Split (1)

train · 42.2k rows

Context stringlengths 285 157k	file_name stringlengths 21 79	start int64 14 3.67k	end int64 18 3.69k	theorem stringlengths 25 2.71k	proof stringlengths 5 10.6k
/- Copyright (c) 2022 Junyan Xu. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Junyan Xu -/ import Mathlib.Data.Sym.Sym2 import Mathlib.Logic.Relation #align_import order.game_add from "leanprover-community/mathlib"@"fee218fb033b2fd390c447f8be27754bc9093be9" /-! # Game addition relation This file defines, given relations `rα : α → α → Prop` and `rβ : β → β → Prop`, a relation `Prod.GameAdd` on pairs, such that `GameAdd rα rβ x y` iff `x` can be reached from `y` by decreasing either entry (with respect to `rα` and `rβ`). It is so called since it models the subsequency relation on the addition of combinatorial games. We also define `Sym2.GameAdd`, which is the unordered pair analog of `Prod.GameAdd`. ## Main definitions and results - `Prod.GameAdd`: the game addition relation on ordered pairs. - `WellFounded.prod_gameAdd`: formalizes induction on ordered pairs, where exactly one entry decreases at a time. - `Sym2.GameAdd`: the game addition relation on unordered pairs. - `WellFounded.sym2_gameAdd`: formalizes induction on unordered pairs, where exactly one entry decreases at a time. -/ set_option autoImplicit true variable {α β : Type} {rα : α → α → Prop} {rβ : β → β → Prop} /-! ### `Prod.GameAdd` -/ namespace Prod variable (rα rβ) /-- `Prod.GameAdd rα rβ x y` means that `x` can be reached from `y` by decreasing either entry with respect to the relations `rα` and `rβ`. It is so called, as it models game addition within combinatorial game theory. If `rα a₁ a₂` means that `a₂ ⟶ a₁` is a valid move in game `α`, and `rβ b₁ b₂` means that `b₂ ⟶ b₁` is a valid move in game `β`, then `GameAdd rα rβ` specifies the valid moves in the juxtaposition of `α` and `β`: the player is free to choose one of the games and make a move in it, while leaving the other game unchanged. See `Sym2.GameAdd` for the unordered pair analog. -/ inductive GameAdd : α × β → α × β → Prop \| fst {a₁ a₂ b} : rα a₁ a₂ → GameAdd (a₁, b) (a₂, b) \| snd {a b₁ b₂} : rβ b₁ b₂ → GameAdd (a, b₁) (a, b₂) #align prod.game_add Prod.GameAdd theorem gameAdd_iff {rα rβ} {x y : α × β} : GameAdd rα rβ x y ↔ rα x.1 y.1 ∧ x.2 = y.2 ∨ rβ x.2 y.2 ∧ x.1 = y.1 := by constructor · rintro (@⟨a₁, a₂, b, h⟩ \| @⟨a, b₁, b₂, h⟩) exacts [Or.inl ⟨h, rfl⟩, Or.inr ⟨h, rfl⟩] · revert x y rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (⟨h, rfl : b₁ = b₂⟩ \| ⟨h, rfl : a₁ = a₂⟩) exacts [GameAdd.fst h, GameAdd.snd h] #align prod.game_add_iff Prod.gameAdd_iff theorem gameAdd_mk_iff {rα rβ} {a₁ a₂ : α} {b₁ b₂ : β} : GameAdd rα rβ (a₁, b₁) (a₂, b₂) ↔ rα a₁ a₂ ∧ b₁ = b₂ ∨ rβ b₁ b₂ ∧ a₁ = a₂ := gameAdd_iff #align prod.game_add_mk_iff Prod.gameAdd_mk_iff @[simp] theorem gameAdd_swap_swap : ∀ a b : α × β, GameAdd rβ rα a.swap b.swap ↔ GameAdd rα rβ a b := fun ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ => by rw [Prod.swap, Prod.swap, gameAdd_mk_iff, gameAdd_mk_iff, or_comm] #align prod.game_add_swap_swap Prod.gameAdd_swap_swap theorem gameAdd_swap_swap_mk (a₁ a₂ : α) (b₁ b₂ : β) : GameAdd rα rβ (a₁, b₁) (a₂, b₂) ↔ GameAdd rβ rα (b₁, a₁) (b₂, a₂) := gameAdd_swap_swap rβ rα (b₁, a₁) (b₂, a₂) #align prod.game_add_swap_swap_mk Prod.gameAdd_swap_swap_mk /-- `Prod.GameAdd` is a subrelation of `Prod.Lex`. -/ theorem gameAdd_le_lex : GameAdd rα rβ ≤ Prod.Lex rα rβ := fun _ _ h => h.rec (Prod.Lex.left _ _) (Prod.Lex.right _) #align prod.game_add_le_lex Prod.gameAdd_le_lex /-- `Prod.RProd` is a subrelation of the transitive closure of `Prod.GameAdd`. -/ theorem rprod_le_transGen_gameAdd : RProd rα rβ ≤ Relation.TransGen (GameAdd rα rβ) \| _, _, h => h.rec (by intro _ _ _ _ hα hβ exact Relation.TransGen.tail (Relation.TransGen.single <\| GameAdd.fst hα) (GameAdd.snd hβ)) #align prod.rprod_le_trans_gen_game_add Prod.rprod_le_transGen_gameAdd end Prod /-- If `a` is accessible under `rα` and `b` is accessible under `rβ`, then `(a, b)` is accessible under `Prod.GameAdd rα rβ`. Notice that `Prod.lexAccessible` requires the stronger condition `∀ b, Acc rβ b`. -/ theorem Acc.prod_gameAdd (ha : Acc rα a) (hb : Acc rβ b) : Acc (Prod.GameAdd rα rβ) (a, b) := by induction' ha with a _ iha generalizing b induction' hb with b hb ihb refine Acc.intro _ fun h => ?_ rintro (⟨ra⟩ \| ⟨rb⟩) exacts [iha _ ra (Acc.intro b hb), ihb _ rb] #align acc.prod_game_add Acc.prod_gameAdd /-- The `Prod.GameAdd` relation on well-founded inputs is well-founded. In particular, the sum of two well-founded games is well-founded. -/ theorem WellFounded.prod_gameAdd (hα : WellFounded rα) (hβ : WellFounded rβ) : WellFounded (Prod.GameAdd rα rβ) := ⟨fun ⟨a, b⟩ => (hα.apply a).prod_gameAdd (hβ.apply b)⟩ #align well_founded.prod_game_add WellFounded.prod_gameAdd namespace Prod /-- Recursion on the well-founded `Prod.GameAdd` relation. Note that it's strictly more general to recurse on the lexicographic order instead. -/ def GameAdd.fix {C : α → β → Sort} (hα : WellFounded rα) (hβ : WellFounded rβ) (IH : ∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) (a : α) (b : β) : C a b := @WellFounded.fix (α × β) (fun x => C x.1 x.2) _ (hα.prod_gameAdd hβ) (fun ⟨x₁, x₂⟩ IH' => IH x₁ x₂ fun a' b' => IH' ⟨a', b'⟩) ⟨a, b⟩ #align prod.game_add.fix Prod.GameAdd.fix theorem GameAdd.fix_eq {C : α → β → Sort*} (hα : WellFounded rα) (hβ : WellFounded rβ) (IH : ∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) (a : α) (b : β) : GameAdd.fix hα hβ IH a b = IH a b fun a' b' _ => GameAdd.fix hα hβ IH a' b' := WellFounded.fix_eq _ _ _ #align prod.game_add.fix_eq Prod.GameAdd.fix_eq /-- Induction on the well-founded `Prod.GameAdd` relation. Note that it's strictly more general to induct on the lexicographic order instead. -/ theorem GameAdd.induction {C : α → β → Prop} : WellFounded rα → WellFounded rβ → (∀ a₁ b₁, (∀ a₂ b₂, GameAdd rα rβ (a₂, b₂) (a₁, b₁) → C a₂ b₂) → C a₁ b₁) → ∀ a b, C a b := GameAdd.fix #align prod.game_add.induction Prod.GameAdd.induction end Prod /-! ### `Sym2.GameAdd` -/ namespace Sym2 /-- `Sym2.GameAdd rα x y` means that `x` can be reached from `y` by decreasing either entry with respect to the relation `rα`. See `Prod.GameAdd` for the ordered pair analog. -/ def GameAdd (rα : α → α → Prop) : Sym2 α → Sym2 α → Prop := Sym2.lift₂ ⟨fun a₁ b₁ a₂ b₂ => Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂), fun a₁ b₁ a₂ b₂ => by dsimp rw [Prod.gameAdd_swap_swap_mk _ _ b₁ b₂ a₁ a₂, Prod.gameAdd_swap_swap_mk _ _ a₁ b₂ b₁ a₂] simp [or_comm]⟩ #align sym2.game_add Sym2.GameAdd theorem gameAdd_iff : ∀ {x y : α × α}, GameAdd rα (Sym2.mk x) (Sym2.mk y) ↔ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y := by rintro ⟨_, _⟩ ⟨_, _⟩ rfl #align sym2.game_add_iff Sym2.gameAdd_iff theorem gameAdd_mk'_iff {a₁ a₂ b₁ b₂ : α} : GameAdd rα s(a₁, b₁) s(a₂, b₂) ↔ Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂) := Iff.rfl #align sym2.game_add_mk_iff Sym2.gameAdd_mk'_iff theorem _root_.Prod.GameAdd.to_sym2 {a₁ a₂ b₁ b₂ : α} (h : Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂)) : Sym2.GameAdd rα s(a₁, b₁) s(a₂, b₂) := gameAdd_mk'_iff.2 <\| Or.inl <\| h #align prod.game_add.to_sym2 Prod.GameAdd.to_sym2 theorem GameAdd.fst {a₁ a₂ b : α} (h : rα a₁ a₂) : GameAdd rα s(a₁, b) s(a₂, b) := (Prod.GameAdd.fst h).to_sym2 #align sym2.game_add.fst Sym2.GameAdd.fst theorem GameAdd.snd {a b₁ b₂ : α} (h : rα b₁ b₂) : GameAdd rα s(a, b₁) s(a, b₂) := (Prod.GameAdd.snd h).to_sym2 #align sym2.game_add.snd Sym2.GameAdd.snd theorem GameAdd.fst_snd {a₁ a₂ b : α} (h : rα a₁ a₂) : GameAdd rα s(a₁, b) s(b, a₂) := by rw [Sym2.eq_swap] exact GameAdd.snd h #align sym2.game_add.fst_snd Sym2.GameAdd.fst_snd theorem GameAdd.snd_fst {a₁ a₂ b : α} (h : rα a₁ a₂) : GameAdd rα s(b, a₁) s(a₂, b) := by rw [Sym2.eq_swap] exact GameAdd.fst h #align sym2.game_add.snd_fst Sym2.GameAdd.snd_fst end Sym2	Mathlib/Order/GameAdd.lean	201	215	theorem Acc.sym2_gameAdd {a b} (ha : Acc rα a) (hb : Acc rα b) : Acc (Sym2.GameAdd rα) s(a, b) := by	induction' ha with a _ iha generalizing b induction' hb with b hb ihb refine Acc.intro _ fun s => ?_ induction' s using Sym2.inductionOn with c d rw [Sym2.GameAdd] dsimp rintro ((rc \| rd) \| (rd \| rc)) · exact iha c rc ⟨b, hb⟩ · exact ihb d rd · rw [Sym2.eq_swap] exact iha d rd ⟨b, hb⟩ · rw [Sym2.eq_swap] exact ihb c rc
/- Copyright (c) 2021 Rémy Degenne. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne -/ import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp #align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Extension of a linear function from indicators to L1 Let `T : Set α → E →L[ℝ] F` be additive for measurable sets with finite measure, in the sense that for `s, t` two such sets, `s ∩ t = ∅ → T (s ∪ t) = T s + T t`. `T` is akin to a bilinear map on `Set α × E`, or a linear map on indicator functions. This file constructs an extension of `T` to integrable simple functions, which are finite sums of indicators of measurable sets with finite measure, then to integrable functions, which are limits of integrable simple functions. The main result is a continuous linear map `(α →₁[μ] E) →L[ℝ] F`. This extension process is used to define the Bochner integral in the `MeasureTheory.Integral.Bochner` file and the conditional expectation of an integrable function in `MeasureTheory.Function.ConditionalExpectation`. ## Main Definitions - `FinMeasAdditive μ T`: the property that `T` is additive on measurable sets with finite measure. For two such sets, `s ∩ t = ∅ → T (s ∪ t) = T s + T t`. - `DominatedFinMeasAdditive μ T C`: `FinMeasAdditive μ T ∧ ∀ s, ‖T s‖ ≤ C * (μ s).toReal`. This is the property needed to perform the extension from indicators to L1. - `setToL1 (hT : DominatedFinMeasAdditive μ T C) : (α →₁[μ] E) →L[ℝ] F`: the extension of `T` from indicators to L1. - `setToFun μ T (hT : DominatedFinMeasAdditive μ T C) (f : α → E) : F`: a version of the extension which applies to functions (with value 0 if the function is not integrable). ## Properties For most properties of `setToFun`, we provide two lemmas. One version uses hypotheses valid on all sets, like `T = T'`, and a second version which uses a primed name uses hypotheses on measurable sets with finite measure, like `∀ s, MeasurableSet s → μ s < ∞ → T s = T' s`. The lemmas listed here don't show all hypotheses. Refer to the actual lemmas for details. Linearity: - `setToFun_zero_left : setToFun μ 0 hT f = 0` - `setToFun_add_left : setToFun μ (T + T') _ f = setToFun μ T hT f + setToFun μ T' hT' f` - `setToFun_smul_left : setToFun μ (fun s ↦ c • (T s)) (hT.smul c) f = c • setToFun μ T hT f` - `setToFun_zero : setToFun μ T hT (0 : α → E) = 0` - `setToFun_neg : setToFun μ T hT (-f) = - setToFun μ T hT f` If `f` and `g` are integrable: - `setToFun_add : setToFun μ T hT (f + g) = setToFun μ T hT f + setToFun μ T hT g` - `setToFun_sub : setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g` If `T` is verifies `∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x`: - `setToFun_smul : setToFun μ T hT (c • f) = c • setToFun μ T hT f` Other: - `setToFun_congr_ae (h : f =ᵐ[μ] g) : setToFun μ T hT f = setToFun μ T hT g` - `setToFun_measure_zero (h : μ = 0) : setToFun μ T hT f = 0` If the space is a `NormedLatticeAddCommGroup` and `T` is such that `0 ≤ T s x` for `0 ≤ x`, we also prove order-related properties: - `setToFun_mono_left (h : ∀ s x, T s x ≤ T' s x) : setToFun μ T hT f ≤ setToFun μ T' hT' f` - `setToFun_nonneg (hf : 0 ≤ᵐ[μ] f) : 0 ≤ setToFun μ T hT f` - `setToFun_mono (hfg : f ≤ᵐ[μ] g) : setToFun μ T hT f ≤ setToFun μ T hT g` ## Implementation notes The starting object `T : Set α → E →L[ℝ] F` matters only through its restriction on measurable sets with finite measure. Its value on other sets is ignored. -/ noncomputable section open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise open Set Filter TopologicalSpace ENNReal EMetric namespace MeasureTheory variable {α E F F' G 𝕜 : Type} {p : ℝ≥0∞} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] [NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α} local infixr:25 " →ₛ " => SimpleFunc open Finset section FinMeasAdditive /-- A set function is `FinMeasAdditive` if its value on the union of two disjoint measurable sets with finite measure is the sum of its values on each set. -/ def FinMeasAdditive {β} [AddMonoid β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) : Prop := ∀ s t, MeasurableSet s → MeasurableSet t → μ s ≠ ∞ → μ t ≠ ∞ → s ∩ t = ∅ → T (s ∪ t) = T s + T t #align measure_theory.fin_meas_additive MeasureTheory.FinMeasAdditive namespace FinMeasAdditive variable {β : Type} [AddCommMonoid β] {T T' : Set α → β} theorem zero : FinMeasAdditive μ (0 : Set α → β) := fun s t _ _ _ _ _ => by simp #align measure_theory.fin_meas_additive.zero MeasureTheory.FinMeasAdditive.zero theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') : FinMeasAdditive μ (T + T') := by intro s t hs ht hμs hμt hst simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply] abel #align measure_theory.fin_meas_additive.add MeasureTheory.FinMeasAdditive.add theorem smul [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) : FinMeasAdditive μ fun s => c • T s := fun s t hs ht hμs hμt hst => by simp [hT s t hs ht hμs hμt hst] #align measure_theory.fin_meas_additive.smul MeasureTheory.FinMeasAdditive.smul theorem of_eq_top_imp_eq_top {μ' : Measure α} (h : ∀ s, MeasurableSet s → μ s = ∞ → μ' s = ∞) (hT : FinMeasAdditive μ T) : FinMeasAdditive μ' T := fun s t hs ht hμ's hμ't hst => hT s t hs ht (mt (h s hs) hμ's) (mt (h t ht) hμ't) hst #align measure_theory.fin_meas_additive.of_eq_top_imp_eq_top MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) : FinMeasAdditive μ T := by refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs exact hμs.2 #align measure_theory.fin_meas_additive.of_smul_measure MeasureTheory.FinMeasAdditive.of_smul_measure theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) : FinMeasAdditive (c • μ) T := by refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff] exact Or.inl hμs #align measure_theory.fin_meas_additive.smul_measure MeasureTheory.FinMeasAdditive.smul_measure theorem smul_measure_iff (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ∞) : FinMeasAdditive (c • μ) T ↔ FinMeasAdditive μ T := ⟨fun hT => of_smul_measure c hc_ne_top hT, fun hT => smul_measure c hc_ne_zero hT⟩ #align measure_theory.fin_meas_additive.smul_measure_iff MeasureTheory.FinMeasAdditive.smul_measure_iff theorem map_empty_eq_zero {β} [AddCancelMonoid β] {T : Set α → β} (hT : FinMeasAdditive μ T) : T ∅ = 0 := by have h_empty : μ ∅ ≠ ∞ := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne specialize hT ∅ ∅ MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty ∅) rw [Set.union_empty] at hT nth_rw 1 [← add_zero (T ∅)] at hT exact (add_left_cancel hT).symm #align measure_theory.fin_meas_additive.map_empty_eq_zero MeasureTheory.FinMeasAdditive.map_empty_eq_zero theorem map_iUnion_fin_meas_set_eq_sum (T : Set α → β) (T_empty : T ∅ = 0) (h_add : FinMeasAdditive μ T) {ι} (S : ι → Set α) (sι : Finset ι) (hS_meas : ∀ i, MeasurableSet (S i)) (hSp : ∀ i ∈ sι, μ (S i) ≠ ∞) (h_disj : ∀ᵉ (i ∈ sι) (j ∈ sι), i ≠ j → Disjoint (S i) (S j)) : T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i) := by revert hSp h_disj refine Finset.induction_on sι ?_ ?_ · simp only [Finset.not_mem_empty, IsEmpty.forall_iff, iUnion_false, iUnion_empty, sum_empty, forall₂_true_iff, imp_true_iff, forall_true_left, not_false_iff, T_empty] intro a s has h hps h_disj rw [Finset.sum_insert has, ← h] swap; · exact fun i hi => hps i (Finset.mem_insert_of_mem hi) swap; · exact fun i hi j hj hij => h_disj i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj) hij rw [← h_add (S a) (⋃ i ∈ s, S i) (hS_meas a) (measurableSet_biUnion _ fun i _ => hS_meas i) (hps a (Finset.mem_insert_self a s))] · congr; convert Finset.iSup_insert a s S · exact ((measure_biUnion_finset_le _ _).trans_lt <\| ENNReal.sum_lt_top fun i hi => hps i <\| Finset.mem_insert_of_mem hi).ne · simp_rw [Set.inter_iUnion] refine iUnion_eq_empty.mpr fun i => iUnion_eq_empty.mpr fun hi => ?_ rw [← Set.disjoint_iff_inter_eq_empty] refine h_disj a (Finset.mem_insert_self a s) i (Finset.mem_insert_of_mem hi) fun hai => ?_ rw [← hai] at hi exact has hi #align measure_theory.fin_meas_additive.map_Union_fin_meas_set_eq_sum MeasureTheory.FinMeasAdditive.map_iUnion_fin_meas_set_eq_sum end FinMeasAdditive /-- A `FinMeasAdditive` set function whose norm on every set is less than the measure of the set (up to a multiplicative constant). -/ def DominatedFinMeasAdditive {β} [SeminormedAddCommGroup β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) (C : ℝ) : Prop := FinMeasAdditive μ T ∧ ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal #align measure_theory.dominated_fin_meas_additive MeasureTheory.DominatedFinMeasAdditive namespace DominatedFinMeasAdditive variable {β : Type} [SeminormedAddCommGroup β] {T T' : Set α → β} {C C' : ℝ} theorem zero {m : MeasurableSpace α} (μ : Measure α) (hC : 0 ≤ C) : DominatedFinMeasAdditive μ (0 : Set α → β) C := by refine ⟨FinMeasAdditive.zero, fun s _ _ => ?_⟩ rw [Pi.zero_apply, norm_zero] exact mul_nonneg hC toReal_nonneg #align measure_theory.dominated_fin_meas_additive.zero MeasureTheory.DominatedFinMeasAdditive.zero theorem eq_zero_of_measure_zero {β : Type} [NormedAddCommGroup β] {T : Set α → β} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hs_zero : μ s = 0) : T s = 0 := by refine norm_eq_zero.mp ?_ refine ((hT.2 s hs (by simp [hs_zero])).trans (le_of_eq ?_)).antisymm (norm_nonneg _) rw [hs_zero, ENNReal.zero_toReal, mul_zero] #align measure_theory.dominated_fin_meas_additive.eq_zero_of_measure_zero MeasureTheory.DominatedFinMeasAdditive.eq_zero_of_measure_zero theorem eq_zero {β : Type} [NormedAddCommGroup β] {T : Set α → β} {C : ℝ} {m : MeasurableSpace α} (hT : DominatedFinMeasAdditive (0 : Measure α) T C) {s : Set α} (hs : MeasurableSet s) : T s = 0 := eq_zero_of_measure_zero hT hs (by simp only [Measure.coe_zero, Pi.zero_apply]) #align measure_theory.dominated_fin_meas_additive.eq_zero MeasureTheory.DominatedFinMeasAdditive.eq_zero theorem add (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') : DominatedFinMeasAdditive μ (T + T') (C + C') := by refine ⟨hT.1.add hT'.1, fun s hs hμs => ?_⟩ rw [Pi.add_apply, add_mul] exact (norm_add_le _ _).trans (add_le_add (hT.2 s hs hμs) (hT'.2 s hs hμs)) #align measure_theory.dominated_fin_meas_additive.add MeasureTheory.DominatedFinMeasAdditive.add theorem smul [NormedField 𝕜] [NormedSpace 𝕜 β] (hT : DominatedFinMeasAdditive μ T C) (c : 𝕜) : DominatedFinMeasAdditive μ (fun s => c • T s) (‖c‖ C) := by refine ⟨hT.1.smul c, fun s hs hμs => ?_⟩ dsimp only rw [norm_smul, mul_assoc] exact mul_le_mul le_rfl (hT.2 s hs hμs) (norm_nonneg _) (norm_nonneg _) #align measure_theory.dominated_fin_meas_additive.smul MeasureTheory.DominatedFinMeasAdditive.smul theorem of_measure_le {μ' : Measure α} (h : μ ≤ μ') (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : DominatedFinMeasAdditive μ' T C := by have h' : ∀ s, μ s = ∞ → μ' s = ∞ := fun s hs ↦ top_unique <\| hs.symm.trans_le (h _) refine ⟨hT.1.of_eq_top_imp_eq_top fun s _ ↦ h' s, fun s hs hμ's ↦ ?_⟩ have hμs : μ s < ∞ := (h s).trans_lt hμ's calc ‖T s‖ ≤ C * (μ s).toReal := hT.2 s hs hμs _ ≤ C * (μ' s).toReal := by gcongr; exacts [hμ's.ne, h _] #align measure_theory.dominated_fin_meas_additive.of_measure_le MeasureTheory.DominatedFinMeasAdditive.of_measure_le theorem add_measure_right {_ : MeasurableSpace α} (μ ν : Measure α) (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : DominatedFinMeasAdditive (μ + ν) T C := of_measure_le (Measure.le_add_right le_rfl) hT hC #align measure_theory.dominated_fin_meas_additive.add_measure_right MeasureTheory.DominatedFinMeasAdditive.add_measure_right theorem add_measure_left {_ : MeasurableSpace α} (μ ν : Measure α) (hT : DominatedFinMeasAdditive ν T C) (hC : 0 ≤ C) : DominatedFinMeasAdditive (μ + ν) T C := of_measure_le (Measure.le_add_left le_rfl) hT hC #align measure_theory.dominated_fin_meas_additive.add_measure_left MeasureTheory.DominatedFinMeasAdditive.add_measure_left theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : DominatedFinMeasAdditive (c • μ) T C) : DominatedFinMeasAdditive μ T (c.toReal * C) := by have h : ∀ s, MeasurableSet s → c • μ s = ∞ → μ s = ∞ := by intro s _ hcμs simp only [hc_ne_top, Algebra.id.smul_eq_mul, ENNReal.mul_eq_top, or_false_iff, Ne, false_and_iff] at hcμs exact hcμs.2 refine ⟨hT.1.of_eq_top_imp_eq_top (μ := c • μ) h, fun s hs hμs => ?_⟩ have hcμs : c • μ s ≠ ∞ := mt (h s hs) hμs.ne rw [smul_eq_mul] at hcμs simp_rw [DominatedFinMeasAdditive, Measure.smul_apply, smul_eq_mul, toReal_mul] at hT refine (hT.2 s hs hcμs.lt_top).trans (le_of_eq ?_) ring #align measure_theory.dominated_fin_meas_additive.of_smul_measure MeasureTheory.DominatedFinMeasAdditive.of_smul_measure theorem of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (h : μ ≤ c • μ') (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : DominatedFinMeasAdditive μ' T (c.toReal * C) := (hT.of_measure_le h hC).of_smul_measure c hc #align measure_theory.dominated_fin_meas_additive.of_measure_le_smul MeasureTheory.DominatedFinMeasAdditive.of_measure_le_smul end DominatedFinMeasAdditive end FinMeasAdditive namespace SimpleFunc /-- Extend `Set α → (F →L[ℝ] F')` to `(α →ₛ F) → F'`. -/ def setToSimpleFunc {_ : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (f : α →ₛ F) : F' := ∑ x ∈ f.range, T (f ⁻¹' {x}) x #align measure_theory.simple_func.set_to_simple_func MeasureTheory.SimpleFunc.setToSimpleFunc @[simp] theorem setToSimpleFunc_zero {m : MeasurableSpace α} (f : α →ₛ F) : setToSimpleFunc (0 : Set α → F →L[ℝ] F') f = 0 := by simp [setToSimpleFunc] #align measure_theory.simple_func.set_to_simple_func_zero MeasureTheory.SimpleFunc.setToSimpleFunc_zero theorem setToSimpleFunc_zero' {T : Set α → E →L[ℝ] F'} (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →ₛ E) (hf : Integrable f μ) : setToSimpleFunc T f = 0 := by simp_rw [setToSimpleFunc] refine sum_eq_zero fun x _ => ?_ by_cases hx0 : x = 0 · simp [hx0] rw [h_zero (f ⁻¹' ({x} : Set E)) (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable f hf hx0), ContinuousLinearMap.zero_apply] #align measure_theory.simple_func.set_to_simple_func_zero' MeasureTheory.SimpleFunc.setToSimpleFunc_zero' @[simp] theorem setToSimpleFunc_zero_apply {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') : setToSimpleFunc T (0 : α →ₛ F) = 0 := by cases isEmpty_or_nonempty α <;> simp [setToSimpleFunc] #align measure_theory.simple_func.set_to_simple_func_zero_apply MeasureTheory.SimpleFunc.setToSimpleFunc_zero_apply theorem setToSimpleFunc_eq_sum_filter {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (f : α →ₛ F) : setToSimpleFunc T f = ∑ x ∈ f.range.filter fun x => x ≠ 0, (T (f ⁻¹' {x})) x := by symm refine sum_filter_of_ne fun x _ => mt fun hx0 => ?_ rw [hx0] exact ContinuousLinearMap.map_zero _ #align measure_theory.simple_func.set_to_simple_func_eq_sum_filter MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_filter theorem map_setToSimpleFunc (T : Set α → F →L[ℝ] F') (h_add : FinMeasAdditive μ T) {f : α →ₛ G} (hf : Integrable f μ) {g : G → F} (hg : g 0 = 0) : (f.map g).setToSimpleFunc T = ∑ x ∈ f.range, T (f ⁻¹' {x}) (g x) := by have T_empty : T ∅ = 0 := h_add.map_empty_eq_zero have hfp : ∀ x ∈ f.range, x ≠ 0 → μ (f ⁻¹' {x}) ≠ ∞ := fun x _ hx0 => (measure_preimage_lt_top_of_integrable f hf hx0).ne simp only [setToSimpleFunc, range_map] refine Finset.sum_image' _ fun b hb => ?_ rcases mem_range.1 hb with ⟨a, rfl⟩ by_cases h0 : g (f a) = 0 · simp_rw [h0] rw [ContinuousLinearMap.map_zero, Finset.sum_eq_zero fun x hx => ?_] rw [mem_filter] at hx rw [hx.2, ContinuousLinearMap.map_zero] have h_left_eq : T (map g f ⁻¹' {g (f a)}) (g (f a)) = T (f ⁻¹' (f.range.filter fun b => g b = g (f a))) (g (f a)) := by congr; rw [map_preimage_singleton] rw [h_left_eq] have h_left_eq' : T (f ⁻¹' (filter (fun b : G => g b = g (f a)) f.range)) (g (f a)) = T (⋃ y ∈ filter (fun b : G => g b = g (f a)) f.range, f ⁻¹' {y}) (g (f a)) := by congr; rw [← Finset.set_biUnion_preimage_singleton] rw [h_left_eq'] rw [h_add.map_iUnion_fin_meas_set_eq_sum T T_empty] · simp only [sum_apply, ContinuousLinearMap.coe_sum'] refine Finset.sum_congr rfl fun x hx => ?_ rw [mem_filter] at hx rw [hx.2] · exact fun i => measurableSet_fiber _ _ · intro i hi rw [mem_filter] at hi refine hfp i hi.1 fun hi0 => ?_ rw [hi0, hg] at hi exact h0 hi.2.symm · intro i _j hi _ hij rw [Set.disjoint_iff] intro x hx rw [Set.mem_inter_iff, Set.mem_preimage, Set.mem_preimage, Set.mem_singleton_iff, Set.mem_singleton_iff] at hx rw [← hx.1, ← hx.2] at hij exact absurd rfl hij #align measure_theory.simple_func.map_set_to_simple_func MeasureTheory.SimpleFunc.map_setToSimpleFunc theorem setToSimpleFunc_congr' (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E} (hf : Integrable f μ) (hg : Integrable g μ) (h : Pairwise fun x y => T (f ⁻¹' {x} ∩ g ⁻¹' {y}) = 0) : f.setToSimpleFunc T = g.setToSimpleFunc T := show ((pair f g).map Prod.fst).setToSimpleFunc T = ((pair f g).map Prod.snd).setToSimpleFunc T by have h_pair : Integrable (f.pair g) μ := integrable_pair hf hg rw [map_setToSimpleFunc T h_add h_pair Prod.fst_zero] rw [map_setToSimpleFunc T h_add h_pair Prod.snd_zero] refine Finset.sum_congr rfl fun p hp => ?_ rcases mem_range.1 hp with ⟨a, rfl⟩ by_cases eq : f a = g a · dsimp only [pair_apply]; rw [eq] · have : T (pair f g ⁻¹' {(f a, g a)}) = 0 := by have h_eq : T ((⇑(f.pair g)) ⁻¹' {(f a, g a)}) = T (f ⁻¹' {f a} ∩ g ⁻¹' {g a}) := by congr; rw [pair_preimage_singleton f g] rw [h_eq] exact h eq simp only [this, ContinuousLinearMap.zero_apply, pair_apply] #align measure_theory.simple_func.set_to_simple_func_congr' MeasureTheory.SimpleFunc.setToSimpleFunc_congr' theorem setToSimpleFunc_congr (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E} (hf : Integrable f μ) (h : f =ᵐ[μ] g) : f.setToSimpleFunc T = g.setToSimpleFunc T := by refine setToSimpleFunc_congr' T h_add hf ((integrable_congr h).mp hf) ?_ refine fun x y hxy => h_zero _ ((measurableSet_fiber f x).inter (measurableSet_fiber g y)) ?_ rw [EventuallyEq, ae_iff] at h refine measure_mono_null (fun z => ?_) h simp_rw [Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_preimage, Set.mem_singleton_iff] intro h rwa [h.1, h.2] #align measure_theory.simple_func.set_to_simple_func_congr MeasureTheory.SimpleFunc.setToSimpleFunc_congr theorem setToSimpleFunc_congr_left (T T' : Set α → E →L[ℝ] F) (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →ₛ E) (hf : Integrable f μ) : setToSimpleFunc T f = setToSimpleFunc T' f := by simp_rw [setToSimpleFunc] refine sum_congr rfl fun x _ => ?_ by_cases hx0 : x = 0 · simp [hx0] · rw [h (f ⁻¹' {x}) (SimpleFunc.measurableSet_fiber _ _) (SimpleFunc.measure_preimage_lt_top_of_integrable _ hf hx0)] #align measure_theory.simple_func.set_to_simple_func_congr_left MeasureTheory.SimpleFunc.setToSimpleFunc_congr_left theorem setToSimpleFunc_add_left {m : MeasurableSpace α} (T T' : Set α → F →L[ℝ] F') {f : α →ₛ F} : setToSimpleFunc (T + T') f = setToSimpleFunc T f + setToSimpleFunc T' f := by simp_rw [setToSimpleFunc, Pi.add_apply] push_cast simp_rw [Pi.add_apply, sum_add_distrib] #align measure_theory.simple_func.set_to_simple_func_add_left MeasureTheory.SimpleFunc.setToSimpleFunc_add_left theorem setToSimpleFunc_add_left' (T T' T'' : Set α → E →L[ℝ] F) (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) {f : α →ₛ E} (hf : Integrable f μ) : setToSimpleFunc T'' f = setToSimpleFunc T f + setToSimpleFunc T' f := by simp_rw [setToSimpleFunc_eq_sum_filter] suffices ∀ x ∈ filter (fun x : E => x ≠ 0) f.range, T'' (f ⁻¹' {x}) = T (f ⁻¹' {x}) + T' (f ⁻¹' {x}) by rw [← sum_add_distrib] refine Finset.sum_congr rfl fun x hx => ?_ rw [this x hx] push_cast rw [Pi.add_apply] intro x hx refine h_add (f ⁻¹' {x}) (measurableSet_preimage _ _) (measure_preimage_lt_top_of_integrable _ hf ?_) rw [mem_filter] at hx exact hx.2 #align measure_theory.simple_func.set_to_simple_func_add_left' MeasureTheory.SimpleFunc.setToSimpleFunc_add_left' theorem setToSimpleFunc_smul_left {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (c : ℝ) (f : α →ₛ F) : setToSimpleFunc (fun s => c • T s) f = c • setToSimpleFunc T f := by simp_rw [setToSimpleFunc, ContinuousLinearMap.smul_apply, smul_sum] #align measure_theory.simple_func.set_to_simple_func_smul_left MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left theorem setToSimpleFunc_smul_left' (T T' : Set α → E →L[ℝ] F') (c : ℝ) (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) {f : α →ₛ E} (hf : Integrable f μ) : setToSimpleFunc T' f = c • setToSimpleFunc T f := by simp_rw [setToSimpleFunc_eq_sum_filter] suffices ∀ x ∈ filter (fun x : E => x ≠ 0) f.range, T' (f ⁻¹' {x}) = c • T (f ⁻¹' {x}) by rw [smul_sum] refine Finset.sum_congr rfl fun x hx => ?_ rw [this x hx] rfl intro x hx refine h_smul (f ⁻¹' {x}) (measurableSet_preimage _ _) (measure_preimage_lt_top_of_integrable _ hf ?_) rw [mem_filter] at hx exact hx.2 #align measure_theory.simple_func.set_to_simple_func_smul_left' MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left' theorem setToSimpleFunc_add (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E} (hf : Integrable f μ) (hg : Integrable g μ) : setToSimpleFunc T (f + g) = setToSimpleFunc T f + setToSimpleFunc T g := have hp_pair : Integrable (f.pair g) μ := integrable_pair hf hg calc setToSimpleFunc T (f + g) = ∑ x ∈ (pair f g).range, T (pair f g ⁻¹' {x}) (x.fst + x.snd) := by rw [add_eq_map₂, map_setToSimpleFunc T h_add hp_pair]; simp _ = ∑ x ∈ (pair f g).range, (T (pair f g ⁻¹' {x}) x.fst + T (pair f g ⁻¹' {x}) x.snd) := (Finset.sum_congr rfl fun a _ => ContinuousLinearMap.map_add _ _ _) _ = (∑ x ∈ (pair f g).range, T (pair f g ⁻¹' {x}) x.fst) + ∑ x ∈ (pair f g).range, T (pair f g ⁻¹' {x}) x.snd := by rw [Finset.sum_add_distrib] _ = ((pair f g).map Prod.fst).setToSimpleFunc T + ((pair f g).map Prod.snd).setToSimpleFunc T := by rw [map_setToSimpleFunc T h_add hp_pair Prod.snd_zero, map_setToSimpleFunc T h_add hp_pair Prod.fst_zero] #align measure_theory.simple_func.set_to_simple_func_add MeasureTheory.SimpleFunc.setToSimpleFunc_add theorem setToSimpleFunc_neg (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f : α →ₛ E} (hf : Integrable f μ) : setToSimpleFunc T (-f) = -setToSimpleFunc T f := calc setToSimpleFunc T (-f) = setToSimpleFunc T (f.map Neg.neg) := rfl _ = -setToSimpleFunc T f := by rw [map_setToSimpleFunc T h_add hf neg_zero, setToSimpleFunc, ← sum_neg_distrib] exact Finset.sum_congr rfl fun x _ => ContinuousLinearMap.map_neg _ _ #align measure_theory.simple_func.set_to_simple_func_neg MeasureTheory.SimpleFunc.setToSimpleFunc_neg theorem setToSimpleFunc_sub (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E} (hf : Integrable f μ) (hg : Integrable g μ) : setToSimpleFunc T (f - g) = setToSimpleFunc T f - setToSimpleFunc T g := by rw [sub_eq_add_neg, setToSimpleFunc_add T h_add hf, setToSimpleFunc_neg T h_add hg, sub_eq_add_neg] rw [integrable_iff] at hg ⊢ intro x hx_ne change μ (Neg.neg ∘ g ⁻¹' {x}) < ∞ rw [preimage_comp, neg_preimage, Set.neg_singleton] refine hg (-x) ?_ simp [hx_ne] #align measure_theory.simple_func.set_to_simple_func_sub MeasureTheory.SimpleFunc.setToSimpleFunc_sub theorem setToSimpleFunc_smul_real (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) (c : ℝ) {f : α →ₛ E} (hf : Integrable f μ) : setToSimpleFunc T (c • f) = c • setToSimpleFunc T f := calc setToSimpleFunc T (c • f) = ∑ x ∈ f.range, T (f ⁻¹' {x}) (c • x) := by rw [smul_eq_map c f, map_setToSimpleFunc T h_add hf]; dsimp only; rw [smul_zero] _ = ∑ x ∈ f.range, c • T (f ⁻¹' {x}) x := (Finset.sum_congr rfl fun b _ => by rw [ContinuousLinearMap.map_smul (T (f ⁻¹' {b})) c b]) _ = c • setToSimpleFunc T f := by simp only [setToSimpleFunc, smul_sum, smul_smul, mul_comm] #align measure_theory.simple_func.set_to_simple_func_smul_real MeasureTheory.SimpleFunc.setToSimpleFunc_smul_real theorem setToSimpleFunc_smul {E} [NormedAddCommGroup E] [NormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [NormedSpace 𝕜 F] (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜) {f : α →ₛ E} (hf : Integrable f μ) : setToSimpleFunc T (c • f) = c • setToSimpleFunc T f := calc setToSimpleFunc T (c • f) = ∑ x ∈ f.range, T (f ⁻¹' {x}) (c • x) := by rw [smul_eq_map c f, map_setToSimpleFunc T h_add hf]; dsimp only; rw [smul_zero] _ = ∑ x ∈ f.range, c • T (f ⁻¹' {x}) x := Finset.sum_congr rfl fun b _ => by rw [h_smul] _ = c • setToSimpleFunc T f := by simp only [setToSimpleFunc, smul_sum, smul_smul, mul_comm] #align measure_theory.simple_func.set_to_simple_func_smul MeasureTheory.SimpleFunc.setToSimpleFunc_smul section Order variable {G' G'' : Type} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] theorem setToSimpleFunc_mono_left {m : MeasurableSpace α} (T T' : Set α → F →L[ℝ] G'') (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →ₛ F) : setToSimpleFunc T f ≤ setToSimpleFunc T' f := by simp_rw [setToSimpleFunc]; exact sum_le_sum fun i _ => hTT' _ i #align measure_theory.simple_func.set_to_simple_func_mono_left MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left theorem setToSimpleFunc_mono_left' (T T' : Set α → E →L[ℝ] G'') (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →ₛ E) (hf : Integrable f μ) : setToSimpleFunc T f ≤ setToSimpleFunc T' f := by refine sum_le_sum fun i _ => ?_ by_cases h0 : i = 0 · simp [h0] · exact hTT' _ (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable _ hf h0) i #align measure_theory.simple_func.set_to_simple_func_mono_left' MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left' theorem setToSimpleFunc_nonneg {m : MeasurableSpace α} (T : Set α → G' →L[ℝ] G'') (hT_nonneg : ∀ s x, 0 ≤ x → 0 ≤ T s x) (f : α →ₛ G') (hf : 0 ≤ f) : 0 ≤ setToSimpleFunc T f := by refine sum_nonneg fun i hi => hT_nonneg _ i ?_ rw [mem_range] at hi obtain ⟨y, hy⟩ := Set.mem_range.mp hi rw [← hy] refine le_trans ?_ (hf y) simp #align measure_theory.simple_func.set_to_simple_func_nonneg MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg theorem setToSimpleFunc_nonneg' (T : Set α → G' →L[ℝ] G'') (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) (f : α →ₛ G') (hf : 0 ≤ f) (hfi : Integrable f μ) : 0 ≤ setToSimpleFunc T f := by refine sum_nonneg fun i hi => ?_ by_cases h0 : i = 0 · simp [h0] refine hT_nonneg _ (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable _ hfi h0) i ?_ rw [mem_range] at hi obtain ⟨y, hy⟩ := Set.mem_range.mp hi rw [← hy] convert hf y #align measure_theory.simple_func.set_to_simple_func_nonneg' MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg' theorem setToSimpleFunc_mono {T : Set α → G' →L[ℝ] G''} (h_add : FinMeasAdditive μ T) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →ₛ G'} (hfi : Integrable f μ) (hgi : Integrable g μ) (hfg : f ≤ g) : setToSimpleFunc T f ≤ setToSimpleFunc T g := by rw [← sub_nonneg, ← setToSimpleFunc_sub T h_add hgi hfi] refine setToSimpleFunc_nonneg' T hT_nonneg _ ?_ (hgi.sub hfi) intro x simp only [coe_sub, sub_nonneg, coe_zero, Pi.zero_apply, Pi.sub_apply] exact hfg x #align measure_theory.simple_func.set_to_simple_func_mono MeasureTheory.SimpleFunc.setToSimpleFunc_mono end Order theorem norm_setToSimpleFunc_le_sum_opNorm {m : MeasurableSpace α} (T : Set α → F' →L[ℝ] F) (f : α →ₛ F') : ‖f.setToSimpleFunc T‖ ≤ ∑ x ∈ f.range, ‖T (f ⁻¹' {x})‖ ‖x‖ := calc ‖∑ x ∈ f.range, T (f ⁻¹' {x}) x‖ ≤ ∑ x ∈ f.range, ‖T (f ⁻¹' {x}) x‖ := norm_sum_le _ _ _ ≤ ∑ x ∈ f.range, ‖T (f ⁻¹' {x})‖ * ‖x‖ := by refine Finset.sum_le_sum fun b _ => ?_; simp_rw [ContinuousLinearMap.le_opNorm] #align measure_theory.simple_func.norm_set_to_simple_func_le_sum_op_norm MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_opNorm @[deprecated (since := "2024-02-02")] alias norm_setToSimpleFunc_le_sum_op_norm := norm_setToSimpleFunc_le_sum_opNorm theorem norm_setToSimpleFunc_le_sum_mul_norm (T : Set α → F →L[ℝ] F') {C : ℝ} (hT_norm : ∀ s, MeasurableSet s → ‖T s‖ ≤ C * (μ s).toReal) (f : α →ₛ F) : ‖f.setToSimpleFunc T‖ ≤ C * ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal * ‖x‖ := calc ‖f.setToSimpleFunc T‖ ≤ ∑ x ∈ f.range, ‖T (f ⁻¹' {x})‖ * ‖x‖ := norm_setToSimpleFunc_le_sum_opNorm T f _ ≤ ∑ x ∈ f.range, C * (μ (f ⁻¹' {x})).toReal * ‖x‖ := by gcongr exact hT_norm _ <\| SimpleFunc.measurableSet_fiber _ _ _ ≤ C * ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal * ‖x‖ := by simp_rw [mul_sum, ← mul_assoc]; rfl #align measure_theory.simple_func.norm_set_to_simple_func_le_sum_mul_norm MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm theorem norm_setToSimpleFunc_le_sum_mul_norm_of_integrable (T : Set α → E →L[ℝ] F') {C : ℝ} (hT_norm : ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal) (f : α →ₛ E) (hf : Integrable f μ) : ‖f.setToSimpleFunc T‖ ≤ C * ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal * ‖x‖ := calc ‖f.setToSimpleFunc T‖ ≤ ∑ x ∈ f.range, ‖T (f ⁻¹' {x})‖ * ‖x‖ := norm_setToSimpleFunc_le_sum_opNorm T f _ ≤ ∑ x ∈ f.range, C * (μ (f ⁻¹' {x})).toReal * ‖x‖ := by refine Finset.sum_le_sum fun b hb => ?_ obtain rfl \| hb := eq_or_ne b 0 · simp gcongr exact hT_norm _ (SimpleFunc.measurableSet_fiber _ _) <\| SimpleFunc.measure_preimage_lt_top_of_integrable _ hf hb _ ≤ C * ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal * ‖x‖ := by simp_rw [mul_sum, ← mul_assoc]; rfl #align measure_theory.simple_func.norm_set_to_simple_func_le_sum_mul_norm_of_integrable MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm_of_integrable theorem setToSimpleFunc_indicator (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0) {m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (x : F) : SimpleFunc.setToSimpleFunc T (SimpleFunc.piecewise s hs (SimpleFunc.const α x) (SimpleFunc.const α 0)) = T s x := by obtain rfl \| hs_empty := s.eq_empty_or_nonempty · simp only [hT_empty, ContinuousLinearMap.zero_apply, piecewise_empty, const_zero, setToSimpleFunc_zero_apply] simp_rw [setToSimpleFunc] obtain rfl \| hs_univ := eq_or_ne s univ · haveI hα := hs_empty.to_type simp [← Function.const_def] rw [range_indicator hs hs_empty hs_univ] by_cases hx0 : x = 0 · simp_rw [hx0]; simp rw [sum_insert] swap; · rw [Finset.mem_singleton]; exact hx0 rw [sum_singleton, (T _).map_zero, add_zero] congr simp only [coe_piecewise, piecewise_eq_indicator, coe_const, Function.const_zero, piecewise_eq_indicator] rw [indicator_preimage, ← Function.const_def, preimage_const_of_mem] swap; · exact Set.mem_singleton x rw [← Function.const_zero, ← Function.const_def, preimage_const_of_not_mem] swap; · rw [Set.mem_singleton_iff]; exact Ne.symm hx0 simp #align measure_theory.simple_func.set_to_simple_func_indicator MeasureTheory.SimpleFunc.setToSimpleFunc_indicator theorem setToSimpleFunc_const' [Nonempty α] (T : Set α → F →L[ℝ] F') (x : F) {m : MeasurableSpace α} : SimpleFunc.setToSimpleFunc T (SimpleFunc.const α x) = T univ x := by simp only [setToSimpleFunc, range_const, Set.mem_singleton, preimage_const_of_mem, sum_singleton, ← Function.const_def, coe_const] #align measure_theory.simple_func.set_to_simple_func_const' MeasureTheory.SimpleFunc.setToSimpleFunc_const' theorem setToSimpleFunc_const (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0) (x : F) {m : MeasurableSpace α} : SimpleFunc.setToSimpleFunc T (SimpleFunc.const α x) = T univ x := by cases isEmpty_or_nonempty α · have h_univ_empty : (univ : Set α) = ∅ := Subsingleton.elim _ _ rw [h_univ_empty, hT_empty] simp only [setToSimpleFunc, ContinuousLinearMap.zero_apply, sum_empty, range_eq_empty_of_isEmpty] · exact setToSimpleFunc_const' T x #align measure_theory.simple_func.set_to_simple_func_const MeasureTheory.SimpleFunc.setToSimpleFunc_const end SimpleFunc namespace L1 set_option linter.uppercaseLean3 false open AEEqFun Lp.simpleFunc Lp namespace SimpleFunc theorem norm_eq_sum_mul (f : α →₁ₛ[μ] G) : ‖f‖ = ∑ x ∈ (toSimpleFunc f).range, (μ (toSimpleFunc f ⁻¹' {x})).toReal * ‖x‖ := by rw [norm_toSimpleFunc, snorm_one_eq_lintegral_nnnorm] have h_eq := SimpleFunc.map_apply (fun x => (‖x‖₊ : ℝ≥0∞)) (toSimpleFunc f) simp_rw [← h_eq] rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum] · congr ext1 x rw [ENNReal.toReal_mul, mul_comm, ← ofReal_norm_eq_coe_nnnorm, ENNReal.toReal_ofReal (norm_nonneg _)] · intro x _ by_cases hx0 : x = 0 · rw [hx0]; simp · exact ENNReal.mul_ne_top ENNReal.coe_ne_top (SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0).ne #align measure_theory.L1.simple_func.norm_eq_sum_mul MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul section SetToL1S variable [NormedField 𝕜] [NormedSpace 𝕜 E] attribute [local instance] Lp.simpleFunc.module attribute [local instance] Lp.simpleFunc.normedSpace /-- Extend `Set α → (E →L[ℝ] F')` to `(α →₁ₛ[μ] E) → F'`. -/ def setToL1S (T : Set α → E →L[ℝ] F) (f : α →₁ₛ[μ] E) : F := (toSimpleFunc f).setToSimpleFunc T #align measure_theory.L1.simple_func.set_to_L1s MeasureTheory.L1.SimpleFunc.setToL1S theorem setToL1S_eq_setToSimpleFunc (T : Set α → E →L[ℝ] F) (f : α →₁ₛ[μ] E) : setToL1S T f = (toSimpleFunc f).setToSimpleFunc T := rfl #align measure_theory.L1.simple_func.set_to_L1s_eq_set_to_simple_func MeasureTheory.L1.SimpleFunc.setToL1S_eq_setToSimpleFunc @[simp] theorem setToL1S_zero_left (f : α →₁ₛ[μ] E) : setToL1S (0 : Set α → E →L[ℝ] F) f = 0 := SimpleFunc.setToSimpleFunc_zero _ #align measure_theory.L1.simple_func.set_to_L1s_zero_left MeasureTheory.L1.SimpleFunc.setToL1S_zero_left theorem setToL1S_zero_left' {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁ₛ[μ] E) : setToL1S T f = 0 := SimpleFunc.setToSimpleFunc_zero' h_zero _ (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_zero_left' MeasureTheory.L1.SimpleFunc.setToL1S_zero_left' theorem setToL1S_congr (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) {f g : α →₁ₛ[μ] E} (h : toSimpleFunc f =ᵐ[μ] toSimpleFunc g) : setToL1S T f = setToL1S T g := SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable f) h #align measure_theory.L1.simple_func.set_to_L1s_congr MeasureTheory.L1.SimpleFunc.setToL1S_congr theorem setToL1S_congr_left (T T' : Set α → E →L[ℝ] F) (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁ₛ[μ] E) : setToL1S T f = setToL1S T' f := SimpleFunc.setToSimpleFunc_congr_left T T' h (simpleFunc.toSimpleFunc f) (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_congr_left MeasureTheory.L1.SimpleFunc.setToL1S_congr_left /-- `setToL1S` does not change if we replace the measure `μ` by `μ'` with `μ ≪ μ'`. The statement uses two functions `f` and `f'` because they have to belong to different types, but morally these are the same function (we have `f =ᵐ[μ] f'`). -/ theorem setToL1S_congr_measure {μ' : Measure α} (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (hμ : μ ≪ μ') (f : α →₁ₛ[μ] E) (f' : α →₁ₛ[μ'] E) (h : (f : α → E) =ᵐ[μ] f') : setToL1S T f = setToL1S T f' := by refine SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable f) ?