_zero.mpr hlt.ne rw [cpow_def_of_ne_zero hne, cpow_def_of_ne_zero (neg_ne_zero.2 hne), ← exp_add, ← add_mul, log, log, abs.map_neg, arg_ofReal_of_neg hlt, ← ofReal_neg, arg_ofReal_of_nonneg (neg_nonneg.2 hx), ofReal_zero, zero_mul, add_zero] #align complex.of_real_cpow_of_nonpos Complex.ofReal_cpow_of_nonpos lemma cpow_ofReal (x : ℂ) (y : ℝ) : x ^ (y : ℂ) = ↑(abs x ^ y) (Real.cos (arg x * y) + Real.sin (arg x * y) * I) := by rcases eq_or_ne x 0 with rfl \| hx · simp [ofReal_cpow le_rfl] · rw [cpow_def_of_ne_zero hx, exp_eq_exp_re_mul_sin_add_cos, mul_comm (log x)] norm_cast rw [re_ofReal_mul, im_ofReal_mul, log_re, log_im, mul_comm y, mul_comm y, Real.exp_mul, Real.exp_log] rwa [abs.pos_iff] lemma cpow_ofReal_re (x : ℂ) (y : ℝ) : (x ^ (y : ℂ)).re = (abs x) ^ y * Real.cos (arg x * y) := by rw [cpow_ofReal]; generalize arg x * y = z; simp [Real.cos] lemma cpow_ofReal_im (x : ℂ) (y : ℝ) : (x ^ (y : ℂ)).im = (abs x) ^ y * Real.sin (arg x * y) := by rw [cpow_ofReal]; generalize arg x * y = z; simp [Real.sin] theorem abs_cpow_of_ne_zero {z : ℂ} (hz : z ≠ 0) (w : ℂ) : abs (z ^ w) = abs z ^ w.re / Real.exp (arg z * im w) := by rw [cpow_def_of_ne_zero hz, abs_exp, mul_re, log_re, log_im, Real.exp_sub, Real.rpow_def_of_pos (abs.pos hz)] #align complex.abs_cpow_of_ne_zero Complex.abs_cpow_of_ne_zero theorem abs_cpow_of_imp {z w : ℂ} (h : z = 0 → w.re = 0 → w = 0) : abs (z ^ w) = abs z ^ w.re / Real.exp (arg z * im w) := by rcases ne_or_eq z 0 with (hz \| rfl) <;> [exact abs_cpow_of_ne_zero hz w; rw [map_zero]] rcases eq_or_ne w.re 0 with hw \| hw · simp [hw, h rfl hw] · rw [Real.zero_rpow hw, zero_div, zero_cpow, map_zero] exact ne_of_apply_ne re hw #align complex.abs_cpow_of_imp Complex.abs_cpow_of_imp theorem abs_cpow_le (z w : ℂ) : abs (z ^ w) ≤ abs z ^ w.re / Real.exp (arg z * im w) := by by_cases h : z = 0 → w.re = 0 → w = 0 · exact (abs_cpow_of_imp h).le · push_neg at h simp [h] #align complex.abs_cpow_le Complex.abs_cpow_le @[simp] theorem abs_cpow_real (x : ℂ) (y : ℝ) : abs (x ^ (y : ℂ)) = Complex.abs x ^ y := by rw [abs_cpow_of_imp] <;> simp #align complex.abs_cpow_real Complex.abs_cpow_real @[simp] theorem abs_cpow_inv_nat (x : ℂ) (n : ℕ) : abs (x ^ (n⁻¹ : ℂ)) = Complex.abs x ^ (n⁻¹ : ℝ) := by rw [← abs_cpow_real]; simp [-abs_cpow_real] #align complex.abs_cpow_inv_nat Complex.abs_cpow_inv_nat theorem abs_cpow_eq_rpow_re_of_pos {x : ℝ} (hx : 0 < x) (y : ℂ) : abs (x ^ y) = x ^ y.re := by rw [abs_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), arg_ofReal_of_nonneg hx.le, zero_mul, Real.exp_zero, div_one, abs_of_nonneg hx.le] #align complex.abs_cpow_eq_rpow_re_of_pos Complex.abs_cpow_eq_rpow_re_of_pos theorem abs_cpow_eq_rpow_re_of_nonneg {x : ℝ} (hx : 0 ≤ x) {y : ℂ} (hy : re y ≠ 0) : abs (x ^ y) = x ^ re y := by rw [abs_cpow_of_imp] <;> simp [, arg_ofReal_of_nonneg, _root_.abs_of_nonneg] #align complex.abs_cpow_eq_rpow_re_of_nonneg Complex.abs_cpow_eq_rpow_re_of_nonneg lemma norm_natCast_cpow_of_re_ne_zero (n : ℕ) {s : ℂ} (hs : s.re ≠ 0) : ‖(n : ℂ) ^ s‖ = (n : ℝ) ^ (s.re) := by rw [norm_eq_abs, ← ofReal_natCast, abs_cpow_eq_rpow_re_of_nonneg n.cast_nonneg hs] lemma norm_natCast_cpow_of_pos {n : ℕ} (hn : 0 < n) (s : ℂ) : ‖(n : ℂ) ^ s‖ = (n : ℝ) ^ (s.re) := by rw [norm_eq_abs, ← ofReal_natCast, abs_cpow_eq_rpow_re_of_pos (Nat.cast_pos.mpr hn) _] lemma norm_natCast_cpow_pos_of_pos {n : ℕ} (hn : 0 < n) (s : ℂ) : 0 < ‖(n : ℂ) ^ s‖ := (norm_natCast_cpow_of_pos hn _).symm ▸ Real.rpow_pos_of_pos (Nat.cast_pos.mpr hn) _ theorem cpow_mul_ofReal_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (z : ℂ) : (x : ℂ) ^ (↑y z) = (↑(x ^ y) : ℂ) ^ z := by rw [cpow_mul, ofReal_cpow hx] · rw [← ofReal_log hx, ← ofReal_mul, ofReal_im, neg_lt_zero]; exact Real.pi_pos · rw [← ofReal_log hx, ← ofReal_mul, ofReal_im]; exact Real.pi_pos.le #align complex.cpow_mul_of_real_nonneg Complex.cpow_mul_ofReal_nonneg end Complex /-! ### Positivity extension -/ namespace Mathlib.Meta.Positivity open Lean Meta Qq /-- Extension for the `positivity` tactic: exponentiation by a real number is positive (namely 1) when the exponent is zero. The other cases are done in `evalRpow`. -/ @[positivity (_ : ℝ) ^ (0 : ℝ)] def evalRpowZero : PositivityExt where eval {u α} _ _ e := do match u, α, e with \| 0, ~q(ℝ), ~q($a ^ (0 : ℝ)) => assertInstancesCommute pure (.positive q(Real.rpow_zero_pos $a)) \| _, _, _ => throwError "not Real.rpow" /-- Extension for the `positivity` tactic: exponentiation by a real number is nonnegative when the base is nonnegative and positive when the base is positive. -/ @[positivity (_ : ℝ) ^ (_ : ℝ)] def evalRpow : PositivityExt where eval {u α} _zα _pα e := do match u, α, e with \| 0, ~q(ℝ), ~q($a ^ ($b : ℝ)) => let ra ← core q(inferInstance) q(inferInstance) a assertInstancesCommute match ra with \| .positive pa => pure (.positive q(Real.rpow_pos_of_pos $pa $b)) \| .nonnegative pa => pure (.nonnegative q(Real.rpow_nonneg $pa $b)) \| _ => pure .none \| _, _, _ => throwError "not Real.rpow" end Mathlib.Meta.Positivity /-! ## Further algebraic properties of `rpow` -/ namespace Real variable {x y z : ℝ} {n : ℕ} theorem rpow_mul {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z := by rw [← Complex.ofReal_inj, Complex.ofReal_cpow (rpow_nonneg hx _), Complex.ofReal_cpow hx, Complex.ofReal_mul, Complex.cpow_mul, Complex.ofReal_cpow hx] <;> simp only [(Complex.ofReal_mul _ _).symm, (Complex.ofReal_log hx).symm, Complex.ofReal_im, neg_lt_zero, pi_pos, le_of_lt pi_pos] #align real.rpow_mul Real.rpow_mul theorem rpow_add_int {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℤ) : x ^ (y + n) = x ^ y * x ^ n := by rw [rpow_def, rpow_def, Complex.ofReal_add, Complex.cpow_add _ _ (Complex.ofReal_ne_zero.mpr hx), Complex.ofReal_intCast, Complex.cpow_intCast, ← Complex.ofReal_zpow, mul_comm, Complex.re_ofReal_mul, mul_comm] #align real.rpow_add_int Real.rpow_add_int theorem rpow_add_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y + n) = x ^ y * x ^ n := by simpa using rpow_add_int hx y n #align real.rpow_add_nat Real.rpow_add_nat theorem rpow_sub_int {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n := by simpa using rpow_add_int hx y (-n) #align real.rpow_sub_int Real.rpow_sub_int theorem rpow_sub_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n := by simpa using rpow_sub_int hx y n #align real.rpow_sub_nat Real.rpow_sub_nat lemma rpow_add_int' (hx : 0 ≤ x) {n : ℤ} (h : y + n ≠ 0) : x ^ (y + n) = x ^ y * x ^ n := by rw [rpow_add' hx h, rpow_intCast] lemma rpow_add_nat' (hx : 0 ≤ x) (h : y + n ≠ 0) : x ^ (y + n) = x ^ y * x ^ n := by rw [rpow_add' hx h, rpow_natCast] lemma rpow_sub_int' (hx : 0 ≤ x) {n : ℤ} (h : y - n ≠ 0) : x ^ (y - n) = x ^ y / x ^ n := by rw [rpow_sub' hx h, rpow_intCast] lemma rpow_sub_nat' (hx : 0 ≤ x) (h : y - n ≠ 0) : x ^ (y - n) = x ^ y / x ^ n := by rw [rpow_sub' hx h, rpow_natCast] theorem rpow_add_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y + 1) = x ^ y * x := by simpa using rpow_add_nat hx y 1 #align real.rpow_add_one Real.rpow_add_one theorem rpow_sub_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y - 1) = x ^ y / x := by simpa using rpow_sub_nat hx y 1 #align real.rpow_sub_one Real.rpow_sub_one lemma rpow_add_one' (hx : 0 ≤ x) (h : y + 1 ≠ 0) : x ^ (y + 1) = x ^ y * x := by rw [rpow_add' hx h, rpow_one] lemma rpow_one_add' (hx : 0 ≤ x) (h : 1 + y ≠ 0) : x ^ (1 + y) = x * x ^ y := by rw [rpow_add' hx h, rpow_one] lemma rpow_sub_one' (hx : 0 ≤ x) (h : y - 1 ≠ 0) : x ^ (y - 1) = x ^ y / x := by rw [rpow_sub' hx h, rpow_one] lemma rpow_one_sub' (hx : 0 ≤ x) (h : 1 - y ≠ 0) : x ^ (1 - y) = x / x ^ y := by rw [rpow_sub' hx h, rpow_one] @[simp] theorem rpow_two (x : ℝ) : x ^ (2 : ℝ) = x ^ 2 := by rw [← rpow_natCast] simp only [Nat.cast_ofNat] #align real.rpow_two Real.rpow_two theorem rpow_neg_one (x : ℝ) : x ^ (-1 : ℝ) = x⁻¹ := by suffices H : x ^ ((-1 : ℤ) : ℝ) = x⁻¹ by rwa [Int.cast_neg, Int.cast_one] at H simp only [rpow_intCast, zpow_one, zpow_neg] #align real.rpow_neg_one Real.rpow_neg_one theorem mul_rpow (hx : 0 ≤ x) (hy : 0 ≤ y) : (x * y) ^ z = x ^ z * y ^ z := by iterate 2 rw [Real.rpow_def_of_nonneg]; split_ifs with h_ifs <;> simp_all · rw [log_mul ‹_› ‹_›, add_mul, exp_add, rpow_def_of_pos (hy.lt_of_ne' ‹_›)] all_goals positivity #align real.mul_rpow Real.mul_rpow theorem inv_rpow (hx : 0 ≤ x) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ := by simp only [← rpow_neg_one, ← rpow_mul hx, mul_comm] #align real.inv_rpow Real.inv_rpow theorem div_rpow (hx : 0 ≤ x) (hy : 0 ≤ y) (z : ℝ) : (x / y) ^ z = x ^ z / y ^ z := by simp only [div_eq_mul_inv, mul_rpow hx (inv_nonneg.2 hy), inv_rpow hy] #align real.div_rpow Real.div_rpow theorem log_rpow {x : ℝ} (hx : 0 < x) (y : ℝ) : log (x ^ y) = y * log x := by apply exp_injective rw [exp_log (rpow_pos_of_pos hx y), ← exp_log hx, mul_comm, rpow_def_of_pos (exp_pos (log x)) y] #align real.log_rpow Real.log_rpow theorem mul_log_eq_log_iff {x y z : ℝ} (hx : 0 < x) (hz : 0 < z) : y * log x = log z ↔ x ^ y = z := ⟨fun h ↦ log_injOn_pos (rpow_pos_of_pos hx _) hz <\| log_rpow hx _ \|>.trans h, by rintro rfl; rw [log_rpow hx]⟩ @[simp] lemma rpow_rpow_inv (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y) ^ y⁻¹ = x := by rw [← rpow_mul hx, mul_inv_cancel hy, rpow_one] @[simp] lemma rpow_inv_rpow (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y⁻¹) ^ y = x := by rw [← rpow_mul hx, inv_mul_cancel hy, rpow_one] theorem pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x := by have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn rw [← rpow_natCast, ← rpow_mul hx, mul_inv_cancel hn0, rpow_one] #align real.pow_nat_rpow_nat_inv Real.pow_rpow_inv_natCast theorem rpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x := by have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn rw [← rpow_natCast, ← rpow_mul hx, inv_mul_cancel hn0, rpow_one] #align real.rpow_nat_inv_pow_nat Real.rpow_inv_natCast_pow lemma rpow_natCast_mul (hx : 0 ≤ x) (n : ℕ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z := by rw [rpow_mul hx, rpow_natCast] lemma rpow_mul_natCast (hx : 0 ≤ x) (y : ℝ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n := by rw [rpow_mul hx, rpow_natCast] lemma rpow_intCast_mul (hx : 0 ≤ x) (n : ℤ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z := by rw [rpow_mul hx, rpow_intCast] lemma rpow_mul_intCast (hx : 0 ≤ x) (y : ℝ) (n : ℤ) : x ^ (y * n) = (x ^ y) ^ n := by rw [rpow_mul hx, rpow_intCast] /-! Note: lemmas about `(∏ i ∈ s, f i ^ r)` such as `Real.finset_prod_rpow` are proved in `Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean` instead. -/ /-! ## Order and monotonicity -/ @[gcongr] theorem rpow_lt_rpow (hx : 0 ≤ x) (hxy : x < y) (hz : 0 < z) : x ^ z < y ^ z := by rw [le_iff_eq_or_lt] at hx; cases' hx with hx hx · rw [← hx, zero_rpow (ne_of_gt hz)] exact rpow_pos_of_pos (by rwa [← hx] at hxy) _ · rw [rpow_def_of_pos hx, rpow_def_of_pos (lt_trans hx hxy), exp_lt_exp] exact mul_lt_mul_of_pos_right (log_lt_log hx hxy) hz #align real.rpow_lt_rpow Real.rpow_lt_rpow theorem strictMonoOn_rpow_Ici_of_exponent_pos {r : ℝ} (hr : 0 < r) : StrictMonoOn (fun (x : ℝ) => x ^ r) (Set.Ici 0) := fun _ ha _ _ hab => rpow_lt_rpow ha hab hr @[gcongr] theorem rpow_le_rpow {x y z : ℝ} (h : 0 ≤ x) (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z := by rcases eq_or_lt_of_le h₁ with (rfl \| h₁'); · rfl rcases eq_or_lt_of_le h₂ with (rfl \| h₂'); · simp exact le_of_lt (rpow_lt_rpow h h₁' h₂') #align real.rpow_le_rpow Real.rpow_le_rpow theorem monotoneOn_rpow_Ici_of_exponent_nonneg {r : ℝ} (hr : 0 ≤ r) : MonotoneOn (fun (x : ℝ) => x ^ r) (Set.Ici 0) := fun _ ha _ _ hab => rpow_le_rpow ha hab hr lemma rpow_lt_rpow_of_neg (hx : 0 < x) (hxy : x < y) (hz : z < 0) : y ^ z < x ^ z := by have := hx.trans hxy rw [← inv_lt_inv, ← rpow_neg, ← rpow_neg] on_goal 1 => refine rpow_lt_rpow ?_ hxy (neg_pos.2 hz) all_goals positivity lemma rpow_le_rpow_of_nonpos (hx : 0 < x) (hxy : x ≤ y) (hz : z ≤ 0) : y ^ z ≤ x ^ z := by have := hx.trans_le hxy rw [← inv_le_inv, ← rpow_neg, ← rpow_neg] on_goal 1 => refine rpow_le_rpow ?_ hxy (neg_nonneg.2 hz) all_goals positivity theorem rpow_lt_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z < y ^ z ↔ x < y := ⟨lt_imp_lt_of_le_imp_le fun h => rpow_le_rpow hy h (le_of_lt hz), fun h => rpow_lt_rpow hx h hz⟩ #align real.rpow_lt_rpow_iff Real.rpow_lt_rpow_iff theorem rpow_le_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y := le_iff_le_iff_lt_iff_lt.2 <\| rpow_lt_rpow_iff hy hx hz #align real.rpow_le_rpow_iff Real.rpow_le_rpow_iff lemma rpow_lt_rpow_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z < y ^ z ↔ y < x := ⟨lt_imp_lt_of_le_imp_le fun h ↦ rpow_le_rpow_of_nonpos hx h hz.le, fun h ↦ rpow_lt_rpow_of_neg hy h hz⟩ lemma rpow_le_rpow_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z ≤ y ^ z ↔ y ≤ x := le_iff_le_iff_lt_iff_lt.2 <\| rpow_lt_rpow_iff_of_neg hy hx hz lemma le_rpow_inv_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y := by rw [← rpow_le_rpow_iff hx _ hz, rpow_inv_rpow] <;> positivity lemma rpow_inv_le_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z := by rw [← rpow_le_rpow_iff _ hy hz, rpow_inv_rpow] <;> positivity lemma lt_rpow_inv_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x < y ^ z⁻¹ ↔ x ^ z < y := lt_iff_lt_of_le_iff_le <\| rpow_inv_le_iff_of_pos hy hx hz lemma rpow_inv_lt_iff_of_pos (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z⁻¹ < y ↔ x < y ^ z := lt_iff_lt_of_le_iff_le <\| le_rpow_inv_iff_of_pos hy hx hz theorem le_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z := by rw [← rpow_le_rpow_iff_of_neg _ hx hz, rpow_inv_rpow _ hz.ne] <;> positivity #align real.le_rpow_inv_iff_of_neg Real.le_rpow_inv_iff_of_neg theorem lt_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x < y ^ z⁻¹ ↔ y < x ^ z := by rw [← rpow_lt_rpow_iff_of_neg _ hx hz, rpow_inv_rpow _ hz.ne] <;> positivity #align real.lt_rpow_inv_iff_of_neg Real.lt_rpow_inv_iff_of_neg theorem rpow_inv_lt_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z⁻¹ < y ↔ y ^ z < x := by rw [← rpow_lt_rpow_iff_of_neg hy _ hz, rpow_inv_rpow _ hz.ne] <;> positivity #align real.rpow_inv_lt_iff_of_neg Real.rpow_inv_lt_iff_of_neg	Mathlib/Analysis/SpecialFunctions/Pow/Real.lean	616	618	theorem rpow_inv_le_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z⁻¹ ≤ y ↔ y ^ z ≤ x := by	rw [← rpow_le_rpow_iff_of_neg hy _ hz, rpow_inv_rpow _ hz.ne] <;> positivity
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel, Rémy Degenne, David Loeffler -/ import Mathlib.Analysis.SpecialFunctions.Pow.Real #align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8" /-! # Power function on `ℝ≥0` and `ℝ≥0∞` We construct the power functions `x ^ y` where * `x` is a nonnegative real number and `y` is a real number; * `x` is a number from `[0, +∞]` (a.k.a. `ℝ≥0∞`) and `y` is a real number. We also prove basic properties of these functions. -/ noncomputable section open scoped Classical open Real NNReal ENNReal ComplexConjugate open Finset Function Set namespace NNReal variable {w x y z : ℝ} /-- The nonnegative real power function `x^y`, defined for `x : ℝ≥0` and `y : ℝ` as the restriction of the real power function. For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`, one sets `0 ^ 0 = 1` and `0 ^ y = 0` for `y ≠ 0`. -/ noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 := ⟨(x : ℝ) ^ y, Real.rpow_nonneg x.2 y⟩ #align nnreal.rpow NNReal.rpow noncomputable instance : Pow ℝ≥0 ℝ := ⟨rpow⟩ @[simp] theorem rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y := rfl #align nnreal.rpow_eq_pow NNReal.rpow_eq_pow @[simp, norm_cast] theorem coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y := rfl #align nnreal.coe_rpow NNReal.coe_rpow @[simp] theorem rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 := NNReal.eq <\| Real.rpow_zero _ #align nnreal.rpow_zero NNReal.rpow_zero @[simp] theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero] exact Real.rpow_eq_zero_iff_of_nonneg x.2 #align nnreal.rpow_eq_zero_iff NNReal.rpow_eq_zero_iff @[simp] theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 := NNReal.eq <\| Real.zero_rpow h #align nnreal.zero_rpow NNReal.zero_rpow @[simp] theorem rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x := NNReal.eq <\| Real.rpow_one _ #align nnreal.rpow_one NNReal.rpow_one @[simp] theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 := NNReal.eq <\| Real.one_rpow _ #align nnreal.one_rpow NNReal.one_rpow theorem rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z := NNReal.eq <\| Real.rpow_add (pos_iff_ne_zero.2 hx) _ _ #align nnreal.rpow_add NNReal.rpow_add theorem rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := NNReal.eq <\| Real.rpow_add' x.2 h #align nnreal.rpow_add' NNReal.rpow_add' /-- Variant of `NNReal.rpow_add'` that avoids having to prove `y + z = w` twice. -/ lemma rpow_of_add_eq (x : ℝ≥0) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z := by rw [← h, rpow_add']; rwa [h] theorem rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z := NNReal.eq <\| Real.rpow_mul x.2 y z #align nnreal.rpow_mul NNReal.rpow_mul theorem rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ := NNReal.eq <\| Real.rpow_neg x.2 _ #align nnreal.rpow_neg NNReal.rpow_neg theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by simp [rpow_neg] #align nnreal.rpow_neg_one NNReal.rpow_neg_one theorem rpow_sub {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z := NNReal.eq <\| Real.rpow_sub (pos_iff_ne_zero.2 hx) y z #align nnreal.rpow_sub NNReal.rpow_sub theorem rpow_sub' (x : ℝ≥0) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := NNReal.eq <\| Real.rpow_sub' x.2 h #align nnreal.rpow_sub' NNReal.rpow_sub' theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by field_simp [← rpow_mul] #align nnreal.rpow_inv_rpow_self NNReal.rpow_inv_rpow_self theorem rpow_self_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ (1 / y)) ^ y = x := by field_simp [← rpow_mul] #align nnreal.rpow_self_rpow_inv NNReal.rpow_self_rpow_inv theorem inv_rpow (x : ℝ≥0) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ := NNReal.eq <\| Real.inv_rpow x.2 y #align nnreal.inv_rpow NNReal.inv_rpow theorem div_rpow (x y : ℝ≥0) (z : ℝ) : (x / y) ^ z = x ^ z / y ^ z := NNReal.eq <\| Real.div_rpow x.2 y.2 z #align nnreal.div_rpow NNReal.div_rpow theorem sqrt_eq_rpow (x : ℝ≥0) : sqrt x = x ^ (1 / (2 : ℝ)) := by refine NNReal.eq ?_ push_cast exact Real.sqrt_eq_rpow x.1 #align nnreal.sqrt_eq_rpow NNReal.sqrt_eq_rpow @[simp, norm_cast] theorem rpow_natCast (x : ℝ≥0) (n : ℕ) : x ^ (n : ℝ) = x ^ n := NNReal.eq <\| by simpa only [coe_rpow, coe_pow] using Real.rpow_natCast x n #align nnreal.rpow_nat_cast NNReal.rpow_natCast @[deprecated (since := "2024-04-17")] alias rpow_nat_cast := rpow_natCast @[simp] lemma rpow_ofNat (x : ℝ≥0) (n : ℕ) [n.AtLeastTwo] : x ^ (no_index (OfNat.ofNat n) : ℝ) = x ^ (OfNat.ofNat n : ℕ) := rpow_natCast x n theorem rpow_two (x : ℝ≥0) : x ^ (2 : ℝ) = x ^ 2 := rpow_ofNat x 2 #align nnreal.rpow_two NNReal.rpow_two theorem mul_rpow {x y : ℝ≥0} {z : ℝ} : (x * y) ^ z = x ^ z * y ^ z := NNReal.eq <\| Real.mul_rpow x.2 y.2 #align nnreal.mul_rpow NNReal.mul_rpow /-- `rpow` as a `MonoidHom`-/ @[simps] def rpowMonoidHom (r : ℝ) : ℝ≥0 →* ℝ≥0 where toFun := (· ^ r) map_one' := one_rpow _ map_mul' _x _y := mul_rpow /-- `rpow` variant of `List.prod_map_pow` for `ℝ≥0`-/ theorem list_prod_map_rpow (l : List ℝ≥0) (r : ℝ) : (l.map (· ^ r)).prod = l.prod ^ r := l.prod_hom (rpowMonoidHom r) theorem list_prod_map_rpow' {ι} (l : List ι) (f : ι → ℝ≥0) (r : ℝ) : (l.map (f · ^ r)).prod = (l.map f).prod ^ r := by rw [← list_prod_map_rpow, List.map_map]; rfl /-- `rpow` version of `Multiset.prod_map_pow` for `ℝ≥0`. -/ lemma multiset_prod_map_rpow {ι} (s : Multiset ι) (f : ι → ℝ≥0) (r : ℝ) : (s.map (f · ^ r)).prod = (s.map f).prod ^ r := s.prod_hom' (rpowMonoidHom r) _ /-- `rpow` version of `Finset.prod_pow` for `ℝ≥0`. -/ lemma finset_prod_rpow {ι} (s : Finset ι) (f : ι → ℝ≥0) (r : ℝ) : (∏ i ∈ s, f i ^ r) = (∏ i ∈ s, f i) ^ r := multiset_prod_map_rpow _ _ _ -- note: these don't really belong here, but they're much easier to prove in terms of the above section Real /-- `rpow` version of `List.prod_map_pow` for `Real`. -/ theorem _root_.Real.list_prod_map_rpow (l : List ℝ) (hl : ∀ x ∈ l, (0 : ℝ) ≤ x) (r : ℝ) : (l.map (· ^ r)).prod = l.prod ^ r := by lift l to List ℝ≥0 using hl have := congr_arg ((↑) : ℝ≥0 → ℝ) (NNReal.list_prod_map_rpow l r) push_cast at this rw [List.map_map] at this ⊢ exact mod_cast this theorem _root_.Real.list_prod_map_rpow' {ι} (l : List ι) (f : ι → ℝ) (hl : ∀ i ∈ l, (0 : ℝ) ≤ f i) (r : ℝ) : (l.map (f · ^ r)).prod = (l.map f).prod ^ r := by rw [← Real.list_prod_map_rpow (l.map f) _ r, List.map_map] · rfl simpa using hl /-- `rpow` version of `Multiset.prod_map_pow`. -/ theorem _root_.Real.multiset_prod_map_rpow {ι} (s : Multiset ι) (f : ι → ℝ) (hs : ∀ i ∈ s, (0 : ℝ) ≤ f i) (r : ℝ) : (s.map (f · ^ r)).prod = (s.map f).prod ^ r := by induction' s using Quotient.inductionOn with l simpa using Real.list_prod_map_rpow' l f hs r /-- `rpow` version of `Finset.prod_pow`. -/ theorem _root_.Real.finset_prod_rpow {ι} (s : Finset ι) (f : ι → ℝ) (hs : ∀ i ∈ s, 0 ≤ f i) (r : ℝ) : (∏ i ∈ s, f i ^ r) = (∏ i ∈ s, f i) ^ r := Real.multiset_prod_map_rpow s.val f hs r end Real @[gcongr] theorem rpow_le_rpow {x y : ℝ≥0} {z : ℝ} (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z := Real.rpow_le_rpow x.2 h₁ h₂ #align nnreal.rpow_le_rpow NNReal.rpow_le_rpow @[gcongr] theorem rpow_lt_rpow {x y : ℝ≥0} {z : ℝ} (h₁ : x < y) (h₂ : 0 < z) : x ^ z < y ^ z := Real.rpow_lt_rpow x.2 h₁ h₂ #align nnreal.rpow_lt_rpow NNReal.rpow_lt_rpow theorem rpow_lt_rpow_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ z < y ^ z ↔ x < y := Real.rpow_lt_rpow_iff x.2 y.2 hz #align nnreal.rpow_lt_rpow_iff NNReal.rpow_lt_rpow_iff theorem rpow_le_rpow_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y := Real.rpow_le_rpow_iff x.2 y.2 hz #align nnreal.rpow_le_rpow_iff NNReal.rpow_le_rpow_iff theorem le_rpow_one_div_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ≤ y ^ (1 / z) ↔ x ^ z ≤ y := by rw [← rpow_le_rpow_iff hz, rpow_self_rpow_inv hz.ne'] #align nnreal.le_rpow_one_div_iff NNReal.le_rpow_one_div_iff theorem rpow_one_div_le_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ (1 / z) ≤ y ↔ x ≤ y ^ z := by rw [← rpow_le_rpow_iff hz, rpow_self_rpow_inv hz.ne'] #align nnreal.rpow_one_div_le_iff NNReal.rpow_one_div_le_iff @[gcongr] theorem rpow_lt_rpow_of_exponent_lt {x : ℝ≥0} {y z : ℝ} (hx : 1 < x) (hyz : y < z) : x ^ y < x ^ z := Real.rpow_lt_rpow_of_exponent_lt hx hyz #align nnreal.rpow_lt_rpow_of_exponent_lt NNReal.rpow_lt_rpow_of_exponent_lt @[gcongr] theorem rpow_le_rpow_of_exponent_le {x : ℝ≥0} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) : x ^ y ≤ x ^ z := Real.rpow_le_rpow_of_exponent_le hx hyz #align nnreal.rpow_le_rpow_of_exponent_le NNReal.rpow_le_rpow_of_exponent_le theorem rpow_lt_rpow_of_exponent_gt {x : ℝ≥0} {y z : ℝ} (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y) : x ^ y < x ^ z := Real.rpow_lt_rpow_of_exponent_gt hx0 hx1 hyz #align nnreal.rpow_lt_rpow_of_exponent_gt NNReal.rpow_lt_rpow_of_exponent_gt theorem rpow_le_rpow_of_exponent_ge {x : ℝ≥0} {y z : ℝ} (hx0 : 0 < x) (hx1 : x ≤ 1) (hyz : z ≤ y) : x ^ y ≤ x ^ z := Real.rpow_le_rpow_of_exponent_ge hx0 hx1 hyz #align nnreal.rpow_le_rpow_of_exponent_ge NNReal.rpow_le_rpow_of_exponent_ge theorem rpow_pos {p : ℝ} {x : ℝ≥0} (hx_pos : 0 < x) : 0 < x ^ p := by have rpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p := by intro p hp_pos rw [← zero_rpow hp_pos.ne'] exact rpow_lt_rpow hx_pos hp_pos rcases lt_trichotomy (0 : ℝ) p with (hp_pos \| rfl \| hp_neg) · exact rpow_pos_of_nonneg hp_pos · simp only [zero_lt_one, rpow_zero] · rw [← neg_neg p, rpow_neg, inv_pos] exact rpow_pos_of_nonneg (neg_pos.mpr hp_neg) #align nnreal.rpow_pos NNReal.rpow_pos theorem rpow_lt_one {x : ℝ≥0} {z : ℝ} (hx1 : x < 1) (hz : 0 < z) : x ^ z < 1 := Real.rpow_lt_one (coe_nonneg x) hx1 hz #align nnreal.rpow_lt_one NNReal.rpow_lt_one theorem rpow_le_one {x : ℝ≥0} {z : ℝ} (hx2 : x ≤ 1) (hz : 0 ≤ z) : x ^ z ≤ 1 := Real.rpow_le_one x.2 hx2 hz #align nnreal.rpow_le_one NNReal.rpow_le_one theorem rpow_lt_one_of_one_lt_of_neg {x : ℝ≥0} {z : ℝ} (hx : 1 < x) (hz : z < 0) : x ^ z < 1 := Real.rpow_lt_one_of_one_lt_of_neg hx hz #align nnreal.rpow_lt_one_of_one_lt_of_neg NNReal.rpow_lt_one_of_one_lt_of_neg theorem rpow_le_one_of_one_le_of_nonpos {x : ℝ≥0} {z : ℝ} (hx : 1 ≤ x) (hz : z ≤ 0) : x ^ z ≤ 1 := Real.rpow_le_one_of_one_le_of_nonpos hx hz #align nnreal.rpow_le_one_of_one_le_of_nonpos NNReal.rpow_le_one_of_one_le_of_nonpos theorem one_lt_rpow {x : ℝ≥0} {z : ℝ} (hx : 1 < x) (hz : 0 < z) : 1 < x ^ z := Real.one_lt_rpow hx hz #align nnreal.one_lt_rpow NNReal.one_lt_rpow theorem one_le_rpow {x : ℝ≥0} {z : ℝ} (h : 1 ≤ x) (h₁ : 0 ≤ z) : 1 ≤ x ^ z := Real.one_le_rpow h h₁ #align nnreal.one_le_rpow NNReal.one_le_rpow theorem one_lt_rpow_of_pos_of_lt_one_of_neg {x : ℝ≥0} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1) (hz : z < 0) : 1 < x ^ z := Real.one_lt_rpow_of_pos_of_lt_one_of_neg hx1 hx2 hz #align nnreal.one_lt_rpow_of_pos_of_lt_one_of_neg NNReal.one_lt_rpow_of_pos_of_lt_one_of_neg theorem one_le_rpow_of_pos_of_le_one_of_nonpos {x : ℝ≥0} {z : ℝ} (hx1 : 0 < x) (hx2 : x ≤ 1) (hz : z ≤ 0) : 1 ≤ x ^ z := Real.one_le_rpow_of_pos_of_le_one_of_nonpos hx1 hx2 hz #align nnreal.one_le_rpow_of_pos_of_le_one_of_nonpos NNReal.one_le_rpow_of_pos_of_le_one_of_nonpos theorem rpow_le_self_of_le_one {x : ℝ≥0} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x := by rcases eq_bot_or_bot_lt x with (rfl \| (h : 0 < x)) · have : z ≠ 0 := by linarith simp [this] nth_rw 2 [← NNReal.rpow_one x] exact NNReal.rpow_le_rpow_of_exponent_ge h hx h_one_le #align nnreal.rpow_le_self_of_le_one NNReal.rpow_le_self_of_le_one theorem rpow_left_injective {x : ℝ} (hx : x ≠ 0) : Function.Injective fun y : ℝ≥0 => y ^ x := fun y z hyz => by simpa only [rpow_inv_rpow_self hx] using congr_arg (fun y => y ^ (1 / x)) hyz #align nnreal.rpow_left_injective NNReal.rpow_left_injective theorem rpow_eq_rpow_iff {x y : ℝ≥0} {z : ℝ} (hz : z ≠ 0) : x ^ z = y ^ z ↔ x = y := (rpow_left_injective hz).eq_iff #align nnreal.rpow_eq_rpow_iff NNReal.rpow_eq_rpow_iff theorem rpow_left_surjective {x : ℝ} (hx : x ≠ 0) : Function.Surjective fun y : ℝ≥0 => y ^ x := fun y => ⟨y ^ x⁻¹, by simp_rw [← rpow_mul, _root_.inv_mul_cancel hx, rpow_one]⟩ #align nnreal.rpow_left_surjective NNReal.rpow_left_surjective theorem rpow_left_bijective {x : ℝ} (hx : x ≠ 0) : Function.Bijective fun y : ℝ≥0 => y ^ x := ⟨rpow_left_injective hx, rpow_left_surjective hx⟩ #align nnreal.rpow_left_bijective NNReal.rpow_left_bijective theorem eq_rpow_one_div_iff {x y : ℝ≥0} {z : ℝ} (hz : z ≠ 0) : x = y ^ (1 / z) ↔ x ^ z = y := by rw [← rpow_eq_rpow_iff hz, rpow_self_rpow_inv hz] #align nnreal.eq_rpow_one_div_iff NNReal.eq_rpow_one_div_iff theorem rpow_one_div_eq_iff {x y : ℝ≥0} {z : ℝ} (hz : z ≠ 0) : x ^ (1 / z) = y ↔ x = y ^ z := by rw [← rpow_eq_rpow_iff hz, rpow_self_rpow_inv hz] #align nnreal.rpow_one_div_eq_iff NNReal.rpow_one_div_eq_iff @[simp] lemma rpow_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ y⁻¹ = x := by rw [← rpow_mul, mul_inv_cancel hy, rpow_one] @[simp] lemma rpow_inv_rpow {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y⁻¹) ^ y = x := by rw [← rpow_mul, inv_mul_cancel hy, rpow_one] theorem pow_rpow_inv_natCast (x : ℝ≥0) {n : ℕ} (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x := by rw [← NNReal.coe_inj, coe_rpow, NNReal.coe_pow] exact Real.pow_rpow_inv_natCast x.2 hn #align nnreal.pow_nat_rpow_nat_inv NNReal.pow_rpow_inv_natCast theorem rpow_inv_natCast_pow (x : ℝ≥0) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x := by rw [← NNReal.coe_inj, NNReal.coe_pow, coe_rpow] exact Real.rpow_inv_natCast_pow x.2 hn #align nnreal.rpow_nat_inv_pow_nat NNReal.rpow_inv_natCast_pow theorem _root_.Real.toNNReal_rpow_of_nonneg {x y : ℝ} (hx : 0 ≤ x) : Real.toNNReal (x ^ y) = Real.toNNReal x ^ y := by nth_rw 1 [← Real.coe_toNNReal x hx] rw [← NNReal.coe_rpow, Real.toNNReal_coe] #align real.to_nnreal_rpow_of_nonneg Real.toNNReal_rpow_of_nonneg theorem strictMono_rpow_of_pos {z : ℝ} (h : 0 < z) : StrictMono fun x : ℝ≥0 => x ^ z := fun x y hxy => by simp only [NNReal.rpow_lt_rpow hxy h, coe_lt_coe] theorem monotone_rpow_of_nonneg {z : ℝ} (h : 0 ≤ z) : Monotone fun x : ℝ≥0 => x ^ z := h.eq_or_lt.elim (fun h0 => h0 ▸ by simp only [rpow_zero, monotone_const]) fun h0 => (strictMono_rpow_of_pos h0).monotone /-- Bundles `fun x : ℝ≥0 => x ^ y` into an order isomorphism when `y : ℝ` is positive, where the inverse is `fun x : ℝ≥0 => x ^ (1 / y)`. -/ @[simps! apply] def orderIsoRpow (y : ℝ) (hy : 0 < y) : ℝ≥0 ≃o ℝ≥0 := (strictMono_rpow_of_pos hy).orderIsoOfRightInverse (fun x => x ^ y) (fun x => x ^ (1 / y)) fun x => by dsimp rw [← rpow_mul, one_div_mul_cancel hy.ne.symm, rpow_one] theorem orderIsoRpow_symm_eq (y : ℝ) (hy : 0 < y) : (orderIsoRpow y hy).symm = orderIsoRpow (1 / y) (one_div_pos.2 hy) := by simp only [orderIsoRpow, one_div_one_div]; rfl end NNReal namespace ENNReal /-- The real power function `x^y` on extended nonnegative reals, defined for `x : ℝ≥0∞` and `y : ℝ` as the restriction of the real power function if `0 < x < ⊤`, and with the natural values for `0` and `⊤` (i.e., `0 ^ x = 0` for `x > 0`, `1` for `x = 0` and `⊤` for `x < 0`, and `⊤ ^ x = 1 / 0 ^ x`). -/ noncomputable def rpow : ℝ≥0∞ → ℝ → ℝ≥0∞ \| some x, y => if x = 0 ∧ y < 0 then ⊤ else (x ^ y : ℝ≥0) \| none, y => if 0 < y then ⊤ else if y = 0 then 1 else 0 #align ennreal.rpow ENNReal.rpow noncomputable instance : Pow ℝ≥0∞ ℝ := ⟨rpow⟩ @[simp] theorem rpow_eq_pow (x : ℝ≥0∞) (y : ℝ) : rpow x y = x ^ y := rfl #align ennreal.rpow_eq_pow ENNReal.rpow_eq_pow @[simp] theorem rpow_zero {x : ℝ≥0∞} : x ^ (0 : ℝ) = 1 := by cases x <;> · dsimp only [(· ^ ·), Pow.pow, rpow] simp [lt_irrefl] #align ennreal.rpow_zero ENNReal.rpow_zero theorem top_rpow_def (y : ℝ) : (⊤ : ℝ≥0∞) ^ y = if 0 < y then ⊤ else if y = 0 then 1 else 0 := rfl #align ennreal.top_rpow_def ENNReal.top_rpow_def @[simp] theorem top_rpow_of_pos {y : ℝ} (h : 0 < y) : (⊤ : ℝ≥0∞) ^ y = ⊤ := by simp [top_rpow_def, h] #align ennreal.top_rpow_of_pos ENNReal.top_rpow_of_pos @[simp] theorem top_rpow_of_neg {y : ℝ} (h : y < 0) : (⊤ : ℝ≥0∞) ^ y = 0 := by simp [top_rpow_def, asymm h, ne_of_lt h] #align ennreal.top_rpow_of_neg ENNReal.top_rpow_of_neg @[simp] theorem zero_rpow_of_pos {y : ℝ} (h : 0 < y) : (0 : ℝ≥0∞) ^ y = 0 := by rw [← ENNReal.coe_zero, ← ENNReal.some_eq_coe] dsimp only [(· ^ ·), rpow, Pow.pow] simp [h, asymm h, ne_of_gt h] #align ennreal.zero_rpow_of_pos ENNReal.zero_rpow_of_pos @[simp] theorem zero_rpow_of_neg {y : ℝ} (h : y < 0) : (0 : ℝ≥0∞) ^ y = ⊤ := by rw [← ENNReal.coe_zero, ← ENNReal.some_eq_coe] dsimp only [(· ^ ·), rpow, Pow.pow] simp [h, ne_of_gt h] #align ennreal.zero_rpow_of_neg ENNReal.zero_rpow_of_neg theorem zero_rpow_def (y : ℝ) : (0 : ℝ≥0∞) ^ y = if 0 < y then 0 else if y = 0 then 1 else ⊤ := by rcases lt_trichotomy (0 : ℝ) y with (H \| rfl \| H) · simp [H, ne_of_gt, zero_rpow_of_pos, lt_irrefl] · simp [lt_irrefl] · simp [H, asymm H, ne_of_lt, zero_rpow_of_neg] #align ennreal.zero_rpow_def ENNReal.zero_rpow_def @[simp] theorem zero_rpow_mul_self (y : ℝ) : (0 : ℝ≥0∞) ^ y * (0 : ℝ≥0∞) ^ y = (0 : ℝ≥0∞) ^ y := by rw [zero_rpow_def] split_ifs exacts [zero_mul _, one_mul _, top_mul_top] #align ennreal.zero_rpow_mul_self ENNReal.zero_rpow_mul_self @[norm_cast] theorem coe_rpow_of_ne_zero {x : ℝ≥0} (h : x ≠ 0) (y : ℝ) : (x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) := by rw [← ENNReal.some_eq_coe] dsimp only [(· ^ ·), Pow.pow, rpow] simp [h] #align ennreal.coe_rpow_of_ne_zero ENNReal.coe_rpow_of_ne_zero @[norm_cast] theorem coe_rpow_of_nonneg (x : ℝ≥0) {y : ℝ} (h : 0 ≤ y) : (x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) := by by_cases hx : x = 0 · rcases le_iff_eq_or_lt.1 h with (H \| H) · simp [hx, H.symm] · simp [hx, zero_rpow_of_pos H, NNReal.zero_rpow (ne_of_gt H)] · exact coe_rpow_of_ne_zero hx _ #align ennreal.coe_rpow_of_nonneg ENNReal.coe_rpow_of_nonneg theorem coe_rpow_def (x : ℝ≥0) (y : ℝ) : (x : ℝ≥0∞) ^ y = if x = 0 ∧ y < 0 then ⊤ else ↑(x ^ y) := rfl #align ennreal.coe_rpow_def ENNReal.coe_rpow_def @[simp]	Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean	466	471	theorem rpow_one (x : ℝ≥0∞) : x ^ (1 : ℝ) = x := by	cases x · exact dif_pos zero_lt_one · change ite _ _ _ = _ simp only [NNReal.rpow_one, some_eq_coe, ite_eq_right_iff, top_ne_coe, and_imp] exact fun _ => zero_le_one.not_lt
/- Copyright (c) 2021 Rémy Degenne. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Rémy Degenne -/ import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp import Mathlib.MeasureTheory.Integral.Bochner import Mathlib.Order.Filter.IndicatorFunction import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner import Mathlib.MeasureTheory.Function.LpSeminorm.Trim #align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e" /-! # Functions a.e. measurable with respect to a sub-σ-algebra A function `f` verifies `AEStronglyMeasurable' m f μ` if it is `μ`-a.e. equal to an `m`-strongly measurable function. This is similar to `AEStronglyMeasurable`, but the `MeasurableSpace` structures used for the measurability statement and for the measure are different. We define `lpMeas F 𝕜 m p μ`, the subspace of `Lp F p μ` containing functions `f` verifying `AEStronglyMeasurable' m f μ`, i.e. functions which are `μ`-a.e. equal to an `m`-strongly measurable function. ## Main statements We define an `IsometryEquiv` between `lpMeasSubgroup` and the `Lp` space corresponding to the measure `μ.trim hm`. As a consequence, the completeness of `Lp` implies completeness of `lpMeas`. `Lp.induction_stronglyMeasurable` (see also `Memℒp.induction_stronglyMeasurable`): To prove something for an `Lp` function a.e. strongly measurable with respect to a sub-σ-algebra `m` in a normed space, it suffices to show that * the property holds for (multiples of) characteristic functions which are measurable w.r.t. `m`; * is closed under addition; * the set of functions in `Lp` strongly measurable w.r.t. `m` for which the property holds is closed. -/ set_option linter.uppercaseLean3 false open TopologicalSpace Filter open scoped ENNReal MeasureTheory namespace MeasureTheory /-- A function `f` verifies `AEStronglyMeasurable' m f μ` if it is `μ`-a.e. equal to an `m`-strongly measurable function. This is similar to `AEStronglyMeasurable`, but the `MeasurableSpace` structures used for the measurability statement and for the measure are different. -/ def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α) {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop := ∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g #align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable' namespace AEStronglyMeasurable' variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f g : α → β} theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) : AEStronglyMeasurable' m g μ := by obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩ #align measure_theory.ae_strongly_measurable'.congr MeasureTheory.AEStronglyMeasurable'.congr theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') : AEStronglyMeasurable' m' f μ := let ⟨f', hf'_meas, hff'⟩ := hf; ⟨f', hf'_meas.mono hm, hff'⟩ theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ) (hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ := by rcases hf with ⟨f', h_f'_meas, hff'⟩ rcases hg with ⟨g', h_g'_meas, hgg'⟩ exact ⟨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'⟩ #align measure_theory.ae_strongly_measurable'.add MeasureTheory.AEStronglyMeasurable'.add theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) : AEStronglyMeasurable' m (-f) μ := by rcases hfm with ⟨f', hf'_meas, hf_ae⟩ refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩ simp_rw [Pi.neg_apply] rw [hx] #align measure_theory.ae_strongly_measurable'.neg MeasureTheory.AEStronglyMeasurable'.neg	Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean	86	92	theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ) (hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ := by	rcases hfm with ⟨f', hf'_meas, hf_ae⟩ rcases hgm with ⟨g', hg'_meas, hg_ae⟩ refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩ simp_rw [Pi.sub_apply] rw [hx1, hx2]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import Mathlib.Topology.Compactness.SigmaCompact import Mathlib.Topology.Connected.TotallyDisconnected import Mathlib.Topology.Inseparable #align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d" /-! # Separation properties of topological spaces. This file defines the predicate `SeparatedNhds`, and common separation axioms (under the Kolmogorov classification). ## Main definitions * `SeparatedNhds`: Two `Set`s are separated by neighbourhoods if they are contained in disjoint open sets. * `T0Space`: A T₀/Kolmogorov space is a space where, for every two points `x ≠ y`, there is an open set that contains one, but not the other. * `R0Space`: An R₀ space (sometimes called a symmetric space) is a topological space such that the `Specializes` relation is symmetric. * `T1Space`: A T₁/Fréchet space is a space where every singleton set is closed. This is equivalent to, for every pair `x ≠ y`, there existing an open set containing `x` but not `y` (`t1Space_iff_exists_open` shows that these conditions are equivalent.) T₁ implies T₀ and R₀. * `R1Space`: An R₁/preregular space is a space where any two topologically distinguishable points have disjoint neighbourhoods. R₁ implies R₀. * `T2Space`: A T₂/Hausdorff space is a space where, for every two points `x ≠ y`, there is two disjoint open sets, one containing `x`, and the other `y`. T₂ implies T₁ and R₁. * `T25Space`: A T₂.₅/Urysohn space is a space where, for every two points `x ≠ y`, there is two open sets, one containing `x`, and the other `y`, whose closures are disjoint. T₂.₅ implies T₂. * `RegularSpace`: A regular space is one where, given any closed `C` and `x ∉ C`, there are disjoint open sets containing `x` and `C` respectively. Such a space is not necessarily Hausdorff. * `T3Space`: A T₃ space is a regular T₀ space. T₃ implies T₂.₅. * `NormalSpace`: A normal space, is one where given two disjoint closed sets, we can find two open sets that separate them. Such a space is not necessarily Hausdorff, even if it is T₀. * `T4Space`: A T₄ space is a normal T₁ space. T₄ implies T₃. * `CompletelyNormalSpace`: A completely normal space is one in which for any two sets `s`, `t` such that if both `closure s` is disjoint with `t`, and `s` is disjoint with `closure t`, then there exist disjoint neighbourhoods of `s` and `t`. `Embedding.completelyNormalSpace` allows us to conclude that this is equivalent to all subspaces being normal. Such a space is not necessarily Hausdorff or regular, even if it is T₀. * `T5Space`: A T₅ space is a completely normal T₁ space. T₅ implies T₄. Note that `mathlib` adopts the modern convention that `m ≤ n` if and only if `T_m → T_n`, but occasionally the literature swaps definitions for e.g. T₃ and regular. ## Main results ### T₀ spaces * `IsClosed.exists_closed_singleton`: Given a closed set `S` in a compact T₀ space, there is some `x ∈ S` such that `{x}` is closed. * `exists_isOpen_singleton_of_isOpen_finite`: Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. ### T₁ spaces * `isClosedMap_const`: The constant map is a closed map. * `discrete_of_t1_of_finite`: A finite T₁ space must have the discrete topology. ### T₂ spaces * `t2_iff_nhds`: A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. * `t2_iff_isClosed_diagonal`: A space is T₂ iff the `diagonal` of `X` (that is, the set of all points of the form `(a, a) : X × X`) is closed under the product topology. * `separatedNhds_of_finset_finset`: Any two disjoint finsets are `SeparatedNhds`. * Most topological constructions preserve Hausdorffness; these results are part of the typeclass inference system (e.g. `Embedding.t2Space`) * `Set.EqOn.closure`: If two functions are equal on some set `s`, they are equal on its closure. * `IsCompact.isClosed`: All compact sets are closed. * `WeaklyLocallyCompactSpace.locallyCompactSpace`: If a topological space is both weakly locally compact (i.e., each point has a compact neighbourhood) and is T₂, then it is locally compact. * `totallySeparatedSpace_of_t1_of_basis_clopen`: If `X` has a clopen basis, then it is a `TotallySeparatedSpace`. * `loc_compact_t2_tot_disc_iff_tot_sep`: A locally compact T₂ space is totally disconnected iff it is totally separated. * `t2Quotient`: the largest T2 quotient of a given topological space. If the space is also compact: * `normalOfCompactT2`: A compact T₂ space is a `NormalSpace`. * `connectedComponent_eq_iInter_isClopen`: The connected component of a point is the intersection of all its clopen neighbourhoods. * `compact_t2_tot_disc_iff_tot_sep`: Being a `TotallyDisconnectedSpace` is equivalent to being a `TotallySeparatedSpace`. * `ConnectedComponents.t2`: `ConnectedComponents X` is T₂ for `X` T₂ and compact. ### T₃ spaces * `disjoint_nested_nhds`: Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`, with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. ## References https://en.wikipedia.org/wiki/Separation_axiom -/ open Function Set Filter Topology TopologicalSpace open scoped Classical universe u v variable {X : Type} {Y : Type} [TopologicalSpace X] section Separation /-- `SeparatedNhds` is a predicate on pairs of sub`Set`s of a topological space. It holds if the two sub`Set`s are contained in disjoint open sets. -/ def SeparatedNhds : Set X → Set X → Prop := fun s t : Set X => ∃ U V : Set X, IsOpen U ∧ IsOpen V ∧ s ⊆ U ∧ t ⊆ V ∧ Disjoint U V #align separated_nhds SeparatedNhds theorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ← exists_and_left, and_assoc, and_comm, and_left_comm] #align separated_nhds_iff_disjoint separatedNhds_iff_disjoint alias ⟨SeparatedNhds.disjoint_nhdsSet, _⟩ := separatedNhds_iff_disjoint namespace SeparatedNhds variable {s s₁ s₂ t t₁ t₂ u : Set X} @[symm] theorem symm : SeparatedNhds s t → SeparatedNhds t s := fun ⟨U, V, oU, oV, aU, bV, UV⟩ => ⟨V, U, oV, oU, bV, aU, Disjoint.symm UV⟩ #align separated_nhds.symm SeparatedNhds.symm theorem comm (s t : Set X) : SeparatedNhds s t ↔ SeparatedNhds t s := ⟨symm, symm⟩ #align separated_nhds.comm SeparatedNhds.comm theorem preimage [TopologicalSpace Y] {f : X → Y} {s t : Set Y} (h : SeparatedNhds s t) (hf : Continuous f) : SeparatedNhds (f ⁻¹' s) (f ⁻¹' t) := let ⟨U, V, oU, oV, sU, tV, UV⟩ := h ⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV, UV.preimage f⟩ #align separated_nhds.preimage SeparatedNhds.preimage protected theorem disjoint (h : SeparatedNhds s t) : Disjoint s t := let ⟨_, _, _, _, hsU, htV, hd⟩ := h; hd.mono hsU htV #align separated_nhds.disjoint SeparatedNhds.disjoint theorem disjoint_closure_left (h : SeparatedNhds s t) : Disjoint (closure s) t := let ⟨_U, _V, _, hV, hsU, htV, hd⟩ := h (hd.closure_left hV).mono (closure_mono hsU) htV #align separated_nhds.disjoint_closure_left SeparatedNhds.disjoint_closure_left theorem disjoint_closure_right (h : SeparatedNhds s t) : Disjoint s (closure t) := h.symm.disjoint_closure_left.symm #align separated_nhds.disjoint_closure_right SeparatedNhds.disjoint_closure_right @[simp] theorem empty_right (s : Set X) : SeparatedNhds s ∅ := ⟨_, _, isOpen_univ, isOpen_empty, fun a _ => mem_univ a, Subset.rfl, disjoint_empty _⟩ #align separated_nhds.empty_right SeparatedNhds.empty_right @[simp] theorem empty_left (s : Set X) : SeparatedNhds ∅ s := (empty_right _).symm #align separated_nhds.empty_left SeparatedNhds.empty_left theorem mono (h : SeparatedNhds s₂ t₂) (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : SeparatedNhds s₁ t₁ := let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h ⟨U, V, hU, hV, hs.trans hsU, ht.trans htV, hd⟩ #align separated_nhds.mono SeparatedNhds.mono theorem union_left : SeparatedNhds s u → SeparatedNhds t u → SeparatedNhds (s ∪ t) u := by simpa only [separatedNhds_iff_disjoint, nhdsSet_union, disjoint_sup_left] using And.intro #align separated_nhds.union_left SeparatedNhds.union_left theorem union_right (ht : SeparatedNhds s t) (hu : SeparatedNhds s u) : SeparatedNhds s (t ∪ u) := (ht.symm.union_left hu.symm).symm #align separated_nhds.union_right SeparatedNhds.union_right end SeparatedNhds /-- A T₀ space, also known as a Kolmogorov space, is a topological space such that for every pair `x ≠ y`, there is an open set containing one but not the other. We formulate the definition in terms of the `Inseparable` relation. -/ class T0Space (X : Type u) [TopologicalSpace X] : Prop where /-- Two inseparable points in a T₀ space are equal. -/ t0 : ∀ ⦃x y : X⦄, Inseparable x y → x = y #align t0_space T0Space theorem t0Space_iff_inseparable (X : Type u) [TopologicalSpace X] : T0Space X ↔ ∀ x y : X, Inseparable x y → x = y := ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩ #align t0_space_iff_inseparable t0Space_iff_inseparable theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y := by simp only [t0Space_iff_inseparable, Ne, not_imp_not, Pairwise] #align t0_space_iff_not_inseparable t0Space_iff_not_inseparable theorem Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y := T0Space.t0 h #align inseparable.eq Inseparable.eq /-- A topology `Inducing` map from a T₀ space is injective. -/ protected theorem Inducing.injective [TopologicalSpace Y] [T0Space X] {f : X → Y} (hf : Inducing f) : Injective f := fun _ _ h => (hf.inseparable_iff.1 <\| .of_eq h).eq #align inducing.injective Inducing.injective /-- A topology `Inducing` map from a T₀ space is a topological embedding. -/ protected theorem Inducing.embedding [TopologicalSpace Y] [T0Space X] {f : X → Y} (hf : Inducing f) : Embedding f := ⟨hf, hf.injective⟩ #align inducing.embedding Inducing.embedding lemma embedding_iff_inducing [TopologicalSpace Y] [T0Space X] {f : X → Y} : Embedding f ↔ Inducing f := ⟨Embedding.toInducing, Inducing.embedding⟩ #align embedding_iff_inducing embedding_iff_inducing theorem t0Space_iff_nhds_injective (X : Type u) [TopologicalSpace X] : T0Space X ↔ Injective (𝓝 : X → Filter X) := t0Space_iff_inseparable X #align t0_space_iff_nhds_injective t0Space_iff_nhds_injective theorem nhds_injective [T0Space X] : Injective (𝓝 : X → Filter X) := (t0Space_iff_nhds_injective X).1 ‹_› #align nhds_injective nhds_injective theorem inseparable_iff_eq [T0Space X] {x y : X} : Inseparable x y ↔ x = y := nhds_injective.eq_iff #align inseparable_iff_eq inseparable_iff_eq @[simp] theorem nhds_eq_nhds_iff [T0Space X] {a b : X} : 𝓝 a = 𝓝 b ↔ a = b := nhds_injective.eq_iff #align nhds_eq_nhds_iff nhds_eq_nhds_iff @[simp] theorem inseparable_eq_eq [T0Space X] : Inseparable = @Eq X := funext₂ fun _ _ => propext inseparable_iff_eq #align inseparable_eq_eq inseparable_eq_eq theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)} (hb : IsTopologicalBasis b) {x y : X} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) := ⟨fun h s hs ↦ inseparable_iff_forall_open.1 h _ (hb.isOpen hs), fun h ↦ hb.nhds_hasBasis.eq_of_same_basis <\| by convert hb.nhds_hasBasis using 2 exact and_congr_right (h _)⟩ theorem TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space X] {b : Set (Set X)} (hb : IsTopologicalBasis b) {x y : X} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) := inseparable_iff_eq.symm.trans hb.inseparable_iff theorem t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop, inseparable_iff_forall_open, Pairwise] #align t0_space_iff_exists_is_open_xor_mem t0Space_iff_exists_isOpen_xor'_mem theorem exists_isOpen_xor'_mem [T0Space X] {x y : X} (h : x ≠ y) : ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := (t0Space_iff_exists_isOpen_xor'_mem X).1 ‹_› h #align exists_is_open_xor_mem exists_isOpen_xor'_mem /-- Specialization forms a partial order on a t0 topological space. -/ def specializationOrder (X) [TopologicalSpace X] [T0Space X] : PartialOrder X := { specializationPreorder X, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with } #align specialization_order specializationOrder instance SeparationQuotient.instT0Space : T0Space (SeparationQuotient X) := ⟨fun x y => Quotient.inductionOn₂' x y fun _ _ h => SeparationQuotient.mk_eq_mk.2 <\| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩ theorem minimal_nonempty_closed_subsingleton [T0Space X] {s : Set X} (hs : IsClosed s) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : s.Subsingleton := by clear Y -- Porting note: added refine fun x hx y hy => of_not_not fun hxy => ?_ rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩ wlog h : x ∈ U ∧ y ∉ U · refine this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h) cases' h with hxU hyU have : s \ U = s := hmin (s \ U) diff_subset ⟨y, hy, hyU⟩ (hs.sdiff hUo) exact (this.symm.subset hx).2 hxU #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton theorem minimal_nonempty_closed_eq_singleton [T0Space X] {s : Set X} (hs : IsClosed s) (hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} := exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_closed_subsingleton hs hmin⟩ #align minimal_nonempty_closed_eq_singleton minimal_nonempty_closed_eq_singleton /-- Given a closed set `S` in a compact T₀ space, there is some `x ∈ S` such that `{x}` is closed. -/ theorem IsClosed.exists_closed_singleton [T0Space X] [CompactSpace X] {S : Set X} (hS : IsClosed S) (hne : S.Nonempty) : ∃ x : X, x ∈ S ∧ IsClosed ({x} : Set X) := by obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with ⟨x, rfl⟩ exact ⟨x, Vsub (mem_singleton x), Vcls⟩ #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton theorem minimal_nonempty_open_subsingleton [T0Space X] {s : Set X} (hs : IsOpen s) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : s.Subsingleton := by clear Y -- Porting note: added refine fun x hx y hy => of_not_not fun hxy => ?_ rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩ wlog h : x ∈ U ∧ y ∉ U · exact this hs hmin y hy x hx (Ne.symm hxy) U hUo hU.symm (hU.resolve_left h) cases' h with hxU hyU have : s ∩ U = s := hmin (s ∩ U) inter_subset_left ⟨x, hx, hxU⟩ (hs.inter hUo) exact hyU (this.symm.subset hy).2 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton theorem minimal_nonempty_open_eq_singleton [T0Space X] {s : Set X} (hs : IsOpen s) (hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} := exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_open_subsingleton hs hmin⟩ #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/ theorem exists_isOpen_singleton_of_isOpen_finite [T0Space X] {s : Set X} (hfin : s.Finite) (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set X) := by lift s to Finset X using hfin induction' s using Finset.strongInductionOn with s ihs rcases em (∃ t, t ⊂ s ∧ t.Nonempty ∧ IsOpen (t : Set X)) with (⟨t, hts, htne, hto⟩ \| ht) · rcases ihs t hts htne hto with ⟨x, hxt, hxo⟩ exact ⟨x, hts.1 hxt, hxo⟩ · -- Porting note: was `rcases minimal_nonempty_open_eq_singleton ho hne _ with ⟨x, hx⟩` -- https://github.com/leanprover/std4/issues/116 rsuffices ⟨x, hx⟩ : ∃ x, s.toSet = {x} · exact ⟨x, hx.symm ▸ rfl, hx ▸ ho⟩ refine minimal_nonempty_open_eq_singleton ho hne ?_ refine fun t hts htne hto => of_not_not fun hts' => ht ?_ lift t to Finset X using s.finite_toSet.subset hts exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩ #align exists_open_singleton_of_open_finite exists_isOpen_singleton_of_isOpen_finite theorem exists_open_singleton_of_finite [T0Space X] [Finite X] [Nonempty X] : ∃ x : X, IsOpen ({x} : Set X) := let ⟨x, _, h⟩ := exists_isOpen_singleton_of_isOpen_finite (Set.toFinite _) univ_nonempty isOpen_univ ⟨x, h⟩ #align exists_open_singleton_of_fintype exists_open_singleton_of_finite theorem t0Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y} (hf : Function.Injective f) (hf' : Continuous f) [T0Space Y] : T0Space X := ⟨fun _ _ h => hf <\| (h.map hf').eq⟩ #align t0_space_of_injective_of_continuous t0Space_of_injective_of_continuous protected theorem Embedding.t0Space [TopologicalSpace Y] [T0Space Y] {f : X → Y} (hf : Embedding f) : T0Space X := t0Space_of_injective_of_continuous hf.inj hf.continuous #align embedding.t0_space Embedding.t0Space instance Subtype.t0Space [T0Space X] {p : X → Prop} : T0Space (Subtype p) := embedding_subtype_val.t0Space #align subtype.t0_space Subtype.t0Space theorem t0Space_iff_or_not_mem_closure (X : Type u) [TopologicalSpace X] : T0Space X ↔ Pairwise fun a b : X => a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) := by simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or] #align t0_space_iff_or_not_mem_closure t0Space_iff_or_not_mem_closure instance Prod.instT0Space [TopologicalSpace Y] [T0Space X] [T0Space Y] : T0Space (X × Y) := ⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩ instance Pi.instT0Space {ι : Type} {X : ι → Type} [∀ i, TopologicalSpace (X i)] [∀ i, T0Space (X i)] : T0Space (∀ i, X i) := ⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩ #align pi.t0_space Pi.instT0Space instance ULift.instT0Space [T0Space X] : T0Space (ULift X) := embedding_uLift_down.t0Space theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) : T0Space X := by refine ⟨fun x y hxy => ?_⟩ rcases h x y hxy with ⟨s, hxs, hys, hs⟩ lift x to s using hxs; lift y to s using hys rw [← subtype_inseparable_iff] at hxy exact congr_arg Subtype.val hxy.eq #align t0_space.of_cover T0Space.of_cover theorem T0Space.of_open_cover (h : ∀ x, ∃ s : Set X, x ∈ s ∧ IsOpen s ∧ T0Space s) : T0Space X := T0Space.of_cover fun x _ hxy => let ⟨s, hxs, hso, hs⟩ := h x ⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, hs⟩ #align t0_space.of_open_cover T0Space.of_open_cover /-- A topological space is called an R₀ space, if `Specializes` relation is symmetric. In other words, given two points `x y : X`, if every neighborhood of `y` contains `x`, then every neighborhood of `x` contains `y`. -/ @[mk_iff] class R0Space (X : Type u) [TopologicalSpace X] : Prop where /-- In an R₀ space, the `Specializes` relation is symmetric. -/ specializes_symmetric : Symmetric (Specializes : X → X → Prop) export R0Space (specializes_symmetric) section R0Space variable [R0Space X] {x y : X} /-- In an R₀ space, the `Specializes` relation is symmetric, dot notation version. -/ theorem Specializes.symm (h : x ⤳ y) : y ⤳ x := specializes_symmetric h #align specializes.symm Specializes.symm /-- In an R₀ space, the `Specializes` relation is symmetric, `Iff` version. -/ theorem specializes_comm : x ⤳ y ↔ y ⤳ x := ⟨Specializes.symm, Specializes.symm⟩ #align specializes_comm specializes_comm /-- In an R₀ space, `Specializes` is equivalent to `Inseparable`. -/ theorem specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y := ⟨fun h ↦ h.antisymm h.symm, Inseparable.specializes⟩ #align specializes_iff_inseparable specializes_iff_inseparable /-- In an R₀ space, `Specializes` implies `Inseparable`. -/ alias ⟨Specializes.inseparable, _⟩ := specializes_iff_inseparable theorem Inducing.r0Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R0Space Y where specializes_symmetric a b := by simpa only [← hf.specializes_iff] using Specializes.symm instance {p : X → Prop} : R0Space {x // p x} := inducing_subtype_val.r0Space instance [TopologicalSpace Y] [R0Space Y] : R0Space (X × Y) where specializes_symmetric _ _ h := h.fst.symm.prod h.snd.symm instance {ι : Type} {X : ι → Type} [∀ i, TopologicalSpace (X i)] [∀ i, R0Space (X i)] : R0Space (∀ i, X i) where specializes_symmetric _ _ h := specializes_pi.2 fun i ↦ (specializes_pi.1 h i).symm /-- In an R₀ space, the closure of a singleton is a compact set. -/ theorem isCompact_closure_singleton : IsCompact (closure {x}) := by refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_ obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <\| hxU <\| subset_closure rfl refine ⟨{i}, fun y hy ↦ ?_⟩ rw [← specializes_iff_mem_closure, specializes_comm] at hy simpa using hy.mem_open (hUo i) hi theorem Filter.coclosedCompact_le_cofinite : coclosedCompact X ≤ cofinite := le_cofinite_iff_compl_singleton_mem.2 fun _ ↦ compl_mem_coclosedCompact.2 isCompact_closure_singleton #align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofinite variable (X) /-- In an R₀ space, relatively compact sets form a bornology. Its cobounded filter is `Filter.coclosedCompact`. See also `Bornology.inCompact` the bornology of sets contained in a compact set. -/ def Bornology.relativelyCompact : Bornology X where cobounded' := Filter.coclosedCompact X le_cofinite' := Filter.coclosedCompact_le_cofinite #align bornology.relatively_compact Bornology.relativelyCompact variable {X} theorem Bornology.relativelyCompact.isBounded_iff {s : Set X} : @Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) := compl_mem_coclosedCompact #align bornology.relatively_compact.is_bounded_iff Bornology.relativelyCompact.isBounded_iff /-- In an R₀ space, the closure of a finite set is a compact set. -/ theorem Set.Finite.isCompact_closure {s : Set X} (hs : s.Finite) : IsCompact (closure s) := let _ : Bornology X := .relativelyCompact X Bornology.relativelyCompact.isBounded_iff.1 hs.isBounded end R0Space /-- A T₁ space, also known as a Fréchet space, is a topological space where every singleton set is closed. Equivalently, for every pair `x ≠ y`, there is an open set containing `x` and not `y`. -/ class T1Space (X : Type u) [TopologicalSpace X] : Prop where /-- A singleton in a T₁ space is a closed set. -/ t1 : ∀ x, IsClosed ({x} : Set X) #align t1_space T1Space theorem isClosed_singleton [T1Space X] {x : X} : IsClosed ({x} : Set X) := T1Space.t1 x #align is_closed_singleton isClosed_singleton theorem isOpen_compl_singleton [T1Space X] {x : X} : IsOpen ({x}ᶜ : Set X) := isClosed_singleton.isOpen_compl #align is_open_compl_singleton isOpen_compl_singleton theorem isOpen_ne [T1Space X] {x : X} : IsOpen { y \| y ≠ x } := isOpen_compl_singleton #align is_open_ne isOpen_ne @[to_additive] theorem Continuous.isOpen_mulSupport [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X} (hf : Continuous f) : IsOpen (mulSupport f) := isOpen_ne.preimage hf #align continuous.is_open_mul_support Continuous.isOpen_mulSupport #align continuous.is_open_support Continuous.isOpen_support theorem Ne.nhdsWithin_compl_singleton [T1Space X] {x y : X} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x := isOpen_ne.nhdsWithin_eq h #align ne.nhds_within_compl_singleton Ne.nhdsWithin_compl_singleton theorem Ne.nhdsWithin_diff_singleton [T1Space X] {x y : X} (h : x ≠ y) (s : Set X) : 𝓝[s \ {y}] x = 𝓝[s] x := by rw [diff_eq, inter_comm, nhdsWithin_inter_of_mem] exact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_nhds h) #align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singleton lemma nhdsWithin_compl_singleton_le [T1Space X] (x y : X) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x := by rcases eq_or_ne x y with rfl\|hy · exact Eq.le rfl · rw [Ne.nhdsWithin_compl_singleton hy] exact nhdsWithin_le_nhds theorem isOpen_setOf_eventually_nhdsWithin [T1Space X] {p : X → Prop} : IsOpen { x \| ∀ᶠ y in 𝓝[≠] x, p y } := by refine isOpen_iff_mem_nhds.mpr fun a ha => ?_ filter_upwards [eventually_nhds_nhdsWithin.mpr ha] with b hb rcases eq_or_ne a b with rfl \| h · exact hb · rw [h.symm.nhdsWithin_compl_singleton] at hb exact hb.filter_mono nhdsWithin_le_nhds #align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithin protected theorem Set.Finite.isClosed [T1Space X] {s : Set X} (hs : Set.Finite s) : IsClosed s := by rw [← biUnion_of_singleton s] exact hs.isClosed_biUnion fun i _ => isClosed_singleton #align set.finite.is_closed Set.Finite.isClosed theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space X] {b : Set (Set X)} (hb : IsTopologicalBasis b) {x y : X} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a := by rcases hb.isOpen_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩ exact ⟨a, ab, xa, fun h => ha h rfl⟩ #align topological_space.is_topological_basis.exists_mem_of_ne TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne protected theorem Finset.isClosed [T1Space X] (s : Finset X) : IsClosed (s : Set X) := s.finite_toSet.isClosed #align finset.is_closed Finset.isClosed theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] : List.TFAE [T1Space X, ∀ x, IsClosed ({ x } : Set X), ∀ x, IsOpen ({ x }ᶜ : Set X), Continuous (@CofiniteTopology.of X), ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s, ∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y), ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y] := by tfae_have 1 ↔ 2 · exact ⟨fun h => h.1, fun h => ⟨h⟩⟩ tfae_have 2 ↔ 3 · simp only [isOpen_compl_iff] tfae_have 5 ↔ 3 · refine forall_swap.trans ?_ simp only [isOpen_iff_mem_nhds, mem_compl_iff, mem_singleton_iff] tfae_have 5 ↔ 6 · simp only [← subset_compl_singleton_iff, exists_mem_subset_iff] tfae_have 5 ↔ 7 · simp only [(nhds_basis_opens _).mem_iff, subset_compl_singleton_iff, exists_prop, and_assoc, and_left_comm] tfae_have 5 ↔ 8 · simp only [← principal_singleton, disjoint_principal_right] tfae_have 8 ↔ 9 · exact forall_swap.trans (by simp only [disjoint_comm, ne_comm]) tfae_have 1 → 4 · simp only [continuous_def, CofiniteTopology.isOpen_iff'] rintro H s (rfl \| hs) exacts [isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl] tfae_have 4 → 2 · exact fun h x => (CofiniteTopology.isClosed_iff.2 <\| Or.inr (finite_singleton _)).preimage h tfae_have 2 ↔ 10 · simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def, mem_singleton_iff, eq_comm] tfae_finish #align t1_space_tfae t1Space_TFAE theorem t1Space_iff_continuous_cofinite_of : T1Space X ↔ Continuous (@CofiniteTopology.of X) := (t1Space_TFAE X).out 0 3 #align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_of theorem CofiniteTopology.continuous_of [T1Space X] : Continuous (@CofiniteTopology.of X) := t1Space_iff_continuous_cofinite_of.mp ‹_› #align cofinite_topology.continuous_of CofiniteTopology.continuous_of theorem t1Space_iff_exists_open : T1Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U := (t1Space_TFAE X).out 0 6 #align t1_space_iff_exists_open t1Space_iff_exists_open theorem t1Space_iff_disjoint_pure_nhds : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y) := (t1Space_TFAE X).out 0 8 #align t1_space_iff_disjoint_pure_nhds t1Space_iff_disjoint_pure_nhds theorem t1Space_iff_disjoint_nhds_pure : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y) := (t1Space_TFAE X).out 0 7 #align t1_space_iff_disjoint_nhds_pure t1Space_iff_disjoint_nhds_pure theorem t1Space_iff_specializes_imp_eq : T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y := (t1Space_TFAE X).out 0 9 #align t1_space_iff_specializes_imp_eq t1Space_iff_specializes_imp_eq theorem disjoint_pure_nhds [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) := t1Space_iff_disjoint_pure_nhds.mp ‹_› h #align disjoint_pure_nhds disjoint_pure_nhds theorem disjoint_nhds_pure [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) := t1Space_iff_disjoint_nhds_pure.mp ‹_› h #align disjoint_nhds_pure disjoint_nhds_pure theorem Specializes.eq [T1Space X] {x y : X} (h : x ⤳ y) : x = y := t1Space_iff_specializes_imp_eq.1 ‹_› h #align specializes.eq Specializes.eq theorem specializes_iff_eq [T1Space X] {x y : X} : x ⤳ y ↔ x = y := ⟨Specializes.eq, fun h => h ▸ specializes_rfl⟩ #align specializes_iff_eq specializes_iff_eq @[simp] theorem specializes_eq_eq [T1Space X] : (· ⤳ ·) = @Eq X := funext₂ fun _ _ => propext specializes_iff_eq #align specializes_eq_eq specializes_eq_eq @[simp] theorem pure_le_nhds_iff [T1Space X] {a b : X} : pure a ≤ 𝓝 b ↔ a = b := specializes_iff_pure.symm.trans specializes_iff_eq #align pure_le_nhds_iff pure_le_nhds_iff @[simp] theorem nhds_le_nhds_iff [T1Space X] {a b : X} : 𝓝 a ≤ 𝓝 b ↔ a = b := specializes_iff_eq #align nhds_le_nhds_iff nhds_le_nhds_iff instance (priority := 100) [T1Space X] : R0Space X where specializes_symmetric _ _ := by rw [specializes_iff_eq, specializes_iff_eq]; exact Eq.symm instance : T1Space (CofiniteTopology X) := t1Space_iff_continuous_cofinite_of.mpr continuous_id theorem t1Space_antitone : Antitone (@T1Space X) := fun a _ h _ => @T1Space.mk _ a fun x => (T1Space.t1 x).mono h #align t1_space_antitone t1Space_antitone theorem continuousWithinAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x x' : X} {y : Y} (hne : x' ≠ x) : ContinuousWithinAt (Function.update f x y) s x' ↔ ContinuousWithinAt f s x' := EventuallyEq.congr_continuousWithinAt (mem_nhdsWithin_of_mem_nhds <\| mem_of_superset (isOpen_ne.mem_nhds hne) fun _y' hy' => Function.update_noteq hy' _ _) (Function.update_noteq hne _ _) #align continuous_within_at_update_of_ne continuousWithinAt_update_of_ne theorem continuousAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) : ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' := by simp only [← continuousWithinAt_univ, continuousWithinAt_update_of_ne hne] #align continuous_at_update_of_ne continuousAt_update_of_ne theorem continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y} {s : Set X} {x : X} {y : Y} : ContinuousOn (Function.update f x y) s ↔ ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y)) := by rw [ContinuousOn, ← and_forall_ne x, and_comm] refine and_congr ⟨fun H z hz => ?_, fun H z hzx hzs => ?_⟩ (forall_congr' fun _ => ?_) · specialize H z hz.2 hz.1 rw [continuousWithinAt_update_of_ne hz.2] at H exact H.mono diff_subset · rw [continuousWithinAt_update_of_ne hzx] refine (H z ⟨hzs, hzx⟩).mono_of_mem (inter_mem_nhdsWithin _ ?_) exact isOpen_ne.mem_nhds hzx · exact continuousWithinAt_update_same #align continuous_on_update_iff continuousOn_update_iff theorem t1Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y} (hf : Function.Injective f) (hf' : Continuous f) [T1Space Y] : T1Space X := t1Space_iff_specializes_imp_eq.2 fun _ _ h => hf (h.map hf').eq #align t1_space_of_injective_of_continuous t1Space_of_injective_of_continuous protected theorem Embedding.t1Space [TopologicalSpace Y] [T1Space Y] {f : X → Y} (hf : Embedding f) : T1Space X := t1Space_of_injective_of_continuous hf.inj hf.continuous #align embedding.t1_space Embedding.t1Space instance Subtype.t1Space {X : Type u} [TopologicalSpace X] [T1Space X] {p : X → Prop} : T1Space (Subtype p) := embedding_subtype_val.t1Space #align subtype.t1_space Subtype.t1Space instance [TopologicalSpace Y] [T1Space X] [T1Space Y] : T1Space (X × Y) := ⟨fun ⟨a, b⟩ => @singleton_prod_singleton _ _ a b ▸ isClosed_singleton.prod isClosed_singleton⟩ instance {ι : Type} {X : ι → Type} [∀ i, TopologicalSpace (X i)] [∀ i, T1Space (X i)] : T1Space (∀ i, X i) := ⟨fun f => univ_pi_singleton f ▸ isClosed_set_pi fun _ _ => isClosed_singleton⟩ instance ULift.instT1Space [T1Space X] : T1Space (ULift X) := embedding_uLift_down.t1Space -- see Note [lower instance priority] instance (priority := 100) TotallyDisconnectedSpace.t1Space [h: TotallyDisconnectedSpace X] : T1Space X := by rw [((t1Space_TFAE X).out 0 1 :)] intro x rw [← totallyDisconnectedSpace_iff_connectedComponent_singleton.mp h x] exact isClosed_connectedComponent -- see Note [lower instance priority] instance (priority := 100) T1Space.t0Space [T1Space X] : T0Space X := ⟨fun _ _ h => h.specializes.eq⟩ #align t1_space.t0_space T1Space.t0Space @[simp] theorem compl_singleton_mem_nhds_iff [T1Space X] {x y : X} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x := isOpen_compl_singleton.mem_nhds_iff #align compl_singleton_mem_nhds_iff compl_singleton_mem_nhds_iff theorem compl_singleton_mem_nhds [T1Space X] {x y : X} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y := compl_singleton_mem_nhds_iff.mpr h #align compl_singleton_mem_nhds compl_singleton_mem_nhds @[simp] theorem closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} := isClosed_singleton.closure_eq #align closure_singleton closure_singleton -- Porting note (#11215): TODO: the proof was `hs.induction_on (by simp) fun x => by simp` theorem Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) : (closure s).Subsingleton := by rcases hs.eq_empty_or_singleton with (rfl \| ⟨x, rfl⟩) <;> simp #align set.subsingleton.closure Set.Subsingleton.closure @[simp] theorem subsingleton_closure [T1Space X] {s : Set X} : (closure s).Subsingleton ↔ s.Subsingleton := ⟨fun h => h.anti subset_closure, fun h => h.closure⟩ #align subsingleton_closure subsingleton_closure theorem isClosedMap_const {X Y} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y} : IsClosedMap (Function.const X y) := IsClosedMap.of_nonempty fun s _ h2s => by simp_rw [const, h2s.image_const, isClosed_singleton] #align is_closed_map_const isClosedMap_const theorem nhdsWithin_insert_of_ne [T1Space X] {x y : X} {s : Set X} (hxy : x ≠ y) : 𝓝[insert y s] x = 𝓝[s] x := by refine le_antisymm (Filter.le_def.2 fun t ht => ?_) (nhdsWithin_mono x <\| subset_insert y s) obtain ⟨o, ho, hxo, host⟩ := mem_nhdsWithin.mp ht refine mem_nhdsWithin.mpr ⟨o \ {y}, ho.sdiff isClosed_singleton, ⟨hxo, hxy⟩, ?_⟩ rw [inter_insert_of_not_mem <\| not_mem_diff_of_mem (mem_singleton y)] exact (inter_subset_inter diff_subset Subset.rfl).trans host #align nhds_within_insert_of_ne nhdsWithin_insert_of_ne /-- If `t` is a subset of `s`, except for one point, then `insert x s` is a neighborhood of `x` within `t`. -/ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space X] {x y : X} {s t : Set X} (hu : t ⊆ insert y s) : insert x s ∈ 𝓝[t] x := by rcases eq_or_ne x y with (rfl \| h) · exact mem_of_superset self_mem_nhdsWithin hu refine nhdsWithin_mono x hu ?_ rw [nhdsWithin_insert_of_ne h] exact mem_of_superset self_mem_nhdsWithin (subset_insert x s) #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert @[simp] theorem ker_nhds [T1Space X] (x : X) : (𝓝 x).ker = {x} := by simp [ker_nhds_eq_specializes] theorem biInter_basis_nhds [T1Space X] {ι : Sort} {p : ι → Prop} {s : ι → Set X} {x : X} (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by rw [← h.ker, ker_nhds] #align bInter_basis_nhds biInter_basis_nhds @[simp] theorem compl_singleton_mem_nhdsSet_iff [T1Space X] {x : X} {s : Set X} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s := by rw [isOpen_compl_singleton.mem_nhdsSet, subset_compl_singleton_iff] #align compl_singleton_mem_nhds_set_iff compl_singleton_mem_nhdsSet_iff @[simp] theorem nhdsSet_le_iff [T1Space X] {s t : Set X} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t := by refine ⟨?_, fun h => monotone_nhdsSet h⟩ simp_rw [Filter.le_def]; intro h x hx specialize h {x}ᶜ simp_rw [compl_singleton_mem_nhdsSet_iff] at h by_contra hxt exact h hxt hx #align nhds_set_le_iff nhdsSet_le_iff @[simp] theorem nhdsSet_inj_iff [T1Space X] {s t : Set X} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t := by simp_rw [le_antisymm_iff] exact and_congr nhdsSet_le_iff nhdsSet_le_iff #align nhds_set_inj_iff nhdsSet_inj_iff theorem injective_nhdsSet [T1Space X] : Function.Injective (𝓝ˢ : Set X → Filter X) := fun _ _ hst => nhdsSet_inj_iff.mp hst #align injective_nhds_set injective_nhdsSet theorem strictMono_nhdsSet [T1Space X] : StrictMono (𝓝ˢ : Set X → Filter X) := monotone_nhdsSet.strictMono_of_injective injective_nhdsSet #align strict_mono_nhds_set strictMono_nhdsSet @[simp] theorem nhds_le_nhdsSet_iff [T1Space X] {s : Set X} {x : X} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s := by rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff] #align nhds_le_nhds_set_iff nhds_le_nhdsSet_iff /-- Removing a non-isolated point from a dense set, one still obtains a dense set. -/ theorem Dense.diff_singleton [T1Space X] {s : Set X} (hs : Dense s) (x : X) [NeBot (𝓝[≠] x)] : Dense (s \ {x}) := hs.inter_of_isOpen_right (dense_compl_singleton x) isOpen_compl_singleton #align dense.diff_singleton Dense.diff_singleton /-- Removing a finset from a dense set in a space without isolated points, one still obtains a dense set. -/ theorem Dense.diff_finset [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s) (t : Finset X) : Dense (s \ t) := by induction t using Finset.induction_on with \| empty => simpa using hs \| insert _ ih => rw [Finset.coe_insert, ← union_singleton, ← diff_diff] exact ih.diff_singleton _ #align dense.diff_finset Dense.diff_finset /-- Removing a finite set from a dense set in a space without isolated points, one still obtains a dense set. -/ theorem Dense.diff_finite [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s) {t : Set X} (ht : t.Finite) : Dense (s \ t) := by convert hs.diff_finset ht.toFinset exact (Finite.coe_toFinset _).symm #align dense.diff_finite Dense.diff_finite /-- If a function to a `T1Space` tends to some limit `y` at some point `x`, then necessarily `y = f x`. -/ theorem eq_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y} (h : Tendsto f (𝓝 x) (𝓝 y)) : f x = y := by_contra fun hfa : f x ≠ y => have fact₁ : {f x}ᶜ ∈ 𝓝 y := compl_singleton_mem_nhds hfa.symm have fact₂ : Tendsto f (pure x) (𝓝 y) := h.comp (tendsto_id'.2 <\| pure_le_nhds x) fact₂ fact₁ (Eq.refl <\| f x) #align eq_of_tendsto_nhds eq_of_tendsto_nhds theorem Filter.Tendsto.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y} {l : Filter X} {b₁ b₂ : Y} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ := hg.eventually (isOpen_compl_singleton.eventually_mem hb) #align filter.tendsto.eventually_ne Filter.Tendsto.eventually_ne theorem ContinuousAt.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y} (hg1 : ContinuousAt g x) (hg2 : g x ≠ y) : ∀ᶠ z in 𝓝 x, g z ≠ y := hg1.tendsto.eventually_ne hg2 #align continuous_at.eventually_ne ContinuousAt.eventually_ne theorem eventually_ne_nhds [T1Space X] {a b : X} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b := IsOpen.eventually_mem isOpen_ne h theorem eventually_ne_nhdsWithin [T1Space X] {a b : X} {s : Set X} (h : a ≠ b) : ∀ᶠ x in 𝓝[s] a, x ≠ b := Filter.Eventually.filter_mono nhdsWithin_le_nhds <\| eventually_ne_nhds h /-- To prove a function to a `T1Space` is continuous at some point `x`, it suffices to prove that `f` admits some* limit at `x`. -/ theorem continuousAt_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y} (h : Tendsto f (𝓝 x) (𝓝 y)) : ContinuousAt f x := by rwa [ContinuousAt, eq_of_tendsto_nhds h] #align continuous_at_of_tendsto_nhds continuousAt_of_tendsto_nhds @[simp] theorem tendsto_const_nhds_iff [T1Space X] {l : Filter Y} [NeBot l] {c d : X} : Tendsto (fun _ => c) l (𝓝 d) ↔ c = d := by simp_rw [Tendsto, Filter.map_const, pure_le_nhds_iff] #align tendsto_const_nhds_iff tendsto_const_nhds_iff /-- A point with a finite neighborhood has to be isolated. -/ theorem isOpen_singleton_of_finite_mem_nhds [T1Space X] (x : X) {s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set X) := by have A : {x} ⊆ s := by simp only [singleton_subset_iff, mem_of_mem_nhds hs] have B : IsClosed (s \ {x}) := (hsf.subset diff_subset).isClosed have C : (s \ {x})ᶜ ∈ 𝓝 x := B.isOpen_compl.mem_nhds fun h => h.2 rfl have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D #align is_open_singleton_of_finite_mem_nhds isOpen_singleton_of_finite_mem_nhds /-- If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is infinite. -/ theorem infinite_of_mem_nhds {X} [TopologicalSpace X] [T1Space X] (x : X) [hx : NeBot (𝓝[≠] x)] {s : Set X} (hs : s ∈ 𝓝 x) : Set.Infinite s := by refine fun hsf => hx.1 ?_ rw [← isOpen_singleton_iff_punctured_nhds] exact isOpen_singleton_of_finite_mem_nhds x hs hsf #align infinite_of_mem_nhds infinite_of_mem_nhds theorem discrete_of_t1_of_finite [T1Space X] [Finite X] : DiscreteTopology X := by apply singletons_open_iff_discrete.mp intro x rw [← isClosed_compl_iff] exact (Set.toFinite _).isClosed #align discrete_of_t1_of_finite discrete_of_t1_of_finite theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTopology X] : Subsingleton X := by rw [← not_nontrivial_iff_subsingleton] rintro ⟨x, y, hxy⟩ rw [Ne, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <\| singleton_nonempty y] at hxy exact hxy (mem_univ x) #align preconnected_space.trivial_of_discrete PreconnectedSpace.trivial_of_discrete theorem IsPreconnected.infinite_of_nontrivial [T1Space X] {s : Set X} (h : IsPreconnected s) (hs : s.Nontrivial) : s.Infinite := by refine mt (fun hf => (subsingleton_coe s).mp ?_) (not_subsingleton_iff.mpr hs) haveI := @discrete_of_t1_of_finite s _ _ hf.to_subtype exact @PreconnectedSpace.trivial_of_discrete _ _ (Subtype.preconnectedSpace h) _ #align is_preconnected.infinite_of_nontrivial IsPreconnected.infinite_of_nontrivial theorem ConnectedSpace.infinite [ConnectedSpace X] [Nontrivial X] [T1Space X] : Infinite X := infinite_univ_iff.mp <\| isPreconnected_univ.infinite_of_nontrivial nontrivial_univ #align connected_space.infinite ConnectedSpace.infinite /-- A non-trivial connected T1 space has no isolated points. -/ instance (priority := 100) ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space [ConnectedSpace X] [Nontrivial X] [T1Space X] (x : X) : NeBot (𝓝[≠] x) := by by_contra contra rw [not_neBot, ← isOpen_singleton_iff_punctured_nhds] at contra replace contra := nonempty_inter isOpen_compl_singleton contra (compl_union_self _) (Set.nonempty_compl_of_nontrivial _) (singleton_nonempty _) simp [compl_inter_self {x}] at contra theorem SeparationQuotient.t1Space_iff : T1Space (SeparationQuotient X) ↔ R0Space X := by rw [r0Space_iff, ((t1Space_TFAE (SeparationQuotient X)).out 0 9 :)] constructor · intro h x y xspecy rw [← Inducing.specializes_iff inducing_mk, h xspecy] at * · rintro h ⟨x⟩ ⟨y⟩ sxspecsy have xspecy : x ⤳ y := (Inducing.specializes_iff inducing_mk).mp sxspecsy have yspecx : y ⤳ x := h xspecy erw [mk_eq_mk, inseparable_iff_specializes_and] exact ⟨xspecy, yspecx⟩ theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) : {x} ∈ 𝓝[s] x := by have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete] simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using @image_mem_map _ _ _ ((↑) : s → X) _ this #align singleton_mem_nhds_within_of_mem_discrete singleton_mem_nhdsWithin_of_mem_discrete /-- The neighbourhoods filter of `x` within `s`, under the discrete topology, is equal to the pure `x` filter (which is the principal filter at the singleton `{x}`.) -/ theorem nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) : 𝓝[s] x = pure x := le_antisymm (le_pure_iff.2 <\| singleton_mem_nhdsWithin_of_mem_discrete hx) (pure_le_nhdsWithin hx) #align nhds_within_of_mem_discrete nhdsWithin_of_mem_discrete theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {p : ι → Prop} {t : ι → Set X} {s : Set X} [DiscreteTopology s] {x : X} (hb : (𝓝 x).HasBasis p t) (hx : x ∈ s) : ∃ i, p i ∧ t i ∩ s = {x} := by rcases (nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hx) with ⟨i, hi, hix⟩ exact ⟨i, hi, hix.antisymm <\| singleton_subset_iff.2 ⟨mem_of_mem_nhds <\| hb.mem_of_mem hi, hx⟩⟩ #align filter.has_basis.exists_inter_eq_singleton_of_mem_discrete Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete /-- A point `x` in a discrete subset `s` of a topological space admits a neighbourhood that only meets `s` at `x`. -/	Mathlib/Topology/Separation.lean	965	967	theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} := by	simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis -/ import Mathlib.Algebra.DirectSum.Module import Mathlib.Analysis.Complex.Basic import Mathlib.Analysis.Convex.Uniform import Mathlib.Analysis.NormedSpace.Completion import Mathlib.Analysis.NormedSpace.BoundedLinearMaps #align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b" /-! # Inner product space This file defines inner product spaces and proves the basic properties. We do not formally define Hilbert spaces, but they can be obtained using the set of assumptions `[NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]`. An inner product space is a vector space endowed with an inner product. It generalizes the notion of dot product in `ℝ^n` and provides the means of defining the length of a vector and the angle between two vectors. In particular vectors `x` and `y` are orthogonal if their inner product equals zero. We define both the real and complex cases at the same time using the `RCLike` typeclass. This file proves general results on inner product spaces. For the specific construction of an inner product structure on `n → 𝕜` for `𝕜 = ℝ` or `ℂ`, see `EuclideanSpace` in `Analysis.InnerProductSpace.PiL2`. ## Main results - We define the class `InnerProductSpace 𝕜 E` extending `NormedSpace 𝕜 E` with a number of basic properties, most notably the Cauchy-Schwarz inequality. Here `𝕜` is understood to be either `ℝ` or `ℂ`, through the `RCLike` typeclass. - We show that the inner product is continuous, `continuous_inner`, and bundle it as the continuous sesquilinear map `innerSL` (see also `innerₛₗ` for the non-continuous version). - We define `Orthonormal`, a predicate on a function `v : ι → E`, and prove the existence of a maximal orthonormal set, `exists_maximal_orthonormal`. Bessel's inequality, `Orthonormal.tsum_inner_products_le`, states that given an orthonormal set `v` and a vector `x`, the sum of the norm-squares of the inner products `⟪v i, x⟫` is no more than the norm-square of `x`. For the existence of orthonormal bases, Hilbert bases, etc., see the file `Analysis.InnerProductSpace.projection`. ## Notation We globally denote the real and complex inner products by `⟪·, ·⟫_ℝ` and `⟪·, ·⟫_ℂ` respectively. We also provide two notation namespaces: `RealInnerProductSpace`, `ComplexInnerProductSpace`, which respectively introduce the plain notation `⟪·, ·⟫` for the real and complex inner product. ## Implementation notes We choose the convention that inner products are conjugate linear in the first argument and linear in the second. ## Tags inner product space, Hilbert space, norm ## References * [Clément & Martin, The Lax-Milgram Theorem. A detailed proof to be formalized in Coq] * [Clément & Martin, A Coq formal proof of the Lax–Milgram theorem] The Coq code is available at the following address: <http://www.lri.fr/~sboldo/elfic/index.html> -/ noncomputable section open RCLike Real Filter open Topology ComplexConjugate open LinearMap (BilinForm) variable {𝕜 E F : Type} [RCLike 𝕜] /-- Syntactic typeclass for types endowed with an inner product -/ class Inner (𝕜 E : Type) where /-- The inner product function. -/ inner : E → E → 𝕜 #align has_inner Inner export Inner (inner) /-- The inner product with values in `𝕜`. -/ notation3:max "⟪" x ", " y "⟫_" 𝕜:max => @inner 𝕜 _ _ x y section Notations /-- The inner product with values in `ℝ`. -/ scoped[RealInnerProductSpace] notation "⟪" x ", " y "⟫" => @inner ℝ _ _ x y /-- The inner product with values in `ℂ`. -/ scoped[ComplexInnerProductSpace] notation "⟪" x ", " y "⟫" => @inner ℂ _ _ x y end Notations /-- An inner product space is a vector space with an additional operation called inner product. The norm could be derived from the inner product, instead we require the existence of a norm and the fact that `‖x‖^2 = re ⟪x, x⟫` to be able to put instances on `𝕂` or product spaces. To construct a norm from an inner product, see `InnerProductSpace.ofCore`. -/ class InnerProductSpace (𝕜 : Type) (E : Type) [RCLike 𝕜] [NormedAddCommGroup E] extends NormedSpace 𝕜 E, Inner 𝕜 E where /-- The inner product induces the norm. -/ norm_sq_eq_inner : ∀ x : E, ‖x‖ ^ 2 = re (inner x x) /-- The inner product is hermitian, taking the `conj` swaps the arguments. -/ conj_symm : ∀ x y, conj (inner y x) = inner x y /-- The inner product is additive in the first coordinate. -/ add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z /-- The inner product is conjugate linear in the first coordinate. -/ smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y #align inner_product_space InnerProductSpace /-! ### Constructing a normed space structure from an inner product In the definition of an inner product space, we require the existence of a norm, which is equal (but maybe not defeq) to the square root of the scalar product. This makes it possible to put an inner product space structure on spaces with a preexisting norm (for instance `ℝ`), with good properties. However, sometimes, one would like to define the norm starting only from a well-behaved scalar product. This is what we implement in this paragraph, starting from a structure `InnerProductSpace.Core` stating that we have a nice scalar product. Our goal here is not to develop a whole theory with all the supporting API, as this will be done below for `InnerProductSpace`. Instead, we implement the bare minimum to go as directly as possible to the construction of the norm and the proof of the triangular inequality. Warning: Do not use this `Core` structure if the space you are interested in already has a norm instance defined on it, otherwise this will create a second non-defeq norm instance! -/ /-- A structure requiring that a scalar product is positive definite and symmetric, from which one can construct an `InnerProductSpace` instance in `InnerProductSpace.ofCore`. -/ -- @[nolint HasNonemptyInstance] porting note: I don't think we have this linter anymore structure InnerProductSpace.Core (𝕜 : Type) (F : Type) [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] extends Inner 𝕜 F where /-- The inner product is hermitian, taking the `conj` swaps the arguments. -/ conj_symm : ∀ x y, conj (inner y x) = inner x y /-- The inner product is positive (semi)definite. -/ nonneg_re : ∀ x, 0 ≤ re (inner x x) /-- The inner product is positive definite. -/ definite : ∀ x, inner x x = 0 → x = 0 /-- The inner product is additive in the first coordinate. -/ add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z /-- The inner product is conjugate linear in the first coordinate. -/ smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y #align inner_product_space.core InnerProductSpace.Core /- We set `InnerProductSpace.Core` to be a class as we will use it as such in the construction of the normed space structure that it produces. However, all the instances we will use will be local to this proof. -/ attribute [class] InnerProductSpace.Core /-- Define `InnerProductSpace.Core` from `InnerProductSpace`. Defined to reuse lemmas about `InnerProductSpace.Core` for `InnerProductSpace`s. Note that the `Norm` instance provided by `InnerProductSpace.Core.norm` is propositionally but not definitionally equal to the original norm. -/ def InnerProductSpace.toCore [NormedAddCommGroup E] [c : InnerProductSpace 𝕜 E] : InnerProductSpace.Core 𝕜 E := { c with nonneg_re := fun x => by rw [← InnerProductSpace.norm_sq_eq_inner] apply sq_nonneg definite := fun x hx => norm_eq_zero.1 <\| pow_eq_zero (n := 2) <\| by rw [InnerProductSpace.norm_sq_eq_inner (𝕜 := 𝕜) x, hx, map_zero] } #align inner_product_space.to_core InnerProductSpace.toCore namespace InnerProductSpace.Core variable [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] local notation "⟪" x ", " y "⟫" => @inner 𝕜 F _ x y local notation "normSqK" => @RCLike.normSq 𝕜 _ local notation "reK" => @RCLike.re 𝕜 _ local notation "ext_iff" => @RCLike.ext_iff 𝕜 _ local postfix:90 "†" => starRingEnd _ /-- Inner product defined by the `InnerProductSpace.Core` structure. We can't reuse `InnerProductSpace.Core.toInner` because it takes `InnerProductSpace.Core` as an explicit argument. -/ def toInner' : Inner 𝕜 F := c.toInner #align inner_product_space.core.to_has_inner' InnerProductSpace.Core.toInner' attribute [local instance] toInner' /-- The norm squared function for `InnerProductSpace.Core` structure. -/ def normSq (x : F) := reK ⟪x, x⟫ #align inner_product_space.core.norm_sq InnerProductSpace.Core.normSq local notation "normSqF" => @normSq 𝕜 F _ _ _ _ theorem inner_conj_symm (x y : F) : ⟪y, x⟫† = ⟪x, y⟫ := c.conj_symm x y #align inner_product_space.core.inner_conj_symm InnerProductSpace.Core.inner_conj_symm theorem inner_self_nonneg {x : F} : 0 ≤ re ⟪x, x⟫ := c.nonneg_re _ #align inner_product_space.core.inner_self_nonneg InnerProductSpace.Core.inner_self_nonneg theorem inner_self_im (x : F) : im ⟪x, x⟫ = 0 := by rw [← @ofReal_inj 𝕜, im_eq_conj_sub] simp [inner_conj_symm] #align inner_product_space.core.inner_self_im InnerProductSpace.Core.inner_self_im theorem inner_add_left (x y z : F) : ⟪x + y, z⟫ = ⟪x, z⟫ + ⟪y, z⟫ := c.add_left _ _ _ #align inner_product_space.core.inner_add_left InnerProductSpace.Core.inner_add_left theorem inner_add_right (x y z : F) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ := by rw [← inner_conj_symm, inner_add_left, RingHom.map_add]; simp only [inner_conj_symm] #align inner_product_space.core.inner_add_right InnerProductSpace.Core.inner_add_right theorem ofReal_normSq_eq_inner_self (x : F) : (normSqF x : 𝕜) = ⟪x, x⟫ := by rw [ext_iff] exact ⟨by simp only [ofReal_re]; rfl, by simp only [inner_self_im, ofReal_im]⟩ #align inner_product_space.core.coe_norm_sq_eq_inner_self InnerProductSpace.Core.ofReal_normSq_eq_inner_self theorem inner_re_symm (x y : F) : re ⟪x, y⟫ = re ⟪y, x⟫ := by rw [← inner_conj_symm, conj_re] #align inner_product_space.core.inner_re_symm InnerProductSpace.Core.inner_re_symm theorem inner_im_symm (x y : F) : im ⟪x, y⟫ = -im ⟪y, x⟫ := by rw [← inner_conj_symm, conj_im] #align inner_product_space.core.inner_im_symm InnerProductSpace.Core.inner_im_symm theorem inner_smul_left (x y : F) {r : 𝕜} : ⟪r • x, y⟫ = r† * ⟪x, y⟫ := c.smul_left _ _ _ #align inner_product_space.core.inner_smul_left InnerProductSpace.Core.inner_smul_left theorem inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by rw [← inner_conj_symm, inner_smul_left]; simp only [conj_conj, inner_conj_symm, RingHom.map_mul] #align inner_product_space.core.inner_smul_right InnerProductSpace.Core.inner_smul_right theorem inner_zero_left (x : F) : ⟪0, x⟫ = 0 := by rw [← zero_smul 𝕜 (0 : F), inner_smul_left]; simp only [zero_mul, RingHom.map_zero] #align inner_product_space.core.inner_zero_left InnerProductSpace.Core.inner_zero_left theorem inner_zero_right (x : F) : ⟪x, 0⟫ = 0 := by rw [← inner_conj_symm, inner_zero_left]; simp only [RingHom.map_zero] #align inner_product_space.core.inner_zero_right InnerProductSpace.Core.inner_zero_right theorem inner_self_eq_zero {x : F} : ⟪x, x⟫ = 0 ↔ x = 0 := ⟨c.definite _, by rintro rfl exact inner_zero_left _⟩ #align inner_product_space.core.inner_self_eq_zero InnerProductSpace.Core.inner_self_eq_zero theorem normSq_eq_zero {x : F} : normSqF x = 0 ↔ x = 0 := Iff.trans (by simp only [normSq, ext_iff, map_zero, inner_self_im, eq_self_iff_true, and_true_iff]) (@inner_self_eq_zero 𝕜 _ _ _ _ _ x) #align inner_product_space.core.norm_sq_eq_zero InnerProductSpace.Core.normSq_eq_zero theorem inner_self_ne_zero {x : F} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 := inner_self_eq_zero.not #align inner_product_space.core.inner_self_ne_zero InnerProductSpace.Core.inner_self_ne_zero theorem inner_self_ofReal_re (x : F) : (re ⟪x, x⟫ : 𝕜) = ⟪x, x⟫ := by norm_num [ext_iff, inner_self_im] set_option linter.uppercaseLean3 false in #align inner_product_space.core.inner_self_re_to_K InnerProductSpace.Core.inner_self_ofReal_re theorem norm_inner_symm (x y : F) : ‖⟪x, y⟫‖ = ‖⟪y, x⟫‖ := by rw [← inner_conj_symm, norm_conj] #align inner_product_space.core.norm_inner_symm InnerProductSpace.Core.norm_inner_symm theorem inner_neg_left (x y : F) : ⟪-x, y⟫ = -⟪x, y⟫ := by rw [← neg_one_smul 𝕜 x, inner_smul_left] simp #align inner_product_space.core.inner_neg_left InnerProductSpace.Core.inner_neg_left theorem inner_neg_right (x y : F) : ⟪x, -y⟫ = -⟪x, y⟫ := by rw [← inner_conj_symm, inner_neg_left]; simp only [RingHom.map_neg, inner_conj_symm] #align inner_product_space.core.inner_neg_right InnerProductSpace.Core.inner_neg_right theorem inner_sub_left (x y z : F) : ⟪x - y, z⟫ = ⟪x, z⟫ - ⟪y, z⟫ := by simp [sub_eq_add_neg, inner_add_left, inner_neg_left] #align inner_product_space.core.inner_sub_left InnerProductSpace.Core.inner_sub_left theorem inner_sub_right (x y z : F) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ := by simp [sub_eq_add_neg, inner_add_right, inner_neg_right] #align inner_product_space.core.inner_sub_right InnerProductSpace.Core.inner_sub_right theorem inner_mul_symm_re_eq_norm (x y : F) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ := by rw [← inner_conj_symm, mul_comm] exact re_eq_norm_of_mul_conj (inner y x) #align inner_product_space.core.inner_mul_symm_re_eq_norm InnerProductSpace.Core.inner_mul_symm_re_eq_norm /-- Expand `inner (x + y) (x + y)` -/ theorem inner_add_add_self (x y : F) : ⟪x + y, x + y⟫ = ⟪x, x⟫ + ⟪x, y⟫ + ⟪y, x⟫ + ⟪y, y⟫ := by simp only [inner_add_left, inner_add_right]; ring #align inner_product_space.core.inner_add_add_self InnerProductSpace.Core.inner_add_add_self -- Expand `inner (x - y) (x - y)` theorem inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ := by simp only [inner_sub_left, inner_sub_right]; ring #align inner_product_space.core.inner_sub_sub_self InnerProductSpace.Core.inner_sub_sub_self /-- An auxiliary equality useful to prove the Cauchy–Schwarz inequality: the square of the norm of `⟪x, y⟫ • x - ⟪x, x⟫ • y` is equal to `‖x‖ ^ 2 * (‖x‖ ^ 2 * ‖y‖ ^ 2 - ‖⟪x, y⟫‖ ^ 2)`. We use `InnerProductSpace.ofCore.normSq x` etc (defeq to `is_R_or_C.re ⟪x, x⟫`) instead of `‖x‖ ^ 2` etc to avoid extra rewrites when applying it to an `InnerProductSpace`. -/ theorem cauchy_schwarz_aux (x y : F) : normSqF (⟪x, y⟫ • x - ⟪x, x⟫ • y) = normSqF x * (normSqF x * normSqF y - ‖⟪x, y⟫‖ ^ 2) := by rw [← @ofReal_inj 𝕜, ofReal_normSq_eq_inner_self] simp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_ofReal, mul_sub, ← ofReal_normSq_eq_inner_self x, ← ofReal_normSq_eq_inner_self y] rw [← mul_assoc, mul_conj, RCLike.conj_mul, mul_left_comm, ← inner_conj_symm y, mul_conj] push_cast ring #align inner_product_space.core.cauchy_schwarz_aux InnerProductSpace.Core.cauchy_schwarz_aux /-- Cauchy–Schwarz inequality. We need this for the `Core` structure to prove the triangle inequality below when showing the core is a normed group. -/ theorem inner_mul_inner_self_le (x y : F) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ := by rcases eq_or_ne x 0 with (rfl \| hx) · simpa only [inner_zero_left, map_zero, zero_mul, norm_zero] using le_rfl · have hx' : 0 < normSqF x := inner_self_nonneg.lt_of_ne' (mt normSq_eq_zero.1 hx) rw [← sub_nonneg, ← mul_nonneg_iff_right_nonneg_of_pos hx', ← normSq, ← normSq, norm_inner_symm y, ← sq, ← cauchy_schwarz_aux] exact inner_self_nonneg #align inner_product_space.core.inner_mul_inner_self_le InnerProductSpace.Core.inner_mul_inner_self_le /-- Norm constructed from an `InnerProductSpace.Core` structure, defined to be the square root of the scalar product. -/ def toNorm : Norm F where norm x := √(re ⟪x, x⟫) #align inner_product_space.core.to_has_norm InnerProductSpace.Core.toNorm attribute [local instance] toNorm theorem norm_eq_sqrt_inner (x : F) : ‖x‖ = √(re ⟪x, x⟫) := rfl #align inner_product_space.core.norm_eq_sqrt_inner InnerProductSpace.Core.norm_eq_sqrt_inner theorem inner_self_eq_norm_mul_norm (x : F) : re ⟪x, x⟫ = ‖x‖ * ‖x‖ := by rw [norm_eq_sqrt_inner, ← sqrt_mul inner_self_nonneg (re ⟪x, x⟫), sqrt_mul_self inner_self_nonneg] #align inner_product_space.core.inner_self_eq_norm_mul_norm InnerProductSpace.Core.inner_self_eq_norm_mul_norm theorem sqrt_normSq_eq_norm (x : F) : √(normSqF x) = ‖x‖ := rfl #align inner_product_space.core.sqrt_norm_sq_eq_norm InnerProductSpace.Core.sqrt_normSq_eq_norm /-- Cauchy–Schwarz inequality with norm -/ theorem norm_inner_le_norm (x y : F) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := nonneg_le_nonneg_of_sq_le_sq (mul_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) <\| calc ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ = ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ := by rw [norm_inner_symm] _ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ := inner_mul_inner_self_le x y _ = ‖x‖ * ‖y‖ * (‖x‖ * ‖y‖) := by simp only [inner_self_eq_norm_mul_norm]; ring #align inner_product_space.core.norm_inner_le_norm InnerProductSpace.Core.norm_inner_le_norm /-- Normed group structure constructed from an `InnerProductSpace.Core` structure -/ def toNormedAddCommGroup : NormedAddCommGroup F := AddGroupNorm.toNormedAddCommGroup { toFun := fun x => √(re ⟪x, x⟫) map_zero' := by simp only [sqrt_zero, inner_zero_right, map_zero] neg' := fun x => by simp only [inner_neg_left, neg_neg, inner_neg_right] add_le' := fun x y => by have h₁ : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm _ _ have h₂ : re ⟪x, y⟫ ≤ ‖⟪x, y⟫‖ := re_le_norm _ have h₃ : re ⟪x, y⟫ ≤ ‖x‖ * ‖y‖ := h₂.trans h₁ have h₄ : re ⟪y, x⟫ ≤ ‖x‖ * ‖y‖ := by rwa [← inner_conj_symm, conj_re] have : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖) := by simp only [← inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add] linarith exact nonneg_le_nonneg_of_sq_le_sq (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this eq_zero_of_map_eq_zero' := fun x hx => normSq_eq_zero.1 <\| (sqrt_eq_zero inner_self_nonneg).1 hx } #align inner_product_space.core.to_normed_add_comm_group InnerProductSpace.Core.toNormedAddCommGroup attribute [local instance] toNormedAddCommGroup /-- Normed space structure constructed from an `InnerProductSpace.Core` structure -/ def toNormedSpace : NormedSpace 𝕜 F where norm_smul_le r x := by rw [norm_eq_sqrt_inner, inner_smul_left, inner_smul_right, ← mul_assoc] rw [RCLike.conj_mul, ← ofReal_pow, re_ofReal_mul, sqrt_mul, ← ofReal_normSq_eq_inner_self, ofReal_re] · simp [sqrt_normSq_eq_norm, RCLike.sqrt_normSq_eq_norm] · positivity #align inner_product_space.core.to_normed_space InnerProductSpace.Core.toNormedSpace end InnerProductSpace.Core section attribute [local instance] InnerProductSpace.Core.toNormedAddCommGroup /-- Given an `InnerProductSpace.Core` structure on a space, one can use it to turn the space into an inner product space. The `NormedAddCommGroup` structure is expected to already be defined with `InnerProductSpace.ofCore.toNormedAddCommGroup`. -/ def InnerProductSpace.ofCore [AddCommGroup F] [Module 𝕜 F] (c : InnerProductSpace.Core 𝕜 F) : InnerProductSpace 𝕜 F := letI : NormedSpace 𝕜 F := @InnerProductSpace.Core.toNormedSpace 𝕜 F _ _ _ c { c with norm_sq_eq_inner := fun x => by have h₁ : ‖x‖ ^ 2 = √(re (c.inner x x)) ^ 2 := rfl have h₂ : 0 ≤ re (c.inner x x) := InnerProductSpace.Core.inner_self_nonneg simp [h₁, sq_sqrt, h₂] } #align inner_product_space.of_core InnerProductSpace.ofCore end /-! ### Properties of inner product spaces -/ variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] variable [NormedAddCommGroup F] [InnerProductSpace ℝ F] local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y local notation "IK" => @RCLike.I 𝕜 _ local postfix:90 "†" => starRingEnd _ export InnerProductSpace (norm_sq_eq_inner) section BasicProperties @[simp] theorem inner_conj_symm (x y : E) : ⟪y, x⟫† = ⟪x, y⟫ := InnerProductSpace.conj_symm _ _ #align inner_conj_symm inner_conj_symm theorem real_inner_comm (x y : F) : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := @inner_conj_symm ℝ _ _ _ _ x y #align real_inner_comm real_inner_comm theorem inner_eq_zero_symm {x y : E} : ⟪x, y⟫ = 0 ↔ ⟪y, x⟫ = 0 := by rw [← inner_conj_symm] exact star_eq_zero #align inner_eq_zero_symm inner_eq_zero_symm @[simp] theorem inner_self_im (x : E) : im ⟪x, x⟫ = 0 := by rw [← @ofReal_inj 𝕜, im_eq_conj_sub]; simp #align inner_self_im inner_self_im theorem inner_add_left (x y z : E) : ⟪x + y, z⟫ = ⟪x, z⟫ + ⟪y, z⟫ := InnerProductSpace.add_left _ _ _ #align inner_add_left inner_add_left theorem inner_add_right (x y z : E) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ := by rw [← inner_conj_symm, inner_add_left, RingHom.map_add] simp only [inner_conj_symm] #align inner_add_right inner_add_right theorem inner_re_symm (x y : E) : re ⟪x, y⟫ = re ⟪y, x⟫ := by rw [← inner_conj_symm, conj_re] #align inner_re_symm inner_re_symm theorem inner_im_symm (x y : E) : im ⟪x, y⟫ = -im ⟪y, x⟫ := by rw [← inner_conj_symm, conj_im] #align inner_im_symm inner_im_symm theorem inner_smul_left (x y : E) (r : 𝕜) : ⟪r • x, y⟫ = r† * ⟪x, y⟫ := InnerProductSpace.smul_left _ _ _ #align inner_smul_left inner_smul_left theorem real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ := inner_smul_left _ _ _ #align real_inner_smul_left real_inner_smul_left theorem inner_smul_real_left (x y : E) (r : ℝ) : ⟪(r : 𝕜) • x, y⟫ = r • ⟪x, y⟫ := by rw [inner_smul_left, conj_ofReal, Algebra.smul_def] rfl #align inner_smul_real_left inner_smul_real_left theorem inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by rw [← inner_conj_symm, inner_smul_left, RingHom.map_mul, conj_conj, inner_conj_symm] #align inner_smul_right inner_smul_right theorem real_inner_smul_right (x y : F) (r : ℝ) : ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ := inner_smul_right _ _ _ #align real_inner_smul_right real_inner_smul_right theorem inner_smul_real_right (x y : E) (r : ℝ) : ⟪x, (r : 𝕜) • y⟫ = r • ⟪x, y⟫ := by rw [inner_smul_right, Algebra.smul_def] rfl #align inner_smul_real_right inner_smul_real_right /-- The inner product as a sesquilinear form. Note that in the case `𝕜 = ℝ` this is a bilinear form. -/ @[simps!] def sesqFormOfInner : E →ₗ[𝕜] E →ₗ⋆[𝕜] 𝕜 := LinearMap.mk₂'ₛₗ (RingHom.id 𝕜) (starRingEnd _) (fun x y => ⟪y, x⟫) (fun _x _y _z => inner_add_right _ _ _) (fun _r _x _y => inner_smul_right _ _ _) (fun _x _y _z => inner_add_left _ _ _) fun _r _x _y => inner_smul_left _ _ _ #align sesq_form_of_inner sesqFormOfInner /-- The real inner product as a bilinear form. Note that unlike `sesqFormOfInner`, this does not reverse the order of the arguments. -/ @[simps!] def bilinFormOfRealInner : BilinForm ℝ F := sesqFormOfInner.flip #align bilin_form_of_real_inner bilinFormOfRealInner /-- An inner product with a sum on the left. -/ theorem sum_inner {ι : Type} (s : Finset ι) (f : ι → E) (x : E) : ⟪∑ i ∈ s, f i, x⟫ = ∑ i ∈ s, ⟪f i, x⟫ := map_sum (sesqFormOfInner (𝕜 := 𝕜) (E := E) x) _ _ #align sum_inner sum_inner /-- An inner product with a sum on the right. -/ theorem inner_sum {ι : Type} (s : Finset ι) (f : ι → E) (x : E) : ⟪x, ∑ i ∈ s, f i⟫ = ∑ i ∈ s, ⟪x, f i⟫ := map_sum (LinearMap.flip sesqFormOfInner x) _ _ #align inner_sum inner_sum /-- An inner product with a sum on the left, `Finsupp` version. -/ theorem Finsupp.sum_inner {ι : Type} (l : ι →₀ 𝕜) (v : ι → E) (x : E) : ⟪l.sum fun (i : ι) (a : 𝕜) => a • v i, x⟫ = l.sum fun (i : ι) (a : 𝕜) => conj a • ⟪v i, x⟫ := by convert _root_.sum_inner (𝕜 := 𝕜) l.support (fun a => l a • v a) x simp only [inner_smul_left, Finsupp.sum, smul_eq_mul] #align finsupp.sum_inner Finsupp.sum_inner /-- An inner product with a sum on the right, `Finsupp` version. -/ theorem Finsupp.inner_sum {ι : Type} (l : ι →₀ 𝕜) (v : ι → E) (x : E) : ⟪x, l.sum fun (i : ι) (a : 𝕜) => a • v i⟫ = l.sum fun (i : ι) (a : 𝕜) => a • ⟪x, v i⟫ := by convert _root_.inner_sum (𝕜 := 𝕜) l.support (fun a => l a • v a) x simp only [inner_smul_right, Finsupp.sum, smul_eq_mul] #align finsupp.inner_sum Finsupp.inner_sum theorem DFinsupp.sum_inner {ι : Type} [DecidableEq ι] {α : ι → Type} [∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E) (l : Π₀ i, α i) (x : E) : ⟪l.sum f, x⟫ = l.sum fun i a => ⟪f i a, x⟫ := by simp (config := { contextual := true }) only [DFinsupp.sum, _root_.sum_inner, smul_eq_mul] #align dfinsupp.sum_inner DFinsupp.sum_inner theorem DFinsupp.inner_sum {ι : Type} [DecidableEq ι] {α : ι → Type} [∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E) (l : Π₀ i, α i) (x : E) : ⟪x, l.sum f⟫ = l.sum fun i a => ⟪x, f i a⟫ := by simp (config := { contextual := true }) only [DFinsupp.sum, _root_.inner_sum, smul_eq_mul] #align dfinsupp.inner_sum DFinsupp.inner_sum @[simp] theorem inner_zero_left (x : E) : ⟪0, x⟫ = 0 := by rw [← zero_smul 𝕜 (0 : E), inner_smul_left, RingHom.map_zero, zero_mul] #align inner_zero_left inner_zero_left theorem inner_re_zero_left (x : E) : re ⟪0, x⟫ = 0 := by simp only [inner_zero_left, AddMonoidHom.map_zero] #align inner_re_zero_left inner_re_zero_left @[simp] theorem inner_zero_right (x : E) : ⟪x, 0⟫ = 0 := by rw [← inner_conj_symm, inner_zero_left, RingHom.map_zero] #align inner_zero_right inner_zero_right theorem inner_re_zero_right (x : E) : re ⟪x, 0⟫ = 0 := by simp only [inner_zero_right, AddMonoidHom.map_zero] #align inner_re_zero_right inner_re_zero_right theorem inner_self_nonneg {x : E} : 0 ≤ re ⟪x, x⟫ := InnerProductSpace.toCore.nonneg_re x #align inner_self_nonneg inner_self_nonneg theorem real_inner_self_nonneg {x : F} : 0 ≤ ⟪x, x⟫_ℝ := @inner_self_nonneg ℝ F _ _ _ x #align real_inner_self_nonneg real_inner_self_nonneg @[simp] theorem inner_self_ofReal_re (x : E) : (re ⟪x, x⟫ : 𝕜) = ⟪x, x⟫ := ((RCLike.is_real_TFAE (⟪x, x⟫ : 𝕜)).out 2 3).2 (inner_self_im _) set_option linter.uppercaseLean3 false in #align inner_self_re_to_K inner_self_ofReal_re theorem inner_self_eq_norm_sq_to_K (x : E) : ⟪x, x⟫ = (‖x‖ : 𝕜) ^ 2 := by rw [← inner_self_ofReal_re, ← norm_sq_eq_inner, ofReal_pow] set_option linter.uppercaseLean3 false in #align inner_self_eq_norm_sq_to_K inner_self_eq_norm_sq_to_K theorem inner_self_re_eq_norm (x : E) : re ⟪x, x⟫ = ‖⟪x, x⟫‖ := by conv_rhs => rw [← inner_self_ofReal_re] symm exact norm_of_nonneg inner_self_nonneg #align inner_self_re_eq_norm inner_self_re_eq_norm theorem inner_self_ofReal_norm (x : E) : (‖⟪x, x⟫‖ : 𝕜) = ⟪x, x⟫ := by rw [← inner_self_re_eq_norm] exact inner_self_ofReal_re _ set_option linter.uppercaseLean3 false in #align inner_self_norm_to_K inner_self_ofReal_norm theorem real_inner_self_abs (x : F) : \|⟪x, x⟫_ℝ\| = ⟪x, x⟫_ℝ := @inner_self_ofReal_norm ℝ F _ _ _ x #align real_inner_self_abs real_inner_self_abs @[simp] theorem inner_self_eq_zero {x : E} : ⟪x, x⟫ = 0 ↔ x = 0 := by rw [inner_self_eq_norm_sq_to_K, sq_eq_zero_iff, ofReal_eq_zero, norm_eq_zero] #align inner_self_eq_zero inner_self_eq_zero theorem inner_self_ne_zero {x : E} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 := inner_self_eq_zero.not #align inner_self_ne_zero inner_self_ne_zero @[simp] theorem inner_self_nonpos {x : E} : re ⟪x, x⟫ ≤ 0 ↔ x = 0 := by rw [← norm_sq_eq_inner, (sq_nonneg _).le_iff_eq, sq_eq_zero_iff, norm_eq_zero] #align inner_self_nonpos inner_self_nonpos theorem real_inner_self_nonpos {x : F} : ⟪x, x⟫_ℝ ≤ 0 ↔ x = 0 := @inner_self_nonpos ℝ F _ _ _ x #align real_inner_self_nonpos real_inner_self_nonpos theorem norm_inner_symm (x y : E) : ‖⟪x, y⟫‖ = ‖⟪y, x⟫‖ := by rw [← inner_conj_symm, norm_conj] #align norm_inner_symm norm_inner_symm @[simp] theorem inner_neg_left (x y : E) : ⟪-x, y⟫ = -⟪x, y⟫ := by rw [← neg_one_smul 𝕜 x, inner_smul_left] simp #align inner_neg_left inner_neg_left @[simp] theorem inner_neg_right (x y : E) : ⟪x, -y⟫ = -⟪x, y⟫ := by rw [← inner_conj_symm, inner_neg_left]; simp only [RingHom.map_neg, inner_conj_symm] #align inner_neg_right inner_neg_right theorem inner_neg_neg (x y : E) : ⟪-x, -y⟫ = ⟪x, y⟫ := by simp #align inner_neg_neg inner_neg_neg -- Porting note: removed `simp` because it can prove it using `inner_conj_symm` theorem inner_self_conj (x : E) : ⟪x, x⟫† = ⟪x, x⟫ := inner_conj_symm _ _ #align inner_self_conj inner_self_conj theorem inner_sub_left (x y z : E) : ⟪x - y, z⟫ = ⟪x, z⟫ - ⟪y, z⟫ := by simp [sub_eq_add_neg, inner_add_left] #align inner_sub_left inner_sub_left theorem inner_sub_right (x y z : E) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ := by simp [sub_eq_add_neg, inner_add_right] #align inner_sub_right inner_sub_right theorem inner_mul_symm_re_eq_norm (x y : E) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ := by rw [← inner_conj_symm, mul_comm] exact re_eq_norm_of_mul_conj (inner y x) #align inner_mul_symm_re_eq_norm inner_mul_symm_re_eq_norm /-- Expand `⟪x + y, x + y⟫` -/ theorem inner_add_add_self (x y : E) : ⟪x + y, x + y⟫ = ⟪x, x⟫ + ⟪x, y⟫ + ⟪y, x⟫ + ⟪y, y⟫ := by simp only [inner_add_left, inner_add_right]; ring #align inner_add_add_self inner_add_add_self /-- Expand `⟪x + y, x + y⟫_ℝ` -/ theorem real_inner_add_add_self (x y : F) : ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ := by have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [← inner_conj_symm]; rfl simp only [inner_add_add_self, this, add_left_inj] ring #align real_inner_add_add_self real_inner_add_add_self -- Expand `⟪x - y, x - y⟫` theorem inner_sub_sub_self (x y : E) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ := by simp only [inner_sub_left, inner_sub_right]; ring #align inner_sub_sub_self inner_sub_sub_self /-- Expand `⟪x - y, x - y⟫_ℝ` -/ theorem real_inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫_ℝ = ⟪x, x⟫_ℝ - 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ := by have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [← inner_conj_symm]; rfl simp only [inner_sub_sub_self, this, add_left_inj] ring #align real_inner_sub_sub_self real_inner_sub_sub_self variable (𝕜) theorem ext_inner_left {x y : E} (h : ∀ v, ⟪v, x⟫ = ⟪v, y⟫) : x = y := by rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜, inner_sub_right, sub_eq_zero, h (x - y)] #align ext_inner_left ext_inner_left theorem ext_inner_right {x y : E} (h : ∀ v, ⟪x, v⟫ = ⟪y, v⟫) : x = y := by rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜, inner_sub_left, sub_eq_zero, h (x - y)] #align ext_inner_right ext_inner_right variable {𝕜} /-- Parallelogram law -/ theorem parallelogram_law {x y : E} : ⟪x + y, x + y⟫ + ⟪x - y, x - y⟫ = 2 * (⟪x, x⟫ + ⟪y, y⟫) := by simp only [inner_add_add_self, inner_sub_sub_self] ring #align parallelogram_law parallelogram_law /-- Cauchy–Schwarz inequality. -/ theorem inner_mul_inner_self_le (x y : E) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ := letI c : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore InnerProductSpace.Core.inner_mul_inner_self_le x y #align inner_mul_inner_self_le inner_mul_inner_self_le /-- Cauchy–Schwarz inequality for real inner products. -/ theorem real_inner_mul_inner_self_le (x y : F) : ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ := calc ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ‖⟪x, y⟫_ℝ‖ * ‖⟪y, x⟫_ℝ‖ := by rw [real_inner_comm y, ← norm_mul] exact le_abs_self _ _ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ := @inner_mul_inner_self_le ℝ _ _ _ _ x y #align real_inner_mul_inner_self_le real_inner_mul_inner_self_le /-- A family of vectors is linearly independent if they are nonzero and orthogonal. -/ theorem linearIndependent_of_ne_zero_of_inner_eq_zero {ι : Type} {v : ι → E} (hz : ∀ i, v i ≠ 0) (ho : Pairwise fun i j => ⟪v i, v j⟫ = 0) : LinearIndependent 𝕜 v := by rw [linearIndependent_iff'] intro s g hg i hi have h' : g i inner (v i) (v i) = inner (v i) (∑ j ∈ s, g j • v j) := by rw [inner_sum] symm convert Finset.sum_eq_single (β := 𝕜) i ?_ ?_ · rw [inner_smul_right] · intro j _hj hji rw [inner_smul_right, ho hji.symm, mul_zero] · exact fun h => False.elim (h hi) simpa [hg, hz] using h' #align linear_independent_of_ne_zero_of_inner_eq_zero linearIndependent_of_ne_zero_of_inner_eq_zero end BasicProperties section OrthonormalSets variable {ι : Type} (𝕜) /-- An orthonormal set of vectors in an `InnerProductSpace` -/ def Orthonormal (v : ι → E) : Prop := (∀ i, ‖v i‖ = 1) ∧ Pairwise fun i j => ⟪v i, v j⟫ = 0 #align orthonormal Orthonormal variable {𝕜} /-- `if ... then ... else` characterization of an indexed set of vectors being orthonormal. (Inner product equals Kronecker delta.) -/ theorem orthonormal_iff_ite [DecidableEq ι] {v : ι → E} : Orthonormal 𝕜 v ↔ ∀ i j, ⟪v i, v j⟫ = if i = j then (1 : 𝕜) else (0 : 𝕜) := by constructor · intro hv i j split_ifs with h · simp [h, inner_self_eq_norm_sq_to_K, hv.1] · exact hv.2 h · intro h constructor · intro i have h' : ‖v i‖ ^ 2 = 1 ^ 2 := by simp [@norm_sq_eq_inner 𝕜, h i i] have h₁ : 0 ≤ ‖v i‖ := norm_nonneg _ have h₂ : (0 : ℝ) ≤ 1 := zero_le_one rwa [sq_eq_sq h₁ h₂] at h' · intro i j hij simpa [hij] using h i j #align orthonormal_iff_ite orthonormal_iff_ite /-- `if ... then ... else` characterization of a set of vectors being orthonormal. (Inner product equals Kronecker delta.) -/ theorem orthonormal_subtype_iff_ite [DecidableEq E] {s : Set E} : Orthonormal 𝕜 (Subtype.val : s → E) ↔ ∀ v ∈ s, ∀ w ∈ s, ⟪v, w⟫ = if v = w then 1 else 0 := by rw [orthonormal_iff_ite] constructor · intro h v hv w hw convert h ⟨v, hv⟩ ⟨w, hw⟩ using 1 simp · rintro h ⟨v, hv⟩ ⟨w, hw⟩ convert h v hv w hw using 1 simp #align orthonormal_subtype_iff_ite orthonormal_subtype_iff_ite /-- The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector. -/ theorem Orthonormal.inner_right_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) : ⟪v i, Finsupp.total ι E 𝕜 v l⟫ = l i := by classical simpa [Finsupp.total_apply, Finsupp.inner_sum, orthonormal_iff_ite.mp hv] using Eq.symm #align orthonormal.inner_right_finsupp Orthonormal.inner_right_finsupp /-- The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector. -/ theorem Orthonormal.inner_right_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι} {i : ι} (hi : i ∈ s) : ⟪v i, ∑ i ∈ s, l i • v i⟫ = l i := by classical simp [inner_sum, inner_smul_right, orthonormal_iff_ite.mp hv, hi] #align orthonormal.inner_right_sum Orthonormal.inner_right_sum /-- The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector. -/ theorem Orthonormal.inner_right_fintype [Fintype ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι) : ⟪v i, ∑ i : ι, l i • v i⟫ = l i := hv.inner_right_sum l (Finset.mem_univ _) #align orthonormal.inner_right_fintype Orthonormal.inner_right_fintype /-- The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector. -/ theorem Orthonormal.inner_left_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) : ⟪Finsupp.total ι E 𝕜 v l, v i⟫ = conj (l i) := by rw [← inner_conj_symm, hv.inner_right_finsupp] #align orthonormal.inner_left_finsupp Orthonormal.inner_left_finsupp /-- The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector. -/ theorem Orthonormal.inner_left_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι} {i : ι} (hi : i ∈ s) : ⟪∑ i ∈ s, l i • v i, v i⟫ = conj (l i) := by classical simp only [sum_inner, inner_smul_left, orthonormal_iff_ite.mp hv, hi, mul_boole, Finset.sum_ite_eq', if_true] #align orthonormal.inner_left_sum Orthonormal.inner_left_sum /-- The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector. -/ theorem Orthonormal.inner_left_fintype [Fintype ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι) : ⟪∑ i : ι, l i • v i, v i⟫ = conj (l i) := hv.inner_left_sum l (Finset.mem_univ _) #align orthonormal.inner_left_fintype Orthonormal.inner_left_fintype /-- The inner product of two linear combinations of a set of orthonormal vectors, expressed as a sum over the first `Finsupp`. -/ theorem Orthonormal.inner_finsupp_eq_sum_left {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) : ⟪Finsupp.total ι E 𝕜 v l₁, Finsupp.total ι E 𝕜 v l₂⟫ = l₁.sum fun i y => conj y l₂ i := by simp only [l₁.total_apply _, Finsupp.sum_inner, hv.inner_right_finsupp, smul_eq_mul] #align orthonormal.inner_finsupp_eq_sum_left Orthonormal.inner_finsupp_eq_sum_left /-- The inner product of two linear combinations of a set of orthonormal vectors, expressed as a sum over the second `Finsupp`. -/ theorem Orthonormal.inner_finsupp_eq_sum_right {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) : ⟪Finsupp.total ι E 𝕜 v l₁, Finsupp.total ι E 𝕜 v l₂⟫ = l₂.sum fun i y => conj (l₁ i) * y := by simp only [l₂.total_apply _, Finsupp.inner_sum, hv.inner_left_finsupp, mul_comm, smul_eq_mul] #align orthonormal.inner_finsupp_eq_sum_right Orthonormal.inner_finsupp_eq_sum_right /-- The inner product of two linear combinations of a set of orthonormal vectors, expressed as a sum. -/ theorem Orthonormal.inner_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι → 𝕜) (s : Finset ι) : ⟪∑ i ∈ s, l₁ i • v i, ∑ i ∈ s, l₂ i • v i⟫ = ∑ i ∈ s, conj (l₁ i) * l₂ i := by simp_rw [sum_inner, inner_smul_left] refine Finset.sum_congr rfl fun i hi => ?_ rw [hv.inner_right_sum l₂ hi] #align orthonormal.inner_sum Orthonormal.inner_sum /-- The double sum of weighted inner products of pairs of vectors from an orthonormal sequence is the sum of the weights. -/ theorem Orthonormal.inner_left_right_finset {s : Finset ι} {v : ι → E} (hv : Orthonormal 𝕜 v) {a : ι → ι → 𝕜} : (∑ i ∈ s, ∑ j ∈ s, a i j • ⟪v j, v i⟫) = ∑ k ∈ s, a k k := by classical simp [orthonormal_iff_ite.mp hv, Finset.sum_ite_of_true] #align orthonormal.inner_left_right_finset Orthonormal.inner_left_right_finset /-- An orthonormal set is linearly independent. -/ theorem Orthonormal.linearIndependent {v : ι → E} (hv : Orthonormal 𝕜 v) : LinearIndependent 𝕜 v := by rw [linearIndependent_iff] intro l hl ext i have key : ⟪v i, Finsupp.total ι E 𝕜 v l⟫ = ⟪v i, 0⟫ := by rw [hl] simpa only [hv.inner_right_finsupp, inner_zero_right] using key #align orthonormal.linear_independent Orthonormal.linearIndependent /-- A subfamily of an orthonormal family (i.e., a composition with an injective map) is an orthonormal family. -/ theorem Orthonormal.comp {ι' : Type} {v : ι → E} (hv : Orthonormal 𝕜 v) (f : ι' → ι) (hf : Function.Injective f) : Orthonormal 𝕜 (v ∘ f) := by classical rw [orthonormal_iff_ite] at hv ⊢ intro i j convert hv (f i) (f j) using 1 simp [hf.eq_iff] #align orthonormal.comp Orthonormal.comp /-- An injective family `v : ι → E` is orthonormal if and only if `Subtype.val : (range v) → E` is orthonormal. -/ theorem orthonormal_subtype_range {v : ι → E} (hv : Function.Injective v) : Orthonormal 𝕜 (Subtype.val : Set.range v → E) ↔ Orthonormal 𝕜 v := by let f : ι ≃ Set.range v := Equiv.ofInjective v hv refine ⟨fun h => h.comp f f.injective, fun h => ?_⟩ rw [← Equiv.self_comp_ofInjective_symm hv] exact h.comp f.symm f.symm.injective #align orthonormal_subtype_range orthonormal_subtype_range /-- If `v : ι → E` is an orthonormal family, then `Subtype.val : (range v) → E` is an orthonormal family. -/ theorem Orthonormal.toSubtypeRange {v : ι → E} (hv : Orthonormal 𝕜 v) : Orthonormal 𝕜 (Subtype.val : Set.range v → E) := (orthonormal_subtype_range hv.linearIndependent.injective).2 hv #align orthonormal.to_subtype_range Orthonormal.toSubtypeRange /-- A linear combination of some subset of an orthonormal set is orthogonal to other members of the set. -/ theorem Orthonormal.inner_finsupp_eq_zero {v : ι → E} (hv : Orthonormal 𝕜 v) {s : Set ι} {i : ι} (hi : i ∉ s) {l : ι →₀ 𝕜} (hl : l ∈ Finsupp.supported 𝕜 𝕜 s) : ⟪Finsupp.total ι E 𝕜 v l, v i⟫ = 0 := by rw [Finsupp.mem_supported'] at hl simp only [hv.inner_left_finsupp, hl i hi, map_zero] #align orthonormal.inner_finsupp_eq_zero Orthonormal.inner_finsupp_eq_zero /-- Given an orthonormal family, a second family of vectors is orthonormal if every vector equals the corresponding vector in the original family or its negation. -/ theorem Orthonormal.orthonormal_of_forall_eq_or_eq_neg {v w : ι → E} (hv : Orthonormal 𝕜 v) (hw : ∀ i, w i = v i ∨ w i = -v i) : Orthonormal 𝕜 w := by classical rw [orthonormal_iff_ite] at intro i j cases' hw i with hi hi <;> cases' hw j with hj hj <;> replace hv := hv i j <;> split_ifs at hv ⊢ with h <;> simpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv #align orthonormal.orthonormal_of_forall_eq_or_eq_neg Orthonormal.orthonormal_of_forall_eq_or_eq_neg /- The material that follows, culminating in the existence of a maximal orthonormal subset, is adapted from the corresponding development of the theory of linearly independents sets. See `exists_linearIndependent` in particular. -/ variable (𝕜 E) theorem orthonormal_empty : Orthonormal 𝕜 (fun x => x : (∅ : Set E) → E) := by classical simp [orthonormal_subtype_iff_ite] #align orthonormal_empty orthonormal_empty variable {𝕜 E} theorem orthonormal_iUnion_of_directed {η : Type} {s : η → Set E} (hs : Directed (· ⊆ ·) s) (h : ∀ i, Orthonormal 𝕜 (fun x => x : s i → E)) : Orthonormal 𝕜 (fun x => x : (⋃ i, s i) → E) := by classical rw [orthonormal_subtype_iff_ite] rintro x ⟨_, ⟨i, rfl⟩, hxi⟩ y ⟨_, ⟨j, rfl⟩, hyj⟩ obtain ⟨k, hik, hjk⟩ := hs i j have h_orth : Orthonormal 𝕜 (fun x => x : s k → E) := h k rw [orthonormal_subtype_iff_ite] at h_orth exact h_orth x (hik hxi) y (hjk hyj) #align orthonormal_Union_of_directed orthonormal_iUnion_of_directed theorem orthonormal_sUnion_of_directed {s : Set (Set E)} (hs : DirectedOn (· ⊆ ·) s) (h : ∀ a ∈ s, Orthonormal 𝕜 (fun x => ((x : a) : E))) : Orthonormal 𝕜 (fun x => x : ⋃₀ s → E) := by rw [Set.sUnion_eq_iUnion]; exact orthonormal_iUnion_of_directed hs.directed_val (by simpa using h) #align orthonormal_sUnion_of_directed orthonormal_sUnion_of_directed /-- Given an orthonormal set `v` of vectors in `E`, there exists a maximal orthonormal set containing it. -/ theorem exists_maximal_orthonormal {s : Set E} (hs : Orthonormal 𝕜 (Subtype.val : s → E)) : ∃ w ⊇ s, Orthonormal 𝕜 (Subtype.val : w → E) ∧ ∀ u ⊇ w, Orthonormal 𝕜 (Subtype.val : u → E) → u = w := by have := zorn_subset_nonempty { b \| Orthonormal 𝕜 (Subtype.val : b → E) } ?_ _ hs · obtain ⟨b, bi, sb, h⟩ := this refine ⟨b, sb, bi, ?_⟩ exact fun u hus hu => h u hu hus · refine fun c hc cc _c0 => ⟨⋃₀ c, ?_, ?_⟩ · exact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc · exact fun _ => Set.subset_sUnion_of_mem #align exists_maximal_orthonormal exists_maximal_orthonormal theorem Orthonormal.ne_zero {v : ι → E} (hv : Orthonormal 𝕜 v) (i : ι) : v i ≠ 0 := by have : ‖v i‖ ≠ 0 := by rw [hv.1 i] norm_num simpa using this #align orthonormal.ne_zero Orthonormal.ne_zero open FiniteDimensional /-- A family of orthonormal vectors with the correct cardinality forms a basis. -/ def basisOfOrthonormalOfCardEqFinrank [Fintype ι] [Nonempty ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (card_eq : Fintype.card ι = finrank 𝕜 E) : Basis ι 𝕜 E := basisOfLinearIndependentOfCardEqFinrank hv.linearIndependent card_eq #align basis_of_orthonormal_of_card_eq_finrank basisOfOrthonormalOfCardEqFinrank @[simp] theorem coe_basisOfOrthonormalOfCardEqFinrank [Fintype ι] [Nonempty ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (card_eq : Fintype.card ι = finrank 𝕜 E) : (basisOfOrthonormalOfCardEqFinrank hv card_eq : ι → E) = v := coe_basisOfLinearIndependentOfCardEqFinrank _ _ #align coe_basis_of_orthonormal_of_card_eq_finrank coe_basisOfOrthonormalOfCardEqFinrank end OrthonormalSets section Norm theorem norm_eq_sqrt_inner (x : E) : ‖x‖ = √(re ⟪x, x⟫) := calc ‖x‖ = √(‖x‖ ^ 2) := (sqrt_sq (norm_nonneg _)).symm _ = √(re ⟪x, x⟫) := congr_arg _ (norm_sq_eq_inner _) #align norm_eq_sqrt_inner norm_eq_sqrt_inner theorem norm_eq_sqrt_real_inner (x : F) : ‖x‖ = √⟪x, x⟫_ℝ := @norm_eq_sqrt_inner ℝ _ _ _ _ x #align norm_eq_sqrt_real_inner norm_eq_sqrt_real_inner theorem inner_self_eq_norm_mul_norm (x : E) : re ⟪x, x⟫ = ‖x‖ ‖x‖ := by rw [@norm_eq_sqrt_inner 𝕜, ← sqrt_mul inner_self_nonneg (re ⟪x, x⟫), sqrt_mul_self inner_self_nonneg] #align inner_self_eq_norm_mul_norm inner_self_eq_norm_mul_norm theorem inner_self_eq_norm_sq (x : E) : re ⟪x, x⟫ = ‖x‖ ^ 2 := by rw [pow_two, inner_self_eq_norm_mul_norm] #align inner_self_eq_norm_sq inner_self_eq_norm_sq theorem real_inner_self_eq_norm_mul_norm (x : F) : ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖ := by have h := @inner_self_eq_norm_mul_norm ℝ F _ _ _ x simpa using h #align real_inner_self_eq_norm_mul_norm real_inner_self_eq_norm_mul_norm theorem real_inner_self_eq_norm_sq (x : F) : ⟪x, x⟫_ℝ = ‖x‖ ^ 2 := by rw [pow_two, real_inner_self_eq_norm_mul_norm] #align real_inner_self_eq_norm_sq real_inner_self_eq_norm_sq -- Porting note: this was present in mathlib3 but seemingly didn't do anything. -- variable (𝕜) /-- Expand the square -/ theorem norm_add_sq (x y : E) : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * re ⟪x, y⟫ + ‖y‖ ^ 2 := by repeat' rw [sq (M := ℝ), ← @inner_self_eq_norm_mul_norm 𝕜] rw [inner_add_add_self, two_mul] simp only [add_assoc, add_left_inj, add_right_inj, AddMonoidHom.map_add] rw [← inner_conj_symm, conj_re] #align norm_add_sq norm_add_sq alias norm_add_pow_two := norm_add_sq #align norm_add_pow_two norm_add_pow_two /-- Expand the square -/ theorem norm_add_sq_real (x y : F) : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2 := by have h := @norm_add_sq ℝ _ _ _ _ x y simpa using h #align norm_add_sq_real norm_add_sq_real alias norm_add_pow_two_real := norm_add_sq_real #align norm_add_pow_two_real norm_add_pow_two_real /-- Expand the square -/ theorem norm_add_mul_self (x y : E) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ := by repeat' rw [← sq (M := ℝ)] exact norm_add_sq _ _ #align norm_add_mul_self norm_add_mul_self /-- Expand the square -/ theorem norm_add_mul_self_real (x y : F) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ := by have h := @norm_add_mul_self ℝ _ _ _ _ x y simpa using h #align norm_add_mul_self_real norm_add_mul_self_real /-- Expand the square -/ theorem norm_sub_sq (x y : E) : ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * re ⟪x, y⟫ + ‖y‖ ^ 2 := by rw [sub_eq_add_neg, @norm_add_sq 𝕜 _ _ _ _ x (-y), norm_neg, inner_neg_right, map_neg, mul_neg, sub_eq_add_neg] #align norm_sub_sq norm_sub_sq alias norm_sub_pow_two := norm_sub_sq #align norm_sub_pow_two norm_sub_pow_two /-- Expand the square -/ theorem norm_sub_sq_real (x y : F) : ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2 := @norm_sub_sq ℝ _ _ _ _ _ _ #align norm_sub_sq_real norm_sub_sq_real alias norm_sub_pow_two_real := norm_sub_sq_real #align norm_sub_pow_two_real norm_sub_pow_two_real /-- Expand the square -/ theorem norm_sub_mul_self (x y : E) : ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ := by repeat' rw [← sq (M := ℝ)] exact norm_sub_sq _ _ #align norm_sub_mul_self norm_sub_mul_self /-- Expand the square -/ theorem norm_sub_mul_self_real (x y : F) : ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ := by have h := @norm_sub_mul_self ℝ _ _ _ _ x y simpa using h #align norm_sub_mul_self_real norm_sub_mul_self_real /-- Cauchy–Schwarz inequality with norm -/ theorem norm_inner_le_norm (x y : E) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := by rw [norm_eq_sqrt_inner (𝕜 := 𝕜) x, norm_eq_sqrt_inner (𝕜 := 𝕜) y] letI : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore exact InnerProductSpace.Core.norm_inner_le_norm x y #align norm_inner_le_norm norm_inner_le_norm theorem nnnorm_inner_le_nnnorm (x y : E) : ‖⟪x, y⟫‖₊ ≤ ‖x‖₊ * ‖y‖₊ := norm_inner_le_norm x y #align nnnorm_inner_le_nnnorm nnnorm_inner_le_nnnorm theorem re_inner_le_norm (x y : E) : re ⟪x, y⟫ ≤ ‖x‖ * ‖y‖ := le_trans (re_le_norm (inner x y)) (norm_inner_le_norm x y) #align re_inner_le_norm re_inner_le_norm /-- Cauchy–Schwarz inequality with norm -/ theorem abs_real_inner_le_norm (x y : F) : \|⟪x, y⟫_ℝ\| ≤ ‖x‖ * ‖y‖ := (Real.norm_eq_abs _).ge.trans (norm_inner_le_norm x y) #align abs_real_inner_le_norm abs_real_inner_le_norm /-- Cauchy–Schwarz inequality with norm -/ theorem real_inner_le_norm (x y : F) : ⟪x, y⟫_ℝ ≤ ‖x‖ * ‖y‖ := le_trans (le_abs_self _) (abs_real_inner_le_norm _ _) #align real_inner_le_norm real_inner_le_norm variable (𝕜) theorem parallelogram_law_with_norm (x y : E) : ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) := by simp only [← @inner_self_eq_norm_mul_norm 𝕜] rw [← re.map_add, parallelogram_law, two_mul, two_mul] simp only [re.map_add] #align parallelogram_law_with_norm parallelogram_law_with_norm theorem parallelogram_law_with_nnnorm (x y : E) : ‖x + y‖₊ * ‖x + y‖₊ + ‖x - y‖₊ * ‖x - y‖₊ = 2 * (‖x‖₊ * ‖x‖₊ + ‖y‖₊ * ‖y‖₊) := Subtype.ext <\| parallelogram_law_with_norm 𝕜 x y #align parallelogram_law_with_nnnorm parallelogram_law_with_nnnorm variable {𝕜} /-- Polarization identity: The real part of the inner product, in terms of the norm. -/ theorem re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : E) : re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2 := by rw [@norm_add_mul_self 𝕜] ring #align re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two /-- Polarization identity: The real part of the inner product, in terms of the norm. -/ theorem re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : E) : re ⟪x, y⟫ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 := by rw [@norm_sub_mul_self 𝕜] ring #align re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two /-- Polarization identity: The real part of the inner product, in terms of the norm. -/ theorem re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four (x y : E) : re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4 := by rw [@norm_add_mul_self 𝕜, @norm_sub_mul_self 𝕜] ring #align re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four /-- Polarization identity: The imaginary part of the inner product, in terms of the norm. -/ theorem im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four (x y : E) : im ⟪x, y⟫ = (‖x - IK • y‖ * ‖x - IK • y‖ - ‖x + IK • y‖ * ‖x + IK • y‖) / 4 := by simp only [@norm_add_mul_self 𝕜, @norm_sub_mul_self 𝕜, inner_smul_right, I_mul_re] ring set_option linter.uppercaseLean3 false in #align im_inner_eq_norm_sub_I_smul_mul_self_sub_norm_add_I_smul_mul_self_div_four im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four /-- Polarization identity: The inner product, in terms of the norm. -/ theorem inner_eq_sum_norm_sq_div_four (x y : E) : ⟪x, y⟫ = ((‖x + y‖ : 𝕜) ^ 2 - (‖x - y‖ : 𝕜) ^ 2 + ((‖x - IK • y‖ : 𝕜) ^ 2 - (‖x + IK • y‖ : 𝕜) ^ 2) * IK) / 4 := by rw [← re_add_im ⟪x, y⟫, re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four, im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four] push_cast simp only [sq, ← mul_div_right_comm, ← add_div] #align inner_eq_sum_norm_sq_div_four inner_eq_sum_norm_sq_div_four /-- Formula for the distance between the images of two nonzero points under an inversion with center zero. See also `EuclideanGeometry.dist_inversion_inversion` for inversions around a general point. -/ theorem dist_div_norm_sq_smul {x y : F} (hx : x ≠ 0) (hy : y ≠ 0) (R : ℝ) : dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = R ^ 2 / (‖x‖ * ‖y‖) * dist x y := have hx' : ‖x‖ ≠ 0 := norm_ne_zero_iff.2 hx have hy' : ‖y‖ ≠ 0 := norm_ne_zero_iff.2 hy calc dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = √(‖(R / ‖x‖) ^ 2 • x - (R / ‖y‖) ^ 2 • y‖ ^ 2) := by rw [dist_eq_norm, sqrt_sq (norm_nonneg _)] _ = √((R ^ 2 / (‖x‖ * ‖y‖)) ^ 2 * ‖x - y‖ ^ 2) := congr_arg sqrt <\| by field_simp [sq, norm_sub_mul_self_real, norm_smul, real_inner_smul_left, inner_smul_right, Real.norm_of_nonneg (mul_self_nonneg _)] ring _ = R ^ 2 / (‖x‖ * ‖y‖) * dist x y := by rw [sqrt_mul, sqrt_sq, sqrt_sq, dist_eq_norm] <;> positivity #align dist_div_norm_sq_smul dist_div_norm_sq_smul -- See note [lower instance priority] instance (priority := 100) InnerProductSpace.toUniformConvexSpace : UniformConvexSpace F := ⟨fun ε hε => by refine ⟨2 - √(4 - ε ^ 2), sub_pos_of_lt <\| (sqrt_lt' zero_lt_two).2 ?_, fun x hx y hy hxy => ?_⟩ · norm_num exact pow_pos hε _ rw [sub_sub_cancel] refine le_sqrt_of_sq_le ?_ rw [sq, eq_sub_iff_add_eq.2 (parallelogram_law_with_norm ℝ x y), ← sq ‖x - y‖, hx, hy] ring_nf exact sub_le_sub_left (pow_le_pow_left hε.le hxy _) 4⟩ #align inner_product_space.to_uniform_convex_space InnerProductSpace.toUniformConvexSpace section Complex variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℂ V] /-- A complex polarization identity, with a linear map -/ theorem inner_map_polarization (T : V →ₗ[ℂ] V) (x y : V) : ⟪T y, x⟫_ℂ = (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ + Complex.I ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ - Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) / 4 := by simp only [map_add, map_sub, inner_add_left, inner_add_right, LinearMap.map_smul, inner_smul_left, inner_smul_right, Complex.conj_I, ← pow_two, Complex.I_sq, inner_sub_left, inner_sub_right, mul_add, ← mul_assoc, mul_neg, neg_neg, sub_neg_eq_add, one_mul, neg_one_mul, mul_sub, sub_sub] ring #align inner_map_polarization inner_map_polarization theorem inner_map_polarization' (T : V →ₗ[ℂ] V) (x y : V) : ⟪T x, y⟫_ℂ = (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ - Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ + Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) / 4 := by simp only [map_add, map_sub, inner_add_left, inner_add_right, LinearMap.map_smul, inner_smul_left, inner_smul_right, Complex.conj_I, ← pow_two, Complex.I_sq, inner_sub_left, inner_sub_right, mul_add, ← mul_assoc, mul_neg, neg_neg, sub_neg_eq_add, one_mul, neg_one_mul, mul_sub, sub_sub] ring #align inner_map_polarization' inner_map_polarization' /-- A linear map `T` is zero, if and only if the identity `⟪T x, x⟫_ℂ = 0` holds for all `x`. -/ theorem inner_map_self_eq_zero (T : V →ₗ[ℂ] V) : (∀ x : V, ⟪T x, x⟫_ℂ = 0) ↔ T = 0 := by constructor · intro hT ext x rw [LinearMap.zero_apply, ← @inner_self_eq_zero ℂ V, inner_map_polarization] simp only [hT] norm_num · rintro rfl x simp only [LinearMap.zero_apply, inner_zero_left] #align inner_map_self_eq_zero inner_map_self_eq_zero /-- Two linear maps `S` and `T` are equal, if and only if the identity `⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ` holds for all `x`. -/ theorem ext_inner_map (S T : V →ₗ[ℂ] V) : (∀ x : V, ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T := by rw [← sub_eq_zero, ← inner_map_self_eq_zero] refine forall_congr' fun x => ?_ rw [LinearMap.sub_apply, inner_sub_left, sub_eq_zero] #align ext_inner_map ext_inner_map end Complex section variable {ι : Type} {ι' : Type} {ι'' : Type} variable {E' : Type} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] variable {E'' : Type} [NormedAddCommGroup E''] [InnerProductSpace 𝕜 E''] /-- A linear isometry preserves the inner product. -/ @[simp] theorem LinearIsometry.inner_map_map (f : E →ₗᵢ[𝕜] E') (x y : E) : ⟪f x, f y⟫ = ⟪x, y⟫ := by simp [inner_eq_sum_norm_sq_div_four, ← f.norm_map] #align linear_isometry.inner_map_map LinearIsometry.inner_map_map /-- A linear isometric equivalence preserves the inner product. -/ @[simp] theorem LinearIsometryEquiv.inner_map_map (f : E ≃ₗᵢ[𝕜] E') (x y : E) : ⟪f x, f y⟫ = ⟪x, y⟫ := f.toLinearIsometry.inner_map_map x y #align linear_isometry_equiv.inner_map_map LinearIsometryEquiv.inner_map_map /-- The adjoint of a linear isometric equivalence is its inverse. -/ theorem LinearIsometryEquiv.inner_map_eq_flip (f : E ≃ₗᵢ[𝕜] E') (x : E) (y : E') : ⟪f x, y⟫_𝕜 = ⟪x, f.symm y⟫_𝕜 := by conv_lhs => rw [← f.apply_symm_apply y, f.inner_map_map] /-- A linear map that preserves the inner product is a linear isometry. -/ def LinearMap.isometryOfInner (f : E →ₗ[𝕜] E') (h : ∀ x y, ⟪f x, f y⟫ = ⟪x, y⟫) : E →ₗᵢ[𝕜] E' := ⟨f, fun x => by simp only [@norm_eq_sqrt_inner 𝕜, h]⟩ #align linear_map.isometry_of_inner LinearMap.isometryOfInner @[simp] theorem LinearMap.coe_isometryOfInner (f : E →ₗ[𝕜] E') (h) : ⇑(f.isometryOfInner h) = f := rfl #align linear_map.coe_isometry_of_inner LinearMap.coe_isometryOfInner @[simp] theorem LinearMap.isometryOfInner_toLinearMap (f : E →ₗ[𝕜] E') (h) : (f.isometryOfInner h).toLinearMap = f := rfl #align linear_map.isometry_of_inner_to_linear_map LinearMap.isometryOfInner_toLinearMap /-- A linear equivalence that preserves the inner product is a linear isometric equivalence. -/ def LinearEquiv.isometryOfInner (f : E ≃ₗ[𝕜] E') (h : ∀ x y, ⟪f x, f y⟫ = ⟪x, y⟫) : E ≃ₗᵢ[𝕜] E' := ⟨f, ((f : E →ₗ[𝕜] E').isometryOfInner h).norm_map⟩ #align linear_equiv.isometry_of_inner LinearEquiv.isometryOfInner @[simp] theorem LinearEquiv.coe_isometryOfInner (f : E ≃ₗ[𝕜] E') (h) : ⇑(f.isometryOfInner h) = f := rfl #align linear_equiv.coe_isometry_of_inner LinearEquiv.coe_isometryOfInner @[simp] theorem LinearEquiv.isometryOfInner_toLinearEquiv (f : E ≃ₗ[𝕜] E') (h) : (f.isometryOfInner h).toLinearEquiv = f := rfl #align linear_equiv.isometry_of_inner_to_linear_equiv LinearEquiv.isometryOfInner_toLinearEquiv /-- A linear map is an isometry if and it preserves the inner product. -/ theorem LinearMap.norm_map_iff_inner_map_map {F : Type} [FunLike F E E'] [LinearMapClass F 𝕜 E E'] (f : F) : (∀ x, ‖f x‖ = ‖x‖) ↔ (∀ x y, ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜) := ⟨({ toLinearMap := LinearMapClass.linearMap f, norm_map' := · : E →ₗᵢ[𝕜] E' }.inner_map_map), (LinearMapClass.linearMap f \|>.isometryOfInner · \|>.norm_map)⟩ /-- A linear isometry preserves the property of being orthonormal. -/ theorem LinearIsometry.orthonormal_comp_iff {v : ι → E} (f : E →ₗᵢ[𝕜] E') : Orthonormal 𝕜 (f ∘ v) ↔ Orthonormal 𝕜 v := by classical simp_rw [orthonormal_iff_ite, Function.comp_apply, LinearIsometry.inner_map_map] #align linear_isometry.orthonormal_comp_iff LinearIsometry.orthonormal_comp_iff /-- A linear isometry preserves the property of being orthonormal. -/	Mathlib/Analysis/InnerProductSpace/Basic.lean	1,315	1,316	theorem Orthonormal.comp_linearIsometry {v : ι → E} (hv : Orthonormal 𝕜 v) (f : E →ₗᵢ[𝕜] E') : Orthonormal 𝕜 (f ∘ v) := by	rwa [f.orthonormal_comp_iff]
