Split (1)

train · 21.9k rows

Context stringlengths 57 85k	file_name stringlengths 21 79	start int64 14 2.42k	end int64 18 2.43k	theorem stringlengths 25 2.71k	proof stringlengths 5 10.6k
import Mathlib.Analysis.BoxIntegral.Basic import Mathlib.MeasureTheory.Integral.SetIntegral import Mathlib.Tactic.Generalize #align_import analysis.box_integral.integrability from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" open scoped Classical NNReal ENNReal Topology universe u v variable {ι : Type u} {E : Type v} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] open MeasureTheory Metric Set Finset Filter BoxIntegral namespace BoxIntegral	Mathlib/Analysis/BoxIntegral/Integrability.lean	39	99	theorem hasIntegralIndicatorConst (l : IntegrationParams) (hl : l.bRiemann = false) {s : Set (ι → ℝ)} (hs : MeasurableSet s) (I : Box ι) (y : E) (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : HasIntegral.{u, v, v} I l (s.indicator fun _ => y) μ.toBoxAdditive.toSMul ((μ (s ∩ I)).toReal • y) := by	refine HasIntegral.of_mul ‖y‖ fun ε ε0 => ?_ lift ε to ℝ≥0 using ε0.le; rw [NNReal.coe_pos] at ε0 /- First we choose a closed set `F ⊆ s ∩ I.Icc` and an open set `U ⊇ s` such that both `(s ∩ I.Icc) \ F` and `U \ s` have measure less than `ε`. -/ have A : μ (s ∩ Box.Icc I) ≠ ∞ := ((measure_mono Set.inter_subset_right).trans_lt (I.measure_Icc_lt_top μ)).ne have B : μ (s ∩ I) ≠ ∞ := ((measure_mono Set.inter_subset_right).trans_lt (I.measure_coe_lt_top μ)).ne obtain ⟨F, hFs, hFc, hμF⟩ : ∃ F, F ⊆ s ∩ Box.Icc I ∧ IsClosed F ∧ μ ((s ∩ Box.Icc I) \ F) < ε := (hs.inter I.measurableSet_Icc).exists_isClosed_diff_lt A (ENNReal.coe_pos.2 ε0).ne' obtain ⟨U, hsU, hUo, hUt, hμU⟩ : ∃ U, s ∩ Box.Icc I ⊆ U ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ (s ∩ Box.Icc I)) < ε := (hs.inter I.measurableSet_Icc).exists_isOpen_diff_lt A (ENNReal.coe_pos.2 ε0).ne' /- Then we choose `r` so that `closed_ball x (r x) ⊆ U` whenever `x ∈ s ∩ I.Icc` and `closed_ball x (r x)` is disjoint with `F` otherwise. -/ have : ∀ x ∈ s ∩ Box.Icc I, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U := fun x hx => by rcases nhds_basis_closedBall.mem_iff.1 (hUo.mem_nhds <\| hsU hx) with ⟨r, hr₀, hr⟩ exact ⟨⟨r, hr₀⟩, hr⟩ choose! rs hrsU using this have : ∀ x ∈ Box.Icc I \ s, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ Fᶜ := fun x hx => by obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 (hFc.isOpen_compl.mem_nhds fun hx' => hx.2 (hFs hx').1) exact ⟨⟨r, hr₀⟩, hr⟩ choose! rs' hrs'F using this set r : (ι → ℝ) → Ioi (0 : ℝ) := s.piecewise rs rs' refine ⟨fun _ => r, fun c => l.rCond_of_bRiemann_eq_false hl, fun c π hπ hπp => ?_⟩; rw [mul_comm] /- Then the union of boxes `J ∈ π` such that `π.tag ∈ s` includes `F` and is included by `U`, hence its measure is `ε`-close to the measure of `s`. -/ dsimp [integralSum] simp only [mem_closedBall, dist_eq_norm, ← indicator_const_smul_apply, sum_indicator_eq_sum_filter, ← sum_smul, ← sub_smul, norm_smul, Real.norm_eq_abs, ← Prepartition.filter_boxes, ← Prepartition.measure_iUnion_toReal] gcongr set t := (π.filter (π.tag · ∈ s)).iUnion change abs ((μ t).toReal - (μ (s ∩ I)).toReal) ≤ ε have htU : t ⊆ U ∩ I := by simp only [t, TaggedPrepartition.iUnion_def, iUnion_subset_iff, TaggedPrepartition.mem_filter, and_imp] refine fun J hJ hJs x hx => ⟨hrsU _ ⟨hJs, π.tag_mem_Icc J⟩ ?_, π.le_of_mem' J hJ hx⟩ simpa only [r, s.piecewise_eq_of_mem _ _ hJs] using hπ.1 J hJ (Box.coe_subset_Icc hx) refine abs_sub_le_iff.2 ⟨?_, ?_⟩ · refine (ENNReal.le_toReal_sub B).trans (ENNReal.toReal_le_coe_of_le_coe ?_) refine (tsub_le_tsub (measure_mono htU) le_rfl).trans (le_measure_diff.trans ?_) refine (measure_mono fun x hx => ?_).trans hμU.le exact ⟨hx.1.1, fun hx' => hx.2 ⟨hx'.1, hx.1.2⟩⟩ · have hμt : μ t ≠ ∞ := ((measure_mono (htU.trans inter_subset_left)).trans_lt hUt).ne refine (ENNReal.le_toReal_sub hμt).trans (ENNReal.toReal_le_coe_of_le_coe ?_) refine le_measure_diff.trans ((measure_mono ?_).trans hμF.le) rintro x ⟨⟨hxs, hxI⟩, hxt⟩ refine ⟨⟨hxs, Box.coe_subset_Icc hxI⟩, fun hxF => hxt ?_⟩ simp only [t, TaggedPrepartition.iUnion_def, TaggedPrepartition.mem_filter, Set.mem_iUnion] rcases hπp x hxI with ⟨J, hJπ, hxJ⟩ refine ⟨J, ⟨hJπ, ?_⟩, hxJ⟩ contrapose hxF refine hrs'F _ ⟨π.tag_mem_Icc J, hxF⟩ ?_ simpa only [r, s.piecewise_eq_of_not_mem _ _ hxF] using hπ.1 J hJπ (Box.coe_subset_Icc hxJ)
import Mathlib.Analysis.Normed.Field.InfiniteSum import Mathlib.Analysis.SpecificLimits.Normed import Mathlib.NumberTheory.ArithmeticFunction import Mathlib.NumberTheory.SmoothNumbers lemma Summable.norm_lt_one {F : Type} [NormedField F] [CompleteSpace F] {f : ℕ → F} (hsum : Summable f) {p : ℕ} (hp : 1 < p) : ‖f p‖ < 1 := by refine summable_geometric_iff_norm_lt_one.mp ?_ simp_rw [← map_pow] exact hsum.comp_injective <\| Nat.pow_right_injective hp open scoped Topology open Nat Finset section General variable {R : Type} [NormedCommRing R] [CompleteSpace R] {f : ℕ → R} variable (hf₁ : f 1 = 1) (hmul : ∀ {m n}, Nat.Coprime m n → f (m n) = f m * f n) -- local instance to speed up typeclass search @[local instance] private lemma instT0Space : T0Space R := MetricSpace.instT0Space namespace EulerProduct lemma summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum (hsum : ∀ {p : ℕ}, p.Prime → Summable (fun n : ℕ ↦ ‖f (p ^ n)‖)) (s : Finset ℕ) : Summable (fun m : factoredNumbers s ↦ ‖f m‖) ∧ HasSum (fun m : factoredNumbers s ↦ f m) (∏ p ∈ s.filter Nat.Prime, ∑' n : ℕ, f (p ^ n)) := by induction' s using Finset.induction with p s hp ih · rw [factoredNumbers_empty] simp only [not_mem_empty, IsEmpty.forall_iff, forall_const, filter_true_of_mem, prod_empty] exact ⟨(Set.finite_singleton 1).summable (‖f ·‖), hf₁ ▸ hasSum_singleton 1 f⟩ · rw [filter_insert] split_ifs with hpp · constructor · simp only [← (equivProdNatFactoredNumbers hpp hp).summable_iff, Function.comp_def, equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp] refine Summable.of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun _ ↦ norm_mul_le ..) ?_ apply Summable.mul_of_nonneg (hsum hpp) ih.1 <;> exact fun n ↦ norm_nonneg _ · have hp' : p ∉ s.filter Nat.Prime := mt (mem_of_mem_filter p) hp rw [prod_insert hp', ← (equivProdNatFactoredNumbers hpp hp).hasSum_iff, Function.comp_def] conv => enter [1, x] rw [equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp] have : T3Space R := instT3Space -- speeds up the following apply (hsum hpp).of_norm.hasSum.mul ih.2 -- `exact summable_mul_of_summable_norm (hsum hpp) ih.1` gives a time-out apply summable_mul_of_summable_norm (hsum hpp) ih.1 · rwa [factoredNumbers_insert s hpp] lemma prod_filter_prime_tsum_eq_tsum_factoredNumbers (hsum : Summable (‖f ·‖)) (s : Finset ℕ) : ∏ p ∈ s.filter Nat.Prime, ∑' n : ℕ, f (p ^ n) = ∑' m : factoredNumbers s, f m := (summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum hf₁ hmul (fun hp ↦ hsum.comp_injective <\| Nat.pow_right_injective hp.one_lt) _).2.tsum_eq.symm lemma norm_tsum_factoredNumbers_sub_tsum_lt (hsum : Summable f) (hf₀ : f 0 = 0) {ε : ℝ} (εpos : 0 < ε) : ∃ N : ℕ, ∀ s : Finset ℕ, primesBelow N ≤ s → ‖(∑' m : ℕ, f m) - ∑' m : factoredNumbers s, f m‖ < ε := by obtain ⟨N, hN⟩ := summable_iff_nat_tsum_vanishing.mp hsum (Metric.ball 0 ε) <\| Metric.ball_mem_nhds 0 εpos simp_rw [mem_ball_zero_iff] at hN refine ⟨N, fun s hs ↦ ?_⟩ have := hN _ <\| factoredNumbers_compl hs rwa [← tsum_subtype_add_tsum_subtype_compl hsum (factoredNumbers s), add_sub_cancel_left, tsum_eq_tsum_diff_singleton (factoredNumbers s)ᶜ hf₀] -- Versions of the three lemmas above for `smoothNumbers N` lemma summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum (hsum : ∀ {p : ℕ}, p.Prime → Summable (fun n : ℕ ↦ ‖f (p ^ n)‖)) (N : ℕ) : Summable (fun m : N.smoothNumbers ↦ ‖f m‖) ∧ HasSum (fun m : N.smoothNumbers ↦ f m) (∏ p ∈ N.primesBelow, ∑' n : ℕ, f (p ^ n)) := by rw [smoothNumbers_eq_factoredNumbers, primesBelow] exact summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum hf₁ hmul hsum _ lemma prod_primesBelow_tsum_eq_tsum_smoothNumbers (hsum : Summable (‖f ·‖)) (N : ℕ) : ∏ p ∈ N.primesBelow, ∑' n : ℕ, f (p ^ n) = ∑' m : N.smoothNumbers, f m := (summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum hf₁ hmul (fun hp ↦ hsum.comp_injective <\| Nat.pow_right_injective hp.one_lt) _).2.tsum_eq.symm lemma norm_tsum_smoothNumbers_sub_tsum_lt (hsum : Summable f) (hf₀ : f 0 = 0) {ε : ℝ} (εpos : 0 < ε) : ∃ N₀ : ℕ, ∀ N ≥ N₀, ‖(∑' m : ℕ, f m) - ∑' m : N.smoothNumbers, f m‖ < ε := by conv => enter [1, N₀, N]; rw [smoothNumbers_eq_factoredNumbers] obtain ⟨N₀, hN₀⟩ := norm_tsum_factoredNumbers_sub_tsum_lt hsum hf₀ εpos refine ⟨N₀, fun N hN ↦ hN₀ (range N) fun p hp ↦ ?_⟩ exact mem_range.mpr <\| (lt_of_mem_primesBelow hp).trans_le hN theorem eulerProduct_hasProd (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) : HasProd (fun p : Primes ↦ ∑' e, f (p ^ e)) (∑' n, f n) := by let F : ℕ → R := fun n ↦ ∑' e, f (n ^ e) change HasProd (F ∘ Subtype.val) _ rw [hasProd_subtype_iff_mulIndicator, show Set.mulIndicator (fun p : ℕ ↦ Irreducible p) = {p \| Nat.Prime p}.mulIndicator from rfl, HasProd, Metric.tendsto_atTop] intro ε hε obtain ⟨N₀, hN₀⟩ := norm_tsum_factoredNumbers_sub_tsum_lt hsum.of_norm hf₀ hε refine ⟨range N₀, fun s hs ↦ ?_⟩ have : ∏ p ∈ s, {p \| Nat.Prime p}.mulIndicator F p = ∏ p ∈ s.filter Nat.Prime, F p := prod_mulIndicator_eq_prod_filter s (fun _ ↦ F) _ id rw [this, dist_eq_norm, prod_filter_prime_tsum_eq_tsum_factoredNumbers hf₁ hmul hsum, norm_sub_rev] exact hN₀ s fun p hp ↦ hs <\| mem_range.mpr <\| lt_of_mem_primesBelow hp	Mathlib/NumberTheory/EulerProduct/Basic.lean	182	185	theorem eulerProduct_hasProd_mulIndicator (hsum : Summable (‖f ·‖)) (hf₀ : f 0 = 0) : HasProd (Set.mulIndicator {p \| Nat.Prime p} fun p ↦ ∑' e, f (p ^ e)) (∑' n, f n) := by	rw [← hasProd_subtype_iff_mulIndicator] exact eulerProduct_hasProd hf₁ hmul hsum hf₀
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics import Mathlib.NumberTheory.Liouville.Basic import Mathlib.Topology.Instances.Irrational #align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8" open Filter Metric Real Set open scoped Filter Topology def LiouvilleWith (p x : ℝ) : Prop := ∃ C, ∃ᶠ n : ℕ in atTop, ∃ m : ℤ, x ≠ m / n ∧ \|x - m / n\| < C / n ^ p #align liouville_with LiouvilleWith theorem liouvilleWith_one (x : ℝ) : LiouvilleWith 1 x := by use 2 refine ((eventually_gt_atTop 0).mono fun n hn => ?_).frequently have hn' : (0 : ℝ) < n := by simpa have : x < ↑(⌊x * ↑n⌋ + 1) / ↑n := by rw [lt_div_iff hn', Int.cast_add, Int.cast_one]; exact Int.lt_floor_add_one _ refine ⟨⌊x * n⌋ + 1, this.ne, ?_⟩ rw [abs_sub_comm, abs_of_pos (sub_pos.2 this), rpow_one, sub_lt_iff_lt_add', add_div_eq_mul_add_div _ _ hn'.ne'] gcongr calc _ ≤ x * n + 1 := by push_cast; gcongr; apply Int.floor_le _ < x * n + 2 := by linarith #align liouville_with_one liouvilleWith_one namespace LiouvilleWith variable {p q x y : ℝ} {r : ℚ} {m : ℤ} {n : ℕ} theorem exists_pos (h : LiouvilleWith p x) : ∃ (C : ℝ) (_h₀ : 0 < C), ∃ᶠ n : ℕ in atTop, 1 ≤ n ∧ ∃ m : ℤ, x ≠ m / n ∧ \|x - m / n\| < C / n ^ p := by rcases h with ⟨C, hC⟩ refine ⟨max C 1, zero_lt_one.trans_le <\| le_max_right _ _, ?_⟩ refine ((eventually_ge_atTop 1).and_frequently hC).mono ?_ rintro n ⟨hle, m, hne, hlt⟩ refine ⟨hle, m, hne, hlt.trans_le ?_⟩ gcongr apply le_max_left #align liouville_with.exists_pos LiouvilleWith.exists_pos theorem mono (h : LiouvilleWith p x) (hle : q ≤ p) : LiouvilleWith q x := by rcases h.exists_pos with ⟨C, hC₀, hC⟩ refine ⟨C, hC.mono ?_⟩; rintro n ⟨hn, m, hne, hlt⟩ refine ⟨m, hne, hlt.trans_le <\| ?_⟩ gcongr exact_mod_cast hn #align liouville_with.mono LiouvilleWith.mono theorem frequently_lt_rpow_neg (h : LiouvilleWith p x) (hlt : q < p) : ∃ᶠ n : ℕ in atTop, ∃ m : ℤ, x ≠ m / n ∧ \|x - m / n\| < n ^ (-q) := by rcases h.exists_pos with ⟨C, _hC₀, hC⟩ have : ∀ᶠ n : ℕ in atTop, C < n ^ (p - q) := by simpa only [(· ∘ ·), neg_sub, one_div] using ((tendsto_rpow_atTop (sub_pos.2 hlt)).comp tendsto_natCast_atTop_atTop).eventually (eventually_gt_atTop C) refine (this.and_frequently hC).mono ?_ rintro n ⟨hnC, hn, m, hne, hlt⟩ replace hn : (0 : ℝ) < n := Nat.cast_pos.2 hn refine ⟨m, hne, hlt.trans <\| (div_lt_iff <\| rpow_pos_of_pos hn _).2 ?_⟩ rwa [mul_comm, ← rpow_add hn, ← sub_eq_add_neg] #align liouville_with.frequently_lt_rpow_neg LiouvilleWith.frequently_lt_rpow_neg theorem mul_rat (h : LiouvilleWith p x) (hr : r ≠ 0) : LiouvilleWith p (x * r) := by rcases h.exists_pos with ⟨C, _hC₀, hC⟩ refine ⟨r.den ^ p * (\|r\| * C), (tendsto_id.nsmul_atTop r.pos).frequently (hC.mono ?_)⟩ rintro n ⟨_hn, m, hne, hlt⟩ have A : (↑(r.num * m) : ℝ) / ↑(r.den • id n) = m / n * r := by simp [← div_mul_div_comm, ← r.cast_def, mul_comm] refine ⟨r.num * m, ?_, ?_⟩ · rw [A]; simp [hne, hr] · rw [A, ← sub_mul, abs_mul] simp only [smul_eq_mul, id, Nat.cast_mul] calc _ < C / ↑n ^ p * \|↑r\| := by gcongr _ = ↑r.den ^ p * (↑\|r\| * C) / (↑r.den * ↑n) ^ p := ?_ rw [mul_rpow, mul_div_mul_left, mul_comm, mul_div_assoc] · simp only [Rat.cast_abs, le_refl] all_goals positivity #align liouville_with.mul_rat LiouvilleWith.mul_rat theorem mul_rat_iff (hr : r ≠ 0) : LiouvilleWith p (x * r) ↔ LiouvilleWith p x := ⟨fun h => by simpa only [mul_assoc, ← Rat.cast_mul, mul_inv_cancel hr, Rat.cast_one, mul_one] using h.mul_rat (inv_ne_zero hr), fun h => h.mul_rat hr⟩ #align liouville_with.mul_rat_iff LiouvilleWith.mul_rat_iff	Mathlib/NumberTheory/Liouville/LiouvilleWith.lean	142	143	theorem rat_mul_iff (hr : r ≠ 0) : LiouvilleWith p (r * x) ↔ LiouvilleWith p x := by	rw [mul_comm, mul_rat_iff hr]
import Mathlib.Topology.Constructions #align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494" 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 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 #align nhds_within_le_nhds nhdsWithin_le_nhds theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂] #align nhds_within_eq_nhds_within' nhdsWithin_eq_nhdsWithin' theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s) (h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂] #align nhds_within_eq_nhds_within nhdsWithin_eq_nhdsWithin @[simp] theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a := inf_eq_left.trans le_principal_iff #align nhds_within_eq_nhds nhdsWithin_eq_nhds theorem IsOpen.nhdsWithin_eq {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s) : 𝓝[s] a = 𝓝 a := nhdsWithin_eq_nhds.2 <\| h.mem_nhds ha #align is_open.nhds_within_eq IsOpen.nhdsWithin_eq theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t) (ht : IsOpen t) (hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) : π ⁻¹' s ∈ 𝓝 a := by rw [← ht.nhdsWithin_eq h] exact preimage_nhdsWithin_coinduced' h hs #align preimage_nhds_within_coinduced preimage_nhds_within_coinduced @[simp] theorem nhdsWithin_empty (a : α) : 𝓝[∅] a = ⊥ := by rw [nhdsWithin, principal_empty, inf_bot_eq] #align nhds_within_empty nhdsWithin_empty theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a := by delta nhdsWithin rw [← inf_sup_left, sup_principal] #align nhds_within_union nhdsWithin_union theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) : 𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a := Set.Finite.induction_on hI (by simp) fun _ _ hT ↦ by simp only [hT, nhdsWithin_union, iSup_insert, biUnion_insert] #align nhds_within_bUnion nhdsWithin_biUnion theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) : 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a := by rw [sUnion_eq_biUnion, nhdsWithin_biUnion hS] #align nhds_within_sUnion nhdsWithin_sUnion theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a := by rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range] #align nhds_within_Union nhdsWithin_iUnion theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by delta nhdsWithin rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem] #align nhds_within_inter nhdsWithin_inter theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t := by delta nhdsWithin rw [← inf_principal, inf_assoc] #align nhds_within_inter' nhdsWithin_inter' theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a := by rw [nhdsWithin_inter, inf_eq_right] exact nhdsWithin_le_of_mem h #align nhds_within_inter_of_mem nhdsWithin_inter_of_mem theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) : 𝓝[s ∩ t] a = 𝓝[s] a := by rw [inter_comm, nhdsWithin_inter_of_mem h] #align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem' @[simp] theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)] #align nhds_within_singleton nhdsWithin_singleton @[simp] theorem nhdsWithin_insert (a : α) (s : Set α) : 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a := by rw [← singleton_union, nhdsWithin_union, nhdsWithin_singleton] #align nhds_within_insert nhdsWithin_insert theorem mem_nhdsWithin_insert {a : α} {s t : Set α} : t ∈ 𝓝[insert a s] a ↔ a ∈ t ∧ t ∈ 𝓝[s] a := by simp #align mem_nhds_within_insert mem_nhdsWithin_insert theorem insert_mem_nhdsWithin_insert {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) : insert a t ∈ 𝓝[insert a s] a := by simp [mem_of_superset h] #align insert_mem_nhds_within_insert insert_mem_nhdsWithin_insert theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a := by simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left, insert_def] #align insert_mem_nhds_iff insert_mem_nhds_iff @[simp] theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a := by rw [← nhdsWithin_singleton, ← nhdsWithin_union, compl_union_self, nhdsWithin_univ] #align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pure theorem nhdsWithin_prod {α : Type} [TopologicalSpace α] {β : Type} [TopologicalSpace β] {s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) : u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by rw [nhdsWithin_prod_eq] exact prod_mem_prod hu hv #align nhds_within_prod nhdsWithin_prod theorem nhdsWithin_pi_eq' {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] {I : Set ι} (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ← iInf_principal_finite hI, ← iInf_inf_eq] #align nhds_within_pi_eq' nhdsWithin_pi_eq' theorem nhdsWithin_pi_eq {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] {I : Set ι} (hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi I s] x = (⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓ ⨅ (i) (_ : i ∉ I), comap (fun x => x i) (𝓝 (x i)) := by simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← iInf_principal_finite hI, comap_inf, comap_principal, eval] rw [iInf_split _ fun i => i ∈ I, inf_right_comm] simp only [iInf_inf_eq] #align nhds_within_pi_eq nhdsWithin_pi_eq theorem nhdsWithin_pi_univ_eq {ι : Type} {α : ι → Type} [Finite ι] [∀ i, TopologicalSpace (α i)] (s : ∀ i, Set (α i)) (x : ∀ i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) := by simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x #align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eq theorem nhdsWithin_pi_eq_bot {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] {I : Set ι} {s : ∀ i, Set (α i)} {x : ∀ i, α i} : 𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ := by simp only [nhdsWithin, nhds_pi, pi_inf_principal_pi_eq_bot] #align nhds_within_pi_eq_bot nhdsWithin_pi_eq_bot theorem nhdsWithin_pi_neBot {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] {I : Set ι} {s : ∀ i, Set (α i)} {x : ∀ i, α i} : (𝓝[pi I s] x).NeBot ↔ ∀ i ∈ I, (𝓝[s i] x i).NeBot := by simp [neBot_iff, nhdsWithin_pi_eq_bot] #align nhds_within_pi_ne_bot nhdsWithin_pi_neBot theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀ x, Decidable (x ∈ t)] {a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ t] a) l) (h₁ : Tendsto g (𝓝[s ∩ tᶜ] a) l) : Tendsto (piecewise t f g) (𝓝[s] a) l := by apply Tendsto.piecewise <;> rwa [← nhdsWithin_inter'] #align filter.tendsto.piecewise_nhds_within Filter.Tendsto.piecewise_nhdsWithin theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ { x \| p x }] a) l) (h₁ : Tendsto g (𝓝[s ∩ { x \| ¬p x }] a) l) : Tendsto (fun x => if p x then f x else g x) (𝓝[s] a) l := h₀.piecewise_nhdsWithin h₁ #align filter.tendsto.if_nhds_within Filter.Tendsto.if_nhdsWithin theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) : map f (𝓝[s] a) = ⨅ t ∈ { t : Set α \| a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) := ((nhdsWithin_basis_open a s).map f).eq_biInf #align map_nhds_within map_nhdsWithin theorem tendsto_nhdsWithin_mono_left {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t) (h : Tendsto f (𝓝[t] a) l) : Tendsto f (𝓝[s] a) l := h.mono_left <\| nhdsWithin_mono a hst #align tendsto_nhds_within_mono_left tendsto_nhdsWithin_mono_left theorem tendsto_nhdsWithin_mono_right {f : β → α} {l : Filter β} {a : α} {s t : Set α} (hst : s ⊆ t) (h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝[t] a) := h.mono_right (nhdsWithin_mono a hst) #align tendsto_nhds_within_mono_right tendsto_nhdsWithin_mono_right theorem tendsto_nhdsWithin_of_tendsto_nhds {f : α → β} {a : α} {s : Set α} {l : Filter β} (h : Tendsto f (𝓝 a) l) : Tendsto f (𝓝[s] a) l := h.mono_left inf_le_left #align tendsto_nhds_within_of_tendsto_nhds tendsto_nhdsWithin_of_tendsto_nhds theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β} (h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s := by simp_rw [nhdsWithin_eq, tendsto_iInf, mem_setOf_eq, tendsto_principal, mem_inter_iff, eventually_and] at h exact (h univ ⟨mem_univ a, isOpen_univ⟩).2 #align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin theorem tendsto_nhds_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β} (h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝 a) := h.mono_right nhdsWithin_le_nhds #align tendsto_nhds_of_tendsto_nhds_within tendsto_nhds_of_tendsto_nhdsWithin theorem nhdsWithin_neBot_of_mem {s : Set α} {x : α} (hx : x ∈ s) : NeBot (𝓝[s] x) := mem_closure_iff_nhdsWithin_neBot.1 <\| subset_closure hx #align nhds_within_ne_bot_of_mem nhdsWithin_neBot_of_mem theorem IsClosed.mem_of_nhdsWithin_neBot {s : Set α} (hs : IsClosed s) {x : α} (hx : NeBot <\| 𝓝[s] x) : x ∈ s := hs.closure_eq ▸ mem_closure_iff_nhdsWithin_neBot.2 hx #align is_closed.mem_of_nhds_within_ne_bot IsClosed.mem_of_nhdsWithin_neBot theorem DenseRange.nhdsWithin_neBot {ι : Type} {f : ι → α} (h : DenseRange f) (x : α) : NeBot (𝓝[range f] x) := mem_closure_iff_clusterPt.1 (h x) #align dense_range.nhds_within_ne_bot DenseRange.nhdsWithin_neBot theorem mem_closure_pi {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] {I : Set ι} {s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) := by simp only [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_pi_neBot] #align mem_closure_pi mem_closure_pi theorem closure_pi_set {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] (I : Set ι) (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i) := Set.ext fun _ => mem_closure_pi #align closure_pi_set closure_pi_set theorem dense_pi {ι : Type} {α : ι → Type} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)} (I : Set ι) (hs : ∀ i ∈ I, Dense (s i)) : Dense (pi I s) := by simp only [dense_iff_closure_eq, closure_pi_set, pi_congr rfl fun i hi => (hs i hi).closure_eq, pi_univ] #align dense_pi dense_pi theorem eventuallyEq_nhdsWithin_iff {f g : α → β} {s : Set α} {a : α} : f =ᶠ[𝓝[s] a] g ↔ ∀ᶠ x in 𝓝 a, x ∈ s → f x = g x := mem_inf_principal #align eventually_eq_nhds_within_iff eventuallyEq_nhdsWithin_iff theorem eventuallyEq_nhdsWithin_of_eqOn {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) : f =ᶠ[𝓝[s] a] g := mem_inf_of_right h #align eventually_eq_nhds_within_of_eq_on eventuallyEq_nhdsWithin_of_eqOn theorem Set.EqOn.eventuallyEq_nhdsWithin {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) : f =ᶠ[𝓝[s] a] g := eventuallyEq_nhdsWithin_of_eqOn h #align set.eq_on.eventually_eq_nhds_within Set.EqOn.eventuallyEq_nhdsWithin theorem tendsto_nhdsWithin_congr {f g : α → β} {s : Set α} {a : α} {l : Filter β} (hfg : ∀ x ∈ s, f x = g x) (hf : Tendsto f (𝓝[s] a) l) : Tendsto g (𝓝[s] a) l := (tendsto_congr' <\| eventuallyEq_nhdsWithin_of_eqOn hfg).1 hf #align tendsto_nhds_within_congr tendsto_nhdsWithin_congr theorem eventually_nhdsWithin_of_forall {s : Set α} {a : α} {p : α → Prop} (h : ∀ x ∈ s, p x) : ∀ᶠ x in 𝓝[s] a, p x := mem_inf_of_right h #align eventually_nhds_within_of_forall eventually_nhdsWithin_of_forall theorem tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within {a : α} {l : Filter β} {s : Set α} (f : β → α) (h1 : Tendsto f l (𝓝 a)) (h2 : ∀ᶠ x in l, f x ∈ s) : Tendsto f l (𝓝[s] a) := tendsto_inf.2 ⟨h1, tendsto_principal.2 h2⟩ #align tendsto_nhds_within_of_tendsto_nhds_of_eventually_within tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within theorem tendsto_nhdsWithin_iff {a : α} {l : Filter β} {s : Set α} {f : β → α} : Tendsto f l (𝓝[s] a) ↔ Tendsto f l (𝓝 a) ∧ ∀ᶠ n in l, f n ∈ s := ⟨fun h => ⟨tendsto_nhds_of_tendsto_nhdsWithin h, eventually_mem_of_tendsto_nhdsWithin h⟩, fun h => tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ h.1 h.2⟩ #align tendsto_nhds_within_iff tendsto_nhdsWithin_iff @[simp] theorem tendsto_nhdsWithin_range {a : α} {l : Filter β} {f : β → α} : Tendsto f l (𝓝[range f] a) ↔ Tendsto f l (𝓝 a) := ⟨fun h => h.mono_right inf_le_left, fun h => tendsto_inf.2 ⟨h, tendsto_principal.2 <\| eventually_of_forall mem_range_self⟩⟩ #align tendsto_nhds_within_range tendsto_nhdsWithin_range theorem Filter.EventuallyEq.eq_of_nhdsWithin {s : Set α} {f g : α → β} {a : α} (h : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : f a = g a := h.self_of_nhdsWithin hmem #align filter.eventually_eq.eq_of_nhds_within Filter.EventuallyEq.eq_of_nhdsWithin theorem eventually_nhdsWithin_of_eventually_nhds {α : Type} [TopologicalSpace α] {s : Set α} {a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) : ∀ᶠ x in 𝓝[s] a, p x := mem_nhdsWithin_of_mem_nhds h #align eventually_nhds_within_of_eventually_nhds eventually_nhdsWithin_of_eventually_nhds theorem mem_nhdsWithin_subtype {s : Set α} {a : { x // x ∈ s }} {t u : Set { x // x ∈ s }} : t ∈ 𝓝[u] a ↔ t ∈ comap ((↑) : s → α) (𝓝[(↑) '' u] a) := by rw [nhdsWithin, nhds_subtype, principal_subtype, ← comap_inf, ← nhdsWithin] #align mem_nhds_within_subtype mem_nhdsWithin_subtype theorem nhdsWithin_subtype (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x ∈ s }) : 𝓝[t] a = comap ((↑) : s → α) (𝓝[(↑) '' t] a) := Filter.ext fun _ => mem_nhdsWithin_subtype #align nhds_within_subtype nhdsWithin_subtype theorem nhdsWithin_eq_map_subtype_coe {s : Set α} {a : α} (h : a ∈ s) : 𝓝[s] a = map ((↑) : s → α) (𝓝 ⟨a, h⟩) := (map_nhds_subtype_val ⟨a, h⟩).symm #align nhds_within_eq_map_subtype_coe nhdsWithin_eq_map_subtype_coe theorem mem_nhds_subtype_iff_nhdsWithin {s : Set α} {a : s} {t : Set s} : t ∈ 𝓝 a ↔ (↑) '' t ∈ 𝓝[s] (a : α) := by rw [← map_nhds_subtype_val, image_mem_map_iff Subtype.val_injective] #align mem_nhds_subtype_iff_nhds_within mem_nhds_subtype_iff_nhdsWithin theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : (↑) ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a := by rw [← map_nhds_subtype_val, mem_map] #align preimage_coe_mem_nhds_subtype preimage_coe_mem_nhds_subtype theorem eventually_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) : (∀ᶠ x : s in 𝓝 a, P x) ↔ ∀ᶠ x in 𝓝[s] a, P x := preimage_coe_mem_nhds_subtype theorem frequently_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) : (∃ᶠ x : s in 𝓝 a, P x) ↔ ∃ᶠ x in 𝓝[s] a, P x := eventually_nhds_subtype_iff s a (¬ P ·) \|>.not theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) : Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l := by rw [nhdsWithin_eq_map_subtype_coe h, tendsto_map'_iff]; rfl #align tendsto_nhds_within_iff_subtype tendsto_nhdsWithin_iff_subtype variable [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] theorem ContinuousWithinAt.tendsto {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) : Tendsto f (𝓝[s] x) (𝓝 (f x)) := h #align continuous_within_at.tendsto ContinuousWithinAt.tendsto theorem ContinuousOn.continuousWithinAt {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s) (hx : x ∈ s) : ContinuousWithinAt f s x := hf x hx #align continuous_on.continuous_within_at ContinuousOn.continuousWithinAt theorem continuousWithinAt_univ (f : α → β) (x : α) : ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x := by rw [ContinuousAt, ContinuousWithinAt, nhdsWithin_univ] #align continuous_within_at_univ continuousWithinAt_univ theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ := by simp [continuous_iff_continuousAt, ContinuousOn, ContinuousAt, ContinuousWithinAt, nhdsWithin_univ] #align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univ theorem continuousWithinAt_iff_continuousAt_restrict (f : α → β) {x : α} {s : Set α} (h : x ∈ s) : ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩ := tendsto_nhdsWithin_iff_subtype h f _ #align continuous_within_at_iff_continuous_at_restrict continuousWithinAt_iff_continuousAt_restrict theorem ContinuousWithinAt.tendsto_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β} (h : ContinuousWithinAt f s x) (ht : MapsTo f s t) : Tendsto f (𝓝[s] x) (𝓝[t] f x) := tendsto_inf.2 ⟨h, tendsto_principal.2 <\| mem_inf_of_right <\| mem_principal.2 <\| ht⟩ #align continuous_within_at.tendsto_nhds_within ContinuousWithinAt.tendsto_nhdsWithin theorem ContinuousWithinAt.tendsto_nhdsWithin_image {f : α → β} {x : α} {s : Set α} (h : ContinuousWithinAt f s x) : Tendsto f (𝓝[s] x) (𝓝[f '' s] f x) := h.tendsto_nhdsWithin (mapsTo_image _ _) #align continuous_within_at.tendsto_nhds_within_image ContinuousWithinAt.tendsto_nhdsWithin_image theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y) : ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y) := by unfold ContinuousWithinAt at * rw [nhdsWithin_prod_eq, Prod.map, nhds_prod_eq] exact hf.prod_map hg #align continuous_within_at.prod_map ContinuousWithinAt.prod_map theorem continuousWithinAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)} {x : α × β} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨x.1, ·⟩) {b \| (x.1, b) ∈ s} x.2 := by rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, pure_prod, ← map_inf_principal_preimage]; rfl theorem continuousWithinAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} {x : α × β} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨·, x.2⟩) {a \| (a, x.2) ∈ s} x.1 := by rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, prod_pure, ← map_inf_principal_preimage]; rfl theorem continuousAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {x : α × β} : ContinuousAt f x ↔ ContinuousAt (f ⟨x.1, ·⟩) x.2 := by simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_left theorem continuousAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {x : α × β} : ContinuousAt f x ↔ ContinuousAt (f ⟨·, x.2⟩) x.1 := by simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_right theorem continuousOn_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)} : ContinuousOn f s ↔ ∀ a, ContinuousOn (f ⟨a, ·⟩) {b \| (a, b) ∈ s} := by simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_left]; rfl theorem continuousOn_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} : ContinuousOn f s ↔ ∀ b, ContinuousOn (f ⟨·, b⟩) {a \| (a, b) ∈ s} := by simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_right]; apply forall_swap theorem continuous_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} : Continuous f ↔ ∀ a, Continuous (f ⟨a, ·⟩) := by simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_left theorem continuous_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} : Continuous f ↔ ∀ b, Continuous (f ⟨·, b⟩) := by simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_right theorem isOpenMap_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} : IsOpenMap f ↔ ∀ a, IsOpenMap (f ⟨a, ·⟩) := by simp_rw [isOpenMap_iff_nhds_le, Prod.forall, nhds_prod_eq, nhds_discrete, pure_prod, map_map] rfl theorem isOpenMap_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} : IsOpenMap f ↔ ∀ b, IsOpenMap (f ⟨·, b⟩) := by simp_rw [isOpenMap_iff_nhds_le, Prod.forall, forall_swap (α := α) (β := β), nhds_prod_eq, nhds_discrete, prod_pure, map_map]; rfl theorem continuousWithinAt_pi {ι : Type} {π : ι → Type} [∀ i, TopologicalSpace (π i)] {f : α → ∀ i, π i} {s : Set α} {x : α} : ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x := tendsto_pi_nhds #align continuous_within_at_pi continuousWithinAt_pi theorem continuousOn_pi {ι : Type} {π : ι → Type} [∀ i, TopologicalSpace (π i)] {f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s := ⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩ #align continuous_on_pi continuousOn_pi @[fun_prop] theorem continuousOn_pi' {ι : Type} {π : ι → Type} [∀ i, TopologicalSpace (π i)] {f : α → ∀ i, π i} {s : Set α} (hf : ∀ i, ContinuousOn (fun y => f y i) s) : ContinuousOn f s := continuousOn_pi.2 hf theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type} [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α} (hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousWithinAt g s a) : ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a := hf.tendsto.fin_insertNth i hg #align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth nonrec theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type} [∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {s : Set α} (hf : ContinuousOn f s) {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) : ContinuousOn (fun a => i.insertNth (f a) (g a)) s := fun a ha => (hf a ha).fin_insertNth i (hg a ha) #align continuous_on.fin_insert_nth ContinuousOn.fin_insertNth theorem continuousOn_iff {f : α → β} {s : Set α} : ContinuousOn f s ↔ ∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t := by simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin] #align continuous_on_iff continuousOn_iff theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} : ContinuousOn f s ↔ Continuous (s.restrict f) := by rw [ContinuousOn, continuous_iff_continuousAt]; constructor · rintro h ⟨x, xs⟩ exact (continuousWithinAt_iff_continuousAt_restrict f xs).mp (h x xs) intro h x xs exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩) #align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict -- Porting note: 2 new lemmas alias ⟨ContinuousOn.restrict, _⟩ := continuousOn_iff_continuous_restrict theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s) (ht : MapsTo f s t) : Continuous (ht.restrict f s t) := hf.restrict.codRestrict _ theorem continuousOn_iff' {f : α → β} {s : Set α} : ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by have : ∀ t, IsOpen (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by intro t rw [isOpen_induced_iff, Set.restrict_eq, Set.preimage_comp] simp only [Subtype.preimage_coe_eq_preimage_coe_iff] constructor <;> · rintro ⟨u, ou, useq⟩ exact ⟨u, ou, by simpa only [Set.inter_comm, eq_comm] using useq⟩ rw [continuousOn_iff_continuous_restrict, continuous_def]; simp only [this] #align continuous_on_iff' continuousOn_iff' theorem ContinuousOn.mono_dom {α β : Type} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₁) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₃ f s) : @ContinuousOn α β t₂ t₃ f s := fun x hx _u hu => map_mono (inf_le_inf_right _ <\| nhds_mono h₁) (h₂ x hx hu) #align continuous_on.mono_dom ContinuousOn.mono_dom theorem ContinuousOn.mono_rng {α β : Type} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β} (h₁ : t₂ ≤ t₃) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₂ f s) : @ContinuousOn α β t₁ t₃ f s := fun x hx _u hu => h₂ x hx <\| nhds_mono h₁ hu #align continuous_on.mono_rng ContinuousOn.mono_rng theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} : ContinuousOn f s ↔ ∀ t : Set β, IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by have : ∀ t, IsClosed (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by intro t rw [isClosed_induced_iff, Set.restrict_eq, Set.preimage_comp] simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm, Set.inter_comm s] rw [continuousOn_iff_continuous_restrict, continuous_iff_isClosed]; simp only [this] #align continuous_on_iff_is_closed continuousOn_iff_isClosed theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} (hf : ContinuousOn f s) (hg : ContinuousOn g t) : ContinuousOn (Prod.map f g) (s ×ˢ t) := fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy) #align continuous_on.prod_map ContinuousOn.prod_map theorem continuous_of_cover_nhds {ι : Sort} {f : α → β} {s : ι → Set α} (hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) : Continuous f := continuous_iff_continuousAt.mpr fun x ↦ let ⟨i, hi⟩ := hs x; by rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi] exact hf _ _ (mem_of_mem_nhds hi) #align continuous_of_cover_nhds continuous_of_cover_nhds theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun _ => False.elim #align continuous_on_empty continuousOn_empty @[simp] theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} := forall_eq.2 <\| by simpa only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_left] using fun s => mem_of_mem_nhds #align continuous_on_singleton continuousOn_singleton theorem Set.Subsingleton.continuousOn {s : Set α} (hs : s.Subsingleton) (f : α → β) : ContinuousOn f s := hs.induction_on (continuousOn_empty f) (continuousOn_singleton f) #align set.subsingleton.continuous_on Set.Subsingleton.continuousOn theorem nhdsWithin_le_comap {x : α} {s : Set α} {f : α → β} (ctsf : ContinuousWithinAt f s x) : 𝓝[s] x ≤ comap f (𝓝[f '' s] f x) := ctsf.tendsto_nhdsWithin_image.le_comap #align nhds_within_le_comap nhdsWithin_le_comap @[simp] theorem comap_nhdsWithin_range {α} (f : α → β) (y : β) : comap f (𝓝[range f] y) = comap f (𝓝 y) := comap_inf_principal_range #align comap_nhds_within_range comap_nhdsWithin_range theorem ContinuousWithinAt.mono {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x) (hs : s ⊆ t) : ContinuousWithinAt f s x := h.mono_left (nhdsWithin_mono x hs) #align continuous_within_at.mono ContinuousWithinAt.mono theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x) (hs : t ∈ 𝓝[s] x) : ContinuousWithinAt f s x := h.mono_left (nhdsWithin_le_of_mem hs) #align continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_mem theorem continuousWithinAt_congr_nhds {f : α → β} {s t : Set α} {x : α} (h : 𝓝[s] x = 𝓝[t] x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x := by simp only [ContinuousWithinAt, h] theorem continuousWithinAt_inter' {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x) : ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by simp [ContinuousWithinAt, nhdsWithin_restrict'' s h] #align continuous_within_at_inter' continuousWithinAt_inter' theorem continuousWithinAt_inter {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝 x) : ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by simp [ContinuousWithinAt, nhdsWithin_restrict' s h] #align continuous_within_at_inter continuousWithinAt_inter theorem continuousWithinAt_union {f : α → β} {s t : Set α} {x : α} : ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x := by simp only [ContinuousWithinAt, nhdsWithin_union, tendsto_sup] #align continuous_within_at_union continuousWithinAt_union theorem ContinuousWithinAt.union {f : α → β} {s t : Set α} {x : α} (hs : ContinuousWithinAt f s x) (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s ∪ t) x := continuousWithinAt_union.2 ⟨hs, ht⟩ #align continuous_within_at.union ContinuousWithinAt.union theorem ContinuousWithinAt.mem_closure_image {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) := haveI := mem_closure_iff_nhdsWithin_neBot.1 hx mem_closure_of_tendsto h <\| mem_of_superset self_mem_nhdsWithin (subset_preimage_image f s) #align continuous_within_at.mem_closure_image ContinuousWithinAt.mem_closure_image theorem ContinuousWithinAt.mem_closure {f : α → β} {s : Set α} {x : α} {A : Set β} (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) (hA : MapsTo f s A) : f x ∈ closure A := closure_mono (image_subset_iff.2 hA) (h.mem_closure_image hx) #align continuous_within_at.mem_closure ContinuousWithinAt.mem_closure theorem Set.MapsTo.closure_of_continuousWithinAt {f : α → β} {s : Set α} {t : Set β} (h : MapsTo f s t) (hc : ∀ x ∈ closure s, ContinuousWithinAt f s x) : MapsTo f (closure s) (closure t) := fun x hx => (hc x hx).mem_closure hx h #align set.maps_to.closure_of_continuous_within_at Set.MapsTo.closure_of_continuousWithinAt theorem Set.MapsTo.closure_of_continuousOn {f : α → β} {s : Set α} {t : Set β} (h : MapsTo f s t) (hc : ContinuousOn f (closure s)) : MapsTo f (closure s) (closure t) := h.closure_of_continuousWithinAt fun x hx => (hc x hx).mono subset_closure #align set.maps_to.closure_of_continuous_on Set.MapsTo.closure_of_continuousOn theorem ContinuousWithinAt.image_closure {f : α → β} {s : Set α} (hf : ∀ x ∈ closure s, ContinuousWithinAt f s x) : f '' closure s ⊆ closure (f '' s) := ((mapsTo_image f s).closure_of_continuousWithinAt hf).image_subset #align continuous_within_at.image_closure ContinuousWithinAt.image_closure theorem ContinuousOn.image_closure {f : α → β} {s : Set α} (hf : ContinuousOn f (closure s)) : f '' closure s ⊆ closure (f '' s) := ContinuousWithinAt.image_closure fun x hx => (hf x hx).mono subset_closure #align continuous_on.image_closure ContinuousOn.image_closure @[simp] theorem continuousWithinAt_singleton {f : α → β} {x : α} : ContinuousWithinAt f {x} x := by simp only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_nhds] #align continuous_within_at_singleton continuousWithinAt_singleton @[simp] theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} : ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x := by simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton, true_and_iff] #align continuous_within_at_insert_self continuousWithinAt_insert_self alias ⟨_, ContinuousWithinAt.insert_self⟩ := continuousWithinAt_insert_self #align continuous_within_at.insert_self ContinuousWithinAt.insert_self theorem ContinuousWithinAt.diff_iff {f : α → β} {s t : Set α} {x : α} (ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s \ t) x ↔ ContinuousWithinAt f s x := ⟨fun h => (h.union ht).mono <\| by simp only [diff_union_self, subset_union_left], fun h => h.mono diff_subset⟩ #align continuous_within_at.diff_iff ContinuousWithinAt.diff_iff @[simp] theorem continuousWithinAt_diff_self {f : α → β} {s : Set α} {x : α} : ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x := continuousWithinAt_singleton.diff_iff #align continuous_within_at_diff_self continuousWithinAt_diff_self @[simp] theorem continuousWithinAt_compl_self {f : α → β} {a : α} : ContinuousWithinAt f {a}ᶜ a ↔ ContinuousAt f a := by rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ] #align continuous_within_at_compl_self continuousWithinAt_compl_self @[simp] theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β} : ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) := calc ContinuousWithinAt (update f x y) s x ↔ Tendsto (update f x y) (𝓝[s \ {x}] x) (𝓝 y) := by { rw [← continuousWithinAt_diff_self, ContinuousWithinAt, update_same] } _ ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) := tendsto_congr' <\| eventually_nhdsWithin_iff.2 <\| eventually_of_forall fun z hz => update_noteq hz.2 _ _ #align continuous_within_at_update_same continuousWithinAt_update_same @[simp] theorem continuousAt_update_same [DecidableEq α] {f : α → β} {x : α} {y : β} : ContinuousAt (Function.update f x y) x ↔ Tendsto f (𝓝[≠] x) (𝓝 y) := by rw [← continuousWithinAt_univ, continuousWithinAt_update_same, compl_eq_univ_diff] #align continuous_at_update_same continuousAt_update_same theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α} (h : IsOpenMap (s.restrict f)) {finv : β → α} (hleft : LeftInvOn finv f s) : ContinuousOn finv (f '' s) := by refine continuousOn_iff'.2 fun t ht => ⟨f '' (t ∩ s), ?_, ?_⟩ · rw [← image_restrict] exact h _ (ht.preimage continuous_subtype_val) · rw [inter_eq_self_of_subset_left (image_subset f inter_subset_right), hleft.image_inter'] #align is_open_map.continuous_on_image_of_left_inv_on IsOpenMap.continuousOn_image_of_leftInvOn theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α} (hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) := by rw [← image_univ] exact (hf.restrict isOpen_univ).continuousOn_image_of_leftInvOn fun x _ => hleft x #align is_open_map.continuous_on_range_of_left_inverse IsOpenMap.continuousOn_range_of_leftInverse theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s) (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁ := by intro x hx unfold ContinuousWithinAt have A := (h x (h₁ hx)).mono h₁ unfold ContinuousWithinAt at A rw [← h' hx] at A exact A.congr' h'.eventuallyEq_nhdsWithin.symm #align continuous_on.congr_mono ContinuousOn.congr_mono theorem ContinuousOn.congr {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : EqOn g f s) : ContinuousOn g s := h.congr_mono h' (Subset.refl _) #align continuous_on.congr ContinuousOn.congr theorem continuousOn_congr {f g : α → β} {s : Set α} (h' : EqOn g f s) : ContinuousOn g s ↔ ContinuousOn f s := ⟨fun h => ContinuousOn.congr h h'.symm, fun h => h.congr h'⟩ #align continuous_on_congr continuousOn_congr theorem ContinuousAt.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x) : ContinuousWithinAt f s x := ContinuousWithinAt.mono ((continuousWithinAt_univ f x).2 h) (subset_univ _) #align continuous_at.continuous_within_at ContinuousAt.continuousWithinAt theorem continuousWithinAt_iff_continuousAt {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x) : ContinuousWithinAt f s x ↔ ContinuousAt f x := by rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ] #align continuous_within_at_iff_continuous_at continuousWithinAt_iff_continuousAt theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (hs : s ∈ 𝓝 x) : ContinuousAt f x := (continuousWithinAt_iff_continuousAt hs).mp h #align continuous_within_at.continuous_at ContinuousWithinAt.continuousAt theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) : ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a := forall₂_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds #align is_open.continuous_on_iff IsOpen.continuousOn_iff theorem ContinuousOn.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s) (hx : s ∈ 𝓝 x) : ContinuousAt f x := (h x (mem_of_mem_nhds hx)).continuousAt hx #align continuous_on.continuous_at ContinuousOn.continuousAt theorem ContinuousAt.continuousOn {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x) : ContinuousOn f s := fun x hx => (hcont x hx).continuousWithinAt #align continuous_at.continuous_on ContinuousAt.continuousOn theorem ContinuousWithinAt.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α} (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : MapsTo f s t) : ContinuousWithinAt (g ∘ f) s x := hg.tendsto.comp (hf.tendsto_nhdsWithin h) #align continuous_within_at.comp ContinuousWithinAt.comp theorem ContinuousWithinAt.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α} (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x := hg.comp (hf.mono inter_subset_left) inter_subset_right #align continuous_within_at.comp' ContinuousWithinAt.comp' theorem ContinuousAt.comp_continuousWithinAt {g : β → γ} {f : α → β} {s : Set α} {x : α} (hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) s x := hg.continuousWithinAt.comp hf (mapsTo_univ _ _) #align continuous_at.comp_continuous_within_at ContinuousAt.comp_continuousWithinAt theorem ContinuousOn.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t) (hf : ContinuousOn f s) (h : MapsTo f s t) : ContinuousOn (g ∘ f) s := fun x hx => ContinuousWithinAt.comp (hg _ (h hx)) (hf x hx) h #align continuous_on.comp ContinuousOn.comp @[fun_prop] theorem ContinuousOn.comp'' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t) (hf : ContinuousOn f s) (h : Set.MapsTo f s t) : ContinuousOn (fun x => g (f x)) s := ContinuousOn.comp hg hf h theorem ContinuousOn.mono {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s) : ContinuousOn f t := fun x hx => (hf x (h hx)).mono_left (nhdsWithin_mono _ h) #align continuous_on.mono ContinuousOn.mono theorem antitone_continuousOn {f : α → β} : Antitone (ContinuousOn f) := fun _s _t hst hf => hf.mono hst #align antitone_continuous_on antitone_continuousOn @[fun_prop] theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t) (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t) := hg.comp (hf.mono inter_subset_left) inter_subset_right #align continuous_on.comp' ContinuousOn.comp' @[fun_prop] theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s := by rw [continuous_iff_continuousOn_univ] at h exact h.mono (subset_univ _) #align continuous.continuous_on Continuous.continuousOn theorem Continuous.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : Continuous f) : ContinuousWithinAt f s x := h.continuousAt.continuousWithinAt #align continuous.continuous_within_at Continuous.continuousWithinAt theorem Continuous.comp_continuousOn {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s) : ContinuousOn (g ∘ f) s := hg.continuousOn.comp hf (mapsTo_univ _ _) #align continuous.comp_continuous_on Continuous.comp_continuousOn @[fun_prop] theorem Continuous.comp_continuousOn' {α β γ : Type} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s) : ContinuousOn (fun x ↦ g (f x)) s := hg.comp_continuousOn hf theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s) (hf : Continuous f) (hs : ∀ x, f x ∈ s) : Continuous (g ∘ f) := by rw [continuous_iff_continuousOn_univ] at * exact hg.comp hf fun x _ => hs x #align continuous_on.comp_continuous ContinuousOn.comp_continuous @[fun_prop] theorem continuousOn_apply {ι : Type} {π : ι → Type} [∀ i, TopologicalSpace (π i)] (i : ι) (s) : ContinuousOn (fun p : ∀ i, π i => p i) s := Continuous.continuousOn (continuous_apply i) theorem ContinuousWithinAt.preimage_mem_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β} (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x := h ht #align continuous_within_at.preimage_mem_nhds_within ContinuousWithinAt.preimage_mem_nhdsWithin theorem Set.LeftInvOn.map_nhdsWithin_eq {f : α → β} {g : β → α} {x : β} {s : Set β} (h : LeftInvOn f g s) (hx : f (g x) = x) (hf : ContinuousWithinAt f (g '' s) (g x)) (hg : ContinuousWithinAt g s x) : map g (𝓝[s] x) = 𝓝[g '' s] g x := by apply le_antisymm · exact hg.tendsto_nhdsWithin (mapsTo_image _ _) · have A : g ∘ f =ᶠ[𝓝[g '' s] g x] id := h.rightInvOn_image.eqOn.eventuallyEq_of_mem self_mem_nhdsWithin refine le_map_of_right_inverse A ?_ simpa only [hx] using hf.tendsto_nhdsWithin (h.mapsTo (surjOn_image _ _)) #align set.left_inv_on.map_nhds_within_eq Set.LeftInvOn.map_nhdsWithin_eq theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β} (h : Function.LeftInverse f g) (hf : ContinuousWithinAt f (range g) (g x)) (hg : ContinuousAt g x) : map g (𝓝 x) = 𝓝[range g] g x := by simpa only [nhdsWithin_univ, image_univ] using (h.leftInvOn univ).map_nhdsWithin_eq (h x) (by rwa [image_univ]) hg.continuousWithinAt #align function.left_inverse.map_nhds_eq Function.LeftInverse.map_nhds_eq theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s : Set α} {t : Set β} (h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x := h.tendsto_nhdsWithin (mapsTo_image _ _) ht #align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhdsWithin' theorem ContinuousWithinAt.preimage_mem_nhdsWithin'' {f : α → β} {x : α} {y : β} {s t : Set β} (h : ContinuousWithinAt f (f ⁻¹' s) x) (ht : t ∈ 𝓝[s] y) (hxy : y = f x) : f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x := by rw [hxy] at ht exact h.preimage_mem_nhdsWithin' (nhdsWithin_mono _ (image_preimage_subset f s) ht) theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α} (h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x := by rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt] #align filter.eventually_eq.congr_continuous_within_at Filter.EventuallyEq.congr_continuousWithinAt theorem ContinuousWithinAt.congr_of_eventuallyEq {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : ContinuousWithinAt f₁ s x := (h₁.congr_continuousWithinAt hx).2 h #align continuous_within_at.congr_of_eventually_eq ContinuousWithinAt.congr_of_eventuallyEq theorem ContinuousWithinAt.congr {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) : ContinuousWithinAt f₁ s x := h.congr_of_eventuallyEq (mem_of_superset self_mem_nhdsWithin h₁) hx #align continuous_within_at.congr ContinuousWithinAt.congr theorem ContinuousWithinAt.congr_mono {f g : α → β} {s s₁ : Set α} {x : α} (h : ContinuousWithinAt f s x) (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x) : ContinuousWithinAt g s₁ x := (h.mono h₁).congr h' hx #align continuous_within_at.congr_mono ContinuousWithinAt.congr_mono @[fun_prop] theorem continuousOn_const {s : Set α} {c : β} : ContinuousOn (fun _ => c) s := continuous_const.continuousOn #align continuous_on_const continuousOn_const theorem continuousWithinAt_const {b : β} {s : Set α} {x : α} : ContinuousWithinAt (fun _ : α => b) s x := continuous_const.continuousWithinAt #align continuous_within_at_const continuousWithinAt_const theorem continuousOn_id {s : Set α} : ContinuousOn id s := continuous_id.continuousOn #align continuous_on_id continuousOn_id @[fun_prop] theorem continuousOn_id' (s : Set α) : ContinuousOn (fun x : α => x) s := continuousOn_id theorem continuousWithinAt_id {s : Set α} {x : α} : ContinuousWithinAt id s x := continuous_id.continuousWithinAt #align continuous_within_at_id continuousWithinAt_id	Mathlib/Topology/ContinuousOn.lean	1,070	1,080	theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) : ContinuousOn f s ↔ ∀ t, IsOpen t → IsOpen (s ∩ f ⁻¹' t) := by	rw [continuousOn_iff'] constructor · intro h t ht rcases h t ht with ⟨u, u_open, hu⟩ rw [inter_comm, hu] apply IsOpen.inter u_open hs · intro h t ht refine ⟨s ∩ f ⁻¹' t, h t ht, ?_⟩ rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self]
import Mathlib.Topology.Maps import Mathlib.Topology.NhdsSet #align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f" noncomputable section open scoped Classical open Topology TopologicalSpace Set Filter Function universe u v variable {X : Type u} {Y : Type v} {Z W ε ζ : Type} section Constructions instance instTopologicalSpaceSubtype {p : X → Prop} [t : TopologicalSpace X] : TopologicalSpace (Subtype p) := induced (↑) t instance {r : X → X → Prop} [t : TopologicalSpace X] : TopologicalSpace (Quot r) := coinduced (Quot.mk r) t instance instTopologicalSpaceQuotient {s : Setoid X} [t : TopologicalSpace X] : TopologicalSpace (Quotient s) := coinduced Quotient.mk' t instance instTopologicalSpaceProd [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] : TopologicalSpace (X × Y) := induced Prod.fst t₁ ⊓ induced Prod.snd t₂ instance instTopologicalSpaceSum [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] : TopologicalSpace (X ⊕ Y) := coinduced Sum.inl t₁ ⊔ coinduced Sum.inr t₂ instance instTopologicalSpaceSigma {ι : Type} {X : ι → Type v} [t₂ : ∀ i, TopologicalSpace (X i)] : TopologicalSpace (Sigma X) := ⨆ i, coinduced (Sigma.mk i) (t₂ i) instance Pi.topologicalSpace {ι : Type} {Y : ι → Type v} [t₂ : (i : ι) → TopologicalSpace (Y i)] : TopologicalSpace ((i : ι) → Y i) := ⨅ i, induced (fun f => f i) (t₂ i) #align Pi.topological_space Pi.topologicalSpace instance ULift.topologicalSpace [t : TopologicalSpace X] : TopologicalSpace (ULift.{v, u} X) := t.induced ULift.down #align ulift.topological_space ULift.topologicalSpace section variable [TopologicalSpace X] open Additive Multiplicative instance : TopologicalSpace (Additive X) := ‹TopologicalSpace X› instance : TopologicalSpace (Multiplicative X) := ‹TopologicalSpace X› instance [DiscreteTopology X] : DiscreteTopology (Additive X) := ‹DiscreteTopology X› instance [DiscreteTopology X] : DiscreteTopology (Multiplicative X) := ‹DiscreteTopology X› theorem continuous_ofMul : Continuous (ofMul : X → Additive X) := continuous_id #align continuous_of_mul continuous_ofMul theorem continuous_toMul : Continuous (toMul : Additive X → X) := continuous_id #align continuous_to_mul continuous_toMul theorem continuous_ofAdd : Continuous (ofAdd : X → Multiplicative X) := continuous_id #align continuous_of_add continuous_ofAdd theorem continuous_toAdd : Continuous (toAdd : Multiplicative X → X) := continuous_id #align continuous_to_add continuous_toAdd theorem isOpenMap_ofMul : IsOpenMap (ofMul : X → Additive X) := IsOpenMap.id #align is_open_map_of_mul isOpenMap_ofMul theorem isOpenMap_toMul : IsOpenMap (toMul : Additive X → X) := IsOpenMap.id #align is_open_map_to_mul isOpenMap_toMul theorem isOpenMap_ofAdd : IsOpenMap (ofAdd : X → Multiplicative X) := IsOpenMap.id #align is_open_map_of_add isOpenMap_ofAdd theorem isOpenMap_toAdd : IsOpenMap (toAdd : Multiplicative X → X) := IsOpenMap.id #align is_open_map_to_add isOpenMap_toAdd theorem isClosedMap_ofMul : IsClosedMap (ofMul : X → Additive X) := IsClosedMap.id #align is_closed_map_of_mul isClosedMap_ofMul theorem isClosedMap_toMul : IsClosedMap (toMul : Additive X → X) := IsClosedMap.id #align is_closed_map_to_mul isClosedMap_toMul theorem isClosedMap_ofAdd : IsClosedMap (ofAdd : X → Multiplicative X) := IsClosedMap.id #align is_closed_map_of_add isClosedMap_ofAdd theorem isClosedMap_toAdd : IsClosedMap (toAdd : Multiplicative X → X) := IsClosedMap.id #align is_closed_map_to_add isClosedMap_toAdd theorem nhds_ofMul (x : X) : 𝓝 (ofMul x) = map ofMul (𝓝 x) := rfl #align nhds_of_mul nhds_ofMul theorem nhds_ofAdd (x : X) : 𝓝 (ofAdd x) = map ofAdd (𝓝 x) := rfl #align nhds_of_add nhds_ofAdd theorem nhds_toMul (x : Additive X) : 𝓝 (toMul x) = map toMul (𝓝 x) := rfl #align nhds_to_mul nhds_toMul theorem nhds_toAdd (x : Multiplicative X) : 𝓝 (toAdd x) = map toAdd (𝓝 x) := rfl #align nhds_to_add nhds_toAdd end section variable [TopologicalSpace X] open OrderDual instance : TopologicalSpace Xᵒᵈ := ‹TopologicalSpace X› instance [DiscreteTopology X] : DiscreteTopology Xᵒᵈ := ‹DiscreteTopology X› theorem continuous_toDual : Continuous (toDual : X → Xᵒᵈ) := continuous_id #align continuous_to_dual continuous_toDual theorem continuous_ofDual : Continuous (ofDual : Xᵒᵈ → X) := continuous_id #align continuous_of_dual continuous_ofDual theorem isOpenMap_toDual : IsOpenMap (toDual : X → Xᵒᵈ) := IsOpenMap.id #align is_open_map_to_dual isOpenMap_toDual theorem isOpenMap_ofDual : IsOpenMap (ofDual : Xᵒᵈ → X) := IsOpenMap.id #align is_open_map_of_dual isOpenMap_ofDual theorem isClosedMap_toDual : IsClosedMap (toDual : X → Xᵒᵈ) := IsClosedMap.id #align is_closed_map_to_dual isClosedMap_toDual theorem isClosedMap_ofDual : IsClosedMap (ofDual : Xᵒᵈ → X) := IsClosedMap.id #align is_closed_map_of_dual isClosedMap_ofDual theorem nhds_toDual (x : X) : 𝓝 (toDual x) = map toDual (𝓝 x) := rfl #align nhds_to_dual nhds_toDual theorem nhds_ofDual (x : X) : 𝓝 (ofDual x) = map ofDual (𝓝 x) := rfl #align nhds_of_dual nhds_ofDual end theorem Quotient.preimage_mem_nhds [TopologicalSpace X] [s : Setoid X] {V : Set <\| Quotient s} {x : X} (hs : V ∈ 𝓝 (Quotient.mk' x)) : Quotient.mk' ⁻¹' V ∈ 𝓝 x := preimage_nhds_coinduced hs #align quotient.preimage_mem_nhds Quotient.preimage_mem_nhds theorem Dense.quotient [Setoid X] [TopologicalSpace X] {s : Set X} (H : Dense s) : Dense (Quotient.mk' '' s) := Quotient.surjective_Quotient_mk''.denseRange.dense_image continuous_coinduced_rng H #align dense.quotient Dense.quotient theorem DenseRange.quotient [Setoid X] [TopologicalSpace X] {f : Y → X} (hf : DenseRange f) : DenseRange (Quotient.mk' ∘ f) := Quotient.surjective_Quotient_mk''.denseRange.comp hf continuous_coinduced_rng #align dense_range.quotient DenseRange.quotient theorem continuous_map_of_le {α : Type} [TopologicalSpace α] {s t : Setoid α} (h : s ≤ t) : Continuous (Setoid.map_of_le h) := continuous_coinduced_rng theorem continuous_map_sInf {α : Type} [TopologicalSpace α] {S : Set (Setoid α)} {s : Setoid α} (h : s ∈ S) : Continuous (Setoid.map_sInf h) := continuous_coinduced_rng instance {p : X → Prop} [TopologicalSpace X] [DiscreteTopology X] : DiscreteTopology (Subtype p) := ⟨bot_unique fun s _ => ⟨(↑) '' s, isOpen_discrete _, preimage_image_eq _ Subtype.val_injective⟩⟩ instance Sum.discreteTopology [TopologicalSpace X] [TopologicalSpace Y] [h : DiscreteTopology X] [hY : DiscreteTopology Y] : DiscreteTopology (X ⊕ Y) := ⟨sup_eq_bot_iff.2 <\| by simp [h.eq_bot, hY.eq_bot]⟩ #align sum.discrete_topology Sum.discreteTopology instance Sigma.discreteTopology {ι : Type} {Y : ι → Type v} [∀ i, TopologicalSpace (Y i)] [h : ∀ i, DiscreteTopology (Y i)] : DiscreteTopology (Sigma Y) := ⟨iSup_eq_bot.2 fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩ #align sigma.discrete_topology Sigma.discreteTopology def CofiniteTopology (X : Type) := X #align cofinite_topology CofiniteTopology section Prod variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] [TopologicalSpace ε] [TopologicalSpace ζ] -- Porting note (#11215): TODO: Lean 4 fails to deduce implicit args @[simp] theorem continuous_prod_mk {f : X → Y} {g : X → Z} : (Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g := (@continuous_inf_rng X (Y × Z) _ _ (TopologicalSpace.induced Prod.fst _) (TopologicalSpace.induced Prod.snd _)).trans <\| continuous_induced_rng.and continuous_induced_rng #align continuous_prod_mk continuous_prod_mk @[continuity] theorem continuous_fst : Continuous (@Prod.fst X Y) := (continuous_prod_mk.1 continuous_id).1 #align continuous_fst continuous_fst @[fun_prop] theorem Continuous.fst {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).1 := continuous_fst.comp hf #align continuous.fst Continuous.fst theorem Continuous.fst' {f : X → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.fst := hf.comp continuous_fst #align continuous.fst' Continuous.fst' theorem continuousAt_fst {p : X × Y} : ContinuousAt Prod.fst p := continuous_fst.continuousAt #align continuous_at_fst continuousAt_fst @[fun_prop] theorem ContinuousAt.fst {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun x : X => (f x).1) x := continuousAt_fst.comp hf #align continuous_at.fst ContinuousAt.fst theorem ContinuousAt.fst' {f : X → Z} {x : X} {y : Y} (hf : ContinuousAt f x) : ContinuousAt (fun x : X × Y => f x.fst) (x, y) := ContinuousAt.comp hf continuousAt_fst #align continuous_at.fst' ContinuousAt.fst' theorem ContinuousAt.fst'' {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.fst) : ContinuousAt (fun x : X × Y => f x.fst) x := hf.comp continuousAt_fst #align continuous_at.fst'' ContinuousAt.fst'' theorem Filter.Tendsto.fst_nhds {l : Filter X} {f : X → Y × Z} {p : Y × Z} (h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).1) l (𝓝 <\| p.1) := continuousAt_fst.tendsto.comp h @[continuity] theorem continuous_snd : Continuous (@Prod.snd X Y) := (continuous_prod_mk.1 continuous_id).2 #align continuous_snd continuous_snd @[fun_prop] theorem Continuous.snd {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).2 := continuous_snd.comp hf #align continuous.snd Continuous.snd theorem Continuous.snd' {f : Y → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.snd := hf.comp continuous_snd #align continuous.snd' Continuous.snd' theorem continuousAt_snd {p : X × Y} : ContinuousAt Prod.snd p := continuous_snd.continuousAt #align continuous_at_snd continuousAt_snd @[fun_prop] theorem ContinuousAt.snd {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) : ContinuousAt (fun x : X => (f x).2) x := continuousAt_snd.comp hf #align continuous_at.snd ContinuousAt.snd theorem ContinuousAt.snd' {f : Y → Z} {x : X} {y : Y} (hf : ContinuousAt f y) : ContinuousAt (fun x : X × Y => f x.snd) (x, y) := ContinuousAt.comp hf continuousAt_snd #align continuous_at.snd' ContinuousAt.snd' theorem ContinuousAt.snd'' {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.snd) : ContinuousAt (fun x : X × Y => f x.snd) x := hf.comp continuousAt_snd #align continuous_at.snd'' ContinuousAt.snd'' theorem Filter.Tendsto.snd_nhds {l : Filter X} {f : X → Y × Z} {p : Y × Z} (h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).2) l (𝓝 <\| p.2) := continuousAt_snd.tendsto.comp h @[continuity, fun_prop] theorem Continuous.prod_mk {f : Z → X} {g : Z → Y} (hf : Continuous f) (hg : Continuous g) : Continuous fun x => (f x, g x) := continuous_prod_mk.2 ⟨hf, hg⟩ #align continuous.prod_mk Continuous.prod_mk @[continuity] theorem Continuous.Prod.mk (x : X) : Continuous fun y : Y => (x, y) := continuous_const.prod_mk continuous_id #align continuous.prod.mk Continuous.Prod.mk @[continuity] theorem Continuous.Prod.mk_left (y : Y) : Continuous fun x : X => (x, y) := continuous_id.prod_mk continuous_const #align continuous.prod.mk_left Continuous.Prod.mk_left lemma IsClosed.setOf_mapsTo {α : Type} {f : X → α → Z} {s : Set α} {t : Set Z} (ht : IsClosed t) (hf : ∀ a ∈ s, Continuous (f · a)) : IsClosed {x \| MapsTo (f x) s t} := by simpa only [MapsTo, setOf_forall] using isClosed_biInter fun y hy ↦ ht.preimage (hf y hy) theorem Continuous.comp₂ {g : X × Y → Z} (hg : Continuous g) {e : W → X} (he : Continuous e) {f : W → Y} (hf : Continuous f) : Continuous fun w => g (e w, f w) := hg.comp <\| he.prod_mk hf #align continuous.comp₂ Continuous.comp₂ theorem Continuous.comp₃ {g : X × Y × Z → ε} (hg : Continuous g) {e : W → X} (he : Continuous e) {f : W → Y} (hf : Continuous f) {k : W → Z} (hk : Continuous k) : Continuous fun w => g (e w, f w, k w) := hg.comp₂ he <\| hf.prod_mk hk #align continuous.comp₃ Continuous.comp₃ theorem Continuous.comp₄ {g : X × Y × Z × ζ → ε} (hg : Continuous g) {e : W → X} (he : Continuous e) {f : W → Y} (hf : Continuous f) {k : W → Z} (hk : Continuous k) {l : W → ζ} (hl : Continuous l) : Continuous fun w => g (e w, f w, k w, l w) := hg.comp₃ he hf <\| hk.prod_mk hl #align continuous.comp₄ Continuous.comp₄ @[continuity] theorem Continuous.prod_map {f : Z → X} {g : W → Y} (hf : Continuous f) (hg : Continuous g) : Continuous fun p : Z × W => (f p.1, g p.2) := hf.fst'.prod_mk hg.snd' #align continuous.prod_map Continuous.prod_map theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X} {tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z} (h : by haveI := ta1; haveI := tb1; exact Continuous fun p : X × Y => f p.1 p.2) : by haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _)) have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _)) have h_continuous_id := @Continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id #align continuous_inf_dom_left₂ continuous_inf_dom_left₂ theorem continuous_inf_dom_right₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X} {tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z} (h : by haveI := ta2; haveI := tb2; exact Continuous fun p : X × Y => f p.1 p.2) : by haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _)) have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _)) have h_continuous_id := @Continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id #align continuous_inf_dom_right₂ continuous_inf_dom_right₂ theorem continuous_sInf_dom₂ {X Y Z} {f : X → Y → Z} {tas : Set (TopologicalSpace X)} {tbs : Set (TopologicalSpace Y)} {tX : TopologicalSpace X} {tY : TopologicalSpace Y} {tc : TopologicalSpace Z} (hX : tX ∈ tas) (hY : tY ∈ tbs) (hf : Continuous fun p : X × Y => f p.1 p.2) : by haveI := sInf tas; haveI := sInf tbs; exact @Continuous _ _ _ tc fun p : X × Y => f p.1 p.2 := by have hX := continuous_sInf_dom hX continuous_id have hY := continuous_sInf_dom hY continuous_id have h_continuous_id := @Continuous.prod_map _ _ _ _ tX tY (sInf tas) (sInf tbs) _ _ hX hY exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id #align continuous_Inf_dom₂ continuous_sInf_dom₂ theorem Filter.Eventually.prod_inl_nhds {p : X → Prop} {x : X} (h : ∀ᶠ x in 𝓝 x, p x) (y : Y) : ∀ᶠ x in 𝓝 (x, y), p (x : X × Y).1 := continuousAt_fst h #align filter.eventually.prod_inl_nhds Filter.Eventually.prod_inl_nhds theorem Filter.Eventually.prod_inr_nhds {p : Y → Prop} {y : Y} (h : ∀ᶠ x in 𝓝 y, p x) (x : X) : ∀ᶠ x in 𝓝 (x, y), p (x : X × Y).2 := continuousAt_snd h #align filter.eventually.prod_inr_nhds Filter.Eventually.prod_inr_nhds theorem Filter.Eventually.prod_mk_nhds {px : X → Prop} {x} (hx : ∀ᶠ x in 𝓝 x, px x) {py : Y → Prop} {y} (hy : ∀ᶠ y in 𝓝 y, py y) : ∀ᶠ p in 𝓝 (x, y), px (p : X × Y).1 ∧ py p.2 := (hx.prod_inl_nhds y).and (hy.prod_inr_nhds x) #align filter.eventually.prod_mk_nhds Filter.Eventually.prod_mk_nhds theorem continuous_swap : Continuous (Prod.swap : X × Y → Y × X) := continuous_snd.prod_mk continuous_fst #align continuous_swap continuous_swap lemma isClosedMap_swap : IsClosedMap (Prod.swap : X × Y → Y × X) := fun s hs ↦ by rw [image_swap_eq_preimage_swap] exact hs.preimage continuous_swap theorem Continuous.uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) : Continuous (f x) := h.comp (Continuous.Prod.mk _) #align continuous_uncurry_left Continuous.uncurry_left theorem Continuous.uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) : Continuous fun a => f a y := h.comp (Continuous.Prod.mk_left _) #align continuous_uncurry_right Continuous.uncurry_right -- 2024-03-09 @[deprecated] alias continuous_uncurry_left := Continuous.uncurry_left @[deprecated] alias continuous_uncurry_right := Continuous.uncurry_right theorem continuous_curry {g : X × Y → Z} (x : X) (h : Continuous g) : Continuous (curry g x) := Continuous.uncurry_left x h #align continuous_curry continuous_curry theorem IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) := (hs.preimage continuous_fst).inter (ht.preimage continuous_snd) #align is_open.prod IsOpen.prod -- Porting note (#11215): TODO: Lean fails to find `t₁` and `t₂` by unification theorem nhds_prod_eq {x : X} {y : Y} : 𝓝 (x, y) = 𝓝 x ×ˢ 𝓝 y := by dsimp only [SProd.sprod] rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _) (t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced] #align nhds_prod_eq nhds_prod_eq -- Porting note: moved from `Topology.ContinuousOn` theorem nhdsWithin_prod_eq (x : X) (y : Y) (s : Set X) (t : Set Y) : 𝓝[s ×ˢ t] (x, y) = 𝓝[s] x ×ˢ 𝓝[t] y := by simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal] #align nhds_within_prod_eq nhdsWithin_prod_eq #noalign continuous_uncurry_of_discrete_topology theorem mem_nhds_prod_iff {x : X} {y : Y} {s : Set (X × Y)} : s ∈ 𝓝 (x, y) ↔ ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff] #align mem_nhds_prod_iff mem_nhds_prod_iff theorem mem_nhdsWithin_prod_iff {x : X} {y : Y} {s : Set (X × Y)} {tx : Set X} {ty : Set Y} : s ∈ 𝓝[tx ×ˢ ty] (x, y) ↔ ∃ u ∈ 𝓝[tx] x, ∃ v ∈ 𝓝[ty] y, u ×ˢ v ⊆ s := by rw [nhdsWithin_prod_eq, mem_prod_iff] -- Porting note: moved up theorem Filter.HasBasis.prod_nhds {ιX ιY : Type} {px : ιX → Prop} {py : ιY → Prop} {sx : ιX → Set X} {sy : ιY → Set Y} {x : X} {y : Y} (hx : (𝓝 x).HasBasis px sx) (hy : (𝓝 y).HasBasis py sy) : (𝓝 (x, y)).HasBasis (fun i : ιX × ιY => px i.1 ∧ py i.2) fun i => sx i.1 ×ˢ sy i.2 := by rw [nhds_prod_eq] exact hx.prod hy #align filter.has_basis.prod_nhds Filter.HasBasis.prod_nhds -- Porting note: moved up theorem Filter.HasBasis.prod_nhds' {ιX ιY : Type} {pX : ιX → Prop} {pY : ιY → Prop} {sx : ιX → Set X} {sy : ιY → Set Y} {p : X × Y} (hx : (𝓝 p.1).HasBasis pX sx) (hy : (𝓝 p.2).HasBasis pY sy) : (𝓝 p).HasBasis (fun i : ιX × ιY => pX i.1 ∧ pY i.2) fun i => sx i.1 ×ˢ sy i.2 := hx.prod_nhds hy #align filter.has_basis.prod_nhds' Filter.HasBasis.prod_nhds' theorem mem_nhds_prod_iff' {x : X} {y : Y} {s : Set (X × Y)} : s ∈ 𝓝 (x, y) ↔ ∃ u v, IsOpen u ∧ x ∈ u ∧ IsOpen v ∧ y ∈ v ∧ u ×ˢ v ⊆ s := ((nhds_basis_opens x).prod_nhds (nhds_basis_opens y)).mem_iff.trans <\| by simp only [Prod.exists, and_comm, and_assoc, and_left_comm] #align mem_nhds_prod_iff' mem_nhds_prod_iff' theorem Prod.tendsto_iff {X} (seq : X → Y × Z) {f : Filter X} (p : Y × Z) : Tendsto seq f (𝓝 p) ↔ Tendsto (fun n => (seq n).fst) f (𝓝 p.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 p.snd) := by rw [nhds_prod_eq, Filter.tendsto_prod_iff'] #align prod.tendsto_iff Prod.tendsto_iff instance [DiscreteTopology X] [DiscreteTopology Y] : DiscreteTopology (X × Y) := discreteTopology_iff_nhds.2 fun (a, b) => by rw [nhds_prod_eq, nhds_discrete X, nhds_discrete Y, prod_pure_pure] theorem prod_mem_nhds_iff {s : Set X} {t : Set Y} {x : X} {y : Y} : s ×ˢ t ∈ 𝓝 (x, y) ↔ s ∈ 𝓝 x ∧ t ∈ 𝓝 y := by rw [nhds_prod_eq, prod_mem_prod_iff] #align prod_mem_nhds_iff prod_mem_nhds_iff theorem prod_mem_nhds {s : Set X} {t : Set Y} {x : X} {y : Y} (hx : s ∈ 𝓝 x) (hy : t ∈ 𝓝 y) : s ×ˢ t ∈ 𝓝 (x, y) := prod_mem_nhds_iff.2 ⟨hx, hy⟩ #align prod_mem_nhds prod_mem_nhds theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : X × X \| Disjoint (𝓝 p.1) (𝓝 p.2) } := by simp only [isOpen_iff_mem_nhds, Prod.forall, mem_setOf_eq] intro x y h obtain ⟨U, hU, V, hV, hd⟩ := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h exact mem_nhds_prod_iff'.mpr ⟨U, V, hU.2, hU.1, hV.2, hV.1, fun ⟨x', y'⟩ ⟨hx', hy'⟩ => disjoint_of_disjoint_of_mem hd (hU.2.mem_nhds hx') (hV.2.mem_nhds hy')⟩ #align is_open_set_of_disjoint_nhds_nhds isOpen_setOf_disjoint_nhds_nhds theorem Filter.Eventually.prod_nhds {p : X → Prop} {q : Y → Prop} {x : X} {y : Y} (hx : ∀ᶠ x in 𝓝 x, p x) (hy : ∀ᶠ y in 𝓝 y, q y) : ∀ᶠ z : X × Y in 𝓝 (x, y), p z.1 ∧ q z.2 := prod_mem_nhds hx hy #align filter.eventually.prod_nhds Filter.Eventually.prod_nhds theorem nhds_swap (x : X) (y : Y) : 𝓝 (x, y) = (𝓝 (y, x)).map Prod.swap := by rw [nhds_prod_eq, Filter.prod_comm, nhds_prod_eq]; rfl #align nhds_swap nhds_swap theorem Filter.Tendsto.prod_mk_nhds {γ} {x : X} {y : Y} {f : Filter γ} {mx : γ → X} {my : γ → Y} (hx : Tendsto mx f (𝓝 x)) (hy : Tendsto my f (𝓝 y)) : Tendsto (fun c => (mx c, my c)) f (𝓝 (x, y)) := by rw [nhds_prod_eq]; exact Filter.Tendsto.prod_mk hx hy #align filter.tendsto.prod_mk_nhds Filter.Tendsto.prod_mk_nhds theorem Filter.Eventually.curry_nhds {p : X × Y → Prop} {x : X} {y : Y} (h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by rw [nhds_prod_eq] at h exact h.curry #align filter.eventually.curry_nhds Filter.Eventually.curry_nhds @[fun_prop] theorem ContinuousAt.prod {f : X → Y} {g : X → Z} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x := hf.prod_mk_nhds hg #align continuous_at.prod ContinuousAt.prod theorem ContinuousAt.prod_map {f : X → Z} {g : Y → W} {p : X × Y} (hf : ContinuousAt f p.fst) (hg : ContinuousAt g p.snd) : ContinuousAt (fun p : X × Y => (f p.1, g p.2)) p := hf.fst''.prod hg.snd'' #align continuous_at.prod_map ContinuousAt.prod_map theorem ContinuousAt.prod_map' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf : ContinuousAt f x) (hg : ContinuousAt g y) : ContinuousAt (fun p : X × Y => (f p.1, g p.2)) (x, y) := hf.fst'.prod hg.snd' #align continuous_at.prod_map' ContinuousAt.prod_map' theorem ContinuousAt.comp₂ {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousAt g x) (hh : ContinuousAt h x) : ContinuousAt (fun x ↦ f (g x, h x)) x := ContinuousAt.comp hf (hg.prod hh)	Mathlib/Topology/Constructions.lean	663	667	theorem ContinuousAt.comp₂_of_eq {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} {y : Y × Z} (hf : ContinuousAt f y) (hg : ContinuousAt g x) (hh : ContinuousAt h x) (e : (g x, h x) = y) : ContinuousAt (fun x ↦ f (g x, h x)) x := by	rw [← e] at hf exact hf.comp₂ hg hh
import Mathlib.LinearAlgebra.TensorProduct.Graded.External import Mathlib.RingTheory.GradedAlgebra.Basic import Mathlib.GroupTheory.GroupAction.Ring suppress_compilation open scoped TensorProduct variable {R ι A B : Type} variable [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] variable [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] variable (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) variable [GradedAlgebra 𝒜] [GradedAlgebra ℬ] open DirectSum variable (R) in @[nolint unusedArguments] def GradedTensorProduct (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] : Type _ := A ⊗[R] B namespace GradedTensorProduct open TensorProduct @[inherit_doc GradedTensorProduct] scoped[TensorProduct] notation:100 𝒜 " ᵍ⊗[" R "] " ℬ:100 => GradedTensorProduct R 𝒜 ℬ instance instAddCommGroupWithOne : AddCommGroupWithOne (𝒜 ᵍ⊗[R] ℬ) := Algebra.TensorProduct.instAddCommGroupWithOne instance : Module R (𝒜 ᵍ⊗[R] ℬ) := TensorProduct.leftModule variable (R) in def of : A ⊗[R] B ≃ₗ[R] 𝒜 ᵍ⊗[R] ℬ := LinearEquiv.refl _ _ @[simp] theorem of_one : of R 𝒜 ℬ 1 = 1 := rfl @[simp] theorem of_symm_one : (of R 𝒜 ℬ).symm 1 = 1 := rfl -- for dsimp @[simp, nolint simpNF] theorem of_symm_of (x : A ⊗[R] B) : (of R 𝒜 ℬ).symm (of R 𝒜 ℬ x) = x := rfl -- for dsimp @[simp, nolint simpNF] theorem symm_of_of (x : 𝒜 ᵍ⊗[R] ℬ) : of R 𝒜 ℬ ((of R 𝒜 ℬ).symm x) = x := rfl @[ext] theorem hom_ext {M} [AddCommMonoid M] [Module R M] ⦃f g : 𝒜 ᵍ⊗[R] ℬ →ₗ[R] M⦄ (h : f ∘ₗ of R 𝒜 ℬ = (g ∘ₗ of R 𝒜 ℬ : A ⊗[R] B →ₗ[R] M)) : f = g := h variable (R) {𝒜 ℬ} in abbrev tmul (a : A) (b : B) : 𝒜 ᵍ⊗[R] ℬ := of R 𝒜 ℬ (a ⊗ₜ b) @[inherit_doc] notation:100 x " ᵍ⊗ₜ" y:100 => tmul _ x y @[inherit_doc] notation:100 x " ᵍ⊗ₜ[" R "] " y:100 => tmul R x y variable (R) in noncomputable def auxEquiv : (𝒜 ᵍ⊗[R] ℬ) ≃ₗ[R] (⨁ i, 𝒜 i) ⊗[R] (⨁ i, ℬ i) := let fA := (decomposeAlgEquiv 𝒜).toLinearEquiv let fB := (decomposeAlgEquiv ℬ).toLinearEquiv (of R 𝒜 ℬ).symm.trans (TensorProduct.congr fA fB) theorem auxEquiv_tmul (a : A) (b : B) : auxEquiv R 𝒜 ℬ (a ᵍ⊗ₜ b) = decompose 𝒜 a ⊗ₜ decompose ℬ b := rfl theorem auxEquiv_one : auxEquiv R 𝒜 ℬ 1 = 1 := by rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one, DirectSum.decompose_one, Algebra.TensorProduct.one_def] theorem auxEquiv_symm_one : (auxEquiv R 𝒜 ℬ).symm 1 = 1 := (LinearEquiv.symm_apply_eq _).mpr (auxEquiv_one _ _).symm noncomputable def mulHom : (𝒜 ᵍ⊗[R] ℬ) →ₗ[R] (𝒜 ᵍ⊗[R] ℬ) →ₗ[R] (𝒜 ᵍ⊗[R] ℬ) := by letI fAB1 := auxEquiv R 𝒜 ℬ have := ((gradedMul R (𝒜 ·) (ℬ ·)).compl₁₂ fAB1.toLinearMap fAB1.toLinearMap).compr₂ fAB1.symm.toLinearMap exact this theorem mulHom_apply (x y : 𝒜 ᵍ⊗[R] ℬ) : mulHom 𝒜 ℬ x y = (auxEquiv R 𝒜 ℬ).symm (gradedMul R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) (auxEquiv R 𝒜 ℬ y)) := rfl instance : Mul (𝒜 ᵍ⊗[R] ℬ) where mul x y := mulHom 𝒜 ℬ x y theorem mul_def (x y : 𝒜 ᵍ⊗[R] ℬ) : x y = mulHom 𝒜 ℬ x y := rfl -- Before #8386 this was `@[simp]` but it times out when we try to apply it. theorem auxEquiv_mul (x y : 𝒜 ᵍ⊗[R] ℬ) : auxEquiv R 𝒜 ℬ (x * y) = gradedMul R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) (auxEquiv R 𝒜 ℬ y) := LinearEquiv.eq_symm_apply _ \|>.mp rfl instance instMonoid : Monoid (𝒜 ᵍ⊗[R] ℬ) where mul_one x := by rw [mul_def, mulHom_apply, auxEquiv_one, gradedMul_one, LinearEquiv.symm_apply_apply] one_mul x := by rw [mul_def, mulHom_apply, auxEquiv_one, one_gradedMul, LinearEquiv.symm_apply_apply] mul_assoc x y z := by simp_rw [mul_def, mulHom_apply, LinearEquiv.apply_symm_apply] rw [gradedMul_assoc] instance instRing : Ring (𝒜 ᵍ⊗[R] ℬ) where __ := instAddCommGroupWithOne 𝒜 ℬ __ := instMonoid 𝒜 ℬ right_distrib x y z := by simp_rw [mul_def, LinearMap.map_add₂] left_distrib x y z := by simp_rw [mul_def, map_add] mul_zero x := by simp_rw [mul_def, map_zero] zero_mul x := by simp_rw [mul_def, LinearMap.map_zero₂] theorem tmul_coe_mul_coe_tmul {j₁ i₂ : ι} (a₁ : A) (b₁ : ℬ j₁) (a₂ : 𝒜 i₂) (b₂ : B) : (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = (-1 : ℤˣ)^(j₁ * i₂) • ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) := by dsimp only [mul_def, mulHom_apply, of_symm_of] dsimp [auxEquiv, tmul] erw [decompose_coe, decompose_coe] simp_rw [← lof_eq_of R] rw [tmul_of_gradedMul_of_tmul] simp_rw [lof_eq_of R] rw [LinearEquiv.symm_symm] -- Note: #8386 had to specialize `map_smul` to `LinearEquiv.map_smul` rw [@Units.smul_def _ _ (_) (_), zsmul_eq_smul_cast R, LinearEquiv.map_smul, map_smul, ← zsmul_eq_smul_cast R, ← @Units.smul_def _ _ (_) (_)] rw [congr_symm_tmul] dsimp simp_rw [decompose_symm_mul, decompose_symm_of, Equiv.symm_apply_apply] theorem tmul_zero_coe_mul_coe_tmul {i₂ : ι} (a₁ : A) (b₁ : ℬ 0) (a₂ : 𝒜 i₂) (b₂ : B) : (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) := by rw [tmul_coe_mul_coe_tmul, zero_mul, uzpow_zero, one_smul]	Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean	207	210	theorem tmul_coe_mul_zero_coe_tmul {j₁ : ι} (a₁ : A) (b₁ : ℬ j₁) (a₂ : 𝒜 0) (b₂ : B) : (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) := by	rw [tmul_coe_mul_coe_tmul, mul_zero, uzpow_zero, one_smul]
import Mathlib.Algebra.Group.Defs #align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422" assert_not_exists MonoidWithZero assert_not_exists DenselyOrdered universe u variable {α : Type u} class Invertible [Mul α] [One α] (a : α) : Type u where invOf : α invOf_mul_self : invOf * a = 1 mul_invOf_self : a * invOf = 1 #align invertible Invertible prefix:max "⅟" =>-- This notation has the same precedence as `Inv.inv`. Invertible.invOf @[simp] theorem invOf_mul_self' [Mul α] [One α] (a : α) {_ : Invertible a} : ⅟ a * a = 1 := Invertible.invOf_mul_self theorem invOf_mul_self [Mul α] [One α] (a : α) [Invertible a] : ⅟ a * a = 1 := Invertible.invOf_mul_self #align inv_of_mul_self invOf_mul_self @[simp] theorem mul_invOf_self' [Mul α] [One α] (a : α) {_ : Invertible a} : a * ⅟ a = 1 := Invertible.mul_invOf_self theorem mul_invOf_self [Mul α] [One α] (a : α) [Invertible a] : a * ⅟ a = 1 := Invertible.mul_invOf_self #align mul_inv_of_self mul_invOf_self @[simp] theorem invOf_mul_self_assoc' [Monoid α] (a b : α) {_ : Invertible a} : ⅟ a * (a * b) = b := by rw [← mul_assoc, invOf_mul_self, one_mul] theorem invOf_mul_self_assoc [Monoid α] (a b : α) [Invertible a] : ⅟ a * (a * b) = b := by rw [← mul_assoc, invOf_mul_self, one_mul] #align inv_of_mul_self_assoc invOf_mul_self_assoc @[simp] theorem mul_invOf_self_assoc' [Monoid α] (a b : α) {_ : Invertible a} : a * (⅟ a * b) = b := by rw [← mul_assoc, mul_invOf_self, one_mul] theorem mul_invOf_self_assoc [Monoid α] (a b : α) [Invertible a] : a * (⅟ a * b) = b := by rw [← mul_assoc, mul_invOf_self, one_mul] #align mul_inv_of_self_assoc mul_invOf_self_assoc @[simp] theorem mul_invOf_mul_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * ⅟ b * b = a := by simp [mul_assoc] theorem mul_invOf_mul_self_cancel [Monoid α] (a b : α) [Invertible b] : a * ⅟ b * b = a := by simp [mul_assoc] #align mul_inv_of_mul_self_cancel mul_invOf_mul_self_cancel @[simp] theorem mul_mul_invOf_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * b * ⅟ b = a := by simp [mul_assoc] theorem mul_mul_invOf_self_cancel [Monoid α] (a b : α) [Invertible b] : a * b * ⅟ b = a := by simp [mul_assoc] #align mul_mul_inv_of_self_cancel mul_mul_invOf_self_cancel theorem invOf_eq_right_inv [Monoid α] {a b : α} [Invertible a] (hac : a * b = 1) : ⅟ a = b := left_inv_eq_right_inv (invOf_mul_self _) hac #align inv_of_eq_right_inv invOf_eq_right_inv theorem invOf_eq_left_inv [Monoid α] {a b : α} [Invertible a] (hac : b * a = 1) : ⅟ a = b := (left_inv_eq_right_inv hac (mul_invOf_self _)).symm #align inv_of_eq_left_inv invOf_eq_left_inv theorem invertible_unique {α : Type u} [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) : ⅟ a = ⅟ b := by apply invOf_eq_right_inv rw [h, mul_invOf_self] #align invertible_unique invertible_unique instance Invertible.subsingleton [Monoid α] (a : α) : Subsingleton (Invertible a) := ⟨fun ⟨b, hba, hab⟩ ⟨c, _, hac⟩ => by congr exact left_inv_eq_right_inv hba hac⟩ #align invertible.subsingleton Invertible.subsingleton @[congr] theorem Invertible.congr [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) : ⅟a = ⅟b := by subst h; congr; apply Subsingleton.allEq def Invertible.copy' [MulOneClass α] {r : α} (hr : Invertible r) (s : α) (si : α) (hs : s = r) (hsi : si = ⅟ r) : Invertible s where invOf := si invOf_mul_self := by rw [hs, hsi, invOf_mul_self] mul_invOf_self := by rw [hs, hsi, mul_invOf_self] #align invertible.copy' Invertible.copy' abbrev Invertible.copy [MulOneClass α] {r : α} (hr : Invertible r) (s : α) (hs : s = r) : Invertible s := hr.copy' _ _ hs rfl #align invertible.copy Invertible.copy def invertibleOfGroup [Group α] (a : α) : Invertible a := ⟨a⁻¹, inv_mul_self a, mul_inv_self a⟩ #align invertible_of_group invertibleOfGroup @[simp] theorem invOf_eq_group_inv [Group α] (a : α) [Invertible a] : ⅟ a = a⁻¹ := invOf_eq_right_inv (mul_inv_self a) #align inv_of_eq_group_inv invOf_eq_group_inv def invertibleOne [Monoid α] : Invertible (1 : α) := ⟨1, mul_one _, one_mul _⟩ #align invertible_one invertibleOne @[simp] theorem invOf_one' [Monoid α] {_ : Invertible (1 : α)} : ⅟ (1 : α) = 1 := invOf_eq_right_inv (mul_one _) theorem invOf_one [Monoid α] [Invertible (1 : α)] : ⅟ (1 : α) = 1 := invOf_eq_right_inv (mul_one _) #align inv_of_one invOf_one instance invertibleInvOf [One α] [Mul α] {a : α} [Invertible a] : Invertible (⅟ a) := ⟨a, mul_invOf_self a, invOf_mul_self a⟩ #align invertible_inv_of invertibleInvOf @[simp] theorem invOf_invOf [Monoid α] (a : α) [Invertible a] [Invertible (⅟ a)] : ⅟ (⅟ a) = a := invOf_eq_right_inv (invOf_mul_self _) #align inv_of_inv_of invOf_invOf @[simp] theorem invOf_inj [Monoid α] {a b : α} [Invertible a] [Invertible b] : ⅟ a = ⅟ b ↔ a = b := ⟨invertible_unique _ _, invertible_unique _ _⟩ #align inv_of_inj invOf_inj def invertibleMul [Monoid α] (a b : α) [Invertible a] [Invertible b] : Invertible (a * b) := ⟨⅟ b * ⅟ a, by simp [← mul_assoc], by simp [← mul_assoc]⟩ #align invertible_mul invertibleMul @[simp] theorem invOf_mul [Monoid α] (a b : α) [Invertible a] [Invertible b] [Invertible (a * b)] : ⅟ (a * b) = ⅟ b * ⅟ a := invOf_eq_right_inv (by simp [← mul_assoc]) #align inv_of_mul invOf_mul abbrev Invertible.mul [Monoid α] {a b : α} (_ : Invertible a) (_ : Invertible b) : Invertible (a * b) := invertibleMul _ _ #align invertible.mul Invertible.mul section variable [Monoid α] {a b c : α} [Invertible c] variable (c) in theorem mul_right_inj_of_invertible : a * c = b * c ↔ a = b := ⟨fun h => by simpa using congr_arg (· * ⅟c) h, congr_arg (· * _)⟩ variable (c) in theorem mul_left_inj_of_invertible : c * a = c * b ↔ a = b := ⟨fun h => by simpa using congr_arg (⅟c * ·) h, congr_arg (_ * ·)⟩ theorem invOf_mul_eq_iff_eq_mul_left : ⅟c * a = b ↔ a = c * b := by rw [← mul_left_inj_of_invertible (c := c), mul_invOf_self_assoc] theorem mul_left_eq_iff_eq_invOf_mul : c * a = b ↔ a = ⅟c * b := by rw [← mul_left_inj_of_invertible (c := ⅟c), invOf_mul_self_assoc] theorem mul_invOf_eq_iff_eq_mul_right : a * ⅟c = b ↔ a = b * c := by rw [← mul_right_inj_of_invertible (c := c), mul_invOf_mul_self_cancel]	Mathlib/Algebra/Group/Invertible/Defs.lean	262	263	theorem mul_right_eq_iff_eq_mul_invOf : a * c = b ↔ a = b * ⅟c := by	rw [← mul_right_inj_of_invertible (c := ⅟c), mul_mul_invOf_self_cancel]