_ refine (toSimpleFunc_eq_toFun f).trans ?_ suffices (f' : α → E) =ᵐ[μ] simpleFunc.toSimpleFunc f' from h.trans this have goal' : (f' : α → E) =ᵐ[μ'] simpleFunc.toSimpleFunc f' := (toSimpleFunc_eq_toFun f').symm exact hμ.ae_eq goal' #align measure_theory.L1.simple_func.set_to_L1s_congr_measure MeasureTheory.L1.SimpleFunc.setToL1S_congr_measure theorem setToL1S_add_left (T T' : Set α → E →L[ℝ] F) (f : α →₁ₛ[μ] E) : setToL1S (T + T') f = setToL1S T f + setToL1S T' f := SimpleFunc.setToSimpleFunc_add_left T T' #align measure_theory.L1.simple_func.set_to_L1s_add_left MeasureTheory.L1.SimpleFunc.setToL1S_add_left theorem setToL1S_add_left' (T T' T'' : Set α → E →L[ℝ] F) (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁ₛ[μ] E) : setToL1S T'' f = setToL1S T f + setToL1S T' f := SimpleFunc.setToSimpleFunc_add_left' T T' T'' h_add (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_add_left' MeasureTheory.L1.SimpleFunc.setToL1S_add_left' theorem setToL1S_smul_left (T : Set α → E →L[ℝ] F) (c : ℝ) (f : α →₁ₛ[μ] E) : setToL1S (fun s => c • T s) f = c • setToL1S T f := SimpleFunc.setToSimpleFunc_smul_left T c _ #align measure_theory.L1.simple_func.set_to_L1s_smul_left MeasureTheory.L1.SimpleFunc.setToL1S_smul_left theorem setToL1S_smul_left' (T T' : Set α → E →L[ℝ] F) (c : ℝ) (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁ₛ[μ] E) : setToL1S T' f = c • setToL1S T f := SimpleFunc.setToSimpleFunc_smul_left' T T' c h_smul (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_smul_left' MeasureTheory.L1.SimpleFunc.setToL1S_smul_left' theorem setToL1S_add (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (f g : α →₁ₛ[μ] E) : setToL1S T (f + g) = setToL1S T f + setToL1S T g := by simp_rw [setToL1S] rw [← SimpleFunc.setToSimpleFunc_add T h_add (SimpleFunc.integrable f) (SimpleFunc.integrable g)] exact SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) (add_toSimpleFunc f g) #align measure_theory.L1.simple_func.set_to_L1s_add MeasureTheory.L1.SimpleFunc.setToL1S_add theorem setToL1S_neg {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (f : α →₁ₛ[μ] E) : setToL1S T (-f) = -setToL1S T f := by simp_rw [setToL1S] have : simpleFunc.toSimpleFunc (-f) =ᵐ[μ] ⇑(-simpleFunc.toSimpleFunc f) := neg_toSimpleFunc f rw [SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) this] exact SimpleFunc.setToSimpleFunc_neg T h_add (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_neg MeasureTheory.L1.SimpleFunc.setToL1S_neg theorem setToL1S_sub {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (f g : α →₁ₛ[μ] E) : setToL1S T (f - g) = setToL1S T f - setToL1S T g := by rw [sub_eq_add_neg, setToL1S_add T h_zero h_add, setToL1S_neg h_zero h_add, sub_eq_add_neg] #align measure_theory.L1.simple_func.set_to_L1s_sub MeasureTheory.L1.SimpleFunc.setToL1S_sub theorem setToL1S_smul_real (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (c : ℝ) (f : α →₁ₛ[μ] E) : setToL1S T (c • f) = c • setToL1S T f := by simp_rw [setToL1S] rw [← SimpleFunc.setToSimpleFunc_smul_real T h_add c (SimpleFunc.integrable f)] refine SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) ?_ exact smul_toSimpleFunc c f #align measure_theory.L1.simple_func.set_to_L1s_smul_real MeasureTheory.L1.SimpleFunc.setToL1S_smul_real theorem setToL1S_smul {E} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜) (f : α →₁ₛ[μ] E) : setToL1S T (c • f) = c • setToL1S T f := by simp_rw [setToL1S] rw [← SimpleFunc.setToSimpleFunc_smul T h_add h_smul c (SimpleFunc.integrable f)] refine SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) ?_ exact smul_toSimpleFunc c f #align measure_theory.L1.simple_func.set_to_L1s_smul MeasureTheory.L1.SimpleFunc.setToL1S_smul theorem norm_setToL1S_le (T : Set α → E →L[ℝ] F) {C : ℝ} (hT_norm : ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal) (f : α →₁ₛ[μ] E) : ‖setToL1S T f‖ ≤ C * ‖f‖ := by rw [setToL1S, norm_eq_sum_mul f] exact SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm_of_integrable T hT_norm _ (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.norm_set_to_L1s_le MeasureTheory.L1.SimpleFunc.norm_setToL1S_le theorem setToL1S_indicatorConst {T : Set α → E →L[ℝ] F} {s : Set α} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (hs : MeasurableSet s) (hμs : μ s < ∞) (x : E) : setToL1S T (simpleFunc.indicatorConst 1 hs hμs.ne x) = T s x := by have h_empty : T ∅ = 0 := h_zero _ MeasurableSet.empty measure_empty rw [setToL1S_eq_setToSimpleFunc] refine Eq.trans ?_ (SimpleFunc.setToSimpleFunc_indicator T h_empty hs x) refine SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) ?_ exact toSimpleFunc_indicatorConst hs hμs.ne x #align measure_theory.L1.simple_func.set_to_L1s_indicator_const MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst theorem setToL1S_const [IsFiniteMeasure μ] {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (x : E) : setToL1S T (simpleFunc.indicatorConst 1 MeasurableSet.univ (measure_ne_top μ _) x) = T univ x := setToL1S_indicatorConst h_zero h_add MeasurableSet.univ (measure_lt_top _ _) x #align measure_theory.L1.simple_func.set_to_L1s_const MeasureTheory.L1.SimpleFunc.setToL1S_const section Order variable {G'' G' : Type} [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] {T : Set α → G'' →L[ℝ] G'} theorem setToL1S_mono_left {T T' : Set α → E →L[ℝ] G''} (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) : setToL1S T f ≤ setToL1S T' f := SimpleFunc.setToSimpleFunc_mono_left T T' hTT' _ #align measure_theory.L1.simple_func.set_to_L1s_mono_left MeasureTheory.L1.SimpleFunc.setToL1S_mono_left theorem setToL1S_mono_left' {T T' : Set α → E →L[ℝ] G''} (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) : setToL1S T f ≤ setToL1S T' f := SimpleFunc.setToSimpleFunc_mono_left' T T' hTT' _ (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_mono_left' MeasureTheory.L1.SimpleFunc.setToL1S_mono_left' theorem setToL1S_nonneg (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α →₁ₛ[μ] G''} (hf : 0 ≤ f) : 0 ≤ setToL1S T f := by simp_rw [setToL1S] obtain ⟨f', hf', hff'⟩ : ∃ f' : α →ₛ G'', 0 ≤ f' ∧ simpleFunc.toSimpleFunc f =ᵐ[μ] f' := by obtain ⟨f'', hf'', hff''⟩ := exists_simpleFunc_nonneg_ae_eq hf exact ⟨f'', hf'', (Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff''⟩ rw [SimpleFunc.setToSimpleFunc_congr _ h_zero h_add (SimpleFunc.integrable _) hff'] exact SimpleFunc.setToSimpleFunc_nonneg' T hT_nonneg _ hf' ((SimpleFunc.integrable f).congr hff') #align measure_theory.L1.simple_func.set_to_L1s_nonneg MeasureTheory.L1.SimpleFunc.setToL1S_nonneg theorem setToL1S_mono (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁ₛ[μ] G''} (hfg : f ≤ g) : setToL1S T f ≤ setToL1S T g := by rw [← sub_nonneg] at hfg ⊢ rw [← setToL1S_sub h_zero h_add] exact setToL1S_nonneg h_zero h_add hT_nonneg hfg #align measure_theory.L1.simple_func.set_to_L1s_mono MeasureTheory.L1.SimpleFunc.setToL1S_mono end Order variable [NormedSpace 𝕜 F] variable (α E μ 𝕜) /-- Extend `Set α → E →L[ℝ] F` to `(α →₁ₛ[μ] E) →L[𝕜] F`. -/ def setToL1SCLM' {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) : (α →₁ₛ[μ] E) →L[𝕜] F := LinearMap.mkContinuous ⟨⟨setToL1S T, setToL1S_add T (fun _ => hT.eq_zero_of_measure_zero) hT.1⟩, setToL1S_smul T (fun _ => hT.eq_zero_of_measure_zero) hT.1 h_smul⟩ C fun f => norm_setToL1S_le T hT.2 f #align measure_theory.L1.simple_func.set_to_L1s_clm' MeasureTheory.L1.SimpleFunc.setToL1SCLM' /-- Extend `Set α → E →L[ℝ] F` to `(α →₁ₛ[μ] E) →L[ℝ] F`. -/ def setToL1SCLM {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) : (α →₁ₛ[μ] E) →L[ℝ] F := LinearMap.mkContinuous ⟨⟨setToL1S T, setToL1S_add T (fun _ => hT.eq_zero_of_measure_zero) hT.1⟩, setToL1S_smul_real T (fun _ => hT.eq_zero_of_measure_zero) hT.1⟩ C fun f => norm_setToL1S_le T hT.2 f #align measure_theory.L1.simple_func.set_to_L1s_clm MeasureTheory.L1.SimpleFunc.setToL1SCLM variable {α E μ 𝕜} variable {T T' T'' : Set α → E →L[ℝ] F} {C C' C'' : ℝ} @[simp] theorem setToL1SCLM_zero_left (hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT f = 0 := setToL1S_zero_left _ #align measure_theory.L1.simple_func.set_to_L1s_clm_zero_left MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left theorem setToL1SCLM_zero_left' (hT : DominatedFinMeasAdditive μ T C) (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT f = 0 := setToL1S_zero_left' h_zero f #align measure_theory.L1.simple_func.set_to_L1s_clm_zero_left' MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left' theorem setToL1SCLM_congr_left (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT f = setToL1SCLM α E μ hT' f := setToL1S_congr_left T T' (fun _ _ _ => by rw [h]) f #align measure_theory.L1.simple_func.set_to_L1s_clm_congr_left MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left theorem setToL1SCLM_congr_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT f = setToL1SCLM α E μ hT' f := setToL1S_congr_left T T' h f #align measure_theory.L1.simple_func.set_to_L1s_clm_congr_left' MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left' theorem setToL1SCLM_congr_measure {μ' : Measure α} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ' T C') (hμ : μ ≪ μ') (f : α →₁ₛ[μ] E) (f' : α →₁ₛ[μ'] E) (h : (f : α → E) =ᵐ[μ] f') : setToL1SCLM α E μ hT f = setToL1SCLM α E μ' hT' f' := setToL1S_congr_measure T (fun _ => hT.eq_zero_of_measure_zero) hT.1 hμ _ _ h #align measure_theory.L1.simple_func.set_to_L1s_clm_congr_measure MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_measure theorem setToL1SCLM_add_left (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ (hT.add hT') f = setToL1SCLM α E μ hT f + setToL1SCLM α E μ hT' f := setToL1S_add_left T T' f #align measure_theory.L1.simple_func.set_to_L1s_clm_add_left MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left theorem setToL1SCLM_add_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT'' f = setToL1SCLM α E μ hT f + setToL1SCLM α E μ hT' f := setToL1S_add_left' T T' T'' h_add f #align measure_theory.L1.simple_func.set_to_L1s_clm_add_left' MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left' theorem setToL1SCLM_smul_left (c : ℝ) (hT : DominatedFinMeasAdditive μ T C) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ (hT.smul c) f = c • setToL1SCLM α E μ hT f := setToL1S_smul_left T c f #align measure_theory.L1.simple_func.set_to_L1s_clm_smul_left MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left theorem setToL1SCLM_smul_left' (c : ℝ) (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT' f = c • setToL1SCLM α E μ hT f := setToL1S_smul_left' T T' c h_smul f #align measure_theory.L1.simple_func.set_to_L1s_clm_smul_left' MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left' theorem norm_setToL1SCLM_le {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : ‖setToL1SCLM α E μ hT‖ ≤ C := LinearMap.mkContinuous_norm_le _ hC _ #align measure_theory.L1.simple_func.norm_set_to_L1s_clm_le MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le theorem norm_setToL1SCLM_le' {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) : ‖setToL1SCLM α E μ hT‖ ≤ max C 0 := LinearMap.mkContinuous_norm_le' _ _ #align measure_theory.L1.simple_func.norm_set_to_L1s_clm_le' MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le' theorem setToL1SCLM_const [IsFiniteMeasure μ] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (x : E) : setToL1SCLM α E μ hT (simpleFunc.indicatorConst 1 MeasurableSet.univ (measure_ne_top μ _) x) = T univ x := setToL1S_const (fun _ => hT.eq_zero_of_measure_zero) hT.1 x #align measure_theory.L1.simple_func.set_to_L1s_clm_const MeasureTheory.L1.SimpleFunc.setToL1SCLM_const section Order variable {G' G'' : Type} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] theorem setToL1SCLM_mono_left {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT f ≤ setToL1SCLM α E μ hT' f := SimpleFunc.setToSimpleFunc_mono_left T T' hTT' _ #align measure_theory.L1.simple_func.set_to_L1s_clm_mono_left MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left theorem setToL1SCLM_mono_left' {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ hT f ≤ setToL1SCLM α E μ hT' f := SimpleFunc.setToSimpleFunc_mono_left' T T' hTT' _ (SimpleFunc.integrable f) #align measure_theory.L1.simple_func.set_to_L1s_clm_mono_left' MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left' theorem setToL1SCLM_nonneg {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α →₁ₛ[μ] G'} (hf : 0 ≤ f) : 0 ≤ setToL1SCLM α G' μ hT f := setToL1S_nonneg (fun _ => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg hf #align measure_theory.L1.simple_func.set_to_L1s_clm_nonneg MeasureTheory.L1.SimpleFunc.setToL1SCLM_nonneg theorem setToL1SCLM_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁ₛ[μ] G'} (hfg : f ≤ g) : setToL1SCLM α G' μ hT f ≤ setToL1SCLM α G' μ hT g := setToL1S_mono (fun _ => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg hfg #align measure_theory.L1.simple_func.set_to_L1s_clm_mono MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono end Order end SetToL1S end SimpleFunc open SimpleFunc section SetToL1 attribute [local instance] Lp.simpleFunc.module attribute [local instance] Lp.simpleFunc.normedSpace variable (𝕜) [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [CompleteSpace F] {T T' T'' : Set α → E →L[ℝ] F} {C C' C'' : ℝ} /-- Extend `set α → (E →L[ℝ] F)` to `(α →₁[μ] E) →L[𝕜] F`. -/ def setToL1' (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) : (α →₁[μ] E) →L[𝕜] F := (setToL1SCLM' α E 𝕜 μ hT h_smul).extend (coeToLp α E 𝕜) (simpleFunc.denseRange one_ne_top) simpleFunc.uniformInducing #align measure_theory.L1.set_to_L1' MeasureTheory.L1.setToL1' variable {𝕜} /-- Extend `Set α → E →L[ℝ] F` to `(α →₁[μ] E) →L[ℝ] F`. -/ def setToL1 (hT : DominatedFinMeasAdditive μ T C) : (α →₁[μ] E) →L[ℝ] F := (setToL1SCLM α E μ hT).extend (coeToLp α E ℝ) (simpleFunc.denseRange one_ne_top) simpleFunc.uniformInducing #align measure_theory.L1.set_to_L1 MeasureTheory.L1.setToL1 theorem setToL1_eq_setToL1SCLM (hT : DominatedFinMeasAdditive μ T C) (f : α →₁ₛ[μ] E) : setToL1 hT f = setToL1SCLM α E μ hT f := uniformly_extend_of_ind simpleFunc.uniformInducing (simpleFunc.denseRange one_ne_top) (setToL1SCLM α E μ hT).uniformContinuous _ #align measure_theory.L1.set_to_L1_eq_set_to_L1s_clm MeasureTheory.L1.setToL1_eq_setToL1SCLM theorem setToL1_eq_setToL1' (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (f : α →₁[μ] E) : setToL1 hT f = setToL1' 𝕜 hT h_smul f := rfl #align measure_theory.L1.set_to_L1_eq_set_to_L1' MeasureTheory.L1.setToL1_eq_setToL1' @[simp] theorem setToL1_zero_left (hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C) (f : α →₁[μ] E) : setToL1 hT f = 0 := by suffices setToL1 hT = 0 by rw [this]; simp refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_ ext1 f rw [setToL1SCLM_zero_left hT f, ContinuousLinearMap.zero_comp, ContinuousLinearMap.zero_apply] #align measure_theory.L1.set_to_L1_zero_left MeasureTheory.L1.setToL1_zero_left theorem setToL1_zero_left' (hT : DominatedFinMeasAdditive μ T C) (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁[μ] E) : setToL1 hT f = 0 := by suffices setToL1 hT = 0 by rw [this]; simp refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_ ext1 f rw [setToL1SCLM_zero_left' hT h_zero f, ContinuousLinearMap.zero_comp, ContinuousLinearMap.zero_apply] #align measure_theory.L1.set_to_L1_zero_left' MeasureTheory.L1.setToL1_zero_left' theorem setToL1_congr_left (T T' : Set α → E →L[ℝ] F) {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : α →₁[μ] E) : setToL1 hT f = setToL1 hT' f := by suffices setToL1 hT = setToL1 hT' by rw [this] refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_ ext1 f suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; rfl rw [setToL1_eq_setToL1SCLM] exact setToL1SCLM_congr_left hT' hT h.symm f #align measure_theory.L1.set_to_L1_congr_left MeasureTheory.L1.setToL1_congr_left theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) : setToL1 hT f = setToL1 hT' f := by suffices setToL1 hT = setToL1 hT' by rw [this] refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_ ext1 f suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; rfl rw [setToL1_eq_setToL1SCLM] exact (setToL1SCLM_congr_left' hT hT' h f).symm #align measure_theory.L1.set_to_L1_congr_left' MeasureTheory.L1.setToL1_congr_left' theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) : setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f := by suffices setToL1 (hT.add hT') = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply] refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.add hT')) _ _ _ _ ?_ ext1 f suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ (hT.add hT') f by rw [← this]; rfl rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM, setToL1SCLM_add_left hT hT'] #align measure_theory.L1.set_to_L1_add_left MeasureTheory.L1.setToL1_add_left theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) : setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply] refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT'') _ _ _ _ ?_ ext1 f suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ hT'' f by rw [← this]; congr rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL1SCLM, setToL1SCLM_add_left' hT hT' hT'' h_add] #align measure_theory.L1.set_to_L1_add_left' MeasureTheory.L1.setToL1_add_left' theorem setToL1_smul_left (hT : DominatedFinMeasAdditive μ T C) (c : ℝ) (f : α →₁[μ] E) : setToL1 (hT.smul c) f = c • setToL1 hT f := by suffices setToL1 (hT.smul c) = c • setToL1 hT by rw [this, ContinuousLinearMap.smul_apply] refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.smul c)) _ _ _ _ ?_ ext1 f suffices c • setToL1 hT f = setToL1SCLM α E μ (hT.smul c) f by rw [← this]; congr rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left c hT] #align measure_theory.L1.set_to_L1_smul_left MeasureTheory.L1.setToL1_smul_left theorem setToL1_smul_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (c : ℝ) (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁[μ] E) : setToL1 hT' f = c • setToL1 hT f := by suffices setToL1 hT' = c • setToL1 hT by rw [this, ContinuousLinearMap.smul_apply] refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT') _ _ _ _ ?_ ext1 f suffices c • setToL1 hT f = setToL1SCLM α E μ hT' f by rw [← this]; congr rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left' c hT hT' h_smul] #align measure_theory.L1.set_to_L1_smul_left' MeasureTheory.L1.setToL1_smul_left' theorem setToL1_smul (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜) (f : α →₁[μ] E) : setToL1 hT (c • f) = c • setToL1 hT f := by rw [setToL1_eq_setToL1' hT h_smul, setToL1_eq_setToL1' hT h_smul] exact ContinuousLinearMap.map_smul _ _ _ #align measure_theory.L1.set_to_L1_smul MeasureTheory.L1.setToL1_smul theorem setToL1_simpleFunc_indicatorConst (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞) (x : E) : setToL1 hT (simpleFunc.indicatorConst 1 hs hμs.ne x) = T s x := by rw [setToL1_eq_setToL1SCLM] exact setToL1S_indicatorConst (fun s => hT.eq_zero_of_measure_zero) hT.1 hs hμs x #align measure_theory.L1.set_to_L1_simple_func_indicator_const MeasureTheory.L1.setToL1_simpleFunc_indicatorConst theorem setToL1_indicatorConstLp (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) : setToL1 hT (indicatorConstLp 1 hs hμs x) = T s x := by rw [← Lp.simpleFunc.coe_indicatorConst hs hμs x] exact setToL1_simpleFunc_indicatorConst hT hs hμs.lt_top x #align measure_theory.L1.set_to_L1_indicator_const_Lp MeasureTheory.L1.setToL1_indicatorConstLp theorem setToL1_const [IsFiniteMeasure μ] (hT : DominatedFinMeasAdditive μ T C) (x : E) : setToL1 hT (indicatorConstLp 1 MeasurableSet.univ (measure_ne_top _ _) x) = T univ x := setToL1_indicatorConstLp hT MeasurableSet.univ (measure_ne_top _ _) x #align measure_theory.L1.set_to_L1_const MeasureTheory.L1.setToL1_const section Order variable {G' G'' : Type} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] theorem setToL1_mono_left' {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →₁[μ] E) : setToL1 hT f ≤ setToL1 hT' f := by induction f using Lp.induction (hp_ne_top := one_ne_top) with \| @h_ind c s hs hμs => rw [setToL1_simpleFunc_indicatorConst hT hs hμs, setToL1_simpleFunc_indicatorConst hT' hs hμs] exact hTT' s hs hμs c \| @h_add f g hf hg _ hf_le hg_le => rw [(setToL1 hT).map_add, (setToL1 hT').map_add] exact add_le_add hf_le hg_le \| h_closed => exact isClosed_le (setToL1 hT).continuous (setToL1 hT').continuous #align measure_theory.L1.set_to_L1_mono_left' MeasureTheory.L1.setToL1_mono_left' theorem setToL1_mono_left {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁[μ] E) : setToL1 hT f ≤ setToL1 hT' f := setToL1_mono_left' hT hT' (fun s _ _ x => hTT' s x) f #align measure_theory.L1.set_to_L1_mono_left MeasureTheory.L1.setToL1_mono_left theorem setToL1_nonneg {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α →₁[μ] G'} (hf : 0 ≤ f) : 0 ≤ setToL1 hT f := by suffices ∀ f : { g : α →₁[μ] G' // 0 ≤ g }, 0 ≤ setToL1 hT f from this (⟨f, hf⟩ : { g : α →₁[μ] G' // 0 ≤ g }) refine fun g => @isClosed_property { g : α →₁ₛ[μ] G' // 0 ≤ g } { g : α →₁[μ] G' // 0 ≤ g } _ _ (fun g => 0 ≤ setToL1 hT g) (denseRange_coeSimpleFuncNonnegToLpNonneg 1 μ G' one_ne_top) ?_ ?_ g · exact isClosed_le continuous_zero ((setToL1 hT).continuous.comp continuous_induced_dom) · intro g have : (coeSimpleFuncNonnegToLpNonneg 1 μ G' g : α →₁[μ] G') = (g : α →₁ₛ[μ] G') := rfl rw [this, setToL1_eq_setToL1SCLM] exact setToL1S_nonneg (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2 #align measure_theory.L1.set_to_L1_nonneg MeasureTheory.L1.setToL1_nonneg theorem setToL1_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁[μ] G'} (hfg : f ≤ g) : setToL1 hT f ≤ setToL1 hT g := by rw [← sub_nonneg] at hfg ⊢ rw [← (setToL1 hT).map_sub] exact setToL1_nonneg hT hT_nonneg hfg #align measure_theory.L1.set_to_L1_mono MeasureTheory.L1.setToL1_mono end Order theorem norm_setToL1_le_norm_setToL1SCLM (hT : DominatedFinMeasAdditive μ T C) : ‖setToL1 hT‖ ≤ ‖setToL1SCLM α E μ hT‖ := calc ‖setToL1 hT‖ ≤ (1 : ℝ≥0) ‖setToL1SCLM α E μ hT‖ := by refine ContinuousLinearMap.opNorm_extend_le (setToL1SCLM α E μ hT) (coeToLp α E ℝ) (simpleFunc.denseRange one_ne_top) fun x => le_of_eq ?_ rw [NNReal.coe_one, one_mul] rfl _ = ‖setToL1SCLM α E μ hT‖ := by rw [NNReal.coe_one, one_mul] #align measure_theory.L1.norm_set_to_L1_le_norm_set_to_L1s_clm MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM theorem norm_setToL1_le_mul_norm (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) (f : α →₁[μ] E) : ‖setToL1 hT f‖ ≤ C * ‖f‖ := calc ‖setToL1 hT f‖ ≤ ‖setToL1SCLM α E μ hT‖ * ‖f‖ := ContinuousLinearMap.le_of_opNorm_le _ (norm_setToL1_le_norm_setToL1SCLM hT) _ _ ≤ C * ‖f‖ := mul_le_mul (norm_setToL1SCLM_le hT hC) le_rfl (norm_nonneg _) hC #align measure_theory.L1.norm_set_to_L1_le_mul_norm MeasureTheory.L1.norm_setToL1_le_mul_norm theorem norm_setToL1_le_mul_norm' (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) : ‖setToL1 hT f‖ ≤ max C 0 * ‖f‖ := calc ‖setToL1 hT f‖ ≤ ‖setToL1SCLM α E μ hT‖ * ‖f‖ := ContinuousLinearMap.le_of_opNorm_le _ (norm_setToL1_le_norm_setToL1SCLM hT) _ _ ≤ max C 0 * ‖f‖ := mul_le_mul (norm_setToL1SCLM_le' hT) le_rfl (norm_nonneg _) (le_max_right _ _) #align measure_theory.L1.norm_set_to_L1_le_mul_norm' MeasureTheory.L1.norm_setToL1_le_mul_norm' theorem norm_setToL1_le (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : ‖setToL1 hT‖ ≤ C := ContinuousLinearMap.opNorm_le_bound _ hC (norm_setToL1_le_mul_norm hT hC) #align measure_theory.L1.norm_set_to_L1_le MeasureTheory.L1.norm_setToL1_le theorem norm_setToL1_le' (hT : DominatedFinMeasAdditive μ T C) : ‖setToL1 hT‖ ≤ max C 0 := ContinuousLinearMap.opNorm_le_bound _ (le_max_right _ _) (norm_setToL1_le_mul_norm' hT) #align measure_theory.L1.norm_set_to_L1_le' MeasureTheory.L1.norm_setToL1_le' theorem setToL1_lipschitz (hT : DominatedFinMeasAdditive μ T C) : LipschitzWith (Real.toNNReal C) (setToL1 hT) := (setToL1 hT).lipschitz.weaken (norm_setToL1_le' hT) #align measure_theory.L1.set_to_L1_lipschitz MeasureTheory.L1.setToL1_lipschitz /-- If `fs i → f` in `L1`, then `setToL1 hT (fs i) → setToL1 hT f`. -/ theorem tendsto_setToL1 (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) {ι} (fs : ι → α →₁[μ] E) {l : Filter ι} (hfs : Tendsto fs l (𝓝 f)) : Tendsto (fun i => setToL1 hT (fs i)) l (𝓝 <\| setToL1 hT f) := ((setToL1 hT).continuous.tendsto _).comp hfs #align measure_theory.L1.tendsto_set_to_L1 MeasureTheory.L1.tendsto_setToL1 end SetToL1 end L1 section Function set_option linter.uppercaseLean3 false variable [CompleteSpace F] {T T' T'' : Set α → E →L[ℝ] F} {C C' C'' : ℝ} {f g : α → E} variable (μ T) /-- Extend `T : Set α → E →L[ℝ] F` to `(α → E) → F` (for integrable functions `α → E`). We set it to 0 if the function is not integrable. -/ def setToFun (hT : DominatedFinMeasAdditive μ T C) (f : α → E) : F := if hf : Integrable f μ then L1.setToL1 hT (hf.toL1 f) else 0 #align measure_theory.set_to_fun MeasureTheory.setToFun variable {μ T} theorem setToFun_eq (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) : setToFun μ T hT f = L1.setToL1 hT (hf.toL1 f) := dif_pos hf #align measure_theory.set_to_fun_eq MeasureTheory.setToFun_eq theorem L1.setToFun_eq_setToL1 (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) : setToFun μ T hT f = L1.setToL1 hT f := by rw [setToFun_eq hT (L1.integrable_coeFn f), Integrable.toL1_coeFn] #align measure_theory.L1.set_to_fun_eq_set_to_L1 MeasureTheory.L1.setToFun_eq_setToL1 theorem setToFun_undef (hT : DominatedFinMeasAdditive μ T C) (hf : ¬Integrable f μ) : setToFun μ T hT f = 0 := dif_neg hf #align measure_theory.set_to_fun_undef MeasureTheory.setToFun_undef theorem setToFun_non_aEStronglyMeasurable (hT : DominatedFinMeasAdditive μ T C) (hf : ¬AEStronglyMeasurable f μ) : setToFun μ T hT f = 0 := setToFun_undef hT (not_and_of_not_left _ hf) #align measure_theory.set_to_fun_non_ae_strongly_measurable MeasureTheory.setToFun_non_aEStronglyMeasurable theorem setToFun_congr_left (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : α → E) : setToFun μ T hT f = setToFun μ T' hT' f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h] · simp_rw [setToFun_undef _ hf] #align measure_theory.set_to_fun_congr_left MeasureTheory.setToFun_congr_left theorem setToFun_congr_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α → E) : setToFun μ T hT f = setToFun μ T' hT' f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left' T T' hT hT' h] · simp_rw [setToFun_undef _ hf] #align measure_theory.set_to_fun_congr_left' MeasureTheory.setToFun_congr_left' theorem setToFun_add_left (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (f : α → E) : setToFun μ (T + T') (hT.add hT') f = setToFun μ T hT f + setToFun μ T' hT' f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf, L1.setToL1_add_left hT hT'] · simp_rw [setToFun_undef _ hf, add_zero] #align measure_theory.set_to_fun_add_left MeasureTheory.setToFun_add_left theorem setToFun_add_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α → E) : setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf, L1.setToL1_add_left' hT hT' hT'' h_add] · simp_rw [setToFun_undef _ hf, add_zero] #align measure_theory.set_to_fun_add_left' MeasureTheory.setToFun_add_left' theorem setToFun_smul_left (hT : DominatedFinMeasAdditive μ T C) (c : ℝ) (f : α → E) : setToFun μ (fun s => c • T s) (hT.smul c) f = c • setToFun μ T hT f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left hT c] · simp_rw [setToFun_undef _ hf, smul_zero] #align measure_theory.set_to_fun_smul_left MeasureTheory.setToFun_smul_left theorem setToFun_smul_left' (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (c : ℝ) (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α → E) : setToFun μ T' hT' f = c • setToFun μ T hT f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left' hT hT' c h_smul] · simp_rw [setToFun_undef _ hf, smul_zero] #align measure_theory.set_to_fun_smul_left' MeasureTheory.setToFun_smul_left' @[simp] theorem setToFun_zero (hT : DominatedFinMeasAdditive μ T C) : setToFun μ T hT (0 : α → E) = 0 := by erw [setToFun_eq hT (integrable_zero _ _ _), Integrable.toL1_zero, ContinuousLinearMap.map_zero] #align measure_theory.set_to_fun_zero MeasureTheory.setToFun_zero @[simp] theorem setToFun_zero_left {hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C} : setToFun μ 0 hT f = 0 := by by_cases hf : Integrable f μ · rw [setToFun_eq hT hf]; exact L1.setToL1_zero_left hT _ · exact setToFun_undef hT hf #align measure_theory.set_to_fun_zero_left MeasureTheory.setToFun_zero_left theorem setToFun_zero_left' (hT : DominatedFinMeasAdditive μ T C) (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) : setToFun μ T hT f = 0 := by by_cases hf : Integrable f μ · rw [setToFun_eq hT hf]; exact L1.setToL1_zero_left' hT h_zero _ · exact setToFun_undef hT hf #align measure_theory.set_to_fun_zero_left' MeasureTheory.setToFun_zero_left' theorem setToFun_add (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hg : Integrable g μ) : setToFun μ T hT (f + g) = setToFun μ T hT f + setToFun μ T hT g := by rw [setToFun_eq hT (hf.add hg), setToFun_eq hT hf, setToFun_eq hT hg, Integrable.toL1_add, (L1.setToL1 hT).map_add] #align measure_theory.set_to_fun_add MeasureTheory.setToFun_add theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) : setToFun μ T hT (∑ i ∈ s, f i) = ∑ i ∈ s, setToFun μ T hT (f i) := by revert hf refine Finset.induction_on s ?_ ?_ · intro _ simp only [setToFun_zero, Finset.sum_empty] · intro i s his ih hf simp only [his, Finset.sum_insert, not_false_iff] rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _] · rw [ih fun i hi => hf i (Finset.mem_insert_of_mem hi)] · convert integrable_finset_sum s fun i hi => hf i (Finset.mem_insert_of_mem hi) with x simp #align measure_theory.set_to_fun_finset_sum' MeasureTheory.setToFun_finset_sum' theorem setToFun_finset_sum (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) : (setToFun μ T hT fun a => ∑ i ∈ s, f i a) = ∑ i ∈ s, setToFun μ T hT (f i) := by convert setToFun_finset_sum' hT s hf with a; simp #align measure_theory.set_to_fun_finset_sum MeasureTheory.setToFun_finset_sum theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) : setToFun μ T hT (-f) = -setToFun μ T hT f := by by_cases hf : Integrable f μ · rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg, (L1.setToL1 hT).map_neg] · rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero] rwa [← integrable_neg_iff] at hf #align measure_theory.set_to_fun_neg MeasureTheory.setToFun_neg theorem setToFun_sub (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hg : Integrable g μ) : setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g := by rw [sub_eq_add_neg, sub_eq_add_neg, setToFun_add hT hf hg.neg, setToFun_neg hT g] #align measure_theory.set_to_fun_sub MeasureTheory.setToFun_sub theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜) (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by by_cases hf : Integrable f μ · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul', L1.setToL1_smul hT h_smul c _] · by_cases hr : c = 0 · rw [hr]; simp · have hf' : ¬Integrable (c • f) μ := by rwa [integrable_smul_iff hr f] rw [setToFun_undef hT hf, setToFun_undef hT hf', smul_zero] #align measure_theory.set_to_fun_smul MeasureTheory.setToFun_smul theorem setToFun_congr_ae (hT : DominatedFinMeasAdditive μ T C) (h : f =ᵐ[μ] g) : setToFun μ T hT f = setToFun μ T hT g := by by_cases hfi : Integrable f μ · have hgi : Integrable g μ := hfi.congr h rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h] · have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi rw [setToFun_undef hT hfi, setToFun_undef hT hgi] #align measure_theory.set_to_fun_congr_ae MeasureTheory.setToFun_congr_ae theorem setToFun_measure_zero (hT : DominatedFinMeasAdditive μ T C) (h : μ = 0) : setToFun μ T hT f = 0 := by have : f =ᵐ[μ] 0 := by simp [h, EventuallyEq] rw [setToFun_congr_ae hT this, setToFun_zero] #align measure_theory.set_to_fun_measure_zero MeasureTheory.setToFun_measure_zero theorem setToFun_measure_zero' (hT : DominatedFinMeasAdditive μ T C) (h : ∀ s, MeasurableSet s → μ s < ∞ → μ s = 0) : setToFun μ T hT f = 0 := setToFun_zero_left' hT fun s hs hμs => hT.eq_zero_of_measure_zero hs (h s hs hμs) #align measure_theory.set_to_fun_measure_zero' MeasureTheory.setToFun_measure_zero' theorem setToFun_toL1 (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) : setToFun μ T hT (hf.toL1 f) = setToFun μ T hT f := setToFun_congr_ae hT hf.coeFn_toL1 #align measure_theory.set_to_fun_to_L1 MeasureTheory.setToFun_toL1 theorem setToFun_indicator_const (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) : setToFun μ T hT (s.indicator fun _ => x) = T s x := by rw [setToFun_congr_ae hT (@indicatorConstLp_coeFn _ _ _ 1 _ _ _ hs hμs x).symm] rw [L1.setToFun_eq_setToL1 hT] exact L1.setToL1_indicatorConstLp hT hs hμs x #align measure_theory.set_to_fun_indicator_const MeasureTheory.setToFun_indicator_const theorem setToFun_const [IsFiniteMeasure μ] (hT : DominatedFinMeasAdditive μ T C) (x : E) : (setToFun μ T hT fun _ => x) = T univ x := by have : (fun _ : α => x) = Set.indicator univ fun _ => x := (indicator_univ _).symm rw [this] exact setToFun_indicator_const hT MeasurableSet.univ (measure_ne_top _ _) x #align measure_theory.set_to_fun_const MeasureTheory.setToFun_const section Order variable {G' G'' : Type} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] theorem setToFun_mono_left' {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α → E) : setToFun μ T hT f ≤ setToFun μ T' hT' f := by by_cases hf : Integrable f μ · simp_rw [setToFun_eq _ hf]; exact L1.setToL1_mono_left' hT hT' hTT' _ · simp_rw [setToFun_undef _ hf]; rfl #align measure_theory.set_to_fun_mono_left' MeasureTheory.setToFun_mono_left' theorem setToFun_mono_left {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁[μ] E) : setToFun μ T hT f ≤ setToFun μ T' hT' f := setToFun_mono_left' hT hT' (fun s _ _ x => hTT' s x) f #align measure_theory.set_to_fun_mono_left MeasureTheory.setToFun_mono_left theorem setToFun_nonneg {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α → G'} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ setToFun μ T hT f := by by_cases hfi : Integrable f μ · simp_rw [setToFun_eq _ hfi] refine L1.setToL1_nonneg hT hT_nonneg ?_ rw [← Lp.coeFn_le] have h0 := Lp.coeFn_zero G' 1 μ have h := Integrable.coeFn_toL1 hfi filter_upwards [h0, h, hf] with _ h0a ha hfa rw [h0a, ha] exact hfa · simp_rw [setToFun_undef _ hfi]; rfl #align measure_theory.set_to_fun_nonneg MeasureTheory.setToFun_nonneg theorem setToFun_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α → G'} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) : setToFun μ T hT f ≤ setToFun μ T hT g := by rw [← sub_nonneg, ← setToFun_sub hT hg hf] refine setToFun_nonneg hT hT_nonneg (hfg.mono fun a ha => ?_) rw [Pi.sub_apply, Pi.zero_apply, sub_nonneg] exact ha #align measure_theory.set_to_fun_mono MeasureTheory.setToFun_mono end Order @[continuity] theorem continuous_setToFun (hT : DominatedFinMeasAdditive μ T C) : Continuous fun f : α →₁[μ] E => setToFun μ T hT f := by simp_rw [L1.setToFun_eq_setToL1 hT]; exact ContinuousLinearMap.continuous _ #align measure_theory.continuous_set_to_fun MeasureTheory.continuous_setToFun /-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/ theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E) (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ) (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) : Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <\| setToFun μ T hT f) := by classical let f_lp := hfi.toL1 f let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0 have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by rw [Lp.tendsto_Lp_iff_tendsto_ℒp'] simp_rw [snorm_one_eq_lintegral_nnnorm, Pi.sub_apply] refine (tendsto_congr' ?_).mp hfs filter_upwards [hfsi] with i hi refine lintegral_congr_ae ?_ filter_upwards [hi.coeFn_toL1, hfi.coeFn_toL1] with x hxi hxf simp_rw [F_lp, dif_pos hi, hxi, hxf] suffices Tendsto (fun i => setToFun μ T hT (F_lp i)) l (𝓝 (setToFun μ T hT f)) by refine (tendsto_congr' ?_).mp this filter_upwards [hfsi] with i hi suffices h_ae_eq : F_lp i =ᵐ[μ] fs i from setToFun_congr_ae hT h_ae_eq simp_rw [F_lp, dif_pos hi] exact hi.coeFn_toL1 rw [setToFun_congr_ae hT hfi.coeFn_toL1.symm] exact ((continuous_setToFun hT).tendsto f_lp).comp tendsto_L1 #align measure_theory.tendsto_set_to_fun_of_L1 MeasureTheory.tendsto_setToFun_of_L1 theorem tendsto_setToFun_approxOn_of_measurable (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E} {s : Set E} [SeparableSpace s] (hfi : Integrable f μ) (hfm : Measurable f) (hs : ∀ᵐ x ∂μ, f x ∈ closure s) {y₀ : E} (h₀ : y₀ ∈ s) (h₀i : Integrable (fun _ => y₀) μ) : Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f hfm s y₀ h₀ n)) atTop (𝓝 <\| setToFun μ T hT f) := tendsto_setToFun_of_L1 hT _ hfi (eventually_of_forall (SimpleFunc.integrable_approxOn hfm hfi h₀ h₀i)) (SimpleFunc.tendsto_approxOn_L1_nnnorm hfm _ hs (hfi.sub h₀i).2) #align measure_theory.tendsto_set_to_fun_approx_on_of_measurable MeasureTheory.tendsto_setToFun_approxOn_of_measurable theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E} (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s] (hs : range f ∪ {0} ⊆ s) : Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f fmeas s 0 (hs <\| by simp) n)) atTop (𝓝 <\| setToFun μ T hT f) := by refine tendsto_setToFun_approxOn_of_measurable hT hf fmeas ?_ _ (integrable_zero _ _ _) exact eventually_of_forall fun x => subset_closure (hs (Set.mem_union_left _ (mem_range_self _))) #align measure_theory.tendsto_set_to_fun_approx_on_of_measurable_of_range_subset MeasureTheory.tendsto_setToFun_approxOn_of_measurable_of_range_subset /-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to `f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/ theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) : Continuous fun f : α →₁[μ] G => (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by by_cases hc'0 : c' = 0 · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine hμ'_le.trans ?