/- Copyright (c) 2014 Parikshit Khanna. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro -/ import Mathlib.Data.Nat.Defs import Mathlib.Data.Option.Basic import Mathlib.Data.List.Defs import Mathlib.Init.Data.List.Basic import Mathlib.Init.Data.List.Instances import Mathlib.Init.Data.List.Lemmas import Mathlib.Logic.Unique import Mathlib.Order.Basic import Mathlib.Tactic.Common #align_import data.list.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83" /-! # Basic properties of lists -/ assert_not_exists Set.range assert_not_exists GroupWithZero assert_not_exists Ring open Function open Nat hiding one_pos namespace List universe u v w variable {ι : Type} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α} -- Porting note: Delete this attribute -- attribute [inline] List.head! /-- There is only one list of an empty type -/ instance uniqueOfIsEmpty [IsEmpty α] : Unique (List α) := { instInhabitedList with uniq := fun l => match l with \| [] => rfl \| a :: _ => isEmptyElim a } #align list.unique_of_is_empty List.uniqueOfIsEmpty instance : Std.LawfulIdentity (α := List α) Append.append [] where left_id := nil_append right_id := append_nil instance : Std.Associative (α := List α) Append.append where assoc := append_assoc #align list.cons_ne_nil List.cons_ne_nil #align list.cons_ne_self List.cons_ne_self #align list.head_eq_of_cons_eq List.head_eq_of_cons_eqₓ -- implicits order #align list.tail_eq_of_cons_eq List.tail_eq_of_cons_eqₓ -- implicits order @[simp] theorem cons_injective {a : α} : Injective (cons a) := fun _ _ => tail_eq_of_cons_eq #align list.cons_injective List.cons_injective #align list.cons_inj List.cons_inj #align list.cons_eq_cons List.cons_eq_cons theorem singleton_injective : Injective fun a : α => [a] := fun _ _ h => (cons_eq_cons.1 h).1 #align list.singleton_injective List.singleton_injective theorem singleton_inj {a b : α} : [a] = [b] ↔ a = b := singleton_injective.eq_iff #align list.singleton_inj List.singleton_inj #align list.exists_cons_of_ne_nil List.exists_cons_of_ne_nil theorem set_of_mem_cons (l : List α) (a : α) : { x \| x ∈ a :: l } = insert a { x \| x ∈ l } := Set.ext fun _ => mem_cons #align list.set_of_mem_cons List.set_of_mem_cons /-! ### mem -/ #align list.mem_singleton_self List.mem_singleton_self #align list.eq_of_mem_singleton List.eq_of_mem_singleton #align list.mem_singleton List.mem_singleton #align list.mem_of_mem_cons_of_mem List.mem_of_mem_cons_of_mem theorem _root_.Decidable.List.eq_or_ne_mem_of_mem [DecidableEq α] {a b : α} {l : List α} (h : a ∈ b :: l) : a = b ∨ a ≠ b ∧ a ∈ l := by by_cases hab : a = b · exact Or.inl hab · exact ((List.mem_cons.1 h).elim Or.inl (fun h => Or.inr ⟨hab, h⟩)) #align decidable.list.eq_or_ne_mem_of_mem Decidable.List.eq_or_ne_mem_of_mem #align list.eq_or_ne_mem_of_mem List.eq_or_ne_mem_of_mem #align list.not_mem_append List.not_mem_append #align list.ne_nil_of_mem List.ne_nil_of_mem lemma mem_pair {a b c : α} : a ∈ [b, c] ↔ a = b ∨ a = c := by rw [mem_cons, mem_singleton] @[deprecated (since := "2024-03-23")] alias mem_split := append_of_mem #align list.mem_split List.append_of_mem #align list.mem_of_ne_of_mem List.mem_of_ne_of_mem #align list.ne_of_not_mem_cons List.ne_of_not_mem_cons #align list.not_mem_of_not_mem_cons List.not_mem_of_not_mem_cons #align list.not_mem_cons_of_ne_of_not_mem List.not_mem_cons_of_ne_of_not_mem #align list.ne_and_not_mem_of_not_mem_cons List.ne_and_not_mem_of_not_mem_cons #align list.mem_map List.mem_map #align list.exists_of_mem_map List.exists_of_mem_map #align list.mem_map_of_mem List.mem_map_of_memₓ -- implicits order -- The simpNF linter says that the LHS can be simplified via `List.mem_map`. -- However this is a higher priority lemma. -- https://github.com/leanprover/std4/issues/207 @[simp 1100, nolint simpNF] theorem mem_map_of_injective {f : α → β} (H : Injective f) {a : α} {l : List α} : f a ∈ map f l ↔ a ∈ l := ⟨fun m => let ⟨_, m', e⟩ := exists_of_mem_map m; H e ▸ m', mem_map_of_mem _⟩ #align list.mem_map_of_injective List.mem_map_of_injective @[simp] theorem _root_.Function.Involutive.exists_mem_and_apply_eq_iff {f : α → α} (hf : Function.Involutive f) (x : α) (l : List α) : (∃ y : α, y ∈ l ∧ f y = x) ↔ f x ∈ l := ⟨by rintro ⟨y, h, rfl⟩; rwa [hf y], fun h => ⟨f x, h, hf _⟩⟩ #align function.involutive.exists_mem_and_apply_eq_iff Function.Involutive.exists_mem_and_apply_eq_iff theorem mem_map_of_involutive {f : α → α} (hf : Involutive f) {a : α} {l : List α} : a ∈ map f l ↔ f a ∈ l := by rw [mem_map, hf.exists_mem_and_apply_eq_iff] #align list.mem_map_of_involutive List.mem_map_of_involutive #align list.forall_mem_map_iff List.forall_mem_map_iffₓ -- universe order #align list.map_eq_nil List.map_eq_nilₓ -- universe order attribute [simp] List.mem_join #align list.mem_join List.mem_join #align list.exists_of_mem_join List.exists_of_mem_join #align list.mem_join_of_mem List.mem_join_of_memₓ -- implicits order attribute [simp] List.mem_bind #align list.mem_bind List.mem_bindₓ -- implicits order -- Porting note: bExists in Lean3, And in Lean4 #align list.exists_of_mem_bind List.exists_of_mem_bindₓ -- implicits order #align list.mem_bind_of_mem List.mem_bind_of_memₓ -- implicits order #align list.bind_map List.bind_mapₓ -- implicits order theorem map_bind (g : β → List γ) (f : α → β) : ∀ l : List α, (List.map f l).bind g = l.bind fun a => g (f a) \| [] => rfl \| a :: l => by simp only [cons_bind, map_cons, map_bind _ _ l] #align list.map_bind List.map_bind /-! ### length -/ #align list.length_eq_zero List.length_eq_zero #align list.length_singleton List.length_singleton #align list.length_pos_of_mem List.length_pos_of_mem #align list.exists_mem_of_length_pos List.exists_mem_of_length_pos #align list.length_pos_iff_exists_mem List.length_pos_iff_exists_mem alias ⟨ne_nil_of_length_pos, length_pos_of_ne_nil⟩ := length_pos #align list.ne_nil_of_length_pos List.ne_nil_of_length_pos #align list.length_pos_of_ne_nil List.length_pos_of_ne_nil theorem length_pos_iff_ne_nil {l : List α} : 0 < length l ↔ l ≠ [] := ⟨ne_nil_of_length_pos, length_pos_of_ne_nil⟩ #align list.length_pos_iff_ne_nil List.length_pos_iff_ne_nil #align list.exists_mem_of_ne_nil List.exists_mem_of_ne_nil #align list.length_eq_one List.length_eq_one theorem exists_of_length_succ {n} : ∀ l : List α, l.length = n + 1 → ∃ h t, l = h :: t \| [], H => absurd H.symm <\| succ_ne_zero n \| h :: t, _ => ⟨h, t, rfl⟩ #align list.exists_of_length_succ List.exists_of_length_succ @[simp] lemma length_injective_iff : Injective (List.length : List α → ℕ) ↔ Subsingleton α := by constructor · intro h; refine ⟨fun x y => ?_⟩; (suffices [x] = [y] by simpa using this); apply h; rfl · intros hα l1 l2 hl induction l1 generalizing l2 <;> cases l2 · rfl · cases hl · cases hl · next ih _ _ => congr · exact Subsingleton.elim _ _ · apply ih; simpa using hl #align list.length_injective_iff List.length_injective_iff @[simp default+1] -- Porting note: this used to be just @[simp] lemma length_injective [Subsingleton α] : Injective (length : List α → ℕ) := length_injective_iff.mpr inferInstance #align list.length_injective List.length_injective theorem length_eq_two {l : List α} : l.length = 2 ↔ ∃ a b, l = [a, b] := ⟨fun _ => let [a, b] := l; ⟨a, b, rfl⟩, fun ⟨_, _, e⟩ => e ▸ rfl⟩ #align list.length_eq_two List.length_eq_two theorem length_eq_three {l : List α} : l.length = 3 ↔ ∃ a b c, l = [a, b, c] := ⟨fun _ => let [a, b, c] := l; ⟨a, b, c, rfl⟩, fun ⟨_, _, _, e⟩ => e ▸ rfl⟩ #align list.length_eq_three List.length_eq_three #align list.sublist.length_le List.Sublist.length_le /-! ### set-theoretic notation of lists -/ -- ADHOC Porting note: instance from Lean3 core instance instSingletonList : Singleton α (List α) := ⟨fun x => [x]⟩ #align list.has_singleton List.instSingletonList -- ADHOC Porting note: instance from Lean3 core instance [DecidableEq α] : Insert α (List α) := ⟨List.insert⟩ -- ADHOC Porting note: instance from Lean3 core instance [DecidableEq α] : LawfulSingleton α (List α) := { insert_emptyc_eq := fun x => show (if x ∈ ([] : List α) then [] else [x]) = [x] from if_neg (not_mem_nil _) } #align list.empty_eq List.empty_eq theorem singleton_eq (x : α) : ({x} : List α) = [x] := rfl #align list.singleton_eq List.singleton_eq theorem insert_neg [DecidableEq α] {x : α} {l : List α} (h : x ∉ l) : Insert.insert x l = x :: l := insert_of_not_mem h #align list.insert_neg List.insert_neg theorem insert_pos [DecidableEq α] {x : α} {l : List α} (h : x ∈ l) : Insert.insert x l = l := insert_of_mem h #align list.insert_pos List.insert_pos theorem doubleton_eq [DecidableEq α] {x y : α} (h : x ≠ y) : ({x, y} : List α) = [x, y] := by rw [insert_neg, singleton_eq] rwa [singleton_eq, mem_singleton] #align list.doubleton_eq List.doubleton_eq /-! ### bounded quantifiers over lists -/ #align list.forall_mem_nil List.forall_mem_nil #align list.forall_mem_cons List.forall_mem_cons theorem forall_mem_of_forall_mem_cons {p : α → Prop} {a : α} {l : List α} (h : ∀ x ∈ a :: l, p x) : ∀ x ∈ l, p x := (forall_mem_cons.1 h).2 #align list.forall_mem_of_forall_mem_cons List.forall_mem_of_forall_mem_cons #align list.forall_mem_singleton List.forall_mem_singleton #align list.forall_mem_append List.forall_mem_append #align list.not_exists_mem_nil List.not_exists_mem_nilₓ -- bExists change -- Porting note: bExists in Lean3 and And in Lean4 theorem exists_mem_cons_of {p : α → Prop} {a : α} (l : List α) (h : p a) : ∃ x ∈ a :: l, p x := ⟨a, mem_cons_self _ _, h⟩ #align list.exists_mem_cons_of List.exists_mem_cons_ofₓ -- bExists change -- Porting note: bExists in Lean3 and And in Lean4 theorem exists_mem_cons_of_exists {p : α → Prop} {a : α} {l : List α} : (∃ x ∈ l, p x) → ∃ x ∈ a :: l, p x := fun ⟨x, xl, px⟩ => ⟨x, mem_cons_of_mem _ xl, px⟩ #align list.exists_mem_cons_of_exists List.exists_mem_cons_of_existsₓ -- bExists change -- Porting note: bExists in Lean3 and And in Lean4 theorem or_exists_of_exists_mem_cons {p : α → Prop} {a : α} {l : List α} : (∃ x ∈ a :: l, p x) → p a ∨ ∃ x ∈ l, p x := fun ⟨x, xal, px⟩ => Or.elim (eq_or_mem_of_mem_cons xal) (fun h : x = a => by rw [← h]; left; exact px) fun h : x ∈ l => Or.inr ⟨x, h, px⟩ #align list.or_exists_of_exists_mem_cons List.or_exists_of_exists_mem_consₓ -- bExists change theorem exists_mem_cons_iff (p : α → Prop) (a : α) (l : List α) : (∃ x ∈ a :: l, p x) ↔ p a ∨ ∃ x ∈ l, p x := Iff.intro or_exists_of_exists_mem_cons fun h => Or.elim h (exists_mem_cons_of l) exists_mem_cons_of_exists #align list.exists_mem_cons_iff List.exists_mem_cons_iff /-! ### list subset -/ instance : IsTrans (List α) Subset where trans := fun _ _ _ => List.Subset.trans #align list.subset_def List.subset_def #align list.subset_append_of_subset_left List.subset_append_of_subset_left #align list.subset_append_of_subset_right List.subset_append_of_subset_right #align list.cons_subset List.cons_subset theorem cons_subset_of_subset_of_mem {a : α} {l m : List α} (ainm : a ∈ m) (lsubm : l ⊆ m) : a::l ⊆ m := cons_subset.2 ⟨ainm, lsubm⟩ #align list.cons_subset_of_subset_of_mem List.cons_subset_of_subset_of_mem theorem append_subset_of_subset_of_subset {l₁ l₂ l : List α} (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) : l₁ ++ l₂ ⊆ l := fun _ h ↦ (mem_append.1 h).elim (@l₁subl _) (@l₂subl _) #align list.append_subset_of_subset_of_subset List.append_subset_of_subset_of_subset -- Porting note: in Batteries #align list.append_subset_iff List.append_subset alias ⟨eq_nil_of_subset_nil, _⟩ := subset_nil #align list.eq_nil_of_subset_nil List.eq_nil_of_subset_nil #align list.eq_nil_iff_forall_not_mem List.eq_nil_iff_forall_not_mem #align list.map_subset List.map_subset theorem map_subset_iff {l₁ l₂ : List α} (f : α → β) (h : Injective f) : map f l₁ ⊆ map f l₂ ↔ l₁ ⊆ l₂ := by refine ⟨?_, map_subset f⟩; intro h2 x hx rcases mem_map.1 (h2 (mem_map_of_mem f hx)) with ⟨x', hx', hxx'⟩ cases h hxx'; exact hx' #align list.map_subset_iff List.map_subset_iff /-! ### append -/ theorem append_eq_has_append {L₁ L₂ : List α} : List.append L₁ L₂ = L₁ ++ L₂ := rfl #align list.append_eq_has_append List.append_eq_has_append #align list.singleton_append List.singleton_append #align list.append_ne_nil_of_ne_nil_left List.append_ne_nil_of_ne_nil_left #align list.append_ne_nil_of_ne_nil_right List.append_ne_nil_of_ne_nil_right #align list.append_eq_nil List.append_eq_nil -- Porting note: in Batteries #align list.nil_eq_append_iff List.nil_eq_append @[deprecated (since := "2024-03-24")] alias append_eq_cons_iff := append_eq_cons #align list.append_eq_cons_iff List.append_eq_cons @[deprecated (since := "2024-03-24")] alias cons_eq_append_iff := cons_eq_append #align list.cons_eq_append_iff List.cons_eq_append #align list.append_eq_append_iff List.append_eq_append_iff #align list.take_append_drop List.take_append_drop #align list.append_inj List.append_inj #align list.append_inj_right List.append_inj_rightₓ -- implicits order #align list.append_inj_left List.append_inj_leftₓ -- implicits order #align list.append_inj' List.append_inj'ₓ -- implicits order #align list.append_inj_right' List.append_inj_right'ₓ -- implicits order #align list.append_inj_left' List.append_inj_left'ₓ -- implicits order @[deprecated (since := "2024-01-18")] alias append_left_cancel := append_cancel_left #align list.append_left_cancel List.append_cancel_left @[deprecated (since := "2024-01-18")] alias append_right_cancel := append_cancel_right #align list.append_right_cancel List.append_cancel_right @[simp] theorem append_left_eq_self {x y : List α} : x ++ y = y ↔ x = [] := by rw [← append_left_inj (s₁ := x), nil_append] @[simp] theorem self_eq_append_left {x y : List α} : y = x ++ y ↔ x = [] := by rw [eq_comm, append_left_eq_self] @[simp] theorem append_right_eq_self {x y : List α} : x ++ y = x ↔ y = [] := by rw [← append_right_inj (t₁ := y), append_nil] @[simp] theorem self_eq_append_right {x y : List α} : x = x ++ y ↔ y = [] := by rw [eq_comm, append_right_eq_self] theorem append_right_injective (s : List α) : Injective fun t ↦ s ++ t := fun _ _ ↦ append_cancel_left #align list.append_right_injective List.append_right_injective #align list.append_right_inj List.append_right_inj theorem append_left_injective (t : List α) : Injective fun s ↦ s ++ t := fun _ _ ↦ append_cancel_right #align list.append_left_injective List.append_left_injective #align list.append_left_inj List.append_left_inj #align list.map_eq_append_split List.map_eq_append_split /-! ### replicate -/ @[simp] lemma replicate_zero (a : α) : replicate 0 a = [] := rfl #align list.replicate_zero List.replicate_zero attribute [simp] replicate_succ #align list.replicate_succ List.replicate_succ lemma replicate_one (a : α) : replicate 1 a = [a] := rfl #align list.replicate_one List.replicate_one #align list.length_replicate List.length_replicate #align list.mem_replicate List.mem_replicate #align list.eq_of_mem_replicate List.eq_of_mem_replicate theorem eq_replicate_length {a : α} : ∀ {l : List α}, l = replicate l.length a ↔ ∀ b ∈ l, b = a \| [] => by simp \| (b :: l) => by simp [eq_replicate_length] #align list.eq_replicate_length List.eq_replicate_length #align list.eq_replicate_of_mem List.eq_replicate_of_mem #align list.eq_replicate List.eq_replicate theorem replicate_add (m n) (a : α) : replicate (m + n) a = replicate m a ++ replicate n a := by induction m <;> simp [, succ_add, replicate] #align list.replicate_add List.replicate_add theorem replicate_succ' (n) (a : α) : replicate (n + 1) a = replicate n a ++ [a] := replicate_add n 1 a #align list.replicate_succ' List.replicate_succ' theorem replicate_subset_singleton (n) (a : α) : replicate n a ⊆ [a] := fun _ h => mem_singleton.2 (eq_of_mem_replicate h) #align list.replicate_subset_singleton List.replicate_subset_singleton theorem subset_singleton_iff {a : α} {L : List α} : L ⊆ [a] ↔ ∃ n, L = replicate n a := by simp only [eq_replicate, subset_def, mem_singleton, exists_eq_left'] #align list.subset_singleton_iff List.subset_singleton_iff @[simp] theorem map_replicate (f : α → β) (n) (a : α) : map f (replicate n a) = replicate n (f a) := by induction n <;> [rfl; simp only [, replicate, map]] #align list.map_replicate List.map_replicate @[simp] theorem tail_replicate (a : α) (n) : tail (replicate n a) = replicate (n - 1) a := by cases n <;> rfl #align list.tail_replicate List.tail_replicate @[simp] theorem join_replicate_nil (n : ℕ) : join (replicate n []) = @nil α := by induction n <;> [rfl; simp only [, replicate, join, append_nil]] #align list.join_replicate_nil List.join_replicate_nil theorem replicate_right_injective {n : ℕ} (hn : n ≠ 0) : Injective (@replicate α n) := fun _ _ h => (eq_replicate.1 h).2 _ <\| mem_replicate.2 ⟨hn, rfl⟩ #align list.replicate_right_injective List.replicate_right_injective theorem replicate_right_inj {a b : α} {n : ℕ} (hn : n ≠ 0) : replicate n a = replicate n b ↔ a = b := (replicate_right_injective hn).eq_iff #align list.replicate_right_inj List.replicate_right_inj @[simp] theorem replicate_right_inj' {a b : α} : ∀ {n}, replicate n a = replicate n b ↔ n = 0 ∨ a = b \| 0 => by simp \| n + 1 => (replicate_right_inj n.succ_ne_zero).trans <\| by simp only [n.succ_ne_zero, false_or] #align list.replicate_right_inj' List.replicate_right_inj' theorem replicate_left_injective (a : α) : Injective (replicate · a) := LeftInverse.injective (length_replicate · a) #align list.replicate_left_injective List.replicate_left_injective @[simp] theorem replicate_left_inj {a : α} {n m : ℕ} : replicate n a = replicate m a ↔ n = m := (replicate_left_injective a).eq_iff #align list.replicate_left_inj List.replicate_left_inj @[simp] theorem head_replicate (n : ℕ) (a : α) (h) : head (replicate n a) h = a := by cases n <;> simp at h ⊢ /-! ### pure -/ theorem mem_pure (x y : α) : x ∈ (pure y : List α) ↔ x = y := by simp #align list.mem_pure List.mem_pure /-! ### bind -/ @[simp] theorem bind_eq_bind {α β} (f : α → List β) (l : List α) : l >>= f = l.bind f := rfl #align list.bind_eq_bind List.bind_eq_bind #align list.bind_append List.append_bind /-! ### concat -/ #align list.concat_nil List.concat_nil #align list.concat_cons List.concat_cons #align list.concat_eq_append List.concat_eq_append #align list.init_eq_of_concat_eq List.init_eq_of_concat_eq #align list.last_eq_of_concat_eq List.last_eq_of_concat_eq #align list.concat_ne_nil List.concat_ne_nil #align list.concat_append List.concat_append #align list.length_concat List.length_concat #align list.append_concat List.append_concat /-! ### reverse -/ #align list.reverse_nil List.reverse_nil #align list.reverse_core List.reverseAux -- Porting note: Do we need this? attribute [local simp] reverseAux #align list.reverse_cons List.reverse_cons #align list.reverse_core_eq List.reverseAux_eq theorem reverse_cons' (a : α) (l : List α) : reverse (a :: l) = concat (reverse l) a := by simp only [reverse_cons, concat_eq_append] #align list.reverse_cons' List.reverse_cons' theorem reverse_concat' (l : List α) (a : α) : (l ++ [a]).reverse = a :: l.reverse := by rw [reverse_append]; rfl -- Porting note (#10618): simp can prove this -- @[simp] theorem reverse_singleton (a : α) : reverse [a] = [a] := rfl #align list.reverse_singleton List.reverse_singleton #align list.reverse_append List.reverse_append #align list.reverse_concat List.reverse_concat #align list.reverse_reverse List.reverse_reverse @[simp] theorem reverse_involutive : Involutive (@reverse α) := reverse_reverse #align list.reverse_involutive List.reverse_involutive @[simp] theorem reverse_injective : Injective (@reverse α) := reverse_involutive.injective #align list.reverse_injective List.reverse_injective theorem reverse_surjective : Surjective (@reverse α) := reverse_involutive.surjective #align list.reverse_surjective List.reverse_surjective theorem reverse_bijective : Bijective (@reverse α) := reverse_involutive.bijective #align list.reverse_bijective List.reverse_bijective @[simp] theorem reverse_inj {l₁ l₂ : List α} : reverse l₁ = reverse l₂ ↔ l₁ = l₂ := reverse_injective.eq_iff #align list.reverse_inj List.reverse_inj theorem reverse_eq_iff {l l' : List α} : l.reverse = l' ↔ l = l'.reverse := reverse_involutive.eq_iff #align list.reverse_eq_iff List.reverse_eq_iff #align list.reverse_eq_nil List.reverse_eq_nil_iff theorem concat_eq_reverse_cons (a : α) (l : List α) : concat l a = reverse (a :: reverse l) := by simp only [concat_eq_append, reverse_cons, reverse_reverse] #align list.concat_eq_reverse_cons List.concat_eq_reverse_cons #align list.length_reverse List.length_reverse -- Porting note: This one was @[simp] in mathlib 3, -- but Lean contains a competing simp lemma reverse_map. -- For now we remove @[simp] to avoid simplification loops. -- TODO: Change Lean lemma to match mathlib 3? theorem map_reverse (f : α → β) (l : List α) : map f (reverse l) = reverse (map f l) := (reverse_map f l).symm #align list.map_reverse List.map_reverse theorem map_reverseAux (f : α → β) (l₁ l₂ : List α) : map f (reverseAux l₁ l₂) = reverseAux (map f l₁) (map f l₂) := by simp only [reverseAux_eq, map_append, map_reverse] #align list.map_reverse_core List.map_reverseAux #align list.mem_reverse List.mem_reverse @[simp] theorem reverse_replicate (n) (a : α) : reverse (replicate n a) = replicate n a := eq_replicate.2 ⟨by rw [length_reverse, length_replicate], fun b h => eq_of_mem_replicate (mem_reverse.1 h)⟩ #align list.reverse_replicate List.reverse_replicate /-! ### empty -/ -- Porting note: this does not work as desired -- attribute [simp] List.isEmpty theorem isEmpty_iff_eq_nil {l : List α} : l.isEmpty ↔ l = [] := by cases l <;> simp [isEmpty] #align list.empty_iff_eq_nil List.isEmpty_iff_eq_nil /-! ### dropLast -/ #align list.length_init List.length_dropLast /-! ### getLast -/ @[simp] theorem getLast_cons {a : α} {l : List α} : ∀ h : l ≠ nil, getLast (a :: l) (cons_ne_nil a l) = getLast l h := by induction l <;> intros · contradiction · rfl #align list.last_cons List.getLast_cons theorem getLast_append_singleton {a : α} (l : List α) : getLast (l ++ [a]) (append_ne_nil_of_ne_nil_right l _ (cons_ne_nil a _)) = a := by simp only [getLast_append] #align list.last_append_singleton List.getLast_append_singleton -- Porting note: name should be fixed upstream theorem getLast_append' (l₁ l₂ : List α) (h : l₂ ≠ []) : getLast (l₁ ++ l₂) (append_ne_nil_of_ne_nil_right l₁ l₂ h) = getLast l₂ h := by induction' l₁ with _ _ ih · simp · simp only [cons_append] rw [List.getLast_cons] exact ih #align list.last_append List.getLast_append' theorem getLast_concat' {a : α} (l : List α) : getLast (concat l a) (concat_ne_nil a l) = a := getLast_concat .. #align list.last_concat List.getLast_concat' @[simp] theorem getLast_singleton' (a : α) : getLast [a] (cons_ne_nil a []) = a := rfl #align list.last_singleton List.getLast_singleton' -- Porting note (#10618): simp can prove this -- @[simp] theorem getLast_cons_cons (a₁ a₂ : α) (l : List α) : getLast (a₁ :: a₂ :: l) (cons_ne_nil _ _) = getLast (a₂ :: l) (cons_ne_nil a₂ l) := rfl #align list.last_cons_cons List.getLast_cons_cons theorem dropLast_append_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l ++ [getLast l h] = l \| [], h => absurd rfl h \| [a], h => rfl \| a :: b :: l, h => by rw [dropLast_cons₂, cons_append, getLast_cons (cons_ne_nil _ _)] congr exact dropLast_append_getLast (cons_ne_nil b l) #align list.init_append_last List.dropLast_append_getLast theorem getLast_congr {l₁ l₂ : List α} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : getLast l₁ h₁ = getLast l₂ h₂ := by subst l₁; rfl #align list.last_congr List.getLast_congr #align list.last_mem List.getLast_mem theorem getLast_replicate_succ (m : ℕ) (a : α) : (replicate (m + 1) a).getLast (ne_nil_of_length_eq_succ (length_replicate _ _)) = a := by simp only [replicate_succ'] exact getLast_append_singleton _ #align list.last_replicate_succ List.getLast_replicate_succ /-! ### getLast? -/ -- Porting note: Moved earlier in file, for use in subsequent lemmas. @[simp] theorem getLast?_cons_cons (a b : α) (l : List α) : getLast? (a :: b :: l) = getLast? (b :: l) := rfl @[simp] theorem getLast?_isNone : ∀ {l : List α}, (getLast? l).isNone ↔ l = [] \| [] => by simp \| [a] => by simp \| a :: b :: l => by simp [@getLast?_isNone (b :: l)] #align list.last'_is_none List.getLast?_isNone @[simp] theorem getLast?_isSome : ∀ {l : List α}, l.getLast?.isSome ↔ l ≠ [] \| [] => by simp \| [a] => by simp \| a :: b :: l => by simp [@getLast?_isSome (b :: l)] #align list.last'_is_some List.getLast?_isSome theorem mem_getLast?_eq_getLast : ∀ {l : List α} {x : α}, x ∈ l.getLast? → ∃ h, x = getLast l h \| [], x, hx => False.elim <\| by simp at hx \| [a], x, hx => have : a = x := by simpa using hx this ▸ ⟨cons_ne_nil a [], rfl⟩ \| a :: b :: l, x, hx => by rw [getLast?_cons_cons] at hx rcases mem_getLast?_eq_getLast hx with ⟨_, h₂⟩ use cons_ne_nil _ _ assumption #align list.mem_last'_eq_last List.mem_getLast?_eq_getLast theorem getLast?_eq_getLast_of_ne_nil : ∀ {l : List α} (h : l ≠ []), l.getLast? = some (l.getLast h) \| [], h => (h rfl).elim \| [_], _ => rfl \| _ :: b :: l, _ => @getLast?_eq_getLast_of_ne_nil (b :: l) (cons_ne_nil _ _) #align list.last'_eq_last_of_ne_nil List.getLast?_eq_getLast_of_ne_nil theorem mem_getLast?_cons {x y : α} : ∀ {l : List α}, x ∈ l.getLast? → x ∈ (y :: l).getLast? \| [], _ => by contradiction \| _ :: _, h => h #align list.mem_last'_cons List.mem_getLast?_cons theorem mem_of_mem_getLast? {l : List α} {a : α} (ha : a ∈ l.getLast?) : a ∈ l := let ⟨_, h₂⟩ := mem_getLast?_eq_getLast ha h₂.symm ▸ getLast_mem _ #align list.mem_of_mem_last' List.mem_of_mem_getLast? theorem dropLast_append_getLast? : ∀ {l : List α}, ∀ a ∈ l.getLast?, dropLast l ++ [a] = l \| [], a, ha => (Option.not_mem_none a ha).elim \| [a], _, rfl => rfl \| a :: b :: l, c, hc => by rw [getLast?_cons_cons] at hc rw [dropLast_cons₂, cons_append, dropLast_append_getLast? _ hc] #align list.init_append_last' List.dropLast_append_getLast? theorem getLastI_eq_getLast? [Inhabited α] : ∀ l : List α, l.getLastI = l.getLast?.iget \| [] => by simp [getLastI, Inhabited.default] \| [a] => rfl \| [a, b] => rfl \| [a, b, c] => rfl \| _ :: _ :: c :: l => by simp [getLastI, getLastI_eq_getLast? (c :: l)] #align list.ilast_eq_last' List.getLastI_eq_getLast? @[simp] theorem getLast?_append_cons : ∀ (l₁ : List α) (a : α) (l₂ : List α), getLast? (l₁ ++ a :: l₂) = getLast? (a :: l₂) \| [], a, l₂ => rfl \| [b], a, l₂ => rfl \| b :: c :: l₁, a, l₂ => by rw [cons_append, cons_append, getLast?_cons_cons, ← cons_append, getLast?_append_cons (c :: l₁)] #align list.last'_append_cons List.getLast?_append_cons #align list.last'_cons_cons List.getLast?_cons_cons theorem getLast?_append_of_ne_nil (l₁ : List α) : ∀ {l₂ : List α} (_ : l₂ ≠ []), getLast? (l₁ ++ l₂) = getLast? l₂ \| [], hl₂ => by contradiction \| b :: l₂, _ => getLast?_append_cons l₁ b l₂ #align list.last'_append_of_ne_nil List.getLast?_append_of_ne_nil theorem getLast?_append {l₁ l₂ : List α} {x : α} (h : x ∈ l₂.getLast?) : x ∈ (l₁ ++ l₂).getLast? := by cases l₂ · contradiction · rw [List.getLast?_append_cons] exact h #align list.last'_append List.getLast?_append /-! ### head(!?) and tail -/ @[simp] theorem head!_nil [Inhabited α] : ([] : List α).head! = default := rfl @[simp] theorem head_cons_tail (x : List α) (h : x ≠ []) : x.head h :: x.tail = x := by cases x <;> simp at h ⊢ theorem head!_eq_head? [Inhabited α] (l : List α) : head! l = (head? l).iget := by cases l <;> rfl #align list.head_eq_head' List.head!_eq_head? theorem surjective_head! [Inhabited α] : Surjective (@head! α _) := fun x => ⟨[x], rfl⟩ #align list.surjective_head List.surjective_head! theorem surjective_head? : Surjective (@head? α) := Option.forall.2 ⟨⟨[], rfl⟩, fun x => ⟨[x], rfl⟩⟩ #align list.surjective_head' List.surjective_head? theorem surjective_tail : Surjective (@tail α) \| [] => ⟨[], rfl⟩ \| a :: l => ⟨a :: a :: l, rfl⟩ #align list.surjective_tail List.surjective_tail theorem eq_cons_of_mem_head? {x : α} : ∀ {l : List α}, x ∈ l.head? → l = x :: tail l \| [], h => (Option.not_mem_none _ h).elim \| a :: l, h => by simp only [head?, Option.mem_def, Option.some_inj] at h exact h ▸ rfl #align list.eq_cons_of_mem_head' List.eq_cons_of_mem_head? theorem mem_of_mem_head? {x : α} {l : List α} (h : x ∈ l.head?) : x ∈ l := (eq_cons_of_mem_head? h).symm ▸ mem_cons_self _ _ #align list.mem_of_mem_head' List.mem_of_mem_head? @[simp] theorem head!_cons [Inhabited α] (a : α) (l : List α) : head! (a :: l) = a := rfl #align list.head_cons List.head!_cons #align list.tail_nil List.tail_nil #align list.tail_cons List.tail_cons @[simp] theorem head!_append [Inhabited α] (t : List α) {s : List α} (h : s ≠ []) : head! (s ++ t) = head! s := by induction s · contradiction · rfl #align list.head_append List.head!_append theorem head?_append {s t : List α} {x : α} (h : x ∈ s.head?) : x ∈ (s ++ t).head? := by cases s · contradiction · exact h #align list.head'_append List.head?_append theorem head?_append_of_ne_nil : ∀ (l₁ : List α) {l₂ : List α} (_ : l₁ ≠ []), head? (l₁ ++ l₂) = head? l₁ \| _ :: _, _, _ => rfl #align list.head'_append_of_ne_nil List.head?_append_of_ne_nil theorem tail_append_singleton_of_ne_nil {a : α} {l : List α} (h : l ≠ nil) : tail (l ++ [a]) = tail l ++ [a] := by induction l · contradiction · rw [tail, cons_append, tail] #align list.tail_append_singleton_of_ne_nil List.tail_append_singleton_of_ne_nil theorem cons_head?_tail : ∀ {l : List α} {a : α}, a ∈ head? l → a :: tail l = l \| [], a, h => by contradiction \| b :: l, a, h => by simp? at h says simp only [head?_cons, Option.mem_def, Option.some.injEq] at h simp [h] #align list.cons_head'_tail List.cons_head?_tail theorem head!_mem_head? [Inhabited α] : ∀ {l : List α}, l ≠ [] → head! l ∈ head? l \| [], h => by contradiction \| a :: l, _ => rfl #align list.head_mem_head' List.head!_mem_head? theorem cons_head!_tail [Inhabited α] {l : List α} (h : l ≠ []) : head! l :: tail l = l := cons_head?_tail (head!_mem_head? h) #align list.cons_head_tail List.cons_head!_tail theorem head!_mem_self [Inhabited α] {l : List α} (h : l ≠ nil) : l.head! ∈ l := by have h' := mem_cons_self l.head! l.tail rwa [cons_head!_tail h] at h' #align list.head_mem_self List.head!_mem_self theorem head_mem {l : List α} : ∀ (h : l ≠ nil), l.head h ∈ l := by cases l <;> simp @[simp] theorem head?_map (f : α → β) (l) : head? (map f l) = (head? l).map f := by cases l <;> rfl #align list.head'_map List.head?_map theorem tail_append_of_ne_nil (l l' : List α) (h : l ≠ []) : (l ++ l').tail = l.tail ++ l' := by cases l · contradiction · simp #align list.tail_append_of_ne_nil List.tail_append_of_ne_nil #align list.nth_le_eq_iff List.get_eq_iff theorem get_eq_get? (l : List α) (i : Fin l.length) : l.get i = (l.get? i).get (by simp [get?_eq_get]) := by simp [get_eq_iff] #align list.some_nth_le_eq List.get?_eq_get section deprecated set_option linter.deprecated false -- TODO(Mario): make replacements for theorems in this section /-- nth element of a list `l` given `n < l.length`. -/ @[deprecated get (since := "2023-01-05")] def nthLe (l : List α) (n) (h : n < l.length) : α := get l ⟨n, h⟩ #align list.nth_le List.nthLe @[simp] theorem nthLe_tail (l : List α) (i) (h : i < l.tail.length) (h' : i + 1 < l.length := (by simp only [length_tail] at h; omega)) : l.tail.nthLe i h = l.nthLe (i + 1) h' := by cases l <;> [cases h; rfl] #align list.nth_le_tail List.nthLe_tail	Mathlib/Data/List/Basic.lean	885	889	theorem nthLe_cons_aux {l : List α} {a : α} {n} (hn : n ≠ 0) (h : n < (a :: l).length) : n - 1 < l.length := by	contrapose! h rw [length_cons] omega
/- Copyright (c) 2020 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Robert Y. Lewis -/ import Mathlib.RingTheory.WittVector.StructurePolynomial #align_import ring_theory.witt_vector.defs from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a" /-! # Witt vectors In this file we define the type of `p`-typical Witt vectors and ring operations on it. The ring axioms are verified in `Mathlib/RingTheory/WittVector/Basic.lean`. For a fixed commutative ring `R` and prime `p`, a Witt vector `x : 𝕎 R` is an infinite sequence `ℕ → R` of elements of `R`. However, the ring operations `+` and `` are not defined in the obvious component-wise way. Instead, these operations are defined via certain polynomials using the machinery in `Mathlib/RingTheory/WittVector/StructurePolynomial.lean`. The `n`th value of the sum of two Witt vectors can depend on the `0`-th through `n`th values of the summands. This effectively simulates a “carrying” operation. ## Main definitions `WittVector p R`: the type of `p`-typical Witt vectors with coefficients in `R`. * `WittVector.coeff x n`: projects the `n`th value of the Witt vector `x`. ## Notation We use notation `𝕎 R`, entered `\bbW`, for the Witt vectors over `R`. ## References * [Hazewinkel, Witt Vectors][Haze09] * [Commelin and Lewis, Formalizing the Ring of Witt Vectors][CL21] -/ noncomputable section /-- `WittVector p R` is the ring of `p`-typical Witt vectors over the commutative ring `R`, where `p` is a prime number. If `p` is invertible in `R`, this ring is isomorphic to `ℕ → R` (the product of `ℕ` copies of `R`). If `R` is a ring of characteristic `p`, then `WittVector p R` is a ring of characteristic `0`. The canonical example is `WittVector p (ZMod p)`, which is isomorphic to the `p`-adic integers `ℤ_[p]`. -/ structure WittVector (p : ℕ) (R : Type) where mk' :: /-- `x.coeff n` is the `n`th coefficient of the Witt vector `x`. This concept does not have a standard name in the literature. -/ coeff : ℕ → R #align witt_vector WittVector -- Porting note: added to make the `p` argument explicit /-- Construct a Witt vector `mk p x : 𝕎 R` from a sequence `x` of elements of `R`. -/ def WittVector.mk (p : ℕ) {R : Type} (coeff : ℕ → R) : WittVector p R := mk' coeff variable {p : ℕ} /- We cannot make this `localized` notation, because the `p` on the RHS doesn't occur on the left Hiding the `p` in the notation is very convenient, so we opt for repeating the `local notation` in other files that use Witt vectors. -/ local notation "𝕎" => WittVector p -- type as `\bbW` namespace WittVector variable {R : Type} @[ext] theorem ext {x y : 𝕎 R} (h : ∀ n, x.coeff n = y.coeff n) : x = y := by cases x cases y simp only at h simp [Function.funext_iff, h] #align witt_vector.ext WittVector.ext theorem ext_iff {x y : 𝕎 R} : x = y ↔ ∀ n, x.coeff n = y.coeff n := ⟨fun h n => by rw [h], ext⟩ #align witt_vector.ext_iff WittVector.ext_iff variable (p) theorem coeff_mk (x : ℕ → R) : (mk p x).coeff = x := rfl #align witt_vector.coeff_mk WittVector.coeff_mk /- These instances are not needed for the rest of the development, but it is interesting to establish early on that `WittVector p` is a lawful functor. -/ instance : Functor (WittVector p) where map f v := mk p (f ∘ v.coeff) mapConst a _ := mk p fun _ => a instance : LawfulFunctor (WittVector p) where map_const := rfl -- Porting note: no longer needs to deconstruct `v` to conclude `{coeff := v.coeff} = v` id_map _ := rfl comp_map _ _ _ := rfl variable [hp : Fact p.Prime] [CommRing R] open MvPolynomial section RingOperations /-- The polynomials used for defining the element `0` of the ring of Witt vectors. -/ def wittZero : ℕ → MvPolynomial (Fin 0 × ℕ) ℤ := wittStructureInt p 0 #align witt_vector.witt_zero WittVector.wittZero /-- The polynomials used for defining the element `1` of the ring of Witt vectors. -/ def wittOne : ℕ → MvPolynomial (Fin 0 × ℕ) ℤ := wittStructureInt p 1 #align witt_vector.witt_one WittVector.wittOne /-- The polynomials used for defining the addition of the ring of Witt vectors. -/ def wittAdd : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ := wittStructureInt p (X 0 + X 1) #align witt_vector.witt_add WittVector.wittAdd /-- The polynomials used for defining repeated addition of the ring of Witt vectors. -/ def wittNSMul (n : ℕ) : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (n • X (0 : (Fin 1))) #align witt_vector.witt_nsmul WittVector.wittNSMul /-- The polynomials used for defining repeated addition of the ring of Witt vectors. -/ def wittZSMul (n : ℤ) : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (n • X (0 : (Fin 1))) #align witt_vector.witt_zsmul WittVector.wittZSMul /-- The polynomials used for describing the subtraction of the ring of Witt vectors. -/ def wittSub : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ := wittStructureInt p (X 0 - X 1) #align witt_vector.witt_sub WittVector.wittSub /-- The polynomials used for defining the multiplication of the ring of Witt vectors. -/ def wittMul : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ := wittStructureInt p (X 0 X 1) #align witt_vector.witt_mul WittVector.wittMul /-- The polynomials used for defining the negation of the ring of Witt vectors. -/ def wittNeg : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (-X 0) #align witt_vector.witt_neg WittVector.wittNeg /-- The polynomials used for defining repeated addition of the ring of Witt vectors. -/ def wittPow (n : ℕ) : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (X 0 ^ n) #align witt_vector.witt_pow WittVector.wittPow variable {p} /-- An auxiliary definition used in `WittVector.eval`. Evaluates a polynomial whose variables come from the disjoint union of `k` copies of `ℕ`, with a curried evaluation `x`. This can be defined more generally but we use only a specific instance here. -/ def peval {k : ℕ} (φ : MvPolynomial (Fin k × ℕ) ℤ) (x : Fin k → ℕ → R) : R := aeval (Function.uncurry x) φ #align witt_vector.peval WittVector.peval /-- Let `φ` be a family of polynomials, indexed by natural numbers, whose variables come from the disjoint union of `k` copies of `ℕ`, and let `xᵢ` be a Witt vector for `0 ≤ i < k`. `eval φ x` evaluates `φ` mapping the variable `X_(i, n)` to the `n`th coefficient of `xᵢ`. Instantiating `φ` with certain polynomials defined in `Mathlib/RingTheory/WittVector/StructurePolynomial.lean` establishes the ring operations on `𝕎 R`. For example, `WittVector.wittAdd` is such a `φ` with `k = 2`; evaluating this at `(x₀, x₁)` gives us the sum of two Witt vectors `x₀ + x₁`. -/ def eval {k : ℕ} (φ : ℕ → MvPolynomial (Fin k × ℕ) ℤ) (x : Fin k → 𝕎 R) : 𝕎 R := mk p fun n => peval (φ n) fun i => (x i).coeff #align witt_vector.eval WittVector.eval variable (R) [Fact p.Prime] instance : Zero (𝕎 R) := ⟨eval (wittZero p) ![]⟩ instance : Inhabited (𝕎 R) := ⟨0⟩ instance : One (𝕎 R) := ⟨eval (wittOne p) ![]⟩ instance : Add (𝕎 R) := ⟨fun x y => eval (wittAdd p) ![x, y]⟩ instance : Sub (𝕎 R) := ⟨fun x y => eval (wittSub p) ![x, y]⟩ instance hasNatScalar : SMul ℕ (𝕎 R) := ⟨fun n x => eval (wittNSMul p n) ![x]⟩ #align witt_vector.has_nat_scalar WittVector.hasNatScalar instance hasIntScalar : SMul ℤ (𝕎 R) := ⟨fun n x => eval (wittZSMul p n) ![x]⟩ #align witt_vector.has_int_scalar WittVector.hasIntScalar instance : Mul (𝕎 R) := ⟨fun x y => eval (wittMul p) ![x, y]⟩ instance : Neg (𝕎 R) := ⟨fun x => eval (wittNeg p) ![x]⟩ instance hasNatPow : Pow (𝕎 R) ℕ := ⟨fun x n => eval (wittPow p n) ![x]⟩ #align witt_vector.has_nat_pow WittVector.hasNatPow instance : NatCast (𝕎 R) := ⟨Nat.unaryCast⟩ instance : IntCast (𝕎 R) := ⟨Int.castDef⟩ end RingOperations section WittStructureSimplifications @[simp] theorem wittZero_eq_zero (n : ℕ) : wittZero p n = 0 := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittZero, wittStructureRat, bind₁, aeval_zero', constantCoeff_xInTermsOfW, RingHom.map_zero, AlgHom.map_zero, map_wittStructureInt] #align witt_vector.witt_zero_eq_zero WittVector.wittZero_eq_zero @[simp] theorem wittOne_zero_eq_one : wittOne p 0 = 1 := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittOne, wittStructureRat, xInTermsOfW_zero, AlgHom.map_one, RingHom.map_one, bind₁_X_right, map_wittStructureInt] #align witt_vector.witt_one_zero_eq_one WittVector.wittOne_zero_eq_one @[simp] theorem wittOne_pos_eq_zero (n : ℕ) (hn : 0 < n) : wittOne p n = 0 := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittOne, wittStructureRat, RingHom.map_zero, AlgHom.map_one, RingHom.map_one, map_wittStructureInt] induction n using Nat.strong_induction_on with \| h n IH => ?_ rw [xInTermsOfW_eq] simp only [AlgHom.map_mul, AlgHom.map_sub, AlgHom.map_sum, AlgHom.map_pow, bind₁_X_right, bind₁_C_right] rw [sub_mul, one_mul] rw [Finset.sum_eq_single 0] · simp only [invOf_eq_inv, one_mul, inv_pow, tsub_zero, RingHom.map_one, pow_zero] simp only [one_pow, one_mul, xInTermsOfW_zero, sub_self, bind₁_X_right] · intro i hin hi0 rw [Finset.mem_range] at hin rw [IH _ hin (Nat.pos_of_ne_zero hi0), zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero] · rw [Finset.mem_range]; intro; contradiction #align witt_vector.witt_one_pos_eq_zero WittVector.wittOne_pos_eq_zero @[simp] theorem wittAdd_zero : wittAdd p 0 = X (0, 0) + X (1, 0) := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittAdd, wittStructureRat, AlgHom.map_add, RingHom.map_add, rename_X, xInTermsOfW_zero, map_X, wittPolynomial_zero, bind₁_X_right, map_wittStructureInt] #align witt_vector.witt_add_zero WittVector.wittAdd_zero @[simp] theorem wittSub_zero : wittSub p 0 = X (0, 0) - X (1, 0) := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittSub, wittStructureRat, AlgHom.map_sub, RingHom.map_sub, rename_X, xInTermsOfW_zero, map_X, wittPolynomial_zero, bind₁_X_right, map_wittStructureInt] #align witt_vector.witt_sub_zero WittVector.wittSub_zero @[simp] theorem wittMul_zero : wittMul p 0 = X (0, 0) * X (1, 0) := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittMul, wittStructureRat, rename_X, xInTermsOfW_zero, map_X, wittPolynomial_zero, RingHom.map_mul, bind₁_X_right, AlgHom.map_mul, map_wittStructureInt] #align witt_vector.witt_mul_zero WittVector.wittMul_zero @[simp] theorem wittNeg_zero : wittNeg p 0 = -X (0, 0) := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittNeg, wittStructureRat, rename_X, xInTermsOfW_zero, map_X, wittPolynomial_zero, RingHom.map_neg, AlgHom.map_neg, bind₁_X_right, map_wittStructureInt] #align witt_vector.witt_neg_zero WittVector.wittNeg_zero @[simp] theorem constantCoeff_wittAdd (n : ℕ) : constantCoeff (wittAdd p n) = 0 := by apply constantCoeff_wittStructureInt p _ _ n simp only [add_zero, RingHom.map_add, constantCoeff_X] #align witt_vector.constant_coeff_witt_add WittVector.constantCoeff_wittAdd @[simp]	Mathlib/RingTheory/WittVector/Defs.lean	292	294	theorem constantCoeff_wittSub (n : ℕ) : constantCoeff (wittSub p n) = 0 := by	apply constantCoeff_wittStructureInt p _ _ n simp only [sub_zero, RingHom.map_sub, constantCoeff_X]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Yury Kudryashov -/ import Mathlib.MeasureTheory.Constructions.BorelSpace.Order #align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce" /-! # Borel (measurable) spaces ℝ, ℝ≥0, ℝ≥0∞ ## Main statements * `borel_eq_generateFrom_Ixx_rat` (where Ixx is one of {Ioo, Ioi, Iio, Ici, Iic): the Borel sigma algebra on ℝ is generated by intervals with rational endpoints; * `isPiSystem_Ixx_rat` (where Ixx is one of {Ioo, Ioi, Iio, Ici, Iic): intervals with rational endpoints form a pi system on ℝ; * `measurable_real_toNNReal`, `measurable_coe_nnreal_real`, `measurable_coe_nnreal_ennreal`, `ENNReal.measurable_ofReal`, `ENNReal.measurable_toReal`: measurability of various coercions between ℝ, ℝ≥0, and ℝ≥0∞; * `Measurable.real_toNNReal`, `Measurable.coe_nnreal_real`, `Measurable.coe_nnreal_ennreal`, `Measurable.ennreal_ofReal`, `Measurable.ennreal_toNNReal`, `Measurable.ennreal_toReal`: measurability of functions composed with various coercions between ℝ, ℝ≥0, and ℝ≥0∞ (also similar results for a.e.-measurability); * `Measurable.ennreal` : measurability of special cases for arithmetic operations on `ℝ≥0∞`. -/ open Set Filter MeasureTheory MeasurableSpace open scoped Classical Topology NNReal ENNReal MeasureTheory universe u v w x y variable {α β γ δ : Type} {ι : Sort y} {s t u : Set α} namespace Real theorem borel_eq_generateFrom_Ioo_rat : borel ℝ = .generateFrom (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) (b : ℝ)}) := isTopologicalBasis_Ioo_rat.borel_eq_generateFrom #align real.borel_eq_generate_from_Ioo_rat Real.borel_eq_generateFrom_Ioo_rat theorem borel_eq_generateFrom_Iio_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iio (a : ℝ)}) := by rw [borel_eq_generateFrom_Iio] refine le_antisymm (generateFrom_le ?_) (generateFrom_mono <\| iUnion_subset fun q ↦ singleton_subset_iff.mpr <\| mem_range_self _) rintro _ ⟨a, rfl⟩ have : IsLUB (range ((↑) : ℚ → ℝ) ∩ Iio a) a := by simp [isLUB_iff_le_iff, mem_upperBounds, ← le_iff_forall_rat_lt_imp_le] rw [← this.biUnion_Iio_eq, ← image_univ, ← image_inter_preimage, univ_inter, biUnion_image] exact MeasurableSet.biUnion (to_countable _) fun b _ => GenerateMeasurable.basic (Iio (b : ℝ)) (by simp) theorem borel_eq_generateFrom_Ioi_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ioi (a : ℝ)}) := by rw [borel_eq_generateFrom_Ioi] refine le_antisymm (generateFrom_le ?_) (generateFrom_mono <\| iUnion_subset fun q ↦ singleton_subset_iff.mpr <\| mem_range_self _) rintro _ ⟨a, rfl⟩ have : IsGLB (range ((↑) : ℚ → ℝ) ∩ Ioi a) a := by simp [isGLB_iff_le_iff, mem_lowerBounds, ← le_iff_forall_lt_rat_imp_le] rw [← this.biUnion_Ioi_eq, ← image_univ, ← image_inter_preimage, univ_inter, biUnion_image] exact MeasurableSet.biUnion (to_countable _) fun b _ => GenerateMeasurable.basic (Ioi (b : ℝ)) (by simp)	Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean	68	74	theorem borel_eq_generateFrom_Iic_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iic (a : ℝ)}) := by	rw [borel_eq_generateFrom_Ioi_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range] refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;> rintro _ ⟨q, rfl⟩ <;> dsimp only <;> [rw [← compl_Iic]; rw [← compl_Ioi]] <;> exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
/- Copyright (c) 2022 Kexing Ying. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kexing Ying -/ import Mathlib.Order.Interval.Set.Monotone import Mathlib.Probability.Process.HittingTime import Mathlib.Probability.Martingale.Basic import Mathlib.Tactic.AdaptationNote #align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # Doob's upcrossing estimate Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing estimate (also known as Doob's inequality) states that $$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$ Doob's upcrossing estimate is an important inequality and is central in proving the martingale convergence theorems. ## Main definitions * `MeasureTheory.upperCrossingTime a b f N n`: is the stopping time corresponding to `f` crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is taken to be `N`). * `MeasureTheory.lowerCrossingTime a b f N n`: is the stopping time corresponding to `f` crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is taken to be `N`). * `MeasureTheory.upcrossingStrat a b f N`: is the predictable process which is 1 if `n` is between a consecutive pair of lower and upper crossings and is 0 otherwise. Intuitively one might think of the `upcrossingStrat` as the strategy of buying 1 share whenever the process crosses below `a` for the first time after selling and selling 1 share whenever the process crosses above `b` for the first time after buying. * `MeasureTheory.upcrossingsBefore a b f N`: is the number of times `f` crosses from below `a` to above `b` before time `N`. * `MeasureTheory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above `b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`. ## Main results * `MeasureTheory.Adapted.isStoppingTime_upperCrossingTime`: `upperCrossingTime` is a stopping time whenever the process it is associated to is adapted. * `MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime`: `lowerCrossingTime` is a stopping time whenever the process it is associated to is adapted. * `MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part`: Doob's upcrossing estimate. * `MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality obtained by taking the supremum on both sides of Doob's upcrossing estimate. ### References We mostly follow the proof from [Kallenberg, Foundations of modern probability][kallenberg2021] -/ open TopologicalSpace Filter open scoped NNReal ENNReal MeasureTheory ProbabilityTheory Topology namespace MeasureTheory variable {Ω ι : Type} {m0 : MeasurableSpace Ω} {μ : Measure Ω} /-! ## Proof outline In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$ to above $b$ before time $N$. To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses below $a$ and above $b$. Namely, we define $$ \sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N; $$ $$ \tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N. $$ These are `lowerCrossingTime` and `upperCrossingTime` in our formalization which are defined using `MeasureTheory.hitting` allowing us to specify a starting and ending time. Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$. Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that $0 \le f_0$ and $a \le f_N$. In particular, we will show $$ (b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N]. $$ This is `MeasureTheory.integral_mul_upcrossingsBefore_le_integral` in our formalization. To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$ (i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is a submartingale if $(f_n)$ is. Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that $(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$, $(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property, $0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying $$ \mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0]. $$ Furthermore, \begin{align} (C \bullet f)_N & = \sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\ & = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\ & = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1} + \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\ & = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k}) \ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b) \end{align} where the inequality follows since for all $k < U_N(a, b)$, $f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$, $f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and $f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have $$ (b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N], $$ as required. To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$. -/ /-- `lowerCrossingTimeAux a f c N` is the first time `f` reached below `a` after time `c` before time `N`. -/ noncomputable def lowerCrossingTimeAux [Preorder ι] [InfSet ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) : Ω → ι := hitting f (Set.Iic a) c N #align measure_theory.lower_crossing_time_aux MeasureTheory.lowerCrossingTimeAux /-- `upperCrossingTime a b f N n` is the first time before time `N`, `f` reaches above `b` after `f` reached below `a` for the `n - 1`-th time. -/ noncomputable def upperCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ) (N : ι) : ℕ → Ω → ι \| 0 => ⊥ \| n + 1 => fun ω => hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω #align measure_theory.upper_crossing_time MeasureTheory.upperCrossingTime /-- `lowerCrossingTime a b f N n` is the first time before time `N`, `f` reaches below `a` after `f` reached above `b` for the `n`-th time. -/ noncomputable def lowerCrossingTime [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ) (N : ι) (n : ℕ) : Ω → ι := fun ω => hitting f (Set.Iic a) (upperCrossingTime a b f N n ω) N ω #align measure_theory.lower_crossing_time MeasureTheory.lowerCrossingTime section variable [Preorder ι] [OrderBot ι] [InfSet ι] variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω} @[simp] theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ := rfl #align measure_theory.upper_crossing_time_zero MeasureTheory.upperCrossingTime_zero @[simp] theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N := rfl #align measure_theory.lower_crossing_time_zero MeasureTheory.lowerCrossingTime_zero theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω = hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by rw [upperCrossingTime] #align measure_theory.upper_crossing_time_succ MeasureTheory.upperCrossingTime_succ theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω = hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by simp only [upperCrossingTime_succ] rfl #align measure_theory.upper_crossing_time_succ_eq MeasureTheory.upperCrossingTime_succ_eq end section ConditionallyCompleteLinearOrderBot variable [ConditionallyCompleteLinearOrderBot ι] variable {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω} theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by cases n · simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq] · simp only [upperCrossingTime_succ, hitting_le] #align measure_theory.upper_crossing_time_le MeasureTheory.upperCrossingTime_le @[simp] theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ := eq_bot_iff.2 upperCrossingTime_le #align measure_theory.upper_crossing_time_zero' MeasureTheory.upperCrossingTime_zero' theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by simp only [lowerCrossingTime, hitting_le ω] #align measure_theory.lower_crossing_time_le MeasureTheory.lowerCrossingTime_le theorem upperCrossingTime_le_lowerCrossingTime : upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω] #align measure_theory.upper_crossing_time_le_lower_crossing_time MeasureTheory.upperCrossingTime_le_lowerCrossingTime theorem lowerCrossingTime_le_upperCrossingTime_succ : lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by rw [upperCrossingTime_succ] exact le_hitting lowerCrossingTime_le ω #align measure_theory.lower_crossing_time_le_upper_crossing_time_succ MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ theorem lowerCrossingTime_mono (hnm : n ≤ m) : lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm exact monotone_nat_of_le_succ fun n => le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime #align measure_theory.lower_crossing_time_mono MeasureTheory.lowerCrossingTime_mono theorem upperCrossingTime_mono (hnm : n ≤ m) : upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm exact monotone_nat_of_le_succ fun n => le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ #align measure_theory.upper_crossing_time_mono MeasureTheory.upperCrossingTime_mono end ConditionallyCompleteLinearOrderBot variable {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω} theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) : stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩ #align measure_theory.stopped_value_lower_crossing_time MeasureTheory.stoppedValue_lowerCrossingTime theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) : b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩ #align measure_theory.stopped_value_upper_crossing_time MeasureTheory.stoppedValue_upperCrossingTime theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b) (hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) : upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by refine lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h => not_le.2 hab <\| le_trans ?_ (stoppedValue_lowerCrossingTime hn) simp only [stoppedValue] rw [← h] exact stoppedValue_upperCrossingTime (h.symm ▸ hn) #align measure_theory.upper_crossing_time_lt_lower_crossing_time MeasureTheory.upperCrossingTime_lt_lowerCrossingTime theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) : lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := by refine lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h => not_le.2 hab <\| le_trans (stoppedValue_upperCrossingTime hn) ?_ simp only [stoppedValue] rw [← h] exact stoppedValue_lowerCrossingTime (h.symm ▸ hn) #align measure_theory.lower_crossing_time_lt_upper_crossing_time MeasureTheory.lowerCrossingTime_lt_upperCrossingTime theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) : upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω := lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime (lowerCrossingTime_lt_upperCrossingTime hab hn) #align measure_theory.upper_crossing_time_lt_succ MeasureTheory.upperCrossingTime_lt_succ theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) : lowerCrossingTime a b f N m ω = N := le_antisymm lowerCrossingTime_le (le_trans (le_of_eq hn.symm) (lowerCrossingTime_mono hnm)) #align measure_theory.lower_crossing_time_stabilize MeasureTheory.lowerCrossingTime_stabilize theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) : upperCrossingTime a b f N m ω = N := le_antisymm upperCrossingTime_le (le_trans (le_of_eq hn.symm) (upperCrossingTime_mono hnm)) #align measure_theory.upper_crossing_time_stabilize MeasureTheory.upperCrossingTime_stabilize theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) : lowerCrossingTime a b f N m ω = N := lowerCrossingTime_stabilize hnm (le_antisymm lowerCrossingTime_le hn) #align measure_theory.lower_crossing_time_stabilize' MeasureTheory.lowerCrossingTime_stabilize' theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) : upperCrossingTime a b f N m ω = N := upperCrossingTime_stabilize hnm (le_antisymm upperCrossingTime_le hn) #align measure_theory.upper_crossing_time_stabilize' MeasureTheory.upperCrossingTime_stabilize' -- `upperCrossingTime_bound_eq` provides an explicit bound theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) : ∃ n, upperCrossingTime a b f N n ω = N := by by_contra h; push_neg at h have : StrictMono fun n => upperCrossingTime a b f N n ω := strictMono_nat_of_lt_succ fun n => upperCrossingTime_lt_succ hab (h _) obtain ⟨_, ⟨k, rfl⟩, hk⟩ : ∃ (m : _) (_ : m ∈ Set.range fun n => upperCrossingTime a b f N n ω), N < m := ⟨upperCrossingTime a b f N (N + 1) ω, ⟨N + 1, rfl⟩, lt_of_lt_of_le N.lt_succ_self (StrictMono.id_le this (N + 1))⟩ exact not_le.2 hk upperCrossingTime_le #align measure_theory.exists_upper_crossing_time_eq MeasureTheory.exists_upperCrossingTime_eq theorem upperCrossingTime_lt_bddAbove (hab : a < b) : BddAbove {n \| upperCrossingTime a b f N n ω < N} := by obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab refine ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => ?_⟩ by_contra hn' exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk) #align measure_theory.upper_crossing_time_lt_bdd_above MeasureTheory.upperCrossingTime_lt_bddAbove theorem upperCrossingTime_lt_nonempty (hN : 0 < N) : {n \| upperCrossingTime a b f N n ω < N}.Nonempty := ⟨0, hN⟩ #align measure_theory.upper_crossing_time_lt_nonempty MeasureTheory.upperCrossingTime_lt_nonempty theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) : upperCrossingTime a b f N N ω = N := by by_cases hN' : N < Nat.find (exists_upperCrossingTime_eq f N ω hab) · refine le_antisymm upperCrossingTime_le ?_ have hmono : StrictMonoOn (fun n => upperCrossingTime a b f N n ω) (Set.Iic (Nat.find (exists_upperCrossingTime_eq f N ω hab)).pred) := by refine strictMonoOn_Iic_of_lt_succ fun m hm => upperCrossingTime_lt_succ hab ?_ rw [Nat.lt_pred_iff] at hm convert Nat.find_min _ hm convert StrictMonoOn.Iic_id_le hmono N (Nat.le_sub_one_of_lt hN') · rw [not_lt] at hN' exact upperCrossingTime_stabilize hN' (Nat.find_spec (exists_upperCrossingTime_eq f N ω hab)) #align measure_theory.upper_crossing_time_bound_eq MeasureTheory.upperCrossingTime_bound_eq theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) : upperCrossingTime a b f N n ω = N := le_antisymm upperCrossingTime_le (le_trans (upperCrossingTime_bound_eq f N ω hab).symm.le (upperCrossingTime_mono hn)) #align measure_theory.upper_crossing_time_eq_of_bound_le MeasureTheory.upperCrossingTime_eq_of_bound_le variable {ℱ : Filtration ℕ m0} theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) : IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧ IsStoppingTime ℱ (lowerCrossingTime a b f N n) := by induction' n with k ih · refine ⟨isStoppingTime_const _ 0, ?_⟩ simp [hitting_isStoppingTime hf measurableSet_Iic] · obtain ⟨_, ih₂⟩ := ih have : IsStoppingTime ℱ (upperCrossingTime a b f N (k + 1)) := by intro n simp_rw [upperCrossingTime_succ_eq] exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le) measurableSet_Ici hf _ refine ⟨this, ?_⟩ intro n exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le) measurableSet_Iic hf _ #align measure_theory.adapted.is_stopping_time_crossing MeasureTheory.Adapted.isStoppingTime_crossing theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) : IsStoppingTime ℱ (upperCrossingTime a b f N n) := hf.isStoppingTime_crossing.1 #align measure_theory.adapted.is_stopping_time_upper_crossing_time MeasureTheory.Adapted.isStoppingTime_upperCrossingTime theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) : IsStoppingTime ℱ (lowerCrossingTime a b f N n) := hf.isStoppingTime_crossing.2 #align measure_theory.adapted.is_stopping_time_lower_crossing_time MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime /-- `upcrossingStrat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper crossings and is 0 otherwise. `upcrossingStrat` is shifted by one index so that it is adapted rather than predictable. -/ noncomputable def upcrossingStrat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ := ∑ k ∈ Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator 1 n #align measure_theory.upcrossing_strat MeasureTheory.upcrossingStrat theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω := Finset.sum_nonneg fun _ _ => Set.indicator_nonneg (fun _ _ => zero_le_one) _ #align measure_theory.upcrossing_strat_nonneg MeasureTheory.upcrossingStrat_nonneg theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 := by rw [upcrossingStrat, ← Finset.indicator_biUnion_apply] · exact Set.indicator_le_self' (fun _ _ => zero_le_one) _ intro i _ j _ hij simp only [Set.Ico_disjoint_Ico] obtain hij' \| hij' := lt_or_gt_of_ne hij · rw [min_eq_left (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) : upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω), max_eq_right (lowerCrossingTime_mono hij'.le : lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)] refine le_trans upperCrossingTime_le_lowerCrossingTime (lowerCrossingTime_mono (Nat.succ_le_of_lt hij')) · rw [gt_iff_lt] at hij' rw [min_eq_right (upperCrossingTime_mono (Nat.succ_le_succ hij'.le) : upperCrossingTime a b f N _ ω ≤ upperCrossingTime a b f N _ ω), max_eq_left (lowerCrossingTime_mono hij'.le : lowerCrossingTime a b f N _ _ ≤ lowerCrossingTime _ _ _ _ _ _)] refine le_trans upperCrossingTime_le_lowerCrossingTime (lowerCrossingTime_mono (Nat.succ_le_of_lt hij')) #align measure_theory.upcrossing_strat_le_one MeasureTheory.upcrossingStrat_le_one theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) : Adapted ℱ (upcrossingStrat a b f N) := by intro n change StronglyMeasurable[ℱ n] fun ω => ∑ k ∈ Finset.range N, ({n \| lowerCrossingTime a b f N k ω ≤ n} ∩ {n \| n < upperCrossingTime a b f N (k + 1) ω}).indicator 1 n refine Finset.stronglyMeasurable_sum _ fun i _ => stronglyMeasurable_const.indicator ((hf.isStoppingTime_lowerCrossingTime n).inter ?_) simp_rw [← not_le] exact (hf.isStoppingTime_upperCrossingTime n).compl #align measure_theory.adapted.upcrossing_strat_adapted MeasureTheory.Adapted.upcrossingStrat_adapted theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ => ∑ k ∈ Finset.range n, upcrossingStrat a b f N k (f (k + 1) - f k)) ℱ μ := hf.sum_mul_sub hf.adapted.upcrossingStrat_adapted (fun _ _ => upcrossingStrat_le_one) fun _ _ => upcrossingStrat_nonneg #align measure_theory.submartingale.sum_upcrossing_strat_mul MeasureTheory.Submartingale.sum_upcrossingStrat_mul theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ => ∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ := by refine hf.sum_mul_sub (fun n => (adapted_const ℱ 1 n).sub (hf.adapted.upcrossingStrat_adapted n)) (?_ : ∀ n ω, (1 - upcrossingStrat a b f N n) ω ≤ 1) ?_ · exact fun n ω => sub_le_self _ upcrossingStrat_nonneg · intro n ω simp [upcrossingStrat_le_one] #align measure_theory.submartingale.sum_sub_upcrossing_strat_mul MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) : μ[∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by have h₁ : (0 : ℝ) ≤ μ[∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by have := (hf.sum_sub_upcrossingStrat_mul a b N).setIntegral_le (zero_le n) MeasurableSet.univ rw [integral_univ, integral_univ] at this refine le_trans ?_ this simp only [Finset.range_zero, Finset.sum_empty, integral_zero', le_refl] have h₂ : μ[∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] = μ[∑ k ∈ Finset.range n, (f (k + 1) - f k)] - μ[∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by simp only [sub_mul, one_mul, Finset.sum_sub_distrib, Pi.sub_apply, Finset.sum_apply, Pi.mul_apply] refine integral_sub (Integrable.sub (integrable_finset_sum _ fun i _ => hf.integrable _) (integrable_finset_sum _ fun i _ => hf.integrable _)) ?_ convert (hf.sum_upcrossingStrat_mul a b N).integrable n using 1 ext; simp rw [h₂, sub_nonneg] at h₁ refine le_trans h₁ ?_ simp_rw [Finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _), le_refl] #align measure_theory.submartingale.sum_mul_upcrossing_strat_le MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le /-- The number of upcrossings (strictly) before time `N`. -/ noncomputable def upcrossingsBefore [Preorder ι] [OrderBot ι] [InfSet ι] (a b : ℝ) (f : ι → Ω → ℝ) (N : ι) (ω : Ω) : ℕ := sSup {n \| upperCrossingTime a b f N n ω < N} #align measure_theory.upcrossings_before MeasureTheory.upcrossingsBefore @[simp] theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ} {ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ := by simp [upcrossingsBefore] #align measure_theory.upcrossings_before_bot MeasureTheory.upcrossingsBefore_bot theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 := by simp [upcrossingsBefore] #align measure_theory.upcrossings_before_zero MeasureTheory.upcrossingsBefore_zero @[simp] theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 := by ext ω; exact upcrossingsBefore_zero #align measure_theory.upcrossings_before_zero' MeasureTheory.upcrossingsBefore_zero' theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b) (hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N := haveI : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N := (upperCrossingTime_lt_nonempty hN).csSup_mem ((OrderBot.bddBelow _).finite_of_bddAbove (upperCrossingTime_lt_bddAbove hab)) lt_of_le_of_lt (upperCrossingTime_mono hn) this #align measure_theory.upper_crossing_time_lt_of_le_upcrossings_before MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by refine le_antisymm upperCrossingTime_le (not_lt.1 ?_) convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab) #align measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) : upcrossingsBefore a b f N ω ≤ N := by by_cases hN : N = 0 · subst hN rw [upcrossingsBefore_zero] · refine csSup_le ⟨0, zero_lt_iff.2 hN⟩ fun n (hn : _ < N) => ?_ by_contra hnN exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le) #align measure_theory.upcrossings_before_le MeasureTheory.upcrossingsBefore_le theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M) (h : lowerCrossingTime a b f N n ω < N) : upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧ lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by have h' : upperCrossingTime a b f N n ω < N := lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h induction' n with k ih · simp only [Nat.zero_eq, upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true, lowerCrossingTime_zero, true_and_iff, eq_comm] refine hitting_eq_hitting_of_exists hNM ?_ rw [lowerCrossingTime, hitting_lt_iff] at h · obtain ⟨j, hj₁, hj₂⟩ := h exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ · exact le_rfl · specialize ih (lt_of_le_of_lt (lowerCrossingTime_mono (Nat.le_succ _)) h) (lt_of_le_of_lt (upperCrossingTime_mono (Nat.le_succ _)) h') have : upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω := by rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h' · simp only [upperCrossingTime_succ_eq] obtain ⟨j, hj₁, hj₂⟩ := h' rw [eq_comm, ih.2] exact hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ · exact le_rfl refine ⟨this, ?_⟩ simp only [lowerCrossingTime, eq_comm, this, Nat.succ_eq_add_one] refine hitting_eq_hitting_of_exists hNM ?_ rw [lowerCrossingTime, hitting_lt_iff _ le_rfl] at h obtain ⟨j, hj₁, hj₂⟩ := h exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ #align measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M) (h : upperCrossingTime a b f N (n + 1) ω < N) : upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧ lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω := by have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM (lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2 refine ⟨?_, this⟩ rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this] refine hitting_eq_hitting_of_exists hNM ?_ rw [upperCrossingTime_succ_eq, hitting_lt_iff] at h · obtain ⟨j, hj₁, hj₂⟩ := h exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ · exact le_rfl #align measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M) (h : upperCrossingTime a b f N n ω < N) : upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω := by cases n · simp · exact (crossing_eq_crossing_of_upperCrossingTime_lt hNM h).1 #align measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by intro N M hNM ω simp only [upcrossingsBefore] by_cases hemp : {n : ℕ \| upperCrossingTime a b f N n ω < N}.Nonempty · refine csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => ?_ rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn] exact lt_of_lt_of_le hn hNM · rw [Set.not_nonempty_iff_eq_empty] at hemp simp [hemp, csSup_empty, bot_eq_zero', zero_le'] #align measure_theory.upcrossings_before_mono MeasureTheory.upcrossingsBefore_mono theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁) (hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) : upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by refine lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) ?_) rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl] refine ⟨N₂, ⟨?_, Nat.lt_succ_self _⟩, hN₂'.le⟩ rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)] refine ⟨N₁, ⟨le_trans ?_ hN₁, hN₂⟩, hN₁'.le⟩ by_cases hN : 0 < N · have : upperCrossingTime a b f N (upcrossingsBefore a b f N ω) ω < N := Nat.sSup_mem (upperCrossingTime_lt_nonempty hN) (upperCrossingTime_lt_bddAbove hab) rw [upperCrossingTime_eq_upperCrossingTime_of_lt (hN₁.trans (hN₂.trans <\| Nat.le_succ _)) this] exact this.le · rw [not_lt, Nat.le_zero] at hN rw [hN, upcrossingsBefore_zero, upperCrossingTime_zero] rfl #align measure_theory.upcrossings_before_lt_of_exists_upcrossing MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b) (hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N := lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn) #align measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b) (hn : n < upcrossingsBefore a b f N ω) : b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω - stoppedValue f (lowerCrossingTime a b f N n) ω := sub_le_sub (stoppedValue_upperCrossingTime (upperCrossingTime_lt_of_le_upcrossingsBefore hN hab hn).ne) (stoppedValue_lowerCrossingTime (lowerCrossingTime_lt_of_lt_upcrossingsBefore hN hab hn).ne) #align measure_theory.le_sub_of_le_upcrossings_before MeasureTheory.le_sub_of_le_upcrossingsBefore theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) : stoppedValue f (upperCrossingTime a b f N (n + 1)) ω - stoppedValue f (lowerCrossingTime a b f N n) ω = 0 := by have : N ≤ upperCrossingTime a b f N n ω := by rw [upcrossingsBefore] at hn rw [← not_lt] exact fun h => not_le.2 hn (le_csSup (upperCrossingTime_lt_bddAbove hab) h) simp [stoppedValue, upperCrossingTime_stabilize' (Nat.le_succ n) this, lowerCrossingTime_stabilize' le_rfl (le_trans this upperCrossingTime_le_lowerCrossingTime)] #align measure_theory.sub_eq_zero_of_upcrossings_before_lt MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) : (b - a) * upcrossingsBefore a b f N ω ≤ ∑ k ∈ Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω := by classical by_cases hN : N = 0 · simp [hN] simp_rw [upcrossingStrat, Finset.sum_mul, ← Set.indicator_mul_left _ _ (fun x ↦ (f (x + 1) - f x) ω), Pi.one_apply, Pi.sub_apply, one_mul] rw [Finset.sum_comm] have h₁ : ∀ k, ∑ n ∈ Finset.range N, (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator (fun m => f (m + 1) ω - f m ω) n = stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω := by intro k rw [Finset.sum_indicator_eq_sum_filter, (_ : Finset.filter (fun i => i ∈ Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) (Finset.range N) = Finset.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)), Finset.sum_Ico_eq_add_neg _ lowerCrossingTime_le_upperCrossingTime_succ, Finset.sum_range_sub fun n => f n ω, Finset.sum_range_sub fun n => f n ω, neg_sub, sub_add_sub_cancel] · rfl · ext i simp only [Set.mem_Ico, Finset.mem_filter, Finset.mem_range, Finset.mem_Ico, and_iff_right_iff_imp, and_imp] exact fun _ h => lt_of_lt_of_le h upperCrossingTime_le simp_rw [h₁] have h₂ : ∑ _k ∈ Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤ ∑ k ∈ Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω) := by calc ∑ _k ∈ Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤ ∑ k ∈ Finset.range (upcrossingsBefore a b f N ω), (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω) := by refine Finset.sum_le_sum fun i hi => le_sub_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab ?_ rwa [Finset.mem_range] at hi _ ≤ ∑ k ∈ Finset.range N, (stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω) := by refine Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_subset.2 (upcrossingsBefore_le f ω hab)) fun i _ hi => ?_ by_cases hi' : i = upcrossingsBefore a b f N ω · subst hi' simp only [stoppedValue] rw [upperCrossingTime_eq_of_upcrossingsBefore_lt hab (Nat.lt_succ_self _)] by_cases heq : lowerCrossingTime a b f N (upcrossingsBefore a b f N ω) ω = N · rw [heq, sub_self] · rw [sub_nonneg] exact le_trans (stoppedValue_lowerCrossingTime heq) hf · rw [sub_eq_zero_of_upcrossingsBefore_lt hab] rw [Finset.mem_range, not_lt] at hi exact lt_of_le_of_ne hi (Ne.symm hi') refine le_trans ?_ h₂ rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_comm] #align measure_theory.mul_upcrossings_before_le MeasureTheory.mul_upcrossingsBefore_le theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) : (b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] := calc (b - a) * μ[upcrossingsBefore a b f N] ≤ μ[∑ k ∈ Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)] := by rw [← integral_mul_left] refine integral_mono_of_nonneg ?_ ((hf.sum_upcrossingStrat_mul a b N).integrable N) ?_ · exact eventually_of_forall fun ω => mul_nonneg (sub_nonneg.2 hab.le) (Nat.cast_nonneg _) · filter_upwards with ω simpa using mul_upcrossingsBefore_le (hfN ω) hab _ ≤ μ[f N] - μ[f 0] := hf.sum_mul_upcrossingStrat_le _ ≤ μ[f N] := (sub_le_self_iff _).2 (integral_nonneg hfzero) #align measure_theory.integral_mul_upcrossings_before_le_integral MeasureTheory.integral_mul_upcrossingsBefore_le_integral theorem crossing_pos_eq (hab : a < b) : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧ lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by have hab' : 0 < b - a := sub_pos.2 hab have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by intro i ω refine ⟨fun h => ?_, fun h => ?_⟩ · rwa [← sub_le_sub_iff_right a, ← posPart_eq_of_posPart_pos (lt_of_lt_of_le hab' h)] · rw [← sub_le_sub_iff_right a] at h rwa [posPart_eq_self.2 (le_trans hab'.le h)] have hf' (ω i) : (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a := by rw [posPart_nonpos, sub_nonpos] induction' n with k ih · refine ⟨rfl, ?_⟩ #adaptation_note /-- nightly-2024-03-16: simp was simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting, Set.mem_Icc, Set.mem_Iic, Nat.zero_eq] -/ simp (config := { unfoldPartialApp := true }) only [lowerCrossingTime_zero, hitting_def, Set.mem_Icc, Set.mem_Iic, Nat.zero_eq] ext ω split_ifs with h₁ h₂ h₂ · simp_rw [hf'] · simp_rw [Set.mem_Iic, ← hf' _ _] at h₂ exact False.elim (h₂ h₁) · simp_rw [Set.mem_Iic, hf' _ _] at h₁ exact False.elim (h₁ h₂) · rfl · have : upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N (k + 1) = upperCrossingTime a b f N (k + 1) := by ext ω simp only [upperCrossingTime_succ_eq, ← ih.2, hitting, Set.mem_Ici, tsub_le_iff_right] split_ifs with h₁ h₂ h₂ · simp_rw [← sub_le_iff_le_add, hf ω] · refine False.elim (h₂ ?_) simp_all only [Set.mem_Ici, not_true_eq_false] · refine False.elim (h₁ ?_) simp_all only [Set.mem_Ici] · rfl refine ⟨this, ?_⟩ ext ω simp only [lowerCrossingTime, this, hitting, Set.mem_Iic] split_ifs with h₁ h₂ h₂ · simp_rw [hf' ω] · refine False.elim (h₂ ?_) simp_all only [Set.mem_Iic, not_true_eq_false] · refine False.elim (h₁ ?_) simp_all only [Set.mem_Iic] · rfl #align measure_theory.crossing_pos_eq MeasureTheory.crossing_pos_eq theorem upcrossingsBefore_pos_eq (hab : a < b) : upcrossingsBefore 0 (b - a) (fun n ω => (f n ω - a)⁺) N ω = upcrossingsBefore a b f N ω := by simp_rw [upcrossingsBefore, (crossing_pos_eq hab).1] #align measure_theory.upcrossings_before_pos_eq MeasureTheory.upcrossingsBefore_pos_eq theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (hab : a < b) : (b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by refine le_trans (le_of_eq ?_) (integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_const _ _ _)).pos (fun ω => posPart_nonneg _) (fun ω => posPart_nonneg _) (sub_pos.2 hab)) simp_rw [sub_zero, ← upcrossingsBefore_pos_eq hab] rfl #align measure_theory.mul_integral_upcrossings_before_le_integral_pos_part_aux MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux /-- Doob's upcrossing estimate: given a real valued discrete submartingale `f` and real values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where `upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above `b` before the time `N`. -/ theorem Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part [IsFiniteMeasure μ] (a b : ℝ) (hf : Submartingale f ℱ μ) (N : ℕ) : (b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by by_cases hab : a < b · exact mul_integral_upcrossingsBefore_le_integral_pos_part_aux hf hab · rw [not_lt, ← sub_nonpos] at hab exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (by positivity)) (integral_nonneg fun ω => posPart_nonneg _) #align measure_theory.submartingale.mul_integral_upcrossings_before_le_integral_pos_part MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part /-! ### Variant of the upcrossing estimate Now, we would like to prove a variant of the upcrossing estimate obtained by taking the supremum over $N$ of the original upcrossing estimate. Namely, we want the inequality $$ (b - a) \sup_N \mathbb{E}[U_N(a, b)] \le \sup_N \mathbb{E}[f_N]. $$ This inequality is central for the martingale convergence theorem as it provides a uniform bound for the upcrossings. We note that on top of taking the supremum on both sides of the inequality, we had also used the monotone convergence theorem on the left hand side to take the supremum outside of the integral. To do this, we need to make sure $U_N(a, b)$ is measurable and integrable. Integrability is easy to check as $U_N(a, b) ≤ N$ and so it suffices to show measurability. Indeed, by noting that $$ U_N(a, b) = \sum_{i = 1}^N \mathbf{1}_{\{U_N(a, b) < N\}} $$ $U_N(a, b)$ is measurable as $\{U_N(a, b) < N\}$ is a measurable set since $U_N(a, b)$ is a stopping time. -/ theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω = ∑ i ∈ Finset.Ico 1 (N + 1), {n \| upperCrossingTime a b f N n ω < N}.indicator 1 i := by by_cases hN : N = 0 · simp [hN] rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le') (Nat.succ_le_succ (upcrossingsBefore_le f ω hab))] have h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n : ℕ \| upperCrossingTime a b f N n ω < N}.indicator 1 k = 1 := by rintro k hk rw [Finset.mem_Ico] at hk rw [Set.indicator_of_mem] · rfl · exact upperCrossingTime_lt_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab (Nat.lt_succ_iff.1 hk.2) have h₂ : ∀ k ∈ Finset.Ico (upcrossingsBefore a b f N ω + 1) (N + 1), {n : ℕ \| upperCrossingTime a b f N n ω < N}.indicator 1 k = 0 := by rintro k hk rw [Finset.mem_Ico, Nat.succ_le_iff] at hk rw [Set.indicator_of_not_mem] simp only [Set.mem_setOf_eq, not_lt] exact (upperCrossingTime_eq_of_upcrossingsBefore_lt hab hk.1).symm.le rw [Finset.sum_congr rfl h₁, Finset.sum_congr rfl h₂, Finset.sum_const, Finset.sum_const, smul_eq_mul, mul_one, smul_eq_mul, mul_zero, Nat.card_Ico, Nat.add_succ_sub_one, add_zero, add_zero] #align measure_theory.upcrossings_before_eq_sum MeasureTheory.upcrossingsBefore_eq_sum theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) : Measurable (upcrossingsBefore a b f N) := by have : upcrossingsBefore a b f N = fun ω => ∑ i ∈ Finset.Ico 1 (N + 1), {n \| upperCrossingTime a b f N n ω < N}.indicator 1 i := by ext ω exact upcrossingsBefore_eq_sum hab rw [this] exact Finset.measurable_sum _ fun i _ => Measurable.indicator measurable_const <\| ℱ.le N _ (hf.isStoppingTime_upperCrossingTime.measurableSet_lt_of_pred N) #align measure_theory.adapted.measurable_upcrossings_before MeasureTheory.Adapted.measurable_upcrossingsBefore	Mathlib/Probability/Martingale/Upcrossing.lean	815	822	theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) : Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ := haveI : ∀ᵐ ω ∂μ, ‖(upcrossingsBefore a b f N ω : ℝ)‖ ≤ N := by	filter_upwards with ω rw [Real.norm_eq_abs, Nat.abs_cast, Nat.cast_le] exact upcrossingsBefore_le _ _ hab ⟨Measurable.aestronglyMeasurable (measurable_from_top.comp (hf.measurable_upcrossingsBefore hab)), hasFiniteIntegral_of_bounded this⟩
/- Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Violeta Hernández Palacios -/ import Mathlib.SetTheory.Ordinal.Arithmetic import Mathlib.Tactic.Abel #align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7" /-! # Natural operations on ordinals The goal of this file is to define natural addition and multiplication on ordinals, also known as the Hessenberg sum and product, and provide a basic API. The natural addition of two ordinals `a ♯ b` is recursively defined as the least ordinal greater than `a' ♯ b` and `a ♯ b'` for `a' < a` and `b' < b`. The natural multiplication `a ⨳ b` is likewise recursively defined as the least ordinal such that `a ⨳ b ♯ a' ⨳ b'` is greater than `a' ⨳ b ♯ a ⨳ b'` for any `a' < a` and `b' < b`. These operations form a rich algebraic structure: they're commutative, associative, preserve order, have the usual `0` and `1` from ordinals, and distribute over one another. Moreover, these operations are the addition and multiplication of ordinals when viewed as combinatorial `Game`s. This makes them particularly useful for game theory. Finally, both operations admit simple, intuitive descriptions in terms of the Cantor normal form. The natural addition of two ordinals corresponds to adding their Cantor normal forms as if they were polynomials in `ω`. Likewise, their natural multiplication corresponds to multiplying the Cantor normal forms as polynomials. # Implementation notes Given the rich algebraic structure of these two operations, we choose to create a type synonym `NatOrdinal`, where we provide the appropriate instances. However, to avoid casting back and forth between both types, we attempt to prove and state most results on `Ordinal`. # Todo - Prove the characterizations of natural addition and multiplication in terms of the Cantor normal form. -/ set_option autoImplicit true universe u v open Function Order noncomputable section /-! ### Basic casts between `Ordinal` and `NatOrdinal` -/ /-- A type synonym for ordinals with natural addition and multiplication. -/ def NatOrdinal : Type _ := -- Porting note: used to derive LinearOrder & SuccOrder but need to manually define Ordinal deriving Zero, Inhabited, One, WellFoundedRelation #align nat_ordinal NatOrdinal instance NatOrdinal.linearOrder : LinearOrder NatOrdinal := {Ordinal.linearOrder with} instance NatOrdinal.succOrder : SuccOrder NatOrdinal := {Ordinal.succOrder with} /-- The identity function between `Ordinal` and `NatOrdinal`. -/ @[match_pattern] def Ordinal.toNatOrdinal : Ordinal ≃o NatOrdinal := OrderIso.refl _ #align ordinal.to_nat_ordinal Ordinal.toNatOrdinal /-- The identity function between `NatOrdinal` and `Ordinal`. -/ @[match_pattern] def NatOrdinal.toOrdinal : NatOrdinal ≃o Ordinal := OrderIso.refl _ #align nat_ordinal.to_ordinal NatOrdinal.toOrdinal namespace NatOrdinal open Ordinal @[simp] theorem toOrdinal_symm_eq : NatOrdinal.toOrdinal.symm = Ordinal.toNatOrdinal := rfl #align nat_ordinal.to_ordinal_symm_eq NatOrdinal.toOrdinal_symm_eq -- Porting note: used to use dot notation, but doesn't work in Lean 4 with `OrderIso` @[simp] theorem toOrdinal_toNatOrdinal (a : NatOrdinal) : Ordinal.toNatOrdinal (NatOrdinal.toOrdinal a) = a := rfl #align nat_ordinal.to_ordinal_to_nat_ordinal NatOrdinal.toOrdinal_toNatOrdinal theorem lt_wf : @WellFounded NatOrdinal (· < ·) := Ordinal.lt_wf #align nat_ordinal.lt_wf NatOrdinal.lt_wf instance : WellFoundedLT NatOrdinal := Ordinal.wellFoundedLT instance : IsWellOrder NatOrdinal (· < ·) := Ordinal.isWellOrder @[simp] theorem toOrdinal_zero : toOrdinal 0 = 0 := rfl #align nat_ordinal.to_ordinal_zero NatOrdinal.toOrdinal_zero @[simp] theorem toOrdinal_one : toOrdinal 1 = 1 := rfl #align nat_ordinal.to_ordinal_one NatOrdinal.toOrdinal_one @[simp] theorem toOrdinal_eq_zero (a) : toOrdinal a = 0 ↔ a = 0 := Iff.rfl #align nat_ordinal.to_ordinal_eq_zero NatOrdinal.toOrdinal_eq_zero @[simp] theorem toOrdinal_eq_one (a) : toOrdinal a = 1 ↔ a = 1 := Iff.rfl #align nat_ordinal.to_ordinal_eq_one NatOrdinal.toOrdinal_eq_one @[simp] theorem toOrdinal_max : toOrdinal (max a b) = max (toOrdinal a) (toOrdinal b) := rfl #align nat_ordinal.to_ordinal_max NatOrdinal.toOrdinal_max @[simp] theorem toOrdinal_min : toOrdinal (min a b)= min (toOrdinal a) (toOrdinal b) := rfl #align nat_ordinal.to_ordinal_min NatOrdinal.toOrdinal_min theorem succ_def (a : NatOrdinal) : succ a = toNatOrdinal (toOrdinal a + 1) := rfl #align nat_ordinal.succ_def NatOrdinal.succ_def /-- A recursor for `NatOrdinal`. Use as `induction x using NatOrdinal.rec`. -/ protected def rec {β : NatOrdinal → Sort*} (h : ∀ a, β (toNatOrdinal a)) : ∀ a, β a := fun a => h (toOrdinal a) #align nat_ordinal.rec NatOrdinal.rec /-- `Ordinal.induction` but for `NatOrdinal`. -/ theorem induction {p : NatOrdinal → Prop} : ∀ (i) (_ : ∀ j, (∀ k, k < j → p k) → p j), p i := Ordinal.induction #align nat_ordinal.induction NatOrdinal.induction end NatOrdinal namespace Ordinal variable {a b c : Ordinal.{u}} @[simp] theorem toNatOrdinal_symm_eq : toNatOrdinal.symm = NatOrdinal.toOrdinal := rfl #align ordinal.to_nat_ordinal_symm_eq Ordinal.toNatOrdinal_symm_eq @[simp] theorem toNatOrdinal_toOrdinal (a : Ordinal) : NatOrdinal.toOrdinal (toNatOrdinal a) = a := rfl #align ordinal.to_nat_ordinal_to_ordinal Ordinal.toNatOrdinal_toOrdinal @[simp] theorem toNatOrdinal_zero : toNatOrdinal 0 = 0 := rfl #align ordinal.to_nat_ordinal_zero Ordinal.toNatOrdinal_zero @[simp] theorem toNatOrdinal_one : toNatOrdinal 1 = 1 := rfl #align ordinal.to_nat_ordinal_one Ordinal.toNatOrdinal_one @[simp] theorem toNatOrdinal_eq_zero (a) : toNatOrdinal a = 0 ↔ a = 0 := Iff.rfl #align ordinal.to_nat_ordinal_eq_zero Ordinal.toNatOrdinal_eq_zero @[simp] theorem toNatOrdinal_eq_one (a) : toNatOrdinal a = 1 ↔ a = 1 := Iff.rfl #align ordinal.to_nat_ordinal_eq_one Ordinal.toNatOrdinal_eq_one @[simp] theorem toNatOrdinal_max (a b : Ordinal) : toNatOrdinal (max a b) = max (toNatOrdinal a) (toNatOrdinal b) := rfl #align ordinal.to_nat_ordinal_max Ordinal.toNatOrdinal_max @[simp] theorem toNatOrdinal_min (a b : Ordinal) : toNatOrdinal (linearOrder.min a b) = linearOrder.min (toNatOrdinal a) (toNatOrdinal b) := rfl #align ordinal.to_nat_ordinal_min Ordinal.toNatOrdinal_min /-! We place the definitions of `nadd` and `nmul` before actually developing their API, as this guarantees we only need to open the `NaturalOps` locale once. -/ /-- Natural addition on ordinals `a ♯ b`, also known as the Hessenberg sum, is recursively defined as the least ordinal greater than `a' ♯ b` and `a ♯ b'` for all `a' < a` and `b' < b`. In contrast to normal ordinal addition, it is commutative. Natural addition can equivalently be characterized as the ordinal resulting from adding up corresponding coefficients in the Cantor normal forms of `a` and `b`. -/ noncomputable def nadd : Ordinal → Ordinal → Ordinal \| a, b => max (blsub.{u, u} a fun a' _ => nadd a' b) (blsub.{u, u} b fun b' _ => nadd a b') termination_by o₁ o₂ => (o₁, o₂) #align ordinal.nadd Ordinal.nadd @[inherit_doc] scoped[NaturalOps] infixl:65 " ♯ " => Ordinal.nadd open NaturalOps /-- Natural multiplication on ordinals `a ⨳ b`, also known as the Hessenberg product, is recursively defined as the least ordinal such that `a ⨳ b + a' ⨳ b'` is greater than `a' ⨳ b + a ⨳ b'` for all `a' < a` and `b < b'`. In contrast to normal ordinal multiplication, it is commutative and distributive (over natural addition). Natural multiplication can equivalently be characterized as the ordinal resulting from multiplying the Cantor normal forms of `a` and `b` as if they were polynomials in `ω`. Addition of exponents is done via natural addition. -/ noncomputable def nmul : Ordinal.{u} → Ordinal.{u} → Ordinal.{u} \| a, b => sInf {c \| ∀ a' < a, ∀ b' < b, nmul a' b ♯ nmul a b' < c ♯ nmul a' b'} termination_by a b => (a, b) #align ordinal.nmul Ordinal.nmul @[inherit_doc] scoped[NaturalOps] infixl:70 " ⨳ " => Ordinal.nmul /-! ### Natural addition -/ theorem nadd_def (a b : Ordinal) : a ♯ b = max (blsub.{u, u} a fun a' _ => a' ♯ b) (blsub.{u, u} b fun b' _ => a ♯ b') := by rw [nadd] #align ordinal.nadd_def Ordinal.nadd_def theorem lt_nadd_iff : a < b ♯ c ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c' := by rw [nadd_def] simp [lt_blsub_iff] #align ordinal.lt_nadd_iff Ordinal.lt_nadd_iff theorem nadd_le_iff : b ♯ c ≤ a ↔ (∀ b' < b, b' ♯ c < a) ∧ ∀ c' < c, b ♯ c' < a := by rw [nadd_def] simp [blsub_le_iff] #align ordinal.nadd_le_iff Ordinal.nadd_le_iff theorem nadd_lt_nadd_left (h : b < c) (a) : a ♯ b < a ♯ c := lt_nadd_iff.2 (Or.inr ⟨b, h, le_rfl⟩) #align ordinal.nadd_lt_nadd_left Ordinal.nadd_lt_nadd_left theorem nadd_lt_nadd_right (h : b < c) (a) : b ♯ a < c ♯ a := lt_nadd_iff.2 (Or.inl ⟨b, h, le_rfl⟩) #align ordinal.nadd_lt_nadd_right Ordinal.nadd_lt_nadd_right theorem nadd_le_nadd_left (h : b ≤ c) (a) : a ♯ b ≤ a ♯ c := by rcases lt_or_eq_of_le h with (h \| rfl) · exact (nadd_lt_nadd_left h a).le · exact le_rfl #align ordinal.nadd_le_nadd_left Ordinal.nadd_le_nadd_left theorem nadd_le_nadd_right (h : b ≤ c) (a) : b ♯ a ≤ c ♯ a := by rcases lt_or_eq_of_le h with (h \| rfl) · exact (nadd_lt_nadd_right h a).le · exact le_rfl #align ordinal.nadd_le_nadd_right Ordinal.nadd_le_nadd_right variable (a b) theorem nadd_comm : ∀ a b, a ♯ b = b ♯ a \| a, b => by rw [nadd_def, nadd_def, max_comm] congr <;> ext <;> apply nadd_comm termination_by a b => (a,b) #align ordinal.nadd_comm Ordinal.nadd_comm theorem blsub_nadd_of_mono {f : ∀ c < a ♯ b, Ordinal.{max u v}} (hf : ∀ {i j} (hi hj), i ≤ j → f i hi ≤ f j hj) : -- Porting note: needed to add universe hint blsub.{u,v} in the line below blsub.{u,v} _ f = max (blsub.{u, v} a fun a' ha' => f (a' ♯ b) <\| nadd_lt_nadd_right ha' b) (blsub.{u, v} b fun b' hb' => f (a ♯ b') <\| nadd_lt_nadd_left hb' a) := by apply (blsub_le_iff.2 fun i h => _).antisymm (max_le _ _) · intro i h rcases lt_nadd_iff.1 h with (⟨a', ha', hi⟩ \| ⟨b', hb', hi⟩) · exact lt_max_of_lt_left ((hf h (nadd_lt_nadd_right ha' b) hi).trans_lt (lt_blsub _ _ ha')) · exact lt_max_of_lt_right ((hf h (nadd_lt_nadd_left hb' a) hi).trans_lt (lt_blsub _ _ hb')) all_goals apply blsub_le_of_brange_subset.{u, u, v} rintro c ⟨d, hd, rfl⟩ apply mem_brange_self #align ordinal.blsub_nadd_of_mono Ordinal.blsub_nadd_of_mono theorem nadd_assoc (a b c) : a ♯ b ♯ c = a ♯ (b ♯ c) := by rw [nadd_def a (b ♯ c), nadd_def, blsub_nadd_of_mono, blsub_nadd_of_mono, max_assoc] · congr <;> ext <;> apply nadd_assoc · exact fun _ _ h => nadd_le_nadd_left h a · exact fun _ _ h => nadd_le_nadd_right h c termination_by (a, b, c) #align ordinal.nadd_assoc Ordinal.nadd_assoc @[simp] theorem nadd_zero : a ♯ 0 = a := by induction' a using Ordinal.induction with a IH rw [nadd_def, blsub_zero, max_zero_right] convert blsub_id a rename_i hb exact IH _ hb #align ordinal.nadd_zero Ordinal.nadd_zero @[simp] theorem zero_nadd : 0 ♯ a = a := by rw [nadd_comm, nadd_zero] #align ordinal.zero_nadd Ordinal.zero_nadd @[simp] theorem nadd_one : a ♯ 1 = succ a := by induction' a using Ordinal.induction with a IH rw [nadd_def, blsub_one, nadd_zero, max_eq_right_iff, blsub_le_iff] intro i hi rwa [IH i hi, succ_lt_succ_iff] #align ordinal.nadd_one Ordinal.nadd_one @[simp] theorem one_nadd : 1 ♯ a = succ a := by rw [nadd_comm, nadd_one] #align ordinal.one_nadd Ordinal.one_nadd theorem nadd_succ : a ♯ succ b = succ (a ♯ b) := by rw [← nadd_one (a ♯ b), nadd_assoc, nadd_one] #align ordinal.nadd_succ Ordinal.nadd_succ theorem succ_nadd : succ a ♯ b = succ (a ♯ b) := by rw [← one_nadd (a ♯ b), ← nadd_assoc, one_nadd] #align ordinal.succ_nadd Ordinal.succ_nadd @[simp] theorem nadd_nat (n : ℕ) : a ♯ n = a + n := by induction' n with n hn · simp · rw [Nat.cast_succ, add_one_eq_succ, nadd_succ, add_succ, hn] #align ordinal.nadd_nat Ordinal.nadd_nat @[simp] theorem nat_nadd (n : ℕ) : ↑n ♯ a = a + n := by rw [nadd_comm, nadd_nat] #align ordinal.nat_nadd Ordinal.nat_nadd theorem add_le_nadd : a + b ≤ a ♯ b := by induction b using limitRecOn with \| H₁ => simp \| H₂ c h => rwa [add_succ, nadd_succ, succ_le_succ_iff] \| H₃ c hc H => simp_rw [← IsNormal.blsub_eq.{u, u} (add_isNormal a) hc, blsub_le_iff] exact fun i hi => (H i hi).trans_lt (nadd_lt_nadd_left hi a) #align ordinal.add_le_nadd Ordinal.add_le_nadd end Ordinal namespace NatOrdinal open Ordinal NaturalOps instance : Add NatOrdinal := ⟨nadd⟩ instance add_covariantClass_lt : CovariantClass NatOrdinal.{u} NatOrdinal.{u} (· + ·) (· < ·) := ⟨fun a _ _ h => nadd_lt_nadd_left h a⟩ #align nat_ordinal.add_covariant_class_lt NatOrdinal.add_covariantClass_lt instance add_covariantClass_le : CovariantClass NatOrdinal.{u} NatOrdinal.{u} (· + ·) (· ≤ ·) := ⟨fun a _ _ h => nadd_le_nadd_left h a⟩ #align nat_ordinal.add_covariant_class_le NatOrdinal.add_covariantClass_le instance add_contravariantClass_le : ContravariantClass NatOrdinal.{u} NatOrdinal.{u} (· + ·) (· ≤ ·) := ⟨fun a b c h => by by_contra! h' exact h.not_lt (add_lt_add_left h' a)⟩ #align nat_ordinal.add_contravariant_class_le NatOrdinal.add_contravariantClass_le instance orderedCancelAddCommMonoid : OrderedCancelAddCommMonoid NatOrdinal := { NatOrdinal.linearOrder with add := (· + ·) add_assoc := nadd_assoc add_le_add_left := fun a b => add_le_add_left le_of_add_le_add_left := fun a b c => le_of_add_le_add_left zero := 0 zero_add := zero_nadd add_zero := nadd_zero add_comm := nadd_comm nsmul := nsmulRec } instance addMonoidWithOne : AddMonoidWithOne NatOrdinal := AddMonoidWithOne.unary @[simp] theorem add_one_eq_succ : ∀ a : NatOrdinal, a + 1 = succ a := nadd_one #align nat_ordinal.add_one_eq_succ NatOrdinal.add_one_eq_succ @[simp] theorem toOrdinal_cast_nat (n : ℕ) : toOrdinal n = n := by induction' n with n hn · rfl · change (toOrdinal n) ♯ 1 = n + 1 rw [hn]; exact nadd_one n #align nat_ordinal.to_ordinal_cast_nat NatOrdinal.toOrdinal_cast_nat end NatOrdinal open NatOrdinal open NaturalOps namespace Ordinal theorem nadd_eq_add (a b : Ordinal) : a ♯ b = toOrdinal (toNatOrdinal a + toNatOrdinal b) := rfl #align ordinal.nadd_eq_add Ordinal.nadd_eq_add @[simp] theorem toNatOrdinal_cast_nat (n : ℕ) : toNatOrdinal n = n := by rw [← toOrdinal_cast_nat n] rfl #align ordinal.to_nat_ordinal_cast_nat Ordinal.toNatOrdinal_cast_nat theorem lt_of_nadd_lt_nadd_left : ∀ {a b c}, a ♯ b < a ♯ c → b < c := @lt_of_add_lt_add_left NatOrdinal _ _ _ #align ordinal.lt_of_nadd_lt_nadd_left Ordinal.lt_of_nadd_lt_nadd_left theorem lt_of_nadd_lt_nadd_right : ∀ {a b c}, b ♯ a < c ♯ a → b < c := @lt_of_add_lt_add_right NatOrdinal _ _ _ #align ordinal.lt_of_nadd_lt_nadd_right Ordinal.lt_of_nadd_lt_nadd_right theorem le_of_nadd_le_nadd_left : ∀ {a b c}, a ♯ b ≤ a ♯ c → b ≤ c := @le_of_add_le_add_left NatOrdinal _ _ _ #align ordinal.le_of_nadd_le_nadd_left Ordinal.le_of_nadd_le_nadd_left theorem le_of_nadd_le_nadd_right : ∀ {a b c}, b ♯ a ≤ c ♯ a → b ≤ c := @le_of_add_le_add_right NatOrdinal _ _ _ #align ordinal.le_of_nadd_le_nadd_right Ordinal.le_of_nadd_le_nadd_right theorem nadd_lt_nadd_iff_left : ∀ (a) {b c}, a ♯ b < a ♯ c ↔ b < c := @add_lt_add_iff_left NatOrdinal _ _ _ _ #align ordinal.nadd_lt_nadd_iff_left Ordinal.nadd_lt_nadd_iff_left theorem nadd_lt_nadd_iff_right : ∀ (a) {b c}, b ♯ a < c ♯ a ↔ b < c := @add_lt_add_iff_right NatOrdinal _ _ _ _ #align ordinal.nadd_lt_nadd_iff_right Ordinal.nadd_lt_nadd_iff_right theorem nadd_le_nadd_iff_left : ∀ (a) {b c}, a ♯ b ≤ a ♯ c ↔ b ≤ c := @add_le_add_iff_left NatOrdinal _ _ _ _ #align ordinal.nadd_le_nadd_iff_left Ordinal.nadd_le_nadd_iff_left theorem nadd_le_nadd_iff_right : ∀ (a) {b c}, b ♯ a ≤ c ♯ a ↔ b ≤ c := @_root_.add_le_add_iff_right NatOrdinal _ _ _ _ #align ordinal.nadd_le_nadd_iff_right Ordinal.nadd_le_nadd_iff_right theorem nadd_le_nadd : ∀ {a b c d}, a ≤ b → c ≤ d → a ♯ c ≤ b ♯ d := @add_le_add NatOrdinal _ _ _ _ #align ordinal.nadd_le_nadd Ordinal.nadd_le_nadd theorem nadd_lt_nadd : ∀ {a b c d}, a < b → c < d → a ♯ c < b ♯ d := @add_lt_add NatOrdinal _ _ _ _ #align ordinal.nadd_lt_nadd Ordinal.nadd_lt_nadd theorem nadd_lt_nadd_of_lt_of_le : ∀ {a b c d}, a < b → c ≤ d → a ♯ c < b ♯ d := @add_lt_add_of_lt_of_le NatOrdinal _ _ _ _ #align ordinal.nadd_lt_nadd_of_lt_of_le Ordinal.nadd_lt_nadd_of_lt_of_le theorem nadd_lt_nadd_of_le_of_lt : ∀ {a b c d}, a ≤ b → c < d → a ♯ c < b ♯ d := @add_lt_add_of_le_of_lt NatOrdinal _ _ _ _ #align ordinal.nadd_lt_nadd_of_le_of_lt Ordinal.nadd_lt_nadd_of_le_of_lt theorem nadd_left_cancel : ∀ {a b c}, a ♯ b = a ♯ c → b = c := @_root_.add_left_cancel NatOrdinal _ _ #align ordinal.nadd_left_cancel Ordinal.nadd_left_cancel theorem nadd_right_cancel : ∀ {a b c}, a ♯ b = c ♯ b → a = c := @_root_.add_right_cancel NatOrdinal _ _ #align ordinal.nadd_right_cancel Ordinal.nadd_right_cancel theorem nadd_left_cancel_iff : ∀ {a b c}, a ♯ b = a ♯ c ↔ b = c := @add_left_cancel_iff NatOrdinal _ _ #align ordinal.nadd_left_cancel_iff Ordinal.nadd_left_cancel_iff theorem nadd_right_cancel_iff : ∀ {a b c}, b ♯ a = c ♯ a ↔ b = c := @add_right_cancel_iff NatOrdinal _ _ #align ordinal.nadd_right_cancel_iff Ordinal.nadd_right_cancel_iff theorem le_nadd_self {a b} : a ≤ b ♯ a := by simpa using nadd_le_nadd_right (Ordinal.zero_le b) a #align ordinal.le_nadd_self Ordinal.le_nadd_self theorem le_nadd_left {a b c} (h : a ≤ c) : a ≤ b ♯ c := le_nadd_self.trans (nadd_le_nadd_left h b) #align ordinal.le_nadd_left Ordinal.le_nadd_left theorem le_self_nadd {a b} : a ≤ a ♯ b := by simpa using nadd_le_nadd_left (Ordinal.zero_le b) a #align ordinal.le_self_nadd Ordinal.le_self_nadd theorem le_nadd_right {a b c} (h : a ≤ b) : a ≤ b ♯ c := le_self_nadd.trans (nadd_le_nadd_right h c) #align ordinal.le_nadd_right Ordinal.le_nadd_right theorem nadd_left_comm : ∀ a b c, a ♯ (b ♯ c) = b ♯ (a ♯ c) := @add_left_comm NatOrdinal _ #align ordinal.nadd_left_comm Ordinal.nadd_left_comm theorem nadd_right_comm : ∀ a b c, a ♯ b ♯ c = a ♯ c ♯ b := @add_right_comm NatOrdinal _ #align ordinal.nadd_right_comm Ordinal.nadd_right_comm /-! ### Natural multiplication -/ variable {a b c d : Ordinal.{u}} theorem nmul_def (a b : Ordinal) : a ⨳ b = sInf {c \| ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} := by rw [nmul] #align ordinal.nmul_def Ordinal.nmul_def /-- The set in the definition of `nmul` is nonempty. -/ theorem nmul_nonempty (a b : Ordinal.{u}) : {c : Ordinal.{u} \| ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'}.Nonempty := ⟨_, fun _ ha _ hb => (lt_blsub₂.{u, u, u} _ ha hb).trans_le le_self_nadd⟩ #align ordinal.nmul_nonempty Ordinal.nmul_nonempty theorem nmul_nadd_lt {a' b' : Ordinal} (ha : a' < a) (hb : b' < b) : a' ⨳ b ♯ a ⨳ b' < a ⨳ b ♯ a' ⨳ b' := by rw [nmul_def a b] exact csInf_mem (nmul_nonempty a b) a' ha b' hb #align ordinal.nmul_nadd_lt Ordinal.nmul_nadd_lt theorem nmul_nadd_le {a' b' : Ordinal} (ha : a' ≤ a) (hb : b' ≤ b) : a' ⨳ b ♯ a ⨳ b' ≤ a ⨳ b ♯ a' ⨳ b' := by rcases lt_or_eq_of_le ha with (ha \| rfl) · rcases lt_or_eq_of_le hb with (hb \| rfl) · exact (nmul_nadd_lt ha hb).le · rw [nadd_comm] · exact le_rfl #align ordinal.nmul_nadd_le Ordinal.nmul_nadd_le theorem lt_nmul_iff : c < a ⨳ b ↔ ∃ a' < a, ∃ b' < b, c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' := by refine ⟨fun h => ?_, ?_⟩ · rw [nmul] at h simpa using not_mem_of_lt_csInf h ⟨0, fun _ _ => bot_le⟩ · rintro ⟨a', ha, b', hb, h⟩ have := h.trans_lt (nmul_nadd_lt ha hb) rwa [nadd_lt_nadd_iff_right] at this #align ordinal.lt_nmul_iff Ordinal.lt_nmul_iff theorem nmul_le_iff : a ⨳ b ≤ c ↔ ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' := by rw [← not_iff_not]; simp [lt_nmul_iff] #align ordinal.nmul_le_iff Ordinal.nmul_le_iff theorem nmul_comm : ∀ a b, a ⨳ b = b ⨳ a \| a, b => by rw [nmul, nmul] congr; ext x; constructor <;> intro H c hc d hd -- Porting note: had to add additional arguments to `nmul_comm` here -- for the termination checker. · rw [nadd_comm, ← nmul_comm d b, ← nmul_comm a c, ← nmul_comm d] exact H _ hd _ hc · rw [nadd_comm, nmul_comm a d, nmul_comm c, nmul_comm c] exact H _ hd _ hc termination_by a b => (a, b) #align ordinal.nmul_comm Ordinal.nmul_comm @[simp] theorem nmul_zero (a) : a ⨳ 0 = 0 := by rw [← Ordinal.le_zero, nmul_le_iff] exact fun _ _ a ha => (Ordinal.not_lt_zero a ha).elim #align ordinal.nmul_zero Ordinal.nmul_zero @[simp] theorem zero_nmul (a) : 0 ⨳ a = 0 := by rw [nmul_comm, nmul_zero] #align ordinal.zero_nmul Ordinal.zero_nmul @[simp] theorem nmul_one (a : Ordinal) : a ⨳ 1 = a := by rw [nmul] simp only [lt_one_iff_zero, forall_eq, nmul_zero, nadd_zero] convert csInf_Ici (α := Ordinal) ext b -- Porting note: added this `simp` line, as the result from `convert` -- is slightly different. simp only [Set.mem_setOf_eq, Set.mem_Ici] refine ⟨fun H => le_of_forall_lt fun c hc => ?_, fun ha c hc => ?_⟩ -- Porting note: had to add arguments to `nmul_one` in the next two lines -- for the termination checker. · simpa only [nmul_one c] using H c hc · simpa only [nmul_one c] using hc.trans_le ha termination_by a #align ordinal.nmul_one Ordinal.nmul_one @[simp]	Mathlib/SetTheory/Ordinal/NaturalOps.lean	592	592	theorem one_nmul (a) : 1 ⨳ a = a := by	rw [nmul_comm, nmul_one]
/- Copyright (c) 2019 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen -/ import Mathlib.LinearAlgebra.FiniteDimensional import Mathlib.LinearAlgebra.GeneralLinearGroup import Mathlib.LinearAlgebra.Matrix.Reindex import Mathlib.Tactic.FieldSimp import Mathlib.LinearAlgebra.Matrix.NonsingularInverse import Mathlib.LinearAlgebra.Matrix.Basis #align_import linear_algebra.determinant from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79" /-! # Determinant of families of vectors This file defines the determinant of an endomorphism, and of a family of vectors with respect to some basis. For the determinant of a matrix, see the file `LinearAlgebra.Matrix.Determinant`. ## Main definitions In the list below, and in all this file, `R` is a commutative ring (semiring is sometimes enough), `M` and its variations are `R`-modules, `ι`, `κ`, `n` and `m` are finite types used for indexing. * `Basis.det`: the determinant of a family of vectors with respect to a basis, as a multilinear map * `LinearMap.det`: the determinant of an endomorphism `f : End R M` as a multiplicative homomorphism (if `M` does not have a finite `R`-basis, the result is `1` instead) * `LinearEquiv.det`: the determinant of an isomorphism `f : M ≃ₗ[R] M` as a multiplicative homomorphism (if `M` does not have a finite `R`-basis, the result is `1` instead) ## Tags basis, det, determinant -/ noncomputable section open Matrix LinearMap Submodule Set Function universe u v w variable {R : Type} [CommRing R] variable {M : Type} [AddCommGroup M] [Module R M] variable {M' : Type} [AddCommGroup M'] [Module R M'] variable {ι : Type} [DecidableEq ι] [Fintype ι] variable (e : Basis ι R M) section Conjugate variable {A : Type} [CommRing A] variable {m n : Type} /-- If `R^m` and `R^n` are linearly equivalent, then `m` and `n` are also equivalent. -/ def equivOfPiLEquivPi {R : Type} [Finite m] [Finite n] [CommRing R] [Nontrivial R] (e : (m → R) ≃ₗ[R] n → R) : m ≃ n := Basis.indexEquiv (Basis.ofEquivFun e.symm) (Pi.basisFun _ _) #align equiv_of_pi_lequiv_pi equivOfPiLEquivPi namespace Matrix variable [Fintype m] [Fintype n] /-- If `M` and `M'` are each other's inverse matrices, they are square matrices up to equivalence of types. -/ def indexEquivOfInv [Nontrivial A] [DecidableEq m] [DecidableEq n] {M : Matrix m n A} {M' : Matrix n m A} (hMM' : M M' = 1) (hM'M : M' * M = 1) : m ≃ n := equivOfPiLEquivPi (toLin'OfInv hMM' hM'M) #align matrix.index_equiv_of_inv Matrix.indexEquivOfInv theorem det_comm [DecidableEq n] (M N : Matrix n n A) : det (M * N) = det (N * M) := by rw [det_mul, det_mul, mul_comm] #align matrix.det_comm Matrix.det_comm /-- If there exists a two-sided inverse `M'` for `M` (indexed differently), then `det (N * M) = det (M * N)`. -/ theorem det_comm' [DecidableEq m] [DecidableEq n] {M : Matrix n m A} {N : Matrix m n A} {M' : Matrix m n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) : det (M * N) = det (N * M) := by nontriviality A -- Although `m` and `n` are different a priori, we will show they have the same cardinality. -- This turns the problem into one for square matrices, which is easy. let e := indexEquivOfInv hMM' hM'M rw [← det_submatrix_equiv_self e, ← submatrix_mul_equiv _ _ _ (Equiv.refl n) _, det_comm, submatrix_mul_equiv, Equiv.coe_refl, submatrix_id_id] #align matrix.det_comm' Matrix.det_comm' /-- If `M'` is a two-sided inverse for `M` (indexed differently), `det (M * N * M') = det N`. See `Matrix.det_conj` and `Matrix.det_conj'` for the case when `M' = M⁻¹` or vice versa. -/ theorem det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A} {M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) : det (M * N * M') = det N := by rw [← det_comm' hM'M hMM', ← Matrix.mul_assoc, hM'M, Matrix.one_mul] #align matrix.det_conj_of_mul_eq_one Matrix.det_conj_of_mul_eq_one end Matrix end Conjugate namespace LinearMap /-! ### Determinant of a linear map -/ variable {A : Type} [CommRing A] [Module A M] variable {κ : Type} [Fintype κ] /-- The determinant of `LinearMap.toMatrix` does not depend on the choice of basis. -/ theorem det_toMatrix_eq_det_toMatrix [DecidableEq κ] (b : Basis ι A M) (c : Basis κ A M) (f : M →ₗ[A] M) : det (LinearMap.toMatrix b b f) = det (LinearMap.toMatrix c c f) := by rw [← linearMap_toMatrix_mul_basis_toMatrix c b c, ← basis_toMatrix_mul_linearMap_toMatrix b c b, Matrix.det_conj_of_mul_eq_one] <;> rw [Basis.toMatrix_mul_toMatrix, Basis.toMatrix_self] #align linear_map.det_to_matrix_eq_det_to_matrix LinearMap.det_toMatrix_eq_det_toMatrix /-- The determinant of an endomorphism given a basis. See `LinearMap.det` for a version that populates the basis non-computably. Although the `Trunc (Basis ι A M)` parameter makes it slightly more convenient to switch bases, there is no good way to generalize over universe parameters, so we can't fully state in `detAux`'s type that it does not depend on the choice of basis. Instead you can use the `detAux_def''` lemma, or avoid mentioning a basis at all using `LinearMap.det`. -/ irreducible_def detAux : Trunc (Basis ι A M) → (M →ₗ[A] M) →* A := Trunc.lift (fun b : Basis ι A M => detMonoidHom.comp (toMatrixAlgEquiv b : (M →ₗ[A] M) →* Matrix ι ι A)) fun b c => MonoidHom.ext <\| det_toMatrix_eq_det_toMatrix b c #align linear_map.det_aux LinearMap.detAux /-- Unfold lemma for `detAux`. See also `detAux_def''` which allows you to vary the basis. -/ theorem detAux_def' (b : Basis ι A M) (f : M →ₗ[A] M) : LinearMap.detAux (Trunc.mk b) f = Matrix.det (LinearMap.toMatrix b b f) := by rw [detAux] rfl #align linear_map.det_aux_def LinearMap.detAux_def' theorem detAux_def'' {ι' : Type} [Fintype ι'] [DecidableEq ι'] (tb : Trunc <\| Basis ι A M) (b' : Basis ι' A M) (f : M →ₗ[A] M) : LinearMap.detAux tb f = Matrix.det (LinearMap.toMatrix b' b' f) := by induction tb using Trunc.induction_on with \| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b'] #align linear_map.det_aux_def' LinearMap.detAux_def'' @[simp] theorem detAux_id (b : Trunc <\| Basis ι A M) : LinearMap.detAux b LinearMap.id = 1 := (LinearMap.detAux b).map_one #align linear_map.det_aux_id LinearMap.detAux_id @[simp] theorem detAux_comp (b : Trunc <\| Basis ι A M) (f g : M →ₗ[A] M) : LinearMap.detAux b (f.comp g) = LinearMap.detAux b f LinearMap.detAux b g := (LinearMap.detAux b).map_mul f g #align linear_map.det_aux_comp LinearMap.detAux_comp section open scoped Classical in -- Discourage the elaborator from unfolding `det` and producing a huge term by marking it -- as irreducible. /-- The determinant of an endomorphism independent of basis. If there is no finite basis on `M`, the result is `1` instead. -/ protected irreducible_def det : (M →ₗ[A] M) →* A := if H : ∃ s : Finset M, Nonempty (Basis s A M) then LinearMap.detAux (Trunc.mk H.choose_spec.some) else 1 #align linear_map.det LinearMap.det open scoped Classical in theorem coe_det [DecidableEq M] : ⇑(LinearMap.det : (M →ₗ[A] M) →* A) = if H : ∃ s : Finset M, Nonempty (Basis s A M) then LinearMap.detAux (Trunc.mk H.choose_spec.some) else 1 := by ext rw [LinearMap.det_def] split_ifs · congr -- use the correct `DecidableEq` instance rfl #align linear_map.coe_det LinearMap.coe_det end -- Auxiliary lemma, the `simp` normal form goes in the other direction -- (using `LinearMap.det_toMatrix`) theorem det_eq_det_toMatrix_of_finset [DecidableEq M] {s : Finset M} (b : Basis s A M) (f : M →ₗ[A] M) : LinearMap.det f = Matrix.det (LinearMap.toMatrix b b f) := by have : ∃ s : Finset M, Nonempty (Basis s A M) := ⟨s, ⟨b⟩⟩ rw [LinearMap.coe_det, dif_pos, detAux_def'' _ b] <;> assumption #align linear_map.det_eq_det_to_matrix_of_finset LinearMap.det_eq_det_toMatrix_of_finset @[simp] theorem det_toMatrix (b : Basis ι A M) (f : M →ₗ[A] M) : Matrix.det (toMatrix b b f) = LinearMap.det f := by haveI := Classical.decEq M rw [det_eq_det_toMatrix_of_finset b.reindexFinsetRange] -- Porting note: moved out of `rw` due to error -- typeclass instance problem is stuck, it is often due to metavariables `DecidableEq ?m.628881` apply det_toMatrix_eq_det_toMatrix b #align linear_map.det_to_matrix LinearMap.det_toMatrix @[simp] theorem det_toMatrix' {ι : Type} [Fintype ι] [DecidableEq ι] (f : (ι → A) →ₗ[A] ι → A) : Matrix.det (LinearMap.toMatrix' f) = LinearMap.det f := by simp [← toMatrix_eq_toMatrix'] #align linear_map.det_to_matrix' LinearMap.det_toMatrix' @[simp] theorem det_toLin (b : Basis ι R M) (f : Matrix ι ι R) : LinearMap.det (Matrix.toLin b b f) = f.det := by rw [← LinearMap.det_toMatrix b, LinearMap.toMatrix_toLin] #align linear_map.det_to_lin LinearMap.det_toLin @[simp] theorem det_toLin' (f : Matrix ι ι R) : LinearMap.det (Matrix.toLin' f) = Matrix.det f := by simp only [← toLin_eq_toLin', det_toLin] #align linear_map.det_to_lin' LinearMap.det_toLin' /-- To show `P (LinearMap.det f)` it suffices to consider `P (Matrix.det (toMatrix _ _ f))` and `P 1`. -/ -- @[elab_as_elim] -- Porting note: This attr can't be applied. theorem det_cases [DecidableEq M] {P : A → Prop} (f : M →ₗ[A] M) (hb : ∀ (s : Finset M) (b : Basis s A M), P (Matrix.det (toMatrix b b f))) (h1 : P 1) : P (LinearMap.det f) := by rw [LinearMap.det_def] split_ifs with h · convert hb _ h.choose_spec.some -- Porting note: was `apply det_aux_def'` convert detAux_def'' (Trunc.mk h.choose_spec.some) h.choose_spec.some f · exact h1 #align linear_map.det_cases LinearMap.det_cases @[simp] theorem det_comp (f g : M →ₗ[A] M) : LinearMap.det (f.comp g) = LinearMap.det f LinearMap.det g := LinearMap.det.map_mul f g #align linear_map.det_comp LinearMap.det_comp @[simp] theorem det_id : LinearMap.det (LinearMap.id : M →ₗ[A] M) = 1 := LinearMap.det.map_one #align linear_map.det_id LinearMap.det_id /-- Multiplying a map by a scalar `c` multiplies its determinant by `c ^ dim M`. -/ @[simp] theorem det_smul {𝕜 : Type} [Field 𝕜] {M : Type} [AddCommGroup M] [Module 𝕜 M] (c : 𝕜) (f : M →ₗ[𝕜] M) : LinearMap.det (c • f) = c ^ FiniteDimensional.finrank 𝕜 M * LinearMap.det f := by by_cases H : ∃ s : Finset M, Nonempty (Basis s 𝕜 M) · have : FiniteDimensional 𝕜 M := by rcases H with ⟨s, ⟨hs⟩⟩ exact FiniteDimensional.of_fintype_basis hs simp only [← det_toMatrix (FiniteDimensional.finBasis 𝕜 M), LinearEquiv.map_smul, Fintype.card_fin, Matrix.det_smul] · classical have : FiniteDimensional.finrank 𝕜 M = 0 := finrank_eq_zero_of_not_exists_basis H simp [coe_det, H, this] #align linear_map.det_smul LinearMap.det_smul theorem det_zero' {ι : Type} [Finite ι] [Nonempty ι] (b : Basis ι A M) : LinearMap.det (0 : M →ₗ[A] M) = 0 := by haveI := Classical.decEq ι cases nonempty_fintype ι rwa [← det_toMatrix b, LinearEquiv.map_zero, det_zero] #align linear_map.det_zero' LinearMap.det_zero' /-- In a finite-dimensional vector space, the zero map has determinant `1` in dimension `0`, and `0` otherwise. We give a formula that also works in infinite dimension, where we define the determinant to be `1`. -/ @[simp] theorem det_zero {𝕜 : Type} [Field 𝕜] {M : Type} [AddCommGroup M] [Module 𝕜 M] : LinearMap.det (0 : M →ₗ[𝕜] M) = (0 : 𝕜) ^ FiniteDimensional.finrank 𝕜 M := by simp only [← zero_smul 𝕜 (1 : M →ₗ[𝕜] M), det_smul, mul_one, MonoidHom.map_one] #align linear_map.det_zero LinearMap.det_zero theorem det_eq_one_of_subsingleton [Subsingleton M] (f : M →ₗ[R] M) : LinearMap.det (f : M →ₗ[R] M) = 1 := by have b : Basis (Fin 0) R M := Basis.empty M rw [← f.det_toMatrix b] exact Matrix.det_isEmpty #align linear_map.det_eq_one_of_subsingleton LinearMap.det_eq_one_of_subsingleton theorem det_eq_one_of_finrank_eq_zero {𝕜 : Type} [Field 𝕜] {M : Type*} [AddCommGroup M] [Module 𝕜 M] (h : FiniteDimensional.finrank 𝕜 M = 0) (f : M →ₗ[𝕜] M) : LinearMap.det (f : M →ₗ[𝕜] M) = 1 := by classical refine @LinearMap.det_cases M _ 𝕜 _ _ _ (fun t => t = 1) f ?_ rfl intro s b have : IsEmpty s := by rw [← Fintype.card_eq_zero_iff] exact (FiniteDimensional.finrank_eq_card_basis b).symm.trans h exact Matrix.det_isEmpty #align linear_map.det_eq_one_of_finrank_eq_zero LinearMap.det_eq_one_of_finrank_eq_zero /-- Conjugating a linear map by a linear equiv does not change its determinant. -/ @[simp]	Mathlib/LinearAlgebra/Determinant.lean	307	322	theorem det_conj {N : Type*} [AddCommGroup N] [Module A N] (f : M →ₗ[A] M) (e : M ≃ₗ[A] N) : LinearMap.det ((e : M →ₗ[A] N) ∘ₗ f ∘ₗ (e.symm : N →ₗ[A] M)) = LinearMap.det f := by	classical by_cases H : ∃ s : Finset M, Nonempty (Basis s A M) · rcases H with ⟨s, ⟨b⟩⟩ rw [← det_toMatrix b f, ← det_toMatrix (b.map e), toMatrix_comp (b.map e) b (b.map e), toMatrix_comp (b.map e) b b, ← Matrix.mul_assoc, Matrix.det_conj_of_mul_eq_one] · rw [← toMatrix_comp, LinearEquiv.comp_coe, e.symm_trans_self, LinearEquiv.refl_toLinearMap, toMatrix_id] · rw [← toMatrix_comp, LinearEquiv.comp_coe, e.self_trans_symm, LinearEquiv.refl_toLinearMap, toMatrix_id] · have H' : ¬∃ t : Finset N, Nonempty (Basis t A N) := by contrapose! H rcases H with ⟨s, ⟨b⟩⟩ exact ⟨_, ⟨(b.map e.symm).reindexFinsetRange⟩⟩ simp only [coe_det, H, H', MonoidHom.one_apply, dif_neg, not_false_eq_true]