import Mathlib.Init.ZeroOne import Mathlib.Data.Set.Defs import Mathlib.Order.Basic import Mathlib.Order.SymmDiff import Mathlib.Tactic.Tauto import Mathlib.Tactic.ByContra import Mathlib.Util.Delaborators #align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29" open Function universe u v w x namespace Set variable {α : Type u} {s t : Set α} instance instBooleanAlgebraSet : BooleanAlgebra (Set α) := { (inferInstance : BooleanAlgebra (α → Prop)) with sup := (· ∪ ·), le := (· ≤ ·), lt := fun s t => s ⊆ t ∧ ¬t ⊆ s, inf := (· ∩ ·), bot := ∅, compl := (·ᶜ), top := univ, sdiff := (· \ ·) } instance : HasSSubset (Set α) := ⟨(· < ·)⟩ @[simp] theorem top_eq_univ : (⊤ : Set α) = univ := rfl #align set.top_eq_univ Set.top_eq_univ @[simp] theorem bot_eq_empty : (⊥ : Set α) = ∅ := rfl #align set.bot_eq_empty Set.bot_eq_empty @[simp] theorem sup_eq_union : ((· ⊔ ·) : Set α → Set α → Set α) = (· ∪ ·) := rfl #align set.sup_eq_union Set.sup_eq_union @[simp] theorem inf_eq_inter : ((· ⊓ ·) : Set α → Set α → Set α) = (· ∩ ·) := rfl #align set.inf_eq_inter Set.inf_eq_inter @[simp] theorem le_eq_subset : ((· ≤ ·) : Set α → Set α → Prop) = (· ⊆ ·) := rfl #align set.le_eq_subset Set.le_eq_subset @[simp] theorem lt_eq_ssubset : ((· < ·) : Set α → Set α → Prop) = (· ⊂ ·) := rfl #align set.lt_eq_ssubset Set.lt_eq_ssubset theorem le_iff_subset : s ≤ t ↔ s ⊆ t := Iff.rfl #align set.le_iff_subset Set.le_iff_subset theorem lt_iff_ssubset : s < t ↔ s ⊂ t := Iff.rfl #align set.lt_iff_ssubset Set.lt_iff_ssubset alias ⟨_root_.LE.le.subset, _root_.HasSubset.Subset.le⟩ := le_iff_subset #align has_subset.subset.le HasSubset.Subset.le alias ⟨_root_.LT.lt.ssubset, _root_.HasSSubset.SSubset.lt⟩ := lt_iff_ssubset #align has_ssubset.ssubset.lt HasSSubset.SSubset.lt instance PiSetCoe.canLift (ι : Type u) (α : ι → Type v) [∀ i, Nonempty (α i)] (s : Set ι) : CanLift (∀ i : s, α i) (∀ i, α i) (fun f i => f i) fun _ => True := PiSubtype.canLift ι α s #align set.pi_set_coe.can_lift Set.PiSetCoe.canLift instance PiSetCoe.canLift' (ι : Type u) (α : Type v) [Nonempty α] (s : Set ι) : CanLift (s → α) (ι → α) (fun f i => f i) fun _ => True := PiSetCoe.canLift ι (fun _ => α) s #align set.pi_set_coe.can_lift' Set.PiSetCoe.canLift' end Set theorem Subtype.mem {α : Type} {s : Set α} (p : s) : (p : α) ∈ s := p.prop #align subtype.mem Subtype.mem theorem Eq.subset {α} {s t : Set α} : s = t → s ⊆ t := fun h₁ _ h₂ => by rw [← h₁]; exact h₂ #align eq.subset Eq.subset namespace Set variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x} {a b : α} {s s₁ s₂ t t₁ t₂ u : Set α} instance : Inhabited (Set α) := ⟨∅⟩ theorem ext_iff {s t : Set α} : s = t ↔ ∀ x, x ∈ s ↔ x ∈ t := ⟨fun h x => by rw [h], ext⟩ #align set.ext_iff Set.ext_iff @[trans] theorem mem_of_mem_of_subset {x : α} {s t : Set α} (hx : x ∈ s) (h : s ⊆ t) : x ∈ t := h hx #align set.mem_of_mem_of_subset Set.mem_of_mem_of_subset theorem forall_in_swap {p : α → β → Prop} : (∀ a ∈ s, ∀ (b), p a b) ↔ ∀ (b), ∀ a ∈ s, p a b := by tauto #align set.forall_in_swap Set.forall_in_swap theorem mem_setOf {a : α} {p : α → Prop} : a ∈ { x \| p x } ↔ p a := Iff.rfl #align set.mem_set_of Set.mem_setOf theorem _root_.Membership.mem.out {p : α → Prop} {a : α} (h : a ∈ { x \| p x }) : p a := h #align has_mem.mem.out Membership.mem.out theorem nmem_setOf_iff {a : α} {p : α → Prop} : a ∉ { x \| p x } ↔ ¬p a := Iff.rfl #align set.nmem_set_of_iff Set.nmem_setOf_iff @[simp] theorem setOf_mem_eq {s : Set α} : { x \| x ∈ s } = s := rfl #align set.set_of_mem_eq Set.setOf_mem_eq theorem setOf_set {s : Set α} : setOf s = s := rfl #align set.set_of_set Set.setOf_set theorem setOf_app_iff {p : α → Prop} {x : α} : { x \| p x } x ↔ p x := Iff.rfl #align set.set_of_app_iff Set.setOf_app_iff theorem mem_def {a : α} {s : Set α} : a ∈ s ↔ s a := Iff.rfl #align set.mem_def Set.mem_def theorem setOf_bijective : Bijective (setOf : (α → Prop) → Set α) := bijective_id #align set.set_of_bijective Set.setOf_bijective theorem subset_setOf {p : α → Prop} {s : Set α} : s ⊆ setOf p ↔ ∀ x, x ∈ s → p x := Iff.rfl theorem setOf_subset {p : α → Prop} {s : Set α} : setOf p ⊆ s ↔ ∀ x, p x → x ∈ s := Iff.rfl @[simp] theorem setOf_subset_setOf {p q : α → Prop} : { a \| p a } ⊆ { a \| q a } ↔ ∀ a, p a → q a := Iff.rfl #align set.set_of_subset_set_of Set.setOf_subset_setOf theorem setOf_and {p q : α → Prop} : { a \| p a ∧ q a } = { a \| p a } ∩ { a \| q a } := rfl #align set.set_of_and Set.setOf_and theorem setOf_or {p q : α → Prop} : { a \| p a ∨ q a } = { a \| p a } ∪ { a \| q a } := rfl #align set.set_of_or Set.setOf_or instance : IsRefl (Set α) (· ⊆ ·) := show IsRefl (Set α) (· ≤ ·) by infer_instance instance : IsTrans (Set α) (· ⊆ ·) := show IsTrans (Set α) (· ≤ ·) by infer_instance instance : Trans ((· ⊆ ·) : Set α → Set α → Prop) (· ⊆ ·) (· ⊆ ·) := show Trans (· ≤ ·) (· ≤ ·) (· ≤ ·) by infer_instance instance : IsAntisymm (Set α) (· ⊆ ·) := show IsAntisymm (Set α) (· ≤ ·) by infer_instance instance : IsIrrefl (Set α) (· ⊂ ·) := show IsIrrefl (Set α) (· < ·) by infer_instance instance : IsTrans (Set α) (· ⊂ ·) := show IsTrans (Set α) (· < ·) by infer_instance instance : Trans ((· ⊂ ·) : Set α → Set α → Prop) (· ⊂ ·) (· ⊂ ·) := show Trans (· < ·) (· < ·) (· < ·) by infer_instance instance : Trans ((· ⊂ ·) : Set α → Set α → Prop) (· ⊆ ·) (· ⊂ ·) := show Trans (· < ·) (· ≤ ·) (· < ·) by infer_instance instance : Trans ((· ⊆ ·) : Set α → Set α → Prop) (· ⊂ ·) (· ⊂ ·) := show Trans (· ≤ ·) (· < ·) (· < ·) by infer_instance instance : IsAsymm (Set α) (· ⊂ ·) := show IsAsymm (Set α) (· < ·) by infer_instance instance : IsNonstrictStrictOrder (Set α) (· ⊆ ·) (· ⊂ ·) := ⟨fun _ _ => Iff.rfl⟩ -- TODO(Jeremy): write a tactic to unfold specific instances of generic notation? theorem subset_def : (s ⊆ t) = ∀ x, x ∈ s → x ∈ t := rfl #align set.subset_def Set.subset_def theorem ssubset_def : (s ⊂ t) = (s ⊆ t ∧ ¬t ⊆ s) := rfl #align set.ssubset_def Set.ssubset_def @[refl] theorem Subset.refl (a : Set α) : a ⊆ a := fun _ => id #align set.subset.refl Set.Subset.refl theorem Subset.rfl {s : Set α} : s ⊆ s := Subset.refl s #align set.subset.rfl Set.Subset.rfl @[trans] theorem Subset.trans {a b c : Set α} (ab : a ⊆ b) (bc : b ⊆ c) : a ⊆ c := fun _ h => bc <\| ab h #align set.subset.trans Set.Subset.trans @[trans] theorem mem_of_eq_of_mem {x y : α} {s : Set α} (hx : x = y) (h : y ∈ s) : x ∈ s := hx.symm ▸ h #align set.mem_of_eq_of_mem Set.mem_of_eq_of_mem theorem Subset.antisymm {a b : Set α} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b := Set.ext fun _ => ⟨@h₁ _, @h₂ _⟩ #align set.subset.antisymm Set.Subset.antisymm theorem Subset.antisymm_iff {a b : Set α} : a = b ↔ a ⊆ b ∧ b ⊆ a := ⟨fun e => ⟨e.subset, e.symm.subset⟩, fun ⟨h₁, h₂⟩ => Subset.antisymm h₁ h₂⟩ #align set.subset.antisymm_iff Set.Subset.antisymm_iff -- an alternative name theorem eq_of_subset_of_subset {a b : Set α} : a ⊆ b → b ⊆ a → a = b := Subset.antisymm #align set.eq_of_subset_of_subset Set.eq_of_subset_of_subset theorem mem_of_subset_of_mem {s₁ s₂ : Set α} {a : α} (h : s₁ ⊆ s₂) : a ∈ s₁ → a ∈ s₂ := @h _ #align set.mem_of_subset_of_mem Set.mem_of_subset_of_mem theorem not_mem_subset (h : s ⊆ t) : a ∉ t → a ∉ s := mt <\| mem_of_subset_of_mem h #align set.not_mem_subset Set.not_mem_subset theorem not_subset : ¬s ⊆ t ↔ ∃ a ∈ s, a ∉ t := by simp only [subset_def, not_forall, exists_prop] #align set.not_subset Set.not_subset lemma eq_of_forall_subset_iff (h : ∀ u, s ⊆ u ↔ t ⊆ u) : s = t := eq_of_forall_ge_iff h protected theorem eq_or_ssubset_of_subset (h : s ⊆ t) : s = t ∨ s ⊂ t := eq_or_lt_of_le h #align set.eq_or_ssubset_of_subset Set.eq_or_ssubset_of_subset theorem exists_of_ssubset {s t : Set α} (h : s ⊂ t) : ∃ x ∈ t, x ∉ s := not_subset.1 h.2 #align set.exists_of_ssubset Set.exists_of_ssubset protected theorem ssubset_iff_subset_ne {s t : Set α} : s ⊂ t ↔ s ⊆ t ∧ s ≠ t := @lt_iff_le_and_ne (Set α) _ s t #align set.ssubset_iff_subset_ne Set.ssubset_iff_subset_ne theorem ssubset_iff_of_subset {s t : Set α} (h : s ⊆ t) : s ⊂ t ↔ ∃ x ∈ t, x ∉ s := ⟨exists_of_ssubset, fun ⟨_, hxt, hxs⟩ => ⟨h, fun h => hxs <\| h hxt⟩⟩ #align set.ssubset_iff_of_subset Set.ssubset_iff_of_subset protected theorem ssubset_of_ssubset_of_subset {s₁ s₂ s₃ : Set α} (hs₁s₂ : s₁ ⊂ s₂) (hs₂s₃ : s₂ ⊆ s₃) : s₁ ⊂ s₃ := ⟨Subset.trans hs₁s₂.1 hs₂s₃, fun hs₃s₁ => hs₁s₂.2 (Subset.trans hs₂s₃ hs₃s₁)⟩ #align set.ssubset_of_ssubset_of_subset Set.ssubset_of_ssubset_of_subset protected theorem ssubset_of_subset_of_ssubset {s₁ s₂ s₃ : Set α} (hs₁s₂ : s₁ ⊆ s₂) (hs₂s₃ : s₂ ⊂ s₃) : s₁ ⊂ s₃ := ⟨Subset.trans hs₁s₂ hs₂s₃.1, fun hs₃s₁ => hs₂s₃.2 (Subset.trans hs₃s₁ hs₁s₂)⟩ #align set.ssubset_of_subset_of_ssubset Set.ssubset_of_subset_of_ssubset theorem not_mem_empty (x : α) : ¬x ∈ (∅ : Set α) := id #align set.not_mem_empty Set.not_mem_empty -- Porting note (#10618): removed `simp` because `simp` can prove it theorem not_not_mem : ¬a ∉ s ↔ a ∈ s := not_not #align set.not_not_mem Set.not_not_mem -- Porting note: we seem to need parentheses at `(↥s)`, -- even if we increase the right precedence of `↥` in `Mathlib.Tactic.Coe`. -- Porting note: removed `simp` as it is competing with `nonempty_subtype`. -- @[simp] theorem nonempty_coe_sort {s : Set α} : Nonempty (↥s) ↔ s.Nonempty := nonempty_subtype #align set.nonempty_coe_sort Set.nonempty_coe_sort alias ⟨_, Nonempty.coe_sort⟩ := nonempty_coe_sort #align set.nonempty.coe_sort Set.Nonempty.coe_sort theorem nonempty_def : s.Nonempty ↔ ∃ x, x ∈ s := Iff.rfl #align set.nonempty_def Set.nonempty_def theorem nonempty_of_mem {x} (h : x ∈ s) : s.Nonempty := ⟨x, h⟩ #align set.nonempty_of_mem Set.nonempty_of_mem theorem Nonempty.not_subset_empty : s.Nonempty → ¬s ⊆ ∅ \| ⟨_, hx⟩, hs => hs hx #align set.nonempty.not_subset_empty Set.Nonempty.not_subset_empty protected noncomputable def Nonempty.some (h : s.Nonempty) : α := Classical.choose h #align set.nonempty.some Set.Nonempty.some protected theorem Nonempty.some_mem (h : s.Nonempty) : h.some ∈ s := Classical.choose_spec h #align set.nonempty.some_mem Set.Nonempty.some_mem theorem Nonempty.mono (ht : s ⊆ t) (hs : s.Nonempty) : t.Nonempty := hs.imp ht #align set.nonempty.mono Set.Nonempty.mono theorem nonempty_of_not_subset (h : ¬s ⊆ t) : (s \ t).Nonempty := let ⟨x, xs, xt⟩ := not_subset.1 h ⟨x, xs, xt⟩ #align set.nonempty_of_not_subset Set.nonempty_of_not_subset theorem nonempty_of_ssubset (ht : s ⊂ t) : (t \ s).Nonempty := nonempty_of_not_subset ht.2 #align set.nonempty_of_ssubset Set.nonempty_of_ssubset theorem Nonempty.of_diff (h : (s \ t).Nonempty) : s.Nonempty := h.imp fun _ => And.left #align set.nonempty.of_diff Set.Nonempty.of_diff theorem nonempty_of_ssubset' (ht : s ⊂ t) : t.Nonempty := (nonempty_of_ssubset ht).of_diff #align set.nonempty_of_ssubset' Set.nonempty_of_ssubset' theorem Nonempty.inl (hs : s.Nonempty) : (s ∪ t).Nonempty := hs.imp fun _ => Or.inl #align set.nonempty.inl Set.Nonempty.inl theorem Nonempty.inr (ht : t.Nonempty) : (s ∪ t).Nonempty := ht.imp fun _ => Or.inr #align set.nonempty.inr Set.Nonempty.inr @[simp] theorem union_nonempty : (s ∪ t).Nonempty ↔ s.Nonempty ∨ t.Nonempty := exists_or #align set.union_nonempty Set.union_nonempty theorem Nonempty.left (h : (s ∩ t).Nonempty) : s.Nonempty := h.imp fun _ => And.left #align set.nonempty.left Set.Nonempty.left theorem Nonempty.right (h : (s ∩ t).Nonempty) : t.Nonempty := h.imp fun _ => And.right #align set.nonempty.right Set.Nonempty.right theorem inter_nonempty : (s ∩ t).Nonempty ↔ ∃ x, x ∈ s ∧ x ∈ t := Iff.rfl #align set.inter_nonempty Set.inter_nonempty theorem inter_nonempty_iff_exists_left : (s ∩ t).Nonempty ↔ ∃ x ∈ s, x ∈ t := by simp_rw [inter_nonempty] #align set.inter_nonempty_iff_exists_left Set.inter_nonempty_iff_exists_left theorem inter_nonempty_iff_exists_right : (s ∩ t).Nonempty ↔ ∃ x ∈ t, x ∈ s := by simp_rw [inter_nonempty, and_comm] #align set.inter_nonempty_iff_exists_right Set.inter_nonempty_iff_exists_right theorem nonempty_iff_univ_nonempty : Nonempty α ↔ (univ : Set α).Nonempty := ⟨fun ⟨x⟩ => ⟨x, trivial⟩, fun ⟨x, _⟩ => ⟨x⟩⟩ #align set.nonempty_iff_univ_nonempty Set.nonempty_iff_univ_nonempty @[simp] theorem univ_nonempty : ∀ [Nonempty α], (univ : Set α).Nonempty \| ⟨x⟩ => ⟨x, trivial⟩ #align set.univ_nonempty Set.univ_nonempty theorem Nonempty.to_subtype : s.Nonempty → Nonempty (↥s) := nonempty_subtype.2 #align set.nonempty.to_subtype Set.Nonempty.to_subtype theorem Nonempty.to_type : s.Nonempty → Nonempty α := fun ⟨x, _⟩ => ⟨x⟩ #align set.nonempty.to_type Set.Nonempty.to_type instance univ.nonempty [Nonempty α] : Nonempty (↥(Set.univ : Set α)) := Set.univ_nonempty.to_subtype #align set.univ.nonempty Set.univ.nonempty theorem nonempty_of_nonempty_subtype [Nonempty (↥s)] : s.Nonempty := nonempty_subtype.mp ‹_› #align set.nonempty_of_nonempty_subtype Set.nonempty_of_nonempty_subtype theorem empty_def : (∅ : Set α) = { _x : α \| False } := rfl #align set.empty_def Set.empty_def @[simp] theorem mem_empty_iff_false (x : α) : x ∈ (∅ : Set α) ↔ False := Iff.rfl #align set.mem_empty_iff_false Set.mem_empty_iff_false @[simp] theorem setOf_false : { _a : α \| False } = ∅ := rfl #align set.set_of_false Set.setOf_false @[simp] theorem setOf_bot : { _x : α \| ⊥ } = ∅ := rfl @[simp] theorem empty_subset (s : Set α) : ∅ ⊆ s := nofun #align set.empty_subset Set.empty_subset theorem subset_empty_iff {s : Set α} : s ⊆ ∅ ↔ s = ∅ := (Subset.antisymm_iff.trans <\| and_iff_left (empty_subset _)).symm #align set.subset_empty_iff Set.subset_empty_iff theorem eq_empty_iff_forall_not_mem {s : Set α} : s = ∅ ↔ ∀ x, x ∉ s := subset_empty_iff.symm #align set.eq_empty_iff_forall_not_mem Set.eq_empty_iff_forall_not_mem theorem eq_empty_of_forall_not_mem (h : ∀ x, x ∉ s) : s = ∅ := subset_empty_iff.1 h #align set.eq_empty_of_forall_not_mem Set.eq_empty_of_forall_not_mem theorem eq_empty_of_subset_empty {s : Set α} : s ⊆ ∅ → s = ∅ := subset_empty_iff.1 #align set.eq_empty_of_subset_empty Set.eq_empty_of_subset_empty theorem eq_empty_of_isEmpty [IsEmpty α] (s : Set α) : s = ∅ := eq_empty_of_subset_empty fun x _ => isEmptyElim x #align set.eq_empty_of_is_empty Set.eq_empty_of_isEmpty instance uniqueEmpty [IsEmpty α] : Unique (Set α) where default := ∅ uniq := eq_empty_of_isEmpty #align set.unique_empty Set.uniqueEmpty theorem not_nonempty_iff_eq_empty {s : Set α} : ¬s.Nonempty ↔ s = ∅ := by simp only [Set.Nonempty, not_exists, eq_empty_iff_forall_not_mem] #align set.not_nonempty_iff_eq_empty Set.not_nonempty_iff_eq_empty theorem nonempty_iff_ne_empty : s.Nonempty ↔ s ≠ ∅ := not_nonempty_iff_eq_empty.not_right #align set.nonempty_iff_ne_empty Set.nonempty_iff_ne_empty theorem not_nonempty_iff_eq_empty' : ¬Nonempty s ↔ s = ∅ := by rw [nonempty_subtype, not_exists, eq_empty_iff_forall_not_mem] theorem nonempty_iff_ne_empty' : Nonempty s ↔ s ≠ ∅ := not_nonempty_iff_eq_empty'.not_right alias ⟨Nonempty.ne_empty, _⟩ := nonempty_iff_ne_empty #align set.nonempty.ne_empty Set.Nonempty.ne_empty @[simp] theorem not_nonempty_empty : ¬(∅ : Set α).Nonempty := fun ⟨_, hx⟩ => hx #align set.not_nonempty_empty Set.not_nonempty_empty -- Porting note: removing `@[simp]` as it is competing with `isEmpty_subtype`. -- @[simp] theorem isEmpty_coe_sort {s : Set α} : IsEmpty (↥s) ↔ s = ∅ := not_iff_not.1 <\| by simpa using nonempty_iff_ne_empty #align set.is_empty_coe_sort Set.isEmpty_coe_sort theorem eq_empty_or_nonempty (s : Set α) : s = ∅ ∨ s.Nonempty := or_iff_not_imp_left.2 nonempty_iff_ne_empty.2 #align set.eq_empty_or_nonempty Set.eq_empty_or_nonempty theorem subset_eq_empty {s t : Set α} (h : t ⊆ s) (e : s = ∅) : t = ∅ := subset_empty_iff.1 <\| e ▸ h #align set.subset_eq_empty Set.subset_eq_empty theorem forall_mem_empty {p : α → Prop} : (∀ x ∈ (∅ : Set α), p x) ↔ True := iff_true_intro fun _ => False.elim #align set.ball_empty_iff Set.forall_mem_empty @[deprecated (since := "2024-03-23")] alias ball_empty_iff := forall_mem_empty instance (α : Type u) : IsEmpty.{u + 1} (↥(∅ : Set α)) := ⟨fun x => x.2⟩ @[simp] theorem empty_ssubset : ∅ ⊂ s ↔ s.Nonempty := (@bot_lt_iff_ne_bot (Set α) _ _ _).trans nonempty_iff_ne_empty.symm #align set.empty_ssubset Set.empty_ssubset alias ⟨_, Nonempty.empty_ssubset⟩ := empty_ssubset #align set.nonempty.empty_ssubset Set.Nonempty.empty_ssubset @[simp] theorem setOf_true : { _x : α \| True } = univ := rfl #align set.set_of_true Set.setOf_true @[simp] theorem setOf_top : { _x : α \| ⊤ } = univ := rfl @[simp] theorem univ_eq_empty_iff : (univ : Set α) = ∅ ↔ IsEmpty α := eq_empty_iff_forall_not_mem.trans ⟨fun H => ⟨fun x => H x trivial⟩, fun H x _ => @IsEmpty.false α H x⟩ #align set.univ_eq_empty_iff Set.univ_eq_empty_iff theorem empty_ne_univ [Nonempty α] : (∅ : Set α) ≠ univ := fun e => not_isEmpty_of_nonempty α <\| univ_eq_empty_iff.1 e.symm #align set.empty_ne_univ Set.empty_ne_univ @[simp] theorem subset_univ (s : Set α) : s ⊆ univ := fun _ _ => trivial #align set.subset_univ Set.subset_univ @[simp] theorem univ_subset_iff {s : Set α} : univ ⊆ s ↔ s = univ := @top_le_iff _ _ _ s #align set.univ_subset_iff Set.univ_subset_iff alias ⟨eq_univ_of_univ_subset, _⟩ := univ_subset_iff #align set.eq_univ_of_univ_subset Set.eq_univ_of_univ_subset theorem eq_univ_iff_forall {s : Set α} : s = univ ↔ ∀ x, x ∈ s := univ_subset_iff.symm.trans <\| forall_congr' fun _ => imp_iff_right trivial #align set.eq_univ_iff_forall Set.eq_univ_iff_forall theorem eq_univ_of_forall {s : Set α} : (∀ x, x ∈ s) → s = univ := eq_univ_iff_forall.2 #align set.eq_univ_of_forall Set.eq_univ_of_forall theorem Nonempty.eq_univ [Subsingleton α] : s.Nonempty → s = univ := by rintro ⟨x, hx⟩ exact eq_univ_of_forall fun y => by rwa [Subsingleton.elim y x] #align set.nonempty.eq_univ Set.Nonempty.eq_univ theorem eq_univ_of_subset {s t : Set α} (h : s ⊆ t) (hs : s = univ) : t = univ := eq_univ_of_univ_subset <\| (hs ▸ h : univ ⊆ t) #align set.eq_univ_of_subset Set.eq_univ_of_subset theorem exists_mem_of_nonempty (α) : ∀ [Nonempty α], ∃ x : α, x ∈ (univ : Set α) \| ⟨x⟩ => ⟨x, trivial⟩ #align set.exists_mem_of_nonempty Set.exists_mem_of_nonempty theorem ne_univ_iff_exists_not_mem {α : Type} (s : Set α) : s ≠ univ ↔ ∃ a, a ∉ s := by rw [← not_forall, ← eq_univ_iff_forall] #align set.ne_univ_iff_exists_not_mem Set.ne_univ_iff_exists_not_mem theorem not_subset_iff_exists_mem_not_mem {α : Type} {s t : Set α} : ¬s ⊆ t ↔ ∃ x, x ∈ s ∧ x ∉ t := by simp [subset_def] #align set.not_subset_iff_exists_mem_not_mem Set.not_subset_iff_exists_mem_not_mem theorem univ_unique [Unique α] : @Set.univ α = {default} := Set.ext fun x => iff_of_true trivial <\| Subsingleton.elim x default #align set.univ_unique Set.univ_unique theorem ssubset_univ_iff : s ⊂ univ ↔ s ≠ univ := lt_top_iff_ne_top #align set.ssubset_univ_iff Set.ssubset_univ_iff instance nontrivial_of_nonempty [Nonempty α] : Nontrivial (Set α) := ⟨⟨∅, univ, empty_ne_univ⟩⟩ #align set.nontrivial_of_nonempty Set.nontrivial_of_nonempty theorem union_def {s₁ s₂ : Set α} : s₁ ∪ s₂ = { a \| a ∈ s₁ ∨ a ∈ s₂ } := rfl #align set.union_def Set.union_def theorem mem_union_left {x : α} {a : Set α} (b : Set α) : x ∈ a → x ∈ a ∪ b := Or.inl #align set.mem_union_left Set.mem_union_left theorem mem_union_right {x : α} {b : Set α} (a : Set α) : x ∈ b → x ∈ a ∪ b := Or.inr #align set.mem_union_right Set.mem_union_right theorem mem_or_mem_of_mem_union {x : α} {a b : Set α} (H : x ∈ a ∪ b) : x ∈ a ∨ x ∈ b := H #align set.mem_or_mem_of_mem_union Set.mem_or_mem_of_mem_union theorem MemUnion.elim {x : α} {a b : Set α} {P : Prop} (H₁ : x ∈ a ∪ b) (H₂ : x ∈ a → P) (H₃ : x ∈ b → P) : P := Or.elim H₁ H₂ H₃ #align set.mem_union.elim Set.MemUnion.elim @[simp] theorem mem_union (x : α) (a b : Set α) : x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b := Iff.rfl #align set.mem_union Set.mem_union @[simp] theorem union_self (a : Set α) : a ∪ a = a := ext fun _ => or_self_iff #align set.union_self Set.union_self @[simp] theorem union_empty (a : Set α) : a ∪ ∅ = a := ext fun _ => or_false_iff _ #align set.union_empty Set.union_empty @[simp] theorem empty_union (a : Set α) : ∅ ∪ a = a := ext fun _ => false_or_iff _ #align set.empty_union Set.empty_union theorem union_comm (a b : Set α) : a ∪ b = b ∪ a := ext fun _ => or_comm #align set.union_comm Set.union_comm theorem union_assoc (a b c : Set α) : a ∪ b ∪ c = a ∪ (b ∪ c) := ext fun _ => or_assoc #align set.union_assoc Set.union_assoc instance union_isAssoc : Std.Associative (α := Set α) (· ∪ ·) := ⟨union_assoc⟩ #align set.union_is_assoc Set.union_isAssoc instance union_isComm : Std.Commutative (α := Set α) (· ∪ ·) := ⟨union_comm⟩ #align set.union_is_comm Set.union_isComm theorem union_left_comm (s₁ s₂ s₃ : Set α) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) := ext fun _ => or_left_comm #align set.union_left_comm Set.union_left_comm theorem union_right_comm (s₁ s₂ s₃ : Set α) : s₁ ∪ s₂ ∪ s₃ = s₁ ∪ s₃ ∪ s₂ := ext fun _ => or_right_comm #align set.union_right_comm Set.union_right_comm @[simp] theorem union_eq_left {s t : Set α} : s ∪ t = s ↔ t ⊆ s := sup_eq_left #align set.union_eq_left_iff_subset Set.union_eq_left @[simp] theorem union_eq_right {s t : Set α} : s ∪ t = t ↔ s ⊆ t := sup_eq_right #align set.union_eq_right_iff_subset Set.union_eq_right theorem union_eq_self_of_subset_left {s t : Set α} (h : s ⊆ t) : s ∪ t = t := union_eq_right.mpr h #align set.union_eq_self_of_subset_left Set.union_eq_self_of_subset_left theorem union_eq_self_of_subset_right {s t : Set α} (h : t ⊆ s) : s ∪ t = s := union_eq_left.mpr h #align set.union_eq_self_of_subset_right Set.union_eq_self_of_subset_right @[simp] theorem subset_union_left {s t : Set α} : s ⊆ s ∪ t := fun _ => Or.inl #align set.subset_union_left Set.subset_union_left @[simp] theorem subset_union_right {s t : Set α} : t ⊆ s ∪ t := fun _ => Or.inr #align set.subset_union_right Set.subset_union_right theorem union_subset {s t r : Set α} (sr : s ⊆ r) (tr : t ⊆ r) : s ∪ t ⊆ r := fun _ => Or.rec (@sr _) (@tr _) #align set.union_subset Set.union_subset @[simp] theorem union_subset_iff {s t u : Set α} : s ∪ t ⊆ u ↔ s ⊆ u ∧ t ⊆ u := (forall_congr' fun _ => or_imp).trans forall_and #align set.union_subset_iff Set.union_subset_iff @[gcongr] theorem union_subset_union {s₁ s₂ t₁ t₂ : Set α} (h₁ : s₁ ⊆ s₂) (h₂ : t₁ ⊆ t₂) : s₁ ∪ t₁ ⊆ s₂ ∪ t₂ := fun _ => Or.imp (@h₁ _) (@h₂ _) #align set.union_subset_union Set.union_subset_union @[gcongr] theorem union_subset_union_left {s₁ s₂ : Set α} (t) (h : s₁ ⊆ s₂) : s₁ ∪ t ⊆ s₂ ∪ t := union_subset_union h Subset.rfl #align set.union_subset_union_left Set.union_subset_union_left @[gcongr] theorem union_subset_union_right (s) {t₁ t₂ : Set α} (h : t₁ ⊆ t₂) : s ∪ t₁ ⊆ s ∪ t₂ := union_subset_union Subset.rfl h #align set.union_subset_union_right Set.union_subset_union_right theorem subset_union_of_subset_left {s t : Set α} (h : s ⊆ t) (u : Set α) : s ⊆ t ∪ u := h.trans subset_union_left #align set.subset_union_of_subset_left Set.subset_union_of_subset_left theorem subset_union_of_subset_right {s u : Set α} (h : s ⊆ u) (t : Set α) : s ⊆ t ∪ u := h.trans subset_union_right #align set.subset_union_of_subset_right Set.subset_union_of_subset_right -- Porting note: replaced `⊔` in RHS theorem union_congr_left (ht : t ⊆ s ∪ u) (hu : u ⊆ s ∪ t) : s ∪ t = s ∪ u := sup_congr_left ht hu #align set.union_congr_left Set.union_congr_left theorem union_congr_right (hs : s ⊆ t ∪ u) (ht : t ⊆ s ∪ u) : s ∪ u = t ∪ u := sup_congr_right hs ht #align set.union_congr_right Set.union_congr_right theorem union_eq_union_iff_left : s ∪ t = s ∪ u ↔ t ⊆ s ∪ u ∧ u ⊆ s ∪ t := sup_eq_sup_iff_left #align set.union_eq_union_iff_left Set.union_eq_union_iff_left theorem union_eq_union_iff_right : s ∪ u = t ∪ u ↔ s ⊆ t ∪ u ∧ t ⊆ s ∪ u := sup_eq_sup_iff_right #align set.union_eq_union_iff_right Set.union_eq_union_iff_right @[simp] theorem union_empty_iff {s t : Set α} : s ∪ t = ∅ ↔ s = ∅ ∧ t = ∅ := by simp only [← subset_empty_iff] exact union_subset_iff #align set.union_empty_iff Set.union_empty_iff @[simp] theorem union_univ (s : Set α) : s ∪ univ = univ := sup_top_eq _ #align set.union_univ Set.union_univ @[simp] theorem univ_union (s : Set α) : univ ∪ s = univ := top_sup_eq _ #align set.univ_union Set.univ_union theorem inter_def {s₁ s₂ : Set α} : s₁ ∩ s₂ = { a \| a ∈ s₁ ∧ a ∈ s₂ } := rfl #align set.inter_def Set.inter_def @[simp, mfld_simps] theorem mem_inter_iff (x : α) (a b : Set α) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b := Iff.rfl #align set.mem_inter_iff Set.mem_inter_iff theorem mem_inter {x : α} {a b : Set α} (ha : x ∈ a) (hb : x ∈ b) : x ∈ a ∩ b := ⟨ha, hb⟩ #align set.mem_inter Set.mem_inter theorem mem_of_mem_inter_left {x : α} {a b : Set α} (h : x ∈ a ∩ b) : x ∈ a := h.left #align set.mem_of_mem_inter_left Set.mem_of_mem_inter_left theorem mem_of_mem_inter_right {x : α} {a b : Set α} (h : x ∈ a ∩ b) : x ∈ b := h.right #align set.mem_of_mem_inter_right Set.mem_of_mem_inter_right @[simp] theorem inter_self (a : Set α) : a ∩ a = a := ext fun _ => and_self_iff #align set.inter_self Set.inter_self @[simp] theorem inter_empty (a : Set α) : a ∩ ∅ = ∅ := ext fun _ => and_false_iff _ #align set.inter_empty Set.inter_empty @[simp] theorem empty_inter (a : Set α) : ∅ ∩ a = ∅ := ext fun _ => false_and_iff _ #align set.empty_inter Set.empty_inter theorem inter_comm (a b : Set α) : a ∩ b = b ∩ a := ext fun _ => and_comm #align set.inter_comm Set.inter_comm theorem inter_assoc (a b c : Set α) : a ∩ b ∩ c = a ∩ (b ∩ c) := ext fun _ => and_assoc #align set.inter_assoc Set.inter_assoc instance inter_isAssoc : Std.Associative (α := Set α) (· ∩ ·) := ⟨inter_assoc⟩ #align set.inter_is_assoc Set.inter_isAssoc instance inter_isComm : Std.Commutative (α := Set α) (· ∩ ·) := ⟨inter_comm⟩ #align set.inter_is_comm Set.inter_isComm theorem inter_left_comm (s₁ s₂ s₃ : Set α) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) := ext fun _ => and_left_comm #align set.inter_left_comm Set.inter_left_comm theorem inter_right_comm (s₁ s₂ s₃ : Set α) : s₁ ∩ s₂ ∩ s₃ = s₁ ∩ s₃ ∩ s₂ := ext fun _ => and_right_comm #align set.inter_right_comm Set.inter_right_comm @[simp, mfld_simps] theorem inter_subset_left {s t : Set α} : s ∩ t ⊆ s := fun _ => And.left #align set.inter_subset_left Set.inter_subset_left @[simp] theorem inter_subset_right {s t : Set α} : s ∩ t ⊆ t := fun _ => And.right #align set.inter_subset_right Set.inter_subset_right theorem subset_inter {s t r : Set α} (rs : r ⊆ s) (rt : r ⊆ t) : r ⊆ s ∩ t := fun _ h => ⟨rs h, rt h⟩ #align set.subset_inter Set.subset_inter @[simp] theorem subset_inter_iff {s t r : Set α} : r ⊆ s ∩ t ↔ r ⊆ s ∧ r ⊆ t := (forall_congr' fun _ => imp_and).trans forall_and #align set.subset_inter_iff Set.subset_inter_iff @[simp] lemma inter_eq_left : s ∩ t = s ↔ s ⊆ t := inf_eq_left #align set.inter_eq_left_iff_subset Set.inter_eq_left @[simp] lemma inter_eq_right : s ∩ t = t ↔ t ⊆ s := inf_eq_right #align set.inter_eq_right_iff_subset Set.inter_eq_right @[simp] lemma left_eq_inter : s = s ∩ t ↔ s ⊆ t := left_eq_inf @[simp] lemma right_eq_inter : t = s ∩ t ↔ t ⊆ s := right_eq_inf theorem inter_eq_self_of_subset_left {s t : Set α} : s ⊆ t → s ∩ t = s := inter_eq_left.mpr #align set.inter_eq_self_of_subset_left Set.inter_eq_self_of_subset_left theorem inter_eq_self_of_subset_right {s t : Set α} : t ⊆ s → s ∩ t = t := inter_eq_right.mpr #align set.inter_eq_self_of_subset_right Set.inter_eq_self_of_subset_right theorem inter_congr_left (ht : s ∩ u ⊆ t) (hu : s ∩ t ⊆ u) : s ∩ t = s ∩ u := inf_congr_left ht hu #align set.inter_congr_left Set.inter_congr_left theorem inter_congr_right (hs : t ∩ u ⊆ s) (ht : s ∩ u ⊆ t) : s ∩ u = t ∩ u := inf_congr_right hs ht #align set.inter_congr_right Set.inter_congr_right theorem inter_eq_inter_iff_left : s ∩ t = s ∩ u ↔ s ∩ u ⊆ t ∧ s ∩ t ⊆ u := inf_eq_inf_iff_left #align set.inter_eq_inter_iff_left Set.inter_eq_inter_iff_left theorem inter_eq_inter_iff_right : s ∩ u = t ∩ u ↔ t ∩ u ⊆ s ∧ s ∩ u ⊆ t := inf_eq_inf_iff_right #align set.inter_eq_inter_iff_right Set.inter_eq_inter_iff_right @[simp, mfld_simps] theorem inter_univ (a : Set α) : a ∩ univ = a := inf_top_eq _ #align set.inter_univ Set.inter_univ @[simp, mfld_simps] theorem univ_inter (a : Set α) : univ ∩ a = a := top_inf_eq _ #align set.univ_inter Set.univ_inter @[gcongr] theorem inter_subset_inter {s₁ s₂ t₁ t₂ : Set α} (h₁ : s₁ ⊆ t₁) (h₂ : s₂ ⊆ t₂) : s₁ ∩ s₂ ⊆ t₁ ∩ t₂ := fun _ => And.imp (@h₁ _) (@h₂ _) #align set.inter_subset_inter Set.inter_subset_inter @[gcongr] theorem inter_subset_inter_left {s t : Set α} (u : Set α) (H : s ⊆ t) : s ∩ u ⊆ t ∩ u := inter_subset_inter H Subset.rfl #align set.inter_subset_inter_left Set.inter_subset_inter_left @[gcongr] theorem inter_subset_inter_right {s t : Set α} (u : Set α) (H : s ⊆ t) : u ∩ s ⊆ u ∩ t := inter_subset_inter Subset.rfl H #align set.inter_subset_inter_right Set.inter_subset_inter_right theorem union_inter_cancel_left {s t : Set α} : (s ∪ t) ∩ s = s := inter_eq_self_of_subset_right subset_union_left #align set.union_inter_cancel_left Set.union_inter_cancel_left theorem union_inter_cancel_right {s t : Set α} : (s ∪ t) ∩ t = t := inter_eq_self_of_subset_right subset_union_right #align set.union_inter_cancel_right Set.union_inter_cancel_right theorem inter_setOf_eq_sep (s : Set α) (p : α → Prop) : s ∩ {a \| p a} = {a ∈ s \| p a} := rfl #align set.inter_set_of_eq_sep Set.inter_setOf_eq_sep theorem setOf_inter_eq_sep (p : α → Prop) (s : Set α) : {a \| p a} ∩ s = {a ∈ s \| p a} := inter_comm _ _ #align set.set_of_inter_eq_sep Set.setOf_inter_eq_sep theorem inter_union_distrib_left (s t u : Set α) : s ∩ (t ∪ u) = s ∩ t ∪ s ∩ u := inf_sup_left _ _ _ #align set.inter_distrib_left Set.inter_union_distrib_left theorem union_inter_distrib_right (s t u : Set α) : (s ∪ t) ∩ u = s ∩ u ∪ t ∩ u := inf_sup_right _ _ _ #align set.inter_distrib_right Set.union_inter_distrib_right theorem union_inter_distrib_left (s t u : Set α) : s ∪ t ∩ u = (s ∪ t) ∩ (s ∪ u) := sup_inf_left _ _ _ #align set.union_distrib_left Set.union_inter_distrib_left theorem inter_union_distrib_right (s t u : Set α) : s ∩ t ∪ u = (s ∪ u) ∩ (t ∪ u) := sup_inf_right _ _ _ #align set.union_distrib_right Set.inter_union_distrib_right -- 2024-03-22 @[deprecated] alias inter_distrib_left := inter_union_distrib_left @[deprecated] alias inter_distrib_right := union_inter_distrib_right @[deprecated] alias union_distrib_left := union_inter_distrib_left @[deprecated] alias union_distrib_right := inter_union_distrib_right theorem union_union_distrib_left (s t u : Set α) : s ∪ (t ∪ u) = s ∪ t ∪ (s ∪ u) := sup_sup_distrib_left _ _ _ #align set.union_union_distrib_left Set.union_union_distrib_left theorem union_union_distrib_right (s t u : Set α) : s ∪ t ∪ u = s ∪ u ∪ (t ∪ u) := sup_sup_distrib_right _ _ _ #align set.union_union_distrib_right Set.union_union_distrib_right theorem inter_inter_distrib_left (s t u : Set α) : s ∩ (t ∩ u) = s ∩ t ∩ (s ∩ u) := inf_inf_distrib_left _ _ _ #align set.inter_inter_distrib_left Set.inter_inter_distrib_left theorem inter_inter_distrib_right (s t u : Set α) : s ∩ t ∩ u = s ∩ u ∩ (t ∩ u) := inf_inf_distrib_right _ _ _ #align set.inter_inter_distrib_right Set.inter_inter_distrib_right theorem union_union_union_comm (s t u v : Set α) : s ∪ t ∪ (u ∪ v) = s ∪ u ∪ (t ∪ v) := sup_sup_sup_comm _ _ _ _ #align set.union_union_union_comm Set.union_union_union_comm theorem inter_inter_inter_comm (s t u v : Set α) : s ∩ t ∩ (u ∩ v) = s ∩ u ∩ (t ∩ v) := inf_inf_inf_comm _ _ _ _ #align set.inter_inter_inter_comm Set.inter_inter_inter_comm theorem insert_def (x : α) (s : Set α) : insert x s = { y \| y = x ∨ y ∈ s } := rfl #align set.insert_def Set.insert_def @[simp] theorem subset_insert (x : α) (s : Set α) : s ⊆ insert x s := fun _ => Or.inr #align set.subset_insert Set.subset_insert theorem mem_insert (x : α) (s : Set α) : x ∈ insert x s := Or.inl rfl #align set.mem_insert Set.mem_insert theorem mem_insert_of_mem {x : α} {s : Set α} (y : α) : x ∈ s → x ∈ insert y s := Or.inr #align set.mem_insert_of_mem Set.mem_insert_of_mem theorem eq_or_mem_of_mem_insert {x a : α} {s : Set α} : x ∈ insert a s → x = a ∨ x ∈ s := id #align set.eq_or_mem_of_mem_insert Set.eq_or_mem_of_mem_insert theorem mem_of_mem_insert_of_ne : b ∈ insert a s → b ≠ a → b ∈ s := Or.resolve_left #align set.mem_of_mem_insert_of_ne Set.mem_of_mem_insert_of_ne theorem eq_of_not_mem_of_mem_insert : b ∈ insert a s → b ∉ s → b = a := Or.resolve_right #align set.eq_of_not_mem_of_mem_insert Set.eq_of_not_mem_of_mem_insert @[simp] theorem mem_insert_iff {x a : α} {s : Set α} : x ∈ insert a s ↔ x = a ∨ x ∈ s := Iff.rfl #align set.mem_insert_iff Set.mem_insert_iff @[simp] theorem insert_eq_of_mem {a : α} {s : Set α} (h : a ∈ s) : insert a s = s := ext fun _ => or_iff_right_of_imp fun e => e.symm ▸ h #align set.insert_eq_of_mem Set.insert_eq_of_mem theorem ne_insert_of_not_mem {s : Set α} (t : Set α) {a : α} : a ∉ s → s ≠ insert a t := mt fun e => e.symm ▸ mem_insert _ _ #align set.ne_insert_of_not_mem Set.ne_insert_of_not_mem @[simp] theorem insert_eq_self : insert a s = s ↔ a ∈ s := ⟨fun h => h ▸ mem_insert _ _, insert_eq_of_mem⟩ #align set.insert_eq_self Set.insert_eq_self theorem insert_ne_self : insert a s ≠ s ↔ a ∉ s := insert_eq_self.not #align set.insert_ne_self Set.insert_ne_self theorem insert_subset_iff : insert a s ⊆ t ↔ a ∈ t ∧ s ⊆ t := by simp only [subset_def, mem_insert_iff, or_imp, forall_and, forall_eq] #align set.insert_subset Set.insert_subset_iff theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t := insert_subset_iff.mpr ⟨ha, hs⟩ theorem insert_subset_insert (h : s ⊆ t) : insert a s ⊆ insert a t := fun _ => Or.imp_right (@h _) #align set.insert_subset_insert Set.insert_subset_insert @[simp] theorem insert_subset_insert_iff (ha : a ∉ s) : insert a s ⊆ insert a t ↔ s ⊆ t := by refine ⟨fun h x hx => ?_, insert_subset_insert⟩ rcases h (subset_insert _ _ hx) with (rfl \| hxt) exacts [(ha hx).elim, hxt] #align set.insert_subset_insert_iff Set.insert_subset_insert_iff theorem subset_insert_iff_of_not_mem (ha : a ∉ s) : s ⊆ insert a t ↔ s ⊆ t := forall₂_congr fun _ hb => or_iff_right <\| ne_of_mem_of_not_mem hb ha #align set.subset_insert_iff_of_not_mem Set.subset_insert_iff_of_not_mem theorem ssubset_iff_insert {s t : Set α} : s ⊂ t ↔ ∃ a ∉ s, insert a s ⊆ t := by simp only [insert_subset_iff, exists_and_right, ssubset_def, not_subset] aesop #align set.ssubset_iff_insert Set.ssubset_iff_insert theorem ssubset_insert {s : Set α} {a : α} (h : a ∉ s) : s ⊂ insert a s := ssubset_iff_insert.2 ⟨a, h, Subset.rfl⟩ #align set.ssubset_insert Set.ssubset_insert theorem insert_comm (a b : α) (s : Set α) : insert a (insert b s) = insert b (insert a s) := ext fun _ => or_left_comm #align set.insert_comm Set.insert_comm -- Porting note (#10618): removing `simp` attribute because `simp` can prove it theorem insert_idem (a : α) (s : Set α) : insert a (insert a s) = insert a s := insert_eq_of_mem <\| mem_insert _ _ #align set.insert_idem Set.insert_idem theorem insert_union : insert a s ∪ t = insert a (s ∪ t) := ext fun _ => or_assoc #align set.insert_union Set.insert_union @[simp] theorem union_insert : s ∪ insert a t = insert a (s ∪ t) := ext fun _ => or_left_comm #align set.union_insert Set.union_insert @[simp] theorem insert_nonempty (a : α) (s : Set α) : (insert a s).Nonempty := ⟨a, mem_insert a s⟩ #align set.insert_nonempty Set.insert_nonempty instance (a : α) (s : Set α) : Nonempty (insert a s : Set α) := (insert_nonempty a s).to_subtype theorem insert_inter_distrib (a : α) (s t : Set α) : insert a (s ∩ t) = insert a s ∩ insert a t := ext fun _ => or_and_left #align set.insert_inter_distrib Set.insert_inter_distrib theorem insert_union_distrib (a : α) (s t : Set α) : insert a (s ∪ t) = insert a s ∪ insert a t := ext fun _ => or_or_distrib_left #align set.insert_union_distrib Set.insert_union_distrib theorem insert_inj (ha : a ∉ s) : insert a s = insert b s ↔ a = b := ⟨fun h => eq_of_not_mem_of_mem_insert (h.subst <\| mem_insert a s) ha, congr_arg (fun x => insert x s)⟩ #align set.insert_inj Set.insert_inj -- useful in proofs by induction theorem forall_of_forall_insert {P : α → Prop} {a : α} {s : Set α} (H : ∀ x, x ∈ insert a s → P x) (x) (h : x ∈ s) : P x := H _ (Or.inr h) #align set.forall_of_forall_insert Set.forall_of_forall_insert theorem forall_insert_of_forall {P : α → Prop} {a : α} {s : Set α} (H : ∀ x, x ∈ s → P x) (ha : P a) (x) (h : x ∈ insert a s) : P x := h.elim (fun e => e.symm ▸ ha) (H _) #align set.forall_insert_of_forall Set.forall_insert_of_forall theorem exists_mem_insert {P : α → Prop} {a : α} {s : Set α} : (∃ x ∈ insert a s, P x) ↔ (P a ∨ ∃ x ∈ s, P x) := by simp [mem_insert_iff, or_and_right, exists_and_left, exists_or] #align set.bex_insert_iff Set.exists_mem_insert @[deprecated (since := "2024-03-23")] alias bex_insert_iff := exists_mem_insert theorem forall_mem_insert {P : α → Prop} {a : α} {s : Set α} : (∀ x ∈ insert a s, P x) ↔ P a ∧ ∀ x ∈ s, P x := forall₂_or_left.trans <\| and_congr_left' forall_eq #align set.ball_insert_iff Set.forall_mem_insert @[deprecated (since := "2024-03-23")] alias ball_insert_iff := forall_mem_insert instance : LawfulSingleton α (Set α) := ⟨fun x => Set.ext fun a => by simp only [mem_empty_iff_false, mem_insert_iff, or_false] exact Iff.rfl⟩ theorem singleton_def (a : α) : ({a} : Set α) = insert a ∅ := (insert_emptyc_eq a).symm #align set.singleton_def Set.singleton_def @[simp] theorem mem_singleton_iff {a b : α} : a ∈ ({b} : Set α) ↔ a = b := Iff.rfl #align set.mem_singleton_iff Set.mem_singleton_iff @[simp] theorem setOf_eq_eq_singleton {a : α} : { n \| n = a } = {a} := rfl #align set.set_of_eq_eq_singleton Set.setOf_eq_eq_singleton @[simp] theorem setOf_eq_eq_singleton' {a : α} : { x \| a = x } = {a} := ext fun _ => eq_comm #align set.set_of_eq_eq_singleton' Set.setOf_eq_eq_singleton' -- TODO: again, annotation needed --Porting note (#11119): removed `simp` attribute theorem mem_singleton (a : α) : a ∈ ({a} : Set α) := @rfl _ _ #align set.mem_singleton Set.mem_singleton theorem eq_of_mem_singleton {x y : α} (h : x ∈ ({y} : Set α)) : x = y := h #align set.eq_of_mem_singleton Set.eq_of_mem_singleton @[simp] theorem singleton_eq_singleton_iff {x y : α} : {x} = ({y} : Set α) ↔ x = y := ext_iff.trans eq_iff_eq_cancel_left #align set.singleton_eq_singleton_iff Set.singleton_eq_singleton_iff theorem singleton_injective : Injective (singleton : α → Set α) := fun _ _ => singleton_eq_singleton_iff.mp #align set.singleton_injective Set.singleton_injective theorem mem_singleton_of_eq {x y : α} (H : x = y) : x ∈ ({y} : Set α) := H #align set.mem_singleton_of_eq Set.mem_singleton_of_eq theorem insert_eq (x : α) (s : Set α) : insert x s = ({x} : Set α) ∪ s := rfl #align set.insert_eq Set.insert_eq @[simp] theorem singleton_nonempty (a : α) : ({a} : Set α).Nonempty := ⟨a, rfl⟩ #align set.singleton_nonempty Set.singleton_nonempty @[simp] theorem singleton_ne_empty (a : α) : ({a} : Set α) ≠ ∅ := (singleton_nonempty _).ne_empty #align set.singleton_ne_empty Set.singleton_ne_empty --Porting note (#10618): removed `simp` attribute because `simp` can prove it theorem empty_ssubset_singleton : (∅ : Set α) ⊂ {a} := (singleton_nonempty _).empty_ssubset #align set.empty_ssubset_singleton Set.empty_ssubset_singleton @[simp] theorem singleton_subset_iff {a : α} {s : Set α} : {a} ⊆ s ↔ a ∈ s := forall_eq #align set.singleton_subset_iff Set.singleton_subset_iff theorem singleton_subset_singleton : ({a} : Set α) ⊆ {b} ↔ a = b := by simp #align set.singleton_subset_singleton Set.singleton_subset_singleton theorem set_compr_eq_eq_singleton {a : α} : { b \| b = a } = {a} := rfl #align set.set_compr_eq_eq_singleton Set.set_compr_eq_eq_singleton @[simp] theorem singleton_union : {a} ∪ s = insert a s := rfl #align set.singleton_union Set.singleton_union @[simp] theorem union_singleton : s ∪ {a} = insert a s := union_comm _ _ #align set.union_singleton Set.union_singleton @[simp] theorem singleton_inter_nonempty : ({a} ∩ s).Nonempty ↔ a ∈ s := by simp only [Set.Nonempty, mem_inter_iff, mem_singleton_iff, exists_eq_left] #align set.singleton_inter_nonempty Set.singleton_inter_nonempty @[simp] theorem inter_singleton_nonempty : (s ∩ {a}).Nonempty ↔ a ∈ s := by rw [inter_comm, singleton_inter_nonempty] #align set.inter_singleton_nonempty Set.inter_singleton_nonempty @[simp] theorem singleton_inter_eq_empty : {a} ∩ s = ∅ ↔ a ∉ s := not_nonempty_iff_eq_empty.symm.trans singleton_inter_nonempty.not #align set.singleton_inter_eq_empty Set.singleton_inter_eq_empty @[simp] theorem inter_singleton_eq_empty : s ∩ {a} = ∅ ↔ a ∉ s := by rw [inter_comm, singleton_inter_eq_empty] #align set.inter_singleton_eq_empty Set.inter_singleton_eq_empty theorem nmem_singleton_empty {s : Set α} : s ∉ ({∅} : Set (Set α)) ↔ s.Nonempty := nonempty_iff_ne_empty.symm #align set.nmem_singleton_empty Set.nmem_singleton_empty instance uniqueSingleton (a : α) : Unique (↥({a} : Set α)) := ⟨⟨⟨a, mem_singleton a⟩⟩, fun ⟨_, h⟩ => Subtype.eq h⟩ #align set.unique_singleton Set.uniqueSingleton theorem eq_singleton_iff_unique_mem : s = {a} ↔ a ∈ s ∧ ∀ x ∈ s, x = a := Subset.antisymm_iff.trans <\| and_comm.trans <\| and_congr_left' singleton_subset_iff #align set.eq_singleton_iff_unique_mem Set.eq_singleton_iff_unique_mem theorem eq_singleton_iff_nonempty_unique_mem : s = {a} ↔ s.Nonempty ∧ ∀ x ∈ s, x = a := eq_singleton_iff_unique_mem.trans <\| and_congr_left fun H => ⟨fun h' => ⟨_, h'⟩, fun ⟨x, h⟩ => H x h ▸ h⟩ #align set.eq_singleton_iff_nonempty_unique_mem Set.eq_singleton_iff_nonempty_unique_mem set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532 -- while `simp` is capable of proving this, it is not capable of turning the LHS into the RHS. @[simp] theorem default_coe_singleton (x : α) : (default : ({x} : Set α)) = ⟨x, rfl⟩ := rfl #align set.default_coe_singleton Set.default_coe_singleton @[simp] theorem subset_singleton_iff {α : Type} {s : Set α} {x : α} : s ⊆ {x} ↔ ∀ y ∈ s, y = x := Iff.rfl #align set.subset_singleton_iff Set.subset_singleton_iff theorem subset_singleton_iff_eq {s : Set α} {x : α} : s ⊆ {x} ↔ s = ∅ ∨ s = {x} := by obtain rfl \| hs := s.eq_empty_or_nonempty · exact ⟨fun _ => Or.inl rfl, fun _ => empty_subset _⟩ · simp [eq_singleton_iff_nonempty_unique_mem, hs, hs.ne_empty] #align set.subset_singleton_iff_eq Set.subset_singleton_iff_eq theorem Nonempty.subset_singleton_iff (h : s.Nonempty) : s ⊆ {a} ↔ s = {a} := subset_singleton_iff_eq.trans <\| or_iff_right h.ne_empty #align set.nonempty.subset_singleton_iff Set.Nonempty.subset_singleton_iff theorem ssubset_singleton_iff {s : Set α} {x : α} : s ⊂ {x} ↔ s = ∅ := by rw [ssubset_iff_subset_ne, subset_singleton_iff_eq, or_and_right, and_not_self_iff, or_false_iff, and_iff_left_iff_imp] exact fun h => h ▸ (singleton_ne_empty _).symm #align set.ssubset_singleton_iff Set.ssubset_singleton_iff theorem eq_empty_of_ssubset_singleton {s : Set α} {x : α} (hs : s ⊂ {x}) : s = ∅ := ssubset_singleton_iff.1 hs #align set.eq_empty_of_ssubset_singleton Set.eq_empty_of_ssubset_singleton theorem eq_of_nonempty_of_subsingleton {α} [Subsingleton α] (s t : Set α) [Nonempty s] [Nonempty t] : s = t := nonempty_of_nonempty_subtype.eq_univ.trans nonempty_of_nonempty_subtype.eq_univ.symm theorem eq_of_nonempty_of_subsingleton' {α} [Subsingleton α] {s : Set α} (t : Set α) (hs : s.Nonempty) [Nonempty t] : s = t := have := hs.to_subtype; eq_of_nonempty_of_subsingleton s t set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532 theorem Nonempty.eq_zero [Subsingleton α] [Zero α] {s : Set α} (h : s.Nonempty) : s = {0} := eq_of_nonempty_of_subsingleton' {0} h set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532 theorem Nonempty.eq_one [Subsingleton α] [One α] {s : Set α} (h : s.Nonempty) : s = {1} := eq_of_nonempty_of_subsingleton' {1} h protected theorem disjoint_iff : Disjoint s t ↔ s ∩ t ⊆ ∅ := disjoint_iff_inf_le #align set.disjoint_iff Set.disjoint_iff theorem disjoint_iff_inter_eq_empty : Disjoint s t ↔ s ∩ t = ∅ := disjoint_iff #align set.disjoint_iff_inter_eq_empty Set.disjoint_iff_inter_eq_empty theorem _root_.Disjoint.inter_eq : Disjoint s t → s ∩ t = ∅ := Disjoint.eq_bot #align disjoint.inter_eq Disjoint.inter_eq theorem disjoint_left : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → a ∉ t := disjoint_iff_inf_le.trans <\| forall_congr' fun _ => not_and #align set.disjoint_left Set.disjoint_left theorem disjoint_right : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ t → a ∉ s := by rw [disjoint_comm, disjoint_left] #align set.disjoint_right Set.disjoint_right lemma not_disjoint_iff : ¬Disjoint s t ↔ ∃ x, x ∈ s ∧ x ∈ t := Set.disjoint_iff.not.trans <\| not_forall.trans <\| exists_congr fun _ ↦ not_not #align set.not_disjoint_iff Set.not_disjoint_iff lemma not_disjoint_iff_nonempty_inter : ¬ Disjoint s t ↔ (s ∩ t).Nonempty := not_disjoint_iff #align set.not_disjoint_iff_nonempty_inter Set.not_disjoint_iff_nonempty_inter alias ⟨_, Nonempty.not_disjoint⟩ := not_disjoint_iff_nonempty_inter #align set.nonempty.not_disjoint Set.Nonempty.not_disjoint lemma disjoint_or_nonempty_inter (s t : Set α) : Disjoint s t ∨ (s ∩ t).Nonempty := (em _).imp_right not_disjoint_iff_nonempty_inter.1 #align set.disjoint_or_nonempty_inter Set.disjoint_or_nonempty_inter lemma disjoint_iff_forall_ne : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ t → a ≠ b := by simp only [Ne, disjoint_left, @imp_not_comm _ (_ = _), forall_eq'] #align set.disjoint_iff_forall_ne Set.disjoint_iff_forall_ne alias ⟨_root_.Disjoint.ne_of_mem, _⟩ := disjoint_iff_forall_ne #align disjoint.ne_of_mem Disjoint.ne_of_mem lemma disjoint_of_subset_left (h : s ⊆ u) (d : Disjoint u t) : Disjoint s t := d.mono_left h #align set.disjoint_of_subset_left Set.disjoint_of_subset_left lemma disjoint_of_subset_right (h : t ⊆ u) (d : Disjoint s u) : Disjoint s t := d.mono_right h #align set.disjoint_of_subset_right Set.disjoint_of_subset_right lemma disjoint_of_subset (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (h : Disjoint s₂ t₂) : Disjoint s₁ t₁ := h.mono hs ht #align set.disjoint_of_subset Set.disjoint_of_subset @[simp] lemma disjoint_union_left : Disjoint (s ∪ t) u ↔ Disjoint s u ∧ Disjoint t u := disjoint_sup_left #align set.disjoint_union_left Set.disjoint_union_left @[simp] lemma disjoint_union_right : Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u := disjoint_sup_right #align set.disjoint_union_right Set.disjoint_union_right @[simp] lemma disjoint_empty (s : Set α) : Disjoint s ∅ := disjoint_bot_right #align set.disjoint_empty Set.disjoint_empty @[simp] lemma empty_disjoint (s : Set α) : Disjoint ∅ s := disjoint_bot_left #align set.empty_disjoint Set.empty_disjoint @[simp] lemma univ_disjoint : Disjoint univ s ↔ s = ∅ := top_disjoint #align set.univ_disjoint Set.univ_disjoint @[simp] lemma disjoint_univ : Disjoint s univ ↔ s = ∅ := disjoint_top #align set.disjoint_univ Set.disjoint_univ lemma disjoint_sdiff_left : Disjoint (t \ s) s := disjoint_sdiff_self_left #align set.disjoint_sdiff_left Set.disjoint_sdiff_left lemma disjoint_sdiff_right : Disjoint s (t \ s) := disjoint_sdiff_self_right #align set.disjoint_sdiff_right Set.disjoint_sdiff_right -- TODO: prove this in terms of a lattice lemma theorem disjoint_sdiff_inter : Disjoint (s \ t) (s ∩ t) := disjoint_of_subset_right inter_subset_right disjoint_sdiff_left #align set.disjoint_sdiff_inter Set.disjoint_sdiff_inter theorem diff_union_diff_cancel (hts : t ⊆ s) (hut : u ⊆ t) : s \ t ∪ t \ u = s \ u := sdiff_sup_sdiff_cancel hts hut #align set.diff_union_diff_cancel Set.diff_union_diff_cancel theorem diff_diff_eq_sdiff_union (h : u ⊆ s) : s \ (t \ u) = s \ t ∪ u := sdiff_sdiff_eq_sdiff_sup h #align set.diff_diff_eq_sdiff_union Set.diff_diff_eq_sdiff_union @[simp default+1] lemma disjoint_singleton_left : Disjoint {a} s ↔ a ∉ s := by simp [Set.disjoint_iff, subset_def] #align set.disjoint_singleton_left Set.disjoint_singleton_left @[simp] lemma disjoint_singleton_right : Disjoint s {a} ↔ a ∉ s := disjoint_comm.trans disjoint_singleton_left #align set.disjoint_singleton_right Set.disjoint_singleton_right lemma disjoint_singleton : Disjoint ({a} : Set α) {b} ↔ a ≠ b := by simp #align set.disjoint_singleton Set.disjoint_singleton lemma subset_diff : s ⊆ t \ u ↔ s ⊆ t ∧ Disjoint s u := le_iff_subset.symm.trans le_sdiff #align set.subset_diff Set.subset_diff lemma ssubset_iff_sdiff_singleton : s ⊂ t ↔ ∃ a ∈ t, s ⊆ t \ {a} := by simp [ssubset_iff_insert, subset_diff, insert_subset_iff]; aesop theorem inter_diff_distrib_left (s t u : Set α) : s ∩ (t \ u) = (s ∩ t) \ (s ∩ u) := inf_sdiff_distrib_left _ _ _ #align set.inter_diff_distrib_left Set.inter_diff_distrib_left theorem inter_diff_distrib_right (s t u : Set α) : s \ t ∩ u = (s ∩ u) \ (t ∩ u) := inf_sdiff_distrib_right _ _ _ #align set.inter_diff_distrib_right Set.inter_diff_distrib_right theorem compl_def (s : Set α) : sᶜ = { x \| x ∉ s } := rfl #align set.compl_def Set.compl_def theorem mem_compl {s : Set α} {x : α} (h : x ∉ s) : x ∈ sᶜ := h #align set.mem_compl Set.mem_compl theorem compl_setOf {α} (p : α → Prop) : { a \| p a }ᶜ = { a \| ¬p a } := rfl #align set.compl_set_of Set.compl_setOf theorem not_mem_of_mem_compl {s : Set α} {x : α} (h : x ∈ sᶜ) : x ∉ s := h #align set.not_mem_of_mem_compl Set.not_mem_of_mem_compl theorem not_mem_compl_iff {x : α} : x ∉ sᶜ ↔ x ∈ s := not_not #align set.not_mem_compl_iff Set.not_mem_compl_iff @[simp] theorem inter_compl_self (s : Set α) : s ∩ sᶜ = ∅ := inf_compl_eq_bot #align set.inter_compl_self Set.inter_compl_self @[simp] theorem compl_inter_self (s : Set α) : sᶜ ∩ s = ∅ := compl_inf_eq_bot #align set.compl_inter_self Set.compl_inter_self @[simp] theorem compl_empty : (∅ : Set α)ᶜ = univ := compl_bot #align set.compl_empty Set.compl_empty @[simp] theorem compl_union (s t : Set α) : (s ∪ t)ᶜ = sᶜ ∩ tᶜ := compl_sup #align set.compl_union Set.compl_union theorem compl_inter (s t : Set α) : (s ∩ t)ᶜ = sᶜ ∪ tᶜ := compl_inf #align set.compl_inter Set.compl_inter @[simp] theorem compl_univ : (univ : Set α)ᶜ = ∅ := compl_top #align set.compl_univ Set.compl_univ @[simp] theorem compl_empty_iff {s : Set α} : sᶜ = ∅ ↔ s = univ := compl_eq_bot #align set.compl_empty_iff Set.compl_empty_iff @[simp] theorem compl_univ_iff {s : Set α} : sᶜ = univ ↔ s = ∅ := compl_eq_top #align set.compl_univ_iff Set.compl_univ_iff theorem compl_ne_univ : sᶜ ≠ univ ↔ s.Nonempty := compl_univ_iff.not.trans nonempty_iff_ne_empty.symm #align set.compl_ne_univ Set.compl_ne_univ theorem nonempty_compl : sᶜ.Nonempty ↔ s ≠ univ := (ne_univ_iff_exists_not_mem s).symm #align set.nonempty_compl Set.nonempty_compl @[simp] lemma nonempty_compl_of_nontrivial [Nontrivial α] (x : α) : Set.Nonempty {x}ᶜ := by obtain ⟨y, hy⟩ := exists_ne x exact ⟨y, by simp [hy]⟩ theorem mem_compl_singleton_iff {a x : α} : x ∈ ({a} : Set α)ᶜ ↔ x ≠ a := Iff.rfl #align set.mem_compl_singleton_iff Set.mem_compl_singleton_iff theorem compl_singleton_eq (a : α) : ({a} : Set α)ᶜ = { x \| x ≠ a } := rfl #align set.compl_singleton_eq Set.compl_singleton_eq @[simp] theorem compl_ne_eq_singleton (a : α) : ({ x \| x ≠ a } : Set α)ᶜ = {a} := compl_compl _ #align set.compl_ne_eq_singleton Set.compl_ne_eq_singleton theorem union_eq_compl_compl_inter_compl (s t : Set α) : s ∪ t = (sᶜ ∩ tᶜ)ᶜ := ext fun _ => or_iff_not_and_not #align set.union_eq_compl_compl_inter_compl Set.union_eq_compl_compl_inter_compl theorem inter_eq_compl_compl_union_compl (s t : Set α) : s ∩ t = (sᶜ ∪ tᶜ)ᶜ := ext fun _ => and_iff_not_or_not #align set.inter_eq_compl_compl_union_compl Set.inter_eq_compl_compl_union_compl @[simp] theorem union_compl_self (s : Set α) : s ∪ sᶜ = univ := eq_univ_iff_forall.2 fun _ => em _ #align set.union_compl_self Set.union_compl_self @[simp] theorem compl_union_self (s : Set α) : sᶜ ∪ s = univ := by rw [union_comm, union_compl_self] #align set.compl_union_self Set.compl_union_self theorem compl_subset_comm : sᶜ ⊆ t ↔ tᶜ ⊆ s := @compl_le_iff_compl_le _ s _ _ #align set.compl_subset_comm Set.compl_subset_comm theorem subset_compl_comm : s ⊆ tᶜ ↔ t ⊆ sᶜ := @le_compl_iff_le_compl _ _ _ t #align set.subset_compl_comm Set.subset_compl_comm @[simp] theorem compl_subset_compl : sᶜ ⊆ tᶜ ↔ t ⊆ s := @compl_le_compl_iff_le (Set α) _ _ _ #align set.compl_subset_compl Set.compl_subset_compl @[gcongr] theorem compl_subset_compl_of_subset (h : t ⊆ s) : sᶜ ⊆ tᶜ := compl_subset_compl.2 h theorem subset_compl_iff_disjoint_left : s ⊆ tᶜ ↔ Disjoint t s := @le_compl_iff_disjoint_left (Set α) _ _ _ #align set.subset_compl_iff_disjoint_left Set.subset_compl_iff_disjoint_left theorem subset_compl_iff_disjoint_right : s ⊆ tᶜ ↔ Disjoint s t := @le_compl_iff_disjoint_right (Set α) _ _ _ #align set.subset_compl_iff_disjoint_right Set.subset_compl_iff_disjoint_right theorem disjoint_compl_left_iff_subset : Disjoint sᶜ t ↔ t ⊆ s := disjoint_compl_left_iff #align set.disjoint_compl_left_iff_subset Set.disjoint_compl_left_iff_subset theorem disjoint_compl_right_iff_subset : Disjoint s tᶜ ↔ s ⊆ t := disjoint_compl_right_iff #align set.disjoint_compl_right_iff_subset Set.disjoint_compl_right_iff_subset alias ⟨_, _root_.Disjoint.subset_compl_right⟩ := subset_compl_iff_disjoint_right #align disjoint.subset_compl_right Disjoint.subset_compl_right alias ⟨_, _root_.Disjoint.subset_compl_left⟩ := subset_compl_iff_disjoint_left #align disjoint.subset_compl_left Disjoint.subset_compl_left alias ⟨_, _root_.HasSubset.Subset.disjoint_compl_left⟩ := disjoint_compl_left_iff_subset #align has_subset.subset.disjoint_compl_left HasSubset.Subset.disjoint_compl_left alias ⟨_, _root_.HasSubset.Subset.disjoint_compl_right⟩ := disjoint_compl_right_iff_subset #align has_subset.subset.disjoint_compl_right HasSubset.Subset.disjoint_compl_right theorem subset_union_compl_iff_inter_subset {s t u : Set α} : s ⊆ t ∪ uᶜ ↔ s ∩ u ⊆ t := (@isCompl_compl _ u _).le_sup_right_iff_inf_left_le #align set.subset_union_compl_iff_inter_subset Set.subset_union_compl_iff_inter_subset theorem compl_subset_iff_union {s t : Set α} : sᶜ ⊆ t ↔ s ∪ t = univ := Iff.symm <\| eq_univ_iff_forall.trans <\| forall_congr' fun _ => or_iff_not_imp_left #align set.compl_subset_iff_union Set.compl_subset_iff_union @[simp] theorem subset_compl_singleton_iff {a : α} {s : Set α} : s ⊆ {a}ᶜ ↔ a ∉ s := subset_compl_comm.trans singleton_subset_iff #align set.subset_compl_singleton_iff Set.subset_compl_singleton_iff theorem inter_subset (a b c : Set α) : a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c := forall_congr' fun _ => and_imp.trans <\| imp_congr_right fun _ => imp_iff_not_or #align set.inter_subset Set.inter_subset theorem inter_compl_nonempty_iff {s t : Set α} : (s ∩ tᶜ).Nonempty ↔ ¬s ⊆ t := (not_subset.trans <\| exists_congr fun x => by simp [mem_compl]).symm #align set.inter_compl_nonempty_iff Set.inter_compl_nonempty_iff theorem not_mem_diff_of_mem {s t : Set α} {x : α} (hx : x ∈ t) : x ∉ s \ t := fun h => h.2 hx #align set.not_mem_diff_of_mem Set.not_mem_diff_of_mem theorem mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∈ s := h.left #align set.mem_of_mem_diff Set.mem_of_mem_diff theorem not_mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∉ t := h.right #align set.not_mem_of_mem_diff Set.not_mem_of_mem_diff theorem diff_eq_compl_inter {s t : Set α} : s \ t = tᶜ ∩ s := by rw [diff_eq, inter_comm] #align set.diff_eq_compl_inter Set.diff_eq_compl_inter theorem nonempty_diff {s t : Set α} : (s \ t).Nonempty ↔ ¬s ⊆ t := inter_compl_nonempty_iff #align set.nonempty_diff Set.nonempty_diff theorem diff_subset {s t : Set α} : s \ t ⊆ s := show s \ t ≤ s from sdiff_le #align set.diff_subset Set.diff_subset theorem diff_subset_compl (s t : Set α) : s \ t ⊆ tᶜ := diff_eq_compl_inter ▸ inter_subset_left theorem union_diff_cancel' {s t u : Set α} (h₁ : s ⊆ t) (h₂ : t ⊆ u) : t ∪ u \ s = u := sup_sdiff_cancel' h₁ h₂ #align set.union_diff_cancel' Set.union_diff_cancel' theorem union_diff_cancel {s t : Set α} (h : s ⊆ t) : s ∪ t \ s = t := sup_sdiff_cancel_right h #align set.union_diff_cancel Set.union_diff_cancel theorem union_diff_cancel_left {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ s = t := Disjoint.sup_sdiff_cancel_left <\| disjoint_iff_inf_le.2 h #align set.union_diff_cancel_left Set.union_diff_cancel_left theorem union_diff_cancel_right {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ t = s := Disjoint.sup_sdiff_cancel_right <\| disjoint_iff_inf_le.2 h #align set.union_diff_cancel_right Set.union_diff_cancel_right @[simp] theorem union_diff_left {s t : Set α} : (s ∪ t) \ s = t \ s := sup_sdiff_left_self #align set.union_diff_left Set.union_diff_left @[simp] theorem union_diff_right {s t : Set α} : (s ∪ t) \ t = s \ t := sup_sdiff_right_self #align set.union_diff_right Set.union_diff_right theorem union_diff_distrib {s t u : Set α} : (s ∪ t) \ u = s \ u ∪ t \ u := sup_sdiff #align set.union_diff_distrib Set.union_diff_distrib theorem inter_diff_assoc (a b c : Set α) : (a ∩ b) \ c = a ∩ (b \ c) := inf_sdiff_assoc #align set.inter_diff_assoc Set.inter_diff_assoc @[simp] theorem inter_diff_self (a b : Set α) : a ∩ (b \ a) = ∅ := inf_sdiff_self_right #align set.inter_diff_self Set.inter_diff_self @[simp] theorem inter_union_diff (s t : Set α) : s ∩ t ∪ s \ t = s := sup_inf_sdiff s t #align set.inter_union_diff Set.inter_union_diff @[simp] theorem diff_union_inter (s t : Set α) : s \ t ∪ s ∩ t = s := by rw [union_comm] exact sup_inf_sdiff _ _ #align set.diff_union_inter Set.diff_union_inter @[simp] theorem inter_union_compl (s t : Set α) : s ∩ t ∪ s ∩ tᶜ = s := inter_union_diff _ _ #align set.inter_union_compl Set.inter_union_compl @[gcongr] theorem diff_subset_diff {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ s₂ → t₂ ⊆ t₁ → s₁ \ t₁ ⊆ s₂ \ t₂ := show s₁ ≤ s₂ → t₂ ≤ t₁ → s₁ \ t₁ ≤ s₂ \ t₂ from sdiff_le_sdiff #align set.diff_subset_diff Set.diff_subset_diff @[gcongr] theorem diff_subset_diff_left {s₁ s₂ t : Set α} (h : s₁ ⊆ s₂) : s₁ \ t ⊆ s₂ \ t := sdiff_le_sdiff_right ‹s₁ ≤ s₂› #align set.diff_subset_diff_left Set.diff_subset_diff_left @[gcongr] theorem diff_subset_diff_right {s t u : Set α} (h : t ⊆ u) : s \ u ⊆ s \ t := sdiff_le_sdiff_left ‹t ≤ u› #align set.diff_subset_diff_right Set.diff_subset_diff_right theorem compl_eq_univ_diff (s : Set α) : sᶜ = univ \ s := top_sdiff.symm #align set.compl_eq_univ_diff Set.compl_eq_univ_diff @[simp] theorem empty_diff (s : Set α) : (∅ \ s : Set α) = ∅ := bot_sdiff #align set.empty_diff Set.empty_diff theorem diff_eq_empty {s t : Set α} : s \ t = ∅ ↔ s ⊆ t := sdiff_eq_bot_iff #align set.diff_eq_empty Set.diff_eq_empty @[simp] theorem diff_empty {s : Set α} : s \ ∅ = s := sdiff_bot #align set.diff_empty Set.diff_empty @[simp] theorem diff_univ (s : Set α) : s \ univ = ∅ := diff_eq_empty.2 (subset_univ s) #align set.diff_univ Set.diff_univ theorem diff_diff {u : Set α} : (s \ t) \ u = s \ (t ∪ u) := sdiff_sdiff_left #align set.diff_diff Set.diff_diff -- the following statement contains parentheses to help the reader theorem diff_diff_comm {s t u : Set α} : (s \ t) \ u = (s \ u) \ t := sdiff_sdiff_comm #align set.diff_diff_comm Set.diff_diff_comm theorem diff_subset_iff {s t u : Set α} : s \ t ⊆ u ↔ s ⊆ t ∪ u := show s \ t ≤ u ↔ s ≤ t ∪ u from sdiff_le_iff #align set.diff_subset_iff Set.diff_subset_iff theorem subset_diff_union (s t : Set α) : s ⊆ s \ t ∪ t := show s ≤ s \ t ∪ t from le_sdiff_sup #align set.subset_diff_union Set.subset_diff_union theorem diff_union_of_subset {s t : Set α} (h : t ⊆ s) : s \ t ∪ t = s := Subset.antisymm (union_subset diff_subset h) (subset_diff_union _ _) #align set.diff_union_of_subset Set.diff_union_of_subset @[simp] theorem diff_singleton_subset_iff {x : α} {s t : Set α} : s \ {x} ⊆ t ↔ s ⊆ insert x t := by rw [← union_singleton, union_comm] apply diff_subset_iff #align set.diff_singleton_subset_iff Set.diff_singleton_subset_iff theorem subset_diff_singleton {x : α} {s t : Set α} (h : s ⊆ t) (hx : x ∉ s) : s ⊆ t \ {x} := subset_inter h <\| subset_compl_comm.1 <\| singleton_subset_iff.2 hx #align set.subset_diff_singleton Set.subset_diff_singleton theorem subset_insert_diff_singleton (x : α) (s : Set α) : s ⊆ insert x (s \ {x}) := by rw [← diff_singleton_subset_iff] #align set.subset_insert_diff_singleton Set.subset_insert_diff_singleton theorem diff_subset_comm {s t u : Set α} : s \ t ⊆ u ↔ s \ u ⊆ t := show s \ t ≤ u ↔ s \ u ≤ t from sdiff_le_comm #align set.diff_subset_comm Set.diff_subset_comm theorem diff_inter {s t u : Set α} : s \ (t ∩ u) = s \ t ∪ s \ u := sdiff_inf #align set.diff_inter Set.diff_inter theorem diff_inter_diff {s t u : Set α} : s \ t ∩ (s \ u) = s \ (t ∪ u) := sdiff_sup.symm #align set.diff_inter_diff Set.diff_inter_diff theorem diff_compl : s \ tᶜ = s ∩ t := sdiff_compl #align set.diff_compl Set.diff_compl theorem diff_diff_right {s t u : Set α} : s \ (t \ u) = s \ t ∪ s ∩ u := sdiff_sdiff_right' #align set.diff_diff_right Set.diff_diff_right @[simp] theorem insert_diff_of_mem (s) (h : a ∈ t) : insert a s \ t = s \ t := by ext constructor <;> simp (config := { contextual := true }) [or_imp, h] #align set.insert_diff_of_mem Set.insert_diff_of_mem theorem insert_diff_of_not_mem (s) (h : a ∉ t) : insert a s \ t = insert a (s \ t) := by classical ext x by_cases h' : x ∈ t · have : x ≠ a := by intro H rw [H] at h' exact h h' simp [h, h', this] · simp [h, h'] #align set.insert_diff_of_not_mem Set.insert_diff_of_not_mem theorem insert_diff_self_of_not_mem {a : α} {s : Set α} (h : a ∉ s) : insert a s \ {a} = s := by ext x simp [and_iff_left_of_imp fun hx : x ∈ s => show x ≠ a from fun hxa => h <\| hxa ▸ hx] #align set.insert_diff_self_of_not_mem Set.insert_diff_self_of_not_mem @[simp] theorem insert_diff_eq_singleton {a : α} {s : Set α} (h : a ∉ s) : insert a s \ s = {a} := by ext rw [Set.mem_diff, Set.mem_insert_iff, Set.mem_singleton_iff, or_and_right, and_not_self_iff, or_false_iff, and_iff_left_iff_imp] rintro rfl exact h #align set.insert_diff_eq_singleton Set.insert_diff_eq_singleton theorem inter_insert_of_mem (h : a ∈ s) : s ∩ insert a t = insert a (s ∩ t) := by rw [insert_inter_distrib, insert_eq_of_mem h] #align set.inter_insert_of_mem Set.inter_insert_of_mem theorem insert_inter_of_mem (h : a ∈ t) : insert a s ∩ t = insert a (s ∩ t) := by rw [insert_inter_distrib, insert_eq_of_mem h] #align set.insert_inter_of_mem Set.insert_inter_of_mem theorem inter_insert_of_not_mem (h : a ∉ s) : s ∩ insert a t = s ∩ t := ext fun _ => and_congr_right fun hx => or_iff_right <\| ne_of_mem_of_not_mem hx h #align set.inter_insert_of_not_mem Set.inter_insert_of_not_mem theorem insert_inter_of_not_mem (h : a ∉ t) : insert a s ∩ t = s ∩ t := ext fun _ => and_congr_left fun hx => or_iff_right <\| ne_of_mem_of_not_mem hx h #align set.insert_inter_of_not_mem Set.insert_inter_of_not_mem @[simp] theorem union_diff_self {s t : Set α} : s ∪ t \ s = s ∪ t := sup_sdiff_self _ _ #align set.union_diff_self Set.union_diff_self @[simp] theorem diff_union_self {s t : Set α} : s \ t ∪ t = s ∪ t := sdiff_sup_self _ _ #align set.diff_union_self Set.diff_union_self @[simp] theorem diff_inter_self {a b : Set α} : b \ a ∩ a = ∅ := inf_sdiff_self_left #align set.diff_inter_self Set.diff_inter_self @[simp] theorem diff_inter_self_eq_diff {s t : Set α} : s \ (t ∩ s) = s \ t := sdiff_inf_self_right _ _ #align set.diff_inter_self_eq_diff Set.diff_inter_self_eq_diff @[simp] theorem diff_self_inter {s t : Set α} : s \ (s ∩ t) = s \ t := sdiff_inf_self_left _ _ #align set.diff_self_inter Set.diff_self_inter @[simp] theorem diff_singleton_eq_self {a : α} {s : Set α} (h : a ∉ s) : s \ {a} = s := sdiff_eq_self_iff_disjoint.2 <\| by simp [h] #align set.diff_singleton_eq_self Set.diff_singleton_eq_self @[simp] theorem diff_singleton_sSubset {s : Set α} {a : α} : s \ {a} ⊂ s ↔ a ∈ s := sdiff_le.lt_iff_ne.trans <\| sdiff_eq_left.not.trans <\| by simp #align set.diff_singleton_ssubset Set.diff_singleton_sSubset @[simp] theorem insert_diff_singleton {a : α} {s : Set α} : insert a (s \ {a}) = insert a s := by simp [insert_eq, union_diff_self, -union_singleton, -singleton_union] #align set.insert_diff_singleton Set.insert_diff_singleton theorem insert_diff_singleton_comm (hab : a ≠ b) (s : Set α) : insert a (s \ {b}) = insert a s \ {b} := by simp_rw [← union_singleton, union_diff_distrib, diff_singleton_eq_self (mem_singleton_iff.not.2 hab.symm)] #align set.insert_diff_singleton_comm Set.insert_diff_singleton_comm --Porting note (#10618): removed `simp` attribute because `simp` can prove it theorem diff_self {s : Set α} : s \ s = ∅ := sdiff_self #align set.diff_self Set.diff_self theorem diff_diff_right_self (s t : Set α) : s \ (s \ t) = s ∩ t := sdiff_sdiff_right_self #align set.diff_diff_right_self Set.diff_diff_right_self theorem diff_diff_cancel_left {s t : Set α} (h : s ⊆ t) : t \ (t \ s) = s := sdiff_sdiff_eq_self h #align set.diff_diff_cancel_left Set.diff_diff_cancel_left theorem mem_diff_singleton {x y : α} {s : Set α} : x ∈ s \ {y} ↔ x ∈ s ∧ x ≠ y := Iff.rfl #align set.mem_diff_singleton Set.mem_diff_singleton theorem mem_diff_singleton_empty {t : Set (Set α)} : s ∈ t \ {∅} ↔ s ∈ t ∧ s.Nonempty := mem_diff_singleton.trans <\| and_congr_right' nonempty_iff_ne_empty.symm #align set.mem_diff_singleton_empty Set.mem_diff_singleton_empty theorem subset_insert_iff {s t : Set α} {x : α} : s ⊆ insert x t ↔ s ⊆ t ∨ (x ∈ s ∧ s \ {x} ⊆ t) := by rw [← diff_singleton_subset_iff] by_cases hx : x ∈ s · rw [and_iff_right hx, or_iff_right_of_imp diff_subset.trans] rw [diff_singleton_eq_self hx, or_iff_left_of_imp And.right] theorem union_eq_diff_union_diff_union_inter (s t : Set α) : s ∪ t = s \ t ∪ t \ s ∪ s ∩ t := sup_eq_sdiff_sup_sdiff_sup_inf #align set.union_eq_diff_union_diff_union_inter Set.union_eq_diff_union_diff_union_inter --Porting note (#10618): removed `simp` attribute because `simp` can prove it theorem pair_eq_singleton (a : α) : ({a, a} : Set α) = {a} := union_self _ #align set.pair_eq_singleton Set.pair_eq_singleton theorem pair_comm (a b : α) : ({a, b} : Set α) = {b, a} := union_comm _ _ #align set.pair_comm Set.pair_comm theorem pair_eq_pair_iff {x y z w : α} : ({x, y} : Set α) = {z, w} ↔ x = z ∧ y = w ∨ x = w ∧ y = z := by simp [subset_antisymm_iff, insert_subset_iff]; aesop #align set.pair_eq_pair_iff Set.pair_eq_pair_iff theorem pair_diff_left (hne : a ≠ b) : ({a, b} : Set α) \ {a} = {b} := by rw [insert_diff_of_mem _ (mem_singleton a), diff_singleton_eq_self (by simpa)] theorem pair_diff_right (hne : a ≠ b) : ({a, b} : Set α) \ {b} = {a} := by rw [pair_comm, pair_diff_left hne.symm] theorem pair_subset_iff : {a, b} ⊆ s ↔ a ∈ s ∧ b ∈ s := by rw [insert_subset_iff, singleton_subset_iff] theorem pair_subset (ha : a ∈ s) (hb : b ∈ s) : {a, b} ⊆ s := pair_subset_iff.2 ⟨ha,hb⟩ theorem subset_pair_iff : s ⊆ {a, b} ↔ ∀ x ∈ s, x = a ∨ x = b := by simp [subset_def] theorem subset_pair_iff_eq {x y : α} : s ⊆ {x, y} ↔ s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y} := by refine ⟨?_, by rintro (rfl \| rfl \| rfl \| rfl) <;> simp [pair_subset_iff]⟩ rw [subset_insert_iff, subset_singleton_iff_eq, subset_singleton_iff_eq, ← subset_empty_iff (s := s \ {x}), diff_subset_iff, union_empty, subset_singleton_iff_eq] have h : x ∈ s → {y} = s \ {x} → s = {x,y} := fun h₁ h₂ ↦ by simp [h₁, h₂] tauto theorem Nonempty.subset_pair_iff_eq (hs : s.Nonempty) : s ⊆ {a, b} ↔ s = {a} ∨ s = {b} ∨ s = {a, b} := by rw [Set.subset_pair_iff_eq, or_iff_right]; exact hs.ne_empty section open scoped symmDiff theorem mem_symmDiff : a ∈ s ∆ t ↔ a ∈ s ∧ a ∉ t ∨ a ∈ t ∧ a ∉ s := Iff.rfl #align set.mem_symm_diff Set.mem_symmDiff protected theorem symmDiff_def (s t : Set α) : s ∆ t = s \ t ∪ t \ s := rfl #align set.symm_diff_def Set.symmDiff_def theorem symmDiff_subset_union : s ∆ t ⊆ s ∪ t := @symmDiff_le_sup (Set α) _ _ _ #align set.symm_diff_subset_union Set.symmDiff_subset_union @[simp] theorem symmDiff_eq_empty : s ∆ t = ∅ ↔ s = t := symmDiff_eq_bot #align set.symm_diff_eq_empty Set.symmDiff_eq_empty @[simp] theorem symmDiff_nonempty : (s ∆ t).Nonempty ↔ s ≠ t := nonempty_iff_ne_empty.trans symmDiff_eq_empty.not #align set.symm_diff_nonempty Set.symmDiff_nonempty theorem inter_symmDiff_distrib_left (s t u : Set α) : s ∩ t ∆ u = (s ∩ t) ∆ (s ∩ u) := inf_symmDiff_distrib_left _ _ _ #align set.inter_symm_diff_distrib_left Set.inter_symmDiff_distrib_left theorem inter_symmDiff_distrib_right (s t u : Set α) : s ∆ t ∩ u = (s ∩ u) ∆ (t ∩ u) := inf_symmDiff_distrib_right _ _ _ #align set.inter_symm_diff_distrib_right Set.inter_symmDiff_distrib_right theorem subset_symmDiff_union_symmDiff_left (h : Disjoint s t) : u ⊆ s ∆ u ∪ t ∆ u := h.le_symmDiff_sup_symmDiff_left #align set.subset_symm_diff_union_symm_diff_left Set.subset_symmDiff_union_symmDiff_left theorem subset_symmDiff_union_symmDiff_right (h : Disjoint t u) : s ⊆ s ∆ t ∪ s ∆ u := h.le_symmDiff_sup_symmDiff_right #align set.subset_symm_diff_union_symm_diff_right Set.subset_symmDiff_union_symmDiff_right end #align set.powerset Set.powerset theorem mem_powerset {x s : Set α} (h : x ⊆ s) : x ∈ 𝒫 s := @h #align set.mem_powerset Set.mem_powerset theorem subset_of_mem_powerset {x s : Set α} (h : x ∈ 𝒫 s) : x ⊆ s := @h #align set.subset_of_mem_powerset Set.subset_of_mem_powerset @[simp] theorem mem_powerset_iff (x s : Set α) : x ∈ 𝒫 s ↔ x ⊆ s := Iff.rfl #align set.mem_powerset_iff Set.mem_powerset_iff theorem powerset_inter (s t : Set α) : 𝒫(s ∩ t) = 𝒫 s ∩ 𝒫 t := ext fun _ => subset_inter_iff #align set.powerset_inter Set.powerset_inter @[simp] theorem powerset_mono : 𝒫 s ⊆ 𝒫 t ↔ s ⊆ t := ⟨fun h => @h _ (fun _ h => h), fun h _ hu _ ha => h (hu ha)⟩ #align set.powerset_mono Set.powerset_mono theorem monotone_powerset : Monotone (powerset : Set α → Set (Set α)) := fun _ _ => powerset_mono.2 #align set.monotone_powerset Set.monotone_powerset @[simp] theorem powerset_nonempty : (𝒫 s).Nonempty := ⟨∅, fun _ h => empty_subset s h⟩ #align set.powerset_nonempty Set.powerset_nonempty @[simp] theorem powerset_empty : 𝒫(∅ : Set α) = {∅} := ext fun _ => subset_empty_iff #align set.powerset_empty Set.powerset_empty @[simp] theorem powerset_univ : 𝒫(univ : Set α) = univ := eq_univ_of_forall subset_univ #align set.powerset_univ Set.powerset_univ theorem powerset_singleton (x : α) : 𝒫({x} : Set α) = {∅, {x}} := by ext y rw [mem_powerset_iff, subset_singleton_iff_eq, mem_insert_iff, mem_singleton_iff] #align set.powerset_singleton Set.powerset_singleton theorem mem_dite (p : Prop) [Decidable p] (s : p → Set α) (t : ¬ p → Set α) (x : α) : (x ∈ if h : p then s h else t h) ↔ (∀ h : p, x ∈ s h) ∧ ∀ h : ¬p, x ∈ t h := by split_ifs with hp · exact ⟨fun hx => ⟨fun _ => hx, fun hnp => (hnp hp).elim⟩, fun hx => hx.1 hp⟩ · exact ⟨fun hx => ⟨fun h => (hp h).elim, fun _ => hx⟩, fun hx => hx.2 hp⟩ theorem mem_dite_univ_right (p : Prop) [Decidable p] (t : p → Set α) (x : α) : (x ∈ if h : p then t h else univ) ↔ ∀ h : p, x ∈ t h := by split_ifs <;> simp_all #align set.mem_dite_univ_right Set.mem_dite_univ_right @[simp] theorem mem_ite_univ_right (p : Prop) [Decidable p] (t : Set α) (x : α) : x ∈ ite p t Set.univ ↔ p → x ∈ t := mem_dite_univ_right p (fun _ => t) x #align set.mem_ite_univ_right Set.mem_ite_univ_right theorem mem_dite_univ_left (p : Prop) [Decidable p] (t : ¬p → Set α) (x : α) : (x ∈ if h : p then univ else t h) ↔ ∀ h : ¬p, x ∈ t h := by split_ifs <;> simp_all #align set.mem_dite_univ_left Set.mem_dite_univ_left @[simp] theorem mem_ite_univ_left (p : Prop) [Decidable p] (t : Set α) (x : α) : x ∈ ite p Set.univ t ↔ ¬p → x ∈ t := mem_dite_univ_left p (fun _ => t) x #align set.mem_ite_univ_left Set.mem_ite_univ_left theorem mem_dite_empty_right (p : Prop) [Decidable p] (t : p → Set α) (x : α) : (x ∈ if h : p then t h else ∅) ↔ ∃ h : p, x ∈ t h := by simp only [mem_dite, mem_empty_iff_false, imp_false, not_not] exact ⟨fun h => ⟨h.2, h.1 h.2⟩, fun ⟨h₁, h₂⟩ => ⟨fun _ => h₂, h₁⟩⟩ #align set.mem_dite_empty_right Set.mem_dite_empty_right @[simp] theorem mem_ite_empty_right (p : Prop) [Decidable p] (t : Set α) (x : α) : x ∈ ite p t ∅ ↔ p ∧ x ∈ t := (mem_dite_empty_right p (fun _ => t) x).trans (by simp) #align set.mem_ite_empty_right Set.mem_ite_empty_right theorem mem_dite_empty_left (p : Prop) [Decidable p] (t : ¬p → Set α) (x : α) : (x ∈ if h : p then ∅ else t h) ↔ ∃ h : ¬p, x ∈ t h := by simp only [mem_dite, mem_empty_iff_false, imp_false] exact ⟨fun h => ⟨h.1, h.2 h.1⟩, fun ⟨h₁, h₂⟩ => ⟨fun h => h₁ h, fun _ => h₂⟩⟩ #align set.mem_dite_empty_left Set.mem_dite_empty_left @[simp] theorem mem_ite_empty_left (p : Prop) [Decidable p] (t : Set α) (x : α) : x ∈ ite p ∅ t ↔ ¬p ∧ x ∈ t := (mem_dite_empty_left p (fun _ => t) x).trans (by simp) #align set.mem_ite_empty_left Set.mem_ite_empty_left protected def ite (t s s' : Set α) : Set α := s ∩ t ∪ s' \ t #align set.ite Set.ite @[simp] theorem ite_inter_self (t s s' : Set α) : t.ite s s' ∩ t = s ∩ t := by rw [Set.ite, union_inter_distrib_right, diff_inter_self, inter_assoc, inter_self, union_empty] #align set.ite_inter_self Set.ite_inter_self @[simp] theorem ite_compl (t s s' : Set α) : tᶜ.ite s s' = t.ite s' s := by rw [Set.ite, Set.ite, diff_compl, union_comm, diff_eq] #align set.ite_compl Set.ite_compl @[simp] theorem ite_inter_compl_self (t s s' : Set α) : t.ite s s' ∩ tᶜ = s' ∩ tᶜ := by rw [← ite_compl, ite_inter_self] #align set.ite_inter_compl_self Set.ite_inter_compl_self @[simp] theorem ite_diff_self (t s s' : Set α) : t.ite s s' \ t = s' \ t := ite_inter_compl_self t s s' #align set.ite_diff_self Set.ite_diff_self @[simp] theorem ite_same (t s : Set α) : t.ite s s = s := inter_union_diff _ _ #align set.ite_same Set.ite_same @[simp] theorem ite_left (s t : Set α) : s.ite s t = s ∪ t := by simp [Set.ite] #align set.ite_left Set.ite_left @[simp] theorem ite_right (s t : Set α) : s.ite t s = t ∩ s := by simp [Set.ite] #align set.ite_right Set.ite_right @[simp] theorem ite_empty (s s' : Set α) : Set.ite ∅ s s' = s' := by simp [Set.ite] #align set.ite_empty Set.ite_empty @[simp] theorem ite_univ (s s' : Set α) : Set.ite univ s s' = s := by simp [Set.ite] #align set.ite_univ Set.ite_univ @[simp] theorem ite_empty_left (t s : Set α) : t.ite ∅ s = s \ t := by simp [Set.ite] #align set.ite_empty_left Set.ite_empty_left @[simp] theorem ite_empty_right (t s : Set α) : t.ite s ∅ = s ∩ t := by simp [Set.ite] #align set.ite_empty_right Set.ite_empty_right theorem ite_mono (t : Set α) {s₁ s₁' s₂ s₂' : Set α} (h : s₁ ⊆ s₂) (h' : s₁' ⊆ s₂') : t.ite s₁ s₁' ⊆ t.ite s₂ s₂' := union_subset_union (inter_subset_inter_left _ h) (inter_subset_inter_left _ h') #align set.ite_mono Set.ite_mono theorem ite_subset_union (t s s' : Set α) : t.ite s s' ⊆ s ∪ s' := union_subset_union inter_subset_left diff_subset #align set.ite_subset_union Set.ite_subset_union theorem inter_subset_ite (t s s' : Set α) : s ∩ s' ⊆ t.ite s s' := ite_same t (s ∩ s') ▸ ite_mono _ inter_subset_left inter_subset_right #align set.inter_subset_ite Set.inter_subset_ite theorem ite_inter_inter (t s₁ s₂ s₁' s₂' : Set α) : t.ite (s₁ ∩ s₂) (s₁' ∩ s₂') = t.ite s₁ s₁' ∩ t.ite s₂ s₂' := by ext x simp only [Set.ite, Set.mem_inter_iff, Set.mem_diff, Set.mem_union] tauto #align set.ite_inter_inter Set.ite_inter_inter theorem ite_inter (t s₁ s₂ s : Set α) : t.ite (s₁ ∩ s) (s₂ ∩ s) = t.ite s₁ s₂ ∩ s := by rw [ite_inter_inter, ite_same] #align set.ite_inter Set.ite_inter theorem ite_inter_of_inter_eq (t : Set α) {s₁ s₂ s : Set α} (h : s₁ ∩ s = s₂ ∩ s) : t.ite s₁ s₂ ∩ s = s₁ ∩ s := by rw [← ite_inter, ← h, ite_same] #align set.ite_inter_of_inter_eq Set.ite_inter_of_inter_eq theorem subset_ite {t s s' u : Set α} : u ⊆ t.ite s s' ↔ u ∩ t ⊆ s ∧ u \ t ⊆ s' := by simp only [subset_def, ← forall_and] refine forall_congr' fun x => ?_ by_cases hx : x ∈ t <;> simp [, Set.ite] #align set.subset_ite Set.subset_ite theorem ite_eq_of_subset_left (t : Set α) {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) : t.ite s₁ s₂ = s₁ ∪ (s₂ \ t) := by ext x by_cases hx : x ∈ t <;> simp [, Set.ite, or_iff_right_of_imp (@h x)] theorem ite_eq_of_subset_right (t : Set α) {s₁ s₂ : Set α} (h : s₂ ⊆ s₁) : t.ite s₁ s₂ = (s₁ ∩ t) ∪ s₂ := by ext x by_cases hx : x ∈ t <;> simp [*, Set.ite, or_iff_left_of_imp (@h x)] section Preorder variable [Preorder α] [Preorder β] {f : α → β} -- Porting note: -- If we decide we want `Elem` to semireducible rather than reducible, we will need: -- instance : Preorder (↑s) := Subtype.instPreorderSubtype _ -- here, along with appropriate lemmas. theorem monotoneOn_iff_monotone : MonotoneOn f s ↔ Monotone fun a : s => f a := by simp [Monotone, MonotoneOn] #align set.monotone_on_iff_monotone Set.monotoneOn_iff_monotone	Mathlib/Data/Set/Basic.lean	2,372	2,374	theorem antitoneOn_iff_antitone : AntitoneOn f s ↔ Antitone fun a : s => f a := by	simp [Antitone, AntitoneOn]
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" open List variable {n : ℕ} @[ext] structure Composition (n : ℕ) where blocks : List ℕ blocks_pos : ∀ {i}, i ∈ blocks → 0 < i blocks_sum : blocks.sum = n #align composition Composition @[ext] structure CompositionAsSet (n : ℕ) where boundaries : Finset (Fin n.succ) zero_mem : (0 : Fin n.succ) ∈ boundaries 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 _⟩⟩ namespace Composition variable (c : Composition n) instance (n : ℕ) : ToString (Composition n) := ⟨fun c => toString c.blocks⟩ 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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]	Mathlib/Combinatorics/Enumerative/Composition.lean	494	495	theorem ones_sizeUpTo (n : ℕ) (i : ℕ) : (ones n).sizeUpTo i = min i n := by	simp [sizeUpTo, ones_blocks, take_replicate]
import Mathlib.RingTheory.Derivation.Basic import Mathlib.RingTheory.Ideal.QuotientOperations #align_import ring_theory.derivation.to_square_zero from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3" section ToSquareZero universe u v w variable {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] variable [Algebra R A] [Algebra R B] (I : Ideal B) (hI : I ^ 2 = ⊥) def diffToIdealOfQuotientCompEq (f₁ f₂ : A →ₐ[R] B) (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) : A →ₗ[R] I := LinearMap.codRestrict (I.restrictScalars _) (f₁.toLinearMap - f₂.toLinearMap) (by intro x change f₁ x - f₂ x ∈ I rw [← Ideal.Quotient.eq, ← Ideal.Quotient.mkₐ_eq_mk R, ← AlgHom.comp_apply, e] rfl) #align diff_to_ideal_of_quotient_comp_eq diffToIdealOfQuotientCompEq @[simp] theorem diffToIdealOfQuotientCompEq_apply (f₁ f₂ : A →ₐ[R] B) (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) (x : A) : ((diffToIdealOfQuotientCompEq I f₁ f₂ e) x : B) = f₁ x - f₂ x := rfl #align diff_to_ideal_of_quotient_comp_eq_apply diffToIdealOfQuotientCompEq_apply variable [Algebra A B] [IsScalarTower R A B] def derivationToSquareZeroOfLift (f : A →ₐ[R] B) (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) : Derivation R A I := by refine { diffToIdealOfQuotientCompEq I f (IsScalarTower.toAlgHom R A B) ?_ with map_one_eq_zero' := ?_ leibniz' := ?_ } · rw [e]; ext; rfl · ext; change f 1 - algebraMap A B 1 = 0; rw [map_one, map_one, sub_self] · intro x y let F := diffToIdealOfQuotientCompEq I f (IsScalarTower.toAlgHom R A B) (by rw [e]; ext; rfl) have : (f x - algebraMap A B x) * (f y - algebraMap A B y) = 0 := by rw [← Ideal.mem_bot, ← hI, pow_two] convert Ideal.mul_mem_mul (F x).2 (F y).2 using 1 ext dsimp only [Submodule.coe_add, Submodule.coe_mk, LinearMap.coe_mk, diffToIdealOfQuotientCompEq_apply, Submodule.coe_smul_of_tower, IsScalarTower.coe_toAlgHom', LinearMap.toFun_eq_coe] simp only [map_mul, sub_mul, mul_sub, Algebra.smul_def] at this ⊢ rw [sub_eq_iff_eq_add, sub_eq_iff_eq_add] at this simp only [LinearMap.coe_toAddHom, diffToIdealOfQuotientCompEq_apply, map_mul, this, IsScalarTower.coe_toAlgHom'] ring #align derivation_to_square_zero_of_lift derivationToSquareZeroOfLift theorem derivationToSquareZeroOfLift_apply (f : A →ₐ[R] B) (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) (x : A) : (derivationToSquareZeroOfLift I hI f e x : B) = f x - algebraMap A B x := rfl #align derivation_to_square_zero_of_lift_apply derivationToSquareZeroOfLift_apply @[simps (config := .lemmasOnly)] def liftOfDerivationToSquareZero (f : Derivation R A I) : A →ₐ[R] B := { ((I.restrictScalars R).subtype.comp f.toLinearMap + (IsScalarTower.toAlgHom R A B).toLinearMap : A →ₗ[R] B) with toFun := fun x => f x + algebraMap A B x map_one' := by dsimp -- Note: added the `(algebraMap _ _)` hint because otherwise it would match `f 1` rw [map_one (algebraMap _ _), f.map_one_eq_zero, Submodule.coe_zero, zero_add] map_mul' := fun x y => by have : (f x : B) * f y = 0 := by rw [← Ideal.mem_bot, ← hI, pow_two] convert Ideal.mul_mem_mul (f x).2 (f y).2 using 1 simp only [map_mul, f.leibniz, add_mul, mul_add, Submodule.coe_add, Submodule.coe_smul_of_tower, Algebra.smul_def, this] ring commutes' := fun r => by simp only [Derivation.map_algebraMap, eq_self_iff_true, zero_add, Submodule.coe_zero, ← IsScalarTower.algebraMap_apply R A B r] map_zero' := ((I.restrictScalars R).subtype.comp f.toLinearMap + (IsScalarTower.toAlgHom R A B).toLinearMap).map_zero } #align lift_of_derivation_to_square_zero liftOfDerivationToSquareZero -- @[simp] -- Porting note: simp normal form is `liftOfDerivationToSquareZero_mk_apply'` theorem liftOfDerivationToSquareZero_mk_apply (d : Derivation R A I) (x : A) : Ideal.Quotient.mk I (liftOfDerivationToSquareZero I hI d x) = algebraMap A (B ⧸ I) x := by rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop, zero_add] rfl #align lift_of_derivation_to_square_zero_mk_apply liftOfDerivationToSquareZero_mk_apply @[simp]	Mathlib/RingTheory/Derivation/ToSquareZero.lean	114	116	theorem liftOfDerivationToSquareZero_mk_apply' (d : Derivation R A I) (x : A) : (Ideal.Quotient.mk I) (d x) + (algebraMap A (B ⧸ I)) x = algebraMap A (B ⧸ I) x := by	simp only [Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop, zero_add]
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.RelIso.Basic #align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432" universe u v w x variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x} open Function OrderDual Set def LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) := ∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x) #align left_ord_continuous LeftOrdContinuous def RightOrdContinuous [Preorder α] [Preorder β] (f : α → β) := ∀ ⦃s : Set α⦄ ⦃x⦄, IsGLB s x → IsGLB (f '' s) (f x) #align right_ord_continuous RightOrdContinuous namespace LeftOrdContinuous section ConditionallyCompleteLattice variable [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] [Nonempty ι] {f : α → β} theorem map_csSup (hf : LeftOrdContinuous f) {s : Set α} (sne : s.Nonempty) (sbdd : BddAbove s) : f (sSup s) = sSup (f '' s) := ((hf <\| isLUB_csSup sne sbdd).csSup_eq <\| sne.image f).symm #align left_ord_continuous.map_cSup LeftOrdContinuous.map_csSup	Mathlib/Order/OrdContinuous.lean	151	154	theorem map_ciSup (hf : LeftOrdContinuous f) {g : ι → α} (hg : BddAbove (range g)) : f (⨆ i, g i) = ⨆ i, f (g i) := by	simp only [iSup, hf.map_csSup (range_nonempty _) hg, ← range_comp] rfl
import Batteries.Data.Rat.Basic import Batteries.Tactic.SeqFocus namespace Rat theorem ext : {p q : Rat} → p.num = q.num → p.den = q.den → p = q \| ⟨_,_,_,_⟩, ⟨_,_,_,_⟩, rfl, rfl => rfl @[simp] theorem mk_den_one {r : Int} : ⟨r, 1, Nat.one_ne_zero, (Nat.coprime_one_right _)⟩ = (r : Rat) := rfl @[simp] theorem zero_num : (0 : Rat).num = 0 := rfl @[simp] theorem zero_den : (0 : Rat).den = 1 := rfl @[simp] theorem one_num : (1 : Rat).num = 1 := rfl @[simp] theorem one_den : (1 : Rat).den = 1 := rfl @[simp] theorem maybeNormalize_eq {num den g} (den_nz reduced) : maybeNormalize num den g den_nz reduced = { num := num.div g, den := den / g, den_nz, reduced } := by unfold maybeNormalize; split · subst g; simp · rfl theorem normalize.reduced' {num : Int} {den g : Nat} (den_nz : den ≠ 0) (e : g = num.natAbs.gcd den) : (num / g).natAbs.Coprime (den / g) := by rw [← Int.div_eq_ediv_of_dvd (e ▸ Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] exact normalize.reduced den_nz e theorem normalize_eq {num den} (den_nz) : normalize num den den_nz = { num := num / num.natAbs.gcd den den := den / num.natAbs.gcd den den_nz := normalize.den_nz den_nz rfl reduced := normalize.reduced' den_nz rfl } := by simp only [normalize, maybeNormalize_eq, Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] @[simp] theorem normalize_zero (nz) : normalize 0 d nz = 0 := by simp [normalize, Int.zero_div, Int.natAbs_zero, Nat.div_self (Nat.pos_of_ne_zero nz)]; rfl theorem mk_eq_normalize (num den nz c) : ⟨num, den, nz, c⟩ = normalize num den nz := by simp [normalize_eq, c.gcd_eq_one] theorem normalize_self (r : Rat) : normalize r.num r.den r.den_nz = r := (mk_eq_normalize ..).symm theorem normalize_mul_left {a : Nat} (d0 : d ≠ 0) (a0 : a ≠ 0) : normalize (↑a * n) (a * d) (Nat.mul_ne_zero a0 d0) = normalize n d d0 := by simp [normalize_eq, mk'.injEq, Int.natAbs_mul, Nat.gcd_mul_left, Nat.mul_div_mul_left _ _ (Nat.pos_of_ne_zero a0), Int.ofNat_mul, Int.mul_ediv_mul_of_pos _ _ (Int.ofNat_pos.2 <\| Nat.pos_of_ne_zero a0)] theorem normalize_mul_right {a : Nat} (d0 : d ≠ 0) (a0 : a ≠ 0) : normalize (n * a) (d * a) (Nat.mul_ne_zero d0 a0) = normalize n d d0 := by rw [← normalize_mul_left (d0 := d0) a0]; congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm] theorem normalize_eq_iff (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) : normalize n₁ d₁ z₁ = normalize n₂ d₂ z₂ ↔ n₁ * d₂ = n₂ * d₁ := by constructor <;> intro h · simp only [normalize_eq, mk'.injEq] at h have' hn₁ := Int.ofNat_dvd_left.2 <\| Nat.gcd_dvd_left n₁.natAbs d₁ have' hn₂ := Int.ofNat_dvd_left.2 <\| Nat.gcd_dvd_left n₂.natAbs d₂ have' hd₁ := Int.ofNat_dvd.2 <\| Nat.gcd_dvd_right n₁.natAbs d₁ have' hd₂ := Int.ofNat_dvd.2 <\| Nat.gcd_dvd_right n₂.natAbs d₂ rw [← Int.ediv_mul_cancel (Int.dvd_trans hd₂ (Int.dvd_mul_left ..)), Int.mul_ediv_assoc _ hd₂, ← Int.ofNat_ediv, ← h.2, Int.ofNat_ediv, ← Int.mul_ediv_assoc _ hd₁, Int.mul_ediv_assoc' _ hn₁, Int.mul_right_comm, h.1, Int.ediv_mul_cancel hn₂] · rw [← normalize_mul_right _ z₂, ← normalize_mul_left z₂ z₁, Int.mul_comm d₁, h] theorem maybeNormalize_eq_normalize {num : Int} {den g : Nat} (den_nz reduced) (hn : ↑g ∣ num) (hd : g ∣ den) : maybeNormalize num den g den_nz reduced = normalize num den (mt (by simp [·]) den_nz) := by simp only [maybeNormalize_eq, mk_eq_normalize, Int.div_eq_ediv_of_dvd hn] have : g ≠ 0 := mt (by simp [·]) den_nz rw [← normalize_mul_right _ this, Int.ediv_mul_cancel hn] congr 1; exact Nat.div_mul_cancel hd @[simp] theorem normalize_eq_zero (d0 : d ≠ 0) : normalize n d d0 = 0 ↔ n = 0 := by have' := normalize_eq_iff d0 Nat.one_ne_zero rw [normalize_zero (d := 1)] at this; rw [this]; simp theorem normalize_num_den' (num den nz) : ∃ d : Nat, d ≠ 0 ∧ num = (normalize num den nz).num * d ∧ den = (normalize num den nz).den * d := by refine ⟨num.natAbs.gcd den, Nat.gcd_ne_zero_right nz, ?_⟩ simp [normalize_eq, Int.ediv_mul_cancel (Int.ofNat_dvd_left.2 <\| Nat.gcd_dvd_left ..), Nat.div_mul_cancel (Nat.gcd_dvd_right ..)] theorem normalize_num_den (h : normalize n d z = ⟨n', d', z', c⟩) : ∃ m : Nat, m ≠ 0 ∧ n = n' * m ∧ d = d' * m := by have := normalize_num_den' n d z; rwa [h] at this theorem normalize_eq_mkRat {num den} (den_nz) : normalize num den den_nz = mkRat num den := by simp [mkRat, den_nz] theorem mkRat_num_den (z : d ≠ 0) (h : mkRat n d = ⟨n', d', z', c⟩) : ∃ m : Nat, m ≠ 0 ∧ n = n' * m ∧ d = d' * m := normalize_num_den ((normalize_eq_mkRat z).symm ▸ h) theorem mkRat_def (n d) : mkRat n d = if d0 : d = 0 then 0 else normalize n d d0 := rfl theorem mkRat_self (a : Rat) : mkRat a.num a.den = a := by rw [← normalize_eq_mkRat a.den_nz, normalize_self] theorem mk_eq_mkRat (num den nz c) : ⟨num, den, nz, c⟩ = mkRat num den := by simp [mk_eq_normalize, normalize_eq_mkRat] @[simp] theorem zero_mkRat (n) : mkRat 0 n = 0 := by simp [mkRat_def] @[simp] theorem mkRat_zero (n) : mkRat n 0 = 0 := by simp [mkRat_def] theorem mkRat_eq_zero (d0 : d ≠ 0) : mkRat n d = 0 ↔ n = 0 := by simp [mkRat_def, d0] theorem mkRat_ne_zero (d0 : d ≠ 0) : mkRat n d ≠ 0 ↔ n ≠ 0 := not_congr (mkRat_eq_zero d0) theorem mkRat_mul_left {a : Nat} (a0 : a ≠ 0) : mkRat (↑a * n) (a * d) = mkRat n d := by if d0 : d = 0 then simp [d0] else rw [← normalize_eq_mkRat d0, ← normalize_mul_left d0 a0, normalize_eq_mkRat] theorem mkRat_mul_right {a : Nat} (a0 : a ≠ 0) : mkRat (n * a) (d * a) = mkRat n d := by rw [← mkRat_mul_left (d := d) a0]; congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm] theorem mkRat_eq_iff (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) : mkRat n₁ d₁ = mkRat n₂ d₂ ↔ n₁ * d₂ = n₂ * d₁ := by rw [← normalize_eq_mkRat z₁, ← normalize_eq_mkRat z₂, normalize_eq_iff] @[simp] theorem divInt_ofNat (num den) : num /. (den : Nat) = mkRat num den := by simp [divInt, normalize_eq_mkRat] theorem mk_eq_divInt (num den nz c) : ⟨num, den, nz, c⟩ = num /. (den : Nat) := by simp [mk_eq_mkRat] theorem divInt_self (a : Rat) : a.num /. a.den = a := by rw [divInt_ofNat, mkRat_self] @[simp] theorem zero_divInt (n) : 0 /. n = 0 := by cases n <;> simp [divInt] @[simp] theorem divInt_zero (n) : n /. 0 = 0 := mkRat_zero n theorem neg_divInt_neg (num den) : -num /. -den = num /. den := by match den with \| Nat.succ n => simp only [divInt, Int.neg_ofNat_succ] simp [normalize_eq_mkRat, Int.neg_neg] \| 0 => rfl \| Int.negSucc n => simp only [divInt, Int.neg_negSucc] simp [normalize_eq_mkRat, Int.neg_neg] theorem divInt_neg' (num den) : num /. -den = -num /. den := by rw [← neg_divInt_neg, Int.neg_neg] theorem divInt_eq_iff (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) : n₁ /. d₁ = n₂ /. d₂ ↔ n₁ * d₂ = n₂ * d₁ := by rcases Int.eq_nat_or_neg d₁ with ⟨_, rfl \| rfl⟩ <;> rcases Int.eq_nat_or_neg d₂ with ⟨_, rfl \| rfl⟩ <;> simp_all [divInt_neg', Int.ofNat_eq_zero, Int.neg_eq_zero, mkRat_eq_iff, Int.neg_mul, Int.mul_neg, Int.eq_neg_comm, eq_comm] theorem divInt_mul_left {a : Int} (a0 : a ≠ 0) : (a * n) /. (a * d) = n /. d := by if d0 : d = 0 then simp [d0] else simp [divInt_eq_iff (Int.mul_ne_zero a0 d0) d0, Int.mul_assoc, Int.mul_left_comm] theorem divInt_mul_right {a : Int} (a0 : a ≠ 0) : (n * a) /. (d * a) = n /. d := by simp [← divInt_mul_left (d := d) a0, Int.mul_comm]	.lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean	170	178	theorem divInt_num_den (z : d ≠ 0) (h : n /. d = ⟨n', d', z', c⟩) : ∃ m, m ≠ 0 ∧ n = n' * m ∧ d = d' * m := by	rcases Int.eq_nat_or_neg d with ⟨_, rfl \| rfl⟩ <;> simp_all [divInt_neg', Int.ofNat_eq_zero, Int.neg_eq_zero] · have ⟨m, h₁, h₂⟩ := mkRat_num_den z h; exists m simp [Int.ofNat_eq_zero, Int.ofNat_mul, h₁, h₂] · have ⟨m, h₁, h₂⟩ := mkRat_num_den z h; exists -m rw [← Int.neg_inj, Int.neg_neg] at h₂ simp [Int.ofNat_eq_zero, Int.ofNat_mul, h₁, h₂, Int.mul_neg, Int.neg_eq_zero]
import Mathlib.Data.Fin.Basic import Mathlib.Order.Chain import Mathlib.Order.Cover import Mathlib.Order.Fin open Set variable {α : Type*} [PartialOrder α] [BoundedOrder α] {n : ℕ} {f : Fin (n + 1) → α}	Mathlib/Data/Fin/FlagRange.lean	32	44	theorem IsMaxChain.range_fin_of_covBy (h0 : f 0 = ⊥) (hlast : f (.last n) = ⊤) (hcovBy : ∀ k : Fin n, f k.castSucc ⩿ f k.succ) : IsMaxChain (· ≤ ·) (range f) := by	have hmono : Monotone f := Fin.monotone_iff_le_succ.2 fun k ↦ (hcovBy k).1 refine ⟨hmono.isChain_range, fun t htc hbt ↦ hbt.antisymm fun x hx ↦ ?_⟩ rw [mem_range]; by_contra! h suffices ∀ k, f k < x by simpa [hlast] using this (.last _) intro k induction k using Fin.induction with \| zero => simpa [h0, bot_lt_iff_ne_bot] using (h 0).symm \| succ k ihk => rw [range_subset_iff] at hbt exact (htc.lt_of_le (hbt k.succ) hx (h _)).resolve_right ((hcovBy k).2 ihk)
import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import geometry.euclidean.angle.unoriented.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" assert_not_exists HasFDerivAt assert_not_exists ConformalAt noncomputable section open Real Set open Real open RealInnerProductSpace namespace InnerProductGeometry variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} def angle (x y : V) : ℝ := Real.arccos (⟪x, y⟫ / (‖x‖ ‖y‖)) #align inner_product_geometry.angle InnerProductGeometry.angle theorem continuousAt_angle {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0) : ContinuousAt (fun y : V × V => angle y.1 y.2) x := Real.continuous_arccos.continuousAt.comp <\| continuous_inner.continuousAt.div ((continuous_norm.comp continuous_fst).mul (continuous_norm.comp continuous_snd)).continuousAt (by simp [hx1, hx2]) #align inner_product_geometry.continuous_at_angle InnerProductGeometry.continuousAt_angle theorem angle_smul_smul {c : ℝ} (hc : c ≠ 0) (x y : V) : angle (c • x) (c • y) = angle x y := by have : c * c ≠ 0 := mul_ne_zero hc hc rw [angle, angle, real_inner_smul_left, inner_smul_right, norm_smul, norm_smul, Real.norm_eq_abs, mul_mul_mul_comm _ ‖x‖, abs_mul_abs_self, ← mul_assoc c c, mul_div_mul_left _ _ this] #align inner_product_geometry.angle_smul_smul InnerProductGeometry.angle_smul_smul @[simp] theorem _root_.LinearIsometry.angle_map {E F : Type} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] (f : E →ₗᵢ[ℝ] F) (u v : E) : angle (f u) (f v) = angle u v := by rw [angle, angle, f.inner_map_map, f.norm_map, f.norm_map] #align linear_isometry.angle_map LinearIsometry.angle_map @[simp, norm_cast] theorem _root_.Submodule.angle_coe {s : Submodule ℝ V} (x y : s) : angle (x : V) (y : V) = angle x y := s.subtypeₗᵢ.angle_map x y #align submodule.angle_coe Submodule.angle_coe theorem cos_angle (x y : V) : Real.cos (angle x y) = ⟪x, y⟫ / (‖x‖ ‖y‖) := Real.cos_arccos (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1 (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2 #align inner_product_geometry.cos_angle InnerProductGeometry.cos_angle theorem angle_comm (x y : V) : angle x y = angle y x := by unfold angle rw [real_inner_comm, mul_comm] #align inner_product_geometry.angle_comm InnerProductGeometry.angle_comm @[simp] theorem angle_neg_neg (x y : V) : angle (-x) (-y) = angle x y := by unfold angle rw [inner_neg_neg, norm_neg, norm_neg] #align inner_product_geometry.angle_neg_neg InnerProductGeometry.angle_neg_neg theorem angle_nonneg (x y : V) : 0 ≤ angle x y := Real.arccos_nonneg _ #align inner_product_geometry.angle_nonneg InnerProductGeometry.angle_nonneg theorem angle_le_pi (x y : V) : angle x y ≤ π := Real.arccos_le_pi _ #align inner_product_geometry.angle_le_pi InnerProductGeometry.angle_le_pi theorem angle_neg_right (x y : V) : angle x (-y) = π - angle x y := by unfold angle rw [← Real.arccos_neg, norm_neg, inner_neg_right, neg_div] #align inner_product_geometry.angle_neg_right InnerProductGeometry.angle_neg_right theorem angle_neg_left (x y : V) : angle (-x) y = π - angle x y := by rw [← angle_neg_neg, neg_neg, angle_neg_right] #align inner_product_geometry.angle_neg_left InnerProductGeometry.angle_neg_left proof_wanted angle_triangle (x y z : V) : angle x z ≤ angle x y + angle y z @[simp] theorem angle_zero_left (x : V) : angle 0 x = π / 2 := by unfold angle rw [inner_zero_left, zero_div, Real.arccos_zero] #align inner_product_geometry.angle_zero_left InnerProductGeometry.angle_zero_left @[simp] theorem angle_zero_right (x : V) : angle x 0 = π / 2 := by unfold angle rw [inner_zero_right, zero_div, Real.arccos_zero] #align inner_product_geometry.angle_zero_right InnerProductGeometry.angle_zero_right @[simp] theorem angle_self {x : V} (hx : x ≠ 0) : angle x x = 0 := by unfold angle rw [← real_inner_self_eq_norm_mul_norm, div_self (inner_self_ne_zero.2 hx : ⟪x, x⟫ ≠ 0), Real.arccos_one] #align inner_product_geometry.angle_self InnerProductGeometry.angle_self @[simp] theorem angle_self_neg_of_nonzero {x : V} (hx : x ≠ 0) : angle x (-x) = π := by rw [angle_neg_right, angle_self hx, sub_zero] #align inner_product_geometry.angle_self_neg_of_nonzero InnerProductGeometry.angle_self_neg_of_nonzero @[simp] theorem angle_neg_self_of_nonzero {x : V} (hx : x ≠ 0) : angle (-x) x = π := by rw [angle_comm, angle_self_neg_of_nonzero hx] #align inner_product_geometry.angle_neg_self_of_nonzero InnerProductGeometry.angle_neg_self_of_nonzero @[simp] theorem angle_smul_right_of_pos (x y : V) {r : ℝ} (hr : 0 < r) : angle x (r • y) = angle x y := by unfold angle rw [inner_smul_right, norm_smul, Real.norm_eq_abs, abs_of_nonneg (le_of_lt hr), ← mul_assoc, mul_comm _ r, mul_assoc, mul_div_mul_left _ _ (ne_of_gt hr)] #align inner_product_geometry.angle_smul_right_of_pos InnerProductGeometry.angle_smul_right_of_pos @[simp] theorem angle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) : angle (r • x) y = angle x y := by rw [angle_comm, angle_smul_right_of_pos y x hr, angle_comm] #align inner_product_geometry.angle_smul_left_of_pos InnerProductGeometry.angle_smul_left_of_pos @[simp] theorem angle_smul_right_of_neg (x y : V) {r : ℝ} (hr : r < 0) : angle x (r • y) = angle x (-y) := by rw [← neg_neg r, neg_smul, angle_neg_right, angle_smul_right_of_pos x y (neg_pos_of_neg hr), angle_neg_right] #align inner_product_geometry.angle_smul_right_of_neg InnerProductGeometry.angle_smul_right_of_neg @[simp] theorem angle_smul_left_of_neg (x y : V) {r : ℝ} (hr : r < 0) : angle (r • x) y = angle (-x) y := by rw [angle_comm, angle_smul_right_of_neg y x hr, angle_comm] #align inner_product_geometry.angle_smul_left_of_neg InnerProductGeometry.angle_smul_left_of_neg theorem cos_angle_mul_norm_mul_norm (x y : V) : Real.cos (angle x y) * (‖x‖ * ‖y‖) = ⟪x, y⟫ := by rw [cos_angle, div_mul_cancel_of_imp] simp (config := { contextual := true }) [or_imp] #align inner_product_geometry.cos_angle_mul_norm_mul_norm InnerProductGeometry.cos_angle_mul_norm_mul_norm theorem sin_angle_mul_norm_mul_norm (x y : V) : Real.sin (angle x y) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) := by unfold angle rw [Real.sin_arccos, ← Real.sqrt_mul_self (mul_nonneg (norm_nonneg x) (norm_nonneg y)), ← Real.sqrt_mul' _ (mul_self_nonneg _), sq, Real.sqrt_mul_self (mul_nonneg (norm_nonneg x) (norm_nonneg y)), real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm] by_cases h : ‖x‖ * ‖y‖ = 0 · rw [show ‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) = ‖x‖ * ‖y‖ * (‖x‖ * ‖y‖) by ring, h, mul_zero, mul_zero, zero_sub] cases' eq_zero_or_eq_zero_of_mul_eq_zero h with hx hy · rw [norm_eq_zero] at hx rw [hx, inner_zero_left, zero_mul, neg_zero] · rw [norm_eq_zero] at hy rw [hy, inner_zero_right, zero_mul, neg_zero] · field_simp [h] ring_nf #align inner_product_geometry.sin_angle_mul_norm_mul_norm InnerProductGeometry.sin_angle_mul_norm_mul_norm theorem angle_eq_zero_iff {x y : V} : angle x y = 0 ↔ x ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • x := by rw [angle, ← real_inner_div_norm_mul_norm_eq_one_iff, Real.arccos_eq_zero, LE.le.le_iff_eq, eq_comm] exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2 #align inner_product_geometry.angle_eq_zero_iff InnerProductGeometry.angle_eq_zero_iff theorem angle_eq_pi_iff {x y : V} : angle x y = π ↔ x ≠ 0 ∧ ∃ r : ℝ, r < 0 ∧ y = r • x := by rw [angle, ← real_inner_div_norm_mul_norm_eq_neg_one_iff, Real.arccos_eq_pi, LE.le.le_iff_eq] exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1 #align inner_product_geometry.angle_eq_pi_iff InnerProductGeometry.angle_eq_pi_iff theorem angle_add_angle_eq_pi_of_angle_eq_pi {x y : V} (z : V) (h : angle x y = π) : angle x z + angle y z = π := by rcases angle_eq_pi_iff.1 h with ⟨_, ⟨r, ⟨hr, rfl⟩⟩⟩ rw [angle_smul_left_of_neg x z hr, angle_neg_left, add_sub_cancel] #align inner_product_geometry.angle_add_angle_eq_pi_of_angle_eq_pi InnerProductGeometry.angle_add_angle_eq_pi_of_angle_eq_pi theorem inner_eq_zero_iff_angle_eq_pi_div_two (x y : V) : ⟪x, y⟫ = 0 ↔ angle x y = π / 2 := Iff.symm <\| by simp (config := { contextual := true }) [angle, or_imp] #align inner_product_geometry.inner_eq_zero_iff_angle_eq_pi_div_two InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two theorem inner_eq_neg_mul_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) : ⟪x, y⟫ = -(‖x‖ * ‖y‖) := by simp [← cos_angle_mul_norm_mul_norm, h] #align inner_product_geometry.inner_eq_neg_mul_norm_of_angle_eq_pi InnerProductGeometry.inner_eq_neg_mul_norm_of_angle_eq_pi theorem inner_eq_mul_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ⟪x, y⟫ = ‖x‖ * ‖y‖ := by simp [← cos_angle_mul_norm_mul_norm, h] #align inner_product_geometry.inner_eq_mul_norm_of_angle_eq_zero InnerProductGeometry.inner_eq_mul_norm_of_angle_eq_zero theorem inner_eq_neg_mul_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : ⟪x, y⟫ = -(‖x‖ * ‖y‖) ↔ angle x y = π := by refine ⟨fun h => ?_, inner_eq_neg_mul_norm_of_angle_eq_pi⟩ have h₁ : ‖x‖ * ‖y‖ ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne' rw [angle, h, neg_div, div_self h₁, Real.arccos_neg_one] #align inner_product_geometry.inner_eq_neg_mul_norm_iff_angle_eq_pi InnerProductGeometry.inner_eq_neg_mul_norm_iff_angle_eq_pi theorem inner_eq_mul_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : ⟪x, y⟫ = ‖x‖ * ‖y‖ ↔ angle x y = 0 := by refine ⟨fun h => ?_, inner_eq_mul_norm_of_angle_eq_zero⟩ have h₁ : ‖x‖ * ‖y‖ ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne' rw [angle, h, div_self h₁, Real.arccos_one] #align inner_product_geometry.inner_eq_mul_norm_iff_angle_eq_zero InnerProductGeometry.inner_eq_mul_norm_iff_angle_eq_zero theorem norm_sub_eq_add_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) : ‖x - y‖ = ‖x‖ + ‖y‖ := by rw [← sq_eq_sq (norm_nonneg (x - y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)), norm_sub_pow_two_real, inner_eq_neg_mul_norm_of_angle_eq_pi h] ring #align inner_product_geometry.norm_sub_eq_add_norm_of_angle_eq_pi InnerProductGeometry.norm_sub_eq_add_norm_of_angle_eq_pi	Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean	279	283	theorem norm_add_eq_add_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ‖x + y‖ = ‖x‖ + ‖y‖ := by	rw [← sq_eq_sq (norm_nonneg (x + y)) (add_nonneg (norm_nonneg x) (norm_nonneg y)), norm_add_pow_two_real, inner_eq_mul_norm_of_angle_eq_zero h] ring
import Mathlib.Algebra.CharZero.Lemmas import Mathlib.Order.Interval.Finset.Basic #align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29" open Finset Int namespace Int instance instLocallyFiniteOrder : LocallyFiniteOrder ℤ where finsetIcc a b := (Finset.range (b + 1 - a).toNat).map <\| Nat.castEmbedding.trans <\| addLeftEmbedding a finsetIco a b := (Finset.range (b - a).toNat).map <\| Nat.castEmbedding.trans <\| addLeftEmbedding a finsetIoc a b := (Finset.range (b - a).toNat).map <\| Nat.castEmbedding.trans <\| addLeftEmbedding (a + 1) finsetIoo a b := (Finset.range (b - a - 1).toNat).map <\| Nat.castEmbedding.trans <\| addLeftEmbedding (a + 1) finset_mem_Icc a b x := by simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply, Nat.castEmbedding_apply, addLeftEmbedding_apply] constructor · rintro ⟨a, h, rfl⟩ rw [lt_sub_iff_add_lt, Int.lt_add_one_iff, add_comm] at h exact ⟨Int.le.intro a rfl, h⟩ · rintro ⟨ha, hb⟩ use (x - a).toNat rw [← lt_add_one_iff] at hb rw [toNat_sub_of_le ha] exact ⟨sub_lt_sub_right hb _, add_sub_cancel _ _⟩ finset_mem_Ico a b x := by simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply, Nat.castEmbedding_apply, addLeftEmbedding_apply] constructor · rintro ⟨a, h, rfl⟩ exact ⟨Int.le.intro a rfl, lt_sub_iff_add_lt'.mp h⟩ · rintro ⟨ha, hb⟩ use (x - a).toNat rw [toNat_sub_of_le ha] exact ⟨sub_lt_sub_right hb _, add_sub_cancel _ _⟩ finset_mem_Ioc a b x := by simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply, Nat.castEmbedding_apply, addLeftEmbedding_apply] constructor · rintro ⟨a, h, rfl⟩ rw [← add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ← add_assoc] at h exact ⟨Int.le.intro a rfl, h⟩ · rintro ⟨ha, hb⟩ use (x - (a + 1)).toNat rw [toNat_sub_of_le ha, ← add_one_le_iff, sub_add, add_sub_cancel_right] exact ⟨sub_le_sub_right hb _, add_sub_cancel _ _⟩ finset_mem_Ioo a b x := by simp_rw [mem_map, mem_range, Int.lt_toNat, Function.Embedding.trans_apply, Nat.castEmbedding_apply, addLeftEmbedding_apply] constructor · rintro ⟨a, h, rfl⟩ rw [sub_sub, lt_sub_iff_add_lt'] at h exact ⟨Int.le.intro a rfl, h⟩ · rintro ⟨ha, hb⟩ use (x - (a + 1)).toNat rw [toNat_sub_of_le ha, sub_sub] exact ⟨sub_lt_sub_right hb _, add_sub_cancel _ _⟩ variable (a b : ℤ) theorem Icc_eq_finset_map : Icc a b = (Finset.range (b + 1 - a).toNat).map (Nat.castEmbedding.trans <\| addLeftEmbedding a) := rfl #align int.Icc_eq_finset_map Int.Icc_eq_finset_map theorem Ico_eq_finset_map : Ico a b = (Finset.range (b - a).toNat).map (Nat.castEmbedding.trans <\| addLeftEmbedding a) := rfl #align int.Ico_eq_finset_map Int.Ico_eq_finset_map theorem Ioc_eq_finset_map : Ioc a b = (Finset.range (b - a).toNat).map (Nat.castEmbedding.trans <\| addLeftEmbedding (a + 1)) := rfl #align int.Ioc_eq_finset_map Int.Ioc_eq_finset_map theorem Ioo_eq_finset_map : Ioo a b = (Finset.range (b - a - 1).toNat).map (Nat.castEmbedding.trans <\| addLeftEmbedding (a + 1)) := rfl #align int.Ioo_eq_finset_map Int.Ioo_eq_finset_map theorem uIcc_eq_finset_map : uIcc a b = (range (max a b + 1 - min a b).toNat).map (Nat.castEmbedding.trans <\| addLeftEmbedding <\| min a b) := rfl #align int.uIcc_eq_finset_map Int.uIcc_eq_finset_map @[simp] theorem card_Icc : (Icc a b).card = (b + 1 - a).toNat := (card_map _).trans <\| card_range _ #align int.card_Icc Int.card_Icc @[simp] theorem card_Ico : (Ico a b).card = (b - a).toNat := (card_map _).trans <\| card_range _ #align int.card_Ico Int.card_Ico @[simp] theorem card_Ioc : (Ioc a b).card = (b - a).toNat := (card_map _).trans <\| card_range _ #align int.card_Ioc Int.card_Ioc @[simp] theorem card_Ioo : (Ioo a b).card = (b - a - 1).toNat := (card_map _).trans <\| card_range _ #align int.card_Ioo Int.card_Ioo @[simp] theorem card_uIcc : (uIcc a b).card = (b - a).natAbs + 1 := (card_map _).trans <\| Int.ofNat.inj <\| by -- Porting note (#11215): TODO: Restore `int.coe_nat_inj` and remove the `change` change ((↑) : ℕ → ℤ) _ = ((↑) : ℕ → ℤ) _ rw [card_range, sup_eq_max, inf_eq_min, Int.toNat_of_nonneg (sub_nonneg_of_le <\| le_add_one min_le_max), Int.ofNat_add, Int.natCast_natAbs, add_comm, add_sub_assoc, max_sub_min_eq_abs, add_comm, Int.ofNat_one] #align int.card_uIcc Int.card_uIcc	Mathlib/Data/Int/Interval.lean	133	134	theorem card_Icc_of_le (h : a ≤ b + 1) : ((Icc a b).card : ℤ) = b + 1 - a := by	rw [card_Icc, toNat_sub_of_le h]
import Batteries.Data.RBMap.Alter import Batteries.Data.List.Lemmas 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	.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean	42	43	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]
import Mathlib.Algebra.Associated import Mathlib.Algebra.BigOperators.Finsupp #align_import algebra.big_operators.associated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c" variable {α β γ δ : Type} -- the same local notation used in `Algebra.Associated` local infixl:50 " ~ᵤ " => Associated theorem Prod.associated_iff {M N : Type} [Monoid M] [Monoid N] {x z : M × N} : x ~ᵤ z ↔ x.1 ~ᵤ z.1 ∧ x.2 ~ᵤ z.2 := ⟨fun ⟨u, hu⟩ => ⟨⟨(MulEquiv.prodUnits.toFun u).1, (Prod.eq_iff_fst_eq_snd_eq.1 hu).1⟩, ⟨(MulEquiv.prodUnits.toFun u).2, (Prod.eq_iff_fst_eq_snd_eq.1 hu).2⟩⟩, fun ⟨⟨u₁, h₁⟩, ⟨u₂, h₂⟩⟩ => ⟨MulEquiv.prodUnits.invFun (u₁, u₂), Prod.eq_iff_fst_eq_snd_eq.2 ⟨h₁, h₂⟩⟩⟩ theorem Associated.prod {M : Type} [CommMonoid M] {ι : Type} (s : Finset ι) (f : ι → M) (g : ι → M) (h : ∀ i, i ∈ s → (f i) ~ᵤ (g i)) : (∏ i ∈ s, f i) ~ᵤ (∏ i ∈ s, g i) := by induction s using Finset.induction with \| empty => simp only [Finset.prod_empty] rfl \| @insert j s hjs IH => classical convert_to (∏ i ∈ insert j s, f i) ~ᵤ (∏ i ∈ insert j s, g i) rw [Finset.prod_insert hjs, Finset.prod_insert hjs] exact Associated.mul_mul (h j (Finset.mem_insert_self j s)) (IH (fun i hi ↦ h i (Finset.mem_insert_of_mem hi))) theorem exists_associated_mem_of_dvd_prod [CancelCommMonoidWithZero α] {p : α} (hp : Prime p) {s : Multiset α} : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q := Multiset.induction_on s (by simp [mt isUnit_iff_dvd_one.2 hp.not_unit]) fun a s ih hs hps => by rw [Multiset.prod_cons] at hps cases' hp.dvd_or_dvd hps with h h · have hap := hs a (Multiset.mem_cons.2 (Or.inl rfl)) exact ⟨a, Multiset.mem_cons_self a _, hp.associated_of_dvd hap h⟩ · rcases ih (fun r hr => hs _ (Multiset.mem_cons.2 (Or.inr hr))) h with ⟨q, hq₁, hq₂⟩ exact ⟨q, Multiset.mem_cons.2 (Or.inr hq₁), hq₂⟩ #align exists_associated_mem_of_dvd_prod exists_associated_mem_of_dvd_prod	Mathlib/Algebra/BigOperators/Associated.lean	82	100	theorem Multiset.prod_primes_dvd [CancelCommMonoidWithZero α] [∀ a : α, DecidablePred (Associated a)] {s : Multiset α} (n : α) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ a, s.countP (Associated a) ≤ 1) : s.prod ∣ n := by	induction' s using Multiset.induction_on with a s induct n primes divs generalizing n · simp only [Multiset.prod_zero, one_dvd] · rw [Multiset.prod_cons] obtain ⟨k, rfl⟩ : a ∣ n := div a (Multiset.mem_cons_self a s) apply mul_dvd_mul_left a refine induct _ (fun a ha => h a (Multiset.mem_cons_of_mem ha)) (fun b b_in_s => ?_) fun a => (Multiset.countP_le_of_le _ (Multiset.le_cons_self _ _)).trans (uniq a) have b_div_n := div b (Multiset.mem_cons_of_mem b_in_s) have a_prime := h a (Multiset.mem_cons_self a s) have b_prime := h b (Multiset.mem_cons_of_mem b_in_s) refine (b_prime.dvd_or_dvd b_div_n).resolve_left fun b_div_a => ?_ have assoc := b_prime.associated_of_dvd a_prime b_div_a have := uniq a rw [Multiset.countP_cons_of_pos _ (Associated.refl _), Nat.succ_le_succ_iff, ← not_lt, Multiset.countP_pos] at this exact this ⟨b, b_in_s, assoc.symm⟩
import Mathlib.Analysis.Calculus.ContDiff.Bounds import Mathlib.Analysis.Calculus.IteratedDeriv.Defs import Mathlib.Analysis.Calculus.LineDeriv.Basic import Mathlib.Analysis.LocallyConvex.WithSeminorms import Mathlib.Analysis.Normed.Group.ZeroAtInfty import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.Analysis.SpecialFunctions.JapaneseBracket import Mathlib.Topology.Algebra.UniformFilterBasis import Mathlib.Tactic.MoveAdd #align_import analysis.schwartz_space from "leanprover-community/mathlib"@"e137999b2c6f2be388f4cd3bbf8523de1910cd2b" noncomputable section open scoped Nat NNReal variable {𝕜 𝕜' D E F G V : Type} variable [NormedAddCommGroup E] [NormedSpace ℝ E] variable [NormedAddCommGroup F] [NormedSpace ℝ F] variable (E F) structure SchwartzMap where toFun : E → F smooth' : ContDiff ℝ ⊤ toFun decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k ‖iteratedFDeriv ℝ n toFun x‖ ≤ C #align schwartz_map SchwartzMap scoped[SchwartzMap] notation "𝓢(" E ", " F ")" => SchwartzMap E F variable {E F} namespace SchwartzMap -- Porting note: removed -- instance : Coe 𝓢(E, F) (E → F) := ⟨toFun⟩ instance instFunLike : FunLike 𝓢(E, F) E F where coe f := f.toFun coe_injective' f g h := by cases f; cases g; congr #align schwartz_map.fun_like SchwartzMap.instFunLike instance instCoeFun : CoeFun 𝓢(E, F) fun _ => E → F := DFunLike.hasCoeToFun #align schwartz_map.has_coe_to_fun SchwartzMap.instCoeFun theorem decay (f : 𝓢(E, F)) (k n : ℕ) : ∃ C : ℝ, 0 < C ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C := by rcases f.decay' k n with ⟨C, hC⟩ exact ⟨max C 1, by positivity, fun x => (hC x).trans (le_max_left _ _)⟩ #align schwartz_map.decay SchwartzMap.decay theorem smooth (f : 𝓢(E, F)) (n : ℕ∞) : ContDiff ℝ n f := f.smooth'.of_le le_top #align schwartz_map.smooth SchwartzMap.smooth @[continuity] protected theorem continuous (f : 𝓢(E, F)) : Continuous f := (f.smooth 0).continuous #align schwartz_map.continuous SchwartzMap.continuous instance instContinuousMapClass : ContinuousMapClass 𝓢(E, F) E F where map_continuous := SchwartzMap.continuous protected theorem differentiable (f : 𝓢(E, F)) : Differentiable ℝ f := (f.smooth 1).differentiable rfl.le #align schwartz_map.differentiable SchwartzMap.differentiable protected theorem differentiableAt (f : 𝓢(E, F)) {x : E} : DifferentiableAt ℝ f x := f.differentiable.differentiableAt #align schwartz_map.differentiable_at SchwartzMap.differentiableAt @[ext] theorem ext {f g : 𝓢(E, F)} (h : ∀ x, (f : E → F) x = g x) : f = g := DFunLike.ext f g h #align schwartz_map.ext SchwartzMap.ext section IsBigO open Asymptotics Filter variable (f : 𝓢(E, F)) theorem isBigO_cocompact_zpow_neg_nat (k : ℕ) : f =O[cocompact E] fun x => ‖x‖ ^ (-k : ℤ) := by obtain ⟨d, _, hd'⟩ := f.decay k 0 simp only [norm_iteratedFDeriv_zero] at hd' simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith] refine ⟨d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite ?_⟩ refine (Filter.eventually_cofinite_ne 0).mono fun x hx => ?_ rw [Real.norm_of_nonneg (zpow_nonneg (norm_nonneg _) _), zpow_neg, ← div_eq_mul_inv, le_div_iff'] exacts [hd' x, zpow_pos_of_pos (norm_pos_iff.mpr hx) _] set_option linter.uppercaseLean3 false in #align schwartz_map.is_O_cocompact_zpow_neg_nat SchwartzMap.isBigO_cocompact_zpow_neg_nat	Mathlib/Analysis/Distribution/SchwartzSpace.lean	157	169	theorem isBigO_cocompact_rpow [ProperSpace E] (s : ℝ) : f =O[cocompact E] fun x => ‖x‖ ^ s := by	let k := ⌈-s⌉₊ have hk : -(k : ℝ) ≤ s := neg_le.mp (Nat.le_ceil (-s)) refine (isBigO_cocompact_zpow_neg_nat f k).trans ?_ suffices (fun x : ℝ => x ^ (-k : ℤ)) =O[atTop] fun x : ℝ => x ^ s from this.comp_tendsto tendsto_norm_cocompact_atTop simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith] refine ⟨1, (Filter.eventually_ge_atTop 1).mono fun x hx => ?_⟩ rw [one_mul, Real.norm_of_nonneg (Real.rpow_nonneg (zero_le_one.trans hx) _), Real.norm_of_nonneg (zpow_nonneg (zero_le_one.trans hx) _), ← Real.rpow_intCast, Int.cast_neg, Int.cast_natCast] exact Real.rpow_le_rpow_of_exponent_le hx hk