_; simp [hc'0] have h_im_zero : (fun f : α →₁[μ] G => (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) = 0 := by ext1 f; ext1; simp_rw [hμ'0]; simp only [ae_zero, EventuallyEq, eventually_bot] rw [h_im_zero] exact continuous_zero rw [Metric.continuous_iff] intro f ε hε_pos use ε / 2 / c'.toReal refine ⟨div_pos (half_pos hε_pos) (toReal_pos hc'0 hc'), ?_⟩ intro g hfg rw [Lp.dist_def] at hfg ⊢ let h_int := fun f' : α →₁[μ] G => (L1.integrable_coeFn f').of_measure_le_smul c' hc' hμ'_le have : snorm (⇑(Integrable.toL1 g (h_int g)) - ⇑(Integrable.toL1 f (h_int f))) 1 μ' = snorm (⇑g - ⇑f) 1 μ' := snorm_congr_ae ((Integrable.coeFn_toL1 _).sub (Integrable.coeFn_toL1 _)) rw [this] have h_snorm_ne_top : snorm (⇑g - ⇑f) 1 μ ≠ ∞ := by rw [← snorm_congr_ae (Lp.coeFn_sub _ _)]; exact Lp.snorm_ne_top _ have h_snorm_ne_top' : snorm (⇑g - ⇑f) 1 μ' ≠ ∞ := by refine ((snorm_mono_measure _ hμ'_le).trans_lt ?_).ne rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul] refine ENNReal.mul_lt_top ?_ h_snorm_ne_top simp [hc', hc'0] calc (snorm (⇑g - ⇑f) 1 μ').toReal ≤ (c' snorm (⇑g - ⇑f) 1 μ).toReal := by rw [toReal_le_toReal h_snorm_ne_top' (ENNReal.mul_ne_top hc' h_snorm_ne_top)] refine (snorm_mono_measure (⇑g - ⇑f) hμ'_le).trans ?_ rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul] simp _ = c'.toReal * (snorm (⇑g - ⇑f) 1 μ).toReal := toReal_mul _ ≤ c'.toReal * (ε / 2 / c'.toReal) := (mul_le_mul le_rfl hfg.le toReal_nonneg toReal_nonneg) _ = ε / 2 := by refine mul_div_cancel₀ (ε / 2) ?_; rw [Ne, toReal_eq_zero_iff]; simp [hc', hc'0] _ < ε := half_lt_self hε_pos #align measure_theory.continuous_L1_to_L1 MeasureTheory.continuous_L1_toL1 theorem setToFun_congr_measure_of_integrable {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : Integrable f μ) : setToFun μ T hT f = setToFun μ' T hT' f := by -- integrability for `μ` implies integrability for `μ'`. have h_int : ∀ g : α → E, Integrable g μ → Integrable g μ' := fun g hg => Integrable.of_measure_le_smul c' hc' hμ'_le hg -- We use `Integrable.induction` apply hfμ.induction (P := fun f => setToFun μ T hT f = setToFun μ' T hT' f) · intro c s hs hμs have hμ's : μ' s ≠ ∞ := by refine ((hμ'_le s).trans_lt ?_).ne rw [Measure.smul_apply, smul_eq_mul] exact ENNReal.mul_lt_top hc' hμs.ne rw [setToFun_indicator_const hT hs hμs.ne, setToFun_indicator_const hT' hs hμ's] · intro f₂ g₂ _ hf₂ hg₂ h_eq_f h_eq_g rw [setToFun_add hT hf₂ hg₂, setToFun_add hT' (h_int f₂ hf₂) (h_int g₂ hg₂), h_eq_f, h_eq_g] · refine isClosed_eq (continuous_setToFun hT) ?_ have : (fun f : α →₁[μ] E => setToFun μ' T hT' f) = fun f : α →₁[μ] E => setToFun μ' T hT' ((h_int f (L1.integrable_coeFn f)).toL1 f) := by ext1 f; exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm rw [this] exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' hμ'_le) · intro f₂ g₂ hfg _ hf_eq have hfg' : f₂ =ᵐ[μ'] g₂ := (Measure.absolutelyContinuous_of_le_smul hμ'_le).ae_eq hfg rw [← setToFun_congr_ae hT hfg, hf_eq, setToFun_congr_ae hT' hfg'] #align measure_theory.set_to_fun_congr_measure_of_integrable MeasureTheory.setToFun_congr_measure_of_integrable theorem setToFun_congr_measure {μ' : Measure α} (c c' : ℝ≥0∞) (hc : c ≠ ∞) (hc' : c' ≠ ∞) (hμ_le : μ ≤ c • μ') (hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) : setToFun μ T hT f = setToFun μ' T hT' f := by by_cases hf : Integrable f μ · exact setToFun_congr_measure_of_integrable c' hc' hμ'_le hT hT' f hf · -- if `f` is not integrable, both `setToFun` are 0. have h_int : ∀ g : α → E, ¬Integrable g μ → ¬Integrable g μ' := fun g => mt fun h => h.of_measure_le_smul _ hc hμ_le simp_rw [setToFun_undef _ hf, setToFun_undef _ (h_int f hf)] #align measure_theory.set_to_fun_congr_measure MeasureTheory.setToFun_congr_measure theorem setToFun_congr_measure_of_add_right {μ' : Measure α} (hT_add : DominatedFinMeasAdditive (μ + μ') T C') (hT : DominatedFinMeasAdditive μ T C) (f : α → E) (hf : Integrable f (μ + μ')) : setToFun (μ + μ') T hT_add f = setToFun μ T hT f := by refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf rw [one_smul] nth_rw 1 [← add_zero μ] exact add_le_add le_rfl bot_le #align measure_theory.set_to_fun_congr_measure_of_add_right MeasureTheory.setToFun_congr_measure_of_add_right theorem setToFun_congr_measure_of_add_left {μ' : Measure α} (hT_add : DominatedFinMeasAdditive (μ + μ') T C') (hT : DominatedFinMeasAdditive μ' T C) (f : α → E) (hf : Integrable f (μ + μ')) : setToFun (μ + μ') T hT_add f = setToFun μ' T hT f := by refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf rw [one_smul] nth_rw 1 [← zero_add μ'] exact add_le_add bot_le le_rfl #align measure_theory.set_to_fun_congr_measure_of_add_left MeasureTheory.setToFun_congr_measure_of_add_left theorem setToFun_top_smul_measure (hT : DominatedFinMeasAdditive (∞ • μ) T C) (f : α → E) : setToFun (∞ • μ) T hT f = 0 := by refine setToFun_measure_zero' hT fun s _ hμs => ?_ rw [lt_top_iff_ne_top] at hμs simp only [true_and_iff, Measure.smul_apply, ENNReal.mul_eq_top, eq_self_iff_true, top_ne_zero, Ne, not_false_iff, not_or, Classical.not_not, smul_eq_mul] at hμs simp only [hμs.right, Measure.smul_apply, mul_zero, smul_eq_mul] #align measure_theory.set_to_fun_top_smul_measure MeasureTheory.setToFun_top_smul_measure theorem setToFun_congr_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : DominatedFinMeasAdditive μ T C) (hT_smul : DominatedFinMeasAdditive (c • μ) T C') (f : α → E) : setToFun μ T hT f = setToFun (c • μ) T hT_smul f := by by_cases hc0 : c = 0 · simp [hc0] at hT_smul have h : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0 := fun s hs _ => hT_smul.eq_zero hs rw [setToFun_zero_left' _ h, setToFun_measure_zero] simp [hc0] refine setToFun_congr_measure c⁻¹ c ?_ hc_ne_top (le_of_eq ?_) le_rfl hT hT_smul f · simp [hc0] · rw [smul_smul, ENNReal.inv_mul_cancel hc0 hc_ne_top, one_smul] #align measure_theory.set_to_fun_congr_smul_measure MeasureTheory.setToFun_congr_smul_measure theorem norm_setToFun_le_mul_norm (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) (hC : 0 ≤ C) : ‖setToFun μ T hT f‖ ≤ C * ‖f‖ := by rw [L1.setToFun_eq_setToL1]; exact L1.norm_setToL1_le_mul_norm hT hC f #align measure_theory.norm_set_to_fun_le_mul_norm MeasureTheory.norm_setToFun_le_mul_norm	Mathlib/MeasureTheory/Integral/SetToL1.lean	1,683	1,685	theorem norm_setToFun_le_mul_norm' (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) : ‖setToFun μ T hT f‖ ≤ max C 0 * ‖f‖ := by	rw [L1.setToFun_eq_setToL1]; exact L1.norm_setToL1_le_mul_norm' hT f
/- Copyright (c) 2017 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax import Mathlib.Algebra.Order.Monoid.WithTop import Mathlib.Data.Finset.Image import Mathlib.Data.Multiset.Fold #align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" /-! # The fold operation for a commutative associative operation over a finset. -/ -- TODO: -- assert_not_exists OrderedCommMonoid assert_not_exists MonoidWithZero namespace Finset open Multiset variable {α β γ : Type} /-! ### fold -/ section Fold variable (op : β → β → β) [hc : Std.Commutative op] [ha : Std.Associative op] local notation a " " b => op a b /-- `fold op b f s` folds the commutative associative operation `op` over the `f`-image of `s`, i.e. `fold (+) b f {1,2,3} = f 1 + f 2 + f 3 + b`. -/ def fold (b : β) (f : α → β) (s : Finset α) : β := (s.1.map f).fold op b #align finset.fold Finset.fold variable {op} {f : α → β} {b : β} {s : Finset α} {a : α} @[simp] theorem fold_empty : (∅ : Finset α).fold op b f = b := rfl #align finset.fold_empty Finset.fold_empty @[simp] theorem fold_cons (h : a ∉ s) : (cons a s h).fold op b f = f a * s.fold op b f := by dsimp only [fold] rw [cons_val, Multiset.map_cons, fold_cons_left] #align finset.fold_cons Finset.fold_cons @[simp] theorem fold_insert [DecidableEq α] (h : a ∉ s) : (insert a s).fold op b f = f a * s.fold op b f := by unfold fold rw [insert_val, ndinsert_of_not_mem h, Multiset.map_cons, fold_cons_left] #align finset.fold_insert Finset.fold_insert @[simp] theorem fold_singleton : ({a} : Finset α).fold op b f = f a * b := rfl #align finset.fold_singleton Finset.fold_singleton @[simp] theorem fold_map {g : γ ↪ α} {s : Finset γ} : (s.map g).fold op b f = s.fold op b (f ∘ g) := by simp only [fold, map, Multiset.map_map] #align finset.fold_map Finset.fold_map @[simp] theorem fold_image [DecidableEq α] {g : γ → α} {s : Finset γ} (H : ∀ x ∈ s, ∀ y ∈ s, g x = g y → x = y) : (s.image g).fold op b f = s.fold op b (f ∘ g) := by simp only [fold, image_val_of_injOn H, Multiset.map_map] #align finset.fold_image Finset.fold_image @[congr] theorem fold_congr {g : α → β} (H : ∀ x ∈ s, f x = g x) : s.fold op b f = s.fold op b g := by rw [fold, fold, map_congr rfl H] #align finset.fold_congr Finset.fold_congr theorem fold_op_distrib {f g : α → β} {b₁ b₂ : β} : (s.fold op (b₁ * b₂) fun x => f x * g x) = s.fold op b₁ f * s.fold op b₂ g := by simp only [fold, fold_distrib] #align finset.fold_op_distrib Finset.fold_op_distrib	Mathlib/Data/Finset/Fold.lean	88	96	theorem fold_const [hd : Decidable (s = ∅)] (c : β) (h : op c (op b c) = op b c) : Finset.fold op b (fun _ => c) s = if s = ∅ then b else op b c := by	classical induction' s using Finset.induction_on with x s hx IH generalizing hd · simp · simp only [Finset.fold_insert hx, IH, if_false, Finset.insert_ne_empty] split_ifs · rw [hc.comm] · exact h
/- Copyright (c) 2018 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison, Markus Himmel -/ import Mathlib.CategoryTheory.EpiMono import Mathlib.CategoryTheory.Limits.HasLimits #align_import category_theory.limits.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba" /-! # Equalizers and coequalizers This file defines (co)equalizers as special cases of (co)limits. An equalizer is the categorical generalization of the subobject {a ∈ A \| f(a) = g(a)} known from abelian groups or modules. It is a limit cone over the diagram formed by `f` and `g`. A coequalizer is the dual concept. ## Main definitions * `WalkingParallelPair` is the indexing category used for (co)equalizer_diagrams * `parallelPair` is a functor from `WalkingParallelPair` to our category `C`. * a `fork` is a cone over a parallel pair. * there is really only one interesting morphism in a fork: the arrow from the vertex of the fork to the domain of f and g. It is called `fork.ι`. * an `equalizer` is now just a `limit (parallelPair f g)` Each of these has a dual. ## Main statements * `equalizer.ι_mono` states that every equalizer map is a monomorphism * `isIso_limit_cone_parallelPair_of_self` states that the identity on the domain of `f` is an equalizer of `f` and `f`. ## Implementation notes As with the other special shapes in the limits library, all the definitions here are given as `abbreviation`s of the general statements for limits, so all the `simp` lemmas and theorems about general limits can be used. ## References * [F. Borceux, Handbook of Categorical Algebra 1][borceux-vol1] -/ /- Porting note: removed global noncomputable since there are things that might be computable value like WalkingPair -/ section open CategoryTheory Opposite namespace CategoryTheory.Limits -- attribute [local tidy] tactic.case_bash -- Porting note: no tidy nor cases_bash universe v v₂ u u₂ /-- The type of objects for the diagram indexing a (co)equalizer. -/ inductive WalkingParallelPair : Type \| zero \| one deriving DecidableEq, Inhabited #align category_theory.limits.walking_parallel_pair CategoryTheory.Limits.WalkingParallelPair open WalkingParallelPair /-- The type family of morphisms for the diagram indexing a (co)equalizer. -/ inductive WalkingParallelPairHom : WalkingParallelPair → WalkingParallelPair → Type \| left : WalkingParallelPairHom zero one \| right : WalkingParallelPairHom zero one \| id (X : WalkingParallelPair) : WalkingParallelPairHom X X deriving DecidableEq #align category_theory.limits.walking_parallel_pair_hom CategoryTheory.Limits.WalkingParallelPairHom /- Porting note: this simplifies using walkingParallelPairHom_id; replacement is below; simpNF still complains of striking this from the simp list -/ attribute [-simp, nolint simpNF] WalkingParallelPairHom.id.sizeOf_spec /-- Satisfying the inhabited linter -/ instance : Inhabited (WalkingParallelPairHom zero one) where default := WalkingParallelPairHom.left open WalkingParallelPairHom /-- Composition of morphisms in the indexing diagram for (co)equalizers. -/ def WalkingParallelPairHom.comp : -- Porting note: changed X Y Z to implicit to match comp fields in precategory ∀ { X Y Z : WalkingParallelPair } (_ : WalkingParallelPairHom X Y) (_ : WalkingParallelPairHom Y Z), WalkingParallelPairHom X Z \| _, _, _, id _, h => h \| _, _, _, left, id one => left \| _, _, _, right, id one => right #align category_theory.limits.walking_parallel_pair_hom.comp CategoryTheory.Limits.WalkingParallelPairHom.comp -- Porting note: adding these since they are simple and aesop couldn't directly prove them theorem WalkingParallelPairHom.id_comp {X Y : WalkingParallelPair} (g : WalkingParallelPairHom X Y) : comp (id X) g = g := rfl theorem WalkingParallelPairHom.comp_id {X Y : WalkingParallelPair} (f : WalkingParallelPairHom X Y) : comp f (id Y) = f := by cases f <;> rfl theorem WalkingParallelPairHom.assoc {X Y Z W : WalkingParallelPair} (f : WalkingParallelPairHom X Y) (g: WalkingParallelPairHom Y Z) (h : WalkingParallelPairHom Z W) : comp (comp f g) h = comp f (comp g h) := by cases f <;> cases g <;> cases h <;> rfl instance walkingParallelPairHomCategory : SmallCategory WalkingParallelPair where Hom := WalkingParallelPairHom id := id comp := comp comp_id := comp_id id_comp := id_comp assoc := assoc #align category_theory.limits.walking_parallel_pair_hom_category CategoryTheory.Limits.walkingParallelPairHomCategory @[simp] theorem walkingParallelPairHom_id (X : WalkingParallelPair) : WalkingParallelPairHom.id X = 𝟙 X := rfl #align category_theory.limits.walking_parallel_pair_hom_id CategoryTheory.Limits.walkingParallelPairHom_id -- Porting note: simpNF asked me to do this because the LHS of the non-primed version reduced @[simp] theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) : (WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X := by cases X <;> rfl /-- The functor `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to right. -/ def walkingParallelPairOp : WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ where obj x := op <\| by cases x; exacts [one, zero] map f := by cases f <;> apply Quiver.Hom.op exacts [left, right, WalkingParallelPairHom.id _] map_comp := by rintro _ _ _ (_\|_\|_) g <;> cases g <;> rfl #align category_theory.limits.walking_parallel_pair_op CategoryTheory.Limits.walkingParallelPairOp @[simp] theorem walkingParallelPairOp_zero : walkingParallelPairOp.obj zero = op one := rfl #align category_theory.limits.walking_parallel_pair_op_zero CategoryTheory.Limits.walkingParallelPairOp_zero @[simp] theorem walkingParallelPairOp_one : walkingParallelPairOp.obj one = op zero := rfl #align category_theory.limits.walking_parallel_pair_op_one CategoryTheory.Limits.walkingParallelPairOp_one @[simp] theorem walkingParallelPairOp_left : walkingParallelPairOp.map left = @Quiver.Hom.op _ _ zero one left := rfl #align category_theory.limits.walking_parallel_pair_op_left CategoryTheory.Limits.walkingParallelPairOp_left @[simp] theorem walkingParallelPairOp_right : walkingParallelPairOp.map right = @Quiver.Hom.op _ _ zero one right := rfl #align category_theory.limits.walking_parallel_pair_op_right CategoryTheory.Limits.walkingParallelPairOp_right /-- The equivalence `WalkingParallelPair ⥤ WalkingParallelPairᵒᵖ` sending left to left and right to right. -/ @[simps functor inverse] def walkingParallelPairOpEquiv : WalkingParallelPair ≌ WalkingParallelPairᵒᵖ where functor := walkingParallelPairOp inverse := walkingParallelPairOp.leftOp unitIso := NatIso.ofComponents (fun j => eqToIso (by cases j <;> rfl)) (by rintro _ _ (_ \| _ \| _) <;> simp) counitIso := NatIso.ofComponents (fun j => eqToIso (by induction' j with X cases X <;> rfl)) (fun {i} {j} f => by induction' i with i induction' j with j let g := f.unop have : f = g.op := rfl rw [this] cases i <;> cases j <;> cases g <;> rfl) functor_unitIso_comp := fun j => by cases j <;> rfl #align category_theory.limits.walking_parallel_pair_op_equiv CategoryTheory.Limits.walkingParallelPairOpEquiv @[simp] theorem walkingParallelPairOpEquiv_unitIso_zero : walkingParallelPairOpEquiv.unitIso.app zero = Iso.refl zero := rfl #align category_theory.limits.walking_parallel_pair_op_equiv_unit_iso_zero CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_zero @[simp] theorem walkingParallelPairOpEquiv_unitIso_one : walkingParallelPairOpEquiv.unitIso.app one = Iso.refl one := rfl #align category_theory.limits.walking_parallel_pair_op_equiv_unit_iso_one CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_one @[simp] theorem walkingParallelPairOpEquiv_counitIso_zero : walkingParallelPairOpEquiv.counitIso.app (op zero) = Iso.refl (op zero) := rfl #align category_theory.limits.walking_parallel_pair_op_equiv_counit_iso_zero CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_zero @[simp] theorem walkingParallelPairOpEquiv_counitIso_one : walkingParallelPairOpEquiv.counitIso.app (op one) = Iso.refl (op one) := rfl #align category_theory.limits.walking_parallel_pair_op_equiv_counit_iso_one CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_one variable {C : Type u} [Category.{v} C] variable {X Y : C} /-- `parallelPair f g` is the diagram in `C` consisting of the two morphisms `f` and `g` with common domain and codomain. -/ def parallelPair (f g : X ⟶ Y) : WalkingParallelPair ⥤ C where obj x := match x with \| zero => X \| one => Y map h := match h with \| WalkingParallelPairHom.id _ => 𝟙 _ \| left => f \| right => g -- `sorry` can cope with this, but it's too slow: map_comp := by rintro _ _ _ ⟨⟩ g <;> cases g <;> {dsimp; simp} #align category_theory.limits.parallel_pair CategoryTheory.Limits.parallelPair @[simp] theorem parallelPair_obj_zero (f g : X ⟶ Y) : (parallelPair f g).obj zero = X := rfl #align category_theory.limits.parallel_pair_obj_zero CategoryTheory.Limits.parallelPair_obj_zero @[simp] theorem parallelPair_obj_one (f g : X ⟶ Y) : (parallelPair f g).obj one = Y := rfl #align category_theory.limits.parallel_pair_obj_one CategoryTheory.Limits.parallelPair_obj_one @[simp] theorem parallelPair_map_left (f g : X ⟶ Y) : (parallelPair f g).map left = f := rfl #align category_theory.limits.parallel_pair_map_left CategoryTheory.Limits.parallelPair_map_left @[simp] theorem parallelPair_map_right (f g : X ⟶ Y) : (parallelPair f g).map right = g := rfl #align category_theory.limits.parallel_pair_map_right CategoryTheory.Limits.parallelPair_map_right @[simp] theorem parallelPair_functor_obj {F : WalkingParallelPair ⥤ C} (j : WalkingParallelPair) : (parallelPair (F.map left) (F.map right)).obj j = F.obj j := by cases j <;> rfl #align category_theory.limits.parallel_pair_functor_obj CategoryTheory.Limits.parallelPair_functor_obj /-- Every functor indexing a (co)equalizer is naturally isomorphic (actually, equal) to a `parallelPair` -/ @[simps!] def diagramIsoParallelPair (F : WalkingParallelPair ⥤ C) : F ≅ parallelPair (F.map left) (F.map right) := NatIso.ofComponents (fun j => eqToIso <\| by cases j <;> rfl) (by rintro _ _ (_\|_\|_) <;> simp) #align category_theory.limits.diagram_iso_parallel_pair CategoryTheory.Limits.diagramIsoParallelPair /-- Construct a morphism between parallel pairs. -/ def parallelPairHom {X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y') (p : X ⟶ X') (q : Y ⟶ Y') (wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') : parallelPair f g ⟶ parallelPair f' g' where app j := match j with \| zero => p \| one => q naturality := by rintro _ _ ⟨⟩ <;> {dsimp; simp [wf,wg]} #align category_theory.limits.parallel_pair_hom CategoryTheory.Limits.parallelPairHom @[simp] theorem parallelPairHom_app_zero {X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y') (p : X ⟶ X') (q : Y ⟶ Y') (wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') : (parallelPairHom f g f' g' p q wf wg).app zero = p := rfl #align category_theory.limits.parallel_pair_hom_app_zero CategoryTheory.Limits.parallelPairHom_app_zero @[simp] theorem parallelPairHom_app_one {X' Y' : C} (f g : X ⟶ Y) (f' g' : X' ⟶ Y') (p : X ⟶ X') (q : Y ⟶ Y') (wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') : (parallelPairHom f g f' g' p q wf wg).app one = q := rfl #align category_theory.limits.parallel_pair_hom_app_one CategoryTheory.Limits.parallelPairHom_app_one /-- Construct a natural isomorphism between functors out of the walking parallel pair from its components. -/ @[simps!] def parallelPair.ext {F G : WalkingParallelPair ⥤ C} (zero : F.obj zero ≅ G.obj zero) (one : F.obj one ≅ G.obj one) (left : F.map left ≫ one.hom = zero.hom ≫ G.map left) (right : F.map right ≫ one.hom = zero.hom ≫ G.map right) : F ≅ G := NatIso.ofComponents (by rintro ⟨j⟩ exacts [zero, one]) (by rintro _ _ ⟨_⟩ <;> simp [left, right]) #align category_theory.limits.parallel_pair.ext CategoryTheory.Limits.parallelPair.ext /-- Construct a natural isomorphism between `parallelPair f g` and `parallelPair f' g'` given equalities `f = f'` and `g = g'`. -/ @[simps!] def parallelPair.eqOfHomEq {f g f' g' : X ⟶ Y} (hf : f = f') (hg : g = g') : parallelPair f g ≅ parallelPair f' g' := parallelPair.ext (Iso.refl _) (Iso.refl _) (by simp [hf]) (by simp [hg]) #align category_theory.limits.parallel_pair.eq_of_hom_eq CategoryTheory.Limits.parallelPair.eqOfHomEq /-- A fork on `f` and `g` is just a `Cone (parallelPair f g)`. -/ abbrev Fork (f g : X ⟶ Y) := Cone (parallelPair f g) #align category_theory.limits.fork CategoryTheory.Limits.Fork /-- A cofork on `f` and `g` is just a `Cocone (parallelPair f g)`. -/ abbrev Cofork (f g : X ⟶ Y) := Cocone (parallelPair f g) #align category_theory.limits.cofork CategoryTheory.Limits.Cofork variable {f g : X ⟶ Y} /-- A fork `t` on the parallel pair `f g : X ⟶ Y` consists of two morphisms `t.π.app zero : t.pt ⟶ X` and `t.π.app one : t.pt ⟶ Y`. Of these, only the first one is interesting, and we give it the shorter name `Fork.ι t`. -/ def Fork.ι (t : Fork f g) := t.π.app zero #align category_theory.limits.fork.ι CategoryTheory.Limits.Fork.ι @[simp] theorem Fork.app_zero_eq_ι (t : Fork f g) : t.π.app zero = t.ι := rfl #align category_theory.limits.fork.app_zero_eq_ι CategoryTheory.Limits.Fork.app_zero_eq_ι /-- A cofork `t` on the parallelPair `f g : X ⟶ Y` consists of two morphisms `t.ι.app zero : X ⟶ t.pt` and `t.ι.app one : Y ⟶ t.pt`. Of these, only the second one is interesting, and we give it the shorter name `Cofork.π t`. -/ def Cofork.π (t : Cofork f g) := t.ι.app one #align category_theory.limits.cofork.π CategoryTheory.Limits.Cofork.π @[simp] theorem Cofork.app_one_eq_π (t : Cofork f g) : t.ι.app one = t.π := rfl #align category_theory.limits.cofork.app_one_eq_π CategoryTheory.Limits.Cofork.app_one_eq_π @[simp] theorem Fork.app_one_eq_ι_comp_left (s : Fork f g) : s.π.app one = s.ι ≫ f := by rw [← s.app_zero_eq_ι, ← s.w left, parallelPair_map_left] #align category_theory.limits.fork.app_one_eq_ι_comp_left CategoryTheory.Limits.Fork.app_one_eq_ι_comp_left @[reassoc] theorem Fork.app_one_eq_ι_comp_right (s : Fork f g) : s.π.app one = s.ι ≫ g := by rw [← s.app_zero_eq_ι, ← s.w right, parallelPair_map_right] #align category_theory.limits.fork.app_one_eq_ι_comp_right CategoryTheory.Limits.Fork.app_one_eq_ι_comp_right @[simp] theorem Cofork.app_zero_eq_comp_π_left (s : Cofork f g) : s.ι.app zero = f ≫ s.π := by rw [← s.app_one_eq_π, ← s.w left, parallelPair_map_left] #align category_theory.limits.cofork.app_zero_eq_comp_π_left CategoryTheory.Limits.Cofork.app_zero_eq_comp_π_left @[reassoc] theorem Cofork.app_zero_eq_comp_π_right (s : Cofork f g) : s.ι.app zero = g ≫ s.π := by rw [← s.app_one_eq_π, ← s.w right, parallelPair_map_right] #align category_theory.limits.cofork.app_zero_eq_comp_π_right CategoryTheory.Limits.Cofork.app_zero_eq_comp_π_right /-- A fork on `f g : X ⟶ Y` is determined by the morphism `ι : P ⟶ X` satisfying `ι ≫ f = ι ≫ g`. -/ @[simps] def Fork.ofι {P : C} (ι : P ⟶ X) (w : ι ≫ f = ι ≫ g) : Fork f g where pt := P π := { app := fun X => by cases X · exact ι · exact ι ≫ f naturality := fun {X} {Y} f => by cases X <;> cases Y <;> cases f <;> dsimp <;> simp; assumption } #align category_theory.limits.fork.of_ι CategoryTheory.Limits.Fork.ofι /-- A cofork on `f g : X ⟶ Y` is determined by the morphism `π : Y ⟶ P` satisfying `f ≫ π = g ≫ π`. -/ @[simps] def Cofork.ofπ {P : C} (π : Y ⟶ P) (w : f ≫ π = g ≫ π) : Cofork f g where pt := P ι := { app := fun X => WalkingParallelPair.casesOn X (f ≫ π) π naturality := fun i j f => by cases f <;> dsimp <;> simp [w] } #align category_theory.limits.cofork.of_π CategoryTheory.Limits.Cofork.ofπ -- See note [dsimp, simp] @[simp] theorem Fork.ι_ofι {P : C} (ι : P ⟶ X) (w : ι ≫ f = ι ≫ g) : (Fork.ofι ι w).ι = ι := rfl #align category_theory.limits.fork.ι_of_ι CategoryTheory.Limits.Fork.ι_ofι @[simp] theorem Cofork.π_ofπ {P : C} (π : Y ⟶ P) (w : f ≫ π = g ≫ π) : (Cofork.ofπ π w).π = π := rfl #align category_theory.limits.cofork.π_of_π CategoryTheory.Limits.Cofork.π_ofπ @[reassoc (attr := simp)] theorem Fork.condition (t : Fork f g) : t.ι ≫ f = t.ι ≫ g := by rw [← t.app_one_eq_ι_comp_left, ← t.app_one_eq_ι_comp_right] #align category_theory.limits.fork.condition CategoryTheory.Limits.Fork.condition @[reassoc (attr := simp)] theorem Cofork.condition (t : Cofork f g) : f ≫ t.π = g ≫ t.π := by rw [← t.app_zero_eq_comp_π_left, ← t.app_zero_eq_comp_π_right] #align category_theory.limits.cofork.condition CategoryTheory.Limits.Cofork.condition /-- To check whether two maps are equalized by both maps of a fork, it suffices to check it for the first map -/ theorem Fork.equalizer_ext (s : Fork f g) {W : C} {k l : W ⟶ s.pt} (h : k ≫ s.ι = l ≫ s.ι) : ∀ j : WalkingParallelPair, k ≫ s.π.app j = l ≫ s.π.app j \| zero => h \| one => by have : k ≫ ι s ≫ f = l ≫ ι s ≫ f := by simp only [← Category.assoc]; exact congrArg (· ≫ f) h rw [s.app_one_eq_ι_comp_left, this] #align category_theory.limits.fork.equalizer_ext CategoryTheory.Limits.Fork.equalizer_ext /-- To check whether two maps are coequalized by both maps of a cofork, it suffices to check it for the second map -/ theorem Cofork.coequalizer_ext (s : Cofork f g) {W : C} {k l : s.pt ⟶ W} (h : Cofork.π s ≫ k = Cofork.π s ≫ l) : ∀ j : WalkingParallelPair, s.ι.app j ≫ k = s.ι.app j ≫ l \| zero => by simp only [s.app_zero_eq_comp_π_left, Category.assoc, h] \| one => h #align category_theory.limits.cofork.coequalizer_ext CategoryTheory.Limits.Cofork.coequalizer_ext theorem Fork.IsLimit.hom_ext {s : Fork f g} (hs : IsLimit s) {W : C} {k l : W ⟶ s.pt} (h : k ≫ Fork.ι s = l ≫ Fork.ι s) : k = l := hs.hom_ext <\| Fork.equalizer_ext _ h #align category_theory.limits.fork.is_limit.hom_ext CategoryTheory.Limits.Fork.IsLimit.hom_ext theorem Cofork.IsColimit.hom_ext {s : Cofork f g} (hs : IsColimit s) {W : C} {k l : s.pt ⟶ W} (h : Cofork.π s ≫ k = Cofork.π s ≫ l) : k = l := hs.hom_ext <\| Cofork.coequalizer_ext _ h #align category_theory.limits.cofork.is_colimit.hom_ext CategoryTheory.Limits.Cofork.IsColimit.hom_ext @[reassoc (attr := simp)] theorem Fork.IsLimit.lift_ι {s t : Fork f g} (hs : IsLimit s) : hs.lift t ≫ s.ι = t.ι := hs.fac _ _ #align category_theory.limits.fork.is_limit.lift_ι CategoryTheory.Limits.Fork.IsLimit.lift_ι @[reassoc (attr := simp)] theorem Cofork.IsColimit.π_desc {s t : Cofork f g} (hs : IsColimit s) : s.π ≫ hs.desc t = t.π := hs.fac _ _ #align category_theory.limits.cofork.is_colimit.π_desc CategoryTheory.Limits.Cofork.IsColimit.π_desc -- Porting note: `Fork.IsLimit.lift` was added in order to ease the port /-- If `s` is a limit fork over `f` and `g`, then a morphism `k : W ⟶ X` satisfying `k ≫ f = k ≫ g` induces a morphism `l : W ⟶ s.pt` such that `l ≫ fork.ι s = k`. -/ def Fork.IsLimit.lift {s : Fork f g} (hs : IsLimit s) {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) : W ⟶ s.pt := hs.lift (Fork.ofι _ h) @[reassoc (attr := simp)] lemma Fork.IsLimit.lift_ι' {s : Fork f g} (hs : IsLimit s) {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) : Fork.IsLimit.lift hs k h ≫ Fork.ι s = k := hs.fac _ _ /-- If `s` is a limit fork over `f` and `g`, then a morphism `k : W ⟶ X` satisfying `k ≫ f = k ≫ g` induces a morphism `l : W ⟶ s.pt` such that `l ≫ fork.ι s = k`. -/ def Fork.IsLimit.lift' {s : Fork f g} (hs : IsLimit s) {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) : { l : W ⟶ s.pt // l ≫ Fork.ι s = k } := ⟨Fork.IsLimit.lift hs k h, by simp⟩ #align category_theory.limits.fork.is_limit.lift' CategoryTheory.Limits.Fork.IsLimit.lift' -- Porting note: `Cofork.IsColimit.desc` was added in order to ease the port /-- If `s` is a colimit cofork over `f` and `g`, then a morphism `k : Y ⟶ W` satisfying `f ≫ k = g ≫ k` induces a morphism `l : s.pt ⟶ W` such that `cofork.π s ≫ l = k`. -/ def Cofork.IsColimit.desc {s : Cofork f g} (hs : IsColimit s) {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) : s.pt ⟶ W := hs.desc (Cofork.ofπ _ h) @[reassoc (attr := simp)] lemma Cofork.IsColimit.π_desc' {s : Cofork f g} (hs : IsColimit s) {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) : Cofork.π s ≫ Cofork.IsColimit.desc hs k h = k := hs.fac _ _ /-- If `s` is a colimit cofork over `f` and `g`, then a morphism `k : Y ⟶ W` satisfying `f ≫ k = g ≫ k` induces a morphism `l : s.pt ⟶ W` such that `cofork.π s ≫ l = k`. -/ def Cofork.IsColimit.desc' {s : Cofork f g} (hs : IsColimit s) {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) : { l : s.pt ⟶ W // Cofork.π s ≫ l = k } := ⟨Cofork.IsColimit.desc hs k h, by simp⟩ #align category_theory.limits.cofork.is_colimit.desc' CategoryTheory.Limits.Cofork.IsColimit.desc' theorem Fork.IsLimit.existsUnique {s : Fork f g} (hs : IsLimit s) {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) : ∃! l : W ⟶ s.pt, l ≫ Fork.ι s = k := ⟨hs.lift <\| Fork.ofι _ h, hs.fac _ _, fun _ hm => Fork.IsLimit.hom_ext hs <\| hm.symm ▸ (hs.fac (Fork.ofι _ h) WalkingParallelPair.zero).symm⟩ #align category_theory.limits.fork.is_limit.exists_unique CategoryTheory.Limits.Fork.IsLimit.existsUnique theorem Cofork.IsColimit.existsUnique {s : Cofork f g} (hs : IsColimit s) {W : C} (k : Y ⟶ W) (h : f ≫ k = g ≫ k) : ∃! d : s.pt ⟶ W, Cofork.π s ≫ d = k := ⟨hs.desc <\| Cofork.ofπ _ h, hs.fac _ _, fun _ hm => Cofork.IsColimit.hom_ext hs <\| hm.symm ▸ (hs.fac (Cofork.ofπ _ h) WalkingParallelPair.one).symm⟩ #align category_theory.limits.cofork.is_colimit.exists_unique CategoryTheory.Limits.Cofork.IsColimit.existsUnique /-- This is a slightly more convenient method to verify that a fork is a limit cone. It only asks for a proof of facts that carry any mathematical content -/ @[simps] def Fork.IsLimit.mk (t : Fork f g) (lift : ∀ s : Fork f g, s.pt ⟶ t.pt) (fac : ∀ s : Fork f g, lift s ≫ Fork.ι t = Fork.ι s) (uniq : ∀ (s : Fork f g) (m : s.pt ⟶ t.pt) (_ : m ≫ t.ι = s.ι), m = lift s) : IsLimit t := { lift fac := fun s j => WalkingParallelPair.casesOn j (fac s) <\| by erw [← s.w left, ← t.w left, ← Category.assoc, fac]; rfl uniq := fun s m j => by aesop} #align category_theory.limits.fork.is_limit.mk CategoryTheory.Limits.Fork.IsLimit.mk /-- This is another convenient method to verify that a fork is a limit cone. It only asks for a proof of facts that carry any mathematical content, and allows access to the same `s` for all parts. -/ def Fork.IsLimit.mk' {X Y : C} {f g : X ⟶ Y} (t : Fork f g) (create : ∀ s : Fork f g, { l // l ≫ t.ι = s.ι ∧ ∀ {m}, m ≫ t.ι = s.ι → m = l }) : IsLimit t := Fork.IsLimit.mk t (fun s => (create s).1) (fun s => (create s).2.1) fun s _ w => (create s).2.2 w #align category_theory.limits.fork.is_limit.mk' CategoryTheory.Limits.Fork.IsLimit.mk' /-- This is a slightly more convenient method to verify that a cofork is a colimit cocone. It only asks for a proof of facts that carry any mathematical content -/ def Cofork.IsColimit.mk (t : Cofork f g) (desc : ∀ s : Cofork f g, t.pt ⟶ s.pt) (fac : ∀ s : Cofork f g, Cofork.π t ≫ desc s = Cofork.π s) (uniq : ∀ (s : Cofork f g) (m : t.pt ⟶ s.pt) (_ : t.π ≫ m = s.π), m = desc s) : IsColimit t := { desc fac := fun s j => WalkingParallelPair.casesOn j (by erw [← s.w left, ← t.w left, Category.assoc, fac]; rfl) (fac s) uniq := by aesop } #align category_theory.limits.cofork.is_colimit.mk CategoryTheory.Limits.Cofork.IsColimit.mk /-- This is another convenient method to verify that a fork is a limit cone. It only asks for a proof of facts that carry any mathematical content, and allows access to the same `s` for all parts. -/ def Cofork.IsColimit.mk' {X Y : C} {f g : X ⟶ Y} (t : Cofork f g) (create : ∀ s : Cofork f g, { l : t.pt ⟶ s.pt // t.π ≫ l = s.π ∧ ∀ {m}, t.π ≫ m = s.π → m = l }) : IsColimit t := Cofork.IsColimit.mk t (fun s => (create s).1) (fun s => (create s).2.1) fun s _ w => (create s).2.2 w #align category_theory.limits.cofork.is_colimit.mk' CategoryTheory.Limits.Cofork.IsColimit.mk' /-- Noncomputably make a limit cone from the existence of unique factorizations. -/ noncomputable def Fork.IsLimit.ofExistsUnique {t : Fork f g} (hs : ∀ s : Fork f g, ∃! l : s.pt ⟶ t.pt, l ≫ Fork.ι t = Fork.ι s) : IsLimit t := by choose d hd hd' using hs exact Fork.IsLimit.mk _ d hd fun s m hm => hd' _ _ hm #align category_theory.limits.fork.is_limit.of_exists_unique CategoryTheory.Limits.Fork.IsLimit.ofExistsUnique /-- Noncomputably make a colimit cocone from the existence of unique factorizations. -/ noncomputable def Cofork.IsColimit.ofExistsUnique {t : Cofork f g} (hs : ∀ s : Cofork f g, ∃! d : t.pt ⟶ s.pt, Cofork.π t ≫ d = Cofork.π s) : IsColimit t := by choose d hd hd' using hs exact Cofork.IsColimit.mk _ d hd fun s m hm => hd' _ _ hm #align category_theory.limits.cofork.is_colimit.of_exists_unique CategoryTheory.Limits.Cofork.IsColimit.ofExistsUnique /-- Given a limit cone for the pair `f g : X ⟶ Y`, for any `Z`, morphisms from `Z` to its point are in bijection with morphisms `h : Z ⟶ X` such that `h ≫ f = h ≫ g`. Further, this bijection is natural in `Z`: see `Fork.IsLimit.homIso_natural`. This is a special case of `IsLimit.homIso'`, often useful to construct adjunctions. -/ @[simps] def Fork.IsLimit.homIso {X Y : C} {f g : X ⟶ Y} {t : Fork f g} (ht : IsLimit t) (Z : C) : (Z ⟶ t.pt) ≃ { h : Z ⟶ X // h ≫ f = h ≫ g } where toFun k := ⟨k ≫ t.ι, by simp only [Category.assoc, t.condition]⟩ invFun h := (Fork.IsLimit.lift' ht _ h.prop).1 left_inv k := Fork.IsLimit.hom_ext ht (Fork.IsLimit.lift' _ _ _).prop right_inv h := Subtype.ext (Fork.IsLimit.lift' ht _ _).prop #align category_theory.limits.fork.is_limit.hom_iso CategoryTheory.Limits.Fork.IsLimit.homIso /-- The bijection of `Fork.IsLimit.homIso` is natural in `Z`. -/ theorem Fork.IsLimit.homIso_natural {X Y : C} {f g : X ⟶ Y} {t : Fork f g} (ht : IsLimit t) {Z Z' : C} (q : Z' ⟶ Z) (k : Z ⟶ t.pt) : (Fork.IsLimit.homIso ht _ (q ≫ k) : Z' ⟶ X) = q ≫ (Fork.IsLimit.homIso ht _ k : Z ⟶ X) := Category.assoc _ _ _ #align category_theory.limits.fork.is_limit.hom_iso_natural CategoryTheory.Limits.Fork.IsLimit.homIso_natural /-- Given a colimit cocone for the pair `f g : X ⟶ Y`, for any `Z`, morphisms from the cocone point to `Z` are in bijection with morphisms `h : Y ⟶ Z` such that `f ≫ h = g ≫ h`. Further, this bijection is natural in `Z`: see `Cofork.IsColimit.homIso_natural`. This is a special case of `IsColimit.homIso'`, often useful to construct adjunctions. -/ @[simps] def Cofork.IsColimit.homIso {X Y : C} {f g : X ⟶ Y} {t : Cofork f g} (ht : IsColimit t) (Z : C) : (t.pt ⟶ Z) ≃ { h : Y ⟶ Z // f ≫ h = g ≫ h } where toFun k := ⟨t.π ≫ k, by simp only [← Category.assoc, t.condition]⟩ invFun h := (Cofork.IsColimit.desc' ht _ h.prop).1 left_inv k := Cofork.IsColimit.hom_ext ht (Cofork.IsColimit.desc' _ _ _).prop right_inv h := Subtype.ext (Cofork.IsColimit.desc' ht _ _).prop #align category_theory.limits.cofork.is_colimit.hom_iso CategoryTheory.Limits.Cofork.IsColimit.homIso /-- The bijection of `Cofork.IsColimit.homIso` is natural in `Z`. -/ theorem Cofork.IsColimit.homIso_natural {X Y : C} {f g : X ⟶ Y} {t : Cofork f g} {Z Z' : C} (q : Z ⟶ Z') (ht : IsColimit t) (k : t.pt ⟶ Z) : (Cofork.IsColimit.homIso ht _ (k ≫ q) : Y ⟶ Z') = (Cofork.IsColimit.homIso ht _ k : Y ⟶ Z) ≫ q := (Category.assoc _ _ _).symm #align category_theory.limits.cofork.is_colimit.hom_iso_natural CategoryTheory.Limits.Cofork.IsColimit.homIso_natural /-- This is a helper construction that can be useful when verifying that a category has all equalizers. Given `F : WalkingParallelPair ⥤ C`, which is really the same as `parallelPair (F.map left) (F.map right)`, and a fork on `F.map left` and `F.map right`, we get a cone on `F`. If you're thinking about using this, have a look at `hasEqualizers_of_hasLimit_parallelPair`, which you may find to be an easier way of achieving your goal. -/ def Cone.ofFork {F : WalkingParallelPair ⥤ C} (t : Fork (F.map left) (F.map right)) : Cone F where pt := t.pt π := { app := fun X => t.π.app X ≫ eqToHom (by aesop) naturality := by rintro _ _ (_\|_\|_) <;> {dsimp; simp [t.condition]}} #align category_theory.limits.cone.of_fork CategoryTheory.Limits.Cone.ofFork /-- This is a helper construction that can be useful when verifying that a category has all coequalizers. Given `F : WalkingParallelPair ⥤ C`, which is really the same as `parallelPair (F.map left) (F.map right)`, and a cofork on `F.map left` and `F.map right`, we get a cocone on `F`. If you're thinking about using this, have a look at `hasCoequalizers_of_hasColimit_parallelPair`, which you may find to be an easier way of achieving your goal. -/ def Cocone.ofCofork {F : WalkingParallelPair ⥤ C} (t : Cofork (F.map left) (F.map right)) : Cocone F where pt := t.pt ι := { app := fun X => eqToHom (by aesop) ≫ t.ι.app X naturality := by rintro _ _ (_\|_\|_) <;> {dsimp; simp [t.condition]}} #align category_theory.limits.cocone.of_cofork CategoryTheory.Limits.Cocone.ofCofork @[simp] theorem Cone.ofFork_π {F : WalkingParallelPair ⥤ C} (t : Fork (F.map left) (F.map right)) (j) : (Cone.ofFork t).π.app j = t.π.app j ≫ eqToHom (by aesop) := rfl #align category_theory.limits.cone.of_fork_π CategoryTheory.Limits.Cone.ofFork_π @[simp] theorem Cocone.ofCofork_ι {F : WalkingParallelPair ⥤ C} (t : Cofork (F.map left) (F.map right)) (j) : (Cocone.ofCofork t).ι.app j = eqToHom (by aesop) ≫ t.ι.app j := rfl #align category_theory.limits.cocone.of_cofork_ι CategoryTheory.Limits.Cocone.ofCofork_ι /-- Given `F : WalkingParallelPair ⥤ C`, which is really the same as `parallelPair (F.map left) (F.map right)` and a cone on `F`, we get a fork on `F.map left` and `F.map right`. -/ def Fork.ofCone {F : WalkingParallelPair ⥤ C} (t : Cone F) : Fork (F.map left) (F.map right) where pt := t.pt π := { app := fun X => t.π.app X ≫ eqToHom (by aesop) naturality := by rintro _ _ (_\|_\|_) <;> {dsimp; simp}} #align category_theory.limits.fork.of_cone CategoryTheory.Limits.Fork.ofCone /-- Given `F : WalkingParallelPair ⥤ C`, which is really the same as `parallelPair (F.map left) (F.map right)` and a cocone on `F`, we get a cofork on `F.map left` and `F.map right`. -/ def Cofork.ofCocone {F : WalkingParallelPair ⥤ C} (t : Cocone F) : Cofork (F.map left) (F.map right) where pt := t.pt ι := { app := fun X => eqToHom (by aesop) ≫ t.ι.app X naturality := by rintro _ _ (_\|_\|_) <;> {dsimp; simp}} #align category_theory.limits.cofork.of_cocone CategoryTheory.Limits.Cofork.ofCocone @[simp] theorem Fork.ofCone_π {F : WalkingParallelPair ⥤ C} (t : Cone F) (j) : (Fork.ofCone t).π.app j = t.π.app j ≫ eqToHom (by aesop) := rfl #align category_theory.limits.fork.of_cone_π CategoryTheory.Limits.Fork.ofCone_π @[simp] theorem Cofork.ofCocone_ι {F : WalkingParallelPair ⥤ C} (t : Cocone F) (j) : (Cofork.ofCocone t).ι.app j = eqToHom (by aesop) ≫ t.ι.app j := rfl #align category_theory.limits.cofork.of_cocone_ι CategoryTheory.Limits.Cofork.ofCocone_ι @[simp] theorem Fork.ι_postcompose {f' g' : X ⟶ Y} {α : parallelPair f g ⟶ parallelPair f' g'} {c : Fork f g} : Fork.ι ((Cones.postcompose α).obj c) = c.ι ≫ α.app _ := rfl #align category_theory.limits.fork.ι_postcompose CategoryTheory.Limits.Fork.ι_postcompose @[simp] theorem Cofork.π_precompose {f' g' : X ⟶ Y} {α : parallelPair f g ⟶ parallelPair f' g'} {c : Cofork f' g'} : Cofork.π ((Cocones.precompose α).obj c) = α.app _ ≫ c.π := rfl #align category_theory.limits.cofork.π_precompose CategoryTheory.Limits.Cofork.π_precompose /-- Helper function for constructing morphisms between equalizer forks. -/ @[simps] def Fork.mkHom {s t : Fork f g} (k : s.pt ⟶ t.pt) (w : k ≫ t.ι = s.ι) : s ⟶ t where hom := k w := by rintro ⟨_ \| _⟩ · exact w · simp only [Fork.app_one_eq_ι_comp_left,← Category.assoc] congr #align category_theory.limits.fork.mk_hom CategoryTheory.Limits.Fork.mkHom /-- To construct an isomorphism between forks, it suffices to give an isomorphism between the cone points and check that it commutes with the `ι` morphisms. -/ @[simps] def Fork.ext {s t : Fork f g} (i : s.pt ≅ t.pt) (w : i.hom ≫ t.ι = s.ι := by aesop_cat) : s ≅ t where hom := Fork.mkHom i.hom w inv := Fork.mkHom i.inv (by rw [← w, Iso.inv_hom_id_assoc]) #align category_theory.limits.fork.ext CategoryTheory.Limits.Fork.ext /-- Two forks of the form `ofι` are isomorphic whenever their `ι`'s are equal. -/ def ForkOfι.ext {P : C} {ι ι' : P ⟶ X} (w : ι ≫ f = ι ≫ g) (w' : ι' ≫ f = ι' ≫ g) (h : ι = ι') : Fork.ofι ι w ≅ Fork.ofι ι' w' := Fork.ext (Iso.refl _) (by simp [h]) /-- Every fork is isomorphic to one of the form `Fork.of_ι _ _`. -/ def Fork.isoForkOfι (c : Fork f g) : c ≅ Fork.ofι c.ι c.condition := Fork.ext (by simp only [Fork.ofι_pt, Functor.const_obj_obj]; rfl) (by simp) #align category_theory.limits.fork.iso_fork_of_ι CategoryTheory.Limits.Fork.isoForkOfι /-- Given two forks with isomorphic components in such a way that the natural diagrams commute, then if one is a limit, then the other one is as well. -/ def Fork.isLimitOfIsos {X' Y' : C} (c : Fork f g) (hc : IsLimit c) {f' g' : X' ⟶ Y'} (c' : Fork f' g') (e₀ : X ≅ X') (e₁ : Y ≅ Y') (e : c.pt ≅ c'.pt) (comm₁ : e₀.hom ≫ f' = f ≫ e₁.hom := by aesop_cat) (comm₂ : e₀.hom ≫ g' = g ≫ e₁.hom := by aesop_cat) (comm₃ : e.hom ≫ c'.ι = c.ι ≫ e₀.hom := by aesop_cat) : IsLimit c' := let i : parallelPair f g ≅ parallelPair f' g' := parallelPair.ext e₀ e₁ comm₁.symm comm₂.symm (IsLimit.equivOfNatIsoOfIso i c c' (Fork.ext e comm₃)) hc /-- Helper function for constructing morphisms between coequalizer coforks. -/ @[simps] def Cofork.mkHom {s t : Cofork f g} (k : s.pt ⟶ t.pt) (w : s.π ≫ k = t.π) : s ⟶ t where hom := k w := by rintro ⟨_ \| _⟩ · simp [Cofork.app_zero_eq_comp_π_left, w] · exact w #align category_theory.limits.cofork.mk_hom CategoryTheory.Limits.Cofork.mkHom @[reassoc (attr := simp)] theorem Fork.hom_comp_ι {s t : Fork f g} (f : s ⟶ t) : f.hom ≫ t.ι = s.ι := by cases s; cases t; cases f; aesop #align category_theory.limits.fork.hom_comp_ι CategoryTheory.Limits.Fork.hom_comp_ι @[reassoc (attr := simp)]	Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean	735	736	theorem Fork.π_comp_hom {s t : Cofork f g} (f : s ⟶ t) : s.π ≫ f.hom = t.π := by	cases s; cases t; cases f; aesop