/- Copyright (c) 2021 Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies [`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/data/finset/sym?range=98e83c3d541c77cdb7da20d79611a780ff8e7d90..02ba8949f486ebecf93fe7460f1ed0564b5e442c) -/ import Mathlib.Data.Finset.Lattice import Mathlib.Data.Fintype.Vector import Mathlib.Data.Multiset.Sym #align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c" /-! # Symmetric powers of a finset This file defines the symmetric powers of a finset as `Finset (Sym α n)` and `Finset (Sym2 α)`. ## Main declarations * `Finset.sym`: The symmetric power of a finset. `s.sym n` is all the multisets of cardinality `n` whose elements are in `s`. * `Finset.sym2`: The symmetric square of a finset. `s.sym2` is all the pairs whose elements are in `s`. * A `Fintype (Sym2 α)` instance that does not require `DecidableEq α`. ## TODO `Finset.sym` forms a Galois connection between `Finset α` and `Finset (Sym α n)`. Similar for `Finset.sym2`. -/ namespace Finset variable {α : Type} /-- `s.sym2` is the finset of all unordered pairs of elements from `s`. It is the image of `s ×ˢ s` under the quotient `α × α → Sym2 α`. -/ @[simps] protected def sym2 (s : Finset α) : Finset (Sym2 α) := ⟨s.1.sym2, s.2.sym2⟩ #align finset.sym2 Finset.sym2 section variable {s t : Finset α} {a b : α} theorem mk_mem_sym2_iff : s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s := by rw [mem_mk, sym2_val, Multiset.mk_mem_sym2_iff, mem_mk, mem_mk] #align finset.mk_mem_sym2_iff Finset.mk_mem_sym2_iff @[simp] theorem mem_sym2_iff {m : Sym2 α} : m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s := by rw [mem_mk, sym2_val, Multiset.mem_sym2_iff] simp only [mem_val] #align finset.mem_sym2_iff Finset.mem_sym2_iff instance _root_.Sym2.instFintype [Fintype α] : Fintype (Sym2 α) where elems := Finset.univ.sym2 complete := fun x ↦ by rw [mem_sym2_iff]; exact (fun a _ ↦ mem_univ a) -- Note(kmill): Using a default argument to make this simp lemma more general. @[simp] theorem sym2_univ [Fintype α] (inst : Fintype (Sym2 α) := Sym2.instFintype) : (univ : Finset α).sym2 = univ := by ext simp only [mem_sym2_iff, mem_univ, implies_true] #align finset.sym2_univ Finset.sym2_univ @[simp, mono] theorem sym2_mono (h : s ⊆ t) : s.sym2 ⊆ t.sym2 := by rw [← val_le_iff, sym2_val, sym2_val] apply Multiset.sym2_mono rwa [val_le_iff] #align finset.sym2_mono Finset.sym2_mono theorem monotone_sym2 : Monotone (Finset.sym2 : Finset α → _) := fun _ _ => sym2_mono theorem injective_sym2 : Function.Injective (Finset.sym2 : Finset α → _) := by intro s t h ext x simpa using congr(s(x, x) ∈ $h) theorem strictMono_sym2 : StrictMono (Finset.sym2 : Finset α → _) := monotone_sym2.strictMono_of_injective injective_sym2 theorem sym2_toFinset [DecidableEq α] (m : Multiset α) : m.toFinset.sym2 = m.sym2.toFinset := by ext z refine z.ind fun x y ↦ ?_ simp only [mk_mem_sym2_iff, Multiset.mem_toFinset, Multiset.mk_mem_sym2_iff] @[simp] theorem sym2_empty : (∅ : Finset α).sym2 = ∅ := rfl #align finset.sym2_empty Finset.sym2_empty @[simp] theorem sym2_eq_empty : s.sym2 = ∅ ↔ s = ∅ := by rw [← val_eq_zero, sym2_val, Multiset.sym2_eq_zero_iff, val_eq_zero] #align finset.sym2_eq_empty Finset.sym2_eq_empty @[simp, aesop safe apply (rule_sets := [finsetNonempty])] theorem sym2_nonempty : s.sym2.Nonempty ↔ s.Nonempty := by rw [← not_iff_not] simp_rw [not_nonempty_iff_eq_empty, sym2_eq_empty] #align finset.sym2_nonempty Finset.sym2_nonempty protected alias ⟨_, Nonempty.sym2⟩ := sym2_nonempty #align finset.nonempty.sym2 Finset.Nonempty.sym2 @[simp] theorem sym2_singleton (a : α) : ({a} : Finset α).sym2 = {Sym2.diag a} := rfl #align finset.sym2_singleton Finset.sym2_singleton /-- Finset stars and bars* for the case `n = 2`. -/ theorem card_sym2 (s : Finset α) : s.sym2.card = Nat.choose (s.card + 1) 2 := by rw [card_def, sym2_val, Multiset.card_sym2, ← card_def] #align finset.card_sym2 Finset.card_sym2 end variable [DecidableEq α] {s t : Finset α} {a b : α} theorem sym2_eq_image : s.sym2 = (s ×ˢ s).image Sym2.mk := by ext z refine z.ind fun x y ↦ ?_ rw [mk_mem_sym2_iff, mem_image] constructor · intro h use (x, y) simp only [mem_product, h, and_self, true_and] · rintro ⟨⟨a, b⟩, h⟩ simp only [mem_product, Sym2.eq_iff] at h obtain ⟨h, (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩)⟩ := h <;> simp [h] theorem isDiag_mk_of_mem_diag {a : α × α} (h : a ∈ s.diag) : (Sym2.mk a).IsDiag := (Sym2.isDiag_iff_proj_eq _).2 (mem_diag.1 h).2 #align finset.is_diag_mk_of_mem_diag Finset.isDiag_mk_of_mem_diag	Mathlib/Data/Finset/Sym.lean	139	142	theorem not_isDiag_mk_of_mem_offDiag {a : α × α} (h : a ∈ s.offDiag) : ¬ (Sym2.mk a).IsDiag := by	rw [Sym2.isDiag_iff_proj_eq] exact (mem_offDiag.1 h).2.2
/- Copyright (c) 2021 Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kyle Miller -/ import Mathlib.Combinatorics.SimpleGraph.Subgraph import Mathlib.Data.List.Rotate #align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4" /-! # Graph connectivity In a simple graph, * A walk is a finite sequence of adjacent vertices, and can be thought of equally well as a sequence of directed edges. * A trail is a walk whose edges each appear no more than once. * A path is a trail whose vertices appear no more than once. * A cycle is a nonempty trail whose first and last vertices are the same and whose vertices except for the first appear no more than once. Warning: graph theorists mean something different by "path" than do homotopy theorists. A "walk" in graph theory is a "path" in homotopy theory. Another warning: some graph theorists use "path" and "simple path" for "walk" and "path." Some definitions and theorems have inspiration from multigraph counterparts in [Chou1994]. ## Main definitions * `SimpleGraph.Walk` (with accompanying pattern definitions `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'`) * `SimpleGraph.Walk.IsTrail`, `SimpleGraph.Walk.IsPath`, and `SimpleGraph.Walk.IsCycle`. * `SimpleGraph.Path` * `SimpleGraph.Walk.map` and `SimpleGraph.Path.map` for the induced map on walks, given an (injective) graph homomorphism. * `SimpleGraph.Reachable` for the relation of whether there exists a walk between a given pair of vertices * `SimpleGraph.Preconnected` and `SimpleGraph.Connected` are predicates on simple graphs for whether every vertex can be reached from every other, and in the latter case, whether the vertex type is nonempty. * `SimpleGraph.ConnectedComponent` is the type of connected components of a given graph. * `SimpleGraph.IsBridge` for whether an edge is a bridge edge ## Main statements * `SimpleGraph.isBridge_iff_mem_and_forall_cycle_not_mem` characterizes bridge edges in terms of there being no cycle containing them. ## Tags walks, trails, paths, circuits, cycles, bridge edges -/ open Function universe u v w namespace SimpleGraph variable {V : Type u} {V' : Type v} {V'' : Type w} variable (G : SimpleGraph V) (G' : SimpleGraph V') (G'' : SimpleGraph V'') /-- A walk is a sequence of adjacent vertices. For vertices `u v : V`, the type `walk u v` consists of all walks starting at `u` and ending at `v`. We say that a walk visits the vertices it contains. The set of vertices a walk visits is `SimpleGraph.Walk.support`. See `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'` for patterns that can be useful in definitions since they make the vertices explicit. -/ inductive Walk : V → V → Type u \| nil {u : V} : Walk u u \| cons {u v w : V} (h : G.Adj u v) (p : Walk v w) : Walk u w deriving DecidableEq #align simple_graph.walk SimpleGraph.Walk attribute [refl] Walk.nil @[simps] instance Walk.instInhabited (v : V) : Inhabited (G.Walk v v) := ⟨Walk.nil⟩ #align simple_graph.walk.inhabited SimpleGraph.Walk.instInhabited /-- The one-edge walk associated to a pair of adjacent vertices. -/ @[match_pattern, reducible] def Adj.toWalk {G : SimpleGraph V} {u v : V} (h : G.Adj u v) : G.Walk u v := Walk.cons h Walk.nil #align simple_graph.adj.to_walk SimpleGraph.Adj.toWalk namespace Walk variable {G} /-- Pattern to get `Walk.nil` with the vertex as an explicit argument. -/ @[match_pattern] abbrev nil' (u : V) : G.Walk u u := Walk.nil #align simple_graph.walk.nil' SimpleGraph.Walk.nil' /-- Pattern to get `Walk.cons` with the vertices as explicit arguments. -/ @[match_pattern] abbrev cons' (u v w : V) (h : G.Adj u v) (p : G.Walk v w) : G.Walk u w := Walk.cons h p #align simple_graph.walk.cons' SimpleGraph.Walk.cons' /-- Change the endpoints of a walk using equalities. This is helpful for relaxing definitional equality constraints and to be able to state otherwise difficult-to-state lemmas. While this is a simple wrapper around `Eq.rec`, it gives a canonical way to write it. The simp-normal form is for the `copy` to be pushed outward. That way calculations can occur within the "copy context." -/ protected def copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : G.Walk u' v' := hu ▸ hv ▸ p #align simple_graph.walk.copy SimpleGraph.Walk.copy @[simp] theorem copy_rfl_rfl {u v} (p : G.Walk u v) : p.copy rfl rfl = p := rfl #align simple_graph.walk.copy_rfl_rfl SimpleGraph.Walk.copy_rfl_rfl @[simp] theorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v) (hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') : (p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by subst_vars rfl #align simple_graph.walk.copy_copy SimpleGraph.Walk.copy_copy @[simp] theorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by subst_vars rfl #align simple_graph.walk.copy_nil SimpleGraph.Walk.copy_nil theorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') : (Walk.cons h p).copy hu hw = Walk.cons (hu ▸ h) (p.copy rfl hw) := by subst_vars rfl #align simple_graph.walk.copy_cons SimpleGraph.Walk.copy_cons @[simp] theorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) : Walk.cons h (p.copy hv hw) = (Walk.cons (hv ▸ h) p).copy rfl hw := by subst_vars rfl #align simple_graph.walk.cons_copy SimpleGraph.Walk.cons_copy theorem exists_eq_cons_of_ne {u v : V} (hne : u ≠ v) : ∀ (p : G.Walk u v), ∃ (w : V) (h : G.Adj u w) (p' : G.Walk w v), p = cons h p' \| nil => (hne rfl).elim \| cons h p' => ⟨_, h, p', rfl⟩ #align simple_graph.walk.exists_eq_cons_of_ne SimpleGraph.Walk.exists_eq_cons_of_ne /-- The length of a walk is the number of edges/darts along it. -/ def length {u v : V} : G.Walk u v → ℕ \| nil => 0 \| cons _ q => q.length.succ #align simple_graph.walk.length SimpleGraph.Walk.length /-- The concatenation of two compatible walks. -/ @[trans] def append {u v w : V} : G.Walk u v → G.Walk v w → G.Walk u w \| nil, q => q \| cons h p, q => cons h (p.append q) #align simple_graph.walk.append SimpleGraph.Walk.append /-- The reversed version of `SimpleGraph.Walk.cons`, concatenating an edge to the end of a walk. -/ def concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : G.Walk u w := p.append (cons h nil) #align simple_graph.walk.concat SimpleGraph.Walk.concat theorem concat_eq_append {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : p.concat h = p.append (cons h nil) := rfl #align simple_graph.walk.concat_eq_append SimpleGraph.Walk.concat_eq_append /-- The concatenation of the reverse of the first walk with the second walk. -/ protected def reverseAux {u v w : V} : G.Walk u v → G.Walk u w → G.Walk v w \| nil, q => q \| cons h p, q => Walk.reverseAux p (cons (G.symm h) q) #align simple_graph.walk.reverse_aux SimpleGraph.Walk.reverseAux /-- The walk in reverse. -/ @[symm] def reverse {u v : V} (w : G.Walk u v) : G.Walk v u := w.reverseAux nil #align simple_graph.walk.reverse SimpleGraph.Walk.reverse /-- Get the `n`th vertex from a walk, where `n` is generally expected to be between `0` and `p.length`, inclusive. If `n` is greater than or equal to `p.length`, the result is the path's endpoint. -/ def getVert {u v : V} : G.Walk u v → ℕ → V \| nil, _ => u \| cons _ _, 0 => u \| cons _ q, n + 1 => q.getVert n #align simple_graph.walk.get_vert SimpleGraph.Walk.getVert @[simp] theorem getVert_zero {u v} (w : G.Walk u v) : w.getVert 0 = u := by cases w <;> rfl #align simple_graph.walk.get_vert_zero SimpleGraph.Walk.getVert_zero theorem getVert_of_length_le {u v} (w : G.Walk u v) {i : ℕ} (hi : w.length ≤ i) : w.getVert i = v := by induction w generalizing i with \| nil => rfl \| cons _ _ ih => cases i · cases hi · exact ih (Nat.succ_le_succ_iff.1 hi) #align simple_graph.walk.get_vert_of_length_le SimpleGraph.Walk.getVert_of_length_le @[simp] theorem getVert_length {u v} (w : G.Walk u v) : w.getVert w.length = v := w.getVert_of_length_le rfl.le #align simple_graph.walk.get_vert_length SimpleGraph.Walk.getVert_length theorem adj_getVert_succ {u v} (w : G.Walk u v) {i : ℕ} (hi : i < w.length) : G.Adj (w.getVert i) (w.getVert (i + 1)) := by induction w generalizing i with \| nil => cases hi \| cons hxy _ ih => cases i · simp [getVert, hxy] · exact ih (Nat.succ_lt_succ_iff.1 hi) #align simple_graph.walk.adj_get_vert_succ SimpleGraph.Walk.adj_getVert_succ @[simp] theorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w x) : (cons h p).append q = cons h (p.append q) := rfl #align simple_graph.walk.cons_append SimpleGraph.Walk.cons_append @[simp] theorem cons_nil_append {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h nil).append p = cons h p := rfl #align simple_graph.walk.cons_nil_append SimpleGraph.Walk.cons_nil_append @[simp] theorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p := by induction p with \| nil => rfl \| cons _ _ ih => rw [cons_append, ih] #align simple_graph.walk.append_nil SimpleGraph.Walk.append_nil @[simp] theorem nil_append {u v : V} (p : G.Walk u v) : nil.append p = p := rfl #align simple_graph.walk.nil_append SimpleGraph.Walk.nil_append theorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) : p.append (q.append r) = (p.append q).append r := by induction p with \| nil => rfl \| cons h p' ih => dsimp only [append] rw [ih] #align simple_graph.walk.append_assoc SimpleGraph.Walk.append_assoc @[simp] theorem append_copy_copy {u v w u' v' w'} (p : G.Walk u v) (q : G.Walk v w) (hu : u = u') (hv : v = v') (hw : w = w') : (p.copy hu hv).append (q.copy hv hw) = (p.append q).copy hu hw := by subst_vars rfl #align simple_graph.walk.append_copy_copy SimpleGraph.Walk.append_copy_copy theorem concat_nil {u v : V} (h : G.Adj u v) : nil.concat h = cons h nil := rfl #align simple_graph.walk.concat_nil SimpleGraph.Walk.concat_nil @[simp] theorem concat_cons {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (h' : G.Adj w x) : (cons h p).concat h' = cons h (p.concat h') := rfl #align simple_graph.walk.concat_cons SimpleGraph.Walk.concat_cons theorem append_concat {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (h : G.Adj w x) : p.append (q.concat h) = (p.append q).concat h := append_assoc _ _ _ #align simple_graph.walk.append_concat SimpleGraph.Walk.append_concat theorem concat_append {u v w x : V} (p : G.Walk u v) (h : G.Adj v w) (q : G.Walk w x) : (p.concat h).append q = p.append (cons h q) := by rw [concat_eq_append, ← append_assoc, cons_nil_append] #align simple_graph.walk.concat_append SimpleGraph.Walk.concat_append /-- A non-trivial `cons` walk is representable as a `concat` walk. -/ theorem exists_cons_eq_concat {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : ∃ (x : V) (q : G.Walk u x) (h' : G.Adj x w), cons h p = q.concat h' := by induction p generalizing u with \| nil => exact ⟨_, nil, h, rfl⟩ \| cons h' p ih => obtain ⟨y, q, h'', hc⟩ := ih h' refine ⟨y, cons h q, h'', ?_⟩ rw [concat_cons, hc] #align simple_graph.walk.exists_cons_eq_concat SimpleGraph.Walk.exists_cons_eq_concat /-- A non-trivial `concat` walk is representable as a `cons` walk. -/ theorem exists_concat_eq_cons {u v w : V} : ∀ (p : G.Walk u v) (h : G.Adj v w), ∃ (x : V) (h' : G.Adj u x) (q : G.Walk x w), p.concat h = cons h' q \| nil, h => ⟨_, h, nil, rfl⟩ \| cons h' p, h => ⟨_, h', Walk.concat p h, concat_cons _ _ _⟩ #align simple_graph.walk.exists_concat_eq_cons SimpleGraph.Walk.exists_concat_eq_cons @[simp] theorem reverse_nil {u : V} : (nil : G.Walk u u).reverse = nil := rfl #align simple_graph.walk.reverse_nil SimpleGraph.Walk.reverse_nil theorem reverse_singleton {u v : V} (h : G.Adj u v) : (cons h nil).reverse = cons (G.symm h) nil := rfl #align simple_graph.walk.reverse_singleton SimpleGraph.Walk.reverse_singleton @[simp] theorem cons_reverseAux {u v w x : V} (p : G.Walk u v) (q : G.Walk w x) (h : G.Adj w u) : (cons h p).reverseAux q = p.reverseAux (cons (G.symm h) q) := rfl #align simple_graph.walk.cons_reverse_aux SimpleGraph.Walk.cons_reverseAux @[simp] protected theorem append_reverseAux {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk u x) : (p.append q).reverseAux r = q.reverseAux (p.reverseAux r) := by induction p with \| nil => rfl \| cons h _ ih => exact ih q (cons (G.symm h) r) #align simple_graph.walk.append_reverse_aux SimpleGraph.Walk.append_reverseAux @[simp] protected theorem reverseAux_append {u v w x : V} (p : G.Walk u v) (q : G.Walk u w) (r : G.Walk w x) : (p.reverseAux q).append r = p.reverseAux (q.append r) := by induction p with \| nil => rfl \| cons h _ ih => simp [ih (cons (G.symm h) q)] #align simple_graph.walk.reverse_aux_append SimpleGraph.Walk.reverseAux_append protected theorem reverseAux_eq_reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk u w) : p.reverseAux q = p.reverse.append q := by simp [reverse] #align simple_graph.walk.reverse_aux_eq_reverse_append SimpleGraph.Walk.reverseAux_eq_reverse_append @[simp] theorem reverse_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).reverse = p.reverse.append (cons (G.symm h) nil) := by simp [reverse] #align simple_graph.walk.reverse_cons SimpleGraph.Walk.reverse_cons @[simp] theorem reverse_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).reverse = p.reverse.copy hv hu := by subst_vars rfl #align simple_graph.walk.reverse_copy SimpleGraph.Walk.reverse_copy @[simp] theorem reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) : (p.append q).reverse = q.reverse.append p.reverse := by simp [reverse] #align simple_graph.walk.reverse_append SimpleGraph.Walk.reverse_append @[simp] theorem reverse_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).reverse = cons (G.symm h) p.reverse := by simp [concat_eq_append] #align simple_graph.walk.reverse_concat SimpleGraph.Walk.reverse_concat @[simp] theorem reverse_reverse {u v : V} (p : G.Walk u v) : p.reverse.reverse = p := by induction p with \| nil => rfl \| cons _ _ ih => simp [ih] #align simple_graph.walk.reverse_reverse SimpleGraph.Walk.reverse_reverse @[simp] theorem length_nil {u : V} : (nil : G.Walk u u).length = 0 := rfl #align simple_graph.walk.length_nil SimpleGraph.Walk.length_nil @[simp] theorem length_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).length = p.length + 1 := rfl #align simple_graph.walk.length_cons SimpleGraph.Walk.length_cons @[simp] theorem length_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : (p.copy hu hv).length = p.length := by subst_vars rfl #align simple_graph.walk.length_copy SimpleGraph.Walk.length_copy @[simp] theorem length_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) : (p.append q).length = p.length + q.length := by induction p with \| nil => simp \| cons _ _ ih => simp [ih, add_comm, add_left_comm, add_assoc] #align simple_graph.walk.length_append SimpleGraph.Walk.length_append @[simp] theorem length_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).length = p.length + 1 := length_append _ _ #align simple_graph.walk.length_concat SimpleGraph.Walk.length_concat @[simp] protected theorem length_reverseAux {u v w : V} (p : G.Walk u v) (q : G.Walk u w) : (p.reverseAux q).length = p.length + q.length := by induction p with \| nil => simp! \| cons _ _ ih => simp [ih, Nat.succ_add, Nat.add_assoc] #align simple_graph.walk.length_reverse_aux SimpleGraph.Walk.length_reverseAux @[simp] theorem length_reverse {u v : V} (p : G.Walk u v) : p.reverse.length = p.length := by simp [reverse] #align simple_graph.walk.length_reverse SimpleGraph.Walk.length_reverse theorem eq_of_length_eq_zero {u v : V} : ∀ {p : G.Walk u v}, p.length = 0 → u = v \| nil, _ => rfl #align simple_graph.walk.eq_of_length_eq_zero SimpleGraph.Walk.eq_of_length_eq_zero theorem adj_of_length_eq_one {u v : V} : ∀ {p : G.Walk u v}, p.length = 1 → G.Adj u v \| cons h nil, _ => h @[simp] theorem exists_length_eq_zero_iff {u v : V} : (∃ p : G.Walk u v, p.length = 0) ↔ u = v := by constructor · rintro ⟨p, hp⟩ exact eq_of_length_eq_zero hp · rintro rfl exact ⟨nil, rfl⟩ #align simple_graph.walk.exists_length_eq_zero_iff SimpleGraph.Walk.exists_length_eq_zero_iff @[simp] theorem length_eq_zero_iff {u : V} {p : G.Walk u u} : p.length = 0 ↔ p = nil := by cases p <;> simp #align simple_graph.walk.length_eq_zero_iff SimpleGraph.Walk.length_eq_zero_iff theorem getVert_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) (i : ℕ) : (p.append q).getVert i = if i < p.length then p.getVert i else q.getVert (i - p.length) := by induction p generalizing i with \| nil => simp \| cons h p ih => cases i <;> simp [getVert, ih, Nat.succ_lt_succ_iff] theorem getVert_reverse {u v : V} (p : G.Walk u v) (i : ℕ) : p.reverse.getVert i = p.getVert (p.length - i) := by induction p with \| nil => rfl \| cons h p ih => simp only [reverse_cons, getVert_append, length_reverse, ih, length_cons] split_ifs next hi => rw [Nat.succ_sub hi.le] simp [getVert] next hi => obtain rfl \| hi' := Nat.eq_or_lt_of_not_lt hi · simp [getVert] · rw [Nat.eq_add_of_sub_eq (Nat.sub_pos_of_lt hi') rfl, Nat.sub_eq_zero_of_le hi'] simp [getVert] section ConcatRec variable {motive : ∀ u v : V, G.Walk u v → Sort*} (Hnil : ∀ {u : V}, motive u u nil) (Hconcat : ∀ {u v w : V} (p : G.Walk u v) (h : G.Adj v w), motive u v p → motive u w (p.concat h)) /-- Auxiliary definition for `SimpleGraph.Walk.concatRec` -/ def concatRecAux {u v : V} : (p : G.Walk u v) → motive v u p.reverse \| nil => Hnil \| cons h p => reverse_cons h p ▸ Hconcat p.reverse h.symm (concatRecAux p) #align simple_graph.walk.concat_rec_aux SimpleGraph.Walk.concatRecAux /-- Recursor on walks by inducting on `SimpleGraph.Walk.concat`. This is inducting from the opposite end of the walk compared to `SimpleGraph.Walk.rec`, which inducts on `SimpleGraph.Walk.cons`. -/ @[elab_as_elim] def concatRec {u v : V} (p : G.Walk u v) : motive u v p := reverse_reverse p ▸ concatRecAux @Hnil @Hconcat p.reverse #align simple_graph.walk.concat_rec SimpleGraph.Walk.concatRec @[simp] theorem concatRec_nil (u : V) : @concatRec _ _ motive @Hnil @Hconcat _ _ (nil : G.Walk u u) = Hnil := rfl #align simple_graph.walk.concat_rec_nil SimpleGraph.Walk.concatRec_nil @[simp] theorem concatRec_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : @concatRec _ _ motive @Hnil @Hconcat _ _ (p.concat h) = Hconcat p h (concatRec @Hnil @Hconcat p) := by simp only [concatRec] apply eq_of_heq apply rec_heq_of_heq trans concatRecAux @Hnil @Hconcat (cons h.symm p.reverse) · congr simp · rw [concatRecAux, rec_heq_iff_heq] congr <;> simp [heq_rec_iff_heq] #align simple_graph.walk.concat_rec_concat SimpleGraph.Walk.concatRec_concat end ConcatRec theorem concat_ne_nil {u v : V} (p : G.Walk u v) (h : G.Adj v u) : p.concat h ≠ nil := by cases p <;> simp [concat] #align simple_graph.walk.concat_ne_nil SimpleGraph.Walk.concat_ne_nil theorem concat_inj {u v v' w : V} {p : G.Walk u v} {h : G.Adj v w} {p' : G.Walk u v'} {h' : G.Adj v' w} (he : p.concat h = p'.concat h') : ∃ hv : v = v', p.copy rfl hv = p' := by induction p with \| nil => cases p' · exact ⟨rfl, rfl⟩ · exfalso simp only [concat_nil, concat_cons, cons.injEq] at he obtain ⟨rfl, he⟩ := he simp only [heq_iff_eq] at he exact concat_ne_nil _ _ he.symm \| cons _ _ ih => rw [concat_cons] at he cases p' · exfalso simp only [concat_nil, cons.injEq] at he obtain ⟨rfl, he⟩ := he rw [heq_iff_eq] at he exact concat_ne_nil _ _ he · rw [concat_cons, cons.injEq] at he obtain ⟨rfl, he⟩ := he rw [heq_iff_eq] at he obtain ⟨rfl, rfl⟩ := ih he exact ⟨rfl, rfl⟩ #align simple_graph.walk.concat_inj SimpleGraph.Walk.concat_inj /-- The `support` of a walk is the list of vertices it visits in order. -/ def support {u v : V} : G.Walk u v → List V \| nil => [u] \| cons _ p => u :: p.support #align simple_graph.walk.support SimpleGraph.Walk.support /-- The `darts` of a walk is the list of darts it visits in order. -/ def darts {u v : V} : G.Walk u v → List G.Dart \| nil => [] \| cons h p => ⟨(u, _), h⟩ :: p.darts #align simple_graph.walk.darts SimpleGraph.Walk.darts /-- The `edges` of a walk is the list of edges it visits in order. This is defined to be the list of edges underlying `SimpleGraph.Walk.darts`. -/ def edges {u v : V} (p : G.Walk u v) : List (Sym2 V) := p.darts.map Dart.edge #align simple_graph.walk.edges SimpleGraph.Walk.edges @[simp] theorem support_nil {u : V} : (nil : G.Walk u u).support = [u] := rfl #align simple_graph.walk.support_nil SimpleGraph.Walk.support_nil @[simp] theorem support_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) : (cons h p).support = u :: p.support := rfl #align simple_graph.walk.support_cons SimpleGraph.Walk.support_cons @[simp]	Mathlib/Combinatorics/SimpleGraph/Connectivity.lean	556	558	theorem support_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : (p.concat h).support = p.support.concat w := by	induction p <;> simp [*, concat_nil]
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Kenny Lau, Scott Morrison, Alex Keizer -/ import Mathlib.Data.List.OfFn import Mathlib.Data.List.Range #align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" /-! # Lists of elements of `Fin n` This file develops some results on `finRange n`. -/ universe u namespace List variable {α : Type u} @[simp] theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = List.range n := by simp_rw [finRange, map_pmap, pmap_eq_map] exact List.map_id _ #align list.map_coe_fin_range List.map_coe_finRange theorem finRange_succ_eq_map (n : ℕ) : finRange n.succ = 0 :: (finRange n).map Fin.succ := by apply map_injective_iff.mpr Fin.val_injective rw [map_cons, map_coe_finRange, range_succ_eq_map, Fin.val_zero, ← map_coe_finRange, map_map, map_map] simp only [Function.comp, Fin.val_succ] #align list.fin_range_succ_eq_map List.finRange_succ_eq_map theorem finRange_succ (n : ℕ) : finRange n.succ = (finRange n \|>.map Fin.castSucc \|>.concat (.last _)) := by apply map_injective_iff.mpr Fin.val_injective simp [range_succ, Function.comp_def] -- Porting note: `map_nth_le` moved to `List.finRange_map_get` in Data.List.Range theorem ofFn_eq_pmap {n} {f : Fin n → α} : ofFn f = pmap (fun i hi => f ⟨i, hi⟩) (range n) fun _ => mem_range.1 := by rw [pmap_eq_map_attach] exact ext_get (by simp) fun i hi1 hi2 => by simp [get_ofFn f ⟨i, hi1⟩] #align list.of_fn_eq_pmap List.ofFn_eq_pmap theorem ofFn_id (n) : ofFn id = finRange n := ofFn_eq_pmap #align list.of_fn_id List.ofFn_id theorem ofFn_eq_map {n} {f : Fin n → α} : ofFn f = (finRange n).map f := by rw [← ofFn_id, map_ofFn, Function.comp_id] #align list.of_fn_eq_map List.ofFn_eq_map theorem nodup_ofFn_ofInjective {n} {f : Fin n → α} (hf : Function.Injective f) : Nodup (ofFn f) := by rw [ofFn_eq_pmap] exact (nodup_range n).pmap fun _ _ _ _ H => Fin.val_eq_of_eq <\| hf H #align list.nodup_of_fn_of_injective List.nodup_ofFn_ofInjective theorem nodup_ofFn {n} {f : Fin n → α} : Nodup (ofFn f) ↔ Function.Injective f := by refine ⟨?_, nodup_ofFn_ofInjective⟩ refine Fin.consInduction ?_ (fun x₀ xs ih => ?_) f · intro _ exact Function.injective_of_subsingleton _ · intro h rw [Fin.cons_injective_iff] simp_rw [ofFn_succ, Fin.cons_succ, nodup_cons, Fin.cons_zero, mem_ofFn] at h exact h.imp_right ih #align list.nodup_of_fn List.nodup_ofFn end List open List	Mathlib/Data/List/FinRange.lean	79	82	theorem Equiv.Perm.map_finRange_perm {n : ℕ} (σ : Equiv.Perm (Fin n)) : map σ (finRange n) ~ finRange n := by	rw [perm_ext_iff_of_nodup ((nodup_finRange n).map σ.injective) <\| nodup_finRange n] simpa [mem_map, mem_finRange, true_and_iff, iff_true_iff] using σ.surjective
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov -/ import Mathlib.Order.Filter.Cofinite import Mathlib.Order.ZornAtoms #align_import order.filter.ultrafilter from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1" /-! # Ultrafilters An ultrafilter is a minimal (maximal in the set order) proper filter. In this file we define * `Ultrafilter.of`: an ultrafilter that is less than or equal to a given filter; * `Ultrafilter`: subtype of ultrafilters; * `pure x : Ultrafilter α`: `pure x` as an `Ultrafilter`; * `Ultrafilter.map`, `Ultrafilter.bind`, `Ultrafilter.comap` : operations on ultrafilters; * `hyperfilter`: the ultrafilter extending the cofinite filter. -/ universe u v variable {α : Type u} {β : Type v} {γ : Type} open Set Filter Function open scoped Classical open Filter /-- `Filter α` is an atomic type: for every filter there exists an ultrafilter that is less than or equal to this filter. -/ instance : IsAtomic (Filter α) := IsAtomic.of_isChain_bounded fun c hc hne hb => ⟨sInf c, (sInf_neBot_of_directed' hne (show IsChain (· ≥ ·) c from hc.symm).directedOn hb).ne, fun _ hx => sInf_le hx⟩ /-- An ultrafilter is a minimal (maximal in the set order) proper filter. -/ structure Ultrafilter (α : Type) extends Filter α where /-- An ultrafilter is nontrivial. -/ protected neBot' : NeBot toFilter /-- If `g` is a nontrivial filter that is less than or equal to an ultrafilter, then it is greater than or equal to the ultrafilter. -/ protected le_of_le : ∀ g, Filter.NeBot g → g ≤ toFilter → toFilter ≤ g #align ultrafilter Ultrafilter namespace Ultrafilter variable {f g : Ultrafilter α} {s t : Set α} {p q : α → Prop} attribute [coe] Ultrafilter.toFilter instance : CoeTC (Ultrafilter α) (Filter α) := ⟨Ultrafilter.toFilter⟩ instance : Membership (Set α) (Ultrafilter α) := ⟨fun s f => s ∈ (f : Filter α)⟩ theorem unique (f : Ultrafilter α) {g : Filter α} (h : g ≤ f) (hne : NeBot g := by infer_instance) : g = f := le_antisymm h <\| f.le_of_le g hne h #align ultrafilter.unique Ultrafilter.unique instance neBot (f : Ultrafilter α) : NeBot (f : Filter α) := f.neBot' #align ultrafilter.ne_bot Ultrafilter.neBot protected theorem isAtom (f : Ultrafilter α) : IsAtom (f : Filter α) := ⟨f.neBot.ne, fun _ hgf => by_contra fun hg => hgf.ne <\| f.unique hgf.le ⟨hg⟩⟩ #align ultrafilter.is_atom Ultrafilter.isAtom @[simp, norm_cast] theorem mem_coe : s ∈ (f : Filter α) ↔ s ∈ f := Iff.rfl #align ultrafilter.mem_coe Ultrafilter.mem_coe theorem coe_injective : Injective ((↑) : Ultrafilter α → Filter α) \| ⟨f, h₁, h₂⟩, ⟨g, _, _⟩, _ => by congr #align ultrafilter.coe_injective Ultrafilter.coe_injective theorem eq_of_le {f g : Ultrafilter α} (h : (f : Filter α) ≤ g) : f = g := coe_injective (g.unique h) #align ultrafilter.eq_of_le Ultrafilter.eq_of_le @[simp, norm_cast] theorem coe_le_coe {f g : Ultrafilter α} : (f : Filter α) ≤ g ↔ f = g := ⟨fun h => eq_of_le h, fun h => h ▸ le_rfl⟩ #align ultrafilter.coe_le_coe Ultrafilter.coe_le_coe @[simp, norm_cast] theorem coe_inj : (f : Filter α) = g ↔ f = g := coe_injective.eq_iff #align ultrafilter.coe_inj Ultrafilter.coe_inj @[ext] theorem ext ⦃f g : Ultrafilter α⦄ (h : ∀ s, s ∈ f ↔ s ∈ g) : f = g := coe_injective <\| Filter.ext h #align ultrafilter.ext Ultrafilter.ext theorem le_of_inf_neBot (f : Ultrafilter α) {g : Filter α} (hg : NeBot (↑f ⊓ g)) : ↑f ≤ g := le_of_inf_eq (f.unique inf_le_left hg) #align ultrafilter.le_of_inf_ne_bot Ultrafilter.le_of_inf_neBot theorem le_of_inf_neBot' (f : Ultrafilter α) {g : Filter α} (hg : NeBot (g ⊓ f)) : ↑f ≤ g := f.le_of_inf_neBot <\| by rwa [inf_comm] #align ultrafilter.le_of_inf_ne_bot' Ultrafilter.le_of_inf_neBot' theorem inf_neBot_iff {f : Ultrafilter α} {g : Filter α} : NeBot (↑f ⊓ g) ↔ ↑f ≤ g := ⟨le_of_inf_neBot f, fun h => (inf_of_le_left h).symm ▸ f.neBot⟩ #align ultrafilter.inf_ne_bot_iff Ultrafilter.inf_neBot_iff theorem disjoint_iff_not_le {f : Ultrafilter α} {g : Filter α} : Disjoint (↑f) g ↔ ¬↑f ≤ g := by rw [← inf_neBot_iff, neBot_iff, Ne, not_not, disjoint_iff] #align ultrafilter.disjoint_iff_not_le Ultrafilter.disjoint_iff_not_le @[simp] theorem compl_not_mem_iff : sᶜ ∉ f ↔ s ∈ f := ⟨fun hsc => le_principal_iff.1 <\| f.le_of_inf_neBot ⟨fun h => hsc <\| mem_of_eq_bot <\| by rwa [compl_compl]⟩, compl_not_mem⟩ #align ultrafilter.compl_not_mem_iff Ultrafilter.compl_not_mem_iff @[simp] theorem frequently_iff_eventually : (∃ᶠ x in f, p x) ↔ ∀ᶠ x in f, p x := compl_not_mem_iff #align ultrafilter.frequently_iff_eventually Ultrafilter.frequently_iff_eventually alias ⟨_root_.Filter.Frequently.eventually, _⟩ := frequently_iff_eventually #align filter.frequently.eventually Filter.Frequently.eventually	Mathlib/Order/Filter/Ultrafilter.lean	135	135	theorem compl_mem_iff_not_mem : sᶜ ∈ f ↔ s ∉ f := by	rw [← compl_not_mem_iff, compl_compl]
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Johannes Hölzl -/ import Mathlib.MeasureTheory.Constructions.BorelSpace.Order #align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf" /-! # Simple functions A function `f` from a measurable space to any type is called simple, if every preimage `f ⁻¹' {x}` is measurable, and the range is finite. In this file, we define simple functions and establish their basic properties; and we construct a sequence of simple functions approximating an arbitrary Borel measurable function `f : α → ℝ≥0∞`. The theorem `Measurable.ennreal_induction` shows that in order to prove something for an arbitrary measurable function into `ℝ≥0∞`, it is sufficient to show that the property holds for (multiples of) characteristic functions and is closed under addition and supremum of increasing sequences of functions. -/ noncomputable section open Set hiding restrict restrict_apply open Filter ENNReal open Function (support) open scoped Classical open Topology NNReal ENNReal MeasureTheory namespace MeasureTheory variable {α β γ δ : Type} /-- A function `f` from a measurable space to any type is called simple, if every preimage `f ⁻¹' {x}` is measurable, and the range is finite. This structure bundles a function with these properties. -/ structure SimpleFunc.{u, v} (α : Type u) [MeasurableSpace α] (β : Type v) where toFun : α → β measurableSet_fiber' : ∀ x, MeasurableSet (toFun ⁻¹' {x}) finite_range' : (Set.range toFun).Finite #align measure_theory.simple_func MeasureTheory.SimpleFunc #align measure_theory.simple_func.to_fun MeasureTheory.SimpleFunc.toFun #align measure_theory.simple_func.measurable_set_fiber' MeasureTheory.SimpleFunc.measurableSet_fiber' #align measure_theory.simple_func.finite_range' MeasureTheory.SimpleFunc.finite_range' local infixr:25 " →ₛ " => SimpleFunc namespace SimpleFunc section Measurable variable [MeasurableSpace α] attribute [coe] toFun instance instCoeFun : CoeFun (α →ₛ β) fun _ => α → β := ⟨toFun⟩ #align measure_theory.simple_func.has_coe_to_fun MeasureTheory.SimpleFunc.instCoeFun theorem coe_injective ⦃f g : α →ₛ β⦄ (H : (f : α → β) = g) : f = g := by cases f; cases g; congr #align measure_theory.simple_func.coe_injective MeasureTheory.SimpleFunc.coe_injective @[ext] theorem ext {f g : α →ₛ β} (H : ∀ a, f a = g a) : f = g := coe_injective <\| funext H #align measure_theory.simple_func.ext MeasureTheory.SimpleFunc.ext theorem finite_range (f : α →ₛ β) : (Set.range f).Finite := f.finite_range' #align measure_theory.simple_func.finite_range MeasureTheory.SimpleFunc.finite_range theorem measurableSet_fiber (f : α →ₛ β) (x : β) : MeasurableSet (f ⁻¹' {x}) := f.measurableSet_fiber' x #align measure_theory.simple_func.measurable_set_fiber MeasureTheory.SimpleFunc.measurableSet_fiber -- @[simp] -- Porting note (#10618): simp can prove this theorem apply_mk (f : α → β) (h h') (x : α) : SimpleFunc.mk f h h' x = f x := rfl #align measure_theory.simple_func.apply_mk MeasureTheory.SimpleFunc.apply_mk /-- Simple function defined on a finite type. -/ def ofFinite [Finite α] [MeasurableSingletonClass α] (f : α → β) : α →ₛ β where toFun := f measurableSet_fiber' x := (toFinite (f ⁻¹' {x})).measurableSet finite_range' := Set.finite_range f @[deprecated (since := "2024-02-05")] alias ofFintype := ofFinite /-- Simple function defined on the empty type. -/ def ofIsEmpty [IsEmpty α] : α →ₛ β := ofFinite isEmptyElim #align measure_theory.simple_func.of_is_empty MeasureTheory.SimpleFunc.ofIsEmpty /-- Range of a simple function `α →ₛ β` as a `Finset β`. -/ protected def range (f : α →ₛ β) : Finset β := f.finite_range.toFinset #align measure_theory.simple_func.range MeasureTheory.SimpleFunc.range @[simp] theorem mem_range {f : α →ₛ β} {b} : b ∈ f.range ↔ b ∈ range f := Finite.mem_toFinset _ #align measure_theory.simple_func.mem_range MeasureTheory.SimpleFunc.mem_range theorem mem_range_self (f : α →ₛ β) (x : α) : f x ∈ f.range := mem_range.2 ⟨x, rfl⟩ #align measure_theory.simple_func.mem_range_self MeasureTheory.SimpleFunc.mem_range_self @[simp] theorem coe_range (f : α →ₛ β) : (↑f.range : Set β) = Set.range f := f.finite_range.coe_toFinset #align measure_theory.simple_func.coe_range MeasureTheory.SimpleFunc.coe_range theorem mem_range_of_measure_ne_zero {f : α →ₛ β} {x : β} {μ : Measure α} (H : μ (f ⁻¹' {x}) ≠ 0) : x ∈ f.range := let ⟨a, ha⟩ := nonempty_of_measure_ne_zero H mem_range.2 ⟨a, ha⟩ #align measure_theory.simple_func.mem_range_of_measure_ne_zero MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zero theorem forall_mem_range {f : α →ₛ β} {p : β → Prop} : (∀ y ∈ f.range, p y) ↔ ∀ x, p (f x) := by simp only [mem_range, Set.forall_mem_range] #align measure_theory.simple_func.forall_mem_range MeasureTheory.SimpleFunc.forall_mem_range theorem exists_range_iff {f : α →ₛ β} {p : β → Prop} : (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x) := by simpa only [mem_range, exists_prop] using Set.exists_range_iff #align measure_theory.simple_func.exists_range_iff MeasureTheory.SimpleFunc.exists_range_iff theorem preimage_eq_empty_iff (f : α →ₛ β) (b : β) : f ⁻¹' {b} = ∅ ↔ b ∉ f.range := preimage_singleton_eq_empty.trans <\| not_congr mem_range.symm #align measure_theory.simple_func.preimage_eq_empty_iff MeasureTheory.SimpleFunc.preimage_eq_empty_iff theorem exists_forall_le [Nonempty β] [Preorder β] [IsDirected β (· ≤ ·)] (f : α →ₛ β) : ∃ C, ∀ x, f x ≤ C := f.range.exists_le.imp fun _ => forall_mem_range.1 #align measure_theory.simple_func.exists_forall_le MeasureTheory.SimpleFunc.exists_forall_le /-- Constant function as a `SimpleFunc`. -/ def const (α) {β} [MeasurableSpace α] (b : β) : α →ₛ β := ⟨fun _ => b, fun _ => MeasurableSet.const _, finite_range_const⟩ #align measure_theory.simple_func.const MeasureTheory.SimpleFunc.const instance instInhabited [Inhabited β] : Inhabited (α →ₛ β) := ⟨const _ default⟩ #align measure_theory.simple_func.inhabited MeasureTheory.SimpleFunc.instInhabited theorem const_apply (a : α) (b : β) : (const α b) a = b := rfl #align measure_theory.simple_func.const_apply MeasureTheory.SimpleFunc.const_apply @[simp] theorem coe_const (b : β) : ⇑(const α b) = Function.const α b := rfl #align measure_theory.simple_func.coe_const MeasureTheory.SimpleFunc.coe_const @[simp] theorem range_const (α) [MeasurableSpace α] [Nonempty α] (b : β) : (const α b).range = {b} := Finset.coe_injective <\| by simp (config := { unfoldPartialApp := true }) [Function.const] #align measure_theory.simple_func.range_const MeasureTheory.SimpleFunc.range_const theorem range_const_subset (α) [MeasurableSpace α] (b : β) : (const α b).range ⊆ {b} := Finset.coe_subset.1 <\| by simp #align measure_theory.simple_func.range_const_subset MeasureTheory.SimpleFunc.range_const_subset theorem simpleFunc_bot {α} (f : @SimpleFunc α ⊥ β) [Nonempty β] : ∃ c, ∀ x, f x = c := by have hf_meas := @SimpleFunc.measurableSet_fiber α _ ⊥ f simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc ↦ congr_fun hc #align measure_theory.simple_func.simple_func_bot MeasureTheory.SimpleFunc.simpleFunc_bot theorem simpleFunc_bot' {α} [Nonempty β] (f : @SimpleFunc α ⊥ β) : ∃ c, f = @SimpleFunc.const α _ ⊥ c := letI : MeasurableSpace α := ⊥; (simpleFunc_bot f).imp fun _ ↦ ext #align measure_theory.simple_func.simple_func_bot' MeasureTheory.SimpleFunc.simpleFunc_bot' theorem measurableSet_cut (r : α → β → Prop) (f : α →ₛ β) (h : ∀ b, MeasurableSet { a \| r a b }) : MeasurableSet { a \| r a (f a) } := by have : { a \| r a (f a) } = ⋃ b ∈ range f, { a \| r a b } ∩ f ⁻¹' {b} := by ext a suffices r a (f a) ↔ ∃ i, r a (f i) ∧ f a = f i by simpa exact ⟨fun h => ⟨a, ⟨h, rfl⟩⟩, fun ⟨a', ⟨h', e⟩⟩ => e.symm ▸ h'⟩ rw [this] exact MeasurableSet.biUnion f.finite_range.countable fun b _ => MeasurableSet.inter (h b) (f.measurableSet_fiber _) #align measure_theory.simple_func.measurable_set_cut MeasureTheory.SimpleFunc.measurableSet_cut @[measurability] theorem measurableSet_preimage (f : α →ₛ β) (s) : MeasurableSet (f ⁻¹' s) := measurableSet_cut (fun _ b => b ∈ s) f fun b => MeasurableSet.const (b ∈ s) #align measure_theory.simple_func.measurable_set_preimage MeasureTheory.SimpleFunc.measurableSet_preimage /-- A simple function is measurable -/ @[measurability] protected theorem measurable [MeasurableSpace β] (f : α →ₛ β) : Measurable f := fun s _ => measurableSet_preimage f s #align measure_theory.simple_func.measurable MeasureTheory.SimpleFunc.measurable @[measurability] protected theorem aemeasurable [MeasurableSpace β] {μ : Measure α} (f : α →ₛ β) : AEMeasurable f μ := f.measurable.aemeasurable #align measure_theory.simple_func.ae_measurable MeasureTheory.SimpleFunc.aemeasurable protected theorem sum_measure_preimage_singleton (f : α →ₛ β) {μ : Measure α} (s : Finset β) : (∑ y ∈ s, μ (f ⁻¹' {y})) = μ (f ⁻¹' ↑s) := sum_measure_preimage_singleton _ fun _ _ => f.measurableSet_fiber _ #align measure_theory.simple_func.sum_measure_preimage_singleton MeasureTheory.SimpleFunc.sum_measure_preimage_singleton theorem sum_range_measure_preimage_singleton (f : α →ₛ β) (μ : Measure α) : (∑ y ∈ f.range, μ (f ⁻¹' {y})) = μ univ := by rw [f.sum_measure_preimage_singleton, coe_range, preimage_range] #align measure_theory.simple_func.sum_range_measure_preimage_singleton MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton /-- If-then-else as a `SimpleFunc`. -/ def piecewise (s : Set α) (hs : MeasurableSet s) (f g : α →ₛ β) : α →ₛ β := ⟨s.piecewise f g, fun _ => letI : MeasurableSpace β := ⊤ f.measurable.piecewise hs g.measurable trivial, (f.finite_range.union g.finite_range).subset range_ite_subset⟩ #align measure_theory.simple_func.piecewise MeasureTheory.SimpleFunc.piecewise @[simp] theorem coe_piecewise {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) : ⇑(piecewise s hs f g) = s.piecewise f g := rfl #align measure_theory.simple_func.coe_piecewise MeasureTheory.SimpleFunc.coe_piecewise theorem piecewise_apply {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) (a) : piecewise s hs f g a = if a ∈ s then f a else g a := rfl #align measure_theory.simple_func.piecewise_apply MeasureTheory.SimpleFunc.piecewise_apply @[simp] theorem piecewise_compl {s : Set α} (hs : MeasurableSet sᶜ) (f g : α →ₛ β) : piecewise sᶜ hs f g = piecewise s hs.of_compl g f := coe_injective <\| by set_option tactic.skipAssignedInstances false in simp [hs]; convert Set.piecewise_compl s f g #align measure_theory.simple_func.piecewise_compl MeasureTheory.SimpleFunc.piecewise_compl @[simp] theorem piecewise_univ (f g : α →ₛ β) : piecewise univ MeasurableSet.univ f g = f := coe_injective <\| by set_option tactic.skipAssignedInstances false in simp; convert Set.piecewise_univ f g #align measure_theory.simple_func.piecewise_univ MeasureTheory.SimpleFunc.piecewise_univ @[simp] theorem piecewise_empty (f g : α →ₛ β) : piecewise ∅ MeasurableSet.empty f g = g := coe_injective <\| by set_option tactic.skipAssignedInstances false in simp; convert Set.piecewise_empty f g #align measure_theory.simple_func.piecewise_empty MeasureTheory.SimpleFunc.piecewise_empty @[simp] theorem piecewise_same (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) : piecewise s hs f f = f := coe_injective <\| Set.piecewise_same _ _ theorem support_indicator [Zero β] {s : Set α} (hs : MeasurableSet s) (f : α →ₛ β) : Function.support (f.piecewise s hs (SimpleFunc.const α 0)) = s ∩ Function.support f := Set.support_indicator #align measure_theory.simple_func.support_indicator MeasureTheory.SimpleFunc.support_indicator theorem range_indicator {s : Set α} (hs : MeasurableSet s) (hs_nonempty : s.Nonempty) (hs_ne_univ : s ≠ univ) (x y : β) : (piecewise s hs (const α x) (const α y)).range = {x, y} := by simp only [← Finset.coe_inj, coe_range, coe_piecewise, range_piecewise, coe_const, Finset.coe_insert, Finset.coe_singleton, hs_nonempty.image_const, (nonempty_compl.2 hs_ne_univ).image_const, singleton_union, Function.const] #align measure_theory.simple_func.range_indicator MeasureTheory.SimpleFunc.range_indicator theorem measurable_bind [MeasurableSpace γ] (f : α →ₛ β) (g : β → α → γ) (hg : ∀ b, Measurable (g b)) : Measurable fun a => g (f a) a := fun s hs => f.measurableSet_cut (fun a b => g b a ∈ s) fun b => hg b hs #align measure_theory.simple_func.measurable_bind MeasureTheory.SimpleFunc.measurable_bind /-- If `f : α →ₛ β` is a simple function and `g : β → α →ₛ γ` is a family of simple functions, then `f.bind g` binds the first argument of `g` to `f`. In other words, `f.bind g a = g (f a) a`. -/ def bind (f : α →ₛ β) (g : β → α →ₛ γ) : α →ₛ γ := ⟨fun a => g (f a) a, fun c => f.measurableSet_cut (fun a b => g b a = c) fun b => (g b).measurableSet_preimage {c}, (f.finite_range.biUnion fun b _ => (g b).finite_range).subset <\| by rintro _ ⟨a, rfl⟩; simp⟩ #align measure_theory.simple_func.bind MeasureTheory.SimpleFunc.bind @[simp] theorem bind_apply (f : α →ₛ β) (g : β → α →ₛ γ) (a) : f.bind g a = g (f a) a := rfl #align measure_theory.simple_func.bind_apply MeasureTheory.SimpleFunc.bind_apply /-- Given a function `g : β → γ` and a simple function `f : α →ₛ β`, `f.map g` return the simple function `g ∘ f : α →ₛ γ` -/ def map (g : β → γ) (f : α →ₛ β) : α →ₛ γ := bind f (const α ∘ g) #align measure_theory.simple_func.map MeasureTheory.SimpleFunc.map theorem map_apply (g : β → γ) (f : α →ₛ β) (a) : f.map g a = g (f a) := rfl #align measure_theory.simple_func.map_apply MeasureTheory.SimpleFunc.map_apply theorem map_map (g : β → γ) (h : γ → δ) (f : α →ₛ β) : (f.map g).map h = f.map (h ∘ g) := rfl #align measure_theory.simple_func.map_map MeasureTheory.SimpleFunc.map_map @[simp] theorem coe_map (g : β → γ) (f : α →ₛ β) : (f.map g : α → γ) = g ∘ f := rfl #align measure_theory.simple_func.coe_map MeasureTheory.SimpleFunc.coe_map @[simp] theorem range_map [DecidableEq γ] (g : β → γ) (f : α →ₛ β) : (f.map g).range = f.range.image g := Finset.coe_injective <\| by simp only [coe_range, coe_map, Finset.coe_image, range_comp] #align measure_theory.simple_func.range_map MeasureTheory.SimpleFunc.range_map @[simp] theorem map_const (g : β → γ) (b : β) : (const α b).map g = const α (g b) := rfl #align measure_theory.simple_func.map_const MeasureTheory.SimpleFunc.map_const theorem map_preimage (f : α →ₛ β) (g : β → γ) (s : Set γ) : f.map g ⁻¹' s = f ⁻¹' ↑(f.range.filter fun b => g b ∈ s) := by simp only [coe_range, sep_mem_eq, coe_map, Finset.coe_filter, ← mem_preimage, inter_comm, preimage_inter_range, ← Finset.mem_coe] exact preimage_comp #align measure_theory.simple_func.map_preimage MeasureTheory.SimpleFunc.map_preimage theorem map_preimage_singleton (f : α →ₛ β) (g : β → γ) (c : γ) : f.map g ⁻¹' {c} = f ⁻¹' ↑(f.range.filter fun b => g b = c) := map_preimage _ _ _ #align measure_theory.simple_func.map_preimage_singleton MeasureTheory.SimpleFunc.map_preimage_singleton /-- Composition of a `SimpleFun` and a measurable function is a `SimpleFunc`. -/ def comp [MeasurableSpace β] (f : β →ₛ γ) (g : α → β) (hgm : Measurable g) : α →ₛ γ where toFun := f ∘ g finite_range' := f.finite_range.subset <\| Set.range_comp_subset_range _ _ measurableSet_fiber' z := hgm (f.measurableSet_fiber z) #align measure_theory.simple_func.comp MeasureTheory.SimpleFunc.comp @[simp] theorem coe_comp [MeasurableSpace β] (f : β →ₛ γ) {g : α → β} (hgm : Measurable g) : ⇑(f.comp g hgm) = f ∘ g := rfl #align measure_theory.simple_func.coe_comp MeasureTheory.SimpleFunc.coe_comp theorem range_comp_subset_range [MeasurableSpace β] (f : β →ₛ γ) {g : α → β} (hgm : Measurable g) : (f.comp g hgm).range ⊆ f.range := Finset.coe_subset.1 <\| by simp only [coe_range, coe_comp, Set.range_comp_subset_range] #align measure_theory.simple_func.range_comp_subset_range MeasureTheory.SimpleFunc.range_comp_subset_range /-- Extend a `SimpleFunc` along a measurable embedding: `f₁.extend g hg f₂` is the function `F : β →ₛ γ` such that `F ∘ g = f₁` and `F y = f₂ y` whenever `y ∉ range g`. -/ def extend [MeasurableSpace β] (f₁ : α →ₛ γ) (g : α → β) (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) : β →ₛ γ where toFun := Function.extend g f₁ f₂ finite_range' := (f₁.finite_range.union <\| f₂.finite_range.subset (image_subset_range _ _)).subset (range_extend_subset _ _ _) measurableSet_fiber' := by letI : MeasurableSpace γ := ⊤; haveI : MeasurableSingletonClass γ := ⟨fun _ => trivial⟩ exact fun x => hg.measurable_extend f₁.measurable f₂.measurable (measurableSet_singleton _) #align measure_theory.simple_func.extend MeasureTheory.SimpleFunc.extend @[simp] theorem extend_apply [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) (x : α) : (f₁.extend g hg f₂) (g x) = f₁ x := hg.injective.extend_apply _ _ _ #align measure_theory.simple_func.extend_apply MeasureTheory.SimpleFunc.extend_apply @[simp] theorem extend_apply' [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) {y : β} (h : ¬∃ x, g x = y) : (f₁.extend g hg f₂) y = f₂ y := Function.extend_apply' _ _ _ h #align measure_theory.simple_func.extend_apply' MeasureTheory.SimpleFunc.extend_apply' @[simp] theorem extend_comp_eq' [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) : f₁.extend g hg f₂ ∘ g = f₁ := funext fun _ => extend_apply _ _ _ _ #align measure_theory.simple_func.extend_comp_eq' MeasureTheory.SimpleFunc.extend_comp_eq' @[simp] theorem extend_comp_eq [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : β →ₛ γ) : (f₁.extend g hg f₂).comp g hg.measurable = f₁ := coe_injective <\| extend_comp_eq' _ hg _ #align measure_theory.simple_func.extend_comp_eq MeasureTheory.SimpleFunc.extend_comp_eq /-- If `f` is a simple function taking values in `β → γ` and `g` is another simple function with the same domain and codomain `β`, then `f.seq g = f a (g a)`. -/ def seq (f : α →ₛ β → γ) (g : α →ₛ β) : α →ₛ γ := f.bind fun f => g.map f #align measure_theory.simple_func.seq MeasureTheory.SimpleFunc.seq @[simp] theorem seq_apply (f : α →ₛ β → γ) (g : α →ₛ β) (a : α) : f.seq g a = f a (g a) := rfl #align measure_theory.simple_func.seq_apply MeasureTheory.SimpleFunc.seq_apply /-- Combine two simple functions `f : α →ₛ β` and `g : α →ₛ β` into `fun a => (f a, g a)`. -/ def pair (f : α →ₛ β) (g : α →ₛ γ) : α →ₛ β × γ := (f.map Prod.mk).seq g #align measure_theory.simple_func.pair MeasureTheory.SimpleFunc.pair @[simp] theorem pair_apply (f : α →ₛ β) (g : α →ₛ γ) (a) : pair f g a = (f a, g a) := rfl #align measure_theory.simple_func.pair_apply MeasureTheory.SimpleFunc.pair_apply theorem pair_preimage (f : α →ₛ β) (g : α →ₛ γ) (s : Set β) (t : Set γ) : pair f g ⁻¹' s ×ˢ t = f ⁻¹' s ∩ g ⁻¹' t := rfl #align measure_theory.simple_func.pair_preimage MeasureTheory.SimpleFunc.pair_preimage -- A special form of `pair_preimage` theorem pair_preimage_singleton (f : α →ₛ β) (g : α →ₛ γ) (b : β) (c : γ) : pair f g ⁻¹' {(b, c)} = f ⁻¹' {b} ∩ g ⁻¹' {c} := by rw [← singleton_prod_singleton] exact pair_preimage _ _ _ _ #align measure_theory.simple_func.pair_preimage_singleton MeasureTheory.SimpleFunc.pair_preimage_singleton theorem bind_const (f : α →ₛ β) : f.bind (const α) = f := by ext; simp #align measure_theory.simple_func.bind_const MeasureTheory.SimpleFunc.bind_const @[to_additive] instance instOne [One β] : One (α →ₛ β) := ⟨const α 1⟩ #align measure_theory.simple_func.has_one MeasureTheory.SimpleFunc.instOne #align measure_theory.simple_func.has_zero MeasureTheory.SimpleFunc.instZero @[to_additive] instance instMul [Mul β] : Mul (α →ₛ β) := ⟨fun f g => (f.map (· ·)).seq g⟩ #align measure_theory.simple_func.has_mul MeasureTheory.SimpleFunc.instMul #align measure_theory.simple_func.has_add MeasureTheory.SimpleFunc.instAdd @[to_additive] instance instDiv [Div β] : Div (α →ₛ β) := ⟨fun f g => (f.map (· / ·)).seq g⟩ #align measure_theory.simple_func.has_div MeasureTheory.SimpleFunc.instDiv #align measure_theory.simple_func.has_sub MeasureTheory.SimpleFunc.instSub @[to_additive] instance instInv [Inv β] : Inv (α →ₛ β) := ⟨fun f => f.map Inv.inv⟩ #align measure_theory.simple_func.has_inv MeasureTheory.SimpleFunc.instInv #align measure_theory.simple_func.has_neg MeasureTheory.SimpleFunc.instNeg instance instSup [Sup β] : Sup (α →ₛ β) := ⟨fun f g => (f.map (· ⊔ ·)).seq g⟩ #align measure_theory.simple_func.has_sup MeasureTheory.SimpleFunc.instSup instance instInf [Inf β] : Inf (α →ₛ β) := ⟨fun f g => (f.map (· ⊓ ·)).seq g⟩ #align measure_theory.simple_func.has_inf MeasureTheory.SimpleFunc.instInf instance instLE [LE β] : LE (α →ₛ β) := ⟨fun f g => ∀ a, f a ≤ g a⟩ #align measure_theory.simple_func.has_le MeasureTheory.SimpleFunc.instLE @[to_additive (attr := simp)] theorem const_one [One β] : const α (1 : β) = 1 := rfl #align measure_theory.simple_func.const_one MeasureTheory.SimpleFunc.const_one #align measure_theory.simple_func.const_zero MeasureTheory.SimpleFunc.const_zero @[to_additive (attr := simp, norm_cast)] theorem coe_one [One β] : ⇑(1 : α →ₛ β) = 1 := rfl #align measure_theory.simple_func.coe_one MeasureTheory.SimpleFunc.coe_one #align measure_theory.simple_func.coe_zero MeasureTheory.SimpleFunc.coe_zero @[to_additive (attr := simp, norm_cast)] theorem coe_mul [Mul β] (f g : α →ₛ β) : ⇑(f * g) = ⇑f * ⇑g := rfl #align measure_theory.simple_func.coe_mul MeasureTheory.SimpleFunc.coe_mul #align measure_theory.simple_func.coe_add MeasureTheory.SimpleFunc.coe_add @[to_additive (attr := simp, norm_cast)] theorem coe_inv [Inv β] (f : α →ₛ β) : ⇑(f⁻¹) = (⇑f)⁻¹ := rfl #align measure_theory.simple_func.coe_inv MeasureTheory.SimpleFunc.coe_inv #align measure_theory.simple_func.coe_neg MeasureTheory.SimpleFunc.coe_neg @[to_additive (attr := simp, norm_cast)] theorem coe_div [Div β] (f g : α →ₛ β) : ⇑(f / g) = ⇑f / ⇑g := rfl #align measure_theory.simple_func.coe_div MeasureTheory.SimpleFunc.coe_div #align measure_theory.simple_func.coe_sub MeasureTheory.SimpleFunc.coe_sub @[simp, norm_cast] theorem coe_le [Preorder β] {f g : α →ₛ β} : (f : α → β) ≤ g ↔ f ≤ g := Iff.rfl #align measure_theory.simple_func.coe_le MeasureTheory.SimpleFunc.coe_le @[simp, norm_cast] theorem coe_sup [Sup β] (f g : α →ₛ β) : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g := rfl #align measure_theory.simple_func.coe_sup MeasureTheory.SimpleFunc.coe_sup @[simp, norm_cast] theorem coe_inf [Inf β] (f g : α →ₛ β) : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g := rfl #align measure_theory.simple_func.coe_inf MeasureTheory.SimpleFunc.coe_inf @[to_additive] theorem mul_apply [Mul β] (f g : α →ₛ β) (a : α) : (f * g) a = f a * g a := rfl #align measure_theory.simple_func.mul_apply MeasureTheory.SimpleFunc.mul_apply #align measure_theory.simple_func.add_apply MeasureTheory.SimpleFunc.add_apply @[to_additive] theorem div_apply [Div β] (f g : α →ₛ β) (x : α) : (f / g) x = f x / g x := rfl #align measure_theory.simple_func.div_apply MeasureTheory.SimpleFunc.div_apply #align measure_theory.simple_func.sub_apply MeasureTheory.SimpleFunc.sub_apply @[to_additive] theorem inv_apply [Inv β] (f : α →ₛ β) (x : α) : f⁻¹ x = (f x)⁻¹ := rfl #align measure_theory.simple_func.inv_apply MeasureTheory.SimpleFunc.inv_apply #align measure_theory.simple_func.neg_apply MeasureTheory.SimpleFunc.neg_apply theorem sup_apply [Sup β] (f g : α →ₛ β) (a : α) : (f ⊔ g) a = f a ⊔ g a := rfl #align measure_theory.simple_func.sup_apply MeasureTheory.SimpleFunc.sup_apply theorem inf_apply [Inf β] (f g : α →ₛ β) (a : α) : (f ⊓ g) a = f a ⊓ g a := rfl #align measure_theory.simple_func.inf_apply MeasureTheory.SimpleFunc.inf_apply @[to_additive (attr := simp)] theorem range_one [Nonempty α] [One β] : (1 : α →ₛ β).range = {1} := Finset.ext fun x => by simp [eq_comm] #align measure_theory.simple_func.range_one MeasureTheory.SimpleFunc.range_one #align measure_theory.simple_func.range_zero MeasureTheory.SimpleFunc.range_zero @[simp] theorem range_eq_empty_of_isEmpty {β} [hα : IsEmpty α] (f : α →ₛ β) : f.range = ∅ := by rw [← Finset.not_nonempty_iff_eq_empty] by_contra h obtain ⟨y, hy_mem⟩ := h rw [SimpleFunc.mem_range, Set.mem_range] at hy_mem obtain ⟨x, hxy⟩ := hy_mem rw [isEmpty_iff] at hα exact hα x #align measure_theory.simple_func.range_eq_empty_of_is_empty MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty theorem eq_zero_of_mem_range_zero [Zero β] : ∀ {y : β}, y ∈ (0 : α →ₛ β).range → y = 0 := @(forall_mem_range.2 fun _ => rfl) #align measure_theory.simple_func.eq_zero_of_mem_range_zero MeasureTheory.SimpleFunc.eq_zero_of_mem_range_zero @[to_additive] theorem mul_eq_map₂ [Mul β] (f g : α →ₛ β) : f * g = (pair f g).map fun p : β × β => p.1 * p.2 := rfl #align measure_theory.simple_func.mul_eq_map₂ MeasureTheory.SimpleFunc.mul_eq_map₂ #align measure_theory.simple_func.add_eq_map₂ MeasureTheory.SimpleFunc.add_eq_map₂ theorem sup_eq_map₂ [Sup β] (f g : α →ₛ β) : f ⊔ g = (pair f g).map fun p : β × β => p.1 ⊔ p.2 := rfl #align measure_theory.simple_func.sup_eq_map₂ MeasureTheory.SimpleFunc.sup_eq_map₂ @[to_additive] theorem const_mul_eq_map [Mul β] (f : α →ₛ β) (b : β) : const α b * f = f.map fun a => b * a := rfl #align measure_theory.simple_func.const_mul_eq_map MeasureTheory.SimpleFunc.const_mul_eq_map #align measure_theory.simple_func.const_add_eq_map MeasureTheory.SimpleFunc.const_add_eq_map @[to_additive] theorem map_mul [Mul β] [Mul γ] {g : β → γ} (hg : ∀ x y, g (x * y) = g x * g y) (f₁ f₂ : α →ₛ β) : (f₁ * f₂).map g = f₁.map g * f₂.map g := ext fun _ => hg _ _ #align measure_theory.simple_func.map_mul MeasureTheory.SimpleFunc.map_mul #align measure_theory.simple_func.map_add MeasureTheory.SimpleFunc.map_add variable {K : Type} @[to_additive] instance instSMul [SMul K β] : SMul K (α →ₛ β) := ⟨fun k f => f.map (k • ·)⟩ #align measure_theory.simple_func.has_smul MeasureTheory.SimpleFunc.instSMul @[to_additive (attr := simp)] theorem coe_smul [SMul K β] (c : K) (f : α →ₛ β) : ⇑(c • f) = c • ⇑f := rfl #align measure_theory.simple_func.coe_smul MeasureTheory.SimpleFunc.coe_smul @[to_additive (attr := simp)] theorem smul_apply [SMul K β] (k : K) (f : α →ₛ β) (a : α) : (k • f) a = k • f a := rfl #align measure_theory.simple_func.smul_apply MeasureTheory.SimpleFunc.smul_apply instance hasNatSMul [AddMonoid β] : SMul ℕ (α →ₛ β) := inferInstance @[to_additive existing hasNatSMul] instance hasNatPow [Monoid β] : Pow (α →ₛ β) ℕ := ⟨fun f n => f.map (· ^ n)⟩ #align measure_theory.simple_func.has_nat_pow MeasureTheory.SimpleFunc.hasNatPow @[simp] theorem coe_pow [Monoid β] (f : α →ₛ β) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n := rfl #align measure_theory.simple_func.coe_pow MeasureTheory.SimpleFunc.coe_pow theorem pow_apply [Monoid β] (n : ℕ) (f : α →ₛ β) (a : α) : (f ^ n) a = f a ^ n := rfl #align measure_theory.simple_func.pow_apply MeasureTheory.SimpleFunc.pow_apply instance hasIntPow [DivInvMonoid β] : Pow (α →ₛ β) ℤ := ⟨fun f n => f.map (· ^ n)⟩ #align measure_theory.simple_func.has_int_pow MeasureTheory.SimpleFunc.hasIntPow @[simp] theorem coe_zpow [DivInvMonoid β] (f : α →ₛ β) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z := rfl #align measure_theory.simple_func.coe_zpow MeasureTheory.SimpleFunc.coe_zpow theorem zpow_apply [DivInvMonoid β] (z : ℤ) (f : α →ₛ β) (a : α) : (f ^ z) a = f a ^ z := rfl #align measure_theory.simple_func.zpow_apply MeasureTheory.SimpleFunc.zpow_apply -- TODO: work out how to generate these instances with `to_additive`, which gets confused by the -- argument order swap between `coe_smul` and `coe_pow`. section Additive instance instAddMonoid [AddMonoid β] : AddMonoid (α →ₛ β) := Function.Injective.addMonoid (fun f => show α → β from f) coe_injective coe_zero coe_add fun _ _ => coe_smul _ _ #align measure_theory.simple_func.add_monoid MeasureTheory.SimpleFunc.instAddMonoid instance instAddCommMonoid [AddCommMonoid β] : AddCommMonoid (α →ₛ β) := Function.Injective.addCommMonoid (fun f => show α → β from f) coe_injective coe_zero coe_add fun _ _ => coe_smul _ _ #align measure_theory.simple_func.add_comm_monoid MeasureTheory.SimpleFunc.instAddCommMonoid instance instAddGroup [AddGroup β] : AddGroup (α →ₛ β) := Function.Injective.addGroup (fun f => show α → β from f) coe_injective coe_zero coe_add coe_neg coe_sub (fun _ _ => coe_smul _ _) fun _ _ => coe_smul _ _ #align measure_theory.simple_func.add_group MeasureTheory.SimpleFunc.instAddGroup instance instAddCommGroup [AddCommGroup β] : AddCommGroup (α →ₛ β) := Function.Injective.addCommGroup (fun f => show α → β from f) coe_injective coe_zero coe_add coe_neg coe_sub (fun _ _ => coe_smul _ _) fun _ _ => coe_smul _ _ #align measure_theory.simple_func.add_comm_group MeasureTheory.SimpleFunc.instAddCommGroup end Additive @[to_additive existing] instance instMonoid [Monoid β] : Monoid (α →ₛ β) := Function.Injective.monoid (fun f => show α → β from f) coe_injective coe_one coe_mul coe_pow #align measure_theory.simple_func.monoid MeasureTheory.SimpleFunc.instMonoid @[to_additive existing] instance instCommMonoid [CommMonoid β] : CommMonoid (α →ₛ β) := Function.Injective.commMonoid (fun f => show α → β from f) coe_injective coe_one coe_mul coe_pow #align measure_theory.simple_func.comm_monoid MeasureTheory.SimpleFunc.instCommMonoid @[to_additive existing] instance instGroup [Group β] : Group (α →ₛ β) := Function.Injective.group (fun f => show α → β from f) coe_injective coe_one coe_mul coe_inv coe_div coe_pow coe_zpow #align measure_theory.simple_func.group MeasureTheory.SimpleFunc.instGroup @[to_additive existing] instance instCommGroup [CommGroup β] : CommGroup (α →ₛ β) := Function.Injective.commGroup (fun f => show α → β from f) coe_injective coe_one coe_mul coe_inv coe_div coe_pow coe_zpow #align measure_theory.simple_func.comm_group MeasureTheory.SimpleFunc.instCommGroup instance instModule [Semiring K] [AddCommMonoid β] [Module K β] : Module K (α →ₛ β) := Function.Injective.module K ⟨⟨fun f => show α → β from f, coe_zero⟩, coe_add⟩ coe_injective coe_smul #align measure_theory.simple_func.module MeasureTheory.SimpleFunc.instModule theorem smul_eq_map [SMul K β] (k : K) (f : α →ₛ β) : k • f = f.map (k • ·) := rfl #align measure_theory.simple_func.smul_eq_map MeasureTheory.SimpleFunc.smul_eq_map instance instPreorder [Preorder β] : Preorder (α →ₛ β) := { SimpleFunc.instLE with le_refl := fun f a => le_rfl le_trans := fun f g h hfg hgh a => le_trans (hfg _) (hgh a) } #align measure_theory.simple_func.preorder MeasureTheory.SimpleFunc.instPreorder instance instPartialOrder [PartialOrder β] : PartialOrder (α →ₛ β) := { SimpleFunc.instPreorder with le_antisymm := fun _f _g hfg hgf => ext fun a => le_antisymm (hfg a) (hgf a) } #align measure_theory.simple_func.partial_order MeasureTheory.SimpleFunc.instPartialOrder instance instOrderBot [LE β] [OrderBot β] : OrderBot (α →ₛ β) where bot := const α ⊥ bot_le _ _ := bot_le #align measure_theory.simple_func.order_bot MeasureTheory.SimpleFunc.instOrderBot instance instOrderTop [LE β] [OrderTop β] : OrderTop (α →ₛ β) where top := const α ⊤ le_top _ _ := le_top #align measure_theory.simple_func.order_top MeasureTheory.SimpleFunc.instOrderTop instance instSemilatticeInf [SemilatticeInf β] : SemilatticeInf (α →ₛ β) := { SimpleFunc.instPartialOrder with inf := (· ⊓ ·) inf_le_left := fun _ _ _ => inf_le_left inf_le_right := fun _ _ _ => inf_le_right le_inf := fun _f _g _h hfh hgh a => le_inf (hfh a) (hgh a) } #align measure_theory.simple_func.semilattice_inf MeasureTheory.SimpleFunc.instSemilatticeInf instance instSemilatticeSup [SemilatticeSup β] : SemilatticeSup (α →ₛ β) := { SimpleFunc.instPartialOrder with sup := (· ⊔ ·) le_sup_left := fun _ _ _ => le_sup_left le_sup_right := fun _ _ _ => le_sup_right sup_le := fun _f _g _h hfh hgh a => sup_le (hfh a) (hgh a) } #align measure_theory.simple_func.semilattice_sup MeasureTheory.SimpleFunc.instSemilatticeSup instance instLattice [Lattice β] : Lattice (α →ₛ β) := { SimpleFunc.instSemilatticeSup, SimpleFunc.instSemilatticeInf with } #align measure_theory.simple_func.lattice MeasureTheory.SimpleFunc.instLattice instance instBoundedOrder [LE β] [BoundedOrder β] : BoundedOrder (α →ₛ β) := { SimpleFunc.instOrderBot, SimpleFunc.instOrderTop with } #align measure_theory.simple_func.bounded_order MeasureTheory.SimpleFunc.instBoundedOrder theorem finset_sup_apply [SemilatticeSup β] [OrderBot β] {f : γ → α →ₛ β} (s : Finset γ) (a : α) : s.sup f a = s.sup fun c => f c a := by refine Finset.induction_on s rfl ?_ intro a s _ ih rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] #align measure_theory.simple_func.finset_sup_apply MeasureTheory.SimpleFunc.finset_sup_apply section Restrict variable [Zero β] /-- Restrict a simple function `f : α →ₛ β` to a set `s`. If `s` is measurable, then `f.restrict s a = if a ∈ s then f a else 0`, otherwise `f.restrict s = const α 0`. -/ def restrict (f : α →ₛ β) (s : Set α) : α →ₛ β := if hs : MeasurableSet s then piecewise s hs f 0 else 0 #align measure_theory.simple_func.restrict MeasureTheory.SimpleFunc.restrict theorem restrict_of_not_measurable {f : α →ₛ β} {s : Set α} (hs : ¬MeasurableSet s) : restrict f s = 0 := dif_neg hs #align measure_theory.simple_func.restrict_of_not_measurable MeasureTheory.SimpleFunc.restrict_of_not_measurable @[simp] theorem coe_restrict (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) : ⇑(restrict f s) = indicator s f := by rw [restrict, dif_pos hs, coe_piecewise, coe_zero, piecewise_eq_indicator] #align measure_theory.simple_func.coe_restrict MeasureTheory.SimpleFunc.coe_restrict @[simp] theorem restrict_univ (f : α →ₛ β) : restrict f univ = f := by simp [restrict] #align measure_theory.simple_func.restrict_univ MeasureTheory.SimpleFunc.restrict_univ @[simp] theorem restrict_empty (f : α →ₛ β) : restrict f ∅ = 0 := by simp [restrict] #align measure_theory.simple_func.restrict_empty MeasureTheory.SimpleFunc.restrict_empty theorem map_restrict_of_zero [Zero γ] {g : β → γ} (hg : g 0 = 0) (f : α →ₛ β) (s : Set α) : (f.restrict s).map g = (f.map g).restrict s := ext fun x => if hs : MeasurableSet s then by simp [hs, Set.indicator_comp_of_zero hg] else by simp [restrict_of_not_measurable hs, hg] #align measure_theory.simple_func.map_restrict_of_zero MeasureTheory.SimpleFunc.map_restrict_of_zero theorem map_coe_ennreal_restrict (f : α →ₛ ℝ≥0) (s : Set α) : (f.restrict s).map ((↑) : ℝ≥0 → ℝ≥0∞) = (f.map (↑)).restrict s := map_restrict_of_zero ENNReal.coe_zero _ _ #align measure_theory.simple_func.map_coe_ennreal_restrict MeasureTheory.SimpleFunc.map_coe_ennreal_restrict theorem map_coe_nnreal_restrict (f : α →ₛ ℝ≥0) (s : Set α) : (f.restrict s).map ((↑) : ℝ≥0 → ℝ) = (f.map (↑)).restrict s := map_restrict_of_zero NNReal.coe_zero _ _ #align measure_theory.simple_func.map_coe_nnreal_restrict MeasureTheory.SimpleFunc.map_coe_nnreal_restrict theorem restrict_apply (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) (a) : restrict f s a = indicator s f a := by simp only [f.coe_restrict hs] #align measure_theory.simple_func.restrict_apply MeasureTheory.SimpleFunc.restrict_apply theorem restrict_preimage (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : (0 : β) ∉ t) : restrict f s ⁻¹' t = s ∩ f ⁻¹' t := by simp [hs, indicator_preimage_of_not_mem _ _ ht, inter_comm] #align measure_theory.simple_func.restrict_preimage MeasureTheory.SimpleFunc.restrict_preimage theorem restrict_preimage_singleton (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) {r : β} (hr : r ≠ 0) : restrict f s ⁻¹' {r} = s ∩ f ⁻¹' {r} := f.restrict_preimage hs hr.symm #align measure_theory.simple_func.restrict_preimage_singleton MeasureTheory.SimpleFunc.restrict_preimage_singleton theorem mem_restrict_range {r : β} {s : Set α} {f : α →ₛ β} (hs : MeasurableSet s) : r ∈ (restrict f s).range ↔ r = 0 ∧ s ≠ univ ∨ r ∈ f '' s := by rw [← Finset.mem_coe, coe_range, coe_restrict _ hs, mem_range_indicator] #align measure_theory.simple_func.mem_restrict_range MeasureTheory.SimpleFunc.mem_restrict_range theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β} (hr : r ∈ (restrict f s).range) (h0 : r ≠ 0) : r ∈ f '' s := if hs : MeasurableSet s then by simpa [mem_restrict_range hs, h0, -mem_range] using hr else by rw [restrict_of_not_measurable hs] at hr exact (h0 <\| eq_zero_of_mem_range_zero hr).elim #align measure_theory.simple_func.mem_image_of_mem_range_restrict MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict @[mono] theorem restrict_mono [Preorder β] (s : Set α) {f g : α →ₛ β} (H : f ≤ g) : f.restrict s ≤ g.restrict s := if hs : MeasurableSet s then fun x => by simp only [coe_restrict _ hs, indicator_le_indicator (H x)] else by simp only [restrict_of_not_measurable hs, le_refl] #align measure_theory.simple_func.restrict_mono MeasureTheory.SimpleFunc.restrict_mono end Restrict section Approx section variable [SemilatticeSup β] [OrderBot β] [Zero β] /-- Fix a sequence `i : ℕ → β`. Given a function `α → β`, its `n`-th approximation by simple functions is defined so that in case `β = ℝ≥0∞` it sends each `a` to the supremum of the set `{i k \| k ≤ n ∧ i k ≤ f a}`, see `approx_apply` and `iSup_approx_apply` for details. -/ def approx (i : ℕ → β) (f : α → β) (n : ℕ) : α →ₛ β := (Finset.range n).sup fun k => restrict (const α (i k)) { a : α \| i k ≤ f a } #align measure_theory.simple_func.approx MeasureTheory.SimpleFunc.approx theorem approx_apply [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β] [OpensMeasurableSpace β] {i : ℕ → β} {f : α → β} {n : ℕ} (a : α) (hf : Measurable f) : (approx i f n : α →ₛ β) a = (Finset.range n).sup fun k => if i k ≤ f a then i k else 0 := by dsimp only [approx] rw [finset_sup_apply] congr funext k rw [restrict_apply] · simp only [coe_const, mem_setOf_eq, indicator_apply, Function.const_apply] · exact hf measurableSet_Ici #align measure_theory.simple_func.approx_apply MeasureTheory.SimpleFunc.approx_apply theorem monotone_approx (i : ℕ → β) (f : α → β) : Monotone (approx i f) := fun _ _ h => Finset.sup_mono <\| Finset.range_subset.2 h #align measure_theory.simple_func.monotone_approx MeasureTheory.SimpleFunc.monotone_approx theorem approx_comp [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β] [OpensMeasurableSpace β] [MeasurableSpace γ] {i : ℕ → β} {f : γ → β} {g : α → γ} {n : ℕ} (a : α) (hf : Measurable f) (hg : Measurable g) : (approx i (f ∘ g) n : α →ₛ β) a = (approx i f n : γ →ₛ β) (g a) := by rw [approx_apply _ hf, approx_apply _ (hf.comp hg), Function.comp_apply] #align measure_theory.simple_func.approx_comp MeasureTheory.SimpleFunc.approx_comp end theorem iSup_approx_apply [TopologicalSpace β] [CompleteLattice β] [OrderClosedTopology β] [Zero β] [MeasurableSpace β] [OpensMeasurableSpace β] (i : ℕ → β) (f : α → β) (a : α) (hf : Measurable f) (h_zero : (0 : β) = ⊥) : ⨆ n, (approx i f n : α →ₛ β) a = ⨆ (k) (_ : i k ≤ f a), i k := by refine le_antisymm (iSup_le fun n => ?_) (iSup_le fun k => iSup_le fun hk => ?_) · rw [approx_apply a hf, h_zero] refine Finset.sup_le fun k _ => ?_ split_ifs with h · exact le_iSup_of_le k (le_iSup (fun _ : i k ≤ f a => i k) h) · exact bot_le · refine le_iSup_of_le (k + 1) ?_ rw [approx_apply a hf] have : k ∈ Finset.range (k + 1) := Finset.mem_range.2 (Nat.lt_succ_self _) refine le_trans (le_of_eq ?_) (Finset.le_sup this) rw [if_pos hk] #align measure_theory.simple_func.supr_approx_apply MeasureTheory.SimpleFunc.iSup_approx_apply end Approx section EApprox /-- A sequence of `ℝ≥0∞`s such that its range is the set of non-negative rational numbers. -/ def ennrealRatEmbed (n : ℕ) : ℝ≥0∞ := ENNReal.ofReal ((Encodable.decode (α := ℚ) n).getD (0 : ℚ)) #align measure_theory.simple_func.ennreal_rat_embed MeasureTheory.SimpleFunc.ennrealRatEmbed theorem ennrealRatEmbed_encode (q : ℚ) : ennrealRatEmbed (Encodable.encode q) = Real.toNNReal q := by rw [ennrealRatEmbed, Encodable.encodek]; rfl #align measure_theory.simple_func.ennreal_rat_embed_encode MeasureTheory.SimpleFunc.ennrealRatEmbed_encode /-- Approximate a function `α → ℝ≥0∞` by a sequence of simple functions. -/ def eapprox : (α → ℝ≥0∞) → ℕ → α →ₛ ℝ≥0∞ := approx ennrealRatEmbed #align measure_theory.simple_func.eapprox MeasureTheory.SimpleFunc.eapprox theorem eapprox_lt_top (f : α → ℝ≥0∞) (n : ℕ) (a : α) : eapprox f n a < ∞ := by simp only [eapprox, approx, finset_sup_apply, Finset.mem_range, ENNReal.bot_eq_zero, restrict] rw [Finset.sup_lt_iff (α := ℝ≥0∞) WithTop.zero_lt_top] intro b _ split_ifs · simp only [coe_zero, coe_piecewise, piecewise_eq_indicator, coe_const] calc { a : α \| ennrealRatEmbed b ≤ f a }.indicator (fun _ => ennrealRatEmbed b) a ≤ ennrealRatEmbed b := indicator_le_self _ _ a _ < ⊤ := ENNReal.coe_lt_top · exact WithTop.zero_lt_top #align measure_theory.simple_func.eapprox_lt_top MeasureTheory.SimpleFunc.eapprox_lt_top @[mono] theorem monotone_eapprox (f : α → ℝ≥0∞) : Monotone (eapprox f) := monotone_approx _ f #align measure_theory.simple_func.monotone_eapprox MeasureTheory.SimpleFunc.monotone_eapprox theorem iSup_eapprox_apply (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) : ⨆ n, (eapprox f n : α →ₛ ℝ≥0∞) a = f a := by rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl] refine le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt ?_) intro h rcases ENNReal.lt_iff_exists_rat_btwn.1 h with ⟨q, _, lt_q, q_lt⟩ have : (Real.toNNReal q : ℝ≥0∞) ≤ ⨆ (k : ℕ) (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k := by refine le_iSup_of_le (Encodable.encode q) ?_ rw [ennrealRatEmbed_encode q] exact le_iSup_of_le (le_of_lt q_lt) le_rfl exact lt_irrefl _ (lt_of_le_of_lt this lt_q) #align measure_theory.simple_func.supr_eapprox_apply MeasureTheory.SimpleFunc.iSup_eapprox_apply theorem eapprox_comp [MeasurableSpace γ] {f : γ → ℝ≥0∞} {g : α → γ} {n : ℕ} (hf : Measurable f) (hg : Measurable g) : (eapprox (f ∘ g) n : α → ℝ≥0∞) = (eapprox f n : γ →ₛ ℝ≥0∞) ∘ g := funext fun a => approx_comp a hf hg #align measure_theory.simple_func.eapprox_comp MeasureTheory.SimpleFunc.eapprox_comp /-- Approximate a function `α → ℝ≥0∞` by a series of simple functions taking their values in `ℝ≥0`. -/ def eapproxDiff (f : α → ℝ≥0∞) : ℕ → α →ₛ ℝ≥0 \| 0 => (eapprox f 0).map ENNReal.toNNReal \| n + 1 => (eapprox f (n + 1) - eapprox f n).map ENNReal.toNNReal #align measure_theory.simple_func.eapprox_diff MeasureTheory.SimpleFunc.eapproxDiff theorem sum_eapproxDiff (f : α → ℝ≥0∞) (n : ℕ) (a : α) : (∑ k ∈ Finset.range (n + 1), (eapproxDiff f k a : ℝ≥0∞)) = eapprox f n a := by induction' n with n IH · simp only [Nat.zero_eq, Nat.zero_add, Finset.sum_singleton, Finset.range_one] rfl · erw [Finset.sum_range_succ, IH, eapproxDiff, coe_map, Function.comp_apply, coe_sub, Pi.sub_apply, ENNReal.coe_toNNReal, add_tsub_cancel_of_le (monotone_eapprox f (Nat.le_succ _) _)] apply (lt_of_le_of_lt _ (eapprox_lt_top f (n + 1) a)).ne rw [tsub_le_iff_right] exact le_self_add #align measure_theory.simple_func.sum_eapprox_diff MeasureTheory.SimpleFunc.sum_eapproxDiff theorem tsum_eapproxDiff (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) : (∑' n, (eapproxDiff f n a : ℝ≥0∞)) = f a := by simp_rw [ENNReal.tsum_eq_iSup_nat' (tendsto_add_atTop_nat 1), sum_eapproxDiff, iSup_eapprox_apply f hf a] #align measure_theory.simple_func.tsum_eapprox_diff MeasureTheory.SimpleFunc.tsum_eapproxDiff end EApprox end Measurable section Measure variable {m : MeasurableSpace α} {μ ν : Measure α} /-- Integral of a simple function whose codomain is `ℝ≥0∞`. -/ def lintegral {_m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ := ∑ x ∈ f.range, x μ (f ⁻¹' {x}) #align measure_theory.simple_func.lintegral MeasureTheory.SimpleFunc.lintegral	Mathlib/MeasureTheory/Function/SimpleFunc.lean	969	981	theorem lintegral_eq_of_subset (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : ∀ x, f x ≠ 0 → μ (f ⁻¹' {f x}) ≠ 0 → f x ∈ s) : f.lintegral μ = ∑ x ∈ s, x * μ (f ⁻¹' {x}) := by	refine Finset.sum_bij_ne_zero (fun r _ _ => r) ?_ ?_ ?_ ?_ · simpa only [forall_mem_range, mul_ne_zero_iff, and_imp] · intros assumption · intro b _ hb refine ⟨b, ?_, hb, rfl⟩ rw [mem_range, ← preimage_singleton_nonempty] exact nonempty_of_measure_ne_zero (mul_ne_zero_iff.1 hb).2 · intros rfl
/- Copyright (c) 2020 Markus Himmel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Markus Himmel, Alex Keizer -/ import Mathlib.Data.List.GetD import Mathlib.Data.Nat.Bits import Mathlib.Algebra.Ring.Nat import Mathlib.Order.Basic import Mathlib.Tactic.AdaptationNote import Mathlib.Tactic.Common #align_import data.nat.bitwise from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2" /-! # Bitwise operations on natural numbers In the first half of this file, we provide theorems for reasoning about natural numbers from their bitwise properties. In the second half of this file, we show properties of the bitwise operations `lor`, `land` and `xor`, which are defined in core. ## Main results * `eq_of_testBit_eq`: two natural numbers are equal if they have equal bits at every position. * `exists_most_significant_bit`: if `n ≠ 0`, then there is some position `i` that contains the most significant `1`-bit of `n`. * `lt_of_testBit`: if `n` and `m` are numbers and `i` is a position such that the `i`-th bit of of `n` is zero, the `i`-th bit of `m` is one, and all more significant bits are equal, then `n < m`. ## Future work There is another way to express bitwise properties of natural number: `digits 2`. The two ways should be connected. ## Keywords bitwise, and, or, xor -/ open Function namespace Nat set_option linter.deprecated false section variable {f : Bool → Bool → Bool} @[simp] lemma bitwise_zero_left (m : Nat) : bitwise f 0 m = if f false true then m else 0 := by simp [bitwise] #align nat.bitwise_zero_left Nat.bitwise_zero_left @[simp] lemma bitwise_zero_right (n : Nat) : bitwise f n 0 = if f true false then n else 0 := by unfold bitwise simp only [ite_self, decide_False, Nat.zero_div, ite_true, ite_eq_right_iff] rintro ⟨⟩ split_ifs <;> rfl #align nat.bitwise_zero_right Nat.bitwise_zero_right lemma bitwise_zero : bitwise f 0 0 = 0 := by simp only [bitwise_zero_right, ite_self] #align nat.bitwise_zero Nat.bitwise_zero lemma bitwise_of_ne_zero {n m : Nat} (hn : n ≠ 0) (hm : m ≠ 0) : bitwise f n m = bit (f (bodd n) (bodd m)) (bitwise f (n / 2) (m / 2)) := by conv_lhs => unfold bitwise have mod_two_iff_bod x : (x % 2 = 1 : Bool) = bodd x := by simp only [mod_two_of_bodd, cond]; cases bodd x <;> rfl simp only [hn, hm, mod_two_iff_bod, ite_false, bit, bit1, bit0, Bool.cond_eq_ite] split_ifs <;> rfl theorem binaryRec_of_ne_zero {C : Nat → Sort} (z : C 0) (f : ∀ b n, C n → C (bit b n)) {n} (h : n ≠ 0) : binaryRec z f n = bit_decomp n ▸ f (bodd n) (div2 n) (binaryRec z f (div2 n)) := by rw [Eq.rec_eq_cast] rw [binaryRec] dsimp only rw [dif_neg h, eq_mpr_eq_cast] @[simp] lemma bitwise_bit {f : Bool → Bool → Bool} (h : f false false = false := by rfl) (a m b n) : bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) := by conv_lhs => unfold bitwise #adaptation_note /-- nightly-2024-03-16: simp was -- simp (config := { unfoldPartialApp := true }) only [bit, bit1, bit0, Bool.cond_eq_ite] -/ simp only [bit, ite_apply, bit1, bit0, Bool.cond_eq_ite] have h1 x : (x + x) % 2 = 0 := by rw [← two_mul, mul_comm]; apply mul_mod_left have h2 x : (x + x + 1) % 2 = 1 := by rw [← two_mul, add_comm]; apply add_mul_mod_self_left have h3 x : (x + x) / 2 = x := by omega have h4 x : (x + x + 1) / 2 = x := by rw [← two_mul, add_comm]; simp [add_mul_div_left] cases a <;> cases b <;> simp [h1, h2, h3, h4] <;> split_ifs <;> simp_all (config := {decide := true}) #align nat.bitwise_bit Nat.bitwise_bit lemma bit_mod_two (a : Bool) (x : ℕ) : bit a x % 2 = if a then 1 else 0 := by #adaptation_note /-- nightly-2024-03-16: simp was -- simp (config := { unfoldPartialApp := true }) only [bit, bit1, bit0, ← mul_two, -- Bool.cond_eq_ite] -/ simp only [bit, ite_apply, bit1, bit0, ← mul_two, Bool.cond_eq_ite] split_ifs <;> simp [Nat.add_mod] @[simp] lemma bit_mod_two_eq_zero_iff (a x) : bit a x % 2 = 0 ↔ !a := by rw [bit_mod_two]; split_ifs <;> simp_all @[simp] lemma bit_mod_two_eq_one_iff (a x) : bit a x % 2 = 1 ↔ a := by rw [bit_mod_two]; split_ifs <;> simp_all @[simp] theorem lor_bit : ∀ a m b n, bit a m \|\|\| bit b n = bit (a \|\| b) (m \|\|\| n) := bitwise_bit #align nat.lor_bit Nat.lor_bit @[simp] theorem land_bit : ∀ a m b n, bit a m &&& bit b n = bit (a && b) (m &&& n) := bitwise_bit #align nat.land_bit Nat.land_bit @[simp] theorem ldiff_bit : ∀ a m b n, ldiff (bit a m) (bit b n) = bit (a && not b) (ldiff m n) := bitwise_bit #align nat.ldiff_bit Nat.ldiff_bit @[simp] theorem xor_bit : ∀ a m b n, bit a m ^^^ bit b n = bit (bne a b) (m ^^^ n) := bitwise_bit #align nat.lxor_bit Nat.xor_bit attribute [simp] Nat.testBit_bitwise #align nat.test_bit_bitwise Nat.testBit_bitwise theorem testBit_lor : ∀ m n k, testBit (m \|\|\| n) k = (testBit m k \|\| testBit n k) := testBit_bitwise rfl #align nat.test_bit_lor Nat.testBit_lor theorem testBit_land : ∀ m n k, testBit (m &&& n) k = (testBit m k && testBit n k) := testBit_bitwise rfl #align nat.test_bit_land Nat.testBit_land @[simp] theorem testBit_ldiff : ∀ m n k, testBit (ldiff m n) k = (testBit m k && not (testBit n k)) := testBit_bitwise rfl #align nat.test_bit_ldiff Nat.testBit_ldiff attribute [simp] testBit_xor #align nat.test_bit_lxor Nat.testBit_xor end @[simp] theorem bit_false : bit false = bit0 := rfl #align nat.bit_ff Nat.bit_false @[simp] theorem bit_true : bit true = bit1 := rfl #align nat.bit_tt Nat.bit_true @[simp] theorem bit_eq_zero {n : ℕ} {b : Bool} : n.bit b = 0 ↔ n = 0 ∧ b = false := by cases b <;> simp [Nat.bit0_eq_zero, Nat.bit1_ne_zero] #align nat.bit_eq_zero Nat.bit_eq_zero theorem bit_ne_zero_iff {n : ℕ} {b : Bool} : n.bit b ≠ 0 ↔ n = 0 → b = true := by simpa only [not_and, Bool.not_eq_false] using (@bit_eq_zero n b).not /-- An alternative for `bitwise_bit` which replaces the `f false false = false` assumption with assumptions that neither `bit a m` nor `bit b n` are `0` (albeit, phrased as the implications `m = 0 → a = true` and `n = 0 → b = true`) -/ lemma bitwise_bit' {f : Bool → Bool → Bool} (a : Bool) (m : Nat) (b : Bool) (n : Nat) (ham : m = 0 → a = true) (hbn : n = 0 → b = true) : bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) := by conv_lhs => unfold bitwise rw [← bit_ne_zero_iff] at ham hbn simp only [ham, hbn, bit_mod_two_eq_one_iff, Bool.decide_coe, ← div2_val, div2_bit, ne_eq, ite_false] conv_rhs => simp only [bit, bit1, bit0, Bool.cond_eq_ite] split_ifs with hf <;> rfl lemma bitwise_eq_binaryRec (f : Bool → Bool → Bool) : bitwise f = binaryRec (fun n => cond (f false true) n 0) fun a m Ia => binaryRec (cond (f true false) (bit a m) 0) fun b n _ => bit (f a b) (Ia n) := by funext x y induction x using binaryRec' generalizing y with \| z => simp only [bitwise_zero_left, binaryRec_zero, Bool.cond_eq_ite] \| f xb x hxb ih => rw [← bit_ne_zero_iff] at hxb simp_rw [binaryRec_of_ne_zero _ _ hxb, bodd_bit, div2_bit, eq_rec_constant] induction y using binaryRec' with \| z => simp only [bitwise_zero_right, binaryRec_zero, Bool.cond_eq_ite] \| f yb y hyb => rw [← bit_ne_zero_iff] at hyb simp_rw [binaryRec_of_ne_zero _ _ hyb, bitwise_of_ne_zero hxb hyb, bodd_bit, ← div2_val, div2_bit, eq_rec_constant, ih] theorem zero_of_testBit_eq_false {n : ℕ} (h : ∀ i, testBit n i = false) : n = 0 := by induction' n using Nat.binaryRec with b n hn · rfl · have : b = false := by simpa using h 0 rw [this, bit_false, bit0_val, hn fun i => by rw [← h (i + 1), testBit_bit_succ], mul_zero] #align nat.zero_of_test_bit_eq_ff Nat.zero_of_testBit_eq_false theorem testBit_eq_false_of_lt {n i} (h : n < 2 ^ i) : n.testBit i = false := by simp [testBit, shiftRight_eq_div_pow, Nat.div_eq_of_lt h] #align nat.zero_test_bit Nat.zero_testBit /-- The ith bit is the ith element of `n.bits`. -/ theorem testBit_eq_inth (n i : ℕ) : n.testBit i = n.bits.getI i := by induction' i with i ih generalizing n · simp only [testBit, zero_eq, shiftRight_zero, one_and_eq_mod_two, mod_two_of_bodd, bodd_eq_bits_head, List.getI_zero_eq_headI] cases List.headI (bits n) <;> rfl conv_lhs => rw [← bit_decomp n] rw [testBit_bit_succ, ih n.div2, div2_bits_eq_tail] cases n.bits <;> simp #align nat.test_bit_eq_inth Nat.testBit_eq_inth #align nat.eq_of_test_bit_eq Nat.eq_of_testBit_eq theorem exists_most_significant_bit {n : ℕ} (h : n ≠ 0) : ∃ i, testBit n i = true ∧ ∀ j, i < j → testBit n j = false := by induction' n using Nat.binaryRec with b n hn · exact False.elim (h rfl) by_cases h' : n = 0 · subst h' rw [show b = true by revert h cases b <;> simp] refine ⟨0, ⟨by rw [testBit_bit_zero], fun j hj => ?_⟩⟩ obtain ⟨j', rfl⟩ := exists_eq_succ_of_ne_zero (ne_of_gt hj) rw [testBit_bit_succ, zero_testBit] · obtain ⟨k, ⟨hk, hk'⟩⟩ := hn h' refine ⟨k + 1, ⟨by rw [testBit_bit_succ, hk], fun j hj => ?_⟩⟩ obtain ⟨j', rfl⟩ := exists_eq_succ_of_ne_zero (show j ≠ 0 by intro x; subst x; simp at hj) exact (testBit_bit_succ _ _ _).trans (hk' _ (lt_of_succ_lt_succ hj)) #align nat.exists_most_significant_bit Nat.exists_most_significant_bit theorem lt_of_testBit {n m : ℕ} (i : ℕ) (hn : testBit n i = false) (hm : testBit m i = true) (hnm : ∀ j, i < j → testBit n j = testBit m j) : n < m := by induction' n using Nat.binaryRec with b n hn' generalizing i m · rw [Nat.pos_iff_ne_zero] rintro rfl simp at hm induction' m using Nat.binaryRec with b' m hm' generalizing i · exact False.elim (Bool.false_ne_true ((zero_testBit i).symm.trans hm)) by_cases hi : i = 0 · subst hi simp only [testBit_bit_zero] at hn hm have : n = m := eq_of_testBit_eq fun i => by convert hnm (i + 1) (Nat.zero_lt_succ _) using 1 <;> rw [testBit_bit_succ] rw [hn, hm, this, bit_false, bit_true, bit0_val, bit1_val] exact Nat.lt_succ_self _ · obtain ⟨i', rfl⟩ := exists_eq_succ_of_ne_zero hi simp only [testBit_bit_succ] at hn hm have := hn' _ hn hm fun j hj => by convert hnm j.succ (succ_lt_succ hj) using 1 <;> rw [testBit_bit_succ] have this' : 2 n < 2 * m := Nat.mul_lt_mul' (le_refl _) this Nat.two_pos cases b <;> cases b' <;> simp only [bit_false, bit_true, bit0_val n, bit1_val n, bit0_val m, bit1_val m] · exact this' · exact Nat.lt_add_right 1 this' · calc 2 * n + 1 < 2 * n + 2 := lt.base _ _ ≤ 2 * m := mul_le_mul_left 2 this · exact Nat.succ_lt_succ this' #align nat.lt_of_test_bit Nat.lt_of_testBit @[simp] theorem testBit_two_pow_self (n : ℕ) : testBit (2 ^ n) n = true := by rw [testBit, shiftRight_eq_div_pow, Nat.div_self (Nat.pow_pos Nat.zero_lt_two)] simp #align nat.test_bit_two_pow_self Nat.testBit_two_pow_self theorem testBit_two_pow_of_ne {n m : ℕ} (hm : n ≠ m) : testBit (2 ^ n) m = false := by rw [testBit, shiftRight_eq_div_pow] cases' hm.lt_or_lt with hm hm · rw [Nat.div_eq_of_lt] · simp · exact Nat.pow_lt_pow_right Nat.one_lt_two hm · rw [Nat.pow_div hm.le Nat.two_pos, ← Nat.sub_add_cancel (succ_le_of_lt <\| Nat.sub_pos_of_lt hm)] -- Porting note: XXX why does this make it work? rw [(rfl : succ 0 = 1)] simp [pow_succ, and_one_is_mod, mul_mod_left] #align nat.test_bit_two_pow_of_ne Nat.testBit_two_pow_of_ne theorem testBit_two_pow (n m : ℕ) : testBit (2 ^ n) m = (n = m) := by by_cases h : n = m · cases h simp · rw [testBit_two_pow_of_ne h] simp [h] #align nat.test_bit_two_pow Nat.testBit_two_pow	Mathlib/Data/Nat/Bitwise.lean	305	314	theorem bitwise_swap {f : Bool → Bool → Bool} : bitwise (Function.swap f) = Function.swap (bitwise f) := by	funext m n simp only [Function.swap] induction' m using Nat.strongInductionOn with m ih generalizing n cases' m with m <;> cases' n with n <;> try rw [bitwise_zero_left, bitwise_zero_right] · specialize ih ((m+1) / 2) (div_lt_self' ..) simp [bitwise_of_ne_zero, ih]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot -/ import Mathlib.Algebra.BigOperators.Group.Finset import Mathlib.Algebra.Order.Field.Defs import Mathlib.Algebra.Order.Group.Instances import Mathlib.Algebra.Order.Group.MinMax import Mathlib.Algebra.Order.Ring.Basic import Mathlib.Data.Finset.Preimage import Mathlib.Order.Interval.Set.Disjoint import Mathlib.Order.Interval.Set.OrderIso import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.Filter.Bases #align_import order.filter.at_top_bot from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58" /-! # `Filter.atTop` and `Filter.atBot` filters on preorders, monoids and groups. In this file we define the filters * `Filter.atTop`: corresponds to `n → +∞`; * `Filter.atBot`: corresponds to `n → -∞`. Then we prove many lemmas like “if `f → +∞`, then `f ± c → +∞`”. -/ set_option autoImplicit true variable {ι ι' α β γ : Type} open Set namespace Filter /-- `atTop` is the filter representing the limit `→ ∞` on an ordered set. It is generated by the collection of up-sets `{b \| a ≤ b}`. (The preorder need not have a top element for this to be well defined, and indeed is trivial when a top element exists.) -/ def atTop [Preorder α] : Filter α := ⨅ a, 𝓟 (Ici a) #align filter.at_top Filter.atTop /-- `atBot` is the filter representing the limit `→ -∞` on an ordered set. It is generated by the collection of down-sets `{b \| b ≤ a}`. (The preorder need not have a bottom element for this to be well defined, and indeed is trivial when a bottom element exists.) -/ def atBot [Preorder α] : Filter α := ⨅ a, 𝓟 (Iic a) #align filter.at_bot Filter.atBot theorem mem_atTop [Preorder α] (a : α) : { b : α \| a ≤ b } ∈ @atTop α _ := mem_iInf_of_mem a <\| Subset.refl _ #align filter.mem_at_top Filter.mem_atTop theorem Ici_mem_atTop [Preorder α] (a : α) : Ici a ∈ (atTop : Filter α) := mem_atTop a #align filter.Ici_mem_at_top Filter.Ici_mem_atTop theorem Ioi_mem_atTop [Preorder α] [NoMaxOrder α] (x : α) : Ioi x ∈ (atTop : Filter α) := let ⟨z, hz⟩ := exists_gt x mem_of_superset (mem_atTop z) fun _ h => lt_of_lt_of_le hz h #align filter.Ioi_mem_at_top Filter.Ioi_mem_atTop theorem mem_atBot [Preorder α] (a : α) : { b : α \| b ≤ a } ∈ @atBot α _ := mem_iInf_of_mem a <\| Subset.refl _ #align filter.mem_at_bot Filter.mem_atBot theorem Iic_mem_atBot [Preorder α] (a : α) : Iic a ∈ (atBot : Filter α) := mem_atBot a #align filter.Iic_mem_at_bot Filter.Iic_mem_atBot theorem Iio_mem_atBot [Preorder α] [NoMinOrder α] (x : α) : Iio x ∈ (atBot : Filter α) := let ⟨z, hz⟩ := exists_lt x mem_of_superset (mem_atBot z) fun _ h => lt_of_le_of_lt h hz #align filter.Iio_mem_at_bot Filter.Iio_mem_atBot theorem disjoint_atBot_principal_Ioi [Preorder α] (x : α) : Disjoint atBot (𝓟 (Ioi x)) := disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl) (Iic_mem_atBot x) (mem_principal_self _) #align filter.disjoint_at_bot_principal_Ioi Filter.disjoint_atBot_principal_Ioi theorem disjoint_atTop_principal_Iio [Preorder α] (x : α) : Disjoint atTop (𝓟 (Iio x)) := @disjoint_atBot_principal_Ioi αᵒᵈ _ _ #align filter.disjoint_at_top_principal_Iio Filter.disjoint_atTop_principal_Iio theorem disjoint_atTop_principal_Iic [Preorder α] [NoMaxOrder α] (x : α) : Disjoint atTop (𝓟 (Iic x)) := disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl).symm (Ioi_mem_atTop x) (mem_principal_self _) #align filter.disjoint_at_top_principal_Iic Filter.disjoint_atTop_principal_Iic theorem disjoint_atBot_principal_Ici [Preorder α] [NoMinOrder α] (x : α) : Disjoint atBot (𝓟 (Ici x)) := @disjoint_atTop_principal_Iic αᵒᵈ _ _ _ #align filter.disjoint_at_bot_principal_Ici Filter.disjoint_atBot_principal_Ici theorem disjoint_pure_atTop [Preorder α] [NoMaxOrder α] (x : α) : Disjoint (pure x) atTop := Disjoint.symm <\| (disjoint_atTop_principal_Iic x).mono_right <\| le_principal_iff.2 <\| mem_pure.2 right_mem_Iic #align filter.disjoint_pure_at_top Filter.disjoint_pure_atTop theorem disjoint_pure_atBot [Preorder α] [NoMinOrder α] (x : α) : Disjoint (pure x) atBot := @disjoint_pure_atTop αᵒᵈ _ _ _ #align filter.disjoint_pure_at_bot Filter.disjoint_pure_atBot theorem not_tendsto_const_atTop [Preorder α] [NoMaxOrder α] (x : α) (l : Filter β) [l.NeBot] : ¬Tendsto (fun _ => x) l atTop := tendsto_const_pure.not_tendsto (disjoint_pure_atTop x) #align filter.not_tendsto_const_at_top Filter.not_tendsto_const_atTop theorem not_tendsto_const_atBot [Preorder α] [NoMinOrder α] (x : α) (l : Filter β) [l.NeBot] : ¬Tendsto (fun _ => x) l atBot := tendsto_const_pure.not_tendsto (disjoint_pure_atBot x) #align filter.not_tendsto_const_at_bot Filter.not_tendsto_const_atBot theorem disjoint_atBot_atTop [PartialOrder α] [Nontrivial α] : Disjoint (atBot : Filter α) atTop := by rcases exists_pair_ne α with ⟨x, y, hne⟩ by_cases hle : x ≤ y · refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot x) (Ici_mem_atTop y) exact Iic_disjoint_Ici.2 (hle.lt_of_ne hne).not_le · refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot y) (Ici_mem_atTop x) exact Iic_disjoint_Ici.2 hle #align filter.disjoint_at_bot_at_top Filter.disjoint_atBot_atTop theorem disjoint_atTop_atBot [PartialOrder α] [Nontrivial α] : Disjoint (atTop : Filter α) atBot := disjoint_atBot_atTop.symm #align filter.disjoint_at_top_at_bot Filter.disjoint_atTop_atBot theorem hasAntitoneBasis_atTop [Nonempty α] [Preorder α] [IsDirected α (· ≤ ·)] : (@atTop α _).HasAntitoneBasis Ici := .iInf_principal fun _ _ ↦ Ici_subset_Ici.2 theorem atTop_basis [Nonempty α] [SemilatticeSup α] : (@atTop α _).HasBasis (fun _ => True) Ici := hasAntitoneBasis_atTop.1 #align filter.at_top_basis Filter.atTop_basis theorem atTop_eq_generate_Ici [SemilatticeSup α] : atTop = generate (range (Ici (α := α))) := by rcases isEmpty_or_nonempty α with hα\|hα · simp only [eq_iff_true_of_subsingleton] · simp [(atTop_basis (α := α)).eq_generate, range] theorem atTop_basis' [SemilatticeSup α] (a : α) : (@atTop α _).HasBasis (fun x => a ≤ x) Ici := ⟨fun _ => (@atTop_basis α ⟨a⟩ _).mem_iff.trans ⟨fun ⟨x, _, hx⟩ => ⟨x ⊔ a, le_sup_right, fun _y hy => hx (le_trans le_sup_left hy)⟩, fun ⟨x, _, hx⟩ => ⟨x, trivial, hx⟩⟩⟩ #align filter.at_top_basis' Filter.atTop_basis' theorem atBot_basis [Nonempty α] [SemilatticeInf α] : (@atBot α _).HasBasis (fun _ => True) Iic := @atTop_basis αᵒᵈ _ _ #align filter.at_bot_basis Filter.atBot_basis theorem atBot_basis' [SemilatticeInf α] (a : α) : (@atBot α _).HasBasis (fun x => x ≤ a) Iic := @atTop_basis' αᵒᵈ _ _ #align filter.at_bot_basis' Filter.atBot_basis' @[instance] theorem atTop_neBot [Nonempty α] [SemilatticeSup α] : NeBot (atTop : Filter α) := atTop_basis.neBot_iff.2 fun _ => nonempty_Ici #align filter.at_top_ne_bot Filter.atTop_neBot @[instance] theorem atBot_neBot [Nonempty α] [SemilatticeInf α] : NeBot (atBot : Filter α) := @atTop_neBot αᵒᵈ _ _ #align filter.at_bot_ne_bot Filter.atBot_neBot @[simp] theorem mem_atTop_sets [Nonempty α] [SemilatticeSup α] {s : Set α} : s ∈ (atTop : Filter α) ↔ ∃ a : α, ∀ b ≥ a, b ∈ s := atTop_basis.mem_iff.trans <\| exists_congr fun _ => true_and_iff _ #align filter.mem_at_top_sets Filter.mem_atTop_sets @[simp] theorem mem_atBot_sets [Nonempty α] [SemilatticeInf α] {s : Set α} : s ∈ (atBot : Filter α) ↔ ∃ a : α, ∀ b ≤ a, b ∈ s := @mem_atTop_sets αᵒᵈ _ _ _ #align filter.mem_at_bot_sets Filter.mem_atBot_sets @[simp] theorem eventually_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} : (∀ᶠ x in atTop, p x) ↔ ∃ a, ∀ b ≥ a, p b := mem_atTop_sets #align filter.eventually_at_top Filter.eventually_atTop @[simp] theorem eventually_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} : (∀ᶠ x in atBot, p x) ↔ ∃ a, ∀ b ≤ a, p b := mem_atBot_sets #align filter.eventually_at_bot Filter.eventually_atBot theorem eventually_ge_atTop [Preorder α] (a : α) : ∀ᶠ x in atTop, a ≤ x := mem_atTop a #align filter.eventually_ge_at_top Filter.eventually_ge_atTop theorem eventually_le_atBot [Preorder α] (a : α) : ∀ᶠ x in atBot, x ≤ a := mem_atBot a #align filter.eventually_le_at_bot Filter.eventually_le_atBot theorem eventually_gt_atTop [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ x in atTop, a < x := Ioi_mem_atTop a #align filter.eventually_gt_at_top Filter.eventually_gt_atTop theorem eventually_ne_atTop [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ x in atTop, x ≠ a := (eventually_gt_atTop a).mono fun _ => ne_of_gt #align filter.eventually_ne_at_top Filter.eventually_ne_atTop protected theorem Tendsto.eventually_gt_atTop [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, c < f x := hf.eventually (eventually_gt_atTop c) #align filter.tendsto.eventually_gt_at_top Filter.Tendsto.eventually_gt_atTop protected theorem Tendsto.eventually_ge_atTop [Preorder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, c ≤ f x := hf.eventually (eventually_ge_atTop c) #align filter.tendsto.eventually_ge_at_top Filter.Tendsto.eventually_ge_atTop protected theorem Tendsto.eventually_ne_atTop [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, f x ≠ c := hf.eventually (eventually_ne_atTop c) #align filter.tendsto.eventually_ne_at_top Filter.Tendsto.eventually_ne_atTop protected theorem Tendsto.eventually_ne_atTop' [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : α) : ∀ᶠ x in l, x ≠ c := (hf.eventually_ne_atTop (f c)).mono fun _ => ne_of_apply_ne f #align filter.tendsto.eventually_ne_at_top' Filter.Tendsto.eventually_ne_atTop' theorem eventually_lt_atBot [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ x in atBot, x < a := Iio_mem_atBot a #align filter.eventually_lt_at_bot Filter.eventually_lt_atBot theorem eventually_ne_atBot [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ x in atBot, x ≠ a := (eventually_lt_atBot a).mono fun _ => ne_of_lt #align filter.eventually_ne_at_bot Filter.eventually_ne_atBot protected theorem Tendsto.eventually_lt_atBot [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x < c := hf.eventually (eventually_lt_atBot c) #align filter.tendsto.eventually_lt_at_bot Filter.Tendsto.eventually_lt_atBot protected theorem Tendsto.eventually_le_atBot [Preorder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x ≤ c := hf.eventually (eventually_le_atBot c) #align filter.tendsto.eventually_le_at_bot Filter.Tendsto.eventually_le_atBot protected theorem Tendsto.eventually_ne_atBot [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x ≠ c := hf.eventually (eventually_ne_atBot c) #align filter.tendsto.eventually_ne_at_bot Filter.Tendsto.eventually_ne_atBot theorem eventually_forall_ge_atTop [Preorder α] {p : α → Prop} : (∀ᶠ x in atTop, ∀ y, x ≤ y → p y) ↔ ∀ᶠ x in atTop, p x := by refine ⟨fun h ↦ h.mono fun x hx ↦ hx x le_rfl, fun h ↦ ?_⟩ rcases (hasBasis_iInf_principal_finite _).eventually_iff.1 h with ⟨S, hSf, hS⟩ refine mem_iInf_of_iInter hSf (V := fun x ↦ Ici x.1) (fun _ ↦ Subset.rfl) fun x hx y hy ↦ ?_ simp only [mem_iInter] at hS hx exact hS fun z hz ↦ le_trans (hx ⟨z, hz⟩) hy theorem eventually_forall_le_atBot [Preorder α] {p : α → Prop} : (∀ᶠ x in atBot, ∀ y, y ≤ x → p y) ↔ ∀ᶠ x in atBot, p x := eventually_forall_ge_atTop (α := αᵒᵈ) theorem Tendsto.eventually_forall_ge_atTop {α β : Type} [Preorder β] {l : Filter α} {p : β → Prop} {f : α → β} (hf : Tendsto f l atTop) (h_evtl : ∀ᶠ x in atTop, p x) : ∀ᶠ x in l, ∀ y, f x ≤ y → p y := by rw [← Filter.eventually_forall_ge_atTop] at h_evtl; exact (h_evtl.comap f).filter_mono hf.le_comap theorem Tendsto.eventually_forall_le_atBot {α β : Type} [Preorder β] {l : Filter α} {p : β → Prop} {f : α → β} (hf : Tendsto f l atBot) (h_evtl : ∀ᶠ x in atBot, p x) : ∀ᶠ x in l, ∀ y, y ≤ f x → p y := by rw [← Filter.eventually_forall_le_atBot] at h_evtl; exact (h_evtl.comap f).filter_mono hf.le_comap theorem atTop_basis_Ioi [Nonempty α] [SemilatticeSup α] [NoMaxOrder α] : (@atTop α _).HasBasis (fun _ => True) Ioi := atTop_basis.to_hasBasis (fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩) fun a ha => (exists_gt a).imp fun _b hb => ⟨ha, Ici_subset_Ioi.2 hb⟩ #align filter.at_top_basis_Ioi Filter.atTop_basis_Ioi lemma atTop_basis_Ioi' [SemilatticeSup α] [NoMaxOrder α] (a : α) : atTop.HasBasis (a < ·) Ioi := have : Nonempty α := ⟨a⟩ atTop_basis_Ioi.to_hasBasis (fun b _ ↦ let ⟨c, hc⟩ := exists_gt (a ⊔ b) ⟨c, le_sup_left.trans_lt hc, Ioi_subset_Ioi <\| le_sup_right.trans hc.le⟩) fun b _ ↦ ⟨b, trivial, Subset.rfl⟩ theorem atTop_countable_basis [Nonempty α] [SemilatticeSup α] [Countable α] : HasCountableBasis (atTop : Filter α) (fun _ => True) Ici := { atTop_basis with countable := to_countable _ } #align filter.at_top_countable_basis Filter.atTop_countable_basis theorem atBot_countable_basis [Nonempty α] [SemilatticeInf α] [Countable α] : HasCountableBasis (atBot : Filter α) (fun _ => True) Iic := { atBot_basis with countable := to_countable _ } #align filter.at_bot_countable_basis Filter.atBot_countable_basis instance (priority := 200) atTop.isCountablyGenerated [Preorder α] [Countable α] : (atTop : Filter <\| α).IsCountablyGenerated := isCountablyGenerated_seq _ #align filter.at_top.is_countably_generated Filter.atTop.isCountablyGenerated instance (priority := 200) atBot.isCountablyGenerated [Preorder α] [Countable α] : (atBot : Filter <\| α).IsCountablyGenerated := isCountablyGenerated_seq _ #align filter.at_bot.is_countably_generated Filter.atBot.isCountablyGenerated theorem _root_.IsTop.atTop_eq [Preorder α] {a : α} (ha : IsTop a) : atTop = 𝓟 (Ici a) := (iInf_le _ _).antisymm <\| le_iInf fun b ↦ principal_mono.2 <\| Ici_subset_Ici.2 <\| ha b theorem _root_.IsBot.atBot_eq [Preorder α] {a : α} (ha : IsBot a) : atBot = 𝓟 (Iic a) := ha.toDual.atTop_eq theorem OrderTop.atTop_eq (α) [PartialOrder α] [OrderTop α] : (atTop : Filter α) = pure ⊤ := by rw [isTop_top.atTop_eq, Ici_top, principal_singleton] #align filter.order_top.at_top_eq Filter.OrderTop.atTop_eq theorem OrderBot.atBot_eq (α) [PartialOrder α] [OrderBot α] : (atBot : Filter α) = pure ⊥ := @OrderTop.atTop_eq αᵒᵈ _ _ #align filter.order_bot.at_bot_eq Filter.OrderBot.atBot_eq @[nontriviality] theorem Subsingleton.atTop_eq (α) [Subsingleton α] [Preorder α] : (atTop : Filter α) = ⊤ := by refine top_unique fun s hs x => ?_ rw [atTop, ciInf_subsingleton x, mem_principal] at hs exact hs left_mem_Ici #align filter.subsingleton.at_top_eq Filter.Subsingleton.atTop_eq @[nontriviality] theorem Subsingleton.atBot_eq (α) [Subsingleton α] [Preorder α] : (atBot : Filter α) = ⊤ := @Subsingleton.atTop_eq αᵒᵈ _ _ #align filter.subsingleton.at_bot_eq Filter.Subsingleton.atBot_eq theorem tendsto_atTop_pure [PartialOrder α] [OrderTop α] (f : α → β) : Tendsto f atTop (pure <\| f ⊤) := (OrderTop.atTop_eq α).symm ▸ tendsto_pure_pure _ _ #align filter.tendsto_at_top_pure Filter.tendsto_atTop_pure theorem tendsto_atBot_pure [PartialOrder α] [OrderBot α] (f : α → β) : Tendsto f atBot (pure <\| f ⊥) := @tendsto_atTop_pure αᵒᵈ _ _ _ _ #align filter.tendsto_at_bot_pure Filter.tendsto_atBot_pure theorem Eventually.exists_forall_of_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} (h : ∀ᶠ x in atTop, p x) : ∃ a, ∀ b ≥ a, p b := eventually_atTop.mp h #align filter.eventually.exists_forall_of_at_top Filter.Eventually.exists_forall_of_atTop theorem Eventually.exists_forall_of_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} (h : ∀ᶠ x in atBot, p x) : ∃ a, ∀ b ≤ a, p b := eventually_atBot.mp h #align filter.eventually.exists_forall_of_at_bot Filter.Eventually.exists_forall_of_atBot lemma exists_eventually_atTop [SemilatticeSup α] [Nonempty α] {r : α → β → Prop} : (∃ b, ∀ᶠ a in atTop, r a b) ↔ ∀ᶠ a₀ in atTop, ∃ b, ∀ a ≥ a₀, r a b := by simp_rw [eventually_atTop, ← exists_swap (α := α)] exact exists_congr fun a ↦ .symm <\| forall_ge_iff <\| Monotone.exists fun _ _ _ hb H n hn ↦ H n (hb.trans hn) lemma exists_eventually_atBot [SemilatticeInf α] [Nonempty α] {r : α → β → Prop} : (∃ b, ∀ᶠ a in atBot, r a b) ↔ ∀ᶠ a₀ in atBot, ∃ b, ∀ a ≤ a₀, r a b := by simp_rw [eventually_atBot, ← exists_swap (α := α)] exact exists_congr fun a ↦ .symm <\| forall_le_iff <\| Antitone.exists fun _ _ _ hb H n hn ↦ H n (hn.trans hb) theorem frequently_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} : (∃ᶠ x in atTop, p x) ↔ ∀ a, ∃ b ≥ a, p b := atTop_basis.frequently_iff.trans <\| by simp #align filter.frequently_at_top Filter.frequently_atTop theorem frequently_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} : (∃ᶠ x in atBot, p x) ↔ ∀ a, ∃ b ≤ a, p b := @frequently_atTop αᵒᵈ _ _ _ #align filter.frequently_at_bot Filter.frequently_atBot theorem frequently_atTop' [SemilatticeSup α] [Nonempty α] [NoMaxOrder α] {p : α → Prop} : (∃ᶠ x in atTop, p x) ↔ ∀ a, ∃ b > a, p b := atTop_basis_Ioi.frequently_iff.trans <\| by simp #align filter.frequently_at_top' Filter.frequently_atTop' theorem frequently_atBot' [SemilatticeInf α] [Nonempty α] [NoMinOrder α] {p : α → Prop} : (∃ᶠ x in atBot, p x) ↔ ∀ a, ∃ b < a, p b := @frequently_atTop' αᵒᵈ _ _ _ _ #align filter.frequently_at_bot' Filter.frequently_atBot' theorem Frequently.forall_exists_of_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} (h : ∃ᶠ x in atTop, p x) : ∀ a, ∃ b ≥ a, p b := frequently_atTop.mp h #align filter.frequently.forall_exists_of_at_top Filter.Frequently.forall_exists_of_atTop theorem Frequently.forall_exists_of_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} (h : ∃ᶠ x in atBot, p x) : ∀ a, ∃ b ≤ a, p b := frequently_atBot.mp h #align filter.frequently.forall_exists_of_at_bot Filter.Frequently.forall_exists_of_atBot theorem map_atTop_eq [Nonempty α] [SemilatticeSup α] {f : α → β} : atTop.map f = ⨅ a, 𝓟 (f '' { a' \| a ≤ a' }) := (atTop_basis.map f).eq_iInf #align filter.map_at_top_eq Filter.map_atTop_eq theorem map_atBot_eq [Nonempty α] [SemilatticeInf α] {f : α → β} : atBot.map f = ⨅ a, 𝓟 (f '' { a' \| a' ≤ a }) := @map_atTop_eq αᵒᵈ _ _ _ _ #align filter.map_at_bot_eq Filter.map_atBot_eq theorem tendsto_atTop [Preorder β] {m : α → β} {f : Filter α} : Tendsto m f atTop ↔ ∀ b, ∀ᶠ a in f, b ≤ m a := by simp only [atTop, tendsto_iInf, tendsto_principal, mem_Ici] #align filter.tendsto_at_top Filter.tendsto_atTop theorem tendsto_atBot [Preorder β] {m : α → β} {f : Filter α} : Tendsto m f atBot ↔ ∀ b, ∀ᶠ a in f, m a ≤ b := @tendsto_atTop α βᵒᵈ _ m f #align filter.tendsto_at_bot Filter.tendsto_atBot theorem tendsto_atTop_mono' [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) (h₁ : Tendsto f₁ l atTop) : Tendsto f₂ l atTop := tendsto_atTop.2 fun b => by filter_upwards [tendsto_atTop.1 h₁ b, h] with x using le_trans #align filter.tendsto_at_top_mono' Filter.tendsto_atTop_mono' theorem tendsto_atBot_mono' [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) : Tendsto f₂ l atBot → Tendsto f₁ l atBot := @tendsto_atTop_mono' _ βᵒᵈ _ _ _ _ h #align filter.tendsto_at_bot_mono' Filter.tendsto_atBot_mono' theorem tendsto_atTop_mono [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ n, f n ≤ g n) : Tendsto f l atTop → Tendsto g l atTop := tendsto_atTop_mono' l <\| eventually_of_forall h #align filter.tendsto_at_top_mono Filter.tendsto_atTop_mono theorem tendsto_atBot_mono [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ n, f n ≤ g n) : Tendsto g l atBot → Tendsto f l atBot := @tendsto_atTop_mono _ βᵒᵈ _ _ _ _ h #align filter.tendsto_at_bot_mono Filter.tendsto_atBot_mono lemma atTop_eq_generate_of_forall_exists_le [LinearOrder α] {s : Set α} (hs : ∀ x, ∃ y ∈ s, x ≤ y) : (atTop : Filter α) = generate (Ici '' s) := by rw [atTop_eq_generate_Ici] apply le_antisymm · rw [le_generate_iff] rintro - ⟨y, -, rfl⟩ exact mem_generate_of_mem ⟨y, rfl⟩ · rw [le_generate_iff] rintro - ⟨x, -, -, rfl⟩ rcases hs x with ⟨y, ys, hy⟩ have A : Ici y ∈ generate (Ici '' s) := mem_generate_of_mem (mem_image_of_mem _ ys) have B : Ici y ⊆ Ici x := Ici_subset_Ici.2 hy exact sets_of_superset (generate (Ici '' s)) A B lemma atTop_eq_generate_of_not_bddAbove [LinearOrder α] {s : Set α} (hs : ¬ BddAbove s) : (atTop : Filter α) = generate (Ici '' s) := by refine atTop_eq_generate_of_forall_exists_le fun x ↦ ?_ obtain ⟨y, hy, hy'⟩ := not_bddAbove_iff.mp hs x exact ⟨y, hy, hy'.le⟩ end Filter namespace OrderIso open Filter variable [Preorder α] [Preorder β] @[simp] theorem comap_atTop (e : α ≃o β) : comap e atTop = atTop := by simp [atTop, ← e.surjective.iInf_comp] #align order_iso.comap_at_top OrderIso.comap_atTop @[simp] theorem comap_atBot (e : α ≃o β) : comap e atBot = atBot := e.dual.comap_atTop #align order_iso.comap_at_bot OrderIso.comap_atBot @[simp] theorem map_atTop (e : α ≃o β) : map (e : α → β) atTop = atTop := by rw [← e.comap_atTop, map_comap_of_surjective e.surjective] #align order_iso.map_at_top OrderIso.map_atTop @[simp] theorem map_atBot (e : α ≃o β) : map (e : α → β) atBot = atBot := e.dual.map_atTop #align order_iso.map_at_bot OrderIso.map_atBot theorem tendsto_atTop (e : α ≃o β) : Tendsto e atTop atTop := e.map_atTop.le #align order_iso.tendsto_at_top OrderIso.tendsto_atTop theorem tendsto_atBot (e : α ≃o β) : Tendsto e atBot atBot := e.map_atBot.le #align order_iso.tendsto_at_bot OrderIso.tendsto_atBot @[simp] theorem tendsto_atTop_iff {l : Filter γ} {f : γ → α} (e : α ≃o β) : Tendsto (fun x => e (f x)) l atTop ↔ Tendsto f l atTop := by rw [← e.comap_atTop, tendsto_comap_iff, Function.comp_def] #align order_iso.tendsto_at_top_iff OrderIso.tendsto_atTop_iff @[simp] theorem tendsto_atBot_iff {l : Filter γ} {f : γ → α} (e : α ≃o β) : Tendsto (fun x => e (f x)) l atBot ↔ Tendsto f l atBot := e.dual.tendsto_atTop_iff #align order_iso.tendsto_at_bot_iff OrderIso.tendsto_atBot_iff end OrderIso namespace Filter /-! ### Sequences -/ theorem inf_map_atTop_neBot_iff [SemilatticeSup α] [Nonempty α] {F : Filter β} {u : α → β} : NeBot (F ⊓ map u atTop) ↔ ∀ U ∈ F, ∀ N, ∃ n ≥ N, u n ∈ U := by simp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]; rfl #align filter.inf_map_at_top_ne_bot_iff Filter.inf_map_atTop_neBot_iff theorem inf_map_atBot_neBot_iff [SemilatticeInf α] [Nonempty α] {F : Filter β} {u : α → β} : NeBot (F ⊓ map u atBot) ↔ ∀ U ∈ F, ∀ N, ∃ n ≤ N, u n ∈ U := @inf_map_atTop_neBot_iff αᵒᵈ _ _ _ _ _ #align filter.inf_map_at_bot_ne_bot_iff Filter.inf_map_atBot_neBot_iff theorem extraction_of_frequently_atTop' {P : ℕ → Prop} (h : ∀ N, ∃ n > N, P n) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P (φ n) := by choose u hu hu' using h refine ⟨fun n => u^[n + 1] 0, strictMono_nat_of_lt_succ fun n => ?_, fun n => ?_⟩ · exact Trans.trans (hu _) (Function.iterate_succ_apply' _ _ _).symm · simpa only [Function.iterate_succ_apply'] using hu' _ #align filter.extraction_of_frequently_at_top' Filter.extraction_of_frequently_atTop' theorem extraction_of_frequently_atTop {P : ℕ → Prop} (h : ∃ᶠ n in atTop, P n) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P (φ n) := by rw [frequently_atTop'] at h exact extraction_of_frequently_atTop' h #align filter.extraction_of_frequently_at_top Filter.extraction_of_frequently_atTop theorem extraction_of_eventually_atTop {P : ℕ → Prop} (h : ∀ᶠ n in atTop, P n) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P (φ n) := extraction_of_frequently_atTop h.frequently #align filter.extraction_of_eventually_at_top Filter.extraction_of_eventually_atTop theorem extraction_forall_of_frequently {P : ℕ → ℕ → Prop} (h : ∀ n, ∃ᶠ k in atTop, P n k) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P n (φ n) := by simp only [frequently_atTop'] at h choose u hu hu' using h use (fun n => Nat.recOn n (u 0 0) fun n v => u (n + 1) v : ℕ → ℕ) constructor · apply strictMono_nat_of_lt_succ intro n apply hu · intro n cases n <;> simp [hu'] #align filter.extraction_forall_of_frequently Filter.extraction_forall_of_frequently theorem extraction_forall_of_eventually {P : ℕ → ℕ → Prop} (h : ∀ n, ∀ᶠ k in atTop, P n k) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P n (φ n) := extraction_forall_of_frequently fun n => (h n).frequently #align filter.extraction_forall_of_eventually Filter.extraction_forall_of_eventually theorem extraction_forall_of_eventually' {P : ℕ → ℕ → Prop} (h : ∀ n, ∃ N, ∀ k ≥ N, P n k) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P n (φ n) := extraction_forall_of_eventually (by simp [eventually_atTop, h]) #align filter.extraction_forall_of_eventually' Filter.extraction_forall_of_eventually' theorem Eventually.atTop_of_arithmetic {p : ℕ → Prop} {n : ℕ} (hn : n ≠ 0) (hp : ∀ k < n, ∀ᶠ a in atTop, p (n a + k)) : ∀ᶠ a in atTop, p a := by simp only [eventually_atTop] at hp ⊢ choose! N hN using hp refine ⟨(Finset.range n).sup (n * N ·), fun b hb => ?_⟩ rw [← Nat.div_add_mod b n] have hlt := Nat.mod_lt b hn.bot_lt refine hN _ hlt _ ?_ rw [ge_iff_le, Nat.le_div_iff_mul_le hn.bot_lt, mul_comm] exact (Finset.le_sup (f := (n * N ·)) (Finset.mem_range.2 hlt)).trans hb theorem exists_le_of_tendsto_atTop [SemilatticeSup α] [Preorder β] {u : α → β} (h : Tendsto u atTop atTop) (a : α) (b : β) : ∃ a' ≥ a, b ≤ u a' := by have : Nonempty α := ⟨a⟩ have : ∀ᶠ x in atTop, a ≤ x ∧ b ≤ u x := (eventually_ge_atTop a).and (h.eventually <\| eventually_ge_atTop b) exact this.exists #align filter.exists_le_of_tendsto_at_top Filter.exists_le_of_tendsto_atTop -- @[nolint ge_or_gt] -- Porting note: restore attribute theorem exists_le_of_tendsto_atBot [SemilatticeSup α] [Preorder β] {u : α → β} (h : Tendsto u atTop atBot) : ∀ a b, ∃ a' ≥ a, u a' ≤ b := @exists_le_of_tendsto_atTop _ βᵒᵈ _ _ _ h #align filter.exists_le_of_tendsto_at_bot Filter.exists_le_of_tendsto_atBot theorem exists_lt_of_tendsto_atTop [SemilatticeSup α] [Preorder β] [NoMaxOrder β] {u : α → β} (h : Tendsto u atTop atTop) (a : α) (b : β) : ∃ a' ≥ a, b < u a' := by cases' exists_gt b with b' hb' rcases exists_le_of_tendsto_atTop h a b' with ⟨a', ha', ha''⟩ exact ⟨a', ha', lt_of_lt_of_le hb' ha''⟩ #align filter.exists_lt_of_tendsto_at_top Filter.exists_lt_of_tendsto_atTop -- @[nolint ge_or_gt] -- Porting note: restore attribute theorem exists_lt_of_tendsto_atBot [SemilatticeSup α] [Preorder β] [NoMinOrder β] {u : α → β} (h : Tendsto u atTop atBot) : ∀ a b, ∃ a' ≥ a, u a' < b := @exists_lt_of_tendsto_atTop _ βᵒᵈ _ _ _ _ h #align filter.exists_lt_of_tendsto_at_bot Filter.exists_lt_of_tendsto_atBot /-- If `u` is a sequence which is unbounded above, then after any point, it reaches a value strictly greater than all previous values. -/ theorem high_scores [LinearOrder β] [NoMaxOrder β] {u : ℕ → β} (hu : Tendsto u atTop atTop) : ∀ N, ∃ n ≥ N, ∀ k < n, u k < u n := by intro N obtain ⟨k : ℕ, - : k ≤ N, hku : ∀ l ≤ N, u l ≤ u k⟩ : ∃ k ≤ N, ∀ l ≤ N, u l ≤ u k := exists_max_image _ u (finite_le_nat N) ⟨N, le_refl N⟩ have ex : ∃ n ≥ N, u k < u n := exists_lt_of_tendsto_atTop hu _ _ obtain ⟨n : ℕ, hnN : n ≥ N, hnk : u k < u n, hn_min : ∀ m, m < n → N ≤ m → u m ≤ u k⟩ : ∃ n ≥ N, u k < u n ∧ ∀ m, m < n → N ≤ m → u m ≤ u k := by rcases Nat.findX ex with ⟨n, ⟨hnN, hnk⟩, hn_min⟩ push_neg at hn_min exact ⟨n, hnN, hnk, hn_min⟩ use n, hnN rintro (l : ℕ) (hl : l < n) have hlk : u l ≤ u k := by cases' (le_total l N : l ≤ N ∨ N ≤ l) with H H · exact hku l H · exact hn_min l hl H calc u l ≤ u k := hlk _ < u n := hnk #align filter.high_scores Filter.high_scores -- see Note [nolint_ge] /-- If `u` is a sequence which is unbounded below, then after any point, it reaches a value strictly smaller than all previous values. -/ -- @[nolint ge_or_gt] Porting note: restore attribute theorem low_scores [LinearOrder β] [NoMinOrder β] {u : ℕ → β} (hu : Tendsto u atTop atBot) : ∀ N, ∃ n ≥ N, ∀ k < n, u n < u k := @high_scores βᵒᵈ _ _ _ hu #align filter.low_scores Filter.low_scores /-- If `u` is a sequence which is unbounded above, then it `Frequently` reaches a value strictly greater than all previous values. -/ theorem frequently_high_scores [LinearOrder β] [NoMaxOrder β] {u : ℕ → β} (hu : Tendsto u atTop atTop) : ∃ᶠ n in atTop, ∀ k < n, u k < u n := by simpa [frequently_atTop] using high_scores hu #align filter.frequently_high_scores Filter.frequently_high_scores /-- If `u` is a sequence which is unbounded below, then it `Frequently` reaches a value strictly smaller than all previous values. -/ theorem frequently_low_scores [LinearOrder β] [NoMinOrder β] {u : ℕ → β} (hu : Tendsto u atTop atBot) : ∃ᶠ n in atTop, ∀ k < n, u n < u k := @frequently_high_scores βᵒᵈ _ _ _ hu #align filter.frequently_low_scores Filter.frequently_low_scores theorem strictMono_subseq_of_tendsto_atTop {β : Type} [LinearOrder β] [NoMaxOrder β] {u : ℕ → β} (hu : Tendsto u atTop atTop) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ StrictMono (u ∘ φ) := let ⟨φ, h, h'⟩ := extraction_of_frequently_atTop (frequently_high_scores hu) ⟨φ, h, fun _ m hnm => h' m _ (h hnm)⟩ #align filter.strict_mono_subseq_of_tendsto_at_top Filter.strictMono_subseq_of_tendsto_atTop theorem strictMono_subseq_of_id_le {u : ℕ → ℕ} (hu : ∀ n, n ≤ u n) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ StrictMono (u ∘ φ) := strictMono_subseq_of_tendsto_atTop (tendsto_atTop_mono hu tendsto_id) #align filter.strict_mono_subseq_of_id_le Filter.strictMono_subseq_of_id_le theorem _root_.StrictMono.tendsto_atTop {φ : ℕ → ℕ} (h : StrictMono φ) : Tendsto φ atTop atTop := tendsto_atTop_mono h.id_le tendsto_id #align strict_mono.tendsto_at_top StrictMono.tendsto_atTop section OrderedAddCommMonoid variable [OrderedAddCommMonoid β] {l : Filter α} {f g : α → β} theorem tendsto_atTop_add_nonneg_left' (hf : ∀ᶠ x in l, 0 ≤ f x) (hg : Tendsto g l atTop) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_mono' l (hf.mono fun _ => le_add_of_nonneg_left) hg #align filter.tendsto_at_top_add_nonneg_left' Filter.tendsto_atTop_add_nonneg_left' theorem tendsto_atBot_add_nonpos_left' (hf : ∀ᶠ x in l, f x ≤ 0) (hg : Tendsto g l atBot) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_nonneg_left' _ βᵒᵈ _ _ _ _ hf hg #align filter.tendsto_at_bot_add_nonpos_left' Filter.tendsto_atBot_add_nonpos_left' theorem tendsto_atTop_add_nonneg_left (hf : ∀ x, 0 ≤ f x) (hg : Tendsto g l atTop) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_add_nonneg_left' (eventually_of_forall hf) hg #align filter.tendsto_at_top_add_nonneg_left Filter.tendsto_atTop_add_nonneg_left theorem tendsto_atBot_add_nonpos_left (hf : ∀ x, f x ≤ 0) (hg : Tendsto g l atBot) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_nonneg_left _ βᵒᵈ _ _ _ _ hf hg #align filter.tendsto_at_bot_add_nonpos_left Filter.tendsto_atBot_add_nonpos_left theorem tendsto_atTop_add_nonneg_right' (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, 0 ≤ g x) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_mono' l (monotone_mem (fun _ => le_add_of_nonneg_right) hg) hf #align filter.tendsto_at_top_add_nonneg_right' Filter.tendsto_atTop_add_nonneg_right' theorem tendsto_atBot_add_nonpos_right' (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ 0) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_nonneg_right' _ βᵒᵈ _ _ _ _ hf hg #align filter.tendsto_at_bot_add_nonpos_right' Filter.tendsto_atBot_add_nonpos_right' theorem tendsto_atTop_add_nonneg_right (hf : Tendsto f l atTop) (hg : ∀ x, 0 ≤ g x) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_add_nonneg_right' hf (eventually_of_forall hg) #align filter.tendsto_at_top_add_nonneg_right Filter.tendsto_atTop_add_nonneg_right theorem tendsto_atBot_add_nonpos_right (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ 0) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_nonneg_right _ βᵒᵈ _ _ _ _ hf hg #align filter.tendsto_at_bot_add_nonpos_right Filter.tendsto_atBot_add_nonpos_right theorem tendsto_atTop_add (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_add_nonneg_left' (tendsto_atTop.mp hf 0) hg #align filter.tendsto_at_top_add Filter.tendsto_atTop_add theorem tendsto_atBot_add (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add _ βᵒᵈ _ _ _ _ hf hg #align filter.tendsto_at_bot_add Filter.tendsto_atBot_add theorem Tendsto.nsmul_atTop (hf : Tendsto f l atTop) {n : ℕ} (hn : 0 < n) : Tendsto (fun x => n • f x) l atTop := tendsto_atTop.2 fun y => (tendsto_atTop.1 hf y).mp <\| (tendsto_atTop.1 hf 0).mono fun x h₀ hy => calc y ≤ f x := hy _ = 1 • f x := (one_nsmul _).symm _ ≤ n • f x := nsmul_le_nsmul_left h₀ hn #align filter.tendsto.nsmul_at_top Filter.Tendsto.nsmul_atTop theorem Tendsto.nsmul_atBot (hf : Tendsto f l atBot) {n : ℕ} (hn : 0 < n) : Tendsto (fun x => n • f x) l atBot := @Tendsto.nsmul_atTop α βᵒᵈ _ l f hf n hn #align filter.tendsto.nsmul_at_bot Filter.Tendsto.nsmul_atBot #noalign filter.tendsto_bit0_at_top #noalign filter.tendsto_bit0_at_bot end OrderedAddCommMonoid section OrderedCancelAddCommMonoid variable [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β} theorem tendsto_atTop_of_add_const_left (C : β) (hf : Tendsto (fun x => C + f x) l atTop) : Tendsto f l atTop := tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (C + b)).mono fun _ => le_of_add_le_add_left #align filter.tendsto_at_top_of_add_const_left Filter.tendsto_atTop_of_add_const_left -- Porting note: the "order dual" trick timeouts theorem tendsto_atBot_of_add_const_left (C : β) (hf : Tendsto (fun x => C + f x) l atBot) : Tendsto f l atBot := tendsto_atBot.2 fun b => (tendsto_atBot.1 hf (C + b)).mono fun _ => le_of_add_le_add_left #align filter.tendsto_at_bot_of_add_const_left Filter.tendsto_atBot_of_add_const_left theorem tendsto_atTop_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atTop) : Tendsto f l atTop := tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (b + C)).mono fun _ => le_of_add_le_add_right #align filter.tendsto_at_top_of_add_const_right Filter.tendsto_atTop_of_add_const_right -- Porting note: the "order dual" trick timeouts theorem tendsto_atBot_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atBot) : Tendsto f l atBot := tendsto_atBot.2 fun b => (tendsto_atBot.1 hf (b + C)).mono fun _ => le_of_add_le_add_right #align filter.tendsto_at_bot_of_add_const_right Filter.tendsto_atBot_of_add_const_right theorem tendsto_atTop_of_add_bdd_above_left' (C) (hC : ∀ᶠ x in l, f x ≤ C) (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto g l atTop := tendsto_atTop_of_add_const_left C (tendsto_atTop_mono' l (hC.mono fun x hx => add_le_add_right hx (g x)) h) #align filter.tendsto_at_top_of_add_bdd_above_left' Filter.tendsto_atTop_of_add_bdd_above_left' -- Porting note: the "order dual" trick timeouts theorem tendsto_atBot_of_add_bdd_below_left' (C) (hC : ∀ᶠ x in l, C ≤ f x) (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto g l atBot := tendsto_atBot_of_add_const_left C (tendsto_atBot_mono' l (hC.mono fun x hx => add_le_add_right hx (g x)) h) #align filter.tendsto_at_bot_of_add_bdd_below_left' Filter.tendsto_atBot_of_add_bdd_below_left' theorem tendsto_atTop_of_add_bdd_above_left (C) (hC : ∀ x, f x ≤ C) : Tendsto (fun x => f x + g x) l atTop → Tendsto g l atTop := tendsto_atTop_of_add_bdd_above_left' C (univ_mem' hC) #align filter.tendsto_at_top_of_add_bdd_above_left Filter.tendsto_atTop_of_add_bdd_above_left -- Porting note: the "order dual" trick timeouts theorem tendsto_atBot_of_add_bdd_below_left (C) (hC : ∀ x, C ≤ f x) : Tendsto (fun x => f x + g x) l atBot → Tendsto g l atBot := tendsto_atBot_of_add_bdd_below_left' C (univ_mem' hC) #align filter.tendsto_at_bot_of_add_bdd_below_left Filter.tendsto_atBot_of_add_bdd_below_left theorem tendsto_atTop_of_add_bdd_above_right' (C) (hC : ∀ᶠ x in l, g x ≤ C) (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto f l atTop := tendsto_atTop_of_add_const_right C (tendsto_atTop_mono' l (hC.mono fun x hx => add_le_add_left hx (f x)) h) #align filter.tendsto_at_top_of_add_bdd_above_right' Filter.tendsto_atTop_of_add_bdd_above_right' -- Porting note: the "order dual" trick timeouts theorem tendsto_atBot_of_add_bdd_below_right' (C) (hC : ∀ᶠ x in l, C ≤ g x) (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto f l atBot := tendsto_atBot_of_add_const_right C (tendsto_atBot_mono' l (hC.mono fun x hx => add_le_add_left hx (f x)) h) #align filter.tendsto_at_bot_of_add_bdd_below_right' Filter.tendsto_atBot_of_add_bdd_below_right' theorem tendsto_atTop_of_add_bdd_above_right (C) (hC : ∀ x, g x ≤ C) : Tendsto (fun x => f x + g x) l atTop → Tendsto f l atTop := tendsto_atTop_of_add_bdd_above_right' C (univ_mem' hC) #align filter.tendsto_at_top_of_add_bdd_above_right Filter.tendsto_atTop_of_add_bdd_above_right -- Porting note: the "order dual" trick timeouts theorem tendsto_atBot_of_add_bdd_below_right (C) (hC : ∀ x, C ≤ g x) : Tendsto (fun x => f x + g x) l atBot → Tendsto f l atBot := tendsto_atBot_of_add_bdd_below_right' C (univ_mem' hC) #align filter.tendsto_at_bot_of_add_bdd_below_right Filter.tendsto_atBot_of_add_bdd_below_right end OrderedCancelAddCommMonoid section OrderedGroup variable [OrderedAddCommGroup β] (l : Filter α) {f g : α → β} theorem tendsto_atTop_add_left_of_le' (C : β) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) : Tendsto (fun x => f x + g x) l atTop := @tendsto_atTop_of_add_bdd_above_left' _ _ _ l (fun x => -f x) (fun x => f x + g x) (-C) (by simpa) (by simpa) #align filter.tendsto_at_top_add_left_of_le' Filter.tendsto_atTop_add_left_of_le' theorem tendsto_atBot_add_left_of_ge' (C : β) (hf : ∀ᶠ x in l, f x ≤ C) (hg : Tendsto g l atBot) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_left_of_le' _ βᵒᵈ _ _ _ _ C hf hg #align filter.tendsto_at_bot_add_left_of_ge' Filter.tendsto_atBot_add_left_of_ge' theorem tendsto_atTop_add_left_of_le (C : β) (hf : ∀ x, C ≤ f x) (hg : Tendsto g l atTop) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_add_left_of_le' l C (univ_mem' hf) hg #align filter.tendsto_at_top_add_left_of_le Filter.tendsto_atTop_add_left_of_le theorem tendsto_atBot_add_left_of_ge (C : β) (hf : ∀ x, f x ≤ C) (hg : Tendsto g l atBot) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_left_of_le _ βᵒᵈ _ _ _ _ C hf hg #align filter.tendsto_at_bot_add_left_of_ge Filter.tendsto_atBot_add_left_of_ge theorem tendsto_atTop_add_right_of_le' (C : β) (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, C ≤ g x) : Tendsto (fun x => f x + g x) l atTop := @tendsto_atTop_of_add_bdd_above_right' _ _ _ l (fun x => f x + g x) (fun x => -g x) (-C) (by simp [hg]) (by simp [hf]) #align filter.tendsto_at_top_add_right_of_le' Filter.tendsto_atTop_add_right_of_le' theorem tendsto_atBot_add_right_of_ge' (C : β) (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ C) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_right_of_le' _ βᵒᵈ _ _ _ _ C hf hg #align filter.tendsto_at_bot_add_right_of_ge' Filter.tendsto_atBot_add_right_of_ge' theorem tendsto_atTop_add_right_of_le (C : β) (hf : Tendsto f l atTop) (hg : ∀ x, C ≤ g x) : Tendsto (fun x => f x + g x) l atTop := tendsto_atTop_add_right_of_le' l C hf (univ_mem' hg) #align filter.tendsto_at_top_add_right_of_le Filter.tendsto_atTop_add_right_of_le theorem tendsto_atBot_add_right_of_ge (C : β) (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ C) : Tendsto (fun x => f x + g x) l atBot := @tendsto_atTop_add_right_of_le _ βᵒᵈ _ _ _ _ C hf hg #align filter.tendsto_at_bot_add_right_of_ge Filter.tendsto_atBot_add_right_of_ge theorem tendsto_atTop_add_const_left (C : β) (hf : Tendsto f l atTop) : Tendsto (fun x => C + f x) l atTop := tendsto_atTop_add_left_of_le' l C (univ_mem' fun _ => le_refl C) hf #align filter.tendsto_at_top_add_const_left Filter.tendsto_atTop_add_const_left theorem tendsto_atBot_add_const_left (C : β) (hf : Tendsto f l atBot) : Tendsto (fun x => C + f x) l atBot := @tendsto_atTop_add_const_left _ βᵒᵈ _ _ _ C hf #align filter.tendsto_at_bot_add_const_left Filter.tendsto_atBot_add_const_left theorem tendsto_atTop_add_const_right (C : β) (hf : Tendsto f l atTop) : Tendsto (fun x => f x + C) l atTop := tendsto_atTop_add_right_of_le' l C hf (univ_mem' fun _ => le_refl C) #align filter.tendsto_at_top_add_const_right Filter.tendsto_atTop_add_const_right theorem tendsto_atBot_add_const_right (C : β) (hf : Tendsto f l atBot) : Tendsto (fun x => f x + C) l atBot := @tendsto_atTop_add_const_right _ βᵒᵈ _ _ _ C hf #align filter.tendsto_at_bot_add_const_right Filter.tendsto_atBot_add_const_right theorem map_neg_atBot : map (Neg.neg : β → β) atBot = atTop := (OrderIso.neg β).map_atBot #align filter.map_neg_at_bot Filter.map_neg_atBot theorem map_neg_atTop : map (Neg.neg : β → β) atTop = atBot := (OrderIso.neg β).map_atTop #align filter.map_neg_at_top Filter.map_neg_atTop theorem comap_neg_atBot : comap (Neg.neg : β → β) atBot = atTop := (OrderIso.neg β).comap_atTop #align filter.comap_neg_at_bot Filter.comap_neg_atBot theorem comap_neg_atTop : comap (Neg.neg : β → β) atTop = atBot := (OrderIso.neg β).comap_atBot #align filter.comap_neg_at_top Filter.comap_neg_atTop theorem tendsto_neg_atTop_atBot : Tendsto (Neg.neg : β → β) atTop atBot := (OrderIso.neg β).tendsto_atTop #align filter.tendsto_neg_at_top_at_bot Filter.tendsto_neg_atTop_atBot theorem tendsto_neg_atBot_atTop : Tendsto (Neg.neg : β → β) atBot atTop := @tendsto_neg_atTop_atBot βᵒᵈ _ #align filter.tendsto_neg_at_bot_at_top Filter.tendsto_neg_atBot_atTop variable {l} @[simp] theorem tendsto_neg_atTop_iff : Tendsto (fun x => -f x) l atTop ↔ Tendsto f l atBot := (OrderIso.neg β).tendsto_atBot_iff #align filter.tendsto_neg_at_top_iff Filter.tendsto_neg_atTop_iff @[simp] theorem tendsto_neg_atBot_iff : Tendsto (fun x => -f x) l atBot ↔ Tendsto f l atTop := (OrderIso.neg β).tendsto_atTop_iff #align filter.tendsto_neg_at_bot_iff Filter.tendsto_neg_atBot_iff end OrderedGroup section OrderedSemiring variable [OrderedSemiring α] {l : Filter β} {f g : β → α} #noalign filter.tendsto_bit1_at_top theorem Tendsto.atTop_mul_atTop (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) : Tendsto (fun x => f x g x) l atTop := by refine tendsto_atTop_mono' _ ?