import Mathlib.Tactic.Qify import Mathlib.Data.ZMod.Basic import Mathlib.NumberTheory.DiophantineApproximation import Mathlib.NumberTheory.Zsqrtd.Basic #align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26" namespace Pell open Zsqrtd theorem is_pell_solution_iff_mem_unitary {d : ℤ} {a : ℤ√d} : a.re ^ 2 - d * a.im ^ 2 = 1 ↔ a ∈ unitary (ℤ√d) := by rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc] #align pell.is_pell_solution_iff_mem_unitary Pell.is_pell_solution_iff_mem_unitary -- We use `solution₁ d` to allow for a more general structure `solution d m` that -- encodes solutions to `x^2 - dy^2 = m` to be added later. def Solution₁ (d : ℤ) : Type := ↥(unitary (ℤ√d)) #align pell.solution₁ Pell.Solution₁ namespace Solution₁ variable {d : ℤ} -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5020): manual deriving instance instCommGroup : CommGroup (Solution₁ d) := inferInstanceAs (CommGroup (unitary (ℤ√d))) #align pell.solution₁.comm_group Pell.Solution₁.instCommGroup instance instHasDistribNeg : HasDistribNeg (Solution₁ d) := inferInstanceAs (HasDistribNeg (unitary (ℤ√d))) #align pell.solution₁.has_distrib_neg Pell.Solution₁.instHasDistribNeg instance instInhabited : Inhabited (Solution₁ d) := inferInstanceAs (Inhabited (unitary (ℤ√d))) #align pell.solution₁.inhabited Pell.Solution₁.instInhabited instance : Coe (Solution₁ d) (ℤ√d) where coe := Subtype.val protected def x (a : Solution₁ d) : ℤ := (a : ℤ√d).re #align pell.solution₁.x Pell.Solution₁.x protected def y (a : Solution₁ d) : ℤ := (a : ℤ√d).im #align pell.solution₁.y Pell.Solution₁.y theorem prop (a : Solution₁ d) : a.x ^ 2 - d a.y ^ 2 = 1 := is_pell_solution_iff_mem_unitary.mpr a.property #align pell.solution₁.prop Pell.Solution₁.prop theorem prop_x (a : Solution₁ d) : a.x ^ 2 = 1 + d * a.y ^ 2 := by rw [← a.prop]; ring #align pell.solution₁.prop_x Pell.Solution₁.prop_x theorem prop_y (a : Solution₁ d) : d * a.y ^ 2 = a.x ^ 2 - 1 := by rw [← a.prop]; ring #align pell.solution₁.prop_y Pell.Solution₁.prop_y @[ext] theorem ext {a b : Solution₁ d} (hx : a.x = b.x) (hy : a.y = b.y) : a = b := Subtype.ext <\| Zsqrtd.ext _ _ hx hy #align pell.solution₁.ext Pell.Solution₁.ext def mk (x y : ℤ) (prop : x ^ 2 - d * y ^ 2 = 1) : Solution₁ d where val := ⟨x, y⟩ property := is_pell_solution_iff_mem_unitary.mp prop #align pell.solution₁.mk Pell.Solution₁.mk @[simp] theorem x_mk (x y : ℤ) (prop : x ^ 2 - d * y ^ 2 = 1) : (mk x y prop).x = x := rfl #align pell.solution₁.x_mk Pell.Solution₁.x_mk @[simp] theorem y_mk (x y : ℤ) (prop : x ^ 2 - d * y ^ 2 = 1) : (mk x y prop).y = y := rfl #align pell.solution₁.y_mk Pell.Solution₁.y_mk @[simp] theorem coe_mk (x y : ℤ) (prop : x ^ 2 - d * y ^ 2 = 1) : (↑(mk x y prop) : ℤ√d) = ⟨x, y⟩ := Zsqrtd.ext _ _ (x_mk x y prop) (y_mk x y prop) #align pell.solution₁.coe_mk Pell.Solution₁.coe_mk @[simp] theorem x_one : (1 : Solution₁ d).x = 1 := rfl #align pell.solution₁.x_one Pell.Solution₁.x_one @[simp] theorem y_one : (1 : Solution₁ d).y = 0 := rfl #align pell.solution₁.y_one Pell.Solution₁.y_one @[simp] theorem x_mul (a b : Solution₁ d) : (a * b).x = a.x * b.x + d * (a.y * b.y) := by rw [← mul_assoc] rfl #align pell.solution₁.x_mul Pell.Solution₁.x_mul @[simp] theorem y_mul (a b : Solution₁ d) : (a * b).y = a.x * b.y + a.y * b.x := rfl #align pell.solution₁.y_mul Pell.Solution₁.y_mul @[simp] theorem x_inv (a : Solution₁ d) : a⁻¹.x = a.x := rfl #align pell.solution₁.x_inv Pell.Solution₁.x_inv @[simp] theorem y_inv (a : Solution₁ d) : a⁻¹.y = -a.y := rfl #align pell.solution₁.y_inv Pell.Solution₁.y_inv @[simp] theorem x_neg (a : Solution₁ d) : (-a).x = -a.x := rfl #align pell.solution₁.x_neg Pell.Solution₁.x_neg @[simp] theorem y_neg (a : Solution₁ d) : (-a).y = -a.y := rfl #align pell.solution₁.y_neg Pell.Solution₁.y_neg theorem eq_zero_of_d_neg (h₀ : d < 0) (a : Solution₁ d) : a.x = 0 ∨ a.y = 0 := by have h := a.prop contrapose! h have h1 := sq_pos_of_ne_zero h.1 have h2 := sq_pos_of_ne_zero h.2 nlinarith #align pell.solution₁.eq_zero_of_d_neg Pell.Solution₁.eq_zero_of_d_neg theorem x_ne_zero (h₀ : 0 ≤ d) (a : Solution₁ d) : a.x ≠ 0 := by intro hx have h : 0 ≤ d * a.y ^ 2 := mul_nonneg h₀ (sq_nonneg _) rw [a.prop_y, hx, sq, zero_mul, zero_sub] at h exact not_le.mpr (neg_one_lt_zero : (-1 : ℤ) < 0) h #align pell.solution₁.x_ne_zero Pell.Solution₁.x_ne_zero theorem y_ne_zero_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : a.y ≠ 0 := by intro hy have prop := a.prop rw [hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero] at prop exact lt_irrefl _ (((one_lt_sq_iff <\| zero_le_one.trans ha.le).mpr ha).trans_eq prop) #align pell.solution₁.y_ne_zero_of_one_lt_x Pell.Solution₁.y_ne_zero_of_one_lt_x theorem d_pos_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : 0 < d := by refine pos_of_mul_pos_left ?_ (sq_nonneg a.y) rw [a.prop_y, sub_pos] exact one_lt_pow ha two_ne_zero #align pell.solution₁.d_pos_of_one_lt_x Pell.Solution₁.d_pos_of_one_lt_x theorem d_nonsquare_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : ¬IsSquare d := by have hp := a.prop rintro ⟨b, rfl⟩ simp_rw [← sq, ← mul_pow, sq_sub_sq, Int.mul_eq_one_iff_eq_one_or_neg_one] at hp rcases hp with (⟨hp₁, hp₂⟩ \| ⟨hp₁, hp₂⟩) <;> omega #align pell.solution₁.d_nonsquare_of_one_lt_x Pell.Solution₁.d_nonsquare_of_one_lt_x theorem eq_one_of_x_eq_one (h₀ : d ≠ 0) {a : Solution₁ d} (ha : a.x = 1) : a = 1 := by have prop := a.prop_y rw [ha, one_pow, sub_self, mul_eq_zero, or_iff_right h₀, sq_eq_zero_iff] at prop exact ext ha prop #align pell.solution₁.eq_one_of_x_eq_one Pell.Solution₁.eq_one_of_x_eq_one	Mathlib/NumberTheory/Pell.lean	256	260	theorem eq_one_or_neg_one_iff_y_eq_zero {a : Solution₁ d} : a = 1 ∨ a = -1 ↔ a.y = 0 := by	refine ⟨fun H => H.elim (fun h => by simp [h]) fun h => by simp [h], fun H => ?_⟩ have prop := a.prop rw [H, sq (0 : ℤ), mul_zero, mul_zero, sub_zero, sq_eq_one_iff] at prop exact prop.imp (fun h => ext h H) fun h => ext h H
import Mathlib.Topology.ContinuousOn import Mathlib.Order.Minimal open Set Classical variable {X : Type} {Y : Type} [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X} section Preirreducible def IsPreirreducible (s : Set X) : Prop := ∀ u v : Set X, IsOpen u → IsOpen v → (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty #align is_preirreducible IsPreirreducible def IsIrreducible (s : Set X) : Prop := s.Nonempty ∧ IsPreirreducible s #align is_irreducible IsIrreducible theorem IsIrreducible.nonempty (h : IsIrreducible s) : s.Nonempty := h.1 #align is_irreducible.nonempty IsIrreducible.nonempty theorem IsIrreducible.isPreirreducible (h : IsIrreducible s) : IsPreirreducible s := h.2 #align is_irreducible.is_preirreducible IsIrreducible.isPreirreducible theorem isPreirreducible_empty : IsPreirreducible (∅ : Set X) := fun _ _ _ _ _ ⟨_, h1, _⟩ => h1.elim #align is_preirreducible_empty isPreirreducible_empty theorem Set.Subsingleton.isPreirreducible (hs : s.Subsingleton) : IsPreirreducible s := fun _u _v _ _ ⟨_x, hxs, hxu⟩ ⟨y, hys, hyv⟩ => ⟨y, hys, hs hxs hys ▸ hxu, hyv⟩ #align set.subsingleton.is_preirreducible Set.Subsingleton.isPreirreducible -- Porting note (#10756): new lemma theorem isPreirreducible_singleton {x} : IsPreirreducible ({x} : Set X) := subsingleton_singleton.isPreirreducible theorem isIrreducible_singleton {x} : IsIrreducible ({x} : Set X) := ⟨singleton_nonempty x, isPreirreducible_singleton⟩ #align is_irreducible_singleton isIrreducible_singleton theorem isPreirreducible_iff_closure : IsPreirreducible (closure s) ↔ IsPreirreducible s := forall₄_congr fun u v hu hv => by iterate 3 rw [closure_inter_open_nonempty_iff] exacts [hu.inter hv, hv, hu] #align is_preirreducible_iff_closure isPreirreducible_iff_closure theorem isIrreducible_iff_closure : IsIrreducible (closure s) ↔ IsIrreducible s := and_congr closure_nonempty_iff isPreirreducible_iff_closure #align is_irreducible_iff_closure isIrreducible_iff_closure protected alias ⟨_, IsPreirreducible.closure⟩ := isPreirreducible_iff_closure #align is_preirreducible.closure IsPreirreducible.closure protected alias ⟨_, IsIrreducible.closure⟩ := isIrreducible_iff_closure #align is_irreducible.closure IsIrreducible.closure theorem exists_preirreducible (s : Set X) (H : IsPreirreducible s) : ∃ t : Set X, IsPreirreducible t ∧ s ⊆ t ∧ ∀ u, IsPreirreducible u → t ⊆ u → u = t := let ⟨m, hm, hsm, hmm⟩ := zorn_subset_nonempty { t : Set X \| IsPreirreducible t } (fun c hc hcc _ => ⟨⋃₀ c, fun u v hu hv ⟨y, hy, hyu⟩ ⟨x, hx, hxv⟩ => let ⟨p, hpc, hyp⟩ := mem_sUnion.1 hy let ⟨q, hqc, hxq⟩ := mem_sUnion.1 hx Or.casesOn (hcc.total hpc hqc) (fun hpq : p ⊆ q => let ⟨x, hxp, hxuv⟩ := hc hqc u v hu hv ⟨y, hpq hyp, hyu⟩ ⟨x, hxq, hxv⟩ ⟨x, mem_sUnion_of_mem hxp hqc, hxuv⟩) fun hqp : q ⊆ p => let ⟨x, hxp, hxuv⟩ := hc hpc u v hu hv ⟨y, hyp, hyu⟩ ⟨x, hqp hxq, hxv⟩ ⟨x, mem_sUnion_of_mem hxp hpc, hxuv⟩, fun _ hxc => subset_sUnion_of_mem hxc⟩) s H ⟨m, hm, hsm, fun _u hu hmu => hmm _ hu hmu⟩ #align exists_preirreducible exists_preirreducible def irreducibleComponents (X : Type) [TopologicalSpace X] : Set (Set X) := maximals (· ≤ ·) { s : Set X \| IsIrreducible s } #align irreducible_components irreducibleComponents theorem isClosed_of_mem_irreducibleComponents (s) (H : s ∈ irreducibleComponents X) : IsClosed s := by rw [← closure_eq_iff_isClosed, eq_comm] exact subset_closure.antisymm (H.2 H.1.closure subset_closure) #align is_closed_of_mem_irreducible_components isClosed_of_mem_irreducibleComponents theorem irreducibleComponents_eq_maximals_closed (X : Type) [TopologicalSpace X] : irreducibleComponents X = maximals (· ≤ ·) { s : Set X \| IsClosed s ∧ IsIrreducible s } := by ext s constructor · intro H exact ⟨⟨isClosed_of_mem_irreducibleComponents _ H, H.1⟩, fun x h e => H.2 h.2 e⟩ · intro H refine ⟨H.1.2, fun x h e => ?_⟩ have : closure x ≤ s := H.2 ⟨isClosed_closure, h.closure⟩ (e.trans subset_closure) exact le_trans subset_closure this #align irreducible_components_eq_maximals_closed irreducibleComponents_eq_maximals_closed def irreducibleComponent (x : X) : Set X := Classical.choose (exists_preirreducible {x} isPreirreducible_singleton) #align irreducible_component irreducibleComponent theorem irreducibleComponent_property (x : X) : IsPreirreducible (irreducibleComponent x) ∧ {x} ⊆ irreducibleComponent x ∧ ∀ u, IsPreirreducible u → irreducibleComponent x ⊆ u → u = irreducibleComponent x := Classical.choose_spec (exists_preirreducible {x} isPreirreducible_singleton) #align irreducible_component_property irreducibleComponent_property theorem mem_irreducibleComponent {x : X} : x ∈ irreducibleComponent x := singleton_subset_iff.1 (irreducibleComponent_property x).2.1 #align mem_irreducible_component mem_irreducibleComponent theorem isIrreducible_irreducibleComponent {x : X} : IsIrreducible (irreducibleComponent x) := ⟨⟨x, mem_irreducibleComponent⟩, (irreducibleComponent_property x).1⟩ #align is_irreducible_irreducible_component isIrreducible_irreducibleComponent theorem eq_irreducibleComponent {x : X} : IsPreirreducible s → irreducibleComponent x ⊆ s → s = irreducibleComponent x := (irreducibleComponent_property x).2.2 _ #align eq_irreducible_component eq_irreducibleComponent theorem irreducibleComponent_mem_irreducibleComponents (x : X) : irreducibleComponent x ∈ irreducibleComponents X := ⟨isIrreducible_irreducibleComponent, fun _ h₁ h₂ => (eq_irreducibleComponent h₁.2 h₂).le⟩ #align irreducible_component_mem_irreducible_components irreducibleComponent_mem_irreducibleComponents theorem isClosed_irreducibleComponent {x : X} : IsClosed (irreducibleComponent x) := isClosed_of_mem_irreducibleComponents _ (irreducibleComponent_mem_irreducibleComponents x) #align is_closed_irreducible_component isClosed_irreducibleComponent class PreirreducibleSpace (X : Type) [TopologicalSpace X] : Prop where isPreirreducible_univ : IsPreirreducible (univ : Set X) #align preirreducible_space PreirreducibleSpace class IrreducibleSpace (X : Type) [TopologicalSpace X] extends PreirreducibleSpace X : Prop where toNonempty : Nonempty X #align irreducible_space IrreducibleSpace -- see Note [lower instance priority] attribute [instance 50] IrreducibleSpace.toNonempty theorem IrreducibleSpace.isIrreducible_univ (X : Type) [TopologicalSpace X] [IrreducibleSpace X] : IsIrreducible (univ : Set X) := ⟨univ_nonempty, PreirreducibleSpace.isPreirreducible_univ⟩ #align irreducible_space.is_irreducible_univ IrreducibleSpace.isIrreducible_univ theorem irreducibleSpace_def (X : Type) [TopologicalSpace X] : IrreducibleSpace X ↔ IsIrreducible (⊤ : Set X) := ⟨@IrreducibleSpace.isIrreducible_univ X _, fun h => haveI : PreirreducibleSpace X := ⟨h.2⟩ ⟨⟨h.1.some⟩⟩⟩ #align irreducible_space_def irreducibleSpace_def theorem nonempty_preirreducible_inter [PreirreducibleSpace X] : IsOpen s → IsOpen t → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty := by simpa only [univ_inter, univ_subset_iff] using @PreirreducibleSpace.isPreirreducible_univ X _ _ s t #align nonempty_preirreducible_inter nonempty_preirreducible_inter protected theorem IsOpen.dense [PreirreducibleSpace X] (ho : IsOpen s) (hne : s.Nonempty) : Dense s := dense_iff_inter_open.2 fun _t hto htne => nonempty_preirreducible_inter hto ho htne hne #align is_open.dense IsOpen.dense theorem IsPreirreducible.image (H : IsPreirreducible s) (f : X → Y) (hf : ContinuousOn f s) : IsPreirreducible (f '' s) := by rintro u v hu hv ⟨_, ⟨⟨x, hx, rfl⟩, hxu⟩⟩ ⟨_, ⟨⟨y, hy, rfl⟩, hyv⟩⟩ rw [← mem_preimage] at hxu hyv rcases continuousOn_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩ rcases continuousOn_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩ have := H u' v' hu' hv' rw [inter_comm s u', ← u'_eq] at this rw [inter_comm s v', ← v'_eq] at this rcases this ⟨x, hxu, hx⟩ ⟨y, hyv, hy⟩ with ⟨x, hxs, hxu', hxv'⟩ refine ⟨f x, mem_image_of_mem f hxs, ?_, ?_⟩ all_goals rw [← mem_preimage] apply mem_of_mem_inter_left show x ∈ _ ∩ s simp [*] #align is_preirreducible.image IsPreirreducible.image theorem IsIrreducible.image (H : IsIrreducible s) (f : X → Y) (hf : ContinuousOn f s) : IsIrreducible (f '' s) := ⟨H.nonempty.image _, H.isPreirreducible.image f hf⟩ #align is_irreducible.image IsIrreducible.image theorem Subtype.preirreducibleSpace (h : IsPreirreducible s) : PreirreducibleSpace s where isPreirreducible_univ := by rintro _ _ ⟨u, hu, rfl⟩ ⟨v, hv, rfl⟩ ⟨⟨x, hxs⟩, -, hxu⟩ ⟨⟨y, hys⟩, -, hyv⟩ rcases h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ with ⟨x, hxs, ⟨hxu, hxv⟩⟩ exact ⟨⟨x, hxs⟩, ⟨Set.mem_univ _, ⟨hxu, hxv⟩⟩⟩ #align subtype.preirreducible_space Subtype.preirreducibleSpace theorem Subtype.irreducibleSpace (h : IsIrreducible s) : IrreducibleSpace s where isPreirreducible_univ := (Subtype.preirreducibleSpace h.isPreirreducible).isPreirreducible_univ toNonempty := h.nonempty.to_subtype #align subtype.irreducible_space Subtype.irreducibleSpace instance (priority := 100) {X} [Infinite X] : IrreducibleSpace (CofiniteTopology X) where isPreirreducible_univ u v := by haveI : Infinite (CofiniteTopology X) := ‹_› simp only [CofiniteTopology.isOpen_iff, univ_inter] intro hu hv hu' hv' simpa only [compl_union, compl_compl] using ((hu hu').union (hv hv')).infinite_compl.nonempty toNonempty := (inferInstance : Nonempty X)	Mathlib/Topology/Irreducible.lean	252	265	theorem isIrreducible_iff_sInter : IsIrreducible s ↔ ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty := by	refine ⟨fun h U hu hU => ?_, fun h => ⟨?_, ?_⟩⟩ · induction U using Finset.induction_on with \| empty => simpa using h.nonempty \| @insert u U _ IH => rw [Finset.coe_insert, sInter_insert] rw [Finset.forall_mem_insert] at hu hU exact h.2 _ _ hu.1 (U.finite_toSet.isOpen_sInter hu.2) hU.1 (IH hu.2 hU.2) · simpa using h ∅ · intro u v hu hv hu' hv' simpa [*] using h {u, v}
import Mathlib.Data.Nat.Choose.Dvd import Mathlib.RingTheory.IntegrallyClosed import Mathlib.RingTheory.Norm import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand #align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf1682e59106d6c942434a32" universe u v w z variable {R : Type u} open Ideal Algebra Finset open scoped Polynomial section Cyclotomic variable (p : ℕ) local notation "𝓟" => Submodule.span ℤ {(p : ℤ)} open Polynomial	Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean	44	73	theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] : ((cyclotomic p ℤ).comp (X + 1)).IsEisensteinAt 𝓟 := by	refine Monic.isEisensteinAt_of_mem_of_not_mem ?_ (Ideal.IsPrime.ne_top <\| (Ideal.span_singleton_prime (mod_cast hp.out.ne_zero)).2 <\| Nat.prime_iff_prime_int.1 hp.out) (fun {i hi} => ?_) ?_ · rw [show (X + 1 : ℤ[X]) = X + C 1 by simp] refine (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => ?_ rw [natDegree_X_add_C] at h exact zero_ne_one h.symm · rw [cyclotomic_prime, geom_sum_X_comp_X_add_one_eq_sum, ← lcoeff_apply, map_sum] conv => congr congr next => skip ext rw [lcoeff_apply, ← C_eq_natCast, C_mul_X_pow_eq_monomial, coeff_monomial] rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one, natDegree_cyclotomic, Nat.totient_prime hp.out] at hi simp only [hi.trans_le (Nat.sub_le _ _), sum_ite_eq', mem_range, if_true, Ideal.submodule_span_eq, Ideal.mem_span_singleton, Int.natCast_dvd_natCast] exact hp.out.dvd_choose_self i.succ_ne_zero (lt_tsub_iff_right.1 hi) · rw [coeff_zero_eq_eval_zero, eval_comp, cyclotomic_prime, eval_add, eval_X, eval_one, zero_add, eval_geom_sum, one_geom_sum, Ideal.submodule_span_eq, Ideal.span_singleton_pow, Ideal.mem_span_singleton] intro h obtain ⟨k, hk⟩ := Int.natCast_dvd_natCast.1 h rw [mul_assoc, mul_comm 1, mul_one] at hk nth_rw 1 [← Nat.mul_one p] at hk rw [mul_right_inj' hp.out.ne_zero] at hk exact Nat.Prime.not_dvd_one hp.out (Dvd.intro k hk.symm)
import Mathlib.Analysis.SpecialFunctions.Complex.Log #align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8" open scoped Classical open Real Topology Filter ComplexConjugate Finset Set namespace Complex noncomputable def cpow (x y : ℂ) : ℂ := if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) #align complex.cpow Complex.cpow noncomputable instance : Pow ℂ ℂ := ⟨cpow⟩ @[simp] theorem cpow_eq_pow (x y : ℂ) : cpow x y = x ^ y := rfl #align complex.cpow_eq_pow Complex.cpow_eq_pow theorem cpow_def (x y : ℂ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) := rfl #align complex.cpow_def Complex.cpow_def theorem cpow_def_of_ne_zero {x : ℂ} (hx : x ≠ 0) (y : ℂ) : x ^ y = exp (log x * y) := if_neg hx #align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zero @[simp] theorem cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 := by simp [cpow_def] #align complex.cpow_zero Complex.cpow_zero @[simp] theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by simp only [cpow_def] split_ifs <;> simp [, exp_ne_zero] #align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iff @[simp] theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, ] #align complex.zero_cpow Complex.zero_cpow theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : (0 : ℂ) ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by constructor · intro hyp simp only [cpow_def, eq_self_iff_true, if_true] at hyp by_cases h : x = 0 · subst h simp only [if_true, eq_self_iff_true] at hyp right exact ⟨rfl, hyp.symm⟩ · rw [if_neg h] at hyp left exact ⟨h, hyp.symm⟩ · rintro (⟨h, rfl⟩ \| ⟨rfl, rfl⟩) · exact zero_cpow h · exact cpow_zero _ #align complex.zero_cpow_eq_iff Complex.zero_cpow_eq_iff theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = (0 : ℂ) ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by rw [← zero_cpow_eq_iff, eq_comm] #align complex.eq_zero_cpow_iff Complex.eq_zero_cpow_iff @[simp] theorem cpow_one (x : ℂ) : x ^ (1 : ℂ) = x := if hx : x = 0 then by simp [hx, cpow_def] else by rw [cpow_def, if_neg (one_ne_zero : (1 : ℂ) ≠ 0), if_neg hx, mul_one, exp_log hx] #align complex.cpow_one Complex.cpow_one @[simp] theorem one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 := by rw [cpow_def] split_ifs <;> simp_all [one_ne_zero] #align complex.one_cpow Complex.one_cpow theorem cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z := by simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole] simp_all [exp_add, mul_add] #align complex.cpow_add Complex.cpow_add	Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean	96	99	theorem cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) : x ^ (y * z) = (x ^ y) ^ z := by	simp only [cpow_def] split_ifs <;> simp_all [exp_ne_zero, log_exp h₁ h₂, mul_assoc]
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral import Mathlib.Analysis.Complex.CauchyIntegral import Mathlib.MeasureTheory.Integral.Pi import Mathlib.Analysis.Fourier.FourierTransform open Real Set MeasureTheory Filter Asymptotics intervalIntegral open scoped Real Topology FourierTransform RealInnerProductSpace open Complex hiding exp continuous_exp abs_of_nonneg sq_abs noncomputable section namespace GaussianFourier variable {b : ℂ} def verticalIntegral (b : ℂ) (c T : ℝ) : ℂ := ∫ y : ℝ in (0 : ℝ)..c, I * (cexp (-b * (T + y * I) ^ 2) - cexp (-b * (T - y * I) ^ 2)) #align gaussian_fourier.vertical_integral GaussianFourier.verticalIntegral theorem norm_cexp_neg_mul_sq_add_mul_I (b : ℂ) (c T : ℝ) : ‖cexp (-b * (T + c * I) ^ 2)‖ = exp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2)) := by rw [Complex.norm_eq_abs, Complex.abs_exp, neg_mul, neg_re, ← re_add_im b] simp only [sq, re_add_im, mul_re, mul_im, add_re, add_im, ofReal_re, ofReal_im, I_re, I_im] ring_nf set_option linter.uppercaseLean3 false in #align gaussian_fourier.norm_cexp_neg_mul_sq_add_mul_I GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I	Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean	59	66	theorem norm_cexp_neg_mul_sq_add_mul_I' (hb : b.re ≠ 0) (c T : ℝ) : ‖cexp (-b * (T + c * I) ^ 2)‖ = exp (-(b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re))) := by	have : b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2 = b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re) := by field_simp; ring rw [norm_cexp_neg_mul_sq_add_mul_I, this]
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic #align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9" open Nat section Euler section Legendre open ZMod variable (p : ℕ) [Fact p.Prime] def legendreSym (a : ℤ) : ℤ := quadraticChar (ZMod p) a #align legendre_sym legendreSym section Values variable {p : ℕ} [Fact p.Prime] open ZMod theorem legendreSym.at_neg_one (hp : p ≠ 2) : legendreSym p (-1) = χ₄ p := by simp only [legendreSym, card p, quadraticChar_neg_one ((ringChar_zmod_n p).substr hp), Int.cast_neg, Int.cast_one] #align legendre_sym.at_neg_one legendreSym.at_neg_one namespace ZMod	Mathlib/NumberTheory/LegendreSymbol/Basic.lean	302	303	theorem exists_sq_eq_neg_one_iff : IsSquare (-1 : ZMod p) ↔ p % 4 ≠ 3 := by	rw [FiniteField.isSquare_neg_one_iff, card p]
import Mathlib.Init.Function import Mathlib.Init.Order.Defs #align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23" namespace Bool @[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true #align bool.to_bool_true decide_true_eq_true @[deprecated (since := "2024-06-07")] alias decide_False := decide_false_eq_false #align bool.to_bool_false decide_false_eq_false #align bool.to_bool_coe Bool.decide_coe @[deprecated (since := "2024-06-07")] alias coe_decide := decide_eq_true_iff #align bool.coe_to_bool decide_eq_true_iff @[deprecated decide_eq_true_iff (since := "2024-06-07")] alias of_decide_iff := decide_eq_true_iff #align bool.of_to_bool_iff decide_eq_true_iff #align bool.tt_eq_to_bool_iff true_eq_decide_iff #align bool.ff_eq_to_bool_iff false_eq_decide_iff @[deprecated (since := "2024-06-07")] alias decide_not := decide_not #align bool.to_bool_not decide_not #align bool.to_bool_and Bool.decide_and #align bool.to_bool_or Bool.decide_or #align bool.to_bool_eq decide_eq_decide @[deprecated (since := "2024-06-07")] alias not_false' := false_ne_true #align bool.not_ff Bool.false_ne_true @[deprecated (since := "2024-06-07")] alias eq_iff_eq_true_iff := eq_iff_iff #align bool.default_bool Bool.default_bool theorem dichotomy (b : Bool) : b = false ∨ b = true := by cases b <;> simp #align bool.dichotomy Bool.dichotomy theorem forall_bool' {p : Bool → Prop} (b : Bool) : (∀ x, p x) ↔ p b ∧ p !b := ⟨fun h ↦ ⟨h _, h _⟩, fun ⟨h₁, h₂⟩ x ↦ by cases b <;> cases x <;> assumption⟩ @[simp] theorem forall_bool {p : Bool → Prop} : (∀ b, p b) ↔ p false ∧ p true := forall_bool' false #align bool.forall_bool Bool.forall_bool theorem exists_bool' {p : Bool → Prop} (b : Bool) : (∃ x, p x) ↔ p b ∨ p !b := ⟨fun ⟨x, hx⟩ ↦ by cases x <;> cases b <;> first \| exact .inl ‹_› \| exact .inr ‹_›, fun h ↦ by cases h <;> exact ⟨_, ‹_›⟩⟩ @[simp] theorem exists_bool {p : Bool → Prop} : (∃ b, p b) ↔ p false ∨ p true := exists_bool' false #align bool.exists_bool Bool.exists_bool #align bool.decidable_forall_bool Bool.instDecidableForallOfDecidablePred #align bool.decidable_exists_bool Bool.instDecidableExistsOfDecidablePred #align bool.cond_eq_ite Bool.cond_eq_ite #align bool.cond_to_bool Bool.cond_decide #align bool.cond_bnot Bool.cond_not theorem not_ne_id : not ≠ id := fun h ↦ false_ne_true <\| congrFun h true #align bool.bnot_ne_id Bool.not_ne_id #align bool.coe_bool_iff Bool.coe_iff_coe @[deprecated (since := "2024-06-07")] alias eq_true_of_ne_false := eq_true_of_ne_false #align bool.eq_tt_of_ne_ff eq_true_of_ne_false @[deprecated (since := "2024-06-07")] alias eq_false_of_ne_true := eq_false_of_ne_true #align bool.eq_ff_of_ne_tt eq_true_of_ne_false #align bool.bor_comm Bool.or_comm #align bool.bor_assoc Bool.or_assoc #align bool.bor_left_comm Bool.or_left_comm theorem or_inl {a b : Bool} (H : a) : a \|\| b := by simp [H] #align bool.bor_inl Bool.or_inl theorem or_inr {a b : Bool} (H : b) : a \|\| b := by cases a <;> simp [H] #align bool.bor_inr Bool.or_inr #align bool.band_comm Bool.and_comm #align bool.band_assoc Bool.and_assoc #align bool.band_left_comm Bool.and_left_comm theorem and_elim_left : ∀ {a b : Bool}, a && b → a := by decide #align bool.band_elim_left Bool.and_elim_left theorem and_intro : ∀ {a b : Bool}, a → b → a && b := by decide #align bool.band_intro Bool.and_intro theorem and_elim_right : ∀ {a b : Bool}, a && b → b := by decide #align bool.band_elim_right Bool.and_elim_right #align bool.band_bor_distrib_left Bool.and_or_distrib_left #align bool.band_bor_distrib_right Bool.and_or_distrib_right #align bool.bor_band_distrib_left Bool.or_and_distrib_left #align bool.bor_band_distrib_right Bool.or_and_distrib_right #align bool.bnot_ff Bool.not_false #align bool.bnot_tt Bool.not_true lemma eq_not_iff : ∀ {a b : Bool}, a = !b ↔ a ≠ b := by decide #align bool.eq_bnot_iff Bool.eq_not_iff lemma not_eq_iff : ∀ {a b : Bool}, !a = b ↔ a ≠ b := by decide #align bool.bnot_eq_iff Bool.not_eq_iff #align bool.not_eq_bnot Bool.not_eq_not #align bool.bnot_not_eq Bool.not_not_eq theorem ne_not {a b : Bool} : a ≠ !b ↔ a = b := not_eq_not #align bool.ne_bnot Bool.ne_not @[deprecated (since := "2024-06-07")] alias not_ne := not_not_eq #align bool.bnot_ne Bool.not_not_eq lemma not_ne_self : ∀ b : Bool, (!b) ≠ b := by decide #align bool.bnot_ne_self Bool.not_ne_self lemma self_ne_not : ∀ b : Bool, b ≠ !b := by decide #align bool.self_ne_bnot Bool.self_ne_not lemma eq_or_eq_not : ∀ a b, a = b ∨ a = !b := by decide #align bool.eq_or_eq_bnot Bool.eq_or_eq_not -- Porting note: naming issue again: these two `not` are different. theorem not_iff_not : ∀ {b : Bool}, !b ↔ ¬b := by simp #align bool.bnot_iff_not Bool.not_iff_not theorem eq_true_of_not_eq_false' {a : Bool} : !a = false → a = true := by cases a <;> decide #align bool.eq_tt_of_bnot_eq_ff Bool.eq_true_of_not_eq_false' theorem eq_false_of_not_eq_true' {a : Bool} : !a = true → a = false := by cases a <;> decide #align bool.eq_ff_of_bnot_eq_tt Bool.eq_false_of_not_eq_true' #align bool.band_bnot_self Bool.and_not_self #align bool.bnot_band_self Bool.not_and_self #align bool.bor_bnot_self Bool.or_not_self #align bool.bnot_bor_self Bool.not_or_self	Mathlib/Data/Bool/Basic.lean	167	167	theorem bne_eq_xor : bne = xor := by	funext a b; revert a b; decide
import Mathlib.Algebra.Group.Equiv.Basic import Mathlib.Data.ENat.Lattice import Mathlib.Data.Part import Mathlib.Tactic.NormNum #align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8" open Part hiding some def PartENat : Type := Part ℕ #align part_enat PartENat namespace PartENat @[coe] def some : ℕ → PartENat := Part.some #align part_enat.some PartENat.some instance : Zero PartENat := ⟨some 0⟩ instance : Inhabited PartENat := ⟨0⟩ instance : One PartENat := ⟨some 1⟩ instance : Add PartENat := ⟨fun x y => ⟨x.Dom ∧ y.Dom, fun h => get x h.1 + get y h.2⟩⟩ instance (n : ℕ) : Decidable (some n).Dom := isTrue trivial @[simp] theorem dom_some (x : ℕ) : (some x).Dom := trivial #align part_enat.dom_some PartENat.dom_some instance addCommMonoid : AddCommMonoid PartENat where add := (· + ·) zero := 0 add_comm x y := Part.ext' and_comm fun _ _ => add_comm _ _ zero_add x := Part.ext' (true_and_iff _) fun _ _ => zero_add _ add_zero x := Part.ext' (and_true_iff _) fun _ _ => add_zero _ add_assoc x y z := Part.ext' and_assoc fun _ _ => add_assoc _ _ _ nsmul := nsmulRec instance : AddCommMonoidWithOne PartENat := { PartENat.addCommMonoid with one := 1 natCast := some natCast_zero := rfl natCast_succ := fun _ => Part.ext' (true_and_iff _).symm fun _ _ => rfl } theorem some_eq_natCast (n : ℕ) : some n = n := rfl #align part_enat.some_eq_coe PartENat.some_eq_natCast instance : CharZero PartENat where cast_injective := Part.some_injective theorem natCast_inj {x y : ℕ} : (x : PartENat) = y ↔ x = y := Nat.cast_inj #align part_enat.coe_inj PartENat.natCast_inj @[simp] theorem dom_natCast (x : ℕ) : (x : PartENat).Dom := trivial #align part_enat.dom_coe PartENat.dom_natCast -- See note [no_index around OfNat.ofNat] @[simp] theorem dom_ofNat (x : ℕ) [x.AtLeastTwo] : (no_index (OfNat.ofNat x : PartENat)).Dom := trivial @[simp] theorem dom_zero : (0 : PartENat).Dom := trivial @[simp] theorem dom_one : (1 : PartENat).Dom := trivial instance : CanLift PartENat ℕ (↑) Dom := ⟨fun n hn => ⟨n.get hn, Part.some_get _⟩⟩ instance : LE PartENat := ⟨fun x y => ∃ h : y.Dom → x.Dom, ∀ hy : y.Dom, x.get (h hy) ≤ y.get hy⟩ instance : Top PartENat := ⟨none⟩ instance : Bot PartENat := ⟨0⟩ instance : Sup PartENat := ⟨fun x y => ⟨x.Dom ∧ y.Dom, fun h => x.get h.1 ⊔ y.get h.2⟩⟩ theorem le_def (x y : PartENat) : x ≤ y ↔ ∃ h : y.Dom → x.Dom, ∀ hy : y.Dom, x.get (h hy) ≤ y.get hy := Iff.rfl #align part_enat.le_def PartENat.le_def @[elab_as_elim] protected theorem casesOn' {P : PartENat → Prop} : ∀ a : PartENat, P ⊤ → (∀ n : ℕ, P (some n)) → P a := Part.induction_on #align part_enat.cases_on' PartENat.casesOn' @[elab_as_elim] protected theorem casesOn {P : PartENat → Prop} : ∀ a : PartENat, P ⊤ → (∀ n : ℕ, P n) → P a := by exact PartENat.casesOn' #align part_enat.cases_on PartENat.casesOn -- not a simp lemma as we will provide a `LinearOrderedAddCommMonoidWithTop` instance later theorem top_add (x : PartENat) : ⊤ + x = ⊤ := Part.ext' (false_and_iff _) fun h => h.left.elim #align part_enat.top_add PartENat.top_add -- not a simp lemma as we will provide a `LinearOrderedAddCommMonoidWithTop` instance later theorem add_top (x : PartENat) : x + ⊤ = ⊤ := by rw [add_comm, top_add] #align part_enat.add_top PartENat.add_top @[simp] theorem natCast_get {x : PartENat} (h : x.Dom) : (x.get h : PartENat) = x := by exact Part.ext' (iff_of_true trivial h) fun _ _ => rfl #align part_enat.coe_get PartENat.natCast_get @[simp, norm_cast] theorem get_natCast' (x : ℕ) (h : (x : PartENat).Dom) : get (x : PartENat) h = x := by rw [← natCast_inj, natCast_get] #align part_enat.get_coe' PartENat.get_natCast' theorem get_natCast {x : ℕ} : get (x : PartENat) (dom_natCast x) = x := get_natCast' _ _ #align part_enat.get_coe PartENat.get_natCast theorem coe_add_get {x : ℕ} {y : PartENat} (h : ((x : PartENat) + y).Dom) : get ((x : PartENat) + y) h = x + get y h.2 := by rfl #align part_enat.coe_add_get PartENat.coe_add_get @[simp] theorem get_add {x y : PartENat} (h : (x + y).Dom) : get (x + y) h = x.get h.1 + y.get h.2 := rfl #align part_enat.get_add PartENat.get_add @[simp] theorem get_zero (h : (0 : PartENat).Dom) : (0 : PartENat).get h = 0 := rfl #align part_enat.get_zero PartENat.get_zero @[simp] theorem get_one (h : (1 : PartENat).Dom) : (1 : PartENat).get h = 1 := rfl #align part_enat.get_one PartENat.get_one -- See note [no_index around OfNat.ofNat] @[simp] theorem get_ofNat' (x : ℕ) [x.AtLeastTwo] (h : (no_index (OfNat.ofNat x : PartENat)).Dom) : Part.get (no_index (OfNat.ofNat x : PartENat)) h = (no_index (OfNat.ofNat x)) := get_natCast' x h nonrec theorem get_eq_iff_eq_some {a : PartENat} {ha : a.Dom} {b : ℕ} : a.get ha = b ↔ a = some b := get_eq_iff_eq_some #align part_enat.get_eq_iff_eq_some PartENat.get_eq_iff_eq_some theorem get_eq_iff_eq_coe {a : PartENat} {ha : a.Dom} {b : ℕ} : a.get ha = b ↔ a = b := by rw [get_eq_iff_eq_some] rfl #align part_enat.get_eq_iff_eq_coe PartENat.get_eq_iff_eq_coe theorem dom_of_le_of_dom {x y : PartENat} : x ≤ y → y.Dom → x.Dom := fun ⟨h, _⟩ => h #align part_enat.dom_of_le_of_dom PartENat.dom_of_le_of_dom theorem dom_of_le_some {x : PartENat} {y : ℕ} (h : x ≤ some y) : x.Dom := dom_of_le_of_dom h trivial #align part_enat.dom_of_le_some PartENat.dom_of_le_some theorem dom_of_le_natCast {x : PartENat} {y : ℕ} (h : x ≤ y) : x.Dom := by exact dom_of_le_some h #align part_enat.dom_of_le_coe PartENat.dom_of_le_natCast instance decidableLe (x y : PartENat) [Decidable x.Dom] [Decidable y.Dom] : Decidable (x ≤ y) := if hx : x.Dom then decidable_of_decidable_of_iff (by rw [le_def]) else if hy : y.Dom then isFalse fun h => hx <\| dom_of_le_of_dom h hy else isTrue ⟨fun h => (hy h).elim, fun h => (hy h).elim⟩ #align part_enat.decidable_le PartENat.decidableLe -- Porting note: Removed. Use `Nat.castAddMonoidHom` instead. #noalign part_enat.coe_hom #noalign part_enat.coe_coe_hom instance partialOrder : PartialOrder PartENat where le := (· ≤ ·) le_refl _ := ⟨id, fun _ => le_rfl⟩ le_trans := fun _ _ _ ⟨hxy₁, hxy₂⟩ ⟨hyz₁, hyz₂⟩ => ⟨hxy₁ ∘ hyz₁, fun _ => le_trans (hxy₂ _) (hyz₂ _)⟩ lt_iff_le_not_le _ _ := Iff.rfl le_antisymm := fun _ _ ⟨hxy₁, hxy₂⟩ ⟨hyx₁, hyx₂⟩ => Part.ext' ⟨hyx₁, hxy₁⟩ fun _ _ => le_antisymm (hxy₂ _) (hyx₂ _) theorem lt_def (x y : PartENat) : x < y ↔ ∃ hx : x.Dom, ∀ hy : y.Dom, x.get hx < y.get hy := by rw [lt_iff_le_not_le, le_def, le_def, not_exists] constructor · rintro ⟨⟨hyx, H⟩, h⟩ by_cases hx : x.Dom · use hx intro hy specialize H hy specialize h fun _ => hy rw [not_forall] at h cases' h with hx' h rw [not_le] at h exact h · specialize h fun hx' => (hx hx').elim rw [not_forall] at h cases' h with hx' h exact (hx hx').elim · rintro ⟨hx, H⟩ exact ⟨⟨fun _ => hx, fun hy => (H hy).le⟩, fun hxy h => not_lt_of_le (h _) (H _)⟩ #align part_enat.lt_def PartENat.lt_def noncomputable instance orderedAddCommMonoid : OrderedAddCommMonoid PartENat := { PartENat.partialOrder, PartENat.addCommMonoid with add_le_add_left := fun a b ⟨h₁, h₂⟩ c => PartENat.casesOn c (by simp [top_add]) fun c => ⟨fun h => And.intro (dom_natCast _) (h₁ h.2), fun h => by simpa only [coe_add_get] using add_le_add_left (h₂ _) c⟩ } instance semilatticeSup : SemilatticeSup PartENat := { PartENat.partialOrder with sup := (· ⊔ ·) le_sup_left := fun _ _ => ⟨And.left, fun _ => le_sup_left⟩ le_sup_right := fun _ _ => ⟨And.right, fun _ => le_sup_right⟩ sup_le := fun _ _ _ ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ => ⟨fun hz => ⟨hx₁ hz, hy₁ hz⟩, fun _ => sup_le (hx₂ _) (hy₂ _)⟩ } #align part_enat.semilattice_sup PartENat.semilatticeSup instance orderBot : OrderBot PartENat where bot := ⊥ bot_le _ := ⟨fun _ => trivial, fun _ => Nat.zero_le _⟩ #align part_enat.order_bot PartENat.orderBot instance orderTop : OrderTop PartENat where top := ⊤ le_top _ := ⟨fun h => False.elim h, fun hy => False.elim hy⟩ #align part_enat.order_top PartENat.orderTop instance : ZeroLEOneClass PartENat where zero_le_one := bot_le theorem coe_le_coe {x y : ℕ} : (x : PartENat) ≤ y ↔ x ≤ y := Nat.cast_le #align part_enat.coe_le_coe PartENat.coe_le_coe theorem coe_lt_coe {x y : ℕ} : (x : PartENat) < y ↔ x < y := Nat.cast_lt #align part_enat.coe_lt_coe PartENat.coe_lt_coe @[simp] theorem get_le_get {x y : PartENat} {hx : x.Dom} {hy : y.Dom} : x.get hx ≤ y.get hy ↔ x ≤ y := by conv => lhs rw [← coe_le_coe, natCast_get, natCast_get] #align part_enat.get_le_get PartENat.get_le_get theorem le_coe_iff (x : PartENat) (n : ℕ) : x ≤ n ↔ ∃ h : x.Dom, x.get h ≤ n := by show (∃ h : True → x.Dom, _) ↔ ∃ h : x.Dom, x.get h ≤ n simp only [forall_prop_of_true, dom_natCast, get_natCast'] #align part_enat.le_coe_iff PartENat.le_coe_iff	Mathlib/Data/Nat/PartENat.lean	329	330	theorem lt_coe_iff (x : PartENat) (n : ℕ) : x < n ↔ ∃ h : x.Dom, x.get h < n := by	simp only [lt_def, forall_prop_of_true, get_natCast', dom_natCast]
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation import Mathlib.LinearAlgebra.CliffordAlgebra.Even import Mathlib.LinearAlgebra.QuadraticForm.Prod import Mathlib.Tactic.LiftLets #align_import linear_algebra.clifford_algebra.even_equiv from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2" namespace CliffordAlgebra variable {R M : Type} [CommRing R] [AddCommGroup M] [Module R M] variable (Q : QuadraticForm R M) namespace EquivEven abbrev Q' : QuadraticForm R (M × R) := Q.prod <\| -@QuadraticForm.sq R _ set_option linter.uppercaseLean3 false in #align clifford_algebra.equiv_even.Q' CliffordAlgebra.EquivEven.Q' theorem Q'_apply (m : M × R) : Q' Q m = Q m.1 - m.2 m.2 := (sub_eq_add_neg _ _).symm set_option linter.uppercaseLean3 false in #align clifford_algebra.equiv_even.Q'_apply CliffordAlgebra.EquivEven.Q'_apply def e0 : CliffordAlgebra (Q' Q) := ι (Q' Q) (0, 1) #align clifford_algebra.equiv_even.e0 CliffordAlgebra.EquivEven.e0 def v : M →ₗ[R] CliffordAlgebra (Q' Q) := ι (Q' Q) ∘ₗ LinearMap.inl _ _ _ #align clifford_algebra.equiv_even.v CliffordAlgebra.EquivEven.v	Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean	69	71	theorem ι_eq_v_add_smul_e0 (m : M) (r : R) : ι (Q' Q) (m, r) = v Q m + r • e0 Q := by	rw [e0, v, LinearMap.comp_apply, LinearMap.inl_apply, ← LinearMap.map_smul, Prod.smul_mk, smul_zero, smul_eq_mul, mul_one, ← LinearMap.map_add, Prod.mk_add_mk, zero_add, add_zero]
import Mathlib.RingTheory.IntegralClosure import Mathlib.RingTheory.FractionalIdeal.Basic #align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7" open IsLocalization Pointwise nonZeroDivisors namespace FractionalIdeal open Set Submodule variable {R : Type} [CommRing R] {S : Submonoid R} {P : Type} [CommRing P] variable [Algebra R P] [loc : IsLocalization S P] section variable {P' : Type} [CommRing P'] [Algebra R P'] [loc' : IsLocalization S P'] variable {P'' : Type} [CommRing P''] [Algebra R P''] [loc'' : IsLocalization S P''] theorem _root_.IsFractional.map (g : P →ₐ[R] P') {I : Submodule R P} : IsFractional S I → IsFractional S (Submodule.map g.toLinearMap I) \| ⟨a, a_nonzero, hI⟩ => ⟨a, a_nonzero, fun b hb => by obtain ⟨b', b'_mem, hb'⟩ := Submodule.mem_map.mp hb rw [AlgHom.toLinearMap_apply] at hb' obtain ⟨x, hx⟩ := hI b' b'_mem use x rw [← g.commutes, hx, g.map_smul, hb']⟩ #align is_fractional.map IsFractional.map def map (g : P →ₐ[R] P') : FractionalIdeal S P → FractionalIdeal S P' := fun I => ⟨Submodule.map g.toLinearMap I, I.isFractional.map g⟩ #align fractional_ideal.map FractionalIdeal.map @[simp, norm_cast] theorem coe_map (g : P →ₐ[R] P') (I : FractionalIdeal S P) : ↑(map g I) = Submodule.map g.toLinearMap I := rfl #align fractional_ideal.coe_map FractionalIdeal.coe_map @[simp] theorem mem_map {I : FractionalIdeal S P} {g : P →ₐ[R] P'} {y : P'} : y ∈ I.map g ↔ ∃ x, x ∈ I ∧ g x = y := Submodule.mem_map #align fractional_ideal.mem_map FractionalIdeal.mem_map variable (I J : FractionalIdeal S P) (g : P →ₐ[R] P') @[simp] theorem map_id : I.map (AlgHom.id _ _) = I := coeToSubmodule_injective (Submodule.map_id (I : Submodule R P)) #align fractional_ideal.map_id FractionalIdeal.map_id @[simp] theorem map_comp (g' : P' →ₐ[R] P'') : I.map (g'.comp g) = (I.map g).map g' := coeToSubmodule_injective (Submodule.map_comp g.toLinearMap g'.toLinearMap I) #align fractional_ideal.map_comp FractionalIdeal.map_comp @[simp, norm_cast] theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by ext x simp only [mem_coeIdeal] constructor · rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩ exact ⟨y, hy, (g.commutes y).symm⟩ · rintro ⟨y, hy, rfl⟩ exact ⟨_, ⟨y, hy, rfl⟩, g.commutes y⟩ #align fractional_ideal.map_coe_ideal FractionalIdeal.map_coeIdeal @[simp] theorem map_one : (1 : FractionalIdeal S P).map g = 1 := map_coeIdeal g ⊤ #align fractional_ideal.map_one FractionalIdeal.map_one @[simp] theorem map_zero : (0 : FractionalIdeal S P).map g = 0 := map_coeIdeal g 0 #align fractional_ideal.map_zero FractionalIdeal.map_zero @[simp] theorem map_add : (I + J).map g = I.map g + J.map g := coeToSubmodule_injective (Submodule.map_sup _ _ _) #align fractional_ideal.map_add FractionalIdeal.map_add @[simp] theorem map_mul : (I * J).map g = I.map g * J.map g := by simp only [mul_def] exact coeToSubmodule_injective (Submodule.map_mul _ _ _) #align fractional_ideal.map_mul FractionalIdeal.map_mul @[simp] theorem map_map_symm (g : P ≃ₐ[R] P') : (I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I := by rw [← map_comp, g.symm_comp, map_id] #align fractional_ideal.map_map_symm FractionalIdeal.map_map_symm @[simp] theorem map_symm_map (I : FractionalIdeal S P') (g : P ≃ₐ[R] P') : (I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I := by rw [← map_comp, g.comp_symm, map_id] #align fractional_ideal.map_symm_map FractionalIdeal.map_symm_map theorem map_mem_map {f : P →ₐ[R] P'} (h : Function.Injective f) {x : P} {I : FractionalIdeal S P} : f x ∈ map f I ↔ x ∈ I := mem_map.trans ⟨fun ⟨_, hx', x'_eq⟩ => h x'_eq ▸ hx', fun h => ⟨x, h, rfl⟩⟩ #align fractional_ideal.map_mem_map FractionalIdeal.map_mem_map theorem map_injective (f : P →ₐ[R] P') (h : Function.Injective f) : Function.Injective (map f : FractionalIdeal S P → FractionalIdeal S P') := fun _ _ hIJ => ext fun _ => (map_mem_map h).symm.trans (hIJ.symm ▸ map_mem_map h) #align fractional_ideal.map_injective FractionalIdeal.map_injective def mapEquiv (g : P ≃ₐ[R] P') : FractionalIdeal S P ≃+* FractionalIdeal S P' where toFun := map g invFun := map g.symm map_add' I J := map_add I J _ map_mul' I J := map_mul I J _ left_inv I := by rw [← map_comp, AlgEquiv.symm_comp, map_id] right_inv I := by rw [← map_comp, AlgEquiv.comp_symm, map_id] #align fractional_ideal.map_equiv FractionalIdeal.mapEquiv @[simp] theorem coeFun_mapEquiv (g : P ≃ₐ[R] P') : (mapEquiv g : FractionalIdeal S P → FractionalIdeal S P') = map g := rfl #align fractional_ideal.coe_fun_map_equiv FractionalIdeal.coeFun_mapEquiv @[simp] theorem mapEquiv_apply (g : P ≃ₐ[R] P') (I : FractionalIdeal S P) : mapEquiv g I = map (↑g) I := rfl #align fractional_ideal.map_equiv_apply FractionalIdeal.mapEquiv_apply @[simp] theorem mapEquiv_symm (g : P ≃ₐ[R] P') : ((mapEquiv g).symm : FractionalIdeal S P' ≃+* _) = mapEquiv g.symm := rfl #align fractional_ideal.map_equiv_symm FractionalIdeal.mapEquiv_symm @[simp] theorem mapEquiv_refl : mapEquiv AlgEquiv.refl = RingEquiv.refl (FractionalIdeal S P) := RingEquiv.ext fun x => by simp #align fractional_ideal.map_equiv_refl FractionalIdeal.mapEquiv_refl theorem isFractional_span_iff {s : Set P} : IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) := ⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => span_induction hb h (by rw [smul_zero] exact isInteger_zero) (fun x y hx hy => by rw [smul_add] exact isInteger_add hx hy) fun s x hx => by rw [smul_comm] exact isInteger_smul hx⟩⟩ #align fractional_ideal.is_fractional_span_iff FractionalIdeal.isFractional_span_iff theorem isFractional_of_fg {I : Submodule R P} (hI : I.FG) : IsFractional S I := by rcases hI with ⟨I, rfl⟩ rcases exist_integer_multiples_of_finset S I with ⟨⟨s, hs1⟩, hs⟩ rw [isFractional_span_iff] exact ⟨s, hs1, hs⟩ #align fractional_ideal.is_fractional_of_fg FractionalIdeal.isFractional_of_fg theorem mem_span_mul_finite_of_mem_mul {I J : FractionalIdeal S P} {x : P} (hx : x ∈ I * J) : ∃ T T' : Finset P, (T : Set P) ⊆ I ∧ (T' : Set P) ⊆ J ∧ x ∈ span R (T * T' : Set P) := Submodule.mem_span_mul_finite_of_mem_mul (by simpa using mem_coe.mpr hx) #align fractional_ideal.mem_span_mul_finite_of_mem_mul FractionalIdeal.mem_span_mul_finite_of_mem_mul variable (S) theorem coeIdeal_fg (inj : Function.Injective (algebraMap R P)) (I : Ideal R) : FG ((I : FractionalIdeal S P) : Submodule R P) ↔ I.FG := coeSubmodule_fg _ inj _ #align fractional_ideal.coe_ideal_fg FractionalIdeal.coeIdeal_fg variable {S} theorem fg_unit (I : (FractionalIdeal S P)ˣ) : FG (I : Submodule R P) := Submodule.fg_unit <\| Units.map (coeSubmoduleHom S P).toMonoidHom I #align fractional_ideal.fg_unit FractionalIdeal.fg_unit theorem fg_of_isUnit (I : FractionalIdeal S P) (h : IsUnit I) : FG (I : Submodule R P) := fg_unit h.unit #align fractional_ideal.fg_of_is_unit FractionalIdeal.fg_of_isUnit theorem _root_.Ideal.fg_of_isUnit (inj : Function.Injective (algebraMap R P)) (I : Ideal R) (h : IsUnit (I : FractionalIdeal S P)) : I.FG := by rw [← coeIdeal_fg S inj I] exact FractionalIdeal.fg_of_isUnit I h #align ideal.fg_of_is_unit Ideal.fg_of_isUnit variable (S P P') noncomputable irreducible_def canonicalEquiv : FractionalIdeal S P ≃+* FractionalIdeal S P' := mapEquiv { ringEquivOfRingEquiv P P' (RingEquiv.refl R) (show S.map _ = S by rw [RingEquiv.toMonoidHom_refl, Submonoid.map_id]) with commutes' := fun r => ringEquivOfRingEquiv_eq _ _ } #align fractional_ideal.canonical_equiv FractionalIdeal.canonicalEquiv @[simp] theorem mem_canonicalEquiv_apply {I : FractionalIdeal S P} {x : P'} : x ∈ canonicalEquiv S P P' I ↔ ∃ y ∈ I, IsLocalization.map P' (RingHom.id R) (fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) (y : P) = x := by rw [canonicalEquiv, mapEquiv_apply, mem_map] exact ⟨fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩, fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩⟩ #align fractional_ideal.mem_canonical_equiv_apply FractionalIdeal.mem_canonicalEquiv_apply @[simp] theorem canonicalEquiv_symm : (canonicalEquiv S P P').symm = canonicalEquiv S P' P := RingEquiv.ext fun I => SetLike.ext_iff.mpr fun x => by rw [mem_canonicalEquiv_apply, canonicalEquiv, mapEquiv_symm, mapEquiv_apply, mem_map] exact ⟨fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩, fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩⟩ #align fractional_ideal.canonical_equiv_symm FractionalIdeal.canonicalEquiv_symm theorem canonicalEquiv_flip (I) : canonicalEquiv S P P' (canonicalEquiv S P' P I) = I := by rw [← canonicalEquiv_symm]; erw [RingEquiv.apply_symm_apply] #align fractional_ideal.canonical_equiv_flip FractionalIdeal.canonicalEquiv_flip @[simp] theorem canonicalEquiv_canonicalEquiv (P'' : Type) [CommRing P''] [Algebra R P''] [IsLocalization S P''] (I : FractionalIdeal S P) : canonicalEquiv S P' P'' (canonicalEquiv S P P' I) = canonicalEquiv S P P'' I := by ext simp only [IsLocalization.map_map, RingHomInvPair.comp_eq₂, mem_canonicalEquiv_apply, exists_prop, exists_exists_and_eq_and] #align fractional_ideal.canonical_equiv_canonical_equiv FractionalIdeal.canonicalEquiv_canonicalEquiv theorem canonicalEquiv_trans_canonicalEquiv (P'' : Type) [CommRing P''] [Algebra R P''] [IsLocalization S P''] : (canonicalEquiv S P P').trans (canonicalEquiv S P' P'') = canonicalEquiv S P P'' := RingEquiv.ext (canonicalEquiv_canonicalEquiv S P P' P'') #align fractional_ideal.canonical_equiv_trans_canonical_equiv FractionalIdeal.canonicalEquiv_trans_canonicalEquiv @[simp] theorem canonicalEquiv_coeIdeal (I : Ideal R) : canonicalEquiv S P P' I = I := by ext simp [IsLocalization.map_eq] #align fractional_ideal.canonical_equiv_coe_ideal FractionalIdeal.canonicalEquiv_coeIdeal @[simp] theorem canonicalEquiv_self : canonicalEquiv S P P = RingEquiv.refl _ := by rw [← canonicalEquiv_trans_canonicalEquiv S P P] convert (canonicalEquiv S P P).symm_trans_self exact (canonicalEquiv_symm S P P).symm #align fractional_ideal.canonical_equiv_self FractionalIdeal.canonicalEquiv_self end section IsFractionRing variable {K K' : Type*} [Field K] [Field K'] variable [Algebra R K] [IsFractionRing R K] [Algebra R K'] [IsFractionRing R K'] variable {I J : FractionalIdeal R⁰ K} (h : K →ₐ[R] K') theorem exists_ne_zero_mem_isInteger [Nontrivial R] (hI : I ≠ 0) : ∃ x, x ≠ 0 ∧ algebraMap R K x ∈ I := by obtain ⟨y : K, y_mem, y_not_mem⟩ := SetLike.exists_of_lt (by simpa only using bot_lt_iff_ne_bot.mpr hI) have y_ne_zero : y ≠ 0 := by simpa using y_not_mem obtain ⟨z, ⟨x, hx⟩⟩ := exists_integer_multiple R⁰ y refine ⟨x, ?_, ?_⟩ · rw [Ne, ← @IsFractionRing.to_map_eq_zero_iff R _ K, hx, Algebra.smul_def] exact mul_ne_zero (IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors z.2) y_ne_zero · rw [hx] exact smul_mem _ _ y_mem #align fractional_ideal.exists_ne_zero_mem_is_integer FractionalIdeal.exists_ne_zero_mem_isInteger theorem map_ne_zero [Nontrivial R] (hI : I ≠ 0) : I.map h ≠ 0 := by obtain ⟨x, x_ne_zero, hx⟩ := exists_ne_zero_mem_isInteger hI contrapose! x_ne_zero with map_eq_zero refine IsFractionRing.to_map_eq_zero_iff.mp (eq_zero_iff.mp map_eq_zero _ (mem_map.mpr ?_)) exact ⟨algebraMap R K x, hx, h.commutes x⟩ #align fractional_ideal.map_ne_zero FractionalIdeal.map_ne_zero @[simp] theorem map_eq_zero_iff [Nontrivial R] : I.map h = 0 ↔ I = 0 := ⟨not_imp_not.mp (map_ne_zero _), fun hI => hI.symm ▸ map_zero h⟩ #align fractional_ideal.map_eq_zero_iff FractionalIdeal.map_eq_zero_iff theorem coeIdeal_injective : Function.Injective (fun (I : Ideal R) ↦ (I : FractionalIdeal R⁰ K)) := coeIdeal_injective' le_rfl #align fractional_ideal.coe_ideal_injective FractionalIdeal.coeIdeal_injective theorem coeIdeal_inj {I J : Ideal R} : (I : FractionalIdeal R⁰ K) = (J : FractionalIdeal R⁰ K) ↔ I = J := coeIdeal_inj' le_rfl #align fractional_ideal.coe_ideal_inj FractionalIdeal.coeIdeal_inj @[simp] theorem coeIdeal_eq_zero {I : Ideal R} : (I : FractionalIdeal R⁰ K) = 0 ↔ I = ⊥ := coeIdeal_eq_zero' le_rfl #align fractional_ideal.coe_ideal_eq_zero FractionalIdeal.coeIdeal_eq_zero theorem coeIdeal_ne_zero {I : Ideal R} : (I : FractionalIdeal R⁰ K) ≠ 0 ↔ I ≠ ⊥ := coeIdeal_ne_zero' le_rfl #align fractional_ideal.coe_ideal_ne_zero FractionalIdeal.coeIdeal_ne_zero @[simp]	Mathlib/RingTheory/FractionalIdeal/Operations.lean	350	351	theorem coeIdeal_eq_one {I : Ideal R} : (I : FractionalIdeal R⁰ K) = 1 ↔ I = 1 := by	simpa only [Ideal.one_eq_top] using coeIdeal_inj
import Mathlib.Probability.Kernel.Composition import Mathlib.MeasureTheory.Integral.SetIntegral #align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113" noncomputable section open scoped Topology ENNReal MeasureTheory ProbabilityTheory open Set Function Real ENNReal MeasureTheory Filter ProbabilityTheory ProbabilityTheory.kernel variable {α β γ E : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [NormedAddCommGroup E] {κ : kernel α β} [IsSFiniteKernel κ] {η : kernel (α × β) γ} [IsSFiniteKernel η] {a : α} namespace ProbabilityTheory theorem hasFiniteIntegral_prod_mk_left (a : α) {s : Set (β × γ)} (h2s : (κ ⊗ₖ η) a s ≠ ∞) : HasFiniteIntegral (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) := by let t := toMeasurable ((κ ⊗ₖ η) a) s simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg] calc ∫⁻ b, ENNReal.ofReal (η (a, b) (Prod.mk b ⁻¹' s)).toReal ∂κ a _ ≤ ∫⁻ b, η (a, b) (Prod.mk b ⁻¹' t) ∂κ a := by refine lintegral_mono_ae ?_ filter_upwards [ae_kernel_lt_top a h2s] with b hb rw [ofReal_toReal hb.ne] exact measure_mono (preimage_mono (subset_toMeasurable _ _)) _ ≤ (κ ⊗ₖ η) a t := le_compProd_apply _ _ _ _ _ = (κ ⊗ₖ η) a s := measure_toMeasurable s _ < ⊤ := h2s.lt_top #align probability_theory.has_finite_integral_prod_mk_left ProbabilityTheory.hasFiniteIntegral_prod_mk_left	Mathlib/Probability/Kernel/IntegralCompProd.lean	64	68	theorem integrable_kernel_prod_mk_left (a : α) {s : Set (β × γ)} (hs : MeasurableSet s) (h2s : (κ ⊗ₖ η) a s ≠ ∞) : Integrable (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) := by	constructor · exact (measurable_kernel_prod_mk_left' hs a).ennreal_toReal.aestronglyMeasurable · exact hasFiniteIntegral_prod_mk_left a h2s
import Mathlib.CategoryTheory.Subobject.Lattice #align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d" universe v u noncomputable section open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite variable {C : Type u} [Category.{v} C] {X Y Z : C} namespace CategoryTheory namespace Limits section Image variable (f : X ⟶ Y) [HasImage f] abbrev imageSubobject : Subobject Y := Subobject.mk (image.ι f) #align category_theory.limits.image_subobject CategoryTheory.Limits.imageSubobject def imageSubobjectIso : (imageSubobject f : C) ≅ image f := Subobject.underlyingIso (image.ι f) #align category_theory.limits.image_subobject_iso CategoryTheory.Limits.imageSubobjectIso @[reassoc (attr := simp)] theorem imageSubobject_arrow : (imageSubobjectIso f).hom ≫ image.ι f = (imageSubobject f).arrow := by simp [imageSubobjectIso] #align category_theory.limits.image_subobject_arrow CategoryTheory.Limits.imageSubobject_arrow @[reassoc (attr := simp)] theorem imageSubobject_arrow' : (imageSubobjectIso f).inv ≫ (imageSubobject f).arrow = image.ι f := by simp [imageSubobjectIso] #align category_theory.limits.image_subobject_arrow' CategoryTheory.Limits.imageSubobject_arrow' def factorThruImageSubobject : X ⟶ imageSubobject f := factorThruImage f ≫ (imageSubobjectIso f).inv #align category_theory.limits.factor_thru_image_subobject CategoryTheory.Limits.factorThruImageSubobject instance [HasEqualizers C] : Epi (factorThruImageSubobject f) := by dsimp [factorThruImageSubobject] apply epi_comp @[reassoc (attr := simp), elementwise (attr := simp)] theorem imageSubobject_arrow_comp : factorThruImageSubobject f ≫ (imageSubobject f).arrow = f := by simp [factorThruImageSubobject, imageSubobject_arrow] #align category_theory.limits.image_subobject_arrow_comp CategoryTheory.Limits.imageSubobject_arrow_comp theorem imageSubobject_arrow_comp_eq_zero [HasZeroMorphisms C] {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [HasImage f] [Epi (factorThruImageSubobject f)] (h : f ≫ g = 0) : (imageSubobject f).arrow ≫ g = 0 := zero_of_epi_comp (factorThruImageSubobject f) <\| by simp [h] #align category_theory.limits.image_subobject_arrow_comp_eq_zero CategoryTheory.Limits.imageSubobject_arrow_comp_eq_zero theorem imageSubobject_factors_comp_self {W : C} (k : W ⟶ X) : (imageSubobject f).Factors (k ≫ f) := ⟨k ≫ factorThruImage f, by simp⟩ #align category_theory.limits.image_subobject_factors_comp_self CategoryTheory.Limits.imageSubobject_factors_comp_self @[simp] theorem factorThruImageSubobject_comp_self {W : C} (k : W ⟶ X) (h) : (imageSubobject f).factorThru (k ≫ f) h = k ≫ factorThruImageSubobject f := by ext simp #align category_theory.limits.factor_thru_image_subobject_comp_self CategoryTheory.Limits.factorThruImageSubobject_comp_self @[simp] theorem factorThruImageSubobject_comp_self_assoc {W W' : C} (k : W ⟶ W') (k' : W' ⟶ X) (h) : (imageSubobject f).factorThru (k ≫ k' ≫ f) h = k ≫ k' ≫ factorThruImageSubobject f := by ext simp #align category_theory.limits.factor_thru_image_subobject_comp_self_assoc CategoryTheory.Limits.factorThruImageSubobject_comp_self_assoc theorem imageSubobject_comp_le {X' : C} (h : X' ⟶ X) (f : X ⟶ Y) [HasImage f] [HasImage (h ≫ f)] : imageSubobject (h ≫ f) ≤ imageSubobject f := Subobject.mk_le_mk_of_comm (image.preComp h f) (by simp) #align category_theory.limits.image_subobject_comp_le CategoryTheory.Limits.imageSubobject_comp_le section open ZeroObject variable [HasZeroMorphisms C] [HasZeroObject C] @[simp] theorem imageSubobject_zero_arrow : (imageSubobject (0 : X ⟶ Y)).arrow = 0 := by rw [← imageSubobject_arrow] simp #align category_theory.limits.image_subobject_zero_arrow CategoryTheory.Limits.imageSubobject_zero_arrow @[simp] theorem imageSubobject_zero {A B : C} : imageSubobject (0 : A ⟶ B) = ⊥ := Subobject.eq_of_comm (imageSubobjectIso _ ≪≫ imageZero ≪≫ Subobject.botCoeIsoZero.symm) (by simp) #align category_theory.limits.image_subobject_zero CategoryTheory.Limits.imageSubobject_zero end section variable [HasEqualizers C] attribute [local instance] epi_comp instance imageSubobject_comp_le_epi_of_epi {X' : C} (h : X' ⟶ X) [Epi h] (f : X ⟶ Y) [HasImage f] [HasImage (h ≫ f)] : Epi (Subobject.ofLE _ _ (imageSubobject_comp_le h f)) := by rw [ofLE_mk_le_mk_of_comm (image.preComp h f)] · infer_instance · simp #align category_theory.limits.image_subobject_comp_le_epi_of_epi CategoryTheory.Limits.imageSubobject_comp_le_epi_of_epi end section variable [HasEqualizers C] def imageSubobjectCompIso (f : X ⟶ Y) [HasImage f] {Y' : C} (h : Y ⟶ Y') [IsIso h] : (imageSubobject (f ≫ h) : C) ≅ (imageSubobject f : C) := imageSubobjectIso _ ≪≫ (image.compIso _ _).symm ≪≫ (imageSubobjectIso _).symm #align category_theory.limits.image_subobject_comp_iso CategoryTheory.Limits.imageSubobjectCompIso @[reassoc (attr := simp)]	Mathlib/CategoryTheory/Subobject/Limits.lean	412	415	theorem imageSubobjectCompIso_hom_arrow (f : X ⟶ Y) [HasImage f] {Y' : C} (h : Y ⟶ Y') [IsIso h] : (imageSubobjectCompIso f h).hom ≫ (imageSubobject f).arrow = (imageSubobject (f ≫ h)).arrow ≫ inv h := by	simp [imageSubobjectCompIso]
import Mathlib.Algebra.MvPolynomial.Funext import Mathlib.Algebra.Ring.ULift import Mathlib.RingTheory.WittVector.Basic #align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4" namespace WittVector universe u variable {p : ℕ} {R S : Type u} {σ idx : Type*} [CommRing R] [CommRing S] local notation "𝕎" => WittVector p -- type as `\bbW` open MvPolynomial open Function (uncurry) variable (p) noncomputable section theorem poly_eq_of_wittPolynomial_bind_eq' [Fact p.Prime] (f g : ℕ → MvPolynomial (idx × ℕ) ℤ) (h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by ext1 n apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective rw [← Function.funext_iff] at h replace h := congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁, bind₁_wittPolynomial_xInTermsOfW, bind₁_X_right] using h #align witt_vector.poly_eq_of_witt_polynomial_bind_eq' WittVector.poly_eq_of_wittPolynomial_bind_eq' theorem poly_eq_of_wittPolynomial_bind_eq [Fact p.Prime] (f g : ℕ → MvPolynomial ℕ ℤ) (h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by ext1 n apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective rw [← Function.funext_iff] at h replace h := congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁, bind₁_wittPolynomial_xInTermsOfW, bind₁_X_right] using h #align witt_vector.poly_eq_of_witt_polynomial_bind_eq WittVector.poly_eq_of_wittPolynomial_bind_eq -- Ideally, we would generalise this to n-ary functions -- But we don't have a good theory of n-ary compositions in mathlib class IsPoly (f : ∀ ⦃R⦄ [CommRing R], WittVector p R → 𝕎 R) : Prop where mk' :: poly : ∃ φ : ℕ → MvPolynomial ℕ ℤ, ∀ ⦃R⦄ [CommRing R] (x : 𝕎 R), (f x).coeff = fun n => aeval x.coeff (φ n) #align witt_vector.is_poly WittVector.IsPoly instance idIsPoly : IsPoly p fun _ _ => id := ⟨⟨X, by intros; simp only [aeval_X, id]⟩⟩ #align witt_vector.id_is_poly WittVector.idIsPoly instance idIsPolyI' : IsPoly p fun _ _ a => a := WittVector.idIsPoly _ #align witt_vector.id_is_poly_i' WittVector.idIsPolyI' namespace IsPoly instance : Inhabited (IsPoly p fun _ _ => id) := ⟨WittVector.idIsPoly p⟩ variable {p}	Mathlib/RingTheory/WittVector/IsPoly.lean	172	195	theorem ext [Fact p.Prime] {f g} (hf : IsPoly p f) (hg : IsPoly p g) (h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ghostComponent n (f x) = ghostComponent n (g x)) : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x := by	obtain ⟨φ, hf⟩ := hf obtain ⟨ψ, hg⟩ := hg intros ext n rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq p φ ψ] intro k apply MvPolynomial.funext intro x simp only [hom_bind₁] specialize h (ULift ℤ) (mk p fun i => ⟨x i⟩) k simp only [ghostComponent_apply, aeval_eq_eval₂Hom] at h apply (ULift.ringEquiv.symm : ℤ ≃+* _).injective simp only [← RingEquiv.coe_toRingHom, map_eval₂Hom] convert h using 1 all_goals simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom] apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl ext1 apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl simp only [coeff_mk]; rfl
import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.LinearAlgebra.SesquilinearForm #align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9" variable {𝕜 E F : Type*} [RCLike 𝕜] variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y namespace Submodule variable (K : Submodule 𝕜 E) def orthogonal : Submodule 𝕜 E where carrier := { v \| ∀ u ∈ K, ⟪u, v⟫ = 0 } zero_mem' _ _ := inner_zero_right _ add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero] smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero] #align submodule.orthogonal Submodule.orthogonal @[inherit_doc] notation:1200 K "ᗮ" => orthogonal K theorem mem_orthogonal (v : E) : v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪u, v⟫ = 0 := Iff.rfl #align submodule.mem_orthogonal Submodule.mem_orthogonal theorem mem_orthogonal' (v : E) : v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪v, u⟫ = 0 := by simp_rw [mem_orthogonal, inner_eq_zero_symm] #align submodule.mem_orthogonal' Submodule.mem_orthogonal' variable {K} theorem inner_right_of_mem_orthogonal {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) : ⟪u, v⟫ = 0 := (K.mem_orthogonal v).1 hv u hu #align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal theorem inner_left_of_mem_orthogonal {u v : E} (hu : u ∈ K) (hv : v ∈ Kᗮ) : ⟪v, u⟫ = 0 := by rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv #align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪u, v⟫ = 0 := by refine ⟨inner_right_of_mem_orthogonal (mem_span_singleton_self u), ?_⟩ intro hv w hw rw [mem_span_singleton] at hw obtain ⟨c, rfl⟩ := hw simp [inner_smul_left, hv] #align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪v, u⟫ = 0 := by rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm] #align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left	Mathlib/Analysis/InnerProductSpace/Orthogonal.lean	86	90	theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : ∀ v : K, ⟪x, v⟫ = ⟪y, v⟫) : x - y ∈ Kᗮ := by	rw [mem_orthogonal'] intro u hu rw [inner_sub_left, sub_eq_zero] exact h ⟨u, hu⟩
import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.Combinatorics.Hall.Basic import Mathlib.Data.Fintype.BigOperators import Mathlib.SetTheory.Cardinal.Finite #align_import combinatorics.configuration from "leanprover-community/mathlib"@"d2d8742b0c21426362a9dacebc6005db895ca963" open Finset namespace Configuration variable (P L : Type*) [Membership P L] def Dual := P #align configuration.dual Configuration.Dual -- Porting note: was `this` instead of `h` instance [h : Inhabited P] : Inhabited (Dual P) := h instance [Finite P] : Finite (Dual P) := ‹Finite P› -- Porting note: was `this` instead of `h` instance [h : Fintype P] : Fintype (Dual P) := h -- Porting note (#11215): TODO: figure out if this is needed. set_option synthInstance.checkSynthOrder false in instance : Membership (Dual L) (Dual P) := ⟨Function.swap (Membership.mem : P → L → Prop)⟩ class Nondegenerate : Prop where exists_point : ∀ l : L, ∃ p, p ∉ l exists_line : ∀ p, ∃ l : L, p ∉ l eq_or_eq : ∀ {p₁ p₂ : P} {l₁ l₂ : L}, p₁ ∈ l₁ → p₂ ∈ l₁ → p₁ ∈ l₂ → p₂ ∈ l₂ → p₁ = p₂ ∨ l₁ = l₂ #align configuration.nondegenerate Configuration.Nondegenerate class HasPoints extends Nondegenerate P L where mkPoint : ∀ {l₁ l₂ : L}, l₁ ≠ l₂ → P mkPoint_ax : ∀ {l₁ l₂ : L} (h : l₁ ≠ l₂), mkPoint h ∈ l₁ ∧ mkPoint h ∈ l₂ #align configuration.has_points Configuration.HasPoints class HasLines extends Nondegenerate P L where mkLine : ∀ {p₁ p₂ : P}, p₁ ≠ p₂ → L mkLine_ax : ∀ {p₁ p₂ : P} (h : p₁ ≠ p₂), p₁ ∈ mkLine h ∧ p₂ ∈ mkLine h #align configuration.has_lines Configuration.HasLines open Nondegenerate open HasPoints (mkPoint mkPoint_ax) open HasLines (mkLine mkLine_ax) instance Dual.Nondegenerate [Nondegenerate P L] : Nondegenerate (Dual L) (Dual P) where exists_point := @exists_line P L _ _ exists_line := @exists_point P L _ _ eq_or_eq := @fun l₁ l₂ p₁ p₂ h₁ h₂ h₃ h₄ => (@eq_or_eq P L _ _ p₁ p₂ l₁ l₂ h₁ h₃ h₂ h₄).symm instance Dual.hasLines [HasPoints P L] : HasLines (Dual L) (Dual P) := { Dual.Nondegenerate _ _ with mkLine := @mkPoint P L _ _ mkLine_ax := @mkPoint_ax P L _ _ } instance Dual.hasPoints [HasLines P L] : HasPoints (Dual L) (Dual P) := { Dual.Nondegenerate _ _ with mkPoint := @mkLine P L _ _ mkPoint_ax := @mkLine_ax P L _ _ } theorem HasPoints.existsUnique_point [HasPoints P L] (l₁ l₂ : L) (hl : l₁ ≠ l₂) : ∃! p, p ∈ l₁ ∧ p ∈ l₂ := ⟨mkPoint hl, mkPoint_ax hl, fun _ hp => (eq_or_eq hp.1 (mkPoint_ax hl).1 hp.2 (mkPoint_ax hl).2).resolve_right hl⟩ #align configuration.has_points.exists_unique_point Configuration.HasPoints.existsUnique_point theorem HasLines.existsUnique_line [HasLines P L] (p₁ p₂ : P) (hp : p₁ ≠ p₂) : ∃! l : L, p₁ ∈ l ∧ p₂ ∈ l := HasPoints.existsUnique_point (Dual L) (Dual P) p₁ p₂ hp #align configuration.has_lines.exists_unique_line Configuration.HasLines.existsUnique_line variable {P L} theorem Nondegenerate.exists_injective_of_card_le [Nondegenerate P L] [Fintype P] [Fintype L] (h : Fintype.card L ≤ Fintype.card P) : ∃ f : L → P, Function.Injective f ∧ ∀ l, f l ∉ l := by classical let t : L → Finset P := fun l => Set.toFinset { p \| p ∉ l } suffices ∀ s : Finset L, s.card ≤ (s.biUnion t).card by -- Hall's marriage theorem obtain ⟨f, hf1, hf2⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this exact ⟨f, hf1, fun l => Set.mem_toFinset.mp (hf2 l)⟩ intro s by_cases hs₀ : s.card = 0 -- If `s = ∅`, then `s.card = 0 ≤ (s.bUnion t).card` · simp_rw [hs₀, zero_le] by_cases hs₁ : s.card = 1 -- If `s = {l}`, then pick a point `p ∉ l` · obtain ⟨l, rfl⟩ := Finset.card_eq_one.mp hs₁ obtain ⟨p, hl⟩ := exists_point l rw [Finset.card_singleton, Finset.singleton_biUnion, Nat.one_le_iff_ne_zero] exact Finset.card_ne_zero_of_mem (Set.mem_toFinset.mpr hl) suffices (s.biUnion t)ᶜ.card ≤ sᶜ.card by -- Rephrase in terms of complements (uses `h`) rw [Finset.card_compl, Finset.card_compl, tsub_le_iff_left] at this replace := h.trans this rwa [← add_tsub_assoc_of_le s.card_le_univ, le_tsub_iff_left (le_add_left s.card_le_univ), add_le_add_iff_right] at this have hs₂ : (s.biUnion t)ᶜ.card ≤ 1 := by -- At most one line through two points of `s` refine Finset.card_le_one_iff.mpr @fun p₁ p₂ hp₁ hp₂ => ?_ simp_rw [t, Finset.mem_compl, Finset.mem_biUnion, not_exists, not_and, Set.mem_toFinset, Set.mem_setOf_eq, Classical.not_not] at hp₁ hp₂ obtain ⟨l₁, l₂, hl₁, hl₂, hl₃⟩ := Finset.one_lt_card_iff.mp (Nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨hs₀, hs₁⟩) exact (eq_or_eq (hp₁ l₁ hl₁) (hp₂ l₁ hl₁) (hp₁ l₂ hl₂) (hp₂ l₂ hl₂)).resolve_right hl₃ by_cases hs₃ : sᶜ.card = 0 · rw [hs₃, Nat.le_zero] rw [Finset.card_compl, tsub_eq_zero_iff_le, LE.le.le_iff_eq (Finset.card_le_univ _), eq_comm, Finset.card_eq_iff_eq_univ] at hs₃ ⊢ rw [hs₃] rw [Finset.eq_univ_iff_forall] at hs₃ ⊢ exact fun p => Exists.elim (exists_line p)-- If `s = univ`, then show `s.bUnion t = univ` fun l hl => Finset.mem_biUnion.mpr ⟨l, Finset.mem_univ l, Set.mem_toFinset.mpr hl⟩ · exact hs₂.trans (Nat.one_le_iff_ne_zero.mpr hs₃) #align configuration.nondegenerate.exists_injective_of_card_le Configuration.Nondegenerate.exists_injective_of_card_le -- If `s < univ`, then consequence of `hs₂` variable (L) noncomputable def lineCount (p : P) : ℕ := Nat.card { l : L // p ∈ l } #align configuration.line_count Configuration.lineCount variable (P) {L} noncomputable def pointCount (l : L) : ℕ := Nat.card { p : P // p ∈ l } #align configuration.point_count Configuration.pointCount variable (L) theorem sum_lineCount_eq_sum_pointCount [Fintype P] [Fintype L] : ∑ p : P, lineCount L p = ∑ l : L, pointCount P l := by classical simp only [lineCount, pointCount, Nat.card_eq_fintype_card, ← Fintype.card_sigma] apply Fintype.card_congr calc (Σp, { l : L // p ∈ l }) ≃ { x : P × L // x.1 ∈ x.2 } := (Equiv.subtypeProdEquivSigmaSubtype (· ∈ ·)).symm _ ≃ { x : L × P // x.2 ∈ x.1 } := (Equiv.prodComm P L).subtypeEquiv fun x => Iff.rfl _ ≃ Σl, { p // p ∈ l } := Equiv.subtypeProdEquivSigmaSubtype fun (l : L) (p : P) => p ∈ l #align configuration.sum_line_count_eq_sum_point_count Configuration.sum_lineCount_eq_sum_pointCount variable {P L}	Mathlib/Combinatorics/Configuration.lean	200	214	theorem HasLines.pointCount_le_lineCount [HasLines P L] {p : P} {l : L} (h : p ∉ l) [Finite { l : L // p ∈ l }] : pointCount P l ≤ lineCount L p := by	by_cases hf : Infinite { p : P // p ∈ l } · exact (le_of_eq Nat.card_eq_zero_of_infinite).trans (zero_le (lineCount L p)) haveI := fintypeOfNotInfinite hf cases nonempty_fintype { l : L // p ∈ l } rw [lineCount, pointCount, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card] have : ∀ p' : { p // p ∈ l }, p ≠ p' := fun p' hp' => h ((congr_arg (· ∈ l) hp').mpr p'.2) exact Fintype.card_le_of_injective (fun p' => ⟨mkLine (this p'), (mkLine_ax (this p')).1⟩) fun p₁ p₂ hp => Subtype.ext ((eq_or_eq p₁.2 p₂.2 (mkLine_ax (this p₁)).2 ((congr_arg _ (Subtype.ext_iff.mp hp)).mpr (mkLine_ax (this p₂)).2)).resolve_right fun h' => (congr_arg (¬p ∈ ·) h').mp h (mkLine_ax (this p₁)).1)
import Mathlib.Analysis.Convolution import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup import Mathlib.Analysis.Analytic.IsolatedZeros import Mathlib.Analysis.Complex.CauchyIntegral #align_import analysis.special_functions.gamma.beta from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090" noncomputable section set_option linter.uppercaseLean3 false open Filter intervalIntegral Set Real MeasureTheory open scoped Nat Topology Real section BetaIntegral namespace Complex noncomputable def betaIntegral (u v : ℂ) : ℂ := ∫ x : ℝ in (0)..1, (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) #align complex.beta_integral Complex.betaIntegral theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) : IntervalIntegrable (fun x => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by apply IntervalIntegrable.mul_continuousOn · refine intervalIntegral.intervalIntegrable_cpow' ?_ rwa [sub_re, one_re, ← zero_sub, sub_lt_sub_iff_right] · apply ContinuousAt.continuousOn intro x hx rw [uIcc_of_le (by positivity : (0 : ℝ) ≤ 1 / 2)] at hx apply ContinuousAt.cpow · exact (continuous_const.sub continuous_ofReal).continuousAt · exact continuousAt_const · norm_cast exact ofReal_mem_slitPlane.2 <\| by linarith only [hx.2] #align complex.beta_integral_convergent_left Complex.betaIntegral_convergent_left theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) : IntervalIntegrable (fun x => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by refine (betaIntegral_convergent_left hu v).trans ?_ rw [IntervalIntegrable.iff_comp_neg] convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1 · ext1 x conv_lhs => rw [mul_comm] congr 2 <;> · push_cast; ring · norm_num · norm_num #align complex.beta_integral_convergent Complex.betaIntegral_convergent theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by rw [betaIntegral, betaIntegral] have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1) (fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1 rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this simp? at this says simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel_right, neg_neg, mul_one, add_left_neg, mul_zero, zero_add] at this conv_lhs at this => arg 1; intro x; rw [add_comm, ← sub_eq_add_neg, mul_comm] exact this #align complex.beta_integral_symm Complex.betaIntegral_symm theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by simp_rw [betaIntegral, sub_self, cpow_zero, mul_one] rw [integral_cpow (Or.inl _)] · rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel] rw [sub_add_cancel] contrapose! hu; rw [hu, zero_re] · rwa [sub_re, one_re, ← sub_pos, sub_neg_eq_add, sub_add_cancel] #align complex.beta_integral_eval_one_right Complex.betaIntegral_eval_one_right theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) : ∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) = (a : ℂ) ^ (s + t - 1) * betaIntegral s t := by have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne' rw [betaIntegral] have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one, mul_assoc] rw [A, mul_assoc, ← intervalIntegral.integral_const_mul, ← real_smul, ← zero_div a, ← div_self ha.ne', ← intervalIntegral.integral_comp_div _ ha.ne', zero_div] simp_rw [intervalIntegral.integral_of_le ha.le] refine setIntegral_congr measurableSet_Ioc fun x hx => ?_ rw [mul_mul_mul_comm] congr 1 · rw [← mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel₀ _ ha'] · rw [(by norm_cast : (1 : ℂ) - ↑(x / a) = ↑(1 - x / a)), ← mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <\| (div_le_one ha).mpr hx.2)] push_cast rw [mul_sub, mul_one, mul_div_cancel₀ _ ha'] #align complex.beta_integral_scaled Complex.betaIntegral_scaled theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) : Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by -- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate -- this as a formula for the Beta function. have conv_int := integral_posConvolution (GammaIntegral_convergent hs) (GammaIntegral_convergent ht) (ContinuousLinearMap.mul ℝ ℂ) simp_rw [ContinuousLinearMap.mul_apply'] at conv_int have hst : 0 < re (s + t) := by rw [add_re]; exact add_pos hs ht rw [Gamma_eq_integral hs, Gamma_eq_integral ht, Gamma_eq_integral hst, GammaIntegral, GammaIntegral, GammaIntegral, ← conv_int, ← integral_mul_right (betaIntegral _ _)] refine setIntegral_congr measurableSet_Ioi fun x hx => ?_ rw [mul_assoc, ← betaIntegral_scaled s t hx, ← intervalIntegral.integral_const_mul] congr 1 with y : 1 push_cast suffices Complex.exp (-x) = Complex.exp (-y) * Complex.exp (-(x - y)) by rw [this]; ring rw [← Complex.exp_add]; congr 1; abel #align complex.Gamma_mul_Gamma_eq_beta_integral Complex.Gamma_mul_Gamma_eq_betaIntegral	Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean	155	209	theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) : u * betaIntegral u (v + 1) = v * betaIntegral (u + 1) v := by	-- NB: If we knew `Gamma (u + v + 1) ≠ 0` this would be an easy consequence of -- `Gamma_mul_Gamma_eq_betaIntegral`; but we don't know that yet. We will prove it later, but -- this lemma is needed in the proof. So we give a (somewhat laborious) direct argument. let F : ℝ → ℂ := fun x => (x : ℂ) ^ u * (1 - (x : ℂ)) ^ v have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity have hc : ContinuousOn F (Icc 0 1) := by refine (ContinuousAt.continuousOn fun x hx => ?_).mul (ContinuousAt.continuousOn fun x hx => ?_) · refine (continuousAt_cpow_const_of_re_pos (Or.inl ?_) hu).comp continuous_ofReal.continuousAt rw [ofReal_re]; exact hx.1 · refine (continuousAt_cpow_const_of_re_pos (Or.inl ?_) hv).comp (continuous_const.sub continuous_ofReal).continuousAt rw [sub_re, one_re, ofReal_re, sub_nonneg] exact hx.2 have hder : ∀ x : ℝ, x ∈ Ioo (0 : ℝ) 1 → HasDerivAt F (u * ((x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ v) - v * ((x : ℂ) ^ u * (1 - (x : ℂ)) ^ (v - 1))) x := by intro x hx have U : HasDerivAt (fun y : ℂ => y ^ u) (u * (x : ℂ) ^ (u - 1)) ↑x := by have := @HasDerivAt.cpow_const _ _ _ u (hasDerivAt_id (x : ℂ)) (Or.inl ?_) · simp only [id_eq, mul_one] at this exact this · rw [id_eq, ofReal_re]; exact hx.1 have V : HasDerivAt (fun y : ℂ => (1 - y) ^ v) (-v * (1 - (x : ℂ)) ^ (v - 1)) ↑x := by have A := @HasDerivAt.cpow_const _ _ _ v (hasDerivAt_id (1 - (x : ℂ))) (Or.inl ?_) swap; · rw [id, sub_re, one_re, ofReal_re, sub_pos]; exact hx.2 simp_rw [id] at A have B : HasDerivAt (fun y : ℂ => 1 - y) (-1) ↑x := by apply HasDerivAt.const_sub; apply hasDerivAt_id convert HasDerivAt.comp (↑x) A B using 1 ring convert (U.mul V).comp_ofReal using 1 ring have h_int := ((betaIntegral_convergent hu hv').const_mul u).sub ((betaIntegral_convergent hu' hv).const_mul v) rw [add_sub_cancel_right, add_sub_cancel_right] at h_int have int_ev := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le zero_le_one hc hder h_int have hF0 : F 0 = 0 := by simp only [F, mul_eq_zero, ofReal_zero, cpow_eq_zero_iff, eq_self_iff_true, Ne, true_and_iff, sub_zero, one_cpow, one_ne_zero, or_false_iff] contrapose! hu; rw [hu, zero_re] have hF1 : F 1 = 0 := by simp only [F, mul_eq_zero, ofReal_one, one_cpow, one_ne_zero, sub_self, cpow_eq_zero_iff, eq_self_iff_true, Ne, true_and_iff, false_or_iff] contrapose! hv; rw [hv, zero_re] rw [hF0, hF1, sub_zero, intervalIntegral.integral_sub, intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at int_ev · rw [betaIntegral, betaIntegral, ← sub_eq_zero] convert int_ev <;> ring · apply IntervalIntegrable.const_mul convert betaIntegral_convergent hu hv'; ring · apply IntervalIntegrable.const_mul convert betaIntegral_convergent hu' hv; ring
import Mathlib.Algebra.Field.Basic import Mathlib.Deprecated.Subring #align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc" variable {F : Type} [Field F] (S : Set F) structure IsSubfield extends IsSubring S : Prop where inv_mem : ∀ {x : F}, x ∈ S → x⁻¹ ∈ S #align is_subfield IsSubfield theorem IsSubfield.div_mem {S : Set F} (hS : IsSubfield S) {x y : F} (hx : x ∈ S) (hy : y ∈ S) : x / y ∈ S := by rw [div_eq_mul_inv] exact hS.toIsSubring.toIsSubmonoid.mul_mem hx (hS.inv_mem hy) #align is_subfield.div_mem IsSubfield.div_mem theorem IsSubfield.pow_mem {a : F} {n : ℤ} {s : Set F} (hs : IsSubfield s) (h : a ∈ s) : a ^ n ∈ s := by cases' n with n n · suffices a ^ (n : ℤ) ∈ s by exact this rw [zpow_natCast] exact hs.toIsSubring.toIsSubmonoid.pow_mem h · rw [zpow_negSucc] exact hs.inv_mem (hs.toIsSubring.toIsSubmonoid.pow_mem h) #align is_subfield.pow_mem IsSubfield.pow_mem theorem Univ.isSubfield : IsSubfield (@Set.univ F) := { Univ.isSubmonoid, IsAddSubgroup.univ_addSubgroup with inv_mem := fun _ ↦ trivial } #align univ.is_subfield Univ.isSubfield theorem Preimage.isSubfield {K : Type} [Field K] (f : F →+* K) {s : Set K} (hs : IsSubfield s) : IsSubfield (f ⁻¹' s) := { f.isSubring_preimage hs.toIsSubring with inv_mem := fun {a} (ha : f a ∈ s) ↦ show f a⁻¹ ∈ s by rw [map_inv₀] exact hs.inv_mem ha } #align preimage.is_subfield Preimage.isSubfield theorem Image.isSubfield {K : Type} [Field K] (f : F →+ K) {s : Set F} (hs : IsSubfield s) : IsSubfield (f '' s) := { f.isSubring_image hs.toIsSubring with inv_mem := fun ⟨x, xmem, ha⟩ ↦ ⟨x⁻¹, hs.inv_mem xmem, ha ▸ map_inv₀ f x⟩ } #align image.is_subfield Image.isSubfield	Mathlib/Deprecated/Subfield.lean	75	77	theorem Range.isSubfield {K : Type} [Field K] (f : F →+ K) : IsSubfield (Set.range f) := by	rw [← Set.image_univ] apply Image.isSubfield _ Univ.isSubfield
import Mathlib.Algebra.Polynomial.Degree.Definitions import Mathlib.Data.ENat.Basic #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836" noncomputable section open Function Polynomial Finsupp Finset open scoped Polynomial namespace Polynomial universe u v variable {R : Type u} {S : Type v} {a b : R} {n m : ℕ} section Semiring variable [Semiring R] {p q r : R[X]} def trailingDegree (p : R[X]) : ℕ∞ := p.support.min #align polynomial.trailing_degree Polynomial.trailingDegree theorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q := InvImage.wf trailingDegree wellFounded_lt #align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wf def natTrailingDegree (p : R[X]) : ℕ := (trailingDegree p).getD 0 #align polynomial.nat_trailing_degree Polynomial.natTrailingDegree def trailingCoeff (p : R[X]) : R := coeff p (natTrailingDegree p) #align polynomial.trailing_coeff Polynomial.trailingCoeff def TrailingMonic (p : R[X]) := trailingCoeff p = (1 : R) #align polynomial.trailing_monic Polynomial.TrailingMonic theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 := Iff.rfl #align polynomial.trailing_monic.def Polynomial.TrailingMonic.def instance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) := inferInstanceAs <\| Decidable (trailingCoeff p = (1 : R)) #align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidable @[simp] theorem TrailingMonic.trailingCoeff {p : R[X]} (hp : p.TrailingMonic) : trailingCoeff p = 1 := hp #align polynomial.trailing_monic.trailing_coeff Polynomial.TrailingMonic.trailingCoeff @[simp] theorem trailingDegree_zero : trailingDegree (0 : R[X]) = ⊤ := rfl #align polynomial.trailing_degree_zero Polynomial.trailingDegree_zero @[simp] theorem trailingCoeff_zero : trailingCoeff (0 : R[X]) = 0 := rfl #align polynomial.trailing_coeff_zero Polynomial.trailingCoeff_zero @[simp] theorem natTrailingDegree_zero : natTrailingDegree (0 : R[X]) = 0 := rfl #align polynomial.nat_trailing_degree_zero Polynomial.natTrailingDegree_zero theorem trailingDegree_eq_top : trailingDegree p = ⊤ ↔ p = 0 := ⟨fun h => support_eq_empty.1 (Finset.min_eq_top.1 h), fun h => by simp [h]⟩ #align polynomial.trailing_degree_eq_top Polynomial.trailingDegree_eq_top theorem trailingDegree_eq_natTrailingDegree (hp : p ≠ 0) : trailingDegree p = (natTrailingDegree p : ℕ∞) := by let ⟨n, hn⟩ := not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt trailingDegree_eq_top.1 hp)) have hn : trailingDegree p = n := Classical.not_not.1 hn rw [natTrailingDegree, hn] rfl #align polynomial.trailing_degree_eq_nat_trailing_degree Polynomial.trailingDegree_eq_natTrailingDegree theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) : p.trailingDegree = n ↔ p.natTrailingDegree = n := by rw [trailingDegree_eq_natTrailingDegree hp] exact WithTop.coe_eq_coe #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (hn : 0 < n) : p.trailingDegree = n ↔ p.natTrailingDegree = n := by constructor · intro H rwa [← trailingDegree_eq_iff_natTrailingDegree_eq] rintro rfl rw [trailingDegree_zero] at H exact Option.noConfusion H · intro H rwa [trailingDegree_eq_iff_natTrailingDegree_eq] rintro rfl rw [natTrailingDegree_zero] at H rw [H] at hn exact lt_irrefl _ hn #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq_of_pos Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ} (h : trailingDegree p = n) : natTrailingDegree p = n := have hp0 : p ≠ 0 := fun hp0 => by rw [hp0] at h; exact Option.noConfusion h Option.some_inj.1 <\| show (natTrailingDegree p : ℕ∞) = n by rwa [← trailingDegree_eq_natTrailingDegree hp0] #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some @[simp]	Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean	141	145	theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p := by	by_cases hp : p = 0; · rw [hp, trailingDegree_zero] exact le_top rw [trailingDegree_eq_natTrailingDegree hp]
import Batteries.Data.Rat.Basic import Batteries.Tactic.SeqFocus namespace Rat theorem ext : {p q : Rat} → p.num = q.num → p.den = q.den → p = q \| ⟨_,_,_,_⟩, ⟨_,_,_,_⟩, rfl, rfl => rfl @[simp] theorem mk_den_one {r : Int} : ⟨r, 1, Nat.one_ne_zero, (Nat.coprime_one_right _)⟩ = (r : Rat) := rfl @[simp] theorem zero_num : (0 : Rat).num = 0 := rfl @[simp] theorem zero_den : (0 : Rat).den = 1 := rfl @[simp] theorem one_num : (1 : Rat).num = 1 := rfl @[simp] theorem one_den : (1 : Rat).den = 1 := rfl @[simp] theorem maybeNormalize_eq {num den g} (den_nz reduced) : maybeNormalize num den g den_nz reduced = { num := num.div g, den := den / g, den_nz, reduced } := by unfold maybeNormalize; split · subst g; simp · rfl theorem normalize.reduced' {num : Int} {den g : Nat} (den_nz : den ≠ 0) (e : g = num.natAbs.gcd den) : (num / g).natAbs.Coprime (den / g) := by rw [← Int.div_eq_ediv_of_dvd (e ▸ Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] exact normalize.reduced den_nz e theorem normalize_eq {num den} (den_nz) : normalize num den den_nz = { num := num / num.natAbs.gcd den den := den / num.natAbs.gcd den den_nz := normalize.den_nz den_nz rfl reduced := normalize.reduced' den_nz rfl } := by simp only [normalize, maybeNormalize_eq, Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))] @[simp] theorem normalize_zero (nz) : normalize 0 d nz = 0 := by simp [normalize, Int.zero_div, Int.natAbs_zero, Nat.div_self (Nat.pos_of_ne_zero nz)]; rfl theorem mk_eq_normalize (num den nz c) : ⟨num, den, nz, c⟩ = normalize num den nz := by simp [normalize_eq, c.gcd_eq_one] theorem normalize_self (r : Rat) : normalize r.num r.den r.den_nz = r := (mk_eq_normalize ..).symm theorem normalize_mul_left {a : Nat} (d0 : d ≠ 0) (a0 : a ≠ 0) : normalize (↑a * n) (a * d) (Nat.mul_ne_zero a0 d0) = normalize n d d0 := by simp [normalize_eq, mk'.injEq, Int.natAbs_mul, Nat.gcd_mul_left, Nat.mul_div_mul_left _ _ (Nat.pos_of_ne_zero a0), Int.ofNat_mul, Int.mul_ediv_mul_of_pos _ _ (Int.ofNat_pos.2 <\| Nat.pos_of_ne_zero a0)] theorem normalize_mul_right {a : Nat} (d0 : d ≠ 0) (a0 : a ≠ 0) : normalize (n * a) (d * a) (Nat.mul_ne_zero d0 a0) = normalize n d d0 := by rw [← normalize_mul_left (d0 := d0) a0]; congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm] theorem normalize_eq_iff (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) : normalize n₁ d₁ z₁ = normalize n₂ d₂ z₂ ↔ n₁ * d₂ = n₂ * d₁ := by constructor <;> intro h · simp only [normalize_eq, mk'.injEq] at h have' hn₁ := Int.ofNat_dvd_left.2 <\| Nat.gcd_dvd_left n₁.natAbs d₁ have' hn₂ := Int.ofNat_dvd_left.2 <\| Nat.gcd_dvd_left n₂.natAbs d₂ have' hd₁ := Int.ofNat_dvd.2 <\| Nat.gcd_dvd_right n₁.natAbs d₁ have' hd₂ := Int.ofNat_dvd.2 <\| Nat.gcd_dvd_right n₂.natAbs d₂ rw [← Int.ediv_mul_cancel (Int.dvd_trans hd₂ (Int.dvd_mul_left ..)), Int.mul_ediv_assoc _ hd₂, ← Int.ofNat_ediv, ← h.2, Int.ofNat_ediv, ← Int.mul_ediv_assoc _ hd₁, Int.mul_ediv_assoc' _ hn₁, Int.mul_right_comm, h.1, Int.ediv_mul_cancel hn₂] · rw [← normalize_mul_right _ z₂, ← normalize_mul_left z₂ z₁, Int.mul_comm d₁, h] theorem maybeNormalize_eq_normalize {num : Int} {den g : Nat} (den_nz reduced) (hn : ↑g ∣ num) (hd : g ∣ den) : maybeNormalize num den g den_nz reduced = normalize num den (mt (by simp [·]) den_nz) := by simp only [maybeNormalize_eq, mk_eq_normalize, Int.div_eq_ediv_of_dvd hn] have : g ≠ 0 := mt (by simp [·]) den_nz rw [← normalize_mul_right _ this, Int.ediv_mul_cancel hn] congr 1; exact Nat.div_mul_cancel hd @[simp] theorem normalize_eq_zero (d0 : d ≠ 0) : normalize n d d0 = 0 ↔ n = 0 := by have' := normalize_eq_iff d0 Nat.one_ne_zero rw [normalize_zero (d := 1)] at this; rw [this]; simp theorem normalize_num_den' (num den nz) : ∃ d : Nat, d ≠ 0 ∧ num = (normalize num den nz).num * d ∧ den = (normalize num den nz).den * d := by refine ⟨num.natAbs.gcd den, Nat.gcd_ne_zero_right nz, ?_⟩ simp [normalize_eq, Int.ediv_mul_cancel (Int.ofNat_dvd_left.2 <\| Nat.gcd_dvd_left ..), Nat.div_mul_cancel (Nat.gcd_dvd_right ..)] theorem normalize_num_den (h : normalize n d z = ⟨n', d', z', c⟩) : ∃ m : Nat, m ≠ 0 ∧ n = n' * m ∧ d = d' * m := by have := normalize_num_den' n d z; rwa [h] at this theorem normalize_eq_mkRat {num den} (den_nz) : normalize num den den_nz = mkRat num den := by simp [mkRat, den_nz] theorem mkRat_num_den (z : d ≠ 0) (h : mkRat n d = ⟨n', d', z', c⟩) : ∃ m : Nat, m ≠ 0 ∧ n = n' * m ∧ d = d' * m := normalize_num_den ((normalize_eq_mkRat z).symm ▸ h) theorem mkRat_def (n d) : mkRat n d = if d0 : d = 0 then 0 else normalize n d d0 := rfl	.lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean	107	108	theorem mkRat_self (a : Rat) : mkRat a.num a.den = a := by	rw [← normalize_eq_mkRat a.den_nz, normalize_self]
import Mathlib.Algebra.BigOperators.Option import Mathlib.Analysis.BoxIntegral.Box.Basic import Mathlib.Data.Set.Pairwise.Lattice #align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219" open Set Finset Function open scoped Classical open NNReal noncomputable section namespace BoxIntegral variable {ι : Type} structure Prepartition (I : Box ι) where boxes : Finset (Box ι) le_of_mem' : ∀ J ∈ boxes, J ≤ I pairwiseDisjoint : Set.Pairwise (↑boxes) (Disjoint on ((↑) : Box ι → Set (ι → ℝ))) #align box_integral.prepartition BoxIntegral.Prepartition namespace Prepartition variable {I J J₁ J₂ : Box ι} (π : Prepartition I) {π₁ π₂ : Prepartition I} {x : ι → ℝ} instance : Membership (Box ι) (Prepartition I) := ⟨fun J π => J ∈ π.boxes⟩ @[simp] theorem mem_boxes : J ∈ π.boxes ↔ J ∈ π := Iff.rfl #align box_integral.prepartition.mem_boxes BoxIntegral.Prepartition.mem_boxes @[simp] theorem mem_mk {s h₁ h₂} : J ∈ (mk s h₁ h₂ : Prepartition I) ↔ J ∈ s := Iff.rfl #align box_integral.prepartition.mem_mk BoxIntegral.Prepartition.mem_mk theorem disjoint_coe_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (h : J₁ ≠ J₂) : Disjoint (J₁ : Set (ι → ℝ)) J₂ := π.pairwiseDisjoint h₁ h₂ h #align box_integral.prepartition.disjoint_coe_of_mem BoxIntegral.Prepartition.disjoint_coe_of_mem theorem eq_of_mem_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hx₁ : x ∈ J₁) (hx₂ : x ∈ J₂) : J₁ = J₂ := by_contra fun H => (π.disjoint_coe_of_mem h₁ h₂ H).le_bot ⟨hx₁, hx₂⟩ #align box_integral.prepartition.eq_of_mem_of_mem BoxIntegral.Prepartition.eq_of_mem_of_mem theorem eq_of_le_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle₁ : J ≤ J₁) (hle₂ : J ≤ J₂) : J₁ = J₂ := π.eq_of_mem_of_mem h₁ h₂ (hle₁ J.upper_mem) (hle₂ J.upper_mem) #align box_integral.prepartition.eq_of_le_of_le BoxIntegral.Prepartition.eq_of_le_of_le theorem eq_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle : J₁ ≤ J₂) : J₁ = J₂ := π.eq_of_le_of_le h₁ h₂ le_rfl hle #align box_integral.prepartition.eq_of_le BoxIntegral.Prepartition.eq_of_le theorem le_of_mem (hJ : J ∈ π) : J ≤ I := π.le_of_mem' J hJ #align box_integral.prepartition.le_of_mem BoxIntegral.Prepartition.le_of_mem theorem lower_le_lower (hJ : J ∈ π) : I.lower ≤ J.lower := Box.antitone_lower (π.le_of_mem hJ) #align box_integral.prepartition.lower_le_lower BoxIntegral.Prepartition.lower_le_lower theorem upper_le_upper (hJ : J ∈ π) : J.upper ≤ I.upper := Box.monotone_upper (π.le_of_mem hJ) #align box_integral.prepartition.upper_le_upper BoxIntegral.Prepartition.upper_le_upper theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) := by rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂) rfl #align box_integral.prepartition.injective_boxes BoxIntegral.Prepartition.injective_boxes @[ext] theorem ext (h : ∀ J, J ∈ π₁ ↔ J ∈ π₂) : π₁ = π₂ := injective_boxes <\| Finset.ext h #align box_integral.prepartition.ext BoxIntegral.Prepartition.ext @[simps] def single (I J : Box ι) (h : J ≤ I) : Prepartition I := ⟨{J}, by simpa, by simp⟩ #align box_integral.prepartition.single BoxIntegral.Prepartition.single @[simp] theorem mem_single {J'} (h : J ≤ I) : J' ∈ single I J h ↔ J' = J := mem_singleton #align box_integral.prepartition.mem_single BoxIntegral.Prepartition.mem_single instance : LE (Prepartition I) := ⟨fun π π' => ∀ ⦃I⦄, I ∈ π → ∃ I' ∈ π', I ≤ I'⟩ instance partialOrder : PartialOrder (Prepartition I) where le := (· ≤ ·) le_refl π I hI := ⟨I, hI, le_rfl⟩ le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ := let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁ let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂ ⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩ le_antisymm := by suffices ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes from fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁)) intro π₁ π₂ h₁ h₂ J hJ rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩ obtain rfl : J = J'' := π₁.eq_of_le hJ hJ'' (hle.trans hle') obtain rfl : J' = J := le_antisymm ‹_› ‹_› assumption instance : OrderTop (Prepartition I) where top := single I I le_rfl le_top π J hJ := ⟨I, by simp, π.le_of_mem hJ⟩ instance : OrderBot (Prepartition I) where bot := ⟨∅, fun _ hJ => (Finset.not_mem_empty _ hJ).elim, fun _ hJ => (Set.not_mem_empty _ <\| Finset.coe_empty ▸ hJ).elim⟩ bot_le _ _ hJ := (Finset.not_mem_empty _ hJ).elim instance : Inhabited (Prepartition I) := ⟨⊤⟩ theorem le_def : π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∃ J' ∈ π₂, J ≤ J' := Iff.rfl #align box_integral.prepartition.le_def BoxIntegral.Prepartition.le_def @[simp] theorem mem_top : J ∈ (⊤ : Prepartition I) ↔ J = I := mem_singleton #align box_integral.prepartition.mem_top BoxIntegral.Prepartition.mem_top @[simp] theorem top_boxes : (⊤ : Prepartition I).boxes = {I} := rfl #align box_integral.prepartition.top_boxes BoxIntegral.Prepartition.top_boxes @[simp] theorem not_mem_bot : J ∉ (⊥ : Prepartition I) := Finset.not_mem_empty _ #align box_integral.prepartition.not_mem_bot BoxIntegral.Prepartition.not_mem_bot @[simp] theorem bot_boxes : (⊥ : Prepartition I).boxes = ∅ := rfl #align box_integral.prepartition.bot_boxes BoxIntegral.Prepartition.bot_boxes theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) : InjOn (fun J : Box ι => { i \| J.lower i = x i }) { J \| J ∈ π ∧ x ∈ Box.Icc J } := by rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i \| J₁.lower i = x i } = { i \| J₂.lower i = x i }) suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by choose y hy₁ hy₂ using this exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂ intro i simp only [Set.ext_iff, mem_setOf] at H rcases (hx₁.1 i).eq_or_lt with hi₁ \| hi₁ · have hi₂ : J₂.lower i = x i := (H _).1 hi₁ have H₁ : x i < J₁.upper i := by simpa only [hi₁] using J₁.lower_lt_upper i have H₂ : x i < J₂.upper i := by simpa only [hi₂] using J₂.lower_lt_upper i rw [Ioc_inter_Ioc, hi₁, hi₂, sup_idem, Set.nonempty_Ioc] exact lt_min H₁ H₂ · have hi₂ : J₂.lower i < x i := (hx₂.1 i).lt_of_ne (mt (H _).2 hi₁.ne) exact ⟨x i, ⟨hi₁, hx₁.2 i⟩, ⟨hi₂, hx₂.2 i⟩⟩ #align box_integral.prepartition.inj_on_set_of_mem_Icc_set_of_lower_eq BoxIntegral.Prepartition.injOn_setOf_mem_Icc_setOf_lower_eq theorem card_filter_mem_Icc_le [Fintype ι] (x : ι → ℝ) : (π.boxes.filter fun J : Box ι => x ∈ Box.Icc J).card ≤ 2 ^ Fintype.card ι := by rw [← Fintype.card_set] refine Finset.card_le_card_of_inj_on (fun J : Box ι => { i \| J.lower i = x i }) (fun _ _ => Finset.mem_univ _) ?_ simpa only [Finset.mem_filter] using π.injOn_setOf_mem_Icc_setOf_lower_eq x #align box_integral.prepartition.card_filter_mem_Icc_le BoxIntegral.Prepartition.card_filter_mem_Icc_le protected def iUnion : Set (ι → ℝ) := ⋃ J ∈ π, ↑J #align box_integral.prepartition.Union BoxIntegral.Prepartition.iUnion theorem iUnion_def : π.iUnion = ⋃ J ∈ π, ↑J := rfl #align box_integral.prepartition.Union_def BoxIntegral.Prepartition.iUnion_def theorem iUnion_def' : π.iUnion = ⋃ J ∈ π.boxes, ↑J := rfl #align box_integral.prepartition.Union_def' BoxIntegral.Prepartition.iUnion_def' -- Porting note: Previous proof was `:= Set.mem_iUnion₂` @[simp] theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J := by convert Set.mem_iUnion₂ rw [Box.mem_coe, exists_prop] #align box_integral.prepartition.mem_Union BoxIntegral.Prepartition.mem_iUnion @[simp] theorem iUnion_single (h : J ≤ I) : (single I J h).iUnion = J := by simp [iUnion_def] #align box_integral.prepartition.Union_single BoxIntegral.Prepartition.iUnion_single @[simp] theorem iUnion_top : (⊤ : Prepartition I).iUnion = I := by simp [Prepartition.iUnion] #align box_integral.prepartition.Union_top BoxIntegral.Prepartition.iUnion_top @[simp] theorem iUnion_eq_empty : π₁.iUnion = ∅ ↔ π₁ = ⊥ := by simp [← injective_boxes.eq_iff, Finset.ext_iff, Prepartition.iUnion, imp_false] #align box_integral.prepartition.Union_eq_empty BoxIntegral.Prepartition.iUnion_eq_empty @[simp] theorem iUnion_bot : (⊥ : Prepartition I).iUnion = ∅ := iUnion_eq_empty.2 rfl #align box_integral.prepartition.Union_bot BoxIntegral.Prepartition.iUnion_bot theorem subset_iUnion (h : J ∈ π) : ↑J ⊆ π.iUnion := subset_biUnion_of_mem h #align box_integral.prepartition.subset_Union BoxIntegral.Prepartition.subset_iUnion theorem iUnion_subset : π.iUnion ⊆ I := iUnion₂_subset π.le_of_mem' #align box_integral.prepartition.Union_subset BoxIntegral.Prepartition.iUnion_subset @[mono] theorem iUnion_mono (h : π₁ ≤ π₂) : π₁.iUnion ⊆ π₂.iUnion := fun _ hx => let ⟨_, hJ₁, hx⟩ := π₁.mem_iUnion.1 hx let ⟨J₂, hJ₂, hle⟩ := h hJ₁ π₂.mem_iUnion.2 ⟨J₂, hJ₂, hle hx⟩ #align box_integral.prepartition.Union_mono BoxIntegral.Prepartition.iUnion_mono theorem disjoint_boxes_of_disjoint_iUnion (h : Disjoint π₁.iUnion π₂.iUnion) : Disjoint π₁.boxes π₂.boxes := Finset.disjoint_left.2 fun J h₁ h₂ => Disjoint.le_bot (h.mono (π₁.subset_iUnion h₁) (π₂.subset_iUnion h₂)) ⟨J.upper_mem, J.upper_mem⟩ #align box_integral.prepartition.disjoint_boxes_of_disjoint_Union BoxIntegral.Prepartition.disjoint_boxes_of_disjoint_iUnion theorem le_iff_nonempty_imp_le_and_iUnion_subset : π₁ ≤ π₂ ↔ (∀ J ∈ π₁, ∀ J' ∈ π₂, (J ∩ J' : Set (ι → ℝ)).Nonempty → J ≤ J') ∧ π₁.iUnion ⊆ π₂.iUnion := by constructor · refine fun H => ⟨fun J hJ J' hJ' Hne => ?_, iUnion_mono H⟩ rcases H hJ with ⟨J'', hJ'', Hle⟩ rcases Hne with ⟨x, hx, hx'⟩ rwa [π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)] · rintro ⟨H, HU⟩ J hJ simp only [Set.subset_def, mem_iUnion] at HU rcases HU J.upper ⟨J, hJ, J.upper_mem⟩ with ⟨J₂, hJ₂, hx⟩ exact ⟨J₂, hJ₂, H _ hJ _ hJ₂ ⟨_, J.upper_mem, hx⟩⟩ #align box_integral.prepartition.le_iff_nonempty_imp_le_and_Union_subset BoxIntegral.Prepartition.le_iff_nonempty_imp_le_and_iUnion_subset theorem eq_of_boxes_subset_iUnion_superset (h₁ : π₁.boxes ⊆ π₂.boxes) (h₂ : π₂.iUnion ⊆ π₁.iUnion) : π₁ = π₂ := le_antisymm (fun J hJ => ⟨J, h₁ hJ, le_rfl⟩) <\| le_iff_nonempty_imp_le_and_iUnion_subset.2 ⟨fun _ hJ₁ _ hJ₂ Hne => (π₂.eq_of_mem_of_mem hJ₁ (h₁ hJ₂) Hne.choose_spec.1 Hne.choose_spec.2).le, h₂⟩ #align box_integral.prepartition.eq_of_boxes_subset_Union_superset BoxIntegral.Prepartition.eq_of_boxes_subset_iUnion_superset @[simps] def biUnion (πi : ∀ J : Box ι, Prepartition J) : Prepartition I where boxes := π.boxes.biUnion fun J => (πi J).boxes le_of_mem' J hJ := by simp only [Finset.mem_biUnion, exists_prop, mem_boxes] at hJ rcases hJ with ⟨J', hJ', hJ⟩ exact ((πi J').le_of_mem hJ).trans (π.le_of_mem hJ') pairwiseDisjoint := by simp only [Set.Pairwise, Finset.mem_coe, Finset.mem_biUnion] rintro J₁' ⟨J₁, hJ₁, hJ₁'⟩ J₂' ⟨J₂, hJ₂, hJ₂'⟩ Hne rw [Function.onFun, Set.disjoint_left] rintro x hx₁ hx₂; apply Hne obtain rfl : J₁ = J₂ := π.eq_of_mem_of_mem hJ₁ hJ₂ ((πi J₁).le_of_mem hJ₁' hx₁) ((πi J₂).le_of_mem hJ₂' hx₂) exact (πi J₁).eq_of_mem_of_mem hJ₁' hJ₂' hx₁ hx₂ #align box_integral.prepartition.bUnion BoxIntegral.Prepartition.biUnion variable {πi πi₁ πi₂ : ∀ J : Box ι, Prepartition J} @[simp] theorem mem_biUnion : J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J' := by simp [biUnion] #align box_integral.prepartition.mem_bUnion BoxIntegral.Prepartition.mem_biUnion theorem biUnion_le (πi : ∀ J, Prepartition J) : π.biUnion πi ≤ π := fun _ hJ => let ⟨J', hJ', hJ⟩ := π.mem_biUnion.1 hJ ⟨J', hJ', (πi J').le_of_mem hJ⟩ #align box_integral.prepartition.bUnion_le BoxIntegral.Prepartition.biUnion_le @[simp] theorem biUnion_top : (π.biUnion fun _ => ⊤) = π := by ext simp #align box_integral.prepartition.bUnion_top BoxIntegral.Prepartition.biUnion_top @[congr] theorem biUnion_congr (h : π₁ = π₂) (hi : ∀ J ∈ π₁, πi₁ J = πi₂ J) : π₁.biUnion πi₁ = π₂.biUnion πi₂ := by subst π₂ ext J simp only [mem_biUnion] constructor <;> exact fun ⟨J', h₁, h₂⟩ => ⟨J', h₁, hi J' h₁ ▸ h₂⟩ #align box_integral.prepartition.bUnion_congr BoxIntegral.Prepartition.biUnion_congr theorem biUnion_congr_of_le (h : π₁ = π₂) (hi : ∀ J ≤ I, πi₁ J = πi₂ J) : π₁.biUnion πi₁ = π₂.biUnion πi₂ := biUnion_congr h fun J hJ => hi J (π₁.le_of_mem hJ) #align box_integral.prepartition.bUnion_congr_of_le BoxIntegral.Prepartition.biUnion_congr_of_le @[simp] theorem iUnion_biUnion (πi : ∀ J : Box ι, Prepartition J) : (π.biUnion πi).iUnion = ⋃ J ∈ π, (πi J).iUnion := by simp [Prepartition.iUnion] #align box_integral.prepartition.Union_bUnion BoxIntegral.Prepartition.iUnion_biUnion @[simp] theorem sum_biUnion_boxes {M : Type} [AddCommMonoid M] (π : Prepartition I) (πi : ∀ J, Prepartition J) (f : Box ι → M) : (∑ J ∈ π.boxes.biUnion fun J => (πi J).boxes, f J) = ∑ J ∈ π.boxes, ∑ J' ∈ (πi J).boxes, f J' := by refine Finset.sum_biUnion fun J₁ h₁ J₂ h₂ hne => Finset.disjoint_left.2 fun J' h₁' h₂' => ?_ exact hne (π.eq_of_le_of_le h₁ h₂ ((πi J₁).le_of_mem h₁') ((πi J₂).le_of_mem h₂')) #align box_integral.prepartition.sum_bUnion_boxes BoxIntegral.Prepartition.sum_biUnion_boxes def biUnionIndex (πi : ∀ (J : Box ι), Prepartition J) (J : Box ι) : Box ι := if hJ : J ∈ π.biUnion πi then (π.mem_biUnion.1 hJ).choose else I #align box_integral.prepartition.bUnion_index BoxIntegral.Prepartition.biUnionIndex theorem biUnionIndex_mem (hJ : J ∈ π.biUnion πi) : π.biUnionIndex πi J ∈ π := by rw [biUnionIndex, dif_pos hJ] exact (π.mem_biUnion.1 hJ).choose_spec.1 #align box_integral.prepartition.bUnion_index_mem BoxIntegral.Prepartition.biUnionIndex_mem theorem biUnionIndex_le (πi : ∀ J, Prepartition J) (J : Box ι) : π.biUnionIndex πi J ≤ I := by by_cases hJ : J ∈ π.biUnion πi · exact π.le_of_mem (π.biUnionIndex_mem hJ) · rw [biUnionIndex, dif_neg hJ] #align box_integral.prepartition.bUnion_index_le BoxIntegral.Prepartition.biUnionIndex_le theorem mem_biUnionIndex (hJ : J ∈ π.biUnion πi) : J ∈ πi (π.biUnionIndex πi J) := by convert (π.mem_biUnion.1 hJ).choose_spec.2 <;> exact dif_pos hJ #align box_integral.prepartition.mem_bUnion_index BoxIntegral.Prepartition.mem_biUnionIndex theorem le_biUnionIndex (hJ : J ∈ π.biUnion πi) : J ≤ π.biUnionIndex πi J := le_of_mem _ (π.mem_biUnionIndex hJ) #align box_integral.prepartition.le_bUnion_index BoxIntegral.Prepartition.le_biUnionIndex theorem biUnionIndex_of_mem (hJ : J ∈ π) {J'} (hJ' : J' ∈ πi J) : π.biUnionIndex πi J' = J := have : J' ∈ π.biUnion πi := π.mem_biUnion.2 ⟨J, hJ, hJ'⟩ π.eq_of_le_of_le (π.biUnionIndex_mem this) hJ (π.le_biUnionIndex this) (le_of_mem _ hJ') #align box_integral.prepartition.bUnion_index_of_mem BoxIntegral.Prepartition.biUnionIndex_of_mem theorem biUnion_assoc (πi : ∀ J, Prepartition J) (πi' : Box ι → ∀ J : Box ι, Prepartition J) : (π.biUnion fun J => (πi J).biUnion (πi' J)) = (π.biUnion πi).biUnion fun J => πi' (π.biUnionIndex πi J) J := by ext J simp only [mem_biUnion, exists_prop] constructor · rintro ⟨J₁, hJ₁, J₂, hJ₂, hJ⟩ refine ⟨J₂, ⟨J₁, hJ₁, hJ₂⟩, ?_⟩ rwa [π.biUnionIndex_of_mem hJ₁ hJ₂] · rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩ refine ⟨J₂, hJ₂, J₁, hJ₁, ?_⟩ rwa [π.biUnionIndex_of_mem hJ₂ hJ₁] at hJ #align box_integral.prepartition.bUnion_assoc BoxIntegral.Prepartition.biUnion_assoc def ofWithBot (boxes : Finset (WithBot (Box ι))) (le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I) (pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) : Prepartition I where boxes := Finset.eraseNone boxes le_of_mem' J hJ := by rw [mem_eraseNone] at hJ simpa only [WithBot.some_eq_coe, WithBot.coe_le_coe] using le_of_mem _ hJ pairwiseDisjoint J₁ h₁ J₂ h₂ hne := by simp only [mem_coe, mem_eraseNone] at h₁ h₂ exact Box.disjoint_coe.1 (pairwise_disjoint h₁ h₂ (mt Option.some_inj.1 hne)) #align box_integral.prepartition.of_with_bot BoxIntegral.Prepartition.ofWithBot @[simp] theorem mem_ofWithBot {boxes : Finset (WithBot (Box ι))} {h₁ h₂} : J ∈ (ofWithBot boxes h₁ h₂ : Prepartition I) ↔ (J : WithBot (Box ι)) ∈ boxes := mem_eraseNone #align box_integral.prepartition.mem_of_with_bot BoxIntegral.Prepartition.mem_ofWithBot @[simp] theorem iUnion_ofWithBot (boxes : Finset (WithBot (Box ι))) (le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I) (pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) : (ofWithBot boxes le_of_mem pairwise_disjoint).iUnion = ⋃ J ∈ boxes, ↑J := by suffices ⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J = ⋃ J ∈ boxes, (J : Set (ι → ℝ)) by simpa [ofWithBot, Prepartition.iUnion] simp only [← Box.biUnion_coe_eq_coe, @iUnion_comm _ _ (Box ι), @iUnion_comm _ _ (@Eq _ _ _), iUnion_iUnion_eq_right] #align box_integral.prepartition.Union_of_with_bot BoxIntegral.Prepartition.iUnion_ofWithBot theorem ofWithBot_le {boxes : Finset (WithBot (Box ι))} {le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I} {pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint} (H : ∀ J ∈ boxes, J ≠ ⊥ → ∃ J' ∈ π, J ≤ ↑J') : ofWithBot boxes le_of_mem pairwise_disjoint ≤ π := by have : ∀ J : Box ι, ↑J ∈ boxes → ∃ J' ∈ π, J ≤ J' := fun J hJ => by simpa only [WithBot.coe_le_coe] using H J hJ WithBot.coe_ne_bot simpa [ofWithBot, le_def] #align box_integral.prepartition.of_with_bot_le BoxIntegral.Prepartition.ofWithBot_le theorem le_ofWithBot {boxes : Finset (WithBot (Box ι))} {le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I} {pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint} (H : ∀ J ∈ π, ∃ J' ∈ boxes, ↑J ≤ J') : π ≤ ofWithBot boxes le_of_mem pairwise_disjoint := by intro J hJ rcases H J hJ with ⟨J', J'mem, hle⟩ lift J' to Box ι using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hle exact ⟨J', mem_ofWithBot.2 J'mem, WithBot.coe_le_coe.1 hle⟩ #align box_integral.prepartition.le_of_with_bot BoxIntegral.Prepartition.le_ofWithBot theorem ofWithBot_mono {boxes₁ : Finset (WithBot (Box ι))} {le_of_mem₁ : ∀ J ∈ boxes₁, (J : WithBot (Box ι)) ≤ I} {pairwise_disjoint₁ : Set.Pairwise (boxes₁ : Set (WithBot (Box ι))) Disjoint} {boxes₂ : Finset (WithBot (Box ι))} {le_of_mem₂ : ∀ J ∈ boxes₂, (J : WithBot (Box ι)) ≤ I} {pairwise_disjoint₂ : Set.Pairwise (boxes₂ : Set (WithBot (Box ι))) Disjoint} (H : ∀ J ∈ boxes₁, J ≠ ⊥ → ∃ J' ∈ boxes₂, J ≤ J') : ofWithBot boxes₁ le_of_mem₁ pairwise_disjoint₁ ≤ ofWithBot boxes₂ le_of_mem₂ pairwise_disjoint₂ := le_ofWithBot _ fun J hJ => H J (mem_ofWithBot.1 hJ) WithBot.coe_ne_bot #align box_integral.prepartition.of_with_bot_mono BoxIntegral.Prepartition.ofWithBot_mono theorem sum_ofWithBot {M : Type*} [AddCommMonoid M] (boxes : Finset (WithBot (Box ι))) (le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I) (pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) (f : Box ι → M) : (∑ J ∈ (ofWithBot boxes le_of_mem pairwise_disjoint).boxes, f J) = ∑ J ∈ boxes, Option.elim' 0 f J := Finset.sum_eraseNone _ _ #align box_integral.prepartition.sum_of_with_bot BoxIntegral.Prepartition.sum_ofWithBot def restrict (π : Prepartition I) (J : Box ι) : Prepartition J := ofWithBot (π.boxes.image fun J' : Box ι => J ⊓ J') (fun J' hJ' => by rcases Finset.mem_image.1 hJ' with ⟨J', -, rfl⟩ exact inf_le_left) (by simp only [Set.Pairwise, onFun, Finset.mem_coe, Finset.mem_image] rintro _ ⟨J₁, h₁, rfl⟩ _ ⟨J₂, h₂, rfl⟩ Hne have : J₁ ≠ J₂ := by rintro rfl exact Hne rfl exact ((Box.disjoint_coe.2 <\| π.disjoint_coe_of_mem h₁ h₂ this).inf_left' _).inf_right' _) #align box_integral.prepartition.restrict BoxIntegral.Prepartition.restrict @[simp] theorem mem_restrict : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : WithBot (Box ι)) = ↑J ⊓ ↑J' := by simp [restrict, eq_comm] #align box_integral.prepartition.mem_restrict BoxIntegral.Prepartition.mem_restrict theorem mem_restrict' : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : Set (ι → ℝ)) = ↑J ∩ ↑J' := by simp only [mem_restrict, ← Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe] #align box_integral.prepartition.mem_restrict' BoxIntegral.Prepartition.mem_restrict' @[mono] theorem restrict_mono {π₁ π₂ : Prepartition I} (Hle : π₁ ≤ π₂) : π₁.restrict J ≤ π₂.restrict J := by refine ofWithBot_mono fun J₁ hJ₁ hne => ?_ rw [Finset.mem_image] at hJ₁; rcases hJ₁ with ⟨J₁, hJ₁, rfl⟩ rcases Hle hJ₁ with ⟨J₂, hJ₂, hle⟩ exact ⟨_, Finset.mem_image_of_mem _ hJ₂, inf_le_inf_left _ <\| WithBot.coe_le_coe.2 hle⟩ #align box_integral.prepartition.restrict_mono BoxIntegral.Prepartition.restrict_mono theorem monotone_restrict : Monotone fun π : Prepartition I => restrict π J := fun _ _ => restrict_mono #align box_integral.prepartition.monotone_restrict BoxIntegral.Prepartition.monotone_restrict theorem restrict_boxes_of_le (π : Prepartition I) (h : I ≤ J) : (π.restrict J).boxes = π.boxes := by simp only [restrict, ofWithBot, eraseNone_eq_biUnion] refine Finset.image_biUnion.trans ?_ refine (Finset.biUnion_congr rfl ?_).trans Finset.biUnion_singleton_eq_self intro J' hJ' rw [inf_of_le_right, ← WithBot.some_eq_coe, Option.toFinset_some] exact WithBot.coe_le_coe.2 ((π.le_of_mem hJ').trans h) #align box_integral.prepartition.restrict_boxes_of_le BoxIntegral.Prepartition.restrict_boxes_of_le @[simp] theorem restrict_self : π.restrict I = π := injective_boxes <\| restrict_boxes_of_le π le_rfl #align box_integral.prepartition.restrict_self BoxIntegral.Prepartition.restrict_self @[simp] theorem iUnion_restrict : (π.restrict J).iUnion = (J : Set (ι → ℝ)) ∩ (π.iUnion) := by simp [restrict, ← inter_iUnion, ← iUnion_def] #align box_integral.prepartition.Union_restrict BoxIntegral.Prepartition.iUnion_restrict @[simp] theorem restrict_biUnion (πi : ∀ J, Prepartition J) (hJ : J ∈ π) : (π.biUnion πi).restrict J = πi J := by refine (eq_of_boxes_subset_iUnion_superset (fun J₁ h₁ => ?_) ?_).symm · refine (mem_restrict _).2 ⟨J₁, π.mem_biUnion.2 ⟨J, hJ, h₁⟩, (inf_of_le_right ?_).symm⟩ exact WithBot.coe_le_coe.2 (le_of_mem _ h₁) · simp only [iUnion_restrict, iUnion_biUnion, Set.subset_def, Set.mem_inter_iff, Set.mem_iUnion] rintro x ⟨hxJ, J₁, h₁, hx⟩ obtain rfl : J = J₁ := π.eq_of_mem_of_mem hJ h₁ hxJ (iUnion_subset _ hx) exact hx #align box_integral.prepartition.restrict_bUnion BoxIntegral.Prepartition.restrict_biUnion theorem biUnion_le_iff {πi : ∀ J, Prepartition J} {π' : Prepartition I} : π.biUnion πi ≤ π' ↔ ∀ J ∈ π, πi J ≤ π'.restrict J := by constructor <;> intro H J hJ · rw [← π.restrict_biUnion πi hJ] exact restrict_mono H · rw [mem_biUnion] at hJ rcases hJ with ⟨J₁, h₁, hJ⟩ rcases H J₁ h₁ hJ with ⟨J₂, h₂, Hle⟩ rcases π'.mem_restrict.mp h₂ with ⟨J₃, h₃, H⟩ exact ⟨J₃, h₃, Hle.trans <\| WithBot.coe_le_coe.1 <\| H.trans_le inf_le_right⟩ #align box_integral.prepartition.bUnion_le_iff BoxIntegral.Prepartition.biUnion_le_iff theorem le_biUnion_iff {πi : ∀ J, Prepartition J} {π' : Prepartition I} : π' ≤ π.biUnion πi ↔ π' ≤ π ∧ ∀ J ∈ π, π'.restrict J ≤ πi J := by refine ⟨fun H => ⟨H.trans (π.biUnion_le πi), fun J hJ => ?_⟩, ?_⟩ · rw [← π.restrict_biUnion πi hJ] exact restrict_mono H · rintro ⟨H, Hi⟩ J' hJ' rcases H hJ' with ⟨J, hJ, hle⟩ have : J' ∈ π'.restrict J := π'.mem_restrict.2 ⟨J', hJ', (inf_of_le_right <\| WithBot.coe_le_coe.2 hle).symm⟩ rcases Hi J hJ this with ⟨Ji, hJi, hlei⟩ exact ⟨Ji, π.mem_biUnion.2 ⟨J, hJ, hJi⟩, hlei⟩ #align box_integral.prepartition.le_bUnion_iff BoxIntegral.Prepartition.le_biUnion_iff instance inf : Inf (Prepartition I) := ⟨fun π₁ π₂ => π₁.biUnion fun J => π₂.restrict J⟩ theorem inf_def (π₁ π₂ : Prepartition I) : π₁ ⊓ π₂ = π₁.biUnion fun J => π₂.restrict J := rfl #align box_integral.prepartition.inf_def BoxIntegral.Prepartition.inf_def @[simp] theorem mem_inf {π₁ π₂ : Prepartition I} : J ∈ π₁ ⊓ π₂ ↔ ∃ J₁ ∈ π₁, ∃ J₂ ∈ π₂, (J : WithBot (Box ι)) = ↑J₁ ⊓ ↑J₂ := by simp only [inf_def, mem_biUnion, mem_restrict] #align box_integral.prepartition.mem_inf BoxIntegral.Prepartition.mem_inf @[simp] theorem iUnion_inf (π₁ π₂ : Prepartition I) : (π₁ ⊓ π₂).iUnion = π₁.iUnion ∩ π₂.iUnion := by simp only [inf_def, iUnion_biUnion, iUnion_restrict, ← iUnion_inter, ← iUnion_def] #align box_integral.prepartition.Union_inf BoxIntegral.Prepartition.iUnion_inf instance : SemilatticeInf (Prepartition I) := { Prepartition.inf, Prepartition.partialOrder with inf_le_left := fun π₁ _ => π₁.biUnion_le _ inf_le_right := fun _ _ => (biUnion_le_iff _).2 fun _ _ => le_rfl le_inf := fun _ π₁ _ h₁ h₂ => π₁.le_biUnion_iff.2 ⟨h₁, fun _ _ => restrict_mono h₂⟩ } @[simps] def filter (π : Prepartition I) (p : Box ι → Prop) : Prepartition I where boxes := π.boxes.filter p le_of_mem' _ hJ := π.le_of_mem (mem_filter.1 hJ).1 pairwiseDisjoint _ h₁ _ h₂ := π.disjoint_coe_of_mem (mem_filter.1 h₁).1 (mem_filter.1 h₂).1 #align box_integral.prepartition.filter BoxIntegral.Prepartition.filter @[simp] theorem mem_filter {p : Box ι → Prop} : J ∈ π.filter p ↔ J ∈ π ∧ p J := Finset.mem_filter #align box_integral.prepartition.mem_filter BoxIntegral.Prepartition.mem_filter theorem filter_le (π : Prepartition I) (p : Box ι → Prop) : π.filter p ≤ π := fun J hJ => let ⟨hπ, _⟩ := π.mem_filter.1 hJ ⟨J, hπ, le_rfl⟩ #align box_integral.prepartition.filter_le BoxIntegral.Prepartition.filter_le	Mathlib/Analysis/BoxIntegral/Partition/Basic.lean	611	613	theorem filter_of_true {p : Box ι → Prop} (hp : ∀ J ∈ π, p J) : π.filter p = π := by	ext J simpa using hp J