/- Copyright (c) 2021 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Riccardo Brasca -/ import Mathlib.Analysis.Normed.Group.Hom import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms import Mathlib.CategoryTheory.ConcreteCategory.BundledHom import Mathlib.CategoryTheory.Elementwise #align_import analysis.normed.group.SemiNormedGroup from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3" /-! # The category of seminormed groups We define `SemiNormedGroupCat`, the category of seminormed groups and normed group homs between them, as well as `SemiNormedGroupCat₁`, the subcategory of norm non-increasing morphisms. -/ set_option linter.uppercaseLean3 false noncomputable section universe u open CategoryTheory /-- The category of seminormed abelian groups and bounded group homomorphisms. -/ def SemiNormedGroupCat : Type (u + 1) := Bundled SeminormedAddCommGroup #align SemiNormedGroup SemiNormedGroupCat namespace SemiNormedGroupCat instance bundledHom : BundledHom @NormedAddGroupHom where toFun := @NormedAddGroupHom.toFun id := @NormedAddGroupHom.id comp := @NormedAddGroupHom.comp #align SemiNormedGroup.bundled_hom SemiNormedGroupCat.bundledHom deriving instance LargeCategory for SemiNormedGroupCat -- Porting note: deriving fails for ConcreteCategory, adding instance manually. -- See https://github.com/leanprover-community/mathlib4/issues/5020 -- deriving instance LargeCategory, ConcreteCategory for SemiRingCat instance : ConcreteCategory SemiNormedGroupCat := by dsimp [SemiNormedGroupCat] infer_instance instance : CoeSort SemiNormedGroupCat Type* where coe X := X.α /-- Construct a bundled `SemiNormedGroupCat` from the underlying type and typeclass. -/ def of (M : Type u) [SeminormedAddCommGroup M] : SemiNormedGroupCat := Bundled.of M #align SemiNormedGroupCat.of SemiNormedGroupCat.of instance (M : SemiNormedGroupCat) : SeminormedAddCommGroup M := M.str -- Porting note (#10754): added instance instance funLike {V W : SemiNormedGroupCat} : FunLike (V ⟶ W) V W where coe := (forget SemiNormedGroupCat).map coe_injective' := fun f g h => by cases f; cases g; congr instance toAddMonoidHomClass {V W : SemiNormedGroupCat} : AddMonoidHomClass (V ⟶ W) V W where map_add f := f.map_add' map_zero f := (AddMonoidHom.mk' f.toFun f.map_add').map_zero -- Porting note (#10688): added to ease automation @[ext] lemma ext {M N : SemiNormedGroupCat} {f₁ f₂ : M ⟶ N} (h : ∀ (x : M), f₁ x = f₂ x) : f₁ = f₂ := DFunLike.ext _ _ h @[simp] theorem coe_of (V : Type u) [SeminormedAddCommGroup V] : (SemiNormedGroupCat.of V : Type u) = V := rfl #align SemiNormedGroup.coe_of SemiNormedGroupCat.coe_of -- Porting note: marked with high priority to short circuit simplifier's path @[simp (high)] theorem coe_id (V : SemiNormedGroupCat) : (𝟙 V : V → V) = id := rfl #align SemiNormedGroup.coe_id SemiNormedGroupCat.coe_id -- Porting note: marked with high priority to short circuit simplifier's path @[simp (high)] theorem coe_comp {M N K : SemiNormedGroupCat} (f : M ⟶ N) (g : N ⟶ K) : (f ≫ g : M → K) = g ∘ f := rfl #align SemiNormedGroup.coe_comp SemiNormedGroupCat.coe_comp instance : Inhabited SemiNormedGroupCat := ⟨of PUnit⟩ instance ofUnique (V : Type u) [SeminormedAddCommGroup V] [i : Unique V] : Unique (SemiNormedGroupCat.of V) := i #align SemiNormedGroup.of_unique SemiNormedGroupCat.ofUnique instance {M N : SemiNormedGroupCat} : Zero (M ⟶ N) := NormedAddGroupHom.zero @[simp] theorem zero_apply {V W : SemiNormedGroupCat} (x : V) : (0 : V ⟶ W) x = 0 := rfl #align SemiNormedGroup.zero_apply SemiNormedGroupCat.zero_apply instance : Limits.HasZeroMorphisms.{u, u + 1} SemiNormedGroupCat where theorem isZero_of_subsingleton (V : SemiNormedGroupCat) [Subsingleton V] : Limits.IsZero V := by refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩ · ext x; have : x = 0 := Subsingleton.elim _ _; simp only [this, map_zero] · ext; apply Subsingleton.elim #align SemiNormedGroup.is_zero_of_subsingleton SemiNormedGroupCat.isZero_of_subsingleton instance hasZeroObject : Limits.HasZeroObject SemiNormedGroupCat.{u} := ⟨⟨of PUnit, isZero_of_subsingleton _⟩⟩ #align SemiNormedGroup.has_zero_object SemiNormedGroupCat.hasZeroObject theorem iso_isometry_of_normNoninc {V W : SemiNormedGroupCat} (i : V ≅ W) (h1 : i.hom.NormNoninc) (h2 : i.inv.NormNoninc) : Isometry i.hom := by apply AddMonoidHomClass.isometry_of_norm intro v apply le_antisymm (h1 v) calc -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 ‖v‖ = ‖i.inv (i.hom v)‖ := by erw [Iso.hom_inv_id_apply] _ ≤ ‖i.hom v‖ := h2 _ #align SemiNormedGroup.iso_isometry_of_norm_noninc SemiNormedGroupCat.iso_isometry_of_normNoninc end SemiNormedGroupCat /-- `SemiNormedGroupCat₁` is a type synonym for `SemiNormedGroupCat`, which we shall equip with the category structure consisting only of the norm non-increasing maps. -/ def SemiNormedGroupCat₁ : Type (u + 1) := Bundled SeminormedAddCommGroup #align SemiNormedGroup₁ SemiNormedGroupCat₁ namespace SemiNormedGroupCat₁ instance : CoeSort SemiNormedGroupCat₁ Type* where coe X := X.α instance : LargeCategory.{u} SemiNormedGroupCat₁ where Hom X Y := { f : NormedAddGroupHom X Y // f.NormNoninc } id X := ⟨NormedAddGroupHom.id X, NormedAddGroupHom.NormNoninc.id⟩ comp {X Y Z} f g := ⟨g.1.comp f.1, g.2.comp f.2⟩ -- Porting note (#10754): added instance instance instFunLike (X Y : SemiNormedGroupCat₁) : FunLike (X ⟶ Y) X Y where coe f := f.1.toFun coe_injective' _ _ h := Subtype.val_inj.mp (NormedAddGroupHom.coe_injective h) @[ext] theorem hom_ext {M N : SemiNormedGroupCat₁} (f g : M ⟶ N) (w : (f : M → N) = (g : M → N)) : f = g := Subtype.eq (NormedAddGroupHom.ext (congr_fun w)) #align SemiNormedGroup₁.hom_ext SemiNormedGroupCat₁.hom_ext instance : ConcreteCategory.{u} SemiNormedGroupCat₁ where forget := { obj := fun X => X map := fun f => f } forget_faithful := { } -- Porting note (#10754): added instance instance toAddMonoidHomClass {V W : SemiNormedGroupCat₁} : AddMonoidHomClass (V ⟶ W) V W where map_add f := f.1.map_add' map_zero f := (AddMonoidHom.mk' f.1 f.1.map_add').map_zero /-- Construct a bundled `SemiNormedGroupCat₁` from the underlying type and typeclass. -/ def of (M : Type u) [SeminormedAddCommGroup M] : SemiNormedGroupCat₁ := Bundled.of M #align SemiNormedGroup₁.of SemiNormedGroupCat₁.of instance (M : SemiNormedGroupCat₁) : SeminormedAddCommGroup M := M.str /-- Promote a morphism in `SemiNormedGroupCat` to a morphism in `SemiNormedGroupCat₁`. -/ def mkHom {M N : SemiNormedGroupCat} (f : M ⟶ N) (i : f.NormNoninc) : SemiNormedGroupCat₁.of M ⟶ SemiNormedGroupCat₁.of N := ⟨f, i⟩ #align SemiNormedGroup₁.mk_hom SemiNormedGroupCat₁.mkHom -- @[simp] -- Porting note: simpNF linter claims LHS simplifies with `SemiNormedGroupCat₁.coe_of` theorem mkHom_apply {M N : SemiNormedGroupCat} (f : M ⟶ N) (i : f.NormNoninc) (x) : mkHom f i x = f x := rfl #align SemiNormedGroup₁.mk_hom_apply SemiNormedGroupCat₁.mkHom_apply /-- Promote an isomorphism in `SemiNormedGroupCat` to an isomorphism in `SemiNormedGroupCat₁`. -/ @[simps] def mkIso {M N : SemiNormedGroupCat} (f : M ≅ N) (i : f.hom.NormNoninc) (i' : f.inv.NormNoninc) : SemiNormedGroupCat₁.of M ≅ SemiNormedGroupCat₁.of N where hom := mkHom f.hom i inv := mkHom f.inv i' hom_inv_id := by apply Subtype.eq; exact f.hom_inv_id inv_hom_id := by apply Subtype.eq; exact f.inv_hom_id #align SemiNormedGroup₁.mk_iso SemiNormedGroupCat₁.mkIso instance : HasForget₂ SemiNormedGroupCat₁ SemiNormedGroupCat where forget₂ := { obj := fun X => X map := fun f => f.1 } @[simp] theorem coe_of (V : Type u) [SeminormedAddCommGroup V] : (SemiNormedGroupCat₁.of V : Type u) = V := rfl #align SemiNormedGroup₁.coe_of SemiNormedGroupCat₁.coe_of -- Porting note: marked with high priority to short circuit simplifier's path @[simp (high)] theorem coe_id (V : SemiNormedGroupCat₁) : ⇑(𝟙 V) = id := rfl #align SemiNormedGroup₁.coe_id SemiNormedGroupCat₁.coe_id -- Porting note: marked with high priority to short circuit simplifier's path @[simp (high)] theorem coe_comp {M N K : SemiNormedGroupCat₁} (f : M ⟶ N) (g : N ⟶ K) : (f ≫ g : M → K) = g ∘ f := rfl #align SemiNormedGroup₁.coe_comp SemiNormedGroupCat₁.coe_comp -- Porting note: deleted `coe_comp'`, as we no longer have the relevant coercion. #noalign SemiNormedGroup₁.coe_comp' instance : Inhabited SemiNormedGroupCat₁ := ⟨of PUnit⟩ instance ofUnique (V : Type u) [SeminormedAddCommGroup V] [i : Unique V] : Unique (SemiNormedGroupCat₁.of V) := i #align SemiNormedGroup₁.of_unique SemiNormedGroupCat₁.ofUnique -- Porting note: extracted from `Limits.HasZeroMorphisms` instance below. instance (X Y : SemiNormedGroupCat₁) : Zero (X ⟶ Y) where zero := ⟨0, NormedAddGroupHom.NormNoninc.zero⟩ @[simp] theorem zero_apply {V W : SemiNormedGroupCat₁} (x : V) : (0 : V ⟶ W) x = 0 := rfl #align SemiNormedGroup₁.zero_apply SemiNormedGroupCat₁.zero_apply instance : Limits.HasZeroMorphisms.{u, u + 1} SemiNormedGroupCat₁ where theorem isZero_of_subsingleton (V : SemiNormedGroupCat₁) [Subsingleton V] : Limits.IsZero V := by refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩ · ext x; have : x = 0 := Subsingleton.elim _ _; simp only [this, map_zero] · ext; apply Subsingleton.elim #align SemiNormedGroup₁.is_zero_of_subsingleton SemiNormedGroupCat₁.isZero_of_subsingleton instance hasZeroObject : Limits.HasZeroObject SemiNormedGroupCat₁.{u} := ⟨⟨of PUnit, isZero_of_subsingleton _⟩⟩ #align SemiNormedGroup₁.has_zero_object SemiNormedGroupCat₁.hasZeroObject	Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean	258	266	theorem iso_isometry {V W : SemiNormedGroupCat₁} (i : V ≅ W) : Isometry i.hom := by	change Isometry (⟨⟨i.hom, map_zero _⟩, fun _ _ => map_add _ _ _⟩ : V →+ W) refine AddMonoidHomClass.isometry_of_norm _ ?_ intro v apply le_antisymm (i.hom.2 v) calc -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 ‖v‖ = ‖i.inv (i.hom v)‖ := by erw [Iso.hom_inv_id_apply] _ ≤ ‖i.hom v‖ := i.inv.2 _
/- Copyright (c) 2020 Bhavik Mehta, E. W. Ayers. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bhavik Mehta, E. W. Ayers -/ import Mathlib.CategoryTheory.Sites.Sieves import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer import Mathlib.CategoryTheory.Category.Preorder import Mathlib.Order.Copy import Mathlib.Data.Set.Subsingleton #align_import category_theory.sites.grothendieck from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7" /-! # Grothendieck topologies Definition and lemmas about Grothendieck topologies. A Grothendieck topology for a category `C` is a set of sieves on each object `X` satisfying certain closure conditions. Alternate versions of the axioms (in arrow form) are also described. Two explicit examples of Grothendieck topologies are given: * The dense topology * The atomic topology as well as the complete lattice structure on Grothendieck topologies (which gives two additional explicit topologies: the discrete and trivial topologies.) A pretopology, or a basis for a topology is defined in `Mathlib/CategoryTheory/Sites/Pretopology.lean`. The topology associated to a topological space is defined in `Mathlib/CategoryTheory/Sites/Spaces.lean`. ## Tags Grothendieck topology, coverage, pretopology, site ## References * [nLab, Grothendieck topology](https://ncatlab.org/nlab/show/Grothendieck+topology) * [S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic][MM92] ## Implementation notes We use the definition of [nlab] and [MM92][] (Chapter III, Section 2), where Grothendieck topologies are saturated collections of morphisms, rather than the notions of the Stacks project (00VG) and the Elephant, in which topologies are allowed to be unsaturated, and are then completed. TODO (BM): Add the definition from Stacks, as a pretopology, and complete to a topology. This is so that we can produce a bijective correspondence between Grothendieck topologies on a small category and Lawvere-Tierney topologies on its presheaf topos, as well as the equivalence between Grothendieck topoi and left exact reflective subcategories of presheaf toposes. -/ universe v₁ u₁ v u namespace CategoryTheory open CategoryTheory Category variable (C : Type u) [Category.{v} C] /-- The definition of a Grothendieck topology: a set of sieves `J X` on each object `X` satisfying three axioms: 1. For every object `X`, the maximal sieve is in `J X`. 2. If `S ∈ J X` then its pullback along any `h : Y ⟶ X` is in `J Y`. 3. If `S ∈ J X` and `R` is a sieve on `X`, then provided that the pullback of `R` along any arrow `f : Y ⟶ X` in `S` is in `J Y`, we have that `R` itself is in `J X`. A sieve `S` on `X` is referred to as `J`-covering, (or just covering), if `S ∈ J X`. See <https://stacks.math.columbia.edu/tag/00Z4>, or [nlab], or [MM92][] Chapter III, Section 2, Definition 1. -/ structure GrothendieckTopology where /-- A Grothendieck topology on `C` consists of a set of sieves for each object `X`, which satisfy some axioms. -/ sieves : ∀ X : C, Set (Sieve X) /-- The sieves associated to each object must contain the top sieve. Use `GrothendieckTopology.top_mem`. -/ top_mem' : ∀ X, ⊤ ∈ sieves X /-- Stability under pullback. Use `GrothendieckTopology.pullback_stable`. -/ pullback_stable' : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves X → S.pullback f ∈ sieves Y /-- Transitivity of sieves in a Grothendieck topology. Use `GrothendieckTopology.transitive`. -/ transitive' : ∀ ⦃X⦄ ⦃S : Sieve X⦄ (_ : S ∈ sieves X) (R : Sieve X), (∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄, S f → R.pullback f ∈ sieves Y) → R ∈ sieves X #align category_theory.grothendieck_topology CategoryTheory.GrothendieckTopology namespace GrothendieckTopology instance : CoeFun (GrothendieckTopology C) fun _ => ∀ X : C, Set (Sieve X) := ⟨sieves⟩ variable {C} variable {X Y : C} {S R : Sieve X} variable (J : GrothendieckTopology C) /-- An extensionality lemma in terms of the coercion to a pi-type. We prove this explicitly rather than deriving it so that it is in terms of the coercion rather than the projection `.sieves`. -/ @[ext] theorem ext {J₁ J₂ : GrothendieckTopology C} (h : (J₁ : ∀ X : C, Set (Sieve X)) = J₂) : J₁ = J₂ := by cases J₁ cases J₂ congr #align category_theory.grothendieck_topology.ext CategoryTheory.GrothendieckTopology.ext /- Porting note: This is now a syntactic tautology. @[simp] theorem mem_sieves_iff_coe : S ∈ J.sieves X ↔ S ∈ J X := Iff.rfl #align category_theory.grothendieck_topology.mem_sieves_iff_coe CategoryTheory.GrothendieckTopology.mem_sieves_iff_coe -/ /-- Also known as the maximality axiom. -/ @[simp] theorem top_mem (X : C) : ⊤ ∈ J X := J.top_mem' X #align category_theory.grothendieck_topology.top_mem CategoryTheory.GrothendieckTopology.top_mem /-- Also known as the stability axiom. -/ @[simp] theorem pullback_stable (f : Y ⟶ X) (hS : S ∈ J X) : S.pullback f ∈ J Y := J.pullback_stable' f hS #align category_theory.grothendieck_topology.pullback_stable CategoryTheory.GrothendieckTopology.pullback_stable theorem transitive (hS : S ∈ J X) (R : Sieve X) (h : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄, S f → R.pullback f ∈ J Y) : R ∈ J X := J.transitive' hS R h #align category_theory.grothendieck_topology.transitive CategoryTheory.GrothendieckTopology.transitive theorem covering_of_eq_top : S = ⊤ → S ∈ J X := fun h => h.symm ▸ J.top_mem X #align category_theory.grothendieck_topology.covering_of_eq_top CategoryTheory.GrothendieckTopology.covering_of_eq_top /-- If `S` is a subset of `R`, and `S` is covering, then `R` is covering as well. See <https://stacks.math.columbia.edu/tag/00Z5> (2), or discussion after [MM92] Chapter III, Section 2, Definition 1. -/ theorem superset_covering (Hss : S ≤ R) (sjx : S ∈ J X) : R ∈ J X := by apply J.transitive sjx R fun Y f hf => _ intros Y f hf apply covering_of_eq_top rw [← top_le_iff, ← S.pullback_eq_top_of_mem hf] apply Sieve.pullback_monotone _ Hss #align category_theory.grothendieck_topology.superset_covering CategoryTheory.GrothendieckTopology.superset_covering /-- The intersection of two covering sieves is covering. See <https://stacks.math.columbia.edu/tag/00Z5> (1), or [MM92] Chapter III, Section 2, Definition 1 (iv). -/ theorem intersection_covering (rj : R ∈ J X) (sj : S ∈ J X) : R ⊓ S ∈ J X := by apply J.transitive rj _ fun Y f Hf => _ intros Y f hf rw [Sieve.pullback_inter, R.pullback_eq_top_of_mem hf] simp [sj] #align category_theory.grothendieck_topology.intersection_covering CategoryTheory.GrothendieckTopology.intersection_covering @[simp] theorem intersection_covering_iff : R ⊓ S ∈ J X ↔ R ∈ J X ∧ S ∈ J X := ⟨fun h => ⟨J.superset_covering inf_le_left h, J.superset_covering inf_le_right h⟩, fun t => intersection_covering _ t.1 t.2⟩ #align category_theory.grothendieck_topology.intersection_covering_iff CategoryTheory.GrothendieckTopology.intersection_covering_iff theorem bind_covering {S : Sieve X} {R : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S f → Sieve Y} (hS : S ∈ J X) (hR : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (H : S f), R H ∈ J Y) : Sieve.bind S R ∈ J X := J.transitive hS _ fun _ f hf => superset_covering J (Sieve.le_pullback_bind S R f hf) (hR hf) #align category_theory.grothendieck_topology.bind_covering CategoryTheory.GrothendieckTopology.bind_covering /-- The sieve `S` on `X` `J`-covers an arrow `f` to `X` if `S.pullback f ∈ J Y`. This definition is an alternate way of presenting a Grothendieck topology. -/ def Covers (S : Sieve X) (f : Y ⟶ X) : Prop := S.pullback f ∈ J Y #align category_theory.grothendieck_topology.covers CategoryTheory.GrothendieckTopology.Covers theorem covers_iff (S : Sieve X) (f : Y ⟶ X) : J.Covers S f ↔ S.pullback f ∈ J Y := Iff.rfl #align category_theory.grothendieck_topology.covers_iff CategoryTheory.GrothendieckTopology.covers_iff theorem covering_iff_covers_id (S : Sieve X) : S ∈ J X ↔ J.Covers S (𝟙 X) := by simp [covers_iff] #align category_theory.grothendieck_topology.covering_iff_covers_id CategoryTheory.GrothendieckTopology.covering_iff_covers_id /-- The maximality axiom in 'arrow' form: Any arrow `f` in `S` is covered by `S`. -/ theorem arrow_max (f : Y ⟶ X) (S : Sieve X) (hf : S f) : J.Covers S f := by rw [Covers, (Sieve.pullback_eq_top_iff_mem f).1 hf] apply J.top_mem #align category_theory.grothendieck_topology.arrow_max CategoryTheory.GrothendieckTopology.arrow_max /-- The stability axiom in 'arrow' form: If `S` covers `f` then `S` covers `g ≫ f` for any `g`. -/ theorem arrow_stable (f : Y ⟶ X) (S : Sieve X) (h : J.Covers S f) {Z : C} (g : Z ⟶ Y) : J.Covers S (g ≫ f) := by rw [covers_iff] at h ⊢ simp [h, Sieve.pullback_comp] #align category_theory.grothendieck_topology.arrow_stable CategoryTheory.GrothendieckTopology.arrow_stable /-- The transitivity axiom in 'arrow' form: If `S` covers `f` and every arrow in `S` is covered by `R`, then `R` covers `f`. -/ theorem arrow_trans (f : Y ⟶ X) (S R : Sieve X) (h : J.Covers S f) : (∀ {Z : C} (g : Z ⟶ X), S g → J.Covers R g) → J.Covers R f := by intro k apply J.transitive h intro Z g hg rw [← Sieve.pullback_comp] apply k (g ≫ f) hg #align category_theory.grothendieck_topology.arrow_trans CategoryTheory.GrothendieckTopology.arrow_trans theorem arrow_intersect (f : Y ⟶ X) (S R : Sieve X) (hS : J.Covers S f) (hR : J.Covers R f) : J.Covers (S ⊓ R) f := by simpa [covers_iff] using And.intro hS hR #align category_theory.grothendieck_topology.arrow_intersect CategoryTheory.GrothendieckTopology.arrow_intersect variable (C) /-- The trivial Grothendieck topology, in which only the maximal sieve is covering. This topology is also known as the indiscrete, coarse, or chaotic topology. See [MM92] Chapter III, Section 2, example (a), or https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies -/ def trivial : GrothendieckTopology C where sieves X := {⊤} top_mem' X := rfl pullback_stable' X Y S f hf := by rw [Set.mem_singleton_iff] at hf ⊢ simp [hf] transitive' X S hS R hR := by rw [Set.mem_singleton_iff, ← Sieve.id_mem_iff_eq_top] at hS simpa using hR hS #align category_theory.grothendieck_topology.trivial CategoryTheory.GrothendieckTopology.trivial /-- The discrete Grothendieck topology, in which every sieve is covering. See https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies. -/ def discrete : GrothendieckTopology C where sieves X := Set.univ top_mem' := by simp pullback_stable' X Y f := by simp transitive' := by simp #align category_theory.grothendieck_topology.discrete CategoryTheory.GrothendieckTopology.discrete variable {C} theorem trivial_covering : S ∈ trivial C X ↔ S = ⊤ := Set.mem_singleton_iff #align category_theory.grothendieck_topology.trivial_covering CategoryTheory.GrothendieckTopology.trivial_covering /-- See <https://stacks.math.columbia.edu/tag/00Z6> -/ instance instLEGrothendieckTopology : LE (GrothendieckTopology C) where le J₁ J₂ := (J₁ : ∀ X : C, Set (Sieve X)) ≤ (J₂ : ∀ X : C, Set (Sieve X)) theorem le_def {J₁ J₂ : GrothendieckTopology C} : J₁ ≤ J₂ ↔ (J₁ : ∀ X : C, Set (Sieve X)) ≤ J₂ := Iff.rfl #align category_theory.grothendieck_topology.le_def CategoryTheory.GrothendieckTopology.le_def /-- See <https://stacks.math.columbia.edu/tag/00Z6> -/ instance : PartialOrder (GrothendieckTopology C) := { instLEGrothendieckTopology with le_refl := fun J₁ => le_def.mpr le_rfl le_trans := fun J₁ J₂ J₃ h₁₂ h₂₃ => le_def.mpr (le_trans h₁₂ h₂₃) le_antisymm := fun J₁ J₂ h₁₂ h₂₁ => GrothendieckTopology.ext (le_antisymm h₁₂ h₂₁) } /-- See <https://stacks.math.columbia.edu/tag/00Z7> -/ instance : InfSet (GrothendieckTopology C) where sInf T := { sieves := sInf (sieves '' T) top_mem' := by rintro X S ⟨⟨_, J, hJ, rfl⟩, rfl⟩ simp pullback_stable' := by rintro X Y S hS f _ ⟨⟨_, J, hJ, rfl⟩, rfl⟩ apply J.pullback_stable _ (f _ ⟨⟨_, _, hJ, rfl⟩, rfl⟩) transitive' := by rintro X S hS R h _ ⟨⟨_, J, hJ, rfl⟩, rfl⟩ apply J.transitive (hS _ ⟨⟨_, _, hJ, rfl⟩, rfl⟩) _ fun Y f hf => h hf _ ⟨⟨_, _, hJ, rfl⟩, rfl⟩ } /-- See <https://stacks.math.columbia.edu/tag/00Z7> -/	Mathlib/CategoryTheory/Sites/Grothendieck.lean	286	290	theorem isGLB_sInf (s : Set (GrothendieckTopology C)) : IsGLB s (sInf s) := by	refine @IsGLB.of_image _ _ _ _ sieves ?_ _ _ ?_ · intros rfl · exact _root_.isGLB_sInf _
/- Copyright (c) 2019 Yury Kudriashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudriashov -/ import Mathlib.Algebra.Order.BigOperators.Ring.Finset import Mathlib.Analysis.Convex.Hull import Mathlib.LinearAlgebra.AffineSpace.Basis #align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d" /-! # Convex combinations This file defines convex combinations of points in a vector space. ## Main declarations * `Finset.centerMass`: Center of mass of a finite family of points. ## Implementation notes We divide by the sum of the weights in the definition of `Finset.centerMass` because of the way mathematical arguments go: one doesn't change weights, but merely adds some. This also makes a few lemmas unconditional on the sum of the weights being `1`. -/ open Set Function open scoped Classical open Pointwise universe u u' variable {R R' E F ι ι' α : Type} [LinearOrderedField R] [LinearOrderedField R'] [AddCommGroup E] [AddCommGroup F] [LinearOrderedAddCommGroup α] [Module R E] [Module R F] [Module R α] [OrderedSMul R α] {s : Set E} /-- Center of mass of a finite collection of points with prescribed weights. Note that we require neither `0 ≤ w i` nor `∑ w = 1`. -/ def Finset.centerMass (t : Finset ι) (w : ι → R) (z : ι → E) : E := (∑ i ∈ t, w i)⁻¹ • ∑ i ∈ t, w i • z i #align finset.center_mass Finset.centerMass variable (i j : ι) (c : R) (t : Finset ι) (w : ι → R) (z : ι → E) open Finset theorem Finset.centerMass_empty : (∅ : Finset ι).centerMass w z = 0 := by simp only [centerMass, sum_empty, smul_zero] #align finset.center_mass_empty Finset.centerMass_empty theorem Finset.centerMass_pair (hne : i ≠ j) : ({i, j} : Finset ι).centerMass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j := by simp only [centerMass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul] #align finset.center_mass_pair Finset.centerMass_pair variable {w} theorem Finset.centerMass_insert (ha : i ∉ t) (hw : ∑ j ∈ t, w j ≠ 0) : (insert i t).centerMass w z = (w i / (w i + ∑ j ∈ t, w j)) • z i + ((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z := by simp only [centerMass, sum_insert ha, smul_add, (mul_smul _ _ _).symm, ← div_eq_inv_mul] congr 2 rw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div] #align finset.center_mass_insert Finset.centerMass_insert theorem Finset.centerMass_singleton (hw : w i ≠ 0) : ({i} : Finset ι).centerMass w z = z i := by rw [centerMass, sum_singleton, sum_singleton, ← mul_smul, inv_mul_cancel hw, one_smul] #align finset.center_mass_singleton Finset.centerMass_singleton @[simp] lemma Finset.centerMass_neg_left : t.centerMass (-w) z = t.centerMass w z := by simp [centerMass, inv_neg] lemma Finset.centerMass_smul_left {c : R'} [Module R' R] [Module R' E] [SMulCommClass R' R R] [IsScalarTower R' R R] [SMulCommClass R R' E] [IsScalarTower R' R E] (hc : c ≠ 0) : t.centerMass (c • w) z = t.centerMass w z := by simp [centerMass, -smul_assoc, smul_assoc c, ← smul_sum, smul_inv₀, smul_smul_smul_comm, hc] theorem Finset.centerMass_eq_of_sum_1 (hw : ∑ i ∈ t, w i = 1) : t.centerMass w z = ∑ i ∈ t, w i • z i := by simp only [Finset.centerMass, hw, inv_one, one_smul] #align finset.center_mass_eq_of_sum_1 Finset.centerMass_eq_of_sum_1 theorem Finset.centerMass_smul : (t.centerMass w fun i => c • z i) = c • t.centerMass w z := by simp only [Finset.centerMass, Finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc] #align finset.center_mass_smul Finset.centerMass_smul /-- A convex combination of two centers of mass is a center of mass as well. This version deals with two different index types. -/ theorem Finset.centerMass_segment' (s : Finset ι) (t : Finset ι') (ws : ι → R) (zs : ι → E) (wt : ι' → R) (zt : ι' → E) (hws : ∑ i ∈ s, ws i = 1) (hwt : ∑ i ∈ t, wt i = 1) (a b : R) (hab : a + b = 1) : a • s.centerMass ws zs + b • t.centerMass wt zt = (s.disjSum t).centerMass (Sum.elim (fun i => a ws i) fun j => b * wt j) (Sum.elim zs zt) := by rw [s.centerMass_eq_of_sum_1 _ hws, t.centerMass_eq_of_sum_1 _ hwt, smul_sum, smul_sum, ← Finset.sum_sum_elim, Finset.centerMass_eq_of_sum_1] · congr with ⟨⟩ <;> simp only [Sum.elim_inl, Sum.elim_inr, mul_smul] · rw [sum_sum_elim, ← mul_sum, ← mul_sum, hws, hwt, mul_one, mul_one, hab] #align finset.center_mass_segment' Finset.centerMass_segment' /-- A convex combination of two centers of mass is a center of mass as well. This version works if two centers of mass share the set of original points. -/ theorem Finset.centerMass_segment (s : Finset ι) (w₁ w₂ : ι → R) (z : ι → E) (hw₁ : ∑ i ∈ s, w₁ i = 1) (hw₂ : ∑ i ∈ s, w₂ i = 1) (a b : R) (hab : a + b = 1) : a • s.centerMass w₁ z + b • s.centerMass w₂ z = s.centerMass (fun i => a * w₁ i + b * w₂ i) z := by have hw : (∑ i ∈ s, (a * w₁ i + b * w₂ i)) = 1 := by simp only [← mul_sum, sum_add_distrib, mul_one, ] simp only [Finset.centerMass_eq_of_sum_1, Finset.centerMass_eq_of_sum_1 _ _ hw, smul_sum, sum_add_distrib, add_smul, mul_smul, ] #align finset.center_mass_segment Finset.centerMass_segment theorem Finset.centerMass_ite_eq (hi : i ∈ t) : t.centerMass (fun j => if i = j then (1 : R) else 0) z = z i := by rw [Finset.centerMass_eq_of_sum_1] · trans ∑ j ∈ t, if i = j then z i else 0 · congr with i split_ifs with h exacts [h ▸ one_smul _ _, zero_smul _ _] · rw [sum_ite_eq, if_pos hi] · rw [sum_ite_eq, if_pos hi] #align finset.center_mass_ite_eq Finset.centerMass_ite_eq variable {t} theorem Finset.centerMass_subset {t' : Finset ι} (ht : t ⊆ t') (h : ∀ i ∈ t', i ∉ t → w i = 0) : t.centerMass w z = t'.centerMass w z := by rw [centerMass, sum_subset ht h, smul_sum, centerMass, smul_sum] apply sum_subset ht intro i hit' hit rw [h i hit' hit, zero_smul, smul_zero] #align finset.center_mass_subset Finset.centerMass_subset theorem Finset.centerMass_filter_ne_zero : (t.filter fun i => w i ≠ 0).centerMass w z = t.centerMass w z := Finset.centerMass_subset z (filter_subset _ _) fun i hit hit' => by simpa only [hit, mem_filter, true_and_iff, Ne, Classical.not_not] using hit' #align finset.center_mass_filter_ne_zero Finset.centerMass_filter_ne_zero namespace Finset theorem centerMass_le_sup {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : 0 < ∑ i ∈ s, w i) : s.centerMass w f ≤ s.sup' (nonempty_of_ne_empty <\| by rintro rfl; simp at hw₁) f := by rw [centerMass, inv_smul_le_iff_of_pos hw₁, sum_smul] exact sum_le_sum fun i hi => smul_le_smul_of_nonneg_left (le_sup' _ hi) <\| hw₀ i hi #align finset.center_mass_le_sup Finset.centerMass_le_sup theorem inf_le_centerMass {s : Finset ι} {f : ι → α} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : 0 < ∑ i ∈ s, w i) : s.inf' (nonempty_of_ne_empty <\| by rintro rfl; simp at hw₁) f ≤ s.centerMass w f := @centerMass_le_sup R _ αᵒᵈ _ _ _ _ _ _ _ hw₀ hw₁ #align finset.inf_le_center_mass Finset.inf_le_centerMass end Finset variable {z} lemma Finset.centerMass_of_sum_add_sum_eq_zero {s t : Finset ι} (hw : ∑ i ∈ s, w i + ∑ i ∈ t, w i = 0) (hz : ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0) : s.centerMass w z = t.centerMass w z := by simp [centerMass, eq_neg_of_add_eq_zero_right hw, eq_neg_of_add_eq_zero_left hz, ← neg_inv] /-- The center of mass of a finite subset of a convex set belongs to the set provided that all weights are non-negative, and the total weight is positive. -/ theorem Convex.centerMass_mem (hs : Convex R s) : (∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i ∈ t, w i) → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s := by induction' t using Finset.induction with i t hi ht · simp [lt_irrefl] intro h₀ hpos hmem have zi : z i ∈ s := hmem _ (mem_insert_self _ _) have hs₀ : ∀ j ∈ t, 0 ≤ w j := fun j hj => h₀ j <\| mem_insert_of_mem hj rw [sum_insert hi] at hpos by_cases hsum_t : ∑ j ∈ t, w j = 0 · have ws : ∀ j ∈ t, w j = 0 := (sum_eq_zero_iff_of_nonneg hs₀).1 hsum_t have wz : ∑ j ∈ t, w j • z j = 0 := sum_eq_zero fun i hi => by simp [ws i hi] simp only [centerMass, sum_insert hi, wz, hsum_t, add_zero] simp only [hsum_t, add_zero] at hpos rw [← mul_smul, inv_mul_cancel (ne_of_gt hpos), one_smul] exact zi · rw [Finset.centerMass_insert _ _ _ hi hsum_t] refine convex_iff_div.1 hs zi (ht hs₀ ?_ ?_) ?_ (sum_nonneg hs₀) hpos · exact lt_of_le_of_ne (sum_nonneg hs₀) (Ne.symm hsum_t) · intro j hj exact hmem j (mem_insert_of_mem hj) · exact h₀ _ (mem_insert_self _ _) #align convex.center_mass_mem Convex.centerMass_mem theorem Convex.sum_mem (hs : Convex R s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hz : ∀ i ∈ t, z i ∈ s) : (∑ i ∈ t, w i • z i) ∈ s := by simpa only [h₁, centerMass, inv_one, one_smul] using hs.centerMass_mem h₀ (h₁.symm ▸ zero_lt_one) hz #align convex.sum_mem Convex.sum_mem /-- A version of `Convex.sum_mem` for `finsum`s. If `s` is a convex set, `w : ι → R` is a family of nonnegative weights with sum one and `z : ι → E` is a family of elements of a module over `R` such that `z i ∈ s` whenever `w i ≠ 0`, then the sum `∑ᶠ i, w i • z i` belongs to `s`. See also `PartitionOfUnity.finsum_smul_mem_convex`. -/ theorem Convex.finsum_mem {ι : Sort} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s) (h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) : (∑ᶠ i, w i • z i) ∈ s := by have hfin_w : (support (w ∘ PLift.down)).Finite := by by_contra H rw [finsum, dif_neg H] at h₁ exact zero_ne_one h₁ have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset := (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge rw [finsum_eq_sum_plift_of_support_subset hsub] refine hs.sum_mem (fun _ _ => h₀ _) ?_ fun i hi => hz _ ?_ · rwa [finsum, dif_pos hfin_w] at h₁ · rwa [hfin_w.mem_toFinset] at hi #align convex.finsum_mem Convex.finsum_mem theorem convex_iff_sum_mem : Convex R s ↔ ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → (∑ x ∈ t, w x • x) ∈ s := by refine ⟨fun hs t w hw₀ hw₁ hts => hs.sum_mem hw₀ hw₁ hts, ?_⟩ intro h x hx y hy a b ha hb hab by_cases h_cases : x = y · rw [h_cases, ← add_smul, hab, one_smul] exact hy · convert h {x, y} (fun z => if z = y then b else a) _ _ _ -- Porting note: Original proof had 2 `simp_intro i hi` · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial] · intro i _ simp only split_ifs <;> assumption · simp only [sum_pair h_cases, if_neg h_cases, if_pos trivial, hab] · intro i hi simp only [Finset.mem_singleton, Finset.mem_insert] at hi cases hi <;> subst i <;> assumption #align convex_iff_sum_mem convex_iff_sum_mem theorem Finset.centerMass_mem_convexHull (t : Finset ι) {w : ι → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hws : 0 < ∑ i ∈ t, w i) {z : ι → E} (hz : ∀ i ∈ t, z i ∈ s) : t.centerMass w z ∈ convexHull R s := (convex_convexHull R s).centerMass_mem hw₀ hws fun i hi => subset_convexHull R s <\| hz i hi #align finset.center_mass_mem_convex_hull Finset.centerMass_mem_convexHull /-- A version of `Finset.centerMass_mem_convexHull` for when the weights are nonpositive. -/ lemma Finset.centerMass_mem_convexHull_of_nonpos (t : Finset ι) (hw₀ : ∀ i ∈ t, w i ≤ 0) (hws : ∑ i ∈ t, w i < 0) (hz : ∀ i ∈ t, z i ∈ s) : t.centerMass w z ∈ convexHull R s := by rw [← centerMass_neg_left] exact Finset.centerMass_mem_convexHull _ (fun _i hi ↦ neg_nonneg.2 <\| hw₀ _ hi) (by simpa) hz /-- A refinement of `Finset.centerMass_mem_convexHull` when the indexed family is a `Finset` of the space. -/ theorem Finset.centerMass_id_mem_convexHull (t : Finset E) {w : E → R} (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hws : 0 < ∑ i ∈ t, w i) : t.centerMass w id ∈ convexHull R (t : Set E) := t.centerMass_mem_convexHull hw₀ hws fun _ => mem_coe.2 #align finset.center_mass_id_mem_convex_hull Finset.centerMass_id_mem_convexHull /-- A version of `Finset.centerMass_mem_convexHull` for when the weights are nonpositive. -/ lemma Finset.centerMass_id_mem_convexHull_of_nonpos (t : Finset E) {w : E → R} (hw₀ : ∀ i ∈ t, w i ≤ 0) (hws : ∑ i ∈ t, w i < 0) : t.centerMass w id ∈ convexHull R (t : Set E) := t.centerMass_mem_convexHull_of_nonpos hw₀ hws fun _ ↦ mem_coe.2 theorem affineCombination_eq_centerMass {ι : Type} {t : Finset ι} {p : ι → E} {w : ι → R} (hw₂ : ∑ i ∈ t, w i = 1) : t.affineCombination R p w = centerMass t w p := by rw [affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ w _ hw₂ (0 : E), Finset.weightedVSubOfPoint_apply, vadd_eq_add, add_zero, t.centerMass_eq_of_sum_1 _ hw₂] simp_rw [vsub_eq_sub, sub_zero] #align affine_combination_eq_center_mass affineCombination_eq_centerMass theorem affineCombination_mem_convexHull {s : Finset ι} {v : ι → E} {w : ι → R} (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1) : s.affineCombination R v w ∈ convexHull R (range v) := by rw [affineCombination_eq_centerMass hw₁] apply s.centerMass_mem_convexHull hw₀ · simp [hw₁] · simp #align affine_combination_mem_convex_hull affineCombination_mem_convexHull /-- The centroid can be regarded as a center of mass. -/ @[simp] theorem Finset.centroid_eq_centerMass (s : Finset ι) (hs : s.Nonempty) (p : ι → E) : s.centroid R p = s.centerMass (s.centroidWeights R) p := affineCombination_eq_centerMass (s.sum_centroidWeights_eq_one_of_nonempty R hs) #align finset.centroid_eq_center_mass Finset.centroid_eq_centerMass theorem Finset.centroid_mem_convexHull (s : Finset E) (hs : s.Nonempty) : s.centroid R id ∈ convexHull R (s : Set E) := by rw [s.centroid_eq_centerMass hs] apply s.centerMass_id_mem_convexHull · simp only [inv_nonneg, imp_true_iff, Nat.cast_nonneg, Finset.centroidWeights_apply] · have hs_card : (s.card : R) ≠ 0 := by simp [Finset.nonempty_iff_ne_empty.mp hs] simp only [hs_card, Finset.sum_const, nsmul_eq_mul, mul_inv_cancel, Ne, not_false_iff, Finset.centroidWeights_apply, zero_lt_one] #align finset.centroid_mem_convex_hull Finset.centroid_mem_convexHull theorem convexHull_range_eq_exists_affineCombination (v : ι → E) : convexHull R (range v) = { x \| ∃ (s : Finset ι) (w : ι → R), (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧ s.affineCombination R v w = x } := by refine Subset.antisymm (convexHull_min ?_ ?_) ?_ · intro x hx obtain ⟨i, hi⟩ := Set.mem_range.mp hx exact ⟨{i}, Function.const ι (1 : R), by simp, by simp, by simp [hi]⟩ · rintro x ⟨s, w, hw₀, hw₁, rfl⟩ y ⟨s', w', hw₀', hw₁', rfl⟩ a b ha hb hab let W : ι → R := fun i => (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0 have hW₁ : (s ∪ s').sum W = 1 := by rw [sum_add_distrib, ← sum_subset subset_union_left, ← sum_subset subset_union_right, sum_ite_of_true _ _ fun i hi => hi, sum_ite_of_true _ _ fun i hi => hi, ← mul_sum, ← mul_sum, hw₁, hw₁', ← add_mul, hab, mul_one] <;> intro i _ hi' <;> simp [hi'] refine ⟨s ∪ s', W, ?_, hW₁, ?_⟩ · rintro i - by_cases hi : i ∈ s <;> by_cases hi' : i ∈ s' <;> simp [W, hi, hi', add_nonneg, mul_nonneg ha (hw₀ i _), mul_nonneg hb (hw₀' i _)] · simp_rw [affineCombination_eq_linear_combination (s ∪ s') v _ hW₁, affineCombination_eq_linear_combination s v w hw₁, affineCombination_eq_linear_combination s' v w' hw₁', add_smul, sum_add_distrib] rw [← sum_subset subset_union_left, ← sum_subset subset_union_right] · simp only [ite_smul, sum_ite_of_true _ _ fun _ hi => hi, mul_smul, ← smul_sum] · intro i _ hi' simp [hi'] · intro i _ hi' simp [hi'] · rintro x ⟨s, w, hw₀, hw₁, rfl⟩ exact affineCombination_mem_convexHull hw₀ hw₁ #align convex_hull_range_eq_exists_affine_combination convexHull_range_eq_exists_affineCombination /-- Convex hull of `s` is equal to the set of all centers of masses of `Finset`s `t`, `z '' t ⊆ s`. For universe reasons, you shouldn't use this lemma to prove that a given center of mass belongs to the convex hull. Use convexity of the convex hull instead. -/ theorem convexHull_eq (s : Set E) : convexHull R s = { x : E \| ∃ (ι : Type) (t : Finset ι) (w : ι → R) (z : ι → E), (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x } := by refine Subset.antisymm (convexHull_min ?