_ hg filter_upwards [hg.eventually (eventually_ge_atTop 0), hf.eventually (eventually_ge_atTop 1)] with _ using le_mul_of_one_le_left #align filter.tendsto.at_top_mul_at_top Filter.Tendsto.atTop_mul_atTop theorem tendsto_mul_self_atTop : Tendsto (fun x : α => x * x) atTop atTop := tendsto_id.atTop_mul_atTop tendsto_id #align filter.tendsto_mul_self_at_top Filter.tendsto_mul_self_atTop /-- The monomial function `x^n` tends to `+∞` at `+∞` for any positive natural `n`. A version for positive real powers exists as `tendsto_rpow_atTop`. -/ theorem tendsto_pow_atTop {n : ℕ} (hn : n ≠ 0) : Tendsto (fun x : α => x ^ n) atTop atTop := tendsto_atTop_mono' _ ((eventually_ge_atTop 1).mono fun _x hx => le_self_pow hx hn) tendsto_id #align filter.tendsto_pow_at_top Filter.tendsto_pow_atTop end OrderedSemiring theorem zero_pow_eventuallyEq [MonoidWithZero α] : (fun n : ℕ => (0 : α) ^ n) =ᶠ[atTop] fun _ => 0 := eventually_atTop.2 ⟨1, fun _n hn ↦ zero_pow $ Nat.one_le_iff_ne_zero.1 hn⟩ #align filter.zero_pow_eventually_eq Filter.zero_pow_eventuallyEq section OrderedRing variable [OrderedRing α] {l : Filter β} {f g : β → α} theorem Tendsto.atTop_mul_atBot (hf : Tendsto f l atTop) (hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atBot := by have := hf.atTop_mul_atTop <\| tendsto_neg_atBot_atTop.comp hg simpa only [(· ∘ ·), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this #align filter.tendsto.at_top_mul_at_bot Filter.Tendsto.atTop_mul_atBot theorem Tendsto.atBot_mul_atTop (hf : Tendsto f l atBot) (hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atBot := by have : Tendsto (fun x => -f x * g x) l atTop := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop hg simpa only [(· ∘ ·), neg_mul_eq_neg_mul, neg_neg] using tendsto_neg_atTop_atBot.comp this #align filter.tendsto.at_bot_mul_at_top Filter.Tendsto.atBot_mul_atTop theorem Tendsto.atBot_mul_atBot (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop := by have : Tendsto (fun x => -f x * -g x) l atTop := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop (tendsto_neg_atBot_atTop.comp hg) simpa only [neg_mul_neg] using this #align filter.tendsto.at_bot_mul_at_bot Filter.Tendsto.atBot_mul_atBot end OrderedRing section LinearOrderedAddCommGroup variable [LinearOrderedAddCommGroup α] /-- $\lim_{x\to+\infty}\|x\|=+\infty$ -/ theorem tendsto_abs_atTop_atTop : Tendsto (abs : α → α) atTop atTop := tendsto_atTop_mono le_abs_self tendsto_id #align filter.tendsto_abs_at_top_at_top Filter.tendsto_abs_atTop_atTop /-- $\lim_{x\to-\infty}\|x\|=+\infty$ -/ theorem tendsto_abs_atBot_atTop : Tendsto (abs : α → α) atBot atTop := tendsto_atTop_mono neg_le_abs tendsto_neg_atBot_atTop #align filter.tendsto_abs_at_bot_at_top Filter.tendsto_abs_atBot_atTop @[simp] theorem comap_abs_atTop : comap (abs : α → α) atTop = atBot ⊔ atTop := by refine le_antisymm (((atTop_basis.comap _).le_basis_iff (atBot_basis.sup atTop_basis)).2 ?_) (sup_le tendsto_abs_atBot_atTop.le_comap tendsto_abs_atTop_atTop.le_comap) rintro ⟨a, b⟩ - refine ⟨max (-a) b, trivial, fun x hx => ?_⟩ rw [mem_preimage, mem_Ici, le_abs', max_le_iff, ← min_neg_neg, le_min_iff, neg_neg] at hx exact hx.imp And.left And.right #align filter.comap_abs_at_top Filter.comap_abs_atTop end LinearOrderedAddCommGroup section LinearOrderedSemiring variable [LinearOrderedSemiring α] {l : Filter β} {f : β → α} theorem Tendsto.atTop_of_const_mul {c : α} (hc : 0 < c) (hf : Tendsto (fun x => c * f x) l atTop) : Tendsto f l atTop := tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (c * b)).mono fun _x hx => le_of_mul_le_mul_left hx hc #align filter.tendsto.at_top_of_const_mul Filter.Tendsto.atTop_of_const_mul theorem Tendsto.atTop_of_mul_const {c : α} (hc : 0 < c) (hf : Tendsto (fun x => f x * c) l atTop) : Tendsto f l atTop := tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (b * c)).mono fun _x hx => le_of_mul_le_mul_right hx hc #align filter.tendsto.at_top_of_mul_const Filter.Tendsto.atTop_of_mul_const @[simp] theorem tendsto_pow_atTop_iff {n : ℕ} : Tendsto (fun x : α => x ^ n) atTop atTop ↔ n ≠ 0 := ⟨fun h hn => by simp only [hn, pow_zero, not_tendsto_const_atTop] at h, tendsto_pow_atTop⟩ #align filter.tendsto_pow_at_top_iff Filter.tendsto_pow_atTop_iff end LinearOrderedSemiring theorem not_tendsto_pow_atTop_atBot [LinearOrderedRing α] : ∀ {n : ℕ}, ¬Tendsto (fun x : α => x ^ n) atTop atBot \| 0 => by simp [not_tendsto_const_atBot] \| n + 1 => (tendsto_pow_atTop n.succ_ne_zero).not_tendsto disjoint_atTop_atBot #align filter.not_tendsto_pow_at_top_at_bot Filter.not_tendsto_pow_atTop_atBot section LinearOrderedSemifield variable [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r c : α} {n : ℕ} /-! ### Multiplication by constant: iff lemmas -/ /-- If `r` is a positive constant, `fun x ↦ r * f x` tends to infinity along a filter if and only if `f` tends to infinity along the same filter. -/ theorem tendsto_const_mul_atTop_of_pos (hr : 0 < r) : Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atTop := ⟨fun h => h.atTop_of_const_mul hr, fun h => Tendsto.atTop_of_const_mul (inv_pos.2 hr) <\| by simpa only [inv_mul_cancel_left₀ hr.ne'] ⟩ #align filter.tendsto_const_mul_at_top_of_pos Filter.tendsto_const_mul_atTop_of_pos /-- If `r` is a positive constant, `fun x ↦ f x * r` tends to infinity along a filter if and only if `f` tends to infinity along the same filter. -/ theorem tendsto_mul_const_atTop_of_pos (hr : 0 < r) : Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atTop := by simpa only [mul_comm] using tendsto_const_mul_atTop_of_pos hr #align filter.tendsto_mul_const_at_top_of_pos Filter.tendsto_mul_const_atTop_of_pos /-- If `r` is a positive constant, `x ↦ f x / r` tends to infinity along a filter if and only if `f` tends to infinity along the same filter. -/ lemma tendsto_div_const_atTop_of_pos (hr : 0 < r) : Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atTop := by simpa only [div_eq_mul_inv] using tendsto_mul_const_atTop_of_pos (inv_pos.2 hr) /-- If `f` tends to infinity along a nontrivial filter `l`, then `fun x ↦ r * f x` tends to infinity if and only if `0 < r. `-/ theorem tendsto_const_mul_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) : Tendsto (fun x => r * f x) l atTop ↔ 0 < r := by refine ⟨fun hrf => not_le.mp fun hr => ?_, fun hr => (tendsto_const_mul_atTop_of_pos hr).mpr h⟩ rcases ((h.eventually_ge_atTop 0).and (hrf.eventually_gt_atTop 0)).exists with ⟨x, hx, hrx⟩ exact (mul_nonpos_of_nonpos_of_nonneg hr hx).not_lt hrx #align filter.tendsto_const_mul_at_top_iff_pos Filter.tendsto_const_mul_atTop_iff_pos /-- If `f` tends to infinity along a nontrivial filter `l`, then `fun x ↦ f x * r` tends to infinity if and only if `0 < r. `-/ theorem tendsto_mul_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) : Tendsto (fun x => f x * r) l atTop ↔ 0 < r := by simp only [mul_comm _ r, tendsto_const_mul_atTop_iff_pos h] #align filter.tendsto_mul_const_at_top_iff_pos Filter.tendsto_mul_const_atTop_iff_pos /-- If `f` tends to infinity along a nontrivial filter `l`, then `x ↦ f x * r` tends to infinity if and only if `0 < r. `-/ lemma tendsto_div_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) : Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r := by simp only [div_eq_mul_inv, tendsto_mul_const_atTop_iff_pos h, inv_pos] /-- If `f` tends to infinity along a filter, then `f` multiplied by a positive constant (on the left) also tends to infinity. For a version working in `ℕ` or `ℤ`, use `Filter.Tendsto.const_mul_atTop'` instead. -/ theorem Tendsto.const_mul_atTop (hr : 0 < r) (hf : Tendsto f l atTop) : Tendsto (fun x => r * f x) l atTop := (tendsto_const_mul_atTop_of_pos hr).2 hf #align filter.tendsto.const_mul_at_top Filter.Tendsto.const_mul_atTop /-- If a function `f` tends to infinity along a filter, then `f` multiplied by a positive constant (on the right) also tends to infinity. For a version working in `ℕ` or `ℤ`, use `Filter.Tendsto.atTop_mul_const'` instead. -/ theorem Tendsto.atTop_mul_const (hr : 0 < r) (hf : Tendsto f l atTop) : Tendsto (fun x => f x * r) l atTop := (tendsto_mul_const_atTop_of_pos hr).2 hf #align filter.tendsto.at_top_mul_const Filter.Tendsto.atTop_mul_const /-- If a function `f` tends to infinity along a filter, then `f` divided by a positive constant also tends to infinity. -/ theorem Tendsto.atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) : Tendsto (fun x => f x / r) l atTop := by simpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr) #align filter.tendsto.at_top_div_const Filter.Tendsto.atTop_div_const theorem tendsto_const_mul_pow_atTop (hn : n ≠ 0) (hc : 0 < c) : Tendsto (fun x => c * x ^ n) atTop atTop := Tendsto.const_mul_atTop hc (tendsto_pow_atTop hn) #align filter.tendsto_const_mul_pow_at_top Filter.tendsto_const_mul_pow_atTop theorem tendsto_const_mul_pow_atTop_iff : Tendsto (fun x => c * x ^ n) atTop atTop ↔ n ≠ 0 ∧ 0 < c := by refine ⟨fun h => ⟨?_, ?_⟩, fun h => tendsto_const_mul_pow_atTop h.1 h.2⟩ · rintro rfl simp only [pow_zero, not_tendsto_const_atTop] at h · rcases ((h.eventually_gt_atTop 0).and (eventually_ge_atTop 0)).exists with ⟨k, hck, hk⟩ exact pos_of_mul_pos_left hck (pow_nonneg hk _) #align filter.tendsto_const_mul_pow_at_top_iff Filter.tendsto_const_mul_pow_atTop_iff lemma tendsto_zpow_atTop_atTop {n : ℤ} (hn : 0 < n) : Tendsto (fun x : α ↦ x ^ n) atTop atTop := by lift n to ℕ+ using hn; simp #align tendsto_zpow_at_top_at_top Filter.tendsto_zpow_atTop_atTop end LinearOrderedSemifield section LinearOrderedField variable [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α} /-- If `r` is a positive constant, `fun x ↦ r * f x` tends to negative infinity along a filter if and only if `f` tends to negative infinity along the same filter. -/ theorem tendsto_const_mul_atBot_of_pos (hr : 0 < r) : Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atBot := by simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr #align filter.tendsto_const_mul_at_bot_of_pos Filter.tendsto_const_mul_atBot_of_pos /-- If `r` is a positive constant, `fun x ↦ f x * r` tends to negative infinity along a filter if and only if `f` tends to negative infinity along the same filter. -/ theorem tendsto_mul_const_atBot_of_pos (hr : 0 < r) : Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atBot := by simpa only [mul_comm] using tendsto_const_mul_atBot_of_pos hr #align filter.tendsto_mul_const_at_bot_of_pos Filter.tendsto_mul_const_atBot_of_pos /-- If `r` is a positive constant, `fun x ↦ f x / r` tends to negative infinity along a filter if and only if `f` tends to negative infinity along the same filter. -/ lemma tendsto_div_const_atBot_of_pos (hr : 0 < r) : Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atBot := by simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_pos, hr] /-- If `r` is a negative constant, `fun x ↦ r * f x` tends to infinity along a filter `l` if and only if `f` tends to negative infinity along `l`. -/ theorem tendsto_const_mul_atTop_of_neg (hr : r < 0) : Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atBot := by simpa only [neg_mul, tendsto_neg_atBot_iff] using tendsto_const_mul_atBot_of_pos (neg_pos.2 hr) #align filter.tendsto_const_mul_at_top_of_neg Filter.tendsto_const_mul_atTop_of_neg /-- If `r` is a negative constant, `fun x ↦ f x * r` tends to infinity along a filter `l` if and only if `f` tends to negative infinity along `l`. -/ theorem tendsto_mul_const_atTop_of_neg (hr : r < 0) : Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atBot := by simpa only [mul_comm] using tendsto_const_mul_atTop_of_neg hr /-- If `r` is a negative constant, `fun x ↦ f x / r` tends to infinity along a filter `l` if and only if `f` tends to negative infinity along `l`. -/ lemma tendsto_div_const_atTop_of_neg (hr : r < 0) : Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atBot := by simp [div_eq_mul_inv, tendsto_mul_const_atTop_of_neg, hr] /-- If `r` is a negative constant, `fun x ↦ r * f x` tends to negative infinity along a filter `l` if and only if `f` tends to infinity along `l`. -/ theorem tendsto_const_mul_atBot_of_neg (hr : r < 0) : Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atTop := by simpa only [neg_mul, tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos (neg_pos.2 hr) #align filter.tendsto_const_mul_at_bot_of_neg Filter.tendsto_const_mul_atBot_of_neg /-- If `r` is a negative constant, `fun x ↦ f x * r` tends to negative infinity along a filter `l` if and only if `f` tends to infinity along `l`. -/ theorem tendsto_mul_const_atBot_of_neg (hr : r < 0) : Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atTop := by simpa only [mul_comm] using tendsto_const_mul_atBot_of_neg hr #align filter.tendsto_mul_const_at_bot_of_neg Filter.tendsto_mul_const_atBot_of_neg /-- If `r` is a negative constant, `fun x ↦ f x / r` tends to negative infinity along a filter `l` if and only if `f` tends to infinity along `l`. -/ lemma tendsto_div_const_atBot_of_neg (hr : r < 0) : Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atTop := by simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_neg, hr] /-- The function `fun x ↦ r * f x` tends to infinity along a nontrivial filter if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/ theorem tendsto_const_mul_atTop_iff [NeBot l] : Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by rcases lt_trichotomy r 0 with (hr \| rfl \| hr) · simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_neg] · simp [not_tendsto_const_atTop] · simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_pos] #align filter.tendsto_const_mul_at_top_iff Filter.tendsto_const_mul_atTop_iff /-- The function `fun x ↦ f x * r` tends to infinity along a nontrivial filter if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/ theorem tendsto_mul_const_atTop_iff [NeBot l] : Tendsto (fun x => f x * r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by simp only [mul_comm _ r, tendsto_const_mul_atTop_iff] #align filter.tendsto_mul_const_at_top_iff Filter.tendsto_mul_const_atTop_iff /-- The function `fun x ↦ f x / r` tends to infinity along a nontrivial filter if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/ lemma tendsto_div_const_atTop_iff [NeBot l] : Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff] /-- The function `fun x ↦ r * f x` tends to negative infinity along a nontrivial filter if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/ theorem tendsto_const_mul_atBot_iff [NeBot l] : Tendsto (fun x => r * f x) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by simp only [← tendsto_neg_atTop_iff, ← mul_neg, tendsto_const_mul_atTop_iff, neg_neg] #align filter.tendsto_const_mul_at_bot_iff Filter.tendsto_const_mul_atBot_iff /-- The function `fun x ↦ f x * r` tends to negative infinity along a nontrivial filter if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/ theorem tendsto_mul_const_atBot_iff [NeBot l] : Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by simp only [mul_comm _ r, tendsto_const_mul_atBot_iff] #align filter.tendsto_mul_const_at_bot_iff Filter.tendsto_mul_const_atBot_iff /-- The function `fun x ↦ f x / r` tends to negative infinity along a nontrivial filter if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/ lemma tendsto_div_const_atBot_iff [NeBot l] : Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff] /-- If `f` tends to negative infinity along a nontrivial filter `l`, then `fun x ↦ r * f x` tends to infinity if and only if `r < 0. `-/	Mathlib/Order/Filter/AtTopBot.lean	1,238	1,240	theorem tendsto_const_mul_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) : Tendsto (fun x => r * f x) l atTop ↔ r < 0 := by	simp [tendsto_const_mul_atTop_iff, h, h.not_tendsto disjoint_atBot_atTop]
/- Copyright (c) 2024 Bolton Bailey. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro -/ import Mathlib.Data.List.Defs import Mathlib.Data.Option.Basic import Mathlib.Data.Nat.Defs import Mathlib.Init.Data.List.Basic import Mathlib.Util.AssertExists /-! # getD and getI This file provides theorems for working with the `getD` and `getI` functions. These are used to access an element of a list by numerical index, with a default value as a fallback when the index is out of range. -/ -- Make sure we haven't imported `Data.Nat.Order.Basic` assert_not_exists OrderedSub namespace List universe u v variable {α : Type u} {β : Type v} (l : List α) (x : α) (xs : List α) (n : ℕ) section getD variable (d : α) #align list.nthd_nil List.getD_nilₓ -- argument order #align list.nthd_cons_zero List.getD_cons_zeroₓ -- argument order #align list.nthd_cons_succ List.getD_cons_succₓ -- argument order theorem getD_eq_get {n : ℕ} (hn : n < l.length) : l.getD n d = l.get ⟨n, hn⟩ := by induction l generalizing n with \| nil => simp at hn \| cons head tail ih => cases n · exact getD_cons_zero · exact ih _ @[simp] theorem getD_map {n : ℕ} (f : α → β) : (map f l).getD n (f d) = f (l.getD n d) := by induction l generalizing n with \| nil => rfl \| cons head tail ih => cases n · rfl · simp [ih] #align list.nthd_eq_nth_le List.getD_eq_get theorem getD_eq_default {n : ℕ} (hn : l.length ≤ n) : l.getD n d = d := by induction l generalizing n with \| nil => exact getD_nil \| cons head tail ih => cases n · simp at hn · exact ih (Nat.le_of_succ_le_succ hn) #align list.nthd_eq_default List.getD_eq_defaultₓ -- argument order /-- An empty list can always be decidably checked for the presence of an element. Not an instance because it would clash with `DecidableEq α`. -/ def decidableGetDNilNe (a : α) : DecidablePred fun i : ℕ => getD ([] : List α) i a ≠ a := fun _ => isFalse fun H => H getD_nil #align list.decidable_nthd_nil_ne List.decidableGetDNilNeₓ -- argument order @[simp] theorem getD_singleton_default_eq (n : ℕ) : [d].getD n d = d := by cases n <;> simp #align list.nthd_singleton_default_eq List.getD_singleton_default_eqₓ -- argument order @[simp] theorem getD_replicate_default_eq (r n : ℕ) : (replicate r d).getD n d = d := by induction r generalizing n with \| zero => simp \| succ n ih => cases n <;> simp [ih] #align list.nthd_replicate_default_eq List.getD_replicate_default_eqₓ -- argument order theorem getD_append (l l' : List α) (d : α) (n : ℕ) (h : n < l.length) : (l ++ l').getD n d = l.getD n d := by rw [getD_eq_get _ _ (Nat.lt_of_lt_of_le h (length_append _ _ ▸ Nat.le_add_right _ _)), get_append _ h, getD_eq_get] #align list.nthd_append List.getD_appendₓ -- argument order theorem getD_append_right (l l' : List α) (d : α) (n : ℕ) (h : l.length ≤ n) : (l ++ l').getD n d = l'.getD (n - l.length) d := by cases Nat.lt_or_ge n (l ++ l').length with \| inl h' => rw [getD_eq_get (l ++ l') d h', get_append_right, getD_eq_get] · rw [length_append] at h' exact Nat.sub_lt_left_of_lt_add h h' · exact Nat.not_lt_of_le h \| inr h' => rw [getD_eq_default _ _ h', getD_eq_default] rwa [Nat.le_sub_iff_add_le' h, ← length_append] #align list.nthd_append_right List.getD_append_rightₓ -- argument order	Mathlib/Data/List/GetD.lean	102	105	theorem getD_eq_getD_get? (n : ℕ) : l.getD n d = (l.get? n).getD d := by	cases Nat.lt_or_ge n l.length with \| inl h => rw [getD_eq_get _ _ h, get?_eq_get h, Option.getD_some] \| inr h => rw [getD_eq_default _ _ h, get?_eq_none.mpr h, Option.getD_none]
/- Copyright (c) 2021 Yury Kudriashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudriashov, Malo Jaffré -/ import Mathlib.Analysis.Convex.Function import Mathlib.Tactic.AdaptationNote import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.Linarith #align_import analysis.convex.slope from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb" /-! # Slopes of convex functions This file relates convexity/concavity of functions in a linearly ordered field and the monotonicity of their slopes. The main use is to show convexity/concavity from monotonicity of the derivative. -/ variable {𝕜 : Type} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} #adaptation_note /-- after v4.7.0-rc1, there is a performance problem in `field_simp`. (Part of the code was ignoring the `maxDischargeDepth` setting: now that we have to increase it, other paths become slow.) -/ /-- If `f : 𝕜 → 𝕜` is convex, then for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/ theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) := by have hxz := hxy.trans hyz rw [← sub_pos] at hxy hxz hyz suffices f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y) by ring_nf at this ⊢ linarith set a := (z - y) / (z - x) set b := (y - x) / (z - x) have hy : a • x + b • z = y := by field_simp [a, b]; ring have key := hf.2 hx hz (show 0 ≤ a by apply div_nonneg <;> linarith) (show 0 ≤ b by apply div_nonneg <;> linarith) (show a + b = 1 by field_simp [a, b]) rw [hy] at key replace key := mul_le_mul_of_nonneg_left key hxz.le field_simp [a, b, mul_comm (z - x) _] at key ⊢ rw [div_le_div_right] · linarith · nlinarith #align convex_on.slope_mono_adjacent ConvexOn.slope_mono_adjacent /-- If `f : 𝕜 → 𝕜` is concave, then for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/ theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) := by have := neg_le_neg (ConvexOn.slope_mono_adjacent hf.neg hx hz hxy hyz) simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this exact this #align concave_on.slope_anti_adjacent ConcaveOn.slope_anti_adjacent /-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on `[x, z]`. -/ theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) < (f z - f y) / (z - y) := by have hxz := hxy.trans hyz have hxz' := hxz.ne rw [← sub_pos] at hxy hxz hyz suffices f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y) by ring_nf at this ⊢ linarith set a := (z - y) / (z - x) set b := (y - x) / (z - x) have hy : a • x + b • z = y := by field_simp [a, b]; ring have key := hf.2 hx hz hxz' (div_pos hyz hxz) (div_pos hxy hxz) (show a + b = 1 by field_simp [a, b]) rw [hy] at key replace key := mul_lt_mul_of_pos_left key hxz field_simp [mul_comm (z - x) _] at key ⊢ rw [div_lt_div_right] · linarith · nlinarith #align strict_convex_on.slope_strict_mono_adjacent StrictConvexOn.slope_strict_mono_adjacent /-- If `f : 𝕜 → 𝕜` is strictly concave, then for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is strictly greater than the slope of the secant line of `f` on `[x, z]`. -/ theorem StrictConcaveOn.slope_anti_adjacent (hf : StrictConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f z - f y) / (z - y) < (f y - f x) / (y - x) := by have := neg_lt_neg (StrictConvexOn.slope_strict_mono_adjacent hf.neg hx hz hxy hyz) simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this exact this #align strict_concave_on.slope_anti_adjacent StrictConcaveOn.slope_anti_adjacent /-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`, then `f` is convex. -/ theorem convexOn_of_slope_mono_adjacent (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) : ConvexOn 𝕜 s f := LinearOrder.convexOn_of_lt hs fun x hx z hz hxz a b ha hb hab => by let y := a x + b * z have hxy : x < y := by rw [← one_mul x, ← hab, add_mul] exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ have hyz : y < z := by rw [← one_mul z, ← hab, add_mul] exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ have : (f y - f x) * (z - y) ≤ (f z - f y) * (y - x) := (div_le_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz) have hxz : 0 < z - x := sub_pos.2 (hxy.trans hyz) have ha : (z - y) / (z - x) = a := by rw [eq_comm, ← sub_eq_iff_eq_add'] at hab dsimp [y] simp_rw [div_eq_iff hxz.ne', ← hab] ring have hb : (y - x) / (z - x) = b := by rw [eq_comm, ← sub_eq_iff_eq_add] at hab dsimp [y] simp_rw [div_eq_iff hxz.ne', ← hab] ring rwa [sub_mul, sub_mul, sub_le_iff_le_add', ← add_sub_assoc, le_sub_iff_add_le, ← mul_add, sub_add_sub_cancel, ← le_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x), mul_comm (f z), ha, hb] at this #align convex_on_of_slope_mono_adjacent convexOn_of_slope_mono_adjacent /-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`, then `f` is concave. -/ theorem concaveOn_of_slope_anti_adjacent (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : ConcaveOn 𝕜 s f := by rw [← neg_convexOn_iff] refine convexOn_of_slope_mono_adjacent hs fun hx hz hxy hyz => ?_ rw [← neg_le_neg_iff] simp_rw [← neg_div, neg_sub, Pi.neg_apply, neg_sub_neg] exact hf hx hz hxy hyz #align concave_on_of_slope_anti_adjacent concaveOn_of_slope_anti_adjacent /-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is strictly less than the slope of the secant line of `f` on `[x, z]`, then `f` is strictly convex. -/ theorem strictConvexOn_of_slope_strict_mono_adjacent (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)) : StrictConvexOn 𝕜 s f := LinearOrder.strictConvexOn_of_lt hs fun x hx z hz hxz a b ha hb hab => by let y := a * x + b * z have hxy : x < y := by rw [← one_mul x, ← hab, add_mul] exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ have hyz : y < z := by rw [← one_mul z, ← hab, add_mul] exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ have : (f y - f x) * (z - y) < (f z - f y) * (y - x) := (div_lt_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz) have hxz : 0 < z - x := sub_pos.2 (hxy.trans hyz) have ha : (z - y) / (z - x) = a := by rw [eq_comm, ← sub_eq_iff_eq_add'] at hab dsimp [y] simp_rw [div_eq_iff hxz.ne', ← hab] ring have hb : (y - x) / (z - x) = b := by rw [eq_comm, ← sub_eq_iff_eq_add] at hab dsimp [y] simp_rw [div_eq_iff hxz.ne', ← hab] ring rwa [sub_mul, sub_mul, sub_lt_iff_lt_add', ← add_sub_assoc, lt_sub_iff_add_lt, ← mul_add, sub_add_sub_cancel, ← lt_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x), mul_comm (f z), ha, hb] at this #align strict_convex_on_of_slope_strict_mono_adjacent strictConvexOn_of_slope_strict_mono_adjacent /-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is strictly greater than the slope of the secant line of `f` on `[x, z]`, then `f` is strictly concave. -/ theorem strictConcaveOn_of_slope_strict_anti_adjacent (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)) : StrictConcaveOn 𝕜 s f := by rw [← neg_strictConvexOn_iff] refine strictConvexOn_of_slope_strict_mono_adjacent hs fun hx hz hxy hyz => ?_ rw [← neg_lt_neg_iff] simp_rw [← neg_div, neg_sub, Pi.neg_apply, neg_sub_neg] exact hf hx hz hxy hyz #align strict_concave_on_of_slope_strict_anti_adjacent strictConcaveOn_of_slope_strict_anti_adjacent /-- A function `f : 𝕜 → 𝕜` is convex iff for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is less than the slope of the secant line of `f` on `[x, z]`. -/ theorem convexOn_iff_slope_mono_adjacent : ConvexOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) := ⟨fun h => ⟨h.1, fun _ _ _ => h.slope_mono_adjacent⟩, fun h => convexOn_of_slope_mono_adjacent h.1 (@fun _ _ _ hx hy => h.2 hx hy)⟩ #align convex_on_iff_slope_mono_adjacent convexOn_iff_slope_mono_adjacent /-- A function `f : 𝕜 → 𝕜` is concave iff for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is greater than the slope of the secant line of `f` on `[x, z]`. -/ theorem concaveOn_iff_slope_anti_adjacent : ConcaveOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) := ⟨fun h => ⟨h.1, fun _ _ _ => h.slope_anti_adjacent⟩, fun h => concaveOn_of_slope_anti_adjacent h.1 (@fun _ _ _ hx hy => h.2 hx hy)⟩ #align concave_on_iff_slope_anti_adjacent concaveOn_iff_slope_anti_adjacent /-- A function `f : 𝕜 → 𝕜` is strictly convex iff for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on `[x, z]`. -/ theorem strictConvexOn_iff_slope_strict_mono_adjacent : StrictConvexOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y) := ⟨fun h => ⟨h.1, fun _ _ _ => h.slope_strict_mono_adjacent⟩, fun h => strictConvexOn_of_slope_strict_mono_adjacent h.1 (@fun _ _ _ hx hy => h.2 hx hy)⟩ #align strict_convex_on_iff_slope_strict_mono_adjacent strictConvexOn_iff_slope_strict_mono_adjacent /-- A function `f : 𝕜 → 𝕜` is strictly concave iff for any three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is strictly greater than the slope of the secant line of `f` on `[x, z]`. -/ theorem strictConcaveOn_iff_slope_strict_anti_adjacent : StrictConcaveOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x) := ⟨fun h => ⟨h.1, fun _ _ _ => h.slope_anti_adjacent⟩, fun h => strictConcaveOn_of_slope_strict_anti_adjacent h.1 (@fun _ _ _ hx hy => h.2 hx hy)⟩ #align strict_concave_on_iff_slope_strict_anti_adjacent strictConcaveOn_iff_slope_strict_anti_adjacent theorem ConvexOn.secant_mono_aux1 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (z - x) * f y ≤ (z - y) * f x + (y - x) * f z := by have hxy' : 0 < y - x := by linarith have hyz' : 0 < z - y := by linarith have hxz' : 0 < z - x := by linarith rw [← le_div_iff' hxz'] have ha : 0 ≤ (z - y) / (z - x) := by positivity have hb : 0 ≤ (y - x) / (z - x) := by positivity calc f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) := ?_ _ ≤ (z - y) / (z - x) * f x + (y - x) / (z - x) * f z := hf.2 hx hz ha hb ?_ _ = ((z - y) * f x + (y - x) * f z) / (z - x) := ?_ · congr 1 field_simp ring · -- Porting note: this `show` wasn't needed in Lean 3 show (z - y) / (z - x) + (y - x) / (z - x) = 1 field_simp · field_simp #align convex_on.secant_mono_aux1 ConvexOn.secant_mono_aux1 theorem ConvexOn.secant_mono_aux2 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f x) / (z - x) := by have hxy' : 0 < y - x := by linarith have hxz' : 0 < z - x := by linarith rw [div_le_div_iff hxy' hxz'] linarith only [hf.secant_mono_aux1 hx hz hxy hyz] #align convex_on.secant_mono_aux2 ConvexOn.secant_mono_aux2	Mathlib/Analysis/Convex/Slope.lean	265	270	theorem ConvexOn.secant_mono_aux3 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f z - f x) / (z - x) ≤ (f z - f y) / (z - y) := by	have hyz' : 0 < z - y := by linarith have hxz' : 0 < z - x := by linarith rw [div_le_div_iff hxz' hyz'] linarith only [hf.secant_mono_aux1 hx hz hxy hyz]
/- Copyright (c) 2022 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import Mathlib.CategoryTheory.Limits.Shapes.Kernels import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero #align_import category_theory.limits.preserves.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d" /-! # Preserving (co)kernels Constructions to relate the notions of preserving (co)kernels and reflecting (co)kernels to concrete (co)forks. In particular, we show that `kernel_comparison f g G` is an isomorphism iff `G` preserves the limit of the parallel pair `f,0`, as well as the dual result. -/ noncomputable section universe v₁ v₂ u₁ u₂ open CategoryTheory CategoryTheory.Category CategoryTheory.Limits variable {C : Type u₁} [Category.{v₁} C] [HasZeroMorphisms C] variable {D : Type u₂} [Category.{v₂} D] [HasZeroMorphisms D] namespace CategoryTheory.Limits namespace KernelFork variable {X Y : C} {f : X ⟶ Y} (c : KernelFork f) (G : C ⥤ D) [Functor.PreservesZeroMorphisms G] @[reassoc (attr := simp)] lemma map_condition : G.map c.ι ≫ G.map f = 0 := by rw [← G.map_comp, c.condition, G.map_zero] /-- A kernel fork for `f` is mapped to a kernel fork for `G.map f` if `G` is a functor which preserves zero morphisms. -/ def map : KernelFork (G.map f) := KernelFork.ofι (G.map c.ι) (c.map_condition G) @[simp] lemma map_ι : (c.map G).ι = G.map c.ι := rfl /-- The underlying cone of a kernel fork is mapped to a limit cone if and only if the mapped kernel fork is limit. -/ def isLimitMapConeEquiv : IsLimit (G.mapCone c) ≃ IsLimit (c.map G) := by refine (IsLimit.postcomposeHomEquiv ?_ _).symm.trans (IsLimit.equivIsoLimit ?_) refine parallelPair.ext (Iso.refl _) (Iso.refl _) ?_ ?_ <;> simp exact Cones.ext (Iso.refl _) (by rintro (_\|_) <;> aesop_cat) /-- A limit kernel fork is mapped to a limit kernel fork by a functor `G` when this functor preserves the corresponding limit. -/ def mapIsLimit (hc : IsLimit c) (G : C ⥤ D) [Functor.PreservesZeroMorphisms G] [PreservesLimit (parallelPair f 0) G] : IsLimit (c.map G) := c.isLimitMapConeEquiv G (isLimitOfPreserves G hc) end KernelFork section Kernels variable (G : C ⥤ D) [Functor.PreservesZeroMorphisms G] {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0) /-- The map of a kernel fork is a limit iff the kernel fork consisting of the mapped morphisms is a limit. This essentially lets us commute `KernelFork.ofι` with `Functor.mapCone`. This is a variant of `isLimitMapConeForkEquiv` for equalizers, which we can't use directly between `G.map 0 = 0` does not hold definitionally. -/ def isLimitMapConeForkEquiv' : IsLimit (G.mapCone (KernelFork.ofι h w)) ≃ IsLimit (KernelFork.ofι (G.map h) (by simp only [← G.map_comp, w, Functor.map_zero]) : Fork (G.map f) 0) := KernelFork.isLimitMapConeEquiv _ _ #align category_theory.limits.is_limit_map_cone_fork_equiv' CategoryTheory.Limits.isLimitMapConeForkEquiv' /-- The property of preserving kernels expressed in terms of kernel forks. This is a variant of `isLimitForkMapOfIsLimit` for equalizers, which we can't use directly between `G.map 0 = 0` does not hold definitionally. -/ def isLimitForkMapOfIsLimit' [PreservesLimit (parallelPair f 0) G] (l : IsLimit (KernelFork.ofι h w)) : IsLimit (KernelFork.ofι (G.map h) (by simp only [← G.map_comp, w, Functor.map_zero]) : Fork (G.map f) 0) := isLimitMapConeForkEquiv' G w (PreservesLimit.preserves l) #align category_theory.limits.is_limit_fork_map_of_is_limit' CategoryTheory.Limits.isLimitForkMapOfIsLimit' variable (f) [HasKernel f] /-- If `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of a kernel fork is a limit. -/ def isLimitOfHasKernelOfPreservesLimit [PreservesLimit (parallelPair f 0) G] : IsLimit (Fork.ofι (G.map (kernel.ι f)) (by simp only [← G.map_comp, kernel.condition, comp_zero, Functor.map_zero]) : Fork (G.map f) 0) := isLimitForkMapOfIsLimit' G (kernel.condition f) (kernelIsKernel f) #align category_theory.limits.is_limit_of_has_kernel_of_preserves_limit CategoryTheory.Limits.isLimitOfHasKernelOfPreservesLimit instance [PreservesLimit (parallelPair f 0) G] : HasKernel (G.map f) where exists_limit := ⟨⟨_, isLimitOfHasKernelOfPreservesLimit G f⟩⟩ variable [HasKernel (G.map f)] /-- If the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the kernel of `f`. -/ def PreservesKernel.ofIsoComparison [i : IsIso (kernelComparison f G)] : PreservesLimit (parallelPair f 0) G := by apply preservesLimitOfPreservesLimitCone (kernelIsKernel f) apply (isLimitMapConeForkEquiv' G (kernel.condition f)).symm _ exact @IsLimit.ofPointIso _ _ _ _ _ _ _ (kernelIsKernel (G.map f)) i #align category_theory.limits.preserves_kernel.of_iso_comparison CategoryTheory.Limits.PreservesKernel.ofIsoComparison variable [PreservesLimit (parallelPair f 0) G] /-- If `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is an isomorphism. -/ def PreservesKernel.iso : G.obj (kernel f) ≅ kernel (G.map f) := IsLimit.conePointUniqueUpToIso (isLimitOfHasKernelOfPreservesLimit G f) (limit.isLimit _) #align category_theory.limits.preserves_kernel.iso CategoryTheory.Limits.PreservesKernel.iso @[simp] theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).hom = kernelComparison f G := by rw [← cancel_mono (kernel.ι _)] simp [PreservesKernel.iso] #align category_theory.limits.preserves_kernel.iso_hom CategoryTheory.Limits.PreservesKernel.iso_hom instance : IsIso (kernelComparison f G) := by rw [← PreservesKernel.iso_hom] infer_instance @[reassoc] theorem kernel_map_comp_preserves_kernel_iso_inv {X' Y' : C} (g : X' ⟶ Y') [HasKernel g] [HasKernel (G.map g)] [PreservesLimit (parallelPair g 0) G] (p : X ⟶ X') (q : Y ⟶ Y') (hpq : f ≫ q = p ≫ g) : kernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by rw [← G.map_comp, hpq, G.map_comp]) ≫ (PreservesKernel.iso G _).inv = (PreservesKernel.iso G _).inv ≫ G.map (kernel.map f g p q hpq) := by rw [Iso.comp_inv_eq, Category.assoc, PreservesKernel.iso_hom, Iso.eq_inv_comp, PreservesKernel.iso_hom, kernelComparison_comp_kernel_map] #align category_theory.limits.kernel_map_comp_preserves_kernel_iso_inv CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv end Kernels namespace CokernelCofork variable {X Y : C} {f : X ⟶ Y} (c : CokernelCofork f) (G : C ⥤ D) [Functor.PreservesZeroMorphisms G] @[reassoc (attr := simp)] lemma map_condition : G.map f ≫ G.map c.π = 0 := by rw [← G.map_comp, c.condition, G.map_zero] /-- A cokernel cofork for `f` is mapped to a cokernel cofork for `G.map f` if `G` is a functor which preserves zero morphisms. -/ def map : CokernelCofork (G.map f) := CokernelCofork.ofπ (G.map c.π) (c.map_condition G) @[simp] lemma map_π : (c.map G).π = G.map c.π := rfl /-- The underlying cocone of a cokernel cofork is mapped to a colimit cocone if and only if the mapped cokernel cofork is colimit. -/ def isColimitMapCoconeEquiv : IsColimit (G.mapCocone c) ≃ IsColimit (c.map G) := by refine (IsColimit.precomposeHomEquiv ?_ _).symm.trans (IsColimit.equivIsoColimit ?_) refine parallelPair.ext (Iso.refl _) (Iso.refl _) ?_ ?_ <;> simp exact Cocones.ext (Iso.refl _) (by rintro (_\|_) <;> aesop_cat) /-- A colimit cokernel cofork is mapped to a colimit cokernel cofork by a functor `G` when this functor preserves the corresponding colimit. -/ def mapIsColimit (hc : IsColimit c) (G : C ⥤ D) [Functor.PreservesZeroMorphisms G] [PreservesColimit (parallelPair f 0) G] : IsColimit (c.map G) := c.isColimitMapCoconeEquiv G (isColimitOfPreserves G hc) end CokernelCofork section Cokernels variable (G : C ⥤ D) [Functor.PreservesZeroMorphisms G] {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0) /-- The map of a cokernel cofork is a colimit iff the cokernel cofork consisting of the mapped morphisms is a colimit. This essentially lets us commute `CokernelCofork.ofπ` with `Functor.mapCocone`. This is a variant of `isColimitMapCoconeCoforkEquiv` for equalizers, which we can't use directly between `G.map 0 = 0` does not hold definitionally. -/ def isColimitMapCoconeCoforkEquiv' : IsColimit (G.mapCocone (CokernelCofork.ofπ h w)) ≃ IsColimit (CokernelCofork.ofπ (G.map h) (by simp only [← G.map_comp, w, Functor.map_zero]) : Cofork (G.map f) 0) := CokernelCofork.isColimitMapCoconeEquiv _ _ #align category_theory.limits.is_colimit_map_cocone_cofork_equiv' CategoryTheory.Limits.isColimitMapCoconeCoforkEquiv' /-- The property of preserving cokernels expressed in terms of cokernel coforks. This is a variant of `isColimitCoforkMapOfIsColimit` for equalizers, which we can't use directly between `G.map 0 = 0` does not hold definitionally. -/ def isColimitCoforkMapOfIsColimit' [PreservesColimit (parallelPair f 0) G] (l : IsColimit (CokernelCofork.ofπ h w)) : IsColimit (CokernelCofork.ofπ (G.map h) (by simp only [← G.map_comp, w, Functor.map_zero]) : Cofork (G.map f) 0) := isColimitMapCoconeCoforkEquiv' G w (PreservesColimit.preserves l) #align category_theory.limits.is_colimit_cofork_map_of_is_colimit' CategoryTheory.Limits.isColimitCoforkMapOfIsColimit' variable (f) [HasCokernel f] /-- If `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of a cokernel cofork is a colimit. -/ def isColimitOfHasCokernelOfPreservesColimit [PreservesColimit (parallelPair f 0) G] : IsColimit (Cofork.ofπ (G.map (cokernel.π f)) (by simp only [← G.map_comp, cokernel.condition, zero_comp, Functor.map_zero]) : Cofork (G.map f) 0) := isColimitCoforkMapOfIsColimit' G (cokernel.condition f) (cokernelIsCokernel f) #align category_theory.limits.is_colimit_of_has_cokernel_of_preserves_colimit CategoryTheory.Limits.isColimitOfHasCokernelOfPreservesColimit instance [PreservesColimit (parallelPair f 0) G] : HasCokernel (G.map f) where exists_colimit := ⟨⟨_, isColimitOfHasCokernelOfPreservesColimit G f⟩⟩ variable [HasCokernel (G.map f)] /-- If the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the cokernel of `f`. -/ def PreservesCokernel.ofIsoComparison [i : IsIso (cokernelComparison f G)] : PreservesColimit (parallelPair f 0) G := by apply preservesColimitOfPreservesColimitCocone (cokernelIsCokernel f) apply (isColimitMapCoconeCoforkEquiv' G (cokernel.condition f)).symm _ exact @IsColimit.ofPointIso _ _ _ _ _ _ _ (cokernelIsCokernel (G.map f)) i #align category_theory.limits.preserves_cokernel.of_iso_comparison CategoryTheory.Limits.PreservesCokernel.ofIsoComparison variable [PreservesColimit (parallelPair f 0) G] /-- If `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is an isomorphism. -/ def PreservesCokernel.iso : G.obj (cokernel f) ≅ cokernel (G.map f) := IsColimit.coconePointUniqueUpToIso (isColimitOfHasCokernelOfPreservesColimit G f) (colimit.isColimit _) #align category_theory.limits.preserves_cokernel.iso CategoryTheory.Limits.PreservesCokernel.iso @[simp]	Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean	267	269	theorem PreservesCokernel.iso_inv : (PreservesCokernel.iso G f).inv = cokernelComparison f G := by	rw [← cancel_epi (cokernel.π _)] simp [PreservesCokernel.iso]
/- Copyright (c) 2020 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel -/ import Mathlib.Order.Interval.Set.Disjoint import Mathlib.MeasureTheory.Integral.SetIntegral import Mathlib.MeasureTheory.Measure.Lebesgue.Basic #align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" /-! # Integral over an interval In this file we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ` if `a ≤ b` and `-∫ x in Ioc b a, f x ∂μ` if `b ≤ a`. ## Implementation notes ### Avoiding `if`, `min`, and `max` In order to avoid `if`s in the definition, we define `IntervalIntegrable f μ a b` as `integrable_on f (Ioc a b) μ ∧ integrable_on f (Ioc b a) μ`. For any `a`, `b` one of these intervals is empty and the other coincides with `Set.uIoc a b = Set.Ioc (min a b) (max a b)`. Similarly, we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`. Again, for any `a`, `b` one of these integrals is zero, and the other gives the expected result. This way some properties can be translated from integrals over sets without dealing with the cases `a ≤ b` and `b ≤ a` separately. ### Choice of the interval We use integral over `Set.uIoc a b = Set.Ioc (min a b) (max a b)` instead of one of the other three possible intervals with the same endpoints for two reasons: * this way `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ` holds whenever `f` is integrable on each interval; in particular, it works even if the measure `μ` has an atom at `b`; this rules out `Set.Ioo` and `Set.Icc` intervals; * with this definition for a probability measure `μ`, the integral `∫ x in a..b, 1 ∂μ` equals the difference $F_μ(b)-F_μ(a)$, where $F_μ(a)=μ(-∞, a]$ is the [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function) of `μ`. ## Tags integral -/ noncomputable section open scoped Classical open MeasureTheory Set Filter Function open scoped Classical Topology Filter ENNReal Interval NNReal variable {ι 𝕜 E F A : Type} [NormedAddCommGroup E] /-! ### Integrability on an interval -/ /-- A function `f` is called interval integrable* with respect to a measure `μ` on an unordered interval `a..b` if it is integrable on both intervals `(a, b]` and `(b, a]`. One of these intervals is always empty, so this property is equivalent to `f` being integrable on `(min a b, max a b]`. -/ def IntervalIntegrable (f : ℝ → E) (μ : Measure ℝ) (a b : ℝ) : Prop := IntegrableOn f (Ioc a b) μ ∧ IntegrableOn f (Ioc b a) μ #align interval_integrable IntervalIntegrable /-! ## Basic iff's for `IntervalIntegrable` -/ section variable {f : ℝ → E} {a b : ℝ} {μ : Measure ℝ} /-- A function is interval integrable with respect to a given measure `μ` on `a..b` if and only if it is integrable on `uIoc a b` with respect to `μ`. This is an equivalent definition of `IntervalIntegrable`. -/ theorem intervalIntegrable_iff : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ := by rw [uIoc_eq_union, integrableOn_union, IntervalIntegrable] #align interval_integrable_iff intervalIntegrable_iff /-- If a function is interval integrable with respect to a given measure `μ` on `a..b` then it is integrable on `uIoc a b` with respect to `μ`. -/ theorem IntervalIntegrable.def' (h : IntervalIntegrable f μ a b) : IntegrableOn f (Ι a b) μ := intervalIntegrable_iff.mp h #align interval_integrable.def IntervalIntegrable.def' theorem intervalIntegrable_iff_integrableOn_Ioc_of_le (hab : a ≤ b) : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ := by rw [intervalIntegrable_iff, uIoc_of_le hab] #align interval_integrable_iff_integrable_Ioc_of_le intervalIntegrable_iff_integrableOn_Ioc_of_le theorem intervalIntegrable_iff' [NoAtoms μ] : IntervalIntegrable f μ a b ↔ IntegrableOn f (uIcc a b) μ := by rw [intervalIntegrable_iff, ← Icc_min_max, uIoc, integrableOn_Icc_iff_integrableOn_Ioc] #align interval_integrable_iff' intervalIntegrable_iff' theorem intervalIntegrable_iff_integrableOn_Icc_of_le {f : ℝ → E} {a b : ℝ} (hab : a ≤ b) {μ : Measure ℝ} [NoAtoms μ] : IntervalIntegrable f μ a b ↔ IntegrableOn f (Icc a b) μ := by rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab, integrableOn_Icc_iff_integrableOn_Ioc] #align interval_integrable_iff_integrable_Icc_of_le intervalIntegrable_iff_integrableOn_Icc_of_le theorem intervalIntegrable_iff_integrableOn_Ico_of_le [NoAtoms μ] (hab : a ≤ b) : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ := by rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ico] theorem intervalIntegrable_iff_integrableOn_Ioo_of_le [NoAtoms μ] (hab : a ≤ b) : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ := by rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ioo] /-- If a function is integrable with respect to a given measure `μ` then it is interval integrable with respect to `μ` on `uIcc a b`. -/ theorem MeasureTheory.Integrable.intervalIntegrable (hf : Integrable f μ) : IntervalIntegrable f μ a b := ⟨hf.integrableOn, hf.integrableOn⟩ #align measure_theory.integrable.interval_integrable MeasureTheory.Integrable.intervalIntegrable theorem MeasureTheory.IntegrableOn.intervalIntegrable (hf : IntegrableOn f [[a, b]] μ) : IntervalIntegrable f μ a b := ⟨MeasureTheory.IntegrableOn.mono_set hf (Ioc_subset_Icc_self.trans Icc_subset_uIcc), MeasureTheory.IntegrableOn.mono_set hf (Ioc_subset_Icc_self.trans Icc_subset_uIcc')⟩ #align measure_theory.integrable_on.interval_integrable MeasureTheory.IntegrableOn.intervalIntegrable theorem intervalIntegrable_const_iff {c : E} : IntervalIntegrable (fun _ => c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ∞ := by simp only [intervalIntegrable_iff, integrableOn_const] #align interval_integrable_const_iff intervalIntegrable_const_iff @[simp] theorem intervalIntegrable_const [IsLocallyFiniteMeasure μ] {c : E} : IntervalIntegrable (fun _ => c) μ a b := intervalIntegrable_const_iff.2 <\| Or.inr measure_Ioc_lt_top #align interval_integrable_const intervalIntegrable_const end /-! ## Basic properties of interval integrability - interval integrability is symmetric, reflexive, transitive - monotonicity and strong measurability of the interval integral - if `f` is interval integrable, so are its absolute value and norm - arithmetic properties -/ namespace IntervalIntegrable section variable {f : ℝ → E} {a b c d : ℝ} {μ ν : Measure ℝ} @[symm] nonrec theorem symm (h : IntervalIntegrable f μ a b) : IntervalIntegrable f μ b a := h.symm #align interval_integrable.symm IntervalIntegrable.symm @[refl, simp] -- Porting note: added `simp` theorem refl : IntervalIntegrable f μ a a := by constructor <;> simp #align interval_integrable.refl IntervalIntegrable.refl @[trans] theorem trans {a b c : ℝ} (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c) : IntervalIntegrable f μ a c := ⟨(hab.1.union hbc.1).mono_set Ioc_subset_Ioc_union_Ioc, (hbc.2.union hab.2).mono_set Ioc_subset_Ioc_union_Ioc⟩ #align interval_integrable.trans IntervalIntegrable.trans theorem trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n) (hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <\| k + 1)) : IntervalIntegrable f μ (a m) (a n) := by revert hint refine Nat.le_induction ?_ ?_ n hmn · simp · intro p hp IH h exact (IH fun k hk => h k (Ico_subset_Ico_right p.le_succ hk)).trans (h p (by simp [hp])) #align interval_integrable.trans_iterate_Ico IntervalIntegrable.trans_iterate_Ico theorem trans_iterate {a : ℕ → ℝ} {n : ℕ} (hint : ∀ k < n, IntervalIntegrable f μ (a k) (a <\| k + 1)) : IntervalIntegrable f μ (a 0) (a n) := trans_iterate_Ico bot_le fun k hk => hint k hk.2 #align interval_integrable.trans_iterate IntervalIntegrable.trans_iterate theorem neg (h : IntervalIntegrable f μ a b) : IntervalIntegrable (-f) μ a b := ⟨h.1.neg, h.2.neg⟩ #align interval_integrable.neg IntervalIntegrable.neg theorem norm (h : IntervalIntegrable f μ a b) : IntervalIntegrable (fun x => ‖f x‖) μ a b := ⟨h.1.norm, h.2.norm⟩ #align interval_integrable.norm IntervalIntegrable.norm theorem intervalIntegrable_norm_iff {f : ℝ → E} {μ : Measure ℝ} {a b : ℝ} (hf : AEStronglyMeasurable f (μ.restrict (Ι a b))) : IntervalIntegrable (fun t => ‖f t‖) μ a b ↔ IntervalIntegrable f μ a b := by simp_rw [intervalIntegrable_iff, IntegrableOn]; exact integrable_norm_iff hf #align interval_integrable.interval_integrable_norm_iff IntervalIntegrable.intervalIntegrable_norm_iff theorem abs {f : ℝ → ℝ} (h : IntervalIntegrable f μ a b) : IntervalIntegrable (fun x => \|f x\|) μ a b := h.norm #align interval_integrable.abs IntervalIntegrable.abs theorem mono (hf : IntervalIntegrable f ν a b) (h1 : [[c, d]] ⊆ [[a, b]]) (h2 : μ ≤ ν) : IntervalIntegrable f μ c d := intervalIntegrable_iff.mpr <\| hf.def'.mono (uIoc_subset_uIoc_of_uIcc_subset_uIcc h1) h2 #align interval_integrable.mono IntervalIntegrable.mono theorem mono_measure (hf : IntervalIntegrable f ν a b) (h : μ ≤ ν) : IntervalIntegrable f μ a b := hf.mono Subset.rfl h #align interval_integrable.mono_measure IntervalIntegrable.mono_measure theorem mono_set (hf : IntervalIntegrable f μ a b) (h : [[c, d]] ⊆ [[a, b]]) : IntervalIntegrable f μ c d := hf.mono h le_rfl #align interval_integrable.mono_set IntervalIntegrable.mono_set theorem mono_set_ae (hf : IntervalIntegrable f μ a b) (h : Ι c d ≤ᵐ[μ] Ι a b) : IntervalIntegrable f μ c d := intervalIntegrable_iff.mpr <\| hf.def'.mono_set_ae h #align interval_integrable.mono_set_ae IntervalIntegrable.mono_set_ae theorem mono_set' (hf : IntervalIntegrable f μ a b) (hsub : Ι c d ⊆ Ι a b) : IntervalIntegrable f μ c d := hf.mono_set_ae <\| eventually_of_forall hsub #align interval_integrable.mono_set' IntervalIntegrable.mono_set' theorem mono_fun [NormedAddCommGroup F] {g : ℝ → F} (hf : IntervalIntegrable f μ a b) (hgm : AEStronglyMeasurable g (μ.restrict (Ι a b))) (hle : (fun x => ‖g x‖) ≤ᵐ[μ.restrict (Ι a b)] fun x => ‖f x‖) : IntervalIntegrable g μ a b := intervalIntegrable_iff.2 <\| hf.def'.integrable.mono hgm hle #align interval_integrable.mono_fun IntervalIntegrable.mono_fun theorem mono_fun' {g : ℝ → ℝ} (hg : IntervalIntegrable g μ a b) (hfm : AEStronglyMeasurable f (μ.restrict (Ι a b))) (hle : (fun x => ‖f x‖) ≤ᵐ[μ.restrict (Ι a b)] g) : IntervalIntegrable f μ a b := intervalIntegrable_iff.2 <\| hg.def'.integrable.mono' hfm hle #align interval_integrable.mono_fun' IntervalIntegrable.mono_fun' protected theorem aestronglyMeasurable (h : IntervalIntegrable f μ a b) : AEStronglyMeasurable f (μ.restrict (Ioc a b)) := h.1.aestronglyMeasurable #align interval_integrable.ae_strongly_measurable IntervalIntegrable.aestronglyMeasurable protected theorem aestronglyMeasurable' (h : IntervalIntegrable f μ a b) : AEStronglyMeasurable f (μ.restrict (Ioc b a)) := h.2.aestronglyMeasurable #align interval_integrable.ae_strongly_measurable' IntervalIntegrable.aestronglyMeasurable' end variable [NormedRing A] {f g : ℝ → E} {a b : ℝ} {μ : Measure ℝ} theorem smul [NormedField 𝕜] [NormedSpace 𝕜 E] {f : ℝ → E} {a b : ℝ} {μ : Measure ℝ} (h : IntervalIntegrable f μ a b) (r : 𝕜) : IntervalIntegrable (r • f) μ a b := ⟨h.1.smul r, h.2.smul r⟩ #align interval_integrable.smul IntervalIntegrable.smul @[simp] theorem add (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) : IntervalIntegrable (fun x => f x + g x) μ a b := ⟨hf.1.add hg.1, hf.2.add hg.2⟩ #align interval_integrable.add IntervalIntegrable.add @[simp] theorem sub (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) : IntervalIntegrable (fun x => f x - g x) μ a b := ⟨hf.1.sub hg.1, hf.2.sub hg.2⟩ #align interval_integrable.sub IntervalIntegrable.sub theorem sum (s : Finset ι) {f : ι → ℝ → E} (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) : IntervalIntegrable (∑ i ∈ s, f i) μ a b := ⟨integrable_finset_sum' s fun i hi => (h i hi).1, integrable_finset_sum' s fun i hi => (h i hi).2⟩ #align interval_integrable.sum IntervalIntegrable.sum theorem mul_continuousOn {f g : ℝ → A} (hf : IntervalIntegrable f μ a b) (hg : ContinuousOn g [[a, b]]) : IntervalIntegrable (fun x => f x * g x) μ a b := by rw [intervalIntegrable_iff] at hf ⊢ exact hf.mul_continuousOn_of_subset hg measurableSet_Ioc isCompact_uIcc Ioc_subset_Icc_self #align interval_integrable.mul_continuous_on IntervalIntegrable.mul_continuousOn theorem continuousOn_mul {f g : ℝ → A} (hf : IntervalIntegrable f μ a b) (hg : ContinuousOn g [[a, b]]) : IntervalIntegrable (fun x => g x * f x) μ a b := by rw [intervalIntegrable_iff] at hf ⊢ exact hf.continuousOn_mul_of_subset hg isCompact_uIcc measurableSet_Ioc Ioc_subset_Icc_self #align interval_integrable.continuous_on_mul IntervalIntegrable.continuousOn_mul @[simp] theorem const_mul {f : ℝ → A} (hf : IntervalIntegrable f μ a b) (c : A) : IntervalIntegrable (fun x => c * f x) μ a b := hf.continuousOn_mul continuousOn_const #align interval_integrable.const_mul IntervalIntegrable.const_mul @[simp] theorem mul_const {f : ℝ → A} (hf : IntervalIntegrable f μ a b) (c : A) : IntervalIntegrable (fun x => f x * c) μ a b := hf.mul_continuousOn continuousOn_const #align interval_integrable.mul_const IntervalIntegrable.mul_const @[simp] theorem div_const {𝕜 : Type} {f : ℝ → 𝕜} [NormedField 𝕜] (h : IntervalIntegrable f μ a b) (c : 𝕜) : IntervalIntegrable (fun x => f x / c) μ a b := by simpa only [div_eq_mul_inv] using mul_const h c⁻¹ #align interval_integrable.div_const IntervalIntegrable.div_const theorem comp_mul_left (hf : IntervalIntegrable f volume a b) (c : ℝ) : IntervalIntegrable (fun x => f (c x)) volume (a / c) (b / c) := by rcases eq_or_ne c 0 with (hc \| hc); · rw [hc]; simp rw [intervalIntegrable_iff'] at hf ⊢ have A : MeasurableEmbedding fun x => x * c⁻¹ := (Homeomorph.mulRight₀ _ (inv_ne_zero hc)).closedEmbedding.measurableEmbedding rw [← Real.smul_map_volume_mul_right (inv_ne_zero hc), IntegrableOn, Measure.restrict_smul, integrable_smul_measure (by simpa : ENNReal.ofReal \|c⁻¹\| ≠ 0) ENNReal.ofReal_ne_top, ← IntegrableOn, MeasurableEmbedding.integrableOn_map_iff A] convert hf using 1 · ext; simp only [comp_apply]; congr 1; field_simp · rw [preimage_mul_const_uIcc (inv_ne_zero hc)]; field_simp [hc] #align interval_integrable.comp_mul_left IntervalIntegrable.comp_mul_left -- Porting note (#10756): new lemma theorem comp_mul_left_iff {c : ℝ} (hc : c ≠ 0) : IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c) ↔ IntervalIntegrable f volume a b := ⟨fun h ↦ by simpa [hc] using h.comp_mul_left c⁻¹, (comp_mul_left · c)⟩ theorem comp_mul_right (hf : IntervalIntegrable f volume a b) (c : ℝ) : IntervalIntegrable (fun x => f (x * c)) volume (a / c) (b / c) := by simpa only [mul_comm] using comp_mul_left hf c #align interval_integrable.comp_mul_right IntervalIntegrable.comp_mul_right theorem comp_add_right (hf : IntervalIntegrable f volume a b) (c : ℝ) : IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) := by wlog h : a ≤ b generalizing a b · exact IntervalIntegrable.symm (this hf.symm (le_of_not_le h)) rw [intervalIntegrable_iff'] at hf ⊢ have A : MeasurableEmbedding fun x => x + c := (Homeomorph.addRight c).closedEmbedding.measurableEmbedding rw [← map_add_right_eq_self volume c] at hf convert (MeasurableEmbedding.integrableOn_map_iff A).mp hf using 1 rw [preimage_add_const_uIcc] #align interval_integrable.comp_add_right IntervalIntegrable.comp_add_right theorem comp_add_left (hf : IntervalIntegrable f volume a b) (c : ℝ) : IntervalIntegrable (fun x => f (c + x)) volume (a - c) (b - c) := by simpa only [add_comm] using IntervalIntegrable.comp_add_right hf c #align interval_integrable.comp_add_left IntervalIntegrable.comp_add_left theorem comp_sub_right (hf : IntervalIntegrable f volume a b) (c : ℝ) : IntervalIntegrable (fun x => f (x - c)) volume (a + c) (b + c) := by simpa only [sub_neg_eq_add] using IntervalIntegrable.comp_add_right hf (-c) #align interval_integrable.comp_sub_right IntervalIntegrable.comp_sub_right theorem iff_comp_neg : IntervalIntegrable f volume a b ↔ IntervalIntegrable (fun x => f (-x)) volume (-a) (-b) := by rw [← comp_mul_left_iff (neg_ne_zero.2 one_ne_zero)]; simp [div_neg] #align interval_integrable.iff_comp_neg IntervalIntegrable.iff_comp_neg theorem comp_sub_left (hf : IntervalIntegrable f volume a b) (c : ℝ) : IntervalIntegrable (fun x => f (c - x)) volume (c - a) (c - b) := by simpa only [neg_sub, ← sub_eq_add_neg] using iff_comp_neg.mp (hf.comp_add_left c) #align interval_integrable.comp_sub_left IntervalIntegrable.comp_sub_left end IntervalIntegrable /-! ## Continuous functions are interval integrable -/ section variable {μ : Measure ℝ} [IsLocallyFiniteMeasure μ] theorem ContinuousOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : ContinuousOn u (uIcc a b)) : IntervalIntegrable u μ a b := (ContinuousOn.integrableOn_Icc hu).intervalIntegrable #align continuous_on.interval_integrable ContinuousOn.intervalIntegrable theorem ContinuousOn.intervalIntegrable_of_Icc {u : ℝ → E} {a b : ℝ} (h : a ≤ b) (hu : ContinuousOn u (Icc a b)) : IntervalIntegrable u μ a b := ContinuousOn.intervalIntegrable ((uIcc_of_le h).symm ▸ hu) #align continuous_on.interval_integrable_of_Icc ContinuousOn.intervalIntegrable_of_Icc /-- A continuous function on `ℝ` is `IntervalIntegrable` with respect to any locally finite measure `ν` on ℝ. -/ theorem Continuous.intervalIntegrable {u : ℝ → E} (hu : Continuous u) (a b : ℝ) : IntervalIntegrable u μ a b := hu.continuousOn.intervalIntegrable #align continuous.interval_integrable Continuous.intervalIntegrable end /-! ## Monotone and antitone functions are integral integrable -/ section variable {μ : Measure ℝ} [IsLocallyFiniteMeasure μ] [ConditionallyCompleteLinearOrder E] [OrderTopology E] [SecondCountableTopology E] theorem MonotoneOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : MonotoneOn u (uIcc a b)) : IntervalIntegrable u μ a b := by rw [intervalIntegrable_iff] exact (hu.integrableOn_isCompact isCompact_uIcc).mono_set Ioc_subset_Icc_self #align monotone_on.interval_integrable MonotoneOn.intervalIntegrable theorem AntitoneOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : AntitoneOn u (uIcc a b)) : IntervalIntegrable u μ a b := hu.dual_right.intervalIntegrable #align antitone_on.interval_integrable AntitoneOn.intervalIntegrable theorem Monotone.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : Monotone u) : IntervalIntegrable u μ a b := (hu.monotoneOn _).intervalIntegrable #align monotone.interval_integrable Monotone.intervalIntegrable theorem Antitone.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : Antitone u) : IntervalIntegrable u μ a b := (hu.antitoneOn _).intervalIntegrable #align antitone.interval_integrable Antitone.intervalIntegrable end /-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'` eventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`. Suppose that `f : ℝ → E` has a finite limit at `l' ⊓ ae μ`. Then `f` is interval integrable on `u..v` provided that both `u` and `v` tend to `l`. Typeclass instances allow Lean to find `l'` based on `l` but not vice versa, so `apply Tendsto.eventually_intervalIntegrable_ae` will generate goals `Filter ℝ` and `TendstoIxxClass Ioc ?m_1 l'`. -/ theorem Filter.Tendsto.eventually_intervalIntegrable_ae {f : ℝ → E} {μ : Measure ℝ} {l l' : Filter ℝ} (hfm : StronglyMeasurableAtFilter f l' μ) [TendstoIxxClass Ioc l l'] [IsMeasurablyGenerated l'] (hμ : μ.FiniteAtFilter l') {c : E} (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) {u v : ι → ℝ} {lt : Filter ι} (hu : Tendsto u lt l) (hv : Tendsto v lt l) : ∀ᶠ t in lt, IntervalIntegrable f μ (u t) (v t) := have := (hf.integrableAtFilter_ae hfm hμ).eventually ((hu.Ioc hv).eventually this).and <\| (hv.Ioc hu).eventually this #align filter.tendsto.eventually_interval_integrable_ae Filter.Tendsto.eventually_intervalIntegrable_ae /-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'` eventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`. Suppose that `f : ℝ → E` has a finite limit at `l`. Then `f` is interval integrable on `u..v` provided that both `u` and `v` tend to `l`. Typeclass instances allow Lean to find `l'` based on `l` but not vice versa, so `apply Tendsto.eventually_intervalIntegrable` will generate goals `Filter ℝ` and `TendstoIxxClass Ioc ?m_1 l'`. -/ theorem Filter.Tendsto.eventually_intervalIntegrable {f : ℝ → E} {μ : Measure ℝ} {l l' : Filter ℝ} (hfm : StronglyMeasurableAtFilter f l' μ) [TendstoIxxClass Ioc l l'] [IsMeasurablyGenerated l'] (hμ : μ.FiniteAtFilter l') {c : E} (hf : Tendsto f l' (𝓝 c)) {u v : ι → ℝ} {lt : Filter ι} (hu : Tendsto u lt l) (hv : Tendsto v lt l) : ∀ᶠ t in lt, IntervalIntegrable f μ (u t) (v t) := (hf.mono_left inf_le_left).eventually_intervalIntegrable_ae hfm hμ hu hv #align filter.tendsto.eventually_interval_integrable Filter.Tendsto.eventually_intervalIntegrable /-! ### Interval integral: definition and basic properties In this section we define `∫ x in a..b, f x ∂μ` as `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ` and prove some basic properties. -/ variable [CompleteSpace E] [NormedSpace ℝ E] /-- The interval integral `∫ x in a..b, f x ∂μ` is defined as `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`. If `a ≤ b`, then it equals `∫ x in Ioc a b, f x ∂μ`, otherwise it equals `-∫ x in Ioc b a, f x ∂μ`. -/ def intervalIntegral (f : ℝ → E) (a b : ℝ) (μ : Measure ℝ) : E := (∫ x in Ioc a b, f x ∂μ) - ∫ x in Ioc b a, f x ∂μ #align interval_integral intervalIntegral notation3"∫ "(...)" in "a".."b", "r:60:(scoped f => f)" ∂"μ:70 => intervalIntegral r a b μ notation3"∫ "(...)" in "a".."b", "r:60:(scoped f => intervalIntegral f a b volume) => r namespace intervalIntegral section Basic variable {a b : ℝ} {f g : ℝ → E} {μ : Measure ℝ} @[simp] theorem integral_zero : (∫ _ in a..b, (0 : E) ∂μ) = 0 := by simp [intervalIntegral] #align interval_integral.integral_zero intervalIntegral.integral_zero theorem integral_of_le (h : a ≤ b) : ∫ x in a..b, f x ∂μ = ∫ x in Ioc a b, f x ∂μ := by simp [intervalIntegral, h] #align interval_integral.integral_of_le intervalIntegral.integral_of_le @[simp] theorem integral_same : ∫ x in a..a, f x ∂μ = 0 := sub_self _ #align interval_integral.integral_same intervalIntegral.integral_same theorem integral_symm (a b) : ∫ x in b..a, f x ∂μ = -∫ x in a..b, f x ∂μ := by simp only [intervalIntegral, neg_sub] #align interval_integral.integral_symm intervalIntegral.integral_symm theorem integral_of_ge (h : b ≤ a) : ∫ x in a..b, f x ∂μ = -∫ x in Ioc b a, f x ∂μ := by simp only [integral_symm b, integral_of_le h] #align interval_integral.integral_of_ge intervalIntegral.integral_of_ge theorem intervalIntegral_eq_integral_uIoc (f : ℝ → E) (a b : ℝ) (μ : Measure ℝ) : ∫ x in a..b, f x ∂μ = (if a ≤ b then 1 else -1 : ℝ) • ∫ x in Ι a b, f x ∂μ := by split_ifs with h · simp only [integral_of_le h, uIoc_of_le h, one_smul] · simp only [integral_of_ge (not_le.1 h).le, uIoc_of_lt (not_le.1 h), neg_one_smul] #align interval_integral.interval_integral_eq_integral_uIoc intervalIntegral.intervalIntegral_eq_integral_uIoc theorem norm_intervalIntegral_eq (f : ℝ → E) (a b : ℝ) (μ : Measure ℝ) : ‖∫ x in a..b, f x ∂μ‖ = ‖∫ x in Ι a b, f x ∂μ‖ := by simp_rw [intervalIntegral_eq_integral_uIoc, norm_smul] split_ifs <;> simp only [norm_neg, norm_one, one_mul] #align interval_integral.norm_interval_integral_eq intervalIntegral.norm_intervalIntegral_eq theorem abs_intervalIntegral_eq (f : ℝ → ℝ) (a b : ℝ) (μ : Measure ℝ) : \|∫ x in a..b, f x ∂μ\| = \|∫ x in Ι a b, f x ∂μ\| := norm_intervalIntegral_eq f a b μ #align interval_integral.abs_interval_integral_eq intervalIntegral.abs_intervalIntegral_eq theorem integral_cases (f : ℝ → E) (a b) : (∫ x in a..b, f x ∂μ) ∈ ({∫ x in Ι a b, f x ∂μ, -∫ x in Ι a b, f x ∂μ} : Set E) := by rw [intervalIntegral_eq_integral_uIoc]; split_ifs <;> simp #align interval_integral.integral_cases intervalIntegral.integral_cases nonrec theorem integral_undef (h : ¬IntervalIntegrable f μ a b) : ∫ x in a..b, f x ∂μ = 0 := by rw [intervalIntegrable_iff] at h rw [intervalIntegral_eq_integral_uIoc, integral_undef h, smul_zero] #align interval_integral.integral_undef intervalIntegral.integral_undef theorem intervalIntegrable_of_integral_ne_zero {a b : ℝ} {f : ℝ → E} {μ : Measure ℝ} (h : (∫ x in a..b, f x ∂μ) ≠ 0) : IntervalIntegrable f μ a b := not_imp_comm.1 integral_undef h #align interval_integral.interval_integrable_of_integral_ne_zero intervalIntegral.intervalIntegrable_of_integral_ne_zero nonrec theorem integral_non_aestronglyMeasurable (hf : ¬AEStronglyMeasurable f (μ.restrict (Ι a b))) : ∫ x in a..b, f x ∂μ = 0 := by rw [intervalIntegral_eq_integral_uIoc, integral_non_aestronglyMeasurable hf, smul_zero] #align interval_integral.integral_non_ae_strongly_measurable intervalIntegral.integral_non_aestronglyMeasurable theorem integral_non_aestronglyMeasurable_of_le (h : a ≤ b) (hf : ¬AEStronglyMeasurable f (μ.restrict (Ioc a b))) : ∫ x in a..b, f x ∂μ = 0 := integral_non_aestronglyMeasurable <\| by rwa [uIoc_of_le h] #align interval_integral.integral_non_ae_strongly_measurable_of_le intervalIntegral.integral_non_aestronglyMeasurable_of_le theorem norm_integral_min_max (f : ℝ → E) : ‖∫ x in min a b..max a b, f x ∂μ‖ = ‖∫ x in a..b, f x ∂μ‖ := by cases le_total a b <;> simp [, integral_symm a b] #align interval_integral.norm_integral_min_max intervalIntegral.norm_integral_min_max theorem norm_integral_eq_norm_integral_Ioc (f : ℝ → E) : ‖∫ x in a..b, f x ∂μ‖ = ‖∫ x in Ι a b, f x ∂μ‖ := by rw [← norm_integral_min_max, integral_of_le min_le_max, uIoc] #align interval_integral.norm_integral_eq_norm_integral_Ioc intervalIntegral.norm_integral_eq_norm_integral_Ioc theorem abs_integral_eq_abs_integral_uIoc (f : ℝ → ℝ) : \|∫ x in a..b, f x ∂μ\| = \|∫ x in Ι a b, f x ∂μ\| := norm_integral_eq_norm_integral_Ioc f #align interval_integral.abs_integral_eq_abs_integral_uIoc intervalIntegral.abs_integral_eq_abs_integral_uIoc theorem norm_integral_le_integral_norm_Ioc : ‖∫ x in a..b, f x ∂μ‖ ≤ ∫ x in Ι a b, ‖f x‖ ∂μ := calc ‖∫ x in a..b, f x ∂μ‖ = ‖∫ x in Ι a b, f x ∂μ‖ := norm_integral_eq_norm_integral_Ioc f _ ≤ ∫ x in Ι a b, ‖f x‖ ∂μ := norm_integral_le_integral_norm f #align interval_integral.norm_integral_le_integral_norm_Ioc intervalIntegral.norm_integral_le_integral_norm_Ioc theorem norm_integral_le_abs_integral_norm : ‖∫ x in a..b, f x ∂μ‖ ≤ \|∫ x in a..b, ‖f x‖ ∂μ\| := by simp only [← Real.norm_eq_abs, norm_integral_eq_norm_integral_Ioc] exact le_trans (norm_integral_le_integral_norm _) (le_abs_self _) #align interval_integral.norm_integral_le_abs_integral_norm intervalIntegral.norm_integral_le_abs_integral_norm theorem norm_integral_le_integral_norm (h : a ≤ b) : ‖∫ x in a..b, f x ∂μ‖ ≤ ∫ x in a..b, ‖f x‖ ∂μ := norm_integral_le_integral_norm_Ioc.trans_eq <\| by rw [uIoc_of_le h, integral_of_le h] #align interval_integral.norm_integral_le_integral_norm intervalIntegral.norm_integral_le_integral_norm nonrec theorem norm_integral_le_of_norm_le {g : ℝ → ℝ} (h : ∀ᵐ t ∂μ.restrict <\| Ι a b, ‖f t‖ ≤ g t) (hbound : IntervalIntegrable g μ a b) : ‖∫ t in a..b, f t ∂μ‖ ≤ \|∫ t in a..b, g t ∂μ\| := by simp_rw [norm_intervalIntegral_eq, abs_intervalIntegral_eq, abs_eq_self.mpr (integral_nonneg_of_ae <\| h.mono fun _t ht => (norm_nonneg _).trans ht), norm_integral_le_of_norm_le hbound.def' h] #align interval_integral.norm_integral_le_of_norm_le intervalIntegral.norm_integral_le_of_norm_le theorem norm_integral_le_of_norm_le_const_ae {a b C : ℝ} {f : ℝ → E} (h : ∀ᵐ x, x ∈ Ι a b → ‖f x‖ ≤ C) : ‖∫ x in a..b, f x‖ ≤ C \|b - a\| := by rw [norm_integral_eq_norm_integral_Ioc] convert norm_setIntegral_le_of_norm_le_const_ae'' _ measurableSet_Ioc h using 1 · rw [Real.volume_Ioc, max_sub_min_eq_abs, ENNReal.toReal_ofReal (abs_nonneg _)] · simp only [Real.volume_Ioc, ENNReal.ofReal_lt_top] #align interval_integral.norm_integral_le_of_norm_le_const_ae intervalIntegral.norm_integral_le_of_norm_le_const_ae theorem norm_integral_le_of_norm_le_const {a b C : ℝ} {f : ℝ → E} (h : ∀ x ∈ Ι a b, ‖f x‖ ≤ C) : ‖∫ x in a..b, f x‖ ≤ C * \|b - a\| := norm_integral_le_of_norm_le_const_ae <\| eventually_of_forall h #align interval_integral.norm_integral_le_of_norm_le_const intervalIntegral.norm_integral_le_of_norm_le_const @[simp] nonrec theorem integral_add (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) : ∫ x in a..b, f x + g x ∂μ = (∫ x in a..b, f x ∂μ) + ∫ x in a..b, g x ∂μ := by simp only [intervalIntegral_eq_integral_uIoc, integral_add hf.def' hg.def', smul_add] #align interval_integral.integral_add intervalIntegral.integral_add nonrec theorem integral_finset_sum {ι} {s : Finset ι} {f : ι → ℝ → E} (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) : ∫ x in a..b, ∑ i ∈ s, f i x ∂μ = ∑ i ∈ s, ∫ x in a..b, f i x ∂μ := by simp only [intervalIntegral_eq_integral_uIoc, integral_finset_sum s fun i hi => (h i hi).def', Finset.smul_sum] #align interval_integral.integral_finset_sum intervalIntegral.integral_finset_sum @[simp] nonrec theorem integral_neg : ∫ x in a..b, -f x ∂μ = -∫ x in a..b, f x ∂μ := by simp only [intervalIntegral, integral_neg]; abel #align interval_integral.integral_neg intervalIntegral.integral_neg @[simp] theorem integral_sub (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) : ∫ x in a..b, f x - g x ∂μ = (∫ x in a..b, f x ∂μ) - ∫ x in a..b, g x ∂μ := by simpa only [sub_eq_add_neg] using (integral_add hf hg.neg).trans (congr_arg _ integral_neg) #align interval_integral.integral_sub intervalIntegral.integral_sub @[simp] nonrec theorem integral_smul {𝕜 : Type} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℝ 𝕜 E] (r : 𝕜) (f : ℝ → E) : ∫ x in a..b, r • f x ∂μ = r • ∫ x in a..b, f x ∂μ := by simp only [intervalIntegral, integral_smul, smul_sub] #align interval_integral.integral_smul intervalIntegral.integral_smul @[simp] nonrec theorem integral_smul_const {𝕜 : Type} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : ℝ → 𝕜) (c : E) : ∫ x in a..b, f x • c ∂μ = (∫ x in a..b, f x ∂μ) • c := by simp only [intervalIntegral_eq_integral_uIoc, integral_smul_const, smul_assoc] #align interval_integral.integral_smul_const intervalIntegral.integral_smul_const @[simp] theorem integral_const_mul {𝕜 : Type} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) : ∫ x in a..b, r f x ∂μ = r * ∫ x in a..b, f x ∂μ := integral_smul r f #align interval_integral.integral_const_mul intervalIntegral.integral_const_mul @[simp] theorem integral_mul_const {𝕜 : Type} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) : ∫ x in a..b, f x r ∂μ = (∫ x in a..b, f x ∂μ) * r := by simpa only [mul_comm r] using integral_const_mul r f #align interval_integral.integral_mul_const intervalIntegral.integral_mul_const @[simp] theorem integral_div {𝕜 : Type} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) : ∫ x in a..b, f x / r ∂μ = (∫ x in a..b, f x ∂μ) / r := by simpa only [div_eq_mul_inv] using integral_mul_const r⁻¹ f #align interval_integral.integral_div intervalIntegral.integral_div theorem integral_const' (c : E) : ∫ _ in a..b, c ∂μ = ((μ <\| Ioc a b).toReal - (μ <\| Ioc b a).toReal) • c := by simp only [intervalIntegral, setIntegral_const, sub_smul] #align interval_integral.integral_const' intervalIntegral.integral_const' @[simp] theorem integral_const (c : E) : ∫ _ in a..b, c = (b - a) • c := by simp only [integral_const', Real.volume_Ioc, ENNReal.toReal_ofReal', ← neg_sub b, max_zero_sub_eq_self] #align interval_integral.integral_const intervalIntegral.integral_const nonrec theorem integral_smul_measure (c : ℝ≥0∞) : ∫ x in a..b, f x ∂c • μ = c.toReal • ∫ x in a..b, f x ∂μ := by simp only [intervalIntegral, Measure.restrict_smul, integral_smul_measure, smul_sub] #align interval_integral.integral_smul_measure intervalIntegral.integral_smul_measure end Basic -- Porting note (#11215): TODO: add `Complex.ofReal` version of `_root_.integral_ofReal` nonrec theorem _root_.RCLike.intervalIntegral_ofReal {𝕜 : Type} [RCLike 𝕜] {a b : ℝ} {μ : Measure ℝ} {f : ℝ → ℝ} : (∫ x in a..b, (f x : 𝕜) ∂μ) = ↑(∫ x in a..b, f x ∂μ) := by simp only [intervalIntegral, integral_ofReal, RCLike.ofReal_sub] @[deprecated (since := "2024-04-06")] alias RCLike.interval_integral_ofReal := RCLike.intervalIntegral_ofReal nonrec theorem integral_ofReal {a b : ℝ} {μ : Measure ℝ} {f : ℝ → ℝ} : (∫ x in a..b, (f x : ℂ) ∂μ) = ↑(∫ x in a..b, f x ∂μ) := RCLike.intervalIntegral_ofReal #align interval_integral.integral_of_real intervalIntegral.integral_ofReal section ContinuousLinearMap variable {a b : ℝ} {μ : Measure ℝ} {f : ℝ → E} variable [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] open ContinuousLinearMap theorem _root_.ContinuousLinearMap.intervalIntegral_apply {a b : ℝ} {φ : ℝ → F →L[𝕜] E} (hφ : IntervalIntegrable φ μ a b) (v : F) : (∫ x in a..b, φ x ∂μ) v = ∫ x in a..b, φ x v ∂μ := by simp_rw [intervalIntegral_eq_integral_uIoc, ← integral_apply hφ.def' v, coe_smul', Pi.smul_apply] #align continuous_linear_map.interval_integral_apply ContinuousLinearMap.intervalIntegral_apply variable [NormedSpace ℝ F] [CompleteSpace F] theorem _root_.ContinuousLinearMap.intervalIntegral_comp_comm (L : E →L[𝕜] F) (hf : IntervalIntegrable f μ a b) : (∫ x in a..b, L (f x) ∂μ) = L (∫ x in a..b, f x ∂μ) := by simp_rw [intervalIntegral, L.integral_comp_comm hf.1, L.integral_comp_comm hf.2, L.map_sub] #align continuous_linear_map.interval_integral_comp_comm ContinuousLinearMap.intervalIntegral_comp_comm end ContinuousLinearMap /-! ## Basic arithmetic Includes addition, scalar multiplication and affine transformations. -/ section Comp variable {a b c d : ℝ} (f : ℝ → E) /-! Porting note: some `@[simp]` attributes in this section were removed to make the `simpNF` linter happy. TODO: find out if these lemmas are actually good or bad `simp` lemmas. -/ -- Porting note (#10618): was @[simp] theorem integral_comp_mul_right (hc : c ≠ 0) : (∫ x in a..b, f (x * c)) = c⁻¹ • ∫ x in a * c..b * c, f x := by have A : MeasurableEmbedding fun x => x * c := (Homeomorph.mulRight₀ c hc).closedEmbedding.measurableEmbedding conv_rhs => rw [← Real.smul_map_volume_mul_right hc] simp_rw [integral_smul_measure, intervalIntegral, A.setIntegral_map, ENNReal.toReal_ofReal (abs_nonneg c)] cases' hc.lt_or_lt with h h · simp [h, mul_div_cancel_right₀, hc, abs_of_neg, Measure.restrict_congr_set (α := ℝ) (μ := volume) Ico_ae_eq_Ioc] · simp [h, mul_div_cancel_right₀, hc, abs_of_pos] #align interval_integral.integral_comp_mul_right intervalIntegral.integral_comp_mul_right -- Porting note (#10618): was @[simp] theorem smul_integral_comp_mul_right (c) : (c • ∫ x in a..b, f (x * c)) = ∫ x in a * c..b * c, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_mul_right] #align interval_integral.smul_integral_comp_mul_right intervalIntegral.smul_integral_comp_mul_right -- Porting note (#10618): was @[simp] theorem integral_comp_mul_left (hc : c ≠ 0) : (∫ x in a..b, f (c * x)) = c⁻¹ • ∫ x in c * a..c * b, f x := by simpa only [mul_comm c] using integral_comp_mul_right f hc #align interval_integral.integral_comp_mul_left intervalIntegral.integral_comp_mul_left -- Porting note (#10618): was @[simp] theorem smul_integral_comp_mul_left (c) : (c • ∫ x in a..b, f (c * x)) = ∫ x in c * a..c * b, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_mul_left] #align interval_integral.smul_integral_comp_mul_left intervalIntegral.smul_integral_comp_mul_left -- Porting note (#10618): was @[simp] theorem integral_comp_div (hc : c ≠ 0) : (∫ x in a..b, f (x / c)) = c • ∫ x in a / c..b / c, f x := by simpa only [inv_inv] using integral_comp_mul_right f (inv_ne_zero hc) #align interval_integral.integral_comp_div intervalIntegral.integral_comp_div -- Porting note (#10618): was @[simp] theorem inv_smul_integral_comp_div (c) : (c⁻¹ • ∫ x in a..b, f (x / c)) = ∫ x in a / c..b / c, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_div] #align interval_integral.inv_smul_integral_comp_div intervalIntegral.inv_smul_integral_comp_div -- Porting note (#10618): was @[simp] theorem integral_comp_add_right (d) : (∫ x in a..b, f (x + d)) = ∫ x in a + d..b + d, f x := have A : MeasurableEmbedding fun x => x + d := (Homeomorph.addRight d).closedEmbedding.measurableEmbedding calc (∫ x in a..b, f (x + d)) = ∫ x in a + d..b + d, f x ∂Measure.map (fun x => x + d) volume := by simp [intervalIntegral, A.setIntegral_map] _ = ∫ x in a + d..b + d, f x := by rw [map_add_right_eq_self] #align interval_integral.integral_comp_add_right intervalIntegral.integral_comp_add_right -- Porting note (#10618): was @[simp] nonrec theorem integral_comp_add_left (d) : (∫ x in a..b, f (d + x)) = ∫ x in d + a..d + b, f x := by simpa only [add_comm d] using integral_comp_add_right f d #align interval_integral.integral_comp_add_left intervalIntegral.integral_comp_add_left -- Porting note (#10618): was @[simp] theorem integral_comp_mul_add (hc : c ≠ 0) (d) : (∫ x in a..b, f (c * x + d)) = c⁻¹ • ∫ x in c * a + d..c * b + d, f x := by rw [← integral_comp_add_right, ← integral_comp_mul_left _ hc] #align interval_integral.integral_comp_mul_add intervalIntegral.integral_comp_mul_add -- Porting note (#10618): was @[simp] theorem smul_integral_comp_mul_add (c d) : (c • ∫ x in a..b, f (c * x + d)) = ∫ x in c * a + d..c * b + d, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_mul_add] #align interval_integral.smul_integral_comp_mul_add intervalIntegral.smul_integral_comp_mul_add -- Porting note (#10618): was @[simp] theorem integral_comp_add_mul (hc : c ≠ 0) (d) : (∫ x in a..b, f (d + c * x)) = c⁻¹ • ∫ x in d + c * a..d + c * b, f x := by rw [← integral_comp_add_left, ← integral_comp_mul_left _ hc] #align interval_integral.integral_comp_add_mul intervalIntegral.integral_comp_add_mul -- Porting note (#10618): was @[simp] theorem smul_integral_comp_add_mul (c d) : (c • ∫ x in a..b, f (d + c * x)) = ∫ x in d + c * a..d + c * b, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_add_mul] #align interval_integral.smul_integral_comp_add_mul intervalIntegral.smul_integral_comp_add_mul -- Porting note (#10618): was @[simp] theorem integral_comp_div_add (hc : c ≠ 0) (d) : (∫ x in a..b, f (x / c + d)) = c • ∫ x in a / c + d..b / c + d, f x := by simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_add f (inv_ne_zero hc) d #align interval_integral.integral_comp_div_add intervalIntegral.integral_comp_div_add -- Porting note (#10618): was @[simp] theorem inv_smul_integral_comp_div_add (c d) : (c⁻¹ • ∫ x in a..b, f (x / c + d)) = ∫ x in a / c + d..b / c + d, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_div_add] #align interval_integral.inv_smul_integral_comp_div_add intervalIntegral.inv_smul_integral_comp_div_add -- Porting note (#10618): was @[simp] theorem integral_comp_add_div (hc : c ≠ 0) (d) : (∫ x in a..b, f (d + x / c)) = c • ∫ x in d + a / c..d + b / c, f x := by simpa only [div_eq_inv_mul, inv_inv] using integral_comp_add_mul f (inv_ne_zero hc) d #align interval_integral.integral_comp_add_div intervalIntegral.integral_comp_add_div -- Porting note (#10618): was @[simp] theorem inv_smul_integral_comp_add_div (c d) : (c⁻¹ • ∫ x in a..b, f (d + x / c)) = ∫ x in d + a / c..d + b / c, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_add_div] #align interval_integral.inv_smul_integral_comp_add_div intervalIntegral.inv_smul_integral_comp_add_div -- Porting note (#10618): was @[simp] theorem integral_comp_mul_sub (hc : c ≠ 0) (d) : (∫ x in a..b, f (c * x - d)) = c⁻¹ • ∫ x in c * a - d..c * b - d, f x := by simpa only [sub_eq_add_neg] using integral_comp_mul_add f hc (-d) #align interval_integral.integral_comp_mul_sub intervalIntegral.integral_comp_mul_sub -- Porting note (#10618): was @[simp] theorem smul_integral_comp_mul_sub (c d) : (c • ∫ x in a..b, f (c * x - d)) = ∫ x in c * a - d..c * b - d, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_mul_sub] #align interval_integral.smul_integral_comp_mul_sub intervalIntegral.smul_integral_comp_mul_sub -- Porting note (#10618): was @[simp] theorem integral_comp_sub_mul (hc : c ≠ 0) (d) : (∫ x in a..b, f (d - c * x)) = c⁻¹ • ∫ x in d - c * b..d - c * a, f x := by simp only [sub_eq_add_neg, neg_mul_eq_neg_mul] rw [integral_comp_add_mul f (neg_ne_zero.mpr hc) d, integral_symm] simp only [inv_neg, smul_neg, neg_neg, neg_smul] #align interval_integral.integral_comp_sub_mul intervalIntegral.integral_comp_sub_mul -- Porting note (#10618): was @[simp] theorem smul_integral_comp_sub_mul (c d) : (c • ∫ x in a..b, f (d - c * x)) = ∫ x in d - c * b..d - c * a, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_sub_mul] #align interval_integral.smul_integral_comp_sub_mul intervalIntegral.smul_integral_comp_sub_mul -- Porting note (#10618): was @[simp] theorem integral_comp_div_sub (hc : c ≠ 0) (d) : (∫ x in a..b, f (x / c - d)) = c • ∫ x in a / c - d..b / c - d, f x := by simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_sub f (inv_ne_zero hc) d #align interval_integral.integral_comp_div_sub intervalIntegral.integral_comp_div_sub -- Porting note (#10618): was @[simp] theorem inv_smul_integral_comp_div_sub (c d) : (c⁻¹ • ∫ x in a..b, f (x / c - d)) = ∫ x in a / c - d..b / c - d, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_div_sub] #align interval_integral.inv_smul_integral_comp_div_sub intervalIntegral.inv_smul_integral_comp_div_sub -- Porting note (#10618): was @[simp] theorem integral_comp_sub_div (hc : c ≠ 0) (d) : (∫ x in a..b, f (d - x / c)) = c • ∫ x in d - b / c..d - a / c, f x := by simpa only [div_eq_inv_mul, inv_inv] using integral_comp_sub_mul f (inv_ne_zero hc) d #align interval_integral.integral_comp_sub_div intervalIntegral.integral_comp_sub_div -- Porting note (#10618): was @[simp] theorem inv_smul_integral_comp_sub_div (c d) : (c⁻¹ • ∫ x in a..b, f (d - x / c)) = ∫ x in d - b / c..d - a / c, f x := by by_cases hc : c = 0 <;> simp [hc, integral_comp_sub_div] #align interval_integral.inv_smul_integral_comp_sub_div intervalIntegral.inv_smul_integral_comp_sub_div -- Porting note (#10618): was @[simp] theorem integral_comp_sub_right (d) : (∫ x in a..b, f (x - d)) = ∫ x in a - d..b - d, f x := by simpa only [sub_eq_add_neg] using integral_comp_add_right f (-d) #align interval_integral.integral_comp_sub_right intervalIntegral.integral_comp_sub_right -- Porting note (#10618): was @[simp] theorem integral_comp_sub_left (d) : (∫ x in a..b, f (d - x)) = ∫ x in d - b..d - a, f x := by simpa only [one_mul, one_smul, inv_one] using integral_comp_sub_mul f one_ne_zero d #align interval_integral.integral_comp_sub_left intervalIntegral.integral_comp_sub_left -- Porting note (#10618): was @[simp] theorem integral_comp_neg : (∫ x in a..b, f (-x)) = ∫ x in -b..-a, f x := by simpa only [zero_sub] using integral_comp_sub_left f 0 #align interval_integral.integral_comp_neg intervalIntegral.integral_comp_neg end Comp /-! ### Integral is an additive function of the interval In this section we prove that `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ` as well as a few other identities trivially equivalent to this one. We also prove that `∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ` provided that `support f ⊆ Ioc a b`. -/ section OrderClosedTopology variable {a b c d : ℝ} {f g : ℝ → E} {μ : Measure ℝ} /-- If two functions are equal in the relevant interval, their interval integrals are also equal. -/ theorem integral_congr {a b : ℝ} (h : EqOn f g [[a, b]]) : ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ := by rcases le_total a b with hab \| hab <;> simpa [hab, integral_of_le, integral_of_ge] using setIntegral_congr measurableSet_Ioc (h.mono Ioc_subset_Icc_self) #align interval_integral.integral_congr intervalIntegral.integral_congr theorem integral_add_adjacent_intervals_cancel (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c) : (((∫ x in a..b, f x ∂μ) + ∫ x in b..c, f x ∂μ) + ∫ x in c..a, f x ∂μ) = 0 := by have hac := hab.trans hbc simp only [intervalIntegral, sub_add_sub_comm, sub_eq_zero] iterate 4 rw [← integral_union] · suffices Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc b a ∪ Ioc c b ∪ Ioc a c by rw [this] rw [Ioc_union_Ioc_union_Ioc_cycle, union_right_comm, Ioc_union_Ioc_union_Ioc_cycle, min_left_comm, max_left_comm] all_goals simp [*, MeasurableSet.union, measurableSet_Ioc, Ioc_disjoint_Ioc_same, Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] #align interval_integral.integral_add_adjacent_intervals_cancel intervalIntegral.integral_add_adjacent_intervals_cancel theorem integral_add_adjacent_intervals (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c) : ((∫ x in a..b, f x ∂μ) + ∫ x in b..c, f x ∂μ) = ∫ x in a..c, f x ∂μ := by rw [← add_neg_eq_zero, ← integral_symm, integral_add_adjacent_intervals_cancel hab hbc] #align interval_integral.integral_add_adjacent_intervals intervalIntegral.integral_add_adjacent_intervals theorem sum_integral_adjacent_intervals_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n) (hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <\| k + 1)) : ∑ k ∈ Finset.Ico m n, ∫ x in a k..a <\| k + 1, f x ∂μ = ∫ x in a m..a n, f x ∂μ := by revert hint refine Nat.le_induction ?_ ?_ n hmn · simp · intro p hmp IH h rw [Finset.sum_Ico_succ_top hmp, IH, integral_add_adjacent_intervals] · refine IntervalIntegrable.trans_iterate_Ico hmp fun k hk => h k ?_ exact (Ico_subset_Ico le_rfl (Nat.le_succ _)) hk · apply h simp [hmp] · intro k hk exact h _ (Ico_subset_Ico_right p.le_succ hk) #align interval_integral.sum_integral_adjacent_intervals_Ico intervalIntegral.sum_integral_adjacent_intervals_Ico theorem sum_integral_adjacent_intervals {a : ℕ → ℝ} {n : ℕ} (hint : ∀ k < n, IntervalIntegrable f μ (a k) (a <\| k + 1)) : ∑ k ∈ Finset.range n, ∫ x in a k..a <\| k + 1, f x ∂μ = ∫ x in (a 0)..(a n), f x ∂μ := by rw [← Nat.Ico_zero_eq_range] exact sum_integral_adjacent_intervals_Ico (zero_le n) fun k hk => hint k hk.2 #align interval_integral.sum_integral_adjacent_intervals intervalIntegral.sum_integral_adjacent_intervals theorem integral_interval_sub_left (hab : IntervalIntegrable f μ a b) (hac : IntervalIntegrable f μ a c) : ((∫ x in a..b, f x ∂μ) - ∫ x in a..c, f x ∂μ) = ∫ x in c..b, f x ∂μ := sub_eq_of_eq_add' <\| Eq.symm <\| integral_add_adjacent_intervals hac (hac.symm.trans hab) #align interval_integral.integral_interval_sub_left intervalIntegral.integral_interval_sub_left theorem integral_interval_add_interval_comm (hab : IntervalIntegrable f μ a b) (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) : ((∫ x in a..b, f x ∂μ) + ∫ x in c..d, f x ∂μ) = (∫ x in a..d, f x ∂μ) + ∫ x in c..b, f x ∂μ := by rw [← integral_add_adjacent_intervals hac hcd, add_assoc, add_left_comm, integral_add_adjacent_intervals hac (hac.symm.trans hab), add_comm] #align interval_integral.integral_interval_add_interval_comm intervalIntegral.integral_interval_add_interval_comm	Mathlib/MeasureTheory/Integral/IntervalIntegral.lean	974	979	theorem integral_interval_sub_interval_comm (hab : IntervalIntegrable f μ a b) (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) : ((∫ x in a..b, f x ∂μ) - ∫ x in c..d, f x ∂μ) = (∫ x in a..c, f x ∂μ) - ∫ x in b..d, f x ∂μ := by	simp only [sub_eq_add_neg, ← integral_symm, integral_interval_add_interval_comm hab hcd.symm (hac.trans hcd)]
/- Copyright (c) 2015 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro -/ import Mathlib.Algebra.BigOperators.Ring.List import Mathlib.Data.Nat.Prime import Mathlib.Data.List.Prime import Mathlib.Data.List.Sort import Mathlib.Data.List.Chain #align_import data.nat.factors from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab" /-! # Prime numbers This file deals with the factors of natural numbers. ## Important declarations - `Nat.factors n`: the prime factorization of `n` - `Nat.factors_unique`: uniqueness of the prime factorisation -/ open Bool Subtype open Nat namespace Nat attribute [instance 0] instBEqNat /-- `factors n` is the prime factorization of `n`, listed in increasing order. -/ def factors : ℕ → List ℕ \| 0 => [] \| 1 => [] \| k + 2 => let m := minFac (k + 2) m :: factors ((k + 2) / m) decreasing_by show (k + 2) / m < (k + 2); exact factors_lemma #align nat.factors Nat.factors @[simp] theorem factors_zero : factors 0 = [] := by rw [factors] #align nat.factors_zero Nat.factors_zero @[simp] theorem factors_one : factors 1 = [] := by rw [factors] #align nat.factors_one Nat.factors_one @[simp] theorem factors_two : factors 2 = [2] := by simp [factors] theorem prime_of_mem_factors {n : ℕ} : ∀ {p : ℕ}, (h : p ∈ factors n) → Prime p := by match n with \| 0 => simp \| 1 => simp \| k + 2 => intro p h let m := minFac (k + 2) have : (k + 2) / m < (k + 2) := factors_lemma have h₁ : p = m ∨ p ∈ factors ((k + 2) / m) := List.mem_cons.1 (by rwa [factors] at h) exact Or.casesOn h₁ (fun h₂ => h₂.symm ▸ minFac_prime (by simp)) prime_of_mem_factors #align nat.prime_of_mem_factors Nat.prime_of_mem_factors theorem pos_of_mem_factors {n p : ℕ} (h : p ∈ factors n) : 0 < p := Prime.pos (prime_of_mem_factors h) #align nat.pos_of_mem_factors Nat.pos_of_mem_factors theorem prod_factors : ∀ {n}, n ≠ 0 → List.prod (factors n) = n \| 0 => by simp \| 1 => by simp \| k + 2 => fun _ => let m := minFac (k + 2) have : (k + 2) / m < (k + 2) := factors_lemma show (factors (k + 2)).prod = (k + 2) by have h₁ : (k + 2) / m ≠ 0 := fun h => by have : (k + 2) = 0 * m := (Nat.div_eq_iff_eq_mul_left (minFac_pos _) (minFac_dvd _)).1 h rw [zero_mul] at this; exact (show k + 2 ≠ 0 by simp) this rw [factors, List.prod_cons, prod_factors h₁, Nat.mul_div_cancel' (minFac_dvd _)] #align nat.prod_factors Nat.prod_factors theorem factors_prime {p : ℕ} (hp : Nat.Prime p) : p.factors = [p] := by have : p = p - 2 + 2 := (tsub_eq_iff_eq_add_of_le hp.two_le).mp rfl rw [this, Nat.factors] simp only [Eq.symm this] have : Nat.minFac p = p := (Nat.prime_def_minFac.mp hp).2 simp only [this, Nat.factors, Nat.div_self (Nat.Prime.pos hp)] #align nat.factors_prime Nat.factors_prime theorem factors_chain {n : ℕ} : ∀ {a}, (∀ p, Prime p → p ∣ n → a ≤ p) → List.Chain (· ≤ ·) a (factors n) := by match n with \| 0 => simp \| 1 => simp \| k + 2 => intro a h let m := minFac (k + 2) have : (k + 2) / m < (k + 2) := factors_lemma rw [factors] refine List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (factors_chain ?_) exact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <\| div_dvd_of_dvd <\| minFac_dvd _) #align nat.factors_chain Nat.factors_chain theorem factors_chain_2 (n) : List.Chain (· ≤ ·) 2 (factors n) := factors_chain fun _ pp _ => pp.two_le #align nat.factors_chain_2 Nat.factors_chain_2 theorem factors_chain' (n) : List.Chain' (· ≤ ·) (factors n) := @List.Chain'.tail _ _ (_ :: _) (factors_chain_2 _) #align nat.factors_chain' Nat.factors_chain' theorem factors_sorted (n : ℕ) : List.Sorted (· ≤ ·) (factors n) := List.chain'_iff_pairwise.1 (factors_chain' _) #align nat.factors_sorted Nat.factors_sorted /-- `factors` can be constructed inductively by extracting `minFac`, for sufficiently large `n`. -/ theorem factors_add_two (n : ℕ) : factors (n + 2) = minFac (n + 2) :: factors ((n + 2) / minFac (n + 2)) := by rw [factors] #align nat.factors_add_two Nat.factors_add_two @[simp] theorem factors_eq_nil (n : ℕ) : n.factors = [] ↔ n = 0 ∨ n = 1 := by constructor <;> intro h · rcases n with (_ \| _ \| n) · exact Or.inl rfl · exact Or.inr rfl · rw [factors] at h injection h · rcases h with (rfl \| rfl) · exact factors_zero · exact factors_one #align nat.factors_eq_nil Nat.factors_eq_nil open scoped List in theorem eq_of_perm_factors {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) (h : a.factors ~ b.factors) : a = b := by simpa [prod_factors ha, prod_factors hb] using List.Perm.prod_eq h #align nat.eq_of_perm_factors Nat.eq_of_perm_factors section open List theorem mem_factors_iff_dvd {n p : ℕ} (hn : n ≠ 0) (hp : Prime p) : p ∈ factors n ↔ p ∣ n := ⟨fun h => prod_factors hn ▸ List.dvd_prod h, fun h => mem_list_primes_of_dvd_prod (prime_iff.mp hp) (fun _ h => prime_iff.mp (prime_of_mem_factors h)) ((prod_factors hn).symm ▸ h)⟩ #align nat.mem_factors_iff_dvd Nat.mem_factors_iff_dvd theorem dvd_of_mem_factors {n p : ℕ} (h : p ∈ n.factors) : p ∣ n := by rcases n.eq_zero_or_pos with (rfl \| hn) · exact dvd_zero p · rwa [← mem_factors_iff_dvd hn.ne' (prime_of_mem_factors h)] #align nat.dvd_of_mem_factors Nat.dvd_of_mem_factors theorem mem_factors {n p} (hn : n ≠ 0) : p ∈ factors n ↔ Prime p ∧ p ∣ n := ⟨fun h => ⟨prime_of_mem_factors h, dvd_of_mem_factors h⟩, fun ⟨hprime, hdvd⟩ => (mem_factors_iff_dvd hn hprime).mpr hdvd⟩ #align nat.mem_factors Nat.mem_factors @[simp] lemma mem_factors' {n p} : p ∈ n.factors ↔ p.Prime ∧ p ∣ n ∧ n ≠ 0 := by cases n <;> simp [mem_factors, ] theorem le_of_mem_factors {n p : ℕ} (h : p ∈ n.factors) : p ≤ n := by rcases n.eq_zero_or_pos with (rfl \| hn) · rw [factors_zero] at h cases h · exact le_of_dvd hn (dvd_of_mem_factors h) #align nat.le_of_mem_factors Nat.le_of_mem_factors /-- Fundamental theorem of arithmetic*-/	Mathlib/Data/Nat/Factors.lean	174	185	theorem factors_unique {n : ℕ} {l : List ℕ} (h₁ : prod l = n) (h₂ : ∀ p ∈ l, Prime p) : l ~ factors n := by	refine perm_of_prod_eq_prod ?_ ?_ ?_ · rw [h₁] refine (prod_factors ?_).symm rintro rfl rw [prod_eq_zero_iff] at h₁ exact Prime.ne_zero (h₂ 0 h₁) rfl · simp_rw [← prime_iff] exact h₂ · simp_rw [← prime_iff] exact fun p => prime_of_mem_factors
/- Copyright (c) 2022 Jiale Miao. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp -/ import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.LinearAlgebra.Matrix.Block #align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0" /-! # Gram-Schmidt Orthogonalization and Orthonormalization In this file we introduce Gram-Schmidt Orthogonalization and Orthonormalization. The Gram-Schmidt process takes a set of vectors as input and outputs a set of orthogonal vectors which have the same span. ## Main results - `gramSchmidt` : the Gram-Schmidt process - `gramSchmidt_orthogonal` : `gramSchmidt` produces an orthogonal system of vectors. - `span_gramSchmidt` : `gramSchmidt` preserves span of vectors. - `gramSchmidt_ne_zero` : If the input vectors of `gramSchmidt` are linearly independent, then the output vectors are non-zero. - `gramSchmidt_basis` : The basis produced by the Gram-Schmidt process when given a basis as input. - `gramSchmidtNormed` : the normalized `gramSchmidt` (i.e each vector in `gramSchmidtNormed` has unit length.) - `gramSchmidt_orthonormal` : `gramSchmidtNormed` produces an orthornormal system of vectors. - `gramSchmidtOrthonormalBasis`: orthonormal basis constructed by the Gram-Schmidt process from an indexed set of vectors of the right size -/ open Finset Submodule FiniteDimensional variable (𝕜 : Type) {E : Type} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] variable {ι : Type} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)] attribute [local instance] IsWellOrder.toHasWellFounded local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y /-- The Gram-Schmidt process takes a set of vectors as input and outputs a set of orthogonal vectors which have the same span. -/ noncomputable def gramSchmidt [IsWellOrder ι (· < ·)] (f : ι → E) (n : ι) : E := f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt f i) (f n) termination_by n decreasing_by exact mem_Iio.1 i.2 #align gram_schmidt gramSchmidt /-- This lemma uses `∑ i in` instead of `∑ i :`. -/ theorem gramSchmidt_def (f : ι → E) (n : ι) : gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by rw [← sum_attach, attach_eq_univ, gramSchmidt] #align gram_schmidt_def gramSchmidt_def theorem gramSchmidt_def' (f : ι → E) (n : ι) : f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by rw [gramSchmidt_def, sub_add_cancel] #align gram_schmidt_def' gramSchmidt_def' theorem gramSchmidt_def'' (f : ι → E) (n : ι) : f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n, (⟪gramSchmidt 𝕜 f i, f n⟫ / (‖gramSchmidt 𝕜 f i‖ : 𝕜) ^ 2) • gramSchmidt 𝕜 f i := by convert gramSchmidt_def' 𝕜 f n rw [orthogonalProjection_singleton, RCLike.ofReal_pow] #align gram_schmidt_def'' gramSchmidt_def'' @[simp] theorem gramSchmidt_zero {ι : Type} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι] [IsWellOrder ι (· < ·)] (f : ι → E) : gramSchmidt 𝕜 f ⊥ = f ⊥ := by rw [gramSchmidt_def, Iio_eq_Ico, Finset.Ico_self, Finset.sum_empty, sub_zero] #align gram_schmidt_zero gramSchmidt_zero /-- Gram-Schmidt Orthogonalisation: `gramSchmidt` produces an orthogonal system of vectors. -/	Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean	83	108	theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) : ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 := by	suffices ∀ a b : ι, a < b → ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 by cases' h₀.lt_or_lt with ha hb · exact this _ _ ha · rw [inner_eq_zero_symm] exact this _ _ hb clear h₀ a b intro a b h₀ revert a apply wellFounded_lt.induction b intro b ih a h₀ simp only [gramSchmidt_def 𝕜 f b, inner_sub_right, inner_sum, orthogonalProjection_singleton, inner_smul_right] rw [Finset.sum_eq_single_of_mem a (Finset.mem_Iio.mpr h₀)] · by_cases h : gramSchmidt 𝕜 f a = 0 · simp only [h, inner_zero_left, zero_div, zero_mul, sub_zero] · rw [RCLike.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel₀, sub_self] rwa [inner_self_ne_zero] intro i hi hia simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero] right cases' hia.lt_or_lt with hia₁ hia₂ · rw [inner_eq_zero_symm] exact ih a h₀ i hia₁ · exact ih i (mem_Iio.1 hi) a hia₂
/- Copyright (c) 2020 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.Data.Finset.Sort import Mathlib.Data.Set.Subsingleton #align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7" /-! # Compositions A composition of a natural number `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum of positive integers. Combinatorially, it corresponds to a decomposition of `{0, ..., n-1}` into non-empty blocks of consecutive integers, where the `iⱼ` are the lengths of the blocks. This notion is closely related to that of a partition of `n`, but in a composition of `n` the order of the `iⱼ`s matters. We implement two different structures covering these two viewpoints on compositions. The first one, made of a list of positive integers summing to `n`, is the main one and is called `Composition n`. The second one is useful for combinatorial arguments (for instance to show that the number of compositions of `n` is `2^(n-1)`). It is given by a subset of `{0, ..., n}` containing `0` and `n`, where the elements of the subset (other than `n`) correspond to the leftmost points of each block. The main API is built on `Composition n`, and we provide an equivalence between the two types. ## Main functions * `c : Composition n` is a structure, made of a list of integers which are all positive and add up to `n`. * `composition_card` states that the cardinality of `Composition n` is exactly `2^(n-1)`, which is proved by constructing an equiv with `CompositionAsSet n` (see below), which is itself in bijection with the subsets of `Fin (n-1)` (this holds even for `n = 0`, where `-` is nat subtraction). Let `c : Composition n` be a composition of `n`. Then * `c.blocks` is the list of blocks in `c`. * `c.length` is the number of blocks in the composition. * `c.blocks_fun : Fin c.length → ℕ` is the realization of `c.blocks` as a function on `Fin c.length`. This is the main object when using compositions to understand the composition of analytic functions. * `c.sizeUpTo : ℕ → ℕ` is the sum of the size of the blocks up to `i`.; * `c.embedding i : Fin (c.blocks_fun i) → Fin n` is the increasing embedding of the `i`-th block in `Fin n`; * `c.index j`, for `j : Fin n`, is the index of the block containing `j`. * `Composition.ones n` is the composition of `n` made of ones, i.e., `[1, ..., 1]`. * `Composition.single n (hn : 0 < n)` is the composition of `n` made of a single block of size `n`. Compositions can also be used to split lists. Let `l` be a list of length `n` and `c` a composition of `n`. * `l.splitWrtComposition c` is a list of lists, made of the slices of `l` corresponding to the blocks of `c`. * `join_splitWrtComposition` states that splitting a list and then joining it gives back the original list. * `joinSplitWrtComposition_join` states that joining a list of lists, and then splitting it back according to the right composition, gives back the original list of lists. We turn to the second viewpoint on compositions, that we realize as a finset of `Fin (n+1)`. `c : CompositionAsSet n` is a structure made of a finset of `Fin (n+1)` called `c.boundaries` and proofs that it contains `0` and `n`. (Taking a finset of `Fin n` containing `0` would not make sense in the edge case `n = 0`, while the previous description works in all cases). The elements of this set (other than `n`) correspond to leftmost points of blocks. Thus, there is an equiv between `Composition n` and `CompositionAsSet n`. We only construct basic API on `CompositionAsSet` (notably `c.length` and `c.blocks`) to be able to construct this equiv, called `compositionEquiv n`. Since there is a straightforward equiv between `CompositionAsSet n` and finsets of `{1, ..., n-1}` (obtained by removing `0` and `n` from a `CompositionAsSet` and called `compositionAsSetEquiv n`), we deduce that `CompositionAsSet n` and `Composition n` are both fintypes of cardinality `2^(n - 1)` (see `compositionAsSet_card` and `composition_card`). ## Implementation details The main motivation for this structure and its API is in the construction of the composition of formal multilinear series, and the proof that the composition of analytic functions is analytic. The representation of a composition as a list is very handy as lists are very flexible and already have a well-developed API. ## Tags Composition, partition ## References <https://en.wikipedia.org/wiki/Composition_(combinatorics)> -/ open List variable {n : ℕ} /-- A composition of `n` is a list of positive integers summing to `n`. -/ @[ext] structure Composition (n : ℕ) where /-- List of positive integers summing to `n`-/ blocks : List ℕ /-- Proof of positivity for `blocks`-/ blocks_pos : ∀ {i}, i ∈ blocks → 0 < i /-- Proof that `blocks` sums to `n`-/ blocks_sum : blocks.sum = n #align composition Composition /-- Combinatorial viewpoint on a composition of `n`, by seeing it as non-empty blocks of consecutive integers in `{0, ..., n-1}`. We register every block by its left end-point, yielding a finset containing `0`. As this does not make sense for `n = 0`, we add `n` to this finset, and get a finset of `{0, ..., n}` containing `0` and `n`. This is the data in the structure `CompositionAsSet n`. -/ @[ext] structure CompositionAsSet (n : ℕ) where /-- Combinatorial viewpoint on a composition of `n` as consecutive integers `{0, ..., n-1}`-/ boundaries : Finset (Fin n.succ) /-- Proof that `0` is a member of `boundaries`-/ zero_mem : (0 : Fin n.succ) ∈ boundaries /-- Last element of the composition-/ getLast_mem : Fin.last n ∈ boundaries #align composition_as_set CompositionAsSet instance {n : ℕ} : Inhabited (CompositionAsSet n) := ⟨⟨Finset.univ, Finset.mem_univ _, Finset.mem_univ _⟩⟩ /-! ### Compositions A composition of an integer `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum of positive integers. -/ namespace Composition variable (c : Composition n) instance (n : ℕ) : ToString (Composition n) := ⟨fun c => toString c.blocks⟩ /-- The length of a composition, i.e., the number of blocks in the composition. -/ abbrev length : ℕ := c.blocks.length #align composition.length Composition.length theorem blocks_length : c.blocks.length = c.length := rfl #align composition.blocks_length Composition.blocks_length /-- The blocks of a composition, seen as a function on `Fin c.length`. When composing analytic functions using compositions, this is the main player. -/ def blocksFun : Fin c.length → ℕ := c.blocks.get #align composition.blocks_fun Composition.blocksFun theorem ofFn_blocksFun : ofFn c.blocksFun = c.blocks := ofFn_get _ #align composition.of_fn_blocks_fun Composition.ofFn_blocksFun theorem sum_blocksFun : ∑ i, c.blocksFun i = n := by conv_rhs => rw [← c.blocks_sum, ← ofFn_blocksFun, sum_ofFn] #align composition.sum_blocks_fun Composition.sum_blocksFun theorem blocksFun_mem_blocks (i : Fin c.length) : c.blocksFun i ∈ c.blocks := get_mem _ _ _ #align composition.blocks_fun_mem_blocks Composition.blocksFun_mem_blocks @[simp] theorem one_le_blocks {i : ℕ} (h : i ∈ c.blocks) : 1 ≤ i := c.blocks_pos h #align composition.one_le_blocks Composition.one_le_blocks @[simp] theorem one_le_blocks' {i : ℕ} (h : i < c.length) : 1 ≤ c.blocks.get ⟨i, h⟩ := c.one_le_blocks (get_mem (blocks c) i h) #align composition.one_le_blocks' Composition.one_le_blocks' @[simp] theorem blocks_pos' (i : ℕ) (h : i < c.length) : 0 < c.blocks.get ⟨i, h⟩ := c.one_le_blocks' h #align composition.blocks_pos' Composition.blocks_pos' theorem one_le_blocksFun (i : Fin c.length) : 1 ≤ c.blocksFun i := c.one_le_blocks (c.blocksFun_mem_blocks i) #align composition.one_le_blocks_fun Composition.one_le_blocksFun theorem length_le : c.length ≤ n := by conv_rhs => rw [← c.blocks_sum] exact length_le_sum_of_one_le _ fun i hi => c.one_le_blocks hi #align composition.length_le Composition.length_le theorem length_pos_of_pos (h : 0 < n) : 0 < c.length := by apply length_pos_of_sum_pos convert h exact c.blocks_sum #align composition.length_pos_of_pos Composition.length_pos_of_pos /-- The sum of the sizes of the blocks in a composition up to `i`. -/ def sizeUpTo (i : ℕ) : ℕ := (c.blocks.take i).sum #align composition.size_up_to Composition.sizeUpTo @[simp] theorem sizeUpTo_zero : c.sizeUpTo 0 = 0 := by simp [sizeUpTo] #align composition.size_up_to_zero Composition.sizeUpTo_zero theorem sizeUpTo_ofLength_le (i : ℕ) (h : c.length ≤ i) : c.sizeUpTo i = n := by dsimp [sizeUpTo] convert c.blocks_sum exact take_all_of_le h #align composition.size_up_to_of_length_le Composition.sizeUpTo_ofLength_le @[simp] theorem sizeUpTo_length : c.sizeUpTo c.length = n := c.sizeUpTo_ofLength_le c.length le_rfl #align composition.size_up_to_length Composition.sizeUpTo_length theorem sizeUpTo_le (i : ℕ) : c.sizeUpTo i ≤ n := by conv_rhs => rw [← c.blocks_sum, ← sum_take_add_sum_drop _ i] exact Nat.le_add_right _ _ #align composition.size_up_to_le Composition.sizeUpTo_le theorem sizeUpTo_succ {i : ℕ} (h : i < c.length) : c.sizeUpTo (i + 1) = c.sizeUpTo i + c.blocks.get ⟨i, h⟩ := by simp only [sizeUpTo] rw [sum_take_succ _ _ h] #align composition.size_up_to_succ Composition.sizeUpTo_succ theorem sizeUpTo_succ' (i : Fin c.length) : c.sizeUpTo ((i : ℕ) + 1) = c.sizeUpTo i + c.blocksFun i := c.sizeUpTo_succ i.2 #align composition.size_up_to_succ' Composition.sizeUpTo_succ' theorem sizeUpTo_strict_mono {i : ℕ} (h : i < c.length) : c.sizeUpTo i < c.sizeUpTo (i + 1) := by rw [c.sizeUpTo_succ h] simp #align composition.size_up_to_strict_mono Composition.sizeUpTo_strict_mono theorem monotone_sizeUpTo : Monotone c.sizeUpTo := monotone_sum_take _ #align composition.monotone_size_up_to Composition.monotone_sizeUpTo /-- The `i`-th boundary of a composition, i.e., the leftmost point of the `i`-th block. We include a virtual point at the right of the last block, to make for a nice equiv with `CompositionAsSet n`. -/ def boundary : Fin (c.length + 1) ↪o Fin (n + 1) := (OrderEmbedding.ofStrictMono fun i => ⟨c.sizeUpTo i, Nat.lt_succ_of_le (c.sizeUpTo_le i)⟩) <\| Fin.strictMono_iff_lt_succ.2 fun ⟨_, hi⟩ => c.sizeUpTo_strict_mono hi #align composition.boundary Composition.boundary @[simp] theorem boundary_zero : c.boundary 0 = 0 := by simp [boundary, Fin.ext_iff] #align composition.boundary_zero Composition.boundary_zero @[simp] theorem boundary_last : c.boundary (Fin.last c.length) = Fin.last n := by simp [boundary, Fin.ext_iff] #align composition.boundary_last Composition.boundary_last /-- The boundaries of a composition, i.e., the leftmost point of all the blocks. We include a virtual point at the right of the last block, to make for a nice equiv with `CompositionAsSet n`. -/ def boundaries : Finset (Fin (n + 1)) := Finset.univ.map c.boundary.toEmbedding #align composition.boundaries Composition.boundaries theorem card_boundaries_eq_succ_length : c.boundaries.card = c.length + 1 := by simp [boundaries] #align composition.card_boundaries_eq_succ_length Composition.card_boundaries_eq_succ_length /-- To `c : Composition n`, one can associate a `CompositionAsSet n` by registering the leftmost point of each block, and adding a virtual point at the right of the last block. -/ def toCompositionAsSet : CompositionAsSet n where boundaries := c.boundaries zero_mem := by simp only [boundaries, Finset.mem_univ, exists_prop_of_true, Finset.mem_map] exact ⟨0, And.intro True.intro rfl⟩ getLast_mem := by simp only [boundaries, Finset.mem_univ, exists_prop_of_true, Finset.mem_map] exact ⟨Fin.last c.length, And.intro True.intro c.boundary_last⟩ #align composition.to_composition_as_set Composition.toCompositionAsSet /-- The canonical increasing bijection between `Fin (c.length + 1)` and `c.boundaries` is exactly `c.boundary`. -/ theorem orderEmbOfFin_boundaries : c.boundaries.orderEmbOfFin c.card_boundaries_eq_succ_length = c.boundary := by refine (Finset.orderEmbOfFin_unique' _ ?_).symm exact fun i => (Finset.mem_map' _).2 (Finset.mem_univ _) #align composition.order_emb_of_fin_boundaries Composition.orderEmbOfFin_boundaries /-- Embedding the `i`-th block of a composition (identified with `Fin (c.blocks_fun i)`) into `Fin n` at the relevant position. -/ def embedding (i : Fin c.length) : Fin (c.blocksFun i) ↪o Fin n := (Fin.natAddOrderEmb <\| c.sizeUpTo i).trans <\| Fin.castLEOrderEmb <\| calc c.sizeUpTo i + c.blocksFun i = c.sizeUpTo (i + 1) := (c.sizeUpTo_succ _).symm _ ≤ c.sizeUpTo c.length := monotone_sum_take _ i.2 _ = n := c.sizeUpTo_length #align composition.embedding Composition.embedding @[simp] theorem coe_embedding (i : Fin c.length) (j : Fin (c.blocksFun i)) : (c.embedding i j : ℕ) = c.sizeUpTo i + j := rfl #align composition.coe_embedding Composition.coe_embedding /-- `index_exists` asserts there is some `i` with `j < c.size_up_to (i+1)`. In the next definition `index` we use `Nat.find` to produce the minimal such index. -/ theorem index_exists {j : ℕ} (h : j < n) : ∃ i : ℕ, j < c.sizeUpTo (i + 1) ∧ i < c.length := by have n_pos : 0 < n := lt_of_le_of_lt (zero_le j) h have : 0 < c.blocks.sum := by rwa [← c.blocks_sum] at n_pos have length_pos : 0 < c.blocks.length := length_pos_of_sum_pos (blocks c) this refine ⟨c.length - 1, ?_, Nat.pred_lt (ne_of_gt length_pos)⟩ have : c.length - 1 + 1 = c.length := Nat.succ_pred_eq_of_pos length_pos simp [this, h] #align composition.index_exists Composition.index_exists /-- `c.index j` is the index of the block in the composition `c` containing `j`. -/ def index (j : Fin n) : Fin c.length := ⟨Nat.find (c.index_exists j.2), (Nat.find_spec (c.index_exists j.2)).2⟩ #align composition.index Composition.index theorem lt_sizeUpTo_index_succ (j : Fin n) : (j : ℕ) < c.sizeUpTo (c.index j).succ := (Nat.find_spec (c.index_exists j.2)).1 #align composition.lt_size_up_to_index_succ Composition.lt_sizeUpTo_index_succ theorem sizeUpTo_index_le (j : Fin n) : c.sizeUpTo (c.index j) ≤ j := by by_contra H set i := c.index j push_neg at H have i_pos : (0 : ℕ) < i := by by_contra! i_pos revert H simp [nonpos_iff_eq_zero.1 i_pos, c.sizeUpTo_zero] let i₁ := (i : ℕ).pred have i₁_lt_i : i₁ < i := Nat.pred_lt (ne_of_gt i_pos) have i₁_succ : i₁ + 1 = i := Nat.succ_pred_eq_of_pos i_pos have := Nat.find_min (c.index_exists j.2) i₁_lt_i simp [lt_trans i₁_lt_i (c.index j).2, i₁_succ] at this exact Nat.lt_le_asymm H this #align composition.size_up_to_index_le Composition.sizeUpTo_index_le /-- Mapping an element `j` of `Fin n` to the element in the block containing it, identified with `Fin (c.blocks_fun (c.index j))` through the canonical increasing bijection. -/ def invEmbedding (j : Fin n) : Fin (c.blocksFun (c.index j)) := ⟨j - c.sizeUpTo (c.index j), by rw [tsub_lt_iff_right, add_comm, ← sizeUpTo_succ'] · exact lt_sizeUpTo_index_succ _ _ · exact sizeUpTo_index_le _ _⟩ #align composition.inv_embedding Composition.invEmbedding @[simp] theorem coe_invEmbedding (j : Fin n) : (c.invEmbedding j : ℕ) = j - c.sizeUpTo (c.index j) := rfl #align composition.coe_inv_embedding Composition.coe_invEmbedding theorem embedding_comp_inv (j : Fin n) : c.embedding (c.index j) (c.invEmbedding j) = j := by rw [Fin.ext_iff] apply add_tsub_cancel_of_le (c.sizeUpTo_index_le j) #align composition.embedding_comp_inv Composition.embedding_comp_inv theorem mem_range_embedding_iff {j : Fin n} {i : Fin c.length} : j ∈ Set.range (c.embedding i) ↔ c.sizeUpTo i ≤ j ∧ (j : ℕ) < c.sizeUpTo (i : ℕ).succ := by constructor · intro h rcases Set.mem_range.2 h with ⟨k, hk⟩ rw [Fin.ext_iff] at hk dsimp at hk rw [← hk] simp [sizeUpTo_succ', k.is_lt] · intro h apply Set.mem_range.2 refine ⟨⟨j - c.sizeUpTo i, ?_⟩, ?_⟩ · rw [tsub_lt_iff_left, ← sizeUpTo_succ'] · exact h.2 · exact h.1 · rw [Fin.ext_iff] exact add_tsub_cancel_of_le h.1 #align composition.mem_range_embedding_iff Composition.mem_range_embedding_iff /-- The embeddings of different blocks of a composition are disjoint. -/ theorem disjoint_range {i₁ i₂ : Fin c.length} (h : i₁ ≠ i₂) : Disjoint (Set.range (c.embedding i₁)) (Set.range (c.embedding i₂)) := by classical wlog h' : i₁ < i₂ · exact (this c h.symm (h.lt_or_lt.resolve_left h')).symm by_contra d obtain ⟨x, hx₁, hx₂⟩ : ∃ x : Fin n, x ∈ Set.range (c.embedding i₁) ∧ x ∈ Set.range (c.embedding i₂) := Set.not_disjoint_iff.1 d have A : (i₁ : ℕ).succ ≤ i₂ := Nat.succ_le_of_lt h' apply lt_irrefl (x : ℕ) calc (x : ℕ) < c.sizeUpTo (i₁ : ℕ).succ := (c.mem_range_embedding_iff.1 hx₁).2 _ ≤ c.sizeUpTo (i₂ : ℕ) := monotone_sum_take _ A _ ≤ x := (c.mem_range_embedding_iff.1 hx₂).1 #align composition.disjoint_range Composition.disjoint_range theorem mem_range_embedding (j : Fin n) : j ∈ Set.range (c.embedding (c.index j)) := by have : c.embedding (c.index j) (c.invEmbedding j) ∈ Set.range (c.embedding (c.index j)) := Set.mem_range_self _ rwa [c.embedding_comp_inv j] at this #align composition.mem_range_embedding Composition.mem_range_embedding theorem mem_range_embedding_iff' {j : Fin n} {i : Fin c.length} : j ∈ Set.range (c.embedding i) ↔ i = c.index j := by constructor · rw [← not_imp_not] intro h exact Set.disjoint_right.1 (c.disjoint_range h) (c.mem_range_embedding j) · intro h rw [h] exact c.mem_range_embedding j #align composition.mem_range_embedding_iff' Composition.mem_range_embedding_iff' theorem index_embedding (i : Fin c.length) (j : Fin (c.blocksFun i)) : c.index (c.embedding i j) = i := by symm rw [← mem_range_embedding_iff'] apply Set.mem_range_self #align composition.index_embedding Composition.index_embedding theorem invEmbedding_comp (i : Fin c.length) (j : Fin (c.blocksFun i)) : (c.invEmbedding (c.embedding i j) : ℕ) = j := by simp_rw [coe_invEmbedding, index_embedding, coe_embedding, add_tsub_cancel_left] #align composition.inv_embedding_comp Composition.invEmbedding_comp /-- Equivalence between the disjoint union of the blocks (each of them seen as `Fin (c.blocks_fun i)`) with `Fin n`. -/ def blocksFinEquiv : (Σi : Fin c.length, Fin (c.blocksFun i)) ≃ Fin n where toFun x := c.embedding x.1 x.2 invFun j := ⟨c.index j, c.invEmbedding j⟩ left_inv x := by rcases x with ⟨i, y⟩ dsimp congr; · exact c.index_embedding _ _ rw [Fin.heq_ext_iff] · exact c.invEmbedding_comp _ _ · rw [c.index_embedding] right_inv j := c.embedding_comp_inv j #align composition.blocks_fin_equiv Composition.blocksFinEquiv theorem blocksFun_congr {n₁ n₂ : ℕ} (c₁ : Composition n₁) (c₂ : Composition n₂) (i₁ : Fin c₁.length) (i₂ : Fin c₂.length) (hn : n₁ = n₂) (hc : c₁.blocks = c₂.blocks) (hi : (i₁ : ℕ) = i₂) : c₁.blocksFun i₁ = c₂.blocksFun i₂ := by cases hn rw [← Composition.ext_iff] at hc cases hc congr rwa [Fin.ext_iff] #align composition.blocks_fun_congr Composition.blocksFun_congr /-- Two compositions (possibly of different integers) coincide if and only if they have the same sequence of blocks. -/ theorem sigma_eq_iff_blocks_eq {c : Σn, Composition n} {c' : Σn, Composition n} : c = c' ↔ c.2.blocks = c'.2.blocks := by refine ⟨fun H => by rw [H], fun H => ?_⟩ rcases c with ⟨n, c⟩ rcases c' with ⟨n', c'⟩ have : n = n' := by rw [← c.blocks_sum, ← c'.blocks_sum, H] induction this congr ext1 exact H #align composition.sigma_eq_iff_blocks_eq Composition.sigma_eq_iff_blocks_eq /-! ### The composition `Composition.ones` -/ /-- The composition made of blocks all of size `1`. -/ def ones (n : ℕ) : Composition n := ⟨replicate n (1 : ℕ), fun {i} hi => by simp [List.eq_of_mem_replicate hi], by simp⟩ #align composition.ones Composition.ones instance {n : ℕ} : Inhabited (Composition n) := ⟨Composition.ones n⟩ @[simp] theorem ones_length (n : ℕ) : (ones n).length = n := List.length_replicate n 1 #align composition.ones_length Composition.ones_length @[simp] theorem ones_blocks (n : ℕ) : (ones n).blocks = replicate n (1 : ℕ) := rfl #align composition.ones_blocks Composition.ones_blocks @[simp] theorem ones_blocksFun (n : ℕ) (i : Fin (ones n).length) : (ones n).blocksFun i = 1 := by simp only [blocksFun, ones, blocks, i.2, List.get_replicate] #align composition.ones_blocks_fun Composition.ones_blocksFun @[simp] theorem ones_sizeUpTo (n : ℕ) (i : ℕ) : (ones n).sizeUpTo i = min i n := by simp [sizeUpTo, ones_blocks, take_replicate] #align composition.ones_size_up_to Composition.ones_sizeUpTo @[simp] theorem ones_embedding (i : Fin (ones n).length) (h : 0 < (ones n).blocksFun i) : (ones n).embedding i ⟨0, h⟩ = ⟨i, lt_of_lt_of_le i.2 (ones n).length_le⟩ := by ext simpa using i.2.le #align composition.ones_embedding Composition.ones_embedding theorem eq_ones_iff {c : Composition n} : c = ones n ↔ ∀ i ∈ c.blocks, i = 1 := by constructor · rintro rfl exact fun i => eq_of_mem_replicate · intro H ext1 have A : c.blocks = replicate c.blocks.length 1 := eq_replicate_of_mem H have : c.blocks.length = n := by conv_rhs => rw [← c.blocks_sum, A] simp rw [A, this, ones_blocks] #align composition.eq_ones_iff Composition.eq_ones_iff theorem ne_ones_iff {c : Composition n} : c ≠ ones n ↔ ∃ i ∈ c.blocks, 1 < i := by refine (not_congr eq_ones_iff).trans ?_ have : ∀ j ∈ c.blocks, j = 1 ↔ j ≤ 1 := fun j hj => by simp [le_antisymm_iff, c.one_le_blocks hj] simp (config := { contextual := true }) [this] #align composition.ne_ones_iff Composition.ne_ones_iff theorem eq_ones_iff_length {c : Composition n} : c = ones n ↔ c.length = n := by constructor · rintro rfl exact ones_length n · contrapose intro H length_n apply lt_irrefl n calc n = ∑ i : Fin c.length, 1 := by simp [length_n] _ < ∑ i : Fin c.length, c.blocksFun i := by { obtain ⟨i, hi, i_blocks⟩ : ∃ i ∈ c.blocks, 1 < i := ne_ones_iff.1 H rw [← ofFn_blocksFun, mem_ofFn c.blocksFun, Set.mem_range] at hi obtain ⟨j : Fin c.length, hj : c.blocksFun j = i⟩ := hi rw [← hj] at i_blocks exact Finset.sum_lt_sum (fun i _ => one_le_blocksFun c i) ⟨j, Finset.mem_univ _, i_blocks⟩ } _ = n := c.sum_blocksFun #align composition.eq_ones_iff_length Composition.eq_ones_iff_length theorem eq_ones_iff_le_length {c : Composition n} : c = ones n ↔ n ≤ c.length := by simp [eq_ones_iff_length, le_antisymm_iff, c.length_le] #align composition.eq_ones_iff_le_length Composition.eq_ones_iff_le_length /-! ### The composition `Composition.single` -/ /-- The composition made of a single block of size `n`. -/ def single (n : ℕ) (h : 0 < n) : Composition n := ⟨[n], by simp [h], by simp⟩ #align composition.single Composition.single @[simp] theorem single_length {n : ℕ} (h : 0 < n) : (single n h).length = 1 := rfl #align composition.single_length Composition.single_length @[simp] theorem single_blocks {n : ℕ} (h : 0 < n) : (single n h).blocks = [n] := rfl #align composition.single_blocks Composition.single_blocks @[simp] theorem single_blocksFun {n : ℕ} (h : 0 < n) (i : Fin (single n h).length) : (single n h).blocksFun i = n := by simp [blocksFun, single, blocks, i.2] #align composition.single_blocks_fun Composition.single_blocksFun @[simp]	Mathlib/Combinatorics/Enumerative/Composition.lean	571	574	theorem single_embedding {n : ℕ} (h : 0 < n) (i : Fin n) : ((single n h).embedding (0 : Fin 1)) i = i := by	ext simp