import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Degree.Lemmas import Mathlib.Algebra.Polynomial.Div #align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8" noncomputable section open Polynomial open Finset namespace Polynomial universe u v w z variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : ℕ} section NoZeroDivisors variable [Semiring R] [NoZeroDivisors R] {p q : R[X]} instance : NoZeroDivisors R[X] where eq_zero_or_eq_zero_of_mul_eq_zero h := by rw [← leadingCoeff_eq_zero, ← leadingCoeff_eq_zero] refine eq_zero_or_eq_zero_of_mul_eq_zero ?_ rw [← leadingCoeff_zero, ← leadingCoeff_mul, h] theorem natDegree_mul (hp : p ≠ 0) (hq : q ≠ 0) : (pq).natDegree = p.natDegree + q.natDegree := by rw [← Nat.cast_inj (R := WithBot ℕ), ← degree_eq_natDegree (mul_ne_zero hp hq), Nat.cast_add, ← degree_eq_natDegree hp, ← degree_eq_natDegree hq, degree_mul] #align polynomial.nat_degree_mul Polynomial.natDegree_mul theorem trailingDegree_mul : (p q).trailingDegree = p.trailingDegree + q.trailingDegree := by by_cases hp : p = 0 · rw [hp, zero_mul, trailingDegree_zero, top_add] by_cases hq : q = 0 · rw [hq, mul_zero, trailingDegree_zero, add_top] · rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq, trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq] apply WithTop.coe_add #align polynomial.trailing_degree_mul Polynomial.trailingDegree_mul @[simp] theorem natDegree_pow (p : R[X]) (n : ℕ) : natDegree (p ^ n) = n * natDegree p := by classical obtain rfl \| hp := eq_or_ne p 0 · obtain rfl \| hn := eq_or_ne n 0 <;> simp [] exact natDegree_pow' $ by rw [← leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp #align polynomial.nat_degree_pow Polynomial.natDegree_pow theorem degree_le_mul_left (p : R[X]) (hq : q ≠ 0) : degree p ≤ degree (p q) := by classical exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl] else by rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq]; exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _) #align polynomial.degree_le_mul_left Polynomial.degree_le_mul_left theorem natDegree_le_of_dvd {p q : R[X]} (h1 : p ∣ q) (h2 : q ≠ 0) : p.natDegree ≤ q.natDegree := by rcases h1 with ⟨q, rfl⟩; rw [mul_ne_zero_iff] at h2 rw [natDegree_mul h2.1 h2.2]; exact Nat.le_add_right _ _ #align polynomial.nat_degree_le_of_dvd Polynomial.natDegree_le_of_dvd theorem degree_le_of_dvd {p q : R[X]} (h1 : p ∣ q) (h2 : q ≠ 0) : degree p ≤ degree q := by rcases h1 with ⟨q, rfl⟩; rw [mul_ne_zero_iff] at h2 exact degree_le_mul_left p h2.2 #align polynomial.degree_le_of_dvd Polynomial.degree_le_of_dvd theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (h₁ : p ∣ q) (h₂ : degree q < degree p) : q = 0 := by by_contra hc exact (lt_iff_not_ge _ _).mp h₂ (degree_le_of_dvd h₁ hc) #align polynomial.eq_zero_of_dvd_of_degree_lt Polynomial.eq_zero_of_dvd_of_degree_lt theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (h₁ : p ∣ q) (h₂ : natDegree q < natDegree p) : q = 0 := by by_contra hc exact (lt_iff_not_ge _ _).mp h₂ (natDegree_le_of_dvd h₁ hc) #align polynomial.eq_zero_of_dvd_of_nat_degree_lt Polynomial.eq_zero_of_dvd_of_natDegree_lt theorem not_dvd_of_degree_lt {p q : R[X]} (h0 : q ≠ 0) (hl : q.degree < p.degree) : ¬p ∣ q := by by_contra hcontra exact h0 (eq_zero_of_dvd_of_degree_lt hcontra hl) #align polynomial.not_dvd_of_degree_lt Polynomial.not_dvd_of_degree_lt theorem not_dvd_of_natDegree_lt {p q : R[X]} (h0 : q ≠ 0) (hl : q.natDegree < p.natDegree) : ¬p ∣ q := by by_contra hcontra exact h0 (eq_zero_of_dvd_of_natDegree_lt hcontra hl) #align polynomial.not_dvd_of_nat_degree_lt Polynomial.not_dvd_of_natDegree_lt theorem natDegree_sub_eq_of_prod_eq {p₁ p₂ q₁ q₂ : R[X]} (hp₁ : p₁ ≠ 0) (hq₁ : q₁ ≠ 0) (hp₂ : p₂ ≠ 0) (hq₂ : q₂ ≠ 0) (h_eq : p₁ * q₂ = p₂ * q₁) : (p₁.natDegree : ℤ) - q₁.natDegree = (p₂.natDegree : ℤ) - q₂.natDegree := by rw [sub_eq_sub_iff_add_eq_add] norm_cast rw [← natDegree_mul hp₁ hq₂, ← natDegree_mul hp₂ hq₁, h_eq] #align polynomial.nat_degree_sub_eq_of_prod_eq Polynomial.natDegree_sub_eq_of_prod_eq theorem natDegree_eq_zero_of_isUnit (h : IsUnit p) : natDegree p = 0 := by nontriviality R obtain ⟨q, hq⟩ := h.exists_right_inv have := natDegree_mul (left_ne_zero_of_mul_eq_one hq) (right_ne_zero_of_mul_eq_one hq) rw [hq, natDegree_one, eq_comm, add_eq_zero_iff] at this exact this.1 #align polynomial.nat_degree_eq_zero_of_is_unit Polynomial.natDegree_eq_zero_of_isUnit theorem degree_eq_zero_of_isUnit [Nontrivial R] (h : IsUnit p) : degree p = 0 := (natDegree_eq_zero_iff_degree_le_zero.mp <\| natDegree_eq_zero_of_isUnit h).antisymm (zero_le_degree_iff.mpr h.ne_zero) #align polynomial.degree_eq_zero_of_is_unit Polynomial.degree_eq_zero_of_isUnit @[simp] theorem degree_coe_units [Nontrivial R] (u : R[X]ˣ) : degree (u : R[X]) = 0 := degree_eq_zero_of_isUnit ⟨u, rfl⟩ #align polynomial.degree_coe_units Polynomial.degree_coe_units theorem isUnit_iff : IsUnit p ↔ ∃ r : R, IsUnit r ∧ C r = p := ⟨fun hp => ⟨p.coeff 0, let h := eq_C_of_natDegree_eq_zero (natDegree_eq_zero_of_isUnit hp) ⟨isUnit_C.1 (h ▸ hp), h.symm⟩⟩, fun ⟨_, hr, hrp⟩ => hrp ▸ isUnit_C.2 hr⟩ #align polynomial.is_unit_iff Polynomial.isUnit_iff	Mathlib/Algebra/Polynomial/RingDivision.lean	226	231	theorem not_isUnit_of_degree_pos (p : R[X]) (hpl : 0 < p.degree) : ¬ IsUnit p := by	cases subsingleton_or_nontrivial R · simp [Subsingleton.elim p 0] at hpl intro h simp [degree_eq_zero_of_isUnit h] at hpl
import Mathlib.Order.Interval.Set.Basic import Mathlib.Order.Hom.Set #align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105" open Set namespace OrderIso section Preorder variable {α β : Type*} [Preorder α] [Preorder β] @[simp] theorem preimage_Iic (e : α ≃o β) (b : β) : e ⁻¹' Iic b = Iic (e.symm b) := by ext x simp [← e.le_iff_le] #align order_iso.preimage_Iic OrderIso.preimage_Iic @[simp] theorem preimage_Ici (e : α ≃o β) (b : β) : e ⁻¹' Ici b = Ici (e.symm b) := by ext x simp [← e.le_iff_le] #align order_iso.preimage_Ici OrderIso.preimage_Ici @[simp] theorem preimage_Iio (e : α ≃o β) (b : β) : e ⁻¹' Iio b = Iio (e.symm b) := by ext x simp [← e.lt_iff_lt] #align order_iso.preimage_Iio OrderIso.preimage_Iio @[simp] theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b) := by ext x simp [← e.lt_iff_lt] #align order_iso.preimage_Ioi OrderIso.preimage_Ioi @[simp] theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by simp [← Ici_inter_Iic] #align order_iso.preimage_Icc OrderIso.preimage_Icc @[simp] theorem preimage_Ico (e : α ≃o β) (a b : β) : e ⁻¹' Ico a b = Ico (e.symm a) (e.symm b) := by simp [← Ici_inter_Iio] #align order_iso.preimage_Ico OrderIso.preimage_Ico @[simp] theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) := by simp [← Ioi_inter_Iic] #align order_iso.preimage_Ioc OrderIso.preimage_Ioc @[simp] theorem preimage_Ioo (e : α ≃o β) (a b : β) : e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b) := by simp [← Ioi_inter_Iio] #align order_iso.preimage_Ioo OrderIso.preimage_Ioo @[simp]	Mathlib/Order/Interval/Set/OrderIso.lean	68	69	theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) := by	rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]
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" 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)] abbrev o := @Module.Oriented.positiveOrientation 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 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 @[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 @[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 @[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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 theorem oangle_rev (p₁ p₂ p₃ : P) : ∡ p₃ p₂ p₁ = -∡ p₁ p₂ p₃ := o.oangle_rev _ _ #align euclidean_geometry.oangle_rev EuclideanGeometry.oangle_rev @[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 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 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	Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean	218	225	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
import Mathlib.Analysis.Calculus.Deriv.Basic import Mathlib.Analysis.Calculus.FDeriv.Comp import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars #align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" universe u v w open scoped Classical open Topology Filter ENNReal open Filter Asymptotics Set open ContinuousLinearMap (smulRight smulRight_one_eq_iff) variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] variable {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] variable {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] variable {f f₀ f₁ g : 𝕜 → F} variable {f' f₀' f₁' g' : F} variable {x : 𝕜} variable {s t : Set 𝕜} variable {L L₁ L₂ : Filter 𝕜} section Composition variable {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {s' t' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h₁ : 𝕜 → 𝕜} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} {h₁' : 𝕜} {g₁ : 𝕜' → F} {g₁' : F} {L' : Filter 𝕜'} {y : 𝕜'} (x) theorem HasDerivAtFilter.scomp (hg : HasDerivAtFilter g₁ g₁' (h x) L') (hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') : HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L := by simpa using ((hg.restrictScalars 𝕜).comp x hh hL).hasDerivAtFilter #align has_deriv_at_filter.scomp HasDerivAtFilter.scomp theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter g₁ g₁' y L') (hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') : HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L := by rw [hy] at hg; exact hg.scomp x hh hL theorem HasDerivWithinAt.scomp_hasDerivAt (hg : HasDerivWithinAt g₁ g₁' s' (h x)) (hh : HasDerivAt h h' x) (hs : ∀ x, h x ∈ s') : HasDerivAt (g₁ ∘ h) (h' • g₁') x := hg.scomp x hh <\| tendsto_inf.2 ⟨hh.continuousAt, tendsto_principal.2 <\| eventually_of_forall hs⟩ #align has_deriv_within_at.scomp_has_deriv_at HasDerivWithinAt.scomp_hasDerivAt theorem HasDerivWithinAt.scomp_hasDerivAt_of_eq (hg : HasDerivWithinAt g₁ g₁' s' y) (hh : HasDerivAt h h' x) (hs : ∀ x, h x ∈ s') (hy : y = h x) : HasDerivAt (g₁ ∘ h) (h' • g₁') x := by rw [hy] at hg; exact hg.scomp_hasDerivAt x hh hs nonrec theorem HasDerivWithinAt.scomp (hg : HasDerivWithinAt g₁ g₁' t' (h x)) (hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') : HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x := hg.scomp x hh <\| hh.continuousWithinAt.tendsto_nhdsWithin hst #align has_deriv_within_at.scomp HasDerivWithinAt.scomp	Mathlib/Analysis/Calculus/Deriv/Comp.lean	101	104	theorem HasDerivWithinAt.scomp_of_eq (hg : HasDerivWithinAt g₁ g₁' t' y) (hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') (hy : y = h x) : HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x := by	rw [hy] at hg; exact hg.scomp x hh hst
import Mathlib.Data.Finset.Lattice #align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α} namespace Finset def nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) : Finset (Finset α) := 𝒜.filter fun s => a ∉ s #align finset.non_member_subfamily Finset.nonMemberSubfamily def memberSubfamily (a : α) (𝒜 : Finset (Finset α)) : Finset (Finset α) := (𝒜.filter fun s => a ∈ s).image fun s => erase s a #align finset.member_subfamily Finset.memberSubfamily @[simp] theorem mem_nonMemberSubfamily : s ∈ 𝒜.nonMemberSubfamily a ↔ s ∈ 𝒜 ∧ a ∉ s := by simp [nonMemberSubfamily] #align finset.mem_non_member_subfamily Finset.mem_nonMemberSubfamily @[simp] theorem mem_memberSubfamily : s ∈ 𝒜.memberSubfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := by simp_rw [memberSubfamily, mem_image, mem_filter] refine ⟨?_, fun h => ⟨insert a s, ⟨h.1, by simp⟩, erase_insert h.2⟩⟩ rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩ rw [insert_erase hs2] exact ⟨hs1, not_mem_erase _ _⟩ #align finset.mem_member_subfamily Finset.mem_memberSubfamily theorem nonMemberSubfamily_inter (a : α) (𝒜 ℬ : Finset (Finset α)) : (𝒜 ∩ ℬ).nonMemberSubfamily a = 𝒜.nonMemberSubfamily a ∩ ℬ.nonMemberSubfamily a := filter_inter_distrib _ _ _ #align finset.non_member_subfamily_inter Finset.nonMemberSubfamily_inter theorem memberSubfamily_inter (a : α) (𝒜 ℬ : Finset (Finset α)) : (𝒜 ∩ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∩ ℬ.memberSubfamily a := by unfold memberSubfamily rw [filter_inter_distrib, image_inter_of_injOn _ _ ((erase_injOn' _).mono _)] simp #align finset.member_subfamily_inter Finset.memberSubfamily_inter theorem nonMemberSubfamily_union (a : α) (𝒜 ℬ : Finset (Finset α)) : (𝒜 ∪ ℬ).nonMemberSubfamily a = 𝒜.nonMemberSubfamily a ∪ ℬ.nonMemberSubfamily a := filter_union _ _ _ #align finset.non_member_subfamily_union Finset.nonMemberSubfamily_union theorem memberSubfamily_union (a : α) (𝒜 ℬ : Finset (Finset α)) : (𝒜 ∪ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∪ ℬ.memberSubfamily a := by simp_rw [memberSubfamily, filter_union, image_union] #align finset.member_subfamily_union Finset.memberSubfamily_union	Mathlib/Combinatorics/SetFamily/Compression/Down.lean	91	96	theorem card_memberSubfamily_add_card_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) : (𝒜.memberSubfamily a).card + (𝒜.nonMemberSubfamily a).card = 𝒜.card := by	rw [memberSubfamily, nonMemberSubfamily, card_image_of_injOn] · conv_rhs => rw [← filter_card_add_filter_neg_card_eq_card (fun s => (a ∈ s))] · apply (erase_injOn' _).mono simp
import Mathlib.Analysis.Calculus.Deriv.Inv import Mathlib.Analysis.NormedSpace.BallAction import Mathlib.Analysis.SpecialFunctions.ExpDeriv import Mathlib.Analysis.InnerProductSpace.Calculus import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.Geometry.Manifold.Algebra.LieGroup import Mathlib.Geometry.Manifold.Instances.Real import Mathlib.Geometry.Manifold.MFDeriv.Basic #align_import geometry.manifold.instances.sphere from "leanprover-community/mathlib"@"0dc4079202c28226b2841a51eb6d3cc2135bb80f" variable {E : Type} [NormedAddCommGroup E] [InnerProductSpace ℝ E] noncomputable section open Metric FiniteDimensional Function open scoped Manifold section StereographicProjection variable (v : E) def stereoToFun (x : E) : (ℝ ∙ v)ᗮ := (2 / ((1 : ℝ) - innerSL ℝ v x)) • orthogonalProjection (ℝ ∙ v)ᗮ x #align stereo_to_fun stereoToFun variable {v} @[simp] theorem stereoToFun_apply (x : E) : stereoToFun v x = (2 / ((1 : ℝ) - innerSL ℝ v x)) • orthogonalProjection (ℝ ∙ v)ᗮ x := rfl #align stereo_to_fun_apply stereoToFun_apply theorem contDiffOn_stereoToFun : ContDiffOn ℝ ⊤ (stereoToFun v) {x : E \| innerSL _ v x ≠ (1 : ℝ)} := by refine ContDiffOn.smul ?_ (orthogonalProjection (ℝ ∙ v)ᗮ).contDiff.contDiffOn refine contDiff_const.contDiffOn.div ?_ ?_ · exact (contDiff_const.sub (innerSL ℝ v).contDiff).contDiffOn · intro x h h' exact h (sub_eq_zero.mp h').symm #align cont_diff_on_stereo_to_fun contDiffOn_stereoToFun theorem continuousOn_stereoToFun : ContinuousOn (stereoToFun v) {x : E \| innerSL _ v x ≠ (1 : ℝ)} := contDiffOn_stereoToFun.continuousOn #align continuous_on_stereo_to_fun continuousOn_stereoToFun variable (v) def stereoInvFunAux (w : E) : E := (‖w‖ ^ 2 + 4)⁻¹ • ((4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v) #align stereo_inv_fun_aux stereoInvFunAux variable {v} @[simp] theorem stereoInvFunAux_apply (w : E) : stereoInvFunAux v w = (‖w‖ ^ 2 + 4)⁻¹ • ((4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v) := rfl #align stereo_inv_fun_aux_apply stereoInvFunAux_apply theorem stereoInvFunAux_mem (hv : ‖v‖ = 1) {w : E} (hw : w ∈ (ℝ ∙ v)ᗮ) : stereoInvFunAux v w ∈ sphere (0 : E) 1 := by have h₁ : (0 : ℝ) < ‖w‖ ^ 2 + 4 := by positivity suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4 by simp only [mem_sphere_zero_iff_norm, norm_smul, Real.norm_eq_abs, abs_inv, this, abs_of_pos h₁, stereoInvFunAux_apply, inv_mul_cancel h₁.ne'] suffices ‖(4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v‖ ^ 2 = (‖w‖ ^ 2 + 4) ^ 2 by simpa [sq_eq_sq_iff_abs_eq_abs, abs_of_pos h₁] using this rw [Submodule.mem_orthogonal_singleton_iff_inner_left] at hw simp [norm_add_sq_real, norm_smul, inner_smul_left, inner_smul_right, hw, mul_pow, Real.norm_eq_abs, hv] ring #align stereo_inv_fun_aux_mem stereoInvFunAux_mem theorem hasFDerivAt_stereoInvFunAux (v : E) : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) 0 := by have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : E →L[ℝ] ℝ) 0 := by convert (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt simp have h₁ : HasFDerivAt (fun w : E => (‖w‖ ^ 2 + 4)⁻¹) (0 : E →L[ℝ] ℝ) 0 := by convert (hasFDerivAt_inv _).comp _ (h₀.add (hasFDerivAt_const 4 0)) <;> simp have h₂ : HasFDerivAt (fun w => (4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v) ((4 : ℝ) • ContinuousLinearMap.id ℝ E) 0 := by convert ((hasFDerivAt_const (4 : ℝ) 0).smul (hasFDerivAt_id 0)).add ((h₀.sub (hasFDerivAt_const (4 : ℝ) 0)).smul (hasFDerivAt_const v 0)) using 1 ext w simp convert h₁.smul h₂ using 1 ext w simp #align has_fderiv_at_stereo_inv_fun_aux hasFDerivAt_stereoInvFunAux theorem hasFDerivAt_stereoInvFunAux_comp_coe (v : E) : HasFDerivAt (stereoInvFunAux v ∘ ((↑) : (ℝ ∙ v)ᗮ → E)) (ℝ ∙ v)ᗮ.subtypeL 0 := by have : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((ℝ ∙ v)ᗮ.subtypeL 0) := hasFDerivAt_stereoInvFunAux v convert this.comp (0 : (ℝ ∙ v)ᗮ) (by apply ContinuousLinearMap.hasFDerivAt) #align has_fderiv_at_stereo_inv_fun_aux_comp_coe hasFDerivAt_stereoInvFunAux_comp_coe theorem contDiff_stereoInvFunAux : ContDiff ℝ ⊤ (stereoInvFunAux v) := by have h₀ : ContDiff ℝ ⊤ fun w : E => ‖w‖ ^ 2 := contDiff_norm_sq ℝ have h₁ : ContDiff ℝ ⊤ fun w : E => (‖w‖ ^ 2 + 4)⁻¹ := by refine (h₀.add contDiff_const).inv ?_ intro x nlinarith have h₂ : ContDiff ℝ ⊤ fun w => (4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v := by refine (contDiff_const.smul contDiff_id).add ?_ exact (h₀.sub contDiff_const).smul contDiff_const exact h₁.smul h₂ #align cont_diff_stereo_inv_fun_aux contDiff_stereoInvFunAux def stereoInvFun (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) : sphere (0 : E) 1 := ⟨stereoInvFunAux v (w : E), stereoInvFunAux_mem hv w.2⟩ #align stereo_inv_fun stereoInvFun @[simp] theorem stereoInvFun_apply (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) : (stereoInvFun hv w : E) = (‖w‖ ^ 2 + 4)⁻¹ • ((4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v) := rfl #align stereo_inv_fun_apply stereoInvFun_apply theorem stereoInvFun_ne_north_pole (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) : stereoInvFun hv w ≠ (⟨v, by simp [hv]⟩ : sphere (0 : E) 1) := by refine Subtype.coe_ne_coe.1 ?_ rw [← inner_lt_one_iff_real_of_norm_one _ hv] · have hw : ⟪v, w⟫_ℝ = 0 := Submodule.mem_orthogonal_singleton_iff_inner_right.mp w.2 have hw' : (‖(w : E)‖ ^ 2 + 4)⁻¹ (‖(w : E)‖ ^ 2 - 4) < 1 := by refine (inv_mul_lt_iff' ?_).mpr ?_ · nlinarith linarith simpa [real_inner_comm, inner_add_right, inner_smul_right, real_inner_self_eq_norm_mul_norm, hw, hv] using hw' · simpa using stereoInvFunAux_mem hv w.2 #align stereo_inv_fun_ne_north_pole stereoInvFun_ne_north_pole theorem continuous_stereoInvFun (hv : ‖v‖ = 1) : Continuous (stereoInvFun hv) := continuous_induced_rng.2 (contDiff_stereoInvFunAux.continuous.comp continuous_subtype_val) #align continuous_stereo_inv_fun continuous_stereoInvFun theorem stereo_left_inv (hv : ‖v‖ = 1) {x : sphere (0 : E) 1} (hx : (x : E) ≠ v) : stereoInvFun hv (stereoToFun v x) = x := by ext simp only [stereoToFun_apply, stereoInvFun_apply, smul_add] -- name two frequently-occuring quantities and write down their basic properties set a : ℝ := innerSL _ v x set y := orthogonalProjection (ℝ ∙ v)ᗮ x have split : ↑x = a • v + ↑y := by convert (orthogonalProjection_add_orthogonalProjection_orthogonal (ℝ ∙ v) x).symm exact (orthogonalProjection_unit_singleton ℝ hv x).symm have hvy : ⟪v, y⟫_ℝ = 0 := Submodule.mem_orthogonal_singleton_iff_inner_right.mp y.2 have pythag : 1 = a ^ 2 + ‖y‖ ^ 2 := by have hvy' : ⟪a • v, y⟫_ℝ = 0 := by simp only [inner_smul_left, hvy, mul_zero] convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero _ _ hvy' using 2 · simp [← split] · simp [norm_smul, hv, ← sq, sq_abs] · exact sq _ -- two facts which will be helpful for clearing denominators in the main calculation have ha : 1 - a ≠ 0 := by have : a < 1 := (inner_lt_one_iff_real_of_norm_one hv (by simp)).mpr hx.symm linarith -- the core of the problem is these two algebraic identities: have h₁ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * 4 * (2 / (1 - a)) = 1 := by field_simp; simp only [Submodule.coe_norm] at ; nlinarith have h₂ : (2 ^ 2 / (1 - a) ^ 2 ‖y‖ ^ 2 + 4)⁻¹ * (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 - 4) = a := by field_simp transitivity (1 - a) ^ 2 * (a * (2 ^ 2 * ‖y‖ ^ 2 + 4 * (1 - a) ^ 2)) · congr simp only [Submodule.coe_norm] at * nlinarith ring! convert congr_arg₂ Add.add (congr_arg (fun t => t • (y : E)) h₁) (congr_arg (fun t => t • v) h₂) using 1 · simp [a, inner_add_right, inner_smul_right, hvy, real_inner_self_eq_norm_mul_norm, hv, mul_smul, mul_pow, Real.norm_eq_abs, sq_abs, norm_smul] -- Porting note: used to be simp only [split, add_comm] but get maxRec errors rw [split, add_comm] ac_rfl -- Porting note: this branch did not exit in ml3 · rw [split, add_comm] congr! dsimp rw [one_smul] #align stereo_left_inv stereo_left_inv	Mathlib/Geometry/Manifold/Instances/Sphere.lean	257	268	theorem stereo_right_inv (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) : stereoToFun v (stereoInvFun hv w) = w := by	have : 2 / (1 - (‖(w : E)‖ ^ 2 + 4)⁻¹ * (‖(w : E)‖ ^ 2 - 4)) * (‖(w : E)‖ ^ 2 + 4)⁻¹ * 4 = 1 := by field_simp; ring convert congr_arg (· • w) this · have h₁ : orthogonalProjection (ℝ ∙ v)ᗮ v = 0 := orthogonalProjection_orthogonalComplement_singleton_eq_zero v -- Porting note: was innerSL _ and now just inner have h₃ : inner v w = (0 : ℝ) := Submodule.mem_orthogonal_singleton_iff_inner_right.mp w.2 -- Porting note: was innerSL _ and now just inner have h₄ : inner v v = (1 : ℝ) := by simp [real_inner_self_eq_norm_mul_norm, hv] simp [h₁, h₃, h₄, ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, mul_smul] · simp
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order #align_import measure_theory.function.floor from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf" open Set section FloorRing variable {α R : Type*} [MeasurableSpace α] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] theorem Int.measurable_floor [OpensMeasurableSpace R] : Measurable (Int.floor : R → ℤ) := measurable_to_countable fun x => by simpa only [Int.preimage_floor_singleton] using measurableSet_Ico #align int.measurable_floor Int.measurable_floor @[measurability] theorem Measurable.floor [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f) : Measurable fun x => ⌊f x⌋ := Int.measurable_floor.comp hf #align measurable.floor Measurable.floor theorem Int.measurable_ceil [OpensMeasurableSpace R] : Measurable (Int.ceil : R → ℤ) := measurable_to_countable fun x => by simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc #align int.measurable_ceil Int.measurable_ceil @[measurability] theorem Measurable.ceil [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f) : Measurable fun x => ⌈f x⌉ := Int.measurable_ceil.comp hf #align measurable.ceil Measurable.ceil	Mathlib/MeasureTheory/Function/Floor.lean	47	50	theorem measurable_fract [BorelSpace R] : Measurable (Int.fract : R → R) := by	intro s hs rw [Int.preimage_fract] exact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico)
import Mathlib.Order.ConditionallyCompleteLattice.Finset import Mathlib.Order.Interval.Finset.Nat #align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54" assert_not_exists MonoidWithZero open Set namespace Nat open scoped Classical noncomputable instance : InfSet ℕ := ⟨fun s ↦ if h : ∃ n, n ∈ s then @Nat.find (fun n ↦ n ∈ s) _ h else 0⟩ noncomputable instance : SupSet ℕ := ⟨fun s ↦ if h : ∃ n, ∀ a ∈ s, a ≤ n then @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h else 0⟩ theorem sInf_def {s : Set ℕ} (h : s.Nonempty) : sInf s = @Nat.find (fun n ↦ n ∈ s) _ h := dif_pos _ #align nat.Inf_def Nat.sInf_def theorem sSup_def {s : Set ℕ} (h : ∃ n, ∀ a ∈ s, a ≤ n) : sSup s = @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h := dif_pos _ #align nat.Sup_def Nat.sSup_def theorem _root_.Set.Infinite.Nat.sSup_eq_zero {s : Set ℕ} (h : s.Infinite) : sSup s = 0 := dif_neg fun ⟨n, hn⟩ ↦ let ⟨k, hks, hk⟩ := h.exists_gt n (hn k hks).not_lt hk #align set.infinite.nat.Sup_eq_zero Set.Infinite.Nat.sSup_eq_zero @[simp] theorem sInf_eq_zero {s : Set ℕ} : sInf s = 0 ↔ 0 ∈ s ∨ s = ∅ := by cases eq_empty_or_nonempty s with \| inl h => subst h simp only [or_true_iff, eq_self_iff_true, iff_true_iff, iInf, InfSet.sInf, mem_empty_iff_false, exists_false, dif_neg, not_false_iff] \| inr h => simp only [h.ne_empty, or_false_iff, Nat.sInf_def, h, Nat.find_eq_zero] #align nat.Inf_eq_zero Nat.sInf_eq_zero @[simp] theorem sInf_empty : sInf ∅ = 0 := by rw [sInf_eq_zero] right rfl #align nat.Inf_empty Nat.sInf_empty @[simp] theorem iInf_of_empty {ι : Sort} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by rw [iInf_of_isEmpty, sInf_empty] #align nat.infi_of_empty Nat.iInf_of_empty @[simp] lemma iInf_const_zero {ι : Sort} : ⨅ i : ι, 0 = 0 := (isEmpty_or_nonempty ι).elim (fun h ↦ by simp) fun h ↦ sInf_eq_zero.2 <\| by simp theorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s := by rw [Nat.sInf_def h] exact Nat.find_spec h #align nat.Inf_mem Nat.sInf_mem theorem not_mem_of_lt_sInf {s : Set ℕ} {m : ℕ} (hm : m < sInf s) : m ∉ s := by cases eq_empty_or_nonempty s with \| inl h => subst h; apply not_mem_empty \| inr h => rw [Nat.sInf_def h] at hm; exact Nat.find_min h hm #align nat.not_mem_of_lt_Inf Nat.not_mem_of_lt_sInf protected theorem sInf_le {s : Set ℕ} {m : ℕ} (hm : m ∈ s) : sInf s ≤ m := by rw [Nat.sInf_def ⟨m, hm⟩] exact Nat.find_min' ⟨m, hm⟩ hm #align nat.Inf_le Nat.sInf_le theorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by by_contra contra rw [Set.not_nonempty_iff_eq_empty] at contra have h' : sInf s ≠ 0 := ne_of_gt h apply h' rw [Nat.sInf_eq_zero] right assumption #align nat.nonempty_of_pos_Inf Nat.nonempty_of_pos_sInf theorem nonempty_of_sInf_eq_succ {s : Set ℕ} {k : ℕ} (h : sInf s = k + 1) : s.Nonempty := nonempty_of_pos_sInf (h.symm ▸ succ_pos k : sInf s > 0) #align nat.nonempty_of_Inf_eq_succ Nat.nonempty_of_sInf_eq_succ theorem eq_Ici_of_nonempty_of_upward_closed {s : Set ℕ} (hs : s.Nonempty) (hs' : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) : s = Ici (sInf s) := ext fun n ↦ ⟨fun H ↦ Nat.sInf_le H, fun H ↦ hs' (sInf s) n H (sInf_mem hs)⟩ #align nat.eq_Ici_of_nonempty_of_upward_closed Nat.eq_Ici_of_nonempty_of_upward_closed theorem sInf_upward_closed_eq_succ_iff {s : Set ℕ} (hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) (k : ℕ) : sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s := by constructor · intro H rw [eq_Ici_of_nonempty_of_upward_closed (nonempty_of_sInf_eq_succ _) hs, H, mem_Ici, mem_Ici] · exact ⟨le_rfl, k.not_succ_le_self⟩; · exact k · assumption · rintro ⟨H, H'⟩ rw [sInf_def (⟨_, H⟩ : s.Nonempty), find_eq_iff] exact ⟨H, fun n hnk hns ↦ H' <\| hs n k (Nat.lt_succ_iff.mp hnk) hns⟩ #align nat.Inf_upward_closed_eq_succ_iff Nat.sInf_upward_closed_eq_succ_iff instance : Lattice ℕ := LinearOrder.toLattice noncomputable instance : ConditionallyCompleteLinearOrderBot ℕ := { (inferInstance : OrderBot ℕ), (LinearOrder.toLattice : Lattice ℕ), (inferInstance : LinearOrder ℕ) with -- sup := sSup -- Porting note: removed, unnecessary? -- inf := sInf -- Porting note: removed, unnecessary? le_csSup := fun s a hb ha ↦ by rw [sSup_def hb]; revert a ha; exact @Nat.find_spec _ _ hb csSup_le := fun s a _ ha ↦ by rw [sSup_def ⟨a, ha⟩]; exact Nat.find_min' _ ha le_csInf := fun s a hs hb ↦ by rw [sInf_def hs]; exact hb (@Nat.find_spec (fun n ↦ n ∈ s) _ _) csInf_le := fun s a _ ha ↦ by rw [sInf_def ⟨a, ha⟩]; exact Nat.find_min' _ ha csSup_empty := by simp only [sSup_def, Set.mem_empty_iff_false, forall_const, forall_prop_of_false, not_false_iff, exists_const] apply bot_unique (Nat.find_min' _ _) trivial csSup_of_not_bddAbove := by intro s hs simp only [mem_univ, forall_true_left, sSup, mem_empty_iff_false, IsEmpty.forall_iff, forall_const, exists_const, dite_true] rw [dif_neg] · exact le_antisymm (zero_le _) (find_le trivial) · exact hs csInf_of_not_bddBelow := fun s hs ↦ by simp at hs } theorem sSup_mem {s : Set ℕ} (h₁ : s.Nonempty) (h₂ : BddAbove s) : sSup s ∈ s := let ⟨k, hk⟩ := h₂ h₁.csSup_mem ((finite_le_nat k).subset hk) #align nat.Sup_mem Nat.sSup_mem theorem sInf_add {n : ℕ} {p : ℕ → Prop} (hn : n ≤ sInf { m \| p m }) : sInf { m \| p (m + n) } + n = sInf { m \| p m } := by obtain h \| ⟨m, hm⟩ := { m \| p (m + n) }.eq_empty_or_nonempty · rw [h, Nat.sInf_empty, zero_add] obtain hnp \| hnp := hn.eq_or_lt · exact hnp suffices hp : p (sInf { m \| p m } - n + n) from (h.subset hp).elim rw [Nat.sub_add_cancel hn] exact csInf_mem (nonempty_of_pos_sInf <\| n.zero_le.trans_lt hnp) · have hp : ∃ n, n ∈ { m \| p m } := ⟨_, hm⟩ rw [Nat.sInf_def ⟨m, hm⟩, Nat.sInf_def hp] rw [Nat.sInf_def hp] at hn exact find_add hn #align nat.Inf_add Nat.sInf_add theorem sInf_add' {n : ℕ} {p : ℕ → Prop} (h : 0 < sInf { m \| p m }) : sInf { m \| p m } + n = sInf { m \| p (m - n) } := by suffices h₁ : n ≤ sInf {m \| p (m - n)} by convert sInf_add h₁ simp_rw [Nat.add_sub_cancel_right] obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h refine le_csInf ⟨m + n, ?_⟩ fun b hb ↦ le_of_not_lt fun hbn ↦ ne_of_mem_of_not_mem ?_ (not_mem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le) · dsimp rwa [Nat.add_sub_cancel_right] · exact hb #align nat.Inf_add' Nat.sInf_add' section variable {α : Type*} [CompleteLattice α] theorem iSup_lt_succ (u : ℕ → α) (n : ℕ) : ⨆ k < n + 1, u k = (⨆ k < n, u k) ⊔ u n := by simp [Nat.lt_succ_iff_lt_or_eq, iSup_or, iSup_sup_eq] #align nat.supr_lt_succ Nat.iSup_lt_succ theorem iSup_lt_succ' (u : ℕ → α) (n : ℕ) : ⨆ k < n + 1, u k = u 0 ⊔ ⨆ k < n, u (k + 1) := by rw [← sup_iSup_nat_succ] simp #align nat.supr_lt_succ' Nat.iSup_lt_succ' theorem iInf_lt_succ (u : ℕ → α) (n : ℕ) : ⨅ k < n + 1, u k = (⨅ k < n, u k) ⊓ u n := @iSup_lt_succ αᵒᵈ _ _ _ #align nat.infi_lt_succ Nat.iInf_lt_succ theorem iInf_lt_succ' (u : ℕ → α) (n : ℕ) : ⨅ k < n + 1, u k = u 0 ⊓ ⨅ k < n, u (k + 1) := @iSup_lt_succ' αᵒᵈ _ _ _ #align nat.infi_lt_succ' Nat.iInf_lt_succ' theorem iSup_le_succ (u : ℕ → α) (n : ℕ) : ⨆ k ≤ n + 1, u k = (⨆ k ≤ n, u k) ⊔ u (n + 1) := by simp_rw [← Nat.lt_succ_iff, iSup_lt_succ]	Mathlib/Data/Nat/Lattice.lean	211	212	theorem iSup_le_succ' (u : ℕ → α) (n : ℕ) : ⨆ k ≤ n + 1, u k = u 0 ⊔ ⨆ k ≤ n, u (k + 1) := by	simp_rw [← Nat.lt_succ_iff, iSup_lt_succ']
import Mathlib.Algebra.Group.Equiv.Basic import Mathlib.Data.ENat.Lattice import Mathlib.Data.Part import Mathlib.Tactic.NormNum #align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8" open Part hiding some def PartENat : Type := Part ℕ #align part_enat PartENat namespace PartENat @[coe] def some : ℕ → PartENat := Part.some #align part_enat.some PartENat.some instance : Zero PartENat := ⟨some 0⟩ instance : Inhabited PartENat := ⟨0⟩ instance : One PartENat := ⟨some 1⟩ instance : Add PartENat := ⟨fun x y => ⟨x.Dom ∧ y.Dom, fun h => get x h.1 + get y h.2⟩⟩ instance (n : ℕ) : Decidable (some n).Dom := isTrue trivial @[simp] theorem dom_some (x : ℕ) : (some x).Dom := trivial #align part_enat.dom_some PartENat.dom_some instance addCommMonoid : AddCommMonoid PartENat where add := (· + ·) zero := 0 add_comm x y := Part.ext' and_comm fun _ _ => add_comm _ _ zero_add x := Part.ext' (true_and_iff _) fun _ _ => zero_add _ add_zero x := Part.ext' (and_true_iff _) fun _ _ => add_zero _ add_assoc x y z := Part.ext' and_assoc fun _ _ => add_assoc _ _ _ nsmul := nsmulRec instance : AddCommMonoidWithOne PartENat := { PartENat.addCommMonoid with one := 1 natCast := some natCast_zero := rfl natCast_succ := fun _ => Part.ext' (true_and_iff _).symm fun _ _ => rfl } theorem some_eq_natCast (n : ℕ) : some n = n := rfl #align part_enat.some_eq_coe PartENat.some_eq_natCast instance : CharZero PartENat where cast_injective := Part.some_injective theorem natCast_inj {x y : ℕ} : (x : PartENat) = y ↔ x = y := Nat.cast_inj #align part_enat.coe_inj PartENat.natCast_inj @[simp] theorem dom_natCast (x : ℕ) : (x : PartENat).Dom := trivial #align part_enat.dom_coe PartENat.dom_natCast -- See note [no_index around OfNat.ofNat] @[simp] theorem dom_ofNat (x : ℕ) [x.AtLeastTwo] : (no_index (OfNat.ofNat x : PartENat)).Dom := trivial @[simp] theorem dom_zero : (0 : PartENat).Dom := trivial @[simp] theorem dom_one : (1 : PartENat).Dom := trivial instance : CanLift PartENat ℕ (↑) Dom := ⟨fun n hn => ⟨n.get hn, Part.some_get _⟩⟩ instance : LE PartENat := ⟨fun x y => ∃ h : y.Dom → x.Dom, ∀ hy : y.Dom, x.get (h hy) ≤ y.get hy⟩ instance : Top PartENat := ⟨none⟩ instance : Bot PartENat := ⟨0⟩ instance : Sup PartENat := ⟨fun x y => ⟨x.Dom ∧ y.Dom, fun h => x.get h.1 ⊔ y.get h.2⟩⟩ theorem le_def (x y : PartENat) : x ≤ y ↔ ∃ h : y.Dom → x.Dom, ∀ hy : y.Dom, x.get (h hy) ≤ y.get hy := Iff.rfl #align part_enat.le_def PartENat.le_def @[elab_as_elim] protected theorem casesOn' {P : PartENat → Prop} : ∀ a : PartENat, P ⊤ → (∀ n : ℕ, P (some n)) → P a := Part.induction_on #align part_enat.cases_on' PartENat.casesOn' @[elab_as_elim] protected theorem casesOn {P : PartENat → Prop} : ∀ a : PartENat, P ⊤ → (∀ n : ℕ, P n) → P a := by exact PartENat.casesOn' #align part_enat.cases_on PartENat.casesOn -- not a simp lemma as we will provide a `LinearOrderedAddCommMonoidWithTop` instance later theorem top_add (x : PartENat) : ⊤ + x = ⊤ := Part.ext' (false_and_iff _) fun h => h.left.elim #align part_enat.top_add PartENat.top_add -- not a simp lemma as we will provide a `LinearOrderedAddCommMonoidWithTop` instance later theorem add_top (x : PartENat) : x + ⊤ = ⊤ := by rw [add_comm, top_add] #align part_enat.add_top PartENat.add_top @[simp] theorem natCast_get {x : PartENat} (h : x.Dom) : (x.get h : PartENat) = x := by exact Part.ext' (iff_of_true trivial h) fun _ _ => rfl #align part_enat.coe_get PartENat.natCast_get @[simp, norm_cast] theorem get_natCast' (x : ℕ) (h : (x : PartENat).Dom) : get (x : PartENat) h = x := by rw [← natCast_inj, natCast_get] #align part_enat.get_coe' PartENat.get_natCast' theorem get_natCast {x : ℕ} : get (x : PartENat) (dom_natCast x) = x := get_natCast' _ _ #align part_enat.get_coe PartENat.get_natCast theorem coe_add_get {x : ℕ} {y : PartENat} (h : ((x : PartENat) + y).Dom) : get ((x : PartENat) + y) h = x + get y h.2 := by rfl #align part_enat.coe_add_get PartENat.coe_add_get @[simp] theorem get_add {x y : PartENat} (h : (x + y).Dom) : get (x + y) h = x.get h.1 + y.get h.2 := rfl #align part_enat.get_add PartENat.get_add @[simp] theorem get_zero (h : (0 : PartENat).Dom) : (0 : PartENat).get h = 0 := rfl #align part_enat.get_zero PartENat.get_zero @[simp] theorem get_one (h : (1 : PartENat).Dom) : (1 : PartENat).get h = 1 := rfl #align part_enat.get_one PartENat.get_one -- See note [no_index around OfNat.ofNat] @[simp] theorem get_ofNat' (x : ℕ) [x.AtLeastTwo] (h : (no_index (OfNat.ofNat x : PartENat)).Dom) : Part.get (no_index (OfNat.ofNat x : PartENat)) h = (no_index (OfNat.ofNat x)) := get_natCast' x h nonrec theorem get_eq_iff_eq_some {a : PartENat} {ha : a.Dom} {b : ℕ} : a.get ha = b ↔ a = some b := get_eq_iff_eq_some #align part_enat.get_eq_iff_eq_some PartENat.get_eq_iff_eq_some theorem get_eq_iff_eq_coe {a : PartENat} {ha : a.Dom} {b : ℕ} : a.get ha = b ↔ a = b := by rw [get_eq_iff_eq_some] rfl #align part_enat.get_eq_iff_eq_coe PartENat.get_eq_iff_eq_coe theorem dom_of_le_of_dom {x y : PartENat} : x ≤ y → y.Dom → x.Dom := fun ⟨h, _⟩ => h #align part_enat.dom_of_le_of_dom PartENat.dom_of_le_of_dom theorem dom_of_le_some {x : PartENat} {y : ℕ} (h : x ≤ some y) : x.Dom := dom_of_le_of_dom h trivial #align part_enat.dom_of_le_some PartENat.dom_of_le_some theorem dom_of_le_natCast {x : PartENat} {y : ℕ} (h : x ≤ y) : x.Dom := by exact dom_of_le_some h #align part_enat.dom_of_le_coe PartENat.dom_of_le_natCast instance decidableLe (x y : PartENat) [Decidable x.Dom] [Decidable y.Dom] : Decidable (x ≤ y) := if hx : x.Dom then decidable_of_decidable_of_iff (by rw [le_def]) else if hy : y.Dom then isFalse fun h => hx <\| dom_of_le_of_dom h hy else isTrue ⟨fun h => (hy h).elim, fun h => (hy h).elim⟩ #align part_enat.decidable_le PartENat.decidableLe -- Porting note: Removed. Use `Nat.castAddMonoidHom` instead. #noalign part_enat.coe_hom #noalign part_enat.coe_coe_hom instance partialOrder : PartialOrder PartENat where le := (· ≤ ·) le_refl _ := ⟨id, fun _ => le_rfl⟩ le_trans := fun _ _ _ ⟨hxy₁, hxy₂⟩ ⟨hyz₁, hyz₂⟩ => ⟨hxy₁ ∘ hyz₁, fun _ => le_trans (hxy₂ _) (hyz₂ _)⟩ lt_iff_le_not_le _ _ := Iff.rfl le_antisymm := fun _ _ ⟨hxy₁, hxy₂⟩ ⟨hyx₁, hyx₂⟩ => Part.ext' ⟨hyx₁, hxy₁⟩ fun _ _ => le_antisymm (hxy₂ _) (hyx₂ _) theorem lt_def (x y : PartENat) : x < y ↔ ∃ hx : x.Dom, ∀ hy : y.Dom, x.get hx < y.get hy := by rw [lt_iff_le_not_le, le_def, le_def, not_exists] constructor · rintro ⟨⟨hyx, H⟩, h⟩ by_cases hx : x.Dom · use hx intro hy specialize H hy specialize h fun _ => hy rw [not_forall] at h cases' h with hx' h rw [not_le] at h exact h · specialize h fun hx' => (hx hx').elim rw [not_forall] at h cases' h with hx' h exact (hx hx').elim · rintro ⟨hx, H⟩ exact ⟨⟨fun _ => hx, fun hy => (H hy).le⟩, fun hxy h => not_lt_of_le (h _) (H _)⟩ #align part_enat.lt_def PartENat.lt_def noncomputable instance orderedAddCommMonoid : OrderedAddCommMonoid PartENat := { PartENat.partialOrder, PartENat.addCommMonoid with add_le_add_left := fun a b ⟨h₁, h₂⟩ c => PartENat.casesOn c (by simp [top_add]) fun c => ⟨fun h => And.intro (dom_natCast _) (h₁ h.2), fun h => by simpa only [coe_add_get] using add_le_add_left (h₂ _) c⟩ } instance semilatticeSup : SemilatticeSup PartENat := { PartENat.partialOrder with sup := (· ⊔ ·) le_sup_left := fun _ _ => ⟨And.left, fun _ => le_sup_left⟩ le_sup_right := fun _ _ => ⟨And.right, fun _ => le_sup_right⟩ sup_le := fun _ _ _ ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ => ⟨fun hz => ⟨hx₁ hz, hy₁ hz⟩, fun _ => sup_le (hx₂ _) (hy₂ _)⟩ } #align part_enat.semilattice_sup PartENat.semilatticeSup instance orderBot : OrderBot PartENat where bot := ⊥ bot_le _ := ⟨fun _ => trivial, fun _ => Nat.zero_le _⟩ #align part_enat.order_bot PartENat.orderBot instance orderTop : OrderTop PartENat where top := ⊤ le_top _ := ⟨fun h => False.elim h, fun hy => False.elim hy⟩ #align part_enat.order_top PartENat.orderTop instance : ZeroLEOneClass PartENat where zero_le_one := bot_le theorem coe_le_coe {x y : ℕ} : (x : PartENat) ≤ y ↔ x ≤ y := Nat.cast_le #align part_enat.coe_le_coe PartENat.coe_le_coe theorem coe_lt_coe {x y : ℕ} : (x : PartENat) < y ↔ x < y := Nat.cast_lt #align part_enat.coe_lt_coe PartENat.coe_lt_coe @[simp]	Mathlib/Data/Nat/PartENat.lean	318	321	theorem get_le_get {x y : PartENat} {hx : x.Dom} {hy : y.Dom} : x.get hx ≤ y.get hy ↔ x ≤ y := by	conv => lhs rw [← coe_le_coe, natCast_get, natCast_get]
import Mathlib.Logic.Relation import Mathlib.Order.GaloisConnection #align_import data.setoid.basic from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821" variable {α : Type} {β : Type} def Setoid.Rel (r : Setoid α) : α → α → Prop := @Setoid.r _ r #align setoid.rel Setoid.Rel instance Setoid.decidableRel (r : Setoid α) [h : DecidableRel r.r] : DecidableRel r.Rel := h #align setoid.decidable_rel Setoid.decidableRel theorem Quotient.eq_rel {r : Setoid α} {x y} : (Quotient.mk' x : Quotient r) = Quotient.mk' y ↔ r.Rel x y := Quotient.eq #align quotient.eq_rel Quotient.eq_rel namespace Setoid @[ext] theorem ext' {r s : Setoid α} (H : ∀ a b, r.Rel a b ↔ s.Rel a b) : r = s := ext H #align setoid.ext' Setoid.ext' theorem ext_iff {r s : Setoid α} : r = s ↔ ∀ a b, r.Rel a b ↔ s.Rel a b := ⟨fun h _ _ => h ▸ Iff.rfl, ext'⟩ #align setoid.ext_iff Setoid.ext_iff theorem eq_iff_rel_eq {r₁ r₂ : Setoid α} : r₁ = r₂ ↔ r₁.Rel = r₂.Rel := ⟨fun h => h ▸ rfl, fun h => Setoid.ext' fun _ _ => h ▸ Iff.rfl⟩ #align setoid.eq_iff_rel_eq Setoid.eq_iff_rel_eq instance : LE (Setoid α) := ⟨fun r s => ∀ ⦃x y⦄, r.Rel x y → s.Rel x y⟩ theorem le_def {r s : Setoid α} : r ≤ s ↔ ∀ {x y}, r.Rel x y → s.Rel x y := Iff.rfl #align setoid.le_def Setoid.le_def @[refl] theorem refl' (r : Setoid α) (x) : r.Rel x x := r.iseqv.refl x #align setoid.refl' Setoid.refl' @[symm] theorem symm' (r : Setoid α) : ∀ {x y}, r.Rel x y → r.Rel y x := r.iseqv.symm #align setoid.symm' Setoid.symm' @[trans] theorem trans' (r : Setoid α) : ∀ {x y z}, r.Rel x y → r.Rel y z → r.Rel x z := r.iseqv.trans #align setoid.trans' Setoid.trans' theorem comm' (s : Setoid α) {x y} : s.Rel x y ↔ s.Rel y x := ⟨s.symm', s.symm'⟩ #align setoid.comm' Setoid.comm' def ker (f : α → β) : Setoid α := ⟨(· = ·) on f, eq_equivalence.comap f⟩ #align setoid.ker Setoid.ker @[simp] theorem ker_mk_eq (r : Setoid α) : ker (@Quotient.mk'' _ r) = r := ext' fun _ _ => Quotient.eq #align setoid.ker_mk_eq Setoid.ker_mk_eq theorem ker_apply_mk_out {f : α → β} (a : α) : f (haveI := Setoid.ker f; ⟦a⟧.out) = f a := @Quotient.mk_out _ (Setoid.ker f) a #align setoid.ker_apply_mk_out Setoid.ker_apply_mk_out theorem ker_apply_mk_out' {f : α → β} (a : α) : f (Quotient.mk _ a : Quotient <\| Setoid.ker f).out' = f a := @Quotient.mk_out' _ (Setoid.ker f) a #align setoid.ker_apply_mk_out' Setoid.ker_apply_mk_out' theorem ker_def {f : α → β} {x y : α} : (ker f).Rel x y ↔ f x = f y := Iff.rfl #align setoid.ker_def Setoid.ker_def protected def prod (r : Setoid α) (s : Setoid β) : Setoid (α × β) where r x y := r.Rel x.1 y.1 ∧ s.Rel x.2 y.2 iseqv := ⟨fun x => ⟨r.refl' x.1, s.refl' x.2⟩, fun h => ⟨r.symm' h.1, s.symm' h.2⟩, fun h₁ h₂ => ⟨r.trans' h₁.1 h₂.1, s.trans' h₁.2 h₂.2⟩⟩ #align setoid.prod Setoid.prod instance : Inf (Setoid α) := ⟨fun r s => ⟨fun x y => r.Rel x y ∧ s.Rel x y, ⟨fun x => ⟨r.refl' x, s.refl' x⟩, fun h => ⟨r.symm' h.1, s.symm' h.2⟩, fun h1 h2 => ⟨r.trans' h1.1 h2.1, s.trans' h1.2 h2.2⟩⟩⟩⟩ theorem inf_def {r s : Setoid α} : (r ⊓ s).Rel = r.Rel ⊓ s.Rel := rfl #align setoid.inf_def Setoid.inf_def theorem inf_iff_and {r s : Setoid α} {x y} : (r ⊓ s).Rel x y ↔ r.Rel x y ∧ s.Rel x y := Iff.rfl #align setoid.inf_iff_and Setoid.inf_iff_and instance : InfSet (Setoid α) := ⟨fun S => { r := fun x y => ∀ r ∈ S, r.Rel x y iseqv := ⟨fun x r _ => r.refl' x, fun h r hr => r.symm' <\| h r hr, fun h1 h2 r hr => r.trans' (h1 r hr) <\| h2 r hr⟩ }⟩ theorem sInf_def {s : Set (Setoid α)} : (sInf s).Rel = sInf (Rel '' s) := by ext simp only [sInf_image, iInf_apply, iInf_Prop_eq] rfl #align setoid.Inf_def Setoid.sInf_def instance : PartialOrder (Setoid α) where le := (· ≤ ·) lt r s := r ≤ s ∧ ¬s ≤ r le_refl _ _ _ := id le_trans _ _ _ hr hs _ _ h := hs <\| hr h lt_iff_le_not_le _ _ := Iff.rfl le_antisymm _ _ h1 h2 := Setoid.ext' fun _ _ => ⟨fun h => h1 h, fun h => h2 h⟩ instance completeLattice : CompleteLattice (Setoid α) := { (completeLatticeOfInf (Setoid α)) fun _ => ⟨fun _ hr _ _ h => h _ hr, fun _ hr _ _ h _ hr' => hr hr' h⟩ with inf := Inf.inf inf_le_left := fun _ _ _ _ h => h.1 inf_le_right := fun _ _ _ _ h => h.2 le_inf := fun _ _ _ h1 h2 _ _ h => ⟨h1 h, h2 h⟩ top := ⟨fun _ _ => True, ⟨fun _ => trivial, fun h => h, fun h1 _ => h1⟩⟩ le_top := fun _ _ _ _ => trivial bot := ⟨(· = ·), ⟨fun _ => rfl, fun h => h.symm, fun h1 h2 => h1.trans h2⟩⟩ bot_le := fun r x _ h => h ▸ r.2.1 x } #align setoid.complete_lattice Setoid.completeLattice @[simp] theorem top_def : (⊤ : Setoid α).Rel = ⊤ := rfl #align setoid.top_def Setoid.top_def @[simp] theorem bot_def : (⊥ : Setoid α).Rel = (· = ·) := rfl #align setoid.bot_def Setoid.bot_def theorem eq_top_iff {s : Setoid α} : s = (⊤ : Setoid α) ↔ ∀ x y : α, s.Rel x y := by rw [_root_.eq_top_iff, Setoid.le_def, Setoid.top_def] simp only [Pi.top_apply, Prop.top_eq_true, forall_true_left] #align setoid.eq_top_iff Setoid.eq_top_iff lemma sInf_equiv {S : Set (Setoid α)} {x y : α} : letI := sInf S x ≈ y ↔ ∀ s ∈ S, s.Rel x y := Iff.rfl lemma quotient_mk_sInf_eq {S : Set (Setoid α)} {x y : α} : Quotient.mk (sInf S) x = Quotient.mk (sInf S) y ↔ ∀ s ∈ S, s.Rel x y := by simp rfl def map_of_le {s t : Setoid α} (h : s ≤ t) : Quotient s → Quotient t := Quotient.map' id h def map_sInf {S : Set (Setoid α)} {s : Setoid α} (h : s ∈ S) : Quotient (sInf S) → Quotient s := Setoid.map_of_le fun _ _ a ↦ a s h theorem eqvGen_eq (r : α → α → Prop) : EqvGen.Setoid r = sInf { s : Setoid α \| ∀ ⦃x y⦄, r x y → s.Rel x y } := le_antisymm (fun _ _ H => EqvGen.rec (fun _ _ h _ hs => hs h) (refl' _) (fun _ _ _ => symm' _) (fun _ _ _ _ _ => trans' _) H) (sInf_le fun _ _ h => EqvGen.rel _ _ h) #align setoid.eqv_gen_eq Setoid.eqvGen_eq theorem sup_eq_eqvGen (r s : Setoid α) : r ⊔ s = EqvGen.Setoid fun x y => r.Rel x y ∨ s.Rel x y := by rw [eqvGen_eq] apply congr_arg sInf simp only [le_def, or_imp, ← forall_and] #align setoid.sup_eq_eqv_gen Setoid.sup_eq_eqvGen	Mathlib/Data/Setoid/Basic.lean	240	241	theorem sup_def {r s : Setoid α} : r ⊔ s = EqvGen.Setoid (r.Rel ⊔ s.Rel) := by	rw [sup_eq_eqvGen]; rfl
import Mathlib.Analysis.Convex.Basic import Mathlib.Order.Closure #align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d" open Set open Pointwise variable {𝕜 E F : Type*} section convexHull section OrderedSemiring variable [OrderedSemiring 𝕜] section AddCommMonoid variable (𝕜) variable [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] @[simps! isClosed] def convexHull : ClosureOperator (Set E) := .ofCompletePred (Convex 𝕜) fun _ ↦ convex_sInter #align convex_hull convexHull variable (s : Set E) theorem subset_convexHull : s ⊆ convexHull 𝕜 s := (convexHull 𝕜).le_closure s #align subset_convex_hull subset_convexHull theorem convex_convexHull : Convex 𝕜 (convexHull 𝕜 s) := (convexHull 𝕜).isClosed_closure s #align convex_convex_hull convex_convexHull theorem convexHull_eq_iInter : convexHull 𝕜 s = ⋂ (t : Set E) (_ : s ⊆ t) (_ : Convex 𝕜 t), t := by simp [convexHull, iInter_subtype, iInter_and] #align convex_hull_eq_Inter convexHull_eq_iInter variable {𝕜 s} {t : Set E} {x y : E}	Mathlib/Analysis/Convex/Hull.lean	62	63	theorem mem_convexHull_iff : x ∈ convexHull 𝕜 s ↔ ∀ t, s ⊆ t → Convex 𝕜 t → x ∈ t := by	simp_rw [convexHull_eq_iInter, mem_iInter]
import Mathlib.Tactic.NormNum import Mathlib.Tactic.TryThis import Mathlib.Util.AtomM set_option autoImplicit true namespace Mathlib.Tactic.Abel open Lean Elab Meta Tactic Qq initialize registerTraceClass `abel initialize registerTraceClass `abel.detail structure Context where α : Expr univ : Level α0 : Expr isGroup : Bool inst : Expr def mkContext (e : Expr) : MetaM Context := do let α ← inferType e let c ← synthInstance (← mkAppM ``AddCommMonoid #[α]) let cg ← synthInstance? (← mkAppM ``AddCommGroup #[α]) let u ← mkFreshLevelMVar _ ← isDefEq (.sort (.succ u)) (← inferType α) let α0 ← Expr.ofNat α 0 match cg with \| some cg => return ⟨α, u, α0, true, cg⟩ \| _ => return ⟨α, u, α0, false, c⟩ abbrev M := ReaderT Context AtomM def Context.app (c : Context) (n : Name) (inst : Expr) : Array Expr → Expr := mkAppN (((@Expr.const n [c.univ]).app c.α).app inst) def Context.mkApp (c : Context) (n inst : Name) (l : Array Expr) : MetaM Expr := do return c.app n (← synthInstance ((Expr.const inst [c.univ]).app c.α)) l def addG : Name → Name \| .str p s => .str p (s ++ "g") \| n => n def iapp (n : Name) (xs : Array Expr) : M Expr := do let c ← read return c.app (if c.isGroup then addG n else n) c.inst xs def term {α} [AddCommMonoid α] (n : ℕ) (x a : α) : α := n • x + a def termg {α} [AddCommGroup α] (n : ℤ) (x a : α) : α := n • x + a def mkTerm (n x a : Expr) : M Expr := iapp ``term #[n, x, a] def intToExpr (n : ℤ) : M Expr := do Expr.ofInt (mkConst (if (← read).isGroup then ``Int else ``Nat) []) n inductive NormalExpr : Type \| zero (e : Expr) : NormalExpr \| nterm (e : Expr) (n : Expr × ℤ) (x : ℕ × Expr) (a : NormalExpr) : NormalExpr deriving Inhabited def NormalExpr.e : NormalExpr → Expr \| .zero e => e \| .nterm e .. => e instance : Coe NormalExpr Expr where coe := NormalExpr.e def NormalExpr.term' (n : Expr × ℤ) (x : ℕ × Expr) (a : NormalExpr) : M NormalExpr := return .nterm (← mkTerm n.1 x.2 a) n x a def NormalExpr.zero' : M NormalExpr := return NormalExpr.zero (← read).α0 open NormalExpr theorem const_add_term {α} [AddCommMonoid α] (k n x a a') (h : k + a = a') : k + @term α _ n x a = term n x a' := by simp [h.symm, term, add_comm, add_assoc] theorem const_add_termg {α} [AddCommGroup α] (k n x a a') (h : k + a = a') : k + @termg α _ n x a = termg n x a' := by simp [h.symm, termg, add_comm, add_assoc] theorem term_add_const {α} [AddCommMonoid α] (n x a k a') (h : a + k = a') : @term α _ n x a + k = term n x a' := by simp [h.symm, term, add_assoc] theorem term_add_constg {α} [AddCommGroup α] (n x a k a') (h : a + k = a') : @termg α _ n x a + k = termg n x a' := by simp [h.symm, termg, add_assoc] theorem term_add_term {α} [AddCommMonoid α] (n₁ x a₁ n₂ a₂ n' a') (h₁ : n₁ + n₂ = n') (h₂ : a₁ + a₂ = a') : @term α _ n₁ x a₁ + @term α _ n₂ x a₂ = term n' x a' := by simp [h₁.symm, h₂.symm, term, add_nsmul, add_assoc, add_left_comm] theorem term_add_termg {α} [AddCommGroup α] (n₁ x a₁ n₂ a₂ n' a') (h₁ : n₁ + n₂ = n') (h₂ : a₁ + a₂ = a') : @termg α _ n₁ x a₁ + @termg α _ n₂ x a₂ = termg n' x a' := by simp only [termg, h₁.symm, add_zsmul, h₂.symm] exact add_add_add_comm (n₁ • x) a₁ (n₂ • x) a₂ theorem zero_term {α} [AddCommMonoid α] (x a) : @term α _ 0 x a = a := by simp [term, zero_nsmul, one_nsmul] theorem zero_termg {α} [AddCommGroup α] (x a) : @termg α _ 0 x a = a := by simp [termg, zero_zsmul] partial def evalAdd : NormalExpr → NormalExpr → M (NormalExpr × Expr) \| zero _, e₂ => do let p ← mkAppM ``zero_add #[e₂] return (e₂, p) \| e₁, zero _ => do let p ← mkAppM ``add_zero #[e₁] return (e₁, p) \| he₁@(nterm e₁ n₁ x₁ a₁), he₂@(nterm e₂ n₂ x₂ a₂) => do if x₁.1 = x₂.1 then let n' ← Mathlib.Meta.NormNum.eval (← mkAppM ``HAdd.hAdd #[n₁.1, n₂.1]) let (a', h₂) ← evalAdd a₁ a₂ let k := n₁.2 + n₂.2 let p₁ ← iapp ``term_add_term #[n₁.1, x₁.2, a₁, n₂.1, a₂, n'.expr, a', ← n'.getProof, h₂] if k = 0 then do let p ← mkEqTrans p₁ (← iapp ``zero_term #[x₁.2, a']) return (a', p) else return (← term' (n'.expr, k) x₁ a', p₁) else if x₁.1 < x₂.1 then do let (a', h) ← evalAdd a₁ he₂ return (← term' n₁ x₁ a', ← iapp ``term_add_const #[n₁.1, x₁.2, a₁, e₂, a', h]) else do let (a', h) ← evalAdd he₁ a₂ return (← term' n₂ x₂ a', ← iapp ``const_add_term #[e₁, n₂.1, x₂.2, a₂, a', h])	Mathlib/Tactic/Abel.lean	188	190	theorem term_neg {α} [AddCommGroup α] (n x a n' a') (h₁ : -n = n') (h₂ : -a = a') : -@termg α _ n x a = termg n' x a' := by	simpa [h₂.symm, h₁.symm, termg] using add_comm _ _
import Mathlib.Algebra.MvPolynomial.Supported import Mathlib.RingTheory.WittVector.Truncated #align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4" noncomputable section namespace WittVector variable (p : ℕ) [hp : Fact p.Prime] variable {k : Type} [CommRing k] local notation "𝕎" => WittVector p -- Porting note: new notation local notation "𝕄" => MvPolynomial (Fin 2 × ℕ) ℤ open Finset MvPolynomial def wittPolyProd (n : ℕ) : 𝕄 := rename (Prod.mk (0 : Fin 2)) (wittPolynomial p ℤ n) rename (Prod.mk (1 : Fin 2)) (wittPolynomial p ℤ n) #align witt_vector.witt_poly_prod WittVector.wittPolyProd theorem wittPolyProd_vars (n : ℕ) : (wittPolyProd p n).vars ⊆ univ ×ˢ range (n + 1) := by rw [wittPolyProd] apply Subset.trans (vars_mul _ _) refine union_subset ?_ ?_ <;> · refine Subset.trans (vars_rename _ _) ?_ simp [wittPolynomial_vars, image_subset_iff] #align witt_vector.witt_poly_prod_vars WittVector.wittPolyProd_vars def wittPolyProdRemainder (n : ℕ) : 𝕄 := ∑ i ∈ range n, (p : 𝕄) ^ i * wittMul p i ^ p ^ (n - i) #align witt_vector.witt_poly_prod_remainder WittVector.wittPolyProdRemainder theorem wittPolyProdRemainder_vars (n : ℕ) : (wittPolyProdRemainder p n).vars ⊆ univ ×ˢ range n := by rw [wittPolyProdRemainder] refine Subset.trans (vars_sum_subset _ _) ?_ rw [biUnion_subset] intro x hx apply Subset.trans (vars_mul _ _) refine union_subset ?_ ?_ · apply Subset.trans (vars_pow _ _) have : (p : 𝕄) = C (p : ℤ) := by simp only [Int.cast_natCast, eq_intCast] rw [this, vars_C] apply empty_subset · apply Subset.trans (vars_pow _ _) apply Subset.trans (wittMul_vars _ _) apply product_subset_product (Subset.refl _) simp only [mem_range, range_subset] at hx ⊢ exact hx #align witt_vector.witt_poly_prod_remainder_vars WittVector.wittPolyProdRemainder_vars def remainder (n : ℕ) : 𝕄 := (∑ x ∈ range (n + 1), (rename (Prod.mk 0)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) ((p : ℤ) ^ x))) * ∑ x ∈ range (n + 1), (rename (Prod.mk 1)) ((monomial (Finsupp.single x (p ^ (n + 1 - x)))) ((p : ℤ) ^ x)) #align witt_vector.remainder WittVector.remainder theorem remainder_vars (n : ℕ) : (remainder p n).vars ⊆ univ ×ˢ range (n + 1) := by rw [remainder] apply Subset.trans (vars_mul _ _) refine union_subset ?_ ?_ <;> · refine Subset.trans (vars_sum_subset _ _) ?_ rw [biUnion_subset] intro x hx rw [rename_monomial, vars_monomial, Finsupp.mapDomain_single] · apply Subset.trans Finsupp.support_single_subset simpa using mem_range.mp hx · apply pow_ne_zero exact mod_cast hp.out.ne_zero #align witt_vector.remainder_vars WittVector.remainder_vars def polyOfInterest (n : ℕ) : 𝕄 := wittMul p (n + 1) + (p : 𝕄) ^ (n + 1) * X (0, n + 1) * X (1, n + 1) - X (0, n + 1) * rename (Prod.mk (1 : Fin 2)) (wittPolynomial p ℤ (n + 1)) - X (1, n + 1) * rename (Prod.mk (0 : Fin 2)) (wittPolynomial p ℤ (n + 1)) #align witt_vector.poly_of_interest WittVector.polyOfInterest theorem mul_polyOfInterest_aux1 (n : ℕ) : ∑ i ∈ range (n + 1), (p : 𝕄) ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n := by simp only [wittPolyProd] convert wittStructureInt_prop p (X (0 : Fin 2) * X 1) n using 1 · simp only [wittPolynomial, wittMul] rw [AlgHom.map_sum] congr 1 with i congr 1 have hsupp : (Finsupp.single i (p ^ (n - i))).support = {i} := by rw [Finsupp.support_eq_singleton] simp only [and_true_iff, Finsupp.single_eq_same, eq_self_iff_true, Ne] exact pow_ne_zero _ hp.out.ne_zero simp only [bind₁_monomial, hsupp, Int.cast_natCast, prod_singleton, eq_intCast, Finsupp.single_eq_same, C_pow, mul_eq_mul_left_iff, true_or_iff, eq_self_iff_true, Int.cast_pow] · simp only [map_mul, bind₁_X_right] #align witt_vector.mul_poly_of_interest_aux1 WittVector.mul_polyOfInterest_aux1 theorem mul_polyOfInterest_aux2 (n : ℕ) : (p : 𝕄) ^ n * wittMul p n + wittPolyProdRemainder p n = wittPolyProd p n := by convert mul_polyOfInterest_aux1 p n rw [sum_range_succ, add_comm, Nat.sub_self, pow_zero, pow_one] rfl #align witt_vector.mul_poly_of_interest_aux2 WittVector.mul_polyOfInterest_aux2 theorem mul_polyOfInterest_aux3 (n : ℕ) : wittPolyProd p (n + 1) = -((p : 𝕄) ^ (n + 1) * X (0, n + 1)) * ((p : 𝕄) ^ (n + 1) * X (1, n + 1)) + (p : 𝕄) ^ (n + 1) * X (0, n + 1) * rename (Prod.mk (1 : Fin 2)) (wittPolynomial p ℤ (n + 1)) + (p : 𝕄) ^ (n + 1) * X (1, n + 1) * rename (Prod.mk (0 : Fin 2)) (wittPolynomial p ℤ (n + 1)) + remainder p n := by -- a useful auxiliary fact have mvpz : (p : 𝕄) ^ (n + 1) = MvPolynomial.C ((p : ℤ) ^ (n + 1)) := by norm_cast -- Porting note: the original proof applies `sum_range_succ` through a non-`conv` rewrite, -- but this does not work in Lean 4; the whole proof also times out very badly. The proof has been -- nearly totally rewritten here and now finishes quite fast. rw [wittPolyProd, wittPolynomial, AlgHom.map_sum, AlgHom.map_sum] conv_lhs => arg 1 rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul, rename_C, rename_X, ← mvpz] conv_lhs => arg 2 rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul, rename_C, rename_X, ← mvpz] conv_rhs => enter [1, 1, 2, 2] rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul, rename_C, rename_X, ← mvpz] conv_rhs => enter [1, 2, 2] rw [sum_range_succ, ← C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul, rename_C, rename_X, ← mvpz] simp only [add_mul, mul_add] rw [add_comm _ (remainder p n)] simp only [add_assoc] apply congrArg (Add.add _) ring #align witt_vector.mul_poly_of_interest_aux3 WittVector.mul_polyOfInterest_aux3	Mathlib/RingTheory/WittVector/MulCoeff.lean	179	186	theorem mul_polyOfInterest_aux4 (n : ℕ) : (p : 𝕄) ^ (n + 1) * wittMul p (n + 1) = -((p : 𝕄) ^ (n + 1) * X (0, n + 1)) * ((p : 𝕄) ^ (n + 1) * X (1, n + 1)) + (p : 𝕄) ^ (n + 1) * X (0, n + 1) * rename (Prod.mk (1 : Fin 2)) (wittPolynomial p ℤ (n + 1)) + (p : 𝕄) ^ (n + 1) * X (1, n + 1) * rename (Prod.mk (0 : Fin 2)) (wittPolynomial p ℤ (n + 1)) + (remainder p n - wittPolyProdRemainder p (n + 1)) := by	rw [← add_sub_assoc, eq_sub_iff_add_eq, mul_polyOfInterest_aux2] exact mul_polyOfInterest_aux3 _ _
import Mathlib.Algebra.Order.ZeroLEOne import Mathlib.Data.List.InsertNth import Mathlib.Logic.Relation import Mathlib.Logic.Small.Defs import Mathlib.Order.GameAdd #align_import set_theory.game.pgame from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618" set_option autoImplicit true namespace SetTheory open Function Relation -- We'd like to be able to use multi-character auto-implicits in this file. set_option relaxedAutoImplicit true inductive PGame : Type (u + 1) \| mk : ∀ α β : Type u, (α → PGame) → (β → PGame) → PGame #align pgame SetTheory.PGame compile_inductive% PGame namespace PGame def LeftMoves : PGame → Type u \| mk l _ _ _ => l #align pgame.left_moves SetTheory.PGame.LeftMoves def RightMoves : PGame → Type u \| mk _ r _ _ => r #align pgame.right_moves SetTheory.PGame.RightMoves def moveLeft : ∀ g : PGame, LeftMoves g → PGame \| mk _l _ L _ => L #align pgame.move_left SetTheory.PGame.moveLeft def moveRight : ∀ g : PGame, RightMoves g → PGame \| mk _ _r _ R => R #align pgame.move_right SetTheory.PGame.moveRight @[simp] theorem leftMoves_mk {xl xr xL xR} : (⟨xl, xr, xL, xR⟩ : PGame).LeftMoves = xl := rfl #align pgame.left_moves_mk SetTheory.PGame.leftMoves_mk @[simp] theorem moveLeft_mk {xl xr xL xR} : (⟨xl, xr, xL, xR⟩ : PGame).moveLeft = xL := rfl #align pgame.move_left_mk SetTheory.PGame.moveLeft_mk @[simp] theorem rightMoves_mk {xl xr xL xR} : (⟨xl, xr, xL, xR⟩ : PGame).RightMoves = xr := rfl #align pgame.right_moves_mk SetTheory.PGame.rightMoves_mk @[simp] theorem moveRight_mk {xl xr xL xR} : (⟨xl, xr, xL, xR⟩ : PGame).moveRight = xR := rfl #align pgame.move_right_mk SetTheory.PGame.moveRight_mk -- TODO define this at the level of games, as well, and perhaps also for finsets of games. def ofLists (L R : List PGame.{u}) : PGame.{u} := mk (ULift (Fin L.length)) (ULift (Fin R.length)) (fun i => L.get i.down) fun j ↦ R.get j.down #align pgame.of_lists SetTheory.PGame.ofLists theorem leftMoves_ofLists (L R : List PGame) : (ofLists L R).LeftMoves = ULift (Fin L.length) := rfl #align pgame.left_moves_of_lists SetTheory.PGame.leftMoves_ofLists theorem rightMoves_ofLists (L R : List PGame) : (ofLists L R).RightMoves = ULift (Fin R.length) := rfl #align pgame.right_moves_of_lists SetTheory.PGame.rightMoves_ofLists def toOfListsLeftMoves {L R : List PGame} : Fin L.length ≃ (ofLists L R).LeftMoves := ((Equiv.cast (leftMoves_ofLists L R).symm).trans Equiv.ulift).symm #align pgame.to_of_lists_left_moves SetTheory.PGame.toOfListsLeftMoves def toOfListsRightMoves {L R : List PGame} : Fin R.length ≃ (ofLists L R).RightMoves := ((Equiv.cast (rightMoves_ofLists L R).symm).trans Equiv.ulift).symm #align pgame.to_of_lists_right_moves SetTheory.PGame.toOfListsRightMoves theorem ofLists_moveLeft {L R : List PGame} (i : Fin L.length) : (ofLists L R).moveLeft (toOfListsLeftMoves i) = L.get i := rfl #align pgame.of_lists_move_left SetTheory.PGame.ofLists_moveLeft @[simp] theorem ofLists_moveLeft' {L R : List PGame} (i : (ofLists L R).LeftMoves) : (ofLists L R).moveLeft i = L.get (toOfListsLeftMoves.symm i) := rfl #align pgame.of_lists_move_left' SetTheory.PGame.ofLists_moveLeft' theorem ofLists_moveRight {L R : List PGame} (i : Fin R.length) : (ofLists L R).moveRight (toOfListsRightMoves i) = R.get i := rfl #align pgame.of_lists_move_right SetTheory.PGame.ofLists_moveRight @[simp] theorem ofLists_moveRight' {L R : List PGame} (i : (ofLists L R).RightMoves) : (ofLists L R).moveRight i = R.get (toOfListsRightMoves.symm i) := rfl #align pgame.of_lists_move_right' SetTheory.PGame.ofLists_moveRight' @[elab_as_elim] def moveRecOn {C : PGame → Sort*} (x : PGame) (IH : ∀ y : PGame, (∀ i, C (y.moveLeft i)) → (∀ j, C (y.moveRight j)) → C y) : C x := x.recOn fun yl yr yL yR => IH (mk yl yr yL yR) #align pgame.move_rec_on SetTheory.PGame.moveRecOn @[mk_iff] inductive IsOption : PGame → PGame → Prop \| moveLeft {x : PGame} (i : x.LeftMoves) : IsOption (x.moveLeft i) x \| moveRight {x : PGame} (i : x.RightMoves) : IsOption (x.moveRight i) x #align pgame.is_option SetTheory.PGame.IsOption theorem IsOption.mk_left {xl xr : Type u} (xL : xl → PGame) (xR : xr → PGame) (i : xl) : (xL i).IsOption (mk xl xr xL xR) := @IsOption.moveLeft (mk _ _ _ _) i #align pgame.is_option.mk_left SetTheory.PGame.IsOption.mk_left theorem IsOption.mk_right {xl xr : Type u} (xL : xl → PGame) (xR : xr → PGame) (i : xr) : (xR i).IsOption (mk xl xr xL xR) := @IsOption.moveRight (mk _ _ _ _) i #align pgame.is_option.mk_right SetTheory.PGame.IsOption.mk_right theorem wf_isOption : WellFounded IsOption := ⟨fun x => moveRecOn x fun x IHl IHr => Acc.intro x fun y h => by induction' h with _ i _ j · exact IHl i · exact IHr j⟩ #align pgame.wf_is_option SetTheory.PGame.wf_isOption def Subsequent : PGame → PGame → Prop := TransGen IsOption #align pgame.subsequent SetTheory.PGame.Subsequent instance : IsTrans _ Subsequent := inferInstanceAs <\| IsTrans _ (TransGen _) @[trans] theorem Subsequent.trans {x y z} : Subsequent x y → Subsequent y z → Subsequent x z := TransGen.trans #align pgame.subsequent.trans SetTheory.PGame.Subsequent.trans theorem wf_subsequent : WellFounded Subsequent := wf_isOption.transGen #align pgame.wf_subsequent SetTheory.PGame.wf_subsequent instance : WellFoundedRelation PGame := ⟨_, wf_subsequent⟩ @[simp] theorem Subsequent.moveLeft {x : PGame} (i : x.LeftMoves) : Subsequent (x.moveLeft i) x := TransGen.single (IsOption.moveLeft i) #align pgame.subsequent.move_left SetTheory.PGame.Subsequent.moveLeft @[simp] theorem Subsequent.moveRight {x : PGame} (j : x.RightMoves) : Subsequent (x.moveRight j) x := TransGen.single (IsOption.moveRight j) #align pgame.subsequent.move_right SetTheory.PGame.Subsequent.moveRight @[simp] theorem Subsequent.mk_left {xl xr} (xL : xl → PGame) (xR : xr → PGame) (i : xl) : Subsequent (xL i) (mk xl xr xL xR) := @Subsequent.moveLeft (mk _ _ _ _) i #align pgame.subsequent.mk_left SetTheory.PGame.Subsequent.mk_left @[simp] theorem Subsequent.mk_right {xl xr} (xL : xl → PGame) (xR : xr → PGame) (j : xr) : Subsequent (xR j) (mk xl xr xL xR) := @Subsequent.moveRight (mk _ _ _ _) j #align pgame.subsequent.mk_right SetTheory.PGame.Subsequent.mk_right macro "pgame_wf_tac" : tactic => `(tactic\| solve_by_elim (config := { maxDepth := 8 }) [Prod.Lex.left, Prod.Lex.right, PSigma.Lex.left, PSigma.Lex.right, Subsequent.moveLeft, Subsequent.moveRight, Subsequent.mk_left, Subsequent.mk_right, Subsequent.trans] ) -- Register some consequences of pgame_wf_tac as simp-lemmas for convenience -- (which are applied by default for WF goals) -- This is different from mk_right from the POV of the simplifier, -- because the unifier can't solve `xr =?= RightMoves (mk xl xr xL xR)` at reducible transparency. @[simp] theorem Subsequent.mk_right' (xL : xl → PGame) (xR : xr → PGame) (j : RightMoves (mk xl xr xL xR)) : Subsequent (xR j) (mk xl xr xL xR) := by pgame_wf_tac @[simp] theorem Subsequent.moveRight_mk_left (xL : xl → PGame) (j) : Subsequent ((xL i).moveRight j) (mk xl xr xL xR) := by pgame_wf_tac @[simp] theorem Subsequent.moveRight_mk_right (xR : xr → PGame) (j) : Subsequent ((xR i).moveRight j) (mk xl xr xL xR) := by pgame_wf_tac @[simp] theorem Subsequent.moveLeft_mk_left (xL : xl → PGame) (j) : Subsequent ((xL i).moveLeft j) (mk xl xr xL xR) := by pgame_wf_tac @[simp] theorem Subsequent.moveLeft_mk_right (xR : xr → PGame) (j) : Subsequent ((xR i).moveLeft j) (mk xl xr xL xR) := by pgame_wf_tac -- Porting note: linter claims these lemmas don't simplify? open Subsequent in attribute [nolint simpNF] mk_left mk_right mk_right' moveRight_mk_left moveRight_mk_right moveLeft_mk_left moveLeft_mk_right instance : Zero PGame := ⟨⟨PEmpty, PEmpty, PEmpty.elim, PEmpty.elim⟩⟩ @[simp] theorem zero_leftMoves : LeftMoves 0 = PEmpty := rfl #align pgame.zero_left_moves SetTheory.PGame.zero_leftMoves @[simp] theorem zero_rightMoves : RightMoves 0 = PEmpty := rfl #align pgame.zero_right_moves SetTheory.PGame.zero_rightMoves instance isEmpty_zero_leftMoves : IsEmpty (LeftMoves 0) := instIsEmptyPEmpty #align pgame.is_empty_zero_left_moves SetTheory.PGame.isEmpty_zero_leftMoves instance isEmpty_zero_rightMoves : IsEmpty (RightMoves 0) := instIsEmptyPEmpty #align pgame.is_empty_zero_right_moves SetTheory.PGame.isEmpty_zero_rightMoves instance : Inhabited PGame := ⟨0⟩ instance instOnePGame : One PGame := ⟨⟨PUnit, PEmpty, fun _ => 0, PEmpty.elim⟩⟩ @[simp] theorem one_leftMoves : LeftMoves 1 = PUnit := rfl #align pgame.one_left_moves SetTheory.PGame.one_leftMoves @[simp] theorem one_moveLeft (x) : moveLeft 1 x = 0 := rfl #align pgame.one_move_left SetTheory.PGame.one_moveLeft @[simp] theorem one_rightMoves : RightMoves 1 = PEmpty := rfl #align pgame.one_right_moves SetTheory.PGame.one_rightMoves instance uniqueOneLeftMoves : Unique (LeftMoves 1) := PUnit.unique #align pgame.unique_one_left_moves SetTheory.PGame.uniqueOneLeftMoves instance isEmpty_one_rightMoves : IsEmpty (RightMoves 1) := instIsEmptyPEmpty #align pgame.is_empty_one_right_moves SetTheory.PGame.isEmpty_one_rightMoves instance le : LE PGame := ⟨Sym2.GameAdd.fix wf_isOption fun x y le => (∀ i, ¬le y (x.moveLeft i) (Sym2.GameAdd.snd_fst <\| IsOption.moveLeft i)) ∧ ∀ j, ¬le (y.moveRight j) x (Sym2.GameAdd.fst_snd <\| IsOption.moveRight j)⟩ def LF (x y : PGame) : Prop := ¬y ≤ x #align pgame.lf SetTheory.PGame.LF @[inherit_doc] scoped infixl:50 " ⧏ " => PGame.LF @[simp] protected theorem not_le {x y : PGame} : ¬x ≤ y ↔ y ⧏ x := Iff.rfl #align pgame.not_le SetTheory.PGame.not_le @[simp] theorem not_lf {x y : PGame} : ¬x ⧏ y ↔ y ≤ x := Classical.not_not #align pgame.not_lf SetTheory.PGame.not_lf theorem _root_.LE.le.not_gf {x y : PGame} : x ≤ y → ¬y ⧏ x := not_lf.2 #align has_le.le.not_gf LE.le.not_gf theorem LF.not_ge {x y : PGame} : x ⧏ y → ¬y ≤ x := id #align pgame.lf.not_ge SetTheory.PGame.LF.not_ge theorem le_iff_forall_lf {x y : PGame} : x ≤ y ↔ (∀ i, x.moveLeft i ⧏ y) ∧ ∀ j, x ⧏ y.moveRight j := by unfold LE.le le simp only rw [Sym2.GameAdd.fix_eq] rfl #align pgame.le_iff_forall_lf SetTheory.PGame.le_iff_forall_lf @[simp] theorem mk_le_mk {xl xr xL xR yl yr yL yR} : mk xl xr xL xR ≤ mk yl yr yL yR ↔ (∀ i, xL i ⧏ mk yl yr yL yR) ∧ ∀ j, mk xl xr xL xR ⧏ yR j := le_iff_forall_lf #align pgame.mk_le_mk SetTheory.PGame.mk_le_mk theorem le_of_forall_lf {x y : PGame} (h₁ : ∀ i, x.moveLeft i ⧏ y) (h₂ : ∀ j, x ⧏ y.moveRight j) : x ≤ y := le_iff_forall_lf.2 ⟨h₁, h₂⟩ #align pgame.le_of_forall_lf SetTheory.PGame.le_of_forall_lf theorem lf_iff_exists_le {x y : PGame} : x ⧏ y ↔ (∃ i, x ≤ y.moveLeft i) ∨ ∃ j, x.moveRight j ≤ y := by rw [LF, le_iff_forall_lf, not_and_or] simp #align pgame.lf_iff_exists_le SetTheory.PGame.lf_iff_exists_le @[simp] theorem mk_lf_mk {xl xr xL xR yl yr yL yR} : mk xl xr xL xR ⧏ mk yl yr yL yR ↔ (∃ i, mk xl xr xL xR ≤ yL i) ∨ ∃ j, xR j ≤ mk yl yr yL yR := lf_iff_exists_le #align pgame.mk_lf_mk SetTheory.PGame.mk_lf_mk theorem le_or_gf (x y : PGame) : x ≤ y ∨ y ⧏ x := by rw [← PGame.not_le] apply em #align pgame.le_or_gf SetTheory.PGame.le_or_gf theorem moveLeft_lf_of_le {x y : PGame} (h : x ≤ y) (i) : x.moveLeft i ⧏ y := (le_iff_forall_lf.1 h).1 i #align pgame.move_left_lf_of_le SetTheory.PGame.moveLeft_lf_of_le alias _root_.LE.le.moveLeft_lf := moveLeft_lf_of_le #align has_le.le.move_left_lf LE.le.moveLeft_lf theorem lf_moveRight_of_le {x y : PGame} (h : x ≤ y) (j) : x ⧏ y.moveRight j := (le_iff_forall_lf.1 h).2 j #align pgame.lf_move_right_of_le SetTheory.PGame.lf_moveRight_of_le alias _root_.LE.le.lf_moveRight := lf_moveRight_of_le #align has_le.le.lf_move_right LE.le.lf_moveRight theorem lf_of_moveRight_le {x y : PGame} {j} (h : x.moveRight j ≤ y) : x ⧏ y := lf_iff_exists_le.2 <\| Or.inr ⟨j, h⟩ #align pgame.lf_of_move_right_le SetTheory.PGame.lf_of_moveRight_le theorem lf_of_le_moveLeft {x y : PGame} {i} (h : x ≤ y.moveLeft i) : x ⧏ y := lf_iff_exists_le.2 <\| Or.inl ⟨i, h⟩ #align pgame.lf_of_le_move_left SetTheory.PGame.lf_of_le_moveLeft theorem lf_of_le_mk {xl xr xL xR y} : mk xl xr xL xR ≤ y → ∀ i, xL i ⧏ y := moveLeft_lf_of_le #align pgame.lf_of_le_mk SetTheory.PGame.lf_of_le_mk theorem lf_of_mk_le {x yl yr yL yR} : x ≤ mk yl yr yL yR → ∀ j, x ⧏ yR j := lf_moveRight_of_le #align pgame.lf_of_mk_le SetTheory.PGame.lf_of_mk_le theorem mk_lf_of_le {xl xr y j} (xL) {xR : xr → PGame} : xR j ≤ y → mk xl xr xL xR ⧏ y := @lf_of_moveRight_le (mk _ _ _ _) y j #align pgame.mk_lf_of_le SetTheory.PGame.mk_lf_of_le theorem lf_mk_of_le {x yl yr} {yL : yl → PGame} (yR) {i} : x ≤ yL i → x ⧏ mk yl yr yL yR := @lf_of_le_moveLeft x (mk _ _ _ _) i #align pgame.lf_mk_of_le SetTheory.PGame.lf_mk_of_le private theorem le_trans_aux {x y z : PGame} (h₁ : ∀ {i}, y ≤ z → z ≤ x.moveLeft i → y ≤ x.moveLeft i) (h₂ : ∀ {j}, z.moveRight j ≤ x → x ≤ y → z.moveRight j ≤ y) (hxy : x ≤ y) (hyz : y ≤ z) : x ≤ z := le_of_forall_lf (fun i => PGame.not_le.1 fun h => (h₁ hyz h).not_gf <\| hxy.moveLeft_lf i) fun j => PGame.not_le.1 fun h => (h₂ h hxy).not_gf <\| hyz.lf_moveRight j instance : Preorder PGame := { PGame.le with le_refl := fun x => by induction' x with _ _ _ _ IHl IHr exact le_of_forall_lf (fun i => lf_of_le_moveLeft (IHl i)) fun i => lf_of_moveRight_le (IHr i) le_trans := by suffices ∀ {x y z : PGame}, (x ≤ y → y ≤ z → x ≤ z) ∧ (y ≤ z → z ≤ x → y ≤ x) ∧ (z ≤ x → x ≤ y → z ≤ y) from fun x y z => this.1 intro x y z induction' x with xl xr xL xR IHxl IHxr generalizing y z induction' y with yl yr yL yR IHyl IHyr generalizing z induction' z with zl zr zL zR IHzl IHzr exact ⟨le_trans_aux (fun {i} => (IHxl i).2.1) fun {j} => (IHzr j).2.2, le_trans_aux (fun {i} => (IHyl i).2.2) fun {j} => (IHxr j).1, le_trans_aux (fun {i} => (IHzl i).1) fun {j} => (IHyr j).2.1⟩ lt := fun x y => x ≤ y ∧ x ⧏ y } theorem lt_iff_le_and_lf {x y : PGame} : x < y ↔ x ≤ y ∧ x ⧏ y := Iff.rfl #align pgame.lt_iff_le_and_lf SetTheory.PGame.lt_iff_le_and_lf theorem lt_of_le_of_lf {x y : PGame} (h₁ : x ≤ y) (h₂ : x ⧏ y) : x < y := ⟨h₁, h₂⟩ #align pgame.lt_of_le_of_lf SetTheory.PGame.lt_of_le_of_lf theorem lf_of_lt {x y : PGame} (h : x < y) : x ⧏ y := h.2 #align pgame.lf_of_lt SetTheory.PGame.lf_of_lt alias _root_.LT.lt.lf := lf_of_lt #align has_lt.lt.lf LT.lt.lf theorem lf_irrefl (x : PGame) : ¬x ⧏ x := le_rfl.not_gf #align pgame.lf_irrefl SetTheory.PGame.lf_irrefl instance : IsIrrefl _ (· ⧏ ·) := ⟨lf_irrefl⟩ @[trans] theorem lf_of_le_of_lf {x y z : PGame} (h₁ : x ≤ y) (h₂ : y ⧏ z) : x ⧏ z := by rw [← PGame.not_le] at h₂ ⊢ exact fun h₃ => h₂ (h₃.trans h₁) #align pgame.lf_of_le_of_lf SetTheory.PGame.lf_of_le_of_lf -- Porting note (#10754): added instance instance : Trans (· ≤ ·) (· ⧏ ·) (· ⧏ ·) := ⟨lf_of_le_of_lf⟩ @[trans] theorem lf_of_lf_of_le {x y z : PGame} (h₁ : x ⧏ y) (h₂ : y ≤ z) : x ⧏ z := by rw [← PGame.not_le] at h₁ ⊢ exact fun h₃ => h₁ (h₂.trans h₃) #align pgame.lf_of_lf_of_le SetTheory.PGame.lf_of_lf_of_le -- Porting note (#10754): added instance instance : Trans (· ⧏ ·) (· ≤ ·) (· ⧏ ·) := ⟨lf_of_lf_of_le⟩ alias _root_.LE.le.trans_lf := lf_of_le_of_lf #align has_le.le.trans_lf LE.le.trans_lf alias LF.trans_le := lf_of_lf_of_le #align pgame.lf.trans_le SetTheory.PGame.LF.trans_le @[trans] theorem lf_of_lt_of_lf {x y z : PGame} (h₁ : x < y) (h₂ : y ⧏ z) : x ⧏ z := h₁.le.trans_lf h₂ #align pgame.lf_of_lt_of_lf SetTheory.PGame.lf_of_lt_of_lf @[trans] theorem lf_of_lf_of_lt {x y z : PGame} (h₁ : x ⧏ y) (h₂ : y < z) : x ⧏ z := h₁.trans_le h₂.le #align pgame.lf_of_lf_of_lt SetTheory.PGame.lf_of_lf_of_lt alias _root_.LT.lt.trans_lf := lf_of_lt_of_lf #align has_lt.lt.trans_lf LT.lt.trans_lf alias LF.trans_lt := lf_of_lf_of_lt #align pgame.lf.trans_lt SetTheory.PGame.LF.trans_lt theorem moveLeft_lf {x : PGame} : ∀ i, x.moveLeft i ⧏ x := le_rfl.moveLeft_lf #align pgame.move_left_lf SetTheory.PGame.moveLeft_lf theorem lf_moveRight {x : PGame} : ∀ j, x ⧏ x.moveRight j := le_rfl.lf_moveRight #align pgame.lf_move_right SetTheory.PGame.lf_moveRight theorem lf_mk {xl xr} (xL : xl → PGame) (xR : xr → PGame) (i) : xL i ⧏ mk xl xr xL xR := @moveLeft_lf (mk _ _ _ _) i #align pgame.lf_mk SetTheory.PGame.lf_mk theorem mk_lf {xl xr} (xL : xl → PGame) (xR : xr → PGame) (j) : mk xl xr xL xR ⧏ xR j := @lf_moveRight (mk _ _ _ _) j #align pgame.mk_lf SetTheory.PGame.mk_lf theorem le_of_forall_lt {x y : PGame} (h₁ : ∀ i, x.moveLeft i < y) (h₂ : ∀ j, x < y.moveRight j) : x ≤ y := le_of_forall_lf (fun i => (h₁ i).lf) fun i => (h₂ i).lf #align pgame.le_of_forall_lt SetTheory.PGame.le_of_forall_lt theorem le_def {x y : PGame} : x ≤ y ↔ (∀ i, (∃ i', x.moveLeft i ≤ y.moveLeft i') ∨ ∃ j, (x.moveLeft i).moveRight j ≤ y) ∧ ∀ j, (∃ i, x ≤ (y.moveRight j).moveLeft i) ∨ ∃ j', x.moveRight j' ≤ y.moveRight j := by rw [le_iff_forall_lf] conv => lhs simp only [lf_iff_exists_le] #align pgame.le_def SetTheory.PGame.le_def theorem lf_def {x y : PGame} : x ⧏ y ↔ (∃ i, (∀ i', x.moveLeft i' ⧏ y.moveLeft i) ∧ ∀ j, x ⧏ (y.moveLeft i).moveRight j) ∨ ∃ j, (∀ i, (x.moveRight j).moveLeft i ⧏ y) ∧ ∀ j', x.moveRight j ⧏ y.moveRight j' := by rw [lf_iff_exists_le] conv => lhs simp only [le_iff_forall_lf] #align pgame.lf_def SetTheory.PGame.lf_def theorem zero_le_lf {x : PGame} : 0 ≤ x ↔ ∀ j, 0 ⧏ x.moveRight j := by rw [le_iff_forall_lf] simp #align pgame.zero_le_lf SetTheory.PGame.zero_le_lf theorem le_zero_lf {x : PGame} : x ≤ 0 ↔ ∀ i, x.moveLeft i ⧏ 0 := by rw [le_iff_forall_lf] simp #align pgame.le_zero_lf SetTheory.PGame.le_zero_lf theorem zero_lf_le {x : PGame} : 0 ⧏ x ↔ ∃ i, 0 ≤ x.moveLeft i := by rw [lf_iff_exists_le] simp #align pgame.zero_lf_le SetTheory.PGame.zero_lf_le theorem lf_zero_le {x : PGame} : x ⧏ 0 ↔ ∃ j, x.moveRight j ≤ 0 := by rw [lf_iff_exists_le] simp #align pgame.lf_zero_le SetTheory.PGame.lf_zero_le theorem zero_le {x : PGame} : 0 ≤ x ↔ ∀ j, ∃ i, 0 ≤ (x.moveRight j).moveLeft i := by rw [le_def] simp #align pgame.zero_le SetTheory.PGame.zero_le	Mathlib/SetTheory/Game/PGame.lean	677	679	theorem le_zero {x : PGame} : x ≤ 0 ↔ ∀ i, ∃ j, (x.moveLeft i).moveRight j ≤ 0 := by	rw [le_def] simp
import Mathlib.RingTheory.RootsOfUnity.Basic import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed import Mathlib.Algebra.GCDMonoid.IntegrallyClosed import Mathlib.FieldTheory.Finite.Basic #align_import ring_theory.roots_of_unity.minpoly from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f" open minpoly Polynomial open scoped Polynomial namespace IsPrimitiveRoot section CommRing variable {n : ℕ} {K : Type*} [CommRing K] {μ : K} (h : IsPrimitiveRoot μ n) -- Porting note: `hpos` was in the `variable` line, with an `omit` in mathlib3 just after this -- declaration. For some reason, in Lean4, `hpos` gets included also in the declarations below, -- even if it is not used in the proof. theorem isIntegral (hpos : 0 < n) : IsIntegral ℤ μ := by use X ^ n - 1 constructor · exact monic_X_pow_sub_C 1 (ne_of_lt hpos).symm · simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, eval₂_one, eval₂_X_pow, eval₂_sub, sub_self] #align is_primitive_root.is_integral IsPrimitiveRoot.isIntegral section IsDomain variable [IsDomain K] [CharZero K] theorem minpoly_dvd_x_pow_sub_one : minpoly ℤ μ ∣ X ^ n - 1 := by rcases n.eq_zero_or_pos with (rfl \| h0) · simp apply minpoly.isIntegrallyClosed_dvd (isIntegral h h0) simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, aeval_X_pow, eq_intCast, Int.cast_one, aeval_one, AlgHom.map_sub, sub_self] set_option linter.uppercaseLean3 false in #align is_primitive_root.minpoly_dvd_X_pow_sub_one IsPrimitiveRoot.minpoly_dvd_x_pow_sub_one theorem separable_minpoly_mod {p : ℕ} [Fact p.Prime] (hdiv : ¬p ∣ n) : Separable (map (Int.castRingHom (ZMod p)) (minpoly ℤ μ)) := by have hdvd : map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ X ^ n - 1 := by convert RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p))) (minpoly_dvd_x_pow_sub_one h) simp only [map_sub, map_pow, coe_mapRingHom, map_X, map_one] refine Separable.of_dvd (separable_X_pow_sub_C 1 ?_ one_ne_zero) hdvd by_contra hzero exact hdiv ((ZMod.natCast_zmod_eq_zero_iff_dvd n p).1 hzero) #align is_primitive_root.separable_minpoly_mod IsPrimitiveRoot.separable_minpoly_mod theorem squarefree_minpoly_mod {p : ℕ} [Fact p.Prime] (hdiv : ¬p ∣ n) : Squarefree (map (Int.castRingHom (ZMod p)) (minpoly ℤ μ)) := (separable_minpoly_mod h hdiv).squarefree #align is_primitive_root.squarefree_minpoly_mod IsPrimitiveRoot.squarefree_minpoly_mod theorem minpoly_dvd_expand {p : ℕ} (hdiv : ¬p ∣ n) : minpoly ℤ μ ∣ expand ℤ p (minpoly ℤ (μ ^ p)) := by rcases n.eq_zero_or_pos with (rfl \| hpos) · simp_all letI : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed refine minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos) ?_ rw [aeval_def, coe_expand, ← comp, eval₂_eq_eval_map, map_comp, Polynomial.map_pow, map_X, eval_comp, eval_pow, eval_X, ← eval₂_eq_eval_map, ← aeval_def] exact minpoly.aeval _ _ #align is_primitive_root.minpoly_dvd_expand IsPrimitiveRoot.minpoly_dvd_expand	Mathlib/RingTheory/RootsOfUnity/Minpoly.lean	95	104	theorem minpoly_dvd_pow_mod {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) : map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) ^ p := by	set Q := minpoly ℤ (μ ^ p) have hfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) (expand ℤ p Q) := by rw [← ZMod.expand_card, map_expand] rw [hfrob] apply RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p))) exact minpoly_dvd_expand h hdiv
import Mathlib.Algebra.GroupWithZero.Divisibility import Mathlib.Algebra.Order.Ring.Nat import Mathlib.Tactic.NthRewrite #align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" namespace Nat theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd : ∀ e : ℕ, e ∣ a → e ∣ b → e ∣ d) : d = a.gcd b := (dvd_antisymm (hd _ (gcd_dvd_left a b) (gcd_dvd_right a b)) (dvd_gcd hda hdb)).symm #align nat.gcd_greatest Nat.gcd_greatest @[simp] theorem gcd_add_mul_right_right (m n k : ℕ) : gcd m (n + k * m) = gcd m n := by simp [gcd_rec m (n + k * m), gcd_rec m n] #align nat.gcd_add_mul_right_right Nat.gcd_add_mul_right_right @[simp] theorem gcd_add_mul_left_right (m n k : ℕ) : gcd m (n + m * k) = gcd m n := by simp [gcd_rec m (n + m * k), gcd_rec m n] #align nat.gcd_add_mul_left_right Nat.gcd_add_mul_left_right @[simp]	Mathlib/Data/Nat/GCD/Basic.lean	45	45	theorem gcd_mul_right_add_right (m n k : ℕ) : gcd m (k * m + n) = gcd m n := by	simp [add_comm _ n]
import Mathlib.CategoryTheory.Subobject.Limits #align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff" universe v u w open CategoryTheory CategoryTheory.Limits variable {ι : Type*} variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V] open scoped Classical noncomputable section section variable {A B C : V} (f : A ⟶ B) [HasImage f] (g : B ⟶ C) [HasKernel g] theorem image_le_kernel (w : f ≫ g = 0) : imageSubobject f ≤ kernelSubobject g := imageSubobject_le_mk _ _ (kernel.lift _ _ w) (by simp) #align image_le_kernel image_le_kernel def imageToKernel (w : f ≫ g = 0) : (imageSubobject f : V) ⟶ (kernelSubobject g : V) := Subobject.ofLE _ _ (image_le_kernel _ _ w) #align image_to_kernel imageToKernel instance (w : f ≫ g = 0) : Mono (imageToKernel f g w) := by dsimp only [imageToKernel] infer_instance @[simp] theorem subobject_ofLE_as_imageToKernel (w : f ≫ g = 0) (h) : Subobject.ofLE (imageSubobject f) (kernelSubobject g) h = imageToKernel f g w := rfl #align subobject_of_le_as_image_to_kernel subobject_ofLE_as_imageToKernel attribute [local instance] ConcreteCategory.instFunLike -- Porting note: removed elementwise attribute which does not seem to be helpful here -- a more suitable lemma is added below @[reassoc (attr := simp)] theorem imageToKernel_arrow (w : f ≫ g = 0) : imageToKernel f g w ≫ (kernelSubobject g).arrow = (imageSubobject f).arrow := by simp [imageToKernel] #align image_to_kernel_arrow imageToKernel_arrow @[simp] lemma imageToKernel_arrow_apply [ConcreteCategory V] (w : f ≫ g = 0) (x : (forget V).obj (Subobject.underlying.obj (imageSubobject f))) : (kernelSubobject g).arrow (imageToKernel f g w x) = (imageSubobject f).arrow x := by rw [← comp_apply, imageToKernel_arrow] -- This is less useful as a `simp` lemma than it initially appears, -- as it "loses" the information the morphism factors through the image. theorem factorThruImageSubobject_comp_imageToKernel (w : f ≫ g = 0) : factorThruImageSubobject f ≫ imageToKernel f g w = factorThruKernelSubobject g f w := by ext simp #align factor_thru_image_subobject_comp_image_to_kernel factorThruImageSubobject_comp_imageToKernel end section variable {A B C : V} (f : A ⟶ B) (g : B ⟶ C) @[simp] theorem imageToKernel_zero_left [HasKernels V] [HasZeroObject V] {w} : imageToKernel (0 : A ⟶ B) g w = 0 := by ext simp #align image_to_kernel_zero_left imageToKernel_zero_left theorem imageToKernel_zero_right [HasImages V] {w} : imageToKernel f (0 : B ⟶ C) w = (imageSubobject f).arrow ≫ inv (kernelSubobject (0 : B ⟶ C)).arrow := by ext simp #align image_to_kernel_zero_right imageToKernel_zero_right section variable [HasKernels V] [HasImages V] theorem imageToKernel_comp_right {D : V} (h : C ⟶ D) (w : f ≫ g = 0) : imageToKernel f (g ≫ h) (by simp [reassoc_of% w]) = imageToKernel f g w ≫ Subobject.ofLE _ _ (kernelSubobject_comp_le g h) := by ext simp #align image_to_kernel_comp_right imageToKernel_comp_right theorem imageToKernel_comp_left {Z : V} (h : Z ⟶ A) (w : f ≫ g = 0) : imageToKernel (h ≫ f) g (by simp [w]) = Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫ imageToKernel f g w := by ext simp #align image_to_kernel_comp_left imageToKernel_comp_left @[simp] theorem imageToKernel_comp_mono {D : V} (h : C ⟶ D) [Mono h] (w) : imageToKernel f (g ≫ h) w = imageToKernel f g ((cancel_mono h).mp (by simpa using w : (f ≫ g) ≫ h = 0 ≫ h)) ≫ (Subobject.isoOfEq _ _ (kernelSubobject_comp_mono g h)).inv := by ext simp #align image_to_kernel_comp_mono imageToKernel_comp_mono @[simp] theorem imageToKernel_epi_comp {Z : V} (h : Z ⟶ A) [Epi h] (w) : imageToKernel (h ≫ f) g w = Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫ imageToKernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0)) := by ext simp #align image_to_kernel_epi_comp imageToKernel_epi_comp end @[simp] theorem imageToKernel_comp_hom_inv_comp [HasEqualizers V] [HasImages V] {Z : V} {i : B ≅ Z} (w) : imageToKernel (f ≫ i.hom) (i.inv ≫ g) w = (imageSubobjectCompIso _ _).hom ≫ imageToKernel f g (by simpa using w) ≫ (kernelSubobjectIsoComp i.inv g).inv := by ext simp #align image_to_kernel_comp_hom_inv_comp imageToKernel_comp_hom_inv_comp open ZeroObject instance imageToKernel_epi_of_zero_of_mono [HasKernels V] [HasZeroObject V] [Mono g] : Epi (imageToKernel (0 : A ⟶ B) g (by simp)) := epi_of_target_iso_zero _ (kernelSubobjectIso g ≪≫ kernel.ofMono g) #align image_to_kernel_epi_of_zero_of_mono imageToKernel_epi_of_zero_of_mono instance imageToKernel_epi_of_epi_of_zero [HasImages V] [Epi f] : Epi (imageToKernel f (0 : B ⟶ C) (by simp)) := by simp only [imageToKernel_zero_right] haveI := epi_image_of_epi f rw [← imageSubobject_arrow] exact @epi_comp _ _ _ _ _ _ (epi_comp _ _) _ _ #align image_to_kernel_epi_of_epi_of_zero imageToKernel_epi_of_epi_of_zero end section variable {A B C : V} (f : A ⟶ B) [HasImage f] (g : B ⟶ C) [HasKernel g] def homology' {A B C : V} (f : A ⟶ B) [HasImage f] (g : B ⟶ C) [HasKernel g] (w : f ≫ g = 0) [HasCokernel (imageToKernel f g w)] : V := cokernel (imageToKernel f g w) #align homology homology' section variable (w : f ≫ g = 0) [HasCokernel (imageToKernel f g w)] def homology'.π : (kernelSubobject g : V) ⟶ homology' f g w := cokernel.π _ #align homology.π homology'.π @[simp] theorem homology'.condition : imageToKernel f g w ≫ homology'.π f g w = 0 := cokernel.condition _ #align homology.condition homology'.condition def homology'.desc {D : V} (k : (kernelSubobject g : V) ⟶ D) (p : imageToKernel f g w ≫ k = 0) : homology' f g w ⟶ D := cokernel.desc _ k p #align homology.desc homology'.desc -- Porting note: removed elementwise attribute which does not seem to be helpful here @[reassoc (attr := simp)] theorem homology'.π_desc {D : V} (k : (kernelSubobject g : V) ⟶ D) (p : imageToKernel f g w ≫ k = 0) : homology'.π f g w ≫ homology'.desc f g w k p = k := by simp [homology'.π, homology'.desc] #align homology.π_desc homology'.π_desc @[ext] theorem homology'.ext {D : V} {k k' : homology' f g w ⟶ D} (p : homology'.π f g w ≫ k = homology'.π f g w ≫ k') : k = k' := coequalizer.hom_ext p #align homology.ext homology'.ext def homology'OfZeroRight [HasCokernel (imageToKernel f (0 : B ⟶ C) comp_zero)] [HasCokernel f] [HasCokernel (image.ι f)] [Epi (factorThruImage f)] : homology' f (0 : B ⟶ C) comp_zero ≅ cokernel f := (cokernel.mapIso _ _ (imageSubobjectIso _) ((kernelSubobjectIso 0).trans kernelZeroIsoSource) (by simp)).trans (cokernelImageι _) #align homology_of_zero_right homology'OfZeroRight def homology'OfZeroLeft [HasZeroObject V] [HasKernels V] [HasImage (0 : A ⟶ B)] [HasCokernel (imageToKernel (0 : A ⟶ B) g zero_comp)] : homology' (0 : A ⟶ B) g zero_comp ≅ kernel g := ((cokernelIsoOfEq <\| imageToKernel_zero_left _).trans cokernelZeroIsoTarget).trans (kernelSubobjectIso _) #align homology_of_zero_left homology'OfZeroLeft @[simps] def homology'ZeroZero [HasZeroObject V] [HasImage (0 : A ⟶ B)] [HasCokernel (imageToKernel (0 : A ⟶ B) (0 : B ⟶ C) zero_comp)] : homology' (0 : A ⟶ B) (0 : B ⟶ C) zero_comp ≅ B where hom := homology'.desc (0 : A ⟶ B) (0 : B ⟶ C) zero_comp (kernelSubobject 0).arrow (by simp) inv := inv (kernelSubobject 0).arrow ≫ homology'.π _ _ _ #align homology_zero_zero homology'ZeroZero end section variable {f g} (w : f ≫ g = 0) {A' B' C' : V} {f' : A' ⟶ B'} [HasImage f'] {g' : B' ⟶ C'} [HasKernel g'] (w' : f' ≫ g' = 0) (α : Arrow.mk f ⟶ Arrow.mk f') [HasImageMap α] (β : Arrow.mk g ⟶ Arrow.mk g') {A₁ B₁ C₁ : V} {f₁ : A₁ ⟶ B₁} [HasImage f₁] {g₁ : B₁ ⟶ C₁} [HasKernel g₁] (w₁ : f₁ ≫ g₁ = 0) {A₂ B₂ C₂ : V} {f₂ : A₂ ⟶ B₂} [HasImage f₂] {g₂ : B₂ ⟶ C₂} [HasKernel g₂] (w₂ : f₂ ≫ g₂ = 0) {A₃ B₃ C₃ : V} {f₃ : A₃ ⟶ B₃} [HasImage f₃] {g₃ : B₃ ⟶ C₃} [HasKernel g₃] (w₃ : f₃ ≫ g₃ = 0) (α₁ : Arrow.mk f₁ ⟶ Arrow.mk f₂) [HasImageMap α₁] (β₁ : Arrow.mk g₁ ⟶ Arrow.mk g₂) (α₂ : Arrow.mk f₂ ⟶ Arrow.mk f₃) [HasImageMap α₂] (β₂ : Arrow.mk g₂ ⟶ Arrow.mk g₃) @[reassoc] theorem imageSubobjectMap_comp_imageToKernel (p : α.right = β.left) : imageToKernel f g w ≫ kernelSubobjectMap β = imageSubobjectMap α ≫ imageToKernel f' g' w' := by ext simp [p] #align image_subobject_map_comp_image_to_kernel imageSubobjectMap_comp_imageToKernel variable [HasCokernel (imageToKernel f g w)] [HasCokernel (imageToKernel f' g' w')] variable [HasCokernel (imageToKernel f₁ g₁ w₁)] variable [HasCokernel (imageToKernel f₂ g₂ w₂)] variable [HasCokernel (imageToKernel f₃ g₃ w₃)] def homology'.map (p : α.right = β.left) : homology' f g w ⟶ homology' f' g' w' := cokernel.desc _ (kernelSubobjectMap β ≫ cokernel.π _) <\| by rw [imageSubobjectMap_comp_imageToKernel_assoc w w' α β p] simp only [cokernel.condition, comp_zero] #align homology.map homology'.map -- Porting note: removed elementwise attribute which does not seem to be helpful here, -- the correct lemma is stated below @[reassoc (attr := simp)] theorem homology'.π_map (p : α.right = β.left) : homology'.π f g w ≫ homology'.map w w' α β p = kernelSubobjectMap β ≫ homology'.π f' g' w' := by simp only [homology'.π, homology'.map, cokernel.π_desc] #align homology.π_map homology'.π_map section attribute [local instance] ConcreteCategory.instFunLike @[simp] lemma homology'.π_map_apply [ConcreteCategory.{w} V] (p : α.right = β.left) (x : (forget V).obj (Subobject.underlying.obj (kernelSubobject g))) : homology'.map w w' α β p (homology'.π f g w x) = homology'.π f' g' w' (kernelSubobjectMap β x) := by simp only [← comp_apply, homology'.π_map w w' α β p] end @[reassoc (attr := simp), elementwise (attr := simp)] theorem homology'.map_desc (p : α.right = β.left) {D : V} (k : (kernelSubobject g' : V) ⟶ D) (z : imageToKernel f' g' w' ≫ k = 0) : homology'.map w w' α β p ≫ homology'.desc f' g' w' k z = homology'.desc f g w (kernelSubobjectMap β ≫ k) (by simp only [imageSubobjectMap_comp_imageToKernel_assoc w w' α β p, z, comp_zero]) := by ext simp only [homology'.π_desc, homology'.π_map_assoc] #align homology.map_desc homology'.map_desc @[simp]	Mathlib/Algebra/Homology/ImageToKernel.lean	324	326	theorem homology'.map_id : homology'.map w w (𝟙 _) (𝟙 _) rfl = 𝟙 _ := by	ext simp only [homology'.π_map, kernelSubobjectMap_id, Category.id_comp, Category.comp_id]
import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Data.Complex.Exponential import Mathlib.Data.Complex.Module import Mathlib.RingTheory.Polynomial.Chebyshev #align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" set_option linter.uppercaseLean3 false namespace Polynomial.Chebyshev open Polynomial variable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] @[simp] theorem aeval_T (x : A) (n : ℤ) : aeval x (T R n) = (T A n).eval x := by rw [aeval_def, eval₂_eq_eval_map, map_T] #align polynomial.chebyshev.aeval_T Polynomial.Chebyshev.aeval_T @[simp] theorem aeval_U (x : A) (n : ℤ) : aeval x (U R n) = (U A n).eval x := by rw [aeval_def, eval₂_eq_eval_map, map_U] #align polynomial.chebyshev.aeval_U Polynomial.Chebyshev.aeval_U @[simp] theorem algebraMap_eval_T (x : R) (n : ℤ) : algebraMap R A ((T R n).eval x) = (T A n).eval (algebraMap R A x) := by rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_T] #align polynomial.chebyshev.algebra_map_eval_T Polynomial.Chebyshev.algebraMap_eval_T @[simp] theorem algebraMap_eval_U (x : R) (n : ℤ) : algebraMap R A ((U R n).eval x) = (U A n).eval (algebraMap R A x) := by rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_U] #align polynomial.chebyshev.algebra_map_eval_U Polynomial.Chebyshev.algebraMap_eval_U -- Porting note: added type ascriptions to the statement @[simp, norm_cast] theorem complex_ofReal_eval_T : ∀ (x : ℝ) n, (((T ℝ n).eval x : ℝ) : ℂ) = (T ℂ n).eval (x : ℂ) := @algebraMap_eval_T ℝ ℂ _ _ _ #align polynomial.chebyshev.complex_of_real_eval_T Polynomial.Chebyshev.complex_ofReal_eval_T -- Porting note: added type ascriptions to the statement @[simp, norm_cast] theorem complex_ofReal_eval_U : ∀ (x : ℝ) n, (((U ℝ n).eval x : ℝ) : ℂ) = (U ℂ n).eval (x : ℂ) := @algebraMap_eval_U ℝ ℂ _ _ _ #align polynomial.chebyshev.complex_of_real_eval_U Polynomial.Chebyshev.complex_ofReal_eval_U section Complex open Complex variable (θ : ℂ) @[simp]	Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean	73	86	theorem T_complex_cos (n : ℤ) : (T ℂ n).eval (cos θ) = cos (n * θ) := by	induction n using Polynomial.Chebyshev.induct with \| zero => simp \| one => simp \| add_two n ih1 ih2 => simp only [T_add_two, eval_sub, eval_mul, eval_X, eval_ofNat, ih1, ih2, sub_eq_iff_eq_add, cos_add_cos] push_cast ring_nf \| neg_add_one n ih1 ih2 => simp only [T_sub_one, eval_sub, eval_mul, eval_X, eval_ofNat, ih1, ih2, sub_eq_iff_eq_add', cos_add_cos] push_cast ring_nf