_ ?_) ?_ · intro x hx use PUnit, {PUnit.unit}, fun _ => 1, fun _ => x, fun _ _ => zero_le_one, sum_singleton _ _, fun _ _ => hx simp only [Finset.centerMass, Finset.sum_singleton, inv_one, one_smul] · rintro x ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ y ⟨ι', sy, wy, zy, hwy₀, hwy₁, hzy, rfl⟩ a b ha hb hab rw [Finset.centerMass_segment' _ _ _ _ _ _ hwx₁ hwy₁ _ _ hab] refine ⟨_, _, _, _, ?_, ?_, ?_, rfl⟩ · rintro i hi rw [Finset.mem_disjSum] at hi rcases hi with (⟨j, hj, rfl⟩ \| ⟨j, hj, rfl⟩) <;> simp only [Sum.elim_inl, Sum.elim_inr] <;> apply_rules [mul_nonneg, hwx₀, hwy₀] · simp [Finset.sum_sum_elim, ← mul_sum, ] · intro i hi rw [Finset.mem_disjSum] at hi rcases hi with (⟨j, hj, rfl⟩ \| ⟨j, hj, rfl⟩) <;> apply_rules [hzx, hzy] · rintro _ ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩ exact t.centerMass_mem_convexHull hw₀ (hw₁.symm ▸ zero_lt_one) hz #align convex_hull_eq convexHull_eq theorem Finset.convexHull_eq (s : Finset E) : convexHull R ↑s = { x : E \| ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x } := by refine Set.Subset.antisymm (convexHull_min ?_ ?_) ?_ · intro x hx rw [Finset.mem_coe] at hx refine ⟨_, ?_, ?_, Finset.centerMass_ite_eq _ _ _ hx⟩ · intros split_ifs exacts [zero_le_one, le_refl 0] · rw [Finset.sum_ite_eq, if_pos hx] · rintro x ⟨wx, hwx₀, hwx₁, rfl⟩ y ⟨wy, hwy₀, hwy₁, rfl⟩ a b ha hb hab rw [Finset.centerMass_segment _ _ _ _ hwx₁ hwy₁ _ _ hab] refine ⟨_, ?_, ?_, rfl⟩ · rintro i hi apply_rules [add_nonneg, mul_nonneg, hwx₀, hwy₀] · simp only [Finset.sum_add_distrib, ← mul_sum, mul_one, ] · rintro _ ⟨w, hw₀, hw₁, rfl⟩ exact s.centerMass_mem_convexHull (fun x hx => hw₀ _ hx) (hw₁.symm ▸ zero_lt_one) fun x hx => hx #align finset.convex_hull_eq Finset.convexHull_eq theorem Finset.mem_convexHull {s : Finset E} {x : E} : x ∈ convexHull R (s : Set E) ↔ ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x := by rw [Finset.convexHull_eq, Set.mem_setOf_eq] #align finset.mem_convex_hull Finset.mem_convexHull /-- This is a version of `Finset.mem_convexHull` stated without `Finset.centerMass`. -/ lemma Finset.mem_convexHull' {s : Finset E} {x : E} : x ∈ convexHull R (s : Set E) ↔ ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ ∑ y ∈ s, w y • y = x := by rw [mem_convexHull] refine exists_congr fun w ↦ and_congr_right' $ and_congr_right fun hw ↦ ?_ simp_rw [centerMass_eq_of_sum_1 _ _ hw, id_eq] theorem Set.Finite.convexHull_eq {s : Set E} (hs : s.Finite) : convexHull R s = { x : E \| ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ hs.toFinset, w y = 1 ∧ hs.toFinset.centerMass w id = x } := by simpa only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, exists_prop] using hs.toFinset.convexHull_eq #align set.finite.convex_hull_eq Set.Finite.convexHull_eq /-- A weak version of Carathéodory's theorem. -/ theorem convexHull_eq_union_convexHull_finite_subsets (s : Set E) : convexHull R s = ⋃ (t : Finset E) (w : ↑t ⊆ s), convexHull R ↑t := by refine Subset.antisymm ?_ ?_ · rw [_root_.convexHull_eq] rintro x ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩ simp only [mem_iUnion] refine ⟨t.image z, ?_, ?_⟩ · rw [coe_image, Set.image_subset_iff] exact hz · apply t.centerMass_mem_convexHull hw₀ · simp only [hw₁, zero_lt_one] · exact fun i hi => Finset.mem_coe.2 (Finset.mem_image_of_mem _ hi) · exact iUnion_subset fun i => iUnion_subset convexHull_mono #align convex_hull_eq_union_convex_hull_finite_subsets convexHull_eq_union_convexHull_finite_subsets theorem mk_mem_convexHull_prod {t : Set F} {x : E} {y : F} (hx : x ∈ convexHull R s) (hy : y ∈ convexHull R t) : (x, y) ∈ convexHull R (s ×ˢ t) := by rw [_root_.convexHull_eq] at hx hy ⊢ obtain ⟨ι, a, w, S, hw, hw', hS, hSp⟩ := hx obtain ⟨κ, b, v, T, hv, hv', hT, hTp⟩ := hy have h_sum : ∑ i ∈ a ×ˢ b, w i.fst * v i.snd = 1 := by rw [Finset.sum_product, ← hw'] congr ext i have : ∑ y ∈ b, w i * v y = ∑ y ∈ b, v y * w i := by congr ext simp [mul_comm] rw [this, ← Finset.sum_mul, hv'] simp refine ⟨ι × κ, a ×ˢ b, fun p => w p.1 * v p.2, fun p => (S p.1, T p.2), fun p hp => ?_, h_sum, fun p hp => ?_, ?_⟩ · rw [mem_product] at hp exact mul_nonneg (hw p.1 hp.1) (hv p.2 hp.2) · rw [mem_product] at hp exact ⟨hS p.1 hp.1, hT p.2 hp.2⟩ ext · rw [← hSp, Finset.centerMass_eq_of_sum_1 _ _ hw', Finset.centerMass_eq_of_sum_1 _ _ h_sum] simp_rw [Prod.fst_sum, Prod.smul_mk] rw [Finset.sum_product] congr ext i have : (∑ j ∈ b, (w i * v j) • S i) = ∑ j ∈ b, v j • w i • S i := by congr ext rw [mul_smul, smul_comm] rw [this, ← Finset.sum_smul, hv', one_smul] · rw [← hTp, Finset.centerMass_eq_of_sum_1 _ _ hv', Finset.centerMass_eq_of_sum_1 _ _ h_sum] simp_rw [Prod.snd_sum, Prod.smul_mk] rw [Finset.sum_product, Finset.sum_comm] congr ext j simp_rw [mul_smul] rw [← Finset.sum_smul, hw', one_smul] #align mk_mem_convex_hull_prod mk_mem_convexHull_prod @[simp] theorem convexHull_prod (s : Set E) (t : Set F) : convexHull R (s ×ˢ t) = convexHull R s ×ˢ convexHull R t := Subset.antisymm (convexHull_min (prod_mono (subset_convexHull _ _) <\| subset_convexHull _ _) <\| (convex_convexHull _ _).prod <\| convex_convexHull _ _) <\| prod_subset_iff.2 fun _ hx _ => mk_mem_convexHull_prod hx #align convex_hull_prod convexHull_prod theorem convexHull_add (s t : Set E) : convexHull R (s + t) = convexHull R s + convexHull R t := by simp_rw [← image2_add, ← image_prod, ← IsLinearMap.isLinearMap_add.image_convexHull, convexHull_prod] #align convex_hull_add convexHull_add variable (R E) /-- `convexHull` is an additive monoid morphism under pointwise addition. -/ @[simps] def convexHullAddMonoidHom : Set E →+ Set E where toFun := convexHull R map_add' := convexHull_add map_zero' := convexHull_zero #align convex_hull_add_monoid_hom convexHullAddMonoidHom variable {R E} theorem convexHull_sub (s t : Set E) : convexHull R (s - t) = convexHull R s - convexHull R t := by simp_rw [sub_eq_add_neg, convexHull_add, ← convexHull_neg] #align convex_hull_sub convexHull_sub theorem convexHull_list_sum (l : List (Set E)) : convexHull R l.sum = (l.map <\| convexHull R).sum := map_list_sum (convexHullAddMonoidHom R E) l #align convex_hull_list_sum convexHull_list_sum theorem convexHull_multiset_sum (s : Multiset (Set E)) : convexHull R s.sum = (s.map <\| convexHull R).sum := map_multiset_sum (convexHullAddMonoidHom R E) s #align convex_hull_multiset_sum convexHull_multiset_sum theorem convexHull_sum {ι} (s : Finset ι) (t : ι → Set E) : convexHull R (∑ i ∈ s, t i) = ∑ i ∈ s, convexHull R (t i) := map_sum (convexHullAddMonoidHom R E) _ _ #align convex_hull_sum convexHull_sum /-! ### `stdSimplex` -/ variable (ι) [Fintype ι] {f : ι → R} /-- `stdSimplex 𝕜 ι` is the convex hull of the canonical basis in `ι → 𝕜`. -/ theorem convexHull_basis_eq_stdSimplex : convexHull R (range fun i j : ι => if i = j then (1 : R) else 0) = stdSimplex R ι := by refine Subset.antisymm (convexHull_min ?_ (convex_stdSimplex R ι)) ?_ · rintro _ ⟨i, rfl⟩ exact ite_eq_mem_stdSimplex R i · rintro w ⟨hw₀, hw₁⟩ rw [pi_eq_sum_univ w, ← Finset.univ.centerMass_eq_of_sum_1 _ hw₁] exact Finset.univ.centerMass_mem_convexHull (fun i _ => hw₀ i) (hw₁.symm ▸ zero_lt_one) fun i _ => mem_range_self i #align convex_hull_basis_eq_std_simplex convexHull_basis_eq_stdSimplex variable {ι} /-- The convex hull of a finite set is the image of the standard simplex in `s → ℝ` under the linear map sending each function `w` to `∑ x ∈ s, w x • x`. Since we have no sums over finite sets, we use sum over `@Finset.univ _ hs.fintype`. The map is defined in terms of operations on `(s → ℝ) →ₗ[ℝ] ℝ` so that later we will not need to prove that this map is linear. -/	Mathlib/Analysis/Convex/Combination.lean	521	529	theorem Set.Finite.convexHull_eq_image {s : Set E} (hs : s.Finite) : convexHull R s = haveI := hs.fintype (⇑(∑ x : s, (@LinearMap.proj R s _ (fun _ => R) _ _ x).smulRight x.1)) '' stdSimplex R s := by	letI := hs.fintype rw [← convexHull_basis_eq_stdSimplex, LinearMap.image_convexHull, ← Set.range_comp] apply congr_arg simp_rw [Function.comp] convert Subtype.range_coe.symm simp [LinearMap.sum_apply, ite_smul, Finset.filter_eq, Finset.mem_univ]
/- Copyright (c) 2021 Andrew Yang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang -/ import Mathlib.CategoryTheory.Limits.KanExtension import Mathlib.CategoryTheory.Sites.CoverPreserving import Mathlib.CategoryTheory.Sites.Sheafification #align_import category_theory.sites.cover_lifting from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7" /-! # Cocontinuous functors between sites. We define cocontinuous functors between sites as functors that pull covering sieves back to covering sieves. This concept is also known as cover-lifting or cover-reflecting functors. We use the original terminology and definition of SGA 4 III 2.1. However, the notion of cocontinuous functor should not be confused with the general definition of cocontinuous functors between categories as functors preserving small colimits. ## Main definitions * `CategoryTheory.Functor.IsCocontinuous`: a functor between sites is cocontinuous if it pulls back covering sieves to covering sieves * `CategoryTheory.Functor.sheafPushforwardCocontinuous`: A cocontinuous functor `G : (C, J) ⥤ (D, K)` induces a functor `Sheaf J A ⥤ Sheaf K A`. ## Main results * `CategoryTheory.ran_isSheaf_of_isCocontinuous`: If `G : C ⥤ D` is cocontinuous, then `Ran G.op` (`ₚu`) as a functor `(Cᵒᵖ ⥤ A) ⥤ (Dᵒᵖ ⥤ A)` of presheaves maps sheaves to sheaves. * `CategoryTheory.Sites.pullbackCopullbackAdjunction`: If `G : (C, J) ⥤ (D, K)` is cocontinuous and continuous, then `G.sheafPushforwardContinuous A J K` and `G.sheafPushforwardCocontinuous A J K` are adjoint. ## References * [Elephant]: Sketches of an Elephant, P. T. Johnstone: C2.3. * [S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic][MM92] * https://stacks.math.columbia.edu/tag/00XI -/ universe w' w v v₁ v₂ v₃ u u₁ u₂ u₃ noncomputable section open CategoryTheory open Opposite open CategoryTheory.Presieve.FamilyOfElements open CategoryTheory.Presieve open CategoryTheory.Limits namespace CategoryTheory section IsCocontinuous variable {C : Type} [Category C] {D : Type} [Category D] {E : Type} [Category E] (G : C ⥤ D) (G' : D ⥤ E) variable (J : GrothendieckTopology C) (K : GrothendieckTopology D) variable {L : GrothendieckTopology E} /-- A functor `G : (C, J) ⥤ (D, K)` between sites is called cocontinuous (SGA 4 III 2.1) if for all covering sieves `R` in `D`, `R.pullback G` is a covering sieve in `C`. -/ -- Porting note(#5171): removed `@[nolint has_nonempty_instance]` class Functor.IsCocontinuous : Prop where cover_lift : ∀ {U : C} {S : Sieve (G.obj U)} (_ : S ∈ K (G.obj U)), S.functorPullback G ∈ J U #align category_theory.cover_lifting CategoryTheory.Functor.IsCocontinuous lemma Functor.cover_lift [G.IsCocontinuous J K] {U : C} {S : Sieve (G.obj U)} (hS : S ∈ K (G.obj U)) : S.functorPullback G ∈ J U := IsCocontinuous.cover_lift hS /-- The identity functor on a site is cocontinuous. -/ instance isCocontinuous_id : Functor.IsCocontinuous (𝟭 C) J J := ⟨fun h => by simpa using h⟩ #align category_theory.id_cover_lifting CategoryTheory.isCocontinuous_id /-- The composition of two cocontinuous functors is cocontinuous. -/ theorem isCocontinuous_comp [G.IsCocontinuous J K] [G'.IsCocontinuous K L] : (G ⋙ G').IsCocontinuous J L where cover_lift h := G.cover_lift J K (G'.cover_lift K L h) #align category_theory.comp_cover_lifting CategoryTheory.isCocontinuous_comp end IsCocontinuous /-! We will now prove that `Ran G.op` (`ₚu`) maps sheaves to sheaves if `G` is cocontinuous (SGA 4 III 2.2). This can also be be found in <https://stacks.math.columbia.edu/tag/00XK>. However, the proof given there uses the amalgamation definition of sheaves, and thus does not require that `C` or `D` has categorical pullbacks. For the following proof sketch, `⊆` denotes the homs on `C` and `D` as in the topological analogy. By definition, the presheaf `𝒢 : Dᵒᵖ ⥤ A` is a sheaf if for every sieve `S` of `U : D`, and every compatible family of morphisms `X ⟶ 𝒢(V)` for each `V ⊆ U : S` with a fixed source `X`, we can glue them into a morphism `X ⟶ 𝒢(U)`. Since the presheaf `𝒢 := (Ran G.op).obj ℱ.val` is defined via `𝒢(U) = lim_{G(V) ⊆ U} ℱ(V)`, for gluing the family `x` into a `X ⟶ 𝒢(U)`, it suffices to provide a `X ⟶ ℱ(Y)` for each `G(Y) ⊆ U`. This can be done since `{ Y' ⊆ Y : G(Y') ⊆ U ∈ S}` is a covering sieve for `Y` on `C` (by the cocontinuity `G`). Thus the morphisms `X ⟶ 𝒢(G(Y')) ⟶ ℱ(Y')` can be glued into a morphism `X ⟶ ℱ(Y)`. This is done in `get_sections`. In `glued_limit_cone`, we verify these obtained sections are indeed compatible, and thus we obtain A `X ⟶ 𝒢(U)`. The remaining work is to verify that this is indeed the amalgamation and is unique. -/ variable {C D : Type} [Category C] [Category D] (G : C ⥤ D) variable {A : Type w} [Category.{w'} A] [∀ X, HasLimitsOfShape (StructuredArrow X G.op) A] variable {J : GrothendieckTopology C} {K : GrothendieckTopology D} [G.IsCocontinuous J K] namespace RanIsSheafOfIsCocontinuous variable {G} variable (ℱ : Sheaf J A) variable {X : A} {U : D} (S : Sieve U) (hS : S ∈ K U) variable (x : S.arrows.FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X))) variable (hx : x.Compatible) /-- The family of morphisms `X ⟶ 𝒢(G(Y')) ⟶ ℱ(Y')` defined on `{ Y' ⊆ Y : G(Y') ⊆ U ∈ S}`. -/ def pulledbackFamily (Y : StructuredArrow (op U) G.op) := ((x.pullback Y.hom.unop).functorPullback G).compPresheafMap (show _ ⟶ _ from whiskerRight ((Ran.adjunction A G.op).counit.app ℱ.val) (coyoneda.obj (op X))) set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.pulledback_family CategoryTheory.RanIsSheafOfIsCocontinuous.pulledbackFamily @[simp] theorem pulledbackFamily_apply (Y : StructuredArrow (op U) G.op) {W} {f : W ⟶ _} (Hf) : pulledbackFamily ℱ S x Y f Hf = x (G.map f ≫ Y.hom.unop) Hf ≫ ((Ran.adjunction A G.op).counit.app ℱ.val).app (op W) := rfl set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.pulledback_family_apply CategoryTheory.RanIsSheafOfIsCocontinuous.pulledbackFamily_apply variable {x} {S} /-- Given a `G(Y) ⊆ U`, we can find a unique section `X ⟶ ℱ(Y)` that agrees with `x`. -/ def getSection (Y : StructuredArrow (op U) G.op) : X ⟶ ℱ.val.obj Y.right := by letI hom_sh := whiskerRight ((Ran.adjunction A G.op).counit.app ℱ.val) (coyoneda.obj (op X)) haveI S' := K.pullback_stable Y.hom.unop hS haveI hs' := ((hx.pullback Y.3.unop).functorPullback G).compPresheafMap hom_sh exact (ℱ.2 X _ (G.cover_lift _ _ S')).amalgamate _ hs' set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.get_section CategoryTheory.RanIsSheafOfIsCocontinuous.getSection theorem getSection_isAmalgamation (Y : StructuredArrow (op U) G.op) : (pulledbackFamily ℱ S x Y).IsAmalgamation (getSection ℱ hS hx Y) := IsSheafFor.isAmalgamation _ _ set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.get_section_is_amalgamation CategoryTheory.RanIsSheafOfIsCocontinuous.getSection_isAmalgamation theorem getSection_is_unique (Y : StructuredArrow (op U) G.op) {y} (H : (pulledbackFamily ℱ S x Y).IsAmalgamation y) : y = getSection ℱ hS hx Y := by apply IsSheafFor.isSeparatedFor _ (pulledbackFamily ℱ S x Y) · exact H · apply getSection_isAmalgamation · exact ℱ.2 X _ (G.cover_lift _ _ (K.pullback_stable Y.hom.unop hS)) set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.get_section_is_unique CategoryTheory.RanIsSheafOfIsCocontinuous.getSection_is_unique @[simp] theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) : getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by apply getSection_is_unique intro V' fV' hV' have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by convert f.w using 1 erw [Category.id_comp] rw [eq] at hV' convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1 · aesop_cat -- Porting note: the below proof was mildly rewritten because `simp` changed behaviour -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of -- motive is not type correct issues) · rw [pulledbackFamily_apply, pulledbackFamily_apply] · congr 2 simp [eq] · change S (G.map _ ≫ Y.hom.unop) simpa only [Functor.map_comp, Category.assoc] using hV' set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.get_section_commute CategoryTheory.RanIsSheafOfIsCocontinuous.getSection_commute /-- The limit cone in order to glue the sections obtained via `get_section`. -/ def gluedLimitCone : Limits.Cone (Ran.diagram G.op ℱ.val (op U)) := { pt := X -- Porting note: autoporter got this wrong π := { app := fun Y => getSection ℱ hS hx Y } } set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.glued_limit_cone CategoryTheory.RanIsSheafOfIsCocontinuous.gluedLimitCone @[simp] theorem gluedLimitCone_π_app (W) : (gluedLimitCone ℱ hS hx).π.app W = getSection ℱ hS hx W := rfl set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.glued_limit_cone_π_app CategoryTheory.RanIsSheafOfIsCocontinuous.gluedLimitCone_π_app /-- The section obtained by passing `glued_limit_cone` into `CategoryTheory.Limits.limit.lift`. -/ def gluedSection : X ⟶ ((ran G.op).obj ℱ.val).obj (op U) := limit.lift _ (gluedLimitCone ℱ hS hx) set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.glued_section CategoryTheory.RanIsSheafOfIsCocontinuous.gluedSection /-- A helper lemma for the following two lemmas. Basically stating that if the section `y : X ⟶ 𝒢(V)` coincides with `x` on `G(V')` for all `G(V') ⊆ V ∈ S`, then `X ⟶ 𝒢(V) ⟶ ℱ(W)` is indeed the section obtained in `get_sections`. That said, this is littered with some more categorical jargon in order to be applied in the following lemmas easier. -/ theorem helper {V} (f : V ⟶ U) (y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)) (W) (H : ∀ {V'} {fV : G.obj V' ⟶ V} (hV), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV) : y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W = (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W) := by dsimp only [gluedLimitCone_π_app] apply getSection_is_unique ℱ hS hx ((StructuredArrow.map f.op).obj W) intro V' fV' hV' dsimp only [Ran.adjunction, Ran.equiv, pulledbackFamily_apply] erw [Adjunction.adjunctionOfEquivRight_counit_app] have : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) (by simpa only using hV') := by convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV') using 2 simp only [Category.assoc] simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp, Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp, ran_obj_map, NatTrans.id_app] erw [Category.id_comp, limit.pre_π] congr convert limit.w (Ran.diagram G.op ℱ.val (op V)) (StructuredArrow.homMk' W fV'.op) rw [StructuredArrow.map_mk] erw [Category.comp_id] simp only [Quiver.Hom.unop_op, Functor.op_map, Quiver.Hom.op_unop] set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.helper CategoryTheory.RanIsSheafOfIsCocontinuous.helper /-- Verify that the `glued_section` is an amalgamation of `x`. -/ theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by intro V fV hV -- Porting note: next line was `ext W` -- Now `ext` can't see that `ran` is defined as a limit. -- See https://github.com/leanprover-community/mathlib4/issues/5229 refine limit.hom_ext (fun (W : StructuredArrow (op V) G.op) ↦ ?_) simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection] erw [limit.lift_π] symm convert helper ℱ hS hx _ (x fV hV) _ _ using 1 intro V' fV' hV' convert hx fV' (𝟙 _) hV hV' (by rw [Category.id_comp]) simp only [op_id, FunctorToTypes.map_id_apply] set_option linter.uppercaseLean3 false in #align category_theory.Ran_is_sheaf_of_cover_lifting.glued_section_is_amalgamation CategoryTheory.RanIsSheafOfIsCocontinuous.gluedSection_isAmalgamation /-- Verify that the amalgamation is indeed unique. -/	Mathlib/CategoryTheory/Sites/CoverLifting.lean	263	274	theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by	unfold gluedSection limit.lift -- Porting note: next line was `ext W` -- Now `ext` can't see that `ran` is defined as a limit. -- See https://github.com/leanprover-community/mathlib4/issues/5229 refine limit.hom_ext (fun (W : StructuredArrow (op U) G.op) ↦ ?_) erw [limit.lift_π] convert helper ℱ hS hx (𝟙 _) y W _ · simp only [op_id, StructuredArrow.map_id] · intro V' fV' hV' convert hy fV' (by simpa only [Category.comp_id] using hV') erw [Category.comp_id]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot -/ import Mathlib.GroupTheory.GroupAction.ConjAct import Mathlib.GroupTheory.GroupAction.Quotient import Mathlib.GroupTheory.QuotientGroup import Mathlib.Topology.Algebra.Monoid import Mathlib.Topology.Algebra.Constructions #align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3b2086ab512c3259a7e9a3ef" /-! # Topological groups This file defines the following typeclasses: * `TopologicalGroup`, `TopologicalAddGroup`: multiplicative and additive topological groups, i.e., groups with continuous `()` and `(⁻¹)` / `(+)` and `(-)`; `ContinuousSub G` means that `G` has a continuous subtraction operation. There is an instance deducing `ContinuousSub` from `TopologicalGroup` but we use a separate typeclass because, e.g., `ℕ` and `ℝ≥0` have continuous subtraction but are not additive groups. We also define `Homeomorph` versions of several `Equiv`s: `Homeomorph.mulLeft`, `Homeomorph.mulRight`, `Homeomorph.inv`, and prove a few facts about neighbourhood filters in groups. ## Tags topological space, group, topological group -/ open scoped Classical open Set Filter TopologicalSpace Function Topology Pointwise MulOpposite universe u v w x variable {G : Type w} {H : Type x} {α : Type u} {β : Type v} section ContinuousMulGroup /-! ### Groups with continuous multiplication In this section we prove a few statements about groups with continuous `()`. -/ variable [TopologicalSpace G] [Group G] [ContinuousMul G] /-- Multiplication from the left in a topological group as a homeomorphism. -/ @[to_additive "Addition from the left in a topological additive group as a homeomorphism."] protected def Homeomorph.mulLeft (a : G) : G ≃ₜ G := { Equiv.mulLeft a with continuous_toFun := continuous_const.mul continuous_id continuous_invFun := continuous_const.mul continuous_id } #align homeomorph.mul_left Homeomorph.mulLeft #align homeomorph.add_left Homeomorph.addLeft @[to_additive (attr := simp)] theorem Homeomorph.coe_mulLeft (a : G) : ⇑(Homeomorph.mulLeft a) = (a ·) := rfl #align homeomorph.coe_mul_left Homeomorph.coe_mulLeft #align homeomorph.coe_add_left Homeomorph.coe_addLeft @[to_additive] theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ := by ext rfl #align homeomorph.mul_left_symm Homeomorph.mulLeft_symm #align homeomorph.add_left_symm Homeomorph.addLeft_symm @[to_additive] lemma isOpenMap_mul_left (a : G) : IsOpenMap (a * ·) := (Homeomorph.mulLeft a).isOpenMap #align is_open_map_mul_left isOpenMap_mul_left #align is_open_map_add_left isOpenMap_add_left @[to_additive IsOpen.left_addCoset] theorem IsOpen.leftCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (x • U) := isOpenMap_mul_left x _ h #align is_open.left_coset IsOpen.leftCoset #align is_open.left_add_coset IsOpen.left_addCoset @[to_additive] lemma isClosedMap_mul_left (a : G) : IsClosedMap (a * ·) := (Homeomorph.mulLeft a).isClosedMap #align is_closed_map_mul_left isClosedMap_mul_left #align is_closed_map_add_left isClosedMap_add_left @[to_additive IsClosed.left_addCoset] theorem IsClosed.leftCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (x • U) := isClosedMap_mul_left x _ h #align is_closed.left_coset IsClosed.leftCoset #align is_closed.left_add_coset IsClosed.left_addCoset /-- Multiplication from the right in a topological group as a homeomorphism. -/ @[to_additive "Addition from the right in a topological additive group as a homeomorphism."] protected def Homeomorph.mulRight (a : G) : G ≃ₜ G := { Equiv.mulRight a with continuous_toFun := continuous_id.mul continuous_const continuous_invFun := continuous_id.mul continuous_const } #align homeomorph.mul_right Homeomorph.mulRight #align homeomorph.add_right Homeomorph.addRight @[to_additive (attr := simp)] lemma Homeomorph.coe_mulRight (a : G) : ⇑(Homeomorph.mulRight a) = (· * a) := rfl #align homeomorph.coe_mul_right Homeomorph.coe_mulRight #align homeomorph.coe_add_right Homeomorph.coe_addRight @[to_additive] theorem Homeomorph.mulRight_symm (a : G) : (Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹ := by ext rfl #align homeomorph.mul_right_symm Homeomorph.mulRight_symm #align homeomorph.add_right_symm Homeomorph.addRight_symm @[to_additive] theorem isOpenMap_mul_right (a : G) : IsOpenMap (· * a) := (Homeomorph.mulRight a).isOpenMap #align is_open_map_mul_right isOpenMap_mul_right #align is_open_map_add_right isOpenMap_add_right @[to_additive IsOpen.right_addCoset] theorem IsOpen.rightCoset {U : Set G} (h : IsOpen U) (x : G) : IsOpen (op x • U) := isOpenMap_mul_right x _ h #align is_open.right_coset IsOpen.rightCoset #align is_open.right_add_coset IsOpen.right_addCoset @[to_additive] theorem isClosedMap_mul_right (a : G) : IsClosedMap (· * a) := (Homeomorph.mulRight a).isClosedMap #align is_closed_map_mul_right isClosedMap_mul_right #align is_closed_map_add_right isClosedMap_add_right @[to_additive IsClosed.right_addCoset] theorem IsClosed.rightCoset {U : Set G} (h : IsClosed U) (x : G) : IsClosed (op x • U) := isClosedMap_mul_right x _ h #align is_closed.right_coset IsClosed.rightCoset #align is_closed.right_add_coset IsClosed.right_addCoset @[to_additive] theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) : DiscreteTopology G := by rw [← singletons_open_iff_discrete] intro g suffices {g} = (g⁻¹ * ·) ⁻¹' {1} by rw [this] exact (continuous_mul_left g⁻¹).isOpen_preimage _ h simp only [mul_one, Set.preimage_mul_left_singleton, eq_self_iff_true, inv_inv, Set.singleton_eq_singleton_iff] #align discrete_topology_of_open_singleton_one discreteTopology_of_isOpen_singleton_one #align discrete_topology_of_open_singleton_zero discreteTopology_of_isOpen_singleton_zero @[to_additive] theorem discreteTopology_iff_isOpen_singleton_one : DiscreteTopology G ↔ IsOpen ({1} : Set G) := ⟨fun h => forall_open_iff_discrete.mpr h {1}, discreteTopology_of_isOpen_singleton_one⟩ #align discrete_topology_iff_open_singleton_one discreteTopology_iff_isOpen_singleton_one #align discrete_topology_iff_open_singleton_zero discreteTopology_iff_isOpen_singleton_zero end ContinuousMulGroup /-! ### `ContinuousInv` and `ContinuousNeg` -/ /-- Basic hypothesis to talk about a topological additive group. A topological additive group over `M`, for example, is obtained by requiring the instances `AddGroup M` and `ContinuousAdd M` and `ContinuousNeg M`. -/ class ContinuousNeg (G : Type u) [TopologicalSpace G] [Neg G] : Prop where continuous_neg : Continuous fun a : G => -a #align has_continuous_neg ContinuousNeg -- Porting note: added attribute [continuity] ContinuousNeg.continuous_neg /-- Basic hypothesis to talk about a topological group. A topological group over `M`, for example, is obtained by requiring the instances `Group M` and `ContinuousMul M` and `ContinuousInv M`. -/ @[to_additive (attr := continuity)] class ContinuousInv (G : Type u) [TopologicalSpace G] [Inv G] : Prop where continuous_inv : Continuous fun a : G => a⁻¹ #align has_continuous_inv ContinuousInv --#align has_continuous_neg ContinuousNeg -- Porting note: added attribute [continuity] ContinuousInv.continuous_inv export ContinuousInv (continuous_inv) export ContinuousNeg (continuous_neg) section ContinuousInv variable [TopologicalSpace G] [Inv G] [ContinuousInv G] @[to_additive] protected theorem Specializes.inv {x y : G} (h : x ⤳ y) : (x⁻¹) ⤳ (y⁻¹) := h.map continuous_inv @[to_additive] protected theorem Inseparable.inv {x y : G} (h : Inseparable x y) : Inseparable (x⁻¹) (y⁻¹) := h.map continuous_inv @[to_additive] protected theorem Specializes.zpow {G : Type} [DivInvMonoid G] [TopologicalSpace G] [ContinuousMul G] [ContinuousInv G] {x y : G} (h : x ⤳ y) : ∀ m : ℤ, (x ^ m) ⤳ (y ^ m) \| .ofNat n => by simpa using h.pow n \| .negSucc n => by simpa using (h.pow (n + 1)).inv @[to_additive] protected theorem Inseparable.zpow {G : Type} [DivInvMonoid G] [TopologicalSpace G] [ContinuousMul G] [ContinuousInv G] {x y : G} (h : Inseparable x y) (m : ℤ) : Inseparable (x ^ m) (y ^ m) := (h.specializes.zpow m).antisymm (h.specializes'.zpow m) @[to_additive] instance : ContinuousInv (ULift G) := ⟨continuous_uLift_up.comp (continuous_inv.comp continuous_uLift_down)⟩ @[to_additive] theorem continuousOn_inv {s : Set G} : ContinuousOn Inv.inv s := continuous_inv.continuousOn #align continuous_on_inv continuousOn_inv #align continuous_on_neg continuousOn_neg @[to_additive] theorem continuousWithinAt_inv {s : Set G} {x : G} : ContinuousWithinAt Inv.inv s x := continuous_inv.continuousWithinAt #align continuous_within_at_inv continuousWithinAt_inv #align continuous_within_at_neg continuousWithinAt_neg @[to_additive] theorem continuousAt_inv {x : G} : ContinuousAt Inv.inv x := continuous_inv.continuousAt #align continuous_at_inv continuousAt_inv #align continuous_at_neg continuousAt_neg @[to_additive] theorem tendsto_inv (a : G) : Tendsto Inv.inv (𝓝 a) (𝓝 a⁻¹) := continuousAt_inv #align tendsto_inv tendsto_inv #align tendsto_neg tendsto_neg /-- If a function converges to a value in a multiplicative topological group, then its inverse converges to the inverse of this value. For the version in normed fields assuming additionally that the limit is nonzero, use `Tendsto.inv'`. -/ @[to_additive "If a function converges to a value in an additive topological group, then its negation converges to the negation of this value."] theorem Filter.Tendsto.inv {f : α → G} {l : Filter α} {y : G} (h : Tendsto f l (𝓝 y)) : Tendsto (fun x => (f x)⁻¹) l (𝓝 y⁻¹) := (continuous_inv.tendsto y).comp h #align filter.tendsto.inv Filter.Tendsto.inv #align filter.tendsto.neg Filter.Tendsto.neg variable [TopologicalSpace α] {f : α → G} {s : Set α} {x : α} @[to_additive (attr := continuity, fun_prop)] theorem Continuous.inv (hf : Continuous f) : Continuous fun x => (f x)⁻¹ := continuous_inv.comp hf #align continuous.inv Continuous.inv #align continuous.neg Continuous.neg @[to_additive (attr := fun_prop)] theorem ContinuousAt.inv (hf : ContinuousAt f x) : ContinuousAt (fun x => (f x)⁻¹) x := continuousAt_inv.comp hf #align continuous_at.inv ContinuousAt.inv #align continuous_at.neg ContinuousAt.neg @[to_additive (attr := fun_prop)] theorem ContinuousOn.inv (hf : ContinuousOn f s) : ContinuousOn (fun x => (f x)⁻¹) s := continuous_inv.comp_continuousOn hf #align continuous_on.inv ContinuousOn.inv #align continuous_on.neg ContinuousOn.neg @[to_additive] theorem ContinuousWithinAt.inv (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (fun x => (f x)⁻¹) s x := Filter.Tendsto.inv hf #align continuous_within_at.inv ContinuousWithinAt.inv #align continuous_within_at.neg ContinuousWithinAt.neg @[to_additive] instance Prod.continuousInv [TopologicalSpace H] [Inv H] [ContinuousInv H] : ContinuousInv (G × H) := ⟨continuous_inv.fst'.prod_mk continuous_inv.snd'⟩ variable {ι : Type} @[to_additive] instance Pi.continuousInv {C : ι → Type} [∀ i, TopologicalSpace (C i)] [∀ i, Inv (C i)] [∀ i, ContinuousInv (C i)] : ContinuousInv (∀ i, C i) where continuous_inv := continuous_pi fun i => (continuous_apply i).inv #align pi.has_continuous_inv Pi.continuousInv #align pi.has_continuous_neg Pi.continuousNeg /-- A version of `Pi.continuousInv` for non-dependent functions. It is needed because sometimes Lean fails to use `Pi.continuousInv` for non-dependent functions. -/ @[to_additive "A version of `Pi.continuousNeg` for non-dependent functions. It is needed because sometimes Lean fails to use `Pi.continuousNeg` for non-dependent functions."] instance Pi.has_continuous_inv' : ContinuousInv (ι → G) := Pi.continuousInv #align pi.has_continuous_inv' Pi.has_continuous_inv' #align pi.has_continuous_neg' Pi.has_continuous_neg' @[to_additive] instance (priority := 100) continuousInv_of_discreteTopology [TopologicalSpace H] [Inv H] [DiscreteTopology H] : ContinuousInv H := ⟨continuous_of_discreteTopology⟩ #align has_continuous_inv_of_discrete_topology continuousInv_of_discreteTopology #align has_continuous_neg_of_discrete_topology continuousNeg_of_discreteTopology section PointwiseLimits variable (G₁ G₂ : Type) [TopologicalSpace G₂] [T2Space G₂] @[to_additive] theorem isClosed_setOf_map_inv [Inv G₁] [Inv G₂] [ContinuousInv G₂] : IsClosed { f : G₁ → G₂ \| ∀ x, f x⁻¹ = (f x)⁻¹ } := by simp only [setOf_forall] exact isClosed_iInter fun i => isClosed_eq (continuous_apply _) (continuous_apply _).inv #align is_closed_set_of_map_inv isClosed_setOf_map_inv #align is_closed_set_of_map_neg isClosed_setOf_map_neg end PointwiseLimits instance [TopologicalSpace H] [Inv H] [ContinuousInv H] : ContinuousNeg (Additive H) where continuous_neg := @continuous_inv H _ _ _ instance [TopologicalSpace H] [Neg H] [ContinuousNeg H] : ContinuousInv (Multiplicative H) where continuous_inv := @continuous_neg H _ _ _ end ContinuousInv section ContinuousInvolutiveInv variable [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] {s : Set G} @[to_additive] theorem IsCompact.inv (hs : IsCompact s) : IsCompact s⁻¹ := by rw [← image_inv] exact hs.image continuous_inv #align is_compact.inv IsCompact.inv #align is_compact.neg IsCompact.neg variable (G) /-- Inversion in a topological group as a homeomorphism. -/ @[to_additive "Negation in a topological group as a homeomorphism."] protected def Homeomorph.inv (G : Type) [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] : G ≃ₜ G := { Equiv.inv G with continuous_toFun := continuous_inv continuous_invFun := continuous_inv } #align homeomorph.inv Homeomorph.inv #align homeomorph.neg Homeomorph.neg @[to_additive (attr := simp)] lemma Homeomorph.coe_inv {G : Type} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] : ⇑(Homeomorph.inv G) = Inv.inv := rfl @[to_additive] theorem isOpenMap_inv : IsOpenMap (Inv.inv : G → G) := (Homeomorph.inv _).isOpenMap #align is_open_map_inv isOpenMap_inv #align is_open_map_neg isOpenMap_neg @[to_additive] theorem isClosedMap_inv : IsClosedMap (Inv.inv : G → G) := (Homeomorph.inv _).isClosedMap #align is_closed_map_inv isClosedMap_inv #align is_closed_map_neg isClosedMap_neg variable {G} @[to_additive] theorem IsOpen.inv (hs : IsOpen s) : IsOpen s⁻¹ := hs.preimage continuous_inv #align is_open.inv IsOpen.inv #align is_open.neg IsOpen.neg @[to_additive] theorem IsClosed.inv (hs : IsClosed s) : IsClosed s⁻¹ := hs.preimage continuous_inv #align is_closed.inv IsClosed.inv #align is_closed.neg IsClosed.neg @[to_additive] theorem inv_closure : ∀ s : Set G, (closure s)⁻¹ = closure s⁻¹ := (Homeomorph.inv G).preimage_closure #align inv_closure inv_closure #align neg_closure neg_closure end ContinuousInvolutiveInv section LatticeOps variable {ι' : Sort} [Inv G] @[to_additive] theorem continuousInv_sInf {ts : Set (TopologicalSpace G)} (h : ∀ t ∈ ts, @ContinuousInv G t _) : @ContinuousInv G (sInf ts) _ := letI := sInf ts { continuous_inv := continuous_sInf_rng.2 fun t ht => continuous_sInf_dom ht (@ContinuousInv.continuous_inv G t _ (h t ht)) } #align has_continuous_inv_Inf continuousInv_sInf #align has_continuous_neg_Inf continuousNeg_sInf @[to_additive] theorem continuousInv_iInf {ts' : ι' → TopologicalSpace G} (h' : ∀ i, @ContinuousInv G (ts' i) _) : @ContinuousInv G (⨅ i, ts' i) _ := by rw [← sInf_range] exact continuousInv_sInf (Set.forall_mem_range.mpr h') #align has_continuous_inv_infi continuousInv_iInf #align has_continuous_neg_infi continuousNeg_iInf @[to_additive] theorem continuousInv_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @ContinuousInv G t₁ _) (h₂ : @ContinuousInv G t₂ _) : @ContinuousInv G (t₁ ⊓ t₂) _ := by rw [inf_eq_iInf] refine continuousInv_iInf fun b => ?