/- Copyright (c) 2023 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes -/ import Mathlib.GroupTheory.CoprodI import Mathlib.GroupTheory.Coprod.Basic import Mathlib.GroupTheory.QuotientGroup import Mathlib.GroupTheory.Complement /-! ## Pushouts of Monoids and Groups This file defines wide pushouts of monoids and groups and proves some properties of the amalgamated product of groups (i.e. the special case where all the maps in the diagram are injective). ## Main definitions - `Monoid.PushoutI`: the pushout of a diagram of monoids indexed by a type `ι` - `Monoid.PushoutI.base`: the map from the amalgamating monoid to the pushout - `Monoid.PushoutI.of`: the map from each Monoid in the family to the pushout - `Monoid.PushoutI.lift`: the universal property used to define homomorphisms out of the pushout. - `Monoid.PushoutI.NormalWord`: a normal form for words in the pushout - `Monoid.PushoutI.of_injective`: if all the maps in the diagram are injective in a pushout of groups then so is `of` - `Monoid.PushoutI.Reduced.eq_empty_of_mem_range`: For any word `w` in the coproduct, if `w` is reduced (i.e none its letters are in the image of the base monoid), and nonempty, then `w` itself is not in the image of the base monoid. ## References * The normal form theorem follows these [notes](https://webspace.maths.qmul.ac.uk/i.m.chiswell/ggt/lecture_notes/lecture2.pdf) from Queen Mary University ## Tags amalgamated product, pushout, group -/ namespace Monoid open CoprodI Subgroup Coprod Function List variable {ι : Type} {G : ι → Type} {H : Type} {K : Type} [Monoid K] /-- The relation we quotient by to form the pushout -/ def PushoutI.con [∀ i, Monoid (G i)] [Monoid H] (φ : ∀ i, H →* G i) : Con (Coprod (CoprodI G) H) := conGen (fun x y : Coprod (CoprodI G) H => ∃ i x', x = inl (of (φ i x')) ∧ y = inr x') /-- The indexed pushout of monoids, which is the pushout in the category of monoids, or the category of groups. -/ def PushoutI [∀ i, Monoid (G i)] [Monoid H] (φ : ∀ i, H →* G i) : Type _ := (PushoutI.con φ).Quotient namespace PushoutI section Monoid variable [∀ i, Monoid (G i)] [Monoid H] {φ : ∀ i, H →* G i} protected instance mul : Mul (PushoutI φ) := by delta PushoutI; infer_instance protected instance one : One (PushoutI φ) := by delta PushoutI; infer_instance instance monoid : Monoid (PushoutI φ) := { Con.monoid _ with toMul := PushoutI.mul toOne := PushoutI.one } /-- The map from each indexing group into the pushout -/ def of (i : ι) : G i →* PushoutI φ := (Con.mk' _).comp <\| inl.comp CoprodI.of variable (φ) in /-- The map from the base monoid into the pushout -/ def base : H →* PushoutI φ := (Con.mk' _).comp inr theorem of_comp_eq_base (i : ι) : (of i).comp (φ i) = (base φ) := by ext x apply (Con.eq _).2 refine ConGen.Rel.of _ _ ?_ simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range] exact ⟨_, _, rfl, rfl⟩ variable (φ) in theorem of_apply_eq_base (i : ι) (x : H) : of i (φ i x) = base φ x := by rw [← MonoidHom.comp_apply, of_comp_eq_base] /-- Define a homomorphism out of the pushout of monoids be defining it on each object in the diagram -/ def lift (f : ∀ i, G i →* K) (k : H →* K) (hf : ∀ i, (f i).comp (φ i) = k) : PushoutI φ →* K := Con.lift _ (Coprod.lift (CoprodI.lift f) k) <\| by apply Con.conGen_le fun x y => ?_ rintro ⟨i, x', rfl, rfl⟩ simp only [DFunLike.ext_iff, MonoidHom.coe_comp, comp_apply] at hf simp [hf] @[simp] theorem lift_of (f : ∀ i, G i →* K) (k : H →* K) (hf : ∀ i, (f i).comp (φ i) = k) {i : ι} (g : G i) : (lift f k hf) (of i g : PushoutI φ) = f i g := by delta PushoutI lift of simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe, lift_apply_inl, CoprodI.lift_of] @[simp] theorem lift_base (f : ∀ i, G i →* K) (k : H →* K) (hf : ∀ i, (f i).comp (φ i) = k) (g : H) : (lift f k hf) (base φ g : PushoutI φ) = k g := by delta PushoutI lift base simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe, lift_apply_inr] -- `ext` attribute should be lower priority then `hom_ext_nonempty` @[ext 1199] theorem hom_ext {f g : PushoutI φ →* K} (h : ∀ i, f.comp (of i : G i →* _) = g.comp (of i : G i →* _)) (hbase : f.comp (base φ) = g.comp (base φ)) : f = g := (MonoidHom.cancel_right Con.mk'_surjective).mp <\| Coprod.hom_ext (CoprodI.ext_hom _ _ h) hbase @[ext high] theorem hom_ext_nonempty [hn : Nonempty ι] {f g : PushoutI φ →* K} (h : ∀ i, f.comp (of i : G i →* _) = g.comp (of i : G i →* _)) : f = g := hom_ext h <\| by cases hn with \| intro i => ext rw [← of_comp_eq_base i, ← MonoidHom.comp_assoc, h, MonoidHom.comp_assoc] /-- The equivalence that is part of the universal property of the pushout. A hom out of the pushout is just a morphism out of all groups in the pushout that satisfies a commutativity condition. -/ @[simps] def homEquiv : (PushoutI φ →* K) ≃ { f : (Π i, G i →* K) × (H →* K) // ∀ i, (f.1 i).comp (φ i) = f.2 } := { toFun := fun f => ⟨(fun i => f.comp (of i), f.comp (base φ)), fun i => by rw [MonoidHom.comp_assoc, of_comp_eq_base]⟩ invFun := fun f => lift f.1.1 f.1.2 f.2, left_inv := fun _ => hom_ext (by simp [DFunLike.ext_iff]) (by simp [DFunLike.ext_iff]) right_inv := fun ⟨⟨_, _⟩, _⟩ => by simp [DFunLike.ext_iff, Function.funext_iff] } /-- The map from the coproduct into the pushout -/ def ofCoprodI : CoprodI G →* PushoutI φ := CoprodI.lift of @[simp] theorem ofCoprodI_of (i : ι) (g : G i) : (ofCoprodI (CoprodI.of g) : PushoutI φ) = of i g := by simp [ofCoprodI] theorem induction_on {motive : PushoutI φ → Prop} (x : PushoutI φ) (of : ∀ (i : ι) (g : G i), motive (of i g)) (base : ∀ h, motive (base φ h)) (mul : ∀ x y, motive x → motive y → motive (x * y)) : motive x := by delta PushoutI PushoutI.of PushoutI.base at * induction x using Con.induction_on with \| H x => induction x using Coprod.induction_on with \| inl g => induction g using CoprodI.induction_on with \| h_of i g => exact of i g \| h_mul x y ihx ihy => rw [map_mul] exact mul _ _ ihx ihy \| h_one => simpa using base 1 \| inr h => exact base h \| mul x y ihx ihy => exact mul _ _ ihx ihy end Monoid variable [∀ i, Group (G i)] [Group H] {φ : ∀ i, H →* G i} instance : Group (PushoutI φ) := { Con.group (PushoutI.con φ) with toMonoid := PushoutI.monoid } namespace NormalWord /- In this section we show that there is a normal form for words in the amalgamated product. To have a normal form, we need to pick canonical choice of element of each right coset of the base group. The choice of element in the base group itself is `1`. Given a choice of element of each right coset, given by the type `Transversal φ` we can find a normal form. The normal form for an element is an element of the base group, multiplied by a word in the coproduct, where each letter in the word is the canonical choice of element of its coset. We then show that all groups in the diagram act faithfully on the normal form. This implies that the maps into the coproduct are injective. We demonstrate the action is faithful using the equivalence `equivPair`. We show that `G i` acts faithfully on `Pair d i` and that `Pair d i` is isomorphic to `NormalWord d`. Here, `d` is a `Transversal`. A `Pair d i` is a word in the coproduct, `Coprod G`, the `tail`, and an element of the group `G i`, the `head`. The first letter of the `tail` must not be an element of `G i`. Note that the `head` may be `1` Every letter in the `tail` must be in the transversal given by `d`. We then show that the equivalence between `NormalWord` and `PushoutI`, between the set of normal words and the elements of the amalgamated product. The key to this is the theorem `prod_smul_empty`, which says that going from `NormalWord` to `PushoutI` and back is the identity. This is proven by induction on the word using `consRecOn`. -/ variable (φ) /-- The data we need to pick a normal form for words in the pushout. We need to pick a canonical element of each coset. We also need all the maps in the diagram to be injective -/ structure Transversal : Type _ where /-- All maps in the diagram are injective -/ injective : ∀ i, Injective (φ i) /-- The underlying set, containing exactly one element of each coset of the base group -/ set : ∀ i, Set (G i) /-- The chosen element of the base group itself is the identity -/ one_mem : ∀ i, 1 ∈ set i /-- We have exactly one element of each coset of the base group -/ compl : ∀ i, IsComplement (φ i).range (set i) theorem transversal_nonempty (hφ : ∀ i, Injective (φ i)) : Nonempty (Transversal φ) := by choose t ht using fun i => (φ i).range.exists_right_transversal 1 apply Nonempty.intro exact { injective := hφ set := t one_mem := fun i => (ht i).2 compl := fun i => (ht i).1 } variable {φ} /-- The normal form for words in the pushout. Every element of the pushout is the product of an element of the base group and a word made up of letters each of which is in the transversal. -/ structure _root_.Monoid.PushoutI.NormalWord (d : Transversal φ) extends CoprodI.Word G where /-- Every `NormalWord` is the product of an element of the base group and a word made up of letters each of which is in the transversal. `head` is that element of the base group. -/ head : H /-- All letter in the word are in the transversal. -/ normalized : ∀ i g, ⟨i, g⟩ ∈ toList → g ∈ d.set i /-- A `Pair d i` is a word in the coproduct, `Coprod G`, the `tail`, and an element of the group `G i`, the `head`. The first letter of the `tail` must not be an element of `G i`. Note that the `head` may be `1` Every letter in the `tail` must be in the transversal given by `d`. Similar to `Monoid.CoprodI.Pair` except every letter must be in the transversal (not including the head letter). -/ structure Pair (d : Transversal φ) (i : ι) extends CoprodI.Word.Pair G i where /-- All letters in the word are in the transversal. -/ normalized : ∀ i g, ⟨i, g⟩ ∈ tail.toList → g ∈ d.set i variable {d : Transversal φ} /-- The empty normalized word, representing the identity element of the group. -/ @[simps!] def empty : NormalWord d := ⟨CoprodI.Word.empty, 1, fun i g => by simp [CoprodI.Word.empty]⟩ instance : Inhabited (NormalWord d) := ⟨NormalWord.empty⟩ instance (i : ι) : Inhabited (Pair d i) := ⟨{ (empty : NormalWord d) with head := 1, fstIdx_ne := fun h => by cases h }⟩ variable [DecidableEq ι] [∀ i, DecidableEq (G i)] @[ext] theorem ext {w₁ w₂ : NormalWord d} (hhead : w₁.head = w₂.head) (hlist : w₁.toList = w₂.toList) : w₁ = w₂ := by rcases w₁ with ⟨⟨_, _, _⟩, _, _⟩ rcases w₂ with ⟨⟨_, _, _⟩, _, _⟩ simp_all theorem ext_iff {w₁ w₂ : NormalWord d} : w₁ = w₂ ↔ w₁.head = w₂.head ∧ w₁.toList = w₂.toList := ⟨fun h => by simp [h], fun ⟨h₁, h₂⟩ => ext h₁ h₂⟩ open Subgroup.IsComplement /-- Given a word in `CoprodI`, if every letter is in the transversal and when we multiply by an element of the base group it still has this property, then the element of the base group we multiplied by was one. -/ theorem eq_one_of_smul_normalized (w : CoprodI.Word G) {i : ι} (h : H) (hw : ∀ i g, ⟨i, g⟩ ∈ w.toList → g ∈ d.set i) (hφw : ∀ j g, ⟨j, g⟩ ∈ (CoprodI.of (φ i h) • w).toList → g ∈ d.set j) : h = 1 := by simp only [← (d.compl _).equiv_snd_eq_self_iff_mem (one_mem _)] at hw hφw have hhead : ((d.compl i).equiv (Word.equivPair i w).head).2 = (Word.equivPair i w).head := by rw [Word.equivPair_head] split_ifs with h · rcases h with ⟨_, rfl⟩ exact hw _ _ (List.head_mem _) · rw [equiv_one (d.compl i) (one_mem _) (d.one_mem _)] by_contra hh1 have := hφw i (φ i h * (Word.equivPair i w).head) ?_ · apply hh1 rw [equiv_mul_left_of_mem (d.compl i) ⟨_, rfl⟩, hhead] at this simpa [((injective_iff_map_eq_one' _).1 (d.injective i))] using this · simp only [Word.mem_smul_iff, not_true, false_and, ne_eq, Option.mem_def, mul_right_inj, exists_eq_right', mul_right_eq_self, exists_prop, true_and, false_or] constructor · intro h apply_fun (d.compl i).equiv at h simp only [Prod.ext_iff, equiv_one (d.compl i) (one_mem _) (d.one_mem _), equiv_mul_left_of_mem (d.compl i) ⟨_, rfl⟩ , hhead, Subtype.ext_iff, Prod.ext_iff, Subgroup.coe_mul] at h rcases h with ⟨h₁, h₂⟩ rw [h₂, equiv_one (d.compl i) (one_mem _) (d.one_mem _), mul_one, ((injective_iff_map_eq_one' _).1 (d.injective i))] at h₁ contradiction · rw [Word.equivPair_head] dsimp split_ifs with hep · rcases hep with ⟨hnil, rfl⟩ rw [head?_eq_head _ hnil] simp_all · push_neg at hep by_cases hw : w.toList = [] · simp [hw, Word.fstIdx] · simp [head?_eq_head _ hw, Word.fstIdx, hep hw] theorem ext_smul {w₁ w₂ : NormalWord d} (i : ι) (h : CoprodI.of (φ i w₁.head) • w₁.toWord = CoprodI.of (φ i w₂.head) • w₂.toWord) : w₁ = w₂ := by rcases w₁ with ⟨w₁, h₁, hw₁⟩ rcases w₂ with ⟨w₂, h₂, hw₂⟩ dsimp at * rw [smul_eq_iff_eq_inv_smul, ← mul_smul] at h subst h simp only [← map_inv, ← map_mul] at hw₁ have : h₁⁻¹ * h₂ = 1 := eq_one_of_smul_normalized w₂ (h₁⁻¹ * h₂) hw₂ hw₁ rw [inv_mul_eq_one] at this; subst this simp /-- A constructor that multiplies a `NormalWord` by an element, with condition to make sure the underlying list does get longer. -/ @[simps!] noncomputable def cons {i} (g : G i) (w : NormalWord d) (hmw : w.fstIdx ≠ some i) (hgr : g ∉ (φ i).range) : NormalWord d := letI n := (d.compl i).equiv (g * (φ i w.head)) letI w' := Word.cons (n.2 : G i) w.toWord hmw (mt (coe_equiv_snd_eq_one_iff_mem _ (d.one_mem _)).1 (mt (mul_mem_cancel_right (by simp)).1 hgr)) { toWord := w' head := (MonoidHom.ofInjective (d.injective i)).symm n.1 normalized := fun i g hg => by simp only [w', Word.cons, mem_cons, Sigma.mk.inj_iff] at hg rcases hg with ⟨rfl, hg \| hg⟩ · simp · exact w.normalized _ _ (by assumption) } /-- Given a pair `(head, tail)`, we can form a word by prepending `head` to `tail`, but putting head into normal form first, by making sure it is expressed as an element of the base group multiplied by an element of the transversal. -/ noncomputable def rcons (i : ι) (p : Pair d i) : NormalWord d := letI n := (d.compl i).equiv p.head let w := (Word.equivPair i).symm { p.toPair with head := n.2 } { toWord := w head := (MonoidHom.ofInjective (d.injective i)).symm n.1 normalized := fun i g hg => by dsimp [w] at hg rw [Word.equivPair_symm, Word.mem_rcons_iff] at hg rcases hg with hg \| ⟨_, rfl, rfl⟩ · exact p.normalized _ _ hg · simp }	Mathlib/GroupTheory/PushoutI.lean	377	386	theorem rcons_injective {i : ι} : Function.Injective (rcons (d := d) i) := by	rintro ⟨⟨head₁, tail₁⟩, _⟩ ⟨⟨head₂, tail₂⟩, _⟩ simp only [rcons, NormalWord.mk.injEq, EmbeddingLike.apply_eq_iff_eq, Word.Pair.mk.injEq, Pair.mk.injEq, and_imp] intro h₁ h₂ h₃ subst h₂ rw [← equiv_fst_mul_equiv_snd (d.compl i) head₁, ← equiv_fst_mul_equiv_snd (d.compl i) head₂, h₁, h₃] simp
/- Copyright (c) 2021 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Analysis.Analytic.Composition #align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228" /-! # Inverse of analytic functions We construct the left and right inverse of a formal multilinear series with invertible linear term, we prove that they coincide and study their properties (notably convergence). ## Main statements * `p.leftInv i`: the formal left inverse of the formal multilinear series `p`, for `i : E ≃L[𝕜] F` which coincides with `p₁`. * `p.rightInv i`: the formal right inverse of the formal multilinear series `p`, for `i : E ≃L[𝕜] F` which coincides with `p₁`. * `p.leftInv_comp` says that `p.leftInv i` is indeed a left inverse to `p` when `p₁ = i`. * `p.rightInv_comp` says that `p.rightInv i` is indeed a right inverse to `p` when `p₁ = i`. * `p.leftInv_eq_rightInv`: the two inverses coincide. * `p.radius_rightInv_pos_of_radius_pos`: if a power series has a positive radius of convergence, then so does its inverse. -/ open scoped Classical Topology open Finset Filter namespace FormalMultilinearSeries variable {𝕜 : Type} [NontriviallyNormedField 𝕜] {E : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] /-! ### The left inverse of a formal multilinear series -/ /-- The left inverse of a formal multilinear series, where the `n`-th term is defined inductively in terms of the previous ones to make sure that `(leftInv p i) ∘ p = id`. For this, the linear term `p₁` in `p` should be invertible. In the definition, `i` is a linear isomorphism that should coincide with `p₁`, so that one can use its inverse in the construction. The definition does not use that `i = p₁`, but proofs that the definition is well-behaved do. The `n`-th term in `q ∘ p` is `∑ qₖ (p_{j₁}, ..., p_{jₖ})` over `j₁ + ... + jₖ = n`. In this expression, `qₙ` appears only once, in `qₙ (p₁, ..., p₁)`. We adjust the definition so that this term compensates the rest of the sum, using `i⁻¹` as an inverse to `p₁`. These formulas only make sense when the constant term `p₀` vanishes. The definition we give is general, but it ignores the value of `p₀`. -/ noncomputable def leftInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : FormalMultilinearSeries 𝕜 F E \| 0 => 0 \| 1 => (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm \| n + 2 => -∑ c : { c : Composition (n + 2) // c.length < n + 2 }, (leftInv p i (c : Composition (n + 2)).length).compAlongComposition (p.compContinuousLinearMap i.symm) c #align formal_multilinear_series.left_inv FormalMultilinearSeries.leftInv @[simp] theorem leftInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : p.leftInv i 0 = 0 := by rw [leftInv] #align formal_multilinear_series.left_inv_coeff_zero FormalMultilinearSeries.leftInv_coeff_zero @[simp] theorem leftInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : p.leftInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [leftInv] #align formal_multilinear_series.left_inv_coeff_one FormalMultilinearSeries.leftInv_coeff_one /-- The left inverse does not depend on the zeroth coefficient of a formal multilinear series. -/ theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : p.removeZero.leftInv i = p.leftInv i := by ext1 n induction' n using Nat.strongRec' with n IH match n with \| 0 => simp -- if one replaces `simp` with `refl`, the proof times out in the kernel. \| 1 => simp -- TODO: why? \| n + 2 => simp only [leftInv, neg_inj] refine Finset.sum_congr rfl fun c cuniv => ?_ rcases c with ⟨c, hc⟩ ext v dsimp simp [IH _ hc] #align formal_multilinear_series.left_inv_remove_zero FormalMultilinearSeries.leftInv_removeZero /-- The left inverse to a formal multilinear series is indeed a left inverse, provided its linear term is invertible. -/ theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : (leftInv p i).comp p = id 𝕜 E := by ext (n v) match n with \| 0 => simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne, not_false_iff, zero_ne_one, comp_coeff_zero'] \| 1 => simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply, ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply] \| n + 2 => have A : (Finset.univ : Finset (Composition (n + 2))) = {c \| Composition.length c < n + 2}.toFinset ∪ {Composition.ones (n + 2)} := by refine Subset.antisymm (fun c _ => ?_) (subset_univ _) by_cases h : c.length < n + 2 · simp [h, Set.mem_toFinset (s := {c \| Composition.length c < n + 2})] · simp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)] have B : Disjoint ({c \| Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset {Composition.ones (n + 2)} := by simp [Set.mem_toFinset (s := {c \| Composition.length c < n + 2})] have C : ((p.leftInv i (Composition.ones (n + 2)).length) fun j : Fin (Composition.ones n.succ.succ).length => p 1 fun _ => v ((Fin.castLE (Composition.length_le _)) j)) = p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j := by apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_ exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr have D : (p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) = -∑ c ∈ {c : Composition (n + 2) \| c.length < n + 2}.toFinset, (p.leftInv i c.length) (p.applyComposition c v) := by simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj, ContinuousMultilinearMap.sum_apply] convert (sum_toFinset_eq_subtype (fun c : Composition (n + 2) => c.length < n + 2) (fun c : Composition (n + 2) => (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap (i.symm : F →L[𝕜] E)) c (p.leftInv i c.length)) fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans _ simp only [compContinuousLinearMap_applyComposition, ContinuousMultilinearMap.compAlongComposition_apply] congr ext c congr ext k simp [h, Function.comp] simp [FormalMultilinearSeries.comp, show n + 2 ≠ 1 by omega, A, Finset.sum_union B, applyComposition_ones, C, D, -Set.toFinset_setOf] #align formal_multilinear_series.left_inv_comp FormalMultilinearSeries.leftInv_comp /-! ### The right inverse of a formal multilinear series -/ /-- The right inverse of a formal multilinear series, where the `n`-th term is defined inductively in terms of the previous ones to make sure that `p ∘ (rightInv p i) = id`. For this, the linear term `p₁` in `p` should be invertible. In the definition, `i` is a linear isomorphism that should coincide with `p₁`, so that one can use its inverse in the construction. The definition does not use that `i = p₁`, but proofs that the definition is well-behaved do. The `n`-th term in `p ∘ q` is `∑ pₖ (q_{j₁}, ..., q_{jₖ})` over `j₁ + ... + jₖ = n`. In this expression, `qₙ` appears only once, in `p₁ (qₙ)`. We adjust the definition of `qₙ` so that this term compensates the rest of the sum, using `i⁻¹` as an inverse to `p₁`. These formulas only make sense when the constant term `p₀` vanishes. The definition we give is general, but it ignores the value of `p₀`. -/ noncomputable def rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : FormalMultilinearSeries 𝕜 F E \| 0 => 0 \| 1 => (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm \| n + 2 => let q : FormalMultilinearSeries 𝕜 F E := fun k => if k < n + 2 then rightInv p i k else 0; -(i.symm : F →L[𝕜] E).compContinuousMultilinearMap ((p.comp q) (n + 2)) #align formal_multilinear_series.right_inv FormalMultilinearSeries.rightInv @[simp] theorem rightInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : p.rightInv i 0 = 0 := by rw [rightInv] #align formal_multilinear_series.right_inv_coeff_zero FormalMultilinearSeries.rightInv_coeff_zero @[simp] theorem rightInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : p.rightInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [rightInv] #align formal_multilinear_series.right_inv_coeff_one FormalMultilinearSeries.rightInv_coeff_one /-- The right inverse does not depend on the zeroth coefficient of a formal multilinear series. -/ theorem rightInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) : p.removeZero.rightInv i = p.rightInv i := by ext1 n induction' n using Nat.strongRec' with n IH match n with \| 0 => simp only [rightInv_coeff_zero] \| 1 => simp only [rightInv_coeff_one] \| n + 2 => simp only [rightInv, neg_inj] rw [removeZero_comp_of_pos _ _ (add_pos_of_nonneg_of_pos n.zero_le zero_lt_two)] congr (config := { closePost := false }) 2 with k by_cases hk : k < n + 2 <;> simp [hk, IH] #align formal_multilinear_series.right_inv_remove_zero FormalMultilinearSeries.rightInv_removeZero theorem comp_rightInv_aux1 {n : ℕ} (hn : 0 < n) (p : FormalMultilinearSeries 𝕜 E F) (q : FormalMultilinearSeries 𝕜 F E) (v : Fin n → F) : p.comp q n v = ∑ c ∈ {c : Composition n \| 1 < c.length}.toFinset, p c.length (q.applyComposition c v) + p 1 fun _ => q n v := by have A : (Finset.univ : Finset (Composition n)) = {c \| 1 < Composition.length c}.toFinset ∪ {Composition.single n hn} := by refine Subset.antisymm (fun c _ => ?_) (subset_univ _) by_cases h : 1 < c.length · simp [h, Set.mem_toFinset (s := {c \| 1 < Composition.length c})] · have : c.length = 1 := by refine (eq_iff_le_not_lt.2 ⟨?_, h⟩).symm; exact c.length_pos_of_pos hn rw [← Composition.eq_single_iff_length hn] at this simp [this] have B : Disjoint ({c \| 1 < Composition.length c} : Set (Composition n)).toFinset {Composition.single n hn} := by simp [Set.mem_toFinset (s := {c \| 1 < Composition.length c})] have C : p (Composition.single n hn).length (q.applyComposition (Composition.single n hn) v) = p 1 fun _ : Fin 1 => q n v := by apply p.congr (Composition.single_length hn) fun j hj1 _ => ?_ simp [applyComposition_single] simp [FormalMultilinearSeries.comp, A, Finset.sum_union B, C, -Set.toFinset_setOf, -add_right_inj, -Composition.single_length] #align formal_multilinear_series.comp_right_inv_aux1 FormalMultilinearSeries.comp_rightInv_aux1 theorem comp_rightInv_aux2 (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (v : Fin (n + 2) → F) : ∑ c ∈ {c : Composition (n + 2) \| 1 < c.length}.toFinset, p c.length (applyComposition (fun k : ℕ => ite (k < n + 2) (p.rightInv i k) 0) c v) = ∑ c ∈ {c : Composition (n + 2) \| 1 < c.length}.toFinset, p c.length ((p.rightInv i).applyComposition c v) := by have N : 0 < n + 2 := by norm_num refine sum_congr rfl fun c hc => p.congr rfl fun j hj1 hj2 => ?_ have : ∀ k, c.blocksFun k < n + 2 := by simp only [Set.mem_toFinset (s := {c : Composition (n + 2) \| 1 < c.length}), Set.mem_setOf_eq] at hc simp [← Composition.ne_single_iff N, Composition.eq_single_iff_length, ne_of_gt hc] simp [applyComposition, this] #align formal_multilinear_series.comp_right_inv_aux2 FormalMultilinearSeries.comp_rightInv_aux2 /-- The right inverse to a formal multilinear series is indeed a right inverse, provided its linear term is invertible and its constant term vanishes. -/ theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) : p.comp (rightInv p i) = id 𝕜 F := by ext (n v) match n with \| 0 => simp only [h0, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne, not_false_iff, zero_ne_one, comp_coeff_zero'] \| 1 => simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply, id_apply_one, ContinuousLinearEquiv.coe_apply, continuousMultilinearCurryFin1_symm_apply] \| n + 2 => have N : 0 < n + 2 := by norm_num simp [comp_rightInv_aux1 N, h, rightInv, lt_irrefl n, show n + 2 ≠ 1 by omega, ← sub_eq_add_neg, sub_eq_zero, comp_rightInv_aux2, -Set.toFinset_setOf] #align formal_multilinear_series.comp_right_inv FormalMultilinearSeries.comp_rightInv theorem rightInv_coeff (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (hn : 2 ≤ n) : p.rightInv i n = -(i.symm : F →L[𝕜] E).compContinuousMultilinearMap (∑ c ∈ ({c \| 1 < Composition.length c}.toFinset : Finset (Composition n)), p.compAlongComposition (p.rightInv i) c) := by match n with \| 0 => exact False.elim (zero_lt_two.not_le hn) \| 1 => exact False.elim (one_lt_two.not_le hn) \| n + 2 => simp only [rightInv, neg_inj] congr (config := { closePost := false }) 1 ext v have N : 0 < n + 2 := by norm_num have : ((p 1) fun i : Fin 1 => 0) = 0 := ContinuousMultilinearMap.map_zero _ simp [comp_rightInv_aux1 N, lt_irrefl n, this, comp_rightInv_aux2, -Set.toFinset_setOf] #align formal_multilinear_series.right_inv_coeff FormalMultilinearSeries.rightInv_coeff /-! ### Coincidence of the left and the right inverse -/ private theorem leftInv_eq_rightInv_aux (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) : leftInv p i = rightInv p i := calc leftInv p i = (leftInv p i).comp (id 𝕜 F) := by simp _ = (leftInv p i).comp (p.comp (rightInv p i)) := by rw [comp_rightInv p i h h0] _ = ((leftInv p i).comp p).comp (rightInv p i) := by rw [comp_assoc] _ = (id 𝕜 E).comp (rightInv p i) := by rw [leftInv_comp p i h] _ = rightInv p i := by simp /-- The left inverse and the right inverse of a formal multilinear series coincide. This is not at all obvious from their definition, but it follows from uniqueness of inverses (which comes from the fact that composition is associative on formal multilinear series). -/ theorem leftInv_eq_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : leftInv p i = rightInv p i := calc leftInv p i = leftInv p.removeZero i := by rw [leftInv_removeZero] _ = rightInv p.removeZero i := by apply leftInv_eq_rightInv_aux <;> simp; exact h _ = rightInv p i := by rw [rightInv_removeZero] #align formal_multilinear_series.left_inv_eq_right_inv FormalMultilinearSeries.leftInv_eq_rightInv /-! ### Convergence of the inverse of a power series Assume that `p` is a convergent multilinear series, and let `q` be its (left or right) inverse. Using the left-inverse formula gives $$ q_n = - (p_1)^{-n} \sum_{k=0}^{n-1} \sum_{i_1 + \dotsc + i_k = n} q_k (p_{i_1}, \dotsc, p_{i_k}). $$ Assume for simplicity that we are in dimension `1` and `p₁ = 1`. In the formula for `qₙ`, the term `q_{n-1}` appears with a multiplicity of `n-1` (choosing the index `i_j` for which `i_j = 2` while all the other indices are equal to `1`), which indicates that `qₙ` might grow like `n!`. This is bad for summability properties. It turns out that the right-inverse formula is better behaved, and should instead be used for this kind of estimate. It reads $$ q_n = - (p_1)^{-1} \sum_{k=2}^n \sum_{i_1 + \dotsc + i_k = n} p_k (q_{i_1}, \dotsc, q_{i_k}). $$ Here, `q_{n-1}` can only appear in the term with `k = 2`, and it only appears twice, so there is hope this formula can lead to an at most geometric behavior. Let `Qₙ = ‖qₙ‖`. Bounding `‖pₖ‖` with `C r^k` gives an inequality $$ Q_n ≤ C' \sum_{k=2}^n r^k \sum_{i_1 + \dotsc + i_k = n} Q_{i_1} \dotsm Q_{i_k}. $$ This formula is not enough to prove by naive induction on `n` a bound of the form `Qₙ ≤ D R^n`. However, assuming that the inequality above were an equality, one could get a formula for the generating series of the `Qₙ`: $$ \begin{align} Q(z) & := \sum Q_n z^n = Q_1 z + C' \sum_{2 \leq k \leq n} \sum_{i_1 + \dotsc + i_k = n} (r z^{i_1} Q_{i_1}) \dotsm (r z^{i_k} Q_{i_k}) \\ & = Q_1 z + C' \sum_{k = 2}^\infty (\sum_{i_1 \geq 1} r z^{i_1} Q_{i_1}) \dotsm (\sum_{i_k \geq 1} r z^{i_k} Q_{i_k}) \\ & = Q_1 z + C' \sum_{k = 2}^\infty (r Q(z))^k = Q_1 z + C' (r Q(z))^2 / (1 - r Q(z)). \end{align} $$ One can solve this formula explicitly. The solution is analytic in a neighborhood of `0` in `ℂ`, hence its coefficients grow at most geometrically (by a contour integral argument), and therefore the original `Qₙ`, which are bounded by these ones, are also at most geometric. This classical argument is not really satisfactory, as it requires an a priori bound on a complex analytic function. Another option would be to compute explicitly its terms (with binomial coefficients) to obtain an explicit geometric bound, but this would be very painful. Instead, we will use the above intuition, but in a slightly different form, with finite sums and an induction. I learnt this trick in [pöschel2017siegelsternberg]. Let $S_n = \sum_{k=1}^n Q_k a^k$ (where `a` is a positive real parameter to be chosen suitably small). The above computation but with finite sums shows that $$ S_n \leq Q_1 a + C' \sum_{k=2}^n (r S_{n-1})^k. $$ In particular, $S_n \leq Q_1 a + C' (r S_{n-1})^2 / (1- r S_{n-1})$. Assume that $S_{n-1} \leq K a$, where `K > Q₁` is fixed and `a` is small enough so that `r K a ≤ 1/2` (to control the denominator). Then this equation gives a bound $S_n \leq Q_1 a + 2 C' r^2 K^2 a^2$. If `a` is small enough, this is bounded by `K a` as the second term is quadratic in `a`, and therefore negligible. By induction, we deduce `Sₙ ≤ K a` for all `n`, which gives in particular the fact that `aⁿ Qₙ` remains bounded. -/ /-- First technical lemma to control the growth of coefficients of the inverse. Bound the explicit expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before, in a general abstract setup. -/	Mathlib/Analysis/Analytic/Inverse.lean	379	435	theorem radius_right_inv_pos_of_radius_pos_aux1 (n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) : ∑ k ∈ Ico 2 (n + 1), a ^ k * ∑ c ∈ ({c \| 1 < Composition.length c}.toFinset : Finset (Composition k)), r ^ c.length * ∏ j, p (c.blocksFun j) ≤ ∑ j ∈ Ico 2 (n + 1), r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j := calc ∑ k ∈ Ico 2 (n + 1), a ^ k * ∑ c ∈ ({c \| 1 < Composition.length c}.toFinset : Finset (Composition k)), r ^ c.length * ∏ j, p (c.blocksFun j) = ∑ k ∈ Ico 2 (n + 1), ∑ c ∈ ({c \| 1 < Composition.length c}.toFinset : Finset (Composition k)), ∏ j, r * (a ^ c.blocksFun j * p (c.blocksFun j)) := by	simp_rw [mul_sum] congr! with k _ c rw [prod_mul_distrib, prod_mul_distrib, prod_pow_eq_pow_sum, Composition.sum_blocksFun, prod_const, card_fin] ring _ ≤ ∑ d ∈ compPartialSumTarget 2 (n + 1) n, ∏ j : Fin d.2.length, r * (a ^ d.2.blocksFun j * p (d.2.blocksFun j)) := by rw [sum_sigma'] refine sum_le_sum_of_subset_of_nonneg ?_ fun x _ _ => prod_nonneg fun j _ => mul_nonneg hr (mul_nonneg (pow_nonneg ha _) (hp _)) rintro ⟨k, c⟩ hd simp only [Set.mem_toFinset (s := {c \| 1 < Composition.length c}), mem_Ico, mem_sigma, Set.mem_setOf_eq] at hd simp only [mem_compPartialSumTarget_iff] refine ⟨hd.2, c.length_le.trans_lt hd.1.2, fun j => ?_⟩ have : c ≠ Composition.single k (zero_lt_two.trans_le hd.1.1) := by simp [Composition.eq_single_iff_length, ne_of_gt hd.2] rw [Composition.ne_single_iff] at this exact (this j).trans_le (Nat.lt_succ_iff.mp hd.1.2) _ = ∑ e ∈ compPartialSumSource 2 (n + 1) n, ∏ j : Fin e.1, r * (a ^ e.2 j * p (e.2 j)) := by symm apply compChangeOfVariables_sum rintro ⟨k, blocks_fun⟩ H have K : (compChangeOfVariables 2 (n + 1) n ⟨k, blocks_fun⟩ H).snd.length = k := by simp congr 2 <;> try rw [K] rw [Fin.heq_fun_iff K.symm] intro j rw [compChangeOfVariables_blocksFun] _ = ∑ j ∈ Ico 2 (n + 1), r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j := by rw [compPartialSumSource, ← sum_sigma' (Ico 2 (n + 1)) (fun k : ℕ => (Fintype.piFinset fun _ : Fin k => Ico 1 n : Finset (Fin k → ℕ))) (fun n e => ∏ j : Fin n, r * (a ^ e j * p (e j)))] congr! with j simp only [← @MultilinearMap.mkPiAlgebra_apply ℝ (Fin j) _ ℝ] simp only [← MultilinearMap.map_sum_finset (MultilinearMap.mkPiAlgebra ℝ (Fin j) ℝ) fun _ (m : ℕ) => r * (a ^ m * p m)] simp only [MultilinearMap.mkPiAlgebra_apply] simp [prod_const, ← mul_sum, mul_pow]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser -/ import Mathlib.Algebra.Algebra.Prod import Mathlib.LinearAlgebra.Basic import Mathlib.LinearAlgebra.Span import Mathlib.Order.PartialSups #align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d" /-! ### Products of modules This file defines constructors for linear maps whose domains or codomains are products. It contains theorems relating these to each other, as well as to `Submodule.prod`, `Submodule.map`, `Submodule.comap`, `LinearMap.range`, and `LinearMap.ker`. ## Main definitions - products in the domain: - `LinearMap.fst` - `LinearMap.snd` - `LinearMap.coprod` - `LinearMap.prod_ext` - products in the codomain: - `LinearMap.inl` - `LinearMap.inr` - `LinearMap.prod` - products in both domain and codomain: - `LinearMap.prodMap` - `LinearEquiv.prodMap` - `LinearEquiv.skewProd` -/ universe u v w x y z u' v' w' y' variable {R : Type u} {K : Type u'} {M : Type v} {V : Type v'} {M₂ : Type w} {V₂ : Type w'} variable {M₃ : Type y} {V₃ : Type y'} {M₄ : Type z} {ι : Type x} variable {M₅ M₆ : Type} section Prod namespace LinearMap variable (S : Type) [Semiring R] [Semiring S] variable [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] variable [AddCommMonoid M₅] [AddCommMonoid M₆] variable [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] variable [Module R M₅] [Module R M₆] variable (f : M →ₗ[R] M₂) section variable (R M M₂) /-- The first projection of a product is a linear map. -/ def fst : M × M₂ →ₗ[R] M where toFun := Prod.fst map_add' _x _y := rfl map_smul' _x _y := rfl #align linear_map.fst LinearMap.fst /-- The second projection of a product is a linear map. -/ def snd : M × M₂ →ₗ[R] M₂ where toFun := Prod.snd map_add' _x _y := rfl map_smul' _x _y := rfl #align linear_map.snd LinearMap.snd end @[simp] theorem fst_apply (x : M × M₂) : fst R M M₂ x = x.1 := rfl #align linear_map.fst_apply LinearMap.fst_apply @[simp] theorem snd_apply (x : M × M₂) : snd R M M₂ x = x.2 := rfl #align linear_map.snd_apply LinearMap.snd_apply theorem fst_surjective : Function.Surjective (fst R M M₂) := fun x => ⟨(x, 0), rfl⟩ #align linear_map.fst_surjective LinearMap.fst_surjective theorem snd_surjective : Function.Surjective (snd R M M₂) := fun x => ⟨(0, x), rfl⟩ #align linear_map.snd_surjective LinearMap.snd_surjective /-- The prod of two linear maps is a linear map. -/ @[simps] def prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : M →ₗ[R] M₂ × M₃ where toFun := Pi.prod f g map_add' x y := by simp only [Pi.prod, Prod.mk_add_mk, map_add] map_smul' c x := by simp only [Pi.prod, Prod.smul_mk, map_smul, RingHom.id_apply] #align linear_map.prod LinearMap.prod theorem coe_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : ⇑(f.prod g) = Pi.prod f g := rfl #align linear_map.coe_prod LinearMap.coe_prod @[simp] theorem fst_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : (fst R M₂ M₃).comp (prod f g) = f := rfl #align linear_map.fst_prod LinearMap.fst_prod @[simp] theorem snd_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : (snd R M₂ M₃).comp (prod f g) = g := rfl #align linear_map.snd_prod LinearMap.snd_prod @[simp] theorem pair_fst_snd : prod (fst R M M₂) (snd R M M₂) = LinearMap.id := rfl #align linear_map.pair_fst_snd LinearMap.pair_fst_snd theorem prod_comp (f : M₂ →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) (h : M →ₗ[R] M₂) : (f.prod g).comp h = (f.comp h).prod (g.comp h) := rfl /-- Taking the product of two maps with the same domain is equivalent to taking the product of their codomains. See note [bundled maps over different rings] for why separate `R` and `S` semirings are used. -/ @[simps] def prodEquiv [Module S M₂] [Module S M₃] [SMulCommClass R S M₂] [SMulCommClass R S M₃] : ((M →ₗ[R] M₂) × (M →ₗ[R] M₃)) ≃ₗ[S] M →ₗ[R] M₂ × M₃ where toFun f := f.1.prod f.2 invFun f := ((fst _ _ _).comp f, (snd _ _ _).comp f) left_inv f := by ext <;> rfl right_inv f := by ext <;> rfl map_add' a b := rfl map_smul' r a := rfl #align linear_map.prod_equiv LinearMap.prodEquiv section variable (R M M₂) /-- The left injection into a product is a linear map. -/ def inl : M →ₗ[R] M × M₂ := prod LinearMap.id 0 #align linear_map.inl LinearMap.inl /-- The right injection into a product is a linear map. -/ def inr : M₂ →ₗ[R] M × M₂ := prod 0 LinearMap.id #align linear_map.inr LinearMap.inr theorem range_inl : range (inl R M M₂) = ker (snd R M M₂) := by ext x simp only [mem_ker, mem_range] constructor · rintro ⟨y, rfl⟩ rfl · intro h exact ⟨x.fst, Prod.ext rfl h.symm⟩ #align linear_map.range_inl LinearMap.range_inl theorem ker_snd : ker (snd R M M₂) = range (inl R M M₂) := Eq.symm <\| range_inl R M M₂ #align linear_map.ker_snd LinearMap.ker_snd theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) := by ext x simp only [mem_ker, mem_range] constructor · rintro ⟨y, rfl⟩ rfl · intro h exact ⟨x.snd, Prod.ext h.symm rfl⟩ #align linear_map.range_inr LinearMap.range_inr theorem ker_fst : ker (fst R M M₂) = range (inr R M M₂) := Eq.symm <\| range_inr R M M₂ #align linear_map.ker_fst LinearMap.ker_fst @[simp] theorem fst_comp_inl : fst R M M₂ ∘ₗ inl R M M₂ = id := rfl @[simp] theorem snd_comp_inl : snd R M M₂ ∘ₗ inl R M M₂ = 0 := rfl @[simp] theorem fst_comp_inr : fst R M M₂ ∘ₗ inr R M M₂ = 0 := rfl @[simp] theorem snd_comp_inr : snd R M M₂ ∘ₗ inr R M M₂ = id := rfl end @[simp] theorem coe_inl : (inl R M M₂ : M → M × M₂) = fun x => (x, 0) := rfl #align linear_map.coe_inl LinearMap.coe_inl theorem inl_apply (x : M) : inl R M M₂ x = (x, 0) := rfl #align linear_map.inl_apply LinearMap.inl_apply @[simp] theorem coe_inr : (inr R M M₂ : M₂ → M × M₂) = Prod.mk 0 := rfl #align linear_map.coe_inr LinearMap.coe_inr theorem inr_apply (x : M₂) : inr R M M₂ x = (0, x) := rfl #align linear_map.inr_apply LinearMap.inr_apply theorem inl_eq_prod : inl R M M₂ = prod LinearMap.id 0 := rfl #align linear_map.inl_eq_prod LinearMap.inl_eq_prod theorem inr_eq_prod : inr R M M₂ = prod 0 LinearMap.id := rfl #align linear_map.inr_eq_prod LinearMap.inr_eq_prod theorem inl_injective : Function.Injective (inl R M M₂) := fun _ => by simp #align linear_map.inl_injective LinearMap.inl_injective theorem inr_injective : Function.Injective (inr R M M₂) := fun _ => by simp #align linear_map.inr_injective LinearMap.inr_injective /-- The coprod function `x : M × M₂ ↦ f x.1 + g x.2` is a linear map. -/ def coprod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : M × M₂ →ₗ[R] M₃ := f.comp (fst _ _ _) + g.comp (snd _ _ _) #align linear_map.coprod LinearMap.coprod @[simp] theorem coprod_apply (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (x : M × M₂) : coprod f g x = f x.1 + g x.2 := rfl #align linear_map.coprod_apply LinearMap.coprod_apply @[simp] theorem coprod_inl (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : (coprod f g).comp (inl R M M₂) = f := by ext; simp only [map_zero, add_zero, coprod_apply, inl_apply, comp_apply] #align linear_map.coprod_inl LinearMap.coprod_inl @[simp] theorem coprod_inr (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : (coprod f g).comp (inr R M M₂) = g := by ext; simp only [map_zero, coprod_apply, inr_apply, zero_add, comp_apply] #align linear_map.coprod_inr LinearMap.coprod_inr @[simp] theorem coprod_inl_inr : coprod (inl R M M₂) (inr R M M₂) = LinearMap.id := by ext <;> simp only [Prod.mk_add_mk, add_zero, id_apply, coprod_apply, inl_apply, inr_apply, zero_add] #align linear_map.coprod_inl_inr LinearMap.coprod_inl_inr theorem coprod_zero_left (g : M₂ →ₗ[R] M₃) : (0 : M →ₗ[R] M₃).coprod g = g.comp (snd R M M₂) := zero_add _ theorem coprod_zero_right (f : M →ₗ[R] M₃) : f.coprod (0 : M₂ →ₗ[R] M₃) = f.comp (fst R M M₂) := add_zero _ theorem comp_coprod (f : M₃ →ₗ[R] M₄) (g₁ : M →ₗ[R] M₃) (g₂ : M₂ →ₗ[R] M₃) : f.comp (g₁.coprod g₂) = (f.comp g₁).coprod (f.comp g₂) := ext fun x => f.map_add (g₁ x.1) (g₂ x.2) #align linear_map.comp_coprod LinearMap.comp_coprod	Mathlib/LinearAlgebra/Prod.lean	256	256	theorem fst_eq_coprod : fst R M M₂ = coprod LinearMap.id 0 := by	ext; simp
/- Copyright (c) 2020 Joseph Myers. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth -/ import Mathlib.Analysis.InnerProductSpace.Projection import Mathlib.Analysis.NormedSpace.PiLp import Mathlib.LinearAlgebra.FiniteDimensional import Mathlib.LinearAlgebra.UnitaryGroup #align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395" /-! # `L²` inner product space structure on finite products of inner product spaces The `L²` norm on a finite product of inner product spaces is compatible with an inner product $$ \langle x, y\rangle = \sum \langle x_i, y_i \rangle. $$ This is recorded in this file as an inner product space instance on `PiLp 2`. This file develops the notion of a finite dimensional Hilbert space over `𝕜 = ℂ, ℝ`, referred to as `E`. We define an `OrthonormalBasis 𝕜 ι E` as a linear isometric equivalence between `E` and `EuclideanSpace 𝕜 ι`. Then `stdOrthonormalBasis` shows that such an equivalence always exists if `E` is finite dimensional. We provide language for converting between a basis that is orthonormal and an orthonormal basis (e.g. `Basis.toOrthonormalBasis`). We show that orthonormal bases for each summand in a direct sum of spaces can be combined into an orthonormal basis for the whole sum in `DirectSum.IsInternal.subordinateOrthonormalBasis`. In the last section, various properties of matrices are explored. ## Main definitions - `EuclideanSpace 𝕜 n`: defined to be `PiLp 2 (n → 𝕜)` for any `Fintype n`, i.e., the space from functions to `n` to `𝕜` with the `L²` norm. We register several instances on it (notably that it is a finite-dimensional inner product space). - `OrthonormalBasis 𝕜 ι`: defined to be an isometry to Euclidean space from a given finite-dimensional inner product space, `E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι`. - `Basis.toOrthonormalBasis`: constructs an `OrthonormalBasis` for a finite-dimensional Euclidean space from a `Basis` which is `Orthonormal`. - `Orthonormal.exists_orthonormalBasis_extension`: provides an existential result of an `OrthonormalBasis` extending a given orthonormal set - `exists_orthonormalBasis`: provides an orthonormal basis on a finite dimensional vector space - `stdOrthonormalBasis`: provides an arbitrarily-chosen `OrthonormalBasis` of a given finite dimensional inner product space For consequences in infinite dimension (Hilbert bases, etc.), see the file `Analysis.InnerProductSpace.L2Space`. -/ set_option linter.uppercaseLean3 false open Real Set Filter RCLike Submodule Function Uniformity Topology NNReal ENNReal ComplexConjugate DirectSum noncomputable section variable {ι ι' 𝕜 : Type} [RCLike 𝕜] variable {E : Type} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] variable {E' : Type} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] variable {F : Type} [NormedAddCommGroup F] [InnerProductSpace ℝ F] variable {F' : Type} [NormedAddCommGroup F'] [InnerProductSpace ℝ F'] local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y /- If `ι` is a finite type and each space `f i`, `i : ι`, is an inner product space, then `Π i, f i` is an inner product space as well. Since `Π i, f i` is endowed with the sup norm, we use instead `PiLp 2 f` for the product space, which is endowed with the `L^2` norm. -/ instance PiLp.innerProductSpace {ι : Type} [Fintype ι] (f : ι → Type) [∀ i, NormedAddCommGroup (f i)] [∀ i, InnerProductSpace 𝕜 (f i)] : InnerProductSpace 𝕜 (PiLp 2 f) where inner x y := ∑ i, inner (x i) (y i) norm_sq_eq_inner x := by simp only [PiLp.norm_sq_eq_of_L2, map_sum, ← norm_sq_eq_inner, one_div] conj_symm := by intro x y unfold inner rw [map_sum] apply Finset.sum_congr rfl rintro z - apply inner_conj_symm add_left x y z := show (∑ i, inner (x i + y i) (z i)) = (∑ i, inner (x i) (z i)) + ∑ i, inner (y i) (z i) by simp only [inner_add_left, Finset.sum_add_distrib] smul_left x y r := show (∑ i : ι, inner (r • x i) (y i)) = conj r ∑ i, inner (x i) (y i) by simp only [Finset.mul_sum, inner_smul_left] #align pi_Lp.inner_product_space PiLp.innerProductSpace @[simp] theorem PiLp.inner_apply {ι : Type} [Fintype ι] {f : ι → Type} [∀ i, NormedAddCommGroup (f i)] [∀ i, InnerProductSpace 𝕜 (f i)] (x y : PiLp 2 f) : ⟪x, y⟫ = ∑ i, ⟪x i, y i⟫ := rfl #align pi_Lp.inner_apply PiLp.inner_apply /-- The standard real/complex Euclidean space, functions on a finite type. For an `n`-dimensional space use `EuclideanSpace 𝕜 (Fin n)`. -/ abbrev EuclideanSpace (𝕜 : Type) (n : Type) : Type _ := PiLp 2 fun _ : n => 𝕜 #align euclidean_space EuclideanSpace theorem EuclideanSpace.nnnorm_eq {𝕜 : Type} [RCLike 𝕜] {n : Type} [Fintype n] (x : EuclideanSpace 𝕜 n) : ‖x‖₊ = NNReal.sqrt (∑ i, ‖x i‖₊ ^ 2) := PiLp.nnnorm_eq_of_L2 x #align euclidean_space.nnnorm_eq EuclideanSpace.nnnorm_eq theorem EuclideanSpace.norm_eq {𝕜 : Type} [RCLike 𝕜] {n : Type} [Fintype n] (x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) := by simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((↑) : ℝ≥0 → ℝ) x.nnnorm_eq #align euclidean_space.norm_eq EuclideanSpace.norm_eq theorem EuclideanSpace.dist_eq {𝕜 : Type} [RCLike 𝕜] {n : Type} [Fintype n] (x y : EuclideanSpace 𝕜 n) : dist x y = √(∑ i, dist (x i) (y i) ^ 2) := PiLp.dist_eq_of_L2 x y #align euclidean_space.dist_eq EuclideanSpace.dist_eq theorem EuclideanSpace.nndist_eq {𝕜 : Type} [RCLike 𝕜] {n : Type} [Fintype n] (x y : EuclideanSpace 𝕜 n) : nndist x y = NNReal.sqrt (∑ i, nndist (x i) (y i) ^ 2) := PiLp.nndist_eq_of_L2 x y #align euclidean_space.nndist_eq EuclideanSpace.nndist_eq theorem EuclideanSpace.edist_eq {𝕜 : Type} [RCLike 𝕜] {n : Type} [Fintype n] (x y : EuclideanSpace 𝕜 n) : edist x y = (∑ i, edist (x i) (y i) ^ 2) ^ (1 / 2 : ℝ) := PiLp.edist_eq_of_L2 x y #align euclidean_space.edist_eq EuclideanSpace.edist_eq theorem EuclideanSpace.ball_zero_eq {n : Type} [Fintype n] (r : ℝ) (hr : 0 ≤ r) : Metric.ball (0 : EuclideanSpace ℝ n) r = {x \| ∑ i, x i ^ 2 < r ^ 2} := by ext x have : (0 : ℝ) ≤ ∑ i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _ simp_rw [mem_setOf, mem_ball_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_lt this hr] theorem EuclideanSpace.closedBall_zero_eq {n : Type} [Fintype n] (r : ℝ) (hr : 0 ≤ r) : Metric.closedBall (0 : EuclideanSpace ℝ n) r = {x \| ∑ i, x i ^ 2 ≤ r ^ 2} := by ext simp_rw [mem_setOf, mem_closedBall_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_le_left hr] theorem EuclideanSpace.sphere_zero_eq {n : Type} [Fintype n] (r : ℝ) (hr : 0 ≤ r) : Metric.sphere (0 : EuclideanSpace ℝ n) r = {x \| ∑ i, x i ^ 2 = r ^ 2} := by ext x have : (0 : ℝ) ≤ ∑ i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _ simp_rw [mem_setOf, mem_sphere_zero_iff_norm, norm_eq, norm_eq_abs, sq_abs, Real.sqrt_eq_iff_sq_eq this hr, eq_comm] section #align euclidean_space.finite_dimensional WithLp.instModuleFinite variable [Fintype ι] #align euclidean_space.inner_product_space PiLp.innerProductSpace @[simp] theorem finrank_euclideanSpace : FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 ι) = Fintype.card ι := by simp [EuclideanSpace, PiLp, WithLp] #align finrank_euclidean_space finrank_euclideanSpace theorem finrank_euclideanSpace_fin {n : ℕ} : FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 (Fin n)) = n := by simp #align finrank_euclidean_space_fin finrank_euclideanSpace_fin theorem EuclideanSpace.inner_eq_star_dotProduct (x y : EuclideanSpace 𝕜 ι) : ⟪x, y⟫ = Matrix.dotProduct (star <\| WithLp.equiv _ _ x) (WithLp.equiv _ _ y) := rfl #align euclidean_space.inner_eq_star_dot_product EuclideanSpace.inner_eq_star_dotProduct theorem EuclideanSpace.inner_piLp_equiv_symm (x y : ι → 𝕜) : ⟪(WithLp.equiv 2 _).symm x, (WithLp.equiv 2 _).symm y⟫ = Matrix.dotProduct (star x) y := rfl #align euclidean_space.inner_pi_Lp_equiv_symm EuclideanSpace.inner_piLp_equiv_symm /-- A finite, mutually orthogonal family of subspaces of `E`, which span `E`, induce an isometry from `E` to `PiLp 2` of the subspaces equipped with the `L2` inner product. -/ def DirectSum.IsInternal.isometryL2OfOrthogonalFamily [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) : E ≃ₗᵢ[𝕜] PiLp 2 fun i => V i := by let e₁ := DirectSum.linearEquivFunOnFintype 𝕜 ι fun i => V i let e₂ := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV refine LinearEquiv.isometryOfInner (e₂.symm.trans e₁) ?_ suffices ∀ (v w : PiLp 2 fun i => V i), ⟪v, w⟫ = ⟪e₂ (e₁.symm v), e₂ (e₁.symm w)⟫ by intro v₀ w₀ convert this (e₁ (e₂.symm v₀)) (e₁ (e₂.symm w₀)) <;> simp only [LinearEquiv.symm_apply_apply, LinearEquiv.apply_symm_apply] intro v w trans ⟪∑ i, (V i).subtypeₗᵢ (v i), ∑ i, (V i).subtypeₗᵢ (w i)⟫ · simp only [sum_inner, hV'.inner_right_fintype, PiLp.inner_apply] · congr <;> simp #align direct_sum.is_internal.isometry_L2_of_orthogonal_family DirectSum.IsInternal.isometryL2OfOrthogonalFamily @[simp] theorem DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) (w : PiLp 2 fun i => V i) : (hV.isometryL2OfOrthogonalFamily hV').symm w = ∑ i, (w i : E) := by classical let e₁ := DirectSum.linearEquivFunOnFintype 𝕜 ι fun i => V i let e₂ := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV suffices ∀ v : ⨁ i, V i, e₂ v = ∑ i, e₁ v i by exact this (e₁.symm w) intro v -- Porting note: added `DFinsupp.lsum` simp [e₁, e₂, DirectSum.coeLinearMap, DirectSum.toModule, DFinsupp.lsum, DFinsupp.sumAddHom_apply] #align direct_sum.is_internal.isometry_L2_of_orthogonal_family_symm_apply DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply end variable (ι 𝕜) /-- A shorthand for `PiLp.continuousLinearEquiv`. -/ abbrev EuclideanSpace.equiv : EuclideanSpace 𝕜 ι ≃L[𝕜] ι → 𝕜 := PiLp.continuousLinearEquiv 2 𝕜 _ #align euclidean_space.equiv EuclideanSpace.equiv #noalign euclidean_space.equiv_to_linear_equiv_apply #noalign euclidean_space.equiv_apply #noalign euclidean_space.equiv_to_linear_equiv_symm_apply #noalign euclidean_space.equiv_symm_apply variable {ι 𝕜} -- TODO : This should be generalized to `PiLp`. /-- The projection on the `i`-th coordinate of `EuclideanSpace 𝕜 ι`, as a linear map. -/ @[simps!] def EuclideanSpace.projₗ (i : ι) : EuclideanSpace 𝕜 ι →ₗ[𝕜] 𝕜 := (LinearMap.proj i).comp (WithLp.linearEquiv 2 𝕜 (ι → 𝕜) : EuclideanSpace 𝕜 ι →ₗ[𝕜] ι → 𝕜) #align euclidean_space.projₗ EuclideanSpace.projₗ #align euclidean_space.projₗ_apply EuclideanSpace.projₗ_apply -- TODO : This should be generalized to `PiLp`. /-- The projection on the `i`-th coordinate of `EuclideanSpace 𝕜 ι`, as a continuous linear map. -/ @[simps! apply coe] def EuclideanSpace.proj (i : ι) : EuclideanSpace 𝕜 ι →L[𝕜] 𝕜 := ⟨EuclideanSpace.projₗ i, continuous_apply i⟩ #align euclidean_space.proj EuclideanSpace.proj #align euclidean_space.proj_coe EuclideanSpace.proj_coe #align euclidean_space.proj_apply EuclideanSpace.proj_apply section DecEq variable [DecidableEq ι] -- TODO : This should be generalized to `PiLp`. /-- The vector given in euclidean space by being `a : 𝕜` at coordinate `i : ι` and `0 : 𝕜` at all other coordinates. -/ def EuclideanSpace.single (i : ι) (a : 𝕜) : EuclideanSpace 𝕜 ι := (WithLp.equiv _ _).symm (Pi.single i a) #align euclidean_space.single EuclideanSpace.single @[simp] theorem WithLp.equiv_single (i : ι) (a : 𝕜) : WithLp.equiv _ _ (EuclideanSpace.single i a) = Pi.single i a := rfl #align pi_Lp.equiv_single WithLp.equiv_single @[simp] theorem WithLp.equiv_symm_single (i : ι) (a : 𝕜) : (WithLp.equiv _ _).symm (Pi.single i a) = EuclideanSpace.single i a := rfl #align pi_Lp.equiv_symm_single WithLp.equiv_symm_single @[simp] theorem EuclideanSpace.single_apply (i : ι) (a : 𝕜) (j : ι) : (EuclideanSpace.single i a) j = ite (j = i) a 0 := by rw [EuclideanSpace.single, WithLp.equiv_symm_pi_apply, ← Pi.single_apply i a j] #align euclidean_space.single_apply EuclideanSpace.single_apply variable [Fintype ι] theorem EuclideanSpace.inner_single_left (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι) : ⟪EuclideanSpace.single i (a : 𝕜), v⟫ = conj a v i := by simp [apply_ite conj] #align euclidean_space.inner_single_left EuclideanSpace.inner_single_left theorem EuclideanSpace.inner_single_right (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι) : ⟪v, EuclideanSpace.single i (a : 𝕜)⟫ = a * conj (v i) := by simp [apply_ite conj, mul_comm] #align euclidean_space.inner_single_right EuclideanSpace.inner_single_right @[simp] theorem EuclideanSpace.norm_single (i : ι) (a : 𝕜) : ‖EuclideanSpace.single i (a : 𝕜)‖ = ‖a‖ := PiLp.norm_equiv_symm_single 2 (fun _ => 𝕜) i a #align euclidean_space.norm_single EuclideanSpace.norm_single @[simp] theorem EuclideanSpace.nnnorm_single (i : ι) (a : 𝕜) : ‖EuclideanSpace.single i (a : 𝕜)‖₊ = ‖a‖₊ := PiLp.nnnorm_equiv_symm_single 2 (fun _ => 𝕜) i a #align euclidean_space.nnnorm_single EuclideanSpace.nnnorm_single @[simp] theorem EuclideanSpace.dist_single_same (i : ι) (a b : 𝕜) : dist (EuclideanSpace.single i (a : 𝕜)) (EuclideanSpace.single i (b : 𝕜)) = dist a b := PiLp.dist_equiv_symm_single_same 2 (fun _ => 𝕜) i a b #align euclidean_space.dist_single_same EuclideanSpace.dist_single_same @[simp] theorem EuclideanSpace.nndist_single_same (i : ι) (a b : 𝕜) : nndist (EuclideanSpace.single i (a : 𝕜)) (EuclideanSpace.single i (b : 𝕜)) = nndist a b := PiLp.nndist_equiv_symm_single_same 2 (fun _ => 𝕜) i a b #align euclidean_space.nndist_single_same EuclideanSpace.nndist_single_same @[simp] theorem EuclideanSpace.edist_single_same (i : ι) (a b : 𝕜) : edist (EuclideanSpace.single i (a : 𝕜)) (EuclideanSpace.single i (b : 𝕜)) = edist a b := PiLp.edist_equiv_symm_single_same 2 (fun _ => 𝕜) i a b #align euclidean_space.edist_single_same EuclideanSpace.edist_single_same /-- `EuclideanSpace.single` forms an orthonormal family. -/ theorem EuclideanSpace.orthonormal_single : Orthonormal 𝕜 fun i : ι => EuclideanSpace.single i (1 : 𝕜) := by simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul, EuclideanSpace.single_apply] intros trivial #align euclidean_space.orthonormal_single EuclideanSpace.orthonormal_single theorem EuclideanSpace.piLpCongrLeft_single {ι' : Type} [Fintype ι'] [DecidableEq ι'] (e : ι' ≃ ι) (i' : ι') (v : 𝕜) : LinearIsometryEquiv.piLpCongrLeft 2 𝕜 𝕜 e (EuclideanSpace.single i' v) = EuclideanSpace.single (e i') v := LinearIsometryEquiv.piLpCongrLeft_single e i' _ #align euclidean_space.pi_Lp_congr_left_single EuclideanSpace.piLpCongrLeft_single end DecEq variable (ι 𝕜 E) variable [Fintype ι] /-- An orthonormal basis on E is an identification of `E` with its dimensional-matching `EuclideanSpace 𝕜 ι`. -/ structure OrthonormalBasis where ofRepr :: /-- Linear isometry between `E` and `EuclideanSpace 𝕜 ι` representing the orthonormal basis. -/ repr : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι #align orthonormal_basis OrthonormalBasis #align orthonormal_basis.of_repr OrthonormalBasis.ofRepr #align orthonormal_basis.repr OrthonormalBasis.repr variable {ι 𝕜 E} namespace OrthonormalBasis theorem repr_injective : Injective (repr : OrthonormalBasis ι 𝕜 E → E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι) := fun f g h => by cases f cases g congr -- Porting note: `CoeFun` → `FunLike` /-- `b i` is the `i`th basis vector. -/ instance instFunLike : FunLike (OrthonormalBasis ι 𝕜 E) ι E where coe b i := by classical exact b.repr.symm (EuclideanSpace.single i (1 : 𝕜)) coe_injective' b b' h := repr_injective <\| LinearIsometryEquiv.toLinearEquiv_injective <\| LinearEquiv.symm_bijective.injective <\| LinearEquiv.toLinearMap_injective <\| by classical rw [← LinearMap.cancel_right (WithLp.linearEquiv 2 𝕜 (_ → 𝕜)).symm.surjective] simp only [LinearIsometryEquiv.toLinearEquiv_symm] refine LinearMap.pi_ext fun i k => ?_ have : k = k • (1 : 𝕜) := by rw [smul_eq_mul, mul_one] rw [this, Pi.single_smul] replace h := congr_fun h i simp only [LinearEquiv.comp_coe, map_smul, LinearEquiv.coe_coe, LinearEquiv.trans_apply, WithLp.linearEquiv_symm_apply, WithLp.equiv_symm_single, LinearIsometryEquiv.coe_toLinearEquiv] at h ⊢ rw [h] #noalign orthonormal_basis.has_coe_to_fun @[simp] theorem coe_ofRepr [DecidableEq ι] (e : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι) : ⇑(OrthonormalBasis.ofRepr e) = fun i => e.symm (EuclideanSpace.single i (1 : 𝕜)) := by -- Porting note: simplified with `congr!` dsimp only [DFunLike.coe] funext congr! #align orthonormal_basis.coe_of_repr OrthonormalBasis.coe_ofRepr @[simp] protected theorem repr_symm_single [DecidableEq ι] (b : OrthonormalBasis ι 𝕜 E) (i : ι) : b.repr.symm (EuclideanSpace.single i (1 : 𝕜)) = b i := by -- Porting note: simplified with `congr!` dsimp only [DFunLike.coe] congr! #align orthonormal_basis.repr_symm_single OrthonormalBasis.repr_symm_single @[simp] protected theorem repr_self [DecidableEq ι] (b : OrthonormalBasis ι 𝕜 E) (i : ι) : b.repr (b i) = EuclideanSpace.single i (1 : 𝕜) := by rw [← b.repr_symm_single i, LinearIsometryEquiv.apply_symm_apply] #align orthonormal_basis.repr_self OrthonormalBasis.repr_self protected theorem repr_apply_apply (b : OrthonormalBasis ι 𝕜 E) (v : E) (i : ι) : b.repr v i = ⟪b i, v⟫ := by classical rw [← b.repr.inner_map_map (b i) v, b.repr_self i, EuclideanSpace.inner_single_left] simp only [one_mul, eq_self_iff_true, map_one] #align orthonormal_basis.repr_apply_apply OrthonormalBasis.repr_apply_apply @[simp] protected theorem orthonormal (b : OrthonormalBasis ι 𝕜 E) : Orthonormal 𝕜 b := by classical rw [orthonormal_iff_ite] intro i j rw [← b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j, EuclideanSpace.inner_single_left, EuclideanSpace.single_apply, map_one, one_mul] #align orthonormal_basis.orthonormal OrthonormalBasis.orthonormal /-- The `Basis ι 𝕜 E` underlying the `OrthonormalBasis` -/ protected def toBasis (b : OrthonormalBasis ι 𝕜 E) : Basis ι 𝕜 E := Basis.ofEquivFun b.repr.toLinearEquiv #align orthonormal_basis.to_basis OrthonormalBasis.toBasis @[simp] protected theorem coe_toBasis (b : OrthonormalBasis ι 𝕜 E) : (⇑b.toBasis : ι → E) = ⇑b := by rw [OrthonormalBasis.toBasis] -- Porting note: was `change` ext j classical rw [Basis.coe_ofEquivFun] congr #align orthonormal_basis.coe_to_basis OrthonormalBasis.coe_toBasis @[simp] protected theorem coe_toBasis_repr (b : OrthonormalBasis ι 𝕜 E) : b.toBasis.equivFun = b.repr.toLinearEquiv := Basis.equivFun_ofEquivFun _ #align orthonormal_basis.coe_to_basis_repr OrthonormalBasis.coe_toBasis_repr @[simp] protected theorem coe_toBasis_repr_apply (b : OrthonormalBasis ι 𝕜 E) (x : E) (i : ι) : b.toBasis.repr x i = b.repr x i := by rw [← Basis.equivFun_apply, OrthonormalBasis.coe_toBasis_repr]; -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [LinearIsometryEquiv.coe_toLinearEquiv] #align orthonormal_basis.coe_to_basis_repr_apply OrthonormalBasis.coe_toBasis_repr_apply protected theorem sum_repr (b : OrthonormalBasis ι 𝕜 E) (x : E) : ∑ i, b.repr x i • b i = x := by simp_rw [← b.coe_toBasis_repr_apply, ← b.coe_toBasis] exact b.toBasis.sum_repr x #align orthonormal_basis.sum_repr OrthonormalBasis.sum_repr protected theorem sum_repr_symm (b : OrthonormalBasis ι 𝕜 E) (v : EuclideanSpace 𝕜 ι) : ∑ i, v i • b i = b.repr.symm v := by simpa using (b.toBasis.equivFun_symm_apply v).symm #align orthonormal_basis.sum_repr_symm OrthonormalBasis.sum_repr_symm protected theorem sum_inner_mul_inner (b : OrthonormalBasis ι 𝕜 E) (x y : E) : ∑ i, ⟪x, b i⟫ ⟪b i, y⟫ = ⟪x, y⟫ := by have := congr_arg (innerSL 𝕜 x) (b.sum_repr y) rw [map_sum] at this convert this rw [map_smul, b.repr_apply_apply, mul_comm] simp only [innerSL_apply, smul_eq_mul] -- Porting note: was `rfl` #align orthonormal_basis.sum_inner_mul_inner OrthonormalBasis.sum_inner_mul_inner protected theorem orthogonalProjection_eq_sum {U : Submodule 𝕜 E} [CompleteSpace U] (b : OrthonormalBasis ι 𝕜 U) (x : E) : orthogonalProjection U x = ∑ i, ⟪(b i : E), x⟫ • b i := by simpa only [b.repr_apply_apply, inner_orthogonalProjection_eq_of_mem_left] using (b.sum_repr (orthogonalProjection U x)).symm #align orthonormal_basis.orthogonal_projection_eq_sum OrthonormalBasis.orthogonalProjection_eq_sum /-- Mapping an orthonormal basis along a `LinearIsometryEquiv`. -/ protected def map {G : Type} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) : OrthonormalBasis ι 𝕜 G where repr := L.symm.trans b.repr #align orthonormal_basis.map OrthonormalBasis.map @[simp] protected theorem map_apply {G : Type} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) (i : ι) : b.map L i = L (b i) := rfl #align orthonormal_basis.map_apply OrthonormalBasis.map_apply @[simp] protected theorem toBasis_map {G : Type} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) : (b.map L).toBasis = b.toBasis.map L.toLinearEquiv := rfl #align orthonormal_basis.to_basis_map OrthonormalBasis.toBasis_map /-- A basis that is orthonormal is an orthonormal basis. -/ def _root_.Basis.toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) : OrthonormalBasis ι 𝕜 E := OrthonormalBasis.ofRepr <\| LinearEquiv.isometryOfInner v.equivFun (by intro x y let p : EuclideanSpace 𝕜 ι := v.equivFun x let q : EuclideanSpace 𝕜 ι := v.equivFun y have key : ⟪p, q⟫ = ⟪∑ i, p i • v i, ∑ i, q i • v i⟫ := by simp [sum_inner, inner_smul_left, hv.inner_right_fintype] convert key · rw [← v.equivFun.symm_apply_apply x, v.equivFun_symm_apply] · rw [← v.equivFun.symm_apply_apply y, v.equivFun_symm_apply]) #align basis.to_orthonormal_basis Basis.toOrthonormalBasis @[simp] theorem _root_.Basis.coe_toOrthonormalBasis_repr (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) : ((v.toOrthonormalBasis hv).repr : E → EuclideanSpace 𝕜 ι) = v.equivFun := rfl #align basis.coe_to_orthonormal_basis_repr Basis.coe_toOrthonormalBasis_repr @[simp] theorem _root_.Basis.coe_toOrthonormalBasis_repr_symm (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) : ((v.toOrthonormalBasis hv).repr.symm : EuclideanSpace 𝕜 ι → E) = v.equivFun.symm := rfl #align basis.coe_to_orthonormal_basis_repr_symm Basis.coe_toOrthonormalBasis_repr_symm @[simp] theorem _root_.Basis.toBasis_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) : (v.toOrthonormalBasis hv).toBasis = v := by simp [Basis.toOrthonormalBasis, OrthonormalBasis.toBasis] #align basis.to_basis_to_orthonormal_basis Basis.toBasis_toOrthonormalBasis @[simp] theorem _root_.Basis.coe_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) : (v.toOrthonormalBasis hv : ι → E) = (v : ι → E) := calc (v.toOrthonormalBasis hv : ι → E) = ((v.toOrthonormalBasis hv).toBasis : ι → E) := by classical rw [OrthonormalBasis.coe_toBasis] _ = (v : ι → E) := by simp #align basis.coe_to_orthonormal_basis Basis.coe_toOrthonormalBasis variable {v : ι → E} /-- A finite orthonormal set that spans is an orthonormal basis -/ protected def mk (hon : Orthonormal 𝕜 v) (hsp : ⊤ ≤ Submodule.span 𝕜 (Set.range v)) : OrthonormalBasis ι 𝕜 E := (Basis.mk (Orthonormal.linearIndependent hon) hsp).toOrthonormalBasis (by rwa [Basis.coe_mk]) #align orthonormal_basis.mk OrthonormalBasis.mk @[simp] protected theorem coe_mk (hon : Orthonormal 𝕜 v) (hsp : ⊤ ≤ Submodule.span 𝕜 (Set.range v)) : ⇑(OrthonormalBasis.mk hon hsp) = v := by classical rw [OrthonormalBasis.mk, _root_.Basis.coe_toOrthonormalBasis, Basis.coe_mk] #align orthonormal_basis.coe_mk OrthonormalBasis.coe_mk /-- Any finite subset of an orthonormal family is an `OrthonormalBasis` for its span. -/ protected def span [DecidableEq E] {v' : ι' → E} (h : Orthonormal 𝕜 v') (s : Finset ι') : OrthonormalBasis s 𝕜 (span 𝕜 (s.image v' : Set E)) := let e₀' : Basis s 𝕜 _ := Basis.span (h.linearIndependent.comp ((↑) : s → ι') Subtype.val_injective) let e₀ : OrthonormalBasis s 𝕜 _ := OrthonormalBasis.mk (by convert orthonormal_span (h.comp ((↑) : s → ι') Subtype.val_injective) simp [e₀', Basis.span_apply]) e₀'.span_eq.ge let φ : span 𝕜 (s.image v' : Set E) ≃ₗᵢ[𝕜] span 𝕜 (range (v' ∘ ((↑) : s → ι'))) := LinearIsometryEquiv.ofEq _ _ (by rw [Finset.coe_image, image_eq_range] rfl) e₀.map φ.symm #align orthonormal_basis.span OrthonormalBasis.span @[simp] protected theorem span_apply [DecidableEq E] {v' : ι' → E} (h : Orthonormal 𝕜 v') (s : Finset ι') (i : s) : (OrthonormalBasis.span h s i : E) = v' i := by simp only [OrthonormalBasis.span, Basis.span_apply, LinearIsometryEquiv.ofEq_symm, OrthonormalBasis.map_apply, OrthonormalBasis.coe_mk, LinearIsometryEquiv.coe_ofEq_apply, comp_apply] #align orthonormal_basis.span_apply OrthonormalBasis.span_apply open Submodule /-- A finite orthonormal family of vectors whose span has trivial orthogonal complement is an orthonormal basis. -/ protected def mkOfOrthogonalEqBot (hon : Orthonormal 𝕜 v) (hsp : (span 𝕜 (Set.range v))ᗮ = ⊥) : OrthonormalBasis ι 𝕜 E := OrthonormalBasis.mk hon (by refine Eq.ge ?_ haveI : FiniteDimensional 𝕜 (span 𝕜 (range v)) := FiniteDimensional.span_of_finite 𝕜 (finite_range v) haveI : CompleteSpace (span 𝕜 (range v)) := FiniteDimensional.complete 𝕜 _ rwa [orthogonal_eq_bot_iff] at hsp) #align orthonormal_basis.mk_of_orthogonal_eq_bot OrthonormalBasis.mkOfOrthogonalEqBot @[simp] protected theorem coe_of_orthogonal_eq_bot_mk (hon : Orthonormal 𝕜 v) (hsp : (span 𝕜 (Set.range v))ᗮ = ⊥) : ⇑(OrthonormalBasis.mkOfOrthogonalEqBot hon hsp) = v := OrthonormalBasis.coe_mk hon _ #align orthonormal_basis.coe_of_orthogonal_eq_bot_mk OrthonormalBasis.coe_of_orthogonal_eq_bot_mk variable [Fintype ι'] /-- `b.reindex (e : ι ≃ ι')` is an `OrthonormalBasis` indexed by `ι'` -/ def reindex (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') : OrthonormalBasis ι' 𝕜 E := OrthonormalBasis.ofRepr (b.repr.trans (LinearIsometryEquiv.piLpCongrLeft 2 𝕜 𝕜 e)) #align orthonormal_basis.reindex OrthonormalBasis.reindex protected theorem reindex_apply (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') (i' : ι') : (b.reindex e) i' = b (e.symm i') := by classical dsimp [reindex] rw [coe_ofRepr] dsimp rw [← b.repr_symm_single, LinearIsometryEquiv.piLpCongrLeft_symm, EuclideanSpace.piLpCongrLeft_single] #align orthonormal_basis.reindex_apply OrthonormalBasis.reindex_apply @[simp] protected theorem coe_reindex (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') : ⇑(b.reindex e) = b ∘ e.symm := funext (b.reindex_apply e) #align orthonormal_basis.coe_reindex OrthonormalBasis.coe_reindex @[simp] protected theorem repr_reindex (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') (x : E) (i' : ι') : (b.reindex e).repr x i' = b.repr x (e.symm i') := by classical rw [OrthonormalBasis.repr_apply_apply, b.repr_apply_apply, OrthonormalBasis.coe_reindex, comp_apply] #align orthonormal_basis.repr_reindex OrthonormalBasis.repr_reindex end OrthonormalBasis namespace EuclideanSpace variable (𝕜 ι) /-- The basis `Pi.basisFun`, bundled as an orthornormal basis of `EuclideanSpace 𝕜 ι`. -/ noncomputable def basisFun : OrthonormalBasis ι 𝕜 (EuclideanSpace 𝕜 ι) := ⟨LinearIsometryEquiv.refl _ _⟩ @[simp] theorem basisFun_apply [DecidableEq ι] (i : ι) : basisFun ι 𝕜 i = EuclideanSpace.single i 1 := PiLp.basisFun_apply _ _ _ _ @[simp] theorem basisFun_repr (x : EuclideanSpace 𝕜 ι) (i : ι) : (basisFun ι 𝕜).repr x i = x i := rfl theorem basisFun_toBasis : (basisFun ι 𝕜).toBasis = PiLp.basisFun _ 𝕜 ι := rfl end EuclideanSpace instance OrthonormalBasis.instInhabited : Inhabited (OrthonormalBasis ι 𝕜 (EuclideanSpace 𝕜 ι)) := ⟨EuclideanSpace.basisFun ι 𝕜⟩ #align orthonormal_basis.inhabited OrthonormalBasis.instInhabited section Complex /-- `![1, I]` is an orthonormal basis for `ℂ` considered as a real inner product space. -/ def Complex.orthonormalBasisOneI : OrthonormalBasis (Fin 2) ℝ ℂ := Complex.basisOneI.toOrthonormalBasis (by rw [orthonormal_iff_ite] intro i; fin_cases i <;> intro j <;> fin_cases j <;> simp [real_inner_eq_re_inner]) #align complex.orthonormal_basis_one_I Complex.orthonormalBasisOneI @[simp] theorem Complex.orthonormalBasisOneI_repr_apply (z : ℂ) : Complex.orthonormalBasisOneI.repr z = ![z.re, z.im] := rfl #align complex.orthonormal_basis_one_I_repr_apply Complex.orthonormalBasisOneI_repr_apply @[simp] theorem Complex.orthonormalBasisOneI_repr_symm_apply (x : EuclideanSpace ℝ (Fin 2)) : Complex.orthonormalBasisOneI.repr.symm x = x 0 + x 1 I := rfl #align complex.orthonormal_basis_one_I_repr_symm_apply Complex.orthonormalBasisOneI_repr_symm_apply @[simp] theorem Complex.toBasis_orthonormalBasisOneI : Complex.orthonormalBasisOneI.toBasis = Complex.basisOneI := Basis.toBasis_toOrthonormalBasis _ _ #align complex.to_basis_orthonormal_basis_one_I Complex.toBasis_orthonormalBasisOneI @[simp] theorem Complex.coe_orthonormalBasisOneI : (Complex.orthonormalBasisOneI : Fin 2 → ℂ) = ![1, I] := by simp [Complex.orthonormalBasisOneI] #align complex.coe_orthonormal_basis_one_I Complex.coe_orthonormalBasisOneI /-- The isometry between `ℂ` and a two-dimensional real inner product space given by a basis. -/ def Complex.isometryOfOrthonormal (v : OrthonormalBasis (Fin 2) ℝ F) : ℂ ≃ₗᵢ[ℝ] F := Complex.orthonormalBasisOneI.repr.trans v.repr.symm #align complex.isometry_of_orthonormal Complex.isometryOfOrthonormal @[simp] theorem Complex.map_isometryOfOrthonormal (v : OrthonormalBasis (Fin 2) ℝ F) (f : F ≃ₗᵢ[ℝ] F') : Complex.isometryOfOrthonormal (v.map f) = (Complex.isometryOfOrthonormal v).trans f := by simp [Complex.isometryOfOrthonormal, LinearIsometryEquiv.trans_assoc, OrthonormalBasis.map] -- Porting note: `LinearIsometryEquiv.trans_assoc` doesn't trigger in the `simp` above rw [LinearIsometryEquiv.trans_assoc] #align complex.map_isometry_of_orthonormal Complex.map_isometryOfOrthonormal theorem Complex.isometryOfOrthonormal_symm_apply (v : OrthonormalBasis (Fin 2) ℝ F) (f : F) : (Complex.isometryOfOrthonormal v).symm f = (v.toBasis.coord 0 f : ℂ) + (v.toBasis.coord 1 f : ℂ) * I := by simp [Complex.isometryOfOrthonormal] #align complex.isometry_of_orthonormal_symm_apply Complex.isometryOfOrthonormal_symm_apply theorem Complex.isometryOfOrthonormal_apply (v : OrthonormalBasis (Fin 2) ℝ F) (z : ℂ) : Complex.isometryOfOrthonormal v z = z.re • v 0 + z.im • v 1 := by -- Porting note: was -- simp [Complex.isometryOfOrthonormal, ← v.sum_repr_symm] rw [Complex.isometryOfOrthonormal, LinearIsometryEquiv.trans_apply] simp [← v.sum_repr_symm] #align complex.isometry_of_orthonormal_apply Complex.isometryOfOrthonormal_apply end Complex open FiniteDimensional /-! ### Matrix representation of an orthonormal basis with respect to another -/ section ToMatrix variable [DecidableEq ι] section variable (a b : OrthonormalBasis ι 𝕜 E) /-- The change-of-basis matrix between two orthonormal bases `a`, `b` is a unitary matrix. -/ theorem OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary : a.toBasis.toMatrix b ∈ Matrix.unitaryGroup ι 𝕜 := by rw [Matrix.mem_unitaryGroup_iff'] ext i j convert a.repr.inner_map_map (b i) (b j) rw [orthonormal_iff_ite.mp b.orthonormal i j] rfl #align orthonormal_basis.to_matrix_orthonormal_basis_mem_unitary OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary /-- The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` has unit length. -/ @[simp] theorem OrthonormalBasis.det_to_matrix_orthonormalBasis : ‖a.toBasis.det b‖ = 1 := by have := (Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)).2 rw [star_def, RCLike.mul_conj] at this norm_cast at this rwa [pow_eq_one_iff_of_nonneg (norm_nonneg _) two_ne_zero] at this #align orthonormal_basis.det_to_matrix_orthonormal_basis OrthonormalBasis.det_to_matrix_orthonormalBasis end section Real variable (a b : OrthonormalBasis ι ℝ F) /-- The change-of-basis matrix between two orthonormal bases `a`, `b` is an orthogonal matrix. -/ theorem OrthonormalBasis.toMatrix_orthonormalBasis_mem_orthogonal : a.toBasis.toMatrix b ∈ Matrix.orthogonalGroup ι ℝ := a.toMatrix_orthonormalBasis_mem_unitary b #align orthonormal_basis.to_matrix_orthonormal_basis_mem_orthogonal OrthonormalBasis.toMatrix_orthonormalBasis_mem_orthogonal /-- The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` is ±1. -/ theorem OrthonormalBasis.det_to_matrix_orthonormalBasis_real : a.toBasis.det b = 1 ∨ a.toBasis.det b = -1 := by rw [← sq_eq_one_iff] simpa [unitary, sq] using Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b) #align orthonormal_basis.det_to_matrix_orthonormal_basis_real OrthonormalBasis.det_to_matrix_orthonormalBasis_real end Real end ToMatrix /-! ### Existence of orthonormal basis, etc. -/ section FiniteDimensional variable {v : Set E} variable {A : ι → Submodule 𝕜 E} /-- Given an internal direct sum decomposition of a module `M`, and an orthonormal basis for each of the components of the direct sum, the disjoint union of these orthonormal bases is an orthonormal basis for `M`. -/ noncomputable def DirectSum.IsInternal.collectedOrthonormalBasis (hV : OrthogonalFamily 𝕜 (fun i => A i) fun i => (A i).subtypeₗᵢ) [DecidableEq ι] (hV_sum : DirectSum.IsInternal fun i => A i) {α : ι → Type} [∀ i, Fintype (α i)] (v_family : ∀ i, OrthonormalBasis (α i) 𝕜 (A i)) : OrthonormalBasis (Σi, α i) 𝕜 E := (hV_sum.collectedBasis fun i => (v_family i).toBasis).toOrthonormalBasis <\| by simpa using hV.orthonormal_sigma_orthonormal (show ∀ i, Orthonormal 𝕜 (v_family i).toBasis by simp) #align direct_sum.is_internal.collected_orthonormal_basis DirectSum.IsInternal.collectedOrthonormalBasis theorem DirectSum.IsInternal.collectedOrthonormalBasis_mem [DecidableEq ι] (h : DirectSum.IsInternal A) {α : ι → Type} [∀ i, Fintype (α i)] (hV : OrthogonalFamily 𝕜 (fun i => A i) fun i => (A i).subtypeₗᵢ) (v : ∀ i, OrthonormalBasis (α i) 𝕜 (A i)) (a : Σi, α i) : h.collectedOrthonormalBasis hV v a ∈ A a.1 := by simp [DirectSum.IsInternal.collectedOrthonormalBasis] #align direct_sum.is_internal.collected_orthonormal_basis_mem DirectSum.IsInternal.collectedOrthonormalBasis_mem variable [FiniteDimensional 𝕜 E] /-- In a finite-dimensional `InnerProductSpace`, any orthonormal subset can be extended to an orthonormal basis. -/ theorem Orthonormal.exists_orthonormalBasis_extension (hv : Orthonormal 𝕜 ((↑) : v → E)) : ∃ (u : Finset E) (b : OrthonormalBasis u 𝕜 E), v ⊆ u ∧ ⇑b = ((↑) : u → E) := by obtain ⟨u₀, hu₀s, hu₀, hu₀_max⟩ := exists_maximal_orthonormal hv rw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hu₀] at hu₀_max have hu₀_finite : u₀.Finite := hu₀.linearIndependent.setFinite let u : Finset E := hu₀_finite.toFinset let fu : ↥u ≃ ↥u₀ := hu₀_finite.subtypeEquivToFinset.symm have hu : Orthonormal 𝕜 ((↑) : u → E) := by simpa using hu₀.comp _ fu.injective refine ⟨u, OrthonormalBasis.mkOfOrthogonalEqBot hu ?_, ?_, ?_⟩ · simpa [u] using hu₀_max · simpa [u] using hu₀s · simp #align orthonormal.exists_orthonormal_basis_extension Orthonormal.exists_orthonormalBasis_extension theorem Orthonormal.exists_orthonormalBasis_extension_of_card_eq {ι : Type} [Fintype ι] (card_ι : finrank 𝕜 E = Fintype.card ι) {v : ι → E} {s : Set ι} (hv : Orthonormal 𝕜 (s.restrict v)) : ∃ b : OrthonormalBasis ι 𝕜 E, ∀ i ∈ s, b i = v i := by have hsv : Injective (s.restrict v) := hv.linearIndependent.injective have hX : Orthonormal 𝕜 ((↑) : Set.range (s.restrict v) → E) := by rwa [orthonormal_subtype_range hsv] obtain ⟨Y, b₀, hX, hb₀⟩ := hX.exists_orthonormalBasis_extension have hιY : Fintype.card ι = Y.card := by refine card_ι.symm.trans ?_ exact FiniteDimensional.finrank_eq_card_finset_basis b₀.toBasis have hvsY : s.MapsTo v Y := (s.mapsTo_image v).mono_right (by rwa [← range_restrict]) have hsv' : Set.InjOn v s := by rw [Set.injOn_iff_injective] exact hsv obtain ⟨g, hg⟩ := hvsY.exists_equiv_extend_of_card_eq hιY hsv' use b₀.reindex g.symm intro i hi simp [hb₀, hg i hi] #align orthonormal.exists_orthonormal_basis_extension_of_card_eq Orthonormal.exists_orthonormalBasis_extension_of_card_eq variable (𝕜 E) /-- A finite-dimensional inner product space admits an orthonormal basis. -/ theorem _root_.exists_orthonormalBasis : ∃ (w : Finset E) (b : OrthonormalBasis w 𝕜 E), ⇑b = ((↑) : w → E) := let ⟨w, hw, _, hw''⟩ := (orthonormal_empty 𝕜 E).exists_orthonormalBasis_extension ⟨w, hw, hw''⟩ #align exists_orthonormal_basis exists_orthonormalBasis /-- A finite-dimensional `InnerProductSpace` has an orthonormal basis. -/ irreducible_def stdOrthonormalBasis : OrthonormalBasis (Fin (finrank 𝕜 E)) 𝕜 E := by let b := Classical.choose (Classical.choose_spec <\| exists_orthonormalBasis 𝕜 E) rw [finrank_eq_card_basis b.toBasis] exact b.reindex (Fintype.equivFinOfCardEq rfl) #align std_orthonormal_basis stdOrthonormalBasis /-- An orthonormal basis of `ℝ` is made either of the vector `1`, or of the vector `-1`. -/ theorem orthonormalBasis_one_dim (b : OrthonormalBasis ι ℝ ℝ) : (⇑b = fun _ => (1 : ℝ)) ∨ ⇑b = fun _ => (-1 : ℝ) := by have : Unique ι := b.toBasis.unique have : b default = 1 ∨ b default = -1 := by have : ‖b default‖ = 1 := b.orthonormal.1 _ rwa [Real.norm_eq_abs, abs_eq (zero_le_one' ℝ)] at this rw [eq_const_of_unique b] refine this.imp ?_ ?_ <;> (intro; ext; simp []) #align orthonormal_basis_one_dim orthonormalBasis_one_dim variable {𝕜 E} section SubordinateOrthonormalBasis open DirectSum variable {n : ℕ} (hn : finrank 𝕜 E = n) [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : IsInternal V) /-- Exhibit a bijection between `Fin n` and the index set of a certain basis of an `n`-dimensional inner product space `E`. This should not be accessed directly, but only via the subsequent API. -/ irreducible_def DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) : (Σi, Fin (finrank 𝕜 (V i))) ≃ Fin n := let b := hV.collectedOrthonormalBasis hV' fun i => stdOrthonormalBasis 𝕜 (V i) Fintype.equivFinOfCardEq <\| (FiniteDimensional.finrank_eq_card_basis b.toBasis).symm.trans hn #align direct_sum.is_internal.sigma_orthonormal_basis_index_equiv DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv /-- An `n`-dimensional `InnerProductSpace` equipped with a decomposition as an internal direct sum has an orthonormal basis indexed by `Fin n` and subordinate to that direct sum. -/ irreducible_def DirectSum.IsInternal.subordinateOrthonormalBasis (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) : OrthonormalBasis (Fin n) 𝕜 E := (hV.collectedOrthonormalBasis hV' fun i => stdOrthonormalBasis 𝕜 (V i)).reindex (hV.sigmaOrthonormalBasisIndexEquiv hn hV') #align direct_sum.is_internal.subordinate_orthonormal_basis DirectSum.IsInternal.subordinateOrthonormalBasis /-- An `n`-dimensional `InnerProductSpace` equipped with a decomposition as an internal direct sum has an orthonormal basis indexed by `Fin n` and subordinate to that direct sum. This function provides the mapping by which it is subordinate. -/ irreducible_def DirectSum.IsInternal.subordinateOrthonormalBasisIndex (a : Fin n) (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) : ι := ((hV.sigmaOrthonormalBasisIndexEquiv hn hV').symm a).1 #align direct_sum.is_internal.subordinate_orthonormal_basis_index DirectSum.IsInternal.subordinateOrthonormalBasisIndex /-- The basis constructed in `DirectSum.IsInternal.subordinateOrthonormalBasis` is subordinate to the `OrthogonalFamily` in question. -/ theorem DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate (a : Fin n) (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) : hV.subordinateOrthonormalBasis hn hV' a ∈ V (hV.subordinateOrthonormalBasisIndex hn a hV') := by simpa only [DirectSum.IsInternal.subordinateOrthonormalBasis, OrthonormalBasis.coe_reindex, DirectSum.IsInternal.subordinateOrthonormalBasisIndex] using hV.collectedOrthonormalBasis_mem hV' (fun i => stdOrthonormalBasis 𝕜 (V i)) ((hV.sigmaOrthonormalBasisIndexEquiv hn hV').symm a) #align direct_sum.is_internal.subordinate_orthonormal_basis_subordinate DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate end SubordinateOrthonormalBasis end FiniteDimensional /-- Given a natural number `n` one less than the `finrank` of a finite-dimensional inner product space, there exists an isometry from the orthogonal complement of a nonzero singleton to `EuclideanSpace 𝕜 (Fin n)`. -/ def OrthonormalBasis.fromOrthogonalSpanSingleton (n : ℕ) [Fact (finrank 𝕜 E = n + 1)] {v : E} (hv : v ≠ 0) : OrthonormalBasis (Fin n) 𝕜 (𝕜 ∙ v)ᗮ := -- Porting note: was `attribute [local instance] FiniteDimensional.of_fact_finrank_eq_succ` haveI : FiniteDimensional 𝕜 E := .of_fact_finrank_eq_succ (K := 𝕜) (V := E) n (stdOrthonormalBasis _ _).reindex <\| finCongr <\| finrank_orthogonal_span_singleton hv #align orthonormal_basis.from_orthogonal_span_singleton OrthonormalBasis.fromOrthogonalSpanSingleton section LinearIsometry variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V] variable {S : Submodule 𝕜 V} {L : S →ₗᵢ[𝕜] V} open FiniteDimensional /-- Let `S` be a subspace of a finite-dimensional complex inner product space `V`. A linear isometry mapping `S` into `V` can be extended to a full isometry of `V`. TODO: The case when `S` is a finite-dimensional subspace of an infinite-dimensional `V`. -/ noncomputable def LinearIsometry.extend (L : S →ₗᵢ[𝕜] V) : V →ₗᵢ[𝕜] V := by -- Build an isometry from Sᗮ to L(S)ᗮ through `EuclideanSpace` let d := finrank 𝕜 Sᗮ let LS := LinearMap.range L.toLinearMap have E : Sᗮ ≃ₗᵢ[𝕜] LSᗮ := by have dim_LS_perp : finrank 𝕜 LSᗮ = d := calc finrank 𝕜 LSᗮ = finrank 𝕜 V - finrank 𝕜 LS := by simp only [← LS.finrank_add_finrank_orthogonal, add_tsub_cancel_left] _ = finrank 𝕜 V - finrank 𝕜 S := by simp only [LinearMap.finrank_range_of_inj L.injective] _ = finrank 𝕜 Sᗮ := by simp only [← S.finrank_add_finrank_orthogonal, add_tsub_cancel_left] exact (stdOrthonormalBasis 𝕜 Sᗮ).repr.trans ((stdOrthonormalBasis 𝕜 LSᗮ).reindex <\| finCongr dim_LS_perp).repr.symm let L3 := LSᗮ.subtypeₗᵢ.comp E.toLinearIsometry -- Project onto S and Sᗮ haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S haveI : CompleteSpace V := FiniteDimensional.complete 𝕜 V let p1 := (orthogonalProjection S).toLinearMap let p2 := (orthogonalProjection Sᗮ).toLinearMap -- Build a linear map from the isometries on S and Sᗮ let M := L.toLinearMap.comp p1 + L3.toLinearMap.comp p2 -- Prove that M is an isometry have M_norm_map : ∀ x : V, ‖M x‖ = ‖x‖ := by intro x -- Apply M to the orthogonal decomposition of x have Mx_decomp : M x = L (p1 x) + L3 (p2 x) := by simp only [M, LinearMap.add_apply, LinearMap.comp_apply, LinearMap.comp_apply, LinearIsometry.coe_toLinearMap] -- Mx_decomp is the orthogonal decomposition of M x have Mx_orth : ⟪L (p1 x), L3 (p2 x)⟫ = 0 := by have Lp1x : L (p1 x) ∈ LinearMap.range L.toLinearMap := LinearMap.mem_range_self L.toLinearMap (p1 x) have Lp2x : L3 (p2 x) ∈ (LinearMap.range L.toLinearMap)ᗮ := by simp only [LinearIsometry.coe_comp, Function.comp_apply, Submodule.coe_subtypeₗᵢ, ← Submodule.range_subtype LSᗮ] apply LinearMap.mem_range_self apply Submodule.inner_right_of_mem_orthogonal Lp1x Lp2x -- Apply the Pythagorean theorem and simplify rw [← sq_eq_sq (norm_nonneg _) (norm_nonneg _), norm_sq_eq_add_norm_sq_projection x S] simp only [sq, Mx_decomp] rw [norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth] simp only [p1, p2, LinearIsometry.norm_map, _root_.add_left_inj, mul_eq_mul_left_iff, norm_eq_zero, true_or_iff, eq_self_iff_true, ContinuousLinearMap.coe_coe, Submodule.coe_norm, Submodule.coe_eq_zero] exact { toLinearMap := M norm_map' := M_norm_map } #align linear_isometry.extend LinearIsometry.extend	Mathlib/Analysis/InnerProductSpace/PiL2.lean	981	990	theorem LinearIsometry.extend_apply (L : S →ₗᵢ[𝕜] V) (s : S) : L.extend s = L s := by	haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S simp only [LinearIsometry.extend, ← LinearIsometry.coe_toLinearMap] simp only [add_right_eq_self, LinearIsometry.coe_toLinearMap, LinearIsometryEquiv.coe_toLinearIsometry, LinearIsometry.coe_comp, Function.comp_apply, orthogonalProjection_mem_subspace_eq_self, LinearMap.coe_comp, ContinuousLinearMap.coe_coe, Submodule.coeSubtype, LinearMap.add_apply, Submodule.coe_eq_zero, LinearIsometryEquiv.map_eq_zero_iff, Submodule.coe_subtypeₗᵢ, orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.orthogonal_orthogonal, Submodule.coe_mem]
/- Copyright (c) 2022 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Batteries.Data.RBMap.Alter import Batteries.Data.List.Lemmas /-! # Additional lemmas for Red-black trees -/ namespace Batteries namespace RBNode open RBColor attribute [simp] fold foldl foldr Any forM foldlM Ordered @[simp] theorem min?_reverse (t : RBNode α) : t.reverse.min? = t.max? := by unfold RBNode.max?; split <;> simp [RBNode.min?] unfold RBNode.min?; rw [min?.match_1.eq_3] · apply min?_reverse · simpa [reverse_eq_iff] @[simp] theorem max?_reverse (t : RBNode α) : t.reverse.max? = t.min? := by rw [← min?_reverse, reverse_reverse] @[simp] theorem mem_nil {x} : ¬x ∈ (.nil : RBNode α) := by simp [(·∈·), EMem] @[simp] theorem mem_node {y c a x b} : y ∈ (.node c a x b : RBNode α) ↔ y = x ∨ y ∈ a ∨ y ∈ b := by simp [(·∈·), EMem] theorem All_def {t : RBNode α} : t.All p ↔ ∀ x ∈ t, p x := by induction t <;> simp [or_imp, forall_and, ] theorem Any_def {t : RBNode α} : t.Any p ↔ ∃ x ∈ t, p x := by induction t <;> simp [or_and_right, exists_or, ] theorem memP_def : MemP cut t ↔ ∃ x ∈ t, cut x = .eq := Any_def theorem mem_def : Mem cmp x t ↔ ∃ y ∈ t, cmp x y = .eq := Any_def theorem mem_congr [@TransCmp α cmp] {t : RBNode α} (h : cmp x y = .eq) : Mem cmp x t ↔ Mem cmp y t := by simp [Mem, TransCmp.cmp_congr_left' h] theorem isOrdered_iff' [@TransCmp α cmp] {t : RBNode α} : isOrdered cmp t L R ↔ (∀ a ∈ L, t.All (cmpLT cmp a ·)) ∧ (∀ a ∈ R, t.All (cmpLT cmp · a)) ∧ (∀ a ∈ L, ∀ b ∈ R, cmpLT cmp a b) ∧ Ordered cmp t := by induction t generalizing L R with \| nil => simp [isOrdered]; split <;> simp [cmpLT_iff] next h => intro _ ha _ hb; cases h _ _ ha hb \| node _ l v r => simp [isOrdered, ] exact ⟨ fun ⟨⟨Ll, lv, Lv, ol⟩, ⟨vr, rR, vR, or⟩⟩ => ⟨ fun _ h => ⟨Lv _ h, Ll _ h, (Lv _ h).trans_l vr⟩, fun _ h => ⟨vR _ h, (vR _ h).trans_r lv, rR _ h⟩, fun _ hL _ hR => (Lv _ hL).trans (vR _ hR), lv, vr, ol, or⟩, fun ⟨hL, hR, _, lv, vr, ol, or⟩ => ⟨ ⟨fun _ h => (hL _ h).2.1, lv, fun _ h => (hL _ h).1, ol⟩, ⟨vr, fun _ h => (hR _ h).2.2, fun _ h => (hR _ h).1, or⟩⟩⟩ theorem isOrdered_iff [@TransCmp α cmp] {t : RBNode α} : isOrdered cmp t ↔ Ordered cmp t := by simp [isOrdered_iff'] instance (cmp) [@TransCmp α cmp] (t) : Decidable (Ordered cmp t) := decidable_of_iff _ isOrdered_iff /-- A cut is like a homomorphism of orderings: it is a monotonic predicate with respect to `cmp`, but it can make things that are distinguished by `cmp` equal. This is sufficient for `find?` to locate an element on which `cut` returns `.eq`, but there may be other elements, not returned by `find?`, on which `cut` also returns `.eq`. -/ class IsCut (cmp : α → α → Ordering) (cut : α → Ordering) : Prop where /-- The set `{x \| cut x = .lt}` is downward-closed. -/ le_lt_trans [TransCmp cmp] : cmp x y ≠ .gt → cut x = .lt → cut y = .lt /-- The set `{x \| cut x = .gt}` is upward-closed. -/ le_gt_trans [TransCmp cmp] : cmp x y ≠ .gt → cut y = .gt → cut x = .gt theorem IsCut.lt_trans [IsCut cmp cut] [TransCmp cmp] (H : cmp x y = .lt) : cut x = .lt → cut y = .lt := IsCut.le_lt_trans <\| TransCmp.gt_asymm <\| OrientedCmp.cmp_eq_gt.2 H theorem IsCut.gt_trans [IsCut cmp cut] [TransCmp cmp] (H : cmp x y = .lt) : cut y = .gt → cut x = .gt := IsCut.le_gt_trans <\| TransCmp.gt_asymm <\| OrientedCmp.cmp_eq_gt.2 H theorem IsCut.congr [IsCut cmp cut] [TransCmp cmp] (H : cmp x y = .eq) : cut x = cut y := by cases ey : cut y · exact IsCut.le_lt_trans (fun h => nomatch H.symm.trans <\| OrientedCmp.cmp_eq_gt.1 h) ey · cases ex : cut x · exact IsCut.le_lt_trans (fun h => nomatch H.symm.trans h) ex \|>.symm.trans ey · rfl · refine IsCut.le_gt_trans (cmp := cmp) (fun h => ?_) ex \|>.symm.trans ey cases H.symm.trans <\| OrientedCmp.cmp_eq_gt.1 h · exact IsCut.le_gt_trans (fun h => nomatch H.symm.trans h) ey instance (cmp cut) [@IsCut α cmp cut] : IsCut (flip cmp) (cut · \|>.swap) where le_lt_trans h₁ h₂ := by have : TransCmp cmp := inferInstanceAs (TransCmp (flip (flip cmp))) rw [IsCut.le_gt_trans (cmp := cmp) h₁ (Ordering.swap_inj.1 h₂)]; rfl le_gt_trans h₁ h₂ := by have : TransCmp cmp := inferInstanceAs (TransCmp (flip (flip cmp))) rw [IsCut.le_lt_trans (cmp := cmp) h₁ (Ordering.swap_inj.1 h₂)]; rfl /-- `IsStrictCut` upgrades the `IsCut` property to ensure that at most one element of the tree can match the cut, and hence `find?` will return the unique such element if one exists. -/ class IsStrictCut (cmp : α → α → Ordering) (cut : α → Ordering) extends IsCut cmp cut : Prop where /-- If `cut = x`, then `cut` and `x` have compare the same with respect to other elements. -/ exact [TransCmp cmp] : cut x = .eq → cmp x y = cut y /-- A "representable cut" is one generated by `cmp a` for some `a`. This is always a valid cut. -/ instance (cmp) (a : α) : IsStrictCut cmp (cmp a) where le_lt_trans h₁ h₂ := TransCmp.lt_le_trans h₂ h₁ le_gt_trans h₁ := Decidable.not_imp_not.1 (TransCmp.le_trans · h₁) exact h := (TransCmp.cmp_congr_left h).symm instance (cmp cut) [@IsStrictCut α cmp cut] : IsStrictCut (flip cmp) (cut · \|>.swap) where exact h := by have : TransCmp cmp := inferInstanceAs (TransCmp (flip (flip cmp))) rw [← IsStrictCut.exact (cmp := cmp) (Ordering.swap_inj.1 h), OrientedCmp.symm]; rfl section fold theorem foldr_cons (t : RBNode α) (l) : t.foldr (·::·) l = t.toList ++ l := by unfold toList induction t generalizing l with \| nil => rfl \| node _ a _ b iha ihb => rw [foldr, foldr, iha, iha (_::_), ihb]; simp @[simp] theorem toList_nil : (.nil : RBNode α).toList = [] := rfl @[simp] theorem toList_node : (.node c a x b : RBNode α).toList = a.toList ++ x :: b.toList := by rw [toList, foldr, foldr_cons]; rfl @[simp] theorem toList_reverse (t : RBNode α) : t.reverse.toList = t.toList.reverse := by induction t <;> simp [] @[simp] theorem mem_toList {t : RBNode α} : x ∈ t.toList ↔ x ∈ t := by induction t <;> simp [, or_left_comm] @[simp] theorem mem_reverse {t : RBNode α} : a ∈ t.reverse ↔ a ∈ t := by rw [← mem_toList]; simp theorem min?_eq_toList_head? {t : RBNode α} : t.min? = t.toList.head? := by induction t with \| nil => rfl \| node _ l _ _ ih => cases l <;> simp [RBNode.min?, ih] next ll _ _ => cases toList ll <;> rfl theorem max?_eq_toList_getLast? {t : RBNode α} : t.max? = t.toList.getLast? := by rw [← min?_reverse, min?_eq_toList_head?]; simp theorem foldr_eq_foldr_toList {t : RBNode α} : t.foldr f init = t.toList.foldr f init := by induction t generalizing init <;> simp [] theorem foldl_eq_foldl_toList {t : RBNode α} : t.foldl f init = t.toList.foldl f init := by induction t generalizing init <;> simp [] theorem foldl_reverse {α β : Type _} {t : RBNode α} {f : β → α → β} {init : β} : t.reverse.foldl f init = t.foldr (flip f) init := by simp (config := {unfoldPartialApp := true}) [foldr_eq_foldr_toList, foldl_eq_foldl_toList, flip] theorem foldr_reverse {α β : Type _} {t : RBNode α} {f : α → β → β} {init : β} : t.reverse.foldr f init = t.foldl (flip f) init := foldl_reverse.symm.trans (by simp; rfl) theorem forM_eq_forM_toList [Monad m] [LawfulMonad m] {t : RBNode α} : t.forM (m := m) f = t.toList.forM f := by induction t <;> simp [] theorem foldlM_eq_foldlM_toList [Monad m] [LawfulMonad m] {t : RBNode α} : t.foldlM (m := m) f init = t.toList.foldlM f init := by induction t generalizing init <;> simp [] theorem forIn_visit_eq_bindList [Monad m] [LawfulMonad m] {t : RBNode α} : forIn.visit (m := m) f t init = (ForInStep.yield init).bindList f t.toList := by induction t generalizing init <;> simp [, forIn.visit] theorem forIn_eq_forIn_toList [Monad m] [LawfulMonad m] {t : RBNode α} : forIn (m := m) t init f = forIn t.toList init f := by conv => lhs; simp only [forIn, RBNode.forIn] rw [List.forIn_eq_bindList, forIn_visit_eq_bindList] end fold namespace Stream attribute [simp] foldl foldr theorem foldr_cons (t : RBNode.Stream α) (l) : t.foldr (·::·) l = t.toList ++ l := by unfold toList; apply Eq.symm; induction t <;> simp [, foldr, RBNode.foldr_cons] @[simp] theorem toList_nil : (.nil : RBNode.Stream α).toList = [] := rfl @[simp] theorem toList_cons : (.cons x r s : RBNode.Stream α).toList = x :: r.toList ++ s.toList := by rw [toList, toList, foldr, RBNode.foldr_cons]; rfl theorem foldr_eq_foldr_toList {s : RBNode.Stream α} : s.foldr f init = s.toList.foldr f init := by induction s <;> simp [, RBNode.foldr_eq_foldr_toList] theorem foldl_eq_foldl_toList {t : RBNode.Stream α} : t.foldl f init = t.toList.foldl f init := by induction t generalizing init <;> simp [, RBNode.foldl_eq_foldl_toList] theorem forIn_eq_forIn_toList [Monad m] [LawfulMonad m] {t : RBNode α} : forIn (m := m) t init f = forIn t.toList init f := by conv => lhs; simp only [forIn, RBNode.forIn] rw [List.forIn_eq_bindList, forIn_visit_eq_bindList] end Stream theorem toStream_toList' {t : RBNode α} {s} : (t.toStream s).toList = t.toList ++ s.toList := by induction t generalizing s <;> simp [, toStream] @[simp] theorem toStream_toList {t : RBNode α} : t.toStream.toList = t.toList := by simp [toStream_toList'] theorem Stream.next?_toList {s : RBNode.Stream α} : (s.next?.map fun (a, b) => (a, b.toList)) = s.toList.next? := by cases s <;> simp [next?, toStream_toList'] theorem ordered_iff {t : RBNode α} : t.Ordered cmp ↔ t.toList.Pairwise (cmpLT cmp) := by induction t with \| nil => simp \| node c l v r ihl ihr => simp [, List.pairwise_append, Ordered, All_def, and_assoc, and_left_comm, and_comm, imp_and, forall_and] exact fun _ _ hl hr a ha b hb => (hl _ ha).trans (hr _ hb) theorem Ordered.toList_sorted {t : RBNode α} : t.Ordered cmp → t.toList.Pairwise (cmpLT cmp) := ordered_iff.1 theorem min?_mem {t : RBNode α} (h : t.min? = some a) : a ∈ t := by rw [min?_eq_toList_head?] at h rw [← mem_toList] revert h; cases toList t <;> rintro ⟨⟩; constructor theorem Ordered.min?_le {t : RBNode α} [TransCmp cmp] (ht : t.Ordered cmp) (h : t.min? = some a) (x) (hx : x ∈ t) : cmp a x ≠ .gt := by rw [min?_eq_toList_head?] at h rw [← mem_toList] at hx have := ht.toList_sorted revert h hx this; cases toList t <;> rintro ⟨⟩ (_ \| ⟨_, hx⟩) (_ \| ⟨h1,h2⟩) · rw [OrientedCmp.cmp_refl (cmp := cmp)]; decide · rw [(h1 _ hx).1]; decide theorem max?_mem {t : RBNode α} (h : t.max? = some a) : a ∈ t := by simpa using min?_mem ((min?_reverse _).trans h) theorem Ordered.le_max? {t : RBNode α} [TransCmp cmp] (ht : t.Ordered cmp) (h : t.max? = some a) (x) (hx : x ∈ t) : cmp x a ≠ .gt := ht.reverse.min?_le ((min?_reverse _).trans h) _ (by simpa using hx) @[simp] theorem setBlack_toList {t : RBNode α} : t.setBlack.toList = t.toList := by cases t <;> simp [setBlack] @[simp] theorem setRed_toList {t : RBNode α} : t.setRed.toList = t.toList := by cases t <;> simp [setRed] @[simp] theorem balance1_toList {l : RBNode α} {v r} : (l.balance1 v r).toList = l.toList ++ v :: r.toList := by unfold balance1; split <;> simp @[simp] theorem balance2_toList {l : RBNode α} {v r} : (l.balance2 v r).toList = l.toList ++ v :: r.toList := by unfold balance2; split <;> simp @[simp] theorem balLeft_toList {l : RBNode α} {v r} : (l.balLeft v r).toList = l.toList ++ v :: r.toList := by unfold balLeft; split <;> (try simp); split <;> simp @[simp] theorem balRight_toList {l : RBNode α} {v r} : (l.balRight v r).toList = l.toList ++ v :: r.toList := by unfold balRight; split <;> (try simp); split <;> simp theorem size_eq {t : RBNode α} : t.size = t.toList.length := by induction t <;> simp [, size]; rfl @[simp] theorem reverse_size (t : RBNode α) : t.reverse.size = t.size := by simp [size_eq] @[simp] theorem Any_reverse {t : RBNode α} : t.reverse.Any p ↔ t.Any p := by simp [Any_def] @[simp] theorem memP_reverse {t : RBNode α} : MemP cut t.reverse ↔ MemP (cut · \|>.swap) t := by simp [MemP]; apply Iff.of_eq; congr; funext x; rw [← Ordering.swap_inj]; rfl theorem Mem_reverse [@OrientedCmp α cmp] {t : RBNode α} : Mem cmp x t.reverse ↔ Mem (flip cmp) x t := by simp [Mem]; apply Iff.of_eq; congr; funext x; rw [OrientedCmp.symm]; rfl section find? theorem find?_some_eq_eq {t : RBNode α} : x ∈ t.find? cut → cut x = .eq := by induction t <;> simp [find?]; split <;> try assumption intro \| rfl => assumption theorem find?_some_mem {t : RBNode α} : x ∈ t.find? cut → x ∈ t := by induction t <;> simp [find?]; split <;> simp (config := {contextual := true}) [] theorem find?_some_memP {t : RBNode α} (h : x ∈ t.find? cut) : MemP cut t := memP_def.2 ⟨_, find?_some_mem h, find?_some_eq_eq h⟩ theorem Ordered.memP_iff_find? [@TransCmp α cmp] [IsCut cmp cut] (ht : Ordered cmp t) : MemP cut t ↔ ∃ x, t.find? cut = some x := by refine ⟨fun H => ?_, fun ⟨x, h⟩ => find?_some_memP h⟩ induction t with simp [find?] at H ⊢ \| nil => cases H \| node _ l _ r ihl ihr => let ⟨lx, xr, hl, hr⟩ := ht split · next ev => refine ihl hl ?_ rcases H with ev' \| hx \| hx · cases ev.symm.trans ev' · exact hx · have ⟨z, hz, ez⟩ := Any_def.1 hx cases ez.symm.trans <\| IsCut.lt_trans (All_def.1 xr _ hz).1 ev · next ev => refine ihr hr ?_ rcases H with ev' \| hx \| hx · cases ev.symm.trans ev' · have ⟨z, hz, ez⟩ := Any_def.1 hx cases ez.symm.trans <\| IsCut.gt_trans (All_def.1 lx _ hz).1 ev · exact hx · exact ⟨_, rfl⟩ theorem Ordered.unique [@TransCmp α cmp] (ht : Ordered cmp t) (hx : x ∈ t) (hy : y ∈ t) (e : cmp x y = .eq) : x = y := by induction t with \| nil => cases hx \| node _ l _ r ihl ihr => let ⟨lx, xr, hl, hr⟩ := ht rcases hx, hy with ⟨rfl \| hx \| hx, rfl \| hy \| hy⟩ · rfl · cases e.symm.trans <\| OrientedCmp.cmp_eq_gt.2 (All_def.1 lx _ hy).1 · cases e.symm.trans (All_def.1 xr _ hy).1 · cases e.symm.trans (All_def.1 lx _ hx).1 · exact ihl hl hx hy · cases e.symm.trans ((All_def.1 lx _ hx).trans (All_def.1 xr _ hy)).1 · cases e.symm.trans <\| OrientedCmp.cmp_eq_gt.2 (All_def.1 xr _ hx).1 · cases e.symm.trans <\| OrientedCmp.cmp_eq_gt.2 ((All_def.1 lx _ hy).trans (All_def.1 xr _ hx)).1 · exact ihr hr hx hy theorem Ordered.find?_some [@TransCmp α cmp] [IsStrictCut cmp cut] (ht : Ordered cmp t) : t.find? cut = some x ↔ x ∈ t ∧ cut x = .eq := by refine ⟨fun h => ⟨find?_some_mem h, find?_some_eq_eq h⟩, fun ⟨hx, e⟩ => ?_⟩ have ⟨y, hy⟩ := ht.memP_iff_find?.1 (memP_def.2 ⟨_, hx, e⟩) exact ht.unique hx (find?_some_mem hy) ((IsStrictCut.exact e).trans (find?_some_eq_eq hy)) ▸ hy @[simp] theorem find?_reverse (t : RBNode α) (cut : α → Ordering) : t.reverse.find? cut = t.find? (cut · \|>.swap) := by induction t <;> simp [, find?] cases cut _ <;> simp [Ordering.swap] /-- Auxiliary definition for `zoom_ins`: set the root of the tree to `v`, creating a node if necessary. -/ def setRoot (v : α) : RBNode α → RBNode α \| nil => node red nil v nil \| node c a _ b => node c a v b /-- Auxiliary definition for `zoom_ins`: set the root of the tree to `v`, creating a node if necessary. -/ def delRoot : RBNode α → RBNode α \| nil => nil \| node _ a _ b => a.append b end find? section «upperBound? and lowerBound?» @[simp] theorem upperBound?_reverse (t : RBNode α) (cut ub) : t.reverse.upperBound? cut ub = t.lowerBound? (cut · \|>.swap) ub := by induction t generalizing ub <;> simp [lowerBound?, upperBound?] split <;> simp [*, Ordering.swap] @[simp] theorem lowerBound?_reverse (t : RBNode α) (cut lb) : t.reverse.lowerBound? cut lb = t.upperBound? (cut · \|>.swap) lb := by simpa using (upperBound?_reverse t.reverse (cut · \|>.swap) lb).symm theorem upperBound?_eq_find? {t : RBNode α} {cut} (ub) (H : t.find? cut = some x) : t.upperBound? cut ub = some x := by induction t generalizing ub with simp [find?] at H \| node c a y b iha ihb => simp [upperBound?]; split at H · apply iha _ H · apply ihb _ H · exact H theorem lowerBound?_eq_find? {t : RBNode α} {cut} (lb) (H : t.find? cut = some x) : t.lowerBound? cut lb = some x := by rw [← reverse_reverse t] at H ⊢; rw [lowerBound?_reverse]; rw [find?_reverse] at H exact upperBound?_eq_find? _ H /-- The value `x` returned by `upperBound?` is greater or equal to the `cut`. -/ theorem upperBound?_ge' {t : RBNode α} (H : ∀ {x}, x ∈ ub → cut x ≠ .gt) : t.upperBound? cut ub = some x → cut x ≠ .gt := by induction t generalizing ub with \| nil => exact H \| node _ _ _ _ ihl ihr => simp [upperBound?]; split · next hv => exact ihl fun \| rfl, e => nomatch hv.symm.trans e · exact ihr H · next hv => intro \| rfl, e => cases hv.symm.trans e /-- The value `x` returned by `upperBound?` is greater or equal to the `cut`. -/ theorem upperBound?_ge {t : RBNode α} : t.upperBound? cut = some x → cut x ≠ .gt := upperBound?_ge' nofun /-- The value `x` returned by `lowerBound?` is less or equal to the `cut`. -/ theorem lowerBound?_le' {t : RBNode α} (H : ∀ {x}, x ∈ lb → cut x ≠ .lt) : t.lowerBound? cut lb = some x → cut x ≠ .lt := by rw [← reverse_reverse t, lowerBound?_reverse, Ne, ← Ordering.swap_inj] exact upperBound?_ge' fun h => by specialize H h; rwa [Ne, ← Ordering.swap_inj] at H /-- The value `x` returned by `lowerBound?` is less or equal to the `cut`. -/ theorem lowerBound?_le {t : RBNode α} : t.lowerBound? cut = some x → cut x ≠ .lt := lowerBound?_le' nofun theorem All.upperBound?_ub {t : RBNode α} (hp : t.All p) (H : ∀ {x}, ub = some x → p x) : t.upperBound? cut ub = some x → p x := by induction t generalizing ub with \| nil => exact H \| node _ _ _ _ ihl ihr => simp [upperBound?]; split · exact ihl hp.2.1 fun \| rfl => hp.1 · exact ihr hp.2.2 H · exact fun \| rfl => hp.1 theorem All.upperBound? {t : RBNode α} (hp : t.All p) : t.upperBound? cut = some x → p x := hp.upperBound?_ub nofun	.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean	442	445	theorem All.lowerBound?_lb {t : RBNode α} (hp : t.All p) (H : ∀ {x}, lb = some x → p x) : t.lowerBound? cut lb = some x → p x := by	rw [← reverse_reverse t, lowerBound?_reverse] exact All.upperBound?_ub (All.reverse.2 hp) H
/- Copyright (c) 2014 Parikshit Khanna. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro, Scott Morrison -/ import Mathlib.Data.List.Basic #align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734" /-! # Lattice structure of lists This files prove basic properties about `List.disjoint`, `List.union`, `List.inter` and `List.bagInter`, which are defined in core Lean and `Data.List.Defs`. `l₁ ∪ l₂` is the list where all elements of `l₁` have been inserted in `l₂` in order. For example, `[0, 0, 1, 2, 2, 3] ∪ [4, 3, 3, 0] = [1, 2, 4, 3, 3, 0]` `l₁ ∩ l₂` is the list of elements of `l₁` in order which are in `l₂`. For example, `[0, 0, 1, 2, 2, 3] ∪ [4, 3, 3, 0] = [0, 0, 3]` `List.bagInter l₁ l₂` is the list of elements that are in both `l₁` and `l₂`, counted with multiplicity and in the order they appear in `l₁`. As opposed to `List.inter`, `List.bagInter` copes well with multiplicity. For example, `bagInter [0, 1, 2, 3, 2, 1, 0] [1, 0, 1, 4, 3] = [0, 1, 3, 1]` -/ open Nat namespace List variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α} /-! ### `Disjoint` -/ section Disjoint @[symm] theorem Disjoint.symm (d : Disjoint l₁ l₂) : Disjoint l₂ l₁ := fun _ i₂ i₁ => d i₁ i₂ #align list.disjoint.symm List.Disjoint.symm #align list.disjoint_comm List.disjoint_comm #align list.disjoint_left List.disjoint_left #align list.disjoint_right List.disjoint_right #align list.disjoint_iff_ne List.disjoint_iff_ne #align list.disjoint_of_subset_left List.disjoint_of_subset_leftₓ #align list.disjoint_of_subset_right List.disjoint_of_subset_right #align list.disjoint_of_disjoint_cons_left List.disjoint_of_disjoint_cons_left #align list.disjoint_of_disjoint_cons_right List.disjoint_of_disjoint_cons_right #align list.disjoint_nil_left List.disjoint_nil_left #align list.disjoint_nil_right List.disjoint_nil_right #align list.singleton_disjoint List.singleton_disjointₓ #align list.disjoint_singleton List.disjoint_singleton #align list.disjoint_append_left List.disjoint_append_leftₓ #align list.disjoint_append_right List.disjoint_append_right #align list.disjoint_cons_left List.disjoint_cons_leftₓ #align list.disjoint_cons_right List.disjoint_cons_right #align list.disjoint_of_disjoint_append_left_left List.disjoint_of_disjoint_append_left_leftₓ #align list.disjoint_of_disjoint_append_left_right List.disjoint_of_disjoint_append_left_rightₓ #align list.disjoint_of_disjoint_append_right_left List.disjoint_of_disjoint_append_right_left #align list.disjoint_of_disjoint_append_right_right List.disjoint_of_disjoint_append_right_right #align list.disjoint_take_drop List.disjoint_take_dropₓ end Disjoint variable [DecidableEq α] /-! ### `union` -/ section Union #align list.nil_union List.nil_union #align list.cons_union List.cons_unionₓ #align list.mem_union List.mem_union_iff theorem mem_union_left (h : a ∈ l₁) (l₂ : List α) : a ∈ l₁ ∪ l₂ := mem_union_iff.2 (Or.inl h) #align list.mem_union_left List.mem_union_left theorem mem_union_right (l₁ : List α) (h : a ∈ l₂) : a ∈ l₁ ∪ l₂ := mem_union_iff.2 (Or.inr h) #align list.mem_union_right List.mem_union_right theorem sublist_suffix_of_union : ∀ l₁ l₂ : List α, ∃ t, t <+ l₁ ∧ t ++ l₂ = l₁ ∪ l₂ \| [], l₂ => ⟨[], by rfl, rfl⟩ \| a :: l₁, l₂ => let ⟨t, s, e⟩ := sublist_suffix_of_union l₁ l₂ if h : a ∈ l₁ ∪ l₂ then ⟨t, sublist_cons_of_sublist _ s, by simp only [e, cons_union, insert_of_mem h]⟩ else ⟨a :: t, s.cons_cons _, by simp only [cons_append, cons_union, e, insert_of_not_mem h]⟩ #align list.sublist_suffix_of_union List.sublist_suffix_of_union theorem suffix_union_right (l₁ l₂ : List α) : l₂ <:+ l₁ ∪ l₂ := (sublist_suffix_of_union l₁ l₂).imp fun _ => And.right #align list.suffix_union_right List.suffix_union_right theorem union_sublist_append (l₁ l₂ : List α) : l₁ ∪ l₂ <+ l₁ ++ l₂ := let ⟨_, s, e⟩ := sublist_suffix_of_union l₁ l₂ e ▸ (append_sublist_append_right _).2 s #align list.union_sublist_append List.union_sublist_append theorem forall_mem_union : (∀ x ∈ l₁ ∪ l₂, p x) ↔ (∀ x ∈ l₁, p x) ∧ ∀ x ∈ l₂, p x := by simp only [mem_union_iff, or_imp, forall_and] #align list.forall_mem_union List.forall_mem_union theorem forall_mem_of_forall_mem_union_left (h : ∀ x ∈ l₁ ∪ l₂, p x) : ∀ x ∈ l₁, p x := (forall_mem_union.1 h).1 #align list.forall_mem_of_forall_mem_union_left List.forall_mem_of_forall_mem_union_left theorem forall_mem_of_forall_mem_union_right (h : ∀ x ∈ l₁ ∪ l₂, p x) : ∀ x ∈ l₂, p x := (forall_mem_union.1 h).2 #align list.forall_mem_of_forall_mem_union_right List.forall_mem_of_forall_mem_union_right end Union /-! ### `inter` -/ section Inter @[simp] theorem inter_nil (l : List α) : [] ∩ l = [] := rfl #align list.inter_nil List.inter_nil @[simp] theorem inter_cons_of_mem (l₁ : List α) (h : a ∈ l₂) : (a :: l₁) ∩ l₂ = a :: l₁ ∩ l₂ := by simp [Inter.inter, List.inter, h] #align list.inter_cons_of_mem List.inter_cons_of_mem @[simp] theorem inter_cons_of_not_mem (l₁ : List α) (h : a ∉ l₂) : (a :: l₁) ∩ l₂ = l₁ ∩ l₂ := by simp [Inter.inter, List.inter, h] #align list.inter_cons_of_not_mem List.inter_cons_of_not_mem theorem mem_of_mem_inter_left : a ∈ l₁ ∩ l₂ → a ∈ l₁ := mem_of_mem_filter #align list.mem_of_mem_inter_left List.mem_of_mem_inter_left theorem mem_of_mem_inter_right (h : a ∈ l₁ ∩ l₂) : a ∈ l₂ := by simpa using of_mem_filter h #align list.mem_of_mem_inter_right List.mem_of_mem_inter_right theorem mem_inter_of_mem_of_mem (h₁ : a ∈ l₁) (h₂ : a ∈ l₂) : a ∈ l₁ ∩ l₂ := mem_filter_of_mem h₁ <\| by simpa using h₂ #align list.mem_inter_of_mem_of_mem List.mem_inter_of_mem_of_mem #align list.mem_inter List.mem_inter_iff theorem inter_subset_left {l₁ l₂ : List α} : l₁ ∩ l₂ ⊆ l₁ := filter_subset _ #align list.inter_subset_left List.inter_subset_left theorem inter_subset_right {l₁ l₂ : List α} : l₁ ∩ l₂ ⊆ l₂ := fun _ => mem_of_mem_inter_right #align list.inter_subset_right List.inter_subset_right theorem subset_inter {l l₁ l₂ : List α} (h₁ : l ⊆ l₁) (h₂ : l ⊆ l₂) : l ⊆ l₁ ∩ l₂ := fun _ h => mem_inter_iff.2 ⟨h₁ h, h₂ h⟩ #align list.subset_inter List.subset_inter	Mathlib/Data/List/Lattice.lean	167	169	theorem inter_eq_nil_iff_disjoint : l₁ ∩ l₂ = [] ↔ Disjoint l₁ l₂ := by	simp only [eq_nil_iff_forall_not_mem, mem_inter_iff, not_and] rfl