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal import Mathlib.Analysis.SpecialFunctions.Pow.Continuity import Mathlib.Analysis.SumOverResidueClass #align_import analysis.p_series from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8" def SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop := ∀ n : ℕ, u (n + 2) - u (n + 1) ≤ C • (u (n + 1) - u n) namespace Finset variable {M : Type*} [OrderedAddCommMonoid M] {f : ℕ → M} {u : ℕ → ℕ} theorem le_sum_schlomilch' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n) (hu : Monotone u) (n : ℕ) : (∑ k ∈ Ico (u 0) (u n), f k) ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) := by induction' n with n ihn · simp suffices (∑ k ∈ Ico (u n) (u (n + 1)), f k) ≤ (u (n + 1) - u n) • f (u n) by rw [sum_range_succ, ← sum_Ico_consecutive] · exact add_le_add ihn this exacts [hu n.zero_le, hu n.le_succ] have : ∀ k ∈ Ico (u n) (u (n + 1)), f k ≤ f (u n) := fun k hk => hf (Nat.succ_le_of_lt (h_pos n)) (mem_Ico.mp hk).1 convert sum_le_sum this simp [pow_succ, mul_two]	Mathlib/Analysis/PSeries.lean	64	68	theorem le_sum_condensed' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) : (∑ k ∈ Ico 1 (2 ^ n), f k) ≤ ∑ k ∈ range n, 2 ^ k • f (2 ^ k) := by	convert le_sum_schlomilch' hf (fun n => pow_pos zero_lt_two n) (fun m n hm => pow_le_pow_right one_le_two hm) n using 2 simp [pow_succ, mul_two, two_mul]
import Mathlib.RingTheory.DedekindDomain.Ideal #align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f" namespace Ideal universe u v variable {R : Type u} [CommRing R] variable {S : Type v} [CommRing S] (f : R →+* S) variable (p : Ideal R) (P : Ideal S) open FiniteDimensional open UniqueFactorizationMonoid section DecEq open scoped Classical noncomputable def ramificationIdx : ℕ := sSup {n \| map f p ≤ P ^ n} #align ideal.ramification_idx Ideal.ramificationIdx variable {f p P} theorem ramificationIdx_eq_find (h : ∃ n, ∀ k, map f p ≤ P ^ k → k ≤ n) : ramificationIdx f p P = Nat.find h := Nat.sSup_def h #align ideal.ramification_idx_eq_find Ideal.ramificationIdx_eq_find theorem ramificationIdx_eq_zero (h : ∀ n : ℕ, ∃ k, map f p ≤ P ^ k ∧ n < k) : ramificationIdx f p P = 0 := dif_neg (by push_neg; exact h) #align ideal.ramification_idx_eq_zero Ideal.ramificationIdx_eq_zero theorem ramificationIdx_spec {n : ℕ} (hle : map f p ≤ P ^ n) (hgt : ¬map f p ≤ P ^ (n + 1)) : ramificationIdx f p P = n := by let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m have : Q n := by intro k hk refine le_of_not_lt fun hnk => ?_ exact hgt (hk.trans (Ideal.pow_le_pow_right hnk)) rw [ramificationIdx_eq_find ⟨n, this⟩] refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_) obtain this' := Nat.find_spec ⟨n, this⟩ exact h.not_le (this' _ hle) #align ideal.ramification_idx_spec Ideal.ramificationIdx_spec theorem ramificationIdx_lt {n : ℕ} (hgt : ¬map f p ≤ P ^ n) : ramificationIdx f p P < n := by cases' n with n n · simp at hgt · rw [Nat.lt_succ_iff] have : ∀ k, map f p ≤ P ^ k → k ≤ n := by refine fun k hk => le_of_not_lt fun hnk => ?_ exact hgt (hk.trans (Ideal.pow_le_pow_right hnk)) rw [ramificationIdx_eq_find ⟨n, this⟩] exact Nat.find_min' ⟨n, this⟩ this #align ideal.ramification_idx_lt Ideal.ramificationIdx_lt @[simp] theorem ramificationIdx_bot : ramificationIdx f ⊥ P = 0 := dif_neg <\| not_exists.mpr fun n hn => n.lt_succ_self.not_le (hn _ (by simp)) #align ideal.ramification_idx_bot Ideal.ramificationIdx_bot @[simp] theorem ramificationIdx_of_not_le (h : ¬map f p ≤ P) : ramificationIdx f p P = 0 := ramificationIdx_spec (by simp) (by simpa using h) #align ideal.ramification_idx_of_not_le Ideal.ramificationIdx_of_not_le	Mathlib/NumberTheory/RamificationInertia.lean	116	118	theorem ramificationIdx_ne_zero {e : ℕ} (he : e ≠ 0) (hle : map f p ≤ P ^ e) (hnle : ¬map f p ≤ P ^ (e + 1)) : ramificationIdx f p P ≠ 0 := by	rwa [ramificationIdx_spec hle hnle]
import Mathlib.Order.Interval.Multiset #align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29" -- TODO -- assert_not_exists Ring open Finset Nat variable (a b c : ℕ) namespace Nat instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩ finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ := rfl #align nat.Icc_eq_range' Nat.Icc_eq_range' theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ := rfl #align nat.Ico_eq_range' Nat.Ico_eq_range' theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ := rfl #align nat.Ioc_eq_range' Nat.Ioc_eq_range' theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ := rfl #align nat.Ioo_eq_range' Nat.Ioo_eq_range' theorem uIcc_eq_range' : uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl #align nat.uIcc_eq_range' Nat.uIcc_eq_range' theorem Iio_eq_range : Iio = range := by ext b x rw [mem_Iio, mem_range] #align nat.Iio_eq_range Nat.Iio_eq_range @[simp] theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range] #align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by ext b simp_all only [mem_Icc, zero_le, true_and, mem_range] exact lt_iff_le_pred (zero_lt_of_ne_zero hn) theorem _root_.Finset.range_eq_Ico : range = Ico 0 := Ico_zero_eq_range.symm #align finset.range_eq_Ico Finset.range_eq_Ico @[simp] theorem card_Icc : (Icc a b).card = b + 1 - a := List.length_range' _ _ _ #align nat.card_Icc Nat.card_Icc @[simp] theorem card_Ico : (Ico a b).card = b - a := List.length_range' _ _ _ #align nat.card_Ico Nat.card_Ico @[simp] theorem card_Ioc : (Ioc a b).card = b - a := List.length_range' _ _ _ #align nat.card_Ioc Nat.card_Ioc @[simp] theorem card_Ioo : (Ioo a b).card = b - a - 1 := List.length_range' _ _ _ #align nat.card_Ioo Nat.card_Ioo @[simp] theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := (card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega #align nat.card_uIcc Nat.card_uIcc @[simp] lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero] #align nat.card_Iic Nat.card_Iic @[simp] theorem card_Iio : (Iio b).card = b := by rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero] #align nat.card_Iio Nat.card_Iio -- Porting note (#10618): simp can prove this -- @[simp] theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by rw [Fintype.card_ofFinset, card_Icc] #align nat.card_fintype_Icc Nat.card_fintypeIcc -- Porting note (#10618): simp can prove this -- @[simp] theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by rw [Fintype.card_ofFinset, card_Ico] #align nat.card_fintype_Ico Nat.card_fintypeIco -- Porting note (#10618): simp can prove this -- @[simp] theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by rw [Fintype.card_ofFinset, card_Ioc] #align nat.card_fintype_Ioc Nat.card_fintypeIoc -- Porting note (#10618): simp can prove this -- @[simp] theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by rw [Fintype.card_ofFinset, card_Ioo] #align nat.card_fintype_Ioo Nat.card_fintypeIoo -- Porting note (#10618): simp can prove this -- @[simp] theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by rw [Fintype.card_ofFinset, card_Iic] #align nat.card_fintype_Iic Nat.card_fintypeIic -- Porting note (#10618): simp can prove this -- @[simp] theorem card_fintypeIio : Fintype.card (Set.Iio b) = b := by rw [Fintype.card_ofFinset, card_Iio] #align nat.card_fintype_Iio Nat.card_fintypeIio -- TODO@Yaël: Generalize all the following lemmas to `SuccOrder` theorem Icc_succ_left : Icc a.succ b = Ioc a b := by ext x rw [mem_Icc, mem_Ioc, succ_le_iff] #align nat.Icc_succ_left Nat.Icc_succ_left theorem Ico_succ_right : Ico a b.succ = Icc a b := by ext x rw [mem_Ico, mem_Icc, Nat.lt_succ_iff] #align nat.Ico_succ_right Nat.Ico_succ_right theorem Ico_succ_left : Ico a.succ b = Ioo a b := by ext x rw [mem_Ico, mem_Ioo, succ_le_iff] #align nat.Ico_succ_left Nat.Ico_succ_left theorem Icc_pred_right {b : ℕ} (h : 0 < b) : Icc a (b - 1) = Ico a b := by ext x rw [mem_Icc, mem_Ico, lt_iff_le_pred h] #align nat.Icc_pred_right Nat.Icc_pred_right theorem Ico_succ_succ : Ico a.succ b.succ = Ioc a b := by ext x rw [mem_Ico, mem_Ioc, succ_le_iff, Nat.lt_succ_iff] #align nat.Ico_succ_succ Nat.Ico_succ_succ @[simp] theorem Ico_succ_singleton : Ico a (a + 1) = {a} := by rw [Ico_succ_right, Icc_self] #align nat.Ico_succ_singleton Nat.Ico_succ_singleton @[simp] theorem Ico_pred_singleton {a : ℕ} (h : 0 < a) : Ico (a - 1) a = {a - 1} := by rw [← Icc_pred_right _ h, Icc_self] #align nat.Ico_pred_singleton Nat.Ico_pred_singleton @[simp]	Mathlib/Order/Interval/Finset/Nat.lean	183	183	theorem Ioc_succ_singleton : Ioc b (b + 1) = {b + 1} := by	rw [← Nat.Icc_succ_left, Icc_self]
import Mathlib.Algebra.Group.Subgroup.Finite import Mathlib.Data.Finset.Fin import Mathlib.Data.Finset.Sort import Mathlib.Data.Int.Order.Units import Mathlib.GroupTheory.Perm.Support import Mathlib.Logic.Equiv.Fin import Mathlib.Tactic.NormNum.Ineq #align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e" universe u v open Equiv Function Fintype Finset variable {α : Type u} [DecidableEq α] {β : Type v} namespace Equiv.Perm def modSwap (i j : α) : Setoid (Perm α) := ⟨fun σ τ => σ = τ ∨ σ = swap i j * τ, fun σ => Or.inl (refl σ), fun {σ τ} h => Or.casesOn h (fun h => Or.inl h.symm) fun h => Or.inr (by rw [h, swap_mul_self_mul]), fun {σ τ υ} hστ hτυ => by cases' hστ with hστ hστ <;> cases' hτυ with hτυ hτυ <;> try rw [hστ, hτυ, swap_mul_self_mul] <;> simp [hστ, hτυ] -- Porting note: should close goals, but doesn't · simp [hστ, hτυ] · simp [hστ, hτυ] · simp [hστ, hτυ]⟩ #align equiv.perm.mod_swap Equiv.Perm.modSwap noncomputable instance {α : Type} [Fintype α] [DecidableEq α] (i j : α) : DecidableRel (modSwap i j).r := fun _ _ => Or.decidable def swapFactorsAux : ∀ (l : List α) (f : Perm α), (∀ {x}, f x ≠ x → x ∈ l) → { l : List (Perm α) // l.prod = f ∧ ∀ g ∈ l, IsSwap g } \| [] => fun f h => ⟨[], Equiv.ext fun x => by rw [List.prod_nil] exact (Classical.not_not.1 (mt h (List.not_mem_nil _))).symm, by simp⟩ \| x::l => fun f h => if hfx : x = f x then swapFactorsAux l f fun {y} hy => List.mem_of_ne_of_mem (fun h : y = x => by simp [h, hfx.symm] at hy) (h hy) else let m := swapFactorsAux l (swap x (f x) f) fun {y} hy => have : f y ≠ y ∧ y ≠ x := ne_and_ne_of_swap_mul_apply_ne_self hy List.mem_of_ne_of_mem this.2 (h this.1) ⟨swap x (f x)::m.1, by rw [List.prod_cons, m.2.1, ← mul_assoc, mul_def (swap x (f x)), swap_swap, ← one_def, one_mul], fun {g} hg => ((List.mem_cons).1 hg).elim (fun h => ⟨x, f x, hfx, h⟩) (m.2.2 _)⟩ #align equiv.perm.swap_factors_aux Equiv.Perm.swapFactorsAux def swapFactors [Fintype α] [LinearOrder α] (f : Perm α) : { l : List (Perm α) // l.prod = f ∧ ∀ g ∈ l, IsSwap g } := swapFactorsAux ((@univ α _).sort (· ≤ ·)) f fun {_ _} => (mem_sort _).2 (mem_univ _) #align equiv.perm.swap_factors Equiv.Perm.swapFactors def truncSwapFactors [Fintype α] (f : Perm α) : Trunc { l : List (Perm α) // l.prod = f ∧ ∀ g ∈ l, IsSwap g } := Quotient.recOnSubsingleton (@univ α _).1 (fun l h => Trunc.mk (swapFactorsAux l f (h _))) (show ∀ x, f x ≠ x → x ∈ (@univ α _).1 from fun _ _ => mem_univ _) #align equiv.perm.trunc_swap_factors Equiv.Perm.truncSwapFactors @[elab_as_elim] theorem swap_induction_on [Finite α] {P : Perm α → Prop} (f : Perm α) : P 1 → (∀ f x y, x ≠ y → P f → P (swap x y * f)) → P f := by cases nonempty_fintype α cases' (truncSwapFactors f).out with l hl induction' l with g l ih generalizing f · simp (config := { contextual := true }) only [hl.left.symm, List.prod_nil, forall_true_iff] · intro h1 hmul_swap rcases hl.2 g (by simp) with ⟨x, y, hxy⟩ rw [← hl.1, List.prod_cons, hxy.2] exact hmul_swap _ _ _ hxy.1 (ih _ ⟨rfl, fun v hv => hl.2 _ (List.mem_cons_of_mem _ hv)⟩ h1 hmul_swap) #align equiv.perm.swap_induction_on Equiv.Perm.swap_induction_on theorem closure_isSwap [Finite α] : Subgroup.closure { σ : Perm α \| IsSwap σ } = ⊤ := by cases nonempty_fintype α refine eq_top_iff.mpr fun x _ => ?_ obtain ⟨h1, h2⟩ := Subtype.mem (truncSwapFactors x).out rw [← h1] exact Subgroup.list_prod_mem _ fun y hy => Subgroup.subset_closure (h2 y hy) #align equiv.perm.closure_is_swap Equiv.Perm.closure_isSwap @[elab_as_elim] theorem swap_induction_on' [Finite α] {P : Perm α → Prop} (f : Perm α) : P 1 → (∀ f x y, x ≠ y → P f → P (f * swap x y)) → P f := fun h1 IH => inv_inv f ▸ swap_induction_on f⁻¹ h1 fun f => IH f⁻¹ #align equiv.perm.swap_induction_on' Equiv.Perm.swap_induction_on' theorem isConj_swap {w x y z : α} (hwx : w ≠ x) (hyz : y ≠ z) : IsConj (swap w x) (swap y z) := isConj_iff.2 (have h : ∀ {y z : α}, y ≠ z → w ≠ z → swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z := fun {y z} hyz hwz => by rw [mul_inv_rev, swap_inv, swap_inv, mul_assoc (swap w y), mul_assoc (swap w y), ← mul_assoc _ (swap x z), swap_mul_swap_mul_swap hwx hwz, ← mul_assoc, swap_mul_swap_mul_swap hwz.symm hyz.symm] if hwz : w = z then have hwy : w ≠ y := by rw [hwz]; exact hyz.symm ⟨swap w z * swap x y, by rw [swap_comm y z, h hyz.symm hwy]⟩ else ⟨swap w y * swap x z, h hyz hwz⟩) #align equiv.perm.is_conj_swap Equiv.Perm.isConj_swap def finPairsLT (n : ℕ) : Finset (Σ_ : Fin n, Fin n) := (univ : Finset (Fin n)).sigma fun a => (range a).attachFin fun _ hm => (mem_range.1 hm).trans a.2 #align equiv.perm.fin_pairs_lt Equiv.Perm.finPairsLT theorem mem_finPairsLT {n : ℕ} {a : Σ_ : Fin n, Fin n} : a ∈ finPairsLT n ↔ a.2 < a.1 := by simp only [finPairsLT, Fin.lt_iff_val_lt_val, true_and_iff, mem_attachFin, mem_range, mem_univ, mem_sigma] #align equiv.perm.mem_fin_pairs_lt Equiv.Perm.mem_finPairsLT def signAux {n : ℕ} (a : Perm (Fin n)) : ℤˣ := ∏ x ∈ finPairsLT n, if a x.1 ≤ a x.2 then -1 else 1 #align equiv.perm.sign_aux Equiv.Perm.signAux @[simp] theorem signAux_one (n : ℕ) : signAux (1 : Perm (Fin n)) = 1 := by unfold signAux conv => rhs; rw [← @Finset.prod_const_one _ _ (finPairsLT n)] exact Finset.prod_congr rfl fun a ha => if_neg (mem_finPairsLT.1 ha).not_le #align equiv.perm.sign_aux_one Equiv.Perm.signAux_one def signBijAux {n : ℕ} (f : Perm (Fin n)) (a : Σ_ : Fin n, Fin n) : Σ_ : Fin n, Fin n := if _ : f a.2 < f a.1 then ⟨f a.1, f a.2⟩ else ⟨f a.2, f a.1⟩ #align equiv.perm.sign_bij_aux Equiv.Perm.signBijAux theorem signBijAux_injOn {n : ℕ} {f : Perm (Fin n)} : (finPairsLT n : Set (Σ _, Fin n)).InjOn (signBijAux f) := by rintro ⟨a₁, a₂⟩ ha ⟨b₁, b₂⟩ hb h dsimp [signBijAux] at h rw [Finset.mem_coe, mem_finPairsLT] at * have : ¬b₁ < b₂ := hb.le.not_lt split_ifs at h <;> simp_all [(Equiv.injective f).eq_iff, eq_self_iff_true, and_self_iff, heq_iff_eq] · exact absurd this (not_le.mpr ha) · exact absurd this (not_le.mpr ha) #align equiv.perm.sign_bij_aux_inj Equiv.Perm.signBijAux_injOn theorem signBijAux_surj {n : ℕ} {f : Perm (Fin n)} : ∀ a ∈ finPairsLT n, ∃ b ∈ finPairsLT n, signBijAux f b = a := fun ⟨a₁, a₂⟩ ha => if hxa : f⁻¹ a₂ < f⁻¹ a₁ then ⟨⟨f⁻¹ a₁, f⁻¹ a₂⟩, mem_finPairsLT.2 hxa, by dsimp [signBijAux] rw [apply_inv_self, apply_inv_self, if_pos (mem_finPairsLT.1 ha)]⟩ else ⟨⟨f⁻¹ a₂, f⁻¹ a₁⟩, mem_finPairsLT.2 <\| (le_of_not_gt hxa).lt_of_ne fun h => by simp [mem_finPairsLT, f⁻¹.injective h, lt_irrefl] at ha, by dsimp [signBijAux] rw [apply_inv_self, apply_inv_self, if_neg (mem_finPairsLT.1 ha).le.not_lt]⟩ #align equiv.perm.sign_bij_aux_surj Equiv.Perm.signBijAux_surj theorem signBijAux_mem {n : ℕ} {f : Perm (Fin n)} : ∀ a : Σ_ : Fin n, Fin n, a ∈ finPairsLT n → signBijAux f a ∈ finPairsLT n := fun ⟨a₁, a₂⟩ ha => by unfold signBijAux split_ifs with h · exact mem_finPairsLT.2 h · exact mem_finPairsLT.2 ((le_of_not_gt h).lt_of_ne fun h => (mem_finPairsLT.1 ha).ne (f.injective h.symm)) #align equiv.perm.sign_bij_aux_mem Equiv.Perm.signBijAux_mem @[simp] theorem signAux_inv {n : ℕ} (f : Perm (Fin n)) : signAux f⁻¹ = signAux f := prod_nbij (signBijAux f⁻¹) signBijAux_mem signBijAux_injOn signBijAux_surj fun ⟨a, b⟩ hab ↦ if h : f⁻¹ b < f⁻¹ a then by simp_all [signBijAux, dif_pos h, if_neg h.not_le, apply_inv_self, apply_inv_self, if_neg (mem_finPairsLT.1 hab).not_le] else by simp_all [signBijAux, if_pos (le_of_not_gt h), dif_neg h, apply_inv_self, apply_inv_self, if_pos (mem_finPairsLT.1 hab).le] #align equiv.perm.sign_aux_inv Equiv.Perm.signAux_inv theorem signAux_mul {n : ℕ} (f g : Perm (Fin n)) : signAux (f * g) = signAux f * signAux g := by rw [← signAux_inv g] unfold signAux rw [← prod_mul_distrib] refine prod_nbij (signBijAux g) signBijAux_mem signBijAux_injOn signBijAux_surj ?_ rintro ⟨a, b⟩ hab dsimp only [signBijAux] rw [mul_apply, mul_apply] rw [mem_finPairsLT] at hab by_cases h : g b < g a · rw [dif_pos h] simp only [not_le_of_gt hab, mul_one, mul_ite, mul_neg, Perm.inv_apply_self, if_false] · rw [dif_neg h, inv_apply_self, inv_apply_self, if_pos hab.le] by_cases h₁ : f (g b) ≤ f (g a) · have : f (g b) ≠ f (g a) := by rw [Ne, f.injective.eq_iff, g.injective.eq_iff] exact ne_of_lt hab rw [if_pos h₁, if_neg (h₁.lt_of_ne this).not_le] rfl · rw [if_neg h₁, if_pos (lt_of_not_ge h₁).le] rfl #align equiv.perm.sign_aux_mul Equiv.Perm.signAux_mul private theorem signAux_swap_zero_one' (n : ℕ) : signAux (swap (0 : Fin (n + 2)) 1) = -1 := show _ = ∏ x ∈ {(⟨1, 0⟩ : Σ a : Fin (n + 2), Fin (n + 2))}, if (Equiv.swap 0 1) x.1 ≤ swap 0 1 x.2 then (-1 : ℤˣ) else 1 by refine Eq.symm (prod_subset (fun ⟨x₁, x₂⟩ => by simp (config := { contextual := true }) [mem_finPairsLT, Fin.one_pos]) fun a ha₁ ha₂ => ?_) rcases a with ⟨a₁, a₂⟩ replace ha₁ : a₂ < a₁ := mem_finPairsLT.1 ha₁ dsimp only rcases a₁.zero_le.eq_or_lt with (rfl \| H) · exact absurd a₂.zero_le ha₁.not_le rcases a₂.zero_le.eq_or_lt with (rfl \| H') · simp only [and_true_iff, eq_self_iff_true, heq_iff_eq, mem_singleton, Sigma.mk.inj_iff] at ha₂ have : 1 < a₁ := lt_of_le_of_ne (Nat.succ_le_of_lt ha₁) (Ne.symm (by intro h; apply ha₂; simp [h])) have h01 : Equiv.swap (0 : Fin (n + 2)) 1 0 = 1 := by simp rw [swap_apply_of_ne_of_ne (ne_of_gt H) ha₂, h01, if_neg this.not_le] · have le : 1 ≤ a₂ := Nat.succ_le_of_lt H' have lt : 1 < a₁ := le.trans_lt ha₁ have h01 : Equiv.swap (0 : Fin (n + 2)) 1 1 = 0 := by simp only [swap_apply_right] rcases le.eq_or_lt with (rfl \| lt') · rw [swap_apply_of_ne_of_ne H.ne' lt.ne', h01, if_neg H.not_le] · rw [swap_apply_of_ne_of_ne (ne_of_gt H) (ne_of_gt lt), swap_apply_of_ne_of_ne (ne_of_gt H') (ne_of_gt lt'), if_neg ha₁.not_le] private theorem signAux_swap_zero_one {n : ℕ} (hn : 2 ≤ n) : signAux (swap (⟨0, lt_of_lt_of_le (by decide) hn⟩ : Fin n) ⟨1, lt_of_lt_of_le (by decide) hn⟩) = -1 := by rcases n with (_ \| _ \| n) · norm_num at hn · norm_num at hn · exact signAux_swap_zero_one' n theorem signAux_swap : ∀ {n : ℕ} {x y : Fin n} (_hxy : x ≠ y), signAux (swap x y) = -1 \| 0, x, y => by intro; exact Fin.elim0 x \| 1, x, y => by dsimp [signAux, swap, swapCore] simp only [eq_iff_true_of_subsingleton, not_true, ite_true, le_refl, prod_const, IsEmpty.forall_iff] \| n + 2, x, y => fun hxy => by have h2n : 2 ≤ n + 2 := by exact le_add_self rw [← isConj_iff_eq, ← signAux_swap_zero_one h2n] exact (MonoidHom.mk' signAux signAux_mul).map_isConj (isConj_swap hxy (by exact of_decide_eq_true rfl)) #align equiv.perm.sign_aux_swap Equiv.Perm.signAux_swap def signAux2 : List α → Perm α → ℤˣ \| [], _ => 1 \| x::l, f => if x = f x then signAux2 l f else -signAux2 l (swap x (f x) * f) #align equiv.perm.sign_aux2 Equiv.Perm.signAux2 theorem signAux_eq_signAux2 {n : ℕ} : ∀ (l : List α) (f : Perm α) (e : α ≃ Fin n) (_h : ∀ x, f x ≠ x → x ∈ l), signAux ((e.symm.trans f).trans e) = signAux2 l f \| [], f, e, h => by have : f = 1 := Equiv.ext fun y => Classical.not_not.1 (mt (h y) (List.not_mem_nil _)) rw [this, one_def, Equiv.trans_refl, Equiv.symm_trans_self, ← one_def, signAux_one, signAux2] \| x::l, f, e, h => by rw [signAux2] by_cases hfx : x = f x · rw [if_pos hfx] exact signAux_eq_signAux2 l f _ fun y (hy : f y ≠ y) => List.mem_of_ne_of_mem (fun h : y = x => by simp [h, hfx.symm] at hy) (h y hy) · have hy : ∀ y : α, (swap x (f x) * f) y ≠ y → y ∈ l := fun y hy => have : f y ≠ y ∧ y ≠ x := ne_and_ne_of_swap_mul_apply_ne_self hy List.mem_of_ne_of_mem this.2 (h _ this.1) have : (e.symm.trans (swap x (f x) * f)).trans e = swap (e x) (e (f x)) * (e.symm.trans f).trans e := by ext rw [← Equiv.symm_trans_swap_trans, mul_def, Equiv.symm_trans_swap_trans, mul_def] repeat (rw [trans_apply]) simp [swap, swapCore] split_ifs <;> rfl have hefx : e x ≠ e (f x) := mt e.injective.eq_iff.1 hfx rw [if_neg hfx, ← signAux_eq_signAux2 _ _ e hy, this, signAux_mul, signAux_swap hefx] simp only [neg_neg, one_mul, neg_mul] #align equiv.perm.sign_aux_eq_sign_aux2 Equiv.Perm.signAux_eq_signAux2 def signAux3 [Finite α] (f : Perm α) {s : Multiset α} : (∀ x, x ∈ s) → ℤˣ := Quotient.hrecOn s (fun l _ => signAux2 l f) fun l₁ l₂ h ↦ by rcases Finite.exists_equiv_fin α with ⟨n, ⟨e⟩⟩ refine Function.hfunext (forall_congr fun _ ↦ propext h.mem_iff) fun h₁ h₂ _ ↦ ?_ rw [← signAux_eq_signAux2 _ _ e fun _ _ => h₁ _, ← signAux_eq_signAux2 _ _ e fun _ _ => h₂ _] #align equiv.perm.sign_aux3 Equiv.Perm.signAux3 theorem signAux3_mul_and_swap [Finite α] (f g : Perm α) (s : Multiset α) (hs : ∀ x, x ∈ s) : signAux3 (f * g) hs = signAux3 f hs * signAux3 g hs ∧ Pairwise fun x y => signAux3 (swap x y) hs = -1 := by obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin α induction s using Quotient.inductionOn with \| _ l => ?_ show signAux2 l (f * g) = signAux2 l f * signAux2 l g ∧ Pairwise fun x y => signAux2 l (swap x y) = -1 have hfg : (e.symm.trans (f * g)).trans e = (e.symm.trans f).trans e * (e.symm.trans g).trans e := Equiv.ext fun h => by simp [mul_apply] constructor · rw [← signAux_eq_signAux2 _ _ e fun _ _ => hs _, ← signAux_eq_signAux2 _ _ e fun _ _ => hs _, ← signAux_eq_signAux2 _ _ e fun _ _ => hs _, hfg, signAux_mul] · intro x y hxy rw [← e.injective.ne_iff] at hxy rw [← signAux_eq_signAux2 _ _ e fun _ _ => hs _, symm_trans_swap_trans, signAux_swap hxy] #align equiv.perm.sign_aux3_mul_and_swap Equiv.Perm.signAux3_mul_and_swap	Mathlib/GroupTheory/Perm/Sign.lean	354	364	theorem signAux3_symm_trans_trans [Finite α] [DecidableEq β] [Finite β] (f : Perm α) (e : α ≃ β) {s : Multiset α} {t : Multiset β} (hs : ∀ x, x ∈ s) (ht : ∀ x, x ∈ t) : signAux3 ((e.symm.trans f).trans e) ht = signAux3 f hs := by	-- Porting note: switched from term mode to tactic mode induction' t, s using Quotient.inductionOn₂ with t s ht hs show signAux2 _ _ = signAux2 _ _ rcases Finite.exists_equiv_fin β with ⟨n, ⟨e'⟩⟩ rw [← signAux_eq_signAux2 _ _ e' fun _ _ => ht _, ← signAux_eq_signAux2 _ _ (e.trans e') fun _ _ => hs _] exact congr_arg signAux (Equiv.ext fun x => by simp [Equiv.coe_trans, apply_eq_iff_eq, symm_trans_apply])
import Mathlib.Algebra.Algebra.Quasispectrum import Mathlib.FieldTheory.IsAlgClosed.Spectrum import Mathlib.Analysis.Complex.Liouville import Mathlib.Analysis.Complex.Polynomial import Mathlib.Analysis.Analytic.RadiusLiminf import Mathlib.Topology.Algebra.Module.CharacterSpace import Mathlib.Analysis.NormedSpace.Exponential import Mathlib.Analysis.NormedSpace.UnitizationL1 #align_import analysis.normed_space.spectrum from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521" open scoped ENNReal NNReal open NormedSpace -- For `NormedSpace.exp`. noncomputable def spectralRadius (𝕜 : Type) {A : Type} [NormedField 𝕜] [Ring A] [Algebra 𝕜 A] (a : A) : ℝ≥0∞ := ⨆ k ∈ spectrum 𝕜 a, ‖k‖₊ #align spectral_radius spectralRadius variable {𝕜 : Type} {A : Type} namespace spectrum section SpectrumCompact open Filter variable [NormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] local notation "σ" => spectrum 𝕜 local notation "ρ" => resolventSet 𝕜 local notation "↑ₐ" => algebraMap 𝕜 A @[simp] theorem SpectralRadius.of_subsingleton [Subsingleton A] (a : A) : spectralRadius 𝕜 a = 0 := by simp [spectralRadius] #align spectrum.spectral_radius.of_subsingleton spectrum.SpectralRadius.of_subsingleton @[simp] theorem spectralRadius_zero : spectralRadius 𝕜 (0 : A) = 0 := by nontriviality A simp [spectralRadius] #align spectrum.spectral_radius_zero spectrum.spectralRadius_zero theorem mem_resolventSet_of_spectralRadius_lt {a : A} {k : 𝕜} (h : spectralRadius 𝕜 a < ‖k‖₊) : k ∈ ρ a := Classical.not_not.mp fun hn => h.not_le <\| le_iSup₂ (α := ℝ≥0∞) k hn #align spectrum.mem_resolvent_set_of_spectral_radius_lt spectrum.mem_resolventSet_of_spectralRadius_lt variable [CompleteSpace A] theorem isOpen_resolventSet (a : A) : IsOpen (ρ a) := Units.isOpen.preimage ((continuous_algebraMap 𝕜 A).sub continuous_const) #align spectrum.is_open_resolvent_set spectrum.isOpen_resolventSet protected theorem isClosed (a : A) : IsClosed (σ a) := (isOpen_resolventSet a).isClosed_compl #align spectrum.is_closed spectrum.isClosed	Mathlib/Analysis/NormedSpace/Spectrum.lean	104	113	theorem mem_resolventSet_of_norm_lt_mul {a : A} {k : 𝕜} (h : ‖a‖ * ‖(1 : A)‖ < ‖k‖) : k ∈ ρ a := by	rw [resolventSet, Set.mem_setOf_eq, Algebra.algebraMap_eq_smul_one] nontriviality A have hk : k ≠ 0 := ne_zero_of_norm_ne_zero ((mul_nonneg (norm_nonneg _) (norm_nonneg _)).trans_lt h).ne' letI ku := Units.map ↑ₐ.toMonoidHom (Units.mk0 k hk) rw [← inv_inv ‖(1 : A)‖, mul_inv_lt_iff (inv_pos.2 <\| norm_pos_iff.2 (one_ne_zero : (1 : A) ≠ 0))] at h have hku : ‖-a‖ < ‖(↑ku⁻¹ : A)‖⁻¹ := by simpa [ku, norm_algebraMap] using h simpa [ku, sub_eq_add_neg, Algebra.algebraMap_eq_smul_one] using (ku.add (-a) hku).isUnit
import Mathlib.Analysis.SpecialFunctions.Exp import Mathlib.Tactic.Positivity.Core import Mathlib.Algebra.Ring.NegOnePow #align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" noncomputable section open scoped Classical open Topology Filter Set namespace Real open Real @[simp] theorem sin_pi : sin π = 0 := by rw [← mul_div_cancel_left₀ π (two_ne_zero' ℝ), two_mul, add_div, sin_add, cos_pi_div_two]; simp #align real.sin_pi Real.sin_pi @[simp] theorem cos_pi : cos π = -1 := by rw [← mul_div_cancel_left₀ π (two_ne_zero' ℝ), mul_div_assoc, cos_two_mul, cos_pi_div_two] norm_num #align real.cos_pi Real.cos_pi @[simp] theorem sin_two_pi : sin (2 * π) = 0 := by simp [two_mul, sin_add] #align real.sin_two_pi Real.sin_two_pi @[simp] theorem cos_two_pi : cos (2 * π) = 1 := by simp [two_mul, cos_add] #align real.cos_two_pi Real.cos_two_pi theorem sin_antiperiodic : Function.Antiperiodic sin π := by simp [sin_add] #align real.sin_antiperiodic Real.sin_antiperiodic theorem sin_periodic : Function.Periodic sin (2 * π) := sin_antiperiodic.periodic_two_mul #align real.sin_periodic Real.sin_periodic @[simp] theorem sin_add_pi (x : ℝ) : sin (x + π) = -sin x := sin_antiperiodic x #align real.sin_add_pi Real.sin_add_pi @[simp] theorem sin_add_two_pi (x : ℝ) : sin (x + 2 * π) = sin x := sin_periodic x #align real.sin_add_two_pi Real.sin_add_two_pi @[simp] theorem sin_sub_pi (x : ℝ) : sin (x - π) = -sin x := sin_antiperiodic.sub_eq x #align real.sin_sub_pi Real.sin_sub_pi @[simp] theorem sin_sub_two_pi (x : ℝ) : sin (x - 2 * π) = sin x := sin_periodic.sub_eq x #align real.sin_sub_two_pi Real.sin_sub_two_pi @[simp] theorem sin_pi_sub (x : ℝ) : sin (π - x) = sin x := neg_neg (sin x) ▸ sin_neg x ▸ sin_antiperiodic.sub_eq' #align real.sin_pi_sub Real.sin_pi_sub @[simp] theorem sin_two_pi_sub (x : ℝ) : sin (2 * π - x) = -sin x := sin_neg x ▸ sin_periodic.sub_eq' #align real.sin_two_pi_sub Real.sin_two_pi_sub @[simp] theorem sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 := sin_antiperiodic.nat_mul_eq_of_eq_zero sin_zero n #align real.sin_nat_mul_pi Real.sin_nat_mul_pi @[simp] theorem sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 := sin_antiperiodic.int_mul_eq_of_eq_zero sin_zero n #align real.sin_int_mul_pi Real.sin_int_mul_pi @[simp] theorem sin_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x + n * (2 * π)) = sin x := sin_periodic.nat_mul n x #align real.sin_add_nat_mul_two_pi Real.sin_add_nat_mul_two_pi @[simp] theorem sin_add_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x + n * (2 * π)) = sin x := sin_periodic.int_mul n x #align real.sin_add_int_mul_two_pi Real.sin_add_int_mul_two_pi @[simp] theorem sin_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x - n * (2 * π)) = sin x := sin_periodic.sub_nat_mul_eq n #align real.sin_sub_nat_mul_two_pi Real.sin_sub_nat_mul_two_pi @[simp] theorem sin_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x - n * (2 * π)) = sin x := sin_periodic.sub_int_mul_eq n #align real.sin_sub_int_mul_two_pi Real.sin_sub_int_mul_two_pi @[simp] theorem sin_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x := sin_neg x ▸ sin_periodic.nat_mul_sub_eq n #align real.sin_nat_mul_two_pi_sub Real.sin_nat_mul_two_pi_sub @[simp] theorem sin_int_mul_two_pi_sub (x : ℝ) (n : ℤ) : sin (n * (2 * π) - x) = -sin x := sin_neg x ▸ sin_periodic.int_mul_sub_eq n #align real.sin_int_mul_two_pi_sub Real.sin_int_mul_two_pi_sub theorem sin_add_int_mul_pi (x : ℝ) (n : ℤ) : sin (x + n * π) = (-1) ^ n * sin x := n.coe_negOnePow ℝ ▸ sin_antiperiodic.add_int_mul_eq n theorem sin_add_nat_mul_pi (x : ℝ) (n : ℕ) : sin (x + n * π) = (-1) ^ n * sin x := sin_antiperiodic.add_nat_mul_eq n theorem sin_sub_int_mul_pi (x : ℝ) (n : ℤ) : sin (x - n * π) = (-1) ^ n * sin x := n.coe_negOnePow ℝ ▸ sin_antiperiodic.sub_int_mul_eq n theorem sin_sub_nat_mul_pi (x : ℝ) (n : ℕ) : sin (x - n * π) = (-1) ^ n * sin x := sin_antiperiodic.sub_nat_mul_eq n theorem sin_int_mul_pi_sub (x : ℝ) (n : ℤ) : sin (n * π - x) = -((-1) ^ n * sin x) := by simpa only [sin_neg, mul_neg, Int.coe_negOnePow] using sin_antiperiodic.int_mul_sub_eq n theorem sin_nat_mul_pi_sub (x : ℝ) (n : ℕ) : sin (n * π - x) = -((-1) ^ n * sin x) := by simpa only [sin_neg, mul_neg] using sin_antiperiodic.nat_mul_sub_eq n theorem cos_antiperiodic : Function.Antiperiodic cos π := by simp [cos_add] #align real.cos_antiperiodic Real.cos_antiperiodic theorem cos_periodic : Function.Periodic cos (2 * π) := cos_antiperiodic.periodic_two_mul #align real.cos_periodic Real.cos_periodic @[simp] theorem cos_add_pi (x : ℝ) : cos (x + π) = -cos x := cos_antiperiodic x #align real.cos_add_pi Real.cos_add_pi @[simp] theorem cos_add_two_pi (x : ℝ) : cos (x + 2 * π) = cos x := cos_periodic x #align real.cos_add_two_pi Real.cos_add_two_pi @[simp] theorem cos_sub_pi (x : ℝ) : cos (x - π) = -cos x := cos_antiperiodic.sub_eq x #align real.cos_sub_pi Real.cos_sub_pi @[simp] theorem cos_sub_two_pi (x : ℝ) : cos (x - 2 * π) = cos x := cos_periodic.sub_eq x #align real.cos_sub_two_pi Real.cos_sub_two_pi @[simp] theorem cos_pi_sub (x : ℝ) : cos (π - x) = -cos x := cos_neg x ▸ cos_antiperiodic.sub_eq' #align real.cos_pi_sub Real.cos_pi_sub @[simp] theorem cos_two_pi_sub (x : ℝ) : cos (2 * π - x) = cos x := cos_neg x ▸ cos_periodic.sub_eq' #align real.cos_two_pi_sub Real.cos_two_pi_sub @[simp] theorem cos_nat_mul_two_pi (n : ℕ) : cos (n * (2 * π)) = 1 := (cos_periodic.nat_mul_eq n).trans cos_zero #align real.cos_nat_mul_two_pi Real.cos_nat_mul_two_pi @[simp] theorem cos_int_mul_two_pi (n : ℤ) : cos (n * (2 * π)) = 1 := (cos_periodic.int_mul_eq n).trans cos_zero #align real.cos_int_mul_two_pi Real.cos_int_mul_two_pi @[simp] theorem cos_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x + n * (2 * π)) = cos x := cos_periodic.nat_mul n x #align real.cos_add_nat_mul_two_pi Real.cos_add_nat_mul_two_pi @[simp] theorem cos_add_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x + n * (2 * π)) = cos x := cos_periodic.int_mul n x #align real.cos_add_int_mul_two_pi Real.cos_add_int_mul_two_pi @[simp] theorem cos_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x - n * (2 * π)) = cos x := cos_periodic.sub_nat_mul_eq n #align real.cos_sub_nat_mul_two_pi Real.cos_sub_nat_mul_two_pi @[simp] theorem cos_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x - n * (2 * π)) = cos x := cos_periodic.sub_int_mul_eq n #align real.cos_sub_int_mul_two_pi Real.cos_sub_int_mul_two_pi @[simp] theorem cos_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : cos (n * (2 * π) - x) = cos x := cos_neg x ▸ cos_periodic.nat_mul_sub_eq n #align real.cos_nat_mul_two_pi_sub Real.cos_nat_mul_two_pi_sub @[simp] theorem cos_int_mul_two_pi_sub (x : ℝ) (n : ℤ) : cos (n * (2 * π) - x) = cos x := cos_neg x ▸ cos_periodic.int_mul_sub_eq n #align real.cos_int_mul_two_pi_sub Real.cos_int_mul_two_pi_sub theorem cos_add_int_mul_pi (x : ℝ) (n : ℤ) : cos (x + n * π) = (-1) ^ n * cos x := n.coe_negOnePow ℝ ▸ cos_antiperiodic.add_int_mul_eq n theorem cos_add_nat_mul_pi (x : ℝ) (n : ℕ) : cos (x + n * π) = (-1) ^ n * cos x := cos_antiperiodic.add_nat_mul_eq n theorem cos_sub_int_mul_pi (x : ℝ) (n : ℤ) : cos (x - n * π) = (-1) ^ n * cos x := n.coe_negOnePow ℝ ▸ cos_antiperiodic.sub_int_mul_eq n theorem cos_sub_nat_mul_pi (x : ℝ) (n : ℕ) : cos (x - n * π) = (-1) ^ n * cos x := cos_antiperiodic.sub_nat_mul_eq n theorem cos_int_mul_pi_sub (x : ℝ) (n : ℤ) : cos (n * π - x) = (-1) ^ n * cos x := n.coe_negOnePow ℝ ▸ cos_neg x ▸ cos_antiperiodic.int_mul_sub_eq n theorem cos_nat_mul_pi_sub (x : ℝ) (n : ℕ) : cos (n * π - x) = (-1) ^ n * cos x := cos_neg x ▸ cos_antiperiodic.nat_mul_sub_eq n -- Porting note (#10618): was @[simp], but simp can prove it theorem cos_nat_mul_two_pi_add_pi (n : ℕ) : cos (n * (2 * π) + π) = -1 := by simpa only [cos_zero] using (cos_periodic.nat_mul n).add_antiperiod_eq cos_antiperiodic #align real.cos_nat_mul_two_pi_add_pi Real.cos_nat_mul_two_pi_add_pi -- Porting note (#10618): was @[simp], but simp can prove it theorem cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 := by simpa only [cos_zero] using (cos_periodic.int_mul n).add_antiperiod_eq cos_antiperiodic #align real.cos_int_mul_two_pi_add_pi Real.cos_int_mul_two_pi_add_pi -- Porting note (#10618): was @[simp], but simp can prove it theorem cos_nat_mul_two_pi_sub_pi (n : ℕ) : cos (n * (2 * π) - π) = -1 := by simpa only [cos_zero] using (cos_periodic.nat_mul n).sub_antiperiod_eq cos_antiperiodic #align real.cos_nat_mul_two_pi_sub_pi Real.cos_nat_mul_two_pi_sub_pi -- Porting note (#10618): was @[simp], but simp can prove it theorem cos_int_mul_two_pi_sub_pi (n : ℤ) : cos (n * (2 * π) - π) = -1 := by simpa only [cos_zero] using (cos_periodic.int_mul n).sub_antiperiod_eq cos_antiperiodic #align real.cos_int_mul_two_pi_sub_pi Real.cos_int_mul_two_pi_sub_pi theorem sin_pos_of_pos_of_lt_pi {x : ℝ} (h0x : 0 < x) (hxp : x < π) : 0 < sin x := if hx2 : x ≤ 2 then sin_pos_of_pos_of_le_two h0x hx2 else have : (2 : ℝ) + 2 = 4 := by norm_num have : π - x ≤ 2 := sub_le_iff_le_add.2 (le_trans pi_le_four (this ▸ add_le_add_left (le_of_not_ge hx2) _)) sin_pi_sub x ▸ sin_pos_of_pos_of_le_two (sub_pos.2 hxp) this #align real.sin_pos_of_pos_of_lt_pi Real.sin_pos_of_pos_of_lt_pi theorem sin_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo 0 π) : 0 < sin x := sin_pos_of_pos_of_lt_pi hx.1 hx.2 #align real.sin_pos_of_mem_Ioo Real.sin_pos_of_mem_Ioo theorem sin_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc 0 π) : 0 ≤ sin x := by rw [← closure_Ioo pi_ne_zero.symm] at hx exact closure_lt_subset_le continuous_const continuous_sin (closure_mono (fun y => sin_pos_of_mem_Ioo) hx) #align real.sin_nonneg_of_mem_Icc Real.sin_nonneg_of_mem_Icc theorem sin_nonneg_of_nonneg_of_le_pi {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π) : 0 ≤ sin x := sin_nonneg_of_mem_Icc ⟨h0x, hxp⟩ #align real.sin_nonneg_of_nonneg_of_le_pi Real.sin_nonneg_of_nonneg_of_le_pi theorem sin_neg_of_neg_of_neg_pi_lt {x : ℝ} (hx0 : x < 0) (hpx : -π < x) : sin x < 0 := neg_pos.1 <\| sin_neg x ▸ sin_pos_of_pos_of_lt_pi (neg_pos.2 hx0) (neg_lt.1 hpx) #align real.sin_neg_of_neg_of_neg_pi_lt Real.sin_neg_of_neg_of_neg_pi_lt theorem sin_nonpos_of_nonnpos_of_neg_pi_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -π ≤ x) : sin x ≤ 0 := neg_nonneg.1 <\| sin_neg x ▸ sin_nonneg_of_nonneg_of_le_pi (neg_nonneg.2 hx0) (neg_le.1 hpx) #align real.sin_nonpos_of_nonnpos_of_neg_pi_le Real.sin_nonpos_of_nonnpos_of_neg_pi_le @[simp] theorem sin_pi_div_two : sin (π / 2) = 1 := have : sin (π / 2) = 1 ∨ sin (π / 2) = -1 := by simpa [sq, mul_self_eq_one_iff] using sin_sq_add_cos_sq (π / 2) this.resolve_right fun h => show ¬(0 : ℝ) < -1 by norm_num <\| h ▸ sin_pos_of_pos_of_lt_pi pi_div_two_pos (half_lt_self pi_pos) #align real.sin_pi_div_two Real.sin_pi_div_two theorem sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x := by simp [sin_add] #align real.sin_add_pi_div_two Real.sin_add_pi_div_two theorem sin_sub_pi_div_two (x : ℝ) : sin (x - π / 2) = -cos x := by simp [sub_eq_add_neg, sin_add] #align real.sin_sub_pi_div_two Real.sin_sub_pi_div_two theorem sin_pi_div_two_sub (x : ℝ) : sin (π / 2 - x) = cos x := by simp [sub_eq_add_neg, sin_add] #align real.sin_pi_div_two_sub Real.sin_pi_div_two_sub theorem cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x := by simp [cos_add] #align real.cos_add_pi_div_two Real.cos_add_pi_div_two theorem cos_sub_pi_div_two (x : ℝ) : cos (x - π / 2) = sin x := by simp [sub_eq_add_neg, cos_add] #align real.cos_sub_pi_div_two Real.cos_sub_pi_div_two theorem cos_pi_div_two_sub (x : ℝ) : cos (π / 2 - x) = sin x := by rw [← cos_neg, neg_sub, cos_sub_pi_div_two] #align real.cos_pi_div_two_sub Real.cos_pi_div_two_sub theorem cos_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo (-(π / 2)) (π / 2)) : 0 < cos x := sin_add_pi_div_two x ▸ sin_pos_of_mem_Ioo ⟨by linarith [hx.1], by linarith [hx.2]⟩ #align real.cos_pos_of_mem_Ioo Real.cos_pos_of_mem_Ioo theorem cos_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : 0 ≤ cos x := sin_add_pi_div_two x ▸ sin_nonneg_of_mem_Icc ⟨by linarith [hx.1], by linarith [hx.2]⟩ #align real.cos_nonneg_of_mem_Icc Real.cos_nonneg_of_mem_Icc theorem cos_nonneg_of_neg_pi_div_two_le_of_le {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) : 0 ≤ cos x := cos_nonneg_of_mem_Icc ⟨hl, hu⟩ #align real.cos_nonneg_of_neg_pi_div_two_le_of_le Real.cos_nonneg_of_neg_pi_div_two_le_of_le theorem cos_neg_of_pi_div_two_lt_of_lt {x : ℝ} (hx₁ : π / 2 < x) (hx₂ : x < π + π / 2) : cos x < 0 := neg_pos.1 <\| cos_pi_sub x ▸ cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩ #align real.cos_neg_of_pi_div_two_lt_of_lt Real.cos_neg_of_pi_div_two_lt_of_lt theorem cos_nonpos_of_pi_div_two_le_of_le {x : ℝ} (hx₁ : π / 2 ≤ x) (hx₂ : x ≤ π + π / 2) : cos x ≤ 0 := neg_nonneg.1 <\| cos_pi_sub x ▸ cos_nonneg_of_mem_Icc ⟨by linarith, by linarith⟩ #align real.cos_nonpos_of_pi_div_two_le_of_le Real.cos_nonpos_of_pi_div_two_le_of_le theorem sin_eq_sqrt_one_sub_cos_sq {x : ℝ} (hl : 0 ≤ x) (hu : x ≤ π) : sin x = √(1 - cos x ^ 2) := by rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)] #align real.sin_eq_sqrt_one_sub_cos_sq Real.sin_eq_sqrt_one_sub_cos_sq theorem cos_eq_sqrt_one_sub_sin_sq {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) : cos x = √(1 - sin x ^ 2) := by rw [← abs_cos_eq_sqrt_one_sub_sin_sq, abs_of_nonneg (cos_nonneg_of_mem_Icc ⟨hl, hu⟩)] #align real.cos_eq_sqrt_one_sub_sin_sq Real.cos_eq_sqrt_one_sub_sin_sq lemma cos_half {x : ℝ} (hl : -π ≤ x) (hr : x ≤ π) : cos (x / 2) = sqrt ((1 + cos x) / 2) := by have : 0 ≤ cos (x / 2) := cos_nonneg_of_mem_Icc <\| by constructor <;> linarith rw [← sqrt_sq this, cos_sq, add_div, two_mul, add_halves] lemma abs_sin_half (x : ℝ) : \|sin (x / 2)\| = sqrt ((1 - cos x) / 2) := by rw [← sqrt_sq_eq_abs, sin_sq_eq_half_sub, two_mul, add_halves, sub_div] lemma sin_half_eq_sqrt {x : ℝ} (hl : 0 ≤ x) (hr : x ≤ 2 * π) : sin (x / 2) = sqrt ((1 - cos x) / 2) := by rw [← abs_sin_half, abs_of_nonneg] apply sin_nonneg_of_nonneg_of_le_pi <;> linarith lemma sin_half_eq_neg_sqrt {x : ℝ} (hl : -(2 * π) ≤ x) (hr : x ≤ 0) : sin (x / 2) = -sqrt ((1 - cos x) / 2) := by rw [← abs_sin_half, abs_of_nonpos, neg_neg] apply sin_nonpos_of_nonnpos_of_neg_pi_le <;> linarith theorem sin_eq_zero_iff_of_lt_of_lt {x : ℝ} (hx₁ : -π < x) (hx₂ : x < π) : sin x = 0 ↔ x = 0 := ⟨fun h => by contrapose! h cases h.lt_or_lt with \| inl h0 => exact (sin_neg_of_neg_of_neg_pi_lt h0 hx₁).ne \| inr h0 => exact (sin_pos_of_pos_of_lt_pi h0 hx₂).ne', fun h => by simp [h]⟩ #align real.sin_eq_zero_iff_of_lt_of_lt Real.sin_eq_zero_iff_of_lt_of_lt theorem sin_eq_zero_iff {x : ℝ} : sin x = 0 ↔ ∃ n : ℤ, (n : ℝ) * π = x := ⟨fun h => ⟨⌊x / π⌋, le_antisymm (sub_nonneg.1 (Int.sub_floor_div_mul_nonneg _ pi_pos)) (sub_nonpos.1 <\| le_of_not_gt fun h₃ => (sin_pos_of_pos_of_lt_pi h₃ (Int.sub_floor_div_mul_lt _ pi_pos)).ne (by simp [sub_eq_add_neg, sin_add, h, sin_int_mul_pi]))⟩, fun ⟨n, hn⟩ => hn ▸ sin_int_mul_pi _⟩ #align real.sin_eq_zero_iff Real.sin_eq_zero_iff theorem sin_ne_zero_iff {x : ℝ} : sin x ≠ 0 ↔ ∀ n : ℤ, (n : ℝ) * π ≠ x := by rw [← not_exists, not_iff_not, sin_eq_zero_iff] #align real.sin_ne_zero_iff Real.sin_ne_zero_iff theorem sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 := by rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq x, sq, sq, ← sub_eq_iff_eq_add, sub_self] exact ⟨fun h => by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ Eq.symm⟩ #align real.sin_eq_zero_iff_cos_eq Real.sin_eq_zero_iff_cos_eq theorem cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * π) = x := ⟨fun h => let ⟨n, hn⟩ := sin_eq_zero_iff.1 (sin_eq_zero_iff_cos_eq.2 (Or.inl h)) ⟨n / 2, (Int.emod_two_eq_zero_or_one n).elim (fun hn0 => by rwa [← mul_assoc, ← @Int.cast_two ℝ, ← Int.cast_mul, Int.ediv_mul_cancel ((Int.dvd_iff_emod_eq_zero _ _).2 hn0)]) fun hn1 => by rw [← Int.emod_add_ediv n 2, hn1, Int.cast_add, Int.cast_one, add_mul, one_mul, add_comm, mul_comm (2 : ℤ), Int.cast_mul, mul_assoc, Int.cast_two] at hn rw [← hn, cos_int_mul_two_pi_add_pi] at h exact absurd h (by norm_num)⟩, fun ⟨n, hn⟩ => hn ▸ cos_int_mul_two_pi _⟩ #align real.cos_eq_one_iff Real.cos_eq_one_iff theorem cos_eq_one_iff_of_lt_of_lt {x : ℝ} (hx₁ : -(2 * π) < x) (hx₂ : x < 2 * π) : cos x = 1 ↔ x = 0 := ⟨fun h => by rcases (cos_eq_one_iff _).1 h with ⟨n, rfl⟩ rw [mul_lt_iff_lt_one_left two_pi_pos] at hx₂ rw [neg_lt, neg_mul_eq_neg_mul, mul_lt_iff_lt_one_left two_pi_pos] at hx₁ norm_cast at hx₁ hx₂ obtain rfl : n = 0 := le_antisymm (by omega) (by omega) simp, fun h => by simp [h]⟩ #align real.cos_eq_one_iff_of_lt_of_lt Real.cos_eq_one_iff_of_lt_of_lt theorem sin_lt_sin_of_lt_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hy₂ : y ≤ π / 2) (hxy : x < y) : sin x < sin y := by rw [← sub_pos, sin_sub_sin] have : 0 < sin ((y - x) / 2) := by apply sin_pos_of_pos_of_lt_pi <;> linarith have : 0 < cos ((y + x) / 2) := by refine cos_pos_of_mem_Ioo ⟨?_, ?_⟩ <;> linarith positivity #align real.sin_lt_sin_of_lt_of_le_pi_div_two Real.sin_lt_sin_of_lt_of_le_pi_div_two theorem strictMonoOn_sin : StrictMonoOn sin (Icc (-(π / 2)) (π / 2)) := fun _ hx _ hy hxy => sin_lt_sin_of_lt_of_le_pi_div_two hx.1 hy.2 hxy #align real.strict_mono_on_sin Real.strictMonoOn_sin theorem cos_lt_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x < y) : cos y < cos x := by rw [← sin_pi_div_two_sub, ← sin_pi_div_two_sub] apply sin_lt_sin_of_lt_of_le_pi_div_two <;> linarith #align real.cos_lt_cos_of_nonneg_of_le_pi Real.cos_lt_cos_of_nonneg_of_le_pi theorem cos_lt_cos_of_nonneg_of_le_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π / 2) (hxy : x < y) : cos y < cos x := cos_lt_cos_of_nonneg_of_le_pi hx₁ (hy₂.trans (by linarith)) hxy #align real.cos_lt_cos_of_nonneg_of_le_pi_div_two Real.cos_lt_cos_of_nonneg_of_le_pi_div_two theorem strictAntiOn_cos : StrictAntiOn cos (Icc 0 π) := fun _ hx _ hy hxy => cos_lt_cos_of_nonneg_of_le_pi hx.1 hy.2 hxy #align real.strict_anti_on_cos Real.strictAntiOn_cos theorem cos_le_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x ≤ y) : cos y ≤ cos x := (strictAntiOn_cos.le_iff_le ⟨hx₁.trans hxy, hy₂⟩ ⟨hx₁, hxy.trans hy₂⟩).2 hxy #align real.cos_le_cos_of_nonneg_of_le_pi Real.cos_le_cos_of_nonneg_of_le_pi theorem sin_le_sin_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hy₂ : y ≤ π / 2) (hxy : x ≤ y) : sin x ≤ sin y := (strictMonoOn_sin.le_iff_le ⟨hx₁, hxy.trans hy₂⟩ ⟨hx₁.trans hxy, hy₂⟩).2 hxy #align real.sin_le_sin_of_le_of_le_pi_div_two Real.sin_le_sin_of_le_of_le_pi_div_two theorem injOn_sin : InjOn sin (Icc (-(π / 2)) (π / 2)) := strictMonoOn_sin.injOn #align real.inj_on_sin Real.injOn_sin theorem injOn_cos : InjOn cos (Icc 0 π) := strictAntiOn_cos.injOn #align real.inj_on_cos Real.injOn_cos theorem surjOn_sin : SurjOn sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) := by simpa only [sin_neg, sin_pi_div_two] using intermediate_value_Icc (neg_le_self pi_div_two_pos.le) continuous_sin.continuousOn #align real.surj_on_sin Real.surjOn_sin theorem surjOn_cos : SurjOn cos (Icc 0 π) (Icc (-1) 1) := by simpa only [cos_zero, cos_pi] using intermediate_value_Icc' pi_pos.le continuous_cos.continuousOn #align real.surj_on_cos Real.surjOn_cos theorem sin_mem_Icc (x : ℝ) : sin x ∈ Icc (-1 : ℝ) 1 := ⟨neg_one_le_sin x, sin_le_one x⟩ #align real.sin_mem_Icc Real.sin_mem_Icc theorem cos_mem_Icc (x : ℝ) : cos x ∈ Icc (-1 : ℝ) 1 := ⟨neg_one_le_cos x, cos_le_one x⟩ #align real.cos_mem_Icc Real.cos_mem_Icc theorem mapsTo_sin (s : Set ℝ) : MapsTo sin s (Icc (-1 : ℝ) 1) := fun x _ => sin_mem_Icc x #align real.maps_to_sin Real.mapsTo_sin theorem mapsTo_cos (s : Set ℝ) : MapsTo cos s (Icc (-1 : ℝ) 1) := fun x _ => cos_mem_Icc x #align real.maps_to_cos Real.mapsTo_cos theorem bijOn_sin : BijOn sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) := ⟨mapsTo_sin _, injOn_sin, surjOn_sin⟩ #align real.bij_on_sin Real.bijOn_sin theorem bijOn_cos : BijOn cos (Icc 0 π) (Icc (-1) 1) := ⟨mapsTo_cos _, injOn_cos, surjOn_cos⟩ #align real.bij_on_cos Real.bijOn_cos @[simp] theorem range_cos : range cos = (Icc (-1) 1 : Set ℝ) := Subset.antisymm (range_subset_iff.2 cos_mem_Icc) surjOn_cos.subset_range #align real.range_cos Real.range_cos @[simp] theorem range_sin : range sin = (Icc (-1) 1 : Set ℝ) := Subset.antisymm (range_subset_iff.2 sin_mem_Icc) surjOn_sin.subset_range #align real.range_sin Real.range_sin theorem range_cos_infinite : (range Real.cos).Infinite := by rw [Real.range_cos] exact Icc_infinite (by norm_num) #align real.range_cos_infinite Real.range_cos_infinite theorem range_sin_infinite : (range Real.sin).Infinite := by rw [Real.range_sin] exact Icc_infinite (by norm_num) #align real.range_sin_infinite Real.range_sin_infinite namespace Complex open Real theorem sin_eq_zero_iff_cos_eq {z : ℂ} : sin z = 0 ↔ cos z = 1 ∨ cos z = -1 := by rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq, sq, sq, ← sub_eq_iff_eq_add, sub_self] exact ⟨fun h => by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ Eq.symm⟩ #align complex.sin_eq_zero_iff_cos_eq Complex.sin_eq_zero_iff_cos_eq @[simp] theorem cos_pi_div_two : cos (π / 2) = 0 := calc cos (π / 2) = Real.cos (π / 2) := by rw [ofReal_cos]; simp _ = 0 := by simp #align complex.cos_pi_div_two Complex.cos_pi_div_two @[simp] theorem sin_pi_div_two : sin (π / 2) = 1 := calc sin (π / 2) = Real.sin (π / 2) := by rw [ofReal_sin]; simp _ = 1 := by simp #align complex.sin_pi_div_two Complex.sin_pi_div_two @[simp] theorem sin_pi : sin π = 0 := by rw [← ofReal_sin, Real.sin_pi]; simp #align complex.sin_pi Complex.sin_pi @[simp] theorem cos_pi : cos π = -1 := by rw [← ofReal_cos, Real.cos_pi]; simp #align complex.cos_pi Complex.cos_pi @[simp] theorem sin_two_pi : sin (2 * π) = 0 := by simp [two_mul, sin_add] #align complex.sin_two_pi Complex.sin_two_pi @[simp] theorem cos_two_pi : cos (2 * π) = 1 := by simp [two_mul, cos_add] #align complex.cos_two_pi Complex.cos_two_pi theorem sin_antiperiodic : Function.Antiperiodic sin π := by simp [sin_add] #align complex.sin_antiperiodic Complex.sin_antiperiodic theorem sin_periodic : Function.Periodic sin (2 * π) := sin_antiperiodic.periodic_two_mul #align complex.sin_periodic Complex.sin_periodic theorem sin_add_pi (x : ℂ) : sin (x + π) = -sin x := sin_antiperiodic x #align complex.sin_add_pi Complex.sin_add_pi theorem sin_add_two_pi (x : ℂ) : sin (x + 2 * π) = sin x := sin_periodic x #align complex.sin_add_two_pi Complex.sin_add_two_pi theorem sin_sub_pi (x : ℂ) : sin (x - π) = -sin x := sin_antiperiodic.sub_eq x #align complex.sin_sub_pi Complex.sin_sub_pi theorem sin_sub_two_pi (x : ℂ) : sin (x - 2 * π) = sin x := sin_periodic.sub_eq x #align complex.sin_sub_two_pi Complex.sin_sub_two_pi theorem sin_pi_sub (x : ℂ) : sin (π - x) = sin x := neg_neg (sin x) ▸ sin_neg x ▸ sin_antiperiodic.sub_eq' #align complex.sin_pi_sub Complex.sin_pi_sub theorem sin_two_pi_sub (x : ℂ) : sin (2 * π - x) = -sin x := sin_neg x ▸ sin_periodic.sub_eq' #align complex.sin_two_pi_sub Complex.sin_two_pi_sub theorem sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 := sin_antiperiodic.nat_mul_eq_of_eq_zero sin_zero n #align complex.sin_nat_mul_pi Complex.sin_nat_mul_pi theorem sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 := sin_antiperiodic.int_mul_eq_of_eq_zero sin_zero n #align complex.sin_int_mul_pi Complex.sin_int_mul_pi theorem sin_add_nat_mul_two_pi (x : ℂ) (n : ℕ) : sin (x + n * (2 * π)) = sin x := sin_periodic.nat_mul n x #align complex.sin_add_nat_mul_two_pi Complex.sin_add_nat_mul_two_pi theorem sin_add_int_mul_two_pi (x : ℂ) (n : ℤ) : sin (x + n * (2 * π)) = sin x := sin_periodic.int_mul n x #align complex.sin_add_int_mul_two_pi Complex.sin_add_int_mul_two_pi theorem sin_sub_nat_mul_two_pi (x : ℂ) (n : ℕ) : sin (x - n * (2 * π)) = sin x := sin_periodic.sub_nat_mul_eq n #align complex.sin_sub_nat_mul_two_pi Complex.sin_sub_nat_mul_two_pi theorem sin_sub_int_mul_two_pi (x : ℂ) (n : ℤ) : sin (x - n * (2 * π)) = sin x := sin_periodic.sub_int_mul_eq n #align complex.sin_sub_int_mul_two_pi Complex.sin_sub_int_mul_two_pi theorem sin_nat_mul_two_pi_sub (x : ℂ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x := sin_neg x ▸ sin_periodic.nat_mul_sub_eq n #align complex.sin_nat_mul_two_pi_sub Complex.sin_nat_mul_two_pi_sub theorem sin_int_mul_two_pi_sub (x : ℂ) (n : ℤ) : sin (n * (2 * π) - x) = -sin x := sin_neg x ▸ sin_periodic.int_mul_sub_eq n #align complex.sin_int_mul_two_pi_sub Complex.sin_int_mul_two_pi_sub	Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean	1,217	1,217	theorem cos_antiperiodic : Function.Antiperiodic cos π := by	simp [cos_add]
import Mathlib.Logic.Equiv.Defs import Mathlib.Tactic.MkIffOfInductiveProp import Mathlib.Tactic.PPWithUniv #align_import logic.small.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105" universe u w v v' @[mk_iff, pp_with_univ] class Small (α : Type v) : Prop where equiv_small : ∃ S : Type w, Nonempty (α ≃ S) #align small Small theorem Small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : Small.{w} α := ⟨⟨S, ⟨e⟩⟩⟩ #align small.mk' Small.mk' @[pp_with_univ] def Shrink (α : Type v) [Small.{w} α] : Type w := Classical.choose (@Small.equiv_small α _) #align shrink Shrink noncomputable def equivShrink (α : Type v) [Small.{w} α] : α ≃ Shrink α := Nonempty.some (Classical.choose_spec (@Small.equiv_small α _)) #align equiv_shrink equivShrink @[ext]	Mathlib/Logic/Small/Defs.lean	56	58	theorem Shrink.ext {α : Type v} [Small.{w} α] {x y : Shrink α} (w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y := by	simpa using w
import Mathlib.Algebra.Algebra.Basic import Mathlib.Algebra.Periodic import Mathlib.Topology.Algebra.Order.Field import Mathlib.Topology.Algebra.UniformMulAction import Mathlib.Topology.Algebra.Star import Mathlib.Topology.Instances.Int import Mathlib.Topology.Order.Bornology #align_import topology.instances.real from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd" noncomputable section open scoped Classical open Filter Int Metric Set TopologicalSpace Bornology open scoped Topology Uniformity Interval universe u v w variable {α : Type u} {β : Type v} {γ : Type w} instance : NoncompactSpace ℝ := Int.closedEmbedding_coe_real.noncompactSpace theorem Real.uniformContinuous_add : UniformContinuous fun p : ℝ × ℝ => p.1 + p.2 := Metric.uniformContinuous_iff.2 fun _ε ε0 => let ⟨δ, δ0, Hδ⟩ := rat_add_continuous_lemma abs ε0 ⟨δ, δ0, fun h => let ⟨h₁, h₂⟩ := max_lt_iff.1 h Hδ h₁ h₂⟩ #align real.uniform_continuous_add Real.uniformContinuous_add theorem Real.uniformContinuous_neg : UniformContinuous (@Neg.neg ℝ _) := Metric.uniformContinuous_iff.2 fun ε ε0 => ⟨_, ε0, fun h => by rw [dist_comm] at h; simpa only [Real.dist_eq, neg_sub_neg] using h⟩ #align real.uniform_continuous_neg Real.uniformContinuous_neg instance : ContinuousStar ℝ := ⟨continuous_id⟩ instance : UniformAddGroup ℝ := UniformAddGroup.mk' Real.uniformContinuous_add Real.uniformContinuous_neg -- short-circuit type class inference instance : TopologicalAddGroup ℝ := by infer_instance instance : TopologicalRing ℝ := inferInstance instance : TopologicalDivisionRing ℝ := inferInstance instance : ProperSpace ℝ where isCompact_closedBall x r := by rw [Real.closedBall_eq_Icc] apply isCompact_Icc instance : SecondCountableTopology ℝ := secondCountable_of_proper theorem Real.isTopologicalBasis_Ioo_rat : @IsTopologicalBasis ℝ _ (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) b}) := isTopologicalBasis_of_isOpen_of_nhds (by simp (config := { contextual := true }) [isOpen_Ioo]) fun a v hav hv => let ⟨l, u, ⟨hl, hu⟩, h⟩ := mem_nhds_iff_exists_Ioo_subset.mp (IsOpen.mem_nhds hv hav) let ⟨q, hlq, hqa⟩ := exists_rat_btwn hl let ⟨p, hap, hpu⟩ := exists_rat_btwn hu ⟨Ioo q p, by simp only [mem_iUnion] exact ⟨q, p, Rat.cast_lt.1 <\| hqa.trans hap, rfl⟩, ⟨hqa, hap⟩, fun a' ⟨hqa', ha'p⟩ => h ⟨hlq.trans hqa', ha'p.trans hpu⟩⟩ #align real.is_topological_basis_Ioo_rat Real.isTopologicalBasis_Ioo_rat @[simp] theorem Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop := by simp only [← comap_dist_right_atTop (0 : ℝ), Real.dist_eq, sub_zero, comap_abs_atTop] @[deprecated] alias Real.cocompact_eq := cocompact_eq_atBot_atTop #align real.cocompact_eq Real.cocompact_eq @[deprecated (since := "2024-02-07")] alias Real.atBot_le_cocompact := atBot_le_cocompact @[deprecated (since := "2024-02-07")] alias Real.atTop_le_cocompact := atTop_le_cocompact	Mathlib/Topology/Instances/Real.lean	92	94	theorem Real.mem_closure_iff {s : Set ℝ} {x : ℝ} : x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, \|y - x\| < ε := by	simp [mem_closure_iff_nhds_basis nhds_basis_ball, Real.dist_eq]
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho import Mathlib.LinearAlgebra.Orientation #align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163" noncomputable section variable {E : Type} [NormedAddCommGroup E] [InnerProductSpace ℝ E] open FiniteDimensional open scoped RealInnerProductSpace namespace OrthonormalBasis variable {ι : Type} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι) theorem det_to_matrix_orthonormalBasis_of_same_orientation (h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right have : 0 < e.toBasis.det f := by rw [e.toBasis.orientation_eq_iff_det_pos] at h simpa using h linarith #align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation theorem det_to_matrix_orthonormalBasis_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by contrapose! h simp [e.toBasis.orientation_eq_iff_det_pos, (e.det_to_matrix_orthonormalBasis_real f).resolve_right h] #align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation variable {e f}	Mathlib/Analysis/InnerProductSpace/Orientation.lean	76	84	theorem same_orientation_iff_det_eq_det : e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by	constructor · intro h dsimp [Basis.orientation] congr · intro h rw [e.toBasis.det.eq_smul_basis_det f.toBasis] simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
import Mathlib.CategoryTheory.Sites.Sieves #align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" universe w v₁ v₂ u₁ u₂ namespace CategoryTheory open Opposite CategoryTheory Category Limits Sieve namespace Presieve variable {C : Type u₁} [Category.{v₁} C] variable {P Q U : Cᵒᵖ ⥤ Type w} variable {X Y : C} {S : Sieve X} {R : Presieve X} def FamilyOfElements (P : Cᵒᵖ ⥤ Type w) (R : Presieve X) := ∀ ⦃Y : C⦄ (f : Y ⟶ X), R f → P.obj (op Y) #align category_theory.presieve.family_of_elements CategoryTheory.Presieve.FamilyOfElements instance : Inhabited (FamilyOfElements P (⊥ : Presieve X)) := ⟨fun _ _ => False.elim⟩ def FamilyOfElements.restrict {R₁ R₂ : Presieve X} (h : R₁ ≤ R₂) : FamilyOfElements P R₂ → FamilyOfElements P R₁ := fun x _ f hf => x f (h _ hf) #align category_theory.presieve.family_of_elements.restrict CategoryTheory.Presieve.FamilyOfElements.restrict def FamilyOfElements.map (p : FamilyOfElements P R) (φ : P ⟶ Q) : FamilyOfElements Q R := fun _ f hf => φ.app _ (p f hf) @[simp] lemma FamilyOfElements.map_apply (p : FamilyOfElements P R) (φ : P ⟶ Q) {Y : C} (f : Y ⟶ X) (hf : R f) : p.map φ f hf = φ.app _ (p f hf) := rfl lemma FamilyOfElements.restrict_map (p : FamilyOfElements P R) (φ : P ⟶ Q) {R' : Presieve X} (h : R' ≤ R) : (p.restrict h).map φ = (p.map φ).restrict h := rfl def FamilyOfElements.Compatible (x : FamilyOfElements P R) : Prop := ∀ ⦃Y₁ Y₂ Z⦄ (g₁ : Z ⟶ Y₁) (g₂ : Z ⟶ Y₂) ⦃f₁ : Y₁ ⟶ X⦄ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂), g₁ ≫ f₁ = g₂ ≫ f₂ → P.map g₁.op (x f₁ h₁) = P.map g₂.op (x f₂ h₂) #align category_theory.presieve.family_of_elements.compatible CategoryTheory.Presieve.FamilyOfElements.Compatible def FamilyOfElements.PullbackCompatible (x : FamilyOfElements P R) [R.hasPullbacks] : Prop := ∀ ⦃Y₁ Y₂⦄ ⦃f₁ : Y₁ ⟶ X⦄ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂), haveI := hasPullbacks.has_pullbacks h₁ h₂ P.map (pullback.fst : Limits.pullback f₁ f₂ ⟶ _).op (x f₁ h₁) = P.map pullback.snd.op (x f₂ h₂) #align category_theory.presieve.family_of_elements.pullback_compatible CategoryTheory.Presieve.FamilyOfElements.PullbackCompatible theorem pullbackCompatible_iff (x : FamilyOfElements P R) [R.hasPullbacks] : x.Compatible ↔ x.PullbackCompatible := by constructor · intro t Y₁ Y₂ f₁ f₂ hf₁ hf₂ apply t haveI := hasPullbacks.has_pullbacks hf₁ hf₂ apply pullback.condition · intro t Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ comm haveI := hasPullbacks.has_pullbacks hf₁ hf₂ rw [← pullback.lift_fst _ _ comm, op_comp, FunctorToTypes.map_comp_apply, t hf₁ hf₂, ← FunctorToTypes.map_comp_apply, ← op_comp, pullback.lift_snd] #align category_theory.presieve.pullback_compatible_iff CategoryTheory.Presieve.pullbackCompatible_iff theorem FamilyOfElements.Compatible.restrict {R₁ R₂ : Presieve X} (h : R₁ ≤ R₂) {x : FamilyOfElements P R₂} : x.Compatible → (x.restrict h).Compatible := fun q _ _ _ g₁ g₂ _ _ h₁ h₂ comm => q g₁ g₂ (h _ h₁) (h _ h₂) comm #align category_theory.presieve.family_of_elements.compatible.restrict CategoryTheory.Presieve.FamilyOfElements.Compatible.restrict noncomputable def FamilyOfElements.sieveExtend (x : FamilyOfElements P R) : FamilyOfElements P (generate R : Presieve X) := fun _ _ hf => P.map hf.choose_spec.choose.op (x _ hf.choose_spec.choose_spec.choose_spec.1) #align category_theory.presieve.family_of_elements.sieve_extend CategoryTheory.Presieve.FamilyOfElements.sieveExtend theorem FamilyOfElements.Compatible.sieveExtend {x : FamilyOfElements P R} (hx : x.Compatible) : x.sieveExtend.Compatible := by intro _ _ _ _ _ _ _ h₁ h₂ comm iterate 2 erw [← FunctorToTypes.map_comp_apply]; rw [← op_comp] apply hx simp [comm, h₁.choose_spec.choose_spec.choose_spec.2, h₂.choose_spec.choose_spec.choose_spec.2] #align category_theory.presieve.family_of_elements.compatible.sieve_extend CategoryTheory.Presieve.FamilyOfElements.Compatible.sieveExtend theorem extend_agrees {x : FamilyOfElements P R} (t : x.Compatible) {f : Y ⟶ X} (hf : R f) : x.sieveExtend f (le_generate R Y hf) = x f hf := by have h := (le_generate R Y hf).choose_spec unfold FamilyOfElements.sieveExtend rw [t h.choose (𝟙 _) _ hf _] · simp · rw [id_comp] exact h.choose_spec.choose_spec.2 #align category_theory.presieve.extend_agrees CategoryTheory.Presieve.extend_agrees @[simp] theorem restrict_extend {x : FamilyOfElements P R} (t : x.Compatible) : x.sieveExtend.restrict (le_generate R) = x := by funext Y f hf exact extend_agrees t hf #align category_theory.presieve.restrict_extend CategoryTheory.Presieve.restrict_extend def FamilyOfElements.SieveCompatible (x : FamilyOfElements P (S : Presieve X)) : Prop := ∀ ⦃Y Z⦄ (f : Y ⟶ X) (g : Z ⟶ Y) (hf), x (g ≫ f) (S.downward_closed hf g) = P.map g.op (x f hf) #align category_theory.presieve.family_of_elements.sieve_compatible CategoryTheory.Presieve.FamilyOfElements.SieveCompatible theorem compatible_iff_sieveCompatible (x : FamilyOfElements P (S : Presieve X)) : x.Compatible ↔ x.SieveCompatible := by constructor · intro h Y Z f g hf simpa using h (𝟙 _) g (S.downward_closed hf g) hf (id_comp _) · intro h Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ k simp_rw [← h f₁ g₁ h₁, ← h f₂ g₂ h₂] congr #align category_theory.presieve.compatible_iff_sieve_compatible CategoryTheory.Presieve.compatible_iff_sieveCompatible theorem FamilyOfElements.Compatible.to_sieveCompatible {x : FamilyOfElements P (S : Presieve X)} (t : x.Compatible) : x.SieveCompatible := (compatible_iff_sieveCompatible x).1 t #align category_theory.presieve.family_of_elements.compatible.to_sieve_compatible CategoryTheory.Presieve.FamilyOfElements.Compatible.to_sieveCompatible @[simp] theorem extend_restrict {x : FamilyOfElements P (generate R)} (t : x.Compatible) : (x.restrict (le_generate R)).sieveExtend = x := by rw [compatible_iff_sieveCompatible] at t funext _ _ h apply (t _ _ _).symm.trans congr exact h.choose_spec.choose_spec.choose_spec.2 #align category_theory.presieve.extend_restrict CategoryTheory.Presieve.extend_restrict theorem restrict_inj {x₁ x₂ : FamilyOfElements P (generate R)} (t₁ : x₁.Compatible) (t₂ : x₂.Compatible) : x₁.restrict (le_generate R) = x₂.restrict (le_generate R) → x₁ = x₂ := fun h => by rw [← extend_restrict t₁, ← extend_restrict t₂] -- Porting note: congr fails to make progress apply congr_arg exact h #align category_theory.presieve.restrict_inj CategoryTheory.Presieve.restrict_inj @[simps] noncomputable def compatibleEquivGenerateSieveCompatible : { x : FamilyOfElements P R // x.Compatible } ≃ { x : FamilyOfElements P (generate R : Presieve X) // x.Compatible } where toFun x := ⟨x.1.sieveExtend, x.2.sieveExtend⟩ invFun x := ⟨x.1.restrict (le_generate R), x.2.restrict _⟩ left_inv x := Subtype.ext (restrict_extend x.2) right_inv x := Subtype.ext (extend_restrict x.2) #align category_theory.presieve.compatible_equiv_generate_sieve_compatible CategoryTheory.Presieve.compatibleEquivGenerateSieveCompatible theorem FamilyOfElements.comp_of_compatible (S : Sieve X) {x : FamilyOfElements P S} (t : x.Compatible) {f : Y ⟶ X} (hf : S f) {Z} (g : Z ⟶ Y) : x (g ≫ f) (S.downward_closed hf g) = P.map g.op (x f hf) := by simpa using t (𝟙 _) g (S.downward_closed hf g) hf (id_comp _) #align category_theory.presieve.family_of_elements.comp_of_compatible CategoryTheory.Presieve.FamilyOfElements.comp_of_compatible noncomputable def FamilyOfElements.functorPushforward {D : Type u₂} [Category.{v₂} D] (F : D ⥤ C) {X : D} {T : Presieve X} (x : FamilyOfElements (F.op ⋙ P) T) : FamilyOfElements P (T.functorPushforward F) := fun Y f h => by obtain ⟨Z, g, h, h₁, _⟩ := getFunctorPushforwardStructure h exact P.map h.op (x g h₁) #align category_theory.presieve.family_of_elements.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.functorPushforward def FamilyOfElements.compPresheafMap (f : P ⟶ Q) (x : FamilyOfElements P R) : FamilyOfElements Q R := fun Y g hg => f.app (op Y) (x g hg) #align category_theory.presieve.family_of_elements.comp_presheaf_map CategoryTheory.Presieve.FamilyOfElements.compPresheafMap @[simp] theorem FamilyOfElements.compPresheafMap_id (x : FamilyOfElements P R) : x.compPresheafMap (𝟙 P) = x := rfl #align category_theory.presieve.family_of_elements.comp_presheaf_map_id CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_id @[simp] theorem FamilyOfElements.compPresheafMap_comp (x : FamilyOfElements P R) (f : P ⟶ Q) (g : Q ⟶ U) : (x.compPresheafMap f).compPresheafMap g = x.compPresheafMap (f ≫ g) := rfl #align category_theory.presieve.family_of_elements.comp_prersheaf_map_comp CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_comp theorem FamilyOfElements.Compatible.compPresheafMap (f : P ⟶ Q) {x : FamilyOfElements P R} (h : x.Compatible) : (x.compPresheafMap f).Compatible := by intro Z₁ Z₂ W g₁ g₂ f₁ f₂ h₁ h₂ eq unfold FamilyOfElements.compPresheafMap rwa [← FunctorToTypes.naturality, ← FunctorToTypes.naturality, h] #align category_theory.presieve.family_of_elements.compatible.comp_presheaf_map CategoryTheory.Presieve.FamilyOfElements.Compatible.compPresheafMap def FamilyOfElements.IsAmalgamation (x : FamilyOfElements P R) (t : P.obj (op X)) : Prop := ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : R f), P.map f.op t = x f h #align category_theory.presieve.family_of_elements.is_amalgamation CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation theorem FamilyOfElements.IsAmalgamation.compPresheafMap {x : FamilyOfElements P R} {t} (f : P ⟶ Q) (h : x.IsAmalgamation t) : (x.compPresheafMap f).IsAmalgamation (f.app (op X) t) := by intro Y g hg dsimp [FamilyOfElements.compPresheafMap] change (f.app _ ≫ Q.map _) _ = _ rw [← f.naturality, types_comp_apply, h g hg] #align category_theory.presieve.family_of_elements.is_amalgamation.comp_presheaf_map CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.compPresheafMap theorem is_compatible_of_exists_amalgamation (x : FamilyOfElements P R) (h : ∃ t, x.IsAmalgamation t) : x.Compatible := by cases' h with t ht intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ comm rw [← ht _ h₁, ← ht _ h₂, ← FunctorToTypes.map_comp_apply, ← op_comp, comm] simp #align category_theory.presieve.is_compatible_of_exists_amalgamation CategoryTheory.Presieve.is_compatible_of_exists_amalgamation theorem isAmalgamation_restrict {R₁ R₂ : Presieve X} (h : R₁ ≤ R₂) (x : FamilyOfElements P R₂) (t : P.obj (op X)) (ht : x.IsAmalgamation t) : (x.restrict h).IsAmalgamation t := fun Y f hf => ht f (h Y hf) #align category_theory.presieve.is_amalgamation_restrict CategoryTheory.Presieve.isAmalgamation_restrict theorem isAmalgamation_sieveExtend {R : Presieve X} (x : FamilyOfElements P R) (t : P.obj (op X)) (ht : x.IsAmalgamation t) : x.sieveExtend.IsAmalgamation t := by intro Y f hf dsimp [FamilyOfElements.sieveExtend] rw [← ht _, ← FunctorToTypes.map_comp_apply, ← op_comp, hf.choose_spec.choose_spec.choose_spec.2] #align category_theory.presieve.is_amalgamation_sieve_extend CategoryTheory.Presieve.isAmalgamation_sieveExtend def IsSeparatedFor (P : Cᵒᵖ ⥤ Type w) (R : Presieve X) : Prop := ∀ (x : FamilyOfElements P R) (t₁ t₂), x.IsAmalgamation t₁ → x.IsAmalgamation t₂ → t₁ = t₂ #align category_theory.presieve.is_separated_for CategoryTheory.Presieve.IsSeparatedFor theorem IsSeparatedFor.ext {R : Presieve X} (hR : IsSeparatedFor P R) {t₁ t₂ : P.obj (op X)} (h : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (_ : R f), P.map f.op t₁ = P.map f.op t₂) : t₁ = t₂ := hR (fun _ f _ => P.map f.op t₂) t₁ t₂ (fun _ _ hf => h hf) fun _ _ _ => rfl #align category_theory.presieve.is_separated_for.ext CategoryTheory.Presieve.IsSeparatedFor.ext theorem isSeparatedFor_iff_generate : IsSeparatedFor P R ↔ IsSeparatedFor P (generate R : Presieve X) := by constructor · intro h x t₁ t₂ ht₁ ht₂ apply h (x.restrict (le_generate R)) t₁ t₂ _ _ · exact isAmalgamation_restrict _ x t₁ ht₁ · exact isAmalgamation_restrict _ x t₂ ht₂ · intro h x t₁ t₂ ht₁ ht₂ apply h x.sieveExtend · exact isAmalgamation_sieveExtend x t₁ ht₁ · exact isAmalgamation_sieveExtend x t₂ ht₂ #align category_theory.presieve.is_separated_for_iff_generate CategoryTheory.Presieve.isSeparatedFor_iff_generate theorem isSeparatedFor_top (P : Cᵒᵖ ⥤ Type w) : IsSeparatedFor P (⊤ : Presieve X) := fun x t₁ t₂ h₁ h₂ => by have q₁ := h₁ (𝟙 X) (by tauto) have q₂ := h₂ (𝟙 X) (by tauto) simp only [op_id, FunctorToTypes.map_id_apply] at q₁ q₂ rw [q₁, q₂] #align category_theory.presieve.is_separated_for_top CategoryTheory.Presieve.isSeparatedFor_top def IsSheafFor (P : Cᵒᵖ ⥤ Type w) (R : Presieve X) : Prop := ∀ x : FamilyOfElements P R, x.Compatible → ∃! t, x.IsAmalgamation t #align category_theory.presieve.is_sheaf_for CategoryTheory.Presieve.IsSheafFor def YonedaSheafCondition (P : Cᵒᵖ ⥤ Type v₁) (S : Sieve X) : Prop := ∀ f : S.functor ⟶ P, ∃! g, S.functorInclusion ≫ g = f #align category_theory.presieve.yoneda_sheaf_condition CategoryTheory.Presieve.YonedaSheafCondition -- TODO: We can generalize the universe parameter v₁ above by composing with -- appropriate `ulift_functor`s. def natTransEquivCompatibleFamily {P : Cᵒᵖ ⥤ Type v₁} : (S.functor ⟶ P) ≃ { x : FamilyOfElements P (S : Presieve X) // x.Compatible } where toFun α := by refine ⟨fun Y f hf => ?_, ?_⟩ · apply α.app (op Y) ⟨_, hf⟩ · rw [compatible_iff_sieveCompatible] intro Y Z f g hf dsimp rw [← FunctorToTypes.naturality _ _ α g.op] rfl invFun t := { app := fun Y f => t.1 _ f.2 naturality := fun Y Z g => by ext ⟨f, hf⟩ apply t.2.to_sieveCompatible _ } left_inv α := by ext X ⟨_, _⟩ rfl right_inv := by rintro ⟨x, hx⟩ rfl #align category_theory.presieve.nat_trans_equiv_compatible_family CategoryTheory.Presieve.natTransEquivCompatibleFamily theorem extension_iff_amalgamation {P : Cᵒᵖ ⥤ Type v₁} (x : S.functor ⟶ P) (g : yoneda.obj X ⟶ P) : S.functorInclusion ≫ g = x ↔ (natTransEquivCompatibleFamily x).1.IsAmalgamation (yonedaEquiv g) := by change _ ↔ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (h : S f), P.map f.op (yonedaEquiv g) = x.app (op Y) ⟨f, h⟩ constructor · rintro rfl Y f hf rw [yonedaEquiv_naturality] dsimp simp [yonedaEquiv_apply] -- See note [dsimp, simp]. · intro h ext Y ⟨f, hf⟩ convert h f hf rw [yonedaEquiv_naturality] dsimp [yonedaEquiv] simp #align category_theory.presieve.extension_iff_amalgamation CategoryTheory.Presieve.extension_iff_amalgamation theorem isSheafFor_iff_yonedaSheafCondition {P : Cᵒᵖ ⥤ Type v₁} : IsSheafFor P (S : Presieve X) ↔ YonedaSheafCondition P S := by rw [IsSheafFor, YonedaSheafCondition] simp_rw [extension_iff_amalgamation] rw [Equiv.forall_congr_left' natTransEquivCompatibleFamily] rw [Subtype.forall] apply forall₂_congr intro x hx rw [Equiv.exists_unique_congr_left _] simp #align category_theory.presieve.is_sheaf_for_iff_yoneda_sheaf_condition CategoryTheory.Presieve.isSheafFor_iff_yonedaSheafCondition noncomputable def IsSheafFor.extend {P : Cᵒᵖ ⥤ Type v₁} (h : IsSheafFor P (S : Presieve X)) (f : S.functor ⟶ P) : yoneda.obj X ⟶ P := (isSheafFor_iff_yonedaSheafCondition.1 h f).exists.choose #align category_theory.presieve.is_sheaf_for.extend CategoryTheory.Presieve.IsSheafFor.extend @[reassoc (attr := simp)] theorem IsSheafFor.functorInclusion_comp_extend {P : Cᵒᵖ ⥤ Type v₁} (h : IsSheafFor P S) (f : S.functor ⟶ P) : S.functorInclusion ≫ h.extend f = f := (isSheafFor_iff_yonedaSheafCondition.1 h f).exists.choose_spec #align category_theory.presieve.is_sheaf_for.functor_inclusion_comp_extend CategoryTheory.Presieve.IsSheafFor.functorInclusion_comp_extend theorem IsSheafFor.unique_extend {P : Cᵒᵖ ⥤ Type v₁} (h : IsSheafFor P S) {f : S.functor ⟶ P} (t : yoneda.obj X ⟶ P) (ht : S.functorInclusion ≫ t = f) : t = h.extend f := (isSheafFor_iff_yonedaSheafCondition.1 h f).unique ht (h.functorInclusion_comp_extend f) #align category_theory.presieve.is_sheaf_for.unique_extend CategoryTheory.Presieve.IsSheafFor.unique_extend theorem IsSheafFor.hom_ext {P : Cᵒᵖ ⥤ Type v₁} (h : IsSheafFor P (S : Presieve X)) (t₁ t₂ : yoneda.obj X ⟶ P) (ht : S.functorInclusion ≫ t₁ = S.functorInclusion ≫ t₂) : t₁ = t₂ := (h.unique_extend t₁ ht).trans (h.unique_extend t₂ rfl).symm #align category_theory.presieve.is_sheaf_for.hom_ext CategoryTheory.Presieve.IsSheafFor.hom_ext theorem isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor : (IsSeparatedFor P R ∧ ∀ x : FamilyOfElements P R, x.Compatible → ∃ t, x.IsAmalgamation t) ↔ IsSheafFor P R := by rw [IsSeparatedFor, ← forall_and] apply forall_congr' intro x constructor · intro z hx exact exists_unique_of_exists_of_unique (z.2 hx) z.1 · intro h refine ⟨?_, ExistsUnique.exists ∘ h⟩ intro t₁ t₂ ht₁ ht₂ apply (h _).unique ht₁ ht₂ exact is_compatible_of_exists_amalgamation x ⟨_, ht₂⟩ #align category_theory.presieve.is_separated_for_and_exists_is_amalgamation_iff_sheaf_for CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor theorem IsSeparatedFor.isSheafFor (t : IsSeparatedFor P R) : (∀ x : FamilyOfElements P R, x.Compatible → ∃ t, x.IsAmalgamation t) → IsSheafFor P R := by rw [← isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] exact And.intro t #align category_theory.presieve.is_separated_for.is_sheaf_for CategoryTheory.Presieve.IsSeparatedFor.isSheafFor theorem IsSheafFor.isSeparatedFor : IsSheafFor P R → IsSeparatedFor P R := fun q => (isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor.2 q).1 #align category_theory.presieve.is_sheaf_for.is_separated_for CategoryTheory.Presieve.IsSheafFor.isSeparatedFor noncomputable def IsSheafFor.amalgamate (t : IsSheafFor P R) (x : FamilyOfElements P R) (hx : x.Compatible) : P.obj (op X) := (t x hx).exists.choose #align category_theory.presieve.is_sheaf_for.amalgamate CategoryTheory.Presieve.IsSheafFor.amalgamate theorem IsSheafFor.isAmalgamation (t : IsSheafFor P R) {x : FamilyOfElements P R} (hx : x.Compatible) : x.IsAmalgamation (t.amalgamate x hx) := (t x hx).exists.choose_spec #align category_theory.presieve.is_sheaf_for.is_amalgamation CategoryTheory.Presieve.IsSheafFor.isAmalgamation @[simp] theorem IsSheafFor.valid_glue (t : IsSheafFor P R) {x : FamilyOfElements P R} (hx : x.Compatible) (f : Y ⟶ X) (Hf : R f) : P.map f.op (t.amalgamate x hx) = x f Hf := t.isAmalgamation hx f Hf #align category_theory.presieve.is_sheaf_for.valid_glue CategoryTheory.Presieve.IsSheafFor.valid_glue theorem isSheafFor_iff_generate (R : Presieve X) : IsSheafFor P R ↔ IsSheafFor P (generate R : Presieve X) := by rw [← isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] rw [← isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] rw [← isSeparatedFor_iff_generate] apply and_congr (Iff.refl _) constructor · intro q x hx apply Exists.imp _ (q _ (hx.restrict (le_generate R))) intro t ht simpa [hx] using isAmalgamation_sieveExtend _ _ ht · intro q x hx apply Exists.imp _ (q _ hx.sieveExtend) intro t ht simpa [hx] using isAmalgamation_restrict (le_generate R) _ _ ht #align category_theory.presieve.is_sheaf_for_iff_generate CategoryTheory.Presieve.isSheafFor_iff_generate	Mathlib/CategoryTheory/Sites/IsSheafFor.lean	644	650	theorem isSheafFor_singleton_iso (P : Cᵒᵖ ⥤ Type w) : IsSheafFor P (Presieve.singleton (𝟙 X)) := by	intro x _ refine ⟨x _ (Presieve.singleton_self _), ?_, ?_⟩ · rintro _ _ ⟨rfl, rfl⟩ simp · intro t ht simpa using ht _ (Presieve.singleton_self _)
import Mathlib.Algebra.Group.Subgroup.Basic import Mathlib.Data.Fintype.Basic import Mathlib.Data.List.Sublists import Mathlib.Data.List.InsertNth #align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6" open Relation universe u v w variable {α : Type u} attribute [local simp] List.append_eq_has_append -- Porting note: to_additive.map_namespace is not supported yet -- worked around it by putting a few extra manual mappings (but not too many all in all) -- run_cmd to_additive.map_namespace `FreeGroup `FreeAddGroup inductive FreeAddGroup.Red.Step : List (α × Bool) → List (α × Bool) → Prop \| not {L₁ L₂ x b} : FreeAddGroup.Red.Step (L₁ ++ (x, b) :: (x, not b) :: L₂) (L₁ ++ L₂) #align free_add_group.red.step FreeAddGroup.Red.Step attribute [simp] FreeAddGroup.Red.Step.not @[to_additive FreeAddGroup.Red.Step] inductive FreeGroup.Red.Step : List (α × Bool) → List (α × Bool) → Prop \| not {L₁ L₂ x b} : FreeGroup.Red.Step (L₁ ++ (x, b) :: (x, not b) :: L₂) (L₁ ++ L₂) #align free_group.red.step FreeGroup.Red.Step attribute [simp] FreeGroup.Red.Step.not namespace FreeGroup variable {L L₁ L₂ L₃ L₄ : List (α × Bool)} @[to_additive FreeAddGroup.Red "Reflexive-transitive closure of `Red.Step`"] def Red : List (α × Bool) → List (α × Bool) → Prop := ReflTransGen Red.Step #align free_group.red FreeGroup.Red #align free_add_group.red FreeAddGroup.Red @[to_additive (attr := refl)] theorem Red.refl : Red L L := ReflTransGen.refl #align free_group.red.refl FreeGroup.Red.refl #align free_add_group.red.refl FreeAddGroup.Red.refl @[to_additive (attr := trans)] theorem Red.trans : Red L₁ L₂ → Red L₂ L₃ → Red L₁ L₃ := ReflTransGen.trans #align free_group.red.trans FreeGroup.Red.trans #align free_add_group.red.trans FreeAddGroup.Red.trans namespace Red @[to_additive "Predicate asserting that the word `w₁` can be reduced to `w₂` in one step, i.e. there are words `w₃ w₄` and letter `x` such that `w₁ = w₃ + x + (-x) + w₄` and `w₂ = w₃w₄`"] theorem Step.length : ∀ {L₁ L₂ : List (α × Bool)}, Step L₁ L₂ → L₂.length + 2 = L₁.length \| _, _, @Red.Step.not _ L1 L2 x b => by rw [List.length_append, List.length_append]; rfl #align free_group.red.step.length FreeGroup.Red.Step.length #align free_add_group.red.step.length FreeAddGroup.Red.Step.length @[to_additive (attr := simp)] theorem Step.not_rev {x b} : Step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂) := by cases b <;> exact Step.not #align free_group.red.step.bnot_rev FreeGroup.Red.Step.not_rev #align free_add_group.red.step.bnot_rev FreeAddGroup.Red.Step.not_rev @[to_additive (attr := simp)] theorem Step.cons_not {x b} : Red.Step ((x, b) :: (x, !b) :: L) L := @Step.not _ [] _ _ _ #align free_group.red.step.cons_bnot FreeGroup.Red.Step.cons_not #align free_add_group.red.step.cons_bnot FreeAddGroup.Red.Step.cons_not @[to_additive (attr := simp)] theorem Step.cons_not_rev {x b} : Red.Step ((x, !b) :: (x, b) :: L) L := @Red.Step.not_rev _ [] _ _ _ #align free_group.red.step.cons_bnot_rev FreeGroup.Red.Step.cons_not_rev #align free_add_group.red.step.cons_bnot_rev FreeAddGroup.Red.Step.cons_not_rev @[to_additive] theorem Step.append_left : ∀ {L₁ L₂ L₃ : List (α × Bool)}, Step L₂ L₃ → Step (L₁ ++ L₂) (L₁ ++ L₃) \| _, _, _, Red.Step.not => by rw [← List.append_assoc, ← List.append_assoc]; constructor #align free_group.red.step.append_left FreeGroup.Red.Step.append_left #align free_add_group.red.step.append_left FreeAddGroup.Red.Step.append_left @[to_additive] theorem Step.cons {x} (H : Red.Step L₁ L₂) : Red.Step (x :: L₁) (x :: L₂) := @Step.append_left _ [x] _ _ H #align free_group.red.step.cons FreeGroup.Red.Step.cons #align free_add_group.red.step.cons FreeAddGroup.Red.Step.cons @[to_additive] theorem Step.append_right : ∀ {L₁ L₂ L₃ : List (α × Bool)}, Step L₁ L₂ → Step (L₁ ++ L₃) (L₂ ++ L₃) \| _, _, _, Red.Step.not => by simp #align free_group.red.step.append_right FreeGroup.Red.Step.append_right #align free_add_group.red.step.append_right FreeAddGroup.Red.Step.append_right @[to_additive] theorem not_step_nil : ¬Step [] L := by generalize h' : [] = L' intro h cases' h with L₁ L₂ simp [List.nil_eq_append] at h' #align free_group.red.not_step_nil FreeGroup.Red.not_step_nil #align free_add_group.red.not_step_nil FreeAddGroup.Red.not_step_nil @[to_additive] theorem Step.cons_left_iff {a : α} {b : Bool} : Step ((a, b) :: L₁) L₂ ↔ (∃ L, Step L₁ L ∧ L₂ = (a, b) :: L) ∨ L₁ = (a, ! b) :: L₂ := by constructor · generalize hL : ((a, b) :: L₁ : List _) = L rintro @⟨_ \| ⟨p, s'⟩, e, a', b'⟩ · simp at hL simp [] · simp at hL rcases hL with ⟨rfl, rfl⟩ refine Or.inl ⟨s' ++ e, Step.not, ?_⟩ simp · rintro (⟨L, h, rfl⟩ \| rfl) · exact Step.cons h · exact Step.cons_not #align free_group.red.step.cons_left_iff FreeGroup.Red.Step.cons_left_iff #align free_add_group.red.step.cons_left_iff FreeAddGroup.Red.Step.cons_left_iff @[to_additive] theorem not_step_singleton : ∀ {p : α × Bool}, ¬Step [p] L \| (a, b) => by simp [Step.cons_left_iff, not_step_nil] #align free_group.red.not_step_singleton FreeGroup.Red.not_step_singleton #align free_add_group.red.not_step_singleton FreeAddGroup.Red.not_step_singleton @[to_additive] theorem Step.cons_cons_iff : ∀ {p : α × Bool}, Step (p :: L₁) (p :: L₂) ↔ Step L₁ L₂ := by simp (config := { contextual := true }) [Step.cons_left_iff, iff_def, or_imp] #align free_group.red.step.cons_cons_iff FreeGroup.Red.Step.cons_cons_iff #align free_add_group.red.step.cons_cons_iff FreeAddGroup.Red.Step.cons_cons_iff @[to_additive] theorem Step.append_left_iff : ∀ L, Step (L ++ L₁) (L ++ L₂) ↔ Step L₁ L₂ \| [] => by simp \| p :: l => by simp [Step.append_left_iff l, Step.cons_cons_iff] #align free_group.red.step.append_left_iff FreeGroup.Red.Step.append_left_iff #align free_add_group.red.step.append_left_iff FreeAddGroup.Red.Step.append_left_iff @[to_additive] theorem Step.diamond_aux : ∀ {L₁ L₂ L₃ L₄ : List (α × Bool)} {x1 b1 x2 b2}, L₁ ++ (x1, b1) :: (x1, !b1) :: L₂ = L₃ ++ (x2, b2) :: (x2, !b2) :: L₄ → L₁ ++ L₂ = L₃ ++ L₄ ∨ ∃ L₅, Red.Step (L₁ ++ L₂) L₅ ∧ Red.Step (L₃ ++ L₄) L₅ \| [], _, [], _, _, _, _, _, H => by injections; subst_vars; simp \| [], _, [(x3, b3)], _, _, _, _, _, H => by injections; subst_vars; simp \| [(x3, b3)], _, [], _, _, _, _, _, H => by injections; subst_vars; simp \| [], _, (x3, b3) :: (x4, b4) :: tl, _, _, _, _, _, H => by injections; subst_vars; simp; right; exact ⟨_, Red.Step.not, Red.Step.cons_not⟩ \| (x3, b3) :: (x4, b4) :: tl, _, [], _, _, _, _, _, H => by injections; subst_vars; simp; right; exact ⟨_, Red.Step.cons_not, Red.Step.not⟩ \| (x3, b3) :: tl, _, (x4, b4) :: tl2, _, _, _, _, _, H => let ⟨H1, H2⟩ := List.cons.inj H match Step.diamond_aux H2 with \| Or.inl H3 => Or.inl <\| by simp [H1, H3] \| Or.inr ⟨L₅, H3, H4⟩ => Or.inr ⟨_, Step.cons H3, by simpa [H1] using Step.cons H4⟩ #align free_group.red.step.diamond_aux FreeGroup.Red.Step.diamond_aux #align free_add_group.red.step.diamond_aux FreeAddGroup.Red.Step.diamond_aux @[to_additive] theorem Step.diamond : ∀ {L₁ L₂ L₃ L₄ : List (α × Bool)}, Red.Step L₁ L₃ → Red.Step L₂ L₄ → L₁ = L₂ → L₃ = L₄ ∨ ∃ L₅, Red.Step L₃ L₅ ∧ Red.Step L₄ L₅ \| _, _, _, _, Red.Step.not, Red.Step.not, H => Step.diamond_aux H #align free_group.red.step.diamond FreeGroup.Red.Step.diamond #align free_add_group.red.step.diamond FreeAddGroup.Red.Step.diamond @[to_additive] theorem Step.to_red : Step L₁ L₂ → Red L₁ L₂ := ReflTransGen.single #align free_group.red.step.to_red FreeGroup.Red.Step.to_red #align free_add_group.red.step.to_red FreeAddGroup.Red.Step.to_red @[to_additive "Church-Rosser theorem* for word reduction: If `w1 w2 w3` are words such that `w1` reduces to `w2` and `w3` respectively, then there is a word `w4` such that `w2` and `w3` reduce to `w4` respectively. This is also known as Newman's diamond lemma."] theorem church_rosser : Red L₁ L₂ → Red L₁ L₃ → Join Red L₂ L₃ := Relation.church_rosser fun a b c hab hac => match b, c, Red.Step.diamond hab hac rfl with \| b, _, Or.inl rfl => ⟨b, by rfl, by rfl⟩ \| b, c, Or.inr ⟨d, hbd, hcd⟩ => ⟨d, ReflGen.single hbd, hcd.to_red⟩ #align free_group.red.church_rosser FreeGroup.Red.church_rosser #align free_add_group.red.church_rosser FreeAddGroup.Red.church_rosser @[to_additive] theorem cons_cons {p} : Red L₁ L₂ → Red (p :: L₁) (p :: L₂) := ReflTransGen.lift (List.cons p) fun _ _ => Step.cons #align free_group.red.cons_cons FreeGroup.Red.cons_cons #align free_add_group.red.cons_cons FreeAddGroup.Red.cons_cons @[to_additive] theorem cons_cons_iff (p) : Red (p :: L₁) (p :: L₂) ↔ Red L₁ L₂ := Iff.intro (by generalize eq₁ : (p :: L₁ : List _) = LL₁ generalize eq₂ : (p :: L₂ : List _) = LL₂ intro h induction' h using Relation.ReflTransGen.head_induction_on with L₁ L₂ h₁₂ h ih generalizing L₁ L₂ · subst_vars cases eq₂ constructor · subst_vars cases' p with a b rw [Step.cons_left_iff] at h₁₂ rcases h₁₂ with (⟨L, h₁₂, rfl⟩ \| rfl) · exact (ih rfl rfl).head h₁₂ · exact (cons_cons h).tail Step.cons_not_rev) cons_cons #align free_group.red.cons_cons_iff FreeGroup.Red.cons_cons_iff #align free_add_group.red.cons_cons_iff FreeAddGroup.Red.cons_cons_iff @[to_additive] theorem append_append_left_iff : ∀ L, Red (L ++ L₁) (L ++ L₂) ↔ Red L₁ L₂ \| [] => Iff.rfl \| p :: L => by simp [append_append_left_iff L, cons_cons_iff] #align free_group.red.append_append_left_iff FreeGroup.Red.append_append_left_iff #align free_add_group.red.append_append_left_iff FreeAddGroup.Red.append_append_left_iff @[to_additive] theorem append_append (h₁ : Red L₁ L₃) (h₂ : Red L₂ L₄) : Red (L₁ ++ L₂) (L₃ ++ L₄) := (h₁.lift (fun L => L ++ L₂) fun _ _ => Step.append_right).trans ((append_append_left_iff _).2 h₂) #align free_group.red.append_append FreeGroup.Red.append_append #align free_add_group.red.append_append FreeAddGroup.Red.append_append @[to_additive] theorem to_append_iff : Red L (L₁ ++ L₂) ↔ ∃ L₃ L₄, L = L₃ ++ L₄ ∧ Red L₃ L₁ ∧ Red L₄ L₂ := Iff.intro (by generalize eq : L₁ ++ L₂ = L₁₂ intro h induction' h with L' L₁₂ hLL' h ih generalizing L₁ L₂ · exact ⟨_, _, eq.symm, by rfl, by rfl⟩ · cases' h with s e a b rcases List.append_eq_append_iff.1 eq with (⟨s', rfl, rfl⟩ \| ⟨e', rfl, rfl⟩) · have : L₁ ++ (s' ++ (a, b) :: (a, not b) :: e) = L₁ ++ s' ++ (a, b) :: (a, not b) :: e := by simp rcases ih this with ⟨w₁, w₂, rfl, h₁, h₂⟩ exact ⟨w₁, w₂, rfl, h₁, h₂.tail Step.not⟩ · have : s ++ (a, b) :: (a, not b) :: e' ++ L₂ = s ++ (a, b) :: (a, not b) :: (e' ++ L₂) := by simp rcases ih this with ⟨w₁, w₂, rfl, h₁, h₂⟩ exact ⟨w₁, w₂, rfl, h₁.tail Step.not, h₂⟩) fun ⟨L₃, L₄, Eq, h₃, h₄⟩ => Eq.symm ▸ append_append h₃ h₄ #align free_group.red.to_append_iff FreeGroup.Red.to_append_iff #align free_add_group.red.to_append_iff FreeAddGroup.Red.to_append_iff @[to_additive "The empty word `[]` only reduces to itself."] theorem nil_iff : Red [] L ↔ L = [] := reflTransGen_iff_eq fun _ => Red.not_step_nil #align free_group.red.nil_iff FreeGroup.Red.nil_iff #align free_add_group.red.nil_iff FreeAddGroup.Red.nil_iff @[to_additive "A letter only reduces to itself."] theorem singleton_iff {x} : Red [x] L₁ ↔ L₁ = [x] := reflTransGen_iff_eq fun _ => not_step_singleton #align free_group.red.singleton_iff FreeGroup.Red.singleton_iff #align free_add_group.red.singleton_iff FreeAddGroup.Red.singleton_iff @[to_additive "If `x` is a letter and `w` is a word such that `x + w` reduces to the empty word, then `w` reduces to `-x`."] theorem cons_nil_iff_singleton {x b} : Red ((x, b) :: L) [] ↔ Red L [(x, not b)] := Iff.intro (fun h => by have h₁ : Red ((x, not b) :: (x, b) :: L) [(x, not b)] := cons_cons h have h₂ : Red ((x, not b) :: (x, b) :: L) L := ReflTransGen.single Step.cons_not_rev let ⟨L', h₁, h₂⟩ := church_rosser h₁ h₂ rw [singleton_iff] at h₁ subst L' assumption) fun h => (cons_cons h).tail Step.cons_not #align free_group.red.cons_nil_iff_singleton FreeGroup.Red.cons_nil_iff_singleton #align free_add_group.red.cons_nil_iff_singleton FreeAddGroup.Red.cons_nil_iff_singleton @[to_additive]	Mathlib/GroupTheory/FreeGroup/Basic.lean	342	350	theorem red_iff_irreducible {x1 b1 x2 b2} (h : (x1, b1) ≠ (x2, b2)) : Red [(x1, !b1), (x2, b2)] L ↔ L = [(x1, !b1), (x2, b2)] := by	apply reflTransGen_iff_eq generalize eq : [(x1, not b1), (x2, b2)] = L' intro L h' cases h' simp [List.cons_eq_append, List.nil_eq_append] at eq rcases eq with ⟨rfl, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl⟩ simp at h