_ cases b <;> assumption #align has_continuous_inv_inf continuousInv_inf #align has_continuous_neg_inf continuousNeg_inf end LatticeOps @[to_additive] theorem Inducing.continuousInv {G H : Type} [Inv G] [Inv H] [TopologicalSpace G] [TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : Inducing f) (hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : ContinuousInv G := ⟨hf.continuous_iff.2 <\| by simpa only [(· ∘ ·), hf_inv] using hf.continuous.inv⟩ #align inducing.has_continuous_inv Inducing.continuousInv #align inducing.has_continuous_neg Inducing.continuousNeg section TopologicalGroup /-! ### Topological groups A topological group is a group in which the multiplication and inversion operations are continuous. Topological additive groups are defined in the same way. Equivalently, we can require that the division operation `x y ↦ x y⁻¹` (resp., subtraction) is continuous. -/ -- Porting note (#11215): TODO should this docstring be extended -- to match the multiplicative version? /-- A topological (additive) group is a group in which the addition and negation operations are continuous. -/ class TopologicalAddGroup (G : Type u) [TopologicalSpace G] [AddGroup G] extends ContinuousAdd G, ContinuousNeg G : Prop #align topological_add_group TopologicalAddGroup /-- A topological group is a group in which the multiplication and inversion operations are continuous. When you declare an instance that does not already have a `UniformSpace` instance, you should also provide an instance of `UniformSpace` and `UniformGroup` using `TopologicalGroup.toUniformSpace` and `topologicalCommGroup_isUniform`. -/ -- Porting note: check that these ↑ names exist once they've been ported in the future. @[to_additive] class TopologicalGroup (G : Type) [TopologicalSpace G] [Group G] extends ContinuousMul G, ContinuousInv G : Prop #align topological_group TopologicalGroup --#align topological_add_group TopologicalAddGroup section Conj instance ConjAct.units_continuousConstSMul {M} [Monoid M] [TopologicalSpace M] [ContinuousMul M] : ContinuousConstSMul (ConjAct Mˣ) M := ⟨fun _ => (continuous_const.mul continuous_id).mul continuous_const⟩ #align conj_act.units_has_continuous_const_smul ConjAct.units_continuousConstSMul variable [TopologicalSpace G] [Inv G] [Mul G] [ContinuousMul G] /-- Conjugation is jointly continuous on `G × G` when both `mul` and `inv` are continuous. -/ @[to_additive "Conjugation is jointly continuous on `G × G` when both `add` and `neg` are continuous."] theorem TopologicalGroup.continuous_conj_prod [ContinuousInv G] : Continuous fun g : G × G => g.fst g.snd * g.fst⁻¹ := continuous_mul.mul (continuous_inv.comp continuous_fst) #align topological_group.continuous_conj_prod TopologicalGroup.continuous_conj_prod #align topological_add_group.continuous_conj_sum TopologicalAddGroup.continuous_conj_sum /-- Conjugation by a fixed element is continuous when `mul` is continuous. -/ @[to_additive (attr := continuity) "Conjugation by a fixed element is continuous when `add` is continuous."] theorem TopologicalGroup.continuous_conj (g : G) : Continuous fun h : G => g * h * g⁻¹ := (continuous_mul_right g⁻¹).comp (continuous_mul_left g) #align topological_group.continuous_conj TopologicalGroup.continuous_conj #align topological_add_group.continuous_conj TopologicalAddGroup.continuous_conj /-- Conjugation acting on fixed element of the group is continuous when both `mul` and `inv` are continuous. -/ @[to_additive (attr := continuity) "Conjugation acting on fixed element of the additive group is continuous when both `add` and `neg` are continuous."] theorem TopologicalGroup.continuous_conj' [ContinuousInv G] (h : G) : Continuous fun g : G => g * h * g⁻¹ := (continuous_mul_right h).mul continuous_inv #align topological_group.continuous_conj' TopologicalGroup.continuous_conj' #align topological_add_group.continuous_conj' TopologicalAddGroup.continuous_conj' end Conj variable [TopologicalSpace G] [Group G] [TopologicalGroup G] [TopologicalSpace α] {f : α → G} {s : Set α} {x : α} instance : TopologicalGroup (ULift G) where section ZPow @[to_additive (attr := continuity)] theorem continuous_zpow : ∀ z : ℤ, Continuous fun a : G => a ^ z \| Int.ofNat n => by simpa using continuous_pow n \| Int.negSucc n => by simpa using (continuous_pow (n + 1)).inv #align continuous_zpow continuous_zpow #align continuous_zsmul continuous_zsmul instance AddGroup.continuousConstSMul_int {A} [AddGroup A] [TopologicalSpace A] [TopologicalAddGroup A] : ContinuousConstSMul ℤ A := ⟨continuous_zsmul⟩ #align add_group.has_continuous_const_smul_int AddGroup.continuousConstSMul_int instance AddGroup.continuousSMul_int {A} [AddGroup A] [TopologicalSpace A] [TopologicalAddGroup A] : ContinuousSMul ℤ A := ⟨continuous_prod_of_discrete_left.mpr continuous_zsmul⟩ #align add_group.has_continuous_smul_int AddGroup.continuousSMul_int @[to_additive (attr := continuity, fun_prop)] theorem Continuous.zpow {f : α → G} (h : Continuous f) (z : ℤ) : Continuous fun b => f b ^ z := (continuous_zpow z).comp h #align continuous.zpow Continuous.zpow #align continuous.zsmul Continuous.zsmul @[to_additive] theorem continuousOn_zpow {s : Set G} (z : ℤ) : ContinuousOn (fun x => x ^ z) s := (continuous_zpow z).continuousOn #align continuous_on_zpow continuousOn_zpow #align continuous_on_zsmul continuousOn_zsmul @[to_additive] theorem continuousAt_zpow (x : G) (z : ℤ) : ContinuousAt (fun x => x ^ z) x := (continuous_zpow z).continuousAt #align continuous_at_zpow continuousAt_zpow #align continuous_at_zsmul continuousAt_zsmul @[to_additive] theorem Filter.Tendsto.zpow {α} {l : Filter α} {f : α → G} {x : G} (hf : Tendsto f l (𝓝 x)) (z : ℤ) : Tendsto (fun x => f x ^ z) l (𝓝 (x ^ z)) := (continuousAt_zpow _ _).tendsto.comp hf #align filter.tendsto.zpow Filter.Tendsto.zpow #align filter.tendsto.zsmul Filter.Tendsto.zsmul @[to_additive] theorem ContinuousWithinAt.zpow {f : α → G} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (z : ℤ) : ContinuousWithinAt (fun x => f x ^ z) s x := Filter.Tendsto.zpow hf z #align continuous_within_at.zpow ContinuousWithinAt.zpow #align continuous_within_at.zsmul ContinuousWithinAt.zsmul @[to_additive (attr := fun_prop)] theorem ContinuousAt.zpow {f : α → G} {x : α} (hf : ContinuousAt f x) (z : ℤ) : ContinuousAt (fun x => f x ^ z) x := Filter.Tendsto.zpow hf z #align continuous_at.zpow ContinuousAt.zpow #align continuous_at.zsmul ContinuousAt.zsmul @[to_additive (attr := fun_prop)] theorem ContinuousOn.zpow {f : α → G} {s : Set α} (hf : ContinuousOn f s) (z : ℤ) : ContinuousOn (fun x => f x ^ z) s := fun x hx => (hf x hx).zpow z #align continuous_on.zpow ContinuousOn.zpow #align continuous_on.zsmul ContinuousOn.zsmul end ZPow section OrderedCommGroup variable [TopologicalSpace H] [OrderedCommGroup H] [ContinuousInv H] @[to_additive] theorem tendsto_inv_nhdsWithin_Ioi {a : H} : Tendsto Inv.inv (𝓝[>] a) (𝓝[<] a⁻¹) := (continuous_inv.tendsto a).inf <\| by simp [tendsto_principal_principal] #align tendsto_inv_nhds_within_Ioi tendsto_inv_nhdsWithin_Ioi #align tendsto_neg_nhds_within_Ioi tendsto_neg_nhdsWithin_Ioi @[to_additive] theorem tendsto_inv_nhdsWithin_Iio {a : H} : Tendsto Inv.inv (𝓝[<] a) (𝓝[>] a⁻¹) := (continuous_inv.tendsto a).inf <\| by simp [tendsto_principal_principal] #align tendsto_inv_nhds_within_Iio tendsto_inv_nhdsWithin_Iio #align tendsto_neg_nhds_within_Iio tendsto_neg_nhdsWithin_Iio @[to_additive] theorem tendsto_inv_nhdsWithin_Ioi_inv {a : H} : Tendsto Inv.inv (𝓝[>] a⁻¹) (𝓝[<] a) := by simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Ioi _ _ _ _ a⁻¹ #align tendsto_inv_nhds_within_Ioi_inv tendsto_inv_nhdsWithin_Ioi_inv #align tendsto_neg_nhds_within_Ioi_neg tendsto_neg_nhdsWithin_Ioi_neg @[to_additive] theorem tendsto_inv_nhdsWithin_Iio_inv {a : H} : Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a) := by simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iio _ _ _ _ a⁻¹ #align tendsto_inv_nhds_within_Iio_inv tendsto_inv_nhdsWithin_Iio_inv #align tendsto_neg_nhds_within_Iio_neg tendsto_neg_nhdsWithin_Iio_neg @[to_additive] theorem tendsto_inv_nhdsWithin_Ici {a : H} : Tendsto Inv.inv (𝓝[≥] a) (𝓝[≤] a⁻¹) := (continuous_inv.tendsto a).inf <\| by simp [tendsto_principal_principal] #align tendsto_inv_nhds_within_Ici tendsto_inv_nhdsWithin_Ici #align tendsto_neg_nhds_within_Ici tendsto_neg_nhdsWithin_Ici @[to_additive] theorem tendsto_inv_nhdsWithin_Iic {a : H} : Tendsto Inv.inv (𝓝[≤] a) (𝓝[≥] a⁻¹) := (continuous_inv.tendsto a).inf <\| by simp [tendsto_principal_principal] #align tendsto_inv_nhds_within_Iic tendsto_inv_nhdsWithin_Iic #align tendsto_neg_nhds_within_Iic tendsto_neg_nhdsWithin_Iic @[to_additive] theorem tendsto_inv_nhdsWithin_Ici_inv {a : H} : Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a) := by simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Ici _ _ _ _ a⁻¹ #align tendsto_inv_nhds_within_Ici_inv tendsto_inv_nhdsWithin_Ici_inv #align tendsto_neg_nhds_within_Ici_neg tendsto_neg_nhdsWithin_Ici_neg @[to_additive] theorem tendsto_inv_nhdsWithin_Iic_inv {a : H} : Tendsto Inv.inv (𝓝[≤] a⁻¹) (𝓝[≥] a) := by simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iic _ _ _ _ a⁻¹ #align tendsto_inv_nhds_within_Iic_inv tendsto_inv_nhdsWithin_Iic_inv #align tendsto_neg_nhds_within_Iic_neg tendsto_neg_nhdsWithin_Iic_neg end OrderedCommGroup @[to_additive] instance [TopologicalSpace H] [Group H] [TopologicalGroup H] : TopologicalGroup (G × H) where continuous_inv := continuous_inv.prod_map continuous_inv @[to_additive] instance Pi.topologicalGroup {C : β → Type} [∀ b, TopologicalSpace (C b)] [∀ b, Group (C b)] [∀ b, TopologicalGroup (C b)] : TopologicalGroup (∀ b, C b) where continuous_inv := continuous_pi fun i => (continuous_apply i).inv #align pi.topological_group Pi.topologicalGroup #align pi.topological_add_group Pi.topologicalAddGroup open MulOpposite @[to_additive] instance [Inv α] [ContinuousInv α] : ContinuousInv αᵐᵒᵖ := opHomeomorph.symm.inducing.continuousInv unop_inv /-- If multiplication is continuous in `α`, then it also is in `αᵐᵒᵖ`. -/ @[to_additive "If addition is continuous in `α`, then it also is in `αᵃᵒᵖ`."] instance [Group α] [TopologicalGroup α] : TopologicalGroup αᵐᵒᵖ where variable (G) @[to_additive] theorem nhds_one_symm : comap Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) := ((Homeomorph.inv G).comap_nhds_eq _).trans (congr_arg nhds inv_one) #align nhds_one_symm nhds_one_symm #align nhds_zero_symm nhds_zero_symm @[to_additive] theorem nhds_one_symm' : map Inv.inv (𝓝 (1 : G)) = 𝓝 (1 : G) := ((Homeomorph.inv G).map_nhds_eq _).trans (congr_arg nhds inv_one) #align nhds_one_symm' nhds_one_symm' #align nhds_zero_symm' nhds_zero_symm' @[to_additive] theorem inv_mem_nhds_one {S : Set G} (hS : S ∈ (𝓝 1 : Filter G)) : S⁻¹ ∈ 𝓝 (1 : G) := by rwa [← nhds_one_symm'] at hS #align inv_mem_nhds_one inv_mem_nhds_one #align neg_mem_nhds_zero neg_mem_nhds_zero /-- The map `(x, y) ↦ (x, x y)` as a homeomorphism. This is a shear mapping. -/ @[to_additive "The map `(x, y) ↦ (x, x + y)` as a homeomorphism. This is a shear mapping."] protected def Homeomorph.shearMulRight : G × G ≃ₜ G × G := { Equiv.prodShear (Equiv.refl _) Equiv.mulLeft with continuous_toFun := continuous_fst.prod_mk continuous_mul continuous_invFun := continuous_fst.prod_mk <\| continuous_fst.inv.mul continuous_snd } #align homeomorph.shear_mul_right Homeomorph.shearMulRight #align homeomorph.shear_add_right Homeomorph.shearAddRight @[to_additive (attr := simp)] theorem Homeomorph.shearMulRight_coe : ⇑(Homeomorph.shearMulRight G) = fun z : G × G => (z.1, z.1 * z.2) := rfl #align homeomorph.shear_mul_right_coe Homeomorph.shearMulRight_coe #align homeomorph.shear_add_right_coe Homeomorph.shearAddRight_coe @[to_additive (attr := simp)] theorem Homeomorph.shearMulRight_symm_coe : ⇑(Homeomorph.shearMulRight G).symm = fun z : G × G => (z.1, z.1⁻¹ * z.2) := rfl #align homeomorph.shear_mul_right_symm_coe Homeomorph.shearMulRight_symm_coe #align homeomorph.shear_add_right_symm_coe Homeomorph.shearAddRight_symm_coe variable {G} @[to_additive] protected theorem Inducing.topologicalGroup {F : Type} [Group H] [TopologicalSpace H] [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Inducing f) : TopologicalGroup H := { toContinuousMul := hf.continuousMul _ toContinuousInv := hf.continuousInv (map_inv f) } #align inducing.topological_group Inducing.topologicalGroup #align inducing.topological_add_group Inducing.topologicalAddGroup @[to_additive] -- Porting note: removed `protected` (needs to be in namespace) theorem topologicalGroup_induced {F : Type} [Group H] [FunLike F H G] [MonoidHomClass F H G] (f : F) : @TopologicalGroup H (induced f ‹_›) _ := letI := induced f ‹_› Inducing.topologicalGroup f ⟨rfl⟩ #align topological_group_induced topologicalGroup_induced #align topological_add_group_induced topologicalAddGroup_induced namespace Subgroup @[to_additive] instance (S : Subgroup G) : TopologicalGroup S := Inducing.topologicalGroup S.subtype inducing_subtype_val end Subgroup /-- The (topological-space) closure of a subgroup of a topological group is itself a subgroup. -/ @[to_additive "The (topological-space) closure of an additive subgroup of an additive topological group is itself an additive subgroup."] def Subgroup.topologicalClosure (s : Subgroup G) : Subgroup G := { s.toSubmonoid.topologicalClosure with carrier := _root_.closure (s : Set G) inv_mem' := fun {g} hg => by simpa only [← Set.mem_inv, inv_closure, inv_coe_set] using hg } #align subgroup.topological_closure Subgroup.topologicalClosure #align add_subgroup.topological_closure AddSubgroup.topologicalClosure @[to_additive (attr := simp)] theorem Subgroup.topologicalClosure_coe {s : Subgroup G} : (s.topologicalClosure : Set G) = _root_.closure s := rfl #align subgroup.topological_closure_coe Subgroup.topologicalClosure_coe #align add_subgroup.topological_closure_coe AddSubgroup.topologicalClosure_coe @[to_additive] theorem Subgroup.le_topologicalClosure (s : Subgroup G) : s ≤ s.topologicalClosure := _root_.subset_closure #align subgroup.le_topological_closure Subgroup.le_topologicalClosure #align add_subgroup.le_topological_closure AddSubgroup.le_topologicalClosure @[to_additive] theorem Subgroup.isClosed_topologicalClosure (s : Subgroup G) : IsClosed (s.topologicalClosure : Set G) := isClosed_closure #align subgroup.is_closed_topological_closure Subgroup.isClosed_topologicalClosure #align add_subgroup.is_closed_topological_closure AddSubgroup.isClosed_topologicalClosure @[to_additive] theorem Subgroup.topologicalClosure_minimal (s : Subgroup G) {t : Subgroup G} (h : s ≤ t) (ht : IsClosed (t : Set G)) : s.topologicalClosure ≤ t := closure_minimal h ht #align subgroup.topological_closure_minimal Subgroup.topologicalClosure_minimal #align add_subgroup.topological_closure_minimal AddSubgroup.topologicalClosure_minimal @[to_additive] theorem DenseRange.topologicalClosure_map_subgroup [Group H] [TopologicalSpace H] [TopologicalGroup H] {f : G →* H} (hf : Continuous f) (hf' : DenseRange f) {s : Subgroup G} (hs : s.topologicalClosure = ⊤) : (s.map f).topologicalClosure = ⊤ := by rw [SetLike.ext'_iff] at hs ⊢ simp only [Subgroup.topologicalClosure_coe, Subgroup.coe_top, ← dense_iff_closure_eq] at hs ⊢ exact hf'.dense_image hf hs #align dense_range.topological_closure_map_subgroup DenseRange.topologicalClosure_map_subgroup #align dense_range.topological_closure_map_add_subgroup DenseRange.topologicalClosure_map_addSubgroup /-- The topological closure of a normal subgroup is normal. -/ @[to_additive "The topological closure of a normal additive subgroup is normal."] theorem Subgroup.is_normal_topologicalClosure {G : Type} [TopologicalSpace G] [Group G] [TopologicalGroup G] (N : Subgroup G) [N.Normal] : (Subgroup.topologicalClosure N).Normal where conj_mem n hn g := by apply map_mem_closure (TopologicalGroup.continuous_conj g) hn exact fun m hm => Subgroup.Normal.conj_mem inferInstance m hm g #align subgroup.is_normal_topological_closure Subgroup.is_normal_topologicalClosure #align add_subgroup.is_normal_topological_closure AddSubgroup.is_normal_topologicalClosure @[to_additive] theorem mul_mem_connectedComponent_one {G : Type} [TopologicalSpace G] [MulOneClass G] [ContinuousMul G] {g h : G} (hg : g ∈ connectedComponent (1 : G)) (hh : h ∈ connectedComponent (1 : G)) : g * h ∈ connectedComponent (1 : G) := by rw [connectedComponent_eq hg] have hmul : g ∈ connectedComponent (g * h) := by apply Continuous.image_connectedComponent_subset (continuous_mul_left g) rw [← connectedComponent_eq hh] exact ⟨(1 : G), mem_connectedComponent, by simp only [mul_one]⟩ simpa [← connectedComponent_eq hmul] using mem_connectedComponent #align mul_mem_connected_component_one mul_mem_connectedComponent_one #align add_mem_connected_component_zero add_mem_connectedComponent_zero @[to_additive] theorem inv_mem_connectedComponent_one {G : Type} [TopologicalSpace G] [Group G] [TopologicalGroup G] {g : G} (hg : g ∈ connectedComponent (1 : G)) : g⁻¹ ∈ connectedComponent (1 : G) := by rw [← inv_one] exact Continuous.image_connectedComponent_subset continuous_inv _ ((Set.mem_image _ _ _).mp ⟨g, hg, rfl⟩) #align inv_mem_connected_component_one inv_mem_connectedComponent_one #align neg_mem_connected_component_zero neg_mem_connectedComponent_zero /-- The connected component of 1 is a subgroup of `G`. -/ @[to_additive "The connected component of 0 is a subgroup of `G`."] def Subgroup.connectedComponentOfOne (G : Type) [TopologicalSpace G] [Group G] [TopologicalGroup G] : Subgroup G where carrier := connectedComponent (1 : G) one_mem' := mem_connectedComponent mul_mem' hg hh := mul_mem_connectedComponent_one hg hh inv_mem' hg := inv_mem_connectedComponent_one hg #align subgroup.connected_component_of_one Subgroup.connectedComponentOfOne #align add_subgroup.connected_component_of_zero AddSubgroup.connectedComponentOfZero /-- If a subgroup of a topological group is commutative, then so is its topological closure. -/ @[to_additive "If a subgroup of an additive topological group is commutative, then so is its topological closure."] def Subgroup.commGroupTopologicalClosure [T2Space G] (s : Subgroup G) (hs : ∀ x y : s, x * y = y * x) : CommGroup s.topologicalClosure := { s.topologicalClosure.toGroup, s.toSubmonoid.commMonoidTopologicalClosure hs with } #align subgroup.comm_group_topological_closure Subgroup.commGroupTopologicalClosure #align add_subgroup.add_comm_group_topological_closure AddSubgroup.addCommGroupTopologicalClosure variable (G) in @[to_additive] lemma Subgroup.coe_topologicalClosure_bot : ((⊥ : Subgroup G).topologicalClosure : Set G) = _root_.closure ({1} : Set G) := by simp @[to_additive exists_nhds_half_neg] theorem exists_nhds_split_inv {s : Set G} (hs : s ∈ 𝓝 (1 : G)) : ∃ V ∈ 𝓝 (1 : G), ∀ v ∈ V, ∀ w ∈ V, v / w ∈ s := by have : (fun p : G × G => p.1 * p.2⁻¹) ⁻¹' s ∈ 𝓝 ((1, 1) : G × G) := continuousAt_fst.mul continuousAt_snd.inv (by simpa) simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage] using this #align exists_nhds_split_inv exists_nhds_split_inv #align exists_nhds_half_neg exists_nhds_half_neg @[to_additive] theorem nhds_translation_mul_inv (x : G) : comap (· * x⁻¹) (𝓝 1) = 𝓝 x := ((Homeomorph.mulRight x⁻¹).comap_nhds_eq 1).trans <\| show 𝓝 (1 * x⁻¹⁻¹) = 𝓝 x by simp #align nhds_translation_mul_inv nhds_translation_mul_inv #align nhds_translation_add_neg nhds_translation_add_neg @[to_additive (attr := simp)] theorem map_mul_left_nhds (x y : G) : map (x * ·) (𝓝 y) = 𝓝 (x * y) := (Homeomorph.mulLeft x).map_nhds_eq y #align map_mul_left_nhds map_mul_left_nhds #align map_add_left_nhds map_add_left_nhds @[to_additive] theorem map_mul_left_nhds_one (x : G) : map (x * ·) (𝓝 1) = 𝓝 x := by simp #align map_mul_left_nhds_one map_mul_left_nhds_one #align map_add_left_nhds_zero map_add_left_nhds_zero @[to_additive (attr := simp)] theorem map_mul_right_nhds (x y : G) : map (· * x) (𝓝 y) = 𝓝 (y * x) := (Homeomorph.mulRight x).map_nhds_eq y #align map_mul_right_nhds map_mul_right_nhds #align map_add_right_nhds map_add_right_nhds @[to_additive] theorem map_mul_right_nhds_one (x : G) : map (· * x) (𝓝 1) = 𝓝 x := by simp #align map_mul_right_nhds_one map_mul_right_nhds_one #align map_add_right_nhds_zero map_add_right_nhds_zero @[to_additive] theorem Filter.HasBasis.nhds_of_one {ι : Sort} {p : ι → Prop} {s : ι → Set G} (hb : HasBasis (𝓝 1 : Filter G) p s) (x : G) : HasBasis (𝓝 x) p fun i => { y \| y / x ∈ s i } := by rw [← nhds_translation_mul_inv] simp_rw [div_eq_mul_inv] exact hb.comap _ #align filter.has_basis.nhds_of_one Filter.HasBasis.nhds_of_one #align filter.has_basis.nhds_of_zero Filter.HasBasis.nhds_of_zero @[to_additive] theorem mem_closure_iff_nhds_one {x : G} {s : Set G} : x ∈ closure s ↔ ∀ U ∈ (𝓝 1 : Filter G), ∃ y ∈ s, y / x ∈ U := by rw [mem_closure_iff_nhds_basis ((𝓝 1 : Filter G).basis_sets.nhds_of_one x)] simp_rw [Set.mem_setOf, id] #align mem_closure_iff_nhds_one mem_closure_iff_nhds_one #align mem_closure_iff_nhds_zero mem_closure_iff_nhds_zero /-- A monoid homomorphism (a bundled morphism of a type that implements `MonoidHomClass`) from a topological group to a topological monoid is continuous provided that it is continuous at one. See also `uniformContinuous_of_continuousAt_one`. -/ @[to_additive "An additive monoid homomorphism (a bundled morphism of a type that implements `AddMonoidHomClass`) from an additive topological group to an additive topological monoid is continuous provided that it is continuous at zero. See also `uniformContinuous_of_continuousAt_zero`."] theorem continuous_of_continuousAt_one {M hom : Type} [MulOneClass M] [TopologicalSpace M] [ContinuousMul M] [FunLike hom G M] [MonoidHomClass hom G M] (f : hom) (hf : ContinuousAt f 1) : Continuous f := continuous_iff_continuousAt.2 fun x => by simpa only [ContinuousAt, ← map_mul_left_nhds_one x, tendsto_map'_iff, (· ∘ ·), map_mul, map_one, mul_one] using hf.tendsto.const_mul (f x) #align continuous_of_continuous_at_one continuous_of_continuousAt_one #align continuous_of_continuous_at_zero continuous_of_continuousAt_zero -- Porting note (#10756): new theorem @[to_additive continuous_of_continuousAt_zero₂] theorem continuous_of_continuousAt_one₂ {H M : Type} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] [Group H] [TopologicalSpace H] [TopologicalGroup H] (f : G → H →* M) (hf : ContinuousAt (fun x : G × H ↦ f x.1 x.2) (1, 1)) (hl : ∀ x, ContinuousAt (f x) 1) (hr : ∀ y, ContinuousAt (f · y) 1) : Continuous (fun x : G × H ↦ f x.1 x.2) := continuous_iff_continuousAt.2 fun (x, y) => by simp only [ContinuousAt, nhds_prod_eq, ← map_mul_left_nhds_one x, ← map_mul_left_nhds_one y, prod_map_map_eq, tendsto_map'_iff, (· ∘ ·), map_mul, MonoidHom.mul_apply] at * refine ((tendsto_const_nhds.mul ((hr y).comp tendsto_fst)).mul (((hl x).comp tendsto_snd).mul hf)).mono_right (le_of_eq ?_) simp only [map_one, mul_one, MonoidHom.one_apply] @[to_additive] theorem TopologicalGroup.ext {G : Type} [Group G] {t t' : TopologicalSpace G} (tg : @TopologicalGroup G t _) (tg' : @TopologicalGroup G t' _) (h : @nhds G t 1 = @nhds G t' 1) : t = t' := TopologicalSpace.ext_nhds fun x ↦ by rw [← @nhds_translation_mul_inv G t _ _ x, ← @nhds_translation_mul_inv G t' _ _ x, ← h] #align topological_group.ext TopologicalGroup.ext #align topological_add_group.ext TopologicalAddGroup.ext @[to_additive] theorem TopologicalGroup.ext_iff {G : Type} [Group G] {t t' : TopologicalSpace G} (tg : @TopologicalGroup G t _) (tg' : @TopologicalGroup G t' _) : t = t' ↔ @nhds G t 1 = @nhds G t' 1 := ⟨fun h => h ▸ rfl, tg.ext tg'⟩ #align topological_group.ext_iff TopologicalGroup.ext_iff #align topological_add_group.ext_iff TopologicalAddGroup.ext_iff @[to_additive] theorem ContinuousInv.of_nhds_one {G : Type} [Group G] [TopologicalSpace G] (hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ x₀ : G, 𝓝 x₀ = map (fun x : G => x₀ x) (𝓝 1)) (hconj : ∀ x₀ : G, Tendsto (fun x : G => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)) : ContinuousInv G := by refine ⟨continuous_iff_continuousAt.2 fun x₀ => ?_⟩ have : Tendsto (fun x => x₀⁻¹ * (x₀ * x⁻¹ * x₀⁻¹)) (𝓝 1) (map (x₀⁻¹ * ·) (𝓝 1)) := (tendsto_map.comp <\| hconj x₀).comp hinv simpa only [ContinuousAt, hleft x₀, hleft x₀⁻¹, tendsto_map'_iff, (· ∘ ·), mul_assoc, mul_inv_rev, inv_mul_cancel_left] using this #align has_continuous_inv.of_nhds_one ContinuousInv.of_nhds_one #align has_continuous_neg.of_nhds_zero ContinuousNeg.of_nhds_zero @[to_additive] theorem TopologicalGroup.of_nhds_one' {G : Type u} [Group G] [TopologicalSpace G] (hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)) (hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ x₀ : G, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) (hright : ∀ x₀ : G, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : TopologicalGroup G := { toContinuousMul := ContinuousMul.of_nhds_one hmul hleft hright toContinuousInv := ContinuousInv.of_nhds_one hinv hleft fun x₀ => le_of_eq (by rw [show (fun x => x₀ * x * x₀⁻¹) = (fun x => x * x₀⁻¹) ∘ fun x => x₀ * x from rfl, ← map_map, ← hleft, hright, map_map] simp [(· ∘ ·)]) } #align topological_group.of_nhds_one' TopologicalGroup.of_nhds_one' #align topological_add_group.of_nhds_zero' TopologicalAddGroup.of_nhds_zero' @[to_additive] theorem TopologicalGroup.of_nhds_one {G : Type u} [Group G] [TopologicalSpace G] (hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)) (hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1)) (hconj : ∀ x₀ : G, Tendsto (x₀ * · * x₀⁻¹) (𝓝 1) (𝓝 1)) : TopologicalGroup G := by refine TopologicalGroup.of_nhds_one' hmul hinv hleft fun x₀ => ?_ replace hconj : ∀ x₀ : G, map (x₀ * · * x₀⁻¹) (𝓝 1) = 𝓝 1 := fun x₀ => map_eq_of_inverse (x₀⁻¹ * · * x₀⁻¹⁻¹) (by ext; simp [mul_assoc]) (hconj _) (hconj _) rw [← hconj x₀] simpa [(· ∘ ·)] using hleft _ #align topological_group.of_nhds_one TopologicalGroup.of_nhds_one #align topological_add_group.of_nhds_zero TopologicalAddGroup.of_nhds_zero @[to_additive] theorem TopologicalGroup.of_comm_of_nhds_one {G : Type u} [CommGroup G] [TopologicalSpace G] (hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)) (hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1)) (hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1)) : TopologicalGroup G := TopologicalGroup.of_nhds_one hmul hinv hleft (by simpa using tendsto_id) #align topological_group.of_comm_of_nhds_one TopologicalGroup.of_comm_of_nhds_one #align topological_add_group.of_comm_of_nhds_zero TopologicalAddGroup.of_comm_of_nhds_zero end TopologicalGroup section QuotientTopologicalGroup variable [TopologicalSpace G] [Group G] [TopologicalGroup G] (N : Subgroup G) (n : N.Normal) @[to_additive] instance QuotientGroup.Quotient.topologicalSpace {G : Type} [Group G] [TopologicalSpace G] (N : Subgroup G) : TopologicalSpace (G ⧸ N) := instTopologicalSpaceQuotient #align quotient_group.quotient.topological_space QuotientGroup.Quotient.topologicalSpace #align quotient_add_group.quotient.topological_space QuotientAddGroup.Quotient.topologicalSpace open QuotientGroup @[to_additive] theorem QuotientGroup.isOpenMap_coe : IsOpenMap ((↑) : G → G ⧸ N) := by intro s s_op change IsOpen (((↑) : G → G ⧸ N) ⁻¹' ((↑) '' s)) rw [QuotientGroup.preimage_image_mk N s] exact isOpen_iUnion fun n => (continuous_mul_right _).isOpen_preimage s s_op #align quotient_group.is_open_map_coe QuotientGroup.isOpenMap_coe #align quotient_add_group.is_open_map_coe QuotientAddGroup.isOpenMap_coe @[to_additive] instance topologicalGroup_quotient [N.Normal] : TopologicalGroup (G ⧸ N) where continuous_mul := by have cont : Continuous (((↑) : G → G ⧸ N) ∘ fun p : G × G ↦ p.fst p.snd) := continuous_quot_mk.comp continuous_mul have quot : QuotientMap fun p : G × G ↦ ((p.1 : G ⧸ N), (p.2 : G ⧸ N)) := by apply IsOpenMap.to_quotientMap · exact (QuotientGroup.isOpenMap_coe N).prod (QuotientGroup.isOpenMap_coe N) · exact continuous_quot_mk.prod_map continuous_quot_mk · exact (surjective_quot_mk _).prodMap (surjective_quot_mk _) exact quot.continuous_iff.2 cont continuous_inv := by have quot := IsOpenMap.to_quotientMap (QuotientGroup.isOpenMap_coe N) continuous_quot_mk (surjective_quot_mk _) rw [quot.continuous_iff] exact continuous_quot_mk.comp continuous_inv #align topological_group_quotient topologicalGroup_quotient #align topological_add_group_quotient topologicalAddGroup_quotient /-- Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. -/ @[to_additive "Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient."] theorem QuotientGroup.nhds_eq (x : G) : 𝓝 (x : G ⧸ N) = Filter.map (↑) (𝓝 x) := le_antisymm ((QuotientGroup.isOpenMap_coe N).nhds_le x) continuous_quot_mk.continuousAt #align quotient_group.nhds_eq QuotientGroup.nhds_eq #align quotient_add_group.nhds_eq QuotientAddGroup.nhds_eq variable (G) variable [FirstCountableTopology G] /-- Any first countable topological group has an antitone neighborhood basis `u : ℕ → Set G` for which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a key tool for `QuotientGroup.completeSpace` -/ @[to_additive "Any first countable topological additive group has an antitone neighborhood basis `u : ℕ → set G` for which `u (n + 1) + u (n + 1) ⊆ u n`. The existence of such a neighborhood basis is a key tool for `QuotientAddGroup.completeSpace`"] theorem TopologicalGroup.exists_antitone_basis_nhds_one : ∃ u : ℕ → Set G, (𝓝 1).HasAntitoneBasis u ∧ ∀ n, u (n + 1) * u (n + 1) ⊆ u n := by rcases (𝓝 (1 : G)).exists_antitone_basis with ⟨u, hu, u_anti⟩ have := ((hu.prod_nhds hu).tendsto_iff hu).mp (by simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G)) simp only [and_self_iff, mem_prod, and_imp, Prod.forall, exists_true_left, Prod.exists, forall_true_left] at this have event_mul : ∀ n : ℕ, ∀ᶠ m in atTop, u m * u m ⊆ u n := by intro n rcases this n with ⟨j, k, -, h⟩ refine atTop_basis.eventually_iff.mpr ⟨max j k, True.intro, fun m hm => ?_⟩ rintro - ⟨a, ha, b, hb, rfl⟩ exact h a b (u_anti ((le_max_left _ _).trans hm) ha) (u_anti ((le_max_right _ _).trans hm) hb) obtain ⟨φ, -, hφ, φ_anti_basis⟩ := HasAntitoneBasis.subbasis_with_rel ⟨hu, u_anti⟩ event_mul exact ⟨u ∘ φ, φ_anti_basis, fun n => hφ n.lt_succ_self⟩ #align topological_group.exists_antitone_basis_nhds_one TopologicalGroup.exists_antitone_basis_nhds_one #align topological_add_group.exists_antitone_basis_nhds_zero TopologicalAddGroup.exists_antitone_basis_nhds_zero /-- In a first countable topological group `G` with normal subgroup `N`, `1 : G ⧸ N` has a countable neighborhood basis. -/ @[to_additive "In a first countable topological additive group `G` with normal additive subgroup `N`, `0 : G ⧸ N` has a countable neighborhood basis."] instance QuotientGroup.nhds_one_isCountablyGenerated : (𝓝 (1 : G ⧸ N)).IsCountablyGenerated := (QuotientGroup.nhds_eq N 1).symm ▸ map.isCountablyGenerated _ _ #align quotient_group.nhds_one_is_countably_generated QuotientGroup.nhds_one_isCountablyGenerated #align quotient_add_group.nhds_zero_is_countably_generated QuotientAddGroup.nhds_zero_isCountablyGenerated end QuotientTopologicalGroup /-- A typeclass saying that `p : G × G ↦ p.1 - p.2` is a continuous function. This property automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. -/ class ContinuousSub (G : Type) [TopologicalSpace G] [Sub G] : Prop where continuous_sub : Continuous fun p : G × G => p.1 - p.2 #align has_continuous_sub ContinuousSub /-- A typeclass saying that `p : G × G ↦ p.1 / p.2` is a continuous function. This property automatically holds for topological groups. Lemmas using this class have primes. The unprimed version is for `GroupWithZero`. -/ @[to_additive existing] class ContinuousDiv (G : Type) [TopologicalSpace G] [Div G] : Prop where continuous_div' : Continuous fun p : G × G => p.1 / p.2 #align has_continuous_div ContinuousDiv -- see Note [lower instance priority] @[to_additive] instance (priority := 100) TopologicalGroup.to_continuousDiv [TopologicalSpace G] [Group G] [TopologicalGroup G] : ContinuousDiv G := ⟨by simp only [div_eq_mul_inv] exact continuous_fst.mul continuous_snd.inv⟩ #align topological_group.to_has_continuous_div TopologicalGroup.to_continuousDiv #align topological_add_group.to_has_continuous_sub TopologicalAddGroup.to_continuousSub export ContinuousSub (continuous_sub) export ContinuousDiv (continuous_div') section ContinuousDiv variable [TopologicalSpace G] [Div G] [ContinuousDiv G] @[to_additive sub] theorem Filter.Tendsto.div' {f g : α → G} {l : Filter α} {a b : G} (hf : Tendsto f l (𝓝 a)) (hg : Tendsto g l (𝓝 b)) : Tendsto (fun x => f x / g x) l (𝓝 (a / b)) := (continuous_div'.tendsto (a, b)).comp (hf.prod_mk_nhds hg) #align filter.tendsto.div' Filter.Tendsto.div' #align filter.tendsto.sub Filter.Tendsto.sub @[to_additive const_sub] theorem Filter.Tendsto.const_div' (b : G) {c : G} {f : α → G} {l : Filter α} (h : Tendsto f l (𝓝 c)) : Tendsto (fun k : α => b / f k) l (𝓝 (b / c)) := tendsto_const_nhds.div' h #align filter.tendsto.const_div' Filter.Tendsto.const_div' #align filter.tendsto.const_sub Filter.Tendsto.const_sub @[to_additive] lemma Filter.tendsto_const_div_iff {G : Type} [CommGroup G] [TopologicalSpace G] [ContinuousDiv G] (b : G) {c : G} {f : α → G} {l : Filter α} : Tendsto (fun k : α ↦ b / f k) l (𝓝 (b / c)) ↔ Tendsto f l (𝓝 c) := by refine ⟨fun h ↦ ?_, Filter.Tendsto.const_div' b⟩ convert h.const_div' b with k <;> rw [div_div_cancel] @[to_additive sub_const] theorem Filter.Tendsto.div_const' {c : G} {f : α → G} {l : Filter α} (h : Tendsto f l (𝓝 c)) (b : G) : Tendsto (f · / b) l (𝓝 (c / b)) := h.div' tendsto_const_nhds #align filter.tendsto.div_const' Filter.Tendsto.div_const' #align filter.tendsto.sub_const Filter.Tendsto.sub_const lemma Filter.tendsto_div_const_iff {G : Type} [CommGroupWithZero G] [TopologicalSpace G] [ContinuousDiv G] {b : G} (hb : b ≠ 0) {c : G} {f : α → G} {l : Filter α} : Tendsto (f · / b) l (𝓝 (c / b)) ↔ Tendsto f l (𝓝 c) := by refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.div_const' h b⟩ convert h.div_const' b⁻¹ with k <;> rw [div_div, mul_inv_cancel hb, div_one] lemma Filter.tendsto_sub_const_iff {G : Type} [AddCommGroup G] [TopologicalSpace G] [ContinuousSub G] (b : G) {c : G} {f : α → G} {l : Filter α} : Tendsto (f · - b) l (𝓝 (c - b)) ↔ Tendsto f l (𝓝 c) := by refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.sub_const h b⟩ convert h.sub_const (-b) with k <;> rw [sub_sub, ← sub_eq_add_neg, sub_self, sub_zero] variable [TopologicalSpace α] {f g : α → G} {s : Set α} {x : α} @[to_additive (attr := continuity, fun_prop) sub] theorem Continuous.div' (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x / g x := continuous_div'.comp (hf.prod_mk hg : _) #align continuous.div' Continuous.div' #align continuous.sub Continuous.sub @[to_additive (attr := continuity) continuous_sub_left] lemma continuous_div_left' (a : G) : Continuous (a / ·) := continuous_const.div' continuous_id #align continuous_div_left' continuous_div_left' #align continuous_sub_left continuous_sub_left @[to_additive (attr := continuity) continuous_sub_right] lemma continuous_div_right' (a : G) : Continuous (· / a) := continuous_id.div' continuous_const #align continuous_div_right' continuous_div_right' #align continuous_sub_right continuous_sub_right @[to_additive (attr := fun_prop) sub] theorem ContinuousAt.div' {f g : α → G} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun x => f x / g x) x := Filter.Tendsto.div' hf hg #align continuous_at.div' ContinuousAt.div' #align continuous_at.sub ContinuousAt.sub @[to_additive sub] theorem ContinuousWithinAt.div' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun x => f x / g x) s x := Filter.Tendsto.div' hf hg #align continuous_within_at.div' ContinuousWithinAt.div' #align continuous_within_at.sub ContinuousWithinAt.sub @[to_additive (attr := fun_prop) sub] theorem ContinuousOn.div' (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun x => f x / g x) s := fun x hx => (hf x hx).div' (hg x hx) #align continuous_on.div' ContinuousOn.div' #align continuous_on.sub ContinuousOn.sub end ContinuousDiv section DivInvTopologicalGroup variable [Group G] [TopologicalSpace G] [TopologicalGroup G] /-- A version of `Homeomorph.mulLeft a b⁻¹` that is defeq to `a / b`. -/ @[to_additive (attr := simps! (config := { simpRhs := true })) " A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. "] def Homeomorph.divLeft (x : G) : G ≃ₜ G := { Equiv.divLeft x with continuous_toFun := continuous_const.div' continuous_id continuous_invFun := continuous_inv.mul continuous_const } #align homeomorph.div_left Homeomorph.divLeft #align homeomorph.sub_left Homeomorph.subLeft @[to_additive] theorem isOpenMap_div_left (a : G) : IsOpenMap (a / ·) := (Homeomorph.divLeft _).isOpenMap #align is_open_map_div_left isOpenMap_div_left #align is_open_map_sub_left isOpenMap_sub_left @[to_additive] theorem isClosedMap_div_left (a : G) : IsClosedMap (a / ·) := (Homeomorph.divLeft _).isClosedMap #align is_closed_map_div_left isClosedMap_div_left #align is_closed_map_sub_left isClosedMap_sub_left /-- A version of `Homeomorph.mulRight a⁻¹ b` that is defeq to `b / a`. -/ @[to_additive (attr := simps! (config := { simpRhs := true })) "A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. "] def Homeomorph.divRight (x : G) : G ≃ₜ G := { Equiv.divRight x with continuous_toFun := continuous_id.div' continuous_const continuous_invFun := continuous_id.mul continuous_const } #align homeomorph.div_right Homeomorph.divRight #align homeomorph.sub_right Homeomorph.subRight @[to_additive] lemma isOpenMap_div_right (a : G) : IsOpenMap (· / a) := (Homeomorph.divRight a).isOpenMap #align is_open_map_div_right isOpenMap_div_right #align is_open_map_sub_right isOpenMap_sub_right @[to_additive] lemma isClosedMap_div_right (a : G) : IsClosedMap (· / a) := (Homeomorph.divRight a).isClosedMap #align is_closed_map_div_right isClosedMap_div_right #align is_closed_map_sub_right isClosedMap_sub_right @[to_additive] theorem tendsto_div_nhds_one_iff {α : Type} {l : Filter α} {x : G} {u : α → G} : Tendsto (u · / x) l (𝓝 1) ↔ Tendsto u l (𝓝 x) := haveI A : Tendsto (fun _ : α => x) l (𝓝 x) := tendsto_const_nhds ⟨fun h => by simpa using h.mul A, fun h => by simpa using h.div' A⟩ #align tendsto_div_nhds_one_iff tendsto_div_nhds_one_iff #align tendsto_sub_nhds_zero_iff tendsto_sub_nhds_zero_iff @[to_additive] theorem nhds_translation_div (x : G) : comap (· / x) (𝓝 1) = 𝓝 x := by simpa only [div_eq_mul_inv] using nhds_translation_mul_inv x #align nhds_translation_div nhds_translation_div #align nhds_translation_sub nhds_translation_sub end DivInvTopologicalGroup /-! ### Topological operations on pointwise sums and products A few results about interior and closure of the pointwise addition/multiplication of sets in groups with continuous addition/multiplication. See also `Submonoid.top_closure_mul_self_eq` in `Topology.Algebra.Monoid`. -/ section ContinuousConstSMul variable [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousConstSMul α β] {s : Set α} {t : Set β} @[to_additive] theorem IsOpen.smul_left (ht : IsOpen t) : IsOpen (s • t) := by rw [← iUnion_smul_set] exact isOpen_biUnion fun a _ => ht.smul _ #align is_open.smul_left IsOpen.smul_left #align is_open.vadd_left IsOpen.vadd_left @[to_additive] theorem subset_interior_smul_right : s • interior t ⊆ interior (s • t) := interior_maximal (Set.smul_subset_smul_left interior_subset) isOpen_interior.smul_left #align subset_interior_smul_right subset_interior_smul_right #align subset_interior_vadd_right subset_interior_vadd_right @[to_additive] theorem smul_mem_nhds (a : α) {x : β} (ht : t ∈ 𝓝 x) : a • t ∈ 𝓝 (a • x) := by rcases mem_nhds_iff.1 ht with ⟨u, ut, u_open, hu⟩ exact mem_nhds_iff.2 ⟨a • u, smul_set_mono ut, u_open.smul a, smul_mem_smul_set hu⟩ #align smul_mem_nhds smul_mem_nhds #align vadd_mem_nhds vadd_mem_nhds variable [TopologicalSpace α] @[to_additive] theorem subset_interior_smul : interior s • interior t ⊆ interior (s • t) := (Set.smul_subset_smul_right interior_subset).trans subset_interior_smul_right #align subset_interior_smul subset_interior_smul #align subset_interior_vadd subset_interior_vadd end ContinuousConstSMul section ContinuousSMul variable [TopologicalSpace α] [TopologicalSpace β] [Group α] [MulAction α β] [ContinuousInv α] [ContinuousSMul α β] {s : Set α} {t : Set β} @[to_additive] theorem IsClosed.smul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) : IsClosed (s • t) := by have : ∀ x ∈ s • t, ∃ g ∈ s, g⁻¹ • x ∈ t := by rintro x ⟨g, hgs, y, hyt, rfl⟩ refine ⟨g, hgs, ?_⟩ rwa [inv_smul_smul] choose! f hf using this refine isClosed_of_closure_subset (fun x hx ↦ ?_) rcases mem_closure_iff_ultrafilter.mp hx with ⟨u, hust, hux⟩ have : Ultrafilter.map f u ≤ 𝓟 s := calc Ultrafilter.map f u ≤ map f (𝓟 (s • t)) := map_mono (le_principal_iff.mpr hust) _ = 𝓟 (f '' (s • t)) := map_principal _ ≤ 𝓟 s := principal_mono.mpr (image_subset_iff.mpr (fun x hx ↦ (hf x hx).1)) rcases hs.ultrafilter_le_nhds (Ultrafilter.map f u) this with ⟨g, hg, hug⟩ suffices g⁻¹ • x ∈ t from ⟨g, hg, g⁻¹ • x, this, smul_inv_smul _ _⟩ exact ht.mem_of_tendsto ((Tendsto.inv hug).smul hux) (Eventually.mono hust (fun y hy ↦ (hf y hy).2)) /-! One may expect a version of `IsClosed.smul_left_of_isCompact` where `t` is compact and `s` is closed, but such a lemma can't be true in this level of generality. For a counterexample, consider `ℚ` acting on `ℝ` by translation, and let `s : Set ℚ := univ`, `t : set ℝ := {0}`. Then `s` is closed and `t` is compact, but `s +ᵥ t` is the set of all rationals, which is definitely not closed in `ℝ`. To fix the proof, we would need to make two additional assumptions: - for any `x ∈ t`, `s • {x}` is closed - for any `x ∈ t`, there is a continuous function `g : s • {x} → s` such that, for all `y ∈ s • {x}`, we have `y = (g y) • x` These are fairly specific hypotheses so we don't state this version of the lemmas, but an interesting fact is that these two assumptions are verified in the case of a `NormedAddTorsor` (or really, any `AddTorsor` with continuous `-ᵥ`). We prove this special case in `IsClosed.vadd_right_of_isCompact`. -/ @[to_additive] theorem MulAction.isClosedMap_quotient [CompactSpace α] : letI := orbitRel α β IsClosedMap (Quotient.mk' : β → Quotient (orbitRel α β)) := by intro t ht rw [← quotientMap_quotient_mk'.isClosed_preimage, MulAction.quotient_preimage_image_eq_union_mul] convert ht.smul_left_of_isCompact (isCompact_univ (X := α)) rw [← biUnion_univ, ← iUnion_smul_left_image] rfl end ContinuousSMul section ContinuousConstSMul variable [TopologicalSpace α] [Group α] [ContinuousConstSMul α α] {s t : Set α} @[to_additive] theorem IsOpen.mul_left : IsOpen t → IsOpen (s * t) := IsOpen.smul_left #align is_open.mul_left IsOpen.mul_left #align is_open.add_left IsOpen.add_left @[to_additive] theorem subset_interior_mul_right : s * interior t ⊆ interior (s * t) := subset_interior_smul_right #align subset_interior_mul_right subset_interior_mul_right #align subset_interior_add_right subset_interior_add_right @[to_additive] theorem subset_interior_mul : interior s * interior t ⊆ interior (s * t) := subset_interior_smul #align subset_interior_mul subset_interior_mul #align subset_interior_add subset_interior_add @[to_additive] theorem singleton_mul_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : {a} * s ∈ 𝓝 (a * b) := by have := smul_mem_nhds a h rwa [← singleton_smul] at this #align singleton_mul_mem_nhds singleton_mul_mem_nhds #align singleton_add_mem_nhds singleton_add_mem_nhds @[to_additive] theorem singleton_mul_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : {a} * s ∈ 𝓝 a := by simpa only [mul_one] using singleton_mul_mem_nhds a h #align singleton_mul_mem_nhds_of_nhds_one singleton_mul_mem_nhds_of_nhds_one #align singleton_add_mem_nhds_of_nhds_zero singleton_add_mem_nhds_of_nhds_zero end ContinuousConstSMul section ContinuousConstSMulOp variable [TopologicalSpace α] [Group α] [ContinuousConstSMul αᵐᵒᵖ α] {s t : Set α} @[to_additive] theorem IsOpen.mul_right (hs : IsOpen s) : IsOpen (s * t) := by rw [← iUnion_op_smul_set] exact isOpen_biUnion fun a _ => hs.smul _ #align is_open.mul_right IsOpen.mul_right #align is_open.add_right IsOpen.add_right @[to_additive] theorem subset_interior_mul_left : interior s * t ⊆ interior (s * t) := interior_maximal (Set.mul_subset_mul_right interior_subset) isOpen_interior.mul_right #align subset_interior_mul_left subset_interior_mul_left #align subset_interior_add_left subset_interior_add_left @[to_additive] theorem subset_interior_mul' : interior s * interior t ⊆ interior (s * t) := (Set.mul_subset_mul_left interior_subset).trans subset_interior_mul_left #align subset_interior_mul' subset_interior_mul' #align subset_interior_add' subset_interior_add' @[to_additive] theorem mul_singleton_mem_nhds (a : α) {b : α} (h : s ∈ 𝓝 b) : s * {a} ∈ 𝓝 (b * a) := by simp only [← iUnion_op_smul_set, mem_singleton_iff, iUnion_iUnion_eq_left] exact smul_mem_nhds _ h #align mul_singleton_mem_nhds mul_singleton_mem_nhds #align add_singleton_mem_nhds add_singleton_mem_nhds @[to_additive] theorem mul_singleton_mem_nhds_of_nhds_one (a : α) (h : s ∈ 𝓝 (1 : α)) : s * {a} ∈ 𝓝 a := by simpa only [one_mul] using mul_singleton_mem_nhds a h #align mul_singleton_mem_nhds_of_nhds_one mul_singleton_mem_nhds_of_nhds_one #align add_singleton_mem_nhds_of_nhds_zero add_singleton_mem_nhds_of_nhds_zero end ContinuousConstSMulOp section TopologicalGroup variable [TopologicalSpace G] [Group G] [TopologicalGroup G] {s t : Set G} @[to_additive] theorem IsOpen.div_left (ht : IsOpen t) : IsOpen (s / t) := by rw [← iUnion_div_left_image] exact isOpen_biUnion fun a _ => isOpenMap_div_left a t ht #align is_open.div_left IsOpen.div_left #align is_open.sub_left IsOpen.sub_left @[to_additive] theorem IsOpen.div_right (hs : IsOpen s) : IsOpen (s / t) := by rw [← iUnion_div_right_image] exact isOpen_biUnion fun a _ => isOpenMap_div_right a s hs #align is_open.div_right IsOpen.div_right #align is_open.sub_right IsOpen.sub_right @[to_additive] theorem subset_interior_div_left : interior s / t ⊆ interior (s / t) := interior_maximal (div_subset_div_right interior_subset) isOpen_interior.div_right #align subset_interior_div_left subset_interior_div_left #align subset_interior_sub_left subset_interior_sub_left @[to_additive] theorem subset_interior_div_right : s / interior t ⊆ interior (s / t) := interior_maximal (div_subset_div_left interior_subset) isOpen_interior.div_left #align subset_interior_div_right subset_interior_div_right #align subset_interior_sub_right subset_interior_sub_right @[to_additive] theorem subset_interior_div : interior s / interior t ⊆ interior (s / t) := (div_subset_div_left interior_subset).trans subset_interior_div_left #align subset_interior_div subset_interior_div #align subset_interior_sub subset_interior_sub @[to_additive] theorem IsOpen.mul_closure (hs : IsOpen s) (t : Set G) : s * closure t = s * t := by refine (mul_subset_iff.2 fun a ha b hb => ?_).antisymm (mul_subset_mul_left subset_closure) rw [mem_closure_iff] at hb have hbU : b ∈ s⁻¹ * {a * b} := ⟨a⁻¹, Set.inv_mem_inv.2 ha, a * b, rfl, inv_mul_cancel_left _ _⟩ obtain ⟨_, ⟨c, hc, d, rfl : d = _, rfl⟩, hcs⟩ := hb _ hs.inv.mul_right hbU exact ⟨c⁻¹, hc, _, hcs, inv_mul_cancel_left _ _⟩ #align is_open.mul_closure IsOpen.mul_closure #align is_open.add_closure IsOpen.add_closure @[to_additive] theorem IsOpen.closure_mul (ht : IsOpen t) (s : Set G) : closure s * t = s * t := by rw [← inv_inv (closure s * t), mul_inv_rev, inv_closure, ht.inv.mul_closure, mul_inv_rev, inv_inv, inv_inv] #align is_open.closure_mul IsOpen.closure_mul #align is_open.closure_add IsOpen.closure_add @[to_additive] theorem IsOpen.div_closure (hs : IsOpen s) (t : Set G) : s / closure t = s / t := by simp_rw [div_eq_mul_inv, inv_closure, hs.mul_closure] #align is_open.div_closure IsOpen.div_closure #align is_open.sub_closure IsOpen.sub_closure @[to_additive] theorem IsOpen.closure_div (ht : IsOpen t) (s : Set G) : closure s / t = s / t := by simp_rw [div_eq_mul_inv, ht.inv.closure_mul] #align is_open.closure_div IsOpen.closure_div #align is_open.closure_sub IsOpen.closure_sub @[to_additive] theorem IsClosed.mul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) : IsClosed (s * t) := ht.smul_left_of_isCompact hs @[to_additive]	Mathlib/Topology/Algebra/Group/Basic.lean	1,501	1,504	theorem IsClosed.mul_right_of_isCompact (ht : IsClosed t) (hs : IsCompact s) : IsClosed (t * s) := by	rw [← image_op_smul] exact IsClosed.smul_left_of_isCompact ht (hs.image continuous_op)