/- Copyright (c) 2022 Joseph Myers. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Joseph Myers -/ import Mathlib.Analysis.Convex.Side import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine #align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" /-! # Oriented angles. This file defines oriented angles in Euclidean affine spaces. ## Main definitions * `EuclideanGeometry.oangle`, with notation `∡`, is the oriented angle determined by three points. -/ noncomputable section open FiniteDimensional Complex open scoped Affine EuclideanGeometry Real RealInnerProductSpace ComplexConjugate namespace EuclideanGeometry variable {V : Type} {P : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] /-- A fixed choice of positive orientation of Euclidean space `ℝ²` -/ abbrev o := @Module.Oriented.positiveOrientation /-- The oriented angle at `p₂` between the line segments to `p₁` and `p₃`, modulo `2 * π`. If either of those points equals `p₂`, this is 0. See `EuclideanGeometry.angle` for the corresponding unoriented angle definition. -/ def oangle (p₁ p₂ p₃ : P) : Real.Angle := o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) #align euclidean_geometry.oangle EuclideanGeometry.oangle @[inherit_doc] scoped notation "∡" => EuclideanGeometry.oangle /-- Oriented angles are continuous when neither end point equals the middle point. -/ theorem continuousAt_oangle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) : ContinuousAt (fun y : P × P × P => ∡ y.1 y.2.1 y.2.2) x := by let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1) have hf1 : (f x).1 ≠ 0 := by simp [hx12] have hf2 : (f x).2 ≠ 0 := by simp [hx32] exact (o.continuousAt_oangle hf1 hf2).comp ((continuous_fst.vsub continuous_snd.fst).prod_mk (continuous_snd.snd.vsub continuous_snd.fst)).continuousAt #align euclidean_geometry.continuous_at_oangle EuclideanGeometry.continuousAt_oangle /-- The angle ∡AAB at a point. -/ @[simp] theorem oangle_self_left (p₁ p₂ : P) : ∡ p₁ p₁ p₂ = 0 := by simp [oangle] #align euclidean_geometry.oangle_self_left EuclideanGeometry.oangle_self_left /-- The angle ∡ABB at a point. -/ @[simp] theorem oangle_self_right (p₁ p₂ : P) : ∡ p₁ p₂ p₂ = 0 := by simp [oangle] #align euclidean_geometry.oangle_self_right EuclideanGeometry.oangle_self_right /-- The angle ∡ABA at a point. -/ @[simp] theorem oangle_self_left_right (p₁ p₂ : P) : ∡ p₁ p₂ p₁ = 0 := o.oangle_self _ #align euclidean_geometry.oangle_self_left_right EuclideanGeometry.oangle_self_left_right /-- If the angle between three points is nonzero, the first two points are not equal. -/ theorem left_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₂ := by rw [← @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h #align euclidean_geometry.left_ne_of_oangle_ne_zero EuclideanGeometry.left_ne_of_oangle_ne_zero /-- If the angle between three points is nonzero, the last two points are not equal. -/ theorem right_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₃ ≠ p₂ := by rw [← @vsub_ne_zero V]; exact o.right_ne_zero_of_oangle_ne_zero h #align euclidean_geometry.right_ne_of_oangle_ne_zero EuclideanGeometry.right_ne_of_oangle_ne_zero /-- If the angle between three points is nonzero, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₃ := by rw [← (vsub_left_injective p₂).ne_iff]; exact o.ne_of_oangle_ne_zero h #align euclidean_geometry.left_ne_right_of_oangle_ne_zero EuclideanGeometry.left_ne_right_of_oangle_ne_zero /-- If the angle between three points is `π`, the first two points are not equal. -/ theorem left_ne_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₁ ≠ p₂ := left_ne_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.left_ne_of_oangle_eq_pi EuclideanGeometry.left_ne_of_oangle_eq_pi /-- If the angle between three points is `π`, the last two points are not equal. -/ theorem right_ne_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₃ ≠ p₂ := right_ne_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.right_ne_of_oangle_eq_pi EuclideanGeometry.right_ne_of_oangle_eq_pi /-- If the angle between three points is `π`, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₁ ≠ p₃ := left_ne_right_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.left_ne_right_of_oangle_eq_pi EuclideanGeometry.left_ne_right_of_oangle_eq_pi /-- If the angle between three points is `π / 2`, the first two points are not equal. -/ theorem left_ne_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (π / 2 : ℝ)) : p₁ ≠ p₂ := left_ne_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_div_two_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.left_ne_of_oangle_eq_pi_div_two EuclideanGeometry.left_ne_of_oangle_eq_pi_div_two /-- If the angle between three points is `π / 2`, the last two points are not equal. -/ theorem right_ne_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (π / 2 : ℝ)) : p₃ ≠ p₂ := right_ne_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_div_two_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.right_ne_of_oangle_eq_pi_div_two EuclideanGeometry.right_ne_of_oangle_eq_pi_div_two /-- If the angle between three points is `π / 2`, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (π / 2 : ℝ)) : p₁ ≠ p₃ := left_ne_right_of_oangle_ne_zero (h.symm ▸ Real.Angle.pi_div_two_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.left_ne_right_of_oangle_eq_pi_div_two EuclideanGeometry.left_ne_right_of_oangle_eq_pi_div_two /-- If the angle between three points is `-π / 2`, the first two points are not equal. -/ theorem left_ne_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (-π / 2 : ℝ)) : p₁ ≠ p₂ := left_ne_of_oangle_ne_zero (h.symm ▸ Real.Angle.neg_pi_div_two_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.left_ne_of_oangle_eq_neg_pi_div_two EuclideanGeometry.left_ne_of_oangle_eq_neg_pi_div_two /-- If the angle between three points is `-π / 2`, the last two points are not equal. -/ theorem right_ne_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (-π / 2 : ℝ)) : p₃ ≠ p₂ := right_ne_of_oangle_ne_zero (h.symm ▸ Real.Angle.neg_pi_div_two_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.right_ne_of_oangle_eq_neg_pi_div_two EuclideanGeometry.right_ne_of_oangle_eq_neg_pi_div_two /-- If the angle between three points is `-π / 2`, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = (-π / 2 : ℝ)) : p₁ ≠ p₃ := left_ne_right_of_oangle_ne_zero (h.symm ▸ Real.Angle.neg_pi_div_two_ne_zero : ∡ p₁ p₂ p₃ ≠ 0) #align euclidean_geometry.left_ne_right_of_oangle_eq_neg_pi_div_two EuclideanGeometry.left_ne_right_of_oangle_eq_neg_pi_div_two /-- If the sign of the angle between three points is nonzero, the first two points are not equal. -/ theorem left_ne_of_oangle_sign_ne_zero {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0) : p₁ ≠ p₂ := left_ne_of_oangle_ne_zero (Real.Angle.sign_ne_zero_iff.1 h).1 #align euclidean_geometry.left_ne_of_oangle_sign_ne_zero EuclideanGeometry.left_ne_of_oangle_sign_ne_zero /-- If the sign of the angle between three points is nonzero, the last two points are not equal. -/ theorem right_ne_of_oangle_sign_ne_zero {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0) : p₃ ≠ p₂ := right_ne_of_oangle_ne_zero (Real.Angle.sign_ne_zero_iff.1 h).1 #align euclidean_geometry.right_ne_of_oangle_sign_ne_zero EuclideanGeometry.right_ne_of_oangle_sign_ne_zero /-- If the sign of the angle between three points is nonzero, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_sign_ne_zero {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign ≠ 0) : p₁ ≠ p₃ := left_ne_right_of_oangle_ne_zero (Real.Angle.sign_ne_zero_iff.1 h).1 #align euclidean_geometry.left_ne_right_of_oangle_sign_ne_zero EuclideanGeometry.left_ne_right_of_oangle_sign_ne_zero /-- If the sign of the angle between three points is positive, the first two points are not equal. -/ theorem left_ne_of_oangle_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : p₁ ≠ p₂ := left_ne_of_oangle_sign_ne_zero (h.symm ▸ by decide : (∡ p₁ p₂ p₃).sign ≠ 0) #align euclidean_geometry.left_ne_of_oangle_sign_eq_one EuclideanGeometry.left_ne_of_oangle_sign_eq_one /-- If the sign of the angle between three points is positive, the last two points are not equal. -/ theorem right_ne_of_oangle_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : p₃ ≠ p₂ := right_ne_of_oangle_sign_ne_zero (h.symm ▸ by decide : (∡ p₁ p₂ p₃).sign ≠ 0) #align euclidean_geometry.right_ne_of_oangle_sign_eq_one EuclideanGeometry.right_ne_of_oangle_sign_eq_one /-- If the sign of the angle between three points is positive, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : p₁ ≠ p₃ := left_ne_right_of_oangle_sign_ne_zero (h.symm ▸ by decide : (∡ p₁ p₂ p₃).sign ≠ 0) #align euclidean_geometry.left_ne_right_of_oangle_sign_eq_one EuclideanGeometry.left_ne_right_of_oangle_sign_eq_one /-- If the sign of the angle between three points is negative, the first two points are not equal. -/ theorem left_ne_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) : p₁ ≠ p₂ := left_ne_of_oangle_sign_ne_zero (h.symm ▸ by decide : (∡ p₁ p₂ p₃).sign ≠ 0) #align euclidean_geometry.left_ne_of_oangle_sign_eq_neg_one EuclideanGeometry.left_ne_of_oangle_sign_eq_neg_one /-- If the sign of the angle between three points is negative, the last two points are not equal. -/ theorem right_ne_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) : p₃ ≠ p₂ := right_ne_of_oangle_sign_ne_zero (h.symm ▸ by decide : (∡ p₁ p₂ p₃).sign ≠ 0) #align euclidean_geometry.right_ne_of_oangle_sign_eq_neg_one EuclideanGeometry.right_ne_of_oangle_sign_eq_neg_one /-- If the sign of the angle between three points is negative, the first and third points are not equal. -/ theorem left_ne_right_of_oangle_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) : p₁ ≠ p₃ := left_ne_right_of_oangle_sign_ne_zero (h.symm ▸ by decide : (∡ p₁ p₂ p₃).sign ≠ 0) #align euclidean_geometry.left_ne_right_of_oangle_sign_eq_neg_one EuclideanGeometry.left_ne_right_of_oangle_sign_eq_neg_one /-- Reversing the order of the points passed to `oangle` negates the angle. -/ theorem oangle_rev (p₁ p₂ p₃ : P) : ∡ p₃ p₂ p₁ = -∡ p₁ p₂ p₃ := o.oangle_rev _ _ #align euclidean_geometry.oangle_rev EuclideanGeometry.oangle_rev /-- Adding an angle to that with the order of the points reversed results in 0. -/ @[simp] theorem oangle_add_oangle_rev (p₁ p₂ p₃ : P) : ∡ p₁ p₂ p₃ + ∡ p₃ p₂ p₁ = 0 := o.oangle_add_oangle_rev _ _ #align euclidean_geometry.oangle_add_oangle_rev EuclideanGeometry.oangle_add_oangle_rev /-- An oriented angle is zero if and only if the angle with the order of the points reversed is zero. -/ theorem oangle_eq_zero_iff_oangle_rev_eq_zero {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = 0 ↔ ∡ p₃ p₂ p₁ = 0 := o.oangle_eq_zero_iff_oangle_rev_eq_zero #align euclidean_geometry.oangle_eq_zero_iff_oangle_rev_eq_zero EuclideanGeometry.oangle_eq_zero_iff_oangle_rev_eq_zero /-- An oriented angle is `π` if and only if the angle with the order of the points reversed is `π`. -/ theorem oangle_eq_pi_iff_oangle_rev_eq_pi {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = π ↔ ∡ p₃ p₂ p₁ = π := o.oangle_eq_pi_iff_oangle_rev_eq_pi #align euclidean_geometry.oangle_eq_pi_iff_oangle_rev_eq_pi EuclideanGeometry.oangle_eq_pi_iff_oangle_rev_eq_pi /-- An oriented angle is not zero or `π` if and only if the three points are affinely independent. -/ theorem oangle_ne_zero_and_ne_pi_iff_affineIndependent {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ ≠ 0 ∧ ∡ p₁ p₂ p₃ ≠ π ↔ AffineIndependent ℝ ![p₁, p₂, p₃] := by rw [oangle, o.oangle_ne_zero_and_ne_pi_iff_linearIndependent, affineIndependent_iff_linearIndependent_vsub ℝ _ (1 : Fin 3), ← linearIndependent_equiv (finSuccAboveEquiv (1 : Fin 3)).toEquiv] convert Iff.rfl ext i fin_cases i <;> rfl #align euclidean_geometry.oangle_ne_zero_and_ne_pi_iff_affine_independent EuclideanGeometry.oangle_ne_zero_and_ne_pi_iff_affineIndependent /-- An oriented angle is zero or `π` if and only if the three points are collinear. -/ theorem oangle_eq_zero_or_eq_pi_iff_collinear {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = 0 ∨ ∡ p₁ p₂ p₃ = π ↔ Collinear ℝ ({p₁, p₂, p₃} : Set P) := by rw [← not_iff_not, not_or, oangle_ne_zero_and_ne_pi_iff_affineIndependent, affineIndependent_iff_not_collinear_set] #align euclidean_geometry.oangle_eq_zero_or_eq_pi_iff_collinear EuclideanGeometry.oangle_eq_zero_or_eq_pi_iff_collinear /-- An oriented angle has a sign zero if and only if the three points are collinear. -/ theorem oangle_sign_eq_zero_iff_collinear {p₁ p₂ p₃ : P} : (∡ p₁ p₂ p₃).sign = 0 ↔ Collinear ℝ ({p₁, p₂, p₃} : Set P) := by rw [Real.Angle.sign_eq_zero_iff, oangle_eq_zero_or_eq_pi_iff_collinear] /-- If twice the oriented angles between two triples of points are equal, one triple is affinely independent if and only if the other is. -/ theorem affineIndependent_iff_of_two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h : (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆) : AffineIndependent ℝ ![p₁, p₂, p₃] ↔ AffineIndependent ℝ ![p₄, p₅, p₆] := by simp_rw [← oangle_ne_zero_and_ne_pi_iff_affineIndependent, ← Real.Angle.two_zsmul_ne_zero_iff, h] #align euclidean_geometry.affine_independent_iff_of_two_zsmul_oangle_eq EuclideanGeometry.affineIndependent_iff_of_two_zsmul_oangle_eq /-- If twice the oriented angles between two triples of points are equal, one triple is collinear if and only if the other is. -/ theorem collinear_iff_of_two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h : (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆) : Collinear ℝ ({p₁, p₂, p₃} : Set P) ↔ Collinear ℝ ({p₄, p₅, p₆} : Set P) := by simp_rw [← oangle_eq_zero_or_eq_pi_iff_collinear, ← Real.Angle.two_zsmul_eq_zero_iff, h] #align euclidean_geometry.collinear_iff_of_two_zsmul_oangle_eq EuclideanGeometry.collinear_iff_of_two_zsmul_oangle_eq /-- If corresponding pairs of points in two angles have the same vector span, twice those angles are equal. -/ theorem two_zsmul_oangle_of_vectorSpan_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h₁₂₄₅ : vectorSpan ℝ ({p₁, p₂} : Set P) = vectorSpan ℝ ({p₄, p₅} : Set P)) (h₃₂₆₅ : vectorSpan ℝ ({p₃, p₂} : Set P) = vectorSpan ℝ ({p₆, p₅} : Set P)) : (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆ := by simp_rw [vectorSpan_pair] at h₁₂₄₅ h₃₂₆₅ exact o.two_zsmul_oangle_of_span_eq_of_span_eq h₁₂₄₅ h₃₂₆₅ #align euclidean_geometry.two_zsmul_oangle_of_vector_span_eq EuclideanGeometry.two_zsmul_oangle_of_vectorSpan_eq /-- If the lines determined by corresponding pairs of points in two angles are parallel, twice those angles are equal. -/ theorem two_zsmul_oangle_of_parallel {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]) (h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]) : (2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₄ p₅ p₆ := by rw [AffineSubspace.affineSpan_pair_parallel_iff_vectorSpan_eq] at h₁₂₄₅ h₃₂₆₅ exact two_zsmul_oangle_of_vectorSpan_eq h₁₂₄₅ h₃₂₆₅ #align euclidean_geometry.two_zsmul_oangle_of_parallel EuclideanGeometry.two_zsmul_oangle_of_parallel /-- Given three points not equal to `p`, the angle between the first and the second at `p` plus the angle between the second and the third equals the angle between the first and the third. -/ @[simp] theorem oangle_add {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) : ∡ p₁ p p₂ + ∡ p₂ p p₃ = ∡ p₁ p p₃ := o.oangle_add (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃) #align euclidean_geometry.oangle_add EuclideanGeometry.oangle_add /-- Given three points not equal to `p`, the angle between the second and the third at `p` plus the angle between the first and the second equals the angle between the first and the third. -/ @[simp] theorem oangle_add_swap {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) : ∡ p₂ p p₃ + ∡ p₁ p p₂ = ∡ p₁ p p₃ := o.oangle_add_swap (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃) #align euclidean_geometry.oangle_add_swap EuclideanGeometry.oangle_add_swap /-- Given three points not equal to `p`, the angle between the first and the third at `p` minus the angle between the first and the second equals the angle between the second and the third. -/ @[simp] theorem oangle_sub_left {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) : ∡ p₁ p p₃ - ∡ p₁ p p₂ = ∡ p₂ p p₃ := o.oangle_sub_left (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃) #align euclidean_geometry.oangle_sub_left EuclideanGeometry.oangle_sub_left /-- Given three points not equal to `p`, the angle between the first and the third at `p` minus the angle between the second and the third equals the angle between the first and the second. -/ @[simp] theorem oangle_sub_right {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) : ∡ p₁ p p₃ - ∡ p₂ p p₃ = ∡ p₁ p p₂ := o.oangle_sub_right (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃) #align euclidean_geometry.oangle_sub_right EuclideanGeometry.oangle_sub_right /-- Given three points not equal to `p`, adding the angles between them at `p` in cyclic order results in 0. -/ @[simp] theorem oangle_add_cyc3 {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) : ∡ p₁ p p₂ + ∡ p₂ p p₃ + ∡ p₃ p p₁ = 0 := o.oangle_add_cyc3 (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃) #align euclidean_geometry.oangle_add_cyc3 EuclideanGeometry.oangle_add_cyc3 /-- Pons asinorum, oriented angle-at-point form. -/ theorem oangle_eq_oangle_of_dist_eq {p₁ p₂ p₃ : P} (h : dist p₁ p₂ = dist p₁ p₃) : ∡ p₁ p₂ p₃ = ∡ p₂ p₃ p₁ := by simp_rw [dist_eq_norm_vsub V] at h rw [oangle, oangle, ← vsub_sub_vsub_cancel_left p₃ p₂ p₁, ← vsub_sub_vsub_cancel_left p₂ p₃ p₁, o.oangle_sub_eq_oangle_sub_rev_of_norm_eq h] #align euclidean_geometry.oangle_eq_oangle_of_dist_eq EuclideanGeometry.oangle_eq_oangle_of_dist_eq /-- The angle at the apex of an isosceles triangle is `π` minus twice a base angle, oriented angle-at-point form. -/ theorem oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq {p₁ p₂ p₃ : P} (hn : p₂ ≠ p₃) (h : dist p₁ p₂ = dist p₁ p₃) : ∡ p₃ p₁ p₂ = π - (2 : ℤ) • ∡ p₁ p₂ p₃ := by simp_rw [dist_eq_norm_vsub V] at h rw [oangle, oangle] convert o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq _ h using 1 · rw [← neg_vsub_eq_vsub_rev p₁ p₃, ← neg_vsub_eq_vsub_rev p₁ p₂, o.oangle_neg_neg] · rw [← o.oangle_sub_eq_oangle_sub_rev_of_norm_eq h]; simp · simpa using hn #align euclidean_geometry.oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq EuclideanGeometry.oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq /-- A base angle of an isosceles triangle is acute, oriented angle-at-point form. -/ theorem abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq {p₁ p₂ p₃ : P} (h : dist p₁ p₂ = dist p₁ p₃) : \|(∡ p₁ p₂ p₃).toReal\| < π / 2 := by simp_rw [dist_eq_norm_vsub V] at h rw [oangle, ← vsub_sub_vsub_cancel_left p₃ p₂ p₁] exact o.abs_oangle_sub_right_toReal_lt_pi_div_two h #align euclidean_geometry.abs_oangle_right_to_real_lt_pi_div_two_of_dist_eq EuclideanGeometry.abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq /-- A base angle of an isosceles triangle is acute, oriented angle-at-point form. -/ theorem abs_oangle_left_toReal_lt_pi_div_two_of_dist_eq {p₁ p₂ p₃ : P} (h : dist p₁ p₂ = dist p₁ p₃) : \|(∡ p₂ p₃ p₁).toReal\| < π / 2 := oangle_eq_oangle_of_dist_eq h ▸ abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq h #align euclidean_geometry.abs_oangle_left_to_real_lt_pi_div_two_of_dist_eq EuclideanGeometry.abs_oangle_left_toReal_lt_pi_div_two_of_dist_eq /-- The cosine of the oriented angle at `p` between two points not equal to `p` equals that of the unoriented angle. -/ theorem cos_oangle_eq_cos_angle {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) : Real.Angle.cos (∡ p₁ p p₂) = Real.cos (∠ p₁ p p₂) := o.cos_oangle_eq_cos_angle (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) #align euclidean_geometry.cos_oangle_eq_cos_angle EuclideanGeometry.cos_oangle_eq_cos_angle /-- The oriented angle at `p` between two points not equal to `p` is plus or minus the unoriented angle. -/ theorem oangle_eq_angle_or_eq_neg_angle {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) : ∡ p₁ p p₂ = ∠ p₁ p p₂ ∨ ∡ p₁ p p₂ = -∠ p₁ p p₂ := o.oangle_eq_angle_or_eq_neg_angle (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) #align euclidean_geometry.oangle_eq_angle_or_eq_neg_angle EuclideanGeometry.oangle_eq_angle_or_eq_neg_angle /-- The unoriented angle at `p` between two points not equal to `p` is the absolute value of the oriented angle. -/ theorem angle_eq_abs_oangle_toReal {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) : ∠ p₁ p p₂ = \|(∡ p₁ p p₂).toReal\| := o.angle_eq_abs_oangle_toReal (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) #align euclidean_geometry.angle_eq_abs_oangle_to_real EuclideanGeometry.angle_eq_abs_oangle_toReal /-- If the sign of the oriented angle at `p` between two points is zero, either one of the points equals `p` or the unoriented angle is 0 or π. -/ theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {p p₁ p₂ : P} (h : (∡ p₁ p p₂).sign = 0) : p₁ = p ∨ p₂ = p ∨ ∠ p₁ p p₂ = 0 ∨ ∠ p₁ p p₂ = π := by convert o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero h <;> simp #align euclidean_geometry.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero EuclideanGeometry.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero /-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are equal, then the oriented angles are equal (even in degenerate cases). -/ theorem oangle_eq_of_angle_eq_of_sign_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (h : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆) (hs : (∡ p₁ p₂ p₃).sign = (∡ p₄ p₅ p₆).sign) : ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆ := o.oangle_eq_of_angle_eq_of_sign_eq h hs #align euclidean_geometry.oangle_eq_of_angle_eq_of_sign_eq EuclideanGeometry.oangle_eq_of_angle_eq_of_sign_eq /-- If the signs of two nondegenerate oriented angles between points are equal, the oriented angles are equal if and only if the unoriented angles are equal. -/ theorem angle_eq_iff_oangle_eq_of_sign_eq {p₁ p₂ p₃ p₄ p₅ p₆ : P} (hp₁ : p₁ ≠ p₂) (hp₃ : p₃ ≠ p₂) (hp₄ : p₄ ≠ p₅) (hp₆ : p₆ ≠ p₅) (hs : (∡ p₁ p₂ p₃).sign = (∡ p₄ p₅ p₆).sign) : ∠ p₁ p₂ p₃ = ∠ p₄ p₅ p₆ ↔ ∡ p₁ p₂ p₃ = ∡ p₄ p₅ p₆ := o.angle_eq_iff_oangle_eq_of_sign_eq (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₃) (vsub_ne_zero.2 hp₄) (vsub_ne_zero.2 hp₆) hs #align euclidean_geometry.angle_eq_iff_oangle_eq_of_sign_eq EuclideanGeometry.angle_eq_iff_oangle_eq_of_sign_eq /-- The oriented angle between three points equals the unoriented angle if the sign is positive. -/ theorem oangle_eq_angle_of_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) : ∡ p₁ p₂ p₃ = ∠ p₁ p₂ p₃ := o.oangle_eq_angle_of_sign_eq_one h #align euclidean_geometry.oangle_eq_angle_of_sign_eq_one EuclideanGeometry.oangle_eq_angle_of_sign_eq_one /-- The oriented angle between three points equals minus the unoriented angle if the sign is negative. -/ theorem oangle_eq_neg_angle_of_sign_eq_neg_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = -1) : ∡ p₁ p₂ p₃ = -∠ p₁ p₂ p₃ := o.oangle_eq_neg_angle_of_sign_eq_neg_one h #align euclidean_geometry.oangle_eq_neg_angle_of_sign_eq_neg_one EuclideanGeometry.oangle_eq_neg_angle_of_sign_eq_neg_one /-- The unoriented angle at `p` between two points not equal to `p` is zero if and only if the unoriented angle is zero. -/ theorem oangle_eq_zero_iff_angle_eq_zero {p p₁ p₂ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) : ∡ p₁ p p₂ = 0 ↔ ∠ p₁ p p₂ = 0 := o.oangle_eq_zero_iff_angle_eq_zero (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) #align euclidean_geometry.oangle_eq_zero_iff_angle_eq_zero EuclideanGeometry.oangle_eq_zero_iff_angle_eq_zero /-- The oriented angle between three points is `π` if and only if the unoriented angle is `π`. -/ theorem oangle_eq_pi_iff_angle_eq_pi {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = π ↔ ∠ p₁ p₂ p₃ = π := o.oangle_eq_pi_iff_angle_eq_pi #align euclidean_geometry.oangle_eq_pi_iff_angle_eq_pi EuclideanGeometry.oangle_eq_pi_iff_angle_eq_pi /-- If the oriented angle between three points is `π / 2`, so is the unoriented angle. -/ theorem angle_eq_pi_div_two_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : ∠ p₁ p₂ p₃ = π / 2 := by rw [angle, ← InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two] exact o.inner_eq_zero_of_oangle_eq_pi_div_two h #align euclidean_geometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_two EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_two /-- If the oriented angle between three points is `π / 2`, so is the unoriented angle (reversed). -/ theorem angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : ∠ p₃ p₂ p₁ = π / 2 := by rw [angle_comm] exact angle_eq_pi_div_two_of_oangle_eq_pi_div_two h #align euclidean_geometry.angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two EuclideanGeometry.angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two /-- If the oriented angle between three points is `-π / 2`, the unoriented angle is `π / 2`. -/ theorem angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(-π / 2)) : ∠ p₁ p₂ p₃ = π / 2 := by rw [angle, ← InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two] exact o.inner_eq_zero_of_oangle_eq_neg_pi_div_two h #align euclidean_geometry.angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two /-- If the oriented angle between three points is `-π / 2`, the unoriented angle (reversed) is `π / 2`. -/ theorem angle_rev_eq_pi_div_two_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(-π / 2)) : ∠ p₃ p₂ p₁ = π / 2 := by rw [angle_comm] exact angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two h #align euclidean_geometry.angle_rev_eq_pi_div_two_of_oangle_eq_neg_pi_div_two EuclideanGeometry.angle_rev_eq_pi_div_two_of_oangle_eq_neg_pi_div_two /-- Swapping the first and second points in an oriented angle negates the sign of that angle. -/ theorem oangle_swap₁₂_sign (p₁ p₂ p₃ : P) : -(∡ p₁ p₂ p₃).sign = (∡ p₂ p₁ p₃).sign := by rw [eq_comm, oangle, oangle, ← o.oangle_neg_neg, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev, ← vsub_sub_vsub_cancel_left p₁ p₃ p₂, ← neg_vsub_eq_vsub_rev p₃ p₂, sub_eq_add_neg, neg_vsub_eq_vsub_rev p₂ p₁, add_comm, ← @neg_one_smul ℝ] nth_rw 2 [← one_smul ℝ (p₁ -ᵥ p₂)] rw [o.oangle_sign_smul_add_smul_right] simp #align euclidean_geometry.oangle_swap₁₂_sign EuclideanGeometry.oangle_swap₁₂_sign /-- Swapping the first and third points in an oriented angle negates the sign of that angle. -/ theorem oangle_swap₁₃_sign (p₁ p₂ p₃ : P) : -(∡ p₁ p₂ p₃).sign = (∡ p₃ p₂ p₁).sign := by rw [oangle_rev, Real.Angle.sign_neg, neg_neg] #align euclidean_geometry.oangle_swap₁₃_sign EuclideanGeometry.oangle_swap₁₃_sign /-- Swapping the second and third points in an oriented angle negates the sign of that angle. -/ theorem oangle_swap₂₃_sign (p₁ p₂ p₃ : P) : -(∡ p₁ p₂ p₃).sign = (∡ p₁ p₃ p₂).sign := by rw [oangle_swap₁₃_sign, ← oangle_swap₁₂_sign, oangle_swap₁₃_sign] #align euclidean_geometry.oangle_swap₂₃_sign EuclideanGeometry.oangle_swap₂₃_sign /-- Rotating the points in an oriented angle does not change the sign of that angle. -/ theorem oangle_rotate_sign (p₁ p₂ p₃ : P) : (∡ p₂ p₃ p₁).sign = (∡ p₁ p₂ p₃).sign := by rw [← oangle_swap₁₂_sign, oangle_swap₁₃_sign] #align euclidean_geometry.oangle_rotate_sign EuclideanGeometry.oangle_rotate_sign /-- The oriented angle between three points is π if and only if the second point is strictly between the other two. -/ theorem oangle_eq_pi_iff_sbtw {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = π ↔ Sbtw ℝ p₁ p₂ p₃ := by rw [oangle_eq_pi_iff_angle_eq_pi, angle_eq_pi_iff_sbtw] #align euclidean_geometry.oangle_eq_pi_iff_sbtw EuclideanGeometry.oangle_eq_pi_iff_sbtw /-- If the second of three points is strictly between the other two, the oriented angle at that point is π. -/ theorem _root_.Sbtw.oangle₁₂₃_eq_pi {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∡ p₁ p₂ p₃ = π := oangle_eq_pi_iff_sbtw.2 h #align sbtw.oangle₁₂₃_eq_pi Sbtw.oangle₁₂₃_eq_pi /-- If the second of three points is strictly between the other two, the oriented angle at that point (reversed) is π. -/	Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean	488	489	theorem _root_.Sbtw.oangle₃₂₁_eq_pi {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∡ p₃ p₂ p₁ = π := by	rw [oangle_eq_pi_iff_oangle_rev_eq_pi, ← h.oangle₁₂₃_eq_pi]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot -/ import Mathlib.Topology.Order #align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d" /-! # Specific classes of maps between topological spaces This file introduces the following properties of a map `f : X → Y` between topological spaces: * `IsOpenMap f` means the image of an open set under `f` is open. * `IsClosedMap f` means the image of a closed set under `f` is closed. (Open and closed maps need not be continuous.) * `Inducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`. These behave like embeddings except they need not be injective. Instead, points of `X` which are identified by `f` are also inseparable in the topology on `X`. * `Embedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with a subspace of `Y`. * `OpenEmbedding f` means `f` is an embedding with open image, so it identifies `X` with an open subspace of `Y`. Equivalently, `f` is an embedding and an open map. * `ClosedEmbedding f` similarly means `f` is an embedding with closed image, so it identifies `X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map. * `QuotientMap f` is the dual condition to `Embedding f`: `f` is surjective and the topology on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies `Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps. ## References * <https://en.wikipedia.org/wiki/Open_and_closed_maps> * <https://en.wikipedia.org/wiki/Embedding#General_topology> * <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map> ## Tags open map, closed map, embedding, quotient map, identification map -/ open Set Filter Function open TopologicalSpace Topology Filter variable {X : Type} {Y : Type} {Z : Type} {ι : Type} {f : X → Y} {g : Y → Z} section Inducing variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] theorem inducing_induced (f : X → Y) : @Inducing X Y (TopologicalSpace.induced f ‹_›) _ f := @Inducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl theorem inducing_id : Inducing (@id X) := ⟨induced_id.symm⟩ #align inducing_id inducing_id protected theorem Inducing.comp (hg : Inducing g) (hf : Inducing f) : Inducing (g ∘ f) := ⟨by rw [hf.induced, hg.induced, induced_compose]⟩ #align inducing.comp Inducing.comp theorem Inducing.of_comp_iff (hg : Inducing g) : Inducing (g ∘ f) ↔ Inducing f := by refine ⟨fun h ↦ ?_, hg.comp⟩ rw [inducing_iff, hg.induced, induced_compose, h.induced] #align inducing.inducing_iff Inducing.of_comp_iff theorem inducing_of_inducing_compose (hf : Continuous f) (hg : Continuous g) (hgf : Inducing (g ∘ f)) : Inducing f := ⟨le_antisymm (by rwa [← continuous_iff_le_induced]) (by rw [hgf.induced, ← induced_compose] exact induced_mono hg.le_induced)⟩ #align inducing_of_inducing_compose inducing_of_inducing_compose theorem inducing_iff_nhds : Inducing f ↔ ∀ x, 𝓝 x = comap f (𝓝 (f x)) := (inducing_iff _).trans (induced_iff_nhds_eq f) #align inducing_iff_nhds inducing_iff_nhds namespace Inducing theorem nhds_eq_comap (hf : Inducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <\| f x) := inducing_iff_nhds.1 hf #align inducing.nhds_eq_comap Inducing.nhds_eq_comap theorem basis_nhds {p : ι → Prop} {s : ι → Set Y} (hf : Inducing f) {x : X} (h_basis : (𝓝 (f x)).HasBasis p s) : (𝓝 x).HasBasis p (preimage f ∘ s) := hf.nhds_eq_comap x ▸ h_basis.comap f theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) : 𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image] #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap theorem map_nhds_eq (hf : Inducing f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x := hf.induced.symm ▸ map_nhds_induced_eq x #align inducing.map_nhds_eq Inducing.map_nhds_eq theorem map_nhds_of_mem (hf : Inducing f) (x : X) (h : range f ∈ 𝓝 (f x)) : (𝓝 x).map f = 𝓝 (f x) := hf.induced.symm ▸ map_nhds_induced_of_mem h #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem -- Porting note (#10756): new lemma theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} : MapClusterPt (f x) l f ↔ ClusterPt x l := by delta MapClusterPt ClusterPt rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff] theorem image_mem_nhdsWithin (hf : Inducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x) : f '' s ∈ 𝓝[range f] f x := hf.map_nhds_eq x ▸ image_mem_map hs #align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin theorem tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : Inducing g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := by rw [hg.nhds_eq_comap, tendsto_comap_iff] #align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff theorem continuousAt_iff (hg : Inducing g) {x : X} : ContinuousAt f x ↔ ContinuousAt (g ∘ f) x := hg.tendsto_nhds_iff #align inducing.continuous_at_iff Inducing.continuousAt_iff theorem continuous_iff (hg : Inducing g) : Continuous f ↔ Continuous (g ∘ f) := by simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff] #align inducing.continuous_iff Inducing.continuous_iff theorem continuousAt_iff' (hf : Inducing f) {x : X} (h : range f ∈ 𝓝 (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp] #align inducing.continuous_at_iff' Inducing.continuousAt_iff' protected theorem continuous (hf : Inducing f) : Continuous f := hf.continuous_iff.mp continuous_id #align inducing.continuous Inducing.continuous theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set X) : closure s = f ⁻¹' closure (f '' s) := by ext x rw [Set.mem_preimage, ← closure_induced, hf.induced] #align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_image theorem isClosed_iff (hf : Inducing f) {s : Set X} : IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.induced, isClosed_induced_iff] #align inducing.is_closed_iff Inducing.isClosed_iff theorem isClosed_iff' (hf : Inducing f) {s : Set X} : IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.induced, isClosed_induced_iff'] #align inducing.is_closed_iff' Inducing.isClosed_iff' theorem isClosed_preimage (h : Inducing f) (s : Set Y) (hs : IsClosed s) : IsClosed (f ⁻¹' s) := (isClosed_iff h).mpr ⟨s, hs, rfl⟩ #align inducing.is_closed_preimage Inducing.isClosed_preimage theorem isOpen_iff (hf : Inducing f) {s : Set X} : IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.induced, isOpen_induced_iff] #align inducing.is_open_iff Inducing.isOpen_iff theorem setOf_isOpen (hf : Inducing f) : {s : Set X \| IsOpen s} = preimage f '' {t \| IsOpen t} := Set.ext fun _ ↦ hf.isOpen_iff theorem dense_iff (hf : Inducing f) {s : Set X} : Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage] #align inducing.dense_iff Inducing.dense_iff theorem of_subsingleton [Subsingleton X] (f : X → Y) : Inducing f := ⟨Subsingleton.elim _ _⟩ end Inducing end Inducing section Embedding theorem Function.Injective.embedding_induced [t : TopologicalSpace Y] (hf : Injective f) : @_root_.Embedding X Y (t.induced f) t f := @_root_.Embedding.mk X Y (t.induced f) t _ (inducing_induced f) hf #align function.injective.embedding_induced Function.Injective.embedding_induced variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] theorem Embedding.mk' (f : X → Y) (inj : Injective f) (induced : ∀ x, comap f (𝓝 (f x)) = 𝓝 x) : Embedding f := ⟨inducing_iff_nhds.2 fun x => (induced x).symm, inj⟩ #align embedding.mk' Embedding.mk' theorem embedding_id : Embedding (@id X) := ⟨inducing_id, fun _ _ h => h⟩ #align embedding_id embedding_id protected theorem Embedding.comp (hg : Embedding g) (hf : Embedding f) : Embedding (g ∘ f) := { hg.toInducing.comp hf.toInducing with inj := fun _ _ h => hf.inj <\| hg.inj h } #align embedding.comp Embedding.comp theorem Embedding.of_comp_iff (hg : Embedding g) : Embedding (g ∘ f) ↔ Embedding f := by simp_rw [embedding_iff, hg.toInducing.of_comp_iff, hg.inj.of_comp_iff f] theorem embedding_of_embedding_compose (hf : Continuous f) (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f := { induced := (inducing_of_inducing_compose hf hg hgf.toInducing).induced inj := fun x₁ x₂ h => hgf.inj <\| by simp [h, (· ∘ ·)] } #align embedding_of_embedding_compose embedding_of_embedding_compose protected theorem Function.LeftInverse.embedding {f : X → Y} {g : Y → X} (h : LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : _root_.Embedding g := embedding_of_embedding_compose hg hf <\| h.comp_eq_id.symm ▸ embedding_id #align function.left_inverse.embedding Function.LeftInverse.embedding theorem Embedding.map_nhds_eq (hf : Embedding f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x := hf.1.map_nhds_eq x #align embedding.map_nhds_eq Embedding.map_nhds_eq theorem Embedding.map_nhds_of_mem (hf : Embedding f) (x : X) (h : range f ∈ 𝓝 (f x)) : (𝓝 x).map f = 𝓝 (f x) := hf.1.map_nhds_of_mem x h #align embedding.map_nhds_of_mem Embedding.map_nhds_of_mem theorem Embedding.tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : Embedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := hg.toInducing.tendsto_nhds_iff #align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iff theorem Embedding.continuous_iff (hg : Embedding g) : Continuous f ↔ Continuous (g ∘ f) := Inducing.continuous_iff hg.1 #align embedding.continuous_iff Embedding.continuous_iff theorem Embedding.continuous (hf : Embedding f) : Continuous f := Inducing.continuous hf.1 #align embedding.continuous Embedding.continuous theorem Embedding.closure_eq_preimage_closure_image (hf : Embedding f) (s : Set X) : closure s = f ⁻¹' closure (f '' s) := hf.1.closure_eq_preimage_closure_image s #align embedding.closure_eq_preimage_closure_image Embedding.closure_eq_preimage_closure_image /-- The topology induced under an inclusion `f : X → Y` from a discrete topological space `Y` is the discrete topology on `X`. See also `DiscreteTopology.of_continuous_injective`. -/ theorem Embedding.discreteTopology [DiscreteTopology Y] (hf : Embedding f) : DiscreteTopology X := .of_continuous_injective hf.continuous hf.inj #align embedding.discrete_topology Embedding.discreteTopology theorem Embedding.of_subsingleton [Subsingleton X] (f : X → Y) : Embedding f := ⟨.of_subsingleton f, f.injective_of_subsingleton⟩ end Embedding section QuotientMap variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] theorem quotientMap_iff : QuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsOpen s ↔ IsOpen (f ⁻¹' s) := and_congr Iff.rfl TopologicalSpace.ext_iff #align quotient_map_iff quotientMap_iff theorem quotientMap_iff_closed : QuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsClosed s ↔ IsClosed (f ⁻¹' s) := quotientMap_iff.trans <\| Iff.rfl.and <\| compl_surjective.forall.trans <\| by simp only [isOpen_compl_iff, preimage_compl] #align quotient_map_iff_closed quotientMap_iff_closed namespace QuotientMap protected theorem id : QuotientMap (@id X) := ⟨fun x => ⟨x, rfl⟩, coinduced_id.symm⟩ #align quotient_map.id QuotientMap.id protected theorem comp (hg : QuotientMap g) (hf : QuotientMap f) : QuotientMap (g ∘ f) := ⟨hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]⟩ #align quotient_map.comp QuotientMap.comp protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g) (hgf : QuotientMap (g ∘ f)) : QuotientMap g := ⟨hgf.1.of_comp, le_antisymm (by rw [hgf.right, ← coinduced_compose]; exact coinduced_mono hf.coinduced_le) hg.coinduced_le⟩ #align quotient_map.of_quotient_map_compose QuotientMap.of_quotientMap_compose theorem of_inverse {g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) : QuotientMap g := QuotientMap.of_quotientMap_compose hf hg <\| h.comp_eq_id.symm ▸ QuotientMap.id #align quotient_map.of_inverse QuotientMap.of_inverse protected theorem continuous_iff (hf : QuotientMap f) : Continuous g ↔ Continuous (g ∘ f) := by rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose] #align quotient_map.continuous_iff QuotientMap.continuous_iff protected theorem continuous (hf : QuotientMap f) : Continuous f := hf.continuous_iff.mp continuous_id #align quotient_map.continuous QuotientMap.continuous protected theorem surjective (hf : QuotientMap f) : Surjective f := hf.1 #align quotient_map.surjective QuotientMap.surjective protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set Y} : IsOpen (f ⁻¹' s) ↔ IsOpen s := ((quotientMap_iff.1 hf).2 s).symm #align quotient_map.is_open_preimage QuotientMap.isOpen_preimage protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set Y} : IsClosed (f ⁻¹' s) ↔ IsClosed s := ((quotientMap_iff_closed.1 hf).2 s).symm #align quotient_map.is_closed_preimage QuotientMap.isClosed_preimage end QuotientMap end QuotientMap section OpenMap variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] namespace IsOpenMap protected theorem id : IsOpenMap (@id X) := fun s hs => by rwa [image_id] #align is_open_map.id IsOpenMap.id protected theorem comp (hg : IsOpenMap g) (hf : IsOpenMap f) : IsOpenMap (g ∘ f) := fun s hs => by rw [image_comp]; exact hg _ (hf _ hs) #align is_open_map.comp IsOpenMap.comp theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by rw [← image_univ] exact hf _ isOpen_univ #align is_open_map.is_open_range IsOpenMap.isOpen_range theorem image_mem_nhds (hf : IsOpenMap f) {x : X} {s : Set X} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) := let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx mem_of_superset (IsOpen.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts) #align is_open_map.image_mem_nhds IsOpenMap.image_mem_nhds theorem range_mem_nhds (hf : IsOpenMap f) (x : X) : range f ∈ 𝓝 (f x) := hf.isOpen_range.mem_nhds <\| mem_range_self _ #align is_open_map.range_mem_nhds IsOpenMap.range_mem_nhds theorem mapsTo_interior (hf : IsOpenMap f) {s : Set X} {t : Set Y} (h : MapsTo f s t) : MapsTo f (interior s) (interior t) := mapsTo'.2 <\| interior_maximal (h.mono interior_subset Subset.rfl).image_subset (hf _ isOpen_interior) #align is_open_map.maps_to_interior IsOpenMap.mapsTo_interior theorem image_interior_subset (hf : IsOpenMap f) (s : Set X) : f '' interior s ⊆ interior (f '' s) := (hf.mapsTo_interior (mapsTo_image f s)).image_subset #align is_open_map.image_interior_subset IsOpenMap.image_interior_subset theorem nhds_le (hf : IsOpenMap f) (x : X) : 𝓝 (f x) ≤ (𝓝 x).map f := le_map fun _ => hf.image_mem_nhds #align is_open_map.nhds_le IsOpenMap.nhds_le theorem of_nhds_le (hf : ∀ x, 𝓝 (f x) ≤ map f (𝓝 x)) : IsOpenMap f := fun _s hs => isOpen_iff_mem_nhds.2 fun _y ⟨_x, hxs, hxy⟩ => hxy ▸ hf _ (image_mem_map <\| hs.mem_nhds hxs) #align is_open_map.of_nhds_le IsOpenMap.of_nhds_le theorem of_sections (h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f := of_nhds_le fun x => let ⟨g, hgc, hgx, hgf⟩ := h x calc 𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id] _ ≤ map f (𝓝 (g (f x))) := map_mono hgc _ = map f (𝓝 x) := by rw [hgx] #align is_open_map.of_sections IsOpenMap.of_sections theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f') (r_inv : RightInverse f f') : IsOpenMap f := of_sections fun _ => ⟨f', h.continuousAt, r_inv _, l_inv⟩ #align is_open_map.of_inverse IsOpenMap.of_inverse /-- A continuous surjective open map is a quotient map. -/ theorem to_quotientMap (open_map : IsOpenMap f) (cont : Continuous f) (surj : Surjective f) : QuotientMap f := quotientMap_iff.2 ⟨surj, fun s => ⟨fun h => h.preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩ #align is_open_map.to_quotient_map IsOpenMap.to_quotientMap theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set Y} : interior (f ⁻¹' s) ⊆ f ⁻¹' interior s := hf.mapsTo_interior (mapsTo_preimage _ _) #align is_open_map.interior_preimage_subset_preimage_interior IsOpenMap.interior_preimage_subset_preimage_interior theorem preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f) (s : Set Y) : f ⁻¹' interior s = interior (f ⁻¹' s) := Subset.antisymm (preimage_interior_subset_interior_preimage hf₂) (interior_preimage_subset_preimage_interior hf₁) #align is_open_map.preimage_interior_eq_interior_preimage IsOpenMap.preimage_interior_eq_interior_preimage theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} : f ⁻¹' closure s ⊆ closure (f ⁻¹' s) := by rw [← compl_subset_compl] simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior] #align is_open_map.preimage_closure_subset_closure_preimage IsOpenMap.preimage_closure_subset_closure_preimage theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) : f ⁻¹' closure s = closure (f ⁻¹' s) := hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s) #align is_open_map.preimage_closure_eq_closure_preimage IsOpenMap.preimage_closure_eq_closure_preimage theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set Y} : f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) := by simpa only [frontier_eq_closure_inter_closure, preimage_inter] using inter_subset_inter hf.preimage_closure_subset_closure_preimage hf.preimage_closure_subset_closure_preimage #align is_open_map.preimage_frontier_subset_frontier_preimage IsOpenMap.preimage_frontier_subset_frontier_preimage theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) : f ⁻¹' frontier s = frontier (f ⁻¹' s) := by simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl, hf.preimage_closure_eq_closure_preimage hfc] #align is_open_map.preimage_frontier_eq_frontier_preimage IsOpenMap.preimage_frontier_eq_frontier_preimage theorem of_isEmpty [h : IsEmpty X] (f : X → Y) : IsOpenMap f := of_nhds_le h.elim end IsOpenMap theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ x : X, 𝓝 (f x) ≤ (𝓝 x).map f := ⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩ #align is_open_map_iff_nhds_le isOpenMap_iff_nhds_le theorem isOpenMap_iff_interior : IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) := ⟨IsOpenMap.image_interior_subset, fun hs u hu => subset_interior_iff_isOpen.mp <\| calc f '' u = f '' interior u := by rw [hu.interior_eq] _ ⊆ interior (f '' u) := hs u⟩ #align is_open_map_iff_interior isOpenMap_iff_interior /-- An inducing map with an open range is an open map. -/ protected theorem Inducing.isOpenMap (hi : Inducing f) (ho : IsOpen (range f)) : IsOpenMap f := IsOpenMap.of_nhds_le fun _ => (hi.map_nhds_of_mem _ <\| IsOpen.mem_nhds ho <\| mem_range_self _).ge #align inducing.is_open_map Inducing.isOpenMap end OpenMap section IsClosedMap variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] namespace IsClosedMap open Function protected theorem id : IsClosedMap (@id X) := fun s hs => by rwa [image_id] #align is_closed_map.id IsClosedMap.id protected theorem comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f) := by intro s hs rw [image_comp] exact hg _ (hf _ hs) #align is_closed_map.comp IsClosedMap.comp theorem closure_image_subset (hf : IsClosedMap f) (s : Set X) : closure (f '' s) ⊆ f '' closure s := closure_minimal (image_subset _ subset_closure) (hf _ isClosed_closure) #align is_closed_map.closure_image_subset IsClosedMap.closure_image_subset theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f') (r_inv : RightInverse f f') : IsClosedMap f := fun s hs => by rw [image_eq_preimage_of_inverse r_inv l_inv] exact hs.preimage h #align is_closed_map.of_inverse IsClosedMap.of_inverse theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) : IsClosedMap f := by intro s hs; rcases eq_empty_or_nonempty s with h2s \| h2s · simp_rw [h2s, image_empty, isClosed_empty] · exact h s hs h2s #align is_closed_map.of_nonempty IsClosedMap.of_nonempty theorem isClosed_range (hf : IsClosedMap f) : IsClosed (range f) := @image_univ _ _ f ▸ hf _ isClosed_univ #align is_closed_map.closed_range IsClosedMap.isClosed_range @[deprecated (since := "2024-03-17")] alias closed_range := isClosed_range theorem to_quotientMap (hcl : IsClosedMap f) (hcont : Continuous f) (hsurj : Surjective f) : QuotientMap f := quotientMap_iff_closed.2 ⟨hsurj, fun s => ⟨fun hs => hs.preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩ #align is_closed_map.to_quotient_map IsClosedMap.to_quotientMap end IsClosedMap theorem Inducing.isClosedMap (hf : Inducing f) (h : IsClosed (range f)) : IsClosedMap f := by intro s hs rcases hf.isClosed_iff.1 hs with ⟨t, ht, rfl⟩ rw [image_preimage_eq_inter_range] exact ht.inter h #align inducing.is_closed_map Inducing.isClosedMap theorem isClosedMap_iff_closure_image : IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s := ⟨IsClosedMap.closure_image_subset, fun hs c hc => isClosed_of_closure_subset <\| calc closure (f '' c) ⊆ f '' closure c := hs c _ = f '' c := by rw [hc.closure_eq]⟩ #align is_closed_map_iff_closure_image isClosedMap_iff_closure_image /-- A map `f : X → Y` is closed if and only if for all sets `s`, any cluster point of `f '' s` is the image by `f` of some cluster point of `s`. If you require this for all filters instead of just principal filters, and also that `f` is continuous, you get the notion of proper map. See `isProperMap_iff_clusterPt`. -/ theorem isClosedMap_iff_clusterPt : IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) := by simp [MapClusterPt, isClosedMap_iff_closure_image, subset_def, mem_closure_iff_clusterPt, and_comm] theorem IsClosedMap.closure_image_eq_of_continuous (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X) : closure (f '' s) = f '' closure s := subset_antisymm (f_closed.closure_image_subset s) (image_closure_subset_closure_image f_cont) theorem IsClosedMap.lift'_closure_map_eq (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) : (map f F).lift' closure = map f (F.lift' closure) := by rw [map_lift'_eq2 (monotone_closure Y), map_lift'_eq (monotone_closure X)] congr ext s : 1 exact f_closed.closure_image_eq_of_continuous f_cont s theorem IsClosedMap.mapClusterPt_iff_lift'_closure {F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y} : MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot := by rw [MapClusterPt, clusterPt_iff_lift'_closure', f_closed.lift'_closure_map_eq f_cont, ← comap_principal, ← map_neBot_iff f, Filter.push_pull, principal_singleton] end IsClosedMap section OpenEmbedding variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] theorem OpenEmbedding.isOpenMap (hf : OpenEmbedding f) : IsOpenMap f := hf.toEmbedding.toInducing.isOpenMap hf.isOpen_range #align open_embedding.is_open_map OpenEmbedding.isOpenMap theorem OpenEmbedding.map_nhds_eq (hf : OpenEmbedding f) (x : X) : map f (𝓝 x) = 𝓝 (f x) := hf.toEmbedding.map_nhds_of_mem _ <\| hf.isOpen_range.mem_nhds <\| mem_range_self _ #align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eq theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set X} : IsOpen s ↔ IsOpen (f '' s) := ⟨hf.isOpenMap s, fun h => by convert ← h.preimage hf.toEmbedding.continuous apply preimage_image_eq _ hf.inj⟩ #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open theorem OpenEmbedding.tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : OpenEmbedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := hg.toEmbedding.tendsto_nhds_iff #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff	Mathlib/Topology/Maps.lean	570	572	theorem OpenEmbedding.tendsto_nhds_iff' (hf : OpenEmbedding f) {l : Filter Z} {x : X} : Tendsto (g ∘ f) (𝓝 x) l ↔ Tendsto g (𝓝 (f x)) l := by	rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl
/- Copyright (c) 2019 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison, Markus Himmel -/ import Mathlib.CategoryTheory.Limits.Shapes.Equalizers import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi import Mathlib.CategoryTheory.MorphismProperty.Factorization #align_import category_theory.limits.shapes.images from "leanprover-community/mathlib"@"563aed347eb59dc4181cb732cda0d124d736eaa3" /-! # Categorical images We define the categorical image of `f` as a factorisation `f = e ≫ m` through a monomorphism `m`, so that `m` factors through the `m'` in any other such factorisation. ## Main definitions * A `MonoFactorisation` is a factorisation `f = e ≫ m`, where `m` is a monomorphism * `IsImage F` means that a given mono factorisation `F` has the universal property of the image. * `HasImage f` means that there is some image factorization for the morphism `f : X ⟶ Y`. * In this case, `image f` is some image object (selected with choice), `image.ι f : image f ⟶ Y` is the monomorphism `m` of the factorisation and `factorThruImage f : X ⟶ image f` is the morphism `e`. * `HasImages C` means that every morphism in `C` has an image. * Let `f : X ⟶ Y` and `g : P ⟶ Q` be morphisms in `C`, which we will represent as objects of the arrow category `arrow C`. Then `sq : f ⟶ g` is a commutative square in `C`. If `f` and `g` have images, then `HasImageMap sq` represents the fact that there is a morphism `i : image f ⟶ image g` making the diagram X ----→ image f ----→ Y \| \| \| \| \| \| ↓ ↓ ↓ P ----→ image g ----→ Q commute, where the top row is the image factorisation of `f`, the bottom row is the image factorisation of `g`, and the outer rectangle is the commutative square `sq`. * If a category `HasImages`, then `HasImageMaps` means that every commutative square admits an image map. * If a category `HasImages`, then `HasStrongEpiImages` means that the morphism to the image is always a strong epimorphism. ## Main statements * When `C` has equalizers, the morphism `e` appearing in an image factorisation is an epimorphism. * When `C` has strong epi images, then these images admit image maps. ## Future work * TODO: coimages, and abelian categories. * TODO: connect this with existing working in the group theory and ring theory libraries. -/ noncomputable section universe v u open CategoryTheory open CategoryTheory.Limits.WalkingParallelPair namespace CategoryTheory.Limits variable {C : Type u} [Category.{v} C] variable {X Y : C} (f : X ⟶ Y) /-- A factorisation of a morphism `f = e ≫ m`, with `m` monic. -/ structure MonoFactorisation (f : X ⟶ Y) where I : C -- Porting note: violates naming conventions but can't think a better replacement m : I ⟶ Y [m_mono : Mono m] e : X ⟶ I fac : e ≫ m = f := by aesop_cat #align category_theory.limits.mono_factorisation CategoryTheory.Limits.MonoFactorisation #align category_theory.limits.mono_factorisation.fac' CategoryTheory.Limits.MonoFactorisation.fac attribute [inherit_doc MonoFactorisation] MonoFactorisation.I MonoFactorisation.m MonoFactorisation.m_mono MonoFactorisation.e MonoFactorisation.fac attribute [reassoc (attr := simp)] MonoFactorisation.fac attribute [instance] MonoFactorisation.m_mono attribute [instance] MonoFactorisation.m_mono namespace MonoFactorisation /-- The obvious factorisation of a monomorphism through itself. -/ def self [Mono f] : MonoFactorisation f where I := X m := f e := 𝟙 X #align category_theory.limits.mono_factorisation.self CategoryTheory.Limits.MonoFactorisation.self -- I'm not sure we really need this, but the linter says that an inhabited instance -- ought to exist... instance [Mono f] : Inhabited (MonoFactorisation f) := ⟨self f⟩ variable {f} /-- The morphism `m` in a factorisation `f = e ≫ m` through a monomorphism is uniquely determined. -/ @[ext] theorem ext {F F' : MonoFactorisation f} (hI : F.I = F'.I) (hm : F.m = eqToHom hI ≫ F'.m) : F = F' := by cases' F with _ Fm _ _ Ffac; cases' F' with _ Fm' _ _ Ffac' cases' hI simp? at hm says simp only [eqToHom_refl, Category.id_comp] at hm congr apply (cancel_mono Fm).1 rw [Ffac, hm, Ffac'] #align category_theory.limits.mono_factorisation.ext CategoryTheory.Limits.MonoFactorisation.ext /-- Any mono factorisation of `f` gives a mono factorisation of `f ≫ g` when `g` is a mono. -/ @[simps] def compMono (F : MonoFactorisation f) {Y' : C} (g : Y ⟶ Y') [Mono g] : MonoFactorisation (f ≫ g) where I := F.I m := F.m ≫ g m_mono := mono_comp _ _ e := F.e #align category_theory.limits.mono_factorisation.comp_mono CategoryTheory.Limits.MonoFactorisation.compMono /-- A mono factorisation of `f ≫ g`, where `g` is an isomorphism, gives a mono factorisation of `f`. -/ @[simps] def ofCompIso {Y' : C} {g : Y ⟶ Y'} [IsIso g] (F : MonoFactorisation (f ≫ g)) : MonoFactorisation f where I := F.I m := F.m ≫ inv g m_mono := mono_comp _ _ e := F.e #align category_theory.limits.mono_factorisation.of_comp_iso CategoryTheory.Limits.MonoFactorisation.ofCompIso /-- Any mono factorisation of `f` gives a mono factorisation of `g ≫ f`. -/ @[simps] def isoComp (F : MonoFactorisation f) {X' : C} (g : X' ⟶ X) : MonoFactorisation (g ≫ f) where I := F.I m := F.m e := g ≫ F.e #align category_theory.limits.mono_factorisation.iso_comp CategoryTheory.Limits.MonoFactorisation.isoComp /-- A mono factorisation of `g ≫ f`, where `g` is an isomorphism, gives a mono factorisation of `f`. -/ @[simps] def ofIsoComp {X' : C} (g : X' ⟶ X) [IsIso g] (F : MonoFactorisation (g ≫ f)) : MonoFactorisation f where I := F.I m := F.m e := inv g ≫ F.e #align category_theory.limits.mono_factorisation.of_iso_comp CategoryTheory.Limits.MonoFactorisation.ofIsoComp /-- If `f` and `g` are isomorphic arrows, then a mono factorisation of `f` gives a mono factorisation of `g` -/ @[simps] def ofArrowIso {f g : Arrow C} (F : MonoFactorisation f.hom) (sq : f ⟶ g) [IsIso sq] : MonoFactorisation g.hom where I := F.I m := F.m ≫ sq.right e := inv sq.left ≫ F.e m_mono := mono_comp _ _ fac := by simp only [fac_assoc, Arrow.w, IsIso.inv_comp_eq, Category.assoc] #align category_theory.limits.mono_factorisation.of_arrow_iso CategoryTheory.Limits.MonoFactorisation.ofArrowIso end MonoFactorisation variable {f} /-- Data exhibiting that a given factorisation through a mono is initial. -/ structure IsImage (F : MonoFactorisation f) where lift : ∀ F' : MonoFactorisation f, F.I ⟶ F'.I lift_fac : ∀ F' : MonoFactorisation f, lift F' ≫ F'.m = F.m := by aesop_cat #align category_theory.limits.is_image CategoryTheory.Limits.IsImage #align category_theory.limits.is_image.lift_fac' CategoryTheory.Limits.IsImage.lift_fac attribute [inherit_doc IsImage] IsImage.lift IsImage.lift_fac attribute [reassoc (attr := simp)] IsImage.lift_fac namespace IsImage @[reassoc (attr := simp)] theorem fac_lift {F : MonoFactorisation f} (hF : IsImage F) (F' : MonoFactorisation f) : F.e ≫ hF.lift F' = F'.e := (cancel_mono F'.m).1 <\| by simp #align category_theory.limits.is_image.fac_lift CategoryTheory.Limits.IsImage.fac_lift variable (f) /-- The trivial factorisation of a monomorphism satisfies the universal property. -/ @[simps] def self [Mono f] : IsImage (MonoFactorisation.self f) where lift F' := F'.e #align category_theory.limits.is_image.self CategoryTheory.Limits.IsImage.self instance [Mono f] : Inhabited (IsImage (MonoFactorisation.self f)) := ⟨self f⟩ variable {f} -- TODO this is another good candidate for a future `UniqueUpToCanonicalIso`. /-- Two factorisations through monomorphisms satisfying the universal property must factor through isomorphic objects. -/ @[simps] def isoExt {F F' : MonoFactorisation f} (hF : IsImage F) (hF' : IsImage F') : F.I ≅ F'.I where hom := hF.lift F' inv := hF'.lift F hom_inv_id := (cancel_mono F.m).1 (by simp) inv_hom_id := (cancel_mono F'.m).1 (by simp) #align category_theory.limits.is_image.iso_ext CategoryTheory.Limits.IsImage.isoExt variable {F F' : MonoFactorisation f} (hF : IsImage F) (hF' : IsImage F') theorem isoExt_hom_m : (isoExt hF hF').hom ≫ F'.m = F.m := by simp #align category_theory.limits.is_image.iso_ext_hom_m CategoryTheory.Limits.IsImage.isoExt_hom_m theorem isoExt_inv_m : (isoExt hF hF').inv ≫ F.m = F'.m := by simp #align category_theory.limits.is_image.iso_ext_inv_m CategoryTheory.Limits.IsImage.isoExt_inv_m theorem e_isoExt_hom : F.e ≫ (isoExt hF hF').hom = F'.e := by simp #align category_theory.limits.is_image.e_iso_ext_hom CategoryTheory.Limits.IsImage.e_isoExt_hom theorem e_isoExt_inv : F'.e ≫ (isoExt hF hF').inv = F.e := by simp #align category_theory.limits.is_image.e_iso_ext_inv CategoryTheory.Limits.IsImage.e_isoExt_inv /-- If `f` and `g` are isomorphic arrows, then a mono factorisation of `f` that is an image gives a mono factorisation of `g` that is an image -/ @[simps] def ofArrowIso {f g : Arrow C} {F : MonoFactorisation f.hom} (hF : IsImage F) (sq : f ⟶ g) [IsIso sq] : IsImage (F.ofArrowIso sq) where lift F' := hF.lift (F'.ofArrowIso (inv sq)) lift_fac F' := by simpa only [MonoFactorisation.ofArrowIso_m, Arrow.inv_right, ← Category.assoc, IsIso.comp_inv_eq] using hF.lift_fac (F'.ofArrowIso (inv sq)) #align category_theory.limits.is_image.of_arrow_iso CategoryTheory.Limits.IsImage.ofArrowIso end IsImage variable (f) /-- Data exhibiting that a morphism `f` has an image. -/ structure ImageFactorisation (f : X ⟶ Y) where F : MonoFactorisation f -- Porting note: another violation of the naming convention isImage : IsImage F #align category_theory.limits.image_factorisation CategoryTheory.Limits.ImageFactorisation #align category_theory.limits.image_factorisation.is_image CategoryTheory.Limits.ImageFactorisation.isImage attribute [inherit_doc ImageFactorisation] ImageFactorisation.F ImageFactorisation.isImage namespace ImageFactorisation instance [Mono f] : Inhabited (ImageFactorisation f) := ⟨⟨_, IsImage.self f⟩⟩ /-- If `f` and `g` are isomorphic arrows, then an image factorisation of `f` gives an image factorisation of `g` -/ @[simps] def ofArrowIso {f g : Arrow C} (F : ImageFactorisation f.hom) (sq : f ⟶ g) [IsIso sq] : ImageFactorisation g.hom where F := F.F.ofArrowIso sq isImage := F.isImage.ofArrowIso sq #align category_theory.limits.image_factorisation.of_arrow_iso CategoryTheory.Limits.ImageFactorisation.ofArrowIso end ImageFactorisation /-- `has_image f` means that there exists an image factorisation of `f`. -/ class HasImage (f : X ⟶ Y) : Prop where mk' :: exists_image : Nonempty (ImageFactorisation f) #align category_theory.limits.has_image CategoryTheory.Limits.HasImage attribute [inherit_doc HasImage] HasImage.exists_image theorem HasImage.mk {f : X ⟶ Y} (F : ImageFactorisation f) : HasImage f := ⟨Nonempty.intro F⟩ #align category_theory.limits.has_image.mk CategoryTheory.Limits.HasImage.mk theorem HasImage.of_arrow_iso {f g : Arrow C} [h : HasImage f.hom] (sq : f ⟶ g) [IsIso sq] : HasImage g.hom := ⟨⟨h.exists_image.some.ofArrowIso sq⟩⟩ #align category_theory.limits.has_image.of_arrow_iso CategoryTheory.Limits.HasImage.of_arrow_iso instance (priority := 100) mono_hasImage (f : X ⟶ Y) [Mono f] : HasImage f := HasImage.mk ⟨_, IsImage.self f⟩ #align category_theory.limits.mono_has_image CategoryTheory.Limits.mono_hasImage section variable [HasImage f] /-- Some factorisation of `f` through a monomorphism (selected with choice). -/ def Image.monoFactorisation : MonoFactorisation f := (Classical.choice HasImage.exists_image).F #align category_theory.limits.image.mono_factorisation CategoryTheory.Limits.Image.monoFactorisation /-- The witness of the universal property for the chosen factorisation of `f` through a monomorphism. -/ def Image.isImage : IsImage (Image.monoFactorisation f) := (Classical.choice HasImage.exists_image).isImage #align category_theory.limits.image.is_image CategoryTheory.Limits.Image.isImage /-- The categorical image of a morphism. -/ def image : C := (Image.monoFactorisation f).I #align category_theory.limits.image CategoryTheory.Limits.image /-- The inclusion of the image of a morphism into the target. -/ def image.ι : image f ⟶ Y := (Image.monoFactorisation f).m #align category_theory.limits.image.ι CategoryTheory.Limits.image.ι @[simp] theorem image.as_ι : (Image.monoFactorisation f).m = image.ι f := rfl #align category_theory.limits.image.as_ι CategoryTheory.Limits.image.as_ι instance : Mono (image.ι f) := (Image.monoFactorisation f).m_mono /-- The map from the source to the image of a morphism. -/ def factorThruImage : X ⟶ image f := (Image.monoFactorisation f).e #align category_theory.limits.factor_thru_image CategoryTheory.Limits.factorThruImage /-- Rewrite in terms of the `factorThruImage` interface. -/ @[simp] theorem as_factorThruImage : (Image.monoFactorisation f).e = factorThruImage f := rfl #align category_theory.limits.as_factor_thru_image CategoryTheory.Limits.as_factorThruImage @[reassoc (attr := simp)] theorem image.fac : factorThruImage f ≫ image.ι f = f := (Image.monoFactorisation f).fac #align category_theory.limits.image.fac CategoryTheory.Limits.image.fac variable {f} /-- Any other factorisation of the morphism `f` through a monomorphism receives a map from the image. -/ def image.lift (F' : MonoFactorisation f) : image f ⟶ F'.I := (Image.isImage f).lift F' #align category_theory.limits.image.lift CategoryTheory.Limits.image.lift @[reassoc (attr := simp)] theorem image.lift_fac (F' : MonoFactorisation f) : image.lift F' ≫ F'.m = image.ι f := (Image.isImage f).lift_fac F' #align category_theory.limits.image.lift_fac CategoryTheory.Limits.image.lift_fac @[reassoc (attr := simp)] theorem image.fac_lift (F' : MonoFactorisation f) : factorThruImage f ≫ image.lift F' = F'.e := (Image.isImage f).fac_lift F' #align category_theory.limits.image.fac_lift CategoryTheory.Limits.image.fac_lift @[simp] theorem image.isImage_lift (F : MonoFactorisation f) : (Image.isImage f).lift F = image.lift F := rfl #align category_theory.limits.image.is_image_lift CategoryTheory.Limits.image.isImage_lift @[reassoc (attr := simp)] theorem IsImage.lift_ι {F : MonoFactorisation f} (hF : IsImage F) : hF.lift (Image.monoFactorisation f) ≫ image.ι f = F.m := hF.lift_fac _ #align category_theory.limits.is_image.lift_ι CategoryTheory.Limits.IsImage.lift_ι -- TODO we could put a category structure on `MonoFactorisation f`, -- with the morphisms being `g : I ⟶ I'` commuting with the `m`s -- (they then automatically commute with the `e`s) -- and show that an `imageOf f` gives an initial object there -- (uniqueness of the lift comes for free). instance image.lift_mono (F' : MonoFactorisation f) : Mono (image.lift F') := by refine @mono_of_mono _ _ _ _ _ _ F'.m ?_ simpa using MonoFactorisation.m_mono _ #align category_theory.limits.image.lift_mono CategoryTheory.Limits.image.lift_mono theorem HasImage.uniq (F' : MonoFactorisation f) (l : image f ⟶ F'.I) (w : l ≫ F'.m = image.ι f) : l = image.lift F' := (cancel_mono F'.m).1 (by simp [w]) #align category_theory.limits.has_image.uniq CategoryTheory.Limits.HasImage.uniq /-- If `has_image g`, then `has_image (f ≫ g)` when `f` is an isomorphism. -/ instance {X Y Z : C} (f : X ⟶ Y) [IsIso f] (g : Y ⟶ Z) [HasImage g] : HasImage (f ≫ g) where exists_image := ⟨{ F := { I := image g m := image.ι g e := f ≫ factorThruImage g } isImage := { lift := fun F' => image.lift { I := F'.I m := F'.m e := inv f ≫ F'.e } } }⟩ end section variable (C) /-- `HasImages` asserts that every morphism has an image. -/ class HasImages : Prop where has_image : ∀ {X Y : C} (f : X ⟶ Y), HasImage f #align category_theory.limits.has_images CategoryTheory.Limits.HasImages attribute [inherit_doc HasImages] HasImages.has_image attribute [instance 100] HasImages.has_image end section /-- The image of a monomorphism is isomorphic to the source. -/ def imageMonoIsoSource [Mono f] : image f ≅ X := IsImage.isoExt (Image.isImage f) (IsImage.self f) #align category_theory.limits.image_mono_iso_source CategoryTheory.Limits.imageMonoIsoSource @[reassoc (attr := simp)] theorem imageMonoIsoSource_inv_ι [Mono f] : (imageMonoIsoSource f).inv ≫ image.ι f = f := by simp [imageMonoIsoSource] #align category_theory.limits.image_mono_iso_source_inv_ι CategoryTheory.Limits.imageMonoIsoSource_inv_ι @[reassoc (attr := simp)] theorem imageMonoIsoSource_hom_self [Mono f] : (imageMonoIsoSource f).hom ≫ f = image.ι f := by simp only [← imageMonoIsoSource_inv_ι f] rw [← Category.assoc, Iso.hom_inv_id, Category.id_comp] #align category_theory.limits.image_mono_iso_source_hom_self CategoryTheory.Limits.imageMonoIsoSource_hom_self -- This is the proof that `factorThruImage f` is an epimorphism -- from https://en.wikipedia.org/wiki/Image_%28category_theory%29, which is in turn taken from: -- Mitchell, Barry (1965), Theory of categories, MR 0202787, p.12, Proposition 10.1 @[ext] theorem image.ext [HasImage f] {W : C} {g h : image f ⟶ W} [HasLimit (parallelPair g h)] (w : factorThruImage f ≫ g = factorThruImage f ≫ h) : g = h := by let q := equalizer.ι g h let e' := equalizer.lift _ w let F' : MonoFactorisation f := { I := equalizer g h m := q ≫ image.ι f m_mono := by apply mono_comp e := e' } let v := image.lift F' have t₀ : v ≫ q ≫ image.ι f = image.ι f := image.lift_fac F' have t : v ≫ q = 𝟙 (image f) := (cancel_mono_id (image.ι f)).1 (by convert t₀ using 1 rw [Category.assoc]) -- The proof from wikipedia next proves `q ≫ v = 𝟙 _`, -- and concludes that `equalizer g h ≅ image f`, -- but this isn't necessary. calc g = 𝟙 (image f) ≫ g := by rw [Category.id_comp] _ = v ≫ q ≫ g := by rw [← t, Category.assoc] _ = v ≫ q ≫ h := by rw [equalizer.condition g h] _ = 𝟙 (image f) ≫ h := by rw [← Category.assoc, t] _ = h := by rw [Category.id_comp] #align category_theory.limits.image.ext CategoryTheory.Limits.image.ext instance [HasImage f] [∀ {Z : C} (g h : image f ⟶ Z), HasLimit (parallelPair g h)] : Epi (factorThruImage f) := ⟨fun _ _ w => image.ext f w⟩ theorem epi_image_of_epi {X Y : C} (f : X ⟶ Y) [HasImage f] [E : Epi f] : Epi (image.ι f) := by rw [← image.fac f] at E exact epi_of_epi (factorThruImage f) (image.ι f) #align category_theory.limits.epi_image_of_epi CategoryTheory.Limits.epi_image_of_epi	Mathlib/CategoryTheory/Limits/Shapes/Images.lean	470	473	theorem epi_of_epi_image {X Y : C} (f : X ⟶ Y) [HasImage f] [Epi (image.ι f)] [Epi (factorThruImage f)] : Epi f := by	rw [← image.fac f] apply epi_comp
/- Copyright (c) 2022 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import Mathlib.Order.Filter.Lift import Mathlib.Topology.Separation import Mathlib.Order.Interval.Set.Monotone #align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514" /-! # Topology on the set of filters on a type This file introduces a topology on `Filter α`. It is generated by the sets `Set.Iic (𝓟 s) = {l : Filter α \| s ∈ l}`, `s : Set α`. A set `s : Set (Filter α)` is open if and only if it is a union of a family of these basic open sets, see `Filter.isOpen_iff`. This topology has the following important properties. * If `X` is a topological space, then the map `𝓝 : X → Filter X` is a topology inducing map. * In particular, it is a continuous map, so `𝓝 ∘ f` tends to `𝓝 (𝓝 a)` whenever `f` tends to `𝓝 a`. * If `X` is an ordered topological space with order topology and no max element, then `𝓝 ∘ f` tends to `𝓝 Filter.atTop` whenever `f` tends to `Filter.atTop`. * It turns `Filter X` into a T₀ space and the order on `Filter X` is the dual of the `specializationOrder (Filter X)`. ## Tags filter, topological space -/ open Set Filter TopologicalSpace open Filter Topology variable {ι : Sort} {α β X Y : Type} namespace Filter /-- The topology on `Filter α` is generated by the sets `Set.Iic (𝓟 s) = {l : Filter α \| s ∈ l}`, `s : Set α`. A set `s : Set (Filter α)` is open if and only if it is a union of a family of these basic open sets, see `Filter.isOpen_iff`. -/ instance : TopologicalSpace (Filter α) := generateFrom <\| range <\| Iic ∘ 𝓟 theorem isOpen_Iic_principal {s : Set α} : IsOpen (Iic (𝓟 s)) := GenerateOpen.basic _ (mem_range_self _) #align filter.is_open_Iic_principal Filter.isOpen_Iic_principal theorem isOpen_setOf_mem {s : Set α} : IsOpen { l : Filter α \| s ∈ l } := by simpa only [Iic_principal] using isOpen_Iic_principal #align filter.is_open_set_of_mem Filter.isOpen_setOf_mem theorem isTopologicalBasis_Iic_principal : IsTopologicalBasis (range (Iic ∘ 𝓟 : Set α → Set (Filter α))) := { exists_subset_inter := by rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩ sUnion_eq := sUnion_eq_univ_iff.2 fun l => ⟨Iic ⊤, ⟨univ, congr_arg Iic principal_univ⟩, mem_Iic.2 le_top⟩ eq_generateFrom := rfl } #align filter.is_topological_basis_Iic_principal Filter.isTopologicalBasis_Iic_principal theorem isOpen_iff {s : Set (Filter α)} : IsOpen s ↔ ∃ T : Set (Set α), s = ⋃ t ∈ T, Iic (𝓟 t) := isTopologicalBasis_Iic_principal.open_iff_eq_sUnion.trans <\| by simp only [exists_subset_range_and_iff, sUnion_image, (· ∘ ·)] #align filter.is_open_iff Filter.isOpen_iff theorem nhds_eq (l : Filter α) : 𝓝 l = l.lift' (Iic ∘ 𝓟) := nhds_generateFrom.trans <\| by simp only [mem_setOf_eq, @and_comm (l ∈ _), iInf_and, iInf_range, Filter.lift', Filter.lift, (· ∘ ·), mem_Iic, le_principal_iff] #align filter.nhds_eq Filter.nhds_eq theorem nhds_eq' (l : Filter α) : 𝓝 l = l.lift' fun s => { l' \| s ∈ l' } := by simpa only [(· ∘ ·), Iic_principal] using nhds_eq l #align filter.nhds_eq' Filter.nhds_eq' protected theorem tendsto_nhds {la : Filter α} {lb : Filter β} {f : α → Filter β} : Tendsto f la (𝓝 lb) ↔ ∀ s ∈ lb, ∀ᶠ a in la, s ∈ f a := by simp only [nhds_eq', tendsto_lift', mem_setOf_eq] #align filter.tendsto_nhds Filter.tendsto_nhds protected theorem HasBasis.nhds {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) : HasBasis (𝓝 l) p fun i => Iic (𝓟 (s i)) := by rw [nhds_eq] exact h.lift' monotone_principal.Iic #align filter.has_basis.nhds Filter.HasBasis.nhds protected theorem tendsto_pure_self (l : Filter X) : Tendsto (pure : X → Filter X) l (𝓝 l) := by rw [Filter.tendsto_nhds] exact fun s hs ↦ Eventually.mono hs fun x ↦ id /-- Neighborhoods of a countably generated filter is a countably generated filter. -/ instance {l : Filter α} [IsCountablyGenerated l] : IsCountablyGenerated (𝓝 l) := let ⟨_b, hb⟩ := l.exists_antitone_basis HasCountableBasis.isCountablyGenerated <\| ⟨hb.nhds, Set.to_countable _⟩ theorem HasBasis.nhds' {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) : HasBasis (𝓝 l) p fun i => { l' \| s i ∈ l' } := by simpa only [Iic_principal] using h.nhds #align filter.has_basis.nhds' Filter.HasBasis.nhds' protected theorem mem_nhds_iff {l : Filter α} {S : Set (Filter α)} : S ∈ 𝓝 l ↔ ∃ t ∈ l, Iic (𝓟 t) ⊆ S := l.basis_sets.nhds.mem_iff #align filter.mem_nhds_iff Filter.mem_nhds_iff theorem mem_nhds_iff' {l : Filter α} {S : Set (Filter α)} : S ∈ 𝓝 l ↔ ∃ t ∈ l, ∀ ⦃l' : Filter α⦄, t ∈ l' → l' ∈ S := l.basis_sets.nhds'.mem_iff #align filter.mem_nhds_iff' Filter.mem_nhds_iff' @[simp] theorem nhds_bot : 𝓝 (⊥ : Filter α) = pure ⊥ := by simp [nhds_eq, (· ∘ ·), lift'_bot monotone_principal.Iic] #align filter.nhds_bot Filter.nhds_bot @[simp] theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by simp [nhds_eq] #align filter.nhds_top Filter.nhds_top @[simp] theorem nhds_principal (s : Set α) : 𝓝 (𝓟 s) = 𝓟 (Iic (𝓟 s)) := (hasBasis_principal s).nhds.eq_of_same_basis (hasBasis_principal _) #align filter.nhds_principal Filter.nhds_principal @[simp] theorem nhds_pure (x : α) : 𝓝 (pure x : Filter α) = 𝓟 {⊥, pure x} := by rw [← principal_singleton, nhds_principal, principal_singleton, Iic_pure] #align filter.nhds_pure Filter.nhds_pure @[simp] theorem nhds_iInf (f : ι → Filter α) : 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i) := by simp only [nhds_eq] apply lift'_iInf_of_map_univ <;> simp #align filter.nhds_infi Filter.nhds_iInf @[simp] theorem nhds_inf (l₁ l₂ : Filter α) : 𝓝 (l₁ ⊓ l₂) = 𝓝 l₁ ⊓ 𝓝 l₂ := by simpa only [iInf_bool_eq] using nhds_iInf fun b => cond b l₁ l₂ #align filter.nhds_inf Filter.nhds_inf theorem monotone_nhds : Monotone (𝓝 : Filter α → Filter (Filter α)) := Monotone.of_map_inf nhds_inf #align filter.monotone_nhds Filter.monotone_nhds theorem sInter_nhds (l : Filter α) : ⋂₀ { s \| s ∈ 𝓝 l } = Iic l := by simp_rw [nhds_eq, (· ∘ ·), sInter_lift'_sets monotone_principal.Iic, Iic, le_principal_iff, ← setOf_forall, ← Filter.le_def] #align filter.Inter_nhds Filter.sInter_nhds @[simp] theorem nhds_mono {l₁ l₂ : Filter α} : 𝓝 l₁ ≤ 𝓝 l₂ ↔ l₁ ≤ l₂ := by refine ⟨fun h => ?_, fun h => monotone_nhds h⟩ rw [← Iic_subset_Iic, ← sInter_nhds, ← sInter_nhds] exact sInter_subset_sInter h #align filter.nhds_mono Filter.nhds_mono protected theorem mem_interior {s : Set (Filter α)} {l : Filter α} : l ∈ interior s ↔ ∃ t ∈ l, Iic (𝓟 t) ⊆ s := by rw [mem_interior_iff_mem_nhds, Filter.mem_nhds_iff] #align filter.mem_interior Filter.mem_interior protected theorem mem_closure {s : Set (Filter α)} {l : Filter α} : l ∈ closure s ↔ ∀ t ∈ l, ∃ l' ∈ s, t ∈ l' := by simp only [closure_eq_compl_interior_compl, Filter.mem_interior, mem_compl_iff, not_exists, not_forall, Classical.not_not, exists_prop, not_and, and_comm, subset_def, mem_Iic, le_principal_iff] #align filter.mem_closure Filter.mem_closure @[simp] protected theorem closure_singleton (l : Filter α) : closure {l} = Ici l := by ext l' simp [Filter.mem_closure, Filter.le_def] #align filter.closure_singleton Filter.closure_singleton @[simp] theorem specializes_iff_le {l₁ l₂ : Filter α} : l₁ ⤳ l₂ ↔ l₁ ≤ l₂ := by simp only [specializes_iff_closure_subset, Filter.closure_singleton, Ici_subset_Ici] #align filter.specializes_iff_le Filter.specializes_iff_le instance : T0Space (Filter α) := ⟨fun _ _ h => (specializes_iff_le.1 h.specializes).antisymm (specializes_iff_le.1 h.symm.specializes)⟩	Mathlib/Topology/Filter.lean	192	193	theorem nhds_atTop [Preorder α] : 𝓝 atTop = ⨅ x : α, 𝓟 (Iic (𝓟 (Ici x))) := by	simp only [atTop, nhds_iInf, nhds_principal]
/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Analysis.SpecificLimits.Basic import Mathlib.Topology.MetricSpace.IsometricSMul #align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156" /-! # Hausdorff distance The Hausdorff distance on subsets of a metric (or emetric) space. Given two subsets `s` and `t` of a metric space, their Hausdorff distance is the smallest `d` such that any point `s` is within `d` of a point in `t`, and conversely. This quantity is often infinite (think of `s` bounded and `t` unbounded), and therefore better expressed in the setting of emetric spaces. ## Main definitions This files introduces: * `EMetric.infEdist x s`, the infimum edistance of a point `x` to a set `s` in an emetric space * `EMetric.hausdorffEdist s t`, the Hausdorff edistance of two sets in an emetric space * Versions of these notions on metric spaces, called respectively `Metric.infDist` and `Metric.hausdorffDist` ## Main results * `infEdist_closure`: the edistance to a set and its closure coincide * `EMetric.mem_closure_iff_infEdist_zero`: a point `x` belongs to the closure of `s` iff `infEdist x s = 0` * `IsCompact.exists_infEdist_eq_edist`: if `s` is compact and non-empty, there exists a point `y` which attains this edistance * `IsOpen.exists_iUnion_isClosed`: every open set `U` can be written as the increasing union of countably many closed subsets of `U` * `hausdorffEdist_closure`: replacing a set by its closure does not change the Hausdorff edistance * `hausdorffEdist_zero_iff_closure_eq_closure`: two sets have Hausdorff edistance zero iff their closures coincide * the Hausdorff edistance is symmetric and satisfies the triangle inequality * in particular, closed sets in an emetric space are an emetric space (this is shown in `EMetricSpace.closeds.emetricspace`) * versions of these notions on metric spaces * `hausdorffEdist_ne_top_of_nonempty_of_bounded`: if two sets in a metric space are nonempty and bounded in a metric space, they are at finite Hausdorff edistance. ## Tags metric space, Hausdorff distance -/ noncomputable section open NNReal ENNReal Topology Set Filter Pointwise Bornology universe u v w variable {ι : Sort*} {α : Type u} {β : Type v} namespace EMetric section InfEdist variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] {x y : α} {s t : Set α} {Φ : α → β} /-! ### Distance of a point to a set as a function into `ℝ≥0∞`. -/ /-- The minimal edistance of a point to a set -/ def infEdist (x : α) (s : Set α) : ℝ≥0∞ := ⨅ y ∈ s, edist x y #align emetric.inf_edist EMetric.infEdist @[simp] theorem infEdist_empty : infEdist x ∅ = ∞ := iInf_emptyset #align emetric.inf_edist_empty EMetric.infEdist_empty	Mathlib/Topology/MetricSpace/HausdorffDistance.lean	80	81	theorem le_infEdist {d} : d ≤ infEdist x s ↔ ∀ y ∈ s, d ≤ edist x y := by	simp only [infEdist, le_iInf_iff]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro -/ import Mathlib.Algebra.Algebra.Tower import Mathlib.Algebra.GroupWithZero.Divisibility import Mathlib.Algebra.Regular.Pow import Mathlib.Algebra.MonoidAlgebra.Support import Mathlib.Data.Finsupp.Antidiagonal import Mathlib.Order.SymmDiff import Mathlib.RingTheory.Adjoin.Basic #align_import data.mv_polynomial.basic from "leanprover-community/mathlib"@"c8734e8953e4b439147bd6f75c2163f6d27cdce6" /-! # Multivariate polynomials This file defines polynomial rings over a base ring (or even semiring), with variables from a general type `σ` (which could be infinite). ## Important definitions Let `R` be a commutative ring (or a semiring) and let `σ` be an arbitrary type. This file creates the type `MvPolynomial σ R`, which mathematicians might denote $R[X_i : i \in σ]$. It is the type of multivariate (a.k.a. multivariable) polynomials, with variables corresponding to the terms in `σ`, and coefficients in `R`. ### Notation In the definitions below, we use the following notation: + `σ : Type` (indexing the variables) + `R : Type` `[CommSemiring R]` (the coefficients) + `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set. This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s` + `a : R` + `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians + `p : MvPolynomial σ R` ### Definitions * `MvPolynomial σ R` : the type of polynomials with variables of type `σ` and coefficients in the commutative semiring `R` * `monomial s a` : the monomial which mathematically would be denoted `a * X^s` * `C a` : the constant polynomial with value `a` * `X i` : the degree one monomial corresponding to i; mathematically this might be denoted `Xᵢ`. * `coeff s p` : the coefficient of `s` in `p`. * `eval₂ (f : R → S₁) (g : σ → S₁) p` : given a semiring homomorphism from `R` to another semiring `S₁`, and a map `σ → S₁`, evaluates `p` at this valuation, returning a term of type `S₁`. Note that `eval₂` can be made using `eval` and `map` (see below), and it has been suggested that sticking to `eval` and `map` might make the code less brittle. * `eval (g : σ → R) p` : given a map `σ → R`, evaluates `p` at this valuation, returning a term of type `R` * `map (f : R → S₁) p` : returns the multivariate polynomial obtained from `p` by the change of coefficient semiring corresponding to `f` ## Implementation notes Recall that if `Y` has a zero, then `X →₀ Y` is the type of functions from `X` to `Y` with finite support, i.e. such that only finitely many elements of `X` get sent to non-zero terms in `Y`. The definition of `MvPolynomial σ R` is `(σ →₀ ℕ) →₀ R`; here `σ →₀ ℕ` denotes the space of all monomials in the variables, and the function to `R` sends a monomial to its coefficient in the polynomial being represented. ## Tags polynomial, multivariate polynomial, multivariable polynomial -/ noncomputable section open Set Function Finsupp AddMonoidAlgebra open scoped Pointwise universe u v w x variable {R : Type u} {S₁ : Type v} {S₂ : Type w} {S₃ : Type x} /-- Multivariate polynomial, where `σ` is the index set of the variables and `R` is the coefficient ring -/ def MvPolynomial (σ : Type) (R : Type) [CommSemiring R] := AddMonoidAlgebra R (σ →₀ ℕ) #align mv_polynomial MvPolynomial namespace MvPolynomial -- Porting note: because of `MvPolynomial.C` and `MvPolynomial.X` this linter throws -- tons of warnings in this file, and it's easier to just disable them globally in the file set_option linter.uppercaseLean3 false variable {σ : Type} {a a' a₁ a₂ : R} {e : ℕ} {n m : σ} {s : σ →₀ ℕ} section CommSemiring section Instances instance decidableEqMvPolynomial [CommSemiring R] [DecidableEq σ] [DecidableEq R] : DecidableEq (MvPolynomial σ R) := Finsupp.instDecidableEq #align mv_polynomial.decidable_eq_mv_polynomial MvPolynomial.decidableEqMvPolynomial instance commSemiring [CommSemiring R] : CommSemiring (MvPolynomial σ R) := AddMonoidAlgebra.commSemiring instance inhabited [CommSemiring R] : Inhabited (MvPolynomial σ R) := ⟨0⟩ instance distribuMulAction [Monoid R] [CommSemiring S₁] [DistribMulAction R S₁] : DistribMulAction R (MvPolynomial σ S₁) := AddMonoidAlgebra.distribMulAction instance smulZeroClass [CommSemiring S₁] [SMulZeroClass R S₁] : SMulZeroClass R (MvPolynomial σ S₁) := AddMonoidAlgebra.smulZeroClass instance faithfulSMul [CommSemiring S₁] [SMulZeroClass R S₁] [FaithfulSMul R S₁] : FaithfulSMul R (MvPolynomial σ S₁) := AddMonoidAlgebra.faithfulSMul instance module [Semiring R] [CommSemiring S₁] [Module R S₁] : Module R (MvPolynomial σ S₁) := AddMonoidAlgebra.module instance isScalarTower [CommSemiring S₂] [SMul R S₁] [SMulZeroClass R S₂] [SMulZeroClass S₁ S₂] [IsScalarTower R S₁ S₂] : IsScalarTower R S₁ (MvPolynomial σ S₂) := AddMonoidAlgebra.isScalarTower instance smulCommClass [CommSemiring S₂] [SMulZeroClass R S₂] [SMulZeroClass S₁ S₂] [SMulCommClass R S₁ S₂] : SMulCommClass R S₁ (MvPolynomial σ S₂) := AddMonoidAlgebra.smulCommClass instance isCentralScalar [CommSemiring S₁] [SMulZeroClass R S₁] [SMulZeroClass Rᵐᵒᵖ S₁] [IsCentralScalar R S₁] : IsCentralScalar R (MvPolynomial σ S₁) := AddMonoidAlgebra.isCentralScalar instance algebra [CommSemiring R] [CommSemiring S₁] [Algebra R S₁] : Algebra R (MvPolynomial σ S₁) := AddMonoidAlgebra.algebra instance isScalarTower_right [CommSemiring S₁] [DistribSMul R S₁] [IsScalarTower R S₁ S₁] : IsScalarTower R (MvPolynomial σ S₁) (MvPolynomial σ S₁) := AddMonoidAlgebra.isScalarTower_self _ #align mv_polynomial.is_scalar_tower_right MvPolynomial.isScalarTower_right instance smulCommClass_right [CommSemiring S₁] [DistribSMul R S₁] [SMulCommClass R S₁ S₁] : SMulCommClass R (MvPolynomial σ S₁) (MvPolynomial σ S₁) := AddMonoidAlgebra.smulCommClass_self _ #align mv_polynomial.smul_comm_class_right MvPolynomial.smulCommClass_right /-- If `R` is a subsingleton, then `MvPolynomial σ R` has a unique element -/ instance unique [CommSemiring R] [Subsingleton R] : Unique (MvPolynomial σ R) := AddMonoidAlgebra.unique #align mv_polynomial.unique MvPolynomial.unique end Instances variable [CommSemiring R] [CommSemiring S₁] {p q : MvPolynomial σ R} /-- `monomial s a` is the monomial with coefficient `a` and exponents given by `s` -/ def monomial (s : σ →₀ ℕ) : R →ₗ[R] MvPolynomial σ R := lsingle s #align mv_polynomial.monomial MvPolynomial.monomial theorem single_eq_monomial (s : σ →₀ ℕ) (a : R) : Finsupp.single s a = monomial s a := rfl #align mv_polynomial.single_eq_monomial MvPolynomial.single_eq_monomial theorem mul_def : p q = p.sum fun m a => q.sum fun n b => monomial (m + n) (a * b) := AddMonoidAlgebra.mul_def #align mv_polynomial.mul_def MvPolynomial.mul_def /-- `C a` is the constant polynomial with value `a` -/ def C : R →+* MvPolynomial σ R := { singleZeroRingHom with toFun := monomial 0 } #align mv_polynomial.C MvPolynomial.C variable (R σ) @[simp] theorem algebraMap_eq : algebraMap R (MvPolynomial σ R) = C := rfl #align mv_polynomial.algebra_map_eq MvPolynomial.algebraMap_eq variable {R σ} /-- `X n` is the degree `1` monomial $X_n$. -/ def X (n : σ) : MvPolynomial σ R := monomial (Finsupp.single n 1) 1 #align mv_polynomial.X MvPolynomial.X theorem monomial_left_injective {r : R} (hr : r ≠ 0) : Function.Injective fun s : σ →₀ ℕ => monomial s r := Finsupp.single_left_injective hr #align mv_polynomial.monomial_left_injective MvPolynomial.monomial_left_injective @[simp] theorem monomial_left_inj {s t : σ →₀ ℕ} {r : R} (hr : r ≠ 0) : monomial s r = monomial t r ↔ s = t := Finsupp.single_left_inj hr #align mv_polynomial.monomial_left_inj MvPolynomial.monomial_left_inj theorem C_apply : (C a : MvPolynomial σ R) = monomial 0 a := rfl #align mv_polynomial.C_apply MvPolynomial.C_apply -- Porting note (#10618): `simp` can prove this theorem C_0 : C 0 = (0 : MvPolynomial σ R) := map_zero _ #align mv_polynomial.C_0 MvPolynomial.C_0 -- Porting note (#10618): `simp` can prove this theorem C_1 : C 1 = (1 : MvPolynomial σ R) := rfl #align mv_polynomial.C_1 MvPolynomial.C_1 theorem C_mul_monomial : C a * monomial s a' = monomial s (a * a') := by -- Porting note: this `show` feels like defeq abuse, but I can't find the appropriate lemmas show AddMonoidAlgebra.single _ _ * AddMonoidAlgebra.single _ _ = AddMonoidAlgebra.single _ _ simp [C_apply, single_mul_single] #align mv_polynomial.C_mul_monomial MvPolynomial.C_mul_monomial -- Porting note (#10618): `simp` can prove this theorem C_add : (C (a + a') : MvPolynomial σ R) = C a + C a' := Finsupp.single_add _ _ _ #align mv_polynomial.C_add MvPolynomial.C_add -- Porting note (#10618): `simp` can prove this theorem C_mul : (C (a * a') : MvPolynomial σ R) = C a * C a' := C_mul_monomial.symm #align mv_polynomial.C_mul MvPolynomial.C_mul -- Porting note (#10618): `simp` can prove this theorem C_pow (a : R) (n : ℕ) : (C (a ^ n) : MvPolynomial σ R) = C a ^ n := map_pow _ _ _ #align mv_polynomial.C_pow MvPolynomial.C_pow theorem C_injective (σ : Type) (R : Type) [CommSemiring R] : Function.Injective (C : R → MvPolynomial σ R) := Finsupp.single_injective _ #align mv_polynomial.C_injective MvPolynomial.C_injective theorem C_surjective {R : Type} [CommSemiring R] (σ : Type) [IsEmpty σ] : Function.Surjective (C : R → MvPolynomial σ R) := by refine fun p => ⟨p.toFun 0, Finsupp.ext fun a => ?_⟩ simp only [C_apply, ← single_eq_monomial, (Finsupp.ext isEmptyElim (α := σ) : a = 0), single_eq_same] rfl #align mv_polynomial.C_surjective MvPolynomial.C_surjective @[simp] theorem C_inj {σ : Type} (R : Type) [CommSemiring R] (r s : R) : (C r : MvPolynomial σ R) = C s ↔ r = s := (C_injective σ R).eq_iff #align mv_polynomial.C_inj MvPolynomial.C_inj instance nontrivial_of_nontrivial (σ : Type) (R : Type) [CommSemiring R] [Nontrivial R] : Nontrivial (MvPolynomial σ R) := inferInstanceAs (Nontrivial <\| AddMonoidAlgebra R (σ →₀ ℕ)) instance infinite_of_infinite (σ : Type) (R : Type) [CommSemiring R] [Infinite R] : Infinite (MvPolynomial σ R) := Infinite.of_injective C (C_injective _ _) #align mv_polynomial.infinite_of_infinite MvPolynomial.infinite_of_infinite instance infinite_of_nonempty (σ : Type) (R : Type) [Nonempty σ] [CommSemiring R] [Nontrivial R] : Infinite (MvPolynomial σ R) := Infinite.of_injective ((fun s : σ →₀ ℕ => monomial s 1) ∘ Finsupp.single (Classical.arbitrary σ)) <\| (monomial_left_injective one_ne_zero).comp (Finsupp.single_injective _) #align mv_polynomial.infinite_of_nonempty MvPolynomial.infinite_of_nonempty theorem C_eq_coe_nat (n : ℕ) : (C ↑n : MvPolynomial σ R) = n := by induction n <;> simp [Nat.succ_eq_add_one, ] #align mv_polynomial.C_eq_coe_nat MvPolynomial.C_eq_coe_nat theorem C_mul' : MvPolynomial.C a p = a • p := (Algebra.smul_def a p).symm #align mv_polynomial.C_mul' MvPolynomial.C_mul' theorem smul_eq_C_mul (p : MvPolynomial σ R) (a : R) : a • p = C a * p := C_mul'.symm #align mv_polynomial.smul_eq_C_mul MvPolynomial.smul_eq_C_mul theorem C_eq_smul_one : (C a : MvPolynomial σ R) = a • (1 : MvPolynomial σ R) := by rw [← C_mul', mul_one] #align mv_polynomial.C_eq_smul_one MvPolynomial.C_eq_smul_one theorem smul_monomial {S₁ : Type} [SMulZeroClass S₁ R] (r : S₁) : r • monomial s a = monomial s (r • a) := Finsupp.smul_single _ _ _ #align mv_polynomial.smul_monomial MvPolynomial.smul_monomial theorem X_injective [Nontrivial R] : Function.Injective (X : σ → MvPolynomial σ R) := (monomial_left_injective one_ne_zero).comp (Finsupp.single_left_injective one_ne_zero) #align mv_polynomial.X_injective MvPolynomial.X_injective @[simp] theorem X_inj [Nontrivial R] (m n : σ) : X m = (X n : MvPolynomial σ R) ↔ m = n := X_injective.eq_iff #align mv_polynomial.X_inj MvPolynomial.X_inj theorem monomial_pow : monomial s a ^ e = monomial (e • s) (a ^ e) := AddMonoidAlgebra.single_pow e #align mv_polynomial.monomial_pow MvPolynomial.monomial_pow @[simp] theorem monomial_mul {s s' : σ →₀ ℕ} {a b : R} : monomial s a monomial s' b = monomial (s + s') (a * b) := AddMonoidAlgebra.single_mul_single #align mv_polynomial.monomial_mul MvPolynomial.monomial_mul variable (σ R) /-- `fun s ↦ monomial s 1` as a homomorphism. -/ def monomialOneHom : Multiplicative (σ →₀ ℕ) →* MvPolynomial σ R := AddMonoidAlgebra.of _ _ #align mv_polynomial.monomial_one_hom MvPolynomial.monomialOneHom variable {σ R} @[simp] theorem monomialOneHom_apply : monomialOneHom R σ s = (monomial s 1 : MvPolynomial σ R) := rfl #align mv_polynomial.monomial_one_hom_apply MvPolynomial.monomialOneHom_apply theorem X_pow_eq_monomial : X n ^ e = monomial (Finsupp.single n e) (1 : R) := by simp [X, monomial_pow] #align mv_polynomial.X_pow_eq_monomial MvPolynomial.X_pow_eq_monomial	Mathlib/Algebra/MvPolynomial/Basic.lean	330	331	theorem monomial_add_single : monomial (s + Finsupp.single n e) a = monomial s a * X n ^ e := by	rw [X_pow_eq_monomial, monomial_mul, mul_one]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import Mathlib.Algebra.Group.Indicator import Mathlib.Data.Finset.Piecewise import Mathlib.Data.Finset.Preimage #align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83" /-! # Big operators In this file we define products and sums indexed by finite sets (specifically, `Finset`). ## Notation We introduce the following notation. Let `s` be a `Finset α`, and `f : α → β` a function. * `∏ x ∈ s, f x` is notation for `Finset.prod s f` (assuming `β` is a `CommMonoid`) * `∑ x ∈ s, f x` is notation for `Finset.sum s f` (assuming `β` is an `AddCommMonoid`) * `∏ x, f x` is notation for `Finset.prod Finset.univ f` (assuming `α` is a `Fintype` and `β` is a `CommMonoid`) * `∑ x, f x` is notation for `Finset.sum Finset.univ f` (assuming `α` is a `Fintype` and `β` is an `AddCommMonoid`) ## Implementation Notes The first arguments in all definitions and lemmas is the codomain of the function of the big operator. This is necessary for the heuristic in `@[to_additive]`. See the documentation of `to_additive.attr` for more information. -/ -- TODO -- assert_not_exists AddCommMonoidWithOne assert_not_exists MonoidWithZero assert_not_exists MulAction variable {ι κ α β γ : Type} open Fin Function namespace Finset /-- `∏ x ∈ s, f x` is the product of `f x` as `x` ranges over the elements of the finite set `s`. -/ @[to_additive "`∑ x ∈ s, f x` is the sum of `f x` as `x` ranges over the elements of the finite set `s`."] protected def prod [CommMonoid β] (s : Finset α) (f : α → β) : β := (s.1.map f).prod #align finset.prod Finset.prod #align finset.sum Finset.sum @[to_additive (attr := simp)] theorem prod_mk [CommMonoid β] (s : Multiset α) (hs : s.Nodup) (f : α → β) : (⟨s, hs⟩ : Finset α).prod f = (s.map f).prod := rfl #align finset.prod_mk Finset.prod_mk #align finset.sum_mk Finset.sum_mk @[to_additive (attr := simp)] theorem prod_val [CommMonoid α] (s : Finset α) : s.1.prod = s.prod id := by rw [Finset.prod, Multiset.map_id] #align finset.prod_val Finset.prod_val #align finset.sum_val Finset.sum_val end Finset library_note "operator precedence of big operators"/-- There is no established mathematical convention for the operator precedence of big operators like `∏` and `∑`. We will have to make a choice. Online discussions, such as https://math.stackexchange.com/q/185538/30839 seem to suggest that `∏` and `∑` should have the same precedence, and that this should be somewhere between `` and `+`. The latter have precedence levels `70` and `65` respectively, and we therefore choose the level `67`. In practice, this means that parentheses should be placed as follows: ```lean ∑ k ∈ K, (a k + b k) = ∑ k ∈ K, a k + ∑ k ∈ K, b k → ∏ k ∈ K, a k * b k = (∏ k ∈ K, a k) * (∏ k ∈ K, b k) ``` (Example taken from page 490 of Knuth's Concrete Mathematics.) -/ namespace BigOperators open Batteries.ExtendedBinder Lean Meta -- TODO: contribute this modification back to `extBinder` /-- A `bigOpBinder` is like an `extBinder` and has the form `x`, `x : ty`, or `x pred` where `pred` is a `binderPred` like `< 2`. Unlike `extBinder`, `x` is a term. -/ syntax bigOpBinder := term:max ((" : " term) <\|> binderPred)? /-- A BigOperator binder in parentheses -/ syntax bigOpBinderParenthesized := " (" bigOpBinder ")" /-- A list of parenthesized binders -/ syntax bigOpBinderCollection := bigOpBinderParenthesized+ /-- A single (unparenthesized) binder, or a list of parenthesized binders -/ syntax bigOpBinders := bigOpBinderCollection <\|> (ppSpace bigOpBinder) /-- Collects additional binder/Finset pairs for the given `bigOpBinder`. Note: this is not extensible at the moment, unlike the usual `bigOpBinder` expansions. -/ def processBigOpBinder (processed : (Array (Term × Term))) (binder : TSyntax ``bigOpBinder) : MacroM (Array (Term × Term)) := set_option hygiene false in withRef binder do match binder with \| `(bigOpBinder\| $x:term) => match x with \| `(($a + $b = $n)) => -- Maybe this is too cute. return processed \|>.push (← `(⟨$a, $b⟩), ← `(Finset.Nat.antidiagonal $n)) \| _ => return processed \|>.push (x, ← ``(Finset.univ)) \| `(bigOpBinder\| $x : $t) => return processed \|>.push (x, ← ``((Finset.univ : Finset $t))) \| `(bigOpBinder\| $x ∈ $s) => return processed \|>.push (x, ← `(finset% $s)) \| `(bigOpBinder\| $x < $n) => return processed \|>.push (x, ← `(Finset.Iio $n)) \| `(bigOpBinder\| $x ≤ $n) => return processed \|>.push (x, ← `(Finset.Iic $n)) \| `(bigOpBinder\| $x > $n) => return processed \|>.push (x, ← `(Finset.Ioi $n)) \| `(bigOpBinder\| $x ≥ $n) => return processed \|>.push (x, ← `(Finset.Ici $n)) \| _ => Macro.throwUnsupported /-- Collects the binder/Finset pairs for the given `bigOpBinders`. -/ def processBigOpBinders (binders : TSyntax ``bigOpBinders) : MacroM (Array (Term × Term)) := match binders with \| `(bigOpBinders\| $b:bigOpBinder) => processBigOpBinder #[] b \| `(bigOpBinders\| $[($bs:bigOpBinder)]) => bs.foldlM processBigOpBinder #[] \| _ => Macro.throwUnsupported /-- Collect the binderIdents into a `⟨...⟩` expression. -/ def bigOpBindersPattern (processed : (Array (Term × Term))) : MacroM Term := do let ts := processed.map Prod.fst if ts.size == 1 then return ts[0]! else `(⟨$ts,⟩) /-- Collect the terms into a product of sets. -/ def bigOpBindersProd (processed : (Array (Term × Term))) : MacroM Term := do if processed.isEmpty then `((Finset.univ : Finset Unit)) else if processed.size == 1 then return processed[0]!.2 else processed.foldrM (fun s p => `(SProd.sprod $(s.2) $p)) processed.back.2 (start := processed.size - 1) /-- - `∑ x, f x` is notation for `Finset.sum Finset.univ f`. It is the sum of `f x`, where `x` ranges over the finite domain of `f`. - `∑ x ∈ s, f x` is notation for `Finset.sum s f`. It is the sum of `f x`, where `x` ranges over the finite set `s` (either a `Finset` or a `Set` with a `Fintype` instance). - `∑ x ∈ s with p x, f x` is notation for `Finset.sum (Finset.filter p s) f`. - `∑ (x ∈ s) (y ∈ t), f x y` is notation for `Finset.sum (s ×ˢ t) (fun ⟨x, y⟩ ↦ f x y)`. These support destructuring, for example `∑ ⟨x, y⟩ ∈ s ×ˢ t, f x y`. Notation: `"∑" bigOpBinders* ("with" term)? "," term` -/ syntax (name := bigsum) "∑ " bigOpBinders ("with " term)? ", " term:67 : term /-- - `∏ x, f x` is notation for `Finset.prod Finset.univ f`. It is the product of `f x`, where `x` ranges over the finite domain of `f`. - `∏ x ∈ s, f x` is notation for `Finset.prod s f`. It is the product of `f x`, where `x` ranges over the finite set `s` (either a `Finset` or a `Set` with a `Fintype` instance). - `∏ x ∈ s with p x, f x` is notation for `Finset.prod (Finset.filter p s) f`. - `∏ (x ∈ s) (y ∈ t), f x y` is notation for `Finset.prod (s ×ˢ t) (fun ⟨x, y⟩ ↦ f x y)`. These support destructuring, for example `∏ ⟨x, y⟩ ∈ s ×ˢ t, f x y`. Notation: `"∏" bigOpBinders* ("with" term)? "," term` -/ syntax (name := bigprod) "∏ " bigOpBinders ("with " term)? ", " term:67 : term macro_rules (kind := bigsum) \| `(∑ $bs:bigOpBinders $[with $p?]?, $v) => do let processed ← processBigOpBinders bs let x ← bigOpBindersPattern processed let s ← bigOpBindersProd processed match p? with \| some p => `(Finset.sum (Finset.filter (fun $x ↦ $p) $s) (fun $x ↦ $v)) \| none => `(Finset.sum $s (fun $x ↦ $v)) macro_rules (kind := bigprod) \| `(∏ $bs:bigOpBinders $[with $p?]?, $v) => do let processed ← processBigOpBinders bs let x ← bigOpBindersPattern processed let s ← bigOpBindersProd processed match p? with \| some p => `(Finset.prod (Finset.filter (fun $x ↦ $p) $s) (fun $x ↦ $v)) \| none => `(Finset.prod $s (fun $x ↦ $v)) /-- (Deprecated, use `∑ x ∈ s, f x`) `∑ x in s, f x` is notation for `Finset.sum s f`. It is the sum of `f x`, where `x` ranges over the finite set `s`. -/ syntax (name := bigsumin) "∑ " extBinder " in " term ", " term:67 : term macro_rules (kind := bigsumin) \| `(∑ $x:ident in $s, $r) => `(∑ $x:ident ∈ $s, $r) \| `(∑ $x:ident : $t in $s, $r) => `(∑ $x:ident ∈ ($s : Finset $t), $r) /-- (Deprecated, use `∏ x ∈ s, f x`) `∏ x in s, f x` is notation for `Finset.prod s f`. It is the product of `f x`, where `x` ranges over the finite set `s`. -/ syntax (name := bigprodin) "∏ " extBinder " in " term ", " term:67 : term macro_rules (kind := bigprodin) \| `(∏ $x:ident in $s, $r) => `(∏ $x:ident ∈ $s, $r) \| `(∏ $x:ident : $t in $s, $r) => `(∏ $x:ident ∈ ($s : Finset $t), $r) open Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr open Batteries.ExtendedBinder /-- Delaborator for `Finset.prod`. The `pp.piBinderTypes` option controls whether to show the domain type when the product is over `Finset.univ`. -/ @[delab app.Finset.prod] def delabFinsetProd : Delab := whenPPOption getPPNotation <\| withOverApp 5 <\| do let #[_, _, _, s, f] := (← getExpr).getAppArgs \| failure guard <\| f.isLambda let ppDomain ← getPPOption getPPPiBinderTypes let (i, body) ← withAppArg <\| withBindingBodyUnusedName fun i => do return (i, ← delab) if s.isAppOfArity ``Finset.univ 2 then let binder ← if ppDomain then let ty ← withNaryArg 0 delab `(bigOpBinder\| $(.mk i):ident : $ty) else `(bigOpBinder\| $(.mk i):ident) `(∏ $binder:bigOpBinder, $body) else let ss ← withNaryArg 3 <\| delab `(∏ $(.mk i):ident ∈ $ss, $body) /-- Delaborator for `Finset.sum`. The `pp.piBinderTypes` option controls whether to show the domain type when the sum is over `Finset.univ`. -/ @[delab app.Finset.sum] def delabFinsetSum : Delab := whenPPOption getPPNotation <\| withOverApp 5 <\| do let #[_, _, _, s, f] := (← getExpr).getAppArgs \| failure guard <\| f.isLambda let ppDomain ← getPPOption getPPPiBinderTypes let (i, body) ← withAppArg <\| withBindingBodyUnusedName fun i => do return (i, ← delab) if s.isAppOfArity ``Finset.univ 2 then let binder ← if ppDomain then let ty ← withNaryArg 0 delab `(bigOpBinder\| $(.mk i):ident : $ty) else `(bigOpBinder\| $(.mk i):ident) `(∑ $binder:bigOpBinder, $body) else let ss ← withNaryArg 3 <\| delab `(∑ $(.mk i):ident ∈ $ss, $body) end BigOperators namespace Finset variable {s s₁ s₂ : Finset α} {a : α} {f g : α → β} @[to_additive] theorem prod_eq_multiset_prod [CommMonoid β] (s : Finset α) (f : α → β) : ∏ x ∈ s, f x = (s.1.map f).prod := rfl #align finset.prod_eq_multiset_prod Finset.prod_eq_multiset_prod #align finset.sum_eq_multiset_sum Finset.sum_eq_multiset_sum @[to_additive (attr := simp)] lemma prod_map_val [CommMonoid β] (s : Finset α) (f : α → β) : (s.1.map f).prod = ∏ a ∈ s, f a := rfl #align finset.prod_map_val Finset.prod_map_val #align finset.sum_map_val Finset.sum_map_val @[to_additive] theorem prod_eq_fold [CommMonoid β] (s : Finset α) (f : α → β) : ∏ x ∈ s, f x = s.fold ((· * ·) : β → β → β) 1 f := rfl #align finset.prod_eq_fold Finset.prod_eq_fold #align finset.sum_eq_fold Finset.sum_eq_fold @[simp] theorem sum_multiset_singleton (s : Finset α) : (s.sum fun x => {x}) = s.val := by simp only [sum_eq_multiset_sum, Multiset.sum_map_singleton] #align finset.sum_multiset_singleton Finset.sum_multiset_singleton end Finset @[to_additive (attr := simp)] theorem map_prod [CommMonoid β] [CommMonoid γ] {G : Type} [FunLike G β γ] [MonoidHomClass G β γ] (g : G) (f : α → β) (s : Finset α) : g (∏ x ∈ s, f x) = ∏ x ∈ s, g (f x) := by simp only [Finset.prod_eq_multiset_prod, map_multiset_prod, Multiset.map_map]; rfl #align map_prod map_prod #align map_sum map_sum @[to_additive] theorem MonoidHom.coe_finset_prod [MulOneClass β] [CommMonoid γ] (f : α → β → γ) (s : Finset α) : ⇑(∏ x ∈ s, f x) = ∏ x ∈ s, ⇑(f x) := map_prod (MonoidHom.coeFn β γ) _ _ #align monoid_hom.coe_finset_prod MonoidHom.coe_finset_prod #align add_monoid_hom.coe_finset_sum AddMonoidHom.coe_finset_sum /-- See also `Finset.prod_apply`, with the same conclusion but with the weaker hypothesis `f : α → β → γ` -/ @[to_additive (attr := simp) "See also `Finset.sum_apply`, with the same conclusion but with the weaker hypothesis `f : α → β → γ`"] theorem MonoidHom.finset_prod_apply [MulOneClass β] [CommMonoid γ] (f : α → β →* γ) (s : Finset α) (b : β) : (∏ x ∈ s, f x) b = ∏ x ∈ s, f x b := map_prod (MonoidHom.eval b) _ _ #align monoid_hom.finset_prod_apply MonoidHom.finset_prod_apply #align add_monoid_hom.finset_sum_apply AddMonoidHom.finset_sum_apply variable {s s₁ s₂ : Finset α} {a : α} {f g : α → β} namespace Finset section CommMonoid variable [CommMonoid β] @[to_additive (attr := simp)] theorem prod_empty : ∏ x ∈ ∅, f x = 1 := rfl #align finset.prod_empty Finset.prod_empty #align finset.sum_empty Finset.sum_empty @[to_additive] theorem prod_of_empty [IsEmpty α] (s : Finset α) : ∏ i ∈ s, f i = 1 := by rw [eq_empty_of_isEmpty s, prod_empty] #align finset.prod_of_empty Finset.prod_of_empty #align finset.sum_of_empty Finset.sum_of_empty @[to_additive (attr := simp)] theorem prod_cons (h : a ∉ s) : ∏ x ∈ cons a s h, f x = f a * ∏ x ∈ s, f x := fold_cons h #align finset.prod_cons Finset.prod_cons #align finset.sum_cons Finset.sum_cons @[to_additive (attr := simp)] theorem prod_insert [DecidableEq α] : a ∉ s → ∏ x ∈ insert a s, f x = f a * ∏ x ∈ s, f x := fold_insert #align finset.prod_insert Finset.prod_insert #align finset.sum_insert Finset.sum_insert /-- The product of `f` over `insert a s` is the same as the product over `s`, as long as `a` is in `s` or `f a = 1`. -/ @[to_additive (attr := simp) "The sum of `f` over `insert a s` is the same as the sum over `s`, as long as `a` is in `s` or `f a = 0`."] theorem prod_insert_of_eq_one_if_not_mem [DecidableEq α] (h : a ∉ s → f a = 1) : ∏ x ∈ insert a s, f x = ∏ x ∈ s, f x := by by_cases hm : a ∈ s · simp_rw [insert_eq_of_mem hm] · rw [prod_insert hm, h hm, one_mul] #align finset.prod_insert_of_eq_one_if_not_mem Finset.prod_insert_of_eq_one_if_not_mem #align finset.sum_insert_of_eq_zero_if_not_mem Finset.sum_insert_of_eq_zero_if_not_mem /-- The product of `f` over `insert a s` is the same as the product over `s`, as long as `f a = 1`. -/ @[to_additive (attr := simp) "The sum of `f` over `insert a s` is the same as the sum over `s`, as long as `f a = 0`."] theorem prod_insert_one [DecidableEq α] (h : f a = 1) : ∏ x ∈ insert a s, f x = ∏ x ∈ s, f x := prod_insert_of_eq_one_if_not_mem fun _ => h #align finset.prod_insert_one Finset.prod_insert_one #align finset.sum_insert_zero Finset.sum_insert_zero @[to_additive] theorem prod_insert_div {M : Type} [CommGroup M] [DecidableEq α] (ha : a ∉ s) {f : α → M} : (∏ x ∈ insert a s, f x) / f a = ∏ x ∈ s, f x := by simp [ha] @[to_additive (attr := simp)] theorem prod_singleton (f : α → β) (a : α) : ∏ x ∈ singleton a, f x = f a := Eq.trans fold_singleton <\| mul_one _ #align finset.prod_singleton Finset.prod_singleton #align finset.sum_singleton Finset.sum_singleton @[to_additive] theorem prod_pair [DecidableEq α] {a b : α} (h : a ≠ b) : (∏ x ∈ ({a, b} : Finset α), f x) = f a f b := by rw [prod_insert (not_mem_singleton.2 h), prod_singleton] #align finset.prod_pair Finset.prod_pair #align finset.sum_pair Finset.sum_pair @[to_additive (attr := simp)] theorem prod_const_one : (∏ _x ∈ s, (1 : β)) = 1 := by simp only [Finset.prod, Multiset.map_const', Multiset.prod_replicate, one_pow] #align finset.prod_const_one Finset.prod_const_one #align finset.sum_const_zero Finset.sum_const_zero @[to_additive (attr := simp)] theorem prod_image [DecidableEq α] {s : Finset γ} {g : γ → α} : (∀ x ∈ s, ∀ y ∈ s, g x = g y → x = y) → ∏ x ∈ s.image g, f x = ∏ x ∈ s, f (g x) := fold_image #align finset.prod_image Finset.prod_image #align finset.sum_image Finset.sum_image @[to_additive (attr := simp)] theorem prod_map (s : Finset α) (e : α ↪ γ) (f : γ → β) : ∏ x ∈ s.map e, f x = ∏ x ∈ s, f (e x) := by rw [Finset.prod, Finset.map_val, Multiset.map_map]; rfl #align finset.prod_map Finset.prod_map #align finset.sum_map Finset.sum_map @[to_additive] lemma prod_attach (s : Finset α) (f : α → β) : ∏ x ∈ s.attach, f x = ∏ x ∈ s, f x := by classical rw [← prod_image Subtype.coe_injective.injOn, attach_image_val] #align finset.prod_attach Finset.prod_attach #align finset.sum_attach Finset.sum_attach @[to_additive (attr := congr)] theorem prod_congr (h : s₁ = s₂) : (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g := by rw [h]; exact fold_congr #align finset.prod_congr Finset.prod_congr #align finset.sum_congr Finset.sum_congr @[to_additive] theorem prod_eq_one {f : α → β} {s : Finset α} (h : ∀ x ∈ s, f x = 1) : ∏ x ∈ s, f x = 1 := calc ∏ x ∈ s, f x = ∏ _x ∈ s, 1 := Finset.prod_congr rfl h _ = 1 := Finset.prod_const_one #align finset.prod_eq_one Finset.prod_eq_one #align finset.sum_eq_zero Finset.sum_eq_zero @[to_additive] theorem prod_disjUnion (h) : ∏ x ∈ s₁.disjUnion s₂ h, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := by refine Eq.trans ?_ (fold_disjUnion h) rw [one_mul] rfl #align finset.prod_disj_union Finset.prod_disjUnion #align finset.sum_disj_union Finset.sum_disjUnion @[to_additive] theorem prod_disjiUnion (s : Finset ι) (t : ι → Finset α) (h) : ∏ x ∈ s.disjiUnion t h, f x = ∏ i ∈ s, ∏ x ∈ t i, f x := by refine Eq.trans ?_ (fold_disjiUnion h) dsimp [Finset.prod, Multiset.prod, Multiset.fold, Finset.disjUnion, Finset.fold] congr exact prod_const_one.symm #align finset.prod_disj_Union Finset.prod_disjiUnion #align finset.sum_disj_Union Finset.sum_disjiUnion @[to_additive] theorem prod_union_inter [DecidableEq α] : (∏ x ∈ s₁ ∪ s₂, f x) * ∏ x ∈ s₁ ∩ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := fold_union_inter #align finset.prod_union_inter Finset.prod_union_inter #align finset.sum_union_inter Finset.sum_union_inter @[to_additive] theorem prod_union [DecidableEq α] (h : Disjoint s₁ s₂) : ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := by rw [← prod_union_inter, disjoint_iff_inter_eq_empty.mp h]; exact (mul_one _).symm #align finset.prod_union Finset.prod_union #align finset.sum_union Finset.sum_union @[to_additive] theorem prod_filter_mul_prod_filter_not (s : Finset α) (p : α → Prop) [DecidablePred p] [∀ x, Decidable (¬p x)] (f : α → β) : (∏ x ∈ s.filter p, f x) * ∏ x ∈ s.filter fun x => ¬p x, f x = ∏ x ∈ s, f x := by have := Classical.decEq α rw [← prod_union (disjoint_filter_filter_neg s s p), filter_union_filter_neg_eq] #align finset.prod_filter_mul_prod_filter_not Finset.prod_filter_mul_prod_filter_not #align finset.sum_filter_add_sum_filter_not Finset.sum_filter_add_sum_filter_not section ToList @[to_additive (attr := simp)] theorem prod_to_list (s : Finset α) (f : α → β) : (s.toList.map f).prod = s.prod f := by rw [Finset.prod, ← Multiset.prod_coe, ← Multiset.map_coe, Finset.coe_toList] #align finset.prod_to_list Finset.prod_to_list #align finset.sum_to_list Finset.sum_to_list end ToList @[to_additive] theorem _root_.Equiv.Perm.prod_comp (σ : Equiv.Perm α) (s : Finset α) (f : α → β) (hs : { a \| σ a ≠ a } ⊆ s) : (∏ x ∈ s, f (σ x)) = ∏ x ∈ s, f x := by convert (prod_map s σ.toEmbedding f).symm exact (map_perm hs).symm #align equiv.perm.prod_comp Equiv.Perm.prod_comp #align equiv.perm.sum_comp Equiv.Perm.sum_comp @[to_additive] theorem _root_.Equiv.Perm.prod_comp' (σ : Equiv.Perm α) (s : Finset α) (f : α → α → β) (hs : { a \| σ a ≠ a } ⊆ s) : (∏ x ∈ s, f (σ x) x) = ∏ x ∈ s, f x (σ.symm x) := by convert σ.prod_comp s (fun x => f x (σ.symm x)) hs rw [Equiv.symm_apply_apply] #align equiv.perm.prod_comp' Equiv.Perm.prod_comp' #align equiv.perm.sum_comp' Equiv.Perm.sum_comp' /-- A product over all subsets of `s ∪ {x}` is obtained by multiplying the product over all subsets of `s`, and over all subsets of `s` to which one adds `x`. -/ @[to_additive "A sum over all subsets of `s ∪ {x}` is obtained by summing the sum over all subsets of `s`, and over all subsets of `s` to which one adds `x`."] lemma prod_powerset_insert [DecidableEq α] (ha : a ∉ s) (f : Finset α → β) : ∏ t ∈ (insert a s).powerset, f t = (∏ t ∈ s.powerset, f t) * ∏ t ∈ s.powerset, f (insert a t) := by rw [powerset_insert, prod_union, prod_image] · exact insert_erase_invOn.2.injOn.mono fun t ht ↦ not_mem_mono (mem_powerset.1 ht) ha · aesop (add simp [disjoint_left, insert_subset_iff]) #align finset.prod_powerset_insert Finset.prod_powerset_insert #align finset.sum_powerset_insert Finset.sum_powerset_insert /-- A product over all subsets of `s ∪ {x}` is obtained by multiplying the product over all subsets of `s`, and over all subsets of `s` to which one adds `x`. -/ @[to_additive "A sum over all subsets of `s ∪ {x}` is obtained by summing the sum over all subsets of `s`, and over all subsets of `s` to which one adds `x`."] lemma prod_powerset_cons (ha : a ∉ s) (f : Finset α → β) : ∏ t ∈ (s.cons a ha).powerset, f t = (∏ t ∈ s.powerset, f t) * ∏ t ∈ s.powerset.attach, f (cons a t $ not_mem_mono (mem_powerset.1 t.2) ha) := by classical simp_rw [cons_eq_insert] rw [prod_powerset_insert ha, prod_attach _ fun t ↦ f (insert a t)] /-- A product over `powerset s` is equal to the double product over sets of subsets of `s` with `card s = k`, for `k = 1, ..., card s`. -/ @[to_additive "A sum over `powerset s` is equal to the double sum over sets of subsets of `s` with `card s = k`, for `k = 1, ..., card s`"] lemma prod_powerset (s : Finset α) (f : Finset α → β) : ∏ t ∈ powerset s, f t = ∏ j ∈ range (card s + 1), ∏ t ∈ powersetCard j s, f t := by rw [powerset_card_disjiUnion, prod_disjiUnion] #align finset.prod_powerset Finset.prod_powerset #align finset.sum_powerset Finset.sum_powerset end CommMonoid end Finset section open Finset variable [Fintype α] [CommMonoid β] @[to_additive] theorem IsCompl.prod_mul_prod {s t : Finset α} (h : IsCompl s t) (f : α → β) : (∏ i ∈ s, f i) * ∏ i ∈ t, f i = ∏ i, f i := (Finset.prod_disjUnion h.disjoint).symm.trans <\| by classical rw [Finset.disjUnion_eq_union, ← Finset.sup_eq_union, h.sup_eq_top]; rfl #align is_compl.prod_mul_prod IsCompl.prod_mul_prod #align is_compl.sum_add_sum IsCompl.sum_add_sum end namespace Finset section CommMonoid variable [CommMonoid β] /-- Multiplying the products of a function over `s` and over `sᶜ` gives the whole product. For a version expressed with subtypes, see `Fintype.prod_subtype_mul_prod_subtype`. -/ @[to_additive "Adding the sums of a function over `s` and over `sᶜ` gives the whole sum. For a version expressed with subtypes, see `Fintype.sum_subtype_add_sum_subtype`. "] theorem prod_mul_prod_compl [Fintype α] [DecidableEq α] (s : Finset α) (f : α → β) : (∏ i ∈ s, f i) * ∏ i ∈ sᶜ, f i = ∏ i, f i := IsCompl.prod_mul_prod isCompl_compl f #align finset.prod_mul_prod_compl Finset.prod_mul_prod_compl #align finset.sum_add_sum_compl Finset.sum_add_sum_compl @[to_additive] theorem prod_compl_mul_prod [Fintype α] [DecidableEq α] (s : Finset α) (f : α → β) : (∏ i ∈ sᶜ, f i) * ∏ i ∈ s, f i = ∏ i, f i := (@isCompl_compl _ s _).symm.prod_mul_prod f #align finset.prod_compl_mul_prod Finset.prod_compl_mul_prod #align finset.sum_compl_add_sum Finset.sum_compl_add_sum @[to_additive] theorem prod_sdiff [DecidableEq α] (h : s₁ ⊆ s₂) : (∏ x ∈ s₂ \ s₁, f x) * ∏ x ∈ s₁, f x = ∏ x ∈ s₂, f x := by rw [← prod_union sdiff_disjoint, sdiff_union_of_subset h] #align finset.prod_sdiff Finset.prod_sdiff #align finset.sum_sdiff Finset.sum_sdiff @[to_additive] theorem prod_subset_one_on_sdiff [DecidableEq α] (h : s₁ ⊆ s₂) (hg : ∀ x ∈ s₂ \ s₁, g x = 1) (hfg : ∀ x ∈ s₁, f x = g x) : ∏ i ∈ s₁, f i = ∏ i ∈ s₂, g i := by rw [← prod_sdiff h, prod_eq_one hg, one_mul] exact prod_congr rfl hfg #align finset.prod_subset_one_on_sdiff Finset.prod_subset_one_on_sdiff #align finset.sum_subset_zero_on_sdiff Finset.sum_subset_zero_on_sdiff @[to_additive] theorem prod_subset (h : s₁ ⊆ s₂) (hf : ∀ x ∈ s₂, x ∉ s₁ → f x = 1) : ∏ x ∈ s₁, f x = ∏ x ∈ s₂, f x := haveI := Classical.decEq α prod_subset_one_on_sdiff h (by simpa) fun _ _ => rfl #align finset.prod_subset Finset.prod_subset #align finset.sum_subset Finset.sum_subset @[to_additive (attr := simp)] theorem prod_disj_sum (s : Finset α) (t : Finset γ) (f : Sum α γ → β) : ∏ x ∈ s.disjSum t, f x = (∏ x ∈ s, f (Sum.inl x)) * ∏ x ∈ t, f (Sum.inr x) := by rw [← map_inl_disjUnion_map_inr, prod_disjUnion, prod_map, prod_map] rfl #align finset.prod_disj_sum Finset.prod_disj_sum #align finset.sum_disj_sum Finset.sum_disj_sum @[to_additive] theorem prod_sum_elim (s : Finset α) (t : Finset γ) (f : α → β) (g : γ → β) : ∏ x ∈ s.disjSum t, Sum.elim f g x = (∏ x ∈ s, f x) * ∏ x ∈ t, g x := by simp #align finset.prod_sum_elim Finset.prod_sum_elim #align finset.sum_sum_elim Finset.sum_sum_elim @[to_additive] theorem prod_biUnion [DecidableEq α] {s : Finset γ} {t : γ → Finset α} (hs : Set.PairwiseDisjoint (↑s) t) : ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i := by rw [← disjiUnion_eq_biUnion _ _ hs, prod_disjiUnion] #align finset.prod_bUnion Finset.prod_biUnion #align finset.sum_bUnion Finset.sum_biUnion /-- Product over a sigma type equals the product of fiberwise products. For rewriting in the reverse direction, use `Finset.prod_sigma'`. -/ @[to_additive "Sum over a sigma type equals the sum of fiberwise sums. For rewriting in the reverse direction, use `Finset.sum_sigma'`"] theorem prod_sigma {σ : α → Type} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : Sigma σ → β) : ∏ x ∈ s.sigma t, f x = ∏ a ∈ s, ∏ s ∈ t a, f ⟨a, s⟩ := by simp_rw [← disjiUnion_map_sigma_mk, prod_disjiUnion, prod_map, Function.Embedding.sigmaMk_apply] #align finset.prod_sigma Finset.prod_sigma #align finset.sum_sigma Finset.sum_sigma @[to_additive] theorem prod_sigma' {σ : α → Type} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : ∀ a, σ a → β) : (∏ a ∈ s, ∏ s ∈ t a, f a s) = ∏ x ∈ s.sigma t, f x.1 x.2 := Eq.symm <\| prod_sigma s t fun x => f x.1 x.2 #align finset.prod_sigma' Finset.prod_sigma' #align finset.sum_sigma' Finset.sum_sigma' section bij variable {ι κ α : Type} [CommMonoid α] {s : Finset ι} {t : Finset κ} {f : ι → α} {g : κ → α} /-- Reorder a product. The difference with `Finset.prod_bij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.prod_nbij` is that the bijection is allowed to use membership of the domain of the product, rather than being a non-dependent function. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_bij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.sum_nbij` is that the bijection is allowed to use membership of the domain of the sum, rather than being a non-dependent function."] theorem prod_bij (i : ∀ a ∈ s, κ) (hi : ∀ a ha, i a ha ∈ t) (i_inj : ∀ a₁ ha₁ a₂ ha₂, i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂) (i_surj : ∀ b ∈ t, ∃ a ha, i a ha = b) (h : ∀ a ha, f a = g (i a ha)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := congr_arg Multiset.prod (Multiset.map_eq_map_of_bij_of_nodup f g s.2 t.2 i hi i_inj i_surj h) #align finset.prod_bij Finset.prod_bij #align finset.sum_bij Finset.sum_bij /-- Reorder a product. The difference with `Finset.prod_bij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.prod_nbij'` is that the bijection and its inverse are allowed to use membership of the domains of the products, rather than being non-dependent functions. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_bij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.sum_nbij'` is that the bijection and its inverse are allowed to use membership of the domains of the sums, rather than being non-dependent functions."] theorem prod_bij' (i : ∀ a ∈ s, κ) (j : ∀ a ∈ t, ι) (hi : ∀ a ha, i a ha ∈ t) (hj : ∀ a ha, j a ha ∈ s) (left_inv : ∀ a ha, j (i a ha) (hi a ha) = a) (right_inv : ∀ a ha, i (j a ha) (hj a ha) = a) (h : ∀ a ha, f a = g (i a ha)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := by refine prod_bij i hi (fun a1 h1 a2 h2 eq ↦ ?_) (fun b hb ↦ ⟨_, hj b hb, right_inv b hb⟩) h rw [← left_inv a1 h1, ← left_inv a2 h2] simp only [eq] #align finset.prod_bij' Finset.prod_bij' #align finset.sum_bij' Finset.sum_bij' /-- Reorder a product. The difference with `Finset.prod_nbij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.prod_bij` is that the bijection is a non-dependent function, rather than being allowed to use membership of the domain of the product. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_nbij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.sum_bij` is that the bijection is a non-dependent function, rather than being allowed to use membership of the domain of the sum."] lemma prod_nbij (i : ι → κ) (hi : ∀ a ∈ s, i a ∈ t) (i_inj : (s : Set ι).InjOn i) (i_surj : (s : Set ι).SurjOn i t) (h : ∀ a ∈ s, f a = g (i a)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := prod_bij (fun a _ ↦ i a) hi i_inj (by simpa using i_surj) h /-- Reorder a product. The difference with `Finset.prod_nbij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.prod_bij'` is that the bijection and its inverse are non-dependent functions, rather than being allowed to use membership of the domains of the products. The difference with `Finset.prod_equiv` is that bijectivity is only required to hold on the domains of the products, rather than on the entire types. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_nbij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.sum_bij'` is that the bijection and its inverse are non-dependent functions, rather than being allowed to use membership of the domains of the sums. The difference with `Finset.sum_equiv` is that bijectivity is only required to hold on the domains of the sums, rather than on the entire types."] lemma prod_nbij' (i : ι → κ) (j : κ → ι) (hi : ∀ a ∈ s, i a ∈ t) (hj : ∀ a ∈ t, j a ∈ s) (left_inv : ∀ a ∈ s, j (i a) = a) (right_inv : ∀ a ∈ t, i (j a) = a) (h : ∀ a ∈ s, f a = g (i a)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := prod_bij' (fun a _ ↦ i a) (fun b _ ↦ j b) hi hj left_inv right_inv h /-- Specialization of `Finset.prod_nbij'` that automatically fills in most arguments. See `Fintype.prod_equiv` for the version where `s` and `t` are `univ`. -/ @[to_additive "`Specialization of `Finset.sum_nbij'` that automatically fills in most arguments. See `Fintype.sum_equiv` for the version where `s` and `t` are `univ`."] lemma prod_equiv (e : ι ≃ κ) (hst : ∀ i, i ∈ s ↔ e i ∈ t) (hfg : ∀ i ∈ s, f i = g (e i)) : ∏ i ∈ s, f i = ∏ i ∈ t, g i := by refine prod_nbij' e e.symm ?_ ?_ ?_ ?_ hfg <;> simp [hst] #align finset.equiv.prod_comp_finset Finset.prod_equiv #align finset.equiv.sum_comp_finset Finset.sum_equiv /-- Specialization of `Finset.prod_bij` that automatically fills in most arguments. See `Fintype.prod_bijective` for the version where `s` and `t` are `univ`. -/ @[to_additive "`Specialization of `Finset.sum_bij` that automatically fills in most arguments. See `Fintype.sum_bijective` for the version where `s` and `t` are `univ`."] lemma prod_bijective (e : ι → κ) (he : e.Bijective) (hst : ∀ i, i ∈ s ↔ e i ∈ t) (hfg : ∀ i ∈ s, f i = g (e i)) : ∏ i ∈ s, f i = ∏ i ∈ t, g i := prod_equiv (.ofBijective e he) hst hfg @[to_additive] lemma prod_of_injOn (e : ι → κ) (he : Set.InjOn e s) (hest : Set.MapsTo e s t) (h' : ∀ i ∈ t, i ∉ e '' s → g i = 1) (h : ∀ i ∈ s, f i = g (e i)) : ∏ i ∈ s, f i = ∏ j ∈ t, g j := by classical exact (prod_nbij e (fun a ↦ mem_image_of_mem e) he (by simp [Set.surjOn_image]) h).trans <\| prod_subset (image_subset_iff.2 hest) <\| by simpa using h' variable [DecidableEq κ] @[to_additive] lemma prod_fiberwise_eq_prod_filter (s : Finset ι) (t : Finset κ) (g : ι → κ) (f : ι → α) : ∏ j ∈ t, ∏ i ∈ s.filter fun i ↦ g i = j, f i = ∏ i ∈ s.filter fun i ↦ g i ∈ t, f i := by rw [← prod_disjiUnion, disjiUnion_filter_eq] @[to_additive] lemma prod_fiberwise_eq_prod_filter' (s : Finset ι) (t : Finset κ) (g : ι → κ) (f : κ → α) : ∏ j ∈ t, ∏ _i ∈ s.filter fun i ↦ g i = j, f j = ∏ i ∈ s.filter fun i ↦ g i ∈ t, f (g i) := by calc _ = ∏ j ∈ t, ∏ i ∈ s.filter fun i ↦ g i = j, f (g i) := prod_congr rfl fun j _ ↦ prod_congr rfl fun i hi ↦ by rw [(mem_filter.1 hi).2] _ = _ := prod_fiberwise_eq_prod_filter _ _ _ _ @[to_additive] lemma prod_fiberwise_of_maps_to {g : ι → κ} (h : ∀ i ∈ s, g i ∈ t) (f : ι → α) : ∏ j ∈ t, ∏ i ∈ s.filter fun i ↦ g i = j, f i = ∏ i ∈ s, f i := by rw [← prod_disjiUnion, disjiUnion_filter_eq_of_maps_to h] #align finset.prod_fiberwise_of_maps_to Finset.prod_fiberwise_of_maps_to #align finset.sum_fiberwise_of_maps_to Finset.sum_fiberwise_of_maps_to @[to_additive] lemma prod_fiberwise_of_maps_to' {g : ι → κ} (h : ∀ i ∈ s, g i ∈ t) (f : κ → α) : ∏ j ∈ t, ∏ _i ∈ s.filter fun i ↦ g i = j, f j = ∏ i ∈ s, f (g i) := by calc _ = ∏ y ∈ t, ∏ x ∈ s.filter fun x ↦ g x = y, f (g x) := prod_congr rfl fun y _ ↦ prod_congr rfl fun x hx ↦ by rw [(mem_filter.1 hx).2] _ = _ := prod_fiberwise_of_maps_to h _ variable [Fintype κ] @[to_additive] lemma prod_fiberwise (s : Finset ι) (g : ι → κ) (f : ι → α) : ∏ j, ∏ i ∈ s.filter fun i ↦ g i = j, f i = ∏ i ∈ s, f i := prod_fiberwise_of_maps_to (fun _ _ ↦ mem_univ _) _ #align finset.prod_fiberwise Finset.prod_fiberwise #align finset.sum_fiberwise Finset.sum_fiberwise @[to_additive] lemma prod_fiberwise' (s : Finset ι) (g : ι → κ) (f : κ → α) : ∏ j, ∏ _i ∈ s.filter fun i ↦ g i = j, f j = ∏ i ∈ s, f (g i) := prod_fiberwise_of_maps_to' (fun _ _ ↦ mem_univ _) _ end bij /-- Taking a product over `univ.pi t` is the same as taking the product over `Fintype.piFinset t`. `univ.pi t` and `Fintype.piFinset t` are essentially the same `Finset`, but differ in the type of their element, `univ.pi t` is a `Finset (Π a ∈ univ, t a)` and `Fintype.piFinset t` is a `Finset (Π a, t a)`. -/ @[to_additive "Taking a sum over `univ.pi t` is the same as taking the sum over `Fintype.piFinset t`. `univ.pi t` and `Fintype.piFinset t` are essentially the same `Finset`, but differ in the type of their element, `univ.pi t` is a `Finset (Π a ∈ univ, t a)` and `Fintype.piFinset t` is a `Finset (Π a, t a)`."] lemma prod_univ_pi [DecidableEq ι] [Fintype ι] {κ : ι → Type} (t : ∀ i, Finset (κ i)) (f : (∀ i ∈ (univ : Finset ι), κ i) → β) : ∏ x ∈ univ.pi t, f x = ∏ x ∈ Fintype.piFinset t, f fun a _ ↦ x a := by apply prod_nbij' (fun x i ↦ x i $ mem_univ _) (fun x i _ ↦ x i) <;> simp #align finset.prod_univ_pi Finset.prod_univ_pi #align finset.sum_univ_pi Finset.sum_univ_pi @[to_additive (attr := simp)] lemma prod_diag [DecidableEq α] (s : Finset α) (f : α × α → β) : ∏ i ∈ s.diag, f i = ∏ i ∈ s, f (i, i) := by apply prod_nbij' Prod.fst (fun i ↦ (i, i)) <;> simp @[to_additive] theorem prod_finset_product (r : Finset (γ × α)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : γ × α, p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1) {f : γ × α → β} : ∏ p ∈ r, f p = ∏ c ∈ s, ∏ a ∈ t c, f (c, a) := by refine Eq.trans ?_ (prod_sigma s t fun p => f (p.1, p.2)) apply prod_equiv (Equiv.sigmaEquivProd _ _).symm <;> simp [h] #align finset.prod_finset_product Finset.prod_finset_product #align finset.sum_finset_product Finset.sum_finset_product @[to_additive] theorem prod_finset_product' (r : Finset (γ × α)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : γ × α, p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1) {f : γ → α → β} : ∏ p ∈ r, f p.1 p.2 = ∏ c ∈ s, ∏ a ∈ t c, f c a := prod_finset_product r s t h #align finset.prod_finset_product' Finset.prod_finset_product' #align finset.sum_finset_product' Finset.sum_finset_product' @[to_additive] theorem prod_finset_product_right (r : Finset (α × γ)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : α × γ, p ∈ r ↔ p.2 ∈ s ∧ p.1 ∈ t p.2) {f : α × γ → β} : ∏ p ∈ r, f p = ∏ c ∈ s, ∏ a ∈ t c, f (a, c) := by refine Eq.trans ?_ (prod_sigma s t fun p => f (p.2, p.1)) apply prod_equiv ((Equiv.prodComm _ _).trans (Equiv.sigmaEquivProd _ _).symm) <;> simp [h] #align finset.prod_finset_product_right Finset.prod_finset_product_right #align finset.sum_finset_product_right Finset.sum_finset_product_right @[to_additive] theorem prod_finset_product_right' (r : Finset (α × γ)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : α × γ, p ∈ r ↔ p.2 ∈ s ∧ p.1 ∈ t p.2) {f : α → γ → β} : ∏ p ∈ r, f p.1 p.2 = ∏ c ∈ s, ∏ a ∈ t c, f a c := prod_finset_product_right r s t h #align finset.prod_finset_product_right' Finset.prod_finset_product_right' #align finset.sum_finset_product_right' Finset.sum_finset_product_right' @[to_additive] theorem prod_image' [DecidableEq α] {s : Finset γ} {g : γ → α} (h : γ → β) (eq : ∀ c ∈ s, f (g c) = ∏ x ∈ s.filter fun c' => g c' = g c, h x) : ∏ x ∈ s.image g, f x = ∏ x ∈ s, h x := calc ∏ x ∈ s.image g, f x = ∏ x ∈ s.image g, ∏ x ∈ s.filter fun c' => g c' = x, h x := (prod_congr rfl) fun _x hx => let ⟨c, hcs, hc⟩ := mem_image.1 hx hc ▸ eq c hcs _ = ∏ x ∈ s, h x := prod_fiberwise_of_maps_to (fun _x => mem_image_of_mem g) _ #align finset.prod_image' Finset.prod_image' #align finset.sum_image' Finset.sum_image' @[to_additive] theorem prod_mul_distrib : ∏ x ∈ s, f x * g x = (∏ x ∈ s, f x) * ∏ x ∈ s, g x := Eq.trans (by rw [one_mul]; rfl) fold_op_distrib #align finset.prod_mul_distrib Finset.prod_mul_distrib #align finset.sum_add_distrib Finset.sum_add_distrib @[to_additive] lemma prod_mul_prod_comm (f g h i : α → β) : (∏ a ∈ s, f a * g a) * ∏ a ∈ s, h a * i a = (∏ a ∈ s, f a * h a) * ∏ a ∈ s, g a * i a := by simp_rw [prod_mul_distrib, mul_mul_mul_comm] @[to_additive] theorem prod_product {s : Finset γ} {t : Finset α} {f : γ × α → β} : ∏ x ∈ s ×ˢ t, f x = ∏ x ∈ s, ∏ y ∈ t, f (x, y) := prod_finset_product (s ×ˢ t) s (fun _a => t) fun _p => mem_product #align finset.prod_product Finset.prod_product #align finset.sum_product Finset.sum_product /-- An uncurried version of `Finset.prod_product`. -/ @[to_additive "An uncurried version of `Finset.sum_product`"] theorem prod_product' {s : Finset γ} {t : Finset α} {f : γ → α → β} : ∏ x ∈ s ×ˢ t, f x.1 x.2 = ∏ x ∈ s, ∏ y ∈ t, f x y := prod_product #align finset.prod_product' Finset.prod_product' #align finset.sum_product' Finset.sum_product' @[to_additive] theorem prod_product_right {s : Finset γ} {t : Finset α} {f : γ × α → β} : ∏ x ∈ s ×ˢ t, f x = ∏ y ∈ t, ∏ x ∈ s, f (x, y) := prod_finset_product_right (s ×ˢ t) t (fun _a => s) fun _p => mem_product.trans and_comm #align finset.prod_product_right Finset.prod_product_right #align finset.sum_product_right Finset.sum_product_right /-- An uncurried version of `Finset.prod_product_right`. -/ @[to_additive "An uncurried version of `Finset.sum_product_right`"] theorem prod_product_right' {s : Finset γ} {t : Finset α} {f : γ → α → β} : ∏ x ∈ s ×ˢ t, f x.1 x.2 = ∏ y ∈ t, ∏ x ∈ s, f x y := prod_product_right #align finset.prod_product_right' Finset.prod_product_right' #align finset.sum_product_right' Finset.sum_product_right' /-- Generalization of `Finset.prod_comm` to the case when the inner `Finset`s depend on the outer variable. -/ @[to_additive "Generalization of `Finset.sum_comm` to the case when the inner `Finset`s depend on the outer variable."] theorem prod_comm' {s : Finset γ} {t : γ → Finset α} {t' : Finset α} {s' : α → Finset γ} (h : ∀ x y, x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t') {f : γ → α → β} : (∏ x ∈ s, ∏ y ∈ t x, f x y) = ∏ y ∈ t', ∏ x ∈ s' y, f x y := by classical have : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <\| Function.Embedding.sectr x _) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1 := by rintro ⟨x, y⟩ simp only [mem_biUnion, mem_map, Function.Embedding.sectr_apply, Prod.mk.injEq, exists_eq_right, ← and_assoc] exact (prod_finset_product' _ _ _ this).symm.trans ((prod_finset_product_right' _ _ _) fun ⟨x, y⟩ => (this _).trans ((h x y).trans and_comm)) #align finset.prod_comm' Finset.prod_comm' #align finset.sum_comm' Finset.sum_comm' @[to_additive] theorem prod_comm {s : Finset γ} {t : Finset α} {f : γ → α → β} : (∏ x ∈ s, ∏ y ∈ t, f x y) = ∏ y ∈ t, ∏ x ∈ s, f x y := prod_comm' fun _ _ => Iff.rfl #align finset.prod_comm Finset.prod_comm #align finset.sum_comm Finset.sum_comm @[to_additive] theorem prod_hom_rel [CommMonoid γ] {r : β → γ → Prop} {f : α → β} {g : α → γ} {s : Finset α} (h₁ : r 1 1) (h₂ : ∀ a b c, r b c → r (f a * b) (g a * c)) : r (∏ x ∈ s, f x) (∏ x ∈ s, g x) := by delta Finset.prod apply Multiset.prod_hom_rel <;> assumption #align finset.prod_hom_rel Finset.prod_hom_rel #align finset.sum_hom_rel Finset.sum_hom_rel @[to_additive] theorem prod_filter_of_ne {p : α → Prop} [DecidablePred p] (hp : ∀ x ∈ s, f x ≠ 1 → p x) : ∏ x ∈ s.filter p, f x = ∏ x ∈ s, f x := (prod_subset (filter_subset _ _)) fun x => by classical rw [not_imp_comm, mem_filter] exact fun h₁ h₂ => ⟨h₁, by simpa using hp _ h₁ h₂⟩ #align finset.prod_filter_of_ne Finset.prod_filter_of_ne #align finset.sum_filter_of_ne Finset.sum_filter_of_ne -- If we use `[DecidableEq β]` here, some rewrites fail because they find a wrong `Decidable` -- instance first; `{∀ x, Decidable (f x ≠ 1)}` doesn't work with `rw ← prod_filter_ne_one` @[to_additive] theorem prod_filter_ne_one (s : Finset α) [∀ x, Decidable (f x ≠ 1)] : ∏ x ∈ s.filter fun x => f x ≠ 1, f x = ∏ x ∈ s, f x := prod_filter_of_ne fun _ _ => id #align finset.prod_filter_ne_one Finset.prod_filter_ne_one #align finset.sum_filter_ne_zero Finset.sum_filter_ne_zero @[to_additive] theorem prod_filter (p : α → Prop) [DecidablePred p] (f : α → β) : ∏ a ∈ s.filter p, f a = ∏ a ∈ s, if p a then f a else 1 := calc ∏ a ∈ s.filter p, f a = ∏ a ∈ s.filter p, if p a then f a else 1 := prod_congr rfl fun a h => by rw [if_pos]; simpa using (mem_filter.1 h).2 _ = ∏ a ∈ s, if p a then f a else 1 := by { refine prod_subset (filter_subset _ s) fun x hs h => ?_ rw [mem_filter, not_and] at h exact if_neg (by simpa using h hs) } #align finset.prod_filter Finset.prod_filter #align finset.sum_filter Finset.sum_filter @[to_additive] theorem prod_eq_single_of_mem {s : Finset α} {f : α → β} (a : α) (h : a ∈ s) (h₀ : ∀ b ∈ s, b ≠ a → f b = 1) : ∏ x ∈ s, f x = f a := by haveI := Classical.decEq α calc ∏ x ∈ s, f x = ∏ x ∈ {a}, f x := by { refine (prod_subset ?_ ?_).symm · intro _ H rwa [mem_singleton.1 H] · simpa only [mem_singleton] } _ = f a := prod_singleton _ _ #align finset.prod_eq_single_of_mem Finset.prod_eq_single_of_mem #align finset.sum_eq_single_of_mem Finset.sum_eq_single_of_mem @[to_additive] theorem prod_eq_single {s : Finset α} {f : α → β} (a : α) (h₀ : ∀ b ∈ s, b ≠ a → f b = 1) (h₁ : a ∉ s → f a = 1) : ∏ x ∈ s, f x = f a := haveI := Classical.decEq α by_cases (prod_eq_single_of_mem a · h₀) fun this => (prod_congr rfl fun b hb => h₀ b hb <\| by rintro rfl; exact this hb).trans <\| prod_const_one.trans (h₁ this).symm #align finset.prod_eq_single Finset.prod_eq_single #align finset.sum_eq_single Finset.sum_eq_single @[to_additive] lemma prod_union_eq_left [DecidableEq α] (hs : ∀ a ∈ s₂, a ∉ s₁ → f a = 1) : ∏ a ∈ s₁ ∪ s₂, f a = ∏ a ∈ s₁, f a := Eq.symm <\| prod_subset subset_union_left fun _a ha ha' ↦ hs _ ((mem_union.1 ha).resolve_left ha') ha' @[to_additive] lemma prod_union_eq_right [DecidableEq α] (hs : ∀ a ∈ s₁, a ∉ s₂ → f a = 1) : ∏ a ∈ s₁ ∪ s₂, f a = ∏ a ∈ s₂, f a := by rw [union_comm, prod_union_eq_left hs] @[to_additive] theorem prod_eq_mul_of_mem {s : Finset α} {f : α → β} (a b : α) (ha : a ∈ s) (hb : b ∈ s) (hn : a ≠ b) (h₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1) : ∏ x ∈ s, f x = f a * f b := by haveI := Classical.decEq α; let s' := ({a, b} : Finset α) have hu : s' ⊆ s := by refine insert_subset_iff.mpr ?_ apply And.intro ha apply singleton_subset_iff.mpr hb have hf : ∀ c ∈ s, c ∉ s' → f c = 1 := by intro c hc hcs apply h₀ c hc apply not_or.mp intro hab apply hcs rw [mem_insert, mem_singleton] exact hab rw [← prod_subset hu hf] exact Finset.prod_pair hn #align finset.prod_eq_mul_of_mem Finset.prod_eq_mul_of_mem #align finset.sum_eq_add_of_mem Finset.sum_eq_add_of_mem @[to_additive] theorem prod_eq_mul {s : Finset α} {f : α → β} (a b : α) (hn : a ≠ b) (h₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1) (ha : a ∉ s → f a = 1) (hb : b ∉ s → f b = 1) : ∏ x ∈ s, f x = f a * f b := by haveI := Classical.decEq α; by_cases h₁ : a ∈ s <;> by_cases h₂ : b ∈ s · exact prod_eq_mul_of_mem a b h₁ h₂ hn h₀ · rw [hb h₂, mul_one] apply prod_eq_single_of_mem a h₁ exact fun c hc hca => h₀ c hc ⟨hca, ne_of_mem_of_not_mem hc h₂⟩ · rw [ha h₁, one_mul] apply prod_eq_single_of_mem b h₂ exact fun c hc hcb => h₀ c hc ⟨ne_of_mem_of_not_mem hc h₁, hcb⟩ · rw [ha h₁, hb h₂, mul_one] exact _root_.trans (prod_congr rfl fun c hc => h₀ c hc ⟨ne_of_mem_of_not_mem hc h₁, ne_of_mem_of_not_mem hc h₂⟩) prod_const_one #align finset.prod_eq_mul Finset.prod_eq_mul #align finset.sum_eq_add Finset.sum_eq_add -- Porting note: simpNF linter complains that LHS doesn't simplify, but it does /-- A product over `s.subtype p` equals one over `s.filter p`. -/ @[to_additive (attr := simp, nolint simpNF) "A sum over `s.subtype p` equals one over `s.filter p`."] theorem prod_subtype_eq_prod_filter (f : α → β) {p : α → Prop} [DecidablePred p] : ∏ x ∈ s.subtype p, f x = ∏ x ∈ s.filter p, f x := by conv_lhs => erw [← prod_map (s.subtype p) (Function.Embedding.subtype _) f] exact prod_congr (subtype_map _) fun x _hx => rfl #align finset.prod_subtype_eq_prod_filter Finset.prod_subtype_eq_prod_filter #align finset.sum_subtype_eq_sum_filter Finset.sum_subtype_eq_sum_filter /-- If all elements of a `Finset` satisfy the predicate `p`, a product over `s.subtype p` equals that product over `s`. -/ @[to_additive "If all elements of a `Finset` satisfy the predicate `p`, a sum over `s.subtype p` equals that sum over `s`."] theorem prod_subtype_of_mem (f : α → β) {p : α → Prop} [DecidablePred p] (h : ∀ x ∈ s, p x) : ∏ x ∈ s.subtype p, f x = ∏ x ∈ s, f x := by rw [prod_subtype_eq_prod_filter, filter_true_of_mem] simpa using h #align finset.prod_subtype_of_mem Finset.prod_subtype_of_mem #align finset.sum_subtype_of_mem Finset.sum_subtype_of_mem /-- A product of a function over a `Finset` in a subtype equals a product in the main type of a function that agrees with the first function on that `Finset`. -/ @[to_additive "A sum of a function over a `Finset` in a subtype equals a sum in the main type of a function that agrees with the first function on that `Finset`."] theorem prod_subtype_map_embedding {p : α → Prop} {s : Finset { x // p x }} {f : { x // p x } → β} {g : α → β} (h : ∀ x : { x // p x }, x ∈ s → g x = f x) : (∏ x ∈ s.map (Function.Embedding.subtype _), g x) = ∏ x ∈ s, f x := by rw [Finset.prod_map] exact Finset.prod_congr rfl h #align finset.prod_subtype_map_embedding Finset.prod_subtype_map_embedding #align finset.sum_subtype_map_embedding Finset.sum_subtype_map_embedding variable (f s) @[to_additive] theorem prod_coe_sort_eq_attach (f : s → β) : ∏ i : s, f i = ∏ i ∈ s.attach, f i := rfl #align finset.prod_coe_sort_eq_attach Finset.prod_coe_sort_eq_attach #align finset.sum_coe_sort_eq_attach Finset.sum_coe_sort_eq_attach @[to_additive] theorem prod_coe_sort : ∏ i : s, f i = ∏ i ∈ s, f i := prod_attach _ _ #align finset.prod_coe_sort Finset.prod_coe_sort #align finset.sum_coe_sort Finset.sum_coe_sort @[to_additive] theorem prod_finset_coe (f : α → β) (s : Finset α) : (∏ i : (s : Set α), f i) = ∏ i ∈ s, f i := prod_coe_sort s f #align finset.prod_finset_coe Finset.prod_finset_coe #align finset.sum_finset_coe Finset.sum_finset_coe variable {f s} @[to_additive]	Mathlib/Algebra/BigOperators/Group/Finset.lean	1,114	1,119	theorem prod_subtype {p : α → Prop} {F : Fintype (Subtype p)} (s : Finset α) (h : ∀ x, x ∈ s ↔ p x) (f : α → β) : ∏ a ∈ s, f a = ∏ a : Subtype p, f a := by	have : (· ∈ s) = p := Set.ext h subst p rw [← prod_coe_sort] congr!