import Mathlib.Logic.Pairwise import Mathlib.Order.CompleteBooleanAlgebra import Mathlib.Order.Directed import Mathlib.Order.GaloisConnection #align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd" open Function Set universe u variable {α β γ : Type} {ι ι' ι₂ : Sort} {κ κ₁ κ₂ : ι → Sort} {κ' : ι' → Sort} namespace Set theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by simp_rw [mem_iUnion] #align set.mem_Union₂ Set.mem_iUnion₂ theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by simp_rw [mem_iInter] #align set.mem_Inter₂ Set.mem_iInter₂ theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i := mem_iUnion.2 ⟨i, ha⟩ #align set.mem_Union_of_mem Set.mem_iUnion_of_mem theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) : a ∈ ⋃ (i) (j), s i j := mem_iUnion₂.2 ⟨i, j, ha⟩ #align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i := mem_iInter.2 h #align set.mem_Inter_of_mem Set.mem_iInter_of_mem theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) : a ∈ ⋂ (i) (j), s i j := mem_iInter₂.2 h #align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem instance completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) := { instBooleanAlgebraSet with le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩ sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in sInf_le := fun s t t_in a h => h _ t_in iInf_iSup_eq := by intros; ext; simp [Classical.skolem] } instance : OrderTop (Set α) where top := univ le_top := by simp @[congr] theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q) (f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ := iSup_congr_Prop pq f #align set.Union_congr_Prop Set.iUnion_congr_Prop @[congr] theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q) (f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ := iInf_congr_Prop pq f #align set.Inter_congr_Prop Set.iInter_congr_Prop theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i := iSup_plift_up _ #align set.Union_plift_up Set.iUnion_plift_up theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i := iSup_plift_down _ #align set.Union_plift_down Set.iUnion_plift_down theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i := iInf_plift_up _ #align set.Inter_plift_up Set.iInter_plift_up theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i := iInf_plift_down _ #align set.Inter_plift_down Set.iInter_plift_down theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ := iSup_eq_if _ #align set.Union_eq_if Set.iUnion_eq_if theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) : ⋃ h : p, s h = if h : p then s h else ∅ := iSup_eq_dif _ #align set.Union_eq_dif Set.iUnion_eq_dif theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ := iInf_eq_if _ #align set.Inter_eq_if Set.iInter_eq_if theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) : ⋂ h : p, s h = if h : p then s h else univ := _root_.iInf_eq_dif _ #align set.Infi_eq_dif Set.iInf_eq_dif theorem exists_set_mem_of_union_eq_top {ι : Type} (t : Set ι) (s : ι → Set β) (w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by have p : x ∈ ⊤ := Set.mem_univ x rw [← w, Set.mem_iUnion] at p simpa using p #align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top theorem nonempty_of_union_eq_top_of_nonempty {ι : Type} (t : Set ι) (s : ι → Set α) (H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some exact ⟨x, m⟩ #align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty theorem nonempty_of_nonempty_iUnion {s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by obtain ⟨x, hx⟩ := h_Union exact ⟨Classical.choose <\| mem_iUnion.mp hx⟩ theorem nonempty_of_nonempty_iUnion_eq_univ {s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι := nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty) theorem setOf_exists (p : ι → β → Prop) : { x \| ∃ i, p i x } = ⋃ i, { x \| p i x } := ext fun _ => mem_iUnion.symm #align set.set_of_exists Set.setOf_exists theorem setOf_forall (p : ι → β → Prop) : { x \| ∀ i, p i x } = ⋂ i, { x \| p i x } := ext fun _ => mem_iInter.symm #align set.set_of_forall Set.setOf_forall theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t := iSup_le h #align set.Union_subset Set.iUnion_subset theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) : ⋃ (i) (j), s i j ⊆ t := iUnion_subset fun x => iUnion_subset (h x) #align set.Union₂_subset Set.iUnion₂_subset theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i := le_iInf h #align set.subset_Inter Set.subset_iInter theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) : s ⊆ ⋂ (i) (j), t i j := subset_iInter fun x => subset_iInter <\| h x #align set.subset_Inter₂ Set.subset_iInter₂ @[simp] theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t := ⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩ #align set.Union_subset_iff Set.iUnion_subset_iff theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} : ⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff] #align set.Union₂_subset_iff Set.iUnion₂_subset_iff @[simp] theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i := le_iInf_iff #align set.subset_Inter_iff Set.subset_iInter_iff -- Porting note (#10618): removing `simp`. `simp` can prove it theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} : (s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff] #align set.subset_Inter₂_iff Set.subset_iInter₂_iff theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i := le_iSup #align set.subset_Union Set.subset_iUnion theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i := iInf_le #align set.Inter_subset Set.iInter_subset theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' := le_iSup₂ i j #align set.subset_Union₂ Set.subset_iUnion₂ theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j := iInf₂_le i j #align set.Inter₂_subset Set.iInter₂_subset theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i := le_iSup_of_le i h #align set.subset_Union_of_subset Set.subset_iUnion_of_subset theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) : ⋂ i, s i ⊆ t := iInf_le_of_le i h #align set.Inter_subset_of_subset Set.iInter_subset_of_subset theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i) (h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j := le_iSup₂_of_le i j h #align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i) (h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t := iInf₂_le_of_le i j h #align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i := iSup_mono h #align set.Union_mono Set.iUnion_mono @[gcongr] theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t := iSup_mono h theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) : ⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j := iSup₂_mono h #align set.Union₂_mono Set.iUnion₂_mono theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i := iInf_mono h #align set.Inter_mono Set.iInter_mono @[gcongr] theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t := iInf_mono h theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) : ⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j := iInf₂_mono h #align set.Inter₂_mono Set.iInter₂_mono theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) : ⋃ i, s i ⊆ ⋃ i, t i := iSup_mono' h #align set.Union_mono' Set.iUnion_mono' theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α} (h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' := iSup₂_mono' h #align set.Union₂_mono' Set.iUnion₂_mono' theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) : ⋂ i, s i ⊆ ⋂ j, t j := Set.subset_iInter fun j => let ⟨i, hi⟩ := h j iInter_subset_of_subset i hi #align set.Inter_mono' Set.iInter_mono' theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α} (h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' := subset_iInter₂_iff.2 fun i' j' => let ⟨_, _, hst⟩ := h i' j' (iInter₂_subset _ _).trans hst #align set.Inter₂_mono' Set.iInter₂_mono' theorem iUnion₂_subset_iUnion (κ : ι → Sort) (s : ι → Set α) : ⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i := iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl #align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion theorem iInter_subset_iInter₂ (κ : ι → Sort) (s : ι → Set α) : ⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i := iInter_mono fun _ => subset_iInter fun _ => Subset.rfl #align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂ theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α \| P i x } = { x : α \| ∃ i, P i x } := by ext exact mem_iUnion #align set.Union_set_of Set.iUnion_setOf theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α \| P i x } = { x : α \| ∀ i, P i x } := by ext exact mem_iInter #align set.Inter_set_of Set.iInter_setOf theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h) (h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y := h1.iSup_congr h h2 #align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h) (h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y := h1.iInf_congr h h2 #align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h #align set.Union_congr Set.iUnion_congr lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h #align set.Inter_congr Set.iInter_congr lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) : ⋃ (i) (j), s i j = ⋃ (i) (j), t i j := iUnion_congr fun i => iUnion_congr <\| h i #align set.Union₂_congr Set.iUnion₂_congr lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) : ⋂ (i) (j), s i j = ⋂ (i) (j), t i j := iInter_congr fun i => iInter_congr <\| h i #align set.Inter₂_congr Set.iInter₂_congr @[simp] theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ := compl_iSup #align set.compl_Union Set.compl_iUnion theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by simp_rw [compl_iUnion] #align set.compl_Union₂ Set.compl_iUnion₂ @[simp] theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ := compl_iInf #align set.compl_Inter Set.compl_iInter theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by simp_rw [compl_iInter] #align set.compl_Inter₂ Set.compl_iInter₂ -- classical -- complete_boolean_algebra theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by simp only [compl_iInter, compl_compl] #align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl -- classical -- complete_boolean_algebra theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by simp only [compl_iUnion, compl_compl] #align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i := inf_iSup_eq _ _ #align set.inter_Union Set.inter_iUnion theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s := iSup_inf_eq _ _ #align set.Union_inter Set.iUnion_inter theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) : ⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i := iSup_sup_eq #align set.Union_union_distrib Set.iUnion_union_distrib theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) : ⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i := iInf_inf_eq #align set.Inter_inter_distrib Set.iInter_inter_distrib theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i := sup_iSup #align set.union_Union Set.union_iUnion theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s := iSup_sup #align set.Union_union Set.iUnion_union theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i := inf_iInf #align set.inter_Inter Set.inter_iInter theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s := iInf_inf #align set.Inter_inter Set.iInter_inter -- classical theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i := sup_iInf_eq _ _ #align set.union_Inter Set.union_iInter theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t := iInf_sup_eq _ _ #align set.Inter_union Set.iInter_union theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s := iUnion_inter _ _ #align set.Union_diff Set.iUnion_diff theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by rw [diff_eq, compl_iUnion, inter_iInter]; rfl #align set.diff_Union Set.diff_iUnion theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by rw [diff_eq, compl_iInter, inter_iUnion]; rfl #align set.diff_Inter Set.diff_iInter theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i := le_iSup_inf_iSup s t #align set.Union_inter_subset Set.iUnion_inter_subset theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α} (hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i := iSup_inf_of_monotone hs ht #align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α} (hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i := iSup_inf_of_antitone hs ht #align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α} (hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i := iInf_sup_of_monotone hs ht #align set.Inter_union_of_monotone Set.iInter_union_of_monotone theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α} (hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i := iInf_sup_of_antitone hs ht #align set.Inter_union_of_antitone Set.iInter_union_of_antitone theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j := iSup_iInf_le_iInf_iSup (flip s) #align set.Union_Inter_subset Set.iUnion_iInter_subset theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) := iSup_option s #align set.Union_option Set.iUnion_option theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) := iInf_option s #align set.Inter_option Set.iInter_option section variable (p : ι → Prop) [DecidablePred p] theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set α) : ⋃ i, (if h : p i then f i h else g i h) = (⋃ (i) (h : p i), f i h) ∪ ⋃ (i) (h : ¬p i), g i h := iSup_dite _ _ _ #align set.Union_dite Set.iUnion_dite theorem iUnion_ite (f g : ι → Set α) : ⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i := iUnion_dite _ _ _ #align set.Union_ite Set.iUnion_ite theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set α) : ⋂ i, (if h : p i then f i h else g i h) = (⋂ (i) (h : p i), f i h) ∩ ⋂ (i) (h : ¬p i), g i h := iInf_dite _ _ _ #align set.Inter_dite Set.iInter_dite theorem iInter_ite (f g : ι → Set α) : ⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i := iInter_dite _ _ _ #align set.Inter_ite Set.iInter_ite end theorem image_projection_prod {ι : Type} {α : ι → Type} {v : ∀ i : ι, Set (α i)} (hv : (pi univ v).Nonempty) (i : ι) : ((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by classical apply Subset.antisymm · simp [iInter_subset] · intro y y_in simp only [mem_image, mem_iInter, mem_preimage] rcases hv with ⟨z, hz⟩ refine ⟨Function.update z i y, ?_, update_same i y z⟩ rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i] exact ⟨y_in, fun j _ => by simpa using hz j⟩ #align set.image_projection_prod Set.image_projection_prod theorem iInter_false {s : False → Set α} : iInter s = univ := iInf_false #align set.Inter_false Set.iInter_false theorem iUnion_false {s : False → Set α} : iUnion s = ∅ := iSup_false #align set.Union_false Set.iUnion_false @[simp] theorem iInter_true {s : True → Set α} : iInter s = s trivial := iInf_true #align set.Inter_true Set.iInter_true @[simp] theorem iUnion_true {s : True → Set α} : iUnion s = s trivial := iSup_true #align set.Union_true Set.iUnion_true @[simp] theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} : ⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ := iInf_exists #align set.Inter_exists Set.iInter_exists @[simp] theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} : ⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ := iSup_exists #align set.Union_exists Set.iUnion_exists @[simp] theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ := iSup_bot #align set.Union_empty Set.iUnion_empty @[simp] theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ := iInf_top #align set.Inter_univ Set.iInter_univ section variable {s : ι → Set α} @[simp] theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ := iSup_eq_bot #align set.Union_eq_empty Set.iUnion_eq_empty @[simp] theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ := iInf_eq_top #align set.Inter_eq_univ Set.iInter_eq_univ @[simp] theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by simp [nonempty_iff_ne_empty] #align set.nonempty_Union Set.nonempty_iUnion -- Porting note (#10618): removing `simp`. `simp` can prove it theorem nonempty_biUnion {t : Set α} {s : α → Set β} : (⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp #align set.nonempty_bUnion Set.nonempty_biUnion theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) : ⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ := iSup_exists #align set.Union_nonempty_index Set.iUnion_nonempty_index end @[simp] theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} : ⋂ (x) (h : x = b), s x h = s b rfl := iInf_iInf_eq_left #align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left @[simp] theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} : ⋂ (x) (h : b = x), s x h = s b rfl := iInf_iInf_eq_right #align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right @[simp] theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} : ⋃ (x) (h : x = b), s x h = s b rfl := iSup_iSup_eq_left #align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left @[simp] theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} : ⋃ (x) (h : b = x), s x h = s b rfl := iSup_iSup_eq_right #align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) : ⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) := iInf_or #align set.Inter_or Set.iInter_or theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) : ⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) := iSup_or #align set.Union_or Set.iUnion_or theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ := iSup_and #align set.Union_and Set.iUnion_and theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ := iInf_and #align set.Inter_and Set.iInter_and theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' := iSup_comm #align set.Union_comm Set.iUnion_comm theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' := iInf_comm #align set.Inter_comm Set.iInter_comm theorem iUnion_sigma {γ : α → Type} (s : Sigma γ → Set β) : ⋃ ia, s ia = ⋃ i, ⋃ a, s ⟨i, a⟩ := iSup_sigma theorem iUnion_sigma' {γ : α → Type} (s : ∀ i, γ i → Set β) : ⋃ i, ⋃ a, s i a = ⋃ ia : Sigma γ, s ia.1 ia.2 := iSup_sigma' _ theorem iInter_sigma {γ : α → Type} (s : Sigma γ → Set β) : ⋂ ia, s ia = ⋂ i, ⋂ a, s ⟨i, a⟩ := iInf_sigma theorem iInter_sigma' {γ : α → Type} (s : ∀ i, γ i → Set β) : ⋂ i, ⋂ a, s i a = ⋂ ia : Sigma γ, s ia.1 ia.2 := iInf_sigma' _ theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) : ⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ := iSup₂_comm _ #align set.Union₂_comm Set.iUnion₂_comm theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) : ⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ := iInf₂_comm _ #align set.Inter₂_comm Set.iInter₂_comm @[simp] theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) : ⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h = ⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ := by simp only [iUnion_and, @iUnion_comm _ ι'] #align set.bUnion_and Set.biUnion_and @[simp] theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) : ⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h = ⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ := by simp only [iUnion_and, @iUnion_comm _ ι] #align set.bUnion_and' Set.biUnion_and' @[simp] theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) : ⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h = ⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ := by simp only [iInter_and, @iInter_comm _ ι'] #align set.bInter_and Set.biInter_and @[simp] theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) : ⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h = ⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ := by simp only [iInter_and, @iInter_comm _ ι] #align set.bInter_and' Set.biInter_and' @[simp] theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} : ⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left] #align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left @[simp] theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} : ⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left] #align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) : y ∈ ⋃ x ∈ s, t x := mem_iUnion₂_of_mem xs ytx #align set.mem_bUnion Set.mem_biUnion theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) : y ∈ ⋂ x ∈ s, t x := mem_iInter₂_of_mem h #align set.mem_bInter Set.mem_biInter theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) : u x ⊆ ⋃ x ∈ s, u x := -- Porting note: Why is this not just `subset_iUnion₂ x xs`? @subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs #align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) : ⋂ x ∈ s, t x ⊆ t x := iInter₂_subset x xs #align set.bInter_subset_of_mem Set.biInter_subset_of_mem theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') : ⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x := iUnion₂_subset fun _ hx => subset_biUnion_of_mem <\| h hx #align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) : ⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x := subset_iInter₂ fun _ hx => biInter_subset_of_mem <\| h hx #align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) : ⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x := (biUnion_subset_biUnion_left hs).trans <\| iUnion₂_mono h #align set.bUnion_mono Set.biUnion_mono theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) : ⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x := (biInter_subset_biInter_left hs).trans <\| iInter₂_mono h #align set.bInter_mono Set.biInter_mono theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) : ⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 := iSup_subtype' #align set.bUnion_eq_Union Set.biUnion_eq_iUnion theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) : ⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 := iInf_subtype' #align set.bInter_eq_Inter Set.biInter_eq_iInter theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) : ⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ := iSup_subtype #align set.Union_subtype Set.iUnion_subtype theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) : ⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ := iInf_subtype #align set.Inter_subtype Set.iInter_subtype theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ := iInf_emptyset #align set.bInter_empty Set.biInter_empty theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x := iInf_univ #align set.bInter_univ Set.biInter_univ @[simp] theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s := Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx #align set.bUnion_self Set.biUnion_self @[simp] theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by rw [iUnion_nonempty_index, biUnion_self] #align set.Union_nonempty_self Set.iUnion_nonempty_self theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a := iInf_singleton #align set.bInter_singleton Set.biInter_singleton theorem biInter_union (s t : Set α) (u : α → Set β) : ⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x := iInf_union #align set.bInter_union Set.biInter_union theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) : ⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp #align set.bInter_insert Set.biInter_insert theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by rw [biInter_insert, biInter_singleton] #align set.bInter_pair Set.biInter_pair theorem biInter_inter {ι α : Type} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) : ⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by haveI : Nonempty s := hs.to_subtype simp [biInter_eq_iInter, ← iInter_inter] #align set.bInter_inter Set.biInter_inter theorem inter_biInter {ι α : Type} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) : ⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by rw [inter_comm, ← biInter_inter hs] simp [inter_comm] #align set.inter_bInter Set.inter_biInter theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ := iSup_emptyset #align set.bUnion_empty Set.biUnion_empty theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x := iSup_univ #align set.bUnion_univ Set.biUnion_univ theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a := iSup_singleton #align set.bUnion_singleton Set.biUnion_singleton @[simp] theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s := ext <\| by simp #align set.bUnion_of_singleton Set.biUnion_of_singleton theorem biUnion_union (s t : Set α) (u : α → Set β) : ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x := iSup_union #align set.bUnion_union Set.biUnion_union @[simp] theorem iUnion_coe_set {α β : Type} (s : Set α) (f : s → Set β) : ⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ := iUnion_subtype _ _ #align set.Union_coe_set Set.iUnion_coe_set @[simp] theorem iInter_coe_set {α β : Type} (s : Set α) (f : s → Set β) : ⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ := iInter_subtype _ _ #align set.Inter_coe_set Set.iInter_coe_set theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) : ⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp #align set.bUnion_insert Set.biUnion_insert theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b := by simp #align set.bUnion_pair Set.biUnion_pair theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) : (s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion] #align set.inter_Union₂ Set.inter_iUnion₂ theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) : (⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter] #align set.Union₂_inter Set.iUnion₂_inter theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) : (s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter] #align set.union_Inter₂ Set.union_iInter₂ theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) : (⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union] #align set.Inter₂_union Set.iInter₂_union theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) : x ∈ ⋃₀S := ⟨t, ht, hx⟩ #align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem -- is this theorem really necessary? theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S) (ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩ #align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t := sInf_le tS #align set.sInter_subset_of_mem Set.sInter_subset_of_mem theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S := le_sSup tS #align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u) (h₂ : u ∈ t) : s ⊆ ⋃₀t := Subset.trans h₁ (subset_sUnion_of_mem h₂) #align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t := sSup_le h #align set.sUnion_subset Set.sUnion_subset @[simp] theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t := sSup_le_iff #align set.sUnion_subset_iff Set.sUnion_subset_iff lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) : ⋃₀ s ⊆ ⋃₀ (f '' s) := fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩ lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) : ⋃₀ (f '' s) ⊆ ⋃₀ s := -- If t ∈ f '' s is arbitrary; t = f u for some u : Set α. fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩ theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S := le_sInf h #align set.subset_sInter Set.subset_sInter @[simp] theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' := le_sInf_iff #align set.subset_sInter_iff Set.subset_sInter_iff @[gcongr] theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T := sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs) #align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion @[gcongr] theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S := subset_sInter fun _ hs => sInter_subset_of_mem (h hs) #align set.sInter_subset_sInter Set.sInter_subset_sInter @[simp] theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) := sSup_empty #align set.sUnion_empty Set.sUnion_empty @[simp] theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) := sInf_empty #align set.sInter_empty Set.sInter_empty @[simp] theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s := sSup_singleton #align set.sUnion_singleton Set.sUnion_singleton @[simp] theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s := sInf_singleton #align set.sInter_singleton Set.sInter_singleton @[simp] theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ := sSup_eq_bot #align set.sUnion_eq_empty Set.sUnion_eq_empty @[simp] theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ := sInf_eq_top #align set.sInter_eq_univ Set.sInter_eq_univ theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t := sUnion_subset_iff.symm theorem sUnion_powerset_gc : GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) := gc_sSup_Iic def sUnion_powerset_gi : GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) := gi_sSup_Iic theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) : ⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall] rintro ⟨s, hs, hne⟩ obtain rfl : s = univ := (h hs).resolve_left hne exact univ_subset_iff.1 <\| subset_sUnion_of_mem hs @[simp] theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by simp [nonempty_iff_ne_empty] #align set.nonempty_sUnion Set.nonempty_sUnion theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty := let ⟨s, hs, _⟩ := nonempty_sUnion.1 h ⟨s, hs⟩ #align set.nonempty.of_sUnion Set.Nonempty.of_sUnion theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty := Nonempty.of_sUnion <\| h.symm ▸ univ_nonempty #align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T := sSup_union #align set.sUnion_union Set.sUnion_union theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T := sInf_union #align set.sInter_union Set.sInter_union @[simp] theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T := sSup_insert #align set.sUnion_insert Set.sUnion_insert @[simp] theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T := sInf_insert #align set.sInter_insert Set.sInter_insert @[simp] theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s := sSup_diff_singleton_bot s #align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty @[simp] theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s := sInf_diff_singleton_top s #align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t := sSup_pair #align set.sUnion_pair Set.sUnion_pair theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t := sInf_pair #align set.sInter_pair Set.sInter_pair @[simp] theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x := sSup_image #align set.sUnion_image Set.sUnion_image @[simp] theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x := sInf_image #align set.sInter_image Set.sInter_image @[simp] theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x := rfl #align set.sUnion_range Set.sUnion_range @[simp] theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x := rfl #align set.sInter_range Set.sInter_range theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by simp only [eq_univ_iff_forall, mem_iUnion] #align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} : ⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j := by simp only [iUnion_eq_univ_iff, mem_iUnion] #align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by simp only [eq_univ_iff_forall, mem_sUnion] #align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff -- classical theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by simp [Set.eq_empty_iff_forall_not_mem] #align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff -- classical theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} : ⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall] #align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff -- classical theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by simp [Set.eq_empty_iff_forall_not_mem] #align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff -- classical @[simp] theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by simp [nonempty_iff_ne_empty, iInter_eq_empty_iff] #align set.nonempty_Inter Set.nonempty_iInter -- classical -- Porting note (#10618): removing `simp`. `simp` can prove it theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} : (⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by simp #align set.nonempty_Inter₂ Set.nonempty_iInter₂ -- classical @[simp] theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by simp [nonempty_iff_ne_empty, sInter_eq_empty_iff] #align set.nonempty_sInter Set.nonempty_sInter -- classical theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) := ext fun x => by simp #align set.compl_sUnion Set.compl_sUnion -- classical theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by rw [← compl_compl (⋃₀S), compl_sUnion] #align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl -- classical theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by rw [sUnion_eq_compl_sInter_compl, compl_compl_image] #align set.compl_sInter Set.compl_sInter -- classical theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by rw [← compl_compl (⋂₀ S), compl_sInter] #align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S) (h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ := eq_empty_of_subset_empty <\| by rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs) #align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty theorem range_sigma_eq_iUnion_range {γ : α → Type} (f : Sigma γ → β) : range f = ⋃ a, range fun b => f ⟨a, b⟩ := Set.ext <\| by simp #align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by simp [Set.ext_iff] #align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by simp [Set.ext_iff] #align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type} {σ : ι → Type} (s : Set (Sigma σ)) : ⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by ext x simp only [mem_iUnion, mem_image, mem_preimage] constructor · rintro ⟨i, a, h, rfl⟩ exact h · intro h cases' x with i a exact ⟨i, a, h, rfl⟩ #align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self theorem Sigma.univ (X : α → Type) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) := Set.ext fun x => iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩ #align set.sigma.univ Set.Sigma.univ alias sUnion_mono := sUnion_subset_sUnion #align set.sUnion_mono Set.sUnion_mono theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s := iSup_const_mono (α := Set α) h #align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const @[simp] theorem iUnion_singleton_eq_range {α β : Type} (f : α → β) : ⋃ x : α, {f x} = range f := by ext x simp [@eq_comm _ x] #align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range theorem iUnion_of_singleton (α : Type) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff] #align set.Union_of_singleton Set.iUnion_of_singleton theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp #align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by rw [← sUnion_image, image_id'] #align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by rw [← sInter_image, image_id'] #align set.sInter_eq_bInter Set.sInter_eq_biInter theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by simp only [← sUnion_range, Subtype.range_coe] #align set.sUnion_eq_Union Set.sUnion_eq_iUnion theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by simp only [← sInter_range, Subtype.range_coe] #align set.sInter_eq_Inter Set.sInter_eq_iInter @[simp] theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ := iSup_of_empty _ #align set.Union_of_empty Set.iUnion_of_empty @[simp] theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ := iInf_of_empty _ #align set.Inter_of_empty Set.iInter_of_empty theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ := sup_eq_iSup s₁ s₂ #align set.union_eq_Union Set.union_eq_iUnion theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ := inf_eq_iInf s₁ s₂ #align set.inter_eq_Inter Set.inter_eq_iInter theorem sInter_union_sInter {S T : Set (Set α)} : ⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 := sInf_sup_sInf #align set.sInter_union_sInter Set.sInter_union_sInter theorem sUnion_inter_sUnion {s t : Set (Set α)} : ⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 := sSup_inf_sSup #align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) : ⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι] #align set.bUnion_Union Set.biUnion_iUnion theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) : ⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι] #align set.bInter_Union Set.biInter_iUnion theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by simp only [sUnion_eq_biUnion, biUnion_iUnion] #align set.sUnion_Union Set.sUnion_iUnion theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by simp only [sInter_eq_biInter, biInter_iUnion] #align set.sInter_Union Set.sInter_iUnion theorem iUnion_range_eq_sUnion {α β : Type} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)} (hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by ext x; constructor · rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩ refine ⟨_, hs, ?_⟩ exact (f ⟨s, hs⟩ y).2 · rintro ⟨s, hs, hx⟩ cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy refine ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, ?_⟩ exact congr_arg Subtype.val hy #align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x} (hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by ext x; rw [mem_iUnion, mem_iUnion]; constructor · rintro ⟨y, i, rfl⟩ exact ⟨i, (f i y).2⟩ · rintro ⟨i, hx⟩ cases' hf i ⟨x, hx⟩ with y hy exact ⟨y, i, congr_arg Subtype.val hy⟩ #align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i := sup_iInf_eq _ _ #align set.union_distrib_Inter_left Set.union_distrib_iInter_left theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) : (s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left] #align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t := iInf_sup_eq _ _ #align set.union_distrib_Inter_right Set.union_distrib_iInter_right theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) : (⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right] #align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right section Pi variable {π : α → Type}	Mathlib/Data/Set/Lattice.lean	1,982	1,984	theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by	ext simp
import Mathlib.Geometry.Euclidean.Circumcenter #align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0" noncomputable section open scoped Classical open scoped RealInnerProductSpace namespace Affine namespace Simplex open Finset AffineSubspace EuclideanGeometry PointsWithCircumcenterIndex variable {V : Type} {P : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] def mongePoint {n : ℕ} (s : Simplex ℝ P n) : P := (((n + 1 : ℕ) : ℝ) / ((n - 1 : ℕ) : ℝ)) • ((univ : Finset (Fin (n + 1))).centroid ℝ s.points -ᵥ s.circumcenter) +ᵥ s.circumcenter #align affine.simplex.monge_point Affine.Simplex.mongePoint theorem mongePoint_eq_smul_vsub_vadd_circumcenter {n : ℕ} (s : Simplex ℝ P n) : s.mongePoint = (((n + 1 : ℕ) : ℝ) / ((n - 1 : ℕ) : ℝ)) • ((univ : Finset (Fin (n + 1))).centroid ℝ s.points -ᵥ s.circumcenter) +ᵥ s.circumcenter := rfl #align affine.simplex.monge_point_eq_smul_vsub_vadd_circumcenter Affine.Simplex.mongePoint_eq_smul_vsub_vadd_circumcenter theorem mongePoint_mem_affineSpan {n : ℕ} (s : Simplex ℝ P n) : s.mongePoint ∈ affineSpan ℝ (Set.range s.points) := smul_vsub_vadd_mem _ _ (centroid_mem_affineSpan_of_card_eq_add_one ℝ _ (card_fin (n + 1))) s.circumcenter_mem_affineSpan s.circumcenter_mem_affineSpan #align affine.simplex.monge_point_mem_affine_span Affine.Simplex.mongePoint_mem_affineSpan theorem mongePoint_eq_of_range_eq {n : ℕ} {s₁ s₂ : Simplex ℝ P n} (h : Set.range s₁.points = Set.range s₂.points) : s₁.mongePoint = s₂.mongePoint := by simp_rw [mongePoint_eq_smul_vsub_vadd_circumcenter, centroid_eq_of_range_eq h, circumcenter_eq_of_range_eq h] #align affine.simplex.monge_point_eq_of_range_eq Affine.Simplex.mongePoint_eq_of_range_eq def mongePointWeightsWithCircumcenter (n : ℕ) : PointsWithCircumcenterIndex (n + 2) → ℝ \| pointIndex _ => ((n + 1 : ℕ) : ℝ)⁻¹ \| circumcenterIndex => -2 / ((n + 1 : ℕ) : ℝ) #align affine.simplex.monge_point_weights_with_circumcenter Affine.Simplex.mongePointWeightsWithCircumcenter @[simp] theorem sum_mongePointWeightsWithCircumcenter (n : ℕ) : ∑ i, mongePointWeightsWithCircumcenter n i = 1 := by simp_rw [sum_pointsWithCircumcenter, mongePointWeightsWithCircumcenter, sum_const, card_fin, nsmul_eq_mul] -- Porting note: replaced -- have hn1 : (n + 1 : ℝ) ≠ 0 := mod_cast Nat.succ_ne_zero _ field_simp [n.cast_add_one_ne_zero] ring #align affine.simplex.sum_monge_point_weights_with_circumcenter Affine.Simplex.sum_mongePointWeightsWithCircumcenter theorem mongePoint_eq_affineCombination_of_pointsWithCircumcenter {n : ℕ} (s : Simplex ℝ P (n + 2)) : s.mongePoint = (univ : Finset (PointsWithCircumcenterIndex (n + 2))).affineCombination ℝ s.pointsWithCircumcenter (mongePointWeightsWithCircumcenter n) := by rw [mongePoint_eq_smul_vsub_vadd_circumcenter, centroid_eq_affineCombination_of_pointsWithCircumcenter, circumcenter_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub, ← LinearMap.map_smul, weightedVSub_vadd_affineCombination] congr with i rw [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Pi.sub_apply] -- Porting note: replaced -- have hn1 : (n + 1 : ℝ) ≠ 0 := mod_cast Nat.succ_ne_zero _ have hn1 : (n + 1 : ℝ) ≠ 0 := n.cast_add_one_ne_zero cases i <;> simp_rw [centroidWeightsWithCircumcenter, circumcenterWeightsWithCircumcenter, mongePointWeightsWithCircumcenter] <;> rw [add_tsub_assoc_of_le (by decide : 1 ≤ 2), (by decide : 2 - 1 = 1)] · rw [if_pos (mem_univ _), sub_zero, add_zero, card_fin] -- Porting note: replaced -- have hn3 : (n + 2 + 1 : ℝ) ≠ 0 := mod_cast Nat.succ_ne_zero _ have hn3 : (n + 2 + 1 : ℝ) ≠ 0 := by norm_cast field_simp [hn1, hn3, mul_comm] · field_simp [hn1] ring #align affine.simplex.monge_point_eq_affine_combination_of_points_with_circumcenter Affine.Simplex.mongePoint_eq_affineCombination_of_pointsWithCircumcenter def mongePointVSubFaceCentroidWeightsWithCircumcenter {n : ℕ} (i₁ i₂ : Fin (n + 3)) : PointsWithCircumcenterIndex (n + 2) → ℝ \| pointIndex i => if i = i₁ ∨ i = i₂ then ((n + 1 : ℕ) : ℝ)⁻¹ else 0 \| circumcenterIndex => -2 / ((n + 1 : ℕ) : ℝ) #align affine.simplex.monge_point_vsub_face_centroid_weights_with_circumcenter Affine.Simplex.mongePointVSubFaceCentroidWeightsWithCircumcenter theorem mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub {n : ℕ} {i₁ i₂ : Fin (n + 3)} (h : i₁ ≠ i₂) : mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ = mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i₁, i₂}ᶜ := by ext i cases' i with i · rw [Pi.sub_apply, mongePointWeightsWithCircumcenter, centroidWeightsWithCircumcenter, mongePointVSubFaceCentroidWeightsWithCircumcenter] have hu : card ({i₁, i₂}ᶜ : Finset (Fin (n + 3))) = n + 1 := by simp [card_compl, Fintype.card_fin, h] rw [hu] by_cases hi : i = i₁ ∨ i = i₂ <;> simp [compl_eq_univ_sdiff, hi] · simp [mongePointWeightsWithCircumcenter, centroidWeightsWithCircumcenter, mongePointVSubFaceCentroidWeightsWithCircumcenter] #align affine.simplex.monge_point_vsub_face_centroid_weights_with_circumcenter_eq_sub Affine.Simplex.mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub @[simp] theorem sum_mongePointVSubFaceCentroidWeightsWithCircumcenter {n : ℕ} {i₁ i₂ : Fin (n + 3)} (h : i₁ ≠ i₂) : ∑ i, mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i = 0 := by rw [mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h] simp_rw [Pi.sub_apply, sum_sub_distrib, sum_mongePointWeightsWithCircumcenter] rw [sum_centroidWeightsWithCircumcenter, sub_self] simp [← card_pos, card_compl, h] #align affine.simplex.sum_monge_point_vsub_face_centroid_weights_with_circumcenter Affine.Simplex.sum_mongePointVSubFaceCentroidWeightsWithCircumcenter theorem mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter {n : ℕ} (s : Simplex ℝ P (n + 2)) {i₁ i₂ : Fin (n + 3)} (h : i₁ ≠ i₂) : s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points = (univ : Finset (PointsWithCircumcenterIndex (n + 2))).weightedVSub s.pointsWithCircumcenter (mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂) := by simp_rw [mongePoint_eq_affineCombination_of_pointsWithCircumcenter, centroid_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub, mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h] #align affine.simplex.monge_point_vsub_face_centroid_eq_weighted_vsub_of_points_with_circumcenter Affine.Simplex.mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter theorem inner_mongePoint_vsub_face_centroid_vsub {n : ℕ} (s : Simplex ℝ P (n + 2)) {i₁ i₂ : Fin (n + 3)} : ⟪s.mongePoint -ᵥ ({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0 := by by_cases h : i₁ = i₂ · simp [h] simp_rw [mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter s h, point_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub] have hs : ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0 := by simp rw [inner_weightedVSub _ (sum_mongePointVSubFaceCentroidWeightsWithCircumcenter h) _ hs, sum_pointsWithCircumcenter, pointsWithCircumcenter_eq_circumcenter] simp only [mongePointVSubFaceCentroidWeightsWithCircumcenter, pointsWithCircumcenter_point] let fs : Finset (Fin (n + 3)) := {i₁, i₂} have hfs : ∀ i : Fin (n + 3), i ∉ fs → i ≠ i₁ ∧ i ≠ i₂ := by intro i hi constructor <;> · intro hj; simp [fs, ← hj] at hi rw [← sum_subset fs.subset_univ _] · simp_rw [sum_pointsWithCircumcenter, pointsWithCircumcenter_eq_circumcenter, pointsWithCircumcenter_point, Pi.sub_apply, pointWeightsWithCircumcenter] rw [← sum_subset fs.subset_univ _] · simp_rw [sum_insert (not_mem_singleton.2 h), sum_singleton] repeat rw [← sum_subset fs.subset_univ _] · simp_rw [sum_insert (not_mem_singleton.2 h), sum_singleton] simp [h, Ne.symm h, dist_comm (s.points i₁)] all_goals intro i _ hi; simp [hfs i hi] · intro i _ hi simp [hfs i hi, pointsWithCircumcenter] · intro i _ hi simp [hfs i hi] #align affine.simplex.inner_monge_point_vsub_face_centroid_vsub Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub def mongePlane {n : ℕ} (s : Simplex ℝ P (n + 2)) (i₁ i₂ : Fin (n + 3)) : AffineSubspace ℝ P := mk' (({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points) (ℝ ∙ s.points i₁ -ᵥ s.points i₂)ᗮ ⊓ affineSpan ℝ (Set.range s.points) #align affine.simplex.monge_plane Affine.Simplex.mongePlane theorem mongePlane_def {n : ℕ} (s : Simplex ℝ P (n + 2)) (i₁ i₂ : Fin (n + 3)) : s.mongePlane i₁ i₂ = mk' (({i₁, i₂}ᶜ : Finset (Fin (n + 3))).centroid ℝ s.points) (ℝ ∙ s.points i₁ -ᵥ s.points i₂)ᗮ ⊓ affineSpan ℝ (Set.range s.points) := rfl #align affine.simplex.monge_plane_def Affine.Simplex.mongePlane_def theorem mongePlane_comm {n : ℕ} (s : Simplex ℝ P (n + 2)) (i₁ i₂ : Fin (n + 3)) : s.mongePlane i₁ i₂ = s.mongePlane i₂ i₁ := by simp_rw [mongePlane_def] congr 3 · congr 1 exact pair_comm _ _ · ext simp_rw [Submodule.mem_span_singleton] constructor all_goals rintro ⟨r, rfl⟩; use -r; rw [neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev] #align affine.simplex.monge_plane_comm Affine.Simplex.mongePlane_comm theorem mongePoint_mem_mongePlane {n : ℕ} (s : Simplex ℝ P (n + 2)) {i₁ i₂ : Fin (n + 3)} : s.mongePoint ∈ s.mongePlane i₁ i₂ := by rw [mongePlane_def, mem_inf_iff, ← vsub_right_mem_direction_iff_mem (self_mem_mk' _ _), direction_mk', Submodule.mem_orthogonal'] refine ⟨?_, s.mongePoint_mem_affineSpan⟩ intro v hv rcases Submodule.mem_span_singleton.mp hv with ⟨r, rfl⟩ rw [inner_smul_right, s.inner_mongePoint_vsub_face_centroid_vsub, mul_zero] #align affine.simplex.monge_point_mem_monge_plane Affine.Simplex.mongePoint_mem_mongePlane theorem direction_mongePlane {n : ℕ} (s : Simplex ℝ P (n + 2)) {i₁ i₂ : Fin (n + 3)} : (s.mongePlane i₁ i₂).direction = (ℝ ∙ s.points i₁ -ᵥ s.points i₂)ᗮ ⊓ vectorSpan ℝ (Set.range s.points) := by rw [mongePlane_def, direction_inf_of_mem_inf s.mongePoint_mem_mongePlane, direction_mk', direction_affineSpan] #align affine.simplex.direction_monge_plane Affine.Simplex.direction_mongePlane theorem eq_mongePoint_of_forall_mem_mongePlane {n : ℕ} {s : Simplex ℝ P (n + 2)} {i₁ : Fin (n + 3)} {p : P} (h : ∀ i₂, i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂) : p = s.mongePoint := by rw [← @vsub_eq_zero_iff_eq V] have h' : ∀ i₂, i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈ (ℝ ∙ s.points i₁ -ᵥ s.points i₂)ᗮ ⊓ vectorSpan ℝ (Set.range s.points) := by intro i₂ hne rw [← s.direction_mongePlane, vsub_right_mem_direction_iff_mem s.mongePoint_mem_mongePlane] exact h i₂ hne have hi : p -ᵥ s.mongePoint ∈ ⨅ i₂ : { i // i₁ ≠ i }, (ℝ ∙ s.points i₁ -ᵥ s.points i₂)ᗮ := by rw [Submodule.mem_iInf] exact fun i => (Submodule.mem_inf.1 (h' i i.property)).1 rw [Submodule.iInf_orthogonal, ← Submodule.span_iUnion] at hi have hu : ⋃ i : { i // i₁ ≠ i }, ({s.points i₁ -ᵥ s.points i} : Set V) = (s.points i₁ -ᵥ ·) '' (s.points '' (Set.univ \ {i₁})) := by rw [Set.image_image] ext x simp_rw [Set.mem_iUnion, Set.mem_image, Set.mem_singleton_iff, Set.mem_diff_singleton] constructor · rintro ⟨i, rfl⟩ use i, ⟨Set.mem_univ _, i.property.symm⟩ · rintro ⟨i, ⟨-, hi⟩, rfl⟩ -- Porting note: was `use ⟨i, hi.symm⟩, rfl` exact ⟨⟨i, hi.symm⟩, rfl⟩ rw [hu, ← vectorSpan_image_eq_span_vsub_set_left_ne ℝ _ (Set.mem_univ _), Set.image_univ] at hi have hv : p -ᵥ s.mongePoint ∈ vectorSpan ℝ (Set.range s.points) := by let s₁ : Finset (Fin (n + 3)) := univ.erase i₁ obtain ⟨i₂, h₂⟩ := card_pos.1 (show 0 < card s₁ by simp [s₁, card_erase_of_mem]) have h₁₂ : i₁ ≠ i₂ := (ne_of_mem_erase h₂).symm exact (Submodule.mem_inf.1 (h' i₂ h₁₂)).2 exact Submodule.disjoint_def.1 (vectorSpan ℝ (Set.range s.points)).orthogonal_disjoint _ hv hi #align affine.simplex.eq_monge_point_of_forall_mem_monge_plane Affine.Simplex.eq_mongePoint_of_forall_mem_mongePlane def altitude {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) : AffineSubspace ℝ P := mk' (s.points i) (affineSpan ℝ (s.points '' ↑(univ.erase i))).directionᗮ ⊓ affineSpan ℝ (Set.range s.points) #align affine.simplex.altitude Affine.Simplex.altitude theorem altitude_def {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) : s.altitude i = mk' (s.points i) (affineSpan ℝ (s.points '' ↑(univ.erase i))).directionᗮ ⊓ affineSpan ℝ (Set.range s.points) := rfl #align affine.simplex.altitude_def Affine.Simplex.altitude_def theorem mem_altitude {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) : s.points i ∈ s.altitude i := (mem_inf_iff _ _ _).2 ⟨self_mem_mk' _ _, mem_affineSpan ℝ (Set.mem_range_self _)⟩ #align affine.simplex.mem_altitude Affine.Simplex.mem_altitude	Mathlib/Geometry/Euclidean/MongePoint.lean	352	357	theorem direction_altitude {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) : (s.altitude i).direction = (vectorSpan ℝ (s.points '' ↑(Finset.univ.erase i)))ᗮ ⊓ vectorSpan ℝ (Set.range s.points) := by	rw [altitude_def, direction_inf_of_mem (self_mem_mk' (s.points i) _) (mem_affineSpan ℝ (Set.mem_range_self _)), direction_mk', direction_affineSpan, direction_affineSpan]
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic import Mathlib.Analysis.Normed.Group.AddCircle import Mathlib.Algebra.CharZero.Quotient import Mathlib.Topology.Instances.Sign #align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829ce42efec" open Real noncomputable section namespace Real -- Porting note: can't derive `NormedAddCommGroup, Inhabited` def Angle : Type := AddCircle (2 * π) #align real.angle Real.Angle namespace Angle -- Porting note (#10754): added due to missing instances due to no deriving instance : NormedAddCommGroup Angle := inferInstanceAs (NormedAddCommGroup (AddCircle (2 * π))) -- Porting note (#10754): added due to missing instances due to no deriving instance : Inhabited Angle := inferInstanceAs (Inhabited (AddCircle (2 * π))) -- Porting note (#10754): added due to missing instances due to no deriving -- also, without this, a plain `QuotientAddGroup.mk` -- causes coerced terms to be of type `ℝ ⧸ AddSubgroup.zmultiples (2 * π)` @[coe] protected def coe (r : ℝ) : Angle := QuotientAddGroup.mk r instance : Coe ℝ Angle := ⟨Angle.coe⟩ instance : CircularOrder Real.Angle := QuotientAddGroup.circularOrder (hp' := ⟨by norm_num [pi_pos]⟩) @[continuity] theorem continuous_coe : Continuous ((↑) : ℝ → Angle) := continuous_quotient_mk' #align real.angle.continuous_coe Real.Angle.continuous_coe def coeHom : ℝ →+ Angle := QuotientAddGroup.mk' _ #align real.angle.coe_hom Real.Angle.coeHom @[simp] theorem coe_coeHom : (coeHom : ℝ → Angle) = ((↑) : ℝ → Angle) := rfl #align real.angle.coe_coe_hom Real.Angle.coe_coeHom @[elab_as_elim] protected theorem induction_on {p : Angle → Prop} (θ : Angle) (h : ∀ x : ℝ, p x) : p θ := Quotient.inductionOn' θ h #align real.angle.induction_on Real.Angle.induction_on @[simp] theorem coe_zero : ↑(0 : ℝ) = (0 : Angle) := rfl #align real.angle.coe_zero Real.Angle.coe_zero @[simp] theorem coe_add (x y : ℝ) : ↑(x + y : ℝ) = (↑x + ↑y : Angle) := rfl #align real.angle.coe_add Real.Angle.coe_add @[simp] theorem coe_neg (x : ℝ) : ↑(-x : ℝ) = -(↑x : Angle) := rfl #align real.angle.coe_neg Real.Angle.coe_neg @[simp] theorem coe_sub (x y : ℝ) : ↑(x - y : ℝ) = (↑x - ↑y : Angle) := rfl #align real.angle.coe_sub Real.Angle.coe_sub theorem coe_nsmul (n : ℕ) (x : ℝ) : ↑(n • x : ℝ) = n • (↑x : Angle) := rfl #align real.angle.coe_nsmul Real.Angle.coe_nsmul theorem coe_zsmul (z : ℤ) (x : ℝ) : ↑(z • x : ℝ) = z • (↑x : Angle) := rfl #align real.angle.coe_zsmul Real.Angle.coe_zsmul @[simp, norm_cast] theorem natCast_mul_eq_nsmul (x : ℝ) (n : ℕ) : ↑((n : ℝ) * x) = n • (↑x : Angle) := by simpa only [nsmul_eq_mul] using coeHom.map_nsmul x n #align real.angle.coe_nat_mul_eq_nsmul Real.Angle.natCast_mul_eq_nsmul @[simp, norm_cast] theorem intCast_mul_eq_zsmul (x : ℝ) (n : ℤ) : ↑((n : ℝ) * x : ℝ) = n • (↑x : Angle) := by simpa only [zsmul_eq_mul] using coeHom.map_zsmul x n #align real.angle.coe_int_mul_eq_zsmul Real.Angle.intCast_mul_eq_zsmul @[deprecated (since := "2024-05-25")] alias coe_nat_mul_eq_nsmul := natCast_mul_eq_nsmul @[deprecated (since := "2024-05-25")] alias coe_int_mul_eq_zsmul := intCast_mul_eq_zsmul theorem angle_eq_iff_two_pi_dvd_sub {ψ θ : ℝ} : (θ : Angle) = ψ ↔ ∃ k : ℤ, θ - ψ = 2 * π * k := by simp only [QuotientAddGroup.eq, AddSubgroup.zmultiples_eq_closure, AddSubgroup.mem_closure_singleton, zsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm] -- Porting note: added `rw`, `simp [Angle.coe, QuotientAddGroup.eq]` doesn't fire otherwise rw [Angle.coe, Angle.coe, QuotientAddGroup.eq] simp only [AddSubgroup.zmultiples_eq_closure, AddSubgroup.mem_closure_singleton, zsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm] #align real.angle.angle_eq_iff_two_pi_dvd_sub Real.Angle.angle_eq_iff_two_pi_dvd_sub @[simp] theorem coe_two_pi : ↑(2 * π : ℝ) = (0 : Angle) := angle_eq_iff_two_pi_dvd_sub.2 ⟨1, by rw [sub_zero, Int.cast_one, mul_one]⟩ #align real.angle.coe_two_pi Real.Angle.coe_two_pi @[simp] theorem neg_coe_pi : -(π : Angle) = π := by rw [← coe_neg, angle_eq_iff_two_pi_dvd_sub] use -1 simp [two_mul, sub_eq_add_neg] #align real.angle.neg_coe_pi Real.Angle.neg_coe_pi @[simp] theorem two_nsmul_coe_div_two (θ : ℝ) : (2 : ℕ) • (↑(θ / 2) : Angle) = θ := by rw [← coe_nsmul, two_nsmul, add_halves] #align real.angle.two_nsmul_coe_div_two Real.Angle.two_nsmul_coe_div_two @[simp] theorem two_zsmul_coe_div_two (θ : ℝ) : (2 : ℤ) • (↑(θ / 2) : Angle) = θ := by rw [← coe_zsmul, two_zsmul, add_halves] #align real.angle.two_zsmul_coe_div_two Real.Angle.two_zsmul_coe_div_two -- Porting note (#10618): @[simp] can prove it theorem two_nsmul_neg_pi_div_two : (2 : ℕ) • (↑(-π / 2) : Angle) = π := by rw [two_nsmul_coe_div_two, coe_neg, neg_coe_pi] #align real.angle.two_nsmul_neg_pi_div_two Real.Angle.two_nsmul_neg_pi_div_two -- Porting note (#10618): @[simp] can prove it theorem two_zsmul_neg_pi_div_two : (2 : ℤ) • (↑(-π / 2) : Angle) = π := by rw [two_zsmul, ← two_nsmul, two_nsmul_neg_pi_div_two] #align real.angle.two_zsmul_neg_pi_div_two Real.Angle.two_zsmul_neg_pi_div_two theorem sub_coe_pi_eq_add_coe_pi (θ : Angle) : θ - π = θ + π := by rw [sub_eq_add_neg, neg_coe_pi] #align real.angle.sub_coe_pi_eq_add_coe_pi Real.Angle.sub_coe_pi_eq_add_coe_pi @[simp] theorem two_nsmul_coe_pi : (2 : ℕ) • (π : Angle) = 0 := by simp [← natCast_mul_eq_nsmul] #align real.angle.two_nsmul_coe_pi Real.Angle.two_nsmul_coe_pi @[simp] theorem two_zsmul_coe_pi : (2 : ℤ) • (π : Angle) = 0 := by simp [← intCast_mul_eq_zsmul] #align real.angle.two_zsmul_coe_pi Real.Angle.two_zsmul_coe_pi @[simp] theorem coe_pi_add_coe_pi : (π : Real.Angle) + π = 0 := by rw [← two_nsmul, two_nsmul_coe_pi] #align real.angle.coe_pi_add_coe_pi Real.Angle.coe_pi_add_coe_pi theorem zsmul_eq_iff {ψ θ : Angle} {z : ℤ} (hz : z ≠ 0) : z • ψ = z • θ ↔ ∃ k : Fin z.natAbs, ψ = θ + (k : ℕ) • (2 * π / z : ℝ) := QuotientAddGroup.zmultiples_zsmul_eq_zsmul_iff hz #align real.angle.zsmul_eq_iff Real.Angle.zsmul_eq_iff theorem nsmul_eq_iff {ψ θ : Angle} {n : ℕ} (hz : n ≠ 0) : n • ψ = n • θ ↔ ∃ k : Fin n, ψ = θ + (k : ℕ) • (2 * π / n : ℝ) := QuotientAddGroup.zmultiples_nsmul_eq_nsmul_iff hz #align real.angle.nsmul_eq_iff Real.Angle.nsmul_eq_iff theorem two_zsmul_eq_iff {ψ θ : Angle} : (2 : ℤ) • ψ = (2 : ℤ) • θ ↔ ψ = θ ∨ ψ = θ + ↑π := by -- Porting note: no `Int.natAbs_bit0` anymore have : Int.natAbs 2 = 2 := rfl rw [zsmul_eq_iff two_ne_zero, this, Fin.exists_fin_two, Fin.val_zero, Fin.val_one, zero_smul, add_zero, one_smul, Int.cast_two, mul_div_cancel_left₀ (_ : ℝ) two_ne_zero] #align real.angle.two_zsmul_eq_iff Real.Angle.two_zsmul_eq_iff theorem two_nsmul_eq_iff {ψ θ : Angle} : (2 : ℕ) • ψ = (2 : ℕ) • θ ↔ ψ = θ ∨ ψ = θ + ↑π := by simp_rw [← natCast_zsmul, Nat.cast_ofNat, two_zsmul_eq_iff] #align real.angle.two_nsmul_eq_iff Real.Angle.two_nsmul_eq_iff theorem two_nsmul_eq_zero_iff {θ : Angle} : (2 : ℕ) • θ = 0 ↔ θ = 0 ∨ θ = π := by convert two_nsmul_eq_iff <;> simp #align real.angle.two_nsmul_eq_zero_iff Real.Angle.two_nsmul_eq_zero_iff theorem two_nsmul_ne_zero_iff {θ : Angle} : (2 : ℕ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π := by rw [← not_or, ← two_nsmul_eq_zero_iff] #align real.angle.two_nsmul_ne_zero_iff Real.Angle.two_nsmul_ne_zero_iff theorem two_zsmul_eq_zero_iff {θ : Angle} : (2 : ℤ) • θ = 0 ↔ θ = 0 ∨ θ = π := by simp_rw [two_zsmul, ← two_nsmul, two_nsmul_eq_zero_iff] #align real.angle.two_zsmul_eq_zero_iff Real.Angle.two_zsmul_eq_zero_iff theorem two_zsmul_ne_zero_iff {θ : Angle} : (2 : ℤ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π := by rw [← not_or, ← two_zsmul_eq_zero_iff] #align real.angle.two_zsmul_ne_zero_iff Real.Angle.two_zsmul_ne_zero_iff theorem eq_neg_self_iff {θ : Angle} : θ = -θ ↔ θ = 0 ∨ θ = π := by rw [← add_eq_zero_iff_eq_neg, ← two_nsmul, two_nsmul_eq_zero_iff] #align real.angle.eq_neg_self_iff Real.Angle.eq_neg_self_iff theorem ne_neg_self_iff {θ : Angle} : θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ π := by rw [← not_or, ← eq_neg_self_iff.not] #align real.angle.ne_neg_self_iff Real.Angle.ne_neg_self_iff theorem neg_eq_self_iff {θ : Angle} : -θ = θ ↔ θ = 0 ∨ θ = π := by rw [eq_comm, eq_neg_self_iff] #align real.angle.neg_eq_self_iff Real.Angle.neg_eq_self_iff theorem neg_ne_self_iff {θ : Angle} : -θ ≠ θ ↔ θ ≠ 0 ∧ θ ≠ π := by rw [← not_or, ← neg_eq_self_iff.not] #align real.angle.neg_ne_self_iff Real.Angle.neg_ne_self_iff theorem two_nsmul_eq_pi_iff {θ : Angle} : (2 : ℕ) • θ = π ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) := by have h : (π : Angle) = ((2 : ℕ) • (π / 2 : ℝ) :) := by rw [two_nsmul, add_halves] nth_rw 1 [h] rw [coe_nsmul, two_nsmul_eq_iff] -- Porting note: `congr` didn't simplify the goal of iff of `Or`s convert Iff.rfl rw [add_comm, ← coe_add, ← sub_eq_zero, ← coe_sub, neg_div, ← neg_sub, sub_neg_eq_add, add_assoc, add_halves, ← two_mul, coe_neg, coe_two_pi, neg_zero] #align real.angle.two_nsmul_eq_pi_iff Real.Angle.two_nsmul_eq_pi_iff theorem two_zsmul_eq_pi_iff {θ : Angle} : (2 : ℤ) • θ = π ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) := by rw [two_zsmul, ← two_nsmul, two_nsmul_eq_pi_iff] #align real.angle.two_zsmul_eq_pi_iff Real.Angle.two_zsmul_eq_pi_iff theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} : cos θ = cos ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) = -ψ := by constructor · intro Hcos rw [← sub_eq_zero, cos_sub_cos, mul_eq_zero, mul_eq_zero, neg_eq_zero, eq_false (two_ne_zero' ℝ), false_or_iff, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos rcases Hcos with (⟨n, hn⟩ \| ⟨n, hn⟩) · right rw [eq_div_iff_mul_eq (two_ne_zero' ℝ), ← sub_eq_iff_eq_add] at hn rw [← hn, coe_sub, eq_neg_iff_add_eq_zero, sub_add_cancel, mul_assoc, intCast_mul_eq_zsmul, mul_comm, coe_two_pi, zsmul_zero] · left rw [eq_div_iff_mul_eq (two_ne_zero' ℝ), eq_sub_iff_add_eq] at hn rw [← hn, coe_add, mul_assoc, intCast_mul_eq_zsmul, mul_comm, coe_two_pi, zsmul_zero, zero_add] · rw [angle_eq_iff_two_pi_dvd_sub, ← coe_neg, angle_eq_iff_two_pi_dvd_sub] rintro (⟨k, H⟩ \| ⟨k, H⟩) · rw [← sub_eq_zero, cos_sub_cos, H, mul_assoc 2 π k, mul_div_cancel_left₀ _ (two_ne_zero' ℝ), mul_comm π _, sin_int_mul_pi, mul_zero] rw [← sub_eq_zero, cos_sub_cos, ← sub_neg_eq_add, H, mul_assoc 2 π k, mul_div_cancel_left₀ _ (two_ne_zero' ℝ), mul_comm π _, sin_int_mul_pi, mul_zero, zero_mul] #align real.angle.cos_eq_iff_coe_eq_or_eq_neg Real.Angle.cos_eq_iff_coe_eq_or_eq_neg theorem sin_eq_iff_coe_eq_or_add_eq_pi {θ ψ : ℝ} : sin θ = sin ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) + ψ = π := by constructor · intro Hsin rw [← cos_pi_div_two_sub, ← cos_pi_div_two_sub] at Hsin cases' cos_eq_iff_coe_eq_or_eq_neg.mp Hsin with h h · left rw [coe_sub, coe_sub] at h exact sub_right_inj.1 h right rw [coe_sub, coe_sub, eq_neg_iff_add_eq_zero, add_sub, sub_add_eq_add_sub, ← coe_add, add_halves, sub_sub, sub_eq_zero] at h exact h.symm · rw [angle_eq_iff_two_pi_dvd_sub, ← eq_sub_iff_add_eq, ← coe_sub, angle_eq_iff_two_pi_dvd_sub] rintro (⟨k, H⟩ \| ⟨k, H⟩) · rw [← sub_eq_zero, sin_sub_sin, H, mul_assoc 2 π k, mul_div_cancel_left₀ _ (two_ne_zero' ℝ), mul_comm π _, sin_int_mul_pi, mul_zero, zero_mul] have H' : θ + ψ = 2 * k * π + π := by rwa [← sub_add, sub_add_eq_add_sub, sub_eq_iff_eq_add, mul_assoc, mul_comm π _, ← mul_assoc] at H rw [← sub_eq_zero, sin_sub_sin, H', add_div, mul_assoc 2 _ π, mul_div_cancel_left₀ _ (two_ne_zero' ℝ), cos_add_pi_div_two, sin_int_mul_pi, neg_zero, mul_zero] #align real.angle.sin_eq_iff_coe_eq_or_add_eq_pi Real.Angle.sin_eq_iff_coe_eq_or_add_eq_pi theorem cos_sin_inj {θ ψ : ℝ} (Hcos : cos θ = cos ψ) (Hsin : sin θ = sin ψ) : (θ : Angle) = ψ := by cases' cos_eq_iff_coe_eq_or_eq_neg.mp Hcos with hc hc; · exact hc cases' sin_eq_iff_coe_eq_or_add_eq_pi.mp Hsin with hs hs; · exact hs rw [eq_neg_iff_add_eq_zero, hs] at hc obtain ⟨n, hn⟩ : ∃ n, n • _ = _ := QuotientAddGroup.leftRel_apply.mp (Quotient.exact' hc) rw [← neg_one_mul, add_zero, ← sub_eq_zero, zsmul_eq_mul, ← mul_assoc, ← sub_mul, mul_eq_zero, eq_false (ne_of_gt pi_pos), or_false_iff, sub_neg_eq_add, ← Int.cast_zero, ← Int.cast_one, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_add, Int.cast_inj] at hn have : (n * 2 + 1) % (2 : ℤ) = 0 % (2 : ℤ) := congr_arg (· % (2 : ℤ)) hn rw [add_comm, Int.add_mul_emod_self] at this exact absurd this one_ne_zero #align real.angle.cos_sin_inj Real.Angle.cos_sin_inj def sin (θ : Angle) : ℝ := sin_periodic.lift θ #align real.angle.sin Real.Angle.sin @[simp] theorem sin_coe (x : ℝ) : sin (x : Angle) = Real.sin x := rfl #align real.angle.sin_coe Real.Angle.sin_coe @[continuity] theorem continuous_sin : Continuous sin := Real.continuous_sin.quotient_liftOn' _ #align real.angle.continuous_sin Real.Angle.continuous_sin def cos (θ : Angle) : ℝ := cos_periodic.lift θ #align real.angle.cos Real.Angle.cos @[simp] theorem cos_coe (x : ℝ) : cos (x : Angle) = Real.cos x := rfl #align real.angle.cos_coe Real.Angle.cos_coe @[continuity] theorem continuous_cos : Continuous cos := Real.continuous_cos.quotient_liftOn' _ #align real.angle.continuous_cos Real.Angle.continuous_cos theorem cos_eq_real_cos_iff_eq_or_eq_neg {θ : Angle} {ψ : ℝ} : cos θ = Real.cos ψ ↔ θ = ψ ∨ θ = -ψ := by induction θ using Real.Angle.induction_on exact cos_eq_iff_coe_eq_or_eq_neg #align real.angle.cos_eq_real_cos_iff_eq_or_eq_neg Real.Angle.cos_eq_real_cos_iff_eq_or_eq_neg theorem cos_eq_iff_eq_or_eq_neg {θ ψ : Angle} : cos θ = cos ψ ↔ θ = ψ ∨ θ = -ψ := by induction ψ using Real.Angle.induction_on exact cos_eq_real_cos_iff_eq_or_eq_neg #align real.angle.cos_eq_iff_eq_or_eq_neg Real.Angle.cos_eq_iff_eq_or_eq_neg theorem sin_eq_real_sin_iff_eq_or_add_eq_pi {θ : Angle} {ψ : ℝ} : sin θ = Real.sin ψ ↔ θ = ψ ∨ θ + ψ = π := by induction θ using Real.Angle.induction_on exact sin_eq_iff_coe_eq_or_add_eq_pi #align real.angle.sin_eq_real_sin_iff_eq_or_add_eq_pi Real.Angle.sin_eq_real_sin_iff_eq_or_add_eq_pi theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : Angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π := by induction ψ using Real.Angle.induction_on exact sin_eq_real_sin_iff_eq_or_add_eq_pi #align real.angle.sin_eq_iff_eq_or_add_eq_pi Real.Angle.sin_eq_iff_eq_or_add_eq_pi @[simp] theorem sin_zero : sin (0 : Angle) = 0 := by rw [← coe_zero, sin_coe, Real.sin_zero] #align real.angle.sin_zero Real.Angle.sin_zero -- Porting note (#10618): @[simp] can prove it theorem sin_coe_pi : sin (π : Angle) = 0 := by rw [sin_coe, Real.sin_pi] #align real.angle.sin_coe_pi Real.Angle.sin_coe_pi theorem sin_eq_zero_iff {θ : Angle} : sin θ = 0 ↔ θ = 0 ∨ θ = π := by nth_rw 1 [← sin_zero] rw [sin_eq_iff_eq_or_add_eq_pi] simp #align real.angle.sin_eq_zero_iff Real.Angle.sin_eq_zero_iff theorem sin_ne_zero_iff {θ : Angle} : sin θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π := by rw [← not_or, ← sin_eq_zero_iff] #align real.angle.sin_ne_zero_iff Real.Angle.sin_ne_zero_iff @[simp] theorem sin_neg (θ : Angle) : sin (-θ) = -sin θ := by induction θ using Real.Angle.induction_on exact Real.sin_neg _ #align real.angle.sin_neg Real.Angle.sin_neg theorem sin_antiperiodic : Function.Antiperiodic sin (π : Angle) := by intro θ induction θ using Real.Angle.induction_on exact Real.sin_antiperiodic _ #align real.angle.sin_antiperiodic Real.Angle.sin_antiperiodic @[simp] theorem sin_add_pi (θ : Angle) : sin (θ + π) = -sin θ := sin_antiperiodic θ #align real.angle.sin_add_pi Real.Angle.sin_add_pi @[simp] theorem sin_sub_pi (θ : Angle) : sin (θ - π) = -sin θ := sin_antiperiodic.sub_eq θ #align real.angle.sin_sub_pi Real.Angle.sin_sub_pi @[simp] theorem cos_zero : cos (0 : Angle) = 1 := by rw [← coe_zero, cos_coe, Real.cos_zero] #align real.angle.cos_zero Real.Angle.cos_zero -- Porting note (#10618): @[simp] can prove it theorem cos_coe_pi : cos (π : Angle) = -1 := by rw [cos_coe, Real.cos_pi] #align real.angle.cos_coe_pi Real.Angle.cos_coe_pi @[simp] theorem cos_neg (θ : Angle) : cos (-θ) = cos θ := by induction θ using Real.Angle.induction_on exact Real.cos_neg _ #align real.angle.cos_neg Real.Angle.cos_neg theorem cos_antiperiodic : Function.Antiperiodic cos (π : Angle) := by intro θ induction θ using Real.Angle.induction_on exact Real.cos_antiperiodic _ #align real.angle.cos_antiperiodic Real.Angle.cos_antiperiodic @[simp] theorem cos_add_pi (θ : Angle) : cos (θ + π) = -cos θ := cos_antiperiodic θ #align real.angle.cos_add_pi Real.Angle.cos_add_pi @[simp] theorem cos_sub_pi (θ : Angle) : cos (θ - π) = -cos θ := cos_antiperiodic.sub_eq θ #align real.angle.cos_sub_pi Real.Angle.cos_sub_pi theorem cos_eq_zero_iff {θ : Angle} : cos θ = 0 ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) := by rw [← cos_pi_div_two, ← cos_coe, cos_eq_iff_eq_or_eq_neg, ← coe_neg, ← neg_div] #align real.angle.cos_eq_zero_iff Real.Angle.cos_eq_zero_iff theorem sin_add (θ₁ θ₂ : Real.Angle) : sin (θ₁ + θ₂) = sin θ₁ * cos θ₂ + cos θ₁ * sin θ₂ := by induction θ₁ using Real.Angle.induction_on induction θ₂ using Real.Angle.induction_on exact Real.sin_add _ _ #align real.angle.sin_add Real.Angle.sin_add theorem cos_add (θ₁ θ₂ : Real.Angle) : cos (θ₁ + θ₂) = cos θ₁ * cos θ₂ - sin θ₁ * sin θ₂ := by induction θ₂ using Real.Angle.induction_on induction θ₁ using Real.Angle.induction_on exact Real.cos_add _ _ #align real.angle.cos_add Real.Angle.cos_add @[simp] theorem cos_sq_add_sin_sq (θ : Real.Angle) : cos θ ^ 2 + sin θ ^ 2 = 1 := by induction θ using Real.Angle.induction_on exact Real.cos_sq_add_sin_sq _ #align real.angle.cos_sq_add_sin_sq Real.Angle.cos_sq_add_sin_sq theorem sin_add_pi_div_two (θ : Angle) : sin (θ + ↑(π / 2)) = cos θ := by induction θ using Real.Angle.induction_on exact Real.sin_add_pi_div_two _ #align real.angle.sin_add_pi_div_two Real.Angle.sin_add_pi_div_two theorem sin_sub_pi_div_two (θ : Angle) : sin (θ - ↑(π / 2)) = -cos θ := by induction θ using Real.Angle.induction_on exact Real.sin_sub_pi_div_two _ #align real.angle.sin_sub_pi_div_two Real.Angle.sin_sub_pi_div_two theorem sin_pi_div_two_sub (θ : Angle) : sin (↑(π / 2) - θ) = cos θ := by induction θ using Real.Angle.induction_on exact Real.sin_pi_div_two_sub _ #align real.angle.sin_pi_div_two_sub Real.Angle.sin_pi_div_two_sub theorem cos_add_pi_div_two (θ : Angle) : cos (θ + ↑(π / 2)) = -sin θ := by induction θ using Real.Angle.induction_on exact Real.cos_add_pi_div_two _ #align real.angle.cos_add_pi_div_two Real.Angle.cos_add_pi_div_two theorem cos_sub_pi_div_two (θ : Angle) : cos (θ - ↑(π / 2)) = sin θ := by induction θ using Real.Angle.induction_on exact Real.cos_sub_pi_div_two _ #align real.angle.cos_sub_pi_div_two Real.Angle.cos_sub_pi_div_two theorem cos_pi_div_two_sub (θ : Angle) : cos (↑(π / 2) - θ) = sin θ := by induction θ using Real.Angle.induction_on exact Real.cos_pi_div_two_sub _ #align real.angle.cos_pi_div_two_sub Real.Angle.cos_pi_div_two_sub theorem abs_sin_eq_of_two_nsmul_eq {θ ψ : Angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) : \|sin θ\| = \|sin ψ\| := by rw [two_nsmul_eq_iff] at h rcases h with (rfl \| rfl) · rfl · rw [sin_add_pi, abs_neg] #align real.angle.abs_sin_eq_of_two_nsmul_eq Real.Angle.abs_sin_eq_of_two_nsmul_eq theorem abs_sin_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : \|sin θ\| = \|sin ψ\| := by simp_rw [two_zsmul, ← two_nsmul] at h exact abs_sin_eq_of_two_nsmul_eq h #align real.angle.abs_sin_eq_of_two_zsmul_eq Real.Angle.abs_sin_eq_of_two_zsmul_eq theorem abs_cos_eq_of_two_nsmul_eq {θ ψ : Angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) : \|cos θ\| = \|cos ψ\| := by rw [two_nsmul_eq_iff] at h rcases h with (rfl \| rfl) · rfl · rw [cos_add_pi, abs_neg] #align real.angle.abs_cos_eq_of_two_nsmul_eq Real.Angle.abs_cos_eq_of_two_nsmul_eq theorem abs_cos_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : \|cos θ\| = \|cos ψ\| := by simp_rw [two_zsmul, ← two_nsmul] at h exact abs_cos_eq_of_two_nsmul_eq h #align real.angle.abs_cos_eq_of_two_zsmul_eq Real.Angle.abs_cos_eq_of_two_zsmul_eq @[simp] theorem coe_toIcoMod (θ ψ : ℝ) : ↑(toIcoMod two_pi_pos ψ θ) = (θ : Angle) := by rw [angle_eq_iff_two_pi_dvd_sub] refine ⟨-toIcoDiv two_pi_pos ψ θ, ?_⟩ rw [toIcoMod_sub_self, zsmul_eq_mul, mul_comm] #align real.angle.coe_to_Ico_mod Real.Angle.coe_toIcoMod @[simp] theorem coe_toIocMod (θ ψ : ℝ) : ↑(toIocMod two_pi_pos ψ θ) = (θ : Angle) := by rw [angle_eq_iff_two_pi_dvd_sub] refine ⟨-toIocDiv two_pi_pos ψ θ, ?_⟩ rw [toIocMod_sub_self, zsmul_eq_mul, mul_comm] #align real.angle.coe_to_Ioc_mod Real.Angle.coe_toIocMod def toReal (θ : Angle) : ℝ := (toIocMod_periodic two_pi_pos (-π)).lift θ #align real.angle.to_real Real.Angle.toReal theorem toReal_coe (θ : ℝ) : (θ : Angle).toReal = toIocMod two_pi_pos (-π) θ := rfl #align real.angle.to_real_coe Real.Angle.toReal_coe theorem toReal_coe_eq_self_iff {θ : ℝ} : (θ : Angle).toReal = θ ↔ -π < θ ∧ θ ≤ π := by rw [toReal_coe, toIocMod_eq_self two_pi_pos] ring_nf rfl #align real.angle.to_real_coe_eq_self_iff Real.Angle.toReal_coe_eq_self_iff theorem toReal_coe_eq_self_iff_mem_Ioc {θ : ℝ} : (θ : Angle).toReal = θ ↔ θ ∈ Set.Ioc (-π) π := by rw [toReal_coe_eq_self_iff, ← Set.mem_Ioc] #align real.angle.to_real_coe_eq_self_iff_mem_Ioc Real.Angle.toReal_coe_eq_self_iff_mem_Ioc theorem toReal_injective : Function.Injective toReal := by intro θ ψ h induction θ using Real.Angle.induction_on induction ψ using Real.Angle.induction_on simpa [toReal_coe, toIocMod_eq_toIocMod, zsmul_eq_mul, mul_comm _ (2 * π), ← angle_eq_iff_two_pi_dvd_sub, eq_comm] using h #align real.angle.to_real_injective Real.Angle.toReal_injective @[simp] theorem toReal_inj {θ ψ : Angle} : θ.toReal = ψ.toReal ↔ θ = ψ := toReal_injective.eq_iff #align real.angle.to_real_inj Real.Angle.toReal_inj @[simp] theorem coe_toReal (θ : Angle) : (θ.toReal : Angle) = θ := by induction θ using Real.Angle.induction_on exact coe_toIocMod _ _ #align real.angle.coe_to_real Real.Angle.coe_toReal theorem neg_pi_lt_toReal (θ : Angle) : -π < θ.toReal := by induction θ using Real.Angle.induction_on exact left_lt_toIocMod _ _ _ #align real.angle.neg_pi_lt_to_real Real.Angle.neg_pi_lt_toReal theorem toReal_le_pi (θ : Angle) : θ.toReal ≤ π := by induction θ using Real.Angle.induction_on convert toIocMod_le_right two_pi_pos _ _ ring #align real.angle.to_real_le_pi Real.Angle.toReal_le_pi theorem abs_toReal_le_pi (θ : Angle) : \|θ.toReal\| ≤ π := abs_le.2 ⟨(neg_pi_lt_toReal _).le, toReal_le_pi _⟩ #align real.angle.abs_to_real_le_pi Real.Angle.abs_toReal_le_pi theorem toReal_mem_Ioc (θ : Angle) : θ.toReal ∈ Set.Ioc (-π) π := ⟨neg_pi_lt_toReal _, toReal_le_pi _⟩ #align real.angle.to_real_mem_Ioc Real.Angle.toReal_mem_Ioc @[simp] theorem toIocMod_toReal (θ : Angle) : toIocMod two_pi_pos (-π) θ.toReal = θ.toReal := by induction θ using Real.Angle.induction_on rw [toReal_coe] exact toIocMod_toIocMod _ _ _ _ #align real.angle.to_Ioc_mod_to_real Real.Angle.toIocMod_toReal @[simp] theorem toReal_zero : (0 : Angle).toReal = 0 := by rw [← coe_zero, toReal_coe_eq_self_iff] exact ⟨Left.neg_neg_iff.2 Real.pi_pos, Real.pi_pos.le⟩ #align real.angle.to_real_zero Real.Angle.toReal_zero @[simp] theorem toReal_eq_zero_iff {θ : Angle} : θ.toReal = 0 ↔ θ = 0 := by nth_rw 1 [← toReal_zero] exact toReal_inj #align real.angle.to_real_eq_zero_iff Real.Angle.toReal_eq_zero_iff @[simp] theorem toReal_pi : (π : Angle).toReal = π := by rw [toReal_coe_eq_self_iff] exact ⟨Left.neg_lt_self Real.pi_pos, le_refl _⟩ #align real.angle.to_real_pi Real.Angle.toReal_pi @[simp] theorem toReal_eq_pi_iff {θ : Angle} : θ.toReal = π ↔ θ = π := by rw [← toReal_inj, toReal_pi] #align real.angle.to_real_eq_pi_iff Real.Angle.toReal_eq_pi_iff theorem pi_ne_zero : (π : Angle) ≠ 0 := by rw [← toReal_injective.ne_iff, toReal_pi, toReal_zero] exact Real.pi_ne_zero #align real.angle.pi_ne_zero Real.Angle.pi_ne_zero @[simp] theorem toReal_pi_div_two : ((π / 2 : ℝ) : Angle).toReal = π / 2 := toReal_coe_eq_self_iff.2 <\| by constructor <;> linarith [pi_pos] #align real.angle.to_real_pi_div_two Real.Angle.toReal_pi_div_two @[simp] theorem toReal_eq_pi_div_two_iff {θ : Angle} : θ.toReal = π / 2 ↔ θ = (π / 2 : ℝ) := by rw [← toReal_inj, toReal_pi_div_two] #align real.angle.to_real_eq_pi_div_two_iff Real.Angle.toReal_eq_pi_div_two_iff @[simp] theorem toReal_neg_pi_div_two : ((-π / 2 : ℝ) : Angle).toReal = -π / 2 := toReal_coe_eq_self_iff.2 <\| by constructor <;> linarith [pi_pos] #align real.angle.to_real_neg_pi_div_two Real.Angle.toReal_neg_pi_div_two @[simp] theorem toReal_eq_neg_pi_div_two_iff {θ : Angle} : θ.toReal = -π / 2 ↔ θ = (-π / 2 : ℝ) := by rw [← toReal_inj, toReal_neg_pi_div_two] #align real.angle.to_real_eq_neg_pi_div_two_iff Real.Angle.toReal_eq_neg_pi_div_two_iff theorem pi_div_two_ne_zero : ((π / 2 : ℝ) : Angle) ≠ 0 := by rw [← toReal_injective.ne_iff, toReal_pi_div_two, toReal_zero] exact div_ne_zero Real.pi_ne_zero two_ne_zero #align real.angle.pi_div_two_ne_zero Real.Angle.pi_div_two_ne_zero theorem neg_pi_div_two_ne_zero : ((-π / 2 : ℝ) : Angle) ≠ 0 := by rw [← toReal_injective.ne_iff, toReal_neg_pi_div_two, toReal_zero] exact div_ne_zero (neg_ne_zero.2 Real.pi_ne_zero) two_ne_zero #align real.angle.neg_pi_div_two_ne_zero Real.Angle.neg_pi_div_two_ne_zero theorem abs_toReal_coe_eq_self_iff {θ : ℝ} : \|(θ : Angle).toReal\| = θ ↔ 0 ≤ θ ∧ θ ≤ π := ⟨fun h => h ▸ ⟨abs_nonneg _, abs_toReal_le_pi _⟩, fun h => (toReal_coe_eq_self_iff.2 ⟨(Left.neg_neg_iff.2 Real.pi_pos).trans_le h.1, h.2⟩).symm ▸ abs_eq_self.2 h.1⟩ #align real.angle.abs_to_real_coe_eq_self_iff Real.Angle.abs_toReal_coe_eq_self_iff theorem abs_toReal_neg_coe_eq_self_iff {θ : ℝ} : \|(-θ : Angle).toReal\| = θ ↔ 0 ≤ θ ∧ θ ≤ π := by refine ⟨fun h => h ▸ ⟨abs_nonneg _, abs_toReal_le_pi _⟩, fun h => ?_⟩ by_cases hnegpi : θ = π; · simp [hnegpi, Real.pi_pos.le] rw [← coe_neg, toReal_coe_eq_self_iff.2 ⟨neg_lt_neg (lt_of_le_of_ne h.2 hnegpi), (neg_nonpos.2 h.1).trans Real.pi_pos.le⟩, abs_neg, abs_eq_self.2 h.1] #align real.angle.abs_to_real_neg_coe_eq_self_iff Real.Angle.abs_toReal_neg_coe_eq_self_iff theorem abs_toReal_eq_pi_div_two_iff {θ : Angle} : \|θ.toReal\| = π / 2 ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) := by rw [abs_eq (div_nonneg Real.pi_pos.le two_pos.le), ← neg_div, toReal_eq_pi_div_two_iff, toReal_eq_neg_pi_div_two_iff] #align real.angle.abs_to_real_eq_pi_div_two_iff Real.Angle.abs_toReal_eq_pi_div_two_iff theorem nsmul_toReal_eq_mul {n : ℕ} (h : n ≠ 0) {θ : Angle} : (n • θ).toReal = n * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / n) (π / n) := by nth_rw 1 [← coe_toReal θ] have h' : 0 < (n : ℝ) := mod_cast Nat.pos_of_ne_zero h rw [← coe_nsmul, nsmul_eq_mul, toReal_coe_eq_self_iff, Set.mem_Ioc, div_lt_iff' h', le_div_iff' h'] #align real.angle.nsmul_to_real_eq_mul Real.Angle.nsmul_toReal_eq_mul theorem two_nsmul_toReal_eq_two_mul {θ : Angle} : ((2 : ℕ) • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2) := mod_cast nsmul_toReal_eq_mul two_ne_zero #align real.angle.two_nsmul_to_real_eq_two_mul Real.Angle.two_nsmul_toReal_eq_two_mul theorem two_zsmul_toReal_eq_two_mul {θ : Angle} : ((2 : ℤ) • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2) := by rw [two_zsmul, ← two_nsmul, two_nsmul_toReal_eq_two_mul] #align real.angle.two_zsmul_to_real_eq_two_mul Real.Angle.two_zsmul_toReal_eq_two_mul theorem toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff {θ : ℝ} {k : ℤ} : (θ : Angle).toReal = θ - 2 * k * π ↔ θ ∈ Set.Ioc ((2 * k - 1 : ℝ) * π) ((2 * k + 1) * π) := by rw [← sub_zero (θ : Angle), ← zsmul_zero k, ← coe_two_pi, ← coe_zsmul, ← coe_sub, zsmul_eq_mul, ← mul_assoc, mul_comm (k : ℝ), toReal_coe_eq_self_iff, Set.mem_Ioc] exact ⟨fun h => ⟨by linarith, by linarith⟩, fun h => ⟨by linarith, by linarith⟩⟩ #align real.angle.to_real_coe_eq_self_sub_two_mul_int_mul_pi_iff Real.Angle.toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff theorem toReal_coe_eq_self_sub_two_pi_iff {θ : ℝ} : (θ : Angle).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π) := by convert @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff θ 1 <;> norm_num #align real.angle.to_real_coe_eq_self_sub_two_pi_iff Real.Angle.toReal_coe_eq_self_sub_two_pi_iff theorem toReal_coe_eq_self_add_two_pi_iff {θ : ℝ} : (θ : Angle).toReal = θ + 2 * π ↔ θ ∈ Set.Ioc (-3 * π) (-π) := by convert @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff θ (-1) using 2 <;> set_option tactic.skipAssignedInstances false in norm_num #align real.angle.to_real_coe_eq_self_add_two_pi_iff Real.Angle.toReal_coe_eq_self_add_two_pi_iff theorem two_nsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} : ((2 : ℕ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal := by nth_rw 1 [← coe_toReal θ] rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_sub_two_pi_iff, Set.mem_Ioc] exact ⟨fun h => by linarith, fun h => ⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩ #align real.angle.two_nsmul_to_real_eq_two_mul_sub_two_pi Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi	Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean	708	710	theorem two_zsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} : ((2 : ℤ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal := by	rw [two_zsmul, ← two_nsmul, two_nsmul_toReal_eq_two_mul_sub_two_pi]
import Mathlib.Analysis.SpecialFunctions.Exponential #align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0" open NormedSpace open scoped Nat section SinCos theorem Complex.hasSum_cos' (z : ℂ) : HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) := by rw [Complex.cos, Complex.exp_eq_exp_ℂ] have := ((expSeries_div_hasSum_exp ℂ (z * Complex.I)).add (expSeries_div_hasSum_exp ℂ (-z * Complex.I))).div_const 2 replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this dsimp [Function.comp_def] at this simp_rw [← mul_comm 2 _] at this refine this.prod_fiberwise fun k => ?_ dsimp only convert hasSum_fintype (_ : Fin 2 → ℂ) using 1 rw [Fin.sum_univ_two] simp_rw [Fin.val_zero, Fin.val_one, add_zero, pow_succ, pow_mul, mul_pow, neg_sq, ← two_mul, neg_mul, mul_neg, neg_div, add_right_neg, zero_div, add_zero, mul_div_cancel_left₀ _ (two_ne_zero : (2 : ℂ) ≠ 0)] #align complex.has_sum_cos' Complex.hasSum_cos'	Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean	49	64	theorem Complex.hasSum_sin' (z : ℂ) : HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / Complex.I) (Complex.sin z) := by	rw [Complex.sin, Complex.exp_eq_exp_ℂ] have := (((expSeries_div_hasSum_exp ℂ (-z * Complex.I)).sub (expSeries_div_hasSum_exp ℂ (z * Complex.I))).mul_right Complex.I).div_const 2 replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this dsimp [Function.comp_def] at this simp_rw [← mul_comm 2 _] at this refine this.prod_fiberwise fun k => ?_ dsimp only convert hasSum_fintype (_ : Fin 2 → ℂ) using 1 rw [Fin.sum_univ_two] simp_rw [Fin.val_zero, Fin.val_one, add_zero, pow_succ, pow_mul, mul_pow, neg_sq, sub_self, zero_mul, zero_div, zero_add, neg_mul, mul_neg, neg_div, ← neg_add', ← two_mul, neg_mul, neg_div, mul_assoc, mul_div_cancel_left₀ _ (two_ne_zero : (2 : ℂ) ≠ 0), Complex.div_I]
import Mathlib.LinearAlgebra.Projectivization.Basic #align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe" open scoped LinearAlgebra.Projectivization variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ℙ K V} namespace Projectivization inductive Independent : (ι → ℙ K V) → Prop \| mk (f : ι → V) (hf : ∀ i : ι, f i ≠ 0) (hl : LinearIndependent K f) : Independent fun i => mk K (f i) (hf i) #align projectivization.independent Projectivization.Independent theorem independent_iff : Independent f ↔ LinearIndependent K (Projectivization.rep ∘ f) := by refine ⟨?_, fun h => ?_⟩ · rintro ⟨ff, hff, hh⟩ choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i) convert hh.units_smul a ext i exact (ha i).symm · convert Independent.mk _ _ h · simp only [mk_rep, Function.comp_apply] · intro i apply rep_nonzero #align projectivization.independent_iff Projectivization.independent_iff	Mathlib/LinearAlgebra/Projectivization/Independence.lean	63	72	theorem independent_iff_completeLattice_independent : Independent f ↔ CompleteLattice.Independent fun i => (f i).submodule := by	refine ⟨?_, fun h => ?_⟩ · rintro ⟨f, hf, hi⟩ simp only [submodule_mk] exact (CompleteLattice.independent_iff_linearIndependent_of_ne_zero (R := K) hf).mpr hi · rw [independent_iff] refine h.linearIndependent (Projectivization.submodule ∘ f) (fun i => ?_) fun i => ?_ · simpa only [Function.comp_apply, submodule_eq] using Submodule.mem_span_singleton_self _ · exact rep_nonzero (f i)
import Mathlib.Data.ENNReal.Inv #align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open Set NNReal ENNReal namespace ENNReal section Real variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0} theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal + b.toReal := by lift a to ℝ≥0 using ha lift b to ℝ≥0 using hb rfl #align ennreal.to_real_add ENNReal.toReal_add theorem toReal_sub_of_le {a b : ℝ≥0∞} (h : b ≤ a) (ha : a ≠ ∞) : (a - b).toReal = a.toReal - b.toReal := by lift b to ℝ≥0 using ne_top_of_le_ne_top ha h lift a to ℝ≥0 using ha simp only [← ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)] #align ennreal.to_real_sub_of_le ENNReal.toReal_sub_of_le theorem le_toReal_sub {a b : ℝ≥0∞} (hb : b ≠ ∞) : a.toReal - b.toReal ≤ (a - b).toReal := by lift b to ℝ≥0 using hb induction a · simp · simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal] exact le_max_left _ _ #align ennreal.le_to_real_sub ENNReal.le_toReal_sub theorem toReal_add_le : (a + b).toReal ≤ a.toReal + b.toReal := if ha : a = ∞ then by simp only [ha, top_add, top_toReal, zero_add, toReal_nonneg] else if hb : b = ∞ then by simp only [hb, add_top, top_toReal, add_zero, toReal_nonneg] else le_of_eq (toReal_add ha hb) #align ennreal.to_real_add_le ENNReal.toReal_add_le theorem ofReal_add {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) : ENNReal.ofReal (p + q) = ENNReal.ofReal p + ENNReal.ofReal q := by rw [ENNReal.ofReal, ENNReal.ofReal, ENNReal.ofReal, ← coe_add, coe_inj, Real.toNNReal_add hp hq] #align ennreal.of_real_add ENNReal.ofReal_add theorem ofReal_add_le {p q : ℝ} : ENNReal.ofReal (p + q) ≤ ENNReal.ofReal p + ENNReal.ofReal q := coe_le_coe.2 Real.toNNReal_add_le #align ennreal.of_real_add_le ENNReal.ofReal_add_le @[simp]	Mathlib/Data/ENNReal/Real.lean	76	79	theorem toReal_le_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal ≤ b.toReal ↔ a ≤ b := by	lift a to ℝ≥0 using ha lift b to ℝ≥0 using hb norm_cast
import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Algebra.Subalgebra.Basic import Mathlib.Algebra.DirectSum.Algebra #align_import algebra.direct_sum.internal from "leanprover-community/mathlib"@"9936c3dfc04e5876f4368aeb2e60f8d8358d095a" open DirectSum variable {ι : Type} {σ S R : Type} instance AddCommMonoid.ofSubmonoidOnSemiring [Semiring R] [SetLike σ R] [AddSubmonoidClass σ R] (A : ι → σ) : ∀ i, AddCommMonoid (A i) := fun i => by infer_instance #align add_comm_monoid.of_submonoid_on_semiring AddCommMonoid.ofSubmonoidOnSemiring instance AddCommGroup.ofSubgroupOnRing [Ring R] [SetLike σ R] [AddSubgroupClass σ R] (A : ι → σ) : ∀ i, AddCommGroup (A i) := fun i => by infer_instance #align add_comm_group.of_subgroup_on_ring AddCommGroup.ofSubgroupOnRing theorem SetLike.algebraMap_mem_graded [Zero ι] [CommSemiring S] [Semiring R] [Algebra S R] (A : ι → Submodule S R) [SetLike.GradedOne A] (s : S) : algebraMap S R s ∈ A 0 := by rw [Algebra.algebraMap_eq_smul_one] exact (A 0).smul_mem s <\| SetLike.one_mem_graded _ #align set_like.algebra_map_mem_graded SetLike.algebraMap_mem_graded	Mathlib/Algebra/DirectSum/Internal.lean	62	68	theorem SetLike.natCast_mem_graded [Zero ι] [AddMonoidWithOne R] [SetLike σ R] [AddSubmonoidClass σ R] (A : ι → σ) [SetLike.GradedOne A] (n : ℕ) : (n : R) ∈ A 0 := by	induction' n with _ n_ih · rw [Nat.cast_zero] exact zero_mem (A 0) · rw [Nat.cast_succ] exact add_mem n_ih (SetLike.one_mem_graded _)
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle #align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" noncomputable section open scoped EuclideanGeometry open scoped Real open scoped RealInnerProductSpace namespace Orientation open FiniteDimensional variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] variable [hd2 : Fact (finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two Orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two theorem oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle (x + y) y = Real.arccos (‖y‖ / ‖x + y‖) := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two h #align orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two Orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two Orientation.oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h #align orientation.oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two Orientation.oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two theorem oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle x (x + y) = Real.arctan (‖y‖ / ‖x‖) := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (o.left_ne_zero_of_oangle_eq_pi_div_two h)] #align orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two Orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h #align orientation.oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two Orientation.oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two theorem cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle x (x + y)) = ‖x‖ / ‖x + y‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe, InnerProductGeometry.cos_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two theorem cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) = ‖y‖ / ‖x + y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).cos_oangle_add_right_of_oangle_eq_pi_div_two h #align orientation.cos_oangle_add_left_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_left_of_oangle_eq_pi_div_two theorem sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle x (x + y)) = ‖y‖ / ‖x + y‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe, InnerProductGeometry.sin_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.sin_oangle_add_right_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_right_of_oangle_eq_pi_div_two theorem sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x + y) y) = ‖x‖ / ‖x + y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).sin_oangle_add_right_of_oangle_eq_pi_div_two h #align orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two theorem tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle x (x + y)) = ‖y‖ / ‖x‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe, InnerProductGeometry.tan_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two theorem tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x + y) y) = ‖x‖ / ‖y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).tan_oangle_add_right_of_oangle_eq_pi_div_two h #align orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two theorem cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle x (x + y)) ‖x + y‖ = ‖x‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe, InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two theorem cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) * ‖x + y‖ = ‖y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h #align orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two theorem sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle x (x + y)) * ‖x + y‖ = ‖y‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe, InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two theorem sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x + y) y) * ‖x + y‖ = ‖x‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h #align orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two theorem tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle x (x + y)) * ‖x‖ = ‖y‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe, InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two theorem tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x + y) y) * ‖y‖ = ‖x‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h #align orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two theorem norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.cos (o.oangle x (x + y)) = ‖x + y‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe, InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two theorem norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle (x + y) y) = ‖x + y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two h #align orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two Orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two theorem norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle x (x + y)) = ‖x + y‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe, InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (Or.inr (o.right_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two theorem norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle (x + y) y) = ‖x + y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two h #align orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two Orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two theorem norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle x (x + y)) = ‖x‖ := by have hs : (o.oangle x (x + y)).sign = 1 := by rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe, InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h) (Or.inr (o.right_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle (x + y) y) = ‖y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ rw [add_comm] exact (-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h #align orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two Orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two theorem oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle y (y - x) = Real.arccos (‖y‖ / ‖y - x‖) := by have hs : (o.oangle y (y - x)).sign = 1 := by rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two Orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two theorem oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle (x - y) x = Real.arccos (‖x‖ / ‖x - y‖) := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ exact (-o).oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two h #align orientation.oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two Orientation.oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two theorem oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle y (y - x) = Real.arcsin (‖x‖ / ‖y - x‖) := by have hs : (o.oangle y (y - x)).sign = 1 := by rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) (Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two Orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two theorem oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle (x - y) x = Real.arcsin (‖y‖ / ‖x - y‖) := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ exact (-o).oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two h #align orientation.oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two Orientation.oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two theorem oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle y (y - x) = Real.arctan (‖x‖ / ‖y‖) := by have hs : (o.oangle y (y - x)).sign = 1 := by rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) (o.right_ne_zero_of_oangle_eq_pi_div_two h)] #align orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two Orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two theorem oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : o.oangle (x - y) x = Real.arctan (‖y‖ / ‖x‖) := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ exact (-o).oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two h #align orientation.oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two Orientation.oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two theorem cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle y (y - x)) = ‖y‖ / ‖y - x‖ := by have hs : (o.oangle y (y - x)).sign = 1 := by rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe, InnerProductGeometry.cos_angle_sub_of_inner_eq_zero (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)] #align orientation.cos_oangle_sub_right_of_oangle_eq_pi_div_two Orientation.cos_oangle_sub_right_of_oangle_eq_pi_div_two theorem cos_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x - y) x) = ‖x‖ / ‖x - y‖ := by rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ exact (-o).cos_oangle_sub_right_of_oangle_eq_pi_div_two h #align orientation.cos_oangle_sub_left_of_oangle_eq_pi_div_two Orientation.cos_oangle_sub_left_of_oangle_eq_pi_div_two theorem sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle y (y - x)) = ‖x‖ / ‖y - x‖ := by have hs : (o.oangle y (y - x)).sign = 1 := by rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two] rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe, InnerProductGeometry.sin_angle_sub_of_inner_eq_zero (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) (Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))] #align orientation.sin_oangle_sub_right_of_oangle_eq_pi_div_two Orientation.sin_oangle_sub_right_of_oangle_eq_pi_div_two	Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean	351	354	theorem sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x - y) x) = ‖y‖ / ‖x - y‖ := by	rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢ exact (-o).sin_oangle_sub_right_of_oangle_eq_pi_div_two h
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero #align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d" noncomputable section universe v v₂ u u' u₂ open CategoryTheory open CategoryTheory.Limits.WalkingParallelPair namespace CategoryTheory.Limits variable {C : Type u} [Category.{v} C] variable [HasZeroMorphisms C] abbrev HasKernel {X Y : C} (f : X ⟶ Y) : Prop := HasLimit (parallelPair f 0) #align category_theory.limits.has_kernel CategoryTheory.Limits.HasKernel abbrev HasCokernel {X Y : C} (f : X ⟶ Y) : Prop := HasColimit (parallelPair f 0) #align category_theory.limits.has_cokernel CategoryTheory.Limits.HasCokernel variable {X Y : C} (f : X ⟶ Y) section abbrev KernelFork := Fork f 0 #align category_theory.limits.kernel_fork CategoryTheory.Limits.KernelFork variable {f} @[reassoc (attr := simp)] theorem KernelFork.condition (s : KernelFork f) : Fork.ι s ≫ f = 0 := by erw [Fork.condition, HasZeroMorphisms.comp_zero] #align category_theory.limits.kernel_fork.condition CategoryTheory.Limits.KernelFork.condition -- Porting note (#10618): simp can prove this, removed simp tag theorem KernelFork.app_one (s : KernelFork f) : s.π.app one = 0 := by simp [Fork.app_one_eq_ι_comp_right] #align category_theory.limits.kernel_fork.app_one CategoryTheory.Limits.KernelFork.app_one abbrev KernelFork.ofι {Z : C} (ι : Z ⟶ X) (w : ι ≫ f = 0) : KernelFork f := Fork.ofι ι <\| by rw [w, HasZeroMorphisms.comp_zero] #align category_theory.limits.kernel_fork.of_ι CategoryTheory.Limits.KernelFork.ofι @[simp] theorem KernelFork.ι_ofι {X Y P : C} (f : X ⟶ Y) (ι : P ⟶ X) (w : ι ≫ f = 0) : Fork.ι (KernelFork.ofι ι w) = ι := rfl #align category_theory.limits.kernel_fork.ι_of_ι CategoryTheory.Limits.KernelFork.ι_ofι section -- attribute [local tidy] tactic.case_bash Porting note: no tidy nor case_bash def isoOfι (s : Fork f 0) : s ≅ Fork.ofι (Fork.ι s) (Fork.condition s) := Cones.ext (Iso.refl _) <\| by rintro ⟨j⟩ <;> simp #align category_theory.limits.iso_of_ι CategoryTheory.Limits.isoOfι def ofιCongr {P : C} {ι ι' : P ⟶ X} {w : ι ≫ f = 0} (h : ι = ι') : KernelFork.ofι ι w ≅ KernelFork.ofι ι' (by rw [← h, w]) := Cones.ext (Iso.refl _) #align category_theory.limits.of_ι_congr CategoryTheory.Limits.ofιCongr def compNatIso {D : Type u'} [Category.{v} D] [HasZeroMorphisms D] (F : C ⥤ D) [F.IsEquivalence] : parallelPair f 0 ⋙ F ≅ parallelPair (F.map f) 0 := let app (j :WalkingParallelPair) : (parallelPair f 0 ⋙ F).obj j ≅ (parallelPair (F.map f) 0).obj j := match j with \| zero => Iso.refl _ \| one => Iso.refl _ NatIso.ofComponents app <\| by rintro ⟨i⟩ ⟨j⟩ <;> intro g <;> cases g <;> simp [app] #align category_theory.limits.comp_nat_iso CategoryTheory.Limits.compNatIso end def KernelFork.IsLimit.lift' {s : KernelFork f} (hs : IsLimit s) {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : { l : W ⟶ s.pt // l ≫ Fork.ι s = k } := ⟨hs.lift <\| KernelFork.ofι _ h, hs.fac _ _⟩ #align category_theory.limits.kernel_fork.is_limit.lift' CategoryTheory.Limits.KernelFork.IsLimit.lift' def isLimitAux (t : KernelFork f) (lift : ∀ s : KernelFork f, s.pt ⟶ t.pt) (fac : ∀ s : KernelFork f, lift s ≫ t.ι = s.ι) (uniq : ∀ (s : KernelFork f) (m : s.pt ⟶ t.pt) (_ : m ≫ t.ι = s.ι), m = lift s) : IsLimit t := { lift fac := fun s j => by cases j · exact fac s · simp uniq := fun s m w => uniq s m (w Limits.WalkingParallelPair.zero) } #align category_theory.limits.is_limit_aux CategoryTheory.Limits.isLimitAux def KernelFork.IsLimit.ofι {W : C} (g : W ⟶ X) (eq : g ≫ f = 0) (lift : ∀ {W' : C} (g' : W' ⟶ X) (_ : g' ≫ f = 0), W' ⟶ W) (fac : ∀ {W' : C} (g' : W' ⟶ X) (eq' : g' ≫ f = 0), lift g' eq' ≫ g = g') (uniq : ∀ {W' : C} (g' : W' ⟶ X) (eq' : g' ≫ f = 0) (m : W' ⟶ W) (_ : m ≫ g = g'), m = lift g' eq') : IsLimit (KernelFork.ofι g eq) := isLimitAux _ (fun s => lift s.ι s.condition) (fun s => fac s.ι s.condition) fun s => uniq s.ι s.condition #align category_theory.limits.kernel_fork.is_limit.of_ι CategoryTheory.Limits.KernelFork.IsLimit.ofι def KernelFork.IsLimit.ofι' {X Y K : C} {f : X ⟶ Y} (i : K ⟶ X) (w : i ≫ f = 0) (h : ∀ {A : C} (k : A ⟶ X) (_ : k ≫ f = 0), { l : A ⟶ K // l ≫ i = k}) [hi : Mono i] : IsLimit (KernelFork.ofι i w) := ofι _ _ (fun {A} k hk => (h k hk).1) (fun {A} k hk => (h k hk).2) (fun {A} k hk m hm => by rw [← cancel_mono i, (h k hk).2, hm]) def isKernelCompMono {c : KernelFork f} (i : IsLimit c) {Z} (g : Y ⟶ Z) [hg : Mono g] {h : X ⟶ Z} (hh : h = f ≫ g) : IsLimit (KernelFork.ofι c.ι (by simp [hh]) : KernelFork h) := Fork.IsLimit.mk' _ fun s => let s' : KernelFork f := Fork.ofι s.ι (by rw [← cancel_mono g]; simp [← hh, s.condition]) let l := KernelFork.IsLimit.lift' i s'.ι s'.condition ⟨l.1, l.2, fun hm => by apply Fork.IsLimit.hom_ext i; rw [Fork.ι_ofι] at hm; rw [hm]; exact l.2.symm⟩ #align category_theory.limits.is_kernel_comp_mono CategoryTheory.Limits.isKernelCompMono theorem isKernelCompMono_lift {c : KernelFork f} (i : IsLimit c) {Z} (g : Y ⟶ Z) [hg : Mono g] {h : X ⟶ Z} (hh : h = f ≫ g) (s : KernelFork h) : (isKernelCompMono i g hh).lift s = i.lift (Fork.ofι s.ι (by rw [← cancel_mono g, Category.assoc, ← hh] simp)) := rfl #align category_theory.limits.is_kernel_comp_mono_lift CategoryTheory.Limits.isKernelCompMono_lift def isKernelOfComp {W : C} (g : Y ⟶ W) (h : X ⟶ W) {c : KernelFork h} (i : IsLimit c) (hf : c.ι ≫ f = 0) (hfg : f ≫ g = h) : IsLimit (KernelFork.ofι c.ι hf) := Fork.IsLimit.mk _ (fun s => i.lift (KernelFork.ofι s.ι (by simp [← hfg]))) (fun s => by simp only [KernelFork.ι_ofι, Fork.IsLimit.lift_ι]) fun s m h => by apply Fork.IsLimit.hom_ext i; simpa using h #align category_theory.limits.is_kernel_of_comp CategoryTheory.Limits.isKernelOfComp def KernelFork.IsLimit.ofId {X Y : C} (f : X ⟶ Y) (hf : f = 0) : IsLimit (KernelFork.ofι (𝟙 X) (show 𝟙 X ≫ f = 0 by rw [hf, comp_zero])) := KernelFork.IsLimit.ofι _ _ (fun x _ => x) (fun _ _ => Category.comp_id _) (fun _ _ _ hb => by simp only [← hb, Category.comp_id]) def KernelFork.IsLimit.ofMonoOfIsZero {X Y : C} {f : X ⟶ Y} (c : KernelFork f) (hf : Mono f) (h : IsZero c.pt) : IsLimit c := isLimitAux _ (fun s => 0) (fun s => by rw [zero_comp, ← cancel_mono f, zero_comp, s.condition]) (fun _ _ _ => h.eq_of_tgt _ _) lemma KernelFork.IsLimit.isIso_ι {X Y : C} {f : X ⟶ Y} (c : KernelFork f) (hc : IsLimit c) (hf : f = 0) : IsIso c.ι := by let e : c.pt ≅ X := IsLimit.conePointUniqueUpToIso hc (KernelFork.IsLimit.ofId (f : X ⟶ Y) hf) have eq : e.inv ≫ c.ι = 𝟙 X := Fork.IsLimit.lift_ι hc haveI : IsIso (e.inv ≫ c.ι) := by rw [eq] infer_instance exact IsIso.of_isIso_comp_left e.inv c.ι end section variable [HasKernel f] abbrev kernel (f : X ⟶ Y) [HasKernel f] : C := equalizer f 0 #align category_theory.limits.kernel CategoryTheory.Limits.kernel abbrev kernel.ι : kernel f ⟶ X := equalizer.ι f 0 #align category_theory.limits.kernel.ι CategoryTheory.Limits.kernel.ι @[simp] theorem equalizer_as_kernel : equalizer.ι f 0 = kernel.ι f := rfl #align category_theory.limits.equalizer_as_kernel CategoryTheory.Limits.equalizer_as_kernel @[reassoc (attr := simp)] theorem kernel.condition : kernel.ι f ≫ f = 0 := KernelFork.condition _ #align category_theory.limits.kernel.condition CategoryTheory.Limits.kernel.condition def kernelIsKernel : IsLimit (Fork.ofι (kernel.ι f) ((kernel.condition f).trans comp_zero.symm)) := IsLimit.ofIsoLimit (limit.isLimit _) (Fork.ext (Iso.refl _) (by aesop_cat)) #align category_theory.limits.kernel_is_kernel CategoryTheory.Limits.kernelIsKernel abbrev kernel.lift {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : W ⟶ kernel f := (kernelIsKernel f).lift (KernelFork.ofι k h) #align category_theory.limits.kernel.lift CategoryTheory.Limits.kernel.lift @[reassoc (attr := simp)] theorem kernel.lift_ι {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : kernel.lift f k h ≫ kernel.ι f = k := (kernelIsKernel f).fac (KernelFork.ofι k h) WalkingParallelPair.zero #align category_theory.limits.kernel.lift_ι CategoryTheory.Limits.kernel.lift_ι @[simp] theorem kernel.lift_zero {W : C} {h} : kernel.lift f (0 : W ⟶ X) h = 0 := by ext; simp #align category_theory.limits.kernel.lift_zero CategoryTheory.Limits.kernel.lift_zero instance kernel.lift_mono {W : C} (k : W ⟶ X) (h : k ≫ f = 0) [Mono k] : Mono (kernel.lift f k h) := ⟨fun {Z} g g' w => by replace w := w =≫ kernel.ι f simp only [Category.assoc, kernel.lift_ι] at w exact (cancel_mono k).1 w⟩ #align category_theory.limits.kernel.lift_mono CategoryTheory.Limits.kernel.lift_mono def kernel.lift' {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : { l : W ⟶ kernel f // l ≫ kernel.ι f = k } := ⟨kernel.lift f k h, kernel.lift_ι _ _ _⟩ #align category_theory.limits.kernel.lift' CategoryTheory.Limits.kernel.lift' abbrev kernel.map {X' Y' : C} (f' : X' ⟶ Y') [HasKernel f'] (p : X ⟶ X') (q : Y ⟶ Y') (w : f ≫ q = p ≫ f') : kernel f ⟶ kernel f' := kernel.lift f' (kernel.ι f ≫ p) (by simp [← w]) #align category_theory.limits.kernel.map CategoryTheory.Limits.kernel.map theorem kernel.lift_map {X Y Z X' Y' Z' : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasKernel g] (w : f ≫ g = 0) (f' : X' ⟶ Y') (g' : Y' ⟶ Z') [HasKernel g'] (w' : f' ≫ g' = 0) (p : X ⟶ X') (q : Y ⟶ Y') (r : Z ⟶ Z') (h₁ : f ≫ q = p ≫ f') (h₂ : g ≫ r = q ≫ g') : kernel.lift g f w ≫ kernel.map g g' q r h₂ = p ≫ kernel.lift g' f' w' := by ext; simp [h₁] #align category_theory.limits.kernel.lift_map CategoryTheory.Limits.kernel.lift_map @[simps] def kernel.mapIso {X' Y' : C} (f' : X' ⟶ Y') [HasKernel f'] (p : X ≅ X') (q : Y ≅ Y') (w : f ≫ q.hom = p.hom ≫ f') : kernel f ≅ kernel f' where hom := kernel.map f f' p.hom q.hom w inv := kernel.map f' f p.inv q.inv (by refine (cancel_mono q.hom).1 ?_ simp [w]) #align category_theory.limits.kernel.map_iso CategoryTheory.Limits.kernel.mapIso instance kernel.ι_zero_isIso : IsIso (kernel.ι (0 : X ⟶ Y)) := equalizer.ι_of_self _ #align category_theory.limits.kernel.ι_zero_is_iso CategoryTheory.Limits.kernel.ι_zero_isIso theorem eq_zero_of_epi_kernel [Epi (kernel.ι f)] : f = 0 := (cancel_epi (kernel.ι f)).1 (by simp) #align category_theory.limits.eq_zero_of_epi_kernel CategoryTheory.Limits.eq_zero_of_epi_kernel def kernelZeroIsoSource : kernel (0 : X ⟶ Y) ≅ X := equalizer.isoSourceOfSelf 0 #align category_theory.limits.kernel_zero_iso_source CategoryTheory.Limits.kernelZeroIsoSource @[simp] theorem kernelZeroIsoSource_hom : kernelZeroIsoSource.hom = kernel.ι (0 : X ⟶ Y) := rfl #align category_theory.limits.kernel_zero_iso_source_hom CategoryTheory.Limits.kernelZeroIsoSource_hom @[simp] theorem kernelZeroIsoSource_inv : kernelZeroIsoSource.inv = kernel.lift (0 : X ⟶ Y) (𝟙 X) (by simp) := by ext simp [kernelZeroIsoSource] #align category_theory.limits.kernel_zero_iso_source_inv CategoryTheory.Limits.kernelZeroIsoSource_inv def kernelIsoOfEq {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) : kernel f ≅ kernel g := HasLimit.isoOfNatIso (by rw [h]) #align category_theory.limits.kernel_iso_of_eq CategoryTheory.Limits.kernelIsoOfEq @[simp] theorem kernelIsoOfEq_refl {h : f = f} : kernelIsoOfEq h = Iso.refl (kernel f) := by ext simp [kernelIsoOfEq] #align category_theory.limits.kernel_iso_of_eq_refl CategoryTheory.Limits.kernelIsoOfEq_refl @[reassoc (attr := simp)] theorem kernelIsoOfEq_hom_comp_ι {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) : (kernelIsoOfEq h).hom ≫ kernel.ι g = kernel.ι f := by cases h; simp #align category_theory.limits.kernel_iso_of_eq_hom_comp_ι CategoryTheory.Limits.kernelIsoOfEq_hom_comp_ι @[reassoc (attr := simp)] theorem kernelIsoOfEq_inv_comp_ι {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) : (kernelIsoOfEq h).inv ≫ kernel.ι _ = kernel.ι _ := by cases h; simp #align category_theory.limits.kernel_iso_of_eq_inv_comp_ι CategoryTheory.Limits.kernelIsoOfEq_inv_comp_ι @[reassoc (attr := simp)] theorem lift_comp_kernelIsoOfEq_hom {Z} {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) (e : Z ⟶ X) (he) : kernel.lift _ e he ≫ (kernelIsoOfEq h).hom = kernel.lift _ e (by simp [← h, he]) := by cases h; simp #align category_theory.limits.lift_comp_kernel_iso_of_eq_hom CategoryTheory.Limits.lift_comp_kernelIsoOfEq_hom @[reassoc (attr := simp)] theorem lift_comp_kernelIsoOfEq_inv {Z} {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) (e : Z ⟶ X) (he) : kernel.lift _ e he ≫ (kernelIsoOfEq h).inv = kernel.lift _ e (by simp [h, he]) := by cases h; simp #align category_theory.limits.lift_comp_kernel_iso_of_eq_inv CategoryTheory.Limits.lift_comp_kernelIsoOfEq_inv @[simp]	Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean	414	416	theorem kernelIsoOfEq_trans {f g h : X ⟶ Y} [HasKernel f] [HasKernel g] [HasKernel h] (w₁ : f = g) (w₂ : g = h) : kernelIsoOfEq w₁ ≪≫ kernelIsoOfEq w₂ = kernelIsoOfEq (w₁.trans w₂) := by	cases w₁; cases w₂; ext; simp [kernelIsoOfEq]
import Mathlib.Algebra.BigOperators.Group.Finset import Mathlib.Data.List.MinMax import Mathlib.Algebra.Tropical.Basic import Mathlib.Order.ConditionallyCompleteLattice.Finset #align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce" variable {R S : Type*} open Tropical Finset theorem List.trop_sum [AddMonoid R] (l : List R) : trop l.sum = List.prod (l.map trop) := by induction' l with hd tl IH · simp · simp [← IH] #align list.trop_sum List.trop_sum theorem Multiset.trop_sum [AddCommMonoid R] (s : Multiset R) : trop s.sum = Multiset.prod (s.map trop) := Quotient.inductionOn s (by simpa using List.trop_sum) #align multiset.trop_sum Multiset.trop_sum theorem trop_sum [AddCommMonoid R] (s : Finset S) (f : S → R) : trop (∑ i ∈ s, f i) = ∏ i ∈ s, trop (f i) := by convert Multiset.trop_sum (s.val.map f) simp only [Multiset.map_map, Function.comp_apply] rfl #align trop_sum trop_sum theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) : untrop l.prod = List.sum (l.map untrop) := by induction' l with hd tl IH · simp · simp [← IH] #align list.untrop_prod List.untrop_prod theorem Multiset.untrop_prod [AddCommMonoid R] (s : Multiset (Tropical R)) : untrop s.prod = Multiset.sum (s.map untrop) := Quotient.inductionOn s (by simpa using List.untrop_prod) #align multiset.untrop_prod Multiset.untrop_prod theorem untrop_prod [AddCommMonoid R] (s : Finset S) (f : S → Tropical R) : untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i) := by convert Multiset.untrop_prod (s.val.map f) simp only [Multiset.map_map, Function.comp_apply] rfl #align untrop_prod untrop_prod -- Porting note: replaced `coe` with `WithTop.some` in statement theorem List.trop_minimum [LinearOrder R] (l : List R) : trop l.minimum = List.sum (l.map (trop ∘ WithTop.some)) := by induction' l with hd tl IH · simp · simp [List.minimum_cons, ← IH] #align list.trop_minimum List.trop_minimum theorem Multiset.trop_inf [LinearOrder R] [OrderTop R] (s : Multiset R) : trop s.inf = Multiset.sum (s.map trop) := by induction' s using Multiset.induction with s x IH · simp · simp [← IH] #align multiset.trop_inf Multiset.trop_inf theorem Finset.trop_inf [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → R) : trop (s.inf f) = ∑ i ∈ s, trop (f i) := by convert Multiset.trop_inf (s.val.map f) simp only [Multiset.map_map, Function.comp_apply] rfl #align finset.trop_inf Finset.trop_inf theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) : trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) := by rcases s.eq_empty_or_nonempty with (rfl \| h) · simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, trop_top] rw [← inf'_eq_csInf_image _ h, inf'_eq_inf, s.trop_inf] #align trop_Inf_image trop_sInf_image theorem trop_iInf [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → WithTop R) : trop (⨅ i : S, f i) = ∑ i : S, trop (f i) := by rw [iInf, ← Set.image_univ, ← coe_univ, trop_sInf_image] #align trop_infi trop_iInf theorem Multiset.untrop_sum [LinearOrder R] [OrderTop R] (s : Multiset (Tropical R)) : untrop s.sum = Multiset.inf (s.map untrop) := by induction' s using Multiset.induction with s x IH · simp · simp only [sum_cons, ge_iff_le, untrop_add, untrop_le_iff, map_cons, inf_cons, ← IH] rfl #align multiset.untrop_sum Multiset.untrop_sum theorem Finset.untrop_sum' [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → Tropical R) : untrop (∑ i ∈ s, f i) = s.inf (untrop ∘ f) := by convert Multiset.untrop_sum (s.val.map f) simp only [Multiset.map_map, Function.comp_apply] rfl #align finset.untrop_sum' Finset.untrop_sum' theorem untrop_sum_eq_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' s) := by rcases s.eq_empty_or_nonempty with (rfl \| h) · simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, untrop_zero] · rw [← inf'_eq_csInf_image _ h, inf'_eq_inf, Finset.untrop_sum'] #align untrop_sum_eq_Inf_image untrop_sum_eq_sInf_image theorem untrop_sum [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → Tropical (WithTop R)) : untrop (∑ i : S, f i) = ⨅ i : S, untrop (f i) := by rw [iInf,← Set.image_univ,← coe_univ, untrop_sum_eq_sInf_image] rfl #align untrop_sum untrop_sum	Mathlib/Algebra/Tropical/BigOperators.lean	141	143	theorem Finset.untrop_sum [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = ⨅ i : s, untrop (f i) := by	simpa [← _root_.untrop_sum] using (sum_attach _ _).symm