/- Copyright (c) 2022 Xavier Roblot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Alex J. Best, Xavier Roblot -/ import Mathlib.Analysis.Complex.Polynomial import Mathlib.NumberTheory.NumberField.Norm import Mathlib.NumberTheory.NumberField.Basic import Mathlib.RingTheory.Norm import Mathlib.Topology.Instances.Complex import Mathlib.RingTheory.RootsOfUnity.Basic #align_import number_theory.number_field.embeddings from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c" /-! # Embeddings of number fields This file defines the embeddings of a number field into an algebraic closed field. ## Main Definitions and Results * `NumberField.Embeddings.range_eval_eq_rootSet_minpoly`: let `x ∈ K` with `K` number field and let `A` be an algebraic closed field of char. 0, then the images of `x` by the embeddings of `K` in `A` are exactly the roots in `A` of the minimal polynomial of `x` over `ℚ`. * `NumberField.Embeddings.pow_eq_one_of_norm_eq_one`: an algebraic integer whose conjugates are all of norm one is a root of unity. * `NumberField.InfinitePlace`: the type of infinite places of a number field `K`. * `NumberField.InfinitePlace.mk_eq_iff`: two complex embeddings define the same infinite place iff they are equal or complex conjugates. * `NumberField.InfinitePlace.prod_eq_abs_norm`: the infinite part of the product formula, that is for `x ∈ K`, we have `Π_w ‖x‖_w = \|norm(x)\|` where the product is over the infinite place `w` and `‖·‖_w` is the normalized absolute value for `w`. ## Tags number field, embeddings, places, infinite places -/ open scoped Classical namespace NumberField.Embeddings section Fintype open FiniteDimensional variable (K : Type) [Field K] [NumberField K] variable (A : Type) [Field A] [CharZero A] /-- There are finitely many embeddings of a number field. -/ noncomputable instance : Fintype (K →+* A) := Fintype.ofEquiv (K →ₐ[ℚ] A) RingHom.equivRatAlgHom.symm variable [IsAlgClosed A] /-- The number of embeddings of a number field is equal to its finrank. -/ theorem card : Fintype.card (K →+* A) = finrank ℚ K := by rw [Fintype.ofEquiv_card RingHom.equivRatAlgHom.symm, AlgHom.card] #align number_field.embeddings.card NumberField.Embeddings.card instance : Nonempty (K →+* A) := by rw [← Fintype.card_pos_iff, NumberField.Embeddings.card K A] exact FiniteDimensional.finrank_pos end Fintype section Roots open Set Polynomial variable (K A : Type) [Field K] [NumberField K] [Field A] [Algebra ℚ A] [IsAlgClosed A] (x : K) /-- Let `A` be an algebraically closed field and let `x ∈ K`, with `K` a number field. The images of `x` by the embeddings of `K` in `A` are exactly the roots in `A` of the minimal polynomial of `x` over `ℚ`. -/ theorem range_eval_eq_rootSet_minpoly : (range fun φ : K →+ A => φ x) = (minpoly ℚ x).rootSet A := by convert (NumberField.isAlgebraic K).range_eval_eq_rootSet_minpoly A x using 1 ext a exact ⟨fun ⟨φ, hφ⟩ => ⟨φ.toRatAlgHom, hφ⟩, fun ⟨φ, hφ⟩ => ⟨φ.toRingHom, hφ⟩⟩ #align number_field.embeddings.range_eval_eq_root_set_minpoly NumberField.Embeddings.range_eval_eq_rootSet_minpoly end Roots section Bounded open FiniteDimensional Polynomial Set variable {K : Type} [Field K] [NumberField K] variable {A : Type} [NormedField A] [IsAlgClosed A] [NormedAlgebra ℚ A] theorem coeff_bdd_of_norm_le {B : ℝ} {x : K} (h : ∀ φ : K →+* A, ‖φ x‖ ≤ B) (i : ℕ) : ‖(minpoly ℚ x).coeff i‖ ≤ max B 1 ^ finrank ℚ K * (finrank ℚ K).choose (finrank ℚ K / 2) := by have hx := IsSeparable.isIntegral ℚ x rw [← norm_algebraMap' A, ← coeff_map (algebraMap ℚ A)] refine coeff_bdd_of_roots_le _ (minpoly.monic hx) (IsAlgClosed.splits_codomain _) (minpoly.natDegree_le x) (fun z hz => ?_) i classical rw [← Multiset.mem_toFinset] at hz obtain ⟨φ, rfl⟩ := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz exact h φ #align number_field.embeddings.coeff_bdd_of_norm_le NumberField.Embeddings.coeff_bdd_of_norm_le variable (K A) /-- Let `B` be a real number. The set of algebraic integers in `K` whose conjugates are all smaller in norm than `B` is finite. -/ theorem finite_of_norm_le (B : ℝ) : {x : K \| IsIntegral ℤ x ∧ ∀ φ : K →+* A, ‖φ x‖ ≤ B}.Finite := by let C := Nat.ceil (max B 1 ^ finrank ℚ K * (finrank ℚ K).choose (finrank ℚ K / 2)) have := bUnion_roots_finite (algebraMap ℤ K) (finrank ℚ K) (finite_Icc (-C : ℤ) C) refine this.subset fun x hx => ?_; simp_rw [mem_iUnion] have h_map_ℚ_minpoly := minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx.1 refine ⟨_, ⟨?_, fun i => ?_⟩, mem_rootSet.2 ⟨minpoly.ne_zero hx.1, minpoly.aeval ℤ x⟩⟩ · rw [← (minpoly.monic hx.1).natDegree_map (algebraMap ℤ ℚ), ← h_map_ℚ_minpoly] exact minpoly.natDegree_le x rw [mem_Icc, ← abs_le, ← @Int.cast_le ℝ] refine (Eq.trans_le ?_ <\| coeff_bdd_of_norm_le hx.2 i).trans (Nat.le_ceil _) rw [h_map_ℚ_minpoly, coeff_map, eq_intCast, Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] #align number_field.embeddings.finite_of_norm_le NumberField.Embeddings.finite_of_norm_le /-- An algebraic integer whose conjugates are all of norm one is a root of unity. -/ theorem pow_eq_one_of_norm_eq_one {x : K} (hxi : IsIntegral ℤ x) (hx : ∀ φ : K →+* A, ‖φ x‖ = 1) : ∃ (n : ℕ) (_ : 0 < n), x ^ n = 1 := by obtain ⟨a, -, b, -, habne, h⟩ := @Set.Infinite.exists_ne_map_eq_of_mapsTo _ _ _ _ (x ^ · : ℕ → K) Set.infinite_univ (by exact fun a _ => ⟨hxi.pow a, fun φ => by simp [hx φ]⟩) (finite_of_norm_le K A (1 : ℝ)) wlog hlt : b < a · exact this K A hxi hx b a habne.symm h.symm (habne.lt_or_lt.resolve_right hlt) refine ⟨a - b, tsub_pos_of_lt hlt, ?_⟩ rw [← Nat.sub_add_cancel hlt.le, pow_add, mul_left_eq_self₀] at h refine h.resolve_right fun hp => ?_ specialize hx (IsAlgClosed.lift (R := ℚ)).toRingHom rw [pow_eq_zero hp, map_zero, norm_zero] at hx; norm_num at hx #align number_field.embeddings.pow_eq_one_of_norm_eq_one NumberField.Embeddings.pow_eq_one_of_norm_eq_one end Bounded end NumberField.Embeddings section Place variable {K : Type} [Field K] {A : Type} [NormedDivisionRing A] [Nontrivial A] (φ : K →+* A) /-- An embedding into a normed division ring defines a place of `K` -/ def NumberField.place : AbsoluteValue K ℝ := (IsAbsoluteValue.toAbsoluteValue (norm : A → ℝ)).comp φ.injective #align number_field.place NumberField.place @[simp] theorem NumberField.place_apply (x : K) : (NumberField.place φ) x = norm (φ x) := rfl #align number_field.place_apply NumberField.place_apply end Place namespace NumberField.ComplexEmbedding open Complex NumberField open scoped ComplexConjugate variable {K : Type} [Field K] {k : Type} [Field k] /-- The conjugate of a complex embedding as a complex embedding. -/ abbrev conjugate (φ : K →+* ℂ) : K →+* ℂ := star φ #align number_field.complex_embedding.conjugate NumberField.ComplexEmbedding.conjugate @[simp] theorem conjugate_coe_eq (φ : K →+* ℂ) (x : K) : (conjugate φ) x = conj (φ x) := rfl #align number_field.complex_embedding.conjugate_coe_eq NumberField.ComplexEmbedding.conjugate_coe_eq theorem place_conjugate (φ : K →+* ℂ) : place (conjugate φ) = place φ := by ext; simp only [place_apply, norm_eq_abs, abs_conj, conjugate_coe_eq] #align number_field.complex_embedding.place_conjugate NumberField.ComplexEmbedding.place_conjugate /-- An embedding into `ℂ` is real if it is fixed by complex conjugation. -/ abbrev IsReal (φ : K →+* ℂ) : Prop := IsSelfAdjoint φ #align number_field.complex_embedding.is_real NumberField.ComplexEmbedding.IsReal theorem isReal_iff {φ : K →+* ℂ} : IsReal φ ↔ conjugate φ = φ := isSelfAdjoint_iff #align number_field.complex_embedding.is_real_iff NumberField.ComplexEmbedding.isReal_iff theorem isReal_conjugate_iff {φ : K →+* ℂ} : IsReal (conjugate φ) ↔ IsReal φ := IsSelfAdjoint.star_iff #align number_field.complex_embedding.is_real_conjugate_iff NumberField.ComplexEmbedding.isReal_conjugate_iff /-- A real embedding as a ring homomorphism from `K` to `ℝ` . -/ def IsReal.embedding {φ : K →+* ℂ} (hφ : IsReal φ) : K →+* ℝ where toFun x := (φ x).re map_one' := by simp only [map_one, one_re] map_mul' := by simp only [Complex.conj_eq_iff_im.mp (RingHom.congr_fun hφ _), map_mul, mul_re, mul_zero, tsub_zero, eq_self_iff_true, forall_const] map_zero' := by simp only [map_zero, zero_re] map_add' := by simp only [map_add, add_re, eq_self_iff_true, forall_const] #align number_field.complex_embedding.is_real.embedding NumberField.ComplexEmbedding.IsReal.embedding @[simp] theorem IsReal.coe_embedding_apply {φ : K →+* ℂ} (hφ : IsReal φ) (x : K) : (hφ.embedding x : ℂ) = φ x := by apply Complex.ext · rfl · rw [ofReal_im, eq_comm, ← Complex.conj_eq_iff_im] exact RingHom.congr_fun hφ x #align number_field.complex_embedding.is_real.coe_embedding_apply NumberField.ComplexEmbedding.IsReal.coe_embedding_apply lemma IsReal.comp (f : k →+* K) {φ : K →+* ℂ} (hφ : IsReal φ) : IsReal (φ.comp f) := by ext1 x; simpa using RingHom.congr_fun hφ (f x) lemma isReal_comp_iff {f : k ≃+* K} {φ : K →+* ℂ} : IsReal (φ.comp (f : k →+* K)) ↔ IsReal φ := ⟨fun H ↦ by convert H.comp f.symm.toRingHom; ext1; simp, IsReal.comp _⟩ lemma exists_comp_symm_eq_of_comp_eq [Algebra k K] [IsGalois k K] (φ ψ : K →+* ℂ) (h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)) : ∃ σ : K ≃ₐ[k] K, φ.comp σ.symm = ψ := by letI := (φ.comp (algebraMap k K)).toAlgebra letI := φ.toAlgebra have : IsScalarTower k K ℂ := IsScalarTower.of_algebraMap_eq' rfl let ψ' : K →ₐ[k] ℂ := { ψ with commutes' := fun r ↦ (RingHom.congr_fun h r).symm } use (AlgHom.restrictNormal' ψ' K).symm ext1 x exact AlgHom.restrictNormal_commutes ψ' K x variable [Algebra k K] (φ : K →+* ℂ) (σ : K ≃ₐ[k] K) /-- `IsConj φ σ` states that `σ : K ≃ₐ[k] K` is the conjugation under the embedding `φ : K →+* ℂ`. -/ def IsConj : Prop := conjugate φ = φ.comp σ variable {φ σ} lemma IsConj.eq (h : IsConj φ σ) (x) : φ (σ x) = star (φ x) := RingHom.congr_fun h.symm x lemma IsConj.ext {σ₁ σ₂ : K ≃ₐ[k] K} (h₁ : IsConj φ σ₁) (h₂ : IsConj φ σ₂) : σ₁ = σ₂ := AlgEquiv.ext fun x ↦ φ.injective ((h₁.eq x).trans (h₂.eq x).symm) lemma IsConj.ext_iff {σ₁ σ₂ : K ≃ₐ[k] K} (h₁ : IsConj φ σ₁) : σ₁ = σ₂ ↔ IsConj φ σ₂ := ⟨fun e ↦ e ▸ h₁, h₁.ext⟩ lemma IsConj.isReal_comp (h : IsConj φ σ) : IsReal (φ.comp (algebraMap k K)) := by ext1 x simp only [conjugate_coe_eq, RingHom.coe_comp, Function.comp_apply, ← h.eq, starRingEnd_apply, AlgEquiv.commutes] lemma isConj_one_iff : IsConj φ (1 : K ≃ₐ[k] K) ↔ IsReal φ := Iff.rfl alias ⟨_, IsReal.isConjGal_one⟩ := ComplexEmbedding.isConj_one_iff lemma IsConj.symm (hσ : IsConj φ σ) : IsConj φ σ.symm := RingHom.ext fun x ↦ by simpa using congr_arg star (hσ.eq (σ.symm x)) lemma isConj_symm : IsConj φ σ.symm ↔ IsConj φ σ := ⟨IsConj.symm, IsConj.symm⟩ end NumberField.ComplexEmbedding section InfinitePlace open NumberField variable {k : Type} [Field k] (K : Type) [Field K] {F : Type} [Field F] /-- An infinite place of a number field `K` is a place associated to a complex embedding. -/ def NumberField.InfinitePlace := { w : AbsoluteValue K ℝ // ∃ φ : K →+ ℂ, place φ = w } #align number_field.infinite_place NumberField.InfinitePlace instance [NumberField K] : Nonempty (NumberField.InfinitePlace K) := Set.instNonemptyRange _ variable {K} /-- Return the infinite place defined by a complex embedding `φ`. -/ noncomputable def NumberField.InfinitePlace.mk (φ : K →+* ℂ) : NumberField.InfinitePlace K := ⟨place φ, ⟨φ, rfl⟩⟩ #align number_field.infinite_place.mk NumberField.InfinitePlace.mk namespace NumberField.InfinitePlace open NumberField instance {K : Type} [Field K] : FunLike (InfinitePlace K) K ℝ where coe w x := w.1 x coe_injective' := fun _ _ h => Subtype.eq (AbsoluteValue.ext fun x => congr_fun h x) instance : MonoidWithZeroHomClass (InfinitePlace K) K ℝ where map_mul w _ _ := w.1.map_mul _ _ map_one w := w.1.map_one map_zero w := w.1.map_zero instance : NonnegHomClass (InfinitePlace K) K ℝ where apply_nonneg w _ := w.1.nonneg _ @[simp] theorem apply (φ : K →+ ℂ) (x : K) : (mk φ) x = Complex.abs (φ x) := rfl #align number_field.infinite_place.apply NumberField.InfinitePlace.apply /-- For an infinite place `w`, return an embedding `φ` such that `w = infinite_place φ` . -/ noncomputable def embedding (w : InfinitePlace K) : K →+* ℂ := w.2.choose #align number_field.infinite_place.embedding NumberField.InfinitePlace.embedding @[simp] theorem mk_embedding (w : InfinitePlace K) : mk (embedding w) = w := Subtype.ext w.2.choose_spec #align number_field.infinite_place.mk_embedding NumberField.InfinitePlace.mk_embedding @[simp] theorem mk_conjugate_eq (φ : K →+* ℂ) : mk (ComplexEmbedding.conjugate φ) = mk φ := by refine DFunLike.ext _ _ (fun x => ?_) rw [apply, apply, ComplexEmbedding.conjugate_coe_eq, Complex.abs_conj] #align number_field.infinite_place.mk_conjugate_eq NumberField.InfinitePlace.mk_conjugate_eq theorem norm_embedding_eq (w : InfinitePlace K) (x : K) : ‖(embedding w) x‖ = w x := by nth_rewrite 2 [← mk_embedding w] rfl theorem eq_iff_eq (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x = r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ = r := ⟨fun hw φ => hw (mk φ), by rintro hφ ⟨w, ⟨φ, rfl⟩⟩; exact hφ φ⟩ #align number_field.infinite_place.eq_iff_eq NumberField.InfinitePlace.eq_iff_eq theorem le_iff_le (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x ≤ r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := ⟨fun hw φ => hw (mk φ), by rintro hφ ⟨w, ⟨φ, rfl⟩⟩; exact hφ φ⟩ #align number_field.infinite_place.le_iff_le NumberField.InfinitePlace.le_iff_le theorem pos_iff {w : InfinitePlace K} {x : K} : 0 < w x ↔ x ≠ 0 := AbsoluteValue.pos_iff w.1 #align number_field.infinite_place.pos_iff NumberField.InfinitePlace.pos_iff @[simp] theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by constructor · -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj` intro h₀ obtain ⟨j, hiφ⟩ := (φ.injective).hasLeftInverse let ι := RingEquiv.ofLeftInverse hiφ have hlip : LipschitzWith 1 (RingHom.comp ψ ι.symm.toRingHom) := by change LipschitzWith 1 (ψ ∘ ι.symm) apply LipschitzWith.of_dist_le_mul intro x y rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply, ← map_sub, ← map_sub] apply le_of_eq suffices ‖φ (ι.symm (x - y))‖ = ‖ψ (ι.symm (x - y))‖ by rw [← this, ← RingEquiv.ofLeftInverse_apply hiφ _, RingEquiv.apply_symm_apply ι _] rfl exact congrFun (congrArg (↑) h₀) _ cases Complex.uniformContinuous_ringHom_eq_id_or_conj φ.fieldRange hlip.uniformContinuous with \| inl h => left; ext1 x conv_rhs => rw [← hiφ x] exact (congrFun h (ι x)).symm \| inr h => right; ext1 x conv_rhs => rw [← hiφ x] exact (congrFun h (ι x)).symm · rintro (⟨h⟩ \| ⟨h⟩) · exact congr_arg mk h · rw [← mk_conjugate_eq] exact congr_arg mk h #align number_field.infinite_place.mk_eq_iff NumberField.InfinitePlace.mk_eq_iff /-- An infinite place is real if it is defined by a real embedding. -/ def IsReal (w : InfinitePlace K) : Prop := ∃ φ : K →+* ℂ, ComplexEmbedding.IsReal φ ∧ mk φ = w #align number_field.infinite_place.is_real NumberField.InfinitePlace.IsReal /-- An infinite place is complex if it is defined by a complex (ie. not real) embedding. -/ def IsComplex (w : InfinitePlace K) : Prop := ∃ φ : K →+* ℂ, ¬ComplexEmbedding.IsReal φ ∧ mk φ = w #align number_field.infinite_place.is_complex NumberField.InfinitePlace.IsComplex theorem embedding_mk_eq (φ : K →+* ℂ) : embedding (mk φ) = φ ∨ embedding (mk φ) = ComplexEmbedding.conjugate φ := by rw [@eq_comm _ _ φ, @eq_comm _ _ (ComplexEmbedding.conjugate φ), ← mk_eq_iff, mk_embedding] @[simp] theorem embedding_mk_eq_of_isReal {φ : K →+* ℂ} (h : ComplexEmbedding.IsReal φ) : embedding (mk φ) = φ := by have := embedding_mk_eq φ rwa [ComplexEmbedding.isReal_iff.mp h, or_self] at this #align number_field.complex_embeddings.is_real.embedding_mk NumberField.InfinitePlace.embedding_mk_eq_of_isReal theorem isReal_iff {w : InfinitePlace K} : IsReal w ↔ ComplexEmbedding.IsReal (embedding w) := by refine ⟨?_, fun h => ⟨embedding w, h, mk_embedding w⟩⟩ rintro ⟨φ, ⟨hφ, rfl⟩⟩ rwa [embedding_mk_eq_of_isReal hφ] #align number_field.infinite_place.is_real_iff NumberField.InfinitePlace.isReal_iff theorem isComplex_iff {w : InfinitePlace K} : IsComplex w ↔ ¬ComplexEmbedding.IsReal (embedding w) := by refine ⟨?_, fun h => ⟨embedding w, h, mk_embedding w⟩⟩ rintro ⟨φ, ⟨hφ, rfl⟩⟩ contrapose! hφ cases mk_eq_iff.mp (mk_embedding (mk φ)) with \| inl h => rwa [h] at hφ \| inr h => rwa [← ComplexEmbedding.isReal_conjugate_iff, h] at hφ #align number_field.infinite_place.is_complex_iff NumberField.InfinitePlace.isComplex_iff @[simp] theorem conjugate_embedding_eq_of_isReal {w : InfinitePlace K} (h : IsReal w) : ComplexEmbedding.conjugate (embedding w) = embedding w := ComplexEmbedding.isReal_iff.mpr (isReal_iff.mp h) @[simp] theorem not_isReal_iff_isComplex {w : InfinitePlace K} : ¬IsReal w ↔ IsComplex w := by rw [isComplex_iff, isReal_iff] #align number_field.infinite_place.not_is_real_iff_is_complex NumberField.InfinitePlace.not_isReal_iff_isComplex @[simp] theorem not_isComplex_iff_isReal {w : InfinitePlace K} : ¬IsComplex w ↔ IsReal w := by rw [isComplex_iff, isReal_iff, not_not] theorem isReal_or_isComplex (w : InfinitePlace K) : IsReal w ∨ IsComplex w := by rw [← not_isReal_iff_isComplex]; exact em _ #align number_field.infinite_place.is_real_or_is_complex NumberField.InfinitePlace.isReal_or_isComplex theorem ne_of_isReal_isComplex {w w' : InfinitePlace K} (h : IsReal w) (h' : IsComplex w') : w ≠ w' := fun h_eq ↦ not_isReal_iff_isComplex.mpr h' (h_eq ▸ h) variable (K) in theorem disjoint_isReal_isComplex : Disjoint {(w : InfinitePlace K) \| IsReal w} {(w : InfinitePlace K) \| IsComplex w} := Set.disjoint_iff.2 <\| fun _ hw ↦ not_isReal_iff_isComplex.2 hw.2 hw.1 /-- The real embedding associated to a real infinite place. -/ noncomputable def embedding_of_isReal {w : InfinitePlace K} (hw : IsReal w) : K →+* ℝ := ComplexEmbedding.IsReal.embedding (isReal_iff.mp hw) #align number_field.infinite_place.is_real.embedding NumberField.InfinitePlace.embedding_of_isReal @[simp] theorem embedding_of_isReal_apply {w : InfinitePlace K} (hw : IsReal w) (x : K) : ((embedding_of_isReal hw) x : ℂ) = (embedding w) x := ComplexEmbedding.IsReal.coe_embedding_apply (isReal_iff.mp hw) x theorem norm_embedding_of_isReal {w : InfinitePlace K} (hw : IsReal w) (x : K) : ‖embedding_of_isReal hw x‖ = w x := by rw [← norm_embedding_eq, ← embedding_of_isReal_apply hw, Complex.norm_real] @[simp] theorem isReal_of_mk_isReal {φ : K →+* ℂ} (h : IsReal (mk φ)) : ComplexEmbedding.IsReal φ := by contrapose! h rw [not_isReal_iff_isComplex] exact ⟨φ, h, rfl⟩ lemma isReal_mk_iff {φ : K →+* ℂ} : IsReal (mk φ) ↔ ComplexEmbedding.IsReal φ := ⟨isReal_of_mk_isReal, fun H ↦ ⟨_, H, rfl⟩⟩ lemma isComplex_mk_iff {φ : K →+* ℂ} : IsComplex (mk φ) ↔ ¬ ComplexEmbedding.IsReal φ := not_isReal_iff_isComplex.symm.trans isReal_mk_iff.not @[simp] theorem not_isReal_of_mk_isComplex {φ : K →+* ℂ} (h : IsComplex (mk φ)) : ¬ ComplexEmbedding.IsReal φ := by rwa [← isComplex_mk_iff] /-- The multiplicity of an infinite place, that is the number of distinct complex embeddings that define it, see `card_filter_mk_eq`. -/ noncomputable def mult (w : InfinitePlace K) : ℕ := if (IsReal w) then 1 else 2 theorem mult_pos {w : InfinitePlace K} : 0 < mult w := by rw [mult] split_ifs <;> norm_num @[simp] theorem mult_ne_zero {w : InfinitePlace K} : mult w ≠ 0 := ne_of_gt mult_pos theorem one_le_mult {w : InfinitePlace K} : (1 : ℝ) ≤ mult w := by rw [← Nat.cast_one, Nat.cast_le] exact mult_pos theorem card_filter_mk_eq [NumberField K] (w : InfinitePlace K) : (Finset.univ.filter fun φ => mk φ = w).card = mult w := by conv_lhs => congr; congr; ext rw [← mk_embedding w, mk_eq_iff, ComplexEmbedding.conjugate, star_involutive.eq_iff] simp_rw [Finset.filter_or, Finset.filter_eq' _ (embedding w), Finset.filter_eq' _ (ComplexEmbedding.conjugate (embedding w)), Finset.mem_univ, ite_true, mult] split_ifs with hw · rw [ComplexEmbedding.isReal_iff.mp (isReal_iff.mp hw), Finset.union_idempotent, Finset.card_singleton] · refine Finset.card_pair ?_ rwa [Ne, eq_comm, ← ComplexEmbedding.isReal_iff, ← isReal_iff] noncomputable instance NumberField.InfinitePlace.fintype [NumberField K] : Fintype (InfinitePlace K) := Set.fintypeRange _ #align number_field.infinite_place.number_field.infinite_place.fintype NumberField.InfinitePlace.NumberField.InfinitePlace.fintype theorem sum_mult_eq [NumberField K] : ∑ w : InfinitePlace K, mult w = FiniteDimensional.finrank ℚ K := by rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise (fun φ => InfinitePlace.mk φ)] exact Finset.sum_congr rfl (fun _ _ => by rw [Finset.sum_const, smul_eq_mul, mul_one, card_filter_mk_eq]) /-- The map from real embeddings to real infinite places as an equiv -/ noncomputable def mkReal : { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } ≃ { w : InfinitePlace K // IsReal w } := by refine (Equiv.ofBijective (fun φ => ⟨mk φ, ?_⟩) ⟨fun φ ψ h => ?_, fun w => ?_⟩) · exact ⟨φ, φ.prop, rfl⟩ · rwa [Subtype.mk.injEq, mk_eq_iff, ComplexEmbedding.isReal_iff.mp φ.prop, or_self, ← Subtype.ext_iff] at h · exact ⟨⟨embedding w, isReal_iff.mp w.prop⟩, by simp⟩ /-- The map from nonreal embeddings to complex infinite places -/ noncomputable def mkComplex : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } → { w : InfinitePlace K // IsComplex w } := Subtype.map mk fun φ hφ => ⟨φ, hφ, rfl⟩ #align number_field.infinite_place.mk_complex NumberField.InfinitePlace.mkComplex @[simp] theorem mkReal_coe (φ : { φ : K →+* ℂ // ComplexEmbedding.IsReal φ }) : (mkReal φ : InfinitePlace K) = mk (φ : K →+* ℂ) := rfl #align number_field.infinite_place.mk_real_coe NumberField.InfinitePlace.mkReal_coe @[simp] theorem mkComplex_coe (φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : (mkComplex φ : InfinitePlace K) = mk (φ : K →+* ℂ) := rfl #align number_field.infinite_place.mk_complex_coe NumberField.InfinitePlace.mkComplex_coe variable [NumberField K] /-- The infinite part of the product formula : for `x ∈ K`, we have `Π_w ‖x‖_w = \|norm(x)\|` where `‖·‖_w` is the normalized absolute value for `w`. -/ theorem prod_eq_abs_norm (x : K) : ∏ w : InfinitePlace K, w x ^ mult w = abs (Algebra.norm ℚ x) := by convert (congr_arg Complex.abs (@Algebra.norm_eq_prod_embeddings ℚ _ _ _ _ ℂ _ _ _ _ _ x)).symm · rw [map_prod, ← Fintype.prod_equiv RingHom.equivRatAlgHom (fun f => Complex.abs (f x)) (fun φ => Complex.abs (φ x)) fun _ => by simp [RingHom.equivRatAlgHom_apply]; rfl] rw [← Finset.prod_fiberwise Finset.univ (fun φ => mk φ) (fun φ => Complex.abs (φ x))] have : ∀ w : InfinitePlace K, ∀ φ ∈ Finset.filter (fun a ↦ mk a = w) Finset.univ, Complex.abs (φ x) = w x := by intro _ _ hφ rw [← (Finset.mem_filter.mp hφ).2] rfl simp_rw [Finset.prod_congr rfl (this _), Finset.prod_const, card_filter_mk_eq] · rw [eq_ratCast, Rat.cast_abs, ← Complex.abs_ofReal, Complex.ofReal_ratCast] #align number_field.infinite_place.prod_eq_abs_norm NumberField.InfinitePlace.prod_eq_abs_norm theorem one_le_of_lt_one {w : InfinitePlace K} {a : (𝓞 K)} (ha : a ≠ 0) (h : ∀ ⦃z⦄, z ≠ w → z a < 1) : 1 ≤ w a := by suffices (1:ℝ) ≤ \|Algebra.norm ℚ (a : K)\| by contrapose! this rw [← InfinitePlace.prod_eq_abs_norm, ← Finset.prod_const_one] refine Finset.prod_lt_prod_of_nonempty (fun _ _ ↦ ?_) (fun z _ ↦ ?_) Finset.univ_nonempty · exact pow_pos (pos_iff.mpr ((Subalgebra.coe_eq_zero _).not.mpr ha)) _ · refine pow_lt_one (apply_nonneg _ _) ?_ (by rw [mult]; split_ifs <;> norm_num) by_cases hz : z = w · rwa [hz] · exact h hz rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_intCast, Int.cast_le] exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr ha) open scoped IntermediateField in theorem _root_.NumberField.is_primitive_element_of_infinitePlace_lt {x : 𝓞 K} {w : InfinitePlace K} (h₁ : x ≠ 0) (h₂ : ∀ ⦃w'⦄, w' ≠ w → w' x < 1) (h₃ : IsReal w ∨ \|(w.embedding x).re\| < 1) : ℚ⟮(x : K)⟯ = ⊤ := by rw [Field.primitive_element_iff_algHom_eq_of_eval ℚ ℂ ?_ _ w.embedding.toRatAlgHom] · intro ψ hψ have h : 1 ≤ w x := one_le_of_lt_one h₁ h₂ have main : w = InfinitePlace.mk ψ.toRingHom := by erw [← norm_embedding_eq, hψ] at h contrapose! h exact h₂ h.symm rw [(mk_embedding w).symm, mk_eq_iff] at main cases h₃ with \| inl hw => rw [conjugate_embedding_eq_of_isReal hw, or_self] at main exact congr_arg RingHom.toRatAlgHom main \| inr hw => refine congr_arg RingHom.toRatAlgHom (main.resolve_right fun h' ↦ hw.not_le ?_) have : (embedding w x).im = 0 := by erw [← Complex.conj_eq_iff_im, RingHom.congr_fun h' x] exact hψ.symm rwa [← norm_embedding_eq, ← Complex.re_add_im (embedding w x), this, Complex.ofReal_zero, zero_mul, add_zero, Complex.norm_eq_abs, Complex.abs_ofReal] at h · exact fun x ↦ IsAlgClosed.splits_codomain (minpoly ℚ x) theorem _root_.NumberField.adjoin_eq_top_of_infinitePlace_lt {x : 𝓞 K} {w : InfinitePlace K} (h₁ : x ≠ 0) (h₂ : ∀ ⦃w'⦄, w' ≠ w → w' x < 1) (h₃ : IsReal w ∨ \|(w.embedding x).re\| < 1) : Algebra.adjoin ℚ {(x : K)} = ⊤ := by rw [← IntermediateField.adjoin_simple_toSubalgebra_of_integral (IsIntegral.of_finite ℚ _)] exact congr_arg IntermediateField.toSubalgebra <\| NumberField.is_primitive_element_of_infinitePlace_lt h₁ h₂ h₃ open Fintype FiniteDimensional variable (K) /-- The number of infinite real places of the number field `K`. -/ noncomputable abbrev NrRealPlaces := card { w : InfinitePlace K // IsReal w } /-- The number of infinite complex places of the number field `K`. -/ noncomputable abbrev NrComplexPlaces := card { w : InfinitePlace K // IsComplex w } theorem card_real_embeddings : card { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } = NrRealPlaces K := Fintype.card_congr mkReal #align number_field.infinite_place.card_real_embeddings NumberField.InfinitePlace.card_real_embeddings theorem card_eq_nrRealPlaces_add_nrComplexPlaces : Fintype.card (InfinitePlace K) = NrRealPlaces K + NrComplexPlaces K := by convert Fintype.card_subtype_or_disjoint (IsReal (K := K)) (IsComplex (K := K)) (disjoint_isReal_isComplex K) using 1 exact (Fintype.card_of_subtype _ (fun w ↦ ⟨fun _ ↦ isReal_or_isComplex w, fun _ ↦ by simp⟩)).symm theorem card_complex_embeddings : card { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } = 2 * NrComplexPlaces K := by suffices ∀ w : { w : InfinitePlace K // IsComplex w }, (Finset.univ.filter fun φ : { φ // ¬ ComplexEmbedding.IsReal φ } => mkComplex φ = w).card = 2 by rw [Fintype.card, Finset.card_eq_sum_ones, ← Finset.sum_fiberwise _ (fun φ => mkComplex φ)] simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones, Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one] rintro ⟨w, hw⟩ convert card_filter_mk_eq w · rw [← Fintype.card_subtype, ← Fintype.card_subtype] refine Fintype.card_congr (Equiv.ofBijective ?_ ⟨fun _ _ h => ?_, fun ⟨φ, hφ⟩ => ?_⟩) · exact fun ⟨φ, hφ⟩ => ⟨φ.val, by rwa [Subtype.ext_iff] at hφ⟩ · rwa [Subtype.mk_eq_mk, ← Subtype.ext_iff, ← Subtype.ext_iff] at h · refine ⟨⟨⟨φ, not_isReal_of_mk_isComplex (hφ.symm ▸ hw)⟩, ?_⟩, rfl⟩ rwa [Subtype.ext_iff, mkComplex_coe] · simp_rw [mult, not_isReal_iff_isComplex.mpr hw, ite_false] #align number_field.infinite_place.card_complex_embeddings NumberField.InfinitePlace.card_complex_embeddings theorem card_add_two_mul_card_eq_rank : NrRealPlaces K + 2 * NrComplexPlaces K = finrank ℚ K := by rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl, ← Embeddings.card K ℂ, Nat.add_sub_of_le] exact Fintype.card_subtype_le _ variable {K}	Mathlib/NumberTheory/NumberField/Embeddings.lean	630	631	theorem nrComplexPlaces_eq_zero_of_finrank_eq_one (h : finrank ℚ K = 1) : NrComplexPlaces K = 0 := by	linarith [card_add_two_mul_card_eq_rank K]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios -/ import Mathlib.SetTheory.Ordinal.Basic import Mathlib.Data.Nat.SuccPred #align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7" /-! # Ordinal arithmetic Ordinals have an addition (corresponding to disjoint union) that turns them into an additive monoid, and a multiplication (corresponding to the lexicographic order on the product) that turns them into a monoid. One can also define correspondingly a subtraction, a division, a successor function, a power function and a logarithm function. We also define limit ordinals and prove the basic induction principle on ordinals separating successor ordinals and limit ordinals, in `limitRecOn`. ## Main definitions and results * `o₁ + o₂` is the order on the disjoint union of `o₁` and `o₂` obtained by declaring that every element of `o₁` is smaller than every element of `o₂`. * `o₁ - o₂` is the unique ordinal `o` such that `o₂ + o = o₁`, when `o₂ ≤ o₁`. * `o₁ * o₂` is the lexicographic order on `o₂ × o₁`. * `o₁ / o₂` is the ordinal `o` such that `o₁ = o₂ * o + o'` with `o' < o₂`. We also define the divisibility predicate, and a modulo operation. * `Order.succ o = o + 1` is the successor of `o`. * `pred o` if the predecessor of `o`. If `o` is not a successor, we set `pred o = o`. We discuss the properties of casts of natural numbers of and of `ω` with respect to these operations. Some properties of the operations are also used to discuss general tools on ordinals: * `IsLimit o`: an ordinal is a limit ordinal if it is neither `0` nor a successor. * `limitRecOn` is the main induction principle of ordinals: if one can prove a property by induction at successor ordinals and at limit ordinals, then it holds for all ordinals. * `IsNormal`: a function `f : Ordinal → Ordinal` satisfies `IsNormal` if it is strictly increasing and order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for `a < o`. * `enumOrd`: enumerates an unbounded set of ordinals by the ordinals themselves. * `sup`, `lsub`: the supremum / least strict upper bound of an indexed family of ordinals in `Type u`, as an ordinal in `Type u`. * `bsup`, `blsub`: the supremum / least strict upper bound of a set of ordinals indexed by ordinals less than a given ordinal `o`. Various other basic arithmetic results are given in `Principal.lean` instead. -/ 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} /-! ### Further properties of addition on ordinals -/ @[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 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] #align ordinal.add_right_cancel Ordinal.add_right_cancel theorem add_eq_zero_iff {a b : Ordinal} : a + b = 0 ↔ a = 0 ∧ b = 0 := inductionOn a fun α r _ => inductionOn b fun β s _ => by simp_rw [← type_sum_lex, type_eq_zero_iff_isEmpty] exact isEmpty_sum #align ordinal.add_eq_zero_iff Ordinal.add_eq_zero_iff theorem left_eq_zero_of_add_eq_zero {a b : Ordinal} (h : a + b = 0) : a = 0 := (add_eq_zero_iff.1 h).1 #align ordinal.left_eq_zero_of_add_eq_zero Ordinal.left_eq_zero_of_add_eq_zero theorem right_eq_zero_of_add_eq_zero {a b : Ordinal} (h : a + b = 0) : b = 0 := (add_eq_zero_iff.1 h).2 #align ordinal.right_eq_zero_of_add_eq_zero Ordinal.right_eq_zero_of_add_eq_zero /-! ### The predecessor of an ordinal -/ /-- The ordinal predecessor of `o` is `o'` if `o = succ o'`, and `o` otherwise. -/ def pred (o : Ordinal) : Ordinal := if h : ∃ a, o = succ a then Classical.choose h else o #align ordinal.pred Ordinal.pred @[simp] theorem pred_succ (o) : pred (succ o) = o := by have h : ∃ a, succ o = succ a := ⟨_, rfl⟩; simpa only [pred, dif_pos h] using (succ_injective <\| Classical.choose_spec h).symm #align ordinal.pred_succ Ordinal.pred_succ theorem pred_le_self (o) : pred o ≤ o := if h : ∃ a, o = succ a then by let ⟨a, e⟩ := h rw [e, pred_succ]; exact le_succ a else by rw [pred, dif_neg h] #align ordinal.pred_le_self Ordinal.pred_le_self theorem pred_eq_iff_not_succ {o} : pred o = o ↔ ¬∃ a, o = succ a := ⟨fun e ⟨a, e'⟩ => by rw [e', pred_succ] at e; exact (lt_succ a).ne e, fun h => dif_neg h⟩ #align ordinal.pred_eq_iff_not_succ Ordinal.pred_eq_iff_not_succ theorem pred_eq_iff_not_succ' {o} : pred o = o ↔ ∀ a, o ≠ succ a := by simpa using pred_eq_iff_not_succ #align ordinal.pred_eq_iff_not_succ' Ordinal.pred_eq_iff_not_succ' theorem pred_lt_iff_is_succ {o} : pred o < o ↔ ∃ a, o = succ a := Iff.trans (by simp only [le_antisymm_iff, pred_le_self, true_and_iff, not_le]) (iff_not_comm.1 pred_eq_iff_not_succ).symm #align ordinal.pred_lt_iff_is_succ Ordinal.pred_lt_iff_is_succ @[simp] theorem pred_zero : pred 0 = 0 := pred_eq_iff_not_succ'.2 fun a => (succ_ne_zero a).symm #align ordinal.pred_zero Ordinal.pred_zero theorem succ_pred_iff_is_succ {o} : succ (pred o) = o ↔ ∃ a, o = succ a := ⟨fun e => ⟨_, e.symm⟩, fun ⟨a, e⟩ => by simp only [e, pred_succ]⟩ #align ordinal.succ_pred_iff_is_succ Ordinal.succ_pred_iff_is_succ theorem succ_lt_of_not_succ {o b : Ordinal} (h : ¬∃ a, o = succ a) : succ b < o ↔ b < o := ⟨(lt_succ b).trans, fun l => lt_of_le_of_ne (succ_le_of_lt l) fun e => h ⟨_, e.symm⟩⟩ #align ordinal.succ_lt_of_not_succ Ordinal.succ_lt_of_not_succ theorem lt_pred {a b} : a < pred b ↔ succ a < b := if h : ∃ a, b = succ a then by let ⟨c, e⟩ := h rw [e, pred_succ, succ_lt_succ_iff] else by simp only [pred, dif_neg h, succ_lt_of_not_succ h] #align ordinal.lt_pred Ordinal.lt_pred theorem pred_le {a b} : pred a ≤ b ↔ a ≤ succ b := le_iff_le_iff_lt_iff_lt.2 lt_pred #align ordinal.pred_le Ordinal.pred_le @[simp] theorem lift_is_succ {o : Ordinal.{v}} : (∃ a, lift.{u} o = succ a) ↔ ∃ a, o = succ a := ⟨fun ⟨a, h⟩ => let ⟨b, e⟩ := lift_down <\| show a ≤ lift.{u} o from le_of_lt <\| h.symm ▸ lt_succ a ⟨b, lift_inj.1 <\| by rw [h, ← e, lift_succ]⟩, fun ⟨a, h⟩ => ⟨lift.{u} a, by simp only [h, lift_succ]⟩⟩ #align ordinal.lift_is_succ Ordinal.lift_is_succ @[simp] theorem lift_pred (o : Ordinal.{v}) : lift.{u} (pred o) = pred (lift.{u} o) := if h : ∃ a, o = succ a then by cases' h with a e; simp only [e, pred_succ, lift_succ] else by rw [pred_eq_iff_not_succ.2 h, pred_eq_iff_not_succ.2 (mt lift_is_succ.1 h)] #align ordinal.lift_pred Ordinal.lift_pred /-! ### Limit ordinals -/ /-- A limit ordinal is an ordinal which is not zero and not a successor. -/ def IsLimit (o : Ordinal) : Prop := o ≠ 0 ∧ ∀ a < o, succ a < o #align ordinal.is_limit Ordinal.IsLimit theorem IsLimit.isSuccLimit {o} (h : IsLimit o) : IsSuccLimit o := isSuccLimit_iff_succ_lt.mpr h.2 theorem IsLimit.succ_lt {o a : Ordinal} (h : IsLimit o) : a < o → succ a < o := h.2 a #align ordinal.is_limit.succ_lt Ordinal.IsLimit.succ_lt theorem isSuccLimit_zero : IsSuccLimit (0 : Ordinal) := isSuccLimit_bot theorem not_zero_isLimit : ¬IsLimit 0 \| ⟨h, _⟩ => h rfl #align ordinal.not_zero_is_limit Ordinal.not_zero_isLimit theorem not_succ_isLimit (o) : ¬IsLimit (succ o) \| ⟨_, h⟩ => lt_irrefl _ (h _ (lt_succ o)) #align ordinal.not_succ_is_limit Ordinal.not_succ_isLimit theorem not_succ_of_isLimit {o} (h : IsLimit o) : ¬∃ a, o = succ a \| ⟨a, e⟩ => not_succ_isLimit a (e ▸ h) #align ordinal.not_succ_of_is_limit Ordinal.not_succ_of_isLimit theorem succ_lt_of_isLimit {o a : Ordinal} (h : IsLimit o) : succ a < o ↔ a < o := ⟨(lt_succ a).trans, h.2 _⟩ #align ordinal.succ_lt_of_is_limit Ordinal.succ_lt_of_isLimit theorem le_succ_of_isLimit {o} (h : IsLimit o) {a} : o ≤ succ a ↔ o ≤ a := le_iff_le_iff_lt_iff_lt.2 <\| succ_lt_of_isLimit h #align ordinal.le_succ_of_is_limit Ordinal.le_succ_of_isLimit theorem limit_le {o} (h : IsLimit o) {a} : o ≤ a ↔ ∀ x < o, x ≤ a := ⟨fun h _x l => l.le.trans h, fun H => (le_succ_of_isLimit h).1 <\| le_of_not_lt fun hn => not_lt_of_le (H _ hn) (lt_succ a)⟩ #align ordinal.limit_le Ordinal.limit_le theorem lt_limit {o} (h : IsLimit o) {a} : a < o ↔ ∃ x < o, a < x := by -- Porting note: `bex_def` is required. simpa only [not_forall₂, not_le, bex_def] using not_congr (@limit_le _ h a) #align ordinal.lt_limit Ordinal.lt_limit @[simp] theorem lift_isLimit (o) : IsLimit (lift o) ↔ IsLimit o := and_congr (not_congr <\| by simpa only [lift_zero] using @lift_inj o 0) ⟨fun H a h => lift_lt.1 <\| by simpa only [lift_succ] using H _ (lift_lt.2 h), fun H a h => by obtain ⟨a', rfl⟩ := lift_down h.le rw [← lift_succ, lift_lt] exact H a' (lift_lt.1 h)⟩ #align ordinal.lift_is_limit Ordinal.lift_isLimit theorem IsLimit.pos {o : Ordinal} (h : IsLimit o) : 0 < o := lt_of_le_of_ne (Ordinal.zero_le _) h.1.symm #align ordinal.is_limit.pos Ordinal.IsLimit.pos theorem IsLimit.one_lt {o : Ordinal} (h : IsLimit o) : 1 < o := by simpa only [succ_zero] using h.2 _ h.pos #align ordinal.is_limit.one_lt Ordinal.IsLimit.one_lt theorem IsLimit.nat_lt {o : Ordinal} (h : IsLimit o) : ∀ n : ℕ, (n : Ordinal) < o \| 0 => h.pos \| n + 1 => h.2 _ (IsLimit.nat_lt h n) #align ordinal.is_limit.nat_lt Ordinal.IsLimit.nat_lt theorem zero_or_succ_or_limit (o : Ordinal) : o = 0 ∨ (∃ a, o = succ a) ∨ IsLimit o := if o0 : o = 0 then Or.inl o0 else if h : ∃ a, o = succ a then Or.inr (Or.inl h) else Or.inr <\| Or.inr ⟨o0, fun _a => (succ_lt_of_not_succ h).2⟩ #align ordinal.zero_or_succ_or_limit Ordinal.zero_or_succ_or_limit /-- Main induction principle of ordinals: if one can prove a property by induction at successor ordinals and at limit ordinals, then it holds for all ordinals. -/ @[elab_as_elim] def limitRecOn {C : Ordinal → Sort} (o : Ordinal) (H₁ : C 0) (H₂ : ∀ o, C o → C (succ o)) (H₃ : ∀ o, IsLimit o → (∀ o' < o, C o') → C o) : C o := SuccOrder.limitRecOn o (fun o _ ↦ H₂ o) fun o hl ↦ if h : o = 0 then fun _ ↦ h ▸ H₁ else H₃ o ⟨h, fun _ ↦ hl.succ_lt⟩ #align ordinal.limit_rec_on Ordinal.limitRecOn @[simp] theorem limitRecOn_zero {C} (H₁ H₂ H₃) : @limitRecOn C 0 H₁ H₂ H₃ = H₁ := by rw [limitRecOn, SuccOrder.limitRecOn_limit _ _ isSuccLimit_zero, dif_pos rfl] #align ordinal.limit_rec_on_zero Ordinal.limitRecOn_zero @[simp] theorem limitRecOn_succ {C} (o H₁ H₂ H₃) : @limitRecOn C (succ o) H₁ H₂ H₃ = H₂ o (@limitRecOn C o H₁ H₂ H₃) := by simp_rw [limitRecOn, SuccOrder.limitRecOn_succ _ _ (not_isMax _)] #align ordinal.limit_rec_on_succ Ordinal.limitRecOn_succ @[simp] theorem limitRecOn_limit {C} (o H₁ H₂ H₃ h) : @limitRecOn C o H₁ H₂ H₃ = H₃ o h fun x _h => @limitRecOn C x H₁ H₂ H₃ := by simp_rw [limitRecOn, SuccOrder.limitRecOn_limit _ _ h.isSuccLimit, dif_neg h.1] #align ordinal.limit_rec_on_limit Ordinal.limitRecOn_limit instance orderTopOutSucc (o : Ordinal) : OrderTop (succ o).out.