/- Copyright (c) 2022 Andrew Yang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang -/ import Mathlib.AlgebraicGeometry.GammaSpecAdjunction import Mathlib.AlgebraicGeometry.Restrict import Mathlib.CategoryTheory.Limits.Opposites import Mathlib.RingTheory.Localization.InvSubmonoid #align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c" /-! # Affine schemes We define the category of `AffineScheme`s as the essential image of `Spec`. We also define predicates about affine schemes and affine open sets. ## Main definitions * `AlgebraicGeometry.AffineScheme`: The category of affine schemes. * `AlgebraicGeometry.IsAffine`: A scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism. * `AlgebraicGeometry.Scheme.isoSpec`: The canonical isomorphism `X ≅ Spec Γ(X)` for an affine scheme. * `AlgebraicGeometry.AffineScheme.equivCommRingCat`: The equivalence of categories `AffineScheme ≌ CommRingᵒᵖ` given by `AffineScheme.Spec : CommRingᵒᵖ ⥤ AffineScheme` and `AffineScheme.Γ : AffineSchemeᵒᵖ ⥤ CommRingCat`. * `AlgebraicGeometry.IsAffineOpen`: An open subset of a scheme is affine if the open subscheme is affine. * `AlgebraicGeometry.IsAffineOpen.fromSpec`: The immersion `Spec 𝒪ₓ(U) ⟶ X` for an affine `U`. -/ -- Explicit universe annotations were used in this file to improve perfomance #12737 set_option linter.uppercaseLean3 false noncomputable section open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace universe u namespace AlgebraicGeometry open Spec (structureSheaf) /-- The category of affine schemes -/ -- Porting note(#5171): linter not ported yet -- @[nolint has_nonempty_instance] def AffineScheme := Scheme.Spec.EssImageSubcategory deriving Category #align algebraic_geometry.AffineScheme AlgebraicGeometry.AffineScheme /-- A Scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism. -/ class IsAffine (X : Scheme) : Prop where affine : IsIso (ΓSpec.adjunction.unit.app X) #align algebraic_geometry.is_affine AlgebraicGeometry.IsAffine attribute [instance] IsAffine.affine /-- The canonical isomorphism `X ≅ Spec Γ(X)` for an affine scheme. -/ def Scheme.isoSpec (X : Scheme) [IsAffine X] : X ≅ Scheme.Spec.obj (op <\| Scheme.Γ.obj <\| op X) := asIso (ΓSpec.adjunction.unit.app X) #align algebraic_geometry.Scheme.iso_Spec AlgebraicGeometry.Scheme.isoSpec /-- Construct an affine scheme from a scheme and the information that it is affine. Also see `AffineScheme.of` for a typeclass version. -/ @[simps] def AffineScheme.mk (X : Scheme) (_ : IsAffine X) : AffineScheme := ⟨X, mem_essImage_of_unit_isIso (adj := ΓSpec.adjunction) _⟩ #align algebraic_geometry.AffineScheme.mk AlgebraicGeometry.AffineScheme.mk /-- Construct an affine scheme from a scheme. Also see `AffineScheme.mk` for a non-typeclass version. -/ def AffineScheme.of (X : Scheme) [h : IsAffine X] : AffineScheme := AffineScheme.mk X h #align algebraic_geometry.AffineScheme.of AlgebraicGeometry.AffineScheme.of /-- Type check a morphism of schemes as a morphism in `AffineScheme`. -/ def AffineScheme.ofHom {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) : AffineScheme.of X ⟶ AffineScheme.of Y := f #align algebraic_geometry.AffineScheme.of_hom AlgebraicGeometry.AffineScheme.ofHom theorem mem_Spec_essImage (X : Scheme) : X ∈ Scheme.Spec.essImage ↔ IsAffine X := ⟨fun h => ⟨Functor.essImage.unit_isIso h⟩, fun _ => mem_essImage_of_unit_isIso (adj := ΓSpec.adjunction) _⟩ #align algebraic_geometry.mem_Spec_ess_image AlgebraicGeometry.mem_Spec_essImage instance isAffineAffineScheme (X : AffineScheme.{u}) : IsAffine X.obj := ⟨Functor.essImage.unit_isIso X.property⟩ #align algebraic_geometry.is_affine_AffineScheme AlgebraicGeometry.isAffineAffineScheme instance SpecIsAffine (R : CommRingCatᵒᵖ) : IsAffine (Scheme.Spec.obj R) := AlgebraicGeometry.isAffineAffineScheme ⟨_, Scheme.Spec.obj_mem_essImage R⟩ #align algebraic_geometry.Spec_is_affine AlgebraicGeometry.SpecIsAffine theorem isAffineOfIso {X Y : Scheme} (f : X ⟶ Y) [IsIso f] [h : IsAffine Y] : IsAffine X := by rw [← mem_Spec_essImage] at h ⊢; exact Functor.essImage.ofIso (asIso f).symm h #align algebraic_geometry.is_affine_of_iso AlgebraicGeometry.isAffineOfIso namespace AffineScheme /-- The `Spec` functor into the category of affine schemes. -/ def Spec : CommRingCatᵒᵖ ⥤ AffineScheme := Scheme.Spec.toEssImage #align algebraic_geometry.AffineScheme.Spec AlgebraicGeometry.AffineScheme.Spec -- Porting note (#11081): cannot automatically derive instance Spec_full : Spec.Full := Functor.Full.toEssImage _ -- Porting note (#11081): cannot automatically derive instance Spec_faithful : Spec.Faithful := Functor.Faithful.toEssImage _ -- Porting note (#11081): cannot automatically derive instance Spec_essSurj : Spec.EssSurj := Functor.EssSurj.toEssImage (F := _) /-- The forgetful functor `AffineScheme ⥤ Scheme`. -/ @[simps!] def forgetToScheme : AffineScheme ⥤ Scheme := Scheme.Spec.essImageInclusion #align algebraic_geometry.AffineScheme.forget_to_Scheme AlgebraicGeometry.AffineScheme.forgetToScheme -- Porting note (#11081): cannot automatically derive instance forgetToScheme_full : forgetToScheme.Full := show (Scheme.Spec.essImageInclusion).Full from inferInstance -- Porting note (#11081): cannot automatically derive instance forgetToScheme_faithful : forgetToScheme.Faithful := show (Scheme.Spec.essImageInclusion).Faithful from inferInstance /-- The global section functor of an affine scheme. -/ def Γ : AffineSchemeᵒᵖ ⥤ CommRingCat := forgetToScheme.op ⋙ Scheme.Γ #align algebraic_geometry.AffineScheme.Γ AlgebraicGeometry.AffineScheme.Γ /-- The category of affine schemes is equivalent to the category of commutative rings. -/ def equivCommRingCat : AffineScheme ≌ CommRingCatᵒᵖ := equivEssImageOfReflective.symm #align algebraic_geometry.AffineScheme.equiv_CommRing AlgebraicGeometry.AffineScheme.equivCommRingCat instance : Γ.{u}.rightOp.IsEquivalence := equivCommRingCat.isEquivalence_functor instance : Γ.{u}.rightOp.op.IsEquivalence := equivCommRingCat.op.isEquivalence_functor instance ΓIsEquiv : Γ.{u}.IsEquivalence := inferInstanceAs (Γ.{u}.rightOp.op ⋙ (opOpEquivalence _).functor).IsEquivalence #align algebraic_geometry.AffineScheme.Γ_is_equiv AlgebraicGeometry.AffineScheme.ΓIsEquiv instance hasColimits : HasColimits AffineScheme.{u} := haveI := Adjunction.has_limits_of_equivalence.{u} Γ.{u} Adjunction.has_colimits_of_equivalence.{u} (opOpEquivalence AffineScheme.{u}).inverse instance hasLimits : HasLimits AffineScheme.{u} := by haveI := Adjunction.has_colimits_of_equivalence Γ.{u} haveI : HasLimits AffineScheme.{u}ᵒᵖᵒᵖ := Limits.hasLimits_op_of_hasColimits exact Adjunction.has_limits_of_equivalence (opOpEquivalence AffineScheme.{u}).inverse noncomputable instance Γ_preservesLimits : PreservesLimits Γ.{u}.rightOp := inferInstance noncomputable instance forgetToScheme_preservesLimits : PreservesLimits forgetToScheme := by apply (config := { allowSynthFailures := true }) @preservesLimitsOfNatIso _ _ _ _ _ _ (isoWhiskerRight equivCommRingCat.unitIso forgetToScheme).symm change PreservesLimits (equivCommRingCat.functor ⋙ Scheme.Spec) infer_instance end AffineScheme /-- An open subset of a scheme is affine if the open subscheme is affine. -/ def IsAffineOpen {X : Scheme} (U : Opens X) : Prop := IsAffine (X ∣_ᵤ U) #align algebraic_geometry.is_affine_open AlgebraicGeometry.IsAffineOpen /-- The set of affine opens as a subset of `opens X`. -/ def Scheme.affineOpens (X : Scheme) : Set (Opens X) := {U : Opens X \| IsAffineOpen U} #align algebraic_geometry.Scheme.affine_opens AlgebraicGeometry.Scheme.affineOpens instance {Y : Scheme.{u}} (U : Y.affineOpens) : IsAffine (Scheme.restrict Y <\| Opens.openEmbedding U.val) := U.property theorem rangeIsAffineOpenOfOpenImmersion {X Y : Scheme} [IsAffine X] (f : X ⟶ Y) [H : IsOpenImmersion f] : IsAffineOpen (Scheme.Hom.opensRange f) := by refine isAffineOfIso (IsOpenImmersion.isoOfRangeEq f (Y.ofRestrict _) ?_).inv exact Subtype.range_val.symm #align algebraic_geometry.range_is_affine_open_of_open_immersion AlgebraicGeometry.rangeIsAffineOpenOfOpenImmersion theorem topIsAffineOpen (X : Scheme) [IsAffine X] : IsAffineOpen (⊤ : Opens X) := by convert rangeIsAffineOpenOfOpenImmersion (𝟙 X) ext1 exact Set.range_id.symm #align algebraic_geometry.top_is_affine_open AlgebraicGeometry.topIsAffineOpen instance Scheme.affineCoverIsAffine (X : Scheme) (i : X.affineCover.J) : IsAffine (X.affineCover.obj i) := AlgebraicGeometry.SpecIsAffine _ #align algebraic_geometry.Scheme.affine_cover_is_affine AlgebraicGeometry.Scheme.affineCoverIsAffine instance Scheme.affineBasisCoverIsAffine (X : Scheme) (i : X.affineBasisCover.J) : IsAffine (X.affineBasisCover.obj i) := AlgebraicGeometry.SpecIsAffine _ #align algebraic_geometry.Scheme.affine_basis_cover_is_affine AlgebraicGeometry.Scheme.affineBasisCoverIsAffine theorem isBasis_affine_open (X : Scheme) : Opens.IsBasis X.affineOpens := by rw [Opens.isBasis_iff_nbhd] rintro U x (hU : x ∈ (U : Set X)) obtain ⟨S, hS, hxS, hSU⟩ := X.affineBasisCover_is_basis.exists_subset_of_mem_open hU U.isOpen refine ⟨⟨S, X.affineBasisCover_is_basis.isOpen hS⟩, ?_, hxS, hSU⟩ rcases hS with ⟨i, rfl⟩ exact rangeIsAffineOpenOfOpenImmersion _ #align algebraic_geometry.is_basis_affine_open AlgebraicGeometry.isBasis_affine_open	Mathlib/AlgebraicGeometry/AffineScheme.lean	218	234	theorem Scheme.map_PrimeSpectrum_basicOpen_of_affine (X : Scheme) [IsAffine X] (f : Scheme.Γ.obj (op X)) : X.isoSpec.hom ⁻¹ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f := by	rw [← basicOpen_eq_of_affine] trans X.isoSpec.hom ⁻¹ᵁ (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))).basicOpen ((inv (X.isoSpec.hom.1.c.app (op ((Opens.map (inv X.isoSpec.hom).val.base).obj ⊤)))) f) · congr rw [← IsIso.inv_eq_inv, IsIso.inv_inv, IsIso.Iso.inv_inv, NatIso.app_hom] -- Porting note: added this `change` to prevent timeout change SpecΓIdentity.hom.app (X.presheaf.obj <\| op ⊤) = _ rw [← ΓSpec.adjunction_unit_app_app_top X] rfl · dsimp refine (Scheme.preimage_basicOpen _ _).trans ?_ congr 1 exact IsIso.inv_hom_id_apply _ _
/- Copyright (c) 2022 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Probability.Variance import Mathlib.MeasureTheory.Function.UniformIntegrable #align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Identically distributed random variables Two random variables defined on two (possibly different) probability spaces but taking value in the same space are identically distributed if their distributions (i.e., the image probability measures on the target space) coincide. We define this concept and establish its basic properties in this file. ## Main definitions and results * `IdentDistrib f g μ ν` registers that the image of `μ` under `f` coincides with the image of `ν` under `g` (and that `f` and `g` are almost everywhere measurable, as otherwise the image measures don't make sense). The measures can be kept implicit as in `IdentDistrib f g` if the spaces are registered as measure spaces. * `IdentDistrib.comp`: being identically distributed is stable under composition with measurable maps. There are two main kinds of lemmas, under the assumption that `f` and `g` are identically distributed: lemmas saying that two quantities computed for `f` and `g` are the same, and lemmas saying that if `f` has some property then `g` also has it. The first kind is registered as `IdentDistrib.foo_fst`, the second one as `IdentDistrib.foo_snd` (in the latter case, to deduce a property of `f` from one of `g`, use `h.symm.foo_snd` where `h : IdentDistrib f g μ ν`). For instance: * `IdentDistrib.measure_mem_eq`: if `f` and `g` are identically distributed, then the probabilities that they belong to a given measurable set are the same. * `IdentDistrib.integral_eq`: if `f` and `g` are identically distributed, then their integrals are the same. * `IdentDistrib.variance_eq`: if `f` and `g` are identically distributed, then their variances are the same. * `IdentDistrib.aestronglyMeasurable_snd`: if `f` and `g` are identically distributed and `f` is almost everywhere strongly measurable, then so is `g`. * `IdentDistrib.memℒp_snd`: if `f` and `g` are identically distributed and `f` belongs to `ℒp`, then so does `g`. We also register several dot notation shortcuts for convenience. For instance, if `h : IdentDistrib f g μ ν`, then `h.sq` states that `f^2` and `g^2` are identically distributed, and `h.norm` states that `‖f‖` and `‖g‖` are identically distributed, and so on. -/ open MeasureTheory Filter Finset noncomputable section open scoped Topology MeasureTheory ENNReal NNReal variable {α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] namespace ProbabilityTheory /-- Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide. This only makes sense when the functions are ae measurable (as otherwise the image measures are not defined), so we require this as well in the definition. -/ structure IdentDistrib (f : α → γ) (g : β → γ) (μ : Measure α := by volume_tac) (ν : Measure β := by volume_tac) : Prop where aemeasurable_fst : AEMeasurable f μ aemeasurable_snd : AEMeasurable g ν map_eq : Measure.map f μ = Measure.map g ν #align probability_theory.ident_distrib ProbabilityTheory.IdentDistrib namespace IdentDistrib open TopologicalSpace variable {μ : Measure α} {ν : Measure β} {f : α → γ} {g : β → γ} protected theorem refl (hf : AEMeasurable f μ) : IdentDistrib f f μ μ := { aemeasurable_fst := hf aemeasurable_snd := hf map_eq := rfl } #align probability_theory.ident_distrib.refl ProbabilityTheory.IdentDistrib.refl protected theorem symm (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ := { aemeasurable_fst := h.aemeasurable_snd aemeasurable_snd := h.aemeasurable_fst map_eq := h.map_eq.symm } #align probability_theory.ident_distrib.symm ProbabilityTheory.IdentDistrib.symm protected theorem trans {ρ : Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν) (h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ := { aemeasurable_fst := h₁.aemeasurable_fst aemeasurable_snd := h₂.aemeasurable_snd map_eq := h₁.map_eq.trans h₂.map_eq } #align probability_theory.ident_distrib.trans ProbabilityTheory.IdentDistrib.trans protected theorem comp_of_aemeasurable {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν := { aemeasurable_fst := hu.comp_aemeasurable h.aemeasurable_fst aemeasurable_snd := by rw [h.map_eq] at hu; exact hu.comp_aemeasurable h.aemeasurable_snd map_eq := by rw [← AEMeasurable.map_map_of_aemeasurable hu h.aemeasurable_fst, ← AEMeasurable.map_map_of_aemeasurable _ h.aemeasurable_snd, h.map_eq] rwa [← h.map_eq] } #align probability_theory.ident_distrib.comp_of_ae_measurable ProbabilityTheory.IdentDistrib.comp_of_aemeasurable protected theorem comp {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) : IdentDistrib (u ∘ f) (u ∘ g) μ ν := h.comp_of_aemeasurable hu.aemeasurable #align probability_theory.ident_distrib.comp ProbabilityTheory.IdentDistrib.comp protected theorem of_ae_eq {g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) : IdentDistrib f g μ μ := { aemeasurable_fst := hf aemeasurable_snd := hf.congr heq map_eq := Measure.map_congr heq } #align probability_theory.ident_distrib.of_ae_eq ProbabilityTheory.IdentDistrib.of_ae_eq lemma _root_.MeasureTheory.AEMeasurable.identDistrib_mk (hf : AEMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ := IdentDistrib.of_ae_eq hf hf.ae_eq_mk lemma _root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ] (hf : AEStronglyMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ := IdentDistrib.of_ae_eq hf.aemeasurable hf.ae_eq_mk theorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) : μ (f ⁻¹' s) = ν (g ⁻¹' s) := by rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ← Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq] #align probability_theory.ident_distrib.measure_mem_eq ProbabilityTheory.IdentDistrib.measure_mem_eq alias measure_preimage_eq := measure_mem_eq #align probability_theory.ident_distrib.measure_preimage_eq ProbabilityTheory.IdentDistrib.measure_preimage_eq	Mathlib/Probability/IdentDistrib.lean	141	145	theorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x \| p x}) (hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by	apply (ae_map_iff h.aemeasurable_snd pmeas).1 rw [← h.map_eq] exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp
/- Copyright (c) 2020 Jannis Limperg. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jannis Limperg -/ import Mathlib.Data.List.OfFn import Mathlib.Data.List.Range #align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1" /-! # Lemmas about List.Idx functions. Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.foldrIdx`. -/ assert_not_exists MonoidWithZero universe u v open Function namespace List variable {α : Type u} {β : Type v} section MapIdx -- Porting note: Add back old definition because it's easier for writing proofs. /-- Lean3 `map_with_index` helper function -/ protected def oldMapIdxCore (f : ℕ → α → β) : ℕ → List α → List β \| _, [] => [] \| k, a :: as => f k a :: List.oldMapIdxCore f (k + 1) as /-- Given a function `f : ℕ → α → β` and `as : List α`, `as = [a₀, a₁, ...]`, returns the list `[f 0 a₀, f 1 a₁, ...]`. -/ protected def oldMapIdx (f : ℕ → α → β) (as : List α) : List β := List.oldMapIdxCore f 0 as @[simp] theorem mapIdx_nil {α β} (f : ℕ → α → β) : mapIdx f [] = [] := rfl #align list.map_with_index_nil List.mapIdx_nil -- Porting note (#10756): new theorem. protected theorem oldMapIdxCore_eq (l : List α) (f : ℕ → α → β) (n : ℕ) : l.oldMapIdxCore f n = l.oldMapIdx fun i a ↦ f (i + n) a := by induction' l with hd tl hl generalizing f n · rfl · rw [List.oldMapIdx] simp only [List.oldMapIdxCore, hl, Nat.add_left_comm, Nat.add_comm, Nat.add_zero] #noalign list.map_with_index_core_eq -- Porting note: convert new definition to old definition. -- A few new theorems are added to achieve this -- 1. Prove that `oldMapIdxCore f (l ++ [e]) = oldMapIdxCore f l ++ [f l.length e]` -- 2. Prove that `oldMapIdx f (l ++ [e]) = oldMapIdx f l ++ [f l.length e]` -- 3. Prove list induction using `∀ l e, p [] → (p l → p (l ++ [e])) → p l` -- Porting note (#10756): new theorem. theorem list_reverse_induction (p : List α → Prop) (base : p []) (ind : ∀ (l : List α) (e : α), p l → p (l ++ [e])) : (∀ (l : List α), p l) := by let q := fun l ↦ p (reverse l) have pq : ∀ l, p (reverse l) → q l := by simp only [q, reverse_reverse]; intro; exact id have qp : ∀ l, q (reverse l) → p l := by simp only [q, reverse_reverse]; intro; exact id intro l apply qp generalize (reverse l) = l induction' l with head tail ih · apply pq; simp only [reverse_nil, base] · apply pq; simp only [reverse_cons]; apply ind; apply qp; rw [reverse_reverse]; exact ih -- Porting note (#10756): new theorem. protected theorem oldMapIdxCore_append : ∀ (f : ℕ → α → β) (n : ℕ) (l₁ l₂ : List α), List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + l₁.length) l₂ := by intros f n l₁ l₂ generalize e : (l₁ ++ l₂).length = len revert n l₁ l₂ induction' len with len ih <;> intros n l₁ l₂ h · have l₁_nil : l₁ = [] := by cases l₁ · rfl · contradiction have l₂_nil : l₂ = [] := by cases l₂ · rfl · rw [List.length_append] at h; contradiction simp only [l₁_nil, l₂_nil]; rfl · cases' l₁ with head tail · rfl · simp only [List.oldMapIdxCore, List.append_eq, length_cons, cons_append,cons.injEq, true_and] suffices n + Nat.succ (length tail) = n + 1 + tail.length by rw [this] apply ih (n + 1) _ _ _ simp only [cons_append, length_cons, length_append, Nat.succ.injEq] at h simp only [length_append, h] rw [Nat.add_assoc]; simp only [Nat.add_comm] -- Porting note (#10756): new theorem. protected theorem oldMapIdx_append : ∀ (f : ℕ → α → β) (l : List α) (e : α), List.oldMapIdx f (l ++ [e]) = List.oldMapIdx f l ++ [f l.length e] := by intros f l e unfold List.oldMapIdx rw [List.oldMapIdxCore_append f 0 l [e]] simp only [Nat.zero_add]; rfl -- Porting note (#10756): new theorem. theorem mapIdxGo_append : ∀ (f : ℕ → α → β) (l₁ l₂ : List α) (arr : Array β), mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (List.toArray (mapIdx.go f l₁ arr)) := by intros f l₁ l₂ arr generalize e : (l₁ ++ l₂).length = len revert l₁ l₂ arr induction' len with len ih <;> intros l₁ l₂ arr h · have l₁_nil : l₁ = [] := by cases l₁ · rfl · contradiction have l₂_nil : l₂ = [] := by cases l₂ · rfl · rw [List.length_append] at h; contradiction rw [l₁_nil, l₂_nil]; simp only [mapIdx.go, Array.toList_eq, Array.toArray_data] · cases' l₁ with head tail <;> simp only [mapIdx.go] · simp only [nil_append, Array.toList_eq, Array.toArray_data] · simp only [List.append_eq] rw [ih] · simp only [cons_append, length_cons, length_append, Nat.succ.injEq] at h simp only [length_append, h] -- Porting note (#10756): new theorem. theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array β), length (mapIdx.go f l arr) = length l + arr.size := by intro f l induction' l with head tail ih · intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add] · intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons]; simp only [Nat.add_succ, add_zero, Nat.add_comm] -- Porting note (#10756): new theorem. theorem mapIdx_append_one : ∀ (f : ℕ → α → β) (l : List α) (e : α), mapIdx f (l ++ [e]) = mapIdx f l ++ [f l.length e] := by intros f l e unfold mapIdx rw [mapIdxGo_append f l [e]] simp only [mapIdx.go, Array.size_toArray, mapIdxGo_length, length_nil, Nat.add_zero, Array.toList_eq, Array.push_data, Array.data_toArray] -- Porting note (#10756): new theorem. protected theorem new_def_eq_old_def : ∀ (f : ℕ → α → β) (l : List α), l.mapIdx f = List.oldMapIdx f l := by intro f apply list_reverse_induction · rfl · intro l e h rw [List.oldMapIdx_append, mapIdx_append_one, h] @[local simp] theorem map_enumFrom_eq_zipWith : ∀ (l : List α) (n : ℕ) (f : ℕ → α → β), map (uncurry f) (enumFrom n l) = zipWith (fun i ↦ f (i + n)) (range (length l)) l := by intro l generalize e : l.length = len revert l induction' len with len ih <;> intros l e n f · have : l = [] := by cases l · rfl · contradiction rw [this]; rfl · cases' l with head tail · contradiction · simp only [map, uncurry_apply_pair, range_succ_eq_map, zipWith, Nat.zero_add, zipWith_map_left] rw [ih] · suffices (fun i ↦ f (i + (n + 1))) = ((fun i ↦ f (i + n)) ∘ Nat.succ) by rw [this] rfl funext n' a simp only [comp, Nat.add_assoc, Nat.add_comm, Nat.add_succ] simp only [length_cons, Nat.succ.injEq] at e; exact e theorem mapIdx_eq_enum_map (l : List α) (f : ℕ → α → β) : l.mapIdx f = l.enum.map (Function.uncurry f) := by rw [List.new_def_eq_old_def] induction' l with hd tl hl generalizing f · rfl · rw [List.oldMapIdx, List.oldMapIdxCore, List.oldMapIdxCore_eq, hl] simp [map, enum_eq_zip_range, map_uncurry_zip_eq_zipWith] #align list.map_with_index_eq_enum_map List.mapIdx_eq_enum_map @[simp] theorem mapIdx_cons (l : List α) (f : ℕ → α → β) (a : α) : mapIdx f (a :: l) = f 0 a :: mapIdx (fun i ↦ f (i + 1)) l := by simp [mapIdx_eq_enum_map, enum_eq_zip_range, map_uncurry_zip_eq_zipWith, range_succ_eq_map, zipWith_map_left] #align list.map_with_index_cons List.mapIdx_cons theorem mapIdx_append (K L : List α) (f : ℕ → α → β) : (K ++ L).mapIdx f = K.mapIdx f ++ L.mapIdx fun i a ↦ f (i + K.length) a := by induction' K with a J IH generalizing f · rfl · simp [IH fun i ↦ f (i + 1), Nat.add_assoc, Nat.succ_eq_add_one] #align list.map_with_index_append List.mapIdx_append @[simp] theorem length_mapIdx (l : List α) (f : ℕ → α → β) : (l.mapIdx f).length = l.length := by induction' l with hd tl IH generalizing f · rfl · simp [IH] #align list.length_map_with_index List.length_mapIdx @[simp] theorem mapIdx_eq_nil {f : ℕ → α → β} {l : List α} : List.mapIdx f l = [] ↔ l = [] := by rw [List.mapIdx_eq_enum_map, List.map_eq_nil, List.enum_eq_nil] set_option linter.deprecated false in @[simp, deprecated (since := "2023-02-11")] theorem nthLe_mapIdx (l : List α) (f : ℕ → α → β) (i : ℕ) (h : i < l.length) (h' : i < (l.mapIdx f).length := h.trans_le (l.length_mapIdx f).ge) : (l.mapIdx f).nthLe i h' = f i (l.nthLe i h) := by simp [mapIdx_eq_enum_map, enum_eq_zip_range] #align list.nth_le_map_with_index List.nthLe_mapIdx theorem mapIdx_eq_ofFn (l : List α) (f : ℕ → α → β) : l.mapIdx f = ofFn fun i : Fin l.length ↦ f (i : ℕ) (l.get i) := by induction l generalizing f with \| nil => simp \| cons _ _ IH => simp [IH] #align list.map_with_index_eq_of_fn List.mapIdx_eq_ofFn end MapIdx section FoldrIdx -- Porting note: Changed argument order of `foldrIdxSpec` to align better with `foldrIdx`. /-- Specification of `foldrIdx`. -/ def foldrIdxSpec (f : ℕ → α → β → β) (b : β) (as : List α) (start : ℕ) : β := foldr (uncurry f) b <\| enumFrom start as #align list.foldr_with_index_aux_spec List.foldrIdxSpecₓ theorem foldrIdxSpec_cons (f : ℕ → α → β → β) (b a as start) : foldrIdxSpec f b (a :: as) start = f start a (foldrIdxSpec f b as (start + 1)) := rfl #align list.foldr_with_index_aux_spec_cons List.foldrIdxSpec_consₓ theorem foldrIdx_eq_foldrIdxSpec (f : ℕ → α → β → β) (b as start) : foldrIdx f b as start = foldrIdxSpec f b as start := by induction as generalizing start · rfl · simp only [foldrIdx, foldrIdxSpec_cons, ] #align list.foldr_with_index_aux_eq_foldr_with_index_aux_spec List.foldrIdx_eq_foldrIdxSpecₓ theorem foldrIdx_eq_foldr_enum (f : ℕ → α → β → β) (b : β) (as : List α) : foldrIdx f b as = foldr (uncurry f) b (enum as) := by simp only [foldrIdx, foldrIdxSpec, foldrIdx_eq_foldrIdxSpec, enum] #align list.foldr_with_index_eq_foldr_enum List.foldrIdx_eq_foldr_enum end FoldrIdx theorem indexesValues_eq_filter_enum (p : α → Prop) [DecidablePred p] (as : List α) : indexesValues p as = filter (p ∘ Prod.snd) (enum as) := by simp (config := { unfoldPartialApp := true }) [indexesValues, foldrIdx_eq_foldr_enum, uncurry, filter_eq_foldr, cond_eq_if] #align list.indexes_values_eq_filter_enum List.indexesValues_eq_filter_enum theorem findIdxs_eq_map_indexesValues (p : α → Prop) [DecidablePred p] (as : List α) : findIdxs p as = map Prod.fst (indexesValues p as) := by simp (config := { unfoldPartialApp := true }) only [indexesValues_eq_filter_enum, map_filter_eq_foldr, findIdxs, uncurry, foldrIdx_eq_foldr_enum, decide_eq_true_eq, comp_apply, Bool.cond_decide] #align list.find_indexes_eq_map_indexes_values List.findIdxs_eq_map_indexesValues section FindIdx -- TODO: upstream to Batteries theorem findIdx_eq_length {p : α → Bool} {xs : List α} : xs.findIdx p = xs.length ↔ ∀ x ∈ xs, ¬p x := by induction xs with \| nil => simp_all \| cons x xs ih => rw [findIdx_cons, length_cons] constructor <;> intro h · have : ¬p x := by contrapose h; simp_all simp_all · simp_rw [h x (mem_cons_self x xs), cond_false, Nat.succ.injEq, ih] exact fun y hy ↦ h y <\| mem_cons.mpr (Or.inr hy) theorem findIdx_le_length (p : α → Bool) {xs : List α} : xs.findIdx p ≤ xs.length := by by_cases e : ∃ x ∈ xs, p x · exact (findIdx_lt_length_of_exists e).le · push_neg at e; exact (findIdx_eq_length.mpr e).le theorem findIdx_lt_length {p : α → Bool} {xs : List α} : xs.findIdx p < xs.length ↔ ∃ x ∈ xs, p x := by rw [← not_iff_not, not_lt] have := @le_antisymm_iff _ _ (xs.findIdx p) xs.length simp only [findIdx_le_length, true_and] at this rw [← this, findIdx_eq_length, not_exists] simp only [Bool.not_eq_true, not_and] /-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/ theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.findIdx p) : ¬p (xs.get ⟨i, h.trans_le (findIdx_le_length p)⟩) := by revert i induction xs with \| nil => intro i h; rw [findIdx_nil] at h; omega \| cons x xs ih => intro i h have ho := h rw [findIdx_cons] at h have npx : ¬p x := by by_contra y; rw [y, cond_true] at h; omega simp_rw [npx, cond_false] at h cases' i.eq_zero_or_pos with e e · simpa only [e, Fin.zero_eta, get_cons_zero] · have ipm := Nat.succ_pred_eq_of_pos e have ilt := ho.trans_le (findIdx_le_length p) rw [(Fin.mk_eq_mk (h' := ipm ▸ ilt)).mpr ipm.symm, get_cons_succ] rw [← ipm, Nat.succ_lt_succ_iff] at h exact ih h theorem le_findIdx_of_not {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.length) (h2 : ∀ j (hji : j < i), ¬p (xs.get ⟨j, hji.trans h⟩)) : i ≤ xs.findIdx p := by by_contra! f exact absurd (@findIdx_get _ p xs (f.trans h)) (h2 (xs.findIdx p) f) theorem lt_findIdx_of_not {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.length) (h2 : ∀ j (hji : j ≤ i), ¬p (xs.get ⟨j, hji.trans_lt h⟩)) : i < xs.findIdx p := by by_contra! f exact absurd (@findIdx_get _ p xs (f.trans_lt h)) (h2 (xs.findIdx p) f) theorem findIdx_eq {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.length) : xs.findIdx p = i ↔ p (xs.get ⟨i, h⟩) ∧ ∀ j (hji : j < i), ¬p (xs.get ⟨j, hji.trans h⟩) := by refine ⟨fun f ↦ ⟨f ▸ (@findIdx_get _ p xs (f ▸ h)), fun _ hji ↦ not_of_lt_findIdx (f ▸ hji)⟩, fun ⟨h1, h2⟩ ↦ ?_⟩ apply Nat.le_antisymm _ (le_findIdx_of_not h h2) contrapose! h1 exact not_of_lt_findIdx h1 end FindIdx section FoldlIdx -- Porting note: Changed argument order of `foldlIdxSpec` to align better with `foldlIdx`. /-- Specification of `foldlIdx`. -/ def foldlIdxSpec (f : ℕ → α → β → α) (a : α) (bs : List β) (start : ℕ) : α := foldl (fun a p ↦ f p.fst a p.snd) a <\| enumFrom start bs #align list.foldl_with_index_aux_spec List.foldlIdxSpecₓ theorem foldlIdxSpec_cons (f : ℕ → α → β → α) (a b bs start) : foldlIdxSpec f a (b :: bs) start = foldlIdxSpec f (f start a b) bs (start + 1) := rfl #align list.foldl_with_index_aux_spec_cons List.foldlIdxSpec_consₓ theorem foldlIdx_eq_foldlIdxSpec (f : ℕ → α → β → α) (a bs start) : foldlIdx f a bs start = foldlIdxSpec f a bs start := by induction bs generalizing start a · rfl · simp [foldlIdxSpec, *] #align list.foldl_with_index_aux_eq_foldl_with_index_aux_spec List.foldlIdx_eq_foldlIdxSpecₓ theorem foldlIdx_eq_foldl_enum (f : ℕ → α → β → α) (a : α) (bs : List β) : foldlIdx f a bs = foldl (fun a p ↦ f p.fst a p.snd) a (enum bs) := by simp only [foldlIdx, foldlIdxSpec, foldlIdx_eq_foldlIdxSpec, enum] #align list.foldl_with_index_eq_foldl_enum List.foldlIdx_eq_foldl_enum end FoldlIdx section FoldIdxM -- Porting note: `foldrM_eq_foldr` now depends on `[LawfulMonad m]` variable {m : Type u → Type v} [Monad m]	Mathlib/Data/List/Indexes.lean	372	375	theorem foldrIdxM_eq_foldrM_enum {β} (f : ℕ → α → β → m β) (b : β) (as : List α) [LawfulMonad m] : foldrIdxM f b as = foldrM (uncurry f) b (enum as) := by	simp (config := { unfoldPartialApp := true }) only [foldrIdxM, foldrM_eq_foldr, foldrIdx_eq_foldr_enum, uncurry]
/- Copyright (c) 2022 Jujian Zhang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang, Jujian Zhang -/ import Mathlib.Algebra.Algebra.Bilinear import Mathlib.RingTheory.Localization.Basic #align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86" /-! # Localized Module Given a commutative semiring `R`, a multiplicative subset `S ⊆ R` and an `R`-module `M`, we can localize `M` by `S`. This gives us a `Localization S`-module. ## Main definitions * `LocalizedModule.r` : the equivalence relation defining this localization, namely `(m, s) ≈ (m', s')` if and only if there is some `u : S` such that `u • s' • m = u • s • m'`. * `LocalizedModule M S` : the localized module by `S`. * `LocalizedModule.mk` : the canonical map sending `(m, s) : M × S ↦ m/s : LocalizedModule M S` * `LocalizedModule.liftOn` : any well defined function `f : M × S → α` respecting `r` descents to a function `LocalizedModule M S → α` * `LocalizedModule.liftOn₂` : any well defined function `f : M × S → M × S → α` respecting `r` descents to a function `LocalizedModule M S → LocalizedModule M S` * `LocalizedModule.mk_add_mk` : in the localized module `mk m s + mk m' s' = mk (s' • m + s • m') (s * s')` * `LocalizedModule.mk_smul_mk` : in the localized module, for any `r : R`, `s t : S`, `m : M`, we have `mk r s • mk m t = mk (r • m) (s * t)` where `mk r s : Localization S` is localized ring by `S`. * `LocalizedModule.isModule` : `LocalizedModule M S` is a `Localization S`-module. ## Future work * Redefine `Localization` for monoids and rings to coincide with `LocalizedModule`. -/ namespace LocalizedModule universe u v variable {R : Type u} [CommSemiring R] (S : Submonoid R) variable (M : Type v) [AddCommMonoid M] [Module R M] variable (T : Type) [CommSemiring T] [Algebra R T] [IsLocalization S T] /-- The equivalence relation on `M × S` where `(m1, s1) ≈ (m2, s2)` if and only if for some (u : S), u (s2 • m1 - s1 • m2) = 0-/ /- Porting note: We use small letter `r` since `R` is used for a ring. -/ def r (a b : M × S) : Prop := ∃ u : S, u • b.2 • a.1 = u • a.2 • b.1 #align localized_module.r LocalizedModule.r theorem r.isEquiv : IsEquiv _ (r S M) := { refl := fun ⟨m, s⟩ => ⟨1, by rw [one_smul]⟩ trans := fun ⟨m1, s1⟩ ⟨m2, s2⟩ ⟨m3, s3⟩ ⟨u1, hu1⟩ ⟨u2, hu2⟩ => by use u1 * u2 * s2 -- Put everything in the same shape, sorting the terms using `simp` have hu1' := congr_arg ((u2 * s3) • ·) hu1.symm have hu2' := congr_arg ((u1 * s1) • ·) hu2.symm simp only [← mul_smul, smul_assoc, mul_assoc, mul_comm, mul_left_comm] at hu1' hu2' ⊢ rw [hu2', hu1'] symm := fun ⟨m1, s1⟩ ⟨m2, s2⟩ ⟨u, hu⟩ => ⟨u, hu.symm⟩ } #align localized_module.r.is_equiv LocalizedModule.r.isEquiv instance r.setoid : Setoid (M × S) where r := r S M iseqv := ⟨(r.isEquiv S M).refl, (r.isEquiv S M).symm _ _, (r.isEquiv S M).trans _ _ _⟩ #align localized_module.r.setoid LocalizedModule.r.setoid -- TODO: change `Localization` to use `r'` instead of `r` so that the two types are also defeq, -- `Localization S = LocalizedModule S R`. example {R} [CommSemiring R] (S : Submonoid R) : ⇑(Localization.r' S) = LocalizedModule.r S R := rfl /-- If `S` is a multiplicative subset of a ring `R` and `M` an `R`-module, then we can localize `M` by `S`. -/ -- Porting note(#5171): @[nolint has_nonempty_instance] def _root_.LocalizedModule : Type max u v := Quotient (r.setoid S M) #align localized_module LocalizedModule section variable {M S} /-- The canonical map sending `(m, s) ↦ m/s`-/ def mk (m : M) (s : S) : LocalizedModule S M := Quotient.mk' ⟨m, s⟩ #align localized_module.mk LocalizedModule.mk theorem mk_eq {m m' : M} {s s' : S} : mk m s = mk m' s' ↔ ∃ u : S, u • s' • m = u • s • m' := Quotient.eq' #align localized_module.mk_eq LocalizedModule.mk_eq @[elab_as_elim] theorem induction_on {β : LocalizedModule S M → Prop} (h : ∀ (m : M) (s : S), β (mk m s)) : ∀ x : LocalizedModule S M, β x := by rintro ⟨⟨m, s⟩⟩ exact h m s #align localized_module.induction_on LocalizedModule.induction_on @[elab_as_elim] theorem induction_on₂ {β : LocalizedModule S M → LocalizedModule S M → Prop} (h : ∀ (m m' : M) (s s' : S), β (mk m s) (mk m' s')) : ∀ x y, β x y := by rintro ⟨⟨m, s⟩⟩ ⟨⟨m', s'⟩⟩ exact h m m' s s' #align localized_module.induction_on₂ LocalizedModule.induction_on₂ /-- If `f : M × S → α` respects the equivalence relation `LocalizedModule.r`, then `f` descents to a map `LocalizedModule M S → α`. -/ def liftOn {α : Type} (x : LocalizedModule S M) (f : M × S → α) (wd : ∀ (p p' : M × S), p ≈ p' → f p = f p') : α := Quotient.liftOn x f wd #align localized_module.lift_on LocalizedModule.liftOn theorem liftOn_mk {α : Type} {f : M × S → α} (wd : ∀ (p p' : M × S), p ≈ p' → f p = f p') (m : M) (s : S) : liftOn (mk m s) f wd = f ⟨m, s⟩ := by convert Quotient.liftOn_mk f wd ⟨m, s⟩ #align localized_module.lift_on_mk LocalizedModule.liftOn_mk /-- If `f : M × S → M × S → α` respects the equivalence relation `LocalizedModule.r`, then `f` descents to a map `LocalizedModule M S → LocalizedModule M S → α`. -/ def liftOn₂ {α : Type} (x y : LocalizedModule S M) (f : M × S → M × S → α) (wd : ∀ (p q p' q' : M × S), p ≈ p' → q ≈ q' → f p q = f p' q') : α := Quotient.liftOn₂ x y f wd #align localized_module.lift_on₂ LocalizedModule.liftOn₂ theorem liftOn₂_mk {α : Type} (f : M × S → M × S → α) (wd : ∀ (p q p' q' : M × S), p ≈ p' → q ≈ q' → f p q = f p' q') (m m' : M) (s s' : S) : liftOn₂ (mk m s) (mk m' s') f wd = f ⟨m, s⟩ ⟨m', s'⟩ := by convert Quotient.liftOn₂_mk f wd _ _ #align localized_module.lift_on₂_mk LocalizedModule.liftOn₂_mk instance : Zero (LocalizedModule S M) := ⟨mk 0 1⟩ /-- If `S` contains `0` then the localization at `S` is trivial. -/ theorem subsingleton (h : 0 ∈ S) : Subsingleton (LocalizedModule S M) := by refine ⟨fun a b ↦ ?_⟩ induction a,b using LocalizedModule.induction_on₂ exact mk_eq.mpr ⟨⟨0, h⟩, by simp only [Submonoid.mk_smul, zero_smul]⟩ @[simp] theorem zero_mk (s : S) : mk (0 : M) s = 0 := mk_eq.mpr ⟨1, by rw [one_smul, smul_zero, smul_zero, one_smul]⟩ #align localized_module.zero_mk LocalizedModule.zero_mk instance : Add (LocalizedModule S M) where add p1 p2 := liftOn₂ p1 p2 (fun x y => mk (y.2 • x.1 + x.2 • y.1) (x.2 * y.2)) <\| fun ⟨m1, s1⟩ ⟨m2, s2⟩ ⟨m1', s1'⟩ ⟨m2', s2'⟩ ⟨u1, hu1⟩ ⟨u2, hu2⟩ => mk_eq.mpr ⟨u1 * u2, by -- Put everything in the same shape, sorting the terms using `simp` have hu1' := congr_arg ((u2 * s2 * s2') • ·) hu1 have hu2' := congr_arg ((u1 * s1 * s1') • ·) hu2 simp only [smul_add, ← mul_smul, smul_assoc, mul_assoc, mul_comm, mul_left_comm] at hu1' hu2' ⊢ rw [hu1', hu2']⟩ theorem mk_add_mk {m1 m2 : M} {s1 s2 : S} : mk m1 s1 + mk m2 s2 = mk (s2 • m1 + s1 • m2) (s1 * s2) := mk_eq.mpr <\| ⟨1, rfl⟩ #align localized_module.mk_add_mk LocalizedModule.mk_add_mk /-- Porting note: Some auxiliary lemmas are declared with `private` in the original mathlib3 file. We take that policy here as well, and remove the `#align` lines accordingly. -/ private theorem add_assoc' (x y z : LocalizedModule S M) : x + y + z = x + (y + z) := by induction' x using LocalizedModule.induction_on with mx sx induction' y using LocalizedModule.induction_on with my sy induction' z using LocalizedModule.induction_on with mz sz simp only [mk_add_mk, smul_add] refine mk_eq.mpr ⟨1, ?_⟩ rw [one_smul, one_smul] congr 1 · rw [mul_assoc] · rw [eq_comm, mul_comm, add_assoc, mul_smul, mul_smul, ← mul_smul sx sz, mul_comm, mul_smul] private theorem add_comm' (x y : LocalizedModule S M) : x + y = y + x := LocalizedModule.induction_on₂ (fun m m' s s' => by rw [mk_add_mk, mk_add_mk, add_comm, mul_comm]) x y private theorem zero_add' (x : LocalizedModule S M) : 0 + x = x := induction_on (fun m s => by rw [← zero_mk s, mk_add_mk, smul_zero, zero_add, mk_eq]; exact ⟨1, by rw [one_smul, mul_smul, one_smul]⟩) x private theorem add_zero' (x : LocalizedModule S M) : x + 0 = x := induction_on (fun m s => by rw [← zero_mk s, mk_add_mk, smul_zero, add_zero, mk_eq]; exact ⟨1, by rw [one_smul, mul_smul, one_smul]⟩) x instance hasNatSMul : SMul ℕ (LocalizedModule S M) where smul n := nsmulRec n #align localized_module.has_nat_smul LocalizedModule.hasNatSMul private theorem nsmul_zero' (x : LocalizedModule S M) : (0 : ℕ) • x = 0 := LocalizedModule.induction_on (fun _ _ => rfl) x private theorem nsmul_succ' (n : ℕ) (x : LocalizedModule S M) : n.succ • x = n • x + x := LocalizedModule.induction_on (fun _ _ => rfl) x instance : AddCommMonoid (LocalizedModule S M) where add := (· + ·) add_assoc := add_assoc' zero := 0 zero_add := zero_add' add_zero := add_zero' nsmul := (· • ·) nsmul_zero := nsmul_zero' nsmul_succ := nsmul_succ' add_comm := add_comm' instance {M : Type} [AddCommGroup M] [Module R M] : Neg (LocalizedModule S M) where neg p := liftOn p (fun x => LocalizedModule.mk (-x.1) x.2) fun ⟨m1, s1⟩ ⟨m2, s2⟩ ⟨u, hu⟩ => by rw [mk_eq] exact ⟨u, by simpa⟩ instance {M : Type} [AddCommGroup M] [Module R M] : AddCommGroup (LocalizedModule S M) := { show AddCommMonoid (LocalizedModule S M) by infer_instance with add_left_neg := by rintro ⟨m, s⟩ change (liftOn (mk m s) (fun x => mk (-x.1) x.2) fun ⟨m1, s1⟩ ⟨m2, s2⟩ ⟨u, hu⟩ => by rw [mk_eq] exact ⟨u, by simpa⟩) + mk m s = 0 rw [liftOn_mk, mk_add_mk] simp -- TODO: fix the diamond zsmul := zsmulRec } theorem mk_neg {M : Type} [AddCommGroup M] [Module R M] {m : M} {s : S} : mk (-m) s = -mk m s := rfl #align localized_module.mk_neg LocalizedModule.mk_neg instance {A : Type} [Semiring A] [Algebra R A] {S : Submonoid R} : Monoid (LocalizedModule S A) := { mul := fun m₁ m₂ => liftOn₂ m₁ m₂ (fun x₁ x₂ => LocalizedModule.mk (x₁.1 * x₂.1) (x₁.2 * x₂.2)) (by rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ ⟨b₁, t₁⟩ ⟨b₂, t₂⟩ ⟨u₁, e₁⟩ ⟨u₂, e₂⟩ rw [mk_eq] use u₁ * u₂ dsimp only at e₁ e₂ ⊢ rw [eq_comm] trans (u₁ • t₁ • a₁) • u₂ • t₂ • a₂ on_goal 1 => rw [e₁, e₂] on_goal 2 => rw [eq_comm] all_goals rw [smul_smul, mul_mul_mul_comm, ← smul_eq_mul, ← smul_eq_mul A, smul_smul_smul_comm, mul_smul, mul_smul]) one := mk 1 (1 : S) one_mul := by rintro ⟨a, s⟩ exact mk_eq.mpr ⟨1, by simp only [one_mul, one_smul]⟩ mul_one := by rintro ⟨a, s⟩ exact mk_eq.mpr ⟨1, by simp only [mul_one, one_smul]⟩ mul_assoc := by rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ ⟨a₃, s₃⟩ apply mk_eq.mpr _ use 1 simp only [one_mul, smul_smul, ← mul_assoc, mul_right_comm] } instance {A : Type} [Semiring A] [Algebra R A] {S : Submonoid R} : Semiring (LocalizedModule S A) := { show (AddCommMonoid (LocalizedModule S A)) by infer_instance, show (Monoid (LocalizedModule S A)) by infer_instance with left_distrib := by rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ ⟨a₃, s₃⟩ apply mk_eq.mpr _ use 1 simp only [one_mul, smul_add, mul_add, mul_smul_comm, smul_smul, ← mul_assoc, mul_right_comm] right_distrib := by rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ ⟨a₃, s₃⟩ apply mk_eq.mpr _ use 1 simp only [one_mul, smul_add, add_mul, smul_smul, ← mul_assoc, smul_mul_assoc, mul_right_comm] zero_mul := by rintro ⟨a, s⟩ exact mk_eq.mpr ⟨1, by simp only [zero_mul, smul_zero]⟩ mul_zero := by rintro ⟨a, s⟩ exact mk_eq.mpr ⟨1, by simp only [mul_zero, smul_zero]⟩ } instance {A : Type} [CommSemiring A] [Algebra R A] {S : Submonoid R} : CommSemiring (LocalizedModule S A) := { show Semiring (LocalizedModule S A) by infer_instance with mul_comm := by rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ exact mk_eq.mpr ⟨1, by simp only [one_smul, mul_comm]⟩ } instance {A : Type} [Ring A] [Algebra R A] {S : Submonoid R} : Ring (LocalizedModule S A) := { inferInstanceAs (AddCommGroup (LocalizedModule S A)), inferInstanceAs (Semiring (LocalizedModule S A)) with } instance {A : Type} [CommRing A] [Algebra R A] {S : Submonoid R} : CommRing (LocalizedModule S A) := { show (Ring (LocalizedModule S A)) by infer_instance with mul_comm := by rintro ⟨a₁, s₁⟩ ⟨a₂, s₂⟩ exact mk_eq.mpr ⟨1, by simp only [one_smul, mul_comm]⟩ } theorem mk_mul_mk {A : Type} [Semiring A] [Algebra R A] {a₁ a₂ : A} {s₁ s₂ : S} : mk a₁ s₁ mk a₂ s₂ = mk (a₁ * a₂) (s₁ * s₂) := rfl #align localized_module.mk_mul_mk LocalizedModule.mk_mul_mk noncomputable instance : SMul T (LocalizedModule S M) where smul x p := let a := IsLocalization.sec S x liftOn p (fun p ↦ mk (a.1 • p.1) (a.2 * p.2)) (by rintro p p' ⟨s, h⟩ refine mk_eq.mpr ⟨s, ?_⟩ calc _ = a.2 • a.1 • s • p'.2 • p.1 := by simp_rw [Submonoid.smul_def, Submonoid.coe_mul, ← mul_smul]; ring_nf _ = a.2 • a.1 • s • p.2 • p'.1 := by rw [h] _ = s • (a.2 * p.2) • a.1 • p'.1 := by simp_rw [Submonoid.smul_def, ← mul_smul, Submonoid.coe_mul]; ring_nf ) theorem smul_def (x : T) (m : M) (s : S) : x • mk m s = mk ((IsLocalization.sec S x).1 • m) ((IsLocalization.sec S x).2 * s) := rfl theorem mk'_smul_mk (r : R) (m : M) (s s' : S) : IsLocalization.mk' T r s • mk m s' = mk (r • m) (s * s') := by rw [smul_def, mk_eq] obtain ⟨c, hc⟩ := IsLocalization.eq.mp <\| IsLocalization.mk'_sec T (IsLocalization.mk' T r s) use c simp_rw [← mul_smul, Submonoid.smul_def, Submonoid.coe_mul, ← mul_smul, ← mul_assoc, mul_comm _ (s':R), mul_assoc, hc] theorem mk_smul_mk (r : R) (m : M) (s t : S) : Localization.mk r s • mk m t = mk (r • m) (s * t) := by rw [Localization.mk_eq_mk'] exact mk'_smul_mk .. #align localized_module.mk_smul_mk LocalizedModule.mk_smul_mk variable {T} private theorem one_smul_aux (p : LocalizedModule S M) : (1 : T) • p = p := by induction' p using LocalizedModule.induction_on with m s rw [show (1:T) = IsLocalization.mk' T (1:R) (1:S) by rw [IsLocalization.mk'_one, map_one]] rw [mk'_smul_mk, one_smul, one_mul] private theorem mul_smul_aux (x y : T) (p : LocalizedModule S M) : (x * y) • p = x • y • p := by induction' p using LocalizedModule.induction_on with m s rw [← IsLocalization.mk'_sec (M := S) T x, ← IsLocalization.mk'_sec (M := S) T y] simp_rw [← IsLocalization.mk'_mul, mk'_smul_mk, ← mul_smul, mul_assoc] private theorem smul_add_aux (x : T) (p q : LocalizedModule S M) : x • (p + q) = x • p + x • q := by induction' p using LocalizedModule.induction_on with m s induction' q using LocalizedModule.induction_on with n t rw [smul_def, smul_def, mk_add_mk, mk_add_mk] rw [show x • _ = IsLocalization.mk' T _ _ • _ by rw [IsLocalization.mk'_sec (M := S) T]] rw [← IsLocalization.mk'_cancel _ _ (IsLocalization.sec S x).2, mk'_smul_mk] congr 1 · simp only [Submonoid.smul_def, smul_add, ← mul_smul, Submonoid.coe_mul]; ring_nf · rw [mul_mul_mul_comm] -- ring does not work here private theorem smul_zero_aux (x : T) : x • (0 : LocalizedModule S M) = 0 := by erw [smul_def, smul_zero, zero_mk] private theorem add_smul_aux (x y : T) (p : LocalizedModule S M) : (x + y) • p = x • p + y • p := by induction' p using LocalizedModule.induction_on with m s rw [smul_def T x, smul_def T y, mk_add_mk, show (x + y) • _ = IsLocalization.mk' T _ _ • _ by rw [← IsLocalization.mk'_sec (M := S) T x, ← IsLocalization.mk'_sec (M := S) T y, ← IsLocalization.mk'_add, IsLocalization.mk'_cancel _ _ s], mk'_smul_mk, ← smul_assoc, ← smul_assoc, ← add_smul] congr 1 · simp only [Submonoid.smul_def, Submonoid.coe_mul, smul_eq_mul]; ring_nf · rw [mul_mul_mul_comm, mul_assoc] -- ring does not work here private theorem zero_smul_aux (p : LocalizedModule S M) : (0 : T) • p = 0 := by induction' p using LocalizedModule.induction_on with m s rw [show (0:T) = IsLocalization.mk' T (0:R) (1:S) by rw [IsLocalization.mk'_zero], mk'_smul_mk, zero_smul, zero_mk] noncomputable instance isModule : Module T (LocalizedModule S M) where smul := (· • ·) one_smul := one_smul_aux mul_smul := mul_smul_aux smul_add := smul_add_aux smul_zero := smul_zero_aux add_smul := add_smul_aux zero_smul := zero_smul_aux @[simp] theorem mk_cancel_common_left (s' s : S) (m : M) : mk (s' • m) (s' * s) = mk m s := mk_eq.mpr ⟨1, by simp only [mul_smul, one_smul] rw [smul_comm]⟩ #align localized_module.mk_cancel_common_left LocalizedModule.mk_cancel_common_left @[simp] theorem mk_cancel (s : S) (m : M) : mk (s • m) s = mk m 1 := mk_eq.mpr ⟨1, by simp⟩ #align localized_module.mk_cancel LocalizedModule.mk_cancel @[simp] theorem mk_cancel_common_right (s s' : S) (m : M) : mk (s' • m) (s * s') = mk m s := mk_eq.mpr ⟨1, by simp [mul_smul]⟩ #align localized_module.mk_cancel_common_right LocalizedModule.mk_cancel_common_right noncomputable instance isModule' : Module R (LocalizedModule S M) := { Module.compHom (LocalizedModule S M) <\| algebraMap R (Localization S) with } #align localized_module.is_module' LocalizedModule.isModule' theorem smul'_mk (r : R) (s : S) (m : M) : r • mk m s = mk (r • m) s := by erw [mk_smul_mk r m 1 s, one_mul] #align localized_module.smul'_mk LocalizedModule.smul'_mk theorem smul'_mul {A : Type} [Semiring A] [Algebra R A] (x : T) (p₁ p₂ : LocalizedModule S A) : x • p₁ p₂ = x • (p₁ * p₂) := by induction p₁, p₂ using induction_on₂ with \| _ a₁ s₁ a₂ s₂ => _ rw [mk_mul_mk, smul_def, smul_def, mk_mul_mk, mul_assoc, smul_mul_assoc] theorem mul_smul' {A : Type} [Semiring A] [Algebra R A] (x : T) (p₁ p₂ : LocalizedModule S A) : p₁ x • p₂ = x • (p₁ * p₂) := by induction p₁, p₂ using induction_on₂ with \| _ a₁ s₁ a₂ s₂ => _ rw [smul_def, mk_mul_mk, mk_mul_mk, smul_def, mul_left_comm, mul_smul_comm] variable (T) noncomputable instance {A : Type} [Semiring A] [Algebra R A] : Algebra T (LocalizedModule S A) := Algebra.ofModule smul'_mul mul_smul' theorem algebraMap_mk' {A : Type} [Semiring A] [Algebra R A] (a : R) (s : S) : algebraMap _ _ (IsLocalization.mk' T a s) = mk (algebraMap R A a) s := by rw [Algebra.algebraMap_eq_smul_one] change _ • mk _ _ = _ rw [mk'_smul_mk, Algebra.algebraMap_eq_smul_one, mul_one] theorem algebraMap_mk {A : Type} [Semiring A] [Algebra R A] (a : R) (s : S) : algebraMap _ _ (Localization.mk a s) = mk (algebraMap R A a) s := by rw [Localization.mk_eq_mk'] exact algebraMap_mk' .. #align localized_module.algebra_map_mk LocalizedModule.algebraMap_mk instance : IsScalarTower R T (LocalizedModule S M) where smul_assoc r x p := by induction' p using LocalizedModule.induction_on with m s rw [← IsLocalization.mk'_sec (M := S) T x, IsLocalization.smul_mk', mk'_smul_mk, mk'_smul_mk, smul'_mk, mul_smul] noncomputable instance algebra' {A : Type} [Semiring A] [Algebra R A] : Algebra R (LocalizedModule S A) := { (algebraMap (Localization S) (LocalizedModule S A)).comp (algebraMap R <\| Localization S), show Module R (LocalizedModule S A) by infer_instance with commutes' := by intro r x induction x using induction_on with \| _ a s => _ dsimp rw [← Localization.mk_one_eq_algebraMap, algebraMap_mk, mk_mul_mk, mk_mul_mk, mul_comm, Algebra.commutes] smul_def' := by intro r x induction x using induction_on with \| _ a s => _ dsimp rw [← Localization.mk_one_eq_algebraMap, algebraMap_mk, mk_mul_mk, smul'_mk, Algebra.smul_def, one_mul] } #align localized_module.algebra' LocalizedModule.algebra' section variable (S M) /-- The function `m ↦ m / 1` as an `R`-linear map. -/ @[simps] def mkLinearMap : M →ₗ[R] LocalizedModule S M where toFun m := mk m 1 map_add' x y := by simp [mk_add_mk] map_smul' r x := (smul'_mk _ _ _).symm #align localized_module.mk_linear_map LocalizedModule.mkLinearMap end /-- For any `s : S`, there is an `R`-linear map given by `a/b ↦ a/(bs)`. -/ @[simps] def divBy (s : S) : LocalizedModule S M →ₗ[R] LocalizedModule S M where toFun p := p.liftOn (fun p => mk p.1 (p.2 s)) fun ⟨a, b⟩ ⟨a', b'⟩ ⟨c, eq1⟩ => mk_eq.mpr ⟨c, by rw [mul_smul, mul_smul, smul_comm _ s, smul_comm _ s, eq1, smul_comm _ s, smul_comm _ s]⟩ map_add' x y := by refine x.induction_on₂ ?_ y intro m₁ m₂ t₁ t₂ simp_rw [mk_add_mk, LocalizedModule.liftOn_mk, mk_add_mk, mul_smul, mul_comm _ s, mul_assoc, smul_comm _ s, ← smul_add, mul_left_comm s t₁ t₂, mk_cancel_common_left s] map_smul' r x := by refine x.induction_on (fun _ _ ↦ ?_) dsimp only change liftOn (mk _ _) _ _ = r • (liftOn (mk _ _) _ _) simp_rw [liftOn_mk, mul_assoc, ← smul_def] congr! #align localized_module.div_by LocalizedModule.divBy theorem divBy_mul_by (s : S) (p : LocalizedModule S M) : divBy s (algebraMap R (Module.End R (LocalizedModule S M)) s p) = p := p.induction_on fun m t => by rw [Module.algebraMap_end_apply, divBy_apply] erw [smul_def] rw [LocalizedModule.liftOn_mk, mul_assoc, ← smul_def] erw [smul'_mk] rw [← Submonoid.smul_def, mk_cancel_common_right _ s] #align localized_module.div_by_mul_by LocalizedModule.divBy_mul_by theorem mul_by_divBy (s : S) (p : LocalizedModule S M) : algebraMap R (Module.End R (LocalizedModule S M)) s (divBy s p) = p := p.induction_on fun m t => by rw [divBy_apply, Module.algebraMap_end_apply, LocalizedModule.liftOn_mk, smul'_mk, ← Submonoid.smul_def, mk_cancel_common_right _ s] #align localized_module.mul_by_div_by LocalizedModule.mul_by_divBy end end LocalizedModule section IsLocalizedModule universe u v variable {R : Type} [CommSemiring R] (S : Submonoid R) variable {M M' M'' : Type} [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid M''] variable {A : Type} [CommSemiring A] [Algebra R A] [Module A M'] [IsLocalization S A] variable [Module R M] [Module R M'] [Module R M''] [IsScalarTower R A M'] variable (f : M →ₗ[R] M') (g : M →ₗ[R] M'') /-- The characteristic predicate for localized module. `IsLocalizedModule S f` describes that `f : M ⟶ M'` is the localization map identifying `M'` as `LocalizedModule S M`. -/ @[mk_iff] class IsLocalizedModule : Prop where map_units : ∀ x : S, IsUnit (algebraMap R (Module.End R M') x) surj' : ∀ y : M', ∃ x : M × S, x.2 • y = f x.1 exists_of_eq : ∀ {x₁ x₂}, f x₁ = f x₂ → ∃ c : S, c • x₁ = c • x₂ #align is_localized_module IsLocalizedModule attribute [nolint docBlame] IsLocalizedModule.map_units IsLocalizedModule.surj' IsLocalizedModule.exists_of_eq -- Porting note: Manually added to make `S` and `f` explicit. lemma IsLocalizedModule.surj [IsLocalizedModule S f] (y : M') : ∃ x : M × S, x.2 • y = f x.1 := surj' y -- Porting note: Manually added to make `S` and `f` explicit. lemma IsLocalizedModule.eq_iff_exists [IsLocalizedModule S f] {x₁ x₂} : f x₁ = f x₂ ↔ ∃ c : S, c • x₁ = c • x₂ := Iff.intro exists_of_eq fun ⟨c, h⟩ ↦ by apply_fun f at h simp_rw [f.map_smul_of_tower, Submonoid.smul_def, ← Module.algebraMap_end_apply R R] at h exact ((Module.End_isUnit_iff _).mp <\| map_units f c).1 h theorem IsLocalizedModule.of_linearEquiv (e : M' ≃ₗ[R] M'') [hf : IsLocalizedModule S f] : IsLocalizedModule S (e ∘ₗ f : M →ₗ[R] M'') where map_units s := by rw [show algebraMap R (Module.End R M'') s = e ∘ₗ (algebraMap R (Module.End R M') s) ∘ₗ e.symm by ext; simp, Module.End_isUnit_iff, LinearMap.coe_comp, LinearMap.coe_comp, LinearEquiv.coe_coe, LinearEquiv.coe_coe, EquivLike.comp_bijective, EquivLike.bijective_comp] exact (Module.End_isUnit_iff _).mp <\| hf.map_units s surj' x := by obtain ⟨p, h⟩ := hf.surj' (e.symm x) exact ⟨p, by rw [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ← e.congr_arg h, Submonoid.smul_def, Submonoid.smul_def, LinearEquiv.map_smul, LinearEquiv.apply_symm_apply]⟩ exists_of_eq h := by simp_rw [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, EmbeddingLike.apply_eq_iff_eq] at h exact hf.exists_of_eq h variable (M) in lemma isLocalizedModule_id (R') [CommSemiring R'] [Algebra R R'] [IsLocalization S R'] [Module R' M] [IsScalarTower R R' M] : IsLocalizedModule S (.id : M →ₗ[R] M) where map_units s := by rw [← (Algebra.lsmul R (A := R') R M).commutes]; exact (IsLocalization.map_units R' s).map _ surj' m := ⟨(m, 1), one_smul _ _⟩ exists_of_eq h := ⟨1, congr_arg _ h⟩ variable {S} in theorem isLocalizedModule_iff_isLocalization {A Aₛ} [CommSemiring A] [Algebra R A] [CommSemiring Aₛ] [Algebra A Aₛ] [Algebra R Aₛ] [IsScalarTower R A Aₛ] : IsLocalizedModule S (IsScalarTower.toAlgHom R A Aₛ).toLinearMap ↔ IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ := by rw [isLocalizedModule_iff, isLocalization_iff] refine and_congr ?_ (and_congr (forall_congr' fun _ ↦ ?_) (forall₂_congr fun _ _ ↦ ?_)) · simp_rw [← (Algebra.lmul R Aₛ).commutes, Algebra.lmul_isUnit_iff, Subtype.forall, Algebra.algebraMapSubmonoid, ← SetLike.mem_coe, Submonoid.coe_map, Set.forall_mem_image, ← IsScalarTower.algebraMap_apply] · simp_rw [Prod.exists, Subtype.exists, Algebra.algebraMapSubmonoid] simp [← IsScalarTower.algebraMap_apply, Submonoid.mk_smul, Algebra.smul_def, mul_comm] · congr!; simp_rw [Subtype.exists, Algebra.algebraMapSubmonoid]; simp [Algebra.smul_def] instance {A Aₛ} [CommSemiring A] [Algebra R A][CommSemiring Aₛ] [Algebra A Aₛ] [Algebra R Aₛ] [IsScalarTower R A Aₛ] [h : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ] : IsLocalizedModule S (IsScalarTower.toAlgHom R A Aₛ).toLinearMap := isLocalizedModule_iff_isLocalization.mpr h lemma isLocalizedModule_iff_isLocalization' (R') [CommSemiring R'] [Algebra R R'] : IsLocalizedModule S (Algebra.ofId R R').toLinearMap ↔ IsLocalization S R' := by convert isLocalizedModule_iff_isLocalization (S := S) (A := R) (Aₛ := R') exact (Submonoid.map_id S).symm namespace LocalizedModule /-- If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then there is a linear map `LocalizedModule S M → M''`. -/ noncomputable def lift' (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit (algebraMap R (Module.End R M'') x)) : LocalizedModule S M → M'' := fun m => m.liftOn (fun p => (h p.2).unit⁻¹.val <\| g p.1) fun ⟨m, s⟩ ⟨m', s'⟩ ⟨c, eq1⟩ => by -- Porting note: We remove `generalize_proofs h1 h2`. This does nothing here. dsimp only simp only [Submonoid.smul_def] at eq1 rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← map_smul, eq_comm, Module.End_algebraMap_isUnit_inv_apply_eq_iff] have : c • s • g m' = c • s' • g m := by simp only [Submonoid.smul_def, ← g.map_smul, eq1] have : Function.Injective (h c).unit.inv := by rw [Function.injective_iff_hasLeftInverse] refine ⟨(h c).unit, ?_⟩ intro x change ((h c).unit.1 (h c).unit.inv) x = x simp only [Units.inv_eq_val_inv, IsUnit.mul_val_inv, LinearMap.one_apply] apply_fun (h c).unit.inv erw [Units.inv_eq_val_inv, Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← (h c).unit⁻¹.val.map_smul] symm rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← g.map_smul, ← g.map_smul, ← g.map_smul, ← g.map_smul, eq1] #align localized_module.lift' LocalizedModule.lift' theorem lift'_mk (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) (m : M) (s : S) : LocalizedModule.lift' S g h (LocalizedModule.mk m s) = (h s).unit⁻¹.val (g m) := rfl #align localized_module.lift'_mk LocalizedModule.lift'_mk theorem lift'_add (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) (x y) : LocalizedModule.lift' S g h (x + y) = LocalizedModule.lift' S g h x + LocalizedModule.lift' S g h y := LocalizedModule.induction_on₂ (by intro a a' b b' erw [LocalizedModule.lift'_mk, LocalizedModule.lift'_mk, LocalizedModule.lift'_mk] -- Porting note: We remove `generalize_proofs h1 h2 h3`. This only generalize `h1`. erw [map_add, Module.End_algebraMap_isUnit_inv_apply_eq_iff, smul_add, ← map_smul, ← map_smul, ← map_smul] congr 1 <;> symm · erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, mul_smul, ← map_smul] rfl · dsimp erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, mul_comm, mul_smul, ← map_smul] rfl) x y #align localized_module.lift'_add LocalizedModule.lift'_add theorem lift'_smul (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) (r : R) (m) : r • LocalizedModule.lift' S g h m = LocalizedModule.lift' S g h (r • m) := m.induction_on fun a b => by rw [LocalizedModule.lift'_mk, LocalizedModule.smul'_mk, LocalizedModule.lift'_mk] -- Porting note: We remove `generalize_proofs h1 h2`. This does nothing here. rw [← map_smul, ← g.map_smul] #align localized_module.lift'_smul LocalizedModule.lift'_smul /-- If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then there is a linear map `LocalizedModule S M → M''`. -/ noncomputable def lift (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) : LocalizedModule S M →ₗ[R] M'' where toFun := LocalizedModule.lift' S g h map_add' := LocalizedModule.lift'_add S g h map_smul' r x := by rw [LocalizedModule.lift'_smul, RingHom.id_apply] #align localized_module.lift LocalizedModule.lift /-- If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then `lift g m s = s⁻¹ • g m`. -/ theorem lift_mk (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit (algebraMap R (Module.End R M'') x)) (m : M) (s : S) : LocalizedModule.lift S g h (LocalizedModule.mk m s) = (h s).unit⁻¹.val (g m) := rfl #align localized_module.lift_mk LocalizedModule.lift_mk /-- If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then there is a linear map `lift g ∘ mkLinearMap = g`. -/ theorem lift_comp (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) : (lift S g h).comp (mkLinearMap S M) = g := by ext x; dsimp; rw [LocalizedModule.lift_mk] erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, one_smul] #align localized_module.lift_comp LocalizedModule.lift_comp /-- If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible and `l` is another linear map `LocalizedModule S M ⟶ M''` such that `l ∘ mkLinearMap = g` then `l = lift g` -/ theorem lift_unique (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) (l : LocalizedModule S M →ₗ[R] M'') (hl : l.comp (LocalizedModule.mkLinearMap S M) = g) : LocalizedModule.lift S g h = l := by ext x; induction' x using LocalizedModule.induction_on with m s rw [LocalizedModule.lift_mk] rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← hl, LinearMap.coe_comp, Function.comp_apply, LocalizedModule.mkLinearMap_apply, ← l.map_smul, LocalizedModule.smul'_mk] congr 1; rw [LocalizedModule.mk_eq] refine ⟨1, ?_⟩; simp only [one_smul, Submonoid.smul_def] #align localized_module.lift_unique LocalizedModule.lift_unique end LocalizedModule instance localizedModuleIsLocalizedModule : IsLocalizedModule S (LocalizedModule.mkLinearMap S M) where map_units s := ⟨⟨algebraMap R (Module.End R (LocalizedModule S M)) s, LocalizedModule.divBy s, DFunLike.ext _ _ <\| LocalizedModule.mul_by_divBy s, DFunLike.ext _ _ <\| LocalizedModule.divBy_mul_by s⟩, DFunLike.ext _ _ fun p => p.induction_on <\| by intros rfl⟩ surj' p := p.induction_on fun m t => by refine ⟨⟨m, t⟩, ?_⟩ erw [LocalizedModule.smul'_mk, LocalizedModule.mkLinearMap_apply, Submonoid.coe_subtype, LocalizedModule.mk_cancel t] exists_of_eq eq1 := by simpa only [eq_comm, one_smul] using LocalizedModule.mk_eq.mp eq1 #align localized_module_is_localized_module localizedModuleIsLocalizedModule namespace IsLocalizedModule variable [IsLocalizedModule S f] /-- If `(M', f : M ⟶ M')` satisfies universal property of localized module, there is a canonical map `LocalizedModule S M ⟶ M'`. -/ noncomputable def fromLocalizedModule' : LocalizedModule S M → M' := fun p => p.liftOn (fun x => (IsLocalizedModule.map_units f x.2).unit⁻¹.val (f x.1)) (by rintro ⟨a, b⟩ ⟨a', b'⟩ ⟨c, eq1⟩ dsimp -- Porting note: We remove `generalize_proofs h1 h2`. rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← map_smul, ← map_smul, Module.End_algebraMap_isUnit_inv_apply_eq_iff', ← map_smul] exact (IsLocalizedModule.eq_iff_exists S f).mpr ⟨c, eq1.symm⟩) #align is_localized_module.from_localized_module' IsLocalizedModule.fromLocalizedModule' @[simp] theorem fromLocalizedModule'_mk (m : M) (s : S) : fromLocalizedModule' S f (LocalizedModule.mk m s) = (IsLocalizedModule.map_units f s).unit⁻¹.val (f m) := rfl #align is_localized_module.from_localized_module'_mk IsLocalizedModule.fromLocalizedModule'_mk theorem fromLocalizedModule'_add (x y : LocalizedModule S M) : fromLocalizedModule' S f (x + y) = fromLocalizedModule' S f x + fromLocalizedModule' S f y := LocalizedModule.induction_on₂ (by intro a a' b b' simp only [LocalizedModule.mk_add_mk, fromLocalizedModule'_mk] -- Porting note: We remove `generalize_proofs h1 h2 h3`. rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, smul_add, ← map_smul, ← map_smul, ← map_smul, map_add] congr 1 all_goals rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff'] · simp [mul_smul, Submonoid.smul_def] · rw [Submonoid.coe_mul, LinearMap.map_smul_of_tower, mul_comm, mul_smul, Submonoid.smul_def]) x y #align is_localized_module.from_localized_module'_add IsLocalizedModule.fromLocalizedModule'_add theorem fromLocalizedModule'_smul (r : R) (x : LocalizedModule S M) : r • fromLocalizedModule' S f x = fromLocalizedModule' S f (r • x) := LocalizedModule.induction_on (by intro a b rw [fromLocalizedModule'_mk, LocalizedModule.smul'_mk, fromLocalizedModule'_mk] -- Porting note: We remove `generalize_proofs h1`. rw [f.map_smul, map_smul]) x #align is_localized_module.from_localized_module'_smul IsLocalizedModule.fromLocalizedModule'_smul /-- If `(M', f : M ⟶ M')` satisfies universal property of localized module, there is a canonical map `LocalizedModule S M ⟶ M'`. -/ noncomputable def fromLocalizedModule : LocalizedModule S M →ₗ[R] M' where toFun := fromLocalizedModule' S f map_add' := fromLocalizedModule'_add S f map_smul' r x := by rw [fromLocalizedModule'_smul, RingHom.id_apply] #align is_localized_module.from_localized_module IsLocalizedModule.fromLocalizedModule theorem fromLocalizedModule_mk (m : M) (s : S) : fromLocalizedModule S f (LocalizedModule.mk m s) = (IsLocalizedModule.map_units f s).unit⁻¹.val (f m) := rfl #align is_localized_module.from_localized_module_mk IsLocalizedModule.fromLocalizedModule_mk theorem fromLocalizedModule.inj : Function.Injective <\| fromLocalizedModule S f := fun x y eq1 => by induction' x using LocalizedModule.induction_on with a b induction' y using LocalizedModule.induction_on with a' b' simp only [fromLocalizedModule_mk] at eq1 -- Porting note: We remove `generalize_proofs h1 h2`. rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← LinearMap.map_smul, Module.End_algebraMap_isUnit_inv_apply_eq_iff'] at eq1 rw [LocalizedModule.mk_eq, ← IsLocalizedModule.eq_iff_exists S f, Submonoid.smul_def, Submonoid.smul_def, f.map_smul, f.map_smul, eq1] #align is_localized_module.from_localized_module.inj IsLocalizedModule.fromLocalizedModule.inj theorem fromLocalizedModule.surj : Function.Surjective <\| fromLocalizedModule S f := fun x => let ⟨⟨m, s⟩, eq1⟩ := IsLocalizedModule.surj S f x ⟨LocalizedModule.mk m s, by rw [fromLocalizedModule_mk, Module.End_algebraMap_isUnit_inv_apply_eq_iff, ← eq1, Submonoid.smul_def]⟩ #align is_localized_module.from_localized_module.surj IsLocalizedModule.fromLocalizedModule.surj theorem fromLocalizedModule.bij : Function.Bijective <\| fromLocalizedModule S f := ⟨fromLocalizedModule.inj _ _, fromLocalizedModule.surj _ _⟩ #align is_localized_module.from_localized_module.bij IsLocalizedModule.fromLocalizedModule.bij /-- If `(M', f : M ⟶ M')` satisfies universal property of localized module, then `M'` is isomorphic to `LocalizedModule S M` as an `R`-module. -/ @[simps!] noncomputable def iso : LocalizedModule S M ≃ₗ[R] M' := { fromLocalizedModule S f, Equiv.ofBijective (fromLocalizedModule S f) <\| fromLocalizedModule.bij _ _ with } #align is_localized_module.iso IsLocalizedModule.iso theorem iso_apply_mk (m : M) (s : S) : iso S f (LocalizedModule.mk m s) = (IsLocalizedModule.map_units f s).unit⁻¹.val (f m) := rfl #align is_localized_module.iso_apply_mk IsLocalizedModule.iso_apply_mk theorem iso_symm_apply_aux (m : M') : (iso S f).symm m = LocalizedModule.mk (IsLocalizedModule.surj S f m).choose.1 (IsLocalizedModule.surj S f m).choose.2 := by -- Porting note: We remove `generalize_proofs _ h2`. apply_fun iso S f using LinearEquiv.injective (iso S f) rw [LinearEquiv.apply_symm_apply] simp only [iso_apply, LinearMap.toFun_eq_coe, fromLocalizedModule_mk] erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff', (surj' _).choose_spec] #align is_localized_module.iso_symm_apply_aux IsLocalizedModule.iso_symm_apply_aux theorem iso_symm_apply' (m : M') (a : M) (b : S) (eq1 : b • m = f a) : (iso S f).symm m = LocalizedModule.mk a b := (iso_symm_apply_aux S f m).trans <\| LocalizedModule.mk_eq.mpr <\| by -- Porting note: We remove `generalize_proofs h1`. rw [← IsLocalizedModule.eq_iff_exists S f, Submonoid.smul_def, Submonoid.smul_def, f.map_smul, f.map_smul, ← (surj' _).choose_spec, ← Submonoid.smul_def, ← Submonoid.smul_def, ← mul_smul, mul_comm, mul_smul, eq1] #align is_localized_module.iso_symm_apply' IsLocalizedModule.iso_symm_apply' theorem iso_symm_comp : (iso S f).symm.toLinearMap.comp f = LocalizedModule.mkLinearMap S M := by ext m rw [LinearMap.comp_apply, LocalizedModule.mkLinearMap_apply, LinearEquiv.coe_coe, iso_symm_apply'] exact one_smul _ _ #align is_localized_module.iso_symm_comp IsLocalizedModule.iso_symm_comp /-- If `M'` is a localized module and `g` is a linear map `M' → M''` such that all scalar multiplication by `s : S` is invertible, then there is a linear map `M' → M''`. -/ noncomputable def lift (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) : M' →ₗ[R] M'' := (LocalizedModule.lift S g h).comp (iso S f).symm.toLinearMap #align is_localized_module.lift IsLocalizedModule.lift theorem lift_comp (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) : (lift S f g h).comp f = g := by dsimp only [IsLocalizedModule.lift] rw [LinearMap.comp_assoc, iso_symm_comp, LocalizedModule.lift_comp S g h] #align is_localized_module.lift_comp IsLocalizedModule.lift_comp @[simp] theorem lift_apply (g : M →ₗ[R] M'') (h) (x) : lift S f g h (f x) = g x := LinearMap.congr_fun (lift_comp S f g h) x theorem lift_unique (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) (l : M' →ₗ[R] M'') (hl : l.comp f = g) : lift S f g h = l := by dsimp only [IsLocalizedModule.lift] rw [LocalizedModule.lift_unique S g h (l.comp (iso S f).toLinearMap), LinearMap.comp_assoc, LinearEquiv.comp_coe, LinearEquiv.symm_trans_self, LinearEquiv.refl_toLinearMap, LinearMap.comp_id] rw [LinearMap.comp_assoc, ← hl] congr 1 ext x rw [LinearMap.comp_apply, LocalizedModule.mkLinearMap_apply, LinearEquiv.coe_coe, iso_apply, fromLocalizedModule'_mk, Module.End_algebraMap_isUnit_inv_apply_eq_iff, OneMemClass.coe_one, one_smul] #align is_localized_module.lift_unique IsLocalizedModule.lift_unique /-- Universal property from localized module: If `(M', f : M ⟶ M')` is a localized module then it satisfies the following universal property: For every `R`-module `M''` which every `s : S`-scalar multiplication is invertible and for every `R`-linear map `g : M ⟶ M''`, there is a unique `R`-linear map `l : M' ⟶ M''` such that `l ∘ f = g`. ``` M -----f----> M' \| / \|g / \| / l v / M'' ``` -/ theorem is_universal : ∀ (g : M →ₗ[R] M'') (_ : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)), ∃! l : M' →ₗ[R] M'', l.comp f = g := fun g h => ⟨lift S f g h, lift_comp S f g h, fun l hl => (lift_unique S f g h l hl).symm⟩ #align is_localized_module.is_universal IsLocalizedModule.is_universal theorem ringHom_ext (map_unit : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) ⦃j k : M' →ₗ[R] M''⦄ (h : j.comp f = k.comp f) : j = k := by rw [← lift_unique S f (k.comp f) map_unit j h, lift_unique] rfl #align is_localized_module.ring_hom_ext IsLocalizedModule.ringHom_ext /-- If `(M', f)` and `(M'', g)` both satisfy universal property of localized module, then `M', M''` are isomorphic as `R`-module -/ noncomputable def linearEquiv [IsLocalizedModule S g] : M' ≃ₗ[R] M'' := (iso S f).symm.trans (iso S g) #align is_localized_module.linear_equiv IsLocalizedModule.linearEquiv variable {S} theorem smul_injective (s : S) : Function.Injective fun m : M' => s • m := ((Module.End_isUnit_iff _).mp (IsLocalizedModule.map_units f s)).injective #align is_localized_module.smul_injective IsLocalizedModule.smul_injective theorem smul_inj (s : S) (m₁ m₂ : M') : s • m₁ = s • m₂ ↔ m₁ = m₂ := (smul_injective f s).eq_iff #align is_localized_module.smul_inj IsLocalizedModule.smul_inj /-- `mk' f m s` is the fraction `m/s` with respect to the localization map `f`. -/ noncomputable def mk' (m : M) (s : S) : M' := fromLocalizedModule S f (LocalizedModule.mk m s) #align is_localized_module.mk' IsLocalizedModule.mk' theorem mk'_smul (r : R) (m : M) (s : S) : mk' f (r • m) s = r • mk' f m s := by delta mk' rw [← LocalizedModule.smul'_mk, LinearMap.map_smul] #align is_localized_module.mk'_smul IsLocalizedModule.mk'_smul theorem mk'_add_mk' (m₁ m₂ : M) (s₁ s₂ : S) : mk' f m₁ s₁ + mk' f m₂ s₂ = mk' f (s₂ • m₁ + s₁ • m₂) (s₁ * s₂) := by delta mk' rw [← map_add, LocalizedModule.mk_add_mk] #align is_localized_module.mk'_add_mk' IsLocalizedModule.mk'_add_mk' @[simp] theorem mk'_zero (s : S) : mk' f 0 s = 0 := by rw [← zero_smul R (0 : M), mk'_smul, zero_smul] #align is_localized_module.mk'_zero IsLocalizedModule.mk'_zero variable (S) @[simp] theorem mk'_one (m : M) : mk' f m (1 : S) = f m := by delta mk' rw [fromLocalizedModule_mk, Module.End_algebraMap_isUnit_inv_apply_eq_iff, Submonoid.coe_one, one_smul] #align is_localized_module.mk'_one IsLocalizedModule.mk'_one variable {S} @[simp] theorem mk'_cancel (m : M) (s : S) : mk' f (s • m) s = f m := by delta mk' rw [LocalizedModule.mk_cancel, ← mk'_one S f, fromLocalizedModule_mk, Module.End_algebraMap_isUnit_inv_apply_eq_iff, OneMemClass.coe_one, mk'_one, one_smul] #align is_localized_module.mk'_cancel IsLocalizedModule.mk'_cancel @[simp] theorem mk'_cancel' (m : M) (s : S) : s • mk' f m s = f m := by rw [Submonoid.smul_def, ← mk'_smul, ← Submonoid.smul_def, mk'_cancel] #align is_localized_module.mk'_cancel' IsLocalizedModule.mk'_cancel' @[simp] theorem mk'_cancel_left (m : M) (s₁ s₂ : S) : mk' f (s₁ • m) (s₁ * s₂) = mk' f m s₂ := by delta mk' rw [LocalizedModule.mk_cancel_common_left] #align is_localized_module.mk'_cancel_left IsLocalizedModule.mk'_cancel_left @[simp] theorem mk'_cancel_right (m : M) (s₁ s₂ : S) : mk' f (s₂ • m) (s₁ * s₂) = mk' f m s₁ := by delta mk' rw [LocalizedModule.mk_cancel_common_right] #align is_localized_module.mk'_cancel_right IsLocalizedModule.mk'_cancel_right theorem mk'_add (m₁ m₂ : M) (s : S) : mk' f (m₁ + m₂) s = mk' f m₁ s + mk' f m₂ s := by rw [mk'_add_mk', ← smul_add, mk'_cancel_left] #align is_localized_module.mk'_add IsLocalizedModule.mk'_add theorem mk'_eq_mk'_iff (m₁ m₂ : M) (s₁ s₂ : S) : mk' f m₁ s₁ = mk' f m₂ s₂ ↔ ∃ s : S, s • s₁ • m₂ = s • s₂ • m₁ := by delta mk' rw [(fromLocalizedModule.inj S f).eq_iff, LocalizedModule.mk_eq] simp_rw [eq_comm] #align is_localized_module.mk'_eq_mk'_iff IsLocalizedModule.mk'_eq_mk'_iff theorem mk'_neg {M M' : Type} [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] (m : M) (s : S) : mk' f (-m) s = -mk' f m s := by delta mk' rw [LocalizedModule.mk_neg, map_neg] #align is_localized_module.mk'_neg IsLocalizedModule.mk'_neg theorem mk'_sub {M M' : Type} [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] (m₁ m₂ : M) (s : S) : mk' f (m₁ - m₂) s = mk' f m₁ s - mk' f m₂ s := by rw [sub_eq_add_neg, sub_eq_add_neg, mk'_add, mk'_neg] #align is_localized_module.mk'_sub IsLocalizedModule.mk'_sub theorem mk'_sub_mk' {M M' : Type} [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : S) : mk' f m₁ s₁ - mk' f m₂ s₂ = mk' f (s₂ • m₁ - s₁ • m₂) (s₁ s₂) := by rw [sub_eq_add_neg, ← mk'_neg, mk'_add_mk', smul_neg, ← sub_eq_add_neg] #align is_localized_module.mk'_sub_mk' IsLocalizedModule.mk'_sub_mk' theorem mk'_mul_mk'_of_map_mul {M M' : Type} [Semiring M] [Semiring M'] [Module R M] [Algebra R M'] (f : M →ₗ[R] M') (hf : ∀ m₁ m₂, f (m₁ m₂) = f m₁ * f m₂) [IsLocalizedModule S f] (m₁ m₂ : M) (s₁ s₂ : S) : mk' f m₁ s₁ * mk' f m₂ s₂ = mk' f (m₁ * m₂) (s₁ * s₂) := by symm apply (Module.End_algebraMap_isUnit_inv_apply_eq_iff _ _ _ _).mpr simp_rw [Submonoid.coe_mul, ← smul_eq_mul] rw [smul_smul_smul_comm, ← mk'_smul, ← mk'_smul] simp_rw [← Submonoid.smul_def, mk'_cancel, smul_eq_mul, hf] #align is_localized_module.mk'_mul_mk'_of_map_mul IsLocalizedModule.mk'_mul_mk'_of_map_mul theorem mk'_mul_mk' {M M' : Type} [Semiring M] [Semiring M'] [Algebra R M] [Algebra R M'] (f : M →ₐ[R] M') [IsLocalizedModule S f.toLinearMap] (m₁ m₂ : M) (s₁ s₂ : S) : mk' f.toLinearMap m₁ s₁ mk' f.toLinearMap m₂ s₂ = mk' f.toLinearMap (m₁ * m₂) (s₁ * s₂) := mk'_mul_mk'_of_map_mul f.toLinearMap f.map_mul m₁ m₂ s₁ s₂ #align is_localized_module.mk'_mul_mk' IsLocalizedModule.mk'_mul_mk' variable {f} /-- Porting note (#10618): simp can prove this @[simp] -/ theorem mk'_eq_iff {m : M} {s : S} {m' : M'} : mk' f m s = m' ↔ f m = s • m' := by rw [← smul_inj f s, Submonoid.smul_def, ← mk'_smul, ← Submonoid.smul_def, mk'_cancel] #align is_localized_module.mk'_eq_iff IsLocalizedModule.mk'_eq_iff @[simp] theorem mk'_eq_zero {m : M} (s : S) : mk' f m s = 0 ↔ f m = 0 := by rw [mk'_eq_iff, smul_zero] #align is_localized_module.mk'_eq_zero IsLocalizedModule.mk'_eq_zero variable (f) theorem mk'_eq_zero' {m : M} (s : S) : mk' f m s = 0 ↔ ∃ s' : S, s' • m = 0 := by simp_rw [← mk'_zero f (1 : S), mk'_eq_mk'_iff, smul_zero, one_smul, eq_comm] #align is_localized_module.mk'_eq_zero' IsLocalizedModule.mk'_eq_zero'	Mathlib/Algebra/Module/LocalizedModule.lean	1,082	1,085	theorem mk_eq_mk' (s : S) (m : M) : LocalizedModule.mk m s = mk' (LocalizedModule.mkLinearMap S M) m s := by	rw [eq_comm, mk'_eq_iff, Submonoid.smul_def, LocalizedModule.smul'_mk, ← Submonoid.smul_def, LocalizedModule.mk_cancel, LocalizedModule.mkLinearMap_apply]
/- Copyright (c) 2021 Thomas Browning. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Thomas Browning -/ import Mathlib.Data.ZMod.Quotient #align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f" /-! # Complements In this file we define the complement of a subgroup. ## Main definitions - `IsComplement S T` where `S` and `T` are subsets of `G` states that every `g : G` can be written uniquely as a product `s * t` for `s ∈ S`, `t ∈ T`. - `leftTransversals T` where `T` is a subset of `G` is the set of all left-complements of `T`, i.e. the set of all `S : Set G` that contain exactly one element of each left coset of `T`. - `rightTransversals S` where `S` is a subset of `G` is the set of all right-complements of `S`, i.e. the set of all `T : Set G` that contain exactly one element of each right coset of `S`. - `transferTransversal H g` is a specific `leftTransversal` of `H` that is used in the computation of the transfer homomorphism evaluated at an element `g : G`. ## Main results - `isComplement'_of_coprime` : Subgroups of coprime order are complements. -/ open Set open scoped Pointwise namespace Subgroup variable {G : Type} [Group G] (H K : Subgroup G) (S T : Set G) /-- `S` and `T` are complements if `() : S × T → G` is a bijection. This notion generalizes left transversals, right transversals, and complementary subgroups. -/ @[to_additive "`S` and `T` are complements if `(+) : S × T → G` is a bijection"] def IsComplement : Prop := Function.Bijective fun x : S × T => x.1.1 * x.2.1 #align subgroup.is_complement Subgroup.IsComplement #align add_subgroup.is_complement AddSubgroup.IsComplement /-- `H` and `K` are complements if `() : H × K → G` is a bijection -/ @[to_additive "`H` and `K` are complements if `(+) : H × K → G` is a bijection"] abbrev IsComplement' := IsComplement (H : Set G) (K : Set G) #align subgroup.is_complement' Subgroup.IsComplement' #align add_subgroup.is_complement' AddSubgroup.IsComplement' /-- The set of left-complements of `T : Set G` -/ @[to_additive "The set of left-complements of `T : Set G`"] def leftTransversals : Set (Set G) := { S : Set G \| IsComplement S T } #align subgroup.left_transversals Subgroup.leftTransversals #align add_subgroup.left_transversals AddSubgroup.leftTransversals /-- The set of right-complements of `S : Set G` -/ @[to_additive "The set of right-complements of `S : Set G`"] def rightTransversals : Set (Set G) := { T : Set G \| IsComplement S T } #align subgroup.right_transversals Subgroup.rightTransversals #align add_subgroup.right_transversals AddSubgroup.rightTransversals variable {H K S T} @[to_additive] theorem isComplement'_def : IsComplement' H K ↔ IsComplement (H : Set G) (K : Set G) := Iff.rfl #align subgroup.is_complement'_def Subgroup.isComplement'_def #align add_subgroup.is_complement'_def AddSubgroup.isComplement'_def @[to_additive] theorem isComplement_iff_existsUnique : IsComplement S T ↔ ∀ g : G, ∃! x : S × T, x.1.1 x.2.1 = g := Function.bijective_iff_existsUnique _ #align subgroup.is_complement_iff_exists_unique Subgroup.isComplement_iff_existsUnique #align add_subgroup.is_complement_iff_exists_unique AddSubgroup.isComplement_iff_existsUnique @[to_additive] theorem IsComplement.existsUnique (h : IsComplement S T) (g : G) : ∃! x : S × T, x.1.1 * x.2.1 = g := isComplement_iff_existsUnique.mp h g #align subgroup.is_complement.exists_unique Subgroup.IsComplement.existsUnique #align add_subgroup.is_complement.exists_unique AddSubgroup.IsComplement.existsUnique @[to_additive] theorem IsComplement'.symm (h : IsComplement' H K) : IsComplement' K H := by let ϕ : H × K ≃ K × H := Equiv.mk (fun x => ⟨x.2⁻¹, x.1⁻¹⟩) (fun x => ⟨x.2⁻¹, x.1⁻¹⟩) (fun x => Prod.ext (inv_inv _) (inv_inv _)) fun x => Prod.ext (inv_inv _) (inv_inv _) let ψ : G ≃ G := Equiv.mk (fun g : G => g⁻¹) (fun g : G => g⁻¹) inv_inv inv_inv suffices hf : (ψ ∘ fun x : H × K => x.1.1 * x.2.1) = (fun x : K × H => x.1.1 * x.2.1) ∘ ϕ by rw [isComplement'_def, IsComplement, ← Equiv.bijective_comp ϕ] apply (congr_arg Function.Bijective hf).mp -- Porting note: This was a `rw` in mathlib3 rwa [ψ.comp_bijective] exact funext fun x => mul_inv_rev _ _ #align subgroup.is_complement'.symm Subgroup.IsComplement'.symm #align add_subgroup.is_complement'.symm AddSubgroup.IsComplement'.symm @[to_additive] theorem isComplement'_comm : IsComplement' H K ↔ IsComplement' K H := ⟨IsComplement'.symm, IsComplement'.symm⟩ #align subgroup.is_complement'_comm Subgroup.isComplement'_comm #align add_subgroup.is_complement'_comm AddSubgroup.isComplement'_comm @[to_additive] theorem isComplement_univ_singleton {g : G} : IsComplement (univ : Set G) {g} := ⟨fun ⟨_, _, rfl⟩ ⟨_, _, rfl⟩ h => Prod.ext (Subtype.ext (mul_right_cancel h)) rfl, fun x => ⟨⟨⟨x * g⁻¹, ⟨⟩⟩, g, rfl⟩, inv_mul_cancel_right x g⟩⟩ #align subgroup.is_complement_top_singleton Subgroup.isComplement_univ_singleton #align add_subgroup.is_complement_top_singleton AddSubgroup.isComplement_univ_singleton @[to_additive] theorem isComplement_singleton_univ {g : G} : IsComplement ({g} : Set G) univ := ⟨fun ⟨⟨_, rfl⟩, _⟩ ⟨⟨_, rfl⟩, _⟩ h => Prod.ext rfl (Subtype.ext (mul_left_cancel h)), fun x => ⟨⟨⟨g, rfl⟩, g⁻¹ * x, ⟨⟩⟩, mul_inv_cancel_left g x⟩⟩ #align subgroup.is_complement_singleton_top Subgroup.isComplement_singleton_univ #align add_subgroup.is_complement_singleton_top AddSubgroup.isComplement_singleton_univ @[to_additive] theorem isComplement_singleton_left {g : G} : IsComplement {g} S ↔ S = univ := by refine ⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => (congr_arg _ h).mpr isComplement_singleton_univ⟩ obtain ⟨⟨⟨z, rfl : z = g⟩, y, _⟩, hy⟩ := h.2 (g * x) rwa [← mul_left_cancel hy] #align subgroup.is_complement_singleton_left Subgroup.isComplement_singleton_left #align add_subgroup.is_complement_singleton_left AddSubgroup.isComplement_singleton_left @[to_additive] theorem isComplement_singleton_right {g : G} : IsComplement S {g} ↔ S = univ := by refine ⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => h ▸ isComplement_univ_singleton⟩ obtain ⟨y, hy⟩ := h.2 (x * g) conv_rhs at hy => rw [← show y.2.1 = g from y.2.2] rw [← mul_right_cancel hy] exact y.1.2 #align subgroup.is_complement_singleton_right Subgroup.isComplement_singleton_right #align add_subgroup.is_complement_singleton_right AddSubgroup.isComplement_singleton_right @[to_additive] theorem isComplement_univ_left : IsComplement univ S ↔ ∃ g : G, S = {g} := by refine ⟨fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr ⟨?_, fun a ha b hb => ?_⟩, ?_⟩ · obtain ⟨a, _⟩ := h.2 1 exact ⟨a.2.1, a.2.2⟩ · have : (⟨⟨_, mem_top a⁻¹⟩, ⟨a, ha⟩⟩ : (⊤ : Set G) × S) = ⟨⟨_, mem_top b⁻¹⟩, ⟨b, hb⟩⟩ := h.1 ((inv_mul_self a).trans (inv_mul_self b).symm) exact Subtype.ext_iff.mp (Prod.ext_iff.mp this).2 · rintro ⟨g, rfl⟩ exact isComplement_univ_singleton #align subgroup.is_complement_top_left Subgroup.isComplement_univ_left #align add_subgroup.is_complement_top_left AddSubgroup.isComplement_univ_left @[to_additive] theorem isComplement_univ_right : IsComplement S univ ↔ ∃ g : G, S = {g} := by refine ⟨fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr ⟨?_, fun a ha b hb => ?_⟩, ?_⟩ · obtain ⟨a, _⟩ := h.2 1 exact ⟨a.1.1, a.1.2⟩ · have : (⟨⟨a, ha⟩, ⟨_, mem_top a⁻¹⟩⟩ : S × (⊤ : Set G)) = ⟨⟨b, hb⟩, ⟨_, mem_top b⁻¹⟩⟩ := h.1 ((mul_inv_self a).trans (mul_inv_self b).symm) exact Subtype.ext_iff.mp (Prod.ext_iff.mp this).1 · rintro ⟨g, rfl⟩ exact isComplement_singleton_univ #align subgroup.is_complement_top_right Subgroup.isComplement_univ_right #align add_subgroup.is_complement_top_right AddSubgroup.isComplement_univ_right @[to_additive] lemma IsComplement.mul_eq (h : IsComplement S T) : S * T = univ := eq_univ_of_forall fun x ↦ by simpa [mem_mul] using (h.existsUnique x).exists @[to_additive AddSubgroup.IsComplement.card_mul_card] lemma IsComplement.card_mul_card (h : IsComplement S T) : Nat.card S * Nat.card T = Nat.card G := (Nat.card_prod _ _).symm.trans <\| Nat.card_congr <\| Equiv.ofBijective _ h @[to_additive] theorem isComplement'_top_bot : IsComplement' (⊤ : Subgroup G) ⊥ := isComplement_univ_singleton #align subgroup.is_complement'_top_bot Subgroup.isComplement'_top_bot #align add_subgroup.is_complement'_top_bot AddSubgroup.isComplement'_top_bot @[to_additive] theorem isComplement'_bot_top : IsComplement' (⊥ : Subgroup G) ⊤ := isComplement_singleton_univ #align subgroup.is_complement'_bot_top Subgroup.isComplement'_bot_top #align add_subgroup.is_complement'_bot_top AddSubgroup.isComplement'_bot_top @[to_additive (attr := simp)] theorem isComplement'_bot_left : IsComplement' ⊥ H ↔ H = ⊤ := isComplement_singleton_left.trans coe_eq_univ #align subgroup.is_complement'_bot_left Subgroup.isComplement'_bot_left #align add_subgroup.is_complement'_bot_left AddSubgroup.isComplement'_bot_left @[to_additive (attr := simp)] theorem isComplement'_bot_right : IsComplement' H ⊥ ↔ H = ⊤ := isComplement_singleton_right.trans coe_eq_univ #align subgroup.is_complement'_bot_right Subgroup.isComplement'_bot_right #align add_subgroup.is_complement'_bot_right AddSubgroup.isComplement'_bot_right @[to_additive (attr := simp)] theorem isComplement'_top_left : IsComplement' ⊤ H ↔ H = ⊥ := isComplement_univ_left.trans coe_eq_singleton #align subgroup.is_complement'_top_left Subgroup.isComplement'_top_left #align add_subgroup.is_complement'_top_left AddSubgroup.isComplement'_top_left @[to_additive (attr := simp)] theorem isComplement'_top_right : IsComplement' H ⊤ ↔ H = ⊥ := isComplement_univ_right.trans coe_eq_singleton #align subgroup.is_complement'_top_right Subgroup.isComplement'_top_right #align add_subgroup.is_complement'_top_right AddSubgroup.isComplement'_top_right @[to_additive] theorem mem_leftTransversals_iff_existsUnique_inv_mul_mem : S ∈ leftTransversals T ↔ ∀ g : G, ∃! s : S, (s : G)⁻¹ * g ∈ T := by rw [leftTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique] refine ⟨fun h g => ?_, fun h g => ?_⟩ · obtain ⟨x, h1, h2⟩ := h g exact ⟨x.1, (congr_arg (· ∈ T) (eq_inv_mul_of_mul_eq h1)).mp x.2.2, fun y hy => (Prod.ext_iff.mp (h2 ⟨y, (↑y)⁻¹ * g, hy⟩ (mul_inv_cancel_left ↑y g))).1⟩ · obtain ⟨x, h1, h2⟩ := h g refine ⟨⟨x, (↑x)⁻¹ * g, h1⟩, mul_inv_cancel_left (↑x) g, fun y hy => ?_⟩ have hf := h2 y.1 ((congr_arg (· ∈ T) (eq_inv_mul_of_mul_eq hy)).mp y.2.2) exact Prod.ext hf (Subtype.ext (eq_inv_mul_of_mul_eq (hf ▸ hy))) #align subgroup.mem_left_transversals_iff_exists_unique_inv_mul_mem Subgroup.mem_leftTransversals_iff_existsUnique_inv_mul_mem #align add_subgroup.mem_left_transversals_iff_exists_unique_neg_add_mem AddSubgroup.mem_leftTransversals_iff_existsUnique_neg_add_mem @[to_additive] theorem mem_rightTransversals_iff_existsUnique_mul_inv_mem : S ∈ rightTransversals T ↔ ∀ g : G, ∃! s : S, g * (s : G)⁻¹ ∈ T := by rw [rightTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique] refine ⟨fun h g => ?_, fun h g => ?_⟩ · obtain ⟨x, h1, h2⟩ := h g exact ⟨x.2, (congr_arg (· ∈ T) (eq_mul_inv_of_mul_eq h1)).mp x.1.2, fun y hy => (Prod.ext_iff.mp (h2 ⟨⟨g * (↑y)⁻¹, hy⟩, y⟩ (inv_mul_cancel_right g y))).2⟩ · obtain ⟨x, h1, h2⟩ := h g refine ⟨⟨⟨g * (↑x)⁻¹, h1⟩, x⟩, inv_mul_cancel_right g x, fun y hy => ?_⟩ have hf := h2 y.2 ((congr_arg (· ∈ T) (eq_mul_inv_of_mul_eq hy)).mp y.1.2) exact Prod.ext (Subtype.ext (eq_mul_inv_of_mul_eq (hf ▸ hy))) hf #align subgroup.mem_right_transversals_iff_exists_unique_mul_inv_mem Subgroup.mem_rightTransversals_iff_existsUnique_mul_inv_mem #align add_subgroup.mem_right_transversals_iff_exists_unique_add_neg_mem AddSubgroup.mem_rightTransversals_iff_existsUnique_add_neg_mem @[to_additive] theorem mem_leftTransversals_iff_existsUnique_quotient_mk''_eq : S ∈ leftTransversals (H : Set G) ↔ ∀ q : Quotient (QuotientGroup.leftRel H), ∃! s : S, Quotient.mk'' s.1 = q := by simp_rw [mem_leftTransversals_iff_existsUnique_inv_mul_mem, SetLike.mem_coe, ← QuotientGroup.eq'] exact ⟨fun h q => Quotient.inductionOn' q h, fun h g => h (Quotient.mk'' g)⟩ #align subgroup.mem_left_transversals_iff_exists_unique_quotient_mk'_eq Subgroup.mem_leftTransversals_iff_existsUnique_quotient_mk''_eq #align add_subgroup.mem_left_transversals_iff_exists_unique_quotient_mk'_eq AddSubgroup.mem_leftTransversals_iff_existsUnique_quotient_mk''_eq @[to_additive] theorem mem_rightTransversals_iff_existsUnique_quotient_mk''_eq : S ∈ rightTransversals (H : Set G) ↔ ∀ q : Quotient (QuotientGroup.rightRel H), ∃! s : S, Quotient.mk'' s.1 = q := by simp_rw [mem_rightTransversals_iff_existsUnique_mul_inv_mem, SetLike.mem_coe, ← QuotientGroup.rightRel_apply, ← Quotient.eq''] exact ⟨fun h q => Quotient.inductionOn' q h, fun h g => h (Quotient.mk'' g)⟩ #align subgroup.mem_right_transversals_iff_exists_unique_quotient_mk'_eq Subgroup.mem_rightTransversals_iff_existsUnique_quotient_mk''_eq #align add_subgroup.mem_right_transversals_iff_exists_unique_quotient_mk'_eq AddSubgroup.mem_rightTransversals_iff_existsUnique_quotient_mk''_eq @[to_additive] theorem mem_leftTransversals_iff_bijective : S ∈ leftTransversals (H : Set G) ↔ Function.Bijective (S.restrict (Quotient.mk'' : G → Quotient (QuotientGroup.leftRel H))) := mem_leftTransversals_iff_existsUnique_quotient_mk''_eq.trans (Function.bijective_iff_existsUnique (S.restrict Quotient.mk'')).symm #align subgroup.mem_left_transversals_iff_bijective Subgroup.mem_leftTransversals_iff_bijective #align add_subgroup.mem_left_transversals_iff_bijective AddSubgroup.mem_leftTransversals_iff_bijective @[to_additive] theorem mem_rightTransversals_iff_bijective : S ∈ rightTransversals (H : Set G) ↔ Function.Bijective (S.restrict (Quotient.mk'' : G → Quotient (QuotientGroup.rightRel H))) := mem_rightTransversals_iff_existsUnique_quotient_mk''_eq.trans (Function.bijective_iff_existsUnique (S.restrict Quotient.mk'')).symm #align subgroup.mem_right_transversals_iff_bijective Subgroup.mem_rightTransversals_iff_bijective #align add_subgroup.mem_right_transversals_iff_bijective AddSubgroup.mem_rightTransversals_iff_bijective @[to_additive] theorem card_left_transversal (h : S ∈ leftTransversals (H : Set G)) : Nat.card S = H.index := Nat.card_congr <\| Equiv.ofBijective _ <\| mem_leftTransversals_iff_bijective.mp h #align subgroup.card_left_transversal Subgroup.card_left_transversal #align add_subgroup.card_left_transversal AddSubgroup.card_left_transversal @[to_additive] theorem card_right_transversal (h : S ∈ rightTransversals (H : Set G)) : Nat.card S = H.index := Nat.card_congr <\| (Equiv.ofBijective _ <\| mem_rightTransversals_iff_bijective.mp h).trans <\| QuotientGroup.quotientRightRelEquivQuotientLeftRel H #align subgroup.card_right_transversal Subgroup.card_right_transversal #align add_subgroup.card_right_transversal AddSubgroup.card_right_transversal @[to_additive] theorem range_mem_leftTransversals {f : G ⧸ H → G} (hf : ∀ q, ↑(f q) = q) : Set.range f ∈ leftTransversals (H : Set G) := mem_leftTransversals_iff_bijective.mpr ⟨by rintro ⟨-, q₁, rfl⟩ ⟨-, q₂, rfl⟩ h exact Subtype.ext <\| congr_arg f <\| ((hf q₁).symm.trans h).trans (hf q₂), fun q => ⟨⟨f q, q, rfl⟩, hf q⟩⟩ #align subgroup.range_mem_left_transversals Subgroup.range_mem_leftTransversals #align add_subgroup.range_mem_left_transversals AddSubgroup.range_mem_leftTransversals @[to_additive] theorem range_mem_rightTransversals {f : Quotient (QuotientGroup.rightRel H) → G} (hf : ∀ q, Quotient.mk'' (f q) = q) : Set.range f ∈ rightTransversals (H : Set G) := mem_rightTransversals_iff_bijective.mpr ⟨by rintro ⟨-, q₁, rfl⟩ ⟨-, q₂, rfl⟩ h exact Subtype.ext <\| congr_arg f <\| ((hf q₁).symm.trans h).trans (hf q₂), fun q => ⟨⟨f q, q, rfl⟩, hf q⟩⟩ #align subgroup.range_mem_right_transversals Subgroup.range_mem_rightTransversals #align add_subgroup.range_mem_right_transversals AddSubgroup.range_mem_rightTransversals @[to_additive] lemma exists_left_transversal (H : Subgroup G) (g : G) : ∃ S ∈ leftTransversals (H : Set G), g ∈ S := by classical refine ⟨Set.range (Function.update Quotient.out' _ g), range_mem_leftTransversals fun q => ?_, Quotient.mk'' g, Function.update_same (Quotient.mk'' g) g Quotient.out'⟩ by_cases hq : q = Quotient.mk'' g · exact hq.symm ▸ congr_arg _ (Function.update_same (Quotient.mk'' g) g Quotient.out') · refine (Function.update_noteq ?_ g Quotient.out') ▸ q.out_eq' exact hq #align subgroup.exists_left_transversal Subgroup.exists_left_transversal #align add_subgroup.exists_left_transversal AddSubgroup.exists_left_transversal @[to_additive] lemma exists_right_transversal (H : Subgroup G) (g : G) : ∃ S ∈ rightTransversals (H : Set G), g ∈ S := by classical refine ⟨Set.range (Function.update Quotient.out' _ g), range_mem_rightTransversals fun q => ?_, Quotient.mk'' g, Function.update_same (Quotient.mk'' g) g Quotient.out'⟩ by_cases hq : q = Quotient.mk'' g · exact hq.symm ▸ congr_arg _ (Function.update_same (Quotient.mk'' g) g Quotient.out') · exact Eq.trans (congr_arg _ (Function.update_noteq hq g Quotient.out')) q.out_eq' #align subgroup.exists_right_transversal Subgroup.exists_right_transversal #align add_subgroup.exists_right_transversal AddSubgroup.exists_right_transversal /-- Given two subgroups `H' ⊆ H`, there exists a left transversal to `H'` inside `H`. -/ @[to_additive "Given two subgroups `H' ⊆ H`, there exists a transversal to `H'` inside `H`"] lemma exists_left_transversal_of_le {H' H : Subgroup G} (h : H' ≤ H) : ∃ S : Set G, S * H' = H ∧ Nat.card S * Nat.card H' = Nat.card H := by let H'' : Subgroup H := H'.comap H.subtype have : H' = H''.map H.subtype := by simp [H'', h] rw [this] obtain ⟨S, cmem, -⟩ := H''.exists_left_transversal 1 refine ⟨H.subtype '' S, ?_, ?_⟩ · have : H.subtype '' (S * H'') = H.subtype '' S * H''.map H.subtype := image_mul H.subtype rw [← this, cmem.mul_eq] simp [Set.ext_iff] · rw [← cmem.card_mul_card] refine congr_arg₂ (· * ·) ?_ ?_ <;> exact Nat.card_congr (Equiv.Set.image _ _ <\| subtype_injective H).symm /-- Given two subgroups `H' ⊆ H`, there exists a right transversal to `H'` inside `H`. -/ @[to_additive "Given two subgroups `H' ⊆ H`, there exists a transversal to `H'` inside `H`"] lemma exists_right_transversal_of_le {H' H : Subgroup G} (h : H' ≤ H) : ∃ S : Set G, H' * S = H ∧ Nat.card H' * Nat.card S = Nat.card H := by let H'' : Subgroup H := H'.comap H.subtype have : H' = H''.map H.subtype := by simp [H'', h] rw [this] obtain ⟨S, cmem, -⟩ := H''.exists_right_transversal 1 refine ⟨H.subtype '' S, ?_, ?_⟩ · have : H.subtype '' (H'' * S) = H''.map H.subtype * H.subtype '' S := image_mul H.subtype rw [← this, cmem.mul_eq] simp [Set.ext_iff] · have : Nat.card H'' * Nat.card S = Nat.card H := cmem.card_mul_card rw [← this] refine congr_arg₂ (· * ·) ?_ ?_ <;> exact Nat.card_congr (Equiv.Set.image _ _ <\| subtype_injective H).symm namespace IsComplement /-- The equivalence `G ≃ S × T`, such that the inverse is `() : S × T → G` -/ noncomputable def equiv {S T : Set G} (hST : IsComplement S T) : G ≃ S × T := (Equiv.ofBijective (fun x : S × T => x.1.1 x.2.1) hST).symm variable (hST : IsComplement S T) (hHT : IsComplement H T) (hSK : IsComplement S K) @[simp] theorem equiv_symm_apply (x : S × T) : (hST.equiv.symm x : G) = x.1.1 * x.2.1 := rfl @[simp] theorem equiv_fst_mul_equiv_snd (g : G) : ↑(hST.equiv g).fst * (hST.equiv g).snd = g := (Equiv.ofBijective (fun x : S × T => x.1.1 * x.2.1) hST).right_inv g theorem equiv_fst_eq_mul_inv (g : G) : ↑(hST.equiv g).fst = g * ((hST.equiv g).snd : G)⁻¹ := eq_mul_inv_of_mul_eq (hST.equiv_fst_mul_equiv_snd g) theorem equiv_snd_eq_inv_mul (g : G) : ↑(hST.equiv g).snd = ((hST.equiv g).fst : G)⁻¹ * g := eq_inv_mul_of_mul_eq (hST.equiv_fst_mul_equiv_snd g) theorem equiv_fst_eq_iff_leftCosetEquivalence {g₁ g₂ : G} : (hSK.equiv g₁).fst = (hSK.equiv g₂).fst ↔ LeftCosetEquivalence K g₁ g₂ := by rw [LeftCosetEquivalence, leftCoset_eq_iff] constructor · intro h rw [← hSK.equiv_fst_mul_equiv_snd g₂, ← hSK.equiv_fst_mul_equiv_snd g₁, ← h, mul_inv_rev, ← mul_assoc, inv_mul_cancel_right, ← coe_inv, ← coe_mul] exact Subtype.property _ · intro h apply (mem_leftTransversals_iff_existsUnique_inv_mul_mem.1 hSK g₁).unique · -- This used to be `simp [...]` before leanprover/lean4#2644 rw [equiv_fst_eq_mul_inv]; simp · rw [SetLike.mem_coe, ← mul_mem_cancel_right h] -- This used to be `simp [...]` before leanprover/lean4#2644 rw [equiv_fst_eq_mul_inv]; simp [equiv_fst_eq_mul_inv, ← mul_assoc] theorem equiv_snd_eq_iff_rightCosetEquivalence {g₁ g₂ : G} : (hHT.equiv g₁).snd = (hHT.equiv g₂).snd ↔ RightCosetEquivalence H g₁ g₂ := by rw [RightCosetEquivalence, rightCoset_eq_iff] constructor · intro h rw [← hHT.equiv_fst_mul_equiv_snd g₂, ← hHT.equiv_fst_mul_equiv_snd g₁, ← h, mul_inv_rev, mul_assoc, mul_inv_cancel_left, ← coe_inv, ← coe_mul] exact Subtype.property _ · intro h apply (mem_rightTransversals_iff_existsUnique_mul_inv_mem.1 hHT g₁).unique · -- This used to be `simp [...]` before leanprover/lean4#2644 rw [equiv_snd_eq_inv_mul]; simp · rw [SetLike.mem_coe, ← mul_mem_cancel_left h] -- This used to be `simp [...]` before leanprover/lean4#2644 rw [equiv_snd_eq_inv_mul, mul_assoc]; simp theorem leftCosetEquivalence_equiv_fst (g : G) : LeftCosetEquivalence K g ((hSK.equiv g).fst : G) := by -- This used to be `simp [...]` before leanprover/lean4#2644 rw [equiv_fst_eq_mul_inv]; simp [LeftCosetEquivalence, leftCoset_eq_iff] theorem rightCosetEquivalence_equiv_snd (g : G) : RightCosetEquivalence H g ((hHT.equiv g).snd : G) := by -- This used to be `simp [...]` before leanprover/lean4#2644 rw [RightCosetEquivalence, rightCoset_eq_iff, equiv_snd_eq_inv_mul]; simp theorem equiv_fst_eq_self_of_mem_of_one_mem {g : G} (h1 : 1 ∈ T) (hg : g ∈ S) : (hST.equiv g).fst = ⟨g, hg⟩ := by have : hST.equiv.symm (⟨g, hg⟩, ⟨1, h1⟩) = g := by rw [equiv, Equiv.ofBijective]; simp conv_lhs => rw [← this, Equiv.apply_symm_apply] theorem equiv_snd_eq_self_of_mem_of_one_mem {g : G} (h1 : 1 ∈ S) (hg : g ∈ T) : (hST.equiv g).snd = ⟨g, hg⟩ := by have : hST.equiv.symm (⟨1, h1⟩, ⟨g, hg⟩) = g := by rw [equiv, Equiv.ofBijective]; simp conv_lhs => rw [← this, Equiv.apply_symm_apply]	Mathlib/GroupTheory/Complement.lean	453	456	theorem equiv_snd_eq_one_of_mem_of_one_mem {g : G} (h1 : 1 ∈ T) (hg : g ∈ S) : (hST.equiv g).snd = ⟨1, h1⟩ := by	ext rw [equiv_snd_eq_inv_mul, equiv_fst_eq_self_of_mem_of_one_mem _ h1 hg, inv_mul_self]