import Mathlib.Data.ENNReal.Inv #align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open Set NNReal ENNReal namespace ENNReal section Real variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0} theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal + b.toReal := by lift a to ℝ≥0 using ha lift b to ℝ≥0 using hb rfl #align ennreal.to_real_add ENNReal.toReal_add theorem toReal_sub_of_le {a b : ℝ≥0∞} (h : b ≤ a) (ha : a ≠ ∞) : (a - b).toReal = a.toReal - b.toReal := by lift b to ℝ≥0 using ne_top_of_le_ne_top ha h lift a to ℝ≥0 using ha simp only [← ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)] #align ennreal.to_real_sub_of_le ENNReal.toReal_sub_of_le theorem le_toReal_sub {a b : ℝ≥0∞} (hb : b ≠ ∞) : a.toReal - b.toReal ≤ (a - b).toReal := by lift b to ℝ≥0 using hb induction a · simp · simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal] exact le_max_left _ _ #align ennreal.le_to_real_sub ENNReal.le_toReal_sub theorem toReal_add_le : (a + b).toReal ≤ a.toReal + b.toReal := if ha : a = ∞ then by simp only [ha, top_add, top_toReal, zero_add, toReal_nonneg] else if hb : b = ∞ then by simp only [hb, add_top, top_toReal, add_zero, toReal_nonneg] else le_of_eq (toReal_add ha hb) #align ennreal.to_real_add_le ENNReal.toReal_add_le theorem ofReal_add {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) : ENNReal.ofReal (p + q) = ENNReal.ofReal p + ENNReal.ofReal q := by rw [ENNReal.ofReal, ENNReal.ofReal, ENNReal.ofReal, ← coe_add, coe_inj, Real.toNNReal_add hp hq] #align ennreal.of_real_add ENNReal.ofReal_add theorem ofReal_add_le {p q : ℝ} : ENNReal.ofReal (p + q) ≤ ENNReal.ofReal p + ENNReal.ofReal q := coe_le_coe.2 Real.toNNReal_add_le #align ennreal.of_real_add_le ENNReal.ofReal_add_le @[simp] theorem toReal_le_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal ≤ b.toReal ↔ a ≤ b := by lift a to ℝ≥0 using ha lift b to ℝ≥0 using hb norm_cast #align ennreal.to_real_le_to_real ENNReal.toReal_le_toReal @[gcongr] theorem toReal_mono (hb : b ≠ ∞) (h : a ≤ b) : a.toReal ≤ b.toReal := (toReal_le_toReal (ne_top_of_le_ne_top hb h) hb).2 h #align ennreal.to_real_mono ENNReal.toReal_mono -- Porting note (#10756): new lemma theorem toReal_mono' (h : a ≤ b) (ht : b = ∞ → a = ∞) : a.toReal ≤ b.toReal := by rcases eq_or_ne a ∞ with rfl \| ha · exact toReal_nonneg · exact toReal_mono (mt ht ha) h @[simp] theorem toReal_lt_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal < b.toReal ↔ a < b := by lift a to ℝ≥0 using ha lift b to ℝ≥0 using hb norm_cast #align ennreal.to_real_lt_to_real ENNReal.toReal_lt_toReal @[gcongr] theorem toReal_strict_mono (hb : b ≠ ∞) (h : a < b) : a.toReal < b.toReal := (toReal_lt_toReal h.ne_top hb).2 h #align ennreal.to_real_strict_mono ENNReal.toReal_strict_mono @[gcongr] theorem toNNReal_mono (hb : b ≠ ∞) (h : a ≤ b) : a.toNNReal ≤ b.toNNReal := toReal_mono hb h #align ennreal.to_nnreal_mono ENNReal.toNNReal_mono -- Porting note (#10756): new lemma theorem toReal_le_add' (hle : a ≤ b + c) (hb : b = ∞ → a = ∞) (hc : c = ∞ → a = ∞) : a.toReal ≤ b.toReal + c.toReal := by refine le_trans (toReal_mono' hle ?_) toReal_add_le simpa only [add_eq_top, or_imp] using And.intro hb hc -- Porting note (#10756): new lemma theorem toReal_le_add (hle : a ≤ b + c) (hb : b ≠ ∞) (hc : c ≠ ∞) : a.toReal ≤ b.toReal + c.toReal := toReal_le_add' hle (flip absurd hb) (flip absurd hc) @[simp] theorem toNNReal_le_toNNReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toNNReal ≤ b.toNNReal ↔ a ≤ b := ⟨fun h => by rwa [← coe_toNNReal ha, ← coe_toNNReal hb, coe_le_coe], toNNReal_mono hb⟩ #align ennreal.to_nnreal_le_to_nnreal ENNReal.toNNReal_le_toNNReal theorem toNNReal_strict_mono (hb : b ≠ ∞) (h : a < b) : a.toNNReal < b.toNNReal := by simpa [← ENNReal.coe_lt_coe, hb, h.ne_top] #align ennreal.to_nnreal_strict_mono ENNReal.toNNReal_strict_mono @[simp] theorem toNNReal_lt_toNNReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toNNReal < b.toNNReal ↔ a < b := ⟨fun h => by rwa [← coe_toNNReal ha, ← coe_toNNReal hb, coe_lt_coe], toNNReal_strict_mono hb⟩ #align ennreal.to_nnreal_lt_to_nnreal ENNReal.toNNReal_lt_toNNReal theorem toReal_max (hr : a ≠ ∞) (hp : b ≠ ∞) : ENNReal.toReal (max a b) = max (ENNReal.toReal a) (ENNReal.toReal b) := (le_total a b).elim (fun h => by simp only [h, (ENNReal.toReal_le_toReal hr hp).2 h, max_eq_right]) fun h => by simp only [h, (ENNReal.toReal_le_toReal hp hr).2 h, max_eq_left] #align ennreal.to_real_max ENNReal.toReal_max theorem toReal_min {a b : ℝ≥0∞} (hr : a ≠ ∞) (hp : b ≠ ∞) : ENNReal.toReal (min a b) = min (ENNReal.toReal a) (ENNReal.toReal b) := (le_total a b).elim (fun h => by simp only [h, (ENNReal.toReal_le_toReal hr hp).2 h, min_eq_left]) fun h => by simp only [h, (ENNReal.toReal_le_toReal hp hr).2 h, min_eq_right] #align ennreal.to_real_min ENNReal.toReal_min theorem toReal_sup {a b : ℝ≥0∞} : a ≠ ∞ → b ≠ ∞ → (a ⊔ b).toReal = a.toReal ⊔ b.toReal := toReal_max #align ennreal.to_real_sup ENNReal.toReal_sup theorem toReal_inf {a b : ℝ≥0∞} : a ≠ ∞ → b ≠ ∞ → (a ⊓ b).toReal = a.toReal ⊓ b.toReal := toReal_min #align ennreal.to_real_inf ENNReal.toReal_inf theorem toNNReal_pos_iff : 0 < a.toNNReal ↔ 0 < a ∧ a < ∞ := by induction a <;> simp #align ennreal.to_nnreal_pos_iff ENNReal.toNNReal_pos_iff theorem toNNReal_pos {a : ℝ≥0∞} (ha₀ : a ≠ 0) (ha_top : a ≠ ∞) : 0 < a.toNNReal := toNNReal_pos_iff.mpr ⟨bot_lt_iff_ne_bot.mpr ha₀, lt_top_iff_ne_top.mpr ha_top⟩ #align ennreal.to_nnreal_pos ENNReal.toNNReal_pos theorem toReal_pos_iff : 0 < a.toReal ↔ 0 < a ∧ a < ∞ := NNReal.coe_pos.trans toNNReal_pos_iff #align ennreal.to_real_pos_iff ENNReal.toReal_pos_iff theorem toReal_pos {a : ℝ≥0∞} (ha₀ : a ≠ 0) (ha_top : a ≠ ∞) : 0 < a.toReal := toReal_pos_iff.mpr ⟨bot_lt_iff_ne_bot.mpr ha₀, lt_top_iff_ne_top.mpr ha_top⟩ #align ennreal.to_real_pos ENNReal.toReal_pos @[gcongr] theorem ofReal_le_ofReal {p q : ℝ} (h : p ≤ q) : ENNReal.ofReal p ≤ ENNReal.ofReal q := by simp [ENNReal.ofReal, Real.toNNReal_le_toNNReal h] #align ennreal.of_real_le_of_real ENNReal.ofReal_le_ofReal theorem ofReal_le_of_le_toReal {a : ℝ} {b : ℝ≥0∞} (h : a ≤ ENNReal.toReal b) : ENNReal.ofReal a ≤ b := (ofReal_le_ofReal h).trans ofReal_toReal_le #align ennreal.of_real_le_of_le_to_real ENNReal.ofReal_le_of_le_toReal @[simp] theorem ofReal_le_ofReal_iff {p q : ℝ} (h : 0 ≤ q) : ENNReal.ofReal p ≤ ENNReal.ofReal q ↔ p ≤ q := by rw [ENNReal.ofReal, ENNReal.ofReal, coe_le_coe, Real.toNNReal_le_toNNReal_iff h] #align ennreal.of_real_le_of_real_iff ENNReal.ofReal_le_ofReal_iff lemma ofReal_le_ofReal_iff' {p q : ℝ} : ENNReal.ofReal p ≤ .ofReal q ↔ p ≤ q ∨ p ≤ 0 := coe_le_coe.trans Real.toNNReal_le_toNNReal_iff' lemma ofReal_lt_ofReal_iff' {p q : ℝ} : ENNReal.ofReal p < .ofReal q ↔ p < q ∧ 0 < q := coe_lt_coe.trans Real.toNNReal_lt_toNNReal_iff' @[simp] theorem ofReal_eq_ofReal_iff {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) : ENNReal.ofReal p = ENNReal.ofReal q ↔ p = q := by rw [ENNReal.ofReal, ENNReal.ofReal, coe_inj, Real.toNNReal_eq_toNNReal_iff hp hq] #align ennreal.of_real_eq_of_real_iff ENNReal.ofReal_eq_ofReal_iff @[simp] theorem ofReal_lt_ofReal_iff {p q : ℝ} (h : 0 < q) : ENNReal.ofReal p < ENNReal.ofReal q ↔ p < q := by rw [ENNReal.ofReal, ENNReal.ofReal, coe_lt_coe, Real.toNNReal_lt_toNNReal_iff h] #align ennreal.of_real_lt_of_real_iff ENNReal.ofReal_lt_ofReal_iff theorem ofReal_lt_ofReal_iff_of_nonneg {p q : ℝ} (hp : 0 ≤ p) : ENNReal.ofReal p < ENNReal.ofReal q ↔ p < q := by rw [ENNReal.ofReal, ENNReal.ofReal, coe_lt_coe, Real.toNNReal_lt_toNNReal_iff_of_nonneg hp] #align ennreal.of_real_lt_of_real_iff_of_nonneg ENNReal.ofReal_lt_ofReal_iff_of_nonneg @[simp] theorem ofReal_pos {p : ℝ} : 0 < ENNReal.ofReal p ↔ 0 < p := by simp [ENNReal.ofReal] #align ennreal.of_real_pos ENNReal.ofReal_pos @[simp]	Mathlib/Data/ENNReal/Real.lean	222	222	theorem ofReal_eq_zero {p : ℝ} : ENNReal.ofReal p = 0 ↔ p ≤ 0 := by	simp [ENNReal.ofReal]
import Mathlib.Algebra.GeomSum import Mathlib.Algebra.Polynomial.Roots import Mathlib.GroupTheory.SpecificGroups.Cyclic #align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e" section open Finset Polynomial Function Nat variable {R : Type} {G : Type} variable [CommRing R] [IsDomain R] [Group G] -- Porting note: Finset doesn't seem to have `{g ∈ univ \| g^n = g₀}` notation anymore, -- so we have to use `Finset.filter` instead theorem card_nthRoots_subgroup_units [Fintype G] [DecidableEq G] (f : G →* R) (hf : Injective f) {n : ℕ} (hn : 0 < n) (g₀ : G) : Finset.card (Finset.univ.filter (fun g ↦ g^n = g₀)) ≤ Multiset.card (nthRoots n (f g₀)) := by haveI : DecidableEq R := Classical.decEq _ refine le_trans ?_ (nthRoots n (f g₀)).toFinset_card_le apply card_le_card_of_inj_on f · intro g hg rw [mem_filter] at hg rw [Multiset.mem_toFinset, mem_nthRoots hn, ← f.map_pow, hg.2] · intros apply hf assumption #align card_nth_roots_subgroup_units card_nthRoots_subgroup_units theorem isCyclic_of_subgroup_isDomain [Finite G] (f : G →* R) (hf : Injective f) : IsCyclic G := by classical cases nonempty_fintype G apply isCyclic_of_card_pow_eq_one_le intro n hn exact le_trans (card_nthRoots_subgroup_units f hf hn 1) (card_nthRoots n (f 1)) #align is_cyclic_of_subgroup_is_domain isCyclic_of_subgroup_isDomain instance [Finite Rˣ] : IsCyclic Rˣ := isCyclic_of_subgroup_isDomain (Units.coeHom R) <\| Units.ext section variable (S : Subgroup Rˣ) [Finite S] instance subgroup_units_cyclic : IsCyclic S := by -- Porting note: the original proof used a `coe`, but I was not able to get it to work. apply isCyclic_of_subgroup_isDomain (R := R) (G := S) _ _ · exact MonoidHom.mk (OneHom.mk (fun s => ↑s.val) rfl) (by simp) · exact Units.ext.comp Subtype.val_injective #align subgroup_units_cyclic subgroup_units_cyclic end section EuclideanDivision namespace Polynomial open Polynomial variable (K : Type) [Field K] [Algebra R[X] K] [IsFractionRing R[X] K]	Mathlib/RingTheory/IntegralDomain.lean	174	185	theorem div_eq_quo_add_rem_div (f : R[X]) {g : R[X]} (hg : g.Monic) : ∃ q r : R[X], r.degree < g.degree ∧ (algebraMap R[X] K f) / (algebraMap R[X] K g) = algebraMap R[X] K q + (algebraMap R[X] K r) / (algebraMap R[X] K g) := by	refine ⟨f /ₘ g, f %ₘ g, ?_, ?_⟩ · exact degree_modByMonic_lt _ hg · have hg' : algebraMap R[X] K g ≠ 0 := -- Porting note: the proof was `by exact_mod_cast Monic.ne_zero hg` (map_ne_zero_iff _ (IsFractionRing.injective R[X] K)).mpr (Monic.ne_zero hg) field_simp [hg'] -- Porting note: `norm_cast` was here, but does nothing. rw [add_comm, mul_comm, ← map_mul, ← map_add, modByMonic_add_div f hg]
import Mathlib.Order.Filter.Bases import Mathlib.Topology.Algebra.Module.Basic #align_import topology.algebra.filter_basis from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" open Filter Set TopologicalSpace Function open Topology Filter Pointwise universe u class GroupFilterBasis (G : Type u) [Group G] extends FilterBasis G where one' : ∀ {U}, U ∈ sets → (1 : G) ∈ U mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U inv' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U #align group_filter_basis GroupFilterBasis class AddGroupFilterBasis (A : Type u) [AddGroup A] extends FilterBasis A where zero' : ∀ {U}, U ∈ sets → (0 : A) ∈ U add' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V + V ⊆ U neg' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ -x) ⁻¹' U conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ + x + -x₀) ⁻¹' U #align add_group_filter_basis AddGroupFilterBasis attribute [to_additive existing] GroupFilterBasis GroupFilterBasis.conj' GroupFilterBasis.toFilterBasis @[to_additive "`AddGroupFilterBasis` constructor in the additive commutative group case."] def groupFilterBasisOfComm {G : Type} [CommGroup G] (sets : Set (Set G)) (nonempty : sets.Nonempty) (inter_sets : ∀ x y, x ∈ sets → y ∈ sets → ∃ z ∈ sets, z ⊆ x ∩ y) (one : ∀ U ∈ sets, (1 : G) ∈ U) (mul : ∀ U ∈ sets, ∃ V ∈ sets, V V ⊆ U) (inv : ∀ U ∈ sets, ∃ V ∈ sets, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U) : GroupFilterBasis G := { sets := sets nonempty := nonempty inter_sets := inter_sets _ _ one' := one _ mul' := mul _ inv' := inv _ conj' := fun x U U_in ↦ ⟨U, U_in, by simp only [mul_inv_cancel_comm, preimage_id']; rfl⟩ } #align group_filter_basis_of_comm groupFilterBasisOfComm #align add_group_filter_basis_of_comm addGroupFilterBasisOfComm namespace GroupFilterBasis variable {G : Type u} [Group G] {B : GroupFilterBasis G} @[to_additive] instance : Membership (Set G) (GroupFilterBasis G) := ⟨fun s f ↦ s ∈ f.sets⟩ @[to_additive] theorem one {U : Set G} : U ∈ B → (1 : G) ∈ U := GroupFilterBasis.one' #align group_filter_basis.one GroupFilterBasis.one #align add_group_filter_basis.zero AddGroupFilterBasis.zero @[to_additive] theorem mul {U : Set G} : U ∈ B → ∃ V ∈ B, V * V ⊆ U := GroupFilterBasis.mul' #align group_filter_basis.mul GroupFilterBasis.mul #align add_group_filter_basis.add AddGroupFilterBasis.add @[to_additive] theorem inv {U : Set G} : U ∈ B → ∃ V ∈ B, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U := GroupFilterBasis.inv' #align group_filter_basis.inv GroupFilterBasis.inv #align add_group_filter_basis.neg AddGroupFilterBasis.neg @[to_additive] theorem conj : ∀ x₀, ∀ {U}, U ∈ B → ∃ V ∈ B, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U := GroupFilterBasis.conj' #align group_filter_basis.conj GroupFilterBasis.conj #align add_group_filter_basis.conj AddGroupFilterBasis.conj @[to_additive "The trivial additive group filter basis consists of `{0}` only. The associated topology is discrete."] instance : Inhabited (GroupFilterBasis G) where default := { sets := {{1}} nonempty := singleton_nonempty _ inter_sets := by simp one' := by simp mul' := by simp inv' := by simp conj' := by simp } @[to_additive] theorem subset_mul_self (B : GroupFilterBasis G) {U : Set G} (h : U ∈ B) : U ⊆ U * U := fun x x_in ↦ ⟨1, one h, x, x_in, one_mul x⟩ #align group_filter_basis.prod_subset_self GroupFilterBasis.subset_mul_self #align add_group_filter_basis.sum_subset_self AddGroupFilterBasis.subset_add_self @[to_additive "The neighborhood function of an `AddGroupFilterBasis`."] def N (B : GroupFilterBasis G) : G → Filter G := fun x ↦ map (fun y ↦ x * y) B.toFilterBasis.filter set_option linter.uppercaseLean3 false in #align group_filter_basis.N GroupFilterBasis.N set_option linter.uppercaseLean3 false in #align add_group_filter_basis.N AddGroupFilterBasis.N @[to_additive (attr := simp)] theorem N_one (B : GroupFilterBasis G) : B.N 1 = B.toFilterBasis.filter := by simp only [N, one_mul, map_id'] set_option linter.uppercaseLean3 false in #align group_filter_basis.N_one GroupFilterBasis.N_one set_option linter.uppercaseLean3 false in #align add_group_filter_basis.N_zero AddGroupFilterBasis.N_zero @[to_additive] protected theorem hasBasis (B : GroupFilterBasis G) (x : G) : HasBasis (B.N x) (fun V : Set G ↦ V ∈ B) fun V ↦ (fun y ↦ x * y) '' V := HasBasis.map (fun y ↦ x * y) toFilterBasis.hasBasis #align group_filter_basis.has_basis GroupFilterBasis.hasBasis #align add_group_filter_basis.has_basis AddGroupFilterBasis.hasBasis @[to_additive "The topological space structure coming from an additive group filter basis."] def topology (B : GroupFilterBasis G) : TopologicalSpace G := TopologicalSpace.mkOfNhds B.N #align group_filter_basis.topology GroupFilterBasis.topology #align add_group_filter_basis.topology AddGroupFilterBasis.topology @[to_additive] theorem nhds_eq (B : GroupFilterBasis G) {x₀ : G} : @nhds G B.topology x₀ = B.N x₀ := by apply TopologicalSpace.nhds_mkOfNhds_of_hasBasis (fun x ↦ (FilterBasis.hasBasis _).map _) · intro a U U_in exact ⟨1, B.one U_in, mul_one a⟩ · intro a U U_in rcases GroupFilterBasis.mul U_in with ⟨V, V_in, hVU⟩ filter_upwards [image_mem_map (B.mem_filter_of_mem V_in)] rintro _ ⟨x, hx, rfl⟩ calc a • U ⊇ a • (V * V) := smul_set_mono hVU _ ⊇ a • x • V := smul_set_mono <\| smul_set_subset_smul hx _ = (a * x) • V := smul_smul .. _ ∈ (a * x) • B.filter := smul_set_mem_smul_filter <\| B.mem_filter_of_mem V_in #align group_filter_basis.nhds_eq GroupFilterBasis.nhds_eq #align add_group_filter_basis.nhds_eq AddGroupFilterBasis.nhds_eq @[to_additive] theorem nhds_one_eq (B : GroupFilterBasis G) : @nhds G B.topology (1 : G) = B.toFilterBasis.filter := by rw [B.nhds_eq] simp only [N, one_mul] exact map_id #align group_filter_basis.nhds_one_eq GroupFilterBasis.nhds_one_eq #align add_group_filter_basis.nhds_zero_eq AddGroupFilterBasis.nhds_zero_eq @[to_additive] theorem nhds_hasBasis (B : GroupFilterBasis G) (x₀ : G) : HasBasis (@nhds G B.topology x₀) (fun V : Set G ↦ V ∈ B) fun V ↦ (fun y ↦ x₀ * y) '' V := by rw [B.nhds_eq] apply B.hasBasis #align group_filter_basis.nhds_has_basis GroupFilterBasis.nhds_hasBasis #align add_group_filter_basis.nhds_has_basis AddGroupFilterBasis.nhds_hasBasis @[to_additive]	Mathlib/Topology/Algebra/FilterBasis.lean	205	208	theorem nhds_one_hasBasis (B : GroupFilterBasis G) : HasBasis (@nhds G B.topology 1) (fun V : Set G ↦ V ∈ B) id := by	rw [B.nhds_one_eq] exact B.toFilterBasis.hasBasis
import Mathlib.Topology.Compactness.Compact open Set Filter Topology TopologicalSpace Classical variable {X : Type} {Y : Type} {ι : Type} variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X} instance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] : WeaklyLocallyCompactSpace (X × Y) where exists_compact_mem_nhds x := let ⟨s₁, hc₁, h₁⟩ := exists_compact_mem_nhds x.1 let ⟨s₂, hc₂, h₂⟩ := exists_compact_mem_nhds x.2 ⟨s₁ ×ˢ s₂, hc₁.prod hc₂, prod_mem_nhds h₁ h₂⟩ instance {ι : Type} [Finite ι] {X : ι → Type} [(i : ι) → TopologicalSpace (X i)] [(i : ι) → WeaklyLocallyCompactSpace (X i)] : WeaklyLocallyCompactSpace ((i : ι) → X i) where exists_compact_mem_nhds := fun f ↦ by choose s hsc hs using fun i ↦ exists_compact_mem_nhds (f i) exact ⟨pi univ s, isCompact_univ_pi hsc, set_pi_mem_nhds univ.toFinite fun i _ ↦ hs i⟩ instance (priority := 100) [CompactSpace X] : WeaklyLocallyCompactSpace X where exists_compact_mem_nhds _ := ⟨univ, isCompact_univ, univ_mem⟩ theorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) : ∃ K', IsCompact K' ∧ K ⊆ interior K' := by choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩ refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩ exact iUnion₂_subset fun x hx ↦ interior_mono <\| subset_iUnion₂ (s := fun x _ ↦ s x) x hx #align exists_compact_superset exists_compact_superset theorem disjoint_nhds_cocompact [WeaklyLocallyCompactSpace X] (x : X) : Disjoint (𝓝 x) (cocompact X) := let ⟨_, hc, hx⟩ := exists_compact_mem_nhds x disjoint_of_disjoint_of_mem disjoint_compl_right hx hc.compl_mem_cocompact theorem compact_basis_nhds [LocallyCompactSpace X] (x : X) : (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s := hasBasis_self.2 <\| by simpa only [and_comm] using LocallyCompactSpace.local_compact_nhds x #align compact_basis_nhds compact_basis_nhds theorem local_compact_nhds [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ 𝓝 x) : ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s := LocallyCompactSpace.local_compact_nhds _ _ h #align local_compact_nhds local_compact_nhds theorem LocallyCompactSpace.of_hasBasis {ι : X → Type} {p : ∀ x, ι x → Prop} {s : ∀ x, ι x → Set X} (h : ∀ x, (𝓝 x).HasBasis (p x) (s x)) (hc : ∀ x i, p x i → IsCompact (s x i)) : LocallyCompactSpace X := ⟨fun x _t ht => let ⟨i, hp, ht⟩ := (h x).mem_iff.1 ht ⟨s x i, (h x).mem_of_mem hp, ht, hc x i hp⟩⟩ #align locally_compact_space_of_has_basis LocallyCompactSpace.of_hasBasis @[deprecated (since := "2023-12-29")] alias locallyCompactSpace_of_hasBasis := LocallyCompactSpace.of_hasBasis instance Prod.locallyCompactSpace (X : Type) (Y : Type) [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace X] [LocallyCompactSpace Y] : LocallyCompactSpace (X × Y) := have := fun x : X × Y => (compact_basis_nhds x.1).prod_nhds' (compact_basis_nhds x.2) .of_hasBasis this fun _ _ ⟨⟨_, h₁⟩, _, h₂⟩ => h₁.prod h₂ #align prod.locally_compact_space Prod.locallyCompactSpace instance (priority := 900) [LocallyCompactSpace X] : LocallyCompactPair X Y where exists_mem_nhds_isCompact_mapsTo hf hs := let ⟨K, hKx, hKs, hKc⟩ := local_compact_nhds (hf.continuousAt hs); ⟨K, hKx, hKc, hKs⟩ instance (priority := 100) [LocallyCompactSpace X] : WeaklyLocallyCompactSpace X where exists_compact_mem_nhds (x : X) := let ⟨K, hx, _, hKc⟩ := local_compact_nhds (x := x) univ_mem; ⟨K, hKc, hx⟩	Mathlib/Topology/Compactness/LocallyCompact.lean	141	144	theorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U) (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U := by	rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩ exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩
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" -- 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]} 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⟩ def natDegree (p : R[X]) : ℕ := (degree p).unbot' 0 #align polynomial.nat_degree Polynomial.natDegree def leadingCoeff (p : R[X]) : R := coeff p (natDegree p) #align polynomial.leading_coeff Polynomial.leadingCoeff 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	Mathlib/Algebra/Polynomial/Degree/Definitions.lean	171	172	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]
import Mathlib.LinearAlgebra.FiniteDimensional import Mathlib.MeasureTheory.Group.Pointwise import Mathlib.MeasureTheory.Measure.Lebesgue.Basic import Mathlib.MeasureTheory.Measure.Haar.Basic import Mathlib.MeasureTheory.Measure.Doubling import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric #align_import measure_theory.measure.lebesgue.eq_haar from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" assert_not_exists MeasureTheory.integral open TopologicalSpace Set Filter Metric Bornology open scoped ENNReal Pointwise Topology NNReal def TopologicalSpace.PositiveCompacts.Icc01 : PositiveCompacts ℝ where carrier := Icc 0 1 isCompact' := isCompact_Icc interior_nonempty' := by simp_rw [interior_Icc, nonempty_Ioo, zero_lt_one] #align topological_space.positive_compacts.Icc01 TopologicalSpace.PositiveCompacts.Icc01 universe u def TopologicalSpace.PositiveCompacts.piIcc01 (ι : Type) [Finite ι] : PositiveCompacts (ι → ℝ) where carrier := pi univ fun _ => Icc 0 1 isCompact' := isCompact_univ_pi fun _ => isCompact_Icc interior_nonempty' := by simp only [interior_pi_set, Set.toFinite, interior_Icc, univ_pi_nonempty_iff, nonempty_Ioo, imp_true_iff, zero_lt_one] #align topological_space.positive_compacts.pi_Icc01 TopologicalSpace.PositiveCompacts.piIcc01 theorem Basis.parallelepiped_basisFun (ι : Type) [Fintype ι] : (Pi.basisFun ℝ ι).parallelepiped = TopologicalSpace.PositiveCompacts.piIcc01 ι := SetLike.coe_injective <\| by refine Eq.trans ?_ ((uIcc_of_le ?_).trans (Set.pi_univ_Icc _ _).symm) · classical convert parallelepiped_single (ι := ι) 1 · exact zero_le_one #align basis.parallelepiped_basis_fun Basis.parallelepiped_basisFun theorem Basis.parallelepiped_eq_map {ι E : Type} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) : b.parallelepiped = (PositiveCompacts.piIcc01 ι).map b.equivFun.symm b.equivFunL.symm.continuous b.equivFunL.symm.isOpenMap := by classical rw [← Basis.parallelepiped_basisFun, ← Basis.parallelepiped_map] congr with x simp open MeasureTheory MeasureTheory.Measure theorem Basis.map_addHaar {ι E F : Type} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E] [BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F] (b : Basis ι ℝ E) (f : E ≃L[ℝ] F) : map f b.addHaar = (b.map f.toLinearEquiv).addHaar := by have : IsAddHaarMeasure (map f b.addHaar) := AddEquiv.isAddHaarMeasure_map b.addHaar f.toAddEquiv f.continuous f.symm.continuous rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable (PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map] erw [← image_parallelepiped, f.toEquiv.preimage_image, addHaar_self] namespace MeasureTheory open Measure TopologicalSpace.PositiveCompacts FiniteDimensional theorem addHaarMeasure_eq_volume : addHaarMeasure Icc01 = volume := by convert (addHaarMeasure_unique volume Icc01).symm; simp [Icc01] #align measure_theory.add_haar_measure_eq_volume MeasureTheory.addHaarMeasure_eq_volume theorem addHaarMeasure_eq_volume_pi (ι : Type) [Fintype ι] : addHaarMeasure (piIcc01 ι) = volume := by convert (addHaarMeasure_unique volume (piIcc01 ι)).symm simp only [piIcc01, volume_pi_pi fun _ => Icc (0 : ℝ) 1, PositiveCompacts.coe_mk, Compacts.coe_mk, Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero] #align measure_theory.add_haar_measure_eq_volume_pi MeasureTheory.addHaarMeasure_eq_volume_pi -- Porting note (#11215): TODO: remove this instance? instance isAddHaarMeasure_volume_pi (ι : Type) [Fintype ι] : IsAddHaarMeasure (volume : Measure (ι → ℝ)) := inferInstance #align measure_theory.is_add_haar_measure_volume_pi MeasureTheory.isAddHaarMeasure_volume_pi namespace Measure theorem addHaar_eq_zero_of_disjoint_translates_aux {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (sb : IsBounded s) (hu : IsBounded (range u)) (hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s) : μ s = 0 := by by_contra h apply lt_irrefl ∞ calc ∞ = ∑' _ : ℕ, μ s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm _ = ∑' n : ℕ, μ ({u n} + s) := by congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add] _ = μ (⋃ n, {u n} + s) := Eq.symm <\| measure_iUnion hs fun n => by simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's _ = μ (range u + s) := by rw [← iUnion_add, iUnion_singleton_eq_range] _ < ∞ := (hu.add sb).measure_lt_top #align measure_theory.measure.add_haar_eq_zero_of_disjoint_translates_aux MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates_aux theorem addHaar_eq_zero_of_disjoint_translates {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (hu : IsBounded (range u)) (hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : MeasurableSet s) : μ s = 0 := by suffices H : ∀ R, μ (s ∩ closedBall 0 R) = 0 by apply le_antisymm _ (zero_le _) calc μ s ≤ ∑' n : ℕ, μ (s ∩ closedBall 0 n) := by conv_lhs => rw [← iUnion_inter_closedBall_nat s 0] exact measure_iUnion_le _ _ = 0 := by simp only [H, tsum_zero] intro R apply addHaar_eq_zero_of_disjoint_translates_aux μ u (isBounded_closedBall.subset inter_subset_right) hu _ (h's.inter measurableSet_closedBall) refine pairwise_disjoint_mono hs fun n => ?_ exact add_subset_add Subset.rfl inter_subset_left #align measure_theory.measure.add_haar_eq_zero_of_disjoint_translates MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates theorem addHaar_submodule {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E) (hs : s ≠ ⊤) : μ s = 0 := by obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by simpa only [Submodule.eq_top_iff', not_exists, Ne, not_forall] using hs obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩ have A : IsBounded (range fun n : ℕ => c ^ n • x) := have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) := (tendsto_pow_atTop_nhds_zero_of_lt_one cpos.le cone).smul_const x isBounded_range_of_tendsto _ this apply addHaar_eq_zero_of_disjoint_translates μ _ A _ (Submodule.closed_of_finiteDimensional s).measurableSet intro m n hmn simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage, SetLike.mem_coe] intro y hym hyn have A : (c ^ n - c ^ m) • x ∈ s := by convert s.sub_mem hym hyn using 1 simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub] have H : c ^ n - c ^ m ≠ 0 := by simpa only [sub_eq_zero, Ne] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm have : x ∈ s := by convert s.smul_mem (c ^ n - c ^ m)⁻¹ A rw [smul_smul, inv_mul_cancel H, one_smul] exact hx this #align measure_theory.measure.add_haar_submodule MeasureTheory.Measure.addHaar_submodule theorem addHaar_affineSubspace {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : AffineSubspace ℝ E) (hs : s ≠ ⊤) : μ s = 0 := by rcases s.eq_bot_or_nonempty with (rfl \| hne) · rw [AffineSubspace.bot_coe, measure_empty] rw [Ne, ← AffineSubspace.direction_eq_top_iff_of_nonempty hne] at hs rcases hne with ⟨x, hx : x ∈ s⟩ simpa only [AffineSubspace.coe_direction_eq_vsub_set_right hx, vsub_eq_sub, sub_eq_add_neg, image_add_right, neg_neg, measure_preimage_add_right] using addHaar_submodule μ s.direction hs #align measure_theory.measure.add_haar_affine_subspace MeasureTheory.Measure.addHaar_affineSubspace theorem map_linearMap_addHaar_pi_eq_smul_addHaar {ι : Type} [Finite ι] {f : (ι → ℝ) →ₗ[ℝ] ι → ℝ} (hf : LinearMap.det f ≠ 0) (μ : Measure (ι → ℝ)) [IsAddHaarMeasure μ] : Measure.map f μ = ENNReal.ofReal (abs (LinearMap.det f)⁻¹) • μ := by cases nonempty_fintype ι have := addHaarMeasure_unique μ (piIcc01 ι) rw [this, addHaarMeasure_eq_volume_pi, Measure.map_smul, Real.map_linearMap_volume_pi_eq_smul_volume_pi hf, smul_comm] #align measure_theory.measure.map_linear_map_add_haar_pi_eq_smul_add_haar MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] {F : Type} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] theorem map_linearMap_addHaar_eq_smul_addHaar {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0) : Measure.map f μ = ENNReal.ofReal \|(LinearMap.det f)⁻¹\| • μ := by -- we reduce to the case of `E = ι → ℝ`, for which we have already proved the result using -- matrices in `map_linearMap_addHaar_pi_eq_smul_addHaar`. let ι := Fin (finrank ℝ E) haveI : FiniteDimensional ℝ (ι → ℝ) := by infer_instance have : finrank ℝ E = finrank ℝ (ι → ℝ) := by simp [ι] have e : E ≃ₗ[ℝ] ι → ℝ := LinearEquiv.ofFinrankEq E (ι → ℝ) this -- next line is to avoid `g` getting reduced by `simp`. obtain ⟨g, hg⟩ : ∃ g, g = (e : E →ₗ[ℝ] ι → ℝ).comp (f.comp (e.symm : (ι → ℝ) →ₗ[ℝ] E)) := ⟨_, rfl⟩ have gdet : LinearMap.det g = LinearMap.det f := by rw [hg]; exact LinearMap.det_conj f e rw [← gdet] at hf ⊢ have fg : f = (e.symm : (ι → ℝ) →ₗ[ℝ] E).comp (g.comp (e : E →ₗ[ℝ] ι → ℝ)) := by ext x simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp, LinearEquiv.symm_apply_apply, hg] simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp] have Ce : Continuous e := (e : E →ₗ[ℝ] ι → ℝ).continuous_of_finiteDimensional have Cg : Continuous g := LinearMap.continuous_of_finiteDimensional g have Cesymm : Continuous e.symm := (e.symm : (ι → ℝ) →ₗ[ℝ] E).continuous_of_finiteDimensional rw [← map_map Cesymm.measurable (Cg.comp Ce).measurable, ← map_map Cg.measurable Ce.measurable] haveI : IsAddHaarMeasure (map e μ) := (e : E ≃+ (ι → ℝ)).isAddHaarMeasure_map μ Ce Cesymm have ecomp : e.symm ∘ e = id := by ext x; simp only [id, Function.comp_apply, LinearEquiv.symm_apply_apply] rw [map_linearMap_addHaar_pi_eq_smul_addHaar hf (map e μ), Measure.map_smul, map_map Cesymm.measurable Ce.measurable, ecomp, Measure.map_id] #align measure_theory.measure.map_linear_map_add_haar_eq_smul_add_haar MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar @[simp] theorem addHaar_preimage_linearMap {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0) (s : Set E) : μ (f ⁻¹' s) = ENNReal.ofReal \|(LinearMap.det f)⁻¹\| μ s := calc μ (f ⁻¹' s) = Measure.map f μ s := ((f.equivOfDetNeZero hf).toContinuousLinearEquiv.toHomeomorph.toMeasurableEquiv.map_apply s).symm _ = ENNReal.ofReal \|(LinearMap.det f)⁻¹\| * μ s := by rw [map_linearMap_addHaar_eq_smul_addHaar μ hf]; rfl #align measure_theory.measure.add_haar_preimage_linear_map MeasureTheory.Measure.addHaar_preimage_linearMap @[simp] theorem addHaar_preimage_continuousLinearMap {f : E →L[ℝ] E} (hf : LinearMap.det (f : E →ₗ[ℝ] E) ≠ 0) (s : Set E) : μ (f ⁻¹' s) = ENNReal.ofReal (abs (LinearMap.det (f : E →ₗ[ℝ] E))⁻¹) * μ s := addHaar_preimage_linearMap μ hf s #align measure_theory.measure.add_haar_preimage_continuous_linear_map MeasureTheory.Measure.addHaar_preimage_continuousLinearMap @[simp] theorem addHaar_preimage_linearEquiv (f : E ≃ₗ[ℝ] E) (s : Set E) : μ (f ⁻¹' s) = ENNReal.ofReal \|LinearMap.det (f.symm : E →ₗ[ℝ] E)\| * μ s := by have A : LinearMap.det (f : E →ₗ[ℝ] E) ≠ 0 := (LinearEquiv.isUnit_det' f).ne_zero convert addHaar_preimage_linearMap μ A s simp only [LinearEquiv.det_coe_symm] #align measure_theory.measure.add_haar_preimage_linear_equiv MeasureTheory.Measure.addHaar_preimage_linearEquiv @[simp] theorem addHaar_preimage_continuousLinearEquiv (f : E ≃L[ℝ] E) (s : Set E) : μ (f ⁻¹' s) = ENNReal.ofReal \|LinearMap.det (f.symm : E →ₗ[ℝ] E)\| * μ s := addHaar_preimage_linearEquiv μ _ s #align measure_theory.measure.add_haar_preimage_continuous_linear_equiv MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv @[simp] theorem addHaar_image_linearMap (f : E →ₗ[ℝ] E) (s : Set E) : μ (f '' s) = ENNReal.ofReal \|LinearMap.det f\| * μ s := by rcases ne_or_eq (LinearMap.det f) 0 with (hf \| hf) · let g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv change μ (g '' s) = _ rw [ContinuousLinearEquiv.image_eq_preimage g s, addHaar_preimage_continuousLinearEquiv] congr · simp only [hf, zero_mul, ENNReal.ofReal_zero, abs_zero] have : μ (LinearMap.range f) = 0 := addHaar_submodule μ _ (LinearMap.range_lt_top_of_det_eq_zero hf).ne exact le_antisymm (le_trans (measure_mono (image_subset_range _ _)) this.le) (zero_le _) #align measure_theory.measure.add_haar_image_linear_map MeasureTheory.Measure.addHaar_image_linearMap @[simp] theorem addHaar_image_continuousLinearMap (f : E →L[ℝ] E) (s : Set E) : μ (f '' s) = ENNReal.ofReal \|LinearMap.det (f : E →ₗ[ℝ] E)\| * μ s := addHaar_image_linearMap μ _ s #align measure_theory.measure.add_haar_image_continuous_linear_map MeasureTheory.Measure.addHaar_image_continuousLinearMap @[simp] theorem addHaar_image_continuousLinearEquiv (f : E ≃L[ℝ] E) (s : Set E) : μ (f '' s) = ENNReal.ofReal \|LinearMap.det (f : E →ₗ[ℝ] E)\| * μ s := μ.addHaar_image_linearMap (f : E →ₗ[ℝ] E) s #align measure_theory.measure.add_haar_image_continuous_linear_equiv MeasureTheory.Measure.addHaar_image_continuousLinearEquiv	Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean	342	346	theorem LinearMap.quasiMeasurePreserving (f : E →ₗ[ℝ] E) (hf : LinearMap.det f ≠ 0) : QuasiMeasurePreserving f μ μ := by	refine ⟨f.continuous_of_finiteDimensional.measurable, ?_⟩ rw [map_linearMap_addHaar_eq_smul_addHaar μ hf] exact smul_absolutelyContinuous
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics #align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8" noncomputable section open scoped Classical open Real Topology NNReal ENNReal Filter ComplexConjugate open Filter Finset Set section CpowLimits open Complex variable {α : Type*} theorem zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) : (fun x : ℂ => (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 := by suffices ∀ᶠ x : ℂ in 𝓝 b, x ≠ 0 from this.mono fun x hx ↦ by dsimp only rw [zero_cpow hx, Pi.zero_apply] exact IsOpen.eventually_mem isOpen_ne hb #align zero_cpow_eq_nhds zero_cpow_eq_nhds	Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean	44	50	theorem cpow_eq_nhds {a b : ℂ} (ha : a ≠ 0) : (fun x => x ^ b) =ᶠ[𝓝 a] fun x => exp (log x * b) := by	suffices ∀ᶠ x : ℂ in 𝓝 a, x ≠ 0 from this.mono fun x hx ↦ by dsimp only rw [cpow_def_of_ne_zero hx] exact IsOpen.eventually_mem isOpen_ne ha
import Mathlib.Data.Fintype.List #align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49" assert_not_exists MonoidWithZero namespace List variable {α : Type*} [DecidableEq α] def nextOr : ∀ (_ : List α) (_ _ : α), α \| [], _, default => default \| [_], _, default => default -- Handles the not-found and the wraparound case \| y :: z :: xs, x, default => if x = y then z else nextOr (z :: xs) x default #align list.next_or List.nextOr @[simp] theorem nextOr_nil (x d : α) : nextOr [] x d = d := rfl #align list.next_or_nil List.nextOr_nil @[simp] theorem nextOr_singleton (x y d : α) : nextOr [y] x d = d := rfl #align list.next_or_singleton List.nextOr_singleton @[simp] theorem nextOr_self_cons_cons (xs : List α) (x y d : α) : nextOr (x :: y :: xs) x d = y := if_pos rfl #align list.next_or_self_cons_cons List.nextOr_self_cons_cons theorem nextOr_cons_of_ne (xs : List α) (y x d : α) (h : x ≠ y) : nextOr (y :: xs) x d = nextOr xs x d := by cases' xs with z zs · rfl · exact if_neg h #align list.next_or_cons_of_ne List.nextOr_cons_of_ne theorem nextOr_eq_nextOr_of_mem_of_ne (xs : List α) (x d d' : α) (x_mem : x ∈ xs) (x_ne : x ≠ xs.getLast (ne_nil_of_mem x_mem)) : nextOr xs x d = nextOr xs x d' := by induction' xs with y ys IH · cases x_mem cases' ys with z zs · simp at x_mem x_ne contradiction by_cases h : x = y · rw [h, nextOr_self_cons_cons, nextOr_self_cons_cons] · rw [nextOr, nextOr, IH] · simpa [h] using x_mem · simpa using x_ne #align list.next_or_eq_next_or_of_mem_of_ne List.nextOr_eq_nextOr_of_mem_of_ne theorem mem_of_nextOr_ne {xs : List α} {x d : α} (h : nextOr xs x d ≠ d) : x ∈ xs := by induction' xs with y ys IH · simp at h cases' ys with z zs · simp at h · by_cases hx : x = y · simp [hx] · rw [nextOr_cons_of_ne _ _ _ _ hx] at h simpa [hx] using IH h #align list.mem_of_next_or_ne List.mem_of_nextOr_ne theorem nextOr_concat {xs : List α} {x : α} (d : α) (h : x ∉ xs) : nextOr (xs ++ [x]) x d = d := by induction' xs with z zs IH · simp · obtain ⟨hz, hzs⟩ := not_or.mp (mt mem_cons.2 h) rw [cons_append, nextOr_cons_of_ne _ _ _ _ hz, IH hzs] #align list.next_or_concat List.nextOr_concat theorem nextOr_mem {xs : List α} {x d : α} (hd : d ∈ xs) : nextOr xs x d ∈ xs := by revert hd suffices ∀ xs' : List α, (∀ x ∈ xs, x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs' by exact this xs fun _ => id intro xs' hxs' hd induction' xs with y ys ih · exact hd cases' ys with z zs · exact hd rw [nextOr] split_ifs with h · exact hxs' _ (mem_cons_of_mem _ (mem_cons_self _ _)) · exact ih fun _ h => hxs' _ (mem_cons_of_mem _ h) #align list.next_or_mem List.nextOr_mem def next (l : List α) (x : α) (h : x ∈ l) : α := nextOr l x (l.get ⟨0, length_pos_of_mem h⟩) #align list.next List.next def prev : ∀ l : List α, ∀ x ∈ l, α \| [], _, h => by simp at h \| [y], _, _ => y \| y :: z :: xs, x, h => if hx : x = y then getLast (z :: xs) (cons_ne_nil _ _) else if x = z then y else prev (z :: xs) x (by simpa [hx] using h) #align list.prev List.prev variable (l : List α) (x : α) @[simp] theorem next_singleton (x y : α) (h : x ∈ [y]) : next [y] x h = y := rfl #align list.next_singleton List.next_singleton @[simp] theorem prev_singleton (x y : α) (h : x ∈ [y]) : prev [y] x h = y := rfl #align list.prev_singleton List.prev_singleton theorem next_cons_cons_eq' (y z : α) (h : x ∈ y :: z :: l) (hx : x = y) : next (y :: z :: l) x h = z := by rw [next, nextOr, if_pos hx] #align list.next_cons_cons_eq' List.next_cons_cons_eq' @[simp] theorem next_cons_cons_eq (z : α) (h : x ∈ x :: z :: l) : next (x :: z :: l) x h = z := next_cons_cons_eq' l x x z h rfl #align list.next_cons_cons_eq List.next_cons_cons_eq theorem next_ne_head_ne_getLast (h : x ∈ l) (y : α) (h : x ∈ y :: l) (hy : x ≠ y) (hx : x ≠ getLast (y :: l) (cons_ne_nil _ _)) : next (y :: l) x h = next l x (by simpa [hy] using h) := by rw [next, next, nextOr_cons_of_ne _ _ _ _ hy, nextOr_eq_nextOr_of_mem_of_ne] · rwa [getLast_cons] at hx exact ne_nil_of_mem (by assumption) · rwa [getLast_cons] at hx #align list.next_ne_head_ne_last List.next_ne_head_ne_getLast theorem next_cons_concat (y : α) (hy : x ≠ y) (hx : x ∉ l) (h : x ∈ y :: l ++ [x] := mem_append_right _ (mem_singleton_self x)) : next (y :: l ++ [x]) x h = y := by rw [next, nextOr_concat] · rfl · simp [hy, hx] #align list.next_cons_concat List.next_cons_concat theorem next_getLast_cons (h : x ∈ l) (y : α) (h : x ∈ y :: l) (hy : x ≠ y) (hx : x = getLast (y :: l) (cons_ne_nil _ _)) (hl : Nodup l) : next (y :: l) x h = y := by rw [next, get, ← dropLast_append_getLast (cons_ne_nil y l), hx, nextOr_concat] subst hx intro H obtain ⟨⟨_ \| k, hk⟩, hk'⟩ := get_of_mem H · rw [← Option.some_inj] at hk' rw [← get?_eq_get, dropLast_eq_take, get?_take, get?_zero, head?_cons, Option.some_inj] at hk' · exact hy (Eq.symm hk') rw [length_cons, Nat.pred_succ] exact length_pos_of_mem (by assumption) suffices k + 1 = l.length by simp [this] at hk cases' l with hd tl · simp at hk · rw [nodup_iff_injective_get] at hl rw [length, Nat.succ_inj'] refine Fin.val_eq_of_eq <\| @hl ⟨k, Nat.lt_of_succ_lt <\| by simpa using hk⟩ ⟨tl.length, by simp⟩ ?_ rw [← Option.some_inj] at hk' rw [← get?_eq_get, dropLast_eq_take, get?_take, get?, get?_eq_get, Option.some_inj] at hk' · rw [hk'] simp only [getLast_eq_get, length_cons, ge_iff_le, Nat.succ_sub_succ_eq_sub, nonpos_iff_eq_zero, add_eq_zero_iff, and_false, Nat.sub_zero, get_cons_succ] simpa using hk #align list.next_last_cons List.next_getLast_cons theorem prev_getLast_cons' (y : α) (hxy : x ∈ y :: l) (hx : x = y) : prev (y :: l) x hxy = getLast (y :: l) (cons_ne_nil _ _) := by cases l <;> simp [prev, hx] #align list.prev_last_cons' List.prev_getLast_cons' @[simp] theorem prev_getLast_cons (h : x ∈ x :: l) : prev (x :: l) x h = getLast (x :: l) (cons_ne_nil _ _) := prev_getLast_cons' l x x h rfl #align list.prev_last_cons List.prev_getLast_cons theorem prev_cons_cons_eq' (y z : α) (h : x ∈ y :: z :: l) (hx : x = y) : prev (y :: z :: l) x h = getLast (z :: l) (cons_ne_nil _ _) := by rw [prev, dif_pos hx] #align list.prev_cons_cons_eq' List.prev_cons_cons_eq' --@[simp] Porting note (#10618): `simp` can prove it theorem prev_cons_cons_eq (z : α) (h : x ∈ x :: z :: l) : prev (x :: z :: l) x h = getLast (z :: l) (cons_ne_nil _ _) := prev_cons_cons_eq' l x x z h rfl #align list.prev_cons_cons_eq List.prev_cons_cons_eq theorem prev_cons_cons_of_ne' (y z : α) (h : x ∈ y :: z :: l) (hy : x ≠ y) (hz : x = z) : prev (y :: z :: l) x h = y := by cases l · simp [prev, hy, hz] · rw [prev, dif_neg hy, if_pos hz] #align list.prev_cons_cons_of_ne' List.prev_cons_cons_of_ne' theorem prev_cons_cons_of_ne (y : α) (h : x ∈ y :: x :: l) (hy : x ≠ y) : prev (y :: x :: l) x h = y := prev_cons_cons_of_ne' _ _ _ _ _ hy rfl #align list.prev_cons_cons_of_ne List.prev_cons_cons_of_ne theorem prev_ne_cons_cons (y z : α) (h : x ∈ y :: z :: l) (hy : x ≠ y) (hz : x ≠ z) : prev (y :: z :: l) x h = prev (z :: l) x (by simpa [hy] using h) := by cases l · simp [hy, hz] at h · rw [prev, dif_neg hy, if_neg hz] #align list.prev_ne_cons_cons List.prev_ne_cons_cons theorem next_mem (h : x ∈ l) : l.next x h ∈ l := nextOr_mem (get_mem _ _ _) #align list.next_mem List.next_mem theorem prev_mem (h : x ∈ l) : l.prev x h ∈ l := by cases' l with hd tl · simp at h induction' tl with hd' tl hl generalizing hd · simp · by_cases hx : x = hd · simp only [hx, prev_cons_cons_eq] exact mem_cons_of_mem _ (getLast_mem _) · rw [prev, dif_neg hx] split_ifs with hm · exact mem_cons_self _ _ · exact mem_cons_of_mem _ (hl _ _) #align list.prev_mem List.prev_mem -- Porting note (#10756): new theorem theorem next_get : ∀ (l : List α) (_h : Nodup l) (i : Fin l.length), next l (l.get i) (get_mem _ _ _) = l.get ⟨(i + 1) % l.length, Nat.mod_lt _ (i.1.zero_le.trans_lt i.2)⟩ \| [], _, i => by simpa using i.2 \| [_], _, _ => by simp \| x::y::l, _h, ⟨0, h0⟩ => by have h₁ : get (x :: y :: l) { val := 0, isLt := h0 } = x := by simp rw [next_cons_cons_eq' _ _ _ _ _ h₁] simp \| x::y::l, hn, ⟨i+1, hi⟩ => by have hx' : (x :: y :: l).get ⟨i+1, hi⟩ ≠ x := by intro H suffices (i + 1 : ℕ) = 0 by simpa rw [nodup_iff_injective_get] at hn refine Fin.val_eq_of_eq (@hn ⟨i + 1, hi⟩ ⟨0, by simp⟩ ?_) simpa using H have hi' : i ≤ l.length := Nat.le_of_lt_succ (Nat.succ_lt_succ_iff.1 hi) rcases hi'.eq_or_lt with (hi' \| hi') · subst hi' rw [next_getLast_cons] · simp [hi', get] · rw [get_cons_succ]; exact get_mem _ _ _ · exact hx' · simp [getLast_eq_get] · exact hn.of_cons · rw [next_ne_head_ne_getLast _ _ _ _ _ hx'] · simp only [get_cons_succ] rw [next_get (y::l), ← get_cons_succ (a := x)] · congr dsimp rw [Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 hi'), Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 (Nat.succ_lt_succ_iff.2 hi'))] · simp [Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 hi'), Nat.succ_eq_add_one, hi'] · exact hn.of_cons · rw [getLast_eq_get] intro h have := nodup_iff_injective_get.1 hn h simp at this; simp [this] at hi' · rw [get_cons_succ]; exact get_mem _ _ _ set_option linter.deprecated false in @[deprecated next_get (since := "2023-01-27")] theorem next_nthLe (l : List α) (h : Nodup l) (n : ℕ) (hn : n < l.length) : next l (l.nthLe n hn) (nthLe_mem _ _ _) = l.nthLe ((n + 1) % l.length) (Nat.mod_lt _ (n.zero_le.trans_lt hn)) := next_get l h ⟨n, hn⟩ #align list.next_nth_le List.next_nthLe set_option linter.deprecated false in theorem prev_nthLe (l : List α) (h : Nodup l) (n : ℕ) (hn : n < l.length) : prev l (l.nthLe n hn) (nthLe_mem _ _ _) = l.nthLe ((n + (l.length - 1)) % l.length) (Nat.mod_lt _ (n.zero_le.trans_lt hn)) := by cases' l with x l · simp at hn induction' l with y l hl generalizing n x · simp · rcases n with (_ \| _ \| n) · simp [Nat.add_succ_sub_one, add_zero, List.prev_cons_cons_eq, Nat.zero_eq, List.length, List.nthLe, Nat.succ_add_sub_one, zero_add, getLast_eq_get, Nat.mod_eq_of_lt (Nat.succ_lt_succ l.length.lt_succ_self)] · simp only [mem_cons, nodup_cons] at h push_neg at h simp only [List.prev_cons_cons_of_ne _ _ _ _ h.left.left.symm, Nat.zero_eq, List.length, List.nthLe, add_comm, eq_self_iff_true, Nat.succ_add_sub_one, Nat.mod_self, zero_add, List.get] · rw [prev_ne_cons_cons] · convert hl n.succ y h.of_cons (Nat.le_of_succ_le_succ hn) using 1 have : ∀ k hk, (y :: l).nthLe k hk = (x :: y :: l).nthLe (k + 1) (Nat.succ_lt_succ hk) := by intros simp [List.nthLe] rw [this] congr simp only [Nat.add_succ_sub_one, add_zero, length] simp only [length, Nat.succ_lt_succ_iff] at hn set k := l.length rw [Nat.succ_add, ← Nat.add_succ, Nat.add_mod_right, Nat.succ_add, ← Nat.add_succ _ k, Nat.add_mod_right, Nat.mod_eq_of_lt, Nat.mod_eq_of_lt] · exact Nat.lt_succ_of_lt hn · exact Nat.succ_lt_succ (Nat.lt_succ_of_lt hn) · intro H suffices n.succ.succ = 0 by simpa rw [nodup_iff_nthLe_inj] at h refine h _ _ hn Nat.succ_pos' ?_ simpa using H · intro H suffices n.succ.succ = 1 by simpa rw [nodup_iff_nthLe_inj] at h refine h _ _ hn (Nat.succ_lt_succ Nat.succ_pos') ?_ simpa using H #align list.prev_nth_le List.prev_nthLe set_option linter.deprecated false in theorem pmap_next_eq_rotate_one (h : Nodup l) : (l.pmap l.next fun _ h => h) = l.rotate 1 := by apply List.ext_nthLe · simp · intros rw [nthLe_pmap, nthLe_rotate, next_nthLe _ h] #align list.pmap_next_eq_rotate_one List.pmap_next_eq_rotate_one set_option linter.deprecated false in theorem pmap_prev_eq_rotate_length_sub_one (h : Nodup l) : (l.pmap l.prev fun _ h => h) = l.rotate (l.length - 1) := by apply List.ext_nthLe · simp · intro n hn hn' rw [nthLe_rotate, nthLe_pmap, prev_nthLe _ h] #align list.pmap_prev_eq_rotate_length_sub_one List.pmap_prev_eq_rotate_length_sub_one set_option linter.deprecated false in	Mathlib/Data/List/Cycle.lean	374	383	theorem prev_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) : prev l (next l x hx) (next_mem _ _ _) = x := by	obtain ⟨n, hn, rfl⟩ := nthLe_of_mem hx simp only [next_nthLe, prev_nthLe, h, Nat.mod_add_mod] cases' l with hd tl · simp at hx · have : (n + 1 + length tl) % (length tl + 1) = n := by rw [length_cons, Nat.succ_eq_add_one] at hn rw [add_assoc, add_comm 1, Nat.add_mod_right, Nat.mod_eq_of_lt hn] simp only [length_cons, Nat.succ_sub_succ_eq_sub, Nat.sub_zero, Nat.succ_eq_add_one, this]
import Mathlib.SetTheory.Ordinal.Arithmetic #align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d" noncomputable section open Function Cardinal Set Equiv Order open scoped Classical open Cardinal Ordinal universe u v w namespace Ordinal instance pow : Pow Ordinal Ordinal := ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩ -- Porting note: Ambiguous notations. -- local infixr:0 "^" => @Pow.pow Ordinal Ordinal Ordinal.instPowOrdinalOrdinal theorem opow_def (a b : Ordinal) : a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b := rfl #align ordinal.opow_def Ordinal.opow_def -- Porting note: `if_pos rfl` → `if_true` theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a := by simp only [opow_def, if_true] #align ordinal.zero_opow' Ordinal.zero_opow' @[simp] theorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0 := by rwa [zero_opow', Ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero] #align ordinal.zero_opow Ordinal.zero_opow @[simp] theorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1 := by by_cases h : a = 0 · simp only [opow_def, if_pos h, sub_zero] · simp only [opow_def, if_neg h, limitRecOn_zero] #align ordinal.opow_zero Ordinal.opow_zero @[simp] theorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a := if h : a = 0 then by subst a; simp only [zero_opow (succ_ne_zero _), mul_zero] else by simp only [opow_def, limitRecOn_succ, if_neg h] #align ordinal.opow_succ Ordinal.opow_succ theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) : a ^ b = bsup.{u, u} b fun c _ => a ^ c := by simp only [opow_def, if_neg a0]; rw [limitRecOn_limit _ _ _ _ h] #align ordinal.opow_limit Ordinal.opow_limit theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) : a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c := by rw [opow_limit a0 h, bsup_le_iff] #align ordinal.opow_le_of_limit Ordinal.opow_le_of_limit theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) : a < b ^ c ↔ ∃ c' < c, a < b ^ c' := by rw [← not_iff_not, not_exists]; simp only [not_lt, opow_le_of_limit b0 h, exists_prop, not_and] #align ordinal.lt_opow_of_limit Ordinal.lt_opow_of_limit @[simp] theorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a := by rw [← succ_zero, opow_succ]; simp only [opow_zero, one_mul] #align ordinal.opow_one Ordinal.opow_one @[simp] theorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1 := by induction a using limitRecOn with \| H₁ => simp only [opow_zero] \| H₂ _ ih => simp only [opow_succ, ih, mul_one] \| H₃ b l IH => refine eq_of_forall_ge_iff fun c => ?_ rw [opow_le_of_limit Ordinal.one_ne_zero l] exact ⟨fun H => by simpa only [opow_zero] using H 0 l.pos, fun H b' h => by rwa [IH _ h]⟩ #align ordinal.one_opow Ordinal.one_opow theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b := by have h0 : 0 < a ^ (0 : Ordinal) := by simp only [opow_zero, zero_lt_one] induction b using limitRecOn with \| H₁ => exact h0 \| H₂ b IH => rw [opow_succ] exact mul_pos IH a0 \| H₃ b l _ => exact (lt_opow_of_limit (Ordinal.pos_iff_ne_zero.1 a0) l).2 ⟨0, l.pos, h0⟩ #align ordinal.opow_pos Ordinal.opow_pos theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0 := Ordinal.pos_iff_ne_zero.1 <\| opow_pos b <\| Ordinal.pos_iff_ne_zero.2 a0 #align ordinal.opow_ne_zero Ordinal.opow_ne_zero theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ ·) := have a0 : 0 < a := zero_lt_one.trans h ⟨fun b => by simpa only [mul_one, opow_succ] using (mul_lt_mul_iff_left (opow_pos b a0)).2 h, fun b l c => opow_le_of_limit (ne_of_gt a0) l⟩ #align ordinal.opow_is_normal Ordinal.opow_isNormal theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c ↔ b < c := (opow_isNormal a1).lt_iff #align ordinal.opow_lt_opow_iff_right Ordinal.opow_lt_opow_iff_right theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b ≤ a ^ c ↔ b ≤ c := (opow_isNormal a1).le_iff #align ordinal.opow_le_opow_iff_right Ordinal.opow_le_opow_iff_right theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c ↔ b = c := (opow_isNormal a1).inj #align ordinal.opow_right_inj Ordinal.opow_right_inj theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b → IsLimit (a ^ b) := (opow_isNormal a1).isLimit #align ordinal.opow_is_limit Ordinal.opow_isLimit theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b ≠ 0) : IsLimit (a ^ b) := by rcases zero_or_succ_or_limit b with (e \| ⟨b, rfl⟩ \| l') · exact absurd e hb · rw [opow_succ] exact mul_isLimit (opow_pos _ l.pos) l · exact opow_isLimit l.one_lt l' #align ordinal.opow_is_limit_left Ordinal.opow_isLimit_left theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c := by rcases lt_or_eq_of_le (one_le_iff_pos.2 h₁) with h₁ \| h₁ · exact (opow_le_opow_iff_right h₁).2 h₂ · subst a -- Porting note: `le_refl` is required. simp only [one_opow, le_refl] #align ordinal.opow_le_opow_right Ordinal.opow_le_opow_right theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c := by by_cases a0 : a = 0 -- Porting note: `le_refl` is required. · subst a by_cases c0 : c = 0 · subst c simp only [opow_zero, le_refl] · simp only [zero_opow c0, Ordinal.zero_le] · induction c using limitRecOn with \| H₁ => simp only [opow_zero, le_refl] \| H₂ c IH => simpa only [opow_succ] using mul_le_mul' IH ab \| H₃ c l IH => exact (opow_le_of_limit a0 l).2 fun b' h => (IH _ h).trans (opow_le_opow_right ((Ordinal.pos_iff_ne_zero.2 a0).trans_le ab) h.le) #align ordinal.opow_le_opow_left Ordinal.opow_le_opow_left theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b := by nth_rw 1 [← opow_one a] cases' le_or_gt a 1 with a1 a1 · rcases lt_or_eq_of_le a1 with a0 \| a1 · rw [lt_one_iff_zero] at a0 rw [a0, zero_opow Ordinal.one_ne_zero] exact Ordinal.zero_le _ rw [a1, one_opow, one_opow] rwa [opow_le_opow_iff_right a1, one_le_iff_pos] #align ordinal.left_le_opow Ordinal.left_le_opow theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b := (opow_isNormal a1).self_le _ #align ordinal.right_le_opow Ordinal.right_le_opow theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c := by rw [opow_succ, opow_succ] exact (mul_le_mul_right' (opow_le_opow_left c ab.le) a).trans_lt (mul_lt_mul_of_pos_left ab (opow_pos c ((Ordinal.zero_le a).trans_lt ab))) #align ordinal.opow_lt_opow_left_of_succ Ordinal.opow_lt_opow_left_of_succ theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c := by rcases eq_or_ne a 0 with (rfl \| a0) · rcases eq_or_ne c 0 with (rfl \| c0) · simp have : b + c ≠ 0 := ((Ordinal.pos_iff_ne_zero.2 c0).trans_le (le_add_left _ _)).ne' simp only [zero_opow c0, zero_opow this, mul_zero] rcases eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with (rfl \| a1) · simp only [one_opow, mul_one] induction c using limitRecOn with \| H₁ => simp \| H₂ c IH => rw [add_succ, opow_succ, IH, opow_succ, mul_assoc] \| H₃ c l IH => refine eq_of_forall_ge_iff fun d => (((opow_isNormal a1).trans (add_isNormal b)).limit_le l).trans ?_ dsimp only [Function.comp_def] simp (config := { contextual := true }) only [IH] exact (((mul_isNormal <\| opow_pos b (Ordinal.pos_iff_ne_zero.2 a0)).trans (opow_isNormal a1)).limit_le l).symm #align ordinal.opow_add Ordinal.opow_add theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b := by rw [opow_add, opow_one] #align ordinal.opow_one_add Ordinal.opow_one_add theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b ≤ c) : a ^ b ∣ a ^ c := ⟨a ^ (c - b), by rw [← opow_add, Ordinal.add_sub_cancel_of_le h]⟩ #align ordinal.opow_dvd_opow Ordinal.opow_dvd_opow theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b ∣ a ^ c ↔ b ≤ c := ⟨fun h => le_of_not_lt fun hn => not_le_of_lt ((opow_lt_opow_iff_right a1).2 hn) <\| le_of_dvd (opow_ne_zero _ <\| one_le_iff_ne_zero.1 <\| a1.le) h, opow_dvd_opow _⟩ #align ordinal.opow_dvd_opow_iff Ordinal.opow_dvd_opow_iff theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c := by by_cases b0 : b = 0; · simp only [b0, zero_mul, opow_zero, one_opow] by_cases a0 : a = 0 · subst a by_cases c0 : c = 0 · simp only [c0, mul_zero, opow_zero] simp only [zero_opow b0, zero_opow c0, zero_opow (mul_ne_zero b0 c0)] cases' eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with a1 a1 · subst a1 simp only [one_opow] induction c using limitRecOn with \| H₁ => simp only [mul_zero, opow_zero] \| H₂ c IH => rw [mul_succ, opow_add, IH, opow_succ] \| H₃ c l IH => refine eq_of_forall_ge_iff fun d => (((opow_isNormal a1).trans (mul_isNormal (Ordinal.pos_iff_ne_zero.2 b0))).limit_le l).trans ?_ dsimp only [Function.comp_def] simp (config := { contextual := true }) only [IH] exact (opow_le_of_limit (opow_ne_zero _ a0) l).symm #align ordinal.opow_mul Ordinal.opow_mul -- @[pp_nodot] -- Porting note: Unknown attribute. def log (b : Ordinal) (x : Ordinal) : Ordinal := if _h : 1 < b then pred (sInf { o \| x < b ^ o }) else 0 #align ordinal.log Ordinal.log theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal \| x < b ^ o }.Nonempty := ⟨_, succ_le_iff.1 (right_le_opow _ h)⟩ #align ordinal.log_nonempty Ordinal.log_nonempty theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) : log b x = pred (sInf { o \| x < b ^ o }) := by simp only [log, dif_pos h] #align ordinal.log_def Ordinal.log_def theorem log_of_not_one_lt_left {b : Ordinal} (h : ¬1 < b) (x : Ordinal) : log b x = 0 := by simp only [log, dif_neg h] #align ordinal.log_of_not_one_lt_left Ordinal.log_of_not_one_lt_left theorem log_of_left_le_one {b : Ordinal} (h : b ≤ 1) : ∀ x, log b x = 0 := log_of_not_one_lt_left h.not_lt #align ordinal.log_of_left_le_one Ordinal.log_of_left_le_one @[simp] theorem log_zero_left : ∀ b, log 0 b = 0 := log_of_left_le_one zero_le_one #align ordinal.log_zero_left Ordinal.log_zero_left @[simp] theorem log_zero_right (b : Ordinal) : log b 0 = 0 := if b1 : 1 < b then by rw [log_def b1, ← Ordinal.le_zero, pred_le] apply csInf_le' dsimp rw [succ_zero, opow_one] exact zero_lt_one.trans b1 else by simp only [log_of_not_one_lt_left b1] #align ordinal.log_zero_right Ordinal.log_zero_right @[simp] theorem log_one_left : ∀ b, log 1 b = 0 := log_of_left_le_one le_rfl #align ordinal.log_one_left Ordinal.log_one_left theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : succ (log b x) = sInf { o : Ordinal \| x < b ^ o } := by let t := sInf { o : Ordinal \| x < b ^ o } have : x < (b^t) := csInf_mem (log_nonempty hb) rcases zero_or_succ_or_limit t with (h \| h \| h) · refine ((one_le_iff_ne_zero.2 hx).not_lt ?_).elim simpa only [h, opow_zero] using this · rw [show log b x = pred t from log_def hb x, succ_pred_iff_is_succ.2 h] · rcases (lt_opow_of_limit (zero_lt_one.trans hb).ne' h).1 this with ⟨a, h₁, h₂⟩ exact h₁.not_le.elim ((le_csInf_iff'' (log_nonempty hb)).1 le_rfl a h₂) #align ordinal.succ_log_def Ordinal.succ_log_def theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) : x < b ^ succ (log b x) := by rcases eq_or_ne x 0 with (rfl \| hx) · apply opow_pos _ (zero_lt_one.trans hb) · rw [succ_log_def hb hx] exact csInf_mem (log_nonempty hb) #align ordinal.lt_opow_succ_log_self Ordinal.lt_opow_succ_log_self theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x := by rcases eq_or_ne b 0 with (rfl \| b0) · rw [zero_opow'] exact (sub_le_self _ _).trans (one_le_iff_ne_zero.2 hx) rcases lt_or_eq_of_le (one_le_iff_ne_zero.2 b0) with (hb \| rfl) · refine le_of_not_lt fun h => (lt_succ (log b x)).not_le ?_ have := @csInf_le' _ _ { o \| x < b ^ o } _ h rwa [← succ_log_def hb hx] at this · rwa [one_opow, one_le_iff_ne_zero] #align ordinal.opow_log_le_self Ordinal.opow_log_le_self theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x := ⟨fun h => le_of_not_lt fun hn => (lt_opow_succ_log_self hb x).not_le <\| ((opow_le_opow_iff_right hb).2 (succ_le_of_lt hn)).trans h, fun h => ((opow_le_opow_iff_right hb).2 h).trans (opow_log_le_self b hx)⟩ #align ordinal.opow_le_iff_le_log Ordinal.opow_le_iff_le_log theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c := lt_iff_lt_of_le_iff_le (opow_le_iff_le_log hb hx) #align ordinal.lt_opow_iff_log_lt Ordinal.lt_opow_iff_log_lt theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o := by rwa [← succ_le_iff, succ_zero, ← opow_le_iff_le_log hb ho, opow_one] #align ordinal.log_pos Ordinal.log_pos theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0 := by rcases eq_or_ne o 0 with (rfl \| ho) · exact log_zero_right b rcases le_or_lt b 1 with hb \| hb · rcases le_one_iff.1 hb with (rfl \| rfl) · exact log_zero_left o · exact log_one_left o · rwa [← Ordinal.le_zero, ← lt_succ_iff, succ_zero, ← lt_opow_iff_log_lt hb ho, opow_one] #align ordinal.log_eq_zero Ordinal.log_eq_zero @[mono] theorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x ≤ y) : log b x ≤ log b y := if hx : x = 0 then by simp only [hx, log_zero_right, Ordinal.zero_le] else if hb : 1 < b then (opow_le_iff_le_log hb (lt_of_lt_of_le (Ordinal.pos_iff_ne_zero.2 hx) xy).ne').1 <\| (opow_log_le_self _ hx).trans xy else by simp only [log_of_not_one_lt_left hb, Ordinal.zero_le] #align ordinal.log_mono_right Ordinal.log_mono_right theorem log_le_self (b x : Ordinal) : log b x ≤ x := if hx : x = 0 then by simp only [hx, log_zero_right, Ordinal.zero_le] else if hb : 1 < b then (right_le_opow _ hb).trans (opow_log_le_self b hx) else by simp only [log_of_not_one_lt_left hb, Ordinal.zero_le] #align ordinal.log_le_self Ordinal.log_le_self @[simp] theorem log_one_right (b : Ordinal) : log b 1 = 0 := if hb : 1 < b then log_eq_zero hb else log_of_not_one_lt_left hb 1 #align ordinal.log_one_right Ordinal.log_one_right theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : o % (b ^ log b o) < o := by rcases eq_or_ne b 0 with (rfl \| hb) · simpa using Ordinal.pos_iff_ne_zero.2 ho · exact (mod_lt _ <\| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) #align ordinal.mod_opow_log_lt_self Ordinal.mod_opow_log_lt_self theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : log b (o % (b ^ log b o)) < log b o := by rcases eq_or_ne (o % (b ^ log b o)) 0 with h \| h · rw [h, log_zero_right] apply log_pos hb ho hbo · rw [← succ_le_iff, succ_log_def hb h] apply csInf_le' apply mod_lt rw [← Ordinal.pos_iff_ne_zero] exact opow_pos _ (zero_lt_one.trans hb) #align ordinal.log_mod_opow_log_lt_log_self Ordinal.log_mod_opow_log_lt_log_self theorem opow_mul_add_pos {b v : Ordinal} (hb : b ≠ 0) (u : Ordinal) (hv : v ≠ 0) (w : Ordinal) : 0 < b ^ u * v + w := (opow_pos u <\| Ordinal.pos_iff_ne_zero.2 hb).trans_le <\| (le_mul_left _ <\| Ordinal.pos_iff_ne_zero.2 hv).trans <\| le_add_right _ _ #align ordinal.opow_mul_add_pos Ordinal.opow_mul_add_pos theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) : b ^ u * v + w < b ^ u * succ v := by rwa [mul_succ, add_lt_add_iff_left] #align ordinal.opow_mul_add_lt_opow_mul_succ Ordinal.opow_mul_add_lt_opow_mul_succ theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) : b ^ u * v + w < b ^ succ u := by convert (opow_mul_add_lt_opow_mul_succ v hw).trans_le (mul_le_mul_left' (succ_le_of_lt hvb) _) using 1 exact opow_succ b u #align ordinal.opow_mul_add_lt_opow_succ Ordinal.opow_mul_add_lt_opow_succ theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b) (hw : w < b ^ u) : log b (b ^ u * v + w) = u := by have hne' := (opow_mul_add_pos (zero_lt_one.trans hb).ne' u hv w).ne' by_contra! hne cases' lt_or_gt_of_ne hne with h h · rw [← lt_opow_iff_log_lt hb hne'] at h exact h.not_le ((le_mul_left _ (Ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _)) · conv at h => change u < log b (b ^ u * v + w) rw [← succ_le_iff, ← opow_le_iff_le_log hb hne'] at h exact (not_lt_of_le h) (opow_mul_add_lt_opow_succ hvb hw) #align ordinal.log_opow_mul_add Ordinal.log_opow_mul_add theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x := by convert log_opow_mul_add hb zero_ne_one.symm hb (opow_pos x (zero_lt_one.trans hb)) using 1 rw [add_zero, mul_one] #align ordinal.log_opow Ordinal.log_opow theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o) := by rcases eq_zero_or_pos b with (rfl \| hb) · simpa using Ordinal.pos_iff_ne_zero.2 ho · rw [div_pos (opow_ne_zero _ hb.ne')] exact opow_log_le_self b ho #align ordinal.div_opow_log_pos Ordinal.div_opow_log_pos theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b := by rw [div_lt (opow_pos _ (zero_lt_one.trans hb)).ne', ← opow_succ] exact lt_opow_succ_log_self hb o #align ordinal.div_opow_log_lt Ordinal.div_opow_log_lt theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x ≠ 0) (hy : y ≠ 0) : log b x + log b y ≤ log b (x * y) := by by_cases hb : 1 < b · rw [← opow_le_iff_le_log hb (mul_ne_zero hx hy), opow_add] exact mul_le_mul' (opow_log_le_self b hx) (opow_log_le_self b hy) -- Porting note: `le_refl` is required. simp only [log_of_not_one_lt_left hb, zero_add, le_refl] #align ordinal.add_log_le_log_mul Ordinal.add_log_le_log_mul @[simp, norm_cast] theorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^ (n : Ordinal) \| 0 => by simp \| n + 1 => by rw [pow_succ, natCast_mul, natCast_opow m n, Nat.cast_succ, add_one_eq_succ, opow_succ] #align ordinal.nat_cast_opow Ordinal.natCast_opow @[deprecated (since := "2024-04-17")] alias nat_cast_opow := natCast_opow	Mathlib/SetTheory/Ordinal/Exponential.lean	465	471	theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ (n : Ordinal)) = o ^ ω := by	rcases lt_or_eq_of_le (one_le_iff_pos.2 ho) with (ho₁ \| rfl) · exact (opow_isNormal ho₁).apply_omega · rw [one_opow] refine le_antisymm (sup_le fun n => by rw [one_opow]) ?_ convert le_sup (fun n : ℕ => 1 ^ (n : Ordinal)) 0 rw [Nat.cast_zero, opow_zero]
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol #align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071ab828a4ee6e26a52afe4" section Lemmas namespace Mathlib.Meta.NormNum def jacobiSymNat (a b : ℕ) : ℤ := jacobiSym a b #align norm_num.jacobi_sym_nat Mathlib.Meta.NormNum.jacobiSymNat theorem jacobiSymNat.zero_right (a : ℕ) : jacobiSymNat a 0 = 1 := by rw [jacobiSymNat, jacobiSym.zero_right] #align norm_num.jacobi_sym_nat.zero_right Mathlib.Meta.NormNum.jacobiSymNat.zero_right theorem jacobiSymNat.one_right (a : ℕ) : jacobiSymNat a 1 = 1 := by rw [jacobiSymNat, jacobiSym.one_right] #align norm_num.jacobi_sym_nat.one_right Mathlib.Meta.NormNum.jacobiSymNat.one_right theorem jacobiSymNat.zero_left (b : ℕ) (hb : Nat.beq (b / 2) 0 = false) : jacobiSymNat 0 b = 0 := by rw [jacobiSymNat, Nat.cast_zero, jacobiSym.zero_left ?_] calc 1 < 2 * 1 := by decide _ ≤ 2 * (b / 2) := Nat.mul_le_mul_left _ (Nat.succ_le.mpr (Nat.pos_of_ne_zero (Nat.ne_of_beq_eq_false hb))) _ ≤ b := Nat.mul_div_le b 2 #align norm_num.jacobi_sym_nat.zero_left_even Mathlib.Meta.NormNum.jacobiSymNat.zero_left #align norm_num.jacobi_sym_nat.zero_left_odd Mathlib.Meta.NormNum.jacobiSymNat.zero_left theorem jacobiSymNat.one_left (b : ℕ) : jacobiSymNat 1 b = 1 := by rw [jacobiSymNat, Nat.cast_one, jacobiSym.one_left] #align norm_num.jacobi_sym_nat.one_left_even Mathlib.Meta.NormNum.jacobiSymNat.one_left #align norm_num.jacobi_sym_nat.one_left_odd Mathlib.Meta.NormNum.jacobiSymNat.one_left theorem LegendreSym.to_jacobiSym (p : ℕ) (pp : Fact p.Prime) (a r : ℤ) (hr : IsInt (jacobiSym a p) r) : IsInt (legendreSym p a) r := by rwa [@jacobiSym.legendreSym.to_jacobiSym p pp a] #align norm_num.legendre_sym.to_jacobi_sym Mathlib.Meta.NormNum.LegendreSym.to_jacobiSym theorem JacobiSym.mod_left (a : ℤ) (b ab' : ℕ) (ab r b' : ℤ) (hb' : (b : ℤ) = b') (hab : a % b' = ab) (h : (ab' : ℤ) = ab) (hr : jacobiSymNat ab' b = r) : jacobiSym a b = r := by rw [← hr, jacobiSymNat, jacobiSym.mod_left, hb', hab, ← h] #align norm_num.jacobi_sym.mod_left Mathlib.Meta.NormNum.JacobiSym.mod_left theorem jacobiSymNat.mod_left (a b ab : ℕ) (r : ℤ) (hab : a % b = ab) (hr : jacobiSymNat ab b = r) : jacobiSymNat a b = r := by rw [← hr, jacobiSymNat, jacobiSymNat, _root_.jacobiSym.mod_left a b, ← hab]; rfl #align norm_num.jacobi_sym_nat.mod_left Mathlib.Meta.NormNum.jacobiSymNat.mod_left theorem jacobiSymNat.even_even (a b : ℕ) (hb₀ : Nat.beq (b / 2) 0 = false) (ha : a % 2 = 0) (hb₁ : b % 2 = 0) : jacobiSymNat a b = 0 := by refine jacobiSym.eq_zero_iff.mpr ⟨ne_of_gt ((Nat.pos_of_ne_zero (Nat.ne_of_beq_eq_false hb₀)).trans_le (Nat.div_le_self b 2)), fun hf => ?_⟩ have h : 2 ∣ a.gcd b := Nat.dvd_gcd (Nat.dvd_of_mod_eq_zero ha) (Nat.dvd_of_mod_eq_zero hb₁) change 2 ∣ (a : ℤ).gcd b at h rw [hf, ← even_iff_two_dvd] at h exact Nat.not_even_one h #align norm_num.jacobi_sym_nat.even_even Mathlib.Meta.NormNum.jacobiSymNat.even_even theorem jacobiSymNat.odd_even (a b c : ℕ) (r : ℤ) (ha : a % 2 = 1) (hb : b % 2 = 0) (hc : b / 2 = c) (hr : jacobiSymNat a c = r) : jacobiSymNat a b = r := by have ha' : legendreSym 2 a = 1 := by simp only [legendreSym.mod 2 a, Int.ofNat_mod_ofNat, ha] decide rcases eq_or_ne c 0 with (rfl \| hc') · rw [← hr, Nat.eq_zero_of_dvd_of_div_eq_zero (Nat.dvd_of_mod_eq_zero hb) hc] · haveI : NeZero c := ⟨hc'⟩ -- for `jacobiSym.mul_right` rwa [← Nat.mod_add_div b 2, hb, hc, Nat.zero_add, jacobiSymNat, jacobiSym.mul_right, ← jacobiSym.legendreSym.to_jacobiSym, ha', one_mul] #align norm_num.jacobi_sym_nat.odd_even Mathlib.Meta.NormNum.jacobiSymNat.odd_even	Mathlib/Tactic/NormNum/LegendreSymbol.lean	135	138	theorem jacobiSymNat.double_even (a b c : ℕ) (r : ℤ) (ha : a % 4 = 0) (hb : b % 2 = 1) (hc : a / 4 = c) (hr : jacobiSymNat c b = r) : jacobiSymNat a b = r := by	simp only [jacobiSymNat, ← hr, ← hc, Int.ofNat_ediv, Nat.cast_ofNat] exact (jacobiSym.div_four_left (mod_cast ha) hb).symm
import Mathlib.Data.ENNReal.Operations #align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open Set NNReal namespace ENNReal noncomputable section Inv variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0} protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [div_eq_mul_inv, mul_comm] #align ennreal.div_eq_inv_mul ENNReal.div_eq_inv_mul @[simp] theorem inv_zero : (0 : ℝ≥0∞)⁻¹ = ∞ := show sInf { b : ℝ≥0∞ \| 1 ≤ 0 * b } = ∞ by simp #align ennreal.inv_zero ENNReal.inv_zero @[simp] theorem inv_top : ∞⁻¹ = 0 := bot_unique <\| le_of_forall_le_of_dense fun a (h : 0 < a) => sInf_le <\| by simp [, h.ne', top_mul] #align ennreal.inv_top ENNReal.inv_top theorem coe_inv_le : (↑r⁻¹ : ℝ≥0∞) ≤ (↑r)⁻¹ := le_sInf fun b (hb : 1 ≤ ↑r b) => coe_le_iff.2 <\| by rintro b rfl apply NNReal.inv_le_of_le_mul rwa [← coe_mul, ← coe_one, coe_le_coe] at hb #align ennreal.coe_inv_le ENNReal.coe_inv_le @[simp, norm_cast] theorem coe_inv (hr : r ≠ 0) : (↑r⁻¹ : ℝ≥0∞) = (↑r)⁻¹ := coe_inv_le.antisymm <\| sInf_le <\| mem_setOf.2 <\| by rw [← coe_mul, mul_inv_cancel hr, coe_one] #align ennreal.coe_inv ENNReal.coe_inv @[norm_cast] theorem coe_inv_two : ((2⁻¹ : ℝ≥0) : ℝ≥0∞) = 2⁻¹ := by rw [coe_inv _root_.two_ne_zero, coe_two] #align ennreal.coe_inv_two ENNReal.coe_inv_two @[simp, norm_cast] theorem coe_div (hr : r ≠ 0) : (↑(p / r) : ℝ≥0∞) = p / r := by rw [div_eq_mul_inv, div_eq_mul_inv, coe_mul, coe_inv hr] #align ennreal.coe_div ENNReal.coe_div lemma coe_div_le : ↑(p / r) ≤ (p / r : ℝ≥0∞) := by simpa only [div_eq_mul_inv, coe_mul] using mul_le_mul_left' coe_inv_le _ theorem div_zero (h : a ≠ 0) : a / 0 = ∞ := by simp [div_eq_mul_inv, h] #align ennreal.div_zero ENNReal.div_zero instance : DivInvOneMonoid ℝ≥0∞ := { inferInstanceAs (DivInvMonoid ℝ≥0∞) with inv_one := by simpa only [coe_inv one_ne_zero, coe_one] using coe_inj.2 inv_one } protected theorem inv_pow : ∀ {a : ℝ≥0∞} {n : ℕ}, (a ^ n)⁻¹ = a⁻¹ ^ n \| _, 0 => by simp only [pow_zero, inv_one] \| ⊤, n + 1 => by simp [top_pow] \| (a : ℝ≥0), n + 1 => by rcases eq_or_ne a 0 with (rfl \| ha) · simp [top_pow] · have := pow_ne_zero (n + 1) ha norm_cast rw [inv_pow] #align ennreal.inv_pow ENNReal.inv_pow protected theorem mul_inv_cancel (h0 : a ≠ 0) (ht : a ≠ ∞) : a * a⁻¹ = 1 := by lift a to ℝ≥0 using ht norm_cast at h0; norm_cast exact mul_inv_cancel h0 #align ennreal.mul_inv_cancel ENNReal.mul_inv_cancel protected theorem inv_mul_cancel (h0 : a ≠ 0) (ht : a ≠ ∞) : a⁻¹ * a = 1 := mul_comm a a⁻¹ ▸ ENNReal.mul_inv_cancel h0 ht #align ennreal.inv_mul_cancel ENNReal.inv_mul_cancel protected theorem div_mul_cancel (h0 : a ≠ 0) (hI : a ≠ ∞) : b / a * a = b := by rw [div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel h0 hI, mul_one] #align ennreal.div_mul_cancel ENNReal.div_mul_cancel protected theorem mul_div_cancel' (h0 : a ≠ 0) (hI : a ≠ ∞) : a * (b / a) = b := by rw [mul_comm, ENNReal.div_mul_cancel h0 hI] #align ennreal.mul_div_cancel' ENNReal.mul_div_cancel' -- Porting note: `simp only [div_eq_mul_inv, mul_comm, mul_assoc]` doesn't work in the following two protected theorem mul_comm_div : a / b * c = a * (c / b) := by simp only [div_eq_mul_inv, mul_right_comm, ← mul_assoc] #align ennreal.mul_comm_div ENNReal.mul_comm_div protected theorem mul_div_right_comm : a * b / c = a / c * b := by simp only [div_eq_mul_inv, mul_right_comm] #align ennreal.mul_div_right_comm ENNReal.mul_div_right_comm instance : InvolutiveInv ℝ≥0∞ where inv_inv a := by by_cases a = 0 <;> cases a <;> simp_all [none_eq_top, some_eq_coe, -coe_inv, (coe_inv _).symm] @[simp] protected lemma inv_eq_one : a⁻¹ = 1 ↔ a = 1 := by rw [← inv_inj, inv_inv, inv_one] @[simp] theorem inv_eq_top : a⁻¹ = ∞ ↔ a = 0 := inv_zero ▸ inv_inj #align ennreal.inv_eq_top ENNReal.inv_eq_top theorem inv_ne_top : a⁻¹ ≠ ∞ ↔ a ≠ 0 := by simp #align ennreal.inv_ne_top ENNReal.inv_ne_top @[simp] theorem inv_lt_top {x : ℝ≥0∞} : x⁻¹ < ∞ ↔ 0 < x := by simp only [lt_top_iff_ne_top, inv_ne_top, pos_iff_ne_zero] #align ennreal.inv_lt_top ENNReal.inv_lt_top theorem div_lt_top {x y : ℝ≥0∞} (h1 : x ≠ ∞) (h2 : y ≠ 0) : x / y < ∞ := mul_lt_top h1 (inv_ne_top.mpr h2) #align ennreal.div_lt_top ENNReal.div_lt_top @[simp] protected theorem inv_eq_zero : a⁻¹ = 0 ↔ a = ∞ := inv_top ▸ inv_inj #align ennreal.inv_eq_zero ENNReal.inv_eq_zero protected theorem inv_ne_zero : a⁻¹ ≠ 0 ↔ a ≠ ∞ := by simp #align ennreal.inv_ne_zero ENNReal.inv_ne_zero protected theorem div_pos (ha : a ≠ 0) (hb : b ≠ ∞) : 0 < a / b := ENNReal.mul_pos ha <\| ENNReal.inv_ne_zero.2 hb #align ennreal.div_pos ENNReal.div_pos protected theorem mul_inv {a b : ℝ≥0∞} (ha : a ≠ 0 ∨ b ≠ ∞) (hb : a ≠ ∞ ∨ b ≠ 0) : (a * b)⁻¹ = a⁻¹ * b⁻¹ := by induction' b with b · replace ha : a ≠ 0 := ha.neg_resolve_right rfl simp [ha] induction' a with a · replace hb : b ≠ 0 := coe_ne_zero.1 (hb.neg_resolve_left rfl) simp [hb] by_cases h'a : a = 0 · simp only [h'a, top_mul, ENNReal.inv_zero, ENNReal.coe_ne_top, zero_mul, Ne, not_false_iff, ENNReal.coe_zero, ENNReal.inv_eq_zero] by_cases h'b : b = 0 · simp only [h'b, ENNReal.inv_zero, ENNReal.coe_ne_top, mul_top, Ne, not_false_iff, mul_zero, ENNReal.coe_zero, ENNReal.inv_eq_zero] rw [← ENNReal.coe_mul, ← ENNReal.coe_inv, ← ENNReal.coe_inv h'a, ← ENNReal.coe_inv h'b, ← ENNReal.coe_mul, mul_inv_rev, mul_comm] simp [h'a, h'b] #align ennreal.mul_inv ENNReal.mul_inv protected theorem mul_div_mul_left (a b : ℝ≥0∞) (hc : c ≠ 0) (hc' : c ≠ ⊤) : c * a / (c * b) = a / b := by rw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inl hc) (Or.inl hc'), mul_mul_mul_comm, ENNReal.mul_inv_cancel hc hc', one_mul] #align ennreal.mul_div_mul_left ENNReal.mul_div_mul_left protected theorem mul_div_mul_right (a b : ℝ≥0∞) (hc : c ≠ 0) (hc' : c ≠ ⊤) : a * c / (b * c) = a / b := by rw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inr hc') (Or.inr hc), mul_mul_mul_comm, ENNReal.mul_inv_cancel hc hc', mul_one] #align ennreal.mul_div_mul_right ENNReal.mul_div_mul_right protected theorem sub_div (h : 0 < b → b < a → c ≠ 0) : (a - b) / c = a / c - b / c := by simp_rw [div_eq_mul_inv] exact ENNReal.sub_mul (by simpa using h) #align ennreal.sub_div ENNReal.sub_div @[simp] protected theorem inv_pos : 0 < a⁻¹ ↔ a ≠ ∞ := pos_iff_ne_zero.trans ENNReal.inv_ne_zero #align ennreal.inv_pos ENNReal.inv_pos theorem inv_strictAnti : StrictAnti (Inv.inv : ℝ≥0∞ → ℝ≥0∞) := by intro a b h lift a to ℝ≥0 using h.ne_top induction b; · simp rw [coe_lt_coe] at h rcases eq_or_ne a 0 with (rfl \| ha); · simp [h] rw [← coe_inv h.ne_bot, ← coe_inv ha, coe_lt_coe] exact NNReal.inv_lt_inv ha h #align ennreal.inv_strict_anti ENNReal.inv_strictAnti @[simp] protected theorem inv_lt_inv : a⁻¹ < b⁻¹ ↔ b < a := inv_strictAnti.lt_iff_lt #align ennreal.inv_lt_inv ENNReal.inv_lt_inv theorem inv_lt_iff_inv_lt : a⁻¹ < b ↔ b⁻¹ < a := by simpa only [inv_inv] using @ENNReal.inv_lt_inv a b⁻¹ #align ennreal.inv_lt_iff_inv_lt ENNReal.inv_lt_iff_inv_lt theorem lt_inv_iff_lt_inv : a < b⁻¹ ↔ b < a⁻¹ := by simpa only [inv_inv] using @ENNReal.inv_lt_inv a⁻¹ b #align ennreal.lt_inv_iff_lt_inv ENNReal.lt_inv_iff_lt_inv @[simp] protected theorem inv_le_inv : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := inv_strictAnti.le_iff_le #align ennreal.inv_le_inv ENNReal.inv_le_inv theorem inv_le_iff_inv_le : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := by simpa only [inv_inv] using @ENNReal.inv_le_inv a b⁻¹ #align ennreal.inv_le_iff_inv_le ENNReal.inv_le_iff_inv_le theorem le_inv_iff_le_inv : a ≤ b⁻¹ ↔ b ≤ a⁻¹ := by simpa only [inv_inv] using @ENNReal.inv_le_inv a⁻¹ b #align ennreal.le_inv_iff_le_inv ENNReal.le_inv_iff_le_inv @[gcongr] protected theorem inv_le_inv' (h : a ≤ b) : b⁻¹ ≤ a⁻¹ := ENNReal.inv_strictAnti.antitone h @[gcongr] protected theorem inv_lt_inv' (h : a < b) : b⁻¹ < a⁻¹ := ENNReal.inv_strictAnti h @[simp] protected theorem inv_le_one : a⁻¹ ≤ 1 ↔ 1 ≤ a := by rw [inv_le_iff_inv_le, inv_one] #align ennreal.inv_le_one ENNReal.inv_le_one protected theorem one_le_inv : 1 ≤ a⁻¹ ↔ a ≤ 1 := by rw [le_inv_iff_le_inv, inv_one] #align ennreal.one_le_inv ENNReal.one_le_inv @[simp] protected theorem inv_lt_one : a⁻¹ < 1 ↔ 1 < a := by rw [inv_lt_iff_inv_lt, inv_one] #align ennreal.inv_lt_one ENNReal.inv_lt_one @[simp] protected theorem one_lt_inv : 1 < a⁻¹ ↔ a < 1 := by rw [lt_inv_iff_lt_inv, inv_one] #align ennreal.one_lt_inv ENNReal.one_lt_inv @[simps! apply] def _root_.OrderIso.invENNReal : ℝ≥0∞ ≃o ℝ≥0∞ᵒᵈ where map_rel_iff' := ENNReal.inv_le_inv toEquiv := (Equiv.inv ℝ≥0∞).trans OrderDual.toDual #align order_iso.inv_ennreal OrderIso.invENNReal #align order_iso.inv_ennreal_apply OrderIso.invENNReal_apply @[simp] theorem _root_.OrderIso.invENNReal_symm_apply (a : ℝ≥0∞ᵒᵈ) : OrderIso.invENNReal.symm a = (OrderDual.ofDual a)⁻¹ := rfl #align order_iso.inv_ennreal_symm_apply OrderIso.invENNReal_symm_apply @[simp] theorem div_top : a / ∞ = 0 := by rw [div_eq_mul_inv, inv_top, mul_zero] #align ennreal.div_top ENNReal.div_top -- Porting note: reordered 4 lemmas theorem top_div : ∞ / a = if a = ∞ then 0 else ∞ := by simp [div_eq_mul_inv, top_mul'] #align ennreal.top_div ENNReal.top_div theorem top_div_of_ne_top (h : a ≠ ∞) : ∞ / a = ∞ := by simp [top_div, h] #align ennreal.top_div_of_ne_top ENNReal.top_div_of_ne_top @[simp] theorem top_div_coe : ∞ / p = ∞ := top_div_of_ne_top coe_ne_top #align ennreal.top_div_coe ENNReal.top_div_coe theorem top_div_of_lt_top (h : a < ∞) : ∞ / a = ∞ := top_div_of_ne_top h.ne #align ennreal.top_div_of_lt_top ENNReal.top_div_of_lt_top @[simp] protected theorem zero_div : 0 / a = 0 := zero_mul a⁻¹ #align ennreal.zero_div ENNReal.zero_div theorem div_eq_top : a / b = ∞ ↔ a ≠ 0 ∧ b = 0 ∨ a = ∞ ∧ b ≠ ∞ := by simp [div_eq_mul_inv, ENNReal.mul_eq_top] #align ennreal.div_eq_top ENNReal.div_eq_top protected theorem le_div_iff_mul_le (h0 : b ≠ 0 ∨ c ≠ 0) (ht : b ≠ ∞ ∨ c ≠ ∞) : a ≤ c / b ↔ a * b ≤ c := by induction' b with b · lift c to ℝ≥0 using ht.neg_resolve_left rfl rw [div_top, nonpos_iff_eq_zero] rcases eq_or_ne a 0 with (rfl \| ha) <;> simp [] rcases eq_or_ne b 0 with (rfl \| hb) · have hc : c ≠ 0 := h0.neg_resolve_left rfl simp [div_zero hc] · rw [← coe_ne_zero] at hb rw [← ENNReal.mul_le_mul_right hb coe_ne_top, ENNReal.div_mul_cancel hb coe_ne_top] #align ennreal.le_div_iff_mul_le ENNReal.le_div_iff_mul_le protected theorem div_le_iff_le_mul (hb0 : b ≠ 0 ∨ c ≠ ∞) (hbt : b ≠ ∞ ∨ c ≠ 0) : a / b ≤ c ↔ a ≤ c b := by suffices a * b⁻¹ ≤ c ↔ a ≤ c / b⁻¹ by simpa [div_eq_mul_inv] refine (ENNReal.le_div_iff_mul_le ?_ ?_).symm <;> simpa #align ennreal.div_le_iff_le_mul ENNReal.div_le_iff_le_mul protected theorem lt_div_iff_mul_lt (hb0 : b ≠ 0 ∨ c ≠ ∞) (hbt : b ≠ ∞ ∨ c ≠ 0) : c < a / b ↔ c * b < a := lt_iff_lt_of_le_iff_le (ENNReal.div_le_iff_le_mul hb0 hbt) #align ennreal.lt_div_iff_mul_lt ENNReal.lt_div_iff_mul_lt theorem div_le_of_le_mul (h : a ≤ b * c) : a / c ≤ b := by by_cases h0 : c = 0 · have : a = 0 := by simpa [h0] using h simp [] by_cases hinf : c = ∞; · simp [hinf] exact (ENNReal.div_le_iff_le_mul (Or.inl h0) (Or.inl hinf)).2 h #align ennreal.div_le_of_le_mul ENNReal.div_le_of_le_mul theorem div_le_of_le_mul' (h : a ≤ b c) : a / b ≤ c := div_le_of_le_mul <\| mul_comm b c ▸ h #align ennreal.div_le_of_le_mul' ENNReal.div_le_of_le_mul' protected theorem div_self_le_one : a / a ≤ 1 := div_le_of_le_mul <\| by rw [one_mul] theorem mul_le_of_le_div (h : a ≤ b / c) : a * c ≤ b := by rw [← inv_inv c] exact div_le_of_le_mul h #align ennreal.mul_le_of_le_div ENNReal.mul_le_of_le_div theorem mul_le_of_le_div' (h : a ≤ b / c) : c * a ≤ b := mul_comm a c ▸ mul_le_of_le_div h #align ennreal.mul_le_of_le_div' ENNReal.mul_le_of_le_div' protected theorem div_lt_iff (h0 : b ≠ 0 ∨ c ≠ 0) (ht : b ≠ ∞ ∨ c ≠ ∞) : c / b < a ↔ c < a * b := lt_iff_lt_of_le_iff_le <\| ENNReal.le_div_iff_mul_le h0 ht #align ennreal.div_lt_iff ENNReal.div_lt_iff theorem mul_lt_of_lt_div (h : a < b / c) : a * c < b := by contrapose! h exact ENNReal.div_le_of_le_mul h #align ennreal.mul_lt_of_lt_div ENNReal.mul_lt_of_lt_div theorem mul_lt_of_lt_div' (h : a < b / c) : c * a < b := mul_comm a c ▸ mul_lt_of_lt_div h #align ennreal.mul_lt_of_lt_div' ENNReal.mul_lt_of_lt_div' theorem div_lt_of_lt_mul (h : a < b * c) : a / c < b := mul_lt_of_lt_div <\| by rwa [div_eq_mul_inv, inv_inv] theorem div_lt_of_lt_mul' (h : a < b * c) : a / b < c := div_lt_of_lt_mul <\| by rwa [mul_comm]	Mathlib/Data/ENNReal/Inv.lean	358	360	theorem inv_le_iff_le_mul (h₁ : b = ∞ → a ≠ 0) (h₂ : a = ∞ → b ≠ 0) : a⁻¹ ≤ b ↔ 1 ≤ a * b := by	rw [← one_div, ENNReal.div_le_iff_le_mul, mul_comm] exacts [or_not_of_imp h₁, not_or_of_imp h₂]
import Batteries.Classes.Order @[ext] theorem UInt8.ext : {x y : UInt8} → x.toNat = y.toNat → x = y \| ⟨⟨_,_⟩⟩, ⟨⟨_,_⟩⟩, rfl => rfl theorem UInt8.ext_iff {x y : UInt8} : x = y ↔ x.toNat = y.toNat := ⟨congrArg _, UInt8.ext⟩ @[simp] theorem UInt8.val_val_eq_toNat (x : UInt8) : x.val.val = x.toNat := rfl theorem UInt8.toNat_lt (x : UInt8) : x.toNat < 2 ^ 8 := x.val.isLt @[simp] theorem UInt8.toUInt16_toNat (x : UInt8) : x.toUInt16.toNat = x.toNat := rfl @[simp] theorem UInt8.toUInt32_toNat (x : UInt8) : x.toUInt32.toNat = x.toNat := rfl @[simp] theorem UInt8.toUInt64_toNat (x : UInt8) : x.toUInt64.toNat = x.toNat := rfl theorem UInt8.le_antisymm_iff {x y : UInt8} : x = y ↔ x ≤ y ∧ y ≤ x := UInt8.ext_iff.trans Nat.le_antisymm_iff theorem UInt8.le_antisymm {x y : UInt8} (h1 : x ≤ y) (h2 : y ≤ x) : x = y := UInt8.le_antisymm_iff.2 ⟨h1, h2⟩ instance : Batteries.LawfulOrd UInt8 := .compareOfLessAndEq (fun _ => Nat.lt_irrefl _) Nat.lt_trans Nat.not_lt UInt8.le_antisymm @[ext] theorem UInt16.ext : {x y : UInt16} → x.toNat = y.toNat → x = y \| ⟨⟨_,_⟩⟩, ⟨⟨_,_⟩⟩, rfl => rfl theorem UInt16.ext_iff {x y : UInt16} : x = y ↔ x.toNat = y.toNat := ⟨congrArg _, UInt16.ext⟩ theorem UInt16.toNat_lt (x : UInt16) : x.toNat < 2 ^ 16 := x.val.isLt @[simp] theorem UInt16.val_val_eq_toNat (x : UInt16) : x.val.val = x.toNat := rfl @[simp] theorem UInt16.toUInt8_toNat (x : UInt16) : x.toUInt8.toNat = x.toNat % 2 ^ 8 := rfl @[simp] theorem UInt16.toUInt32_toNat (x : UInt16) : x.toUInt32.toNat = x.toNat := rfl @[simp] theorem UInt16.toUInt64_toNat (x : UInt16) : x.toUInt64.toNat = x.toNat := rfl theorem UInt16.le_antisymm_iff {x y : UInt16} : x = y ↔ x ≤ y ∧ y ≤ x := UInt16.ext_iff.trans Nat.le_antisymm_iff theorem UInt16.le_antisymm {x y : UInt16} (h1 : x ≤ y) (h2 : y ≤ x) : x = y := UInt16.le_antisymm_iff.2 ⟨h1, h2⟩ instance : Batteries.LawfulOrd UInt16 := .compareOfLessAndEq (fun _ => Nat.lt_irrefl _) Nat.lt_trans Nat.not_lt UInt16.le_antisymm @[ext] theorem UInt32.ext : {x y : UInt32} → x.toNat = y.toNat → x = y \| ⟨⟨_,_⟩⟩, ⟨⟨_,_⟩⟩, rfl => rfl theorem UInt32.ext_iff {x y : UInt32} : x = y ↔ x.toNat = y.toNat := ⟨congrArg _, UInt32.ext⟩ @[simp] theorem UInt32.val_val_eq_toNat (x : UInt32) : x.val.val = x.toNat := rfl theorem UInt32.toNat_lt (x : UInt32) : x.toNat < 2 ^ 32 := x.val.isLt @[simp] theorem UInt32.toUInt8_toNat (x : UInt32) : x.toUInt8.toNat = x.toNat % 2 ^ 8 := rfl @[simp] theorem UInt32.toUInt16_toNat (x : UInt32) : x.toUInt16.toNat = x.toNat % 2 ^ 16 := rfl @[simp] theorem UInt32.toUInt64_toNat (x : UInt32) : x.toUInt64.toNat = x.toNat := rfl theorem UInt32.le_antisymm_iff {x y : UInt32} : x = y ↔ x ≤ y ∧ y ≤ x := UInt32.ext_iff.trans Nat.le_antisymm_iff theorem UInt32.le_antisymm {x y : UInt32} (h1 : x ≤ y) (h2 : y ≤ x) : x = y := UInt32.le_antisymm_iff.2 ⟨h1, h2⟩ instance : Batteries.LawfulOrd UInt32 := .compareOfLessAndEq (fun _ => Nat.lt_irrefl _) Nat.lt_trans Nat.not_lt UInt32.le_antisymm @[ext] theorem UInt64.ext : {x y : UInt64} → x.toNat = y.toNat → x = y \| ⟨⟨_,_⟩⟩, ⟨⟨_,_⟩⟩, rfl => rfl theorem UInt64.ext_iff {x y : UInt64} : x = y ↔ x.toNat = y.toNat := ⟨congrArg _, UInt64.ext⟩ @[simp] theorem UInt64.val_val_eq_toNat (x : UInt64) : x.val.val = x.toNat := rfl theorem UInt64.toNat_lt (x : UInt64) : x.toNat < 2 ^ 64 := x.val.isLt @[simp] theorem UInt64.toUInt8_toNat (x : UInt64) : x.toUInt8.toNat = x.toNat % 2 ^ 8 := rfl @[simp] theorem UInt64.toUInt16_toNat (x : UInt64) : x.toUInt16.toNat = x.toNat % 2 ^ 16 := rfl @[simp] theorem UInt64.toUInt32_toNat (x : UInt64) : x.toUInt32.toNat = x.toNat % 2 ^ 32 := rfl theorem UInt64.le_antisymm_iff {x y : UInt64} : x = y ↔ x ≤ y ∧ y ≤ x := UInt64.ext_iff.trans Nat.le_antisymm_iff theorem UInt64.le_antisymm {x y : UInt64} (h1 : x ≤ y) (h2 : y ≤ x) : x = y := UInt64.le_antisymm_iff.2 ⟨h1, h2⟩ instance : Batteries.LawfulOrd UInt64 := .compareOfLessAndEq (fun _ => Nat.lt_irrefl _) Nat.lt_trans Nat.not_lt UInt64.le_antisymm @[ext] theorem USize.ext : {x y : USize} → x.toNat = y.toNat → x = y \| ⟨⟨_,_⟩⟩, ⟨⟨_,_⟩⟩, rfl => rfl theorem USize.ext_iff {x y : USize} : x = y ↔ x.toNat = y.toNat := ⟨congrArg _, USize.ext⟩ @[simp] theorem USize.val_val_eq_toNat (x : USize) : x.val.val = x.toNat := rfl theorem USize.size_eq : USize.size = 2 ^ System.Platform.numBits := by have : 1 ≤ 2 ^ System.Platform.numBits := Nat.succ_le_of_lt (Nat.two_pow_pos _) rw [USize.size, Nat.sub_add_cancel this]	.lake/packages/batteries/Batteries/Data/UInt.lean	125	128	theorem USize.le_size : 2 ^ 32 ≤ USize.size := by	rw [size_eq] apply Nat.pow_le_pow_of_le_right (by decide) cases System.Platform.numBits_eq <;> simp_arith [*]
import Batteries.Control.ForInStep.Lemmas import Batteries.Data.List.Basic import Batteries.Tactic.Init import Batteries.Tactic.Alias namespace List open Nat @[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by simp [Array.mem_def] @[simp] theorem drop_one : ∀ l : List α, drop 1 l = tail l \| [] \| _ :: _ => rfl theorem zipWith_distrib_tail : (zipWith f l l').tail = zipWith f l.tail l'.tail := by rw [← drop_one]; simp [zipWith_distrib_drop] theorem subset_def {l₁ l₂ : List α} : l₁ ⊆ l₂ ↔ ∀ {a : α}, a ∈ l₁ → a ∈ l₂ := .rfl @[simp] theorem nil_subset (l : List α) : [] ⊆ l := nofun @[simp] theorem Subset.refl (l : List α) : l ⊆ l := fun _ i => i theorem Subset.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ := fun _ i => h₂ (h₁ i) instance : Trans (Membership.mem : α → List α → Prop) Subset Membership.mem := ⟨fun h₁ h₂ => h₂ h₁⟩ instance : Trans (Subset : List α → List α → Prop) Subset Subset := ⟨Subset.trans⟩ @[simp] theorem subset_cons (a : α) (l : List α) : l ⊆ a :: l := fun _ => Mem.tail _ theorem subset_of_cons_subset {a : α} {l₁ l₂ : List α} : a :: l₁ ⊆ l₂ → l₁ ⊆ l₂ := fun s _ i => s (mem_cons_of_mem _ i) theorem subset_cons_of_subset (a : α) {l₁ l₂ : List α} : l₁ ⊆ l₂ → l₁ ⊆ a :: l₂ := fun s _ i => .tail _ (s i) theorem cons_subset_cons {l₁ l₂ : List α} (a : α) (s : l₁ ⊆ l₂) : a :: l₁ ⊆ a :: l₂ := fun _ => by simp only [mem_cons]; exact Or.imp_right (@s _) @[simp] theorem subset_append_left (l₁ l₂ : List α) : l₁ ⊆ l₁ ++ l₂ := fun _ => mem_append_left _ @[simp] theorem subset_append_right (l₁ l₂ : List α) : l₂ ⊆ l₁ ++ l₂ := fun _ => mem_append_right _ theorem subset_append_of_subset_left (l₂ : List α) : l ⊆ l₁ → l ⊆ l₁ ++ l₂ := fun s => Subset.trans s <\| subset_append_left _ _ theorem subset_append_of_subset_right (l₁ : List α) : l ⊆ l₂ → l ⊆ l₁ ++ l₂ := fun s => Subset.trans s <\| subset_append_right _ _ @[simp] theorem cons_subset : a :: l ⊆ m ↔ a ∈ m ∧ l ⊆ m := by simp only [subset_def, mem_cons, or_imp, forall_and, forall_eq] @[simp] theorem append_subset {l₁ l₂ l : List α} : l₁ ++ l₂ ⊆ l ↔ l₁ ⊆ l ∧ l₂ ⊆ l := by simp [subset_def, or_imp, forall_and] theorem subset_nil {l : List α} : l ⊆ [] ↔ l = [] := ⟨fun h => match l with \| [] => rfl \| _::_ => (nomatch h (.head ..)), fun \| rfl => Subset.refl _⟩ theorem map_subset {l₁ l₂ : List α} (f : α → β) (H : l₁ ⊆ l₂) : map f l₁ ⊆ map f l₂ := fun x => by simp only [mem_map]; exact .imp fun a => .imp_left (@H _) @[simp] theorem nil_sublist : ∀ l : List α, [] <+ l \| [] => .slnil \| a :: l => (nil_sublist l).cons a @[simp] theorem Sublist.refl : ∀ l : List α, l <+ l \| [] => .slnil \| a :: l => (Sublist.refl l).cons₂ a	.lake/packages/batteries/Batteries/Data/List/Lemmas.lean	91	100	theorem Sublist.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ <+ l₂) (h₂ : l₂ <+ l₃) : l₁ <+ l₃ := by	induction h₂ generalizing l₁ with \| slnil => exact h₁ \| cons _ _ IH => exact (IH h₁).cons _ \| @cons₂ l₂ _ a _ IH => generalize e : a :: l₂ = l₂' match e ▸ h₁ with \| .slnil => apply nil_sublist \| .cons a' h₁' => cases e; apply (IH h₁').cons \| .cons₂ a' h₁' => cases e; apply (IH h₁').cons₂
import Mathlib.Probability.Kernel.MeasurableIntegral #align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b" open MeasureTheory open scoped ENNReal namespace ProbabilityTheory namespace kernel variable {α β ι : Type} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} section CompositionProduct variable {γ : Type} {mγ : MeasurableSpace γ} {s : Set (β × γ)} noncomputable def compProdFun (κ : kernel α β) (η : kernel (α × β) γ) (a : α) (s : Set (β × γ)) : ℝ≥0∞ := ∫⁻ b, η (a, b) {c \| (b, c) ∈ s} ∂κ a #align probability_theory.kernel.comp_prod_fun ProbabilityTheory.kernel.compProdFun theorem compProdFun_empty (κ : kernel α β) (η : kernel (α × β) γ) (a : α) : compProdFun κ η a ∅ = 0 := by simp only [compProdFun, Set.mem_empty_iff_false, Set.setOf_false, measure_empty, MeasureTheory.lintegral_const, zero_mul] #align probability_theory.kernel.comp_prod_fun_empty ProbabilityTheory.kernel.compProdFun_empty theorem compProdFun_iUnion (κ : kernel α β) (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (f : ℕ → Set (β × γ)) (hf_meas : ∀ i, MeasurableSet (f i)) (hf_disj : Pairwise (Disjoint on f)) : compProdFun κ η a (⋃ i, f i) = ∑' i, compProdFun κ η a (f i) := by have h_Union : (fun b => η (a, b) {c : γ \| (b, c) ∈ ⋃ i, f i}) = fun b => η (a, b) (⋃ i, {c : γ \| (b, c) ∈ f i}) := by ext1 b congr with c simp only [Set.mem_iUnion, Set.iSup_eq_iUnion, Set.mem_setOf_eq] rw [compProdFun, h_Union] have h_tsum : (fun b => η (a, b) (⋃ i, {c : γ \| (b, c) ∈ f i})) = fun b => ∑' i, η (a, b) {c : γ \| (b, c) ∈ f i} := by ext1 b rw [measure_iUnion] · intro i j hij s hsi hsj c hcs have hbci : {(b, c)} ⊆ f i := by rw [Set.singleton_subset_iff]; exact hsi hcs have hbcj : {(b, c)} ⊆ f j := by rw [Set.singleton_subset_iff]; exact hsj hcs simpa only [Set.bot_eq_empty, Set.le_eq_subset, Set.singleton_subset_iff, Set.mem_empty_iff_false] using hf_disj hij hbci hbcj · -- Porting note: behavior of `@` changed relative to lean 3, was -- exact fun i => (@measurable_prod_mk_left β γ _ _ b) _ (hf_meas i) exact fun i => (@measurable_prod_mk_left β γ _ _ b) (hf_meas i) rw [h_tsum, lintegral_tsum] · rfl · intro i have hm : MeasurableSet {p : (α × β) × γ \| (p.1.2, p.2) ∈ f i} := measurable_fst.snd.prod_mk measurable_snd (hf_meas i) exact ((measurable_kernel_prod_mk_left hm).comp measurable_prod_mk_left).aemeasurable #align probability_theory.kernel.comp_prod_fun_Union ProbabilityTheory.kernel.compProdFun_iUnion theorem compProdFun_tsum_right (κ : kernel α β) (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) : compProdFun κ η a s = ∑' n, compProdFun κ (seq η n) a s := by simp_rw [compProdFun, (measure_sum_seq η _).symm] have : ∫⁻ b, Measure.sum (fun n => seq η n (a, b)) {c : γ \| (b, c) ∈ s} ∂κ a = ∫⁻ b, ∑' n, seq η n (a, b) {c : γ \| (b, c) ∈ s} ∂κ a := by congr ext1 b rw [Measure.sum_apply] exact measurable_prod_mk_left hs rw [this, lintegral_tsum] exact fun n => ((measurable_kernel_prod_mk_left (κ := (seq η n)) ((measurable_fst.snd.prod_mk measurable_snd) hs)).comp measurable_prod_mk_left).aemeasurable #align probability_theory.kernel.comp_prod_fun_tsum_right ProbabilityTheory.kernel.compProdFun_tsum_right theorem compProdFun_tsum_left (κ : kernel α β) (η : kernel (α × β) γ) [IsSFiniteKernel κ] (a : α) (s : Set (β × γ)) : compProdFun κ η a s = ∑' n, compProdFun (seq κ n) η a s := by simp_rw [compProdFun, (measure_sum_seq κ _).symm, lintegral_sum_measure] #align probability_theory.kernel.comp_prod_fun_tsum_left ProbabilityTheory.kernel.compProdFun_tsum_left theorem compProdFun_eq_tsum (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) : compProdFun κ η a s = ∑' (n) (m), compProdFun (seq κ n) (seq η m) a s := by simp_rw [compProdFun_tsum_left κ η a s, compProdFun_tsum_right _ η a hs] #align probability_theory.kernel.comp_prod_fun_eq_tsum ProbabilityTheory.kernel.compProdFun_eq_tsum theorem measurable_compProdFun_of_finite (κ : kernel α β) [IsFiniteKernel κ] (η : kernel (α × β) γ) [IsFiniteKernel η] (hs : MeasurableSet s) : Measurable fun a => compProdFun κ η a s := by simp only [compProdFun] have h_meas : Measurable (Function.uncurry fun a b => η (a, b) {c : γ \| (b, c) ∈ s}) := by have : (Function.uncurry fun a b => η (a, b) {c : γ \| (b, c) ∈ s}) = fun p => η p {c : γ \| (p.2, c) ∈ s} := by ext1 p rw [Function.uncurry_apply_pair] rw [this] exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs) exact h_meas.lintegral_kernel_prod_right #align probability_theory.kernel.measurable_comp_prod_fun_of_finite ProbabilityTheory.kernel.measurable_compProdFun_of_finite theorem measurable_compProdFun (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] (hs : MeasurableSet s) : Measurable fun a => compProdFun κ η a s := by simp_rw [compProdFun_tsum_right κ η _ hs] refine Measurable.ennreal_tsum fun n => ?_ simp only [compProdFun] have h_meas : Measurable (Function.uncurry fun a b => seq η n (a, b) {c : γ \| (b, c) ∈ s}) := by have : (Function.uncurry fun a b => seq η n (a, b) {c : γ \| (b, c) ∈ s}) = fun p => seq η n p {c : γ \| (p.2, c) ∈ s} := by ext1 p rw [Function.uncurry_apply_pair] rw [this] exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs) exact h_meas.lintegral_kernel_prod_right #align probability_theory.kernel.measurable_comp_prod_fun ProbabilityTheory.kernel.measurable_compProdFun open scoped Classical noncomputable def compProd (κ : kernel α β) (η : kernel (α × β) γ) : kernel α (β × γ) := if h : IsSFiniteKernel κ ∧ IsSFiniteKernel η then { val := fun a ↦ Measure.ofMeasurable (fun s _ => compProdFun κ η a s) (compProdFun_empty κ η a) (@compProdFun_iUnion _ _ _ _ _ _ κ η h.2 a) property := by have : IsSFiniteKernel κ := h.1 have : IsSFiniteKernel η := h.2 refine Measure.measurable_of_measurable_coe _ fun s hs => ?_ have : (fun a => Measure.ofMeasurable (fun s _ => compProdFun κ η a s) (compProdFun_empty κ η a) (compProdFun_iUnion κ η a) s) = fun a => compProdFun κ η a s := by ext1 a; rwa [Measure.ofMeasurable_apply] rw [this] exact measurable_compProdFun κ η hs } else 0 #align probability_theory.kernel.comp_prod ProbabilityTheory.kernel.compProd scoped[ProbabilityTheory] infixl:100 " ⊗ₖ " => ProbabilityTheory.kernel.compProd theorem compProd_apply_eq_compProdFun (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) : (κ ⊗ₖ η) a s = compProdFun κ η a s := by rw [compProd, dif_pos] swap · constructor <;> infer_instance change Measure.ofMeasurable (fun s _ => compProdFun κ η a s) (compProdFun_empty κ η a) (compProdFun_iUnion κ η a) s = ∫⁻ b, η (a, b) {c \| (b, c) ∈ s} ∂κ a rw [Measure.ofMeasurable_apply _ hs] rfl #align probability_theory.kernel.comp_prod_apply_eq_comp_prod_fun ProbabilityTheory.kernel.compProd_apply_eq_compProdFun theorem compProd_of_not_isSFiniteKernel_left (κ : kernel α β) (η : kernel (α × β) γ) (h : ¬ IsSFiniteKernel κ) : κ ⊗ₖ η = 0 := by rw [compProd, dif_neg] simp [h] theorem compProd_of_not_isSFiniteKernel_right (κ : kernel α β) (η : kernel (α × β) γ) (h : ¬ IsSFiniteKernel η) : κ ⊗ₖ η = 0 := by rw [compProd, dif_neg] simp [h] theorem compProd_apply (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) : (κ ⊗ₖ η) a s = ∫⁻ b, η (a, b) {c \| (b, c) ∈ s} ∂κ a := compProd_apply_eq_compProdFun κ η a hs #align probability_theory.kernel.comp_prod_apply ProbabilityTheory.kernel.compProd_apply theorem le_compProd_apply (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (s : Set (β × γ)) : ∫⁻ b, η (a, b) {c \| (b, c) ∈ s} ∂κ a ≤ (κ ⊗ₖ η) a s := calc ∫⁻ b, η (a, b) {c \| (b, c) ∈ s} ∂κ a ≤ ∫⁻ b, η (a, b) {c \| (b, c) ∈ toMeasurable ((κ ⊗ₖ η) a) s} ∂κ a := lintegral_mono fun _ => measure_mono fun _ h_mem => subset_toMeasurable _ _ h_mem _ = (κ ⊗ₖ η) a (toMeasurable ((κ ⊗ₖ η) a) s) := (kernel.compProd_apply_eq_compProdFun κ η a (measurableSet_toMeasurable _ _)).symm _ = (κ ⊗ₖ η) a s := measure_toMeasurable s #align probability_theory.kernel.le_comp_prod_apply ProbabilityTheory.kernel.le_compProd_apply @[simp] lemma compProd_zero_left (κ : kernel (α × β) γ) : (0 : kernel α β) ⊗ₖ κ = 0 := by by_cases h : IsSFiniteKernel κ · ext a s hs rw [kernel.compProd_apply _ _ _ hs] simp · rw [kernel.compProd_of_not_isSFiniteKernel_right _ _ h] @[simp] lemma compProd_zero_right (κ : kernel α β) (γ : Type*) [MeasurableSpace γ] : κ ⊗ₖ (0 : kernel (α × β) γ) = 0 := by by_cases h : IsSFiniteKernel κ · ext a s hs rw [kernel.compProd_apply _ _ _ hs] simp · rw [kernel.compProd_of_not_isSFiniteKernel_left _ _ h] theorem compProd_eq_tsum_compProd (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) : (κ ⊗ₖ η) a s = ∑' (n : ℕ) (m : ℕ), (seq κ n ⊗ₖ seq η m) a s := by simp_rw [compProd_apply_eq_compProdFun _ _ _ hs]; exact compProdFun_eq_tsum κ η a hs #align probability_theory.kernel.comp_prod_eq_tsum_comp_prod ProbabilityTheory.kernel.compProd_eq_tsum_compProd theorem compProd_eq_sum_compProd (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) [IsSFiniteKernel η] : κ ⊗ₖ η = kernel.sum fun n => kernel.sum fun m => seq κ n ⊗ₖ seq η m := by ext a s hs; simp_rw [kernel.sum_apply' _ a hs]; rw [compProd_eq_tsum_compProd κ η a hs] #align probability_theory.kernel.comp_prod_eq_sum_comp_prod ProbabilityTheory.kernel.compProd_eq_sum_compProd theorem compProd_eq_sum_compProd_left (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) : κ ⊗ₖ η = kernel.sum fun n => seq κ n ⊗ₖ η := by by_cases h : IsSFiniteKernel η swap · simp_rw [compProd_of_not_isSFiniteKernel_right _ _ h] simp rw [compProd_eq_sum_compProd] congr with n a s hs simp_rw [kernel.sum_apply' _ _ hs, compProd_apply_eq_compProdFun _ _ _ hs, compProdFun_tsum_right _ η a hs] #align probability_theory.kernel.comp_prod_eq_sum_comp_prod_left ProbabilityTheory.kernel.compProd_eq_sum_compProd_left theorem compProd_eq_sum_compProd_right (κ : kernel α β) (η : kernel (α × β) γ) [IsSFiniteKernel η] : κ ⊗ₖ η = kernel.sum fun n => κ ⊗ₖ seq η n := by by_cases hκ : IsSFiniteKernel κ swap · simp_rw [compProd_of_not_isSFiniteKernel_left _ _ hκ] simp rw [compProd_eq_sum_compProd] simp_rw [compProd_eq_sum_compProd_left κ _] rw [kernel.sum_comm] #align probability_theory.kernel.comp_prod_eq_sum_comp_prod_right ProbabilityTheory.kernel.compProd_eq_sum_compProd_right instance IsMarkovKernel.compProd (κ : kernel α β) [IsMarkovKernel κ] (η : kernel (α × β) γ) [IsMarkovKernel η] : IsMarkovKernel (κ ⊗ₖ η) := ⟨fun a => ⟨by rw [compProd_apply κ η a MeasurableSet.univ] simp only [Set.mem_univ, Set.setOf_true, measure_univ, lintegral_one]⟩⟩ #align probability_theory.kernel.is_markov_kernel.comp_prod ProbabilityTheory.kernel.IsMarkovKernel.compProd	Mathlib/Probability/Kernel/Composition.lean	534	547	theorem compProd_apply_univ_le (κ : kernel α β) (η : kernel (α × β) γ) [IsFiniteKernel η] (a : α) : (κ ⊗ₖ η) a Set.univ ≤ κ a Set.univ * IsFiniteKernel.bound η := by	by_cases hκ : IsSFiniteKernel κ swap · rw [compProd_of_not_isSFiniteKernel_left _ _ hκ] simp rw [compProd_apply κ η a MeasurableSet.univ] simp only [Set.mem_univ, Set.setOf_true] let Cη := IsFiniteKernel.bound η calc ∫⁻ b, η (a, b) Set.univ ∂κ a ≤ ∫⁻ _, Cη ∂κ a := lintegral_mono fun b => measure_le_bound η (a, b) Set.univ _ = Cη * κ a Set.univ := MeasureTheory.lintegral_const Cη _ = κ a Set.univ * Cη := mul_comm _ _