/- Copyright (c) 2023 Peter Nelson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Peter Nelson -/ import Mathlib.SetTheory.Cardinal.Finite #align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04" /-! # Noncomputable Set Cardinality We define the cardinality of set `s` as a term `Set.encard s : ℕ∞` and a term `Set.ncard s : ℕ`. The latter takes the junk value of zero if `s` is infinite. Both functions are noncomputable, and are defined in terms of `PartENat.card` (which takes a type as its argument); this file can be seen as an API for the same function in the special case where the type is a coercion of a `Set`, allowing for smoother interactions with the `Set` API. `Set.encard` never takes junk values, so is more mathematically natural than `Set.ncard`, even though it takes values in a less convenient type. It is probably the right choice in settings where one is concerned with the cardinalities of sets that may or may not be infinite. `Set.ncard` has a nicer codomain, but when using it, `Set.Finite` hypotheses are normally needed to make sure its values are meaningful. More generally, `Set.ncard` is intended to be used over the obvious alternative `Finset.card` when finiteness is 'propositional' rather than 'structural'. When working with sets that are finite by virtue of their definition, then `Finset.card` probably makes more sense. One setting where `Set.ncard` works nicely is in a type `α` with `[Finite α]`, where every set is automatically finite. In this setting, we use default arguments and a simple tactic so that finiteness goals are discharged automatically in `Set.ncard` theorems. ## Main Definitions * `Set.encard s` is the cardinality of the set `s` as an extended natural number, with value `⊤` if `s` is infinite. * `Set.ncard s` is the cardinality of the set `s` as a natural number, provided `s` is Finite. If `s` is Infinite, then `Set.ncard s = 0`. * `toFinite_tac` is a tactic that tries to synthesize a `Set.Finite s` argument with `Set.toFinite`. This will work for `s : Set α` where there is a `Finite α` instance. ## Implementation Notes The theorems in this file are very similar to those in `Data.Finset.Card`, but with `Set` operations instead of `Finset`. We first prove all the theorems for `Set.encard`, and then derive most of the `Set.ncard` results as a consequence. Things are done this way to avoid reliance on the `Finset` API for theorems about infinite sets, and to allow for a refactor that removes or modifies `Set.ncard` in the future. Nearly all the theorems for `Set.ncard` require finiteness of one or more of their arguments. We provide this assumption with a default argument of the form `(hs : s.Finite := by toFinite_tac)`, where `toFinite_tac` will find an `s.Finite` term in the cases where `s` is a set in a `Finite` type. Often, where there are two set arguments `s` and `t`, the finiteness of one follows from the other in the context of the theorem, in which case we only include the ones that are needed, and derive the other inside the proof. A few of the theorems, such as `ncard_union_le` do not require finiteness arguments; they are true by coincidence due to junk values. -/ namespace Set variable {α β : Type} {s t : Set α} /-- The cardinality of a set as a term in `ℕ∞` -/ noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s) @[simp] theorem encard_univ_coe (s : Set α) : encard (univ : Set s) = encard s := by rw [encard, encard, PartENat.card_congr (Equiv.Set.univ ↑s)] theorem encard_univ (α : Type) : encard (univ : Set α) = PartENat.withTopEquiv (PartENat.card α) := by rw [encard, PartENat.card_congr (Equiv.Set.univ α)] theorem Finite.encard_eq_coe_toFinset_card (h : s.Finite) : s.encard = h.toFinset.card := by have := h.fintype rw [encard, PartENat.card_eq_coe_fintype_card, PartENat.withTopEquiv_natCast, toFinite_toFinset, toFinset_card] theorem encard_eq_coe_toFinset_card (s : Set α) [Fintype s] : encard s = s.toFinset.card := by have h := toFinite s rw [h.encard_eq_coe_toFinset_card, toFinite_toFinset] theorem encard_coe_eq_coe_finsetCard (s : Finset α) : encard (s : Set α) = s.card := by rw [Finite.encard_eq_coe_toFinset_card (Finset.finite_toSet s)]; simp theorem Infinite.encard_eq {s : Set α} (h : s.Infinite) : s.encard = ⊤ := by have := h.to_subtype rw [encard, ← PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply, PartENat.withTopEquiv_symm_top, PartENat.card_eq_top_of_infinite] @[simp] theorem encard_eq_zero : s.encard = 0 ↔ s = ∅ := by rw [encard, ← PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply, PartENat.withTopEquiv_symm_zero, PartENat.card_eq_zero_iff_empty, isEmpty_subtype, eq_empty_iff_forall_not_mem] @[simp] theorem encard_empty : (∅ : Set α).encard = 0 := by rw [encard_eq_zero] theorem nonempty_of_encard_ne_zero (h : s.encard ≠ 0) : s.Nonempty := by rwa [nonempty_iff_ne_empty, Ne, ← encard_eq_zero] theorem encard_ne_zero : s.encard ≠ 0 ↔ s.Nonempty := by rw [ne_eq, encard_eq_zero, nonempty_iff_ne_empty] @[simp] theorem encard_pos : 0 < s.encard ↔ s.Nonempty := by rw [pos_iff_ne_zero, encard_ne_zero] @[simp] theorem encard_singleton (e : α) : ({e} : Set α).encard = 1 := by rw [encard, ← PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply, PartENat.card_eq_coe_fintype_card, Fintype.card_ofSubsingleton, Nat.cast_one]; rfl theorem encard_union_eq (h : Disjoint s t) : (s ∪ t).encard = s.encard + t.encard := by classical have e := (Equiv.Set.union (by rwa [subset_empty_iff, ← disjoint_iff_inter_eq_empty])).symm simp [encard, ← PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv] theorem encard_insert_of_not_mem {a : α} (has : a ∉ s) : (insert a s).encard = s.encard + 1 := by rw [← union_singleton, encard_union_eq (by simpa), encard_singleton] theorem Finite.encard_lt_top (h : s.Finite) : s.encard < ⊤ := by refine h.induction_on (by simp) ?_ rintro a t hat _ ht' rw [encard_insert_of_not_mem hat] exact lt_tsub_iff_right.1 ht' theorem Finite.encard_eq_coe (h : s.Finite) : s.encard = ENat.toNat s.encard := (ENat.coe_toNat h.encard_lt_top.ne).symm theorem Finite.exists_encard_eq_coe (h : s.Finite) : ∃ (n : ℕ), s.encard = n := ⟨_, h.encard_eq_coe⟩ @[simp] theorem encard_lt_top_iff : s.encard < ⊤ ↔ s.Finite := ⟨fun h ↦ by_contra fun h' ↦ h.ne (Infinite.encard_eq h'), Finite.encard_lt_top⟩ @[simp] theorem encard_eq_top_iff : s.encard = ⊤ ↔ s.Infinite := by rw [← not_iff_not, ← Ne, ← lt_top_iff_ne_top, encard_lt_top_iff, not_infinite] theorem encard_ne_top_iff : s.encard ≠ ⊤ ↔ s.Finite := by simp theorem finite_of_encard_le_coe {k : ℕ} (h : s.encard ≤ k) : s.Finite := by rw [← encard_lt_top_iff]; exact h.trans_lt (WithTop.coe_lt_top _) theorem finite_of_encard_eq_coe {k : ℕ} (h : s.encard = k) : s.Finite := finite_of_encard_le_coe h.le theorem encard_le_coe_iff {k : ℕ} : s.encard ≤ k ↔ s.Finite ∧ ∃ (n₀ : ℕ), s.encard = n₀ ∧ n₀ ≤ k := ⟨fun h ↦ ⟨finite_of_encard_le_coe h, by rwa [ENat.le_coe_iff] at h⟩, fun ⟨_,⟨n₀,hs, hle⟩⟩ ↦ by rwa [hs, Nat.cast_le]⟩ section Lattice theorem encard_le_card (h : s ⊆ t) : s.encard ≤ t.encard := by rw [← union_diff_cancel h, encard_union_eq disjoint_sdiff_right]; exact le_self_add theorem encard_mono {α : Type*} : Monotone (encard : Set α → ℕ∞) := fun _ _ ↦ encard_le_card theorem encard_diff_add_encard_of_subset (h : s ⊆ t) : (t \ s).encard + s.encard = t.encard := by rw [← encard_union_eq disjoint_sdiff_left, diff_union_self, union_eq_self_of_subset_right h] @[simp] theorem one_le_encard_iff_nonempty : 1 ≤ s.encard ↔ s.Nonempty := by rw [nonempty_iff_ne_empty, Ne, ← encard_eq_zero, ENat.one_le_iff_ne_zero] theorem encard_diff_add_encard_inter (s t : Set α) : (s \ t).encard + (s ∩ t).encard = s.encard := by rw [← encard_union_eq (disjoint_of_subset_right inter_subset_right disjoint_sdiff_left), diff_union_inter] theorem encard_union_add_encard_inter (s t : Set α) : (s ∪ t).encard + (s ∩ t).encard = s.encard + t.encard := by rw [← diff_union_self, encard_union_eq disjoint_sdiff_left, add_right_comm, encard_diff_add_encard_inter] theorem encard_eq_encard_iff_encard_diff_eq_encard_diff (h : (s ∩ t).Finite) : s.encard = t.encard ↔ (s \ t).encard = (t \ s).encard := by rw [← encard_diff_add_encard_inter s t, ← encard_diff_add_encard_inter t s, inter_comm t s, WithTop.add_right_cancel_iff h.encard_lt_top.ne] theorem encard_le_encard_iff_encard_diff_le_encard_diff (h : (s ∩ t).Finite) : s.encard ≤ t.encard ↔ (s \ t).encard ≤ (t \ s).encard := by rw [← encard_diff_add_encard_inter s t, ← encard_diff_add_encard_inter t s, inter_comm t s, WithTop.add_le_add_iff_right h.encard_lt_top.ne] theorem encard_lt_encard_iff_encard_diff_lt_encard_diff (h : (s ∩ t).Finite) : s.encard < t.encard ↔ (s \ t).encard < (t \ s).encard := by rw [← encard_diff_add_encard_inter s t, ← encard_diff_add_encard_inter t s, inter_comm t s, WithTop.add_lt_add_iff_right h.encard_lt_top.ne] theorem encard_union_le (s t : Set α) : (s ∪ t).encard ≤ s.encard + t.encard := by rw [← encard_union_add_encard_inter]; exact le_self_add theorem finite_iff_finite_of_encard_eq_encard (h : s.encard = t.encard) : s.Finite ↔ t.Finite := by rw [← encard_lt_top_iff, ← encard_lt_top_iff, h] theorem infinite_iff_infinite_of_encard_eq_encard (h : s.encard = t.encard) : s.Infinite ↔ t.Infinite := by rw [← encard_eq_top_iff, h, encard_eq_top_iff] theorem Finite.finite_of_encard_le {s : Set α} {t : Set β} (hs : s.Finite) (h : t.encard ≤ s.encard) : t.Finite := encard_lt_top_iff.1 (h.trans_lt hs.encard_lt_top) theorem Finite.eq_of_subset_of_encard_le (ht : t.Finite) (hst : s ⊆ t) (hts : t.encard ≤ s.encard) : s = t := by rw [← zero_add (a := encard s), ← encard_diff_add_encard_of_subset hst] at hts have hdiff := WithTop.le_of_add_le_add_right (ht.subset hst).encard_lt_top.ne hts rw [nonpos_iff_eq_zero, encard_eq_zero, diff_eq_empty] at hdiff exact hst.antisymm hdiff theorem Finite.eq_of_subset_of_encard_le' (hs : s.Finite) (hst : s ⊆ t) (hts : t.encard ≤ s.encard) : s = t := (hs.finite_of_encard_le hts).eq_of_subset_of_encard_le hst hts theorem Finite.encard_lt_encard (ht : t.Finite) (h : s ⊂ t) : s.encard < t.encard := (encard_mono h.subset).lt_of_ne (fun he ↦ h.ne (ht.eq_of_subset_of_encard_le h.subset he.symm.le)) theorem encard_strictMono [Finite α] : StrictMono (encard : Set α → ℕ∞) := fun _ _ h ↦ (toFinite _).encard_lt_encard h theorem encard_diff_add_encard (s t : Set α) : (s \ t).encard + t.encard = (s ∪ t).encard := by rw [← encard_union_eq disjoint_sdiff_left, diff_union_self] theorem encard_le_encard_diff_add_encard (s t : Set α) : s.encard ≤ (s \ t).encard + t.encard := (encard_mono subset_union_left).trans_eq (encard_diff_add_encard _ _).symm theorem tsub_encard_le_encard_diff (s t : Set α) : s.encard - t.encard ≤ (s \ t).encard := by rw [tsub_le_iff_left, add_comm]; apply encard_le_encard_diff_add_encard theorem encard_add_encard_compl (s : Set α) : s.encard + sᶜ.encard = (univ : Set α).encard := by rw [← encard_union_eq disjoint_compl_right, union_compl_self] end Lattice section InsertErase variable {a b : α} theorem encard_insert_le (s : Set α) (x : α) : (insert x s).encard ≤ s.encard + 1 := by rw [← union_singleton, ← encard_singleton x]; apply encard_union_le theorem encard_singleton_inter (s : Set α) (x : α) : ({x} ∩ s).encard ≤ 1 := by rw [← encard_singleton x]; exact encard_le_card inter_subset_left theorem encard_diff_singleton_add_one (h : a ∈ s) : (s \ {a}).encard + 1 = s.encard := by rw [← encard_insert_of_not_mem (fun h ↦ h.2 rfl), insert_diff_singleton, insert_eq_of_mem h] theorem encard_diff_singleton_of_mem (h : a ∈ s) : (s \ {a}).encard = s.encard - 1 := by rw [← encard_diff_singleton_add_one h, ← WithTop.add_right_cancel_iff WithTop.one_ne_top, tsub_add_cancel_of_le (self_le_add_left _ _)] theorem encard_tsub_one_le_encard_diff_singleton (s : Set α) (x : α) : s.encard - 1 ≤ (s \ {x}).encard := by rw [← encard_singleton x]; apply tsub_encard_le_encard_diff	Mathlib/Data/Set/Card.lean	256	258	theorem encard_exchange (ha : a ∉ s) (hb : b ∈ s) : (insert a (s \ {b})).encard = s.encard := by	rw [encard_insert_of_not_mem, encard_diff_singleton_add_one hb] simp_all only [not_true, mem_diff, mem_singleton_iff, false_and, not_false_eq_true]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot -/ import Mathlib.Order.Filter.SmallSets import Mathlib.Tactic.Monotonicity import Mathlib.Topology.Compactness.Compact import Mathlib.Topology.NhdsSet import Mathlib.Algebra.Group.Defs #align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c" /-! # Uniform spaces Uniform spaces are a generalization of metric spaces and topological groups. Many concepts directly generalize to uniform spaces, e.g. * uniform continuity (in this file) * completeness (in `Cauchy.lean`) * extension of uniform continuous functions to complete spaces (in `UniformEmbedding.lean`) * totally bounded sets (in `Cauchy.lean`) * totally bounded complete sets are compact (in `Cauchy.lean`) A uniform structure on a type `X` is a filter `𝓤 X` on `X × X` satisfying some conditions which makes it reasonable to say that `∀ᶠ (p : X × X) in 𝓤 X, ...` means "for all p.1 and p.2 in X close enough, ...". Elements of this filter are called entourages of `X`. The two main examples are: * If `X` is a metric space, `V ∈ 𝓤 X ↔ ∃ ε > 0, { p \| dist p.1 p.2 < ε } ⊆ V` * If `G` is an additive topological group, `V ∈ 𝓤 G ↔ ∃ U ∈ 𝓝 (0 : G), {p \| p.2 - p.1 ∈ U} ⊆ V` Those examples are generalizations in two different directions of the elementary example where `X = ℝ` and `V ∈ 𝓤 ℝ ↔ ∃ ε > 0, { p \| \|p.2 - p.1\| < ε } ⊆ V` which features both the topological group structure on `ℝ` and its metric space structure. Each uniform structure on `X` induces a topology on `X` characterized by > `nhds_eq_comap_uniformity : ∀ {x : X}, 𝓝 x = comap (Prod.mk x) (𝓤 X)` where `Prod.mk x : X → X × X := (fun y ↦ (x, y))` is the partial evaluation of the product constructor. The dictionary with metric spaces includes: * an upper bound for `dist x y` translates into `(x, y) ∈ V` for some `V ∈ 𝓤 X` * a ball `ball x r` roughly corresponds to `UniformSpace.ball x V := {y \| (x, y) ∈ V}` for some `V ∈ 𝓤 X`, but the later is more general (it includes in particular both open and closed balls for suitable `V`). In particular we have: `isOpen_iff_ball_subset {s : Set X} : IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 X, ball x V ⊆ s` The triangle inequality is abstracted to a statement involving the composition of relations in `X`. First note that the triangle inequality in a metric space is equivalent to `∀ (x y z : X) (r r' : ℝ), dist x y ≤ r → dist y z ≤ r' → dist x z ≤ r + r'`. Then, for any `V` and `W` with type `Set (X × X)`, the composition `V ○ W : Set (X × X)` is defined as `{ p : X × X \| ∃ z, (p.1, z) ∈ V ∧ (z, p.2) ∈ W }`. In the metric space case, if `V = { p \| dist p.1 p.2 ≤ r }` and `W = { p \| dist p.1 p.2 ≤ r' }` then the triangle inequality, as reformulated above, says `V ○ W` is contained in `{p \| dist p.1 p.2 ≤ r + r'}` which is the entourage associated to the radius `r + r'`. In general we have `mem_ball_comp (h : y ∈ ball x V) (h' : z ∈ ball y W) : z ∈ ball x (V ○ W)`. Note that this discussion does not depend on any axiom imposed on the uniformity filter, it is simply captured by the definition of composition. The uniform space axioms ask the filter `𝓤 X` to satisfy the following: * every `V ∈ 𝓤 X` contains the diagonal `idRel = { p \| p.1 = p.2 }`. This abstracts the fact that `dist x x ≤ r` for every non-negative radius `r` in the metric space case and also that `x - x` belongs to every neighborhood of zero in the topological group case. * `V ∈ 𝓤 X → Prod.swap '' V ∈ 𝓤 X`. This is tightly related the fact that `dist x y = dist y x` in a metric space, and to continuity of negation in the topological group case. * `∀ V ∈ 𝓤 X, ∃ W ∈ 𝓤 X, W ○ W ⊆ V`. In the metric space case, it corresponds to cutting the radius of a ball in half and applying the triangle inequality. In the topological group case, it comes from continuity of addition at `(0, 0)`. These three axioms are stated more abstractly in the definition below, in terms of operations on filters, without directly manipulating entourages. ## Main definitions * `UniformSpace X` is a uniform space structure on a type `X` * `UniformContinuous f` is a predicate saying a function `f : α → β` between uniform spaces is uniformly continuous : `∀ r ∈ 𝓤 β, ∀ᶠ (x : α × α) in 𝓤 α, (f x.1, f x.2) ∈ r` In this file we also define a complete lattice structure on the type `UniformSpace X` of uniform structures on `X`, as well as the pullback (`UniformSpace.comap`) of uniform structures coming from the pullback of filters. Like distance functions, uniform structures cannot be pushed forward in general. ## Notations Localized in `Uniformity`, we have the notation `𝓤 X` for the uniformity on a uniform space `X`, and `○` for composition of relations, seen as terms with type `Set (X × X)`. ## Implementation notes There is already a theory of relations in `Data/Rel.lean` where the main definition is `def Rel (α β : Type) := α → β → Prop`. The relations used in the current file involve only one type, but this is not the reason why we don't reuse `Data/Rel.lean`. We use `Set (α × α)` instead of `Rel α α` because we really need sets to use the filter library, and elements of filters on `α × α` have type `Set (α × α)`. The structure `UniformSpace X` bundles a uniform structure on `X`, a topology on `X` and an assumption saying those are compatible. This may not seem mathematically reasonable at first, but is in fact an instance of the forgetful inheritance pattern. See Note [forgetful inheritance] below. ## References The formalization uses the books: [N. Bourbaki, General Topology][bourbaki1966] * [I. M. James, Topologies and Uniformities][james1999] But it makes a more systematic use of the filter library. -/ open Set Filter Topology universe u v ua ub uc ud /-! ### Relations, seen as `Set (α × α)` -/ variable {α : Type ua} {β : Type ub} {γ : Type uc} {δ : Type ud} {ι : Sort} /-- The identity relation, or the graph of the identity function -/ def idRel {α : Type} := { p : α × α \| p.1 = p.2 } #align id_rel idRel @[simp] theorem mem_idRel {a b : α} : (a, b) ∈ @idRel α ↔ a = b := Iff.rfl #align mem_id_rel mem_idRel @[simp] theorem idRel_subset {s : Set (α × α)} : idRel ⊆ s ↔ ∀ a, (a, a) ∈ s := by simp [subset_def] #align id_rel_subset idRel_subset /-- The composition of relations -/ def compRel (r₁ r₂ : Set (α × α)) := { p : α × α \| ∃ z : α, (p.1, z) ∈ r₁ ∧ (z, p.2) ∈ r₂ } #align comp_rel compRel @[inherit_doc] scoped[Uniformity] infixl:62 " ○ " => compRel open Uniformity @[simp] theorem mem_compRel {α : Type u} {r₁ r₂ : Set (α × α)} {x y : α} : (x, y) ∈ r₁ ○ r₂ ↔ ∃ z, (x, z) ∈ r₁ ∧ (z, y) ∈ r₂ := Iff.rfl #align mem_comp_rel mem_compRel @[simp] theorem swap_idRel : Prod.swap '' idRel = @idRel α := Set.ext fun ⟨a, b⟩ => by simpa [image_swap_eq_preimage_swap] using eq_comm #align swap_id_rel swap_idRel theorem Monotone.compRel [Preorder β] {f g : β → Set (α × α)} (hf : Monotone f) (hg : Monotone g) : Monotone fun x => f x ○ g x := fun _ _ h _ ⟨z, h₁, h₂⟩ => ⟨z, hf h h₁, hg h h₂⟩ #align monotone.comp_rel Monotone.compRel @[mono] theorem compRel_mono {f g h k : Set (α × α)} (h₁ : f ⊆ h) (h₂ : g ⊆ k) : f ○ g ⊆ h ○ k := fun _ ⟨z, h, h'⟩ => ⟨z, h₁ h, h₂ h'⟩ #align comp_rel_mono compRel_mono theorem prod_mk_mem_compRel {a b c : α} {s t : Set (α × α)} (h₁ : (a, c) ∈ s) (h₂ : (c, b) ∈ t) : (a, b) ∈ s ○ t := ⟨c, h₁, h₂⟩ #align prod_mk_mem_comp_rel prod_mk_mem_compRel @[simp] theorem id_compRel {r : Set (α × α)} : idRel ○ r = r := Set.ext fun ⟨a, b⟩ => by simp #align id_comp_rel id_compRel theorem compRel_assoc {r s t : Set (α × α)} : r ○ s ○ t = r ○ (s ○ t) := by ext ⟨a, b⟩; simp only [mem_compRel]; tauto #align comp_rel_assoc compRel_assoc theorem left_subset_compRel {s t : Set (α × α)} (h : idRel ⊆ t) : s ⊆ s ○ t := fun ⟨_x, y⟩ xy_in => ⟨y, xy_in, h <\| rfl⟩ #align left_subset_comp_rel left_subset_compRel theorem right_subset_compRel {s t : Set (α × α)} (h : idRel ⊆ s) : t ⊆ s ○ t := fun ⟨x, _y⟩ xy_in => ⟨x, h <\| rfl, xy_in⟩ #align right_subset_comp_rel right_subset_compRel theorem subset_comp_self {s : Set (α × α)} (h : idRel ⊆ s) : s ⊆ s ○ s := left_subset_compRel h #align subset_comp_self subset_comp_self theorem subset_iterate_compRel {s t : Set (α × α)} (h : idRel ⊆ s) (n : ℕ) : t ⊆ (s ○ ·)^[n] t := by induction' n with n ihn generalizing t exacts [Subset.rfl, (right_subset_compRel h).trans ihn] #align subset_iterate_comp_rel subset_iterate_compRel /-- The relation is invariant under swapping factors. -/ def SymmetricRel (V : Set (α × α)) : Prop := Prod.swap ⁻¹' V = V #align symmetric_rel SymmetricRel /-- The maximal symmetric relation contained in a given relation. -/ def symmetrizeRel (V : Set (α × α)) : Set (α × α) := V ∩ Prod.swap ⁻¹' V #align symmetrize_rel symmetrizeRel theorem symmetric_symmetrizeRel (V : Set (α × α)) : SymmetricRel (symmetrizeRel V) := by simp [SymmetricRel, symmetrizeRel, preimage_inter, inter_comm, ← preimage_comp] #align symmetric_symmetrize_rel symmetric_symmetrizeRel theorem symmetrizeRel_subset_self (V : Set (α × α)) : symmetrizeRel V ⊆ V := sep_subset _ _ #align symmetrize_rel_subset_self symmetrizeRel_subset_self @[mono] theorem symmetrize_mono {V W : Set (α × α)} (h : V ⊆ W) : symmetrizeRel V ⊆ symmetrizeRel W := inter_subset_inter h <\| preimage_mono h #align symmetrize_mono symmetrize_mono theorem SymmetricRel.mk_mem_comm {V : Set (α × α)} (hV : SymmetricRel V) {x y : α} : (x, y) ∈ V ↔ (y, x) ∈ V := Set.ext_iff.1 hV (y, x) #align symmetric_rel.mk_mem_comm SymmetricRel.mk_mem_comm theorem SymmetricRel.eq {U : Set (α × α)} (hU : SymmetricRel U) : Prod.swap ⁻¹' U = U := hU #align symmetric_rel.eq SymmetricRel.eq theorem SymmetricRel.inter {U V : Set (α × α)} (hU : SymmetricRel U) (hV : SymmetricRel V) : SymmetricRel (U ∩ V) := by rw [SymmetricRel, preimage_inter, hU.eq, hV.eq] #align symmetric_rel.inter SymmetricRel.inter /-- This core description of a uniform space is outside of the type class hierarchy. It is useful for constructions of uniform spaces, when the topology is derived from the uniform space. -/ structure UniformSpace.Core (α : Type u) where /-- The uniformity filter. Once `UniformSpace` is defined, `𝓤 α` (`_root_.uniformity`) becomes the normal form. -/ uniformity : Filter (α × α) /-- Every set in the uniformity filter includes the diagonal. -/ refl : 𝓟 idRel ≤ uniformity /-- If `s ∈ uniformity`, then `Prod.swap ⁻¹' s ∈ uniformity`. -/ symm : Tendsto Prod.swap uniformity uniformity /-- For every set `u ∈ uniformity`, there exists `v ∈ uniformity` such that `v ○ v ⊆ u`. -/ comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity #align uniform_space.core UniformSpace.Core protected theorem UniformSpace.Core.comp_mem_uniformity_sets {c : Core α} {s : Set (α × α)} (hs : s ∈ c.uniformity) : ∃ t ∈ c.uniformity, t ○ t ⊆ s := (mem_lift'_sets <\| monotone_id.compRel monotone_id).mp <\| c.comp hs /-- An alternative constructor for `UniformSpace.Core`. This version unfolds various `Filter`-related definitions. -/ def UniformSpace.Core.mk' {α : Type u} (U : Filter (α × α)) (refl : ∀ r ∈ U, ∀ (x), (x, x) ∈ r) (symm : ∀ r ∈ U, Prod.swap ⁻¹' r ∈ U) (comp : ∀ r ∈ U, ∃ t ∈ U, t ○ t ⊆ r) : UniformSpace.Core α := ⟨U, fun _r ru => idRel_subset.2 (refl _ ru), symm, fun _r ru => let ⟨_s, hs, hsr⟩ := comp _ ru mem_of_superset (mem_lift' hs) hsr⟩ #align uniform_space.core.mk' UniformSpace.Core.mk' /-- Defining a `UniformSpace.Core` from a filter basis satisfying some uniformity-like axioms. -/ def UniformSpace.Core.mkOfBasis {α : Type u} (B : FilterBasis (α × α)) (refl : ∀ r ∈ B, ∀ (x), (x, x) ∈ r) (symm : ∀ r ∈ B, ∃ t ∈ B, t ⊆ Prod.swap ⁻¹' r) (comp : ∀ r ∈ B, ∃ t ∈ B, t ○ t ⊆ r) : UniformSpace.Core α where uniformity := B.filter refl := B.hasBasis.ge_iff.mpr fun _r ru => idRel_subset.2 <\| refl _ ru symm := (B.hasBasis.tendsto_iff B.hasBasis).mpr symm comp := (HasBasis.le_basis_iff (B.hasBasis.lift' (monotone_id.compRel monotone_id)) B.hasBasis).2 comp #align uniform_space.core.mk_of_basis UniformSpace.Core.mkOfBasis /-- A uniform space generates a topological space -/ def UniformSpace.Core.toTopologicalSpace {α : Type u} (u : UniformSpace.Core α) : TopologicalSpace α := .mkOfNhds fun x ↦ .comap (Prod.mk x) u.uniformity #align uniform_space.core.to_topological_space UniformSpace.Core.toTopologicalSpace theorem UniformSpace.Core.ext : ∀ {u₁ u₂ : UniformSpace.Core α}, u₁.uniformity = u₂.uniformity → u₁ = u₂ \| ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl #align uniform_space.core_eq UniformSpace.Core.ext theorem UniformSpace.Core.nhds_toTopologicalSpace {α : Type u} (u : Core α) (x : α) : @nhds α u.toTopologicalSpace x = comap (Prod.mk x) u.uniformity := by apply TopologicalSpace.nhds_mkOfNhds_of_hasBasis (fun _ ↦ (basis_sets _).comap _) · exact fun a U hU ↦ u.refl hU rfl · intro a U hU rcases u.comp_mem_uniformity_sets hU with ⟨V, hV, hVU⟩ filter_upwards [preimage_mem_comap hV] with b hb filter_upwards [preimage_mem_comap hV] with c hc exact hVU ⟨b, hb, hc⟩ -- the topological structure is embedded in the uniform structure -- to avoid instance diamond issues. See Note [forgetful inheritance]. /-- A uniform space is a generalization of the "uniform" topological aspects of a metric space. It consists of a filter on `α × α` called the "uniformity", which satisfies properties analogous to the reflexivity, symmetry, and triangle properties of a metric. A metric space has a natural uniformity, and a uniform space has a natural topology. A topological group also has a natural uniformity, even when it is not metrizable. -/ class UniformSpace (α : Type u) extends TopologicalSpace α where /-- The uniformity filter. -/ protected uniformity : Filter (α × α) /-- If `s ∈ uniformity`, then `Prod.swap ⁻¹' s ∈ uniformity`. -/ protected symm : Tendsto Prod.swap uniformity uniformity /-- For every set `u ∈ uniformity`, there exists `v ∈ uniformity` such that `v ○ v ⊆ u`. -/ protected comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity /-- The uniformity agrees with the topology: the neighborhoods filter of each point `x` is equal to `Filter.comap (Prod.mk x) (𝓤 α)`. -/ protected nhds_eq_comap_uniformity (x : α) : 𝓝 x = comap (Prod.mk x) uniformity #align uniform_space UniformSpace #noalign uniform_space.mk' -- Can't be a `match_pattern`, so not useful anymore /-- The uniformity is a filter on α × α (inferred from an ambient uniform space structure on α). -/ def uniformity (α : Type u) [UniformSpace α] : Filter (α × α) := @UniformSpace.uniformity α _ #align uniformity uniformity /-- Notation for the uniformity filter with respect to a non-standard `UniformSpace` instance. -/ scoped[Uniformity] notation "𝓤[" u "]" => @uniformity _ u @[inherit_doc] -- Porting note (#11215): TODO: should we drop the `uniformity` def? scoped[Uniformity] notation "𝓤" => uniformity /-- Construct a `UniformSpace` from a `u : UniformSpace.Core` and a `TopologicalSpace` structure that is equal to `u.toTopologicalSpace`. -/ abbrev UniformSpace.ofCoreEq {α : Type u} (u : UniformSpace.Core α) (t : TopologicalSpace α) (h : t = u.toTopologicalSpace) : UniformSpace α where __ := u toTopologicalSpace := t nhds_eq_comap_uniformity x := by rw [h, u.nhds_toTopologicalSpace] #align uniform_space.of_core_eq UniformSpace.ofCoreEq /-- Construct a `UniformSpace` from a `UniformSpace.Core`. -/ abbrev UniformSpace.ofCore {α : Type u} (u : UniformSpace.Core α) : UniformSpace α := .ofCoreEq u _ rfl #align uniform_space.of_core UniformSpace.ofCore /-- Construct a `UniformSpace.Core` from a `UniformSpace`. -/ abbrev UniformSpace.toCore (u : UniformSpace α) : UniformSpace.Core α where __ := u refl := by rintro U hU ⟨x, y⟩ (rfl : x = y) have : Prod.mk x ⁻¹' U ∈ 𝓝 x := by rw [UniformSpace.nhds_eq_comap_uniformity] exact preimage_mem_comap hU convert mem_of_mem_nhds this theorem UniformSpace.toCore_toTopologicalSpace (u : UniformSpace α) : u.toCore.toTopologicalSpace = u.toTopologicalSpace := TopologicalSpace.ext_nhds fun a ↦ by rw [u.nhds_eq_comap_uniformity, u.toCore.nhds_toTopologicalSpace] #align uniform_space.to_core_to_topological_space UniformSpace.toCore_toTopologicalSpace /-- Build a `UniformSpace` from a `UniformSpace.Core` and a compatible topology. Use `UniformSpace.mk` instead to avoid proving the unnecessary assumption `UniformSpace.Core.refl`. The main constructor used to use a different compatibility assumption. This definition was created as a step towards porting to a new definition. Now the main definition is ported, so this constructor will be removed in a few months. -/ @[deprecated UniformSpace.mk (since := "2024-03-20")] def UniformSpace.ofNhdsEqComap (u : UniformSpace.Core α) (_t : TopologicalSpace α) (h : ∀ x, 𝓝 x = u.uniformity.comap (Prod.mk x)) : UniformSpace α where __ := u nhds_eq_comap_uniformity := h @[ext] protected theorem UniformSpace.ext {u₁ u₂ : UniformSpace α} (h : 𝓤[u₁] = 𝓤[u₂]) : u₁ = u₂ := by have : u₁.toTopologicalSpace = u₂.toTopologicalSpace := TopologicalSpace.ext_nhds fun x ↦ by rw [u₁.nhds_eq_comap_uniformity, u₂.nhds_eq_comap_uniformity] exact congr_arg (comap _) h cases u₁; cases u₂; congr #align uniform_space_eq UniformSpace.ext protected theorem UniformSpace.ext_iff {u₁ u₂ : UniformSpace α} : u₁ = u₂ ↔ ∀ s, s ∈ 𝓤[u₁] ↔ s ∈ 𝓤[u₂] := ⟨fun h _ => h ▸ Iff.rfl, fun h => by ext; exact h _⟩ theorem UniformSpace.ofCoreEq_toCore (u : UniformSpace α) (t : TopologicalSpace α) (h : t = u.toCore.toTopologicalSpace) : .ofCoreEq u.toCore t h = u := UniformSpace.ext rfl #align uniform_space.of_core_eq_to_core UniformSpace.ofCoreEq_toCore /-- Replace topology in a `UniformSpace` instance with a propositionally (but possibly not definitionally) equal one. -/ abbrev UniformSpace.replaceTopology {α : Type} [i : TopologicalSpace α] (u : UniformSpace α) (h : i = u.toTopologicalSpace) : UniformSpace α where __ := u toTopologicalSpace := i nhds_eq_comap_uniformity x := by rw [h, u.nhds_eq_comap_uniformity] #align uniform_space.replace_topology UniformSpace.replaceTopology theorem UniformSpace.replaceTopology_eq {α : Type} [i : TopologicalSpace α] (u : UniformSpace α) (h : i = u.toTopologicalSpace) : u.replaceTopology h = u := UniformSpace.ext rfl #align uniform_space.replace_topology_eq UniformSpace.replaceTopology_eq -- Porting note: rfc: use `UniformSpace.Core.mkOfBasis`? This will change defeq here and there /-- Define a `UniformSpace` using a "distance" function. The function can be, e.g., the distance in a (usual or extended) metric space or an absolute value on a ring. -/ def UniformSpace.ofFun {α : Type u} {β : Type v} [OrderedAddCommMonoid β] (d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x) (triangle : ∀ x y z, d x z ≤ d x y + d y z) (half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) : UniformSpace α := .ofCore { uniformity := ⨅ r > 0, 𝓟 { x \| d x.1 x.2 < r } refl := le_iInf₂ fun r hr => principal_mono.2 <\| idRel_subset.2 fun x => by simpa [refl] symm := tendsto_iInf_iInf fun r => tendsto_iInf_iInf fun _ => tendsto_principal_principal.2 fun x hx => by rwa [mem_setOf, symm] comp := le_iInf₂ fun r hr => let ⟨δ, h0, hδr⟩ := half r hr; le_principal_iff.2 <\| mem_of_superset (mem_lift' <\| mem_iInf_of_mem δ <\| mem_iInf_of_mem h0 <\| mem_principal_self _) fun (x, z) ⟨y, h₁, h₂⟩ => (triangle _ _ _).trans_lt (hδr _ h₁ _ h₂) } #align uniform_space.of_fun UniformSpace.ofFun theorem UniformSpace.hasBasis_ofFun {α : Type u} {β : Type v} [LinearOrderedAddCommMonoid β] (h₀ : ∃ x : β, 0 < x) (d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x) (triangle : ∀ x y z, d x z ≤ d x y + d y z) (half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) : 𝓤[.ofFun d refl symm triangle half].HasBasis ((0 : β) < ·) (fun ε => { x \| d x.1 x.2 < ε }) := hasBasis_biInf_principal' (fun ε₁ h₁ ε₂ h₂ => ⟨min ε₁ ε₂, lt_min h₁ h₂, fun _x hx => lt_of_lt_of_le hx (min_le_left _ _), fun _x hx => lt_of_lt_of_le hx (min_le_right _ _)⟩) h₀ #align uniform_space.has_basis_of_fun UniformSpace.hasBasis_ofFun section UniformSpace variable [UniformSpace α] theorem nhds_eq_comap_uniformity {x : α} : 𝓝 x = (𝓤 α).comap (Prod.mk x) := UniformSpace.nhds_eq_comap_uniformity x #align nhds_eq_comap_uniformity nhds_eq_comap_uniformity theorem isOpen_uniformity {s : Set α} : IsOpen s ↔ ∀ x ∈ s, { p : α × α \| p.1 = x → p.2 ∈ s } ∈ 𝓤 α := by simp only [isOpen_iff_mem_nhds, nhds_eq_comap_uniformity, mem_comap_prod_mk] #align is_open_uniformity isOpen_uniformity theorem refl_le_uniformity : 𝓟 idRel ≤ 𝓤 α := (@UniformSpace.toCore α _).refl #align refl_le_uniformity refl_le_uniformity instance uniformity.neBot [Nonempty α] : NeBot (𝓤 α) := diagonal_nonempty.principal_neBot.mono refl_le_uniformity #align uniformity.ne_bot uniformity.neBot theorem refl_mem_uniformity {x : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) : (x, x) ∈ s := refl_le_uniformity h rfl #align refl_mem_uniformity refl_mem_uniformity theorem mem_uniformity_of_eq {x y : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) (hx : x = y) : (x, y) ∈ s := refl_le_uniformity h hx #align mem_uniformity_of_eq mem_uniformity_of_eq theorem symm_le_uniformity : map (@Prod.swap α α) (𝓤 _) ≤ 𝓤 _ := UniformSpace.symm #align symm_le_uniformity symm_le_uniformity theorem comp_le_uniformity : ((𝓤 α).lift' fun s : Set (α × α) => s ○ s) ≤ 𝓤 α := UniformSpace.comp #align comp_le_uniformity comp_le_uniformity theorem lift'_comp_uniformity : ((𝓤 α).lift' fun s : Set (α × α) => s ○ s) = 𝓤 α := comp_le_uniformity.antisymm <\| le_lift'.2 fun _s hs ↦ mem_of_superset hs <\| subset_comp_self <\| idRel_subset.2 fun _ ↦ refl_mem_uniformity hs theorem tendsto_swap_uniformity : Tendsto (@Prod.swap α α) (𝓤 α) (𝓤 α) := symm_le_uniformity #align tendsto_swap_uniformity tendsto_swap_uniformity theorem comp_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, t ○ t ⊆ s := (mem_lift'_sets <\| monotone_id.compRel monotone_id).mp <\| comp_le_uniformity hs #align comp_mem_uniformity_sets comp_mem_uniformity_sets /-- If `s ∈ 𝓤 α`, then for any natural `n`, for a subset `t` of a sufficiently small set in `𝓤 α`, we have `t ○ t ○ ... ○ t ⊆ s` (`n` compositions). -/ theorem eventually_uniformity_iterate_comp_subset {s : Set (α × α)} (hs : s ∈ 𝓤 α) (n : ℕ) : ∀ᶠ t in (𝓤 α).smallSets, (t ○ ·)^[n] t ⊆ s := by suffices ∀ᶠ t in (𝓤 α).smallSets, t ⊆ s ∧ (t ○ ·)^[n] t ⊆ s from (eventually_and.1 this).2 induction' n with n ihn generalizing s · simpa rcases comp_mem_uniformity_sets hs with ⟨t, htU, hts⟩ refine (ihn htU).mono fun U hU => ?_ rw [Function.iterate_succ_apply'] exact ⟨hU.1.trans <\| (subset_comp_self <\| refl_le_uniformity htU).trans hts, (compRel_mono hU.1 hU.2).trans hts⟩ #align eventually_uniformity_iterate_comp_subset eventually_uniformity_iterate_comp_subset /-- If `s ∈ 𝓤 α`, then for a subset `t` of a sufficiently small set in `𝓤 α`, we have `t ○ t ⊆ s`. -/ theorem eventually_uniformity_comp_subset {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∀ᶠ t in (𝓤 α).smallSets, t ○ t ⊆ s := eventually_uniformity_iterate_comp_subset hs 1 #align eventually_uniformity_comp_subset eventually_uniformity_comp_subset /-- Relation `fun f g ↦ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α)` is transitive. -/ theorem Filter.Tendsto.uniformity_trans {l : Filter β} {f₁ f₂ f₃ : β → α} (h₁₂ : Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α)) (h₂₃ : Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α)) : Tendsto (fun x => (f₁ x, f₃ x)) l (𝓤 α) := by refine le_trans (le_lift'.2 fun s hs => mem_map.2 ?_) comp_le_uniformity filter_upwards [mem_map.1 (h₁₂ hs), mem_map.1 (h₂₃ hs)] with x hx₁₂ hx₂₃ using ⟨_, hx₁₂, hx₂₃⟩ #align filter.tendsto.uniformity_trans Filter.Tendsto.uniformity_trans /-- Relation `fun f g ↦ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α)` is symmetric. -/ theorem Filter.Tendsto.uniformity_symm {l : Filter β} {f : β → α × α} (h : Tendsto f l (𝓤 α)) : Tendsto (fun x => ((f x).2, (f x).1)) l (𝓤 α) := tendsto_swap_uniformity.comp h #align filter.tendsto.uniformity_symm Filter.Tendsto.uniformity_symm /-- Relation `fun f g ↦ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α)` is reflexive. -/ theorem tendsto_diag_uniformity (f : β → α) (l : Filter β) : Tendsto (fun x => (f x, f x)) l (𝓤 α) := fun _s hs => mem_map.2 <\| univ_mem' fun _ => refl_mem_uniformity hs #align tendsto_diag_uniformity tendsto_diag_uniformity theorem tendsto_const_uniformity {a : α} {f : Filter β} : Tendsto (fun _ => (a, a)) f (𝓤 α) := tendsto_diag_uniformity (fun _ => a) f #align tendsto_const_uniformity tendsto_const_uniformity theorem symm_of_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, (∀ a b, (a, b) ∈ t → (b, a) ∈ t) ∧ t ⊆ s := have : preimage Prod.swap s ∈ 𝓤 α := symm_le_uniformity hs ⟨s ∩ preimage Prod.swap s, inter_mem hs this, fun _ _ ⟨h₁, h₂⟩ => ⟨h₂, h₁⟩, inter_subset_left⟩ #align symm_of_uniformity symm_of_uniformity theorem comp_symm_of_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, (∀ {a b}, (a, b) ∈ t → (b, a) ∈ t) ∧ t ○ t ⊆ s := let ⟨_t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs let ⟨t', ht', ht'₁, ht'₂⟩ := symm_of_uniformity ht₁ ⟨t', ht', ht'₁ _ _, Subset.trans (monotone_id.compRel monotone_id ht'₂) ht₂⟩ #align comp_symm_of_uniformity comp_symm_of_uniformity theorem uniformity_le_symm : 𝓤 α ≤ @Prod.swap α α <$> 𝓤 α := by rw [map_swap_eq_comap_swap]; exact tendsto_swap_uniformity.le_comap #align uniformity_le_symm uniformity_le_symm theorem uniformity_eq_symm : 𝓤 α = @Prod.swap α α <$> 𝓤 α := le_antisymm uniformity_le_symm symm_le_uniformity #align uniformity_eq_symm uniformity_eq_symm @[simp] theorem comap_swap_uniformity : comap (@Prod.swap α α) (𝓤 α) = 𝓤 α := (congr_arg _ uniformity_eq_symm).trans <\| comap_map Prod.swap_injective #align comap_swap_uniformity comap_swap_uniformity theorem symmetrize_mem_uniformity {V : Set (α × α)} (h : V ∈ 𝓤 α) : symmetrizeRel V ∈ 𝓤 α := by apply (𝓤 α).inter_sets h rw [← image_swap_eq_preimage_swap, uniformity_eq_symm] exact image_mem_map h #align symmetrize_mem_uniformity symmetrize_mem_uniformity /-- Symmetric entourages form a basis of `𝓤 α` -/ theorem UniformSpace.hasBasis_symmetric : (𝓤 α).HasBasis (fun s : Set (α × α) => s ∈ 𝓤 α ∧ SymmetricRel s) id := hasBasis_self.2 fun t t_in => ⟨symmetrizeRel t, symmetrize_mem_uniformity t_in, symmetric_symmetrizeRel t, symmetrizeRel_subset_self t⟩ #align uniform_space.has_basis_symmetric UniformSpace.hasBasis_symmetric theorem uniformity_lift_le_swap {g : Set (α × α) → Filter β} {f : Filter β} (hg : Monotone g) (h : ((𝓤 α).lift fun s => g (preimage Prod.swap s)) ≤ f) : (𝓤 α).lift g ≤ f := calc (𝓤 α).lift g ≤ (Filter.map (@Prod.swap α α) <\| 𝓤 α).lift g := lift_mono uniformity_le_symm le_rfl _ ≤ _ := by rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]; exact h #align uniformity_lift_le_swap uniformity_lift_le_swap theorem uniformity_lift_le_comp {f : Set (α × α) → Filter β} (h : Monotone f) : ((𝓤 α).lift fun s => f (s ○ s)) ≤ (𝓤 α).lift f := calc ((𝓤 α).lift fun s => f (s ○ s)) = ((𝓤 α).lift' fun s : Set (α × α) => s ○ s).lift f := by rw [lift_lift'_assoc] · exact monotone_id.compRel monotone_id · exact h _ ≤ (𝓤 α).lift f := lift_mono comp_le_uniformity le_rfl #align uniformity_lift_le_comp uniformity_lift_le_comp -- Porting note (#10756): new lemma theorem comp3_mem_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, t ○ (t ○ t) ⊆ s := let ⟨_t', ht', ht's⟩ := comp_mem_uniformity_sets hs let ⟨t, ht, htt'⟩ := comp_mem_uniformity_sets ht' ⟨t, ht, (compRel_mono ((subset_comp_self (refl_le_uniformity ht)).trans htt') htt').trans ht's⟩ /-- See also `comp3_mem_uniformity`. -/ theorem comp_le_uniformity3 : ((𝓤 α).lift' fun s : Set (α × α) => s ○ (s ○ s)) ≤ 𝓤 α := fun _ h => let ⟨_t, htU, ht⟩ := comp3_mem_uniformity h mem_of_superset (mem_lift' htU) ht #align comp_le_uniformity3 comp_le_uniformity3 /-- See also `comp_open_symm_mem_uniformity_sets`. -/ theorem comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ⊆ s := by obtain ⟨w, w_in, w_sub⟩ : ∃ w ∈ 𝓤 α, w ○ w ⊆ s := comp_mem_uniformity_sets hs use symmetrizeRel w, symmetrize_mem_uniformity w_in, symmetric_symmetrizeRel w have : symmetrizeRel w ⊆ w := symmetrizeRel_subset_self w calc symmetrizeRel w ○ symmetrizeRel w _ ⊆ w ○ w := by mono _ ⊆ s := w_sub #align comp_symm_mem_uniformity_sets comp_symm_mem_uniformity_sets theorem subset_comp_self_of_mem_uniformity {s : Set (α × α)} (h : s ∈ 𝓤 α) : s ⊆ s ○ s := subset_comp_self (refl_le_uniformity h) #align subset_comp_self_of_mem_uniformity subset_comp_self_of_mem_uniformity theorem comp_comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ○ t ⊆ s := by rcases comp_symm_mem_uniformity_sets hs with ⟨w, w_in, _, w_sub⟩ rcases comp_symm_mem_uniformity_sets w_in with ⟨t, t_in, t_symm, t_sub⟩ use t, t_in, t_symm have : t ⊆ t ○ t := subset_comp_self_of_mem_uniformity t_in -- Porting note: Needed the following `have`s to make `mono` work have ht := Subset.refl t have hw := Subset.refl w calc t ○ t ○ t ⊆ w ○ t := by mono _ ⊆ w ○ (t ○ t) := by mono _ ⊆ w ○ w := by mono _ ⊆ s := w_sub #align comp_comp_symm_mem_uniformity_sets comp_comp_symm_mem_uniformity_sets /-! ### Balls in uniform spaces -/ /-- The ball around `(x : β)` with respect to `(V : Set (β × β))`. Intended to be used for `V ∈ 𝓤 β`, but this is not needed for the definition. Recovers the notions of metric space ball when `V = {p \| dist p.1 p.2 < r }`. -/ def UniformSpace.ball (x : β) (V : Set (β × β)) : Set β := Prod.mk x ⁻¹' V #align uniform_space.ball UniformSpace.ball open UniformSpace (ball) theorem UniformSpace.mem_ball_self (x : α) {V : Set (α × α)} (hV : V ∈ 𝓤 α) : x ∈ ball x V := refl_mem_uniformity hV #align uniform_space.mem_ball_self UniformSpace.mem_ball_self /-- The triangle inequality for `UniformSpace.ball` -/ theorem mem_ball_comp {V W : Set (β × β)} {x y z} (h : y ∈ ball x V) (h' : z ∈ ball y W) : z ∈ ball x (V ○ W) := prod_mk_mem_compRel h h' #align mem_ball_comp mem_ball_comp theorem ball_subset_of_comp_subset {V W : Set (β × β)} {x y} (h : x ∈ ball y W) (h' : W ○ W ⊆ V) : ball x W ⊆ ball y V := fun _z z_in => h' (mem_ball_comp h z_in) #align ball_subset_of_comp_subset ball_subset_of_comp_subset theorem ball_mono {V W : Set (β × β)} (h : V ⊆ W) (x : β) : ball x V ⊆ ball x W := preimage_mono h #align ball_mono ball_mono theorem ball_inter (x : β) (V W : Set (β × β)) : ball x (V ∩ W) = ball x V ∩ ball x W := preimage_inter #align ball_inter ball_inter theorem ball_inter_left (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x V := ball_mono inter_subset_left x #align ball_inter_left ball_inter_left theorem ball_inter_right (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x W := ball_mono inter_subset_right x #align ball_inter_right ball_inter_right theorem mem_ball_symmetry {V : Set (β × β)} (hV : SymmetricRel V) {x y} : x ∈ ball y V ↔ y ∈ ball x V := show (x, y) ∈ Prod.swap ⁻¹' V ↔ (x, y) ∈ V by unfold SymmetricRel at hV rw [hV] #align mem_ball_symmetry mem_ball_symmetry theorem ball_eq_of_symmetry {V : Set (β × β)} (hV : SymmetricRel V) {x} : ball x V = { y \| (y, x) ∈ V } := by ext y rw [mem_ball_symmetry hV] exact Iff.rfl #align ball_eq_of_symmetry ball_eq_of_symmetry theorem mem_comp_of_mem_ball {V W : Set (β × β)} {x y z : β} (hV : SymmetricRel V) (hx : x ∈ ball z V) (hy : y ∈ ball z W) : (x, y) ∈ V ○ W := by rw [mem_ball_symmetry hV] at hx exact ⟨z, hx, hy⟩ #align mem_comp_of_mem_ball mem_comp_of_mem_ball theorem UniformSpace.isOpen_ball (x : α) {V : Set (α × α)} (hV : IsOpen V) : IsOpen (ball x V) := hV.preimage <\| continuous_const.prod_mk continuous_id #align uniform_space.is_open_ball UniformSpace.isOpen_ball theorem UniformSpace.isClosed_ball (x : α) {V : Set (α × α)} (hV : IsClosed V) : IsClosed (ball x V) := hV.preimage <\| continuous_const.prod_mk continuous_id theorem mem_comp_comp {V W M : Set (β × β)} (hW' : SymmetricRel W) {p : β × β} : p ∈ V ○ M ○ W ↔ (ball p.1 V ×ˢ ball p.2 W ∩ M).Nonempty := by cases' p with x y constructor · rintro ⟨z, ⟨w, hpw, hwz⟩, hzy⟩ exact ⟨(w, z), ⟨hpw, by rwa [mem_ball_symmetry hW']⟩, hwz⟩ · rintro ⟨⟨w, z⟩, ⟨w_in, z_in⟩, hwz⟩ rw [mem_ball_symmetry hW'] at z_in exact ⟨z, ⟨w, w_in, hwz⟩, z_in⟩ #align mem_comp_comp mem_comp_comp /-! ### Neighborhoods in uniform spaces -/ theorem mem_nhds_uniformity_iff_right {x : α} {s : Set α} : s ∈ 𝓝 x ↔ { p : α × α \| p.1 = x → p.2 ∈ s } ∈ 𝓤 α := by simp only [nhds_eq_comap_uniformity, mem_comap_prod_mk] #align mem_nhds_uniformity_iff_right mem_nhds_uniformity_iff_right theorem mem_nhds_uniformity_iff_left {x : α} {s : Set α} : s ∈ 𝓝 x ↔ { p : α × α \| p.2 = x → p.1 ∈ s } ∈ 𝓤 α := by rw [uniformity_eq_symm, mem_nhds_uniformity_iff_right] simp only [map_def, mem_map, preimage_setOf_eq, Prod.snd_swap, Prod.fst_swap] #align mem_nhds_uniformity_iff_left mem_nhds_uniformity_iff_left theorem nhdsWithin_eq_comap_uniformity_of_mem {x : α} {T : Set α} (hx : x ∈ T) (S : Set α) : 𝓝[S] x = (𝓤 α ⊓ 𝓟 (T ×ˢ S)).comap (Prod.mk x) := by simp [nhdsWithin, nhds_eq_comap_uniformity, hx] theorem nhdsWithin_eq_comap_uniformity {x : α} (S : Set α) : 𝓝[S] x = (𝓤 α ⊓ 𝓟 (univ ×ˢ S)).comap (Prod.mk x) := nhdsWithin_eq_comap_uniformity_of_mem (mem_univ _) S /-- See also `isOpen_iff_open_ball_subset`. -/ theorem isOpen_iff_ball_subset {s : Set α} : IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, ball x V ⊆ s := by simp_rw [isOpen_iff_mem_nhds, nhds_eq_comap_uniformity, mem_comap, ball] #align is_open_iff_ball_subset isOpen_iff_ball_subset theorem nhds_basis_uniformity' {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {x : α} : (𝓝 x).HasBasis p fun i => ball x (s i) := by rw [nhds_eq_comap_uniformity] exact h.comap (Prod.mk x) #align nhds_basis_uniformity' nhds_basis_uniformity' theorem nhds_basis_uniformity {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) {x : α} : (𝓝 x).HasBasis p fun i => { y \| (y, x) ∈ s i } := by replace h := h.comap Prod.swap rw [comap_swap_uniformity] at h exact nhds_basis_uniformity' h #align nhds_basis_uniformity nhds_basis_uniformity theorem nhds_eq_comap_uniformity' {x : α} : 𝓝 x = (𝓤 α).comap fun y => (y, x) := (nhds_basis_uniformity (𝓤 α).basis_sets).eq_of_same_basis <\| (𝓤 α).basis_sets.comap _ #align nhds_eq_comap_uniformity' nhds_eq_comap_uniformity' theorem UniformSpace.mem_nhds_iff {x : α} {s : Set α} : s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, ball x V ⊆ s := by rw [nhds_eq_comap_uniformity, mem_comap] simp_rw [ball] #align uniform_space.mem_nhds_iff UniformSpace.mem_nhds_iff theorem UniformSpace.ball_mem_nhds (x : α) ⦃V : Set (α × α)⦄ (V_in : V ∈ 𝓤 α) : ball x V ∈ 𝓝 x := by rw [UniformSpace.mem_nhds_iff] exact ⟨V, V_in, Subset.rfl⟩ #align uniform_space.ball_mem_nhds UniformSpace.ball_mem_nhds theorem UniformSpace.ball_mem_nhdsWithin {x : α} {S : Set α} ⦃V : Set (α × α)⦄ (x_in : x ∈ S) (V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S)) : ball x V ∈ 𝓝[S] x := by rw [nhdsWithin_eq_comap_uniformity_of_mem x_in, mem_comap] exact ⟨V, V_in, Subset.rfl⟩ theorem UniformSpace.mem_nhds_iff_symm {x : α} {s : Set α} : s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, SymmetricRel V ∧ ball x V ⊆ s := by rw [UniformSpace.mem_nhds_iff] constructor · rintro ⟨V, V_in, V_sub⟩ use symmetrizeRel V, symmetrize_mem_uniformity V_in, symmetric_symmetrizeRel V exact Subset.trans (ball_mono (symmetrizeRel_subset_self V) x) V_sub · rintro ⟨V, V_in, _, V_sub⟩ exact ⟨V, V_in, V_sub⟩ #align uniform_space.mem_nhds_iff_symm UniformSpace.mem_nhds_iff_symm theorem UniformSpace.hasBasis_nhds (x : α) : HasBasis (𝓝 x) (fun s : Set (α × α) => s ∈ 𝓤 α ∧ SymmetricRel s) fun s => ball x s := ⟨fun t => by simp [UniformSpace.mem_nhds_iff_symm, and_assoc]⟩ #align uniform_space.has_basis_nhds UniformSpace.hasBasis_nhds open UniformSpace	Mathlib/Topology/UniformSpace/Basic.lean	796	798	theorem UniformSpace.mem_closure_iff_symm_ball {s : Set α} {x} : x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (s ∩ ball x V).Nonempty := by	simp [mem_closure_iff_nhds_basis (hasBasis_nhds x), Set.Nonempty]
/- Copyright (c) 2021 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov -/ import Mathlib.Topology.MetricSpace.Lipschitz import Mathlib.Analysis.SpecialFunctions.Pow.Continuity #align_import topology.metric_space.holder from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8" /-! # Hölder continuous functions In this file we define Hölder continuity on a set and on the whole space. We also prove some basic properties of Hölder continuous functions. ## Main definitions * `HolderOnWith`: `f : X → Y` is said to be Hölder continuous with constant `C : ℝ≥0` and exponent `r : ℝ≥0` on a set `s`, if `edist (f x) (f y) ≤ C * edist x y ^ r` for all `x y ∈ s`; * `HolderWith`: `f : X → Y` is said to be Hölder continuous with constant `C : ℝ≥0` and exponent `r : ℝ≥0`, if `edist (f x) (f y) ≤ C * edist x y ^ r` for all `x y : X`. ## Implementation notes We use the type `ℝ≥0` (a.k.a. `NNReal`) for `C` because this type has coercion both to `ℝ` and `ℝ≥0∞`, so it can be easily used both in inequalities about `dist` and `edist`. We also use `ℝ≥0` for `r` to ensure that `d ^ r` is monotone in `d`. It might be a good idea to use `ℝ>0` for `r` but we don't have this type in `mathlib` (yet). ## Tags Hölder continuity, Lipschitz continuity -/ variable {X Y Z : Type} open Filter Set open NNReal ENNReal Topology section Emetric variable [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] /-- A function `f : X → Y` between two `PseudoEMetricSpace`s is Hölder continuous with constant `C : ℝ≥0` and exponent `r : ℝ≥0`, if `edist (f x) (f y) ≤ C edist x y ^ r` for all `x y : X`. -/ def HolderWith (C r : ℝ≥0) (f : X → Y) : Prop := ∀ x y, edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) #align holder_with HolderWith /-- A function `f : X → Y` between two `PseudoEMetricSpace`s is Hölder continuous with constant `C : ℝ≥0` and exponent `r : ℝ≥0` on a set `s : Set X`, if `edist (f x) (f y) ≤ C * edist x y ^ r` for all `x y ∈ s`. -/ def HolderOnWith (C r : ℝ≥0) (f : X → Y) (s : Set X) : Prop := ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) #align holder_on_with HolderOnWith @[simp] theorem holderOnWith_empty (C r : ℝ≥0) (f : X → Y) : HolderOnWith C r f ∅ := fun _ hx => hx.elim #align holder_on_with_empty holderOnWith_empty @[simp] theorem holderOnWith_singleton (C r : ℝ≥0) (f : X → Y) (x : X) : HolderOnWith C r f {x} := by rintro a (rfl : a = x) b (rfl : b = a) rw [edist_self] exact zero_le _ #align holder_on_with_singleton holderOnWith_singleton theorem Set.Subsingleton.holderOnWith {s : Set X} (hs : s.Subsingleton) (C r : ℝ≥0) (f : X → Y) : HolderOnWith C r f s := hs.induction_on (holderOnWith_empty C r f) (holderOnWith_singleton C r f) #align set.subsingleton.holder_on_with Set.Subsingleton.holderOnWith theorem holderOnWith_univ {C r : ℝ≥0} {f : X → Y} : HolderOnWith C r f univ ↔ HolderWith C r f := by simp only [HolderOnWith, HolderWith, mem_univ, true_imp_iff] #align holder_on_with_univ holderOnWith_univ @[simp] theorem holderOnWith_one {C : ℝ≥0} {f : X → Y} {s : Set X} : HolderOnWith C 1 f s ↔ LipschitzOnWith C f s := by simp only [HolderOnWith, LipschitzOnWith, NNReal.coe_one, ENNReal.rpow_one] #align holder_on_with_one holderOnWith_one alias ⟨_, LipschitzOnWith.holderOnWith⟩ := holderOnWith_one #align lipschitz_on_with.holder_on_with LipschitzOnWith.holderOnWith @[simp] theorem holderWith_one {C : ℝ≥0} {f : X → Y} : HolderWith C 1 f ↔ LipschitzWith C f := holderOnWith_univ.symm.trans <\| holderOnWith_one.trans lipschitzOn_univ #align holder_with_one holderWith_one alias ⟨_, LipschitzWith.holderWith⟩ := holderWith_one #align lipschitz_with.holder_with LipschitzWith.holderWith theorem holderWith_id : HolderWith 1 1 (id : X → X) := LipschitzWith.id.holderWith #align holder_with_id holderWith_id protected theorem HolderWith.holderOnWith {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) (s : Set X) : HolderOnWith C r f s := fun x _ y _ => h x y #align holder_with.holder_on_with HolderWith.holderOnWith namespace HolderOnWith variable {C r : ℝ≥0} {f : X → Y} {s t : Set X} theorem edist_le (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) := h x hx y hy #align holder_on_with.edist_le HolderOnWith.edist_le theorem edist_le_of_le (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) {d : ℝ≥0∞} (hd : edist x y ≤ d) : edist (f x) (f y) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := (h.edist_le hx hy).trans <\| by gcongr #align holder_on_with.edist_le_of_le HolderOnWith.edist_le_of_le theorem comp {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderOnWith Cf rf f s) (hst : MapsTo f s t) : HolderOnWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s := by intro x hx y hy rw [ENNReal.coe_mul, mul_comm rg, NNReal.coe_mul, ENNReal.rpow_mul, mul_assoc, ← ENNReal.coe_rpow_of_nonneg _ rg.coe_nonneg, ← ENNReal.mul_rpow_of_nonneg _ _ rg.coe_nonneg] exact hg.edist_le_of_le (hst hx) (hst hy) (hf.edist_le hx hy) #align holder_on_with.comp HolderOnWith.comp theorem comp_holderWith {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderWith Cf rf f) (ht : ∀ x, f x ∈ t) : HolderWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) := holderOnWith_univ.mp <\| hg.comp (hf.holderOnWith univ) fun x _ => ht x #align holder_on_with.comp_holder_with HolderOnWith.comp_holderWith /-- A Hölder continuous function is uniformly continuous -/ protected theorem uniformContinuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : UniformContinuousOn f s := by refine EMetric.uniformContinuousOn_iff.2 fun ε εpos => ?_ have : Tendsto (fun d : ℝ≥0∞ => (C : ℝ≥0∞) * d ^ (r : ℝ)) (𝓝 0) (𝓝 0) := ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos ENNReal.coe_ne_top h0 rcases ENNReal.nhds_zero_basis.mem_iff.1 (this (gt_mem_nhds εpos)) with ⟨δ, δ0, H⟩ exact ⟨δ, δ0, fun hx y hy h => (hf.edist_le hx hy).trans_lt (H h)⟩ #align holder_on_with.uniform_continuous_on HolderOnWith.uniformContinuousOn protected theorem continuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : ContinuousOn f s := (hf.uniformContinuousOn h0).continuousOn #align holder_on_with.continuous_on HolderOnWith.continuousOn protected theorem mono (hf : HolderOnWith C r f s) (ht : t ⊆ s) : HolderOnWith C r f t := fun _ hx _ hy => hf.edist_le (ht hx) (ht hy) #align holder_on_with.mono HolderOnWith.mono theorem ediam_image_le_of_le (hf : HolderOnWith C r f s) {d : ℝ≥0∞} (hd : EMetric.diam s ≤ d) : EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := EMetric.diam_image_le_iff.2 fun _ hx _ hy => hf.edist_le_of_le hx hy <\| (EMetric.edist_le_diam_of_mem hx hy).trans hd #align holder_on_with.ediam_image_le_of_le HolderOnWith.ediam_image_le_of_le theorem ediam_image_le (hf : HolderOnWith C r f s) : EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) := hf.ediam_image_le_of_le le_rfl #align holder_on_with.ediam_image_le HolderOnWith.ediam_image_le theorem ediam_image_le_of_subset (hf : HolderOnWith C r f s) (ht : t ⊆ s) : EMetric.diam (f '' t) ≤ (C : ℝ≥0∞) * EMetric.diam t ^ (r : ℝ) := (hf.mono ht).ediam_image_le #align holder_on_with.ediam_image_le_of_subset HolderOnWith.ediam_image_le_of_subset theorem ediam_image_le_of_subset_of_le (hf : HolderOnWith C r f s) (ht : t ⊆ s) {d : ℝ≥0∞} (hd : EMetric.diam t ≤ d) : EMetric.diam (f '' t) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := (hf.mono ht).ediam_image_le_of_le hd #align holder_on_with.ediam_image_le_of_subset_of_le HolderOnWith.ediam_image_le_of_subset_of_le theorem ediam_image_inter_le_of_le (hf : HolderOnWith C r f s) {d : ℝ≥0∞} (hd : EMetric.diam t ≤ d) : EMetric.diam (f '' (t ∩ s)) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := hf.ediam_image_le_of_subset_of_le inter_subset_right <\| (EMetric.diam_mono inter_subset_left).trans hd #align holder_on_with.ediam_image_inter_le_of_le HolderOnWith.ediam_image_inter_le_of_le theorem ediam_image_inter_le (hf : HolderOnWith C r f s) (t : Set X) : EMetric.diam (f '' (t ∩ s)) ≤ (C : ℝ≥0∞) * EMetric.diam t ^ (r : ℝ) := hf.ediam_image_inter_le_of_le le_rfl #align holder_on_with.ediam_image_inter_le HolderOnWith.ediam_image_inter_le end HolderOnWith namespace HolderWith variable {C r : ℝ≥0} {f : X → Y} theorem edist_le (h : HolderWith C r f) (x y : X) : edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) := h x y #align holder_with.edist_le HolderWith.edist_le theorem edist_le_of_le (h : HolderWith C r f) {x y : X} {d : ℝ≥0∞} (hd : edist x y ≤ d) : edist (f x) (f y) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) := (h.holderOnWith univ).edist_le_of_le trivial trivial hd #align holder_with.edist_le_of_le HolderWith.edist_le_of_le theorem comp {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderWith Cf rf f) : HolderWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) := (hg.holderOnWith univ).comp_holderWith hf fun _ => trivial #align holder_with.comp HolderWith.comp theorem comp_holderOnWith {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith Cf rf f s) : HolderOnWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s := (hg.holderOnWith univ).comp hf fun _ _ => trivial #align holder_with.comp_holder_on_with HolderWith.comp_holderOnWith /-- A Hölder continuous function is uniformly continuous -/ protected theorem uniformContinuous (hf : HolderWith C r f) (h0 : 0 < r) : UniformContinuous f := uniformContinuousOn_univ.mp <\| (hf.holderOnWith univ).uniformContinuousOn h0 #align holder_with.uniform_continuous HolderWith.uniformContinuous protected theorem continuous (hf : HolderWith C r f) (h0 : 0 < r) : Continuous f := (hf.uniformContinuous h0).continuous #align holder_with.continuous HolderWith.continuous theorem ediam_image_le (hf : HolderWith C r f) (s : Set X) : EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) := EMetric.diam_image_le_iff.2 fun _ hx _ hy => hf.edist_le_of_le <\| EMetric.edist_le_diam_of_mem hx hy #align holder_with.ediam_image_le HolderWith.ediam_image_le end HolderWith end Emetric section Metric variable [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y} namespace HolderWith theorem nndist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ≥0} (hd : nndist x y ≤ d) : nndist (f x) (f y) ≤ C * d ^ (r : ℝ) := by rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul, ← ENNReal.coe_rpow_of_nonneg _ r.coe_nonneg] apply hf.edist_le_of_le rwa [edist_nndist, ENNReal.coe_le_coe] #align holder_with.nndist_le_of_le HolderWith.nndist_le_of_le theorem nndist_le (hf : HolderWith C r f) (x y : X) : nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) := hf.nndist_le_of_le le_rfl #align holder_with.nndist_le HolderWith.nndist_le	Mathlib/Topology/MetricSpace/Holder.lean	250	255	theorem dist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ} (hd : dist x y ≤ d) : dist (f x) (f y) ≤ C * d ^ (r : ℝ) := by	lift d to ℝ≥0 using dist_nonneg.trans hd rw [dist_nndist] at hd ⊢ norm_cast at hd ⊢ exact hf.nndist_le_of_le hd
/- Copyright (c) 2020 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Robert Y. Lewis -/ import Mathlib.RingTheory.WittVector.StructurePolynomial #align_import ring_theory.witt_vector.defs from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a" /-! # Witt vectors In this file we define the type of `p`-typical Witt vectors and ring operations on it. The ring axioms are verified in `Mathlib/RingTheory/WittVector/Basic.lean`. For a fixed commutative ring `R` and prime `p`, a Witt vector `x : 𝕎 R` is an infinite sequence `ℕ → R` of elements of `R`. However, the ring operations `+` and `` are not defined in the obvious component-wise way. Instead, these operations are defined via certain polynomials using the machinery in `Mathlib/RingTheory/WittVector/StructurePolynomial.lean`. The `n`th value of the sum of two Witt vectors can depend on the `0`-th through `n`th values of the summands. This effectively simulates a “carrying” operation. ## Main definitions `WittVector p R`: the type of `p`-typical Witt vectors with coefficients in `R`. * `WittVector.coeff x n`: projects the `n`th value of the Witt vector `x`. ## Notation We use notation `𝕎 R`, entered `\bbW`, for the Witt vectors over `R`. ## References * [Hazewinkel, Witt Vectors][Haze09] * [Commelin and Lewis, Formalizing the Ring of Witt Vectors][CL21] -/ noncomputable section /-- `WittVector p R` is the ring of `p`-typical Witt vectors over the commutative ring `R`, where `p` is a prime number. If `p` is invertible in `R`, this ring is isomorphic to `ℕ → R` (the product of `ℕ` copies of `R`). If `R` is a ring of characteristic `p`, then `WittVector p R` is a ring of characteristic `0`. The canonical example is `WittVector p (ZMod p)`, which is isomorphic to the `p`-adic integers `ℤ_[p]`. -/ structure WittVector (p : ℕ) (R : Type) where mk' :: /-- `x.coeff n` is the `n`th coefficient of the Witt vector `x`. This concept does not have a standard name in the literature. -/ coeff : ℕ → R #align witt_vector WittVector -- Porting note: added to make the `p` argument explicit /-- Construct a Witt vector `mk p x : 𝕎 R` from a sequence `x` of elements of `R`. -/ def WittVector.mk (p : ℕ) {R : Type} (coeff : ℕ → R) : WittVector p R := mk' coeff variable {p : ℕ} /- We cannot make this `localized` notation, because the `p` on the RHS doesn't occur on the left Hiding the `p` in the notation is very convenient, so we opt for repeating the `local notation` in other files that use Witt vectors. -/ local notation "𝕎" => WittVector p -- type as `\bbW` namespace WittVector variable {R : Type} @[ext] theorem ext {x y : 𝕎 R} (h : ∀ n, x.coeff n = y.coeff n) : x = y := by cases x cases y simp only at h simp [Function.funext_iff, h] #align witt_vector.ext WittVector.ext theorem ext_iff {x y : 𝕎 R} : x = y ↔ ∀ n, x.coeff n = y.coeff n := ⟨fun h n => by rw [h], ext⟩ #align witt_vector.ext_iff WittVector.ext_iff variable (p) theorem coeff_mk (x : ℕ → R) : (mk p x).coeff = x := rfl #align witt_vector.coeff_mk WittVector.coeff_mk /- These instances are not needed for the rest of the development, but it is interesting to establish early on that `WittVector p` is a lawful functor. -/ instance : Functor (WittVector p) where map f v := mk p (f ∘ v.coeff) mapConst a _ := mk p fun _ => a instance : LawfulFunctor (WittVector p) where map_const := rfl -- Porting note: no longer needs to deconstruct `v` to conclude `{coeff := v.coeff} = v` id_map _ := rfl comp_map _ _ _ := rfl variable [hp : Fact p.Prime] [CommRing R] open MvPolynomial section RingOperations /-- The polynomials used for defining the element `0` of the ring of Witt vectors. -/ def wittZero : ℕ → MvPolynomial (Fin 0 × ℕ) ℤ := wittStructureInt p 0 #align witt_vector.witt_zero WittVector.wittZero /-- The polynomials used for defining the element `1` of the ring of Witt vectors. -/ def wittOne : ℕ → MvPolynomial (Fin 0 × ℕ) ℤ := wittStructureInt p 1 #align witt_vector.witt_one WittVector.wittOne /-- The polynomials used for defining the addition of the ring of Witt vectors. -/ def wittAdd : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ := wittStructureInt p (X 0 + X 1) #align witt_vector.witt_add WittVector.wittAdd /-- The polynomials used for defining repeated addition of the ring of Witt vectors. -/ def wittNSMul (n : ℕ) : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (n • X (0 : (Fin 1))) #align witt_vector.witt_nsmul WittVector.wittNSMul /-- The polynomials used for defining repeated addition of the ring of Witt vectors. -/ def wittZSMul (n : ℤ) : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (n • X (0 : (Fin 1))) #align witt_vector.witt_zsmul WittVector.wittZSMul /-- The polynomials used for describing the subtraction of the ring of Witt vectors. -/ def wittSub : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ := wittStructureInt p (X 0 - X 1) #align witt_vector.witt_sub WittVector.wittSub /-- The polynomials used for defining the multiplication of the ring of Witt vectors. -/ def wittMul : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ := wittStructureInt p (X 0 X 1) #align witt_vector.witt_mul WittVector.wittMul /-- The polynomials used for defining the negation of the ring of Witt vectors. -/ def wittNeg : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (-X 0) #align witt_vector.witt_neg WittVector.wittNeg /-- The polynomials used for defining repeated addition of the ring of Witt vectors. -/ def wittPow (n : ℕ) : ℕ → MvPolynomial (Fin 1 × ℕ) ℤ := wittStructureInt p (X 0 ^ n) #align witt_vector.witt_pow WittVector.wittPow variable {p} /-- An auxiliary definition used in `WittVector.eval`. Evaluates a polynomial whose variables come from the disjoint union of `k` copies of `ℕ`, with a curried evaluation `x`. This can be defined more generally but we use only a specific instance here. -/ def peval {k : ℕ} (φ : MvPolynomial (Fin k × ℕ) ℤ) (x : Fin k → ℕ → R) : R := aeval (Function.uncurry x) φ #align witt_vector.peval WittVector.peval /-- Let `φ` be a family of polynomials, indexed by natural numbers, whose variables come from the disjoint union of `k` copies of `ℕ`, and let `xᵢ` be a Witt vector for `0 ≤ i < k`. `eval φ x` evaluates `φ` mapping the variable `X_(i, n)` to the `n`th coefficient of `xᵢ`. Instantiating `φ` with certain polynomials defined in `Mathlib/RingTheory/WittVector/StructurePolynomial.lean` establishes the ring operations on `𝕎 R`. For example, `WittVector.wittAdd` is such a `φ` with `k = 2`; evaluating this at `(x₀, x₁)` gives us the sum of two Witt vectors `x₀ + x₁`. -/ def eval {k : ℕ} (φ : ℕ → MvPolynomial (Fin k × ℕ) ℤ) (x : Fin k → 𝕎 R) : 𝕎 R := mk p fun n => peval (φ n) fun i => (x i).coeff #align witt_vector.eval WittVector.eval variable (R) [Fact p.Prime] instance : Zero (𝕎 R) := ⟨eval (wittZero p) ![]⟩ instance : Inhabited (𝕎 R) := ⟨0⟩ instance : One (𝕎 R) := ⟨eval (wittOne p) ![]⟩ instance : Add (𝕎 R) := ⟨fun x y => eval (wittAdd p) ![x, y]⟩ instance : Sub (𝕎 R) := ⟨fun x y => eval (wittSub p) ![x, y]⟩ instance hasNatScalar : SMul ℕ (𝕎 R) := ⟨fun n x => eval (wittNSMul p n) ![x]⟩ #align witt_vector.has_nat_scalar WittVector.hasNatScalar instance hasIntScalar : SMul ℤ (𝕎 R) := ⟨fun n x => eval (wittZSMul p n) ![x]⟩ #align witt_vector.has_int_scalar WittVector.hasIntScalar instance : Mul (𝕎 R) := ⟨fun x y => eval (wittMul p) ![x, y]⟩ instance : Neg (𝕎 R) := ⟨fun x => eval (wittNeg p) ![x]⟩ instance hasNatPow : Pow (𝕎 R) ℕ := ⟨fun x n => eval (wittPow p n) ![x]⟩ #align witt_vector.has_nat_pow WittVector.hasNatPow instance : NatCast (𝕎 R) := ⟨Nat.unaryCast⟩ instance : IntCast (𝕎 R) := ⟨Int.castDef⟩ end RingOperations section WittStructureSimplifications @[simp] theorem wittZero_eq_zero (n : ℕ) : wittZero p n = 0 := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittZero, wittStructureRat, bind₁, aeval_zero', constantCoeff_xInTermsOfW, RingHom.map_zero, AlgHom.map_zero, map_wittStructureInt] #align witt_vector.witt_zero_eq_zero WittVector.wittZero_eq_zero @[simp] theorem wittOne_zero_eq_one : wittOne p 0 = 1 := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittOne, wittStructureRat, xInTermsOfW_zero, AlgHom.map_one, RingHom.map_one, bind₁_X_right, map_wittStructureInt] #align witt_vector.witt_one_zero_eq_one WittVector.wittOne_zero_eq_one @[simp] theorem wittOne_pos_eq_zero (n : ℕ) (hn : 0 < n) : wittOne p n = 0 := by apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittOne, wittStructureRat, RingHom.map_zero, AlgHom.map_one, RingHom.map_one, map_wittStructureInt] induction n using Nat.strong_induction_on with \| h n IH => ?_ rw [xInTermsOfW_eq] simp only [AlgHom.map_mul, AlgHom.map_sub, AlgHom.map_sum, AlgHom.map_pow, bind₁_X_right, bind₁_C_right] rw [sub_mul, one_mul] rw [Finset.sum_eq_single 0] · simp only [invOf_eq_inv, one_mul, inv_pow, tsub_zero, RingHom.map_one, pow_zero] simp only [one_pow, one_mul, xInTermsOfW_zero, sub_self, bind₁_X_right] · intro i hin hi0 rw [Finset.mem_range] at hin rw [IH _ hin (Nat.pos_of_ne_zero hi0), zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero] · rw [Finset.mem_range]; intro; contradiction #align witt_vector.witt_one_pos_eq_zero WittVector.wittOne_pos_eq_zero @[simp]	Mathlib/RingTheory/WittVector/Defs.lean	258	261	theorem wittAdd_zero : wittAdd p 0 = X (0, 0) + X (1, 0) := by	apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective simp only [wittAdd, wittStructureRat, AlgHom.map_add, RingHom.map_add, rename_X, xInTermsOfW_zero, map_X, wittPolynomial_zero, bind₁_X_right, map_wittStructureInt]
/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Mathlib.Algebra.Order.Ring.Cast import Mathlib.Data.Int.Cast.Lemmas import Mathlib.Data.Nat.Bitwise import Mathlib.Data.Nat.PSub import Mathlib.Data.Nat.Size import Mathlib.Data.Num.Bitwise #align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2" /-! # Properties of the binary representation of integers -/ /- Porting note: `bit0` and `bit1` are deprecated because it is mainly used to represent number literal in Lean3 but not in Lean4 anymore. However, this file uses them for encoding numbers so this linter is unnecessary. -/ set_option linter.deprecated false -- Porting note: Required for the notation `-[n+1]`. open Int Function attribute [local simp] add_assoc namespace PosNum variable {α : Type} @[simp, norm_cast] theorem cast_one [One α] [Add α] : ((1 : PosNum) : α) = 1 := rfl #align pos_num.cast_one PosNum.cast_one @[simp] theorem cast_one' [One α] [Add α] : (PosNum.one : α) = 1 := rfl #align pos_num.cast_one' PosNum.cast_one' @[simp, norm_cast] theorem cast_bit0 [One α] [Add α] (n : PosNum) : (n.bit0 : α) = _root_.bit0 (n : α) := rfl #align pos_num.cast_bit0 PosNum.cast_bit0 @[simp, norm_cast] theorem cast_bit1 [One α] [Add α] (n : PosNum) : (n.bit1 : α) = _root_.bit1 (n : α) := rfl #align pos_num.cast_bit1 PosNum.cast_bit1 @[simp, norm_cast] theorem cast_to_nat [AddMonoidWithOne α] : ∀ n : PosNum, ((n : ℕ) : α) = n \| 1 => Nat.cast_one \| bit0 p => (Nat.cast_bit0 _).trans <\| congr_arg _root_.bit0 p.cast_to_nat \| bit1 p => (Nat.cast_bit1 _).trans <\| congr_arg _root_.bit1 p.cast_to_nat #align pos_num.cast_to_nat PosNum.cast_to_nat @[norm_cast] -- @[simp] -- Porting note (#10618): simp can prove this theorem to_nat_to_int (n : PosNum) : ((n : ℕ) : ℤ) = n := cast_to_nat _ #align pos_num.to_nat_to_int PosNum.to_nat_to_int @[simp, norm_cast] theorem cast_to_int [AddGroupWithOne α] (n : PosNum) : ((n : ℤ) : α) = n := by rw [← to_nat_to_int, Int.cast_natCast, cast_to_nat] #align pos_num.cast_to_int PosNum.cast_to_int theorem succ_to_nat : ∀ n, (succ n : ℕ) = n + 1 \| 1 => rfl \| bit0 p => rfl \| bit1 p => (congr_arg _root_.bit0 (succ_to_nat p)).trans <\| show ↑p + 1 + ↑p + 1 = ↑p + ↑p + 1 + 1 by simp [add_left_comm] #align pos_num.succ_to_nat PosNum.succ_to_nat theorem one_add (n : PosNum) : 1 + n = succ n := by cases n <;> rfl #align pos_num.one_add PosNum.one_add theorem add_one (n : PosNum) : n + 1 = succ n := by cases n <;> rfl #align pos_num.add_one PosNum.add_one @[norm_cast] theorem add_to_nat : ∀ m n, ((m + n : PosNum) : ℕ) = m + n \| 1, b => by rw [one_add b, succ_to_nat, add_comm, cast_one] \| a, 1 => by rw [add_one a, succ_to_nat, cast_one] \| bit0 a, bit0 b => (congr_arg _root_.bit0 (add_to_nat a b)).trans <\| add_add_add_comm _ _ _ _ \| bit0 a, bit1 b => (congr_arg _root_.bit1 (add_to_nat a b)).trans <\| show (a + b + (a + b) + 1 : ℕ) = a + a + (b + b + 1) by simp [add_left_comm] \| bit1 a, bit0 b => (congr_arg _root_.bit1 (add_to_nat a b)).trans <\| show (a + b + (a + b) + 1 : ℕ) = a + a + 1 + (b + b) by simp [add_comm, add_left_comm] \| bit1 a, bit1 b => show (succ (a + b) + succ (a + b) : ℕ) = a + a + 1 + (b + b + 1) by rw [succ_to_nat, add_to_nat a b]; simp [add_left_comm] #align pos_num.add_to_nat PosNum.add_to_nat theorem add_succ : ∀ m n : PosNum, m + succ n = succ (m + n) \| 1, b => by simp [one_add] \| bit0 a, 1 => congr_arg bit0 (add_one a) \| bit1 a, 1 => congr_arg bit1 (add_one a) \| bit0 a, bit0 b => rfl \| bit0 a, bit1 b => congr_arg bit0 (add_succ a b) \| bit1 a, bit0 b => rfl \| bit1 a, bit1 b => congr_arg bit1 (add_succ a b) #align pos_num.add_succ PosNum.add_succ theorem bit0_of_bit0 : ∀ n, _root_.bit0 n = bit0 n \| 1 => rfl \| bit0 p => congr_arg bit0 (bit0_of_bit0 p) \| bit1 p => show bit0 (succ (_root_.bit0 p)) = _ by rw [bit0_of_bit0 p, succ] #align pos_num.bit0_of_bit0 PosNum.bit0_of_bit0 theorem bit1_of_bit1 (n : PosNum) : _root_.bit1 n = bit1 n := show _root_.bit0 n + 1 = bit1 n by rw [add_one, bit0_of_bit0, succ] #align pos_num.bit1_of_bit1 PosNum.bit1_of_bit1 @[norm_cast] theorem mul_to_nat (m) : ∀ n, ((m n : PosNum) : ℕ) = m * n \| 1 => (mul_one _).symm \| bit0 p => show (↑(m * p) + ↑(m * p) : ℕ) = ↑m * (p + p) by rw [mul_to_nat m p, left_distrib] \| bit1 p => (add_to_nat (bit0 (m * p)) m).trans <\| show (↑(m * p) + ↑(m * p) + ↑m : ℕ) = ↑m * (p + p) + m by rw [mul_to_nat m p, left_distrib] #align pos_num.mul_to_nat PosNum.mul_to_nat theorem to_nat_pos : ∀ n : PosNum, 0 < (n : ℕ) \| 1 => Nat.zero_lt_one \| bit0 p => let h := to_nat_pos p add_pos h h \| bit1 _p => Nat.succ_pos _ #align pos_num.to_nat_pos PosNum.to_nat_pos theorem cmp_to_nat_lemma {m n : PosNum} : (m : ℕ) < n → (bit1 m : ℕ) < bit0 n := show (m : ℕ) < n → (m + m + 1 + 1 : ℕ) ≤ n + n by intro h; rw [Nat.add_right_comm m m 1, add_assoc]; exact Nat.add_le_add h h #align pos_num.cmp_to_nat_lemma PosNum.cmp_to_nat_lemma theorem cmp_swap (m) : ∀ n, (cmp m n).swap = cmp n m := by induction' m with m IH m IH <;> intro n <;> cases' n with n n <;> unfold cmp <;> try { rfl } <;> rw [← IH] <;> cases cmp m n <;> rfl #align pos_num.cmp_swap PosNum.cmp_swap theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop) \| 1, 1 => rfl \| bit0 a, 1 => let h : (1 : ℕ) ≤ a := to_nat_pos a Nat.add_le_add h h \| bit1 a, 1 => Nat.succ_lt_succ <\| to_nat_pos <\| bit0 a \| 1, bit0 b => let h : (1 : ℕ) ≤ b := to_nat_pos b Nat.add_le_add h h \| 1, bit1 b => Nat.succ_lt_succ <\| to_nat_pos <\| bit0 b \| bit0 a, bit0 b => by dsimp [cmp] have := cmp_to_nat a b; revert this; cases cmp a b <;> dsimp <;> intro this · exact Nat.add_lt_add this this · rw [this] · exact Nat.add_lt_add this this \| bit0 a, bit1 b => by dsimp [cmp] have := cmp_to_nat a b; revert this; cases cmp a b <;> dsimp <;> intro this · exact Nat.le_succ_of_le (Nat.add_lt_add this this) · rw [this] apply Nat.lt_succ_self · exact cmp_to_nat_lemma this \| bit1 a, bit0 b => by dsimp [cmp] have := cmp_to_nat a b; revert this; cases cmp a b <;> dsimp <;> intro this · exact cmp_to_nat_lemma this · rw [this] apply Nat.lt_succ_self · exact Nat.le_succ_of_le (Nat.add_lt_add this this) \| bit1 a, bit1 b => by dsimp [cmp] have := cmp_to_nat a b; revert this; cases cmp a b <;> dsimp <;> intro this · exact Nat.succ_lt_succ (Nat.add_lt_add this this) · rw [this] · exact Nat.succ_lt_succ (Nat.add_lt_add this this) #align pos_num.cmp_to_nat PosNum.cmp_to_nat @[norm_cast] theorem lt_to_nat {m n : PosNum} : (m : ℕ) < n ↔ m < n := show (m : ℕ) < n ↔ cmp m n = Ordering.lt from match cmp m n, cmp_to_nat m n with \| Ordering.lt, h => by simp only at h; simp [h] \| Ordering.eq, h => by simp only at h; simp [h, lt_irrefl] \| Ordering.gt, h => by simp [not_lt_of_gt h] #align pos_num.lt_to_nat PosNum.lt_to_nat @[norm_cast] theorem le_to_nat {m n : PosNum} : (m : ℕ) ≤ n ↔ m ≤ n := by rw [← not_lt]; exact not_congr lt_to_nat #align pos_num.le_to_nat PosNum.le_to_nat end PosNum namespace Num variable {α : Type} open PosNum theorem add_zero (n : Num) : n + 0 = n := by cases n <;> rfl #align num.add_zero Num.add_zero theorem zero_add (n : Num) : 0 + n = n := by cases n <;> rfl #align num.zero_add Num.zero_add theorem add_one : ∀ n : Num, n + 1 = succ n \| 0 => rfl \| pos p => by cases p <;> rfl #align num.add_one Num.add_one theorem add_succ : ∀ m n : Num, m + succ n = succ (m + n) \| 0, n => by simp [zero_add] \| pos p, 0 => show pos (p + 1) = succ (pos p + 0) by rw [PosNum.add_one, add_zero, succ, succ'] \| pos p, pos q => congr_arg pos (PosNum.add_succ _ _) #align num.add_succ Num.add_succ theorem bit0_of_bit0 : ∀ n : Num, bit0 n = n.bit0 \| 0 => rfl \| pos p => congr_arg pos p.bit0_of_bit0 #align num.bit0_of_bit0 Num.bit0_of_bit0 theorem bit1_of_bit1 : ∀ n : Num, bit1 n = n.bit1 \| 0 => rfl \| pos p => congr_arg pos p.bit1_of_bit1 #align num.bit1_of_bit1 Num.bit1_of_bit1 @[simp] theorem ofNat'_zero : Num.ofNat' 0 = 0 := by simp [Num.ofNat'] #align num.of_nat'_zero Num.ofNat'_zero theorem ofNat'_bit (b n) : ofNat' (Nat.bit b n) = cond b Num.bit1 Num.bit0 (ofNat' n) := Nat.binaryRec_eq rfl _ _ #align num.of_nat'_bit Num.ofNat'_bit @[simp] theorem ofNat'_one : Num.ofNat' 1 = 1 := by erw [ofNat'_bit true 0, cond, ofNat'_zero]; rfl #align num.of_nat'_one Num.ofNat'_one theorem bit1_succ : ∀ n : Num, n.bit1.succ = n.succ.bit0 \| 0 => rfl \| pos _n => rfl #align num.bit1_succ Num.bit1_succ theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 := @(Nat.binaryRec (by simp [zero_add]) fun b n ih => by cases b · erw [ofNat'_bit true n, ofNat'_bit] simp only [← bit1_of_bit1, ← bit0_of_bit0, cond, _root_.bit1] -- Porting note: `cc` was not ported yet so `exact Nat.add_left_comm n 1 1` is used. · erw [show n.bit true + 1 = (n + 1).bit false by simpa [Nat.bit, _root_.bit1, _root_.bit0] using Nat.add_left_comm n 1 1, ofNat'_bit, ofNat'_bit, ih] simp only [cond, add_one, bit1_succ]) #align num.of_nat'_succ Num.ofNat'_succ @[simp] theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by induction n · simp only [Nat.add_zero, ofNat'_zero, add_zero] · simp only [Nat.add_succ, Nat.add_zero, ofNat'_succ, add_one, add_succ, ] #align num.add_of_nat' Num.add_ofNat' @[simp, norm_cast] theorem cast_zero [Zero α] [One α] [Add α] : ((0 : Num) : α) = 0 := rfl #align num.cast_zero Num.cast_zero @[simp] theorem cast_zero' [Zero α] [One α] [Add α] : (Num.zero : α) = 0 := rfl #align num.cast_zero' Num.cast_zero' @[simp, norm_cast] theorem cast_one [Zero α] [One α] [Add α] : ((1 : Num) : α) = 1 := rfl #align num.cast_one Num.cast_one @[simp] theorem cast_pos [Zero α] [One α] [Add α] (n : PosNum) : (Num.pos n : α) = n := rfl #align num.cast_pos Num.cast_pos theorem succ'_to_nat : ∀ n, (succ' n : ℕ) = n + 1 \| 0 => (Nat.zero_add _).symm \| pos _p => PosNum.succ_to_nat _ #align num.succ'_to_nat Num.succ'_to_nat theorem succ_to_nat (n) : (succ n : ℕ) = n + 1 := succ'_to_nat n #align num.succ_to_nat Num.succ_to_nat @[simp, norm_cast] theorem cast_to_nat [AddMonoidWithOne α] : ∀ n : Num, ((n : ℕ) : α) = n \| 0 => Nat.cast_zero \| pos p => p.cast_to_nat #align num.cast_to_nat Num.cast_to_nat @[norm_cast] theorem add_to_nat : ∀ m n, ((m + n : Num) : ℕ) = m + n \| 0, 0 => rfl \| 0, pos _q => (Nat.zero_add _).symm \| pos _p, 0 => rfl \| pos _p, pos _q => PosNum.add_to_nat _ _ #align num.add_to_nat Num.add_to_nat @[norm_cast] theorem mul_to_nat : ∀ m n, ((m * n : Num) : ℕ) = m * n \| 0, 0 => rfl \| 0, pos _q => (zero_mul _).symm \| pos _p, 0 => rfl \| pos _p, pos _q => PosNum.mul_to_nat _ _ #align num.mul_to_nat Num.mul_to_nat theorem cmp_to_nat : ∀ m n, (Ordering.casesOn (cmp m n) ((m : ℕ) < n) (m = n) ((n : ℕ) < m) : Prop) \| 0, 0 => rfl \| 0, pos b => to_nat_pos _ \| pos a, 0 => to_nat_pos _ \| pos a, pos b => by have := PosNum.cmp_to_nat a b; revert this; dsimp [cmp]; cases PosNum.cmp a b exacts [id, congr_arg pos, id] #align num.cmp_to_nat Num.cmp_to_nat @[norm_cast] theorem lt_to_nat {m n : Num} : (m : ℕ) < n ↔ m < n := show (m : ℕ) < n ↔ cmp m n = Ordering.lt from match cmp m n, cmp_to_nat m n with \| Ordering.lt, h => by simp only at h; simp [h] \| Ordering.eq, h => by simp only at h; simp [h, lt_irrefl] \| Ordering.gt, h => by simp [not_lt_of_gt h] #align num.lt_to_nat Num.lt_to_nat @[norm_cast] theorem le_to_nat {m n : Num} : (m : ℕ) ≤ n ↔ m ≤ n := by rw [← not_lt]; exact not_congr lt_to_nat #align num.le_to_nat Num.le_to_nat end Num namespace PosNum @[simp] theorem of_to_nat' : ∀ n : PosNum, Num.ofNat' (n : ℕ) = Num.pos n \| 1 => by erw [@Num.ofNat'_bit true 0, Num.ofNat'_zero]; rfl \| bit0 p => by erw [@Num.ofNat'_bit false, of_to_nat' p]; rfl \| bit1 p => by erw [@Num.ofNat'_bit true, of_to_nat' p]; rfl #align pos_num.of_to_nat' PosNum.of_to_nat' end PosNum namespace Num @[simp, norm_cast] theorem of_to_nat' : ∀ n : Num, Num.ofNat' (n : ℕ) = n \| 0 => ofNat'_zero \| pos p => p.of_to_nat' #align num.of_to_nat' Num.of_to_nat' lemma toNat_injective : Injective (castNum : Num → ℕ) := LeftInverse.injective of_to_nat' @[norm_cast] theorem to_nat_inj {m n : Num} : (m : ℕ) = n ↔ m = n := toNat_injective.eq_iff #align num.to_nat_inj Num.to_nat_inj /-- This tactic tries to turn an (in)equality about `Num`s to one about `Nat`s by rewriting. ```lean example (n : Num) (m : Num) : n ≤ n + m := by transfer_rw exact Nat.le_add_right _ _ ``` -/ scoped macro (name := transfer_rw) "transfer_rw" : tactic => `(tactic\| (repeat first \| rw [← to_nat_inj] \| rw [← lt_to_nat] \| rw [← le_to_nat] repeat first \| rw [add_to_nat] \| rw [mul_to_nat] \| rw [cast_one] \| rw [cast_zero])) /-- This tactic tries to prove (in)equalities about `Num`s by transferring them to the `Nat` world and then trying to call `simp`. ```lean example (n : Num) (m : Num) : n ≤ n + m := by transfer ``` -/ scoped macro (name := transfer) "transfer" : tactic => `(tactic\| (intros; transfer_rw; try simp)) instance addMonoid : AddMonoid Num where add := (· + ·) zero := 0 zero_add := zero_add add_zero := add_zero add_assoc := by transfer nsmul := nsmulRec #align num.add_monoid Num.addMonoid instance addMonoidWithOne : AddMonoidWithOne Num := { Num.addMonoid with natCast := Num.ofNat' one := 1 natCast_zero := ofNat'_zero natCast_succ := fun _ => ofNat'_succ } #align num.add_monoid_with_one Num.addMonoidWithOne instance commSemiring : CommSemiring Num where __ := Num.addMonoid __ := Num.addMonoidWithOne mul := (· * ·) npow := @npowRec Num ⟨1⟩ ⟨(· * ·)⟩ mul_zero _ := by rw [← to_nat_inj, mul_to_nat, cast_zero, mul_zero] zero_mul _ := by rw [← to_nat_inj, mul_to_nat, cast_zero, zero_mul] mul_one _ := by rw [← to_nat_inj, mul_to_nat, cast_one, mul_one] one_mul _ := by rw [← to_nat_inj, mul_to_nat, cast_one, one_mul] add_comm _ _ := by simp_rw [← to_nat_inj, add_to_nat, add_comm] mul_comm _ _ := by simp_rw [← to_nat_inj, mul_to_nat, mul_comm] mul_assoc _ _ _ := by simp_rw [← to_nat_inj, mul_to_nat, mul_assoc] left_distrib _ _ _ := by simp only [← to_nat_inj, mul_to_nat, add_to_nat, mul_add] right_distrib _ _ _ := by simp only [← to_nat_inj, mul_to_nat, add_to_nat, add_mul] #align num.comm_semiring Num.commSemiring instance orderedCancelAddCommMonoid : OrderedCancelAddCommMonoid Num where le := (· ≤ ·) lt := (· < ·) lt_iff_le_not_le a b := by simp only [← lt_to_nat, ← le_to_nat, lt_iff_le_not_le] le_refl := by transfer le_trans a b c := by transfer_rw; apply le_trans le_antisymm a b := by transfer_rw; apply le_antisymm add_le_add_left a b h c := by revert h; transfer_rw; exact fun h => add_le_add_left h c le_of_add_le_add_left a b c := by transfer_rw; apply le_of_add_le_add_left #align num.ordered_cancel_add_comm_monoid Num.orderedCancelAddCommMonoid instance linearOrderedSemiring : LinearOrderedSemiring Num := { Num.commSemiring, Num.orderedCancelAddCommMonoid with le_total := by intro a b transfer_rw apply le_total zero_le_one := by decide mul_lt_mul_of_pos_left := by intro a b c transfer_rw apply mul_lt_mul_of_pos_left mul_lt_mul_of_pos_right := by intro a b c transfer_rw apply mul_lt_mul_of_pos_right decidableLT := Num.decidableLT decidableLE := Num.decidableLE -- This is relying on an automatically generated instance name, -- generated in a `deriving` handler. -- See https://github.com/leanprover/lean4/issues/2343 decidableEq := instDecidableEqNum exists_pair_ne := ⟨0, 1, by decide⟩ } #align num.linear_ordered_semiring Num.linearOrderedSemiring @[norm_cast] -- @[simp] -- Porting note (#10618): simp can prove this theorem add_of_nat (m n) : ((m + n : ℕ) : Num) = m + n := add_ofNat' _ _ #align num.add_of_nat Num.add_of_nat @[norm_cast] -- @[simp] -- Porting note (#10618): simp can prove this theorem to_nat_to_int (n : Num) : ((n : ℕ) : ℤ) = n := cast_to_nat _ #align num.to_nat_to_int Num.to_nat_to_int @[simp, norm_cast] theorem cast_to_int {α} [AddGroupWithOne α] (n : Num) : ((n : ℤ) : α) = n := by rw [← to_nat_to_int, Int.cast_natCast, cast_to_nat] #align num.cast_to_int Num.cast_to_int theorem to_of_nat : ∀ n : ℕ, ((n : Num) : ℕ) = n \| 0 => by rw [Nat.cast_zero, cast_zero] \| n + 1 => by rw [Nat.cast_succ, add_one, succ_to_nat, to_of_nat n] #align num.to_of_nat Num.to_of_nat @[simp, norm_cast] theorem of_natCast {α} [AddMonoidWithOne α] (n : ℕ) : ((n : Num) : α) = n := by rw [← cast_to_nat, to_of_nat] #align num.of_nat_cast Num.of_natCast @[deprecated (since := "2024-04-17")] alias of_nat_cast := of_natCast @[norm_cast] -- @[simp] -- Porting note (#10618): simp can prove this theorem of_nat_inj {m n : ℕ} : (m : Num) = n ↔ m = n := ⟨fun h => Function.LeftInverse.injective to_of_nat h, congr_arg _⟩ #align num.of_nat_inj Num.of_nat_inj -- Porting note: The priority should be `high`er than `cast_to_nat`. @[simp high, norm_cast] theorem of_to_nat : ∀ n : Num, ((n : ℕ) : Num) = n := of_to_nat' #align num.of_to_nat Num.of_to_nat @[norm_cast] theorem dvd_to_nat (m n : Num) : (m : ℕ) ∣ n ↔ m ∣ n := ⟨fun ⟨k, e⟩ => ⟨k, by rw [← of_to_nat n, e]; simp⟩, fun ⟨k, e⟩ => ⟨k, by simp [e, mul_to_nat]⟩⟩ #align num.dvd_to_nat Num.dvd_to_nat end Num namespace PosNum variable {α : Type} open Num -- Porting note: The priority should be `high`er than `cast_to_nat`. @[simp high, norm_cast] theorem of_to_nat : ∀ n : PosNum, ((n : ℕ) : Num) = Num.pos n := of_to_nat' #align pos_num.of_to_nat PosNum.of_to_nat @[norm_cast] theorem to_nat_inj {m n : PosNum} : (m : ℕ) = n ↔ m = n := ⟨fun h => Num.pos.inj <\| by rw [← PosNum.of_to_nat, ← PosNum.of_to_nat, h], congr_arg _⟩ #align pos_num.to_nat_inj PosNum.to_nat_inj theorem pred'_to_nat : ∀ n, (pred' n : ℕ) = Nat.pred n \| 1 => rfl \| bit0 n => have : Nat.succ ↑(pred' n) = ↑n := by rw [pred'_to_nat n, Nat.succ_pred_eq_of_pos (to_nat_pos n)] match (motive := ∀ k : Num, Nat.succ ↑k = ↑n → ↑(Num.casesOn k 1 bit1 : PosNum) = Nat.pred (_root_.bit0 n)) pred' n, this with \| 0, (h : ((1 : Num) : ℕ) = n) => by rw [← to_nat_inj.1 h]; rfl \| Num.pos p, (h : Nat.succ ↑p = n) => by rw [← h]; exact (Nat.succ_add p p).symm \| bit1 n => rfl #align pos_num.pred'_to_nat PosNum.pred'_to_nat @[simp] theorem pred'_succ' (n) : pred' (succ' n) = n := Num.to_nat_inj.1 <\| by rw [pred'_to_nat, succ'_to_nat, Nat.add_one, Nat.pred_succ] #align pos_num.pred'_succ' PosNum.pred'_succ' @[simp] theorem succ'_pred' (n) : succ' (pred' n) = n := to_nat_inj.1 <\| by rw [succ'_to_nat, pred'_to_nat, Nat.add_one, Nat.succ_pred_eq_of_pos (to_nat_pos _)] #align pos_num.succ'_pred' PosNum.succ'_pred' instance dvd : Dvd PosNum := ⟨fun m n => pos m ∣ pos n⟩ #align pos_num.has_dvd PosNum.dvd @[norm_cast] theorem dvd_to_nat {m n : PosNum} : (m : ℕ) ∣ n ↔ m ∣ n := Num.dvd_to_nat (pos m) (pos n) #align pos_num.dvd_to_nat PosNum.dvd_to_nat theorem size_to_nat : ∀ n, (size n : ℕ) = Nat.size n \| 1 => Nat.size_one.symm \| bit0 n => by rw [size, succ_to_nat, size_to_nat n, cast_bit0, Nat.size_bit0 <\| ne_of_gt <\| to_nat_pos n] \| bit1 n => by rw [size, succ_to_nat, size_to_nat n, cast_bit1, Nat.size_bit1] #align pos_num.size_to_nat PosNum.size_to_nat theorem size_eq_natSize : ∀ n, (size n : ℕ) = natSize n \| 1 => rfl \| bit0 n => by rw [size, succ_to_nat, natSize, size_eq_natSize n] \| bit1 n => by rw [size, succ_to_nat, natSize, size_eq_natSize n] #align pos_num.size_eq_nat_size PosNum.size_eq_natSize theorem natSize_to_nat (n) : natSize n = Nat.size n := by rw [← size_eq_natSize, size_to_nat] #align pos_num.nat_size_to_nat PosNum.natSize_to_nat theorem natSize_pos (n) : 0 < natSize n := by cases n <;> apply Nat.succ_pos #align pos_num.nat_size_pos PosNum.natSize_pos /-- This tactic tries to turn an (in)equality about `PosNum`s to one about `Nat`s by rewriting. ```lean example (n : PosNum) (m : PosNum) : n ≤ n + m := by transfer_rw exact Nat.le_add_right _ _ ``` -/ scoped macro (name := transfer_rw) "transfer_rw" : tactic => `(tactic\| (repeat first \| rw [← to_nat_inj] \| rw [← lt_to_nat] \| rw [← le_to_nat] repeat first \| rw [add_to_nat] \| rw [mul_to_nat] \| rw [cast_one] \| rw [cast_zero])) /-- This tactic tries to prove (in)equalities about `PosNum`s by transferring them to the `Nat` world and then trying to call `simp`. ```lean example (n : PosNum) (m : PosNum) : n ≤ n + m := by transfer ``` -/ scoped macro (name := transfer) "transfer" : tactic => `(tactic\| (intros; transfer_rw; try simp [add_comm, add_left_comm, mul_comm, mul_left_comm])) instance addCommSemigroup : AddCommSemigroup PosNum where add := (· + ·) add_assoc := by transfer add_comm := by transfer #align pos_num.add_comm_semigroup PosNum.addCommSemigroup instance commMonoid : CommMonoid PosNum where mul := (· ·) one := (1 : PosNum) npow := @npowRec PosNum ⟨1⟩ ⟨(· * ·)⟩ mul_assoc := by transfer one_mul := by transfer mul_one := by transfer mul_comm := by transfer #align pos_num.comm_monoid PosNum.commMonoid instance distrib : Distrib PosNum where add := (· + ·) mul := (· * ·) left_distrib := by transfer; simp [mul_add] right_distrib := by transfer; simp [mul_add, mul_comm] #align pos_num.distrib PosNum.distrib instance linearOrder : LinearOrder PosNum where lt := (· < ·) lt_iff_le_not_le := by intro a b transfer_rw apply lt_iff_le_not_le le := (· ≤ ·) le_refl := by transfer le_trans := by intro a b c transfer_rw apply le_trans le_antisymm := by intro a b transfer_rw apply le_antisymm le_total := by intro a b transfer_rw apply le_total decidableLT := by infer_instance decidableLE := by infer_instance decidableEq := by infer_instance #align pos_num.linear_order PosNum.linearOrder @[simp] theorem cast_to_num (n : PosNum) : ↑n = Num.pos n := by rw [← cast_to_nat, ← of_to_nat n] #align pos_num.cast_to_num PosNum.cast_to_num @[simp, norm_cast] theorem bit_to_nat (b n) : (bit b n : ℕ) = Nat.bit b n := by cases b <;> rfl #align pos_num.bit_to_nat PosNum.bit_to_nat @[simp, norm_cast] theorem cast_add [AddMonoidWithOne α] (m n) : ((m + n : PosNum) : α) = m + n := by rw [← cast_to_nat, add_to_nat, Nat.cast_add, cast_to_nat, cast_to_nat] #align pos_num.cast_add PosNum.cast_add @[simp 500, norm_cast] theorem cast_succ [AddMonoidWithOne α] (n : PosNum) : (succ n : α) = n + 1 := by rw [← add_one, cast_add, cast_one] #align pos_num.cast_succ PosNum.cast_succ @[simp, norm_cast] theorem cast_inj [AddMonoidWithOne α] [CharZero α] {m n : PosNum} : (m : α) = n ↔ m = n := by rw [← cast_to_nat m, ← cast_to_nat n, Nat.cast_inj, to_nat_inj] #align pos_num.cast_inj PosNum.cast_inj @[simp] theorem one_le_cast [LinearOrderedSemiring α] (n : PosNum) : (1 : α) ≤ n := by rw [← cast_to_nat, ← Nat.cast_one, Nat.cast_le (α := α)]; apply to_nat_pos #align pos_num.one_le_cast PosNum.one_le_cast @[simp] theorem cast_pos [LinearOrderedSemiring α] (n : PosNum) : 0 < (n : α) := lt_of_lt_of_le zero_lt_one (one_le_cast n) #align pos_num.cast_pos PosNum.cast_pos @[simp, norm_cast] theorem cast_mul [Semiring α] (m n) : ((m * n : PosNum) : α) = m * n := by rw [← cast_to_nat, mul_to_nat, Nat.cast_mul, cast_to_nat, cast_to_nat] #align pos_num.cast_mul PosNum.cast_mul @[simp] theorem cmp_eq (m n) : cmp m n = Ordering.eq ↔ m = n := by have := cmp_to_nat m n -- Porting note: `cases` didn't rewrite at `this`, so `revert` & `intro` are required. revert this; cases cmp m n <;> intro this <;> simp at this ⊢ <;> try { exact this } <;> simp [show m ≠ n from fun e => by rw [e] at this;exact lt_irrefl _ this] #align pos_num.cmp_eq PosNum.cmp_eq @[simp, norm_cast] theorem cast_lt [LinearOrderedSemiring α] {m n : PosNum} : (m : α) < n ↔ m < n := by rw [← cast_to_nat m, ← cast_to_nat n, Nat.cast_lt (α := α), lt_to_nat] #align pos_num.cast_lt PosNum.cast_lt @[simp, norm_cast] theorem cast_le [LinearOrderedSemiring α] {m n : PosNum} : (m : α) ≤ n ↔ m ≤ n := by rw [← not_lt]; exact not_congr cast_lt #align pos_num.cast_le PosNum.cast_le end PosNum namespace Num variable {α : Type} open PosNum theorem bit_to_nat (b n) : (bit b n : ℕ) = Nat.bit b n := by cases b <;> cases n <;> rfl #align num.bit_to_nat Num.bit_to_nat theorem cast_succ' [AddMonoidWithOne α] (n) : (succ' n : α) = n + 1 := by rw [← PosNum.cast_to_nat, succ'_to_nat, Nat.cast_add_one, cast_to_nat] #align num.cast_succ' Num.cast_succ' theorem cast_succ [AddMonoidWithOne α] (n) : (succ n : α) = n + 1 := cast_succ' n #align num.cast_succ Num.cast_succ @[simp, norm_cast] theorem cast_add [Semiring α] (m n) : ((m + n : Num) : α) = m + n := by rw [← cast_to_nat, add_to_nat, Nat.cast_add, cast_to_nat, cast_to_nat] #align num.cast_add Num.cast_add @[simp, norm_cast] theorem cast_bit0 [Semiring α] (n : Num) : (n.bit0 : α) = _root_.bit0 (n : α) := by rw [← bit0_of_bit0, _root_.bit0, cast_add]; rfl #align num.cast_bit0 Num.cast_bit0 @[simp, norm_cast] theorem cast_bit1 [Semiring α] (n : Num) : (n.bit1 : α) = _root_.bit1 (n : α) := by rw [← bit1_of_bit1, _root_.bit1, bit0_of_bit0, cast_add, cast_bit0]; rfl #align num.cast_bit1 Num.cast_bit1 @[simp, norm_cast] theorem cast_mul [Semiring α] : ∀ m n, ((m n : Num) : α) = m * n \| 0, 0 => (zero_mul _).symm \| 0, pos _q => (zero_mul _).symm \| pos _p, 0 => (mul_zero _).symm \| pos _p, pos _q => PosNum.cast_mul _ _ #align num.cast_mul Num.cast_mul theorem size_to_nat : ∀ n, (size n : ℕ) = Nat.size n \| 0 => Nat.size_zero.symm \| pos p => p.size_to_nat #align num.size_to_nat Num.size_to_nat theorem size_eq_natSize : ∀ n, (size n : ℕ) = natSize n \| 0 => rfl \| pos p => p.size_eq_natSize #align num.size_eq_nat_size Num.size_eq_natSize theorem natSize_to_nat (n) : natSize n = Nat.size n := by rw [← size_eq_natSize, size_to_nat] #align num.nat_size_to_nat Num.natSize_to_nat @[simp 999] theorem ofNat'_eq : ∀ n, Num.ofNat' n = n := Nat.binaryRec (by simp) fun b n IH => by rw [ofNat'] at IH ⊢ rw [Nat.binaryRec_eq, IH] -- Porting note: `Nat.cast_bit0` & `Nat.cast_bit1` are not `simp` theorems anymore. · cases b <;> simp [Nat.bit, bit0_of_bit0, bit1_of_bit1, Nat.cast_bit0, Nat.cast_bit1] · rfl #align num.of_nat'_eq Num.ofNat'_eq theorem zneg_toZNum (n : Num) : -n.toZNum = n.toZNumNeg := by cases n <;> rfl #align num.zneg_to_znum Num.zneg_toZNum theorem zneg_toZNumNeg (n : Num) : -n.toZNumNeg = n.toZNum := by cases n <;> rfl #align num.zneg_to_znum_neg Num.zneg_toZNumNeg theorem toZNum_inj {m n : Num} : m.toZNum = n.toZNum ↔ m = n := ⟨fun h => by cases m <;> cases n <;> cases h <;> rfl, congr_arg _⟩ #align num.to_znum_inj Num.toZNum_inj @[simp] theorem cast_toZNum [Zero α] [One α] [Add α] [Neg α] : ∀ n : Num, (n.toZNum : α) = n \| 0 => rfl \| Num.pos _p => rfl #align num.cast_to_znum Num.cast_toZNum @[simp] theorem cast_toZNumNeg [AddGroup α] [One α] : ∀ n : Num, (n.toZNumNeg : α) = -n \| 0 => neg_zero.symm \| Num.pos _p => rfl #align num.cast_to_znum_neg Num.cast_toZNumNeg @[simp] theorem add_toZNum (m n : Num) : Num.toZNum (m + n) = m.toZNum + n.toZNum := by cases m <;> cases n <;> rfl #align num.add_to_znum Num.add_toZNum end Num namespace PosNum open Num theorem pred_to_nat {n : PosNum} (h : 1 < n) : (pred n : ℕ) = Nat.pred n := by unfold pred cases e : pred' n · have : (1 : ℕ) ≤ Nat.pred n := Nat.pred_le_pred ((@cast_lt ℕ _ _ _).2 h) rw [← pred'_to_nat, e] at this exact absurd this (by decide) · rw [← pred'_to_nat, e] rfl #align pos_num.pred_to_nat PosNum.pred_to_nat theorem sub'_one (a : PosNum) : sub' a 1 = (pred' a).toZNum := by cases a <;> rfl #align pos_num.sub'_one PosNum.sub'_one theorem one_sub' (a : PosNum) : sub' 1 a = (pred' a).toZNumNeg := by cases a <;> rfl #align pos_num.one_sub' PosNum.one_sub' theorem lt_iff_cmp {m n} : m < n ↔ cmp m n = Ordering.lt := Iff.rfl #align pos_num.lt_iff_cmp PosNum.lt_iff_cmp theorem le_iff_cmp {m n} : m ≤ n ↔ cmp m n ≠ Ordering.gt := not_congr <\| lt_iff_cmp.trans <\| by rw [← cmp_swap]; cases cmp m n <;> decide #align pos_num.le_iff_cmp PosNum.le_iff_cmp end PosNum namespace Num variable {α : Type*} open PosNum theorem pred_to_nat : ∀ n : Num, (pred n : ℕ) = Nat.pred n \| 0 => rfl \| pos p => by rw [pred, PosNum.pred'_to_nat]; rfl #align num.pred_to_nat Num.pred_to_nat theorem ppred_to_nat : ∀ n : Num, (↑) <$> ppred n = Nat.ppred n \| 0 => rfl \| pos p => by rw [ppred, Option.map_some, Nat.ppred_eq_some.2] rw [PosNum.pred'_to_nat, Nat.succ_pred_eq_of_pos (PosNum.to_nat_pos _)] rfl #align num.ppred_to_nat Num.ppred_to_nat	Mathlib/Data/Num/Lemmas.lean	848	849	theorem cmp_swap (m n) : (cmp m n).swap = cmp n m := by	cases m <;> cases n <;> try { rfl }; apply PosNum.cmp_swap

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