import Mathlib.SetTheory.Ordinal.Basic import Mathlib.Data.Nat.SuccPred #align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7" assert_not_exists Field assert_not_exists Module noncomputable section open Function Cardinal Set Equiv Order open scoped Classical open Cardinal Ordinal universe u v w namespace Ordinal variable {α : Type} {β : Type} {γ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} @[simp] theorem lift_add (a b : Ordinal.{v}) : lift.{u} (a + b) = lift.{u} a + lift.{u} b := Quotient.inductionOn₂ a b fun ⟨_α, _r, _⟩ ⟨_β, _s, _⟩ => Quotient.sound ⟨(RelIso.preimage Equiv.ulift _).trans (RelIso.sumLexCongr (RelIso.preimage Equiv.ulift _) (RelIso.preimage Equiv.ulift _)).symm⟩ #align ordinal.lift_add Ordinal.lift_add @[simp] theorem lift_succ (a : Ordinal.{v}) : lift.{u} (succ a) = succ (lift.{u} a) := by rw [← add_one_eq_succ, lift_add, lift_one] rfl #align ordinal.lift_succ Ordinal.lift_succ instance add_contravariantClass_le : ContravariantClass Ordinal.{u} Ordinal.{u} (· + ·) (· ≤ ·) := ⟨fun a b c => inductionOn a fun α r hr => inductionOn b fun β₁ s₁ hs₁ => inductionOn c fun β₂ s₂ hs₂ ⟨f⟩ => ⟨have fl : ∀ a, f (Sum.inl a) = Sum.inl a := fun a => by simpa only [InitialSeg.trans_apply, InitialSeg.leAdd_apply] using @InitialSeg.eq _ _ _ _ _ ((InitialSeg.leAdd r s₁).trans f) (InitialSeg.leAdd r s₂) a have : ∀ b, { b' // f (Sum.inr b) = Sum.inr b' } := by intro b; cases e : f (Sum.inr b) · rw [← fl] at e have := f.inj' e contradiction · exact ⟨_, rfl⟩ let g (b) := (this b).1 have fr : ∀ b, f (Sum.inr b) = Sum.inr (g b) := fun b => (this b).2 ⟨⟨⟨g, fun x y h => by injection f.inj' (by rw [fr, fr, h] : f (Sum.inr x) = f (Sum.inr y))⟩, @fun a b => by -- Porting note: -- `relEmbedding.coe_fn_to_embedding` & `initial_seg.coe_fn_to_rel_embedding` -- → `InitialSeg.coe_coe_fn` simpa only [Sum.lex_inr_inr, fr, InitialSeg.coe_coe_fn, Embedding.coeFn_mk] using @RelEmbedding.map_rel_iff _ _ _ _ f.toRelEmbedding (Sum.inr a) (Sum.inr b)⟩, fun a b H => by rcases f.init (by rw [fr] <;> exact Sum.lex_inr_inr.2 H) with ⟨a' \| a', h⟩ · rw [fl] at h cases h · rw [fr] at h exact ⟨a', Sum.inr.inj h⟩⟩⟩⟩ #align ordinal.add_contravariant_class_le Ordinal.add_contravariantClass_le theorem add_left_cancel (a) {b c : Ordinal} : a + b = a + c ↔ b = c := by simp only [le_antisymm_iff, add_le_add_iff_left] #align ordinal.add_left_cancel Ordinal.add_left_cancel private theorem add_lt_add_iff_left' (a) {b c : Ordinal} : a + b < a + c ↔ b < c := by rw [← not_le, ← not_le, add_le_add_iff_left] instance add_covariantClass_lt : CovariantClass Ordinal.{u} Ordinal.{u} (· + ·) (· < ·) := ⟨fun a _b _c => (add_lt_add_iff_left' a).2⟩ #align ordinal.add_covariant_class_lt Ordinal.add_covariantClass_lt instance add_contravariantClass_lt : ContravariantClass Ordinal.{u} Ordinal.{u} (· + ·) (· < ·) := ⟨fun a _b _c => (add_lt_add_iff_left' a).1⟩ #align ordinal.add_contravariant_class_lt Ordinal.add_contravariantClass_lt instance add_swap_contravariantClass_lt : ContravariantClass Ordinal.{u} Ordinal.{u} (swap (· + ·)) (· < ·) := ⟨fun _a _b _c => lt_imp_lt_of_le_imp_le fun h => add_le_add_right h _⟩ #align ordinal.add_swap_contravariant_class_lt Ordinal.add_swap_contravariantClass_lt theorem add_le_add_iff_right {a b : Ordinal} : ∀ n : ℕ, a + n ≤ b + n ↔ a ≤ b \| 0 => by simp \| n + 1 => by simp only [natCast_succ, add_succ, add_succ, succ_le_succ_iff, add_le_add_iff_right] #align ordinal.add_le_add_iff_right Ordinal.add_le_add_iff_right	Mathlib/SetTheory/Ordinal/Arithmetic.lean	145	146	theorem add_right_cancel {a b : Ordinal} (n : ℕ) : a + n = b + n ↔ a = b := by	simp only [le_antisymm_iff, add_le_add_iff_right]
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic import Mathlib.Topology.Order.ProjIcc #align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable section open scoped Classical open Topology Filter open Set Filter open Real namespace Real variable {x y : ℝ} -- @[pp_nodot] Porting note: not implemented noncomputable def arcsin : ℝ → ℝ := Subtype.val ∘ IccExtend (neg_le_self zero_le_one) sinOrderIso.symm #align real.arcsin Real.arcsin theorem arcsin_mem_Icc (x : ℝ) : arcsin x ∈ Icc (-(π / 2)) (π / 2) := Subtype.coe_prop _ #align real.arcsin_mem_Icc Real.arcsin_mem_Icc @[simp] theorem range_arcsin : range arcsin = Icc (-(π / 2)) (π / 2) := by rw [arcsin, range_comp Subtype.val] simp [Icc] #align real.range_arcsin Real.range_arcsin theorem arcsin_le_pi_div_two (x : ℝ) : arcsin x ≤ π / 2 := (arcsin_mem_Icc x).2 #align real.arcsin_le_pi_div_two Real.arcsin_le_pi_div_two theorem neg_pi_div_two_le_arcsin (x : ℝ) : -(π / 2) ≤ arcsin x := (arcsin_mem_Icc x).1 #align real.neg_pi_div_two_le_arcsin Real.neg_pi_div_two_le_arcsin theorem arcsin_projIcc (x : ℝ) : arcsin (projIcc (-1) 1 (neg_le_self zero_le_one) x) = arcsin x := by rw [arcsin, Function.comp_apply, IccExtend_val, Function.comp_apply, IccExtend, Function.comp_apply] #align real.arcsin_proj_Icc Real.arcsin_projIcc theorem sin_arcsin' {x : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) : sin (arcsin x) = x := by simpa [arcsin, IccExtend_of_mem _ _ hx, -OrderIso.apply_symm_apply] using Subtype.ext_iff.1 (sinOrderIso.apply_symm_apply ⟨x, hx⟩) #align real.sin_arcsin' Real.sin_arcsin' theorem sin_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arcsin x) = x := sin_arcsin' ⟨hx₁, hx₂⟩ #align real.sin_arcsin Real.sin_arcsin theorem arcsin_sin' {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : arcsin (sin x) = x := injOn_sin (arcsin_mem_Icc _) hx <\| by rw [sin_arcsin (neg_one_le_sin _) (sin_le_one _)] #align real.arcsin_sin' Real.arcsin_sin' theorem arcsin_sin {x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) : arcsin (sin x) = x := arcsin_sin' ⟨hx₁, hx₂⟩ #align real.arcsin_sin Real.arcsin_sin theorem strictMonoOn_arcsin : StrictMonoOn arcsin (Icc (-1) 1) := (Subtype.strictMono_coe _).comp_strictMonoOn <\| sinOrderIso.symm.strictMono.strictMonoOn_IccExtend _ #align real.strict_mono_on_arcsin Real.strictMonoOn_arcsin theorem monotone_arcsin : Monotone arcsin := (Subtype.mono_coe _).comp <\| sinOrderIso.symm.monotone.IccExtend _ #align real.monotone_arcsin Real.monotone_arcsin theorem injOn_arcsin : InjOn arcsin (Icc (-1) 1) := strictMonoOn_arcsin.injOn #align real.inj_on_arcsin Real.injOn_arcsin theorem arcsin_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) : arcsin x = arcsin y ↔ x = y := injOn_arcsin.eq_iff ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ #align real.arcsin_inj Real.arcsin_inj @[continuity] theorem continuous_arcsin : Continuous arcsin := continuous_subtype_val.comp sinOrderIso.symm.continuous.Icc_extend' #align real.continuous_arcsin Real.continuous_arcsin theorem continuousAt_arcsin {x : ℝ} : ContinuousAt arcsin x := continuous_arcsin.continuousAt #align real.continuous_at_arcsin Real.continuousAt_arcsin theorem arcsin_eq_of_sin_eq {x y : ℝ} (h₁ : sin x = y) (h₂ : x ∈ Icc (-(π / 2)) (π / 2)) : arcsin y = x := by subst y exact injOn_sin (arcsin_mem_Icc _) h₂ (sin_arcsin' (sin_mem_Icc x)) #align real.arcsin_eq_of_sin_eq Real.arcsin_eq_of_sin_eq @[simp] theorem arcsin_zero : arcsin 0 = 0 := arcsin_eq_of_sin_eq sin_zero ⟨neg_nonpos.2 pi_div_two_pos.le, pi_div_two_pos.le⟩ #align real.arcsin_zero Real.arcsin_zero @[simp] theorem arcsin_one : arcsin 1 = π / 2 := arcsin_eq_of_sin_eq sin_pi_div_two <\| right_mem_Icc.2 (neg_le_self pi_div_two_pos.le) #align real.arcsin_one Real.arcsin_one theorem arcsin_of_one_le {x : ℝ} (hx : 1 ≤ x) : arcsin x = π / 2 := by rw [← arcsin_projIcc, projIcc_of_right_le _ hx, Subtype.coe_mk, arcsin_one] #align real.arcsin_of_one_le Real.arcsin_of_one_le theorem arcsin_neg_one : arcsin (-1) = -(π / 2) := arcsin_eq_of_sin_eq (by rw [sin_neg, sin_pi_div_two]) <\| left_mem_Icc.2 (neg_le_self pi_div_two_pos.le) #align real.arcsin_neg_one Real.arcsin_neg_one theorem arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) := by rw [← arcsin_projIcc, projIcc_of_le_left _ hx, Subtype.coe_mk, arcsin_neg_one] #align real.arcsin_of_le_neg_one Real.arcsin_of_le_neg_one @[simp] theorem arcsin_neg (x : ℝ) : arcsin (-x) = -arcsin x := by rcases le_total x (-1) with hx₁ \| hx₁ · rw [arcsin_of_le_neg_one hx₁, neg_neg, arcsin_of_one_le (le_neg.2 hx₁)] rcases le_total 1 x with hx₂ \| hx₂ · rw [arcsin_of_one_le hx₂, arcsin_of_le_neg_one (neg_le_neg hx₂)] refine arcsin_eq_of_sin_eq ?_ ?_ · rw [sin_neg, sin_arcsin hx₁ hx₂] · exact ⟨neg_le_neg (arcsin_le_pi_div_two _), neg_le.2 (neg_pi_div_two_le_arcsin _)⟩ #align real.arcsin_neg Real.arcsin_neg theorem arcsin_le_iff_le_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) : arcsin x ≤ y ↔ x ≤ sin y := by rw [← arcsin_sin' hy, strictMonoOn_arcsin.le_iff_le hx (sin_mem_Icc _), arcsin_sin' hy] #align real.arcsin_le_iff_le_sin Real.arcsin_le_iff_le_sin theorem arcsin_le_iff_le_sin' {x y : ℝ} (hy : y ∈ Ico (-(π / 2)) (π / 2)) : arcsin x ≤ y ↔ x ≤ sin y := by rcases le_total x (-1) with hx₁ \| hx₁ · simp [arcsin_of_le_neg_one hx₁, hy.1, hx₁.trans (neg_one_le_sin _)] cases' lt_or_le 1 x with hx₂ hx₂ · simp [arcsin_of_one_le hx₂.le, hy.2.not_le, (sin_le_one y).trans_lt hx₂] exact arcsin_le_iff_le_sin ⟨hx₁, hx₂⟩ (mem_Icc_of_Ico hy) #align real.arcsin_le_iff_le_sin' Real.arcsin_le_iff_le_sin' theorem le_arcsin_iff_sin_le {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) : x ≤ arcsin y ↔ sin x ≤ y := by rw [← neg_le_neg_iff, ← arcsin_neg, arcsin_le_iff_le_sin ⟨neg_le_neg hy.2, neg_le.2 hy.1⟩ ⟨neg_le_neg hx.2, neg_le.2 hx.1⟩, sin_neg, neg_le_neg_iff] #align real.le_arcsin_iff_sin_le Real.le_arcsin_iff_sin_le theorem le_arcsin_iff_sin_le' {x y : ℝ} (hx : x ∈ Ioc (-(π / 2)) (π / 2)) : x ≤ arcsin y ↔ sin x ≤ y := by rw [← neg_le_neg_iff, ← arcsin_neg, arcsin_le_iff_le_sin' ⟨neg_le_neg hx.2, neg_lt.2 hx.1⟩, sin_neg, neg_le_neg_iff] #align real.le_arcsin_iff_sin_le' Real.le_arcsin_iff_sin_le' theorem arcsin_lt_iff_lt_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) : arcsin x < y ↔ x < sin y := not_le.symm.trans <\| (not_congr <\| le_arcsin_iff_sin_le hy hx).trans not_le #align real.arcsin_lt_iff_lt_sin Real.arcsin_lt_iff_lt_sin theorem arcsin_lt_iff_lt_sin' {x y : ℝ} (hy : y ∈ Ioc (-(π / 2)) (π / 2)) : arcsin x < y ↔ x < sin y := not_le.symm.trans <\| (not_congr <\| le_arcsin_iff_sin_le' hy).trans not_le #align real.arcsin_lt_iff_lt_sin' Real.arcsin_lt_iff_lt_sin' theorem lt_arcsin_iff_sin_lt {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) : x < arcsin y ↔ sin x < y := not_le.symm.trans <\| (not_congr <\| arcsin_le_iff_le_sin hy hx).trans not_le #align real.lt_arcsin_iff_sin_lt Real.lt_arcsin_iff_sin_lt theorem lt_arcsin_iff_sin_lt' {x y : ℝ} (hx : x ∈ Ico (-(π / 2)) (π / 2)) : x < arcsin y ↔ sin x < y := not_le.symm.trans <\| (not_congr <\| arcsin_le_iff_le_sin' hx).trans not_le #align real.lt_arcsin_iff_sin_lt' Real.lt_arcsin_iff_sin_lt' theorem arcsin_eq_iff_eq_sin {x y : ℝ} (hy : y ∈ Ioo (-(π / 2)) (π / 2)) : arcsin x = y ↔ x = sin y := by simp only [le_antisymm_iff, arcsin_le_iff_le_sin' (mem_Ico_of_Ioo hy), le_arcsin_iff_sin_le' (mem_Ioc_of_Ioo hy)] #align real.arcsin_eq_iff_eq_sin Real.arcsin_eq_iff_eq_sin @[simp] theorem arcsin_nonneg {x : ℝ} : 0 ≤ arcsin x ↔ 0 ≤ x := (le_arcsin_iff_sin_le' ⟨neg_lt_zero.2 pi_div_two_pos, pi_div_two_pos.le⟩).trans <\| by rw [sin_zero] #align real.arcsin_nonneg Real.arcsin_nonneg @[simp] theorem arcsin_nonpos {x : ℝ} : arcsin x ≤ 0 ↔ x ≤ 0 := neg_nonneg.symm.trans <\| arcsin_neg x ▸ arcsin_nonneg.trans neg_nonneg #align real.arcsin_nonpos Real.arcsin_nonpos @[simp] theorem arcsin_eq_zero_iff {x : ℝ} : arcsin x = 0 ↔ x = 0 := by simp [le_antisymm_iff] #align real.arcsin_eq_zero_iff Real.arcsin_eq_zero_iff @[simp] theorem zero_eq_arcsin_iff {x} : 0 = arcsin x ↔ x = 0 := eq_comm.trans arcsin_eq_zero_iff #align real.zero_eq_arcsin_iff Real.zero_eq_arcsin_iff @[simp] theorem arcsin_pos {x : ℝ} : 0 < arcsin x ↔ 0 < x := lt_iff_lt_of_le_iff_le arcsin_nonpos #align real.arcsin_pos Real.arcsin_pos @[simp] theorem arcsin_lt_zero {x : ℝ} : arcsin x < 0 ↔ x < 0 := lt_iff_lt_of_le_iff_le arcsin_nonneg #align real.arcsin_lt_zero Real.arcsin_lt_zero @[simp] theorem arcsin_lt_pi_div_two {x : ℝ} : arcsin x < π / 2 ↔ x < 1 := (arcsin_lt_iff_lt_sin' (right_mem_Ioc.2 <\| neg_lt_self pi_div_two_pos)).trans <\| by rw [sin_pi_div_two] #align real.arcsin_lt_pi_div_two Real.arcsin_lt_pi_div_two @[simp] theorem neg_pi_div_two_lt_arcsin {x : ℝ} : -(π / 2) < arcsin x ↔ -1 < x := (lt_arcsin_iff_sin_lt' <\| left_mem_Ico.2 <\| neg_lt_self pi_div_two_pos).trans <\| by rw [sin_neg, sin_pi_div_two] #align real.neg_pi_div_two_lt_arcsin Real.neg_pi_div_two_lt_arcsin @[simp] theorem arcsin_eq_pi_div_two {x : ℝ} : arcsin x = π / 2 ↔ 1 ≤ x := ⟨fun h => not_lt.1 fun h' => (arcsin_lt_pi_div_two.2 h').ne h, arcsin_of_one_le⟩ #align real.arcsin_eq_pi_div_two Real.arcsin_eq_pi_div_two @[simp] theorem pi_div_two_eq_arcsin {x} : π / 2 = arcsin x ↔ 1 ≤ x := eq_comm.trans arcsin_eq_pi_div_two #align real.pi_div_two_eq_arcsin Real.pi_div_two_eq_arcsin @[simp] theorem pi_div_two_le_arcsin {x} : π / 2 ≤ arcsin x ↔ 1 ≤ x := (arcsin_le_pi_div_two x).le_iff_eq.trans pi_div_two_eq_arcsin #align real.pi_div_two_le_arcsin Real.pi_div_two_le_arcsin @[simp] theorem arcsin_eq_neg_pi_div_two {x : ℝ} : arcsin x = -(π / 2) ↔ x ≤ -1 := ⟨fun h => not_lt.1 fun h' => (neg_pi_div_two_lt_arcsin.2 h').ne' h, arcsin_of_le_neg_one⟩ #align real.arcsin_eq_neg_pi_div_two Real.arcsin_eq_neg_pi_div_two @[simp] theorem neg_pi_div_two_eq_arcsin {x} : -(π / 2) = arcsin x ↔ x ≤ -1 := eq_comm.trans arcsin_eq_neg_pi_div_two #align real.neg_pi_div_two_eq_arcsin Real.neg_pi_div_two_eq_arcsin @[simp] theorem arcsin_le_neg_pi_div_two {x} : arcsin x ≤ -(π / 2) ↔ x ≤ -1 := (neg_pi_div_two_le_arcsin x).le_iff_eq.trans arcsin_eq_neg_pi_div_two #align real.arcsin_le_neg_pi_div_two Real.arcsin_le_neg_pi_div_two @[simp] theorem pi_div_four_le_arcsin {x} : π / 4 ≤ arcsin x ↔ √2 / 2 ≤ x := by rw [← sin_pi_div_four, le_arcsin_iff_sin_le'] have := pi_pos constructor <;> linarith #align real.pi_div_four_le_arcsin Real.pi_div_four_le_arcsin theorem mapsTo_sin_Ioo : MapsTo sin (Ioo (-(π / 2)) (π / 2)) (Ioo (-1) 1) := fun x h => by rwa [mem_Ioo, ← arcsin_lt_pi_div_two, ← neg_pi_div_two_lt_arcsin, arcsin_sin h.1.le h.2.le] #align real.maps_to_sin_Ioo Real.mapsTo_sin_Ioo @[simp] def sinPartialHomeomorph : PartialHomeomorph ℝ ℝ where toFun := sin invFun := arcsin source := Ioo (-(π / 2)) (π / 2) target := Ioo (-1) 1 map_source' := mapsTo_sin_Ioo map_target' _ hy := ⟨neg_pi_div_two_lt_arcsin.2 hy.1, arcsin_lt_pi_div_two.2 hy.2⟩ left_inv' _ hx := arcsin_sin hx.1.le hx.2.le right_inv' _ hy := sin_arcsin hy.1.le hy.2.le open_source := isOpen_Ioo open_target := isOpen_Ioo continuousOn_toFun := continuous_sin.continuousOn continuousOn_invFun := continuous_arcsin.continuousOn #align real.sin_local_homeomorph Real.sinPartialHomeomorph theorem cos_arcsin_nonneg (x : ℝ) : 0 ≤ cos (arcsin x) := cos_nonneg_of_mem_Icc ⟨neg_pi_div_two_le_arcsin _, arcsin_le_pi_div_two _⟩ #align real.cos_arcsin_nonneg Real.cos_arcsin_nonneg -- The junk values for `arcsin` and `sqrt` make this true even outside `[-1, 1]`. theorem cos_arcsin (x : ℝ) : cos (arcsin x) = √(1 - x ^ 2) := by by_cases hx₁ : -1 ≤ x; swap · rw [not_le] at hx₁ rw [arcsin_of_le_neg_one hx₁.le, cos_neg, cos_pi_div_two, sqrt_eq_zero_of_nonpos] nlinarith by_cases hx₂ : x ≤ 1; swap · rw [not_le] at hx₂ rw [arcsin_of_one_le hx₂.le, cos_pi_div_two, sqrt_eq_zero_of_nonpos] nlinarith have : sin (arcsin x) ^ 2 + cos (arcsin x) ^ 2 = 1 := sin_sq_add_cos_sq (arcsin x) rw [← eq_sub_iff_add_eq', ← sqrt_inj (sq_nonneg _) (sub_nonneg.2 (sin_sq_le_one (arcsin x))), sq, sqrt_mul_self (cos_arcsin_nonneg _)] at this rw [this, sin_arcsin hx₁ hx₂] #align real.cos_arcsin Real.cos_arcsin -- The junk values for `arcsin` and `sqrt` make this true even outside `[-1, 1]`. theorem tan_arcsin (x : ℝ) : tan (arcsin x) = x / √(1 - x ^ 2) := by rw [tan_eq_sin_div_cos, cos_arcsin] by_cases hx₁ : -1 ≤ x; swap · have h : √(1 - x ^ 2) = 0 := sqrt_eq_zero_of_nonpos (by nlinarith) rw [h] simp by_cases hx₂ : x ≤ 1; swap · have h : √(1 - x ^ 2) = 0 := sqrt_eq_zero_of_nonpos (by nlinarith) rw [h] simp rw [sin_arcsin hx₁ hx₂] #align real.tan_arcsin Real.tan_arcsin -- @[pp_nodot] Porting note: not implemented noncomputable def arccos (x : ℝ) : ℝ := π / 2 - arcsin x #align real.arccos Real.arccos theorem arccos_eq_pi_div_two_sub_arcsin (x : ℝ) : arccos x = π / 2 - arcsin x := rfl #align real.arccos_eq_pi_div_two_sub_arcsin Real.arccos_eq_pi_div_two_sub_arcsin theorem arcsin_eq_pi_div_two_sub_arccos (x : ℝ) : arcsin x = π / 2 - arccos x := by simp [arccos] #align real.arcsin_eq_pi_div_two_sub_arccos Real.arcsin_eq_pi_div_two_sub_arccos theorem arccos_le_pi (x : ℝ) : arccos x ≤ π := by unfold arccos; linarith [neg_pi_div_two_le_arcsin x] #align real.arccos_le_pi Real.arccos_le_pi theorem arccos_nonneg (x : ℝ) : 0 ≤ arccos x := by unfold arccos; linarith [arcsin_le_pi_div_two x] #align real.arccos_nonneg Real.arccos_nonneg @[simp] theorem arccos_pos {x : ℝ} : 0 < arccos x ↔ x < 1 := by simp [arccos] #align real.arccos_pos Real.arccos_pos theorem cos_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arccos x) = x := by rw [arccos, cos_pi_div_two_sub, sin_arcsin hx₁ hx₂] #align real.cos_arccos Real.cos_arccos theorem arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x := by rw [arccos, ← sin_pi_div_two_sub, arcsin_sin] <;> simp [sub_eq_add_neg] <;> linarith #align real.arccos_cos Real.arccos_cos lemma arccos_eq_of_eq_cos (hy₀ : 0 ≤ y) (hy₁ : y ≤ π) (hxy : x = cos y) : arccos x = y := by rw [hxy, arccos_cos hy₀ hy₁] theorem strictAntiOn_arccos : StrictAntiOn arccos (Icc (-1) 1) := fun _ hx _ hy h => sub_lt_sub_left (strictMonoOn_arcsin hx hy h) _ #align real.strict_anti_on_arccos Real.strictAntiOn_arccos theorem arccos_injOn : InjOn arccos (Icc (-1) 1) := strictAntiOn_arccos.injOn #align real.arccos_inj_on Real.arccos_injOn theorem arccos_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) : arccos x = arccos y ↔ x = y := arccos_injOn.eq_iff ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ #align real.arccos_inj Real.arccos_inj @[simp] theorem arccos_zero : arccos 0 = π / 2 := by simp [arccos] #align real.arccos_zero Real.arccos_zero @[simp] theorem arccos_one : arccos 1 = 0 := by simp [arccos] #align real.arccos_one Real.arccos_one @[simp] theorem arccos_neg_one : arccos (-1) = π := by simp [arccos, add_halves] #align real.arccos_neg_one Real.arccos_neg_one @[simp] theorem arccos_eq_zero {x} : arccos x = 0 ↔ 1 ≤ x := by simp [arccos, sub_eq_zero] #align real.arccos_eq_zero Real.arccos_eq_zero @[simp] theorem arccos_eq_pi_div_two {x} : arccos x = π / 2 ↔ x = 0 := by simp [arccos] #align real.arccos_eq_pi_div_two Real.arccos_eq_pi_div_two @[simp] theorem arccos_eq_pi {x} : arccos x = π ↔ x ≤ -1 := by rw [arccos, sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', div_two_sub_self, neg_pi_div_two_eq_arcsin] #align real.arccos_eq_pi Real.arccos_eq_pi	Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean	410	411	theorem arccos_neg (x : ℝ) : arccos (-x) = π - arccos x := by	rw [← add_halves π, arccos, arcsin_neg, arccos, add_sub_assoc, sub_sub_self, sub_neg_eq_add]
import Mathlib.Data.Nat.Multiplicity import Mathlib.Data.ZMod.Algebra import Mathlib.RingTheory.WittVector.Basic import Mathlib.RingTheory.WittVector.IsPoly import Mathlib.FieldTheory.Perfect #align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472" namespace WittVector variable {p : ℕ} {R S : Type} [hp : Fact p.Prime] [CommRing R] [CommRing S] local notation "𝕎" => WittVector p -- type as `\bbW` noncomputable section open MvPolynomial Finset variable (p) def frobeniusPolyRat (n : ℕ) : MvPolynomial ℕ ℚ := bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1) (xInTermsOfW p ℚ n) #align witt_vector.frobenius_poly_rat WittVector.frobeniusPolyRat theorem bind₁_frobeniusPolyRat_wittPolynomial (n : ℕ) : bind₁ (frobeniusPolyRat p) (wittPolynomial p ℚ n) = wittPolynomial p ℚ (n + 1) := by delta frobeniusPolyRat rw [← bind₁_bind₁, bind₁_xInTermsOfW_wittPolynomial, bind₁_X_right, Function.comp_apply] #align witt_vector.bind₁_frobenius_poly_rat_witt_polynomial WittVector.bind₁_frobeniusPolyRat_wittPolynomial private def pnat_multiplicity (n : ℕ+) : ℕ := (multiplicity p n).get <\| multiplicity.finite_nat_iff.mpr <\| ⟨ne_of_gt hp.1.one_lt, n.2⟩ local notation "v" => pnat_multiplicity noncomputable def frobeniusPolyAux : ℕ → MvPolynomial ℕ ℤ \| n => X (n + 1) - ∑ i : Fin n, have _ := i.is_lt ∑ j ∈ range (p ^ (n - i)), (((X (i : ℕ) ^ p) ^ (p ^ (n - (i : ℕ)) - (j + 1)) : MvPolynomial ℕ ℤ) (frobeniusPolyAux i) ^ (j + 1)) * C (((p ^ (n - i)).choose (j + 1) / (p ^ (n - i - v p ⟨j + 1, Nat.succ_pos j⟩)) * ↑p ^ (j - v p ⟨j + 1, Nat.succ_pos j⟩) : ℕ) : ℤ) #align witt_vector.frobenius_poly_aux WittVector.frobeniusPolyAux theorem frobeniusPolyAux_eq (n : ℕ) : frobeniusPolyAux p n = X (n + 1) - ∑ i ∈ range n, ∑ j ∈ range (p ^ (n - i)), (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) * C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p ⟨j + 1, Nat.succ_pos j⟩) * ↑p ^ (j - v p ⟨j + 1, Nat.succ_pos j⟩) : ℕ) := by rw [frobeniusPolyAux, ← Fin.sum_univ_eq_sum_range] #align witt_vector.frobenius_poly_aux_eq WittVector.frobeniusPolyAux_eq def frobeniusPoly (n : ℕ) : MvPolynomial ℕ ℤ := X n ^ p + C (p : ℤ) * frobeniusPolyAux p n #align witt_vector.frobenius_poly WittVector.frobeniusPoly theorem map_frobeniusPoly.key₁ (n j : ℕ) (hj : j < p ^ n) : p ^ (n - v p ⟨j + 1, j.succ_pos⟩) ∣ (p ^ n).choose (j + 1) := by apply multiplicity.pow_dvd_of_le_multiplicity rw [hp.out.multiplicity_choose_prime_pow hj j.succ_ne_zero] rfl #align witt_vector.map_frobenius_poly.key₁ WittVector.map_frobeniusPoly.key₁ theorem map_frobeniusPoly.key₂ {n i j : ℕ} (hi : i ≤ n) (hj : j < p ^ (n - i)) : j - v p ⟨j + 1, j.succ_pos⟩ + n = i + j + (n - i - v p ⟨j + 1, j.succ_pos⟩) := by generalize h : v p ⟨j + 1, j.succ_pos⟩ = m rsuffices ⟨h₁, h₂⟩ : m ≤ n - i ∧ m ≤ j · rw [tsub_add_eq_add_tsub h₂, add_comm i j, add_tsub_assoc_of_le (h₁.trans (Nat.sub_le n i)), add_assoc, tsub_right_comm, add_comm i, tsub_add_cancel_of_le (le_tsub_of_add_le_right ((le_tsub_iff_left hi).mp h₁))] have hle : p ^ m ≤ j + 1 := h ▸ Nat.le_of_dvd j.succ_pos (multiplicity.pow_multiplicity_dvd _) exact ⟨(pow_le_pow_iff_right hp.1.one_lt).1 (hle.trans hj), Nat.le_of_lt_succ ((Nat.lt_pow_self hp.1.one_lt m).trans_le hle)⟩ #align witt_vector.map_frobenius_poly.key₂ WittVector.map_frobeniusPoly.key₂	Mathlib/RingTheory/WittVector/Frobenius.lean	143	193	theorem map_frobeniusPoly (n : ℕ) : MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by	rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast, Int.cast_natCast, frobeniusPolyRat] refine Nat.strong_induction_on n ?_; clear n intro n IH rw [xInTermsOfW_eq] simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right] have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow] rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ, sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul, add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ', mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc, ← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg, add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub, add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm] simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib] apply sum_congr rfl intro i hi rw [mem_range] at hi rw [← IH i hi] clear IH rw [add_comm (X i ^ p), add_pow, sum_range_succ', pow_zero, tsub_zero, Nat.choose_zero_right, one_mul, Nat.cast_one, mul_one, mul_add, add_mul, Nat.succ_sub (le_of_lt hi), Nat.succ_eq_add_one (n - i), pow_succ', pow_mul, add_sub_cancel_right, mul_sum, sum_mul] apply sum_congr rfl intro j hj rw [mem_range] at hj rw [RingHom.map_mul, RingHom.map_mul, RingHom.map_pow, RingHom.map_pow, RingHom.map_pow, RingHom.map_pow, RingHom.map_pow, map_C, map_X, mul_pow] rw [mul_comm (C (p : ℚ) ^ i), mul_comm _ ((X i ^ p) ^ _), mul_comm (C (p : ℚ) ^ (j + 1)), mul_comm (C (p : ℚ))] simp only [mul_assoc] apply congr_arg apply congr_arg rw [← C_eq_coe_nat] simp only [← RingHom.map_pow, ← C_mul] rw [C_inj] simp only [invOf_eq_inv, eq_intCast, inv_pow, Int.cast_natCast, Nat.cast_mul, Int.cast_mul] rw [Rat.natCast_div _ _ (map_frobeniusPoly.key₁ p (n - i) j hj)] simp only [Nat.cast_pow, pow_add, pow_one] suffices (((p ^ (n - i)).choose (j + 1): ℚ) * (p : ℚ) ^ (j - v p ⟨j + 1, j.succ_pos⟩) * ↑p * (p ^ n : ℚ)) = (p : ℚ) ^ j * p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * (p : ℚ) ^ (n - i - v p ⟨j + 1, j.succ_pos⟩) by have aux : ∀ k : ℕ, (p : ℚ)^ k ≠ 0 := by intro; apply pow_ne_zero; exact mod_cast hp.1.ne_zero simpa [aux, -one_div, -pow_eq_zero_iff', field_simps] using this.symm rw [mul_comm _ (p : ℚ), mul_assoc, mul_assoc, ← pow_add, map_frobeniusPoly.key₂ p hi.le hj, Nat.cast_mul, Nat.cast_pow] ring
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence import Mathlib.Algebra.ContinuedFractions.TerminatedStable import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.Ring #align_import algebra.continued_fractions.convergents_equiv from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad" variable {K : Type} {n : ℕ} namespace GeneralizedContinuedFraction variable {g : GeneralizedContinuedFraction K} {s : Stream'.Seq <\| Pair K} section Squash section WithDivisionRing variable [DivisionRing K] def squashSeq (s : Stream'.Seq <\| Pair K) (n : ℕ) : Stream'.Seq (Pair K) := match Prod.mk (s.get? n) (s.get? (n + 1)) with \| ⟨some gp_n, some gp_succ_n⟩ => Stream'.Seq.nats.zipWith -- return the squashed value at position `n`; otherwise, do nothing. (fun n' gp => if n' = n then ⟨gp_n.a, gp_n.b + gp_succ_n.a / gp_succ_n.b⟩ else gp) s \| _ => s #align generalized_continued_fraction.squash_seq GeneralizedContinuedFraction.squashSeq theorem squashSeq_eq_self_of_terminated (terminated_at_succ_n : s.TerminatedAt (n + 1)) : squashSeq s n = s := by change s.get? (n + 1) = none at terminated_at_succ_n cases s_nth_eq : s.get? n <;> simp only [, squashSeq] #align generalized_continued_fraction.squash_seq_eq_self_of_terminated GeneralizedContinuedFraction.squashSeq_eq_self_of_terminated theorem squashSeq_nth_of_not_terminated {gp_n gp_succ_n : Pair K} (s_nth_eq : s.get? n = some gp_n) (s_succ_nth_eq : s.get? (n + 1) = some gp_succ_n) : (squashSeq s n).get? n = some ⟨gp_n.a, gp_n.b + gp_succ_n.a / gp_succ_n.b⟩ := by simp [, squashSeq] #align generalized_continued_fraction.squash_seq_nth_of_not_terminated GeneralizedContinuedFraction.squashSeq_nth_of_not_terminated theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by cases s_succ_nth_eq : s.get? (n + 1) with \| none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq] \| some => obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n := s.ge_stable n.le_succ s_succ_nth_eq obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.get? m = some gp_m := s.ge_stable (le_of_lt m_lt_n) s_nth_eq simp [, squashSeq, m_lt_n.ne] #align generalized_continued_fraction.squash_seq_nth_of_lt GeneralizedContinuedFraction.squashSeq_nth_of_lt theorem squashSeq_succ_n_tail_eq_squashSeq_tail_n : (squashSeq s (n + 1)).tail = squashSeq s.tail n := by cases s_succ_succ_nth_eq : s.get? (n + 2) with \| none => cases s_succ_nth_eq : s.get? (n + 1) <;> simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq] \| some gp_succ_succ_n => obtain ⟨gp_succ_n, s_succ_nth_eq⟩ : ∃ gp_succ_n, s.get? (n + 1) = some gp_succ_n := s.ge_stable (n + 1).le_succ s_succ_succ_nth_eq -- apply extensionality with `m` and continue by cases `m = n`. ext1 m cases' Decidable.em (m = n) with m_eq_n m_ne_n · simp [, squashSeq] · cases s_succ_mth_eq : s.get? (m + 1) · simp only [, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith, Option.map₂_none_right] · simp [*, squashSeq] #align generalized_continued_fraction.squash_seq_succ_n_tail_eq_squash_seq_tail_n GeneralizedContinuedFraction.squashSeq_succ_n_tail_eq_squashSeq_tail_n theorem succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq : convergents'Aux s (n + 2) = convergents'Aux (squashSeq s n) (n + 1) := by cases s_succ_nth_eq : s.get? <\| n + 1 with \| none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq, convergents'Aux_stable_step_of_terminated s_succ_nth_eq] \| some gp_succ_n => induction n generalizing s gp_succ_n with \| zero => obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head := s.ge_stable zero_le_one s_succ_nth_eq have : (squashSeq s 0).head = some ⟨gp_head.a, gp_head.b + gp_succ_n.a / gp_succ_n.b⟩ := squashSeq_nth_of_not_terminated s_head_eq s_succ_nth_eq simp_all [convergents'Aux, Stream'.Seq.head, Stream'.Seq.get?_tail] \| succ m IH => obtain ⟨gp_head, s_head_eq⟩ : ∃ gp_head, s.head = some gp_head := s.ge_stable (m + 2).zero_le s_succ_nth_eq suffices gp_head.a / (gp_head.b + convergents'Aux s.tail (m + 2)) = convergents'Aux (squashSeq s (m + 1)) (m + 2) by simpa only [convergents'Aux, s_head_eq] have : convergents'Aux s.tail (m + 2) = convergents'Aux (squashSeq s.tail m) (m + 1) := by refine IH gp_succ_n ?_ simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq have : (squashSeq s (m + 1)).head = some gp_head := (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq simp_all [convergents'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n] #align generalized_continued_fraction.succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squash_seq GeneralizedContinuedFraction.succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq def squashGCF (g : GeneralizedContinuedFraction K) : ℕ → GeneralizedContinuedFraction K \| 0 => match g.s.get? 0 with \| none => g \| some gp => ⟨g.h + gp.a / gp.b, g.s⟩ \| n + 1 => ⟨g.h, squashSeq g.s n⟩ #align generalized_continued_fraction.squash_gcf GeneralizedContinuedFraction.squashGCF	Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean	204	213	theorem squashGCF_eq_self_of_terminated (terminated_at_n : TerminatedAt g n) : squashGCF g n = g := by	cases n with \| zero => change g.s.get? 0 = none at terminated_at_n simp only [convergents', squashGCF, convergents'Aux, terminated_at_n] \| succ => cases g simp only [squashGCF, mk.injEq, true_and] exact squashSeq_eq_self_of_terminated terminated_at_n
import Mathlib.Analysis.Convex.Cone.Basic import Mathlib.Analysis.InnerProductSpace.Projection #align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4" open Set LinearMap open scoped Classical open Pointwise variable {𝕜 E F G : Type} section Dual variable {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s t : Set H) open RealInnerProductSpace def Set.innerDualCone (s : Set H) : ConvexCone ℝ H where carrier := { y \| ∀ x ∈ s, 0 ≤ ⟪x, y⟫ } smul_mem' c hc y hy x hx := by rw [real_inner_smul_right] exact mul_nonneg hc.le (hy x hx) add_mem' u hu v hv x hx := by rw [inner_add_right] exact add_nonneg (hu x hx) (hv x hx) #align set.inner_dual_cone Set.innerDualCone @[simp] theorem mem_innerDualCone (y : H) (s : Set H) : y ∈ s.innerDualCone ↔ ∀ x ∈ s, 0 ≤ ⟪x, y⟫ := Iff.rfl #align mem_inner_dual_cone mem_innerDualCone @[simp] theorem innerDualCone_empty : (∅ : Set H).innerDualCone = ⊤ := eq_top_iff.mpr fun _ _ _ => False.elim #align inner_dual_cone_empty innerDualCone_empty @[simp] theorem innerDualCone_zero : (0 : Set H).innerDualCone = ⊤ := eq_top_iff.mpr fun _ _ y (hy : y = 0) => hy.symm ▸ (inner_zero_left _).ge #align inner_dual_cone_zero innerDualCone_zero @[simp] theorem innerDualCone_univ : (univ : Set H).innerDualCone = 0 := by suffices ∀ x : H, x ∈ (univ : Set H).innerDualCone → x = 0 by apply SetLike.coe_injective exact eq_singleton_iff_unique_mem.mpr ⟨fun x _ => (inner_zero_right _).ge, this⟩ exact fun x hx => by simpa [← real_inner_self_nonpos] using hx (-x) (mem_univ _) #align inner_dual_cone_univ innerDualCone_univ theorem innerDualCone_le_innerDualCone (h : t ⊆ s) : s.innerDualCone ≤ t.innerDualCone := fun _ hy x hx => hy x (h hx) #align inner_dual_cone_le_inner_dual_cone innerDualCone_le_innerDualCone theorem pointed_innerDualCone : s.innerDualCone.Pointed := fun x _ => by rw [inner_zero_right] #align pointed_inner_dual_cone pointed_innerDualCone theorem innerDualCone_singleton (x : H) : ({x} : Set H).innerDualCone = (ConvexCone.positive ℝ ℝ).comap (innerₛₗ ℝ x) := ConvexCone.ext fun _ => forall_eq #align inner_dual_cone_singleton innerDualCone_singleton theorem innerDualCone_union (s t : Set H) : (s ∪ t).innerDualCone = s.innerDualCone ⊓ t.innerDualCone := le_antisymm (le_inf (fun _ hx _ hy => hx _ <\| Or.inl hy) fun _ hx _ hy => hx _ <\| Or.inr hy) fun _ hx _ => Or.rec (hx.1 _) (hx.2 _) #align inner_dual_cone_union innerDualCone_union theorem innerDualCone_insert (x : H) (s : Set H) : (insert x s).innerDualCone = Set.innerDualCone {x} ⊓ s.innerDualCone := by rw [insert_eq, innerDualCone_union] #align inner_dual_cone_insert innerDualCone_insert	Mathlib/Analysis/Convex/Cone/InnerDual.lean	110	116	theorem innerDualCone_iUnion {ι : Sort*} (f : ι → Set H) : (⋃ i, f i).innerDualCone = ⨅ i, (f i).innerDualCone := by	refine le_antisymm (le_iInf fun i x hx y hy => hx _ <\| mem_iUnion_of_mem _ hy) ?_ intro x hx y hy rw [ConvexCone.mem_iInf] at hx obtain ⟨j, hj⟩ := mem_iUnion.mp hy exact hx _ _ hj
import Mathlib.SetTheory.Game.Basic import Mathlib.SetTheory.Ordinal.NaturalOps #align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834" universe u open SetTheory PGame open scoped NaturalOps PGame namespace Ordinal noncomputable def toPGame : Ordinal.{u} → PGame.{u} \| o => have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o ⟨o.out.α, PEmpty, fun x => have := Ordinal.typein_lt_self x (typein (· < ·) x).toPGame, PEmpty.elim⟩ termination_by x => x #align ordinal.to_pgame Ordinal.toPGame @[nolint unusedHavesSuffices] theorem toPGame_def (o : Ordinal) : have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o o.toPGame = ⟨o.out.α, PEmpty, fun x => (typein (· < ·) x).toPGame, PEmpty.elim⟩ := by rw [toPGame] #align ordinal.to_pgame_def Ordinal.toPGame_def @[simp, nolint unusedHavesSuffices] theorem toPGame_leftMoves (o : Ordinal) : o.toPGame.LeftMoves = o.out.α := by rw [toPGame, LeftMoves] #align ordinal.to_pgame_left_moves Ordinal.toPGame_leftMoves @[simp, nolint unusedHavesSuffices] theorem toPGame_rightMoves (o : Ordinal) : o.toPGame.RightMoves = PEmpty := by rw [toPGame, RightMoves] #align ordinal.to_pgame_right_moves Ordinal.toPGame_rightMoves instance isEmpty_zero_toPGame_leftMoves : IsEmpty (toPGame 0).LeftMoves := by rw [toPGame_leftMoves]; infer_instance #align ordinal.is_empty_zero_to_pgame_left_moves Ordinal.isEmpty_zero_toPGame_leftMoves instance isEmpty_toPGame_rightMoves (o : Ordinal) : IsEmpty o.toPGame.RightMoves := by rw [toPGame_rightMoves]; infer_instance #align ordinal.is_empty_to_pgame_right_moves Ordinal.isEmpty_toPGame_rightMoves noncomputable def toLeftMovesToPGame {o : Ordinal} : Set.Iio o ≃ o.toPGame.LeftMoves := (enumIsoOut o).toEquiv.trans (Equiv.cast (toPGame_leftMoves o).symm) #align ordinal.to_left_moves_to_pgame Ordinal.toLeftMovesToPGame @[simp] theorem toLeftMovesToPGame_symm_lt {o : Ordinal} (i : o.toPGame.LeftMoves) : ↑(toLeftMovesToPGame.symm i) < o := (toLeftMovesToPGame.symm i).prop #align ordinal.to_left_moves_to_pgame_symm_lt Ordinal.toLeftMovesToPGame_symm_lt @[nolint unusedHavesSuffices] theorem toPGame_moveLeft_hEq {o : Ordinal} : have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o HEq o.toPGame.moveLeft fun x : o.out.α => (typein (· < ·) x).toPGame := by rw [toPGame] rfl #align ordinal.to_pgame_move_left_heq Ordinal.toPGame_moveLeft_hEq @[simp] theorem toPGame_moveLeft' {o : Ordinal} (i) : o.toPGame.moveLeft i = (toLeftMovesToPGame.symm i).val.toPGame := (congr_heq toPGame_moveLeft_hEq.symm (cast_heq _ i)).symm #align ordinal.to_pgame_move_left' Ordinal.toPGame_moveLeft' theorem toPGame_moveLeft {o : Ordinal} (i) : o.toPGame.moveLeft (toLeftMovesToPGame i) = i.val.toPGame := by simp #align ordinal.to_pgame_move_left Ordinal.toPGame_moveLeft noncomputable def zeroToPGameRelabelling : toPGame 0 ≡r 0 := Relabelling.isEmpty _ #align ordinal.zero_to_pgame_relabelling Ordinal.zeroToPGameRelabelling noncomputable instance uniqueOneToPGameLeftMoves : Unique (toPGame 1).LeftMoves := (Equiv.cast <\| toPGame_leftMoves 1).unique #align ordinal.unique_one_to_pgame_left_moves Ordinal.uniqueOneToPGameLeftMoves @[simp] theorem one_toPGame_leftMoves_default_eq : (default : (toPGame 1).LeftMoves) = @toLeftMovesToPGame 1 ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := rfl #align ordinal.one_to_pgame_left_moves_default_eq Ordinal.one_toPGame_leftMoves_default_eq @[simp] theorem to_leftMoves_one_toPGame_symm (i) : (@toLeftMovesToPGame 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by simp [eq_iff_true_of_subsingleton] #align ordinal.to_left_moves_one_to_pgame_symm Ordinal.to_leftMoves_one_toPGame_symm	Mathlib/SetTheory/Game/Ordinal.lean	121	121	theorem one_toPGame_moveLeft (x) : (toPGame 1).moveLeft x = toPGame 0 := by	simp
import Mathlib.Algebra.Polynomial.Reverse import Mathlib.Algebra.Regular.SMul #align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5" noncomputable section open Finset open Polynomial namespace Polynomial universe u v y variable {R : Type u} {S : Type v} {a b : R} {m n : ℕ} {ι : Type y} section Semiring variable [Semiring R] {p q r : R[X]} theorem monic_zero_iff_subsingleton : Monic (0 : R[X]) ↔ Subsingleton R := subsingleton_iff_zero_eq_one #align polynomial.monic_zero_iff_subsingleton Polynomial.monic_zero_iff_subsingleton theorem not_monic_zero_iff : ¬Monic (0 : R[X]) ↔ (0 : R) ≠ 1 := (monic_zero_iff_subsingleton.trans subsingleton_iff_zero_eq_one.symm).not #align polynomial.not_monic_zero_iff Polynomial.not_monic_zero_iff theorem monic_zero_iff_subsingleton' : Monic (0 : R[X]) ↔ (∀ f g : R[X], f = g) ∧ ∀ a b : R, a = b := Polynomial.monic_zero_iff_subsingleton.trans ⟨by intro simp [eq_iff_true_of_subsingleton], fun h => subsingleton_iff.mpr h.2⟩ #align polynomial.monic_zero_iff_subsingleton' Polynomial.monic_zero_iff_subsingleton' theorem Monic.as_sum (hp : p.Monic) : p = X ^ p.natDegree + ∑ i ∈ range p.natDegree, C (p.coeff i) * X ^ i := by conv_lhs => rw [p.as_sum_range_C_mul_X_pow, sum_range_succ_comm] suffices C (p.coeff p.natDegree) = 1 by rw [this, one_mul] exact congr_arg C hp #align polynomial.monic.as_sum Polynomial.Monic.as_sum theorem ne_zero_of_ne_zero_of_monic (hp : p ≠ 0) (hq : Monic q) : q ≠ 0 := by rintro rfl rw [Monic.def, leadingCoeff_zero] at hq rw [← mul_one p, ← C_1, ← hq, C_0, mul_zero] at hp exact hp rfl #align polynomial.ne_zero_of_ne_zero_of_monic Polynomial.ne_zero_of_ne_zero_of_monic theorem Monic.map [Semiring S] (f : R →+* S) (hp : Monic p) : Monic (p.map f) := by unfold Monic nontriviality have : f p.leadingCoeff ≠ 0 := by rw [show _ = _ from hp, f.map_one] exact one_ne_zero rw [Polynomial.leadingCoeff, coeff_map] suffices p.coeff (p.map f).natDegree = 1 by simp [this] rwa [natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f this)] #align polynomial.monic.map Polynomial.Monic.map	Mathlib/Algebra/Polynomial/Monic.lean	76	80	theorem monic_C_mul_of_mul_leadingCoeff_eq_one {b : R} (hp : b * p.leadingCoeff = 1) : Monic (C b * p) := by	unfold Monic nontriviality rw [leadingCoeff_mul' _] <;> simp [leadingCoeff_C b, hp]
import Mathlib.Init.ZeroOne import Mathlib.Data.Set.Defs import Mathlib.Order.Basic import Mathlib.Order.SymmDiff import Mathlib.Tactic.Tauto import Mathlib.Tactic.ByContra import Mathlib.Util.Delaborators #align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29" open Function universe u v w x namespace Set variable {α : Type u} {s t : Set α} instance instBooleanAlgebraSet : BooleanAlgebra (Set α) := { (inferInstance : BooleanAlgebra (α → Prop)) with sup := (· ∪ ·), le := (· ≤ ·), lt := fun s t => s ⊆ t ∧ ¬t ⊆ s, inf := (· ∩ ·), bot := ∅, compl := (·ᶜ), top := univ, sdiff := (· \ ·) } instance : HasSSubset (Set α) := ⟨(· < ·)⟩ @[simp] theorem top_eq_univ : (⊤ : Set α) = univ := rfl #align set.top_eq_univ Set.top_eq_univ @[simp] theorem bot_eq_empty : (⊥ : Set α) = ∅ := rfl #align set.bot_eq_empty Set.bot_eq_empty @[simp] theorem sup_eq_union : ((· ⊔ ·) : Set α → Set α → Set α) = (· ∪ ·) := rfl #align set.sup_eq_union Set.sup_eq_union @[simp] theorem inf_eq_inter : ((· ⊓ ·) : Set α → Set α → Set α) = (· ∩ ·) := rfl #align set.inf_eq_inter Set.inf_eq_inter @[simp] theorem le_eq_subset : ((· ≤ ·) : Set α → Set α → Prop) = (· ⊆ ·) := rfl #align set.le_eq_subset Set.le_eq_subset @[simp] theorem lt_eq_ssubset : ((· < ·) : Set α → Set α → Prop) = (· ⊂ ·) := rfl #align set.lt_eq_ssubset Set.lt_eq_ssubset theorem le_iff_subset : s ≤ t ↔ s ⊆ t := Iff.rfl #align set.le_iff_subset Set.le_iff_subset theorem lt_iff_ssubset : s < t ↔ s ⊂ t := Iff.rfl #align set.lt_iff_ssubset Set.lt_iff_ssubset alias ⟨_root_.LE.le.subset, _root_.HasSubset.Subset.le⟩ := le_iff_subset #align has_subset.subset.le HasSubset.Subset.le alias ⟨_root_.LT.lt.ssubset, _root_.HasSSubset.SSubset.lt⟩ := lt_iff_ssubset #align has_ssubset.ssubset.lt HasSSubset.SSubset.lt instance PiSetCoe.canLift (ι : Type u) (α : ι → Type v) [∀ i, Nonempty (α i)] (s : Set ι) : CanLift (∀ i : s, α i) (∀ i, α i) (fun f i => f i) fun _ => True := PiSubtype.canLift ι α s #align set.pi_set_coe.can_lift Set.PiSetCoe.canLift instance PiSetCoe.canLift' (ι : Type u) (α : Type v) [Nonempty α] (s : Set ι) : CanLift (s → α) (ι → α) (fun f i => f i) fun _ => True := PiSetCoe.canLift ι (fun _ => α) s #align set.pi_set_coe.can_lift' Set.PiSetCoe.canLift' end Set theorem Subtype.mem {α : Type} {s : Set α} (p : s) : (p : α) ∈ s := p.prop #align subtype.mem Subtype.mem theorem Eq.subset {α} {s t : Set α} : s = t → s ⊆ t := fun h₁ _ h₂ => by rw [← h₁]; exact h₂ #align eq.subset Eq.subset namespace Set variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x} {a b : α} {s s₁ s₂ t t₁ t₂ u : Set α} instance : Inhabited (Set α) := ⟨∅⟩ theorem ext_iff {s t : Set α} : s = t ↔ ∀ x, x ∈ s ↔ x ∈ t := ⟨fun h x => by rw [h], ext⟩ #align set.ext_iff Set.ext_iff @[trans] theorem mem_of_mem_of_subset {x : α} {s t : Set α} (hx : x ∈ s) (h : s ⊆ t) : x ∈ t := h hx #align set.mem_of_mem_of_subset Set.mem_of_mem_of_subset theorem forall_in_swap {p : α → β → Prop} : (∀ a ∈ s, ∀ (b), p a b) ↔ ∀ (b), ∀ a ∈ s, p a b := by tauto #align set.forall_in_swap Set.forall_in_swap theorem mem_setOf {a : α} {p : α → Prop} : a ∈ { x \| p x } ↔ p a := Iff.rfl #align set.mem_set_of Set.mem_setOf theorem _root_.Membership.mem.out {p : α → Prop} {a : α} (h : a ∈ { x \| p x }) : p a := h #align has_mem.mem.out Membership.mem.out theorem nmem_setOf_iff {a : α} {p : α → Prop} : a ∉ { x \| p x } ↔ ¬p a := Iff.rfl #align set.nmem_set_of_iff Set.nmem_setOf_iff @[simp] theorem setOf_mem_eq {s : Set α} : { x \| x ∈ s } = s := rfl #align set.set_of_mem_eq Set.setOf_mem_eq theorem setOf_set {s : Set α} : setOf s = s := rfl #align set.set_of_set Set.setOf_set theorem setOf_app_iff {p : α → Prop} {x : α} : { x \| p x } x ↔ p x := Iff.rfl #align set.set_of_app_iff Set.setOf_app_iff theorem mem_def {a : α} {s : Set α} : a ∈ s ↔ s a := Iff.rfl #align set.mem_def Set.mem_def theorem setOf_bijective : Bijective (setOf : (α → Prop) → Set α) := bijective_id #align set.set_of_bijective Set.setOf_bijective theorem subset_setOf {p : α → Prop} {s : Set α} : s ⊆ setOf p ↔ ∀ x, x ∈ s → p x := Iff.rfl theorem setOf_subset {p : α → Prop} {s : Set α} : setOf p ⊆ s ↔ ∀ x, p x → x ∈ s := Iff.rfl @[simp] theorem setOf_subset_setOf {p q : α → Prop} : { a \| p a } ⊆ { a \| q a } ↔ ∀ a, p a → q a := Iff.rfl #align set.set_of_subset_set_of Set.setOf_subset_setOf theorem setOf_and {p q : α → Prop} : { a \| p a ∧ q a } = { a \| p a } ∩ { a \| q a } := rfl #align set.set_of_and Set.setOf_and theorem setOf_or {p q : α → Prop} : { a \| p a ∨ q a } = { a \| p a } ∪ { a \| q a } := rfl #align set.set_of_or Set.setOf_or instance : IsRefl (Set α) (· ⊆ ·) := show IsRefl (Set α) (· ≤ ·) by infer_instance instance : IsTrans (Set α) (· ⊆ ·) := show IsTrans (Set α) (· ≤ ·) by infer_instance instance : Trans ((· ⊆ ·) : Set α → Set α → Prop) (· ⊆ ·) (· ⊆ ·) := show Trans (· ≤ ·) (· ≤ ·) (· ≤ ·) by infer_instance instance : IsAntisymm (Set α) (· ⊆ ·) := show IsAntisymm (Set α) (· ≤ ·) by infer_instance instance : IsIrrefl (Set α) (· ⊂ ·) := show IsIrrefl (Set α) (· < ·) by infer_instance instance : IsTrans (Set α) (· ⊂ ·) := show IsTrans (Set α) (· < ·) by infer_instance instance : Trans ((· ⊂ ·) : Set α → Set α → Prop) (· ⊂ ·) (· ⊂ ·) := show Trans (· < ·) (· < ·) (· < ·) by infer_instance instance : Trans ((· ⊂ ·) : Set α → Set α → Prop) (· ⊆ ·) (· ⊂ ·) := show Trans (· < ·) (· ≤ ·) (· < ·) by infer_instance instance : Trans ((· ⊆ ·) : Set α → Set α → Prop) (· ⊂ ·) (· ⊂ ·) := show Trans (· ≤ ·) (· < ·) (· < ·) by infer_instance instance : IsAsymm (Set α) (· ⊂ ·) := show IsAsymm (Set α) (· < ·) by infer_instance instance : IsNonstrictStrictOrder (Set α) (· ⊆ ·) (· ⊂ ·) := ⟨fun _ _ => Iff.rfl⟩ -- TODO(Jeremy): write a tactic to unfold specific instances of generic notation? theorem subset_def : (s ⊆ t) = ∀ x, x ∈ s → x ∈ t := rfl #align set.subset_def Set.subset_def theorem ssubset_def : (s ⊂ t) = (s ⊆ t ∧ ¬t ⊆ s) := rfl #align set.ssubset_def Set.ssubset_def @[refl] theorem Subset.refl (a : Set α) : a ⊆ a := fun _ => id #align set.subset.refl Set.Subset.refl theorem Subset.rfl {s : Set α} : s ⊆ s := Subset.refl s #align set.subset.rfl Set.Subset.rfl @[trans] theorem Subset.trans {a b c : Set α} (ab : a ⊆ b) (bc : b ⊆ c) : a ⊆ c := fun _ h => bc <\| ab h #align set.subset.trans Set.Subset.trans @[trans] theorem mem_of_eq_of_mem {x y : α} {s : Set α} (hx : x = y) (h : y ∈ s) : x ∈ s := hx.symm ▸ h #align set.mem_of_eq_of_mem Set.mem_of_eq_of_mem theorem Subset.antisymm {a b : Set α} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b := Set.ext fun _ => ⟨@h₁ _, @h₂ _⟩ #align set.subset.antisymm Set.Subset.antisymm theorem Subset.antisymm_iff {a b : Set α} : a = b ↔ a ⊆ b ∧ b ⊆ a := ⟨fun e => ⟨e.subset, e.symm.subset⟩, fun ⟨h₁, h₂⟩ => Subset.antisymm h₁ h₂⟩ #align set.subset.antisymm_iff Set.Subset.antisymm_iff -- an alternative name theorem eq_of_subset_of_subset {a b : Set α} : a ⊆ b → b ⊆ a → a = b := Subset.antisymm #align set.eq_of_subset_of_subset Set.eq_of_subset_of_subset theorem mem_of_subset_of_mem {s₁ s₂ : Set α} {a : α} (h : s₁ ⊆ s₂) : a ∈ s₁ → a ∈ s₂ := @h _ #align set.mem_of_subset_of_mem Set.mem_of_subset_of_mem theorem not_mem_subset (h : s ⊆ t) : a ∉ t → a ∉ s := mt <\| mem_of_subset_of_mem h #align set.not_mem_subset Set.not_mem_subset theorem not_subset : ¬s ⊆ t ↔ ∃ a ∈ s, a ∉ t := by simp only [subset_def, not_forall, exists_prop] #align set.not_subset Set.not_subset lemma eq_of_forall_subset_iff (h : ∀ u, s ⊆ u ↔ t ⊆ u) : s = t := eq_of_forall_ge_iff h protected theorem eq_or_ssubset_of_subset (h : s ⊆ t) : s = t ∨ s ⊂ t := eq_or_lt_of_le h #align set.eq_or_ssubset_of_subset Set.eq_or_ssubset_of_subset theorem exists_of_ssubset {s t : Set α} (h : s ⊂ t) : ∃ x ∈ t, x ∉ s := not_subset.1 h.2 #align set.exists_of_ssubset Set.exists_of_ssubset protected theorem ssubset_iff_subset_ne {s t : Set α} : s ⊂ t ↔ s ⊆ t ∧ s ≠ t := @lt_iff_le_and_ne (Set α) _ s t #align set.ssubset_iff_subset_ne Set.ssubset_iff_subset_ne theorem ssubset_iff_of_subset {s t : Set α} (h : s ⊆ t) : s ⊂ t ↔ ∃ x ∈ t, x ∉ s := ⟨exists_of_ssubset, fun ⟨_, hxt, hxs⟩ => ⟨h, fun h => hxs <\| h hxt⟩⟩ #align set.ssubset_iff_of_subset Set.ssubset_iff_of_subset protected theorem ssubset_of_ssubset_of_subset {s₁ s₂ s₃ : Set α} (hs₁s₂ : s₁ ⊂ s₂) (hs₂s₃ : s₂ ⊆ s₃) : s₁ ⊂ s₃ := ⟨Subset.trans hs₁s₂.1 hs₂s₃, fun hs₃s₁ => hs₁s₂.2 (Subset.trans hs₂s₃ hs₃s₁)⟩ #align set.ssubset_of_ssubset_of_subset Set.ssubset_of_ssubset_of_subset protected theorem ssubset_of_subset_of_ssubset {s₁ s₂ s₃ : Set α} (hs₁s₂ : s₁ ⊆ s₂) (hs₂s₃ : s₂ ⊂ s₃) : s₁ ⊂ s₃ := ⟨Subset.trans hs₁s₂ hs₂s₃.1, fun hs₃s₁ => hs₂s₃.2 (Subset.trans hs₃s₁ hs₁s₂)⟩ #align set.ssubset_of_subset_of_ssubset Set.ssubset_of_subset_of_ssubset theorem not_mem_empty (x : α) : ¬x ∈ (∅ : Set α) := id #align set.not_mem_empty Set.not_mem_empty -- Porting note (#10618): removed `simp` because `simp` can prove it theorem not_not_mem : ¬a ∉ s ↔ a ∈ s := not_not #align set.not_not_mem Set.not_not_mem -- Porting note: we seem to need parentheses at `(↥s)`, -- even if we increase the right precedence of `↥` in `Mathlib.Tactic.Coe`. -- Porting note: removed `simp` as it is competing with `nonempty_subtype`. -- @[simp] theorem nonempty_coe_sort {s : Set α} : Nonempty (↥s) ↔ s.Nonempty := nonempty_subtype #align set.nonempty_coe_sort Set.nonempty_coe_sort alias ⟨_, Nonempty.coe_sort⟩ := nonempty_coe_sort #align set.nonempty.coe_sort Set.Nonempty.coe_sort theorem nonempty_def : s.Nonempty ↔ ∃ x, x ∈ s := Iff.rfl #align set.nonempty_def Set.nonempty_def theorem nonempty_of_mem {x} (h : x ∈ s) : s.Nonempty := ⟨x, h⟩ #align set.nonempty_of_mem Set.nonempty_of_mem theorem Nonempty.not_subset_empty : s.Nonempty → ¬s ⊆ ∅ \| ⟨_, hx⟩, hs => hs hx #align set.nonempty.not_subset_empty Set.Nonempty.not_subset_empty protected noncomputable def Nonempty.some (h : s.Nonempty) : α := Classical.choose h #align set.nonempty.some Set.Nonempty.some protected theorem Nonempty.some_mem (h : s.Nonempty) : h.some ∈ s := Classical.choose_spec h #align set.nonempty.some_mem Set.Nonempty.some_mem theorem Nonempty.mono (ht : s ⊆ t) (hs : s.Nonempty) : t.Nonempty := hs.imp ht #align set.nonempty.mono Set.Nonempty.mono theorem nonempty_of_not_subset (h : ¬s ⊆ t) : (s \ t).Nonempty := let ⟨x, xs, xt⟩ := not_subset.1 h ⟨x, xs, xt⟩ #align set.nonempty_of_not_subset Set.nonempty_of_not_subset theorem nonempty_of_ssubset (ht : s ⊂ t) : (t \ s).Nonempty := nonempty_of_not_subset ht.2 #align set.nonempty_of_ssubset Set.nonempty_of_ssubset theorem Nonempty.of_diff (h : (s \ t).Nonempty) : s.Nonempty := h.imp fun _ => And.left #align set.nonempty.of_diff Set.Nonempty.of_diff theorem nonempty_of_ssubset' (ht : s ⊂ t) : t.Nonempty := (nonempty_of_ssubset ht).of_diff #align set.nonempty_of_ssubset' Set.nonempty_of_ssubset' theorem Nonempty.inl (hs : s.Nonempty) : (s ∪ t).Nonempty := hs.imp fun _ => Or.inl #align set.nonempty.inl Set.Nonempty.inl theorem Nonempty.inr (ht : t.Nonempty) : (s ∪ t).Nonempty := ht.imp fun _ => Or.inr #align set.nonempty.inr Set.Nonempty.inr @[simp] theorem union_nonempty : (s ∪ t).Nonempty ↔ s.Nonempty ∨ t.Nonempty := exists_or #align set.union_nonempty Set.union_nonempty theorem Nonempty.left (h : (s ∩ t).Nonempty) : s.Nonempty := h.imp fun _ => And.left #align set.nonempty.left Set.Nonempty.left theorem Nonempty.right (h : (s ∩ t).Nonempty) : t.Nonempty := h.imp fun _ => And.right #align set.nonempty.right Set.Nonempty.right theorem inter_nonempty : (s ∩ t).Nonempty ↔ ∃ x, x ∈ s ∧ x ∈ t := Iff.rfl #align set.inter_nonempty Set.inter_nonempty theorem inter_nonempty_iff_exists_left : (s ∩ t).Nonempty ↔ ∃ x ∈ s, x ∈ t := by simp_rw [inter_nonempty] #align set.inter_nonempty_iff_exists_left Set.inter_nonempty_iff_exists_left theorem inter_nonempty_iff_exists_right : (s ∩ t).Nonempty ↔ ∃ x ∈ t, x ∈ s := by simp_rw [inter_nonempty, and_comm] #align set.inter_nonempty_iff_exists_right Set.inter_nonempty_iff_exists_right theorem nonempty_iff_univ_nonempty : Nonempty α ↔ (univ : Set α).Nonempty := ⟨fun ⟨x⟩ => ⟨x, trivial⟩, fun ⟨x, _⟩ => ⟨x⟩⟩ #align set.nonempty_iff_univ_nonempty Set.nonempty_iff_univ_nonempty @[simp] theorem univ_nonempty : ∀ [Nonempty α], (univ : Set α).Nonempty \| ⟨x⟩ => ⟨x, trivial⟩ #align set.univ_nonempty Set.univ_nonempty theorem Nonempty.to_subtype : s.Nonempty → Nonempty (↥s) := nonempty_subtype.2 #align set.nonempty.to_subtype Set.Nonempty.to_subtype theorem Nonempty.to_type : s.Nonempty → Nonempty α := fun ⟨x, _⟩ => ⟨x⟩ #align set.nonempty.to_type Set.Nonempty.to_type instance univ.nonempty [Nonempty α] : Nonempty (↥(Set.univ : Set α)) := Set.univ_nonempty.to_subtype #align set.univ.nonempty Set.univ.nonempty theorem nonempty_of_nonempty_subtype [Nonempty (↥s)] : s.Nonempty := nonempty_subtype.mp ‹_› #align set.nonempty_of_nonempty_subtype Set.nonempty_of_nonempty_subtype theorem empty_def : (∅ : Set α) = { _x : α \| False } := rfl #align set.empty_def Set.empty_def @[simp] theorem mem_empty_iff_false (x : α) : x ∈ (∅ : Set α) ↔ False := Iff.rfl #align set.mem_empty_iff_false Set.mem_empty_iff_false @[simp] theorem setOf_false : { _a : α \| False } = ∅ := rfl #align set.set_of_false Set.setOf_false @[simp] theorem setOf_bot : { _x : α \| ⊥ } = ∅ := rfl @[simp] theorem empty_subset (s : Set α) : ∅ ⊆ s := nofun #align set.empty_subset Set.empty_subset theorem subset_empty_iff {s : Set α} : s ⊆ ∅ ↔ s = ∅ := (Subset.antisymm_iff.trans <\| and_iff_left (empty_subset _)).symm #align set.subset_empty_iff Set.subset_empty_iff theorem eq_empty_iff_forall_not_mem {s : Set α} : s = ∅ ↔ ∀ x, x ∉ s := subset_empty_iff.symm #align set.eq_empty_iff_forall_not_mem Set.eq_empty_iff_forall_not_mem theorem eq_empty_of_forall_not_mem (h : ∀ x, x ∉ s) : s = ∅ := subset_empty_iff.1 h #align set.eq_empty_of_forall_not_mem Set.eq_empty_of_forall_not_mem theorem eq_empty_of_subset_empty {s : Set α} : s ⊆ ∅ → s = ∅ := subset_empty_iff.1 #align set.eq_empty_of_subset_empty Set.eq_empty_of_subset_empty theorem eq_empty_of_isEmpty [IsEmpty α] (s : Set α) : s = ∅ := eq_empty_of_subset_empty fun x _ => isEmptyElim x #align set.eq_empty_of_is_empty Set.eq_empty_of_isEmpty instance uniqueEmpty [IsEmpty α] : Unique (Set α) where default := ∅ uniq := eq_empty_of_isEmpty #align set.unique_empty Set.uniqueEmpty theorem not_nonempty_iff_eq_empty {s : Set α} : ¬s.Nonempty ↔ s = ∅ := by simp only [Set.Nonempty, not_exists, eq_empty_iff_forall_not_mem] #align set.not_nonempty_iff_eq_empty Set.not_nonempty_iff_eq_empty theorem nonempty_iff_ne_empty : s.Nonempty ↔ s ≠ ∅ := not_nonempty_iff_eq_empty.not_right #align set.nonempty_iff_ne_empty Set.nonempty_iff_ne_empty theorem not_nonempty_iff_eq_empty' : ¬Nonempty s ↔ s = ∅ := by rw [nonempty_subtype, not_exists, eq_empty_iff_forall_not_mem] theorem nonempty_iff_ne_empty' : Nonempty s ↔ s ≠ ∅ := not_nonempty_iff_eq_empty'.not_right alias ⟨Nonempty.ne_empty, _⟩ := nonempty_iff_ne_empty #align set.nonempty.ne_empty Set.Nonempty.ne_empty @[simp] theorem not_nonempty_empty : ¬(∅ : Set α).Nonempty := fun ⟨_, hx⟩ => hx #align set.not_nonempty_empty Set.not_nonempty_empty -- Porting note: removing `@[simp]` as it is competing with `isEmpty_subtype`. -- @[simp] theorem isEmpty_coe_sort {s : Set α} : IsEmpty (↥s) ↔ s = ∅ := not_iff_not.1 <\| by simpa using nonempty_iff_ne_empty #align set.is_empty_coe_sort Set.isEmpty_coe_sort theorem eq_empty_or_nonempty (s : Set α) : s = ∅ ∨ s.Nonempty := or_iff_not_imp_left.2 nonempty_iff_ne_empty.2 #align set.eq_empty_or_nonempty Set.eq_empty_or_nonempty theorem subset_eq_empty {s t : Set α} (h : t ⊆ s) (e : s = ∅) : t = ∅ := subset_empty_iff.1 <\| e ▸ h #align set.subset_eq_empty Set.subset_eq_empty theorem forall_mem_empty {p : α → Prop} : (∀ x ∈ (∅ : Set α), p x) ↔ True := iff_true_intro fun _ => False.elim #align set.ball_empty_iff Set.forall_mem_empty @[deprecated (since := "2024-03-23")] alias ball_empty_iff := forall_mem_empty instance (α : Type u) : IsEmpty.{u + 1} (↥(∅ : Set α)) := ⟨fun x => x.2⟩ @[simp] theorem empty_ssubset : ∅ ⊂ s ↔ s.Nonempty := (@bot_lt_iff_ne_bot (Set α) _ _ _).trans nonempty_iff_ne_empty.symm #align set.empty_ssubset Set.empty_ssubset alias ⟨_, Nonempty.empty_ssubset⟩ := empty_ssubset #align set.nonempty.empty_ssubset Set.Nonempty.empty_ssubset @[simp] theorem setOf_true : { _x : α \| True } = univ := rfl #align set.set_of_true Set.setOf_true @[simp] theorem setOf_top : { _x : α \| ⊤ } = univ := rfl @[simp] theorem univ_eq_empty_iff : (univ : Set α) = ∅ ↔ IsEmpty α := eq_empty_iff_forall_not_mem.trans ⟨fun H => ⟨fun x => H x trivial⟩, fun H x _ => @IsEmpty.false α H x⟩ #align set.univ_eq_empty_iff Set.univ_eq_empty_iff theorem empty_ne_univ [Nonempty α] : (∅ : Set α) ≠ univ := fun e => not_isEmpty_of_nonempty α <\| univ_eq_empty_iff.1 e.symm #align set.empty_ne_univ Set.empty_ne_univ @[simp] theorem subset_univ (s : Set α) : s ⊆ univ := fun _ _ => trivial #align set.subset_univ Set.subset_univ @[simp] theorem univ_subset_iff {s : Set α} : univ ⊆ s ↔ s = univ := @top_le_iff _ _ _ s #align set.univ_subset_iff Set.univ_subset_iff alias ⟨eq_univ_of_univ_subset, _⟩ := univ_subset_iff #align set.eq_univ_of_univ_subset Set.eq_univ_of_univ_subset theorem eq_univ_iff_forall {s : Set α} : s = univ ↔ ∀ x, x ∈ s := univ_subset_iff.symm.trans <\| forall_congr' fun _ => imp_iff_right trivial #align set.eq_univ_iff_forall Set.eq_univ_iff_forall theorem eq_univ_of_forall {s : Set α} : (∀ x, x ∈ s) → s = univ := eq_univ_iff_forall.2 #align set.eq_univ_of_forall Set.eq_univ_of_forall theorem Nonempty.eq_univ [Subsingleton α] : s.Nonempty → s = univ := by rintro ⟨x, hx⟩ exact eq_univ_of_forall fun y => by rwa [Subsingleton.elim y x] #align set.nonempty.eq_univ Set.Nonempty.eq_univ theorem eq_univ_of_subset {s t : Set α} (h : s ⊆ t) (hs : s = univ) : t = univ := eq_univ_of_univ_subset <\| (hs ▸ h : univ ⊆ t) #align set.eq_univ_of_subset Set.eq_univ_of_subset theorem exists_mem_of_nonempty (α) : ∀ [Nonempty α], ∃ x : α, x ∈ (univ : Set α) \| ⟨x⟩ => ⟨x, trivial⟩ #align set.exists_mem_of_nonempty Set.exists_mem_of_nonempty theorem ne_univ_iff_exists_not_mem {α : Type} (s : Set α) : s ≠ univ ↔ ∃ a, a ∉ s := by rw [← not_forall, ← eq_univ_iff_forall] #align set.ne_univ_iff_exists_not_mem Set.ne_univ_iff_exists_not_mem theorem not_subset_iff_exists_mem_not_mem {α : Type} {s t : Set α} : ¬s ⊆ t ↔ ∃ x, x ∈ s ∧ x ∉ t := by simp [subset_def] #align set.not_subset_iff_exists_mem_not_mem Set.not_subset_iff_exists_mem_not_mem theorem univ_unique [Unique α] : @Set.univ α = {default} := Set.ext fun x => iff_of_true trivial <\| Subsingleton.elim x default #align set.univ_unique Set.univ_unique theorem ssubset_univ_iff : s ⊂ univ ↔ s ≠ univ := lt_top_iff_ne_top #align set.ssubset_univ_iff Set.ssubset_univ_iff instance nontrivial_of_nonempty [Nonempty α] : Nontrivial (Set α) := ⟨⟨∅, univ, empty_ne_univ⟩⟩ #align set.nontrivial_of_nonempty Set.nontrivial_of_nonempty theorem union_def {s₁ s₂ : Set α} : s₁ ∪ s₂ = { a \| a ∈ s₁ ∨ a ∈ s₂ } := rfl #align set.union_def Set.union_def theorem mem_union_left {x : α} {a : Set α} (b : Set α) : x ∈ a → x ∈ a ∪ b := Or.inl #align set.mem_union_left Set.mem_union_left theorem mem_union_right {x : α} {b : Set α} (a : Set α) : x ∈ b → x ∈ a ∪ b := Or.inr #align set.mem_union_right Set.mem_union_right theorem mem_or_mem_of_mem_union {x : α} {a b : Set α} (H : x ∈ a ∪ b) : x ∈ a ∨ x ∈ b := H #align set.mem_or_mem_of_mem_union Set.mem_or_mem_of_mem_union theorem MemUnion.elim {x : α} {a b : Set α} {P : Prop} (H₁ : x ∈ a ∪ b) (H₂ : x ∈ a → P) (H₃ : x ∈ b → P) : P := Or.elim H₁ H₂ H₃ #align set.mem_union.elim Set.MemUnion.elim @[simp] theorem mem_union (x : α) (a b : Set α) : x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b := Iff.rfl #align set.mem_union Set.mem_union @[simp] theorem union_self (a : Set α) : a ∪ a = a := ext fun _ => or_self_iff #align set.union_self Set.union_self @[simp] theorem union_empty (a : Set α) : a ∪ ∅ = a := ext fun _ => or_false_iff _ #align set.union_empty Set.union_empty @[simp] theorem empty_union (a : Set α) : ∅ ∪ a = a := ext fun _ => false_or_iff _ #align set.empty_union Set.empty_union theorem union_comm (a b : Set α) : a ∪ b = b ∪ a := ext fun _ => or_comm #align set.union_comm Set.union_comm theorem union_assoc (a b c : Set α) : a ∪ b ∪ c = a ∪ (b ∪ c) := ext fun _ => or_assoc #align set.union_assoc Set.union_assoc instance union_isAssoc : Std.Associative (α := Set α) (· ∪ ·) := ⟨union_assoc⟩ #align set.union_is_assoc Set.union_isAssoc instance union_isComm : Std.Commutative (α := Set α) (· ∪ ·) := ⟨union_comm⟩ #align set.union_is_comm Set.union_isComm theorem union_left_comm (s₁ s₂ s₃ : Set α) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) := ext fun _ => or_left_comm #align set.union_left_comm Set.union_left_comm theorem union_right_comm (s₁ s₂ s₃ : Set α) : s₁ ∪ s₂ ∪ s₃ = s₁ ∪ s₃ ∪ s₂ := ext fun _ => or_right_comm #align set.union_right_comm Set.union_right_comm @[simp] theorem union_eq_left {s t : Set α} : s ∪ t = s ↔ t ⊆ s := sup_eq_left #align set.union_eq_left_iff_subset Set.union_eq_left @[simp] theorem union_eq_right {s t : Set α} : s ∪ t = t ↔ s ⊆ t := sup_eq_right #align set.union_eq_right_iff_subset Set.union_eq_right theorem union_eq_self_of_subset_left {s t : Set α} (h : s ⊆ t) : s ∪ t = t := union_eq_right.mpr h #align set.union_eq_self_of_subset_left Set.union_eq_self_of_subset_left theorem union_eq_self_of_subset_right {s t : Set α} (h : t ⊆ s) : s ∪ t = s := union_eq_left.mpr h #align set.union_eq_self_of_subset_right Set.union_eq_self_of_subset_right @[simp] theorem subset_union_left {s t : Set α} : s ⊆ s ∪ t := fun _ => Or.inl #align set.subset_union_left Set.subset_union_left @[simp] theorem subset_union_right {s t : Set α} : t ⊆ s ∪ t := fun _ => Or.inr #align set.subset_union_right Set.subset_union_right theorem union_subset {s t r : Set α} (sr : s ⊆ r) (tr : t ⊆ r) : s ∪ t ⊆ r := fun _ => Or.rec (@sr _) (@tr _) #align set.union_subset Set.union_subset @[simp] theorem union_subset_iff {s t u : Set α} : s ∪ t ⊆ u ↔ s ⊆ u ∧ t ⊆ u := (forall_congr' fun _ => or_imp).trans forall_and #align set.union_subset_iff Set.union_subset_iff @[gcongr] theorem union_subset_union {s₁ s₂ t₁ t₂ : Set α} (h₁ : s₁ ⊆ s₂) (h₂ : t₁ ⊆ t₂) : s₁ ∪ t₁ ⊆ s₂ ∪ t₂ := fun _ => Or.imp (@h₁ _) (@h₂ _) #align set.union_subset_union Set.union_subset_union @[gcongr] theorem union_subset_union_left {s₁ s₂ : Set α} (t) (h : s₁ ⊆ s₂) : s₁ ∪ t ⊆ s₂ ∪ t := union_subset_union h Subset.rfl #align set.union_subset_union_left Set.union_subset_union_left @[gcongr] theorem union_subset_union_right (s) {t₁ t₂ : Set α} (h : t₁ ⊆ t₂) : s ∪ t₁ ⊆ s ∪ t₂ := union_subset_union Subset.rfl h #align set.union_subset_union_right Set.union_subset_union_right theorem subset_union_of_subset_left {s t : Set α} (h : s ⊆ t) (u : Set α) : s ⊆ t ∪ u := h.trans subset_union_left #align set.subset_union_of_subset_left Set.subset_union_of_subset_left theorem subset_union_of_subset_right {s u : Set α} (h : s ⊆ u) (t : Set α) : s ⊆ t ∪ u := h.trans subset_union_right #align set.subset_union_of_subset_right Set.subset_union_of_subset_right -- Porting note: replaced `⊔` in RHS theorem union_congr_left (ht : t ⊆ s ∪ u) (hu : u ⊆ s ∪ t) : s ∪ t = s ∪ u := sup_congr_left ht hu #align set.union_congr_left Set.union_congr_left theorem union_congr_right (hs : s ⊆ t ∪ u) (ht : t ⊆ s ∪ u) : s ∪ u = t ∪ u := sup_congr_right hs ht #align set.union_congr_right Set.union_congr_right theorem union_eq_union_iff_left : s ∪ t = s ∪ u ↔ t ⊆ s ∪ u ∧ u ⊆ s ∪ t := sup_eq_sup_iff_left #align set.union_eq_union_iff_left Set.union_eq_union_iff_left theorem union_eq_union_iff_right : s ∪ u = t ∪ u ↔ s ⊆ t ∪ u ∧ t ⊆ s ∪ u := sup_eq_sup_iff_right #align set.union_eq_union_iff_right Set.union_eq_union_iff_right @[simp] theorem union_empty_iff {s t : Set α} : s ∪ t = ∅ ↔ s = ∅ ∧ t = ∅ := by simp only [← subset_empty_iff] exact union_subset_iff #align set.union_empty_iff Set.union_empty_iff @[simp] theorem union_univ (s : Set α) : s ∪ univ = univ := sup_top_eq _ #align set.union_univ Set.union_univ @[simp] theorem univ_union (s : Set α) : univ ∪ s = univ := top_sup_eq _ #align set.univ_union Set.univ_union theorem inter_def {s₁ s₂ : Set α} : s₁ ∩ s₂ = { a \| a ∈ s₁ ∧ a ∈ s₂ } := rfl #align set.inter_def Set.inter_def @[simp, mfld_simps] theorem mem_inter_iff (x : α) (a b : Set α) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b := Iff.rfl #align set.mem_inter_iff Set.mem_inter_iff theorem mem_inter {x : α} {a b : Set α} (ha : x ∈ a) (hb : x ∈ b) : x ∈ a ∩ b := ⟨ha, hb⟩ #align set.mem_inter Set.mem_inter theorem mem_of_mem_inter_left {x : α} {a b : Set α} (h : x ∈ a ∩ b) : x ∈ a := h.left #align set.mem_of_mem_inter_left Set.mem_of_mem_inter_left theorem mem_of_mem_inter_right {x : α} {a b : Set α} (h : x ∈ a ∩ b) : x ∈ b := h.right #align set.mem_of_mem_inter_right Set.mem_of_mem_inter_right @[simp] theorem inter_self (a : Set α) : a ∩ a = a := ext fun _ => and_self_iff #align set.inter_self Set.inter_self @[simp] theorem inter_empty (a : Set α) : a ∩ ∅ = ∅ := ext fun _ => and_false_iff _ #align set.inter_empty Set.inter_empty @[simp] theorem empty_inter (a : Set α) : ∅ ∩ a = ∅ := ext fun _ => false_and_iff _ #align set.empty_inter Set.empty_inter theorem inter_comm (a b : Set α) : a ∩ b = b ∩ a := ext fun _ => and_comm #align set.inter_comm Set.inter_comm theorem inter_assoc (a b c : Set α) : a ∩ b ∩ c = a ∩ (b ∩ c) := ext fun _ => and_assoc #align set.inter_assoc Set.inter_assoc instance inter_isAssoc : Std.Associative (α := Set α) (· ∩ ·) := ⟨inter_assoc⟩ #align set.inter_is_assoc Set.inter_isAssoc instance inter_isComm : Std.Commutative (α := Set α) (· ∩ ·) := ⟨inter_comm⟩ #align set.inter_is_comm Set.inter_isComm theorem inter_left_comm (s₁ s₂ s₃ : Set α) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) := ext fun _ => and_left_comm #align set.inter_left_comm Set.inter_left_comm theorem inter_right_comm (s₁ s₂ s₃ : Set α) : s₁ ∩ s₂ ∩ s₃ = s₁ ∩ s₃ ∩ s₂ := ext fun _ => and_right_comm #align set.inter_right_comm Set.inter_right_comm @[simp, mfld_simps] theorem inter_subset_left {s t : Set α} : s ∩ t ⊆ s := fun _ => And.left #align set.inter_subset_left Set.inter_subset_left @[simp] theorem inter_subset_right {s t : Set α} : s ∩ t ⊆ t := fun _ => And.right #align set.inter_subset_right Set.inter_subset_right theorem subset_inter {s t r : Set α} (rs : r ⊆ s) (rt : r ⊆ t) : r ⊆ s ∩ t := fun _ h => ⟨rs h, rt h⟩ #align set.subset_inter Set.subset_inter @[simp] theorem subset_inter_iff {s t r : Set α} : r ⊆ s ∩ t ↔ r ⊆ s ∧ r ⊆ t := (forall_congr' fun _ => imp_and).trans forall_and #align set.subset_inter_iff Set.subset_inter_iff @[simp] lemma inter_eq_left : s ∩ t = s ↔ s ⊆ t := inf_eq_left #align set.inter_eq_left_iff_subset Set.inter_eq_left @[simp] lemma inter_eq_right : s ∩ t = t ↔ t ⊆ s := inf_eq_right #align set.inter_eq_right_iff_subset Set.inter_eq_right @[simp] lemma left_eq_inter : s = s ∩ t ↔ s ⊆ t := left_eq_inf @[simp] lemma right_eq_inter : t = s ∩ t ↔ t ⊆ s := right_eq_inf theorem inter_eq_self_of_subset_left {s t : Set α} : s ⊆ t → s ∩ t = s := inter_eq_left.mpr #align set.inter_eq_self_of_subset_left Set.inter_eq_self_of_subset_left theorem inter_eq_self_of_subset_right {s t : Set α} : t ⊆ s → s ∩ t = t := inter_eq_right.mpr #align set.inter_eq_self_of_subset_right Set.inter_eq_self_of_subset_right theorem inter_congr_left (ht : s ∩ u ⊆ t) (hu : s ∩ t ⊆ u) : s ∩ t = s ∩ u := inf_congr_left ht hu #align set.inter_congr_left Set.inter_congr_left theorem inter_congr_right (hs : t ∩ u ⊆ s) (ht : s ∩ u ⊆ t) : s ∩ u = t ∩ u := inf_congr_right hs ht #align set.inter_congr_right Set.inter_congr_right theorem inter_eq_inter_iff_left : s ∩ t = s ∩ u ↔ s ∩ u ⊆ t ∧ s ∩ t ⊆ u := inf_eq_inf_iff_left #align set.inter_eq_inter_iff_left Set.inter_eq_inter_iff_left theorem inter_eq_inter_iff_right : s ∩ u = t ∩ u ↔ t ∩ u ⊆ s ∧ s ∩ u ⊆ t := inf_eq_inf_iff_right #align set.inter_eq_inter_iff_right Set.inter_eq_inter_iff_right @[simp, mfld_simps] theorem inter_univ (a : Set α) : a ∩ univ = a := inf_top_eq _ #align set.inter_univ Set.inter_univ @[simp, mfld_simps] theorem univ_inter (a : Set α) : univ ∩ a = a := top_inf_eq _ #align set.univ_inter Set.univ_inter @[gcongr] theorem inter_subset_inter {s₁ s₂ t₁ t₂ : Set α} (h₁ : s₁ ⊆ t₁) (h₂ : s₂ ⊆ t₂) : s₁ ∩ s₂ ⊆ t₁ ∩ t₂ := fun _ => And.imp (@h₁ _) (@h₂ _) #align set.inter_subset_inter Set.inter_subset_inter @[gcongr] theorem inter_subset_inter_left {s t : Set α} (u : Set α) (H : s ⊆ t) : s ∩ u ⊆ t ∩ u := inter_subset_inter H Subset.rfl #align set.inter_subset_inter_left Set.inter_subset_inter_left @[gcongr] theorem inter_subset_inter_right {s t : Set α} (u : Set α) (H : s ⊆ t) : u ∩ s ⊆ u ∩ t := inter_subset_inter Subset.rfl H #align set.inter_subset_inter_right Set.inter_subset_inter_right theorem union_inter_cancel_left {s t : Set α} : (s ∪ t) ∩ s = s := inter_eq_self_of_subset_right subset_union_left #align set.union_inter_cancel_left Set.union_inter_cancel_left theorem union_inter_cancel_right {s t : Set α} : (s ∪ t) ∩ t = t := inter_eq_self_of_subset_right subset_union_right #align set.union_inter_cancel_right Set.union_inter_cancel_right theorem inter_setOf_eq_sep (s : Set α) (p : α → Prop) : s ∩ {a \| p a} = {a ∈ s \| p a} := rfl #align set.inter_set_of_eq_sep Set.inter_setOf_eq_sep theorem setOf_inter_eq_sep (p : α → Prop) (s : Set α) : {a \| p a} ∩ s = {a ∈ s \| p a} := inter_comm _ _ #align set.set_of_inter_eq_sep Set.setOf_inter_eq_sep theorem inter_union_distrib_left (s t u : Set α) : s ∩ (t ∪ u) = s ∩ t ∪ s ∩ u := inf_sup_left _ _ _ #align set.inter_distrib_left Set.inter_union_distrib_left theorem union_inter_distrib_right (s t u : Set α) : (s ∪ t) ∩ u = s ∩ u ∪ t ∩ u := inf_sup_right _ _ _ #align set.inter_distrib_right Set.union_inter_distrib_right theorem union_inter_distrib_left (s t u : Set α) : s ∪ t ∩ u = (s ∪ t) ∩ (s ∪ u) := sup_inf_left _ _ _ #align set.union_distrib_left Set.union_inter_distrib_left theorem inter_union_distrib_right (s t u : Set α) : s ∩ t ∪ u = (s ∪ u) ∩ (t ∪ u) := sup_inf_right _ _ _ #align set.union_distrib_right Set.inter_union_distrib_right -- 2024-03-22 @[deprecated] alias inter_distrib_left := inter_union_distrib_left @[deprecated] alias inter_distrib_right := union_inter_distrib_right @[deprecated] alias union_distrib_left := union_inter_distrib_left @[deprecated] alias union_distrib_right := inter_union_distrib_right theorem union_union_distrib_left (s t u : Set α) : s ∪ (t ∪ u) = s ∪ t ∪ (s ∪ u) := sup_sup_distrib_left _ _ _ #align set.union_union_distrib_left Set.union_union_distrib_left theorem union_union_distrib_right (s t u : Set α) : s ∪ t ∪ u = s ∪ u ∪ (t ∪ u) := sup_sup_distrib_right _ _ _ #align set.union_union_distrib_right Set.union_union_distrib_right theorem inter_inter_distrib_left (s t u : Set α) : s ∩ (t ∩ u) = s ∩ t ∩ (s ∩ u) := inf_inf_distrib_left _ _ _ #align set.inter_inter_distrib_left Set.inter_inter_distrib_left theorem inter_inter_distrib_right (s t u : Set α) : s ∩ t ∩ u = s ∩ u ∩ (t ∩ u) := inf_inf_distrib_right _ _ _ #align set.inter_inter_distrib_right Set.inter_inter_distrib_right theorem union_union_union_comm (s t u v : Set α) : s ∪ t ∪ (u ∪ v) = s ∪ u ∪ (t ∪ v) := sup_sup_sup_comm _ _ _ _ #align set.union_union_union_comm Set.union_union_union_comm theorem inter_inter_inter_comm (s t u v : Set α) : s ∩ t ∩ (u ∩ v) = s ∩ u ∩ (t ∩ v) := inf_inf_inf_comm _ _ _ _ #align set.inter_inter_inter_comm Set.inter_inter_inter_comm theorem insert_def (x : α) (s : Set α) : insert x s = { y \| y = x ∨ y ∈ s } := rfl #align set.insert_def Set.insert_def @[simp] theorem subset_insert (x : α) (s : Set α) : s ⊆ insert x s := fun _ => Or.inr #align set.subset_insert Set.subset_insert theorem mem_insert (x : α) (s : Set α) : x ∈ insert x s := Or.inl rfl #align set.mem_insert Set.mem_insert theorem mem_insert_of_mem {x : α} {s : Set α} (y : α) : x ∈ s → x ∈ insert y s := Or.inr #align set.mem_insert_of_mem Set.mem_insert_of_mem theorem eq_or_mem_of_mem_insert {x a : α} {s : Set α} : x ∈ insert a s → x = a ∨ x ∈ s := id #align set.eq_or_mem_of_mem_insert Set.eq_or_mem_of_mem_insert theorem mem_of_mem_insert_of_ne : b ∈ insert a s → b ≠ a → b ∈ s := Or.resolve_left #align set.mem_of_mem_insert_of_ne Set.mem_of_mem_insert_of_ne theorem eq_of_not_mem_of_mem_insert : b ∈ insert a s → b ∉ s → b = a := Or.resolve_right #align set.eq_of_not_mem_of_mem_insert Set.eq_of_not_mem_of_mem_insert @[simp] theorem mem_insert_iff {x a : α} {s : Set α} : x ∈ insert a s ↔ x = a ∨ x ∈ s := Iff.rfl #align set.mem_insert_iff Set.mem_insert_iff @[simp] theorem insert_eq_of_mem {a : α} {s : Set α} (h : a ∈ s) : insert a s = s := ext fun _ => or_iff_right_of_imp fun e => e.symm ▸ h #align set.insert_eq_of_mem Set.insert_eq_of_mem theorem ne_insert_of_not_mem {s : Set α} (t : Set α) {a : α} : a ∉ s → s ≠ insert a t := mt fun e => e.symm ▸ mem_insert _ _ #align set.ne_insert_of_not_mem Set.ne_insert_of_not_mem @[simp] theorem insert_eq_self : insert a s = s ↔ a ∈ s := ⟨fun h => h ▸ mem_insert _ _, insert_eq_of_mem⟩ #align set.insert_eq_self Set.insert_eq_self theorem insert_ne_self : insert a s ≠ s ↔ a ∉ s := insert_eq_self.not #align set.insert_ne_self Set.insert_ne_self theorem insert_subset_iff : insert a s ⊆ t ↔ a ∈ t ∧ s ⊆ t := by simp only [subset_def, mem_insert_iff, or_imp, forall_and, forall_eq] #align set.insert_subset Set.insert_subset_iff theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t := insert_subset_iff.mpr ⟨ha, hs⟩ theorem insert_subset_insert (h : s ⊆ t) : insert a s ⊆ insert a t := fun _ => Or.imp_right (@h _) #align set.insert_subset_insert Set.insert_subset_insert @[simp] theorem insert_subset_insert_iff (ha : a ∉ s) : insert a s ⊆ insert a t ↔ s ⊆ t := by refine ⟨fun h x hx => ?_, insert_subset_insert⟩ rcases h (subset_insert _ _ hx) with (rfl \| hxt) exacts [(ha hx).elim, hxt] #align set.insert_subset_insert_iff Set.insert_subset_insert_iff theorem subset_insert_iff_of_not_mem (ha : a ∉ s) : s ⊆ insert a t ↔ s ⊆ t := forall₂_congr fun _ hb => or_iff_right <\| ne_of_mem_of_not_mem hb ha #align set.subset_insert_iff_of_not_mem Set.subset_insert_iff_of_not_mem theorem ssubset_iff_insert {s t : Set α} : s ⊂ t ↔ ∃ a ∉ s, insert a s ⊆ t := by simp only [insert_subset_iff, exists_and_right, ssubset_def, not_subset] aesop #align set.ssubset_iff_insert Set.ssubset_iff_insert theorem ssubset_insert {s : Set α} {a : α} (h : a ∉ s) : s ⊂ insert a s := ssubset_iff_insert.2 ⟨a, h, Subset.rfl⟩ #align set.ssubset_insert Set.ssubset_insert theorem insert_comm (a b : α) (s : Set α) : insert a (insert b s) = insert b (insert a s) := ext fun _ => or_left_comm #align set.insert_comm Set.insert_comm -- Porting note (#10618): removing `simp` attribute because `simp` can prove it theorem insert_idem (a : α) (s : Set α) : insert a (insert a s) = insert a s := insert_eq_of_mem <\| mem_insert _ _ #align set.insert_idem Set.insert_idem theorem insert_union : insert a s ∪ t = insert a (s ∪ t) := ext fun _ => or_assoc #align set.insert_union Set.insert_union @[simp] theorem union_insert : s ∪ insert a t = insert a (s ∪ t) := ext fun _ => or_left_comm #align set.union_insert Set.union_insert @[simp] theorem insert_nonempty (a : α) (s : Set α) : (insert a s).Nonempty := ⟨a, mem_insert a s⟩ #align set.insert_nonempty Set.insert_nonempty instance (a : α) (s : Set α) : Nonempty (insert a s : Set α) := (insert_nonempty a s).to_subtype theorem insert_inter_distrib (a : α) (s t : Set α) : insert a (s ∩ t) = insert a s ∩ insert a t := ext fun _ => or_and_left #align set.insert_inter_distrib Set.insert_inter_distrib theorem insert_union_distrib (a : α) (s t : Set α) : insert a (s ∪ t) = insert a s ∪ insert a t := ext fun _ => or_or_distrib_left #align set.insert_union_distrib Set.insert_union_distrib theorem insert_inj (ha : a ∉ s) : insert a s = insert b s ↔ a = b := ⟨fun h => eq_of_not_mem_of_mem_insert (h.subst <\| mem_insert a s) ha, congr_arg (fun x => insert x s)⟩ #align set.insert_inj Set.insert_inj -- useful in proofs by induction theorem forall_of_forall_insert {P : α → Prop} {a : α} {s : Set α} (H : ∀ x, x ∈ insert a s → P x) (x) (h : x ∈ s) : P x := H _ (Or.inr h) #align set.forall_of_forall_insert Set.forall_of_forall_insert theorem forall_insert_of_forall {P : α → Prop} {a : α} {s : Set α} (H : ∀ x, x ∈ s → P x) (ha : P a) (x) (h : x ∈ insert a s) : P x := h.elim (fun e => e.symm ▸ ha) (H _) #align set.forall_insert_of_forall Set.forall_insert_of_forall theorem exists_mem_insert {P : α → Prop} {a : α} {s : Set α} : (∃ x ∈ insert a s, P x) ↔ (P a ∨ ∃ x ∈ s, P x) := by simp [mem_insert_iff, or_and_right, exists_and_left, exists_or] #align set.bex_insert_iff Set.exists_mem_insert @[deprecated (since := "2024-03-23")] alias bex_insert_iff := exists_mem_insert theorem forall_mem_insert {P : α → Prop} {a : α} {s : Set α} : (∀ x ∈ insert a s, P x) ↔ P a ∧ ∀ x ∈ s, P x := forall₂_or_left.trans <\| and_congr_left' forall_eq #align set.ball_insert_iff Set.forall_mem_insert @[deprecated (since := "2024-03-23")] alias ball_insert_iff := forall_mem_insert instance : LawfulSingleton α (Set α) := ⟨fun x => Set.ext fun a => by simp only [mem_empty_iff_false, mem_insert_iff, or_false] exact Iff.rfl⟩ theorem singleton_def (a : α) : ({a} : Set α) = insert a ∅ := (insert_emptyc_eq a).symm #align set.singleton_def Set.singleton_def @[simp] theorem mem_singleton_iff {a b : α} : a ∈ ({b} : Set α) ↔ a = b := Iff.rfl #align set.mem_singleton_iff Set.mem_singleton_iff @[simp] theorem setOf_eq_eq_singleton {a : α} : { n \| n = a } = {a} := rfl #align set.set_of_eq_eq_singleton Set.setOf_eq_eq_singleton @[simp] theorem setOf_eq_eq_singleton' {a : α} : { x \| a = x } = {a} := ext fun _ => eq_comm #align set.set_of_eq_eq_singleton' Set.setOf_eq_eq_singleton' -- TODO: again, annotation needed --Porting note (#11119): removed `simp` attribute theorem mem_singleton (a : α) : a ∈ ({a} : Set α) := @rfl _ _ #align set.mem_singleton Set.mem_singleton theorem eq_of_mem_singleton {x y : α} (h : x ∈ ({y} : Set α)) : x = y := h #align set.eq_of_mem_singleton Set.eq_of_mem_singleton @[simp] theorem singleton_eq_singleton_iff {x y : α} : {x} = ({y} : Set α) ↔ x = y := ext_iff.trans eq_iff_eq_cancel_left #align set.singleton_eq_singleton_iff Set.singleton_eq_singleton_iff theorem singleton_injective : Injective (singleton : α → Set α) := fun _ _ => singleton_eq_singleton_iff.mp #align set.singleton_injective Set.singleton_injective theorem mem_singleton_of_eq {x y : α} (H : x = y) : x ∈ ({y} : Set α) := H #align set.mem_singleton_of_eq Set.mem_singleton_of_eq theorem insert_eq (x : α) (s : Set α) : insert x s = ({x} : Set α) ∪ s := rfl #align set.insert_eq Set.insert_eq @[simp] theorem singleton_nonempty (a : α) : ({a} : Set α).Nonempty := ⟨a, rfl⟩ #align set.singleton_nonempty Set.singleton_nonempty @[simp] theorem singleton_ne_empty (a : α) : ({a} : Set α) ≠ ∅ := (singleton_nonempty _).ne_empty #align set.singleton_ne_empty Set.singleton_ne_empty --Porting note (#10618): removed `simp` attribute because `simp` can prove it theorem empty_ssubset_singleton : (∅ : Set α) ⊂ {a} := (singleton_nonempty _).empty_ssubset #align set.empty_ssubset_singleton Set.empty_ssubset_singleton @[simp] theorem singleton_subset_iff {a : α} {s : Set α} : {a} ⊆ s ↔ a ∈ s := forall_eq #align set.singleton_subset_iff Set.singleton_subset_iff theorem singleton_subset_singleton : ({a} : Set α) ⊆ {b} ↔ a = b := by simp #align set.singleton_subset_singleton Set.singleton_subset_singleton theorem set_compr_eq_eq_singleton {a : α} : { b \| b = a } = {a} := rfl #align set.set_compr_eq_eq_singleton Set.set_compr_eq_eq_singleton @[simp] theorem singleton_union : {a} ∪ s = insert a s := rfl #align set.singleton_union Set.singleton_union @[simp] theorem union_singleton : s ∪ {a} = insert a s := union_comm _ _ #align set.union_singleton Set.union_singleton @[simp] theorem singleton_inter_nonempty : ({a} ∩ s).Nonempty ↔ a ∈ s := by simp only [Set.Nonempty, mem_inter_iff, mem_singleton_iff, exists_eq_left] #align set.singleton_inter_nonempty Set.singleton_inter_nonempty @[simp] theorem inter_singleton_nonempty : (s ∩ {a}).Nonempty ↔ a ∈ s := by rw [inter_comm, singleton_inter_nonempty] #align set.inter_singleton_nonempty Set.inter_singleton_nonempty @[simp] theorem singleton_inter_eq_empty : {a} ∩ s = ∅ ↔ a ∉ s := not_nonempty_iff_eq_empty.symm.trans singleton_inter_nonempty.not #align set.singleton_inter_eq_empty Set.singleton_inter_eq_empty @[simp] theorem inter_singleton_eq_empty : s ∩ {a} = ∅ ↔ a ∉ s := by rw [inter_comm, singleton_inter_eq_empty] #align set.inter_singleton_eq_empty Set.inter_singleton_eq_empty theorem nmem_singleton_empty {s : Set α} : s ∉ ({∅} : Set (Set α)) ↔ s.Nonempty := nonempty_iff_ne_empty.symm #align set.nmem_singleton_empty Set.nmem_singleton_empty instance uniqueSingleton (a : α) : Unique (↥({a} : Set α)) := ⟨⟨⟨a, mem_singleton a⟩⟩, fun ⟨_, h⟩ => Subtype.eq h⟩ #align set.unique_singleton Set.uniqueSingleton theorem eq_singleton_iff_unique_mem : s = {a} ↔ a ∈ s ∧ ∀ x ∈ s, x = a := Subset.antisymm_iff.trans <\| and_comm.trans <\| and_congr_left' singleton_subset_iff #align set.eq_singleton_iff_unique_mem Set.eq_singleton_iff_unique_mem theorem eq_singleton_iff_nonempty_unique_mem : s = {a} ↔ s.Nonempty ∧ ∀ x ∈ s, x = a := eq_singleton_iff_unique_mem.trans <\| and_congr_left fun H => ⟨fun h' => ⟨_, h'⟩, fun ⟨x, h⟩ => H x h ▸ h⟩ #align set.eq_singleton_iff_nonempty_unique_mem Set.eq_singleton_iff_nonempty_unique_mem set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532 -- while `simp` is capable of proving this, it is not capable of turning the LHS into the RHS. @[simp] theorem default_coe_singleton (x : α) : (default : ({x} : Set α)) = ⟨x, rfl⟩ := rfl #align set.default_coe_singleton Set.default_coe_singleton @[simp] theorem subset_singleton_iff {α : Type} {s : Set α} {x : α} : s ⊆ {x} ↔ ∀ y ∈ s, y = x := Iff.rfl #align set.subset_singleton_iff Set.subset_singleton_iff theorem subset_singleton_iff_eq {s : Set α} {x : α} : s ⊆ {x} ↔ s = ∅ ∨ s = {x} := by obtain rfl \| hs := s.eq_empty_or_nonempty · exact ⟨fun _ => Or.inl rfl, fun _ => empty_subset _⟩ · simp [eq_singleton_iff_nonempty_unique_mem, hs, hs.ne_empty] #align set.subset_singleton_iff_eq Set.subset_singleton_iff_eq theorem Nonempty.subset_singleton_iff (h : s.Nonempty) : s ⊆ {a} ↔ s = {a} := subset_singleton_iff_eq.trans <\| or_iff_right h.ne_empty #align set.nonempty.subset_singleton_iff Set.Nonempty.subset_singleton_iff theorem ssubset_singleton_iff {s : Set α} {x : α} : s ⊂ {x} ↔ s = ∅ := by rw [ssubset_iff_subset_ne, subset_singleton_iff_eq, or_and_right, and_not_self_iff, or_false_iff, and_iff_left_iff_imp] exact fun h => h ▸ (singleton_ne_empty _).symm #align set.ssubset_singleton_iff Set.ssubset_singleton_iff theorem eq_empty_of_ssubset_singleton {s : Set α} {x : α} (hs : s ⊂ {x}) : s = ∅ := ssubset_singleton_iff.1 hs #align set.eq_empty_of_ssubset_singleton Set.eq_empty_of_ssubset_singleton theorem eq_of_nonempty_of_subsingleton {α} [Subsingleton α] (s t : Set α) [Nonempty s] [Nonempty t] : s = t := nonempty_of_nonempty_subtype.eq_univ.trans nonempty_of_nonempty_subtype.eq_univ.symm theorem eq_of_nonempty_of_subsingleton' {α} [Subsingleton α] {s : Set α} (t : Set α) (hs : s.Nonempty) [Nonempty t] : s = t := have := hs.to_subtype; eq_of_nonempty_of_subsingleton s t set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532 theorem Nonempty.eq_zero [Subsingleton α] [Zero α] {s : Set α} (h : s.Nonempty) : s = {0} := eq_of_nonempty_of_subsingleton' {0} h set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532 theorem Nonempty.eq_one [Subsingleton α] [One α] {s : Set α} (h : s.Nonempty) : s = {1} := eq_of_nonempty_of_subsingleton' {1} h protected theorem disjoint_iff : Disjoint s t ↔ s ∩ t ⊆ ∅ := disjoint_iff_inf_le #align set.disjoint_iff Set.disjoint_iff theorem disjoint_iff_inter_eq_empty : Disjoint s t ↔ s ∩ t = ∅ := disjoint_iff #align set.disjoint_iff_inter_eq_empty Set.disjoint_iff_inter_eq_empty theorem _root_.Disjoint.inter_eq : Disjoint s t → s ∩ t = ∅ := Disjoint.eq_bot #align disjoint.inter_eq Disjoint.inter_eq theorem disjoint_left : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → a ∉ t := disjoint_iff_inf_le.trans <\| forall_congr' fun _ => not_and #align set.disjoint_left Set.disjoint_left theorem disjoint_right : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ t → a ∉ s := by rw [disjoint_comm, disjoint_left] #align set.disjoint_right Set.disjoint_right lemma not_disjoint_iff : ¬Disjoint s t ↔ ∃ x, x ∈ s ∧ x ∈ t := Set.disjoint_iff.not.trans <\| not_forall.trans <\| exists_congr fun _ ↦ not_not #align set.not_disjoint_iff Set.not_disjoint_iff lemma not_disjoint_iff_nonempty_inter : ¬ Disjoint s t ↔ (s ∩ t).Nonempty := not_disjoint_iff #align set.not_disjoint_iff_nonempty_inter Set.not_disjoint_iff_nonempty_inter alias ⟨_, Nonempty.not_disjoint⟩ := not_disjoint_iff_nonempty_inter #align set.nonempty.not_disjoint Set.Nonempty.not_disjoint lemma disjoint_or_nonempty_inter (s t : Set α) : Disjoint s t ∨ (s ∩ t).Nonempty := (em _).imp_right not_disjoint_iff_nonempty_inter.1 #align set.disjoint_or_nonempty_inter Set.disjoint_or_nonempty_inter lemma disjoint_iff_forall_ne : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ t → a ≠ b := by simp only [Ne, disjoint_left, @imp_not_comm _ (_ = _), forall_eq'] #align set.disjoint_iff_forall_ne Set.disjoint_iff_forall_ne alias ⟨_root_.Disjoint.ne_of_mem, _⟩ := disjoint_iff_forall_ne #align disjoint.ne_of_mem Disjoint.ne_of_mem lemma disjoint_of_subset_left (h : s ⊆ u) (d : Disjoint u t) : Disjoint s t := d.mono_left h #align set.disjoint_of_subset_left Set.disjoint_of_subset_left lemma disjoint_of_subset_right (h : t ⊆ u) (d : Disjoint s u) : Disjoint s t := d.mono_right h #align set.disjoint_of_subset_right Set.disjoint_of_subset_right lemma disjoint_of_subset (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (h : Disjoint s₂ t₂) : Disjoint s₁ t₁ := h.mono hs ht #align set.disjoint_of_subset Set.disjoint_of_subset @[simp] lemma disjoint_union_left : Disjoint (s ∪ t) u ↔ Disjoint s u ∧ Disjoint t u := disjoint_sup_left #align set.disjoint_union_left Set.disjoint_union_left @[simp] lemma disjoint_union_right : Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u := disjoint_sup_right #align set.disjoint_union_right Set.disjoint_union_right @[simp] lemma disjoint_empty (s : Set α) : Disjoint s ∅ := disjoint_bot_right #align set.disjoint_empty Set.disjoint_empty @[simp] lemma empty_disjoint (s : Set α) : Disjoint ∅ s := disjoint_bot_left #align set.empty_disjoint Set.empty_disjoint @[simp] lemma univ_disjoint : Disjoint univ s ↔ s = ∅ := top_disjoint #align set.univ_disjoint Set.univ_disjoint @[simp] lemma disjoint_univ : Disjoint s univ ↔ s = ∅ := disjoint_top #align set.disjoint_univ Set.disjoint_univ lemma disjoint_sdiff_left : Disjoint (t \ s) s := disjoint_sdiff_self_left #align set.disjoint_sdiff_left Set.disjoint_sdiff_left lemma disjoint_sdiff_right : Disjoint s (t \ s) := disjoint_sdiff_self_right #align set.disjoint_sdiff_right Set.disjoint_sdiff_right -- TODO: prove this in terms of a lattice lemma theorem disjoint_sdiff_inter : Disjoint (s \ t) (s ∩ t) := disjoint_of_subset_right inter_subset_right disjoint_sdiff_left #align set.disjoint_sdiff_inter Set.disjoint_sdiff_inter theorem diff_union_diff_cancel (hts : t ⊆ s) (hut : u ⊆ t) : s \ t ∪ t \ u = s \ u := sdiff_sup_sdiff_cancel hts hut #align set.diff_union_diff_cancel Set.diff_union_diff_cancel theorem diff_diff_eq_sdiff_union (h : u ⊆ s) : s \ (t \ u) = s \ t ∪ u := sdiff_sdiff_eq_sdiff_sup h #align set.diff_diff_eq_sdiff_union Set.diff_diff_eq_sdiff_union @[simp default+1] lemma disjoint_singleton_left : Disjoint {a} s ↔ a ∉ s := by simp [Set.disjoint_iff, subset_def] #align set.disjoint_singleton_left Set.disjoint_singleton_left @[simp] lemma disjoint_singleton_right : Disjoint s {a} ↔ a ∉ s := disjoint_comm.trans disjoint_singleton_left #align set.disjoint_singleton_right Set.disjoint_singleton_right lemma disjoint_singleton : Disjoint ({a} : Set α) {b} ↔ a ≠ b := by simp #align set.disjoint_singleton Set.disjoint_singleton lemma subset_diff : s ⊆ t \ u ↔ s ⊆ t ∧ Disjoint s u := le_iff_subset.symm.trans le_sdiff #align set.subset_diff Set.subset_diff lemma ssubset_iff_sdiff_singleton : s ⊂ t ↔ ∃ a ∈ t, s ⊆ t \ {a} := by simp [ssubset_iff_insert, subset_diff, insert_subset_iff]; aesop theorem inter_diff_distrib_left (s t u : Set α) : s ∩ (t \ u) = (s ∩ t) \ (s ∩ u) := inf_sdiff_distrib_left _ _ _ #align set.inter_diff_distrib_left Set.inter_diff_distrib_left theorem inter_diff_distrib_right (s t u : Set α) : s \ t ∩ u = (s ∩ u) \ (t ∩ u) := inf_sdiff_distrib_right _ _ _ #align set.inter_diff_distrib_right Set.inter_diff_distrib_right theorem compl_def (s : Set α) : sᶜ = { x \| x ∉ s } := rfl #align set.compl_def Set.compl_def theorem mem_compl {s : Set α} {x : α} (h : x ∉ s) : x ∈ sᶜ := h #align set.mem_compl Set.mem_compl theorem compl_setOf {α} (p : α → Prop) : { a \| p a }ᶜ = { a \| ¬p a } := rfl #align set.compl_set_of Set.compl_setOf theorem not_mem_of_mem_compl {s : Set α} {x : α} (h : x ∈ sᶜ) : x ∉ s := h #align set.not_mem_of_mem_compl Set.not_mem_of_mem_compl theorem not_mem_compl_iff {x : α} : x ∉ sᶜ ↔ x ∈ s := not_not #align set.not_mem_compl_iff Set.not_mem_compl_iff @[simp] theorem inter_compl_self (s : Set α) : s ∩ sᶜ = ∅ := inf_compl_eq_bot #align set.inter_compl_self Set.inter_compl_self @[simp] theorem compl_inter_self (s : Set α) : sᶜ ∩ s = ∅ := compl_inf_eq_bot #align set.compl_inter_self Set.compl_inter_self @[simp] theorem compl_empty : (∅ : Set α)ᶜ = univ := compl_bot #align set.compl_empty Set.compl_empty @[simp] theorem compl_union (s t : Set α) : (s ∪ t)ᶜ = sᶜ ∩ tᶜ := compl_sup #align set.compl_union Set.compl_union theorem compl_inter (s t : Set α) : (s ∩ t)ᶜ = sᶜ ∪ tᶜ := compl_inf #align set.compl_inter Set.compl_inter @[simp] theorem compl_univ : (univ : Set α)ᶜ = ∅ := compl_top #align set.compl_univ Set.compl_univ @[simp] theorem compl_empty_iff {s : Set α} : sᶜ = ∅ ↔ s = univ := compl_eq_bot #align set.compl_empty_iff Set.compl_empty_iff @[simp] theorem compl_univ_iff {s : Set α} : sᶜ = univ ↔ s = ∅ := compl_eq_top #align set.compl_univ_iff Set.compl_univ_iff theorem compl_ne_univ : sᶜ ≠ univ ↔ s.Nonempty := compl_univ_iff.not.trans nonempty_iff_ne_empty.symm #align set.compl_ne_univ Set.compl_ne_univ theorem nonempty_compl : sᶜ.Nonempty ↔ s ≠ univ := (ne_univ_iff_exists_not_mem s).symm #align set.nonempty_compl Set.nonempty_compl @[simp] lemma nonempty_compl_of_nontrivial [Nontrivial α] (x : α) : Set.Nonempty {x}ᶜ := by obtain ⟨y, hy⟩ := exists_ne x exact ⟨y, by simp [hy]⟩ theorem mem_compl_singleton_iff {a x : α} : x ∈ ({a} : Set α)ᶜ ↔ x ≠ a := Iff.rfl #align set.mem_compl_singleton_iff Set.mem_compl_singleton_iff theorem compl_singleton_eq (a : α) : ({a} : Set α)ᶜ = { x \| x ≠ a } := rfl #align set.compl_singleton_eq Set.compl_singleton_eq @[simp] theorem compl_ne_eq_singleton (a : α) : ({ x \| x ≠ a } : Set α)ᶜ = {a} := compl_compl _ #align set.compl_ne_eq_singleton Set.compl_ne_eq_singleton theorem union_eq_compl_compl_inter_compl (s t : Set α) : s ∪ t = (sᶜ ∩ tᶜ)ᶜ := ext fun _ => or_iff_not_and_not #align set.union_eq_compl_compl_inter_compl Set.union_eq_compl_compl_inter_compl theorem inter_eq_compl_compl_union_compl (s t : Set α) : s ∩ t = (sᶜ ∪ tᶜ)ᶜ := ext fun _ => and_iff_not_or_not #align set.inter_eq_compl_compl_union_compl Set.inter_eq_compl_compl_union_compl @[simp] theorem union_compl_self (s : Set α) : s ∪ sᶜ = univ := eq_univ_iff_forall.2 fun _ => em _ #align set.union_compl_self Set.union_compl_self @[simp] theorem compl_union_self (s : Set α) : sᶜ ∪ s = univ := by rw [union_comm, union_compl_self] #align set.compl_union_self Set.compl_union_self theorem compl_subset_comm : sᶜ ⊆ t ↔ tᶜ ⊆ s := @compl_le_iff_compl_le _ s _ _ #align set.compl_subset_comm Set.compl_subset_comm theorem subset_compl_comm : s ⊆ tᶜ ↔ t ⊆ sᶜ := @le_compl_iff_le_compl _ _ _ t #align set.subset_compl_comm Set.subset_compl_comm @[simp] theorem compl_subset_compl : sᶜ ⊆ tᶜ ↔ t ⊆ s := @compl_le_compl_iff_le (Set α) _ _ _ #align set.compl_subset_compl Set.compl_subset_compl @[gcongr] theorem compl_subset_compl_of_subset (h : t ⊆ s) : sᶜ ⊆ tᶜ := compl_subset_compl.2 h theorem subset_compl_iff_disjoint_left : s ⊆ tᶜ ↔ Disjoint t s := @le_compl_iff_disjoint_left (Set α) _ _ _ #align set.subset_compl_iff_disjoint_left Set.subset_compl_iff_disjoint_left theorem subset_compl_iff_disjoint_right : s ⊆ tᶜ ↔ Disjoint s t := @le_compl_iff_disjoint_right (Set α) _ _ _ #align set.subset_compl_iff_disjoint_right Set.subset_compl_iff_disjoint_right theorem disjoint_compl_left_iff_subset : Disjoint sᶜ t ↔ t ⊆ s := disjoint_compl_left_iff #align set.disjoint_compl_left_iff_subset Set.disjoint_compl_left_iff_subset theorem disjoint_compl_right_iff_subset : Disjoint s tᶜ ↔ s ⊆ t := disjoint_compl_right_iff #align set.disjoint_compl_right_iff_subset Set.disjoint_compl_right_iff_subset alias ⟨_, _root_.Disjoint.subset_compl_right⟩ := subset_compl_iff_disjoint_right #align disjoint.subset_compl_right Disjoint.subset_compl_right alias ⟨_, _root_.Disjoint.subset_compl_left⟩ := subset_compl_iff_disjoint_left #align disjoint.subset_compl_left Disjoint.subset_compl_left alias ⟨_, _root_.HasSubset.Subset.disjoint_compl_left⟩ := disjoint_compl_left_iff_subset #align has_subset.subset.disjoint_compl_left HasSubset.Subset.disjoint_compl_left alias ⟨_, _root_.HasSubset.Subset.disjoint_compl_right⟩ := disjoint_compl_right_iff_subset #align has_subset.subset.disjoint_compl_right HasSubset.Subset.disjoint_compl_right theorem subset_union_compl_iff_inter_subset {s t u : Set α} : s ⊆ t ∪ uᶜ ↔ s ∩ u ⊆ t := (@isCompl_compl _ u _).le_sup_right_iff_inf_left_le #align set.subset_union_compl_iff_inter_subset Set.subset_union_compl_iff_inter_subset theorem compl_subset_iff_union {s t : Set α} : sᶜ ⊆ t ↔ s ∪ t = univ := Iff.symm <\| eq_univ_iff_forall.trans <\| forall_congr' fun _ => or_iff_not_imp_left #align set.compl_subset_iff_union Set.compl_subset_iff_union @[simp] theorem subset_compl_singleton_iff {a : α} {s : Set α} : s ⊆ {a}ᶜ ↔ a ∉ s := subset_compl_comm.trans singleton_subset_iff #align set.subset_compl_singleton_iff Set.subset_compl_singleton_iff theorem inter_subset (a b c : Set α) : a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c := forall_congr' fun _ => and_imp.trans <\| imp_congr_right fun _ => imp_iff_not_or #align set.inter_subset Set.inter_subset theorem inter_compl_nonempty_iff {s t : Set α} : (s ∩ tᶜ).Nonempty ↔ ¬s ⊆ t := (not_subset.trans <\| exists_congr fun x => by simp [mem_compl]).symm #align set.inter_compl_nonempty_iff Set.inter_compl_nonempty_iff theorem not_mem_diff_of_mem {s t : Set α} {x : α} (hx : x ∈ t) : x ∉ s \ t := fun h => h.2 hx #align set.not_mem_diff_of_mem Set.not_mem_diff_of_mem theorem mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∈ s := h.left #align set.mem_of_mem_diff Set.mem_of_mem_diff theorem not_mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∉ t := h.right #align set.not_mem_of_mem_diff Set.not_mem_of_mem_diff theorem diff_eq_compl_inter {s t : Set α} : s \ t = tᶜ ∩ s := by rw [diff_eq, inter_comm] #align set.diff_eq_compl_inter Set.diff_eq_compl_inter theorem nonempty_diff {s t : Set α} : (s \ t).Nonempty ↔ ¬s ⊆ t := inter_compl_nonempty_iff #align set.nonempty_diff Set.nonempty_diff theorem diff_subset {s t : Set α} : s \ t ⊆ s := show s \ t ≤ s from sdiff_le #align set.diff_subset Set.diff_subset theorem diff_subset_compl (s t : Set α) : s \ t ⊆ tᶜ := diff_eq_compl_inter ▸ inter_subset_left theorem union_diff_cancel' {s t u : Set α} (h₁ : s ⊆ t) (h₂ : t ⊆ u) : t ∪ u \ s = u := sup_sdiff_cancel' h₁ h₂ #align set.union_diff_cancel' Set.union_diff_cancel' theorem union_diff_cancel {s t : Set α} (h : s ⊆ t) : s ∪ t \ s = t := sup_sdiff_cancel_right h #align set.union_diff_cancel Set.union_diff_cancel theorem union_diff_cancel_left {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ s = t := Disjoint.sup_sdiff_cancel_left <\| disjoint_iff_inf_le.2 h #align set.union_diff_cancel_left Set.union_diff_cancel_left theorem union_diff_cancel_right {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ t = s := Disjoint.sup_sdiff_cancel_right <\| disjoint_iff_inf_le.2 h #align set.union_diff_cancel_right Set.union_diff_cancel_right @[simp] theorem union_diff_left {s t : Set α} : (s ∪ t) \ s = t \ s := sup_sdiff_left_self #align set.union_diff_left Set.union_diff_left @[simp] theorem union_diff_right {s t : Set α} : (s ∪ t) \ t = s \ t := sup_sdiff_right_self #align set.union_diff_right Set.union_diff_right theorem union_diff_distrib {s t u : Set α} : (s ∪ t) \ u = s \ u ∪ t \ u := sup_sdiff #align set.union_diff_distrib Set.union_diff_distrib theorem inter_diff_assoc (a b c : Set α) : (a ∩ b) \ c = a ∩ (b \ c) := inf_sdiff_assoc #align set.inter_diff_assoc Set.inter_diff_assoc @[simp] theorem inter_diff_self (a b : Set α) : a ∩ (b \ a) = ∅ := inf_sdiff_self_right #align set.inter_diff_self Set.inter_diff_self @[simp] theorem inter_union_diff (s t : Set α) : s ∩ t ∪ s \ t = s := sup_inf_sdiff s t #align set.inter_union_diff Set.inter_union_diff @[simp] theorem diff_union_inter (s t : Set α) : s \ t ∪ s ∩ t = s := by rw [union_comm] exact sup_inf_sdiff _ _ #align set.diff_union_inter Set.diff_union_inter @[simp] theorem inter_union_compl (s t : Set α) : s ∩ t ∪ s ∩ tᶜ = s := inter_union_diff _ _ #align set.inter_union_compl Set.inter_union_compl @[gcongr] theorem diff_subset_diff {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ s₂ → t₂ ⊆ t₁ → s₁ \ t₁ ⊆ s₂ \ t₂ := show s₁ ≤ s₂ → t₂ ≤ t₁ → s₁ \ t₁ ≤ s₂ \ t₂ from sdiff_le_sdiff #align set.diff_subset_diff Set.diff_subset_diff @[gcongr] theorem diff_subset_diff_left {s₁ s₂ t : Set α} (h : s₁ ⊆ s₂) : s₁ \ t ⊆ s₂ \ t := sdiff_le_sdiff_right ‹s₁ ≤ s₂› #align set.diff_subset_diff_left Set.diff_subset_diff_left @[gcongr] theorem diff_subset_diff_right {s t u : Set α} (h : t ⊆ u) : s \ u ⊆ s \ t := sdiff_le_sdiff_left ‹t ≤ u› #align set.diff_subset_diff_right Set.diff_subset_diff_right theorem compl_eq_univ_diff (s : Set α) : sᶜ = univ \ s := top_sdiff.symm #align set.compl_eq_univ_diff Set.compl_eq_univ_diff @[simp] theorem empty_diff (s : Set α) : (∅ \ s : Set α) = ∅ := bot_sdiff #align set.empty_diff Set.empty_diff theorem diff_eq_empty {s t : Set α} : s \ t = ∅ ↔ s ⊆ t := sdiff_eq_bot_iff #align set.diff_eq_empty Set.diff_eq_empty @[simp] theorem diff_empty {s : Set α} : s \ ∅ = s := sdiff_bot #align set.diff_empty Set.diff_empty @[simp] theorem diff_univ (s : Set α) : s \ univ = ∅ := diff_eq_empty.2 (subset_univ s) #align set.diff_univ Set.diff_univ theorem diff_diff {u : Set α} : (s \ t) \ u = s \ (t ∪ u) := sdiff_sdiff_left #align set.diff_diff Set.diff_diff -- the following statement contains parentheses to help the reader theorem diff_diff_comm {s t u : Set α} : (s \ t) \ u = (s \ u) \ t := sdiff_sdiff_comm #align set.diff_diff_comm Set.diff_diff_comm theorem diff_subset_iff {s t u : Set α} : s \ t ⊆ u ↔ s ⊆ t ∪ u := show s \ t ≤ u ↔ s ≤ t ∪ u from sdiff_le_iff #align set.diff_subset_iff Set.diff_subset_iff theorem subset_diff_union (s t : Set α) : s ⊆ s \ t ∪ t := show s ≤ s \ t ∪ t from le_sdiff_sup #align set.subset_diff_union Set.subset_diff_union theorem diff_union_of_subset {s t : Set α} (h : t ⊆ s) : s \ t ∪ t = s := Subset.antisymm (union_subset diff_subset h) (subset_diff_union _ _) #align set.diff_union_of_subset Set.diff_union_of_subset @[simp] theorem diff_singleton_subset_iff {x : α} {s t : Set α} : s \ {x} ⊆ t ↔ s ⊆ insert x t := by rw [← union_singleton, union_comm] apply diff_subset_iff #align set.diff_singleton_subset_iff Set.diff_singleton_subset_iff theorem subset_diff_singleton {x : α} {s t : Set α} (h : s ⊆ t) (hx : x ∉ s) : s ⊆ t \ {x} := subset_inter h <\| subset_compl_comm.1 <\| singleton_subset_iff.2 hx #align set.subset_diff_singleton Set.subset_diff_singleton theorem subset_insert_diff_singleton (x : α) (s : Set α) : s ⊆ insert x (s \ {x}) := by rw [← diff_singleton_subset_iff] #align set.subset_insert_diff_singleton Set.subset_insert_diff_singleton theorem diff_subset_comm {s t u : Set α} : s \ t ⊆ u ↔ s \ u ⊆ t := show s \ t ≤ u ↔ s \ u ≤ t from sdiff_le_comm #align set.diff_subset_comm Set.diff_subset_comm theorem diff_inter {s t u : Set α} : s \ (t ∩ u) = s \ t ∪ s \ u := sdiff_inf #align set.diff_inter Set.diff_inter theorem diff_inter_diff {s t u : Set α} : s \ t ∩ (s \ u) = s \ (t ∪ u) := sdiff_sup.symm #align set.diff_inter_diff Set.diff_inter_diff theorem diff_compl : s \ tᶜ = s ∩ t := sdiff_compl #align set.diff_compl Set.diff_compl theorem diff_diff_right {s t u : Set α} : s \ (t \ u) = s \ t ∪ s ∩ u := sdiff_sdiff_right' #align set.diff_diff_right Set.diff_diff_right @[simp] theorem insert_diff_of_mem (s) (h : a ∈ t) : insert a s \ t = s \ t := by ext constructor <;> simp (config := { contextual := true }) [or_imp, h] #align set.insert_diff_of_mem Set.insert_diff_of_mem theorem insert_diff_of_not_mem (s) (h : a ∉ t) : insert a s \ t = insert a (s \ t) := by classical ext x by_cases h' : x ∈ t · have : x ≠ a := by intro H rw [H] at h' exact h h' simp [h, h', this] · simp [h, h'] #align set.insert_diff_of_not_mem Set.insert_diff_of_not_mem	Mathlib/Data/Set/Basic.lean	1,958	1,960	theorem insert_diff_self_of_not_mem {a : α} {s : Set α} (h : a ∉ s) : insert a s \ {a} = s := by	ext x simp [and_iff_left_of_imp fun hx : x ∈ s => show x ≠ a from fun hxa => h <\| hxa ▸ hx]
import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Degree.Lemmas import Mathlib.LinearAlgebra.Matrix.Determinant.Basic import Mathlib.Tactic.ComputeDegree #align_import linear_algebra.matrix.polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" set_option linter.uppercaseLean3 false open Matrix Polynomial variable {n α : Type*} [DecidableEq n] [Fintype n] [CommRing α] open Polynomial Matrix Equiv.Perm namespace Polynomial	Mathlib/LinearAlgebra/Matrix/Polynomial.lean	39	59	theorem natDegree_det_X_add_C_le (A B : Matrix n n α) : natDegree (det ((X : α[X]) • A.map C + B.map C : Matrix n n α[X])) ≤ Fintype.card n := by	rw [det_apply] refine (natDegree_sum_le _ _).trans ?_ refine Multiset.max_le_of_forall_le _ _ ?_ simp only [forall_apply_eq_imp_iff, true_and_iff, Function.comp_apply, Multiset.map_map, Multiset.mem_map, exists_imp, Finset.mem_univ_val] intro g calc natDegree (sign g • ∏ i : n, (X • A.map C + B.map C : Matrix n n α[X]) (g i) i) ≤ natDegree (∏ i : n, (X • A.map C + B.map C : Matrix n n α[X]) (g i) i) := by cases' Int.units_eq_one_or (sign g) with sg sg · rw [sg, one_smul] · rw [sg, Units.neg_smul, one_smul, natDegree_neg] _ ≤ ∑ i : n, natDegree (((X : α[X]) • A.map C + B.map C : Matrix n n α[X]) (g i) i) := (natDegree_prod_le (Finset.univ : Finset n) fun i : n => (X • A.map C + B.map C : Matrix n n α[X]) (g i) i) _ ≤ Finset.univ.card • 1 := (Finset.sum_le_card_nsmul _ _ 1 fun (i : n) _ => ?_) _ ≤ Fintype.card n := by simp [mul_one, Algebra.id.smul_eq_mul, Finset.card_univ] dsimp only [add_apply, smul_apply, map_apply, smul_eq_mul] compute_degree

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.