α := @OrderTop.mk _ _ (Top.mk _) le_enum_succ #align ordinal.order_top_out_succ Ordinal.orderTopOutSucc theorem enum_succ_eq_top {o : Ordinal} : enum (· < ·) o (by rw [type_lt] exact lt_succ o) = (⊤ : (succ o).out.α) := rfl #align ordinal.enum_succ_eq_top Ordinal.enum_succ_eq_top theorem has_succ_of_type_succ_lt {α} {r : α → α → Prop} [wo : IsWellOrder α r] (h : ∀ a < type r, succ a < type r) (x : α) : ∃ y, r x y := by use enum r (succ (typein r x)) (h _ (typein_lt_type r x)) convert (enum_lt_enum (typein_lt_type r x) (h _ (typein_lt_type r x))).mpr (lt_succ _); rw [enum_typein] #align ordinal.has_succ_of_type_succ_lt Ordinal.has_succ_of_type_succ_lt theorem out_no_max_of_succ_lt {o : Ordinal} (ho : ∀ a < o, succ a < o) : NoMaxOrder o.out.α := ⟨has_succ_of_type_succ_lt (by rwa [type_lt])⟩ #align ordinal.out_no_max_of_succ_lt Ordinal.out_no_max_of_succ_lt theorem bounded_singleton {r : α → α → Prop} [IsWellOrder α r] (hr : (type r).IsLimit) (x) : Bounded r {x} := by refine ⟨enum r (succ (typein r x)) (hr.2 _ (typein_lt_type r x)), ?_⟩ intro b hb rw [mem_singleton_iff.1 hb] nth_rw 1 [← enum_typein r x] rw [@enum_lt_enum _ r] apply lt_succ #align ordinal.bounded_singleton Ordinal.bounded_singleton -- Porting note: `· < ·` requires a type ascription for an `IsWellOrder` instance. theorem type_subrel_lt (o : Ordinal.{u}) : type (Subrel ((· < ·) : Ordinal → Ordinal → Prop) { o' : Ordinal \| o' < o }) = Ordinal.lift.{u + 1} o := by refine Quotient.inductionOn o ?_ rintro ⟨α, r, wo⟩; apply Quotient.sound -- Porting note: `symm; refine' [term]` → `refine' [term].symm` constructor; refine ((RelIso.preimage Equiv.ulift r).trans (enumIso r).symm).symm #align ordinal.type_subrel_lt Ordinal.type_subrel_lt theorem mk_initialSeg (o : Ordinal.{u}) : #{ o' : Ordinal \| o' < o } = Cardinal.lift.{u + 1} o.card := by rw [lift_card, ← type_subrel_lt, card_type] #align ordinal.mk_initial_seg Ordinal.mk_initialSeg /-! ### Normal ordinal functions -/ /-- A normal ordinal function is a strictly increasing function which is order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for `a < o`. -/ def IsNormal (f : Ordinal → Ordinal) : Prop := (∀ o, f o < f (succ o)) ∧ ∀ o, IsLimit o → ∀ a, f o ≤ a ↔ ∀ b < o, f b ≤ a #align ordinal.is_normal Ordinal.IsNormal theorem IsNormal.limit_le {f} (H : IsNormal f) : ∀ {o}, IsLimit o → ∀ {a}, f o ≤ a ↔ ∀ b < o, f b ≤ a := @H.2 #align ordinal.is_normal.limit_le Ordinal.IsNormal.limit_le theorem IsNormal.limit_lt {f} (H : IsNormal f) {o} (h : IsLimit o) {a} : a < f o ↔ ∃ b < o, a < f b := not_iff_not.1 <\| by simpa only [exists_prop, not_exists, not_and, not_lt] using H.2 _ h a #align ordinal.is_normal.limit_lt Ordinal.IsNormal.limit_lt theorem IsNormal.strictMono {f} (H : IsNormal f) : StrictMono f := fun a b => limitRecOn b (Not.elim (not_lt_of_le <\| Ordinal.zero_le _)) (fun _b IH h => (lt_or_eq_of_le (le_of_lt_succ h)).elim (fun h => (IH h).trans (H.1 _)) fun e => e ▸ H.1 _) fun _b l _IH h => lt_of_lt_of_le (H.1 a) ((H.2 _ l _).1 le_rfl _ (l.2 _ h)) #align ordinal.is_normal.strict_mono Ordinal.IsNormal.strictMono theorem IsNormal.monotone {f} (H : IsNormal f) : Monotone f := H.strictMono.monotone #align ordinal.is_normal.monotone Ordinal.IsNormal.monotone theorem isNormal_iff_strictMono_limit (f : Ordinal → Ordinal) : IsNormal f ↔ StrictMono f ∧ ∀ o, IsLimit o → ∀ a, (∀ b < o, f b ≤ a) → f o ≤ a := ⟨fun hf => ⟨hf.strictMono, fun a ha c => (hf.2 a ha c).2⟩, fun ⟨hs, hl⟩ => ⟨fun a => hs (lt_succ a), fun a ha c => ⟨fun hac _b hba => ((hs hba).trans_le hac).le, hl a ha c⟩⟩⟩ #align ordinal.is_normal_iff_strict_mono_limit Ordinal.isNormal_iff_strictMono_limit theorem IsNormal.lt_iff {f} (H : IsNormal f) {a b} : f a < f b ↔ a < b := StrictMono.lt_iff_lt <\| H.strictMono #align ordinal.is_normal.lt_iff Ordinal.IsNormal.lt_iff theorem IsNormal.le_iff {f} (H : IsNormal f) {a b} : f a ≤ f b ↔ a ≤ b := le_iff_le_iff_lt_iff_lt.2 H.lt_iff #align ordinal.is_normal.le_iff Ordinal.IsNormal.le_iff theorem IsNormal.inj {f} (H : IsNormal f) {a b} : f a = f b ↔ a = b := by simp only [le_antisymm_iff, H.le_iff] #align ordinal.is_normal.inj Ordinal.IsNormal.inj theorem IsNormal.self_le {f} (H : IsNormal f) (a) : a ≤ f a := lt_wf.self_le_of_strictMono H.strictMono a #align ordinal.is_normal.self_le Ordinal.IsNormal.self_le theorem IsNormal.le_set {f o} (H : IsNormal f) (p : Set Ordinal) (p0 : p.Nonempty) (b) (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f a ≤ o := ⟨fun h a pa => (H.le_iff.2 ((H₂ _).1 le_rfl _ pa)).trans h, fun h => by -- Porting note: `refine'` didn't work well so `induction` is used induction b using limitRecOn with \| H₁ => cases' p0 with x px have := Ordinal.le_zero.1 ((H₂ _).1 (Ordinal.zero_le _) _ px) rw [this] at px exact h _ px \| H₂ S _ => rcases not_forall₂.1 (mt (H₂ S).2 <\| (lt_succ S).not_le) with ⟨a, h₁, h₂⟩ exact (H.le_iff.2 <\| succ_le_of_lt <\| not_le.1 h₂).trans (h _ h₁) \| H₃ S L _ => refine (H.2 _ L _).2 fun a h' => ?_ rcases not_forall₂.1 (mt (H₂ a).2 h'.not_le) with ⟨b, h₁, h₂⟩ exact (H.le_iff.2 <\| (not_le.1 h₂).le).trans (h _ h₁)⟩ #align ordinal.is_normal.le_set Ordinal.IsNormal.le_set theorem IsNormal.le_set' {f o} (H : IsNormal f) (p : Set α) (p0 : p.Nonempty) (g : α → Ordinal) (b) (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, g a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o := by simpa [H₂] using H.le_set (g '' p) (p0.image g) b #align ordinal.is_normal.le_set' Ordinal.IsNormal.le_set' theorem IsNormal.refl : IsNormal id := ⟨lt_succ, fun _o l _a => Ordinal.limit_le l⟩ #align ordinal.is_normal.refl Ordinal.IsNormal.refl theorem IsNormal.trans {f g} (H₁ : IsNormal f) (H₂ : IsNormal g) : IsNormal (f ∘ g) := ⟨fun _x => H₁.lt_iff.2 (H₂.1 _), fun o l _a => H₁.le_set' (· < o) ⟨0, l.pos⟩ g _ fun _c => H₂.2 _ l _⟩ #align ordinal.is_normal.trans Ordinal.IsNormal.trans theorem IsNormal.isLimit {f} (H : IsNormal f) {o} (l : IsLimit o) : IsLimit (f o) := ⟨ne_of_gt <\| (Ordinal.zero_le _).trans_lt <\| H.lt_iff.2 l.pos, fun _ h => let ⟨_b, h₁, h₂⟩ := (H.limit_lt l).1 h (succ_le_of_lt h₂).trans_lt (H.lt_iff.2 h₁)⟩ #align ordinal.is_normal.is_limit Ordinal.IsNormal.isLimit theorem IsNormal.le_iff_eq {f} (H : IsNormal f) {a} : f a ≤ a ↔ f a = a := (H.self_le a).le_iff_eq #align ordinal.is_normal.le_iff_eq Ordinal.IsNormal.le_iff_eq theorem add_le_of_limit {a b c : Ordinal} (h : IsLimit b) : a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c := ⟨fun h b' l => (add_le_add_left l.le _).trans h, fun H => le_of_not_lt <\| by -- Porting note: `induction` tactics are required because of the parser bug. induction a using inductionOn with \| H α r => induction b using inductionOn with \| H β s => intro l suffices ∀ x : β, Sum.Lex r s (Sum.inr x) (enum _ _ l) by -- Porting note: `revert` & `intro` is required because `cases'` doesn't replace -- `enum _ _ l` in `this`. revert this; cases' enum _ _ l with x x <;> intro this · cases this (enum s 0 h.pos) · exact irrefl _ (this _) intro x rw [← typein_lt_typein (Sum.Lex r s), typein_enum] have := H _ (h.2 _ (typein_lt_type s x)) rw [add_succ, succ_le_iff] at this refine (RelEmbedding.ofMonotone (fun a => ?_) fun a b => ?_).ordinal_type_le.trans_lt this · rcases a with ⟨a \| b, h⟩ · exact Sum.inl a · exact Sum.inr ⟨b, by cases h; assumption⟩ · rcases a with ⟨a \| a, h₁⟩ <;> rcases b with ⟨b \| b, h₂⟩ <;> cases h₁ <;> cases h₂ <;> rintro ⟨⟩ <;> constructor <;> assumption⟩ #align ordinal.add_le_of_limit Ordinal.add_le_of_limit theorem add_isNormal (a : Ordinal) : IsNormal (a + ·) := ⟨fun b => (add_lt_add_iff_left a).2 (lt_succ b), fun _b l _c => add_le_of_limit l⟩ #align ordinal.add_is_normal Ordinal.add_isNormal theorem add_isLimit (a) {b} : IsLimit b → IsLimit (a + b) := (add_isNormal a).isLimit #align ordinal.add_is_limit Ordinal.add_isLimit alias IsLimit.add := add_isLimit #align ordinal.is_limit.add Ordinal.IsLimit.add /-! ### Subtraction on ordinals-/ /-- The set in the definition of subtraction is nonempty. -/ theorem sub_nonempty {a b : Ordinal} : { o \| a ≤ b + o }.Nonempty := ⟨a, le_add_left _ _⟩ #align ordinal.sub_nonempty Ordinal.sub_nonempty /-- `a - b` is the unique ordinal satisfying `b + (a - b) = a` when `b ≤ a`. -/ instance sub : Sub Ordinal := ⟨fun a b => sInf { o \| a ≤ b + o }⟩ theorem le_add_sub (a b : Ordinal) : a ≤ b + (a - b) := csInf_mem sub_nonempty #align ordinal.le_add_sub Ordinal.le_add_sub theorem sub_le {a b c : Ordinal} : a - b ≤ c ↔ a ≤ b + c := ⟨fun h => (le_add_sub a b).trans (add_le_add_left h _), fun h => csInf_le' h⟩ #align ordinal.sub_le Ordinal.sub_le theorem lt_sub {a b c : Ordinal} : a < b - c ↔ c + a < b := lt_iff_lt_of_le_iff_le sub_le #align ordinal.lt_sub Ordinal.lt_sub theorem add_sub_cancel (a b : Ordinal) : a + b - a = b := le_antisymm (sub_le.2 <\| le_rfl) ((add_le_add_iff_left a).1 <\| le_add_sub _ _) #align ordinal.add_sub_cancel Ordinal.add_sub_cancel theorem sub_eq_of_add_eq {a b c : Ordinal} (h : a + b = c) : c - a = b := h ▸ add_sub_cancel _ _ #align ordinal.sub_eq_of_add_eq Ordinal.sub_eq_of_add_eq theorem sub_le_self (a b : Ordinal) : a - b ≤ a := sub_le.2 <\| le_add_left _ _ #align ordinal.sub_le_self Ordinal.sub_le_self protected theorem add_sub_cancel_of_le {a b : Ordinal} (h : b ≤ a) : b + (a - b) = a := (le_add_sub a b).antisymm' (by rcases zero_or_succ_or_limit (a - b) with (e \| ⟨c, e⟩ \| l) · simp only [e, add_zero, h] · rw [e, add_succ, succ_le_iff, ← lt_sub, e] exact lt_succ c · exact (add_le_of_limit l).2 fun c l => (lt_sub.1 l).le) #align ordinal.add_sub_cancel_of_le Ordinal.add_sub_cancel_of_le theorem le_sub_of_le {a b c : Ordinal} (h : b ≤ a) : c ≤ a - b ↔ b + c ≤ a := by rw [← add_le_add_iff_left b, Ordinal.add_sub_cancel_of_le h] #align ordinal.le_sub_of_le Ordinal.le_sub_of_le theorem sub_lt_of_le {a b c : Ordinal} (h : b ≤ a) : a - b < c ↔ a < b + c := lt_iff_lt_of_le_iff_le (le_sub_of_le h) #align ordinal.sub_lt_of_le Ordinal.sub_lt_of_le instance existsAddOfLE : ExistsAddOfLE Ordinal := ⟨fun h => ⟨_, (Ordinal.add_sub_cancel_of_le h).symm⟩⟩ @[simp] theorem sub_zero (a : Ordinal) : a - 0 = a := by simpa only [zero_add] using add_sub_cancel 0 a #align ordinal.sub_zero Ordinal.sub_zero @[simp] theorem zero_sub (a : Ordinal) : 0 - a = 0 := by rw [← Ordinal.le_zero]; apply sub_le_self #align ordinal.zero_sub Ordinal.zero_sub @[simp] theorem sub_self (a : Ordinal) : a - a = 0 := by simpa only [add_zero] using add_sub_cancel a 0 #align ordinal.sub_self Ordinal.sub_self protected theorem sub_eq_zero_iff_le {a b : Ordinal} : a - b = 0 ↔ a ≤ b := ⟨fun h => by simpa only [h, add_zero] using le_add_sub a b, fun h => by rwa [← Ordinal.le_zero, sub_le, add_zero]⟩ #align ordinal.sub_eq_zero_iff_le Ordinal.sub_eq_zero_iff_le theorem sub_sub (a b c : Ordinal) : a - b - c = a - (b + c) := eq_of_forall_ge_iff fun d => by rw [sub_le, sub_le, sub_le, add_assoc] #align ordinal.sub_sub Ordinal.sub_sub @[simp] theorem add_sub_add_cancel (a b c : Ordinal) : a + b - (a + c) = b - c := by rw [← sub_sub, add_sub_cancel] #align ordinal.add_sub_add_cancel Ordinal.add_sub_add_cancel theorem sub_isLimit {a b} (l : IsLimit a) (h : b < a) : IsLimit (a - b) := ⟨ne_of_gt <\| lt_sub.2 <\| by rwa [add_zero], fun c h => by rw [lt_sub, add_succ]; exact l.2 _ (lt_sub.1 h)⟩ #align ordinal.sub_is_limit Ordinal.sub_isLimit -- @[simp] -- Porting note (#10618): simp can prove this theorem one_add_omega : 1 + ω = ω := by refine le_antisymm ?_ (le_add_left _ _) rw [omega, ← lift_one.{_, 0}, ← lift_add, lift_le, ← type_unit, ← type_sum_lex] refine ⟨RelEmbedding.collapse (RelEmbedding.ofMonotone ?_ ?_)⟩ · apply Sum.rec · exact fun _ => 0 · exact Nat.succ · intro a b cases a <;> cases b <;> intro H <;> cases' H with _ _ H _ _ H <;> [exact H.elim; exact Nat.succ_pos _; exact Nat.succ_lt_succ H] #align ordinal.one_add_omega Ordinal.one_add_omega @[simp] theorem one_add_of_omega_le {o} (h : ω ≤ o) : 1 + o = o := by rw [← Ordinal.add_sub_cancel_of_le h, ← add_assoc, one_add_omega] #align ordinal.one_add_of_omega_le Ordinal.one_add_of_omega_le /-! ### Multiplication of ordinals-/ /-- The multiplication of ordinals `o₁` and `o₂` is the (well founded) lexicographic order on `o₂ × o₁`. -/ instance monoid : Monoid Ordinal.{u} where mul a b := Quotient.liftOn₂ a b (fun ⟨α, r, wo⟩ ⟨β, s, wo'⟩ => ⟦⟨β × α, Prod.Lex s r, inferInstance⟩⟧ : WellOrder → WellOrder → Ordinal) fun ⟨α₁, r₁, o₁⟩ ⟨α₂, r₂, o₂⟩ ⟨β₁, s₁, p₁⟩ ⟨β₂, s₂, p₂⟩ ⟨f⟩ ⟨g⟩ => Quot.sound ⟨RelIso.prodLexCongr g f⟩ one := 1 mul_assoc a b c := Quotient.inductionOn₃ a b c fun ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ => Eq.symm <\| Quotient.sound ⟨⟨prodAssoc _ _ _, @fun a b => by rcases a with ⟨⟨a₁, a₂⟩, a₃⟩ rcases b with ⟨⟨b₁, b₂⟩, b₃⟩ simp [Prod.lex_def, and_or_left, or_assoc, and_assoc]⟩⟩ mul_one a := inductionOn a fun α r _ => Quotient.sound ⟨⟨punitProd _, @fun a b => by rcases a with ⟨⟨⟨⟩⟩, a⟩; rcases b with ⟨⟨⟨⟩⟩, b⟩ simp only [Prod.lex_def, EmptyRelation, false_or_iff] simp only [eq_self_iff_true, true_and_iff] rfl⟩⟩ one_mul a := inductionOn a fun α r _ => Quotient.sound ⟨⟨prodPUnit _, @fun a b => by rcases a with ⟨a, ⟨⟨⟩⟩⟩; rcases b with ⟨b, ⟨⟨⟩⟩⟩ simp only [Prod.lex_def, EmptyRelation, and_false_iff, or_false_iff] rfl⟩⟩ @[simp] theorem type_prod_lex {α β : Type u} (r : α → α → Prop) (s : β → β → Prop) [IsWellOrder α r] [IsWellOrder β s] : type (Prod.Lex s r) = type r * type s := rfl #align ordinal.type_prod_lex Ordinal.type_prod_lex private theorem mul_eq_zero' {a b : Ordinal} : a * b = 0 ↔ a = 0 ∨ b = 0 := inductionOn a fun α _ _ => inductionOn b fun β _ _ => by simp_rw [← type_prod_lex, type_eq_zero_iff_isEmpty] rw [or_comm] exact isEmpty_prod instance monoidWithZero : MonoidWithZero Ordinal := { Ordinal.monoid with zero := 0 mul_zero := fun _a => mul_eq_zero'.2 <\| Or.inr rfl zero_mul := fun _a => mul_eq_zero'.2 <\| Or.inl rfl } instance noZeroDivisors : NoZeroDivisors Ordinal := ⟨fun {_ _} => mul_eq_zero'.1⟩ @[simp] theorem lift_mul (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.prodLexCongr (RelIso.preimage Equiv.ulift _) (RelIso.preimage Equiv.ulift _)).symm⟩ #align ordinal.lift_mul Ordinal.lift_mul @[simp] theorem card_mul (a b) : card (a * b) = card a * card b := Quotient.inductionOn₂ a b fun ⟨α, _r, _⟩ ⟨β, _s, _⟩ => mul_comm #β #α #align ordinal.card_mul Ordinal.card_mul instance leftDistribClass : LeftDistribClass Ordinal.{u} := ⟨fun a b c => Quotient.inductionOn₃ a b c fun ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ => Quotient.sound ⟨⟨sumProdDistrib _ _ _, by rintro ⟨a₁ \| a₁, a₂⟩ ⟨b₁ \| b₁, b₂⟩ <;> simp only [Prod.lex_def, Sum.lex_inl_inl, Sum.Lex.sep, Sum.lex_inr_inl, Sum.lex_inr_inr, sumProdDistrib_apply_left, sumProdDistrib_apply_right] <;> -- Porting note: `Sum.inr.inj_iff` is required. simp only [Sum.inl.inj_iff, Sum.inr.inj_iff, true_or_iff, false_and_iff, false_or_iff]⟩⟩⟩ theorem mul_succ (a b : Ordinal) : a * succ b = a * b + a := mul_add_one a b #align ordinal.mul_succ Ordinal.mul_succ instance mul_covariantClass_le : CovariantClass Ordinal.{u} Ordinal.{u} (· * ·) (· ≤ ·) := ⟨fun c a b => Quotient.inductionOn₃ a b c fun ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩ => by refine (RelEmbedding.ofMonotone (fun a : α × γ => (f a.1, a.2)) fun a b h => ?_).ordinal_type_le cases' h with a₁ b₁ a₂ b₂ h' a b₁ b₂ h' · exact Prod.Lex.left _ _ (f.toRelEmbedding.map_rel_iff.2 h') · exact Prod.Lex.right _ h'⟩ #align ordinal.mul_covariant_class_le Ordinal.mul_covariantClass_le instance mul_swap_covariantClass_le : CovariantClass Ordinal.{u} Ordinal.{u} (swap (· * ·)) (· ≤ ·) := ⟨fun c a b => Quotient.inductionOn₃ a b c fun ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩ => by refine (RelEmbedding.ofMonotone (fun a : γ × α => (a.1, f a.2)) fun a b h => ?_).ordinal_type_le cases' h with a₁ b₁ a₂ b₂ h' a b₁ b₂ h' · exact Prod.Lex.left _ _ h' · exact Prod.Lex.right _ (f.toRelEmbedding.map_rel_iff.2 h')⟩ #align ordinal.mul_swap_covariant_class_le Ordinal.mul_swap_covariantClass_le theorem le_mul_left (a : Ordinal) {b : Ordinal} (hb : 0 < b) : a ≤ a * b := by convert mul_le_mul_left' (one_le_iff_pos.2 hb) a rw [mul_one a] #align ordinal.le_mul_left Ordinal.le_mul_left theorem le_mul_right (a : Ordinal) {b : Ordinal} (hb : 0 < b) : a ≤ b * a := by convert mul_le_mul_right' (one_le_iff_pos.2 hb) a rw [one_mul a] #align ordinal.le_mul_right Ordinal.le_mul_right private theorem mul_le_of_limit_aux {α β r s} [IsWellOrder α r] [IsWellOrder β s] {c} (h : IsLimit (type s)) (H : ∀ b' < type s, type r * b' ≤ c) (l : c < type r * type s) : False := by suffices ∀ a b, Prod.Lex s r (b, a) (enum _ _ l) by cases' enum _ _ l with b a exact irrefl _ (this _ _) intro a b rw [← typein_lt_typein (Prod.Lex s r), typein_enum] have := H _ (h.2 _ (typein_lt_type s b)) rw [mul_succ] at this have := ((add_lt_add_iff_left _).2 (typein_lt_type _ a)).trans_le this refine (RelEmbedding.ofMonotone (fun a => ?_) fun a b => ?_).ordinal_type_le.trans_lt this · rcases a with ⟨⟨b', a'⟩, h⟩ by_cases e : b = b' · refine Sum.inr ⟨a', ?_⟩ subst e cases' h with _ _ _ _ h _ _ _ h · exact (irrefl _ h).elim · exact h · refine Sum.inl (⟨b', ?_⟩, a') cases' h with _ _ _ _ h _ _ _ h · exact h · exact (e rfl).elim · rcases a with ⟨⟨b₁, a₁⟩, h₁⟩ rcases b with ⟨⟨b₂, a₂⟩, h₂⟩ intro h by_cases e₁ : b = b₁ <;> by_cases e₂ : b = b₂ · substs b₁ b₂ simpa only [subrel_val, Prod.lex_def, @irrefl _ s _ b, true_and_iff, false_or_iff, eq_self_iff_true, dif_pos, Sum.lex_inr_inr] using h · subst b₁ simp only [subrel_val, Prod.lex_def, e₂, Prod.lex_def, dif_pos, subrel_val, eq_self_iff_true, or_false_iff, dif_neg, not_false_iff, Sum.lex_inr_inl, false_and_iff] at h ⊢ cases' h₂ with _ _ _ _ h₂_h h₂_h <;> [exact asymm h h₂_h; exact e₂ rfl] -- Porting note: `cc` hadn't ported yet. · simp [e₂, dif_neg e₁, show b₂ ≠ b₁ from e₂ ▸ e₁] · simpa only [dif_neg e₁, dif_neg e₂, Prod.lex_def, subrel_val, Subtype.mk_eq_mk, Sum.lex_inl_inl] using h theorem mul_le_of_limit {a b c : Ordinal} (h : IsLimit b) : a * b ≤ c ↔ ∀ b' < b, a * b' ≤ c := ⟨fun h b' l => (mul_le_mul_left' l.le _).trans h, fun H => -- Porting note: `induction` tactics are required because of the parser bug. le_of_not_lt <\| by induction a using inductionOn with \| H α r => induction b using inductionOn with \| H β s => exact mul_le_of_limit_aux h H⟩ #align ordinal.mul_le_of_limit Ordinal.mul_le_of_limit theorem mul_isNormal {a : Ordinal} (h : 0 < a) : IsNormal (a * ·) := -- Porting note(#12129): additional beta reduction needed ⟨fun b => by beta_reduce rw [mul_succ] simpa only [add_zero] using (add_lt_add_iff_left (a * b)).2 h, fun b l c => mul_le_of_limit l⟩ #align ordinal.mul_is_normal Ordinal.mul_isNormal theorem lt_mul_of_limit {a b c : Ordinal} (h : IsLimit c) : a < b * c ↔ ∃ c' < c, a < b * c' := by -- Porting note: `bex_def` is required. simpa only [not_forall₂, not_le, bex_def] using not_congr (@mul_le_of_limit b c a h) #align ordinal.lt_mul_of_limit Ordinal.lt_mul_of_limit theorem mul_lt_mul_iff_left {a b c : Ordinal} (a0 : 0 < a) : a * b < a * c ↔ b < c := (mul_isNormal a0).lt_iff #align ordinal.mul_lt_mul_iff_left Ordinal.mul_lt_mul_iff_left theorem mul_le_mul_iff_left {a b c : Ordinal} (a0 : 0 < a) : a * b ≤ a * c ↔ b ≤ c := (mul_isNormal a0).le_iff #align ordinal.mul_le_mul_iff_left Ordinal.mul_le_mul_iff_left theorem mul_lt_mul_of_pos_left {a b c : Ordinal} (h : a < b) (c0 : 0 < c) : c * a < c * b := (mul_lt_mul_iff_left c0).2 h #align ordinal.mul_lt_mul_of_pos_left Ordinal.mul_lt_mul_of_pos_left theorem mul_pos {a b : Ordinal} (h₁ : 0 < a) (h₂ : 0 < b) : 0 < a * b := by simpa only [mul_zero] using mul_lt_mul_of_pos_left h₂ h₁ #align ordinal.mul_pos Ordinal.mul_pos theorem mul_ne_zero {a b : Ordinal} : a ≠ 0 → b ≠ 0 → a * b ≠ 0 := by simpa only [Ordinal.pos_iff_ne_zero] using mul_pos #align ordinal.mul_ne_zero Ordinal.mul_ne_zero theorem le_of_mul_le_mul_left {a b c : Ordinal} (h : c * a ≤ c * b) (h0 : 0 < c) : a ≤ b := le_imp_le_of_lt_imp_lt (fun h' => mul_lt_mul_of_pos_left h' h0) h #align ordinal.le_of_mul_le_mul_left Ordinal.le_of_mul_le_mul_left theorem mul_right_inj {a b c : Ordinal} (a0 : 0 < a) : a * b = a * c ↔ b = c := (mul_isNormal a0).inj #align ordinal.mul_right_inj Ordinal.mul_right_inj theorem mul_isLimit {a b : Ordinal} (a0 : 0 < a) : IsLimit b → IsLimit (a * b) := (mul_isNormal a0).isLimit #align ordinal.mul_is_limit Ordinal.mul_isLimit theorem mul_isLimit_left {a b : Ordinal} (l : IsLimit a) (b0 : 0 < b) : IsLimit (a * b) := by rcases zero_or_succ_or_limit b with (rfl \| ⟨b, rfl⟩ \| lb) · exact b0.false.elim · rw [mul_succ] exact add_isLimit _ l · exact mul_isLimit l.pos lb #align ordinal.mul_is_limit_left Ordinal.mul_isLimit_left theorem smul_eq_mul : ∀ (n : ℕ) (a : Ordinal), n • a = a * n \| 0, a => by rw [zero_nsmul, Nat.cast_zero, mul_zero] \| n + 1, a => by rw [succ_nsmul, Nat.cast_add, mul_add, Nat.cast_one, mul_one, smul_eq_mul n] #align ordinal.smul_eq_mul Ordinal.smul_eq_mul /-! ### Division on ordinals -/ /-- The set in the definition of division is nonempty. -/ theorem div_nonempty {a b : Ordinal} (h : b ≠ 0) : { o \| a < b * succ o }.Nonempty := ⟨a, (succ_le_iff (a := a) (b := b * succ a)).1 <\| by simpa only [succ_zero, one_mul] using mul_le_mul_right' (succ_le_of_lt (Ordinal.pos_iff_ne_zero.2 h)) (succ a)⟩ #align ordinal.div_nonempty Ordinal.div_nonempty /-- `a / b` is the unique ordinal `o` satisfying `a = b * o + o'` with `o' < b`. -/ instance div : Div Ordinal := ⟨fun a b => if _h : b = 0 then 0 else sInf { o \| a < b * succ o }⟩ @[simp] theorem div_zero (a : Ordinal) : a / 0 = 0 := dif_pos rfl #align ordinal.div_zero Ordinal.div_zero theorem div_def (a) {b : Ordinal} (h : b ≠ 0) : a / b = sInf { o \| a < b * succ o } := dif_neg h #align ordinal.div_def Ordinal.div_def theorem lt_mul_succ_div (a) {b : Ordinal} (h : b ≠ 0) : a < b * succ (a / b) := by rw [div_def a h]; exact csInf_mem (div_nonempty h) #align ordinal.lt_mul_succ_div Ordinal.lt_mul_succ_div theorem lt_mul_div_add (a) {b : Ordinal} (h : b ≠ 0) : a < b * (a / b) + b := by simpa only [mul_succ] using lt_mul_succ_div a h #align ordinal.lt_mul_div_add Ordinal.lt_mul_div_add theorem div_le {a b c : Ordinal} (b0 : b ≠ 0) : a / b ≤ c ↔ a < b * succ c := ⟨fun h => (lt_mul_succ_div a b0).trans_le (mul_le_mul_left' (succ_le_succ_iff.2 h) _), fun h => by rw [div_def a b0]; exact csInf_le' h⟩ #align ordinal.div_le Ordinal.div_le theorem lt_div {a b c : Ordinal} (h : c ≠ 0) : a < b / c ↔ c * succ a ≤ b := by rw [← not_le, div_le h, not_lt] #align ordinal.lt_div Ordinal.lt_div theorem div_pos {b c : Ordinal} (h : c ≠ 0) : 0 < b / c ↔ c ≤ b := by simp [lt_div h] #align ordinal.div_pos Ordinal.div_pos theorem le_div {a b c : Ordinal} (c0 : c ≠ 0) : a ≤ b / c ↔ c * a ≤ b := by induction a using limitRecOn with \| H₁ => simp only [mul_zero, Ordinal.zero_le] \| H₂ _ _ => rw [succ_le_iff, lt_div c0] \| H₃ _ h₁ h₂ => revert h₁ h₂ simp (config := { contextual := true }) only [mul_le_of_limit, limit_le, iff_self_iff, forall_true_iff] #align ordinal.le_div Ordinal.le_div theorem div_lt {a b c : Ordinal} (b0 : b ≠ 0) : a / b < c ↔ a < b * c := lt_iff_lt_of_le_iff_le <\| le_div b0 #align ordinal.div_lt Ordinal.div_lt theorem div_le_of_le_mul {a b c : Ordinal} (h : a ≤ b * c) : a / b ≤ c := if b0 : b = 0 then by simp only [b0, div_zero, Ordinal.zero_le] else (div_le b0).2 <\| h.trans_lt <\| mul_lt_mul_of_pos_left (lt_succ c) (Ordinal.pos_iff_ne_zero.2 b0) #align ordinal.div_le_of_le_mul Ordinal.div_le_of_le_mul theorem mul_lt_of_lt_div {a b c : Ordinal} : a < b / c → c * a < b := lt_imp_lt_of_le_imp_le div_le_of_le_mul #align ordinal.mul_lt_of_lt_div Ordinal.mul_lt_of_lt_div @[simp] theorem zero_div (a : Ordinal) : 0 / a = 0 := Ordinal.le_zero.1 <\| div_le_of_le_mul <\| Ordinal.zero_le _ #align ordinal.zero_div Ordinal.zero_div theorem mul_div_le (a b : Ordinal) : b * (a / b) ≤ a := if b0 : b = 0 then by simp only [b0, zero_mul, Ordinal.zero_le] else (le_div b0).1 le_rfl #align ordinal.mul_div_le Ordinal.mul_div_le theorem mul_add_div (a) {b : Ordinal} (b0 : b ≠ 0) (c) : (b * a + c) / b = a + c / b := by apply le_antisymm · apply (div_le b0).2 rw [mul_succ, mul_add, add_assoc, add_lt_add_iff_left] apply lt_mul_div_add _ b0 · rw [le_div b0, mul_add, add_le_add_iff_left] apply mul_div_le #align ordinal.mul_add_div Ordinal.mul_add_div theorem div_eq_zero_of_lt {a b : Ordinal} (h : a < b) : a / b = 0 := by rw [← Ordinal.le_zero, div_le <\| Ordinal.pos_iff_ne_zero.1 <\| (Ordinal.zero_le _).trans_lt h] simpa only [succ_zero, mul_one] using h #align ordinal.div_eq_zero_of_lt Ordinal.div_eq_zero_of_lt @[simp] theorem mul_div_cancel (a) {b : Ordinal} (b0 : b ≠ 0) : b * a / b = a := by simpa only [add_zero, zero_div] using mul_add_div a b0 0 #align ordinal.mul_div_cancel Ordinal.mul_div_cancel @[simp] theorem div_one (a : Ordinal) : a / 1 = a := by simpa only [one_mul] using mul_div_cancel a Ordinal.one_ne_zero #align ordinal.div_one Ordinal.div_one @[simp] theorem div_self {a : Ordinal} (h : a ≠ 0) : a / a = 1 := by simpa only [mul_one] using mul_div_cancel 1 h #align ordinal.div_self Ordinal.div_self theorem mul_sub (a b c : Ordinal) : a * (b - c) = a * b - a * c := if a0 : a = 0 then by simp only [a0, zero_mul, sub_self] else eq_of_forall_ge_iff fun d => by rw [sub_le, ← le_div a0, sub_le, ← le_div a0, mul_add_div _ a0] #align ordinal.mul_sub Ordinal.mul_sub theorem isLimit_add_iff {a b} : IsLimit (a + b) ↔ IsLimit b ∨ b = 0 ∧ IsLimit a := by constructor <;> intro h · by_cases h' : b = 0 · rw [h', add_zero] at h right exact ⟨h', h⟩ left rw [← add_sub_cancel a b] apply sub_isLimit h suffices a + 0 < a + b by simpa only [add_zero] using this rwa [add_lt_add_iff_left, Ordinal.pos_iff_ne_zero] rcases h with (h \| ⟨rfl, h⟩) · exact add_isLimit a h · simpa only [add_zero] #align ordinal.is_limit_add_iff Ordinal.isLimit_add_iff theorem dvd_add_iff : ∀ {a b c : Ordinal}, a ∣ b → (a ∣ b + c ↔ a ∣ c) \| a, _, c, ⟨b, rfl⟩ => ⟨fun ⟨d, e⟩ => ⟨d - b, by rw [mul_sub, ← e, add_sub_cancel]⟩, fun ⟨d, e⟩ => by rw [e, ← mul_add] apply dvd_mul_right⟩ #align ordinal.dvd_add_iff Ordinal.dvd_add_iff theorem div_mul_cancel : ∀ {a b : Ordinal}, a ≠ 0 → a ∣ b → a * (b / a) = b \| a, _, a0, ⟨b, rfl⟩ => by rw [mul_div_cancel _ a0] #align ordinal.div_mul_cancel Ordinal.div_mul_cancel theorem le_of_dvd : ∀ {a b : Ordinal}, b ≠ 0 → a ∣ b → a ≤ b -- Porting note: `⟨b, rfl⟩ => by` → `⟨b, e⟩ => by subst e` \| a, _, b0, ⟨b, e⟩ => by subst e -- Porting note: `Ne` is required. simpa only [mul_one] using mul_le_mul_left' (one_le_iff_ne_zero.2 fun h : b = 0 => by simp only [h, mul_zero, Ne, not_true_eq_false] at b0) a #align ordinal.le_of_dvd Ordinal.le_of_dvd theorem dvd_antisymm {a b : Ordinal} (h₁ : a ∣ b) (h₂ : b ∣ a) : a = b := if a0 : a = 0 then by subst a; exact (eq_zero_of_zero_dvd h₁).symm else if b0 : b = 0 then by subst b; exact eq_zero_of_zero_dvd h₂ else (le_of_dvd b0 h₁).antisymm (le_of_dvd a0 h₂) #align ordinal.dvd_antisymm Ordinal.dvd_antisymm instance isAntisymm : IsAntisymm Ordinal (· ∣ ·) := ⟨@dvd_antisymm⟩ /-- `a % b` is the unique ordinal `o'` satisfying `a = b * o + o'` with `o' < b`. -/ instance mod : Mod Ordinal := ⟨fun a b => a - b * (a / b)⟩ theorem mod_def (a b : Ordinal) : a % b = a - b * (a / b) := rfl #align ordinal.mod_def Ordinal.mod_def theorem mod_le (a b : Ordinal) : a % b ≤ a := sub_le_self a _ #align ordinal.mod_le Ordinal.mod_le @[simp] theorem mod_zero (a : Ordinal) : a % 0 = a := by simp only [mod_def, div_zero, zero_mul, sub_zero] #align ordinal.mod_zero Ordinal.mod_zero theorem mod_eq_of_lt {a b : Ordinal} (h : a < b) : a % b = a := by simp only [mod_def, div_eq_zero_of_lt h, mul_zero, sub_zero] #align ordinal.mod_eq_of_lt Ordinal.mod_eq_of_lt @[simp] theorem zero_mod (b : Ordinal) : 0 % b = 0 := by simp only [mod_def, zero_div, mul_zero, sub_self] #align ordinal.zero_mod Ordinal.zero_mod theorem div_add_mod (a b : Ordinal) : b * (a / b) + a % b = a := Ordinal.add_sub_cancel_of_le <\| mul_div_le _ _ #align ordinal.div_add_mod Ordinal.div_add_mod theorem mod_lt (a) {b : Ordinal} (h : b ≠ 0) : a % b < b := (add_lt_add_iff_left (b * (a / b))).1 <\| by rw [div_add_mod]; exact lt_mul_div_add a h #align ordinal.mod_lt Ordinal.mod_lt @[simp] theorem mod_self (a : Ordinal) : a % a = 0 := if a0 : a = 0 then by simp only [a0, zero_mod] else by simp only [mod_def, div_self a0, mul_one, sub_self] #align ordinal.mod_self Ordinal.mod_self @[simp] theorem mod_one (a : Ordinal) : a % 1 = 0 := by simp only [mod_def, div_one, one_mul, sub_self] #align ordinal.mod_one Ordinal.mod_one theorem dvd_of_mod_eq_zero {a b : Ordinal} (H : a % b = 0) : b ∣ a := ⟨a / b, by simpa [H] using (div_add_mod a b).symm⟩ #align ordinal.dvd_of_mod_eq_zero Ordinal.dvd_of_mod_eq_zero theorem mod_eq_zero_of_dvd {a b : Ordinal} (H : b ∣ a) : a % b = 0 := by rcases H with ⟨c, rfl⟩ rcases eq_or_ne b 0 with (rfl \| hb) · simp · simp [mod_def, hb] #align ordinal.mod_eq_zero_of_dvd Ordinal.mod_eq_zero_of_dvd theorem dvd_iff_mod_eq_zero {a b : Ordinal} : b ∣ a ↔ a % b = 0 := ⟨mod_eq_zero_of_dvd, dvd_of_mod_eq_zero⟩ #align ordinal.dvd_iff_mod_eq_zero Ordinal.dvd_iff_mod_eq_zero @[simp] theorem mul_add_mod_self (x y z : Ordinal) : (x * y + z) % x = z % x := by rcases eq_or_ne x 0 with rfl \| hx · simp · rwa [mod_def, mul_add_div, mul_add, ← sub_sub, add_sub_cancel, mod_def] #align ordinal.mul_add_mod_self Ordinal.mul_add_mod_self @[simp] theorem mul_mod (x y : Ordinal) : x * y % x = 0 := by simpa using mul_add_mod_self x y 0 #align ordinal.mul_mod Ordinal.mul_mod theorem mod_mod_of_dvd (a : Ordinal) {b c : Ordinal} (h : c ∣ b) : a % b % c = a % c := by nth_rw 2 [← div_add_mod a b] rcases h with ⟨d, rfl⟩ rw [mul_assoc, mul_add_mod_self] #align ordinal.mod_mod_of_dvd Ordinal.mod_mod_of_dvd @[simp] theorem mod_mod (a b : Ordinal) : a % b % b = a % b := mod_mod_of_dvd a dvd_rfl #align ordinal.mod_mod Ordinal.mod_mod /-! ### Families of ordinals There are two kinds of indexed families that naturally arise when dealing with ordinals: those indexed by some type in the appropriate universe, and those indexed by ordinals less than another. The following API allows one to convert from one kind of family to the other. In many cases, this makes it easy to prove claims about one kind of family via the corresponding claim on the other. -/ /-- Converts a family indexed by a `Type u` to one indexed by an `Ordinal.{u}` using a specified well-ordering. -/ def bfamilyOfFamily' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) : ∀ a < type r, α := fun a ha => f (enum r a ha) #align ordinal.bfamily_of_family' Ordinal.bfamilyOfFamily' /-- Converts a family indexed by a `Type u` to one indexed by an `Ordinal.{u}` using a well-ordering given by the axiom of choice. -/ def bfamilyOfFamily {ι : Type u} : (ι → α) → ∀ a < type (@WellOrderingRel ι), α := bfamilyOfFamily' WellOrderingRel #align ordinal.bfamily_of_family Ordinal.bfamilyOfFamily /-- Converts a family indexed by an `Ordinal.{u}` to one indexed by a `Type u` using a specified well-ordering. -/ def familyOfBFamily' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o} (ho : type r = o) (f : ∀ a < o, α) : ι → α := fun i => f (typein r i) (by rw [← ho] exact typein_lt_type r i) #align ordinal.family_of_bfamily' Ordinal.familyOfBFamily' /-- Converts a family indexed by an `Ordinal.{u}` to one indexed by a `Type u` using a well-ordering given by the axiom of choice. -/ def familyOfBFamily (o : Ordinal) (f : ∀ a < o, α) : o.out.α → α := familyOfBFamily' (· < ·) (type_lt o) f #align ordinal.family_of_bfamily Ordinal.familyOfBFamily @[simp] theorem bfamilyOfFamily'_typein {ι} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) (i) : bfamilyOfFamily' r f (typein r i) (typein_lt_type r i) = f i := by simp only [bfamilyOfFamily', enum_typein] #align ordinal.bfamily_of_family'_typein Ordinal.bfamilyOfFamily'_typein @[simp] theorem bfamilyOfFamily_typein {ι} (f : ι → α) (i) : bfamilyOfFamily f (typein _ i) (typein_lt_type _ i) = f i := bfamilyOfFamily'_typein _ f i #align ordinal.bfamily_of_family_typein Ordinal.bfamilyOfFamily_typein @[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this theorem familyOfBFamily'_enum {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o} (ho : type r = o) (f : ∀ a < o, α) (i hi) : familyOfBFamily' r ho f (enum r i (by rwa [ho])) = f i hi := by simp only [familyOfBFamily', typein_enum] #align ordinal.family_of_bfamily'_enum Ordinal.familyOfBFamily'_enum @[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this theorem familyOfBFamily_enum (o : Ordinal) (f : ∀ a < o, α) (i hi) : familyOfBFamily o f (enum (· < ·) i (by convert hi exact type_lt _)) = f i hi := familyOfBFamily'_enum _ (type_lt o) f _ _ #align ordinal.family_of_bfamily_enum Ordinal.familyOfBFamily_enum /-- The range of a family indexed by ordinals. -/ def brange (o : Ordinal) (f : ∀ a < o, α) : Set α := { a \| ∃ i hi, f i hi = a } #align ordinal.brange Ordinal.brange theorem mem_brange {o : Ordinal} {f : ∀ a < o, α} {a} : a ∈ brange o f ↔ ∃ i hi, f i hi = a := Iff.rfl #align ordinal.mem_brange Ordinal.mem_brange theorem mem_brange_self {o} (f : ∀ a < o, α) (i hi) : f i hi ∈ brange o f := ⟨i, hi, rfl⟩ #align ordinal.mem_brange_self Ordinal.mem_brange_self @[simp] theorem range_familyOfBFamily' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o} (ho : type r = o) (f : ∀ a < o, α) : range (familyOfBFamily' r ho f) = brange o f := by refine Set.ext fun a => ⟨?_, ?_⟩ · rintro ⟨b, rfl⟩ apply mem_brange_self · rintro ⟨i, hi, rfl⟩ exact ⟨_, familyOfBFamily'_enum _ _ _ _ _⟩ #align ordinal.range_family_of_bfamily' Ordinal.range_familyOfBFamily' @[simp] theorem range_familyOfBFamily {o} (f : ∀ a < o, α) : range (familyOfBFamily o f) = brange o f := range_familyOfBFamily' _ _ f #align ordinal.range_family_of_bfamily Ordinal.range_familyOfBFamily @[simp] theorem brange_bfamilyOfFamily' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) : brange _ (bfamilyOfFamily' r f) = range f := by refine Set.ext fun a => ⟨?_, ?_⟩ · rintro ⟨i, hi, rfl⟩ apply mem_range_self · rintro ⟨b, rfl⟩ exact ⟨_, _, bfamilyOfFamily'_typein _ _ _⟩ #align ordinal.brange_bfamily_of_family' Ordinal.brange_bfamilyOfFamily' @[simp] theorem brange_bfamilyOfFamily {ι : Type u} (f : ι → α) : brange _ (bfamilyOfFamily f) = range f := brange_bfamilyOfFamily' _ _ #align ordinal.brange_bfamily_of_family Ordinal.brange_bfamilyOfFamily @[simp] theorem brange_const {o : Ordinal} (ho : o ≠ 0) {c : α} : (brange o fun _ _ => c) = {c} := by rw [← range_familyOfBFamily] exact @Set.range_const _ o.out.α (out_nonempty_iff_ne_zero.2 ho) c #align ordinal.brange_const Ordinal.brange_const theorem comp_bfamilyOfFamily' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) (g : α → β) : (fun i hi => g (bfamilyOfFamily' r f i hi)) = bfamilyOfFamily' r (g ∘ f) := rfl #align ordinal.comp_bfamily_of_family' Ordinal.comp_bfamilyOfFamily' theorem comp_bfamilyOfFamily {ι : Type u} (f : ι → α) (g : α → β) : (fun i hi => g (bfamilyOfFamily f i hi)) = bfamilyOfFamily (g ∘ f) := rfl #align ordinal.comp_bfamily_of_family Ordinal.comp_bfamilyOfFamily theorem comp_familyOfBFamily' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o} (ho : type r = o) (f : ∀ a < o, α) (g : α → β) : g ∘ familyOfBFamily' r ho f = familyOfBFamily' r ho fun i hi => g (f i hi) := rfl #align ordinal.comp_family_of_bfamily' Ordinal.comp_familyOfBFamily' theorem comp_familyOfBFamily {o} (f : ∀ a < o, α) (g : α → β) : g ∘ familyOfBFamily o f = familyOfBFamily o fun i hi => g (f i hi) := rfl #align ordinal.comp_family_of_bfamily Ordinal.comp_familyOfBFamily /-! ### Supremum of a family of ordinals -/ -- Porting note: Universes should be specified in `sup`s. /-- The supremum of a family of ordinals -/ def sup {ι : Type u} (f : ι → Ordinal.{max u v}) : Ordinal.{max u v} := iSup f #align ordinal.sup Ordinal.sup @[simp] theorem sSup_eq_sup {ι : Type u} (f : ι → Ordinal.{max u v}) : sSup (Set.range f) = sup.{_, v} f := rfl #align ordinal.Sup_eq_sup Ordinal.sSup_eq_sup /-- The range of an indexed ordinal function, whose outputs live in a higher universe than the inputs, is always bounded above. See `Ordinal.lsub` for an explicit bound. -/ theorem bddAbove_range {ι : Type u} (f : ι → Ordinal.{max u v}) : BddAbove (Set.range f) := ⟨(iSup (succ ∘ card ∘ f)).ord, by rintro a ⟨i, rfl⟩ exact le_of_lt (Cardinal.lt_ord.2 ((lt_succ _).trans_le (le_ciSup (Cardinal.bddAbove_range.{_, v} _) _)))⟩ #align ordinal.bdd_above_range Ordinal.bddAbove_range theorem le_sup {ι : Type u} (f : ι → Ordinal.{max u v}) : ∀ i, f i ≤ sup.{_, v} f := fun i => le_csSup (bddAbove_range.{_, v} f) (mem_range_self i) #align ordinal.le_sup Ordinal.le_sup theorem sup_le_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : sup.{_, v} f ≤ a ↔ ∀ i, f i ≤ a := (csSup_le_iff' (bddAbove_range.{_, v} f)).trans (by simp) #align ordinal.sup_le_iff Ordinal.sup_le_iff theorem sup_le {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : (∀ i, f i ≤ a) → sup.{_, v} f ≤ a := sup_le_iff.2 #align ordinal.sup_le Ordinal.sup_le theorem lt_sup {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : a < sup.{_, v} f ↔ ∃ i, a < f i := by simpa only [not_forall, not_le] using not_congr (@sup_le_iff.{_, v} _ f a) #align ordinal.lt_sup Ordinal.lt_sup theorem ne_sup_iff_lt_sup {ι : Type u} {f : ι → Ordinal.{max u v}} : (∀ i, f i ≠ sup.{_, v} f) ↔ ∀ i, f i < sup.{_, v} f := ⟨fun hf _ => lt_of_le_of_ne (le_sup _ _) (hf _), fun hf _ => ne_of_lt (hf _)⟩ #align ordinal.ne_sup_iff_lt_sup Ordinal.ne_sup_iff_lt_sup theorem sup_not_succ_of_ne_sup {ι : Type u} {f : ι → Ordinal.{max u v}} (hf : ∀ i, f i ≠ sup.{_, v} f) {a} (hao : a < sup.{_, v} f) : succ a < sup.{_, v} f := by by_contra! hoa exact hao.not_le (sup_le fun i => le_of_lt_succ <\| (lt_of_le_of_ne (le_sup _ _) (hf i)).trans_le hoa) #align ordinal.sup_not_succ_of_ne_sup Ordinal.sup_not_succ_of_ne_sup @[simp] theorem sup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v}} : sup.{_, v} f = 0 ↔ ∀ i, f i = 0 := by refine ⟨fun h i => ?_, fun h => le_antisymm (sup_le fun i => Ordinal.le_zero.2 (h i)) (Ordinal.zero_le _)⟩ rw [← Ordinal.le_zero, ← h] exact le_sup f i #align ordinal.sup_eq_zero_iff Ordinal.sup_eq_zero_iff theorem IsNormal.sup {f : Ordinal.{max u v} → Ordinal.{max u w}} (H : IsNormal f) {ι : Type u} (g : ι → Ordinal.{max u v}) [Nonempty ι] : f (sup.{_, v} g) = sup.{_, w} (f ∘ g) := eq_of_forall_ge_iff fun a => by rw [sup_le_iff]; simp only [comp]; rw [H.le_set' Set.univ Set.univ_nonempty g] <;> simp [sup_le_iff] #align ordinal.is_normal.sup Ordinal.IsNormal.sup @[simp] theorem sup_empty {ι} [IsEmpty ι] (f : ι → Ordinal) : sup f = 0 := ciSup_of_empty f #align ordinal.sup_empty Ordinal.sup_empty @[simp] theorem sup_const {ι} [_hι : Nonempty ι] (o : Ordinal) : (sup fun _ : ι => o) = o := ciSup_const #align ordinal.sup_const Ordinal.sup_const @[simp] theorem sup_unique {ι} [Unique ι] (f : ι → Ordinal) : sup f = f default := ciSup_unique #align ordinal.sup_unique Ordinal.sup_unique theorem sup_le_of_range_subset {ι ι'} {f : ι → Ordinal} {g : ι' → Ordinal} (h : Set.range f ⊆ Set.range g) : sup.{u, max v w} f ≤ sup.{v, max u w} g := sup_le fun i => match h (mem_range_self i) with \| ⟨_j, hj⟩ => hj ▸ le_sup _ _ #align ordinal.sup_le_of_range_subset Ordinal.sup_le_of_range_subset theorem sup_eq_of_range_eq {ι ι'} {f : ι → Ordinal} {g : ι' → Ordinal} (h : Set.range f = Set.range g) : sup.{u, max v w} f = sup.{v, max u w} g := (sup_le_of_range_subset.{u, v, w} h.le).antisymm (sup_le_of_range_subset.{v, u, w} h.ge) #align ordinal.sup_eq_of_range_eq Ordinal.sup_eq_of_range_eq @[simp] theorem sup_sum {α : Type u} {β : Type v} (f : Sum α β → Ordinal) : sup.{max u v, w} f = max (sup.{u, max v w} fun a => f (Sum.inl a)) (sup.{v, max u w} fun b => f (Sum.inr b)) := by apply (sup_le_iff.2 _).antisymm (max_le_iff.2 ⟨_, _⟩) · rintro (i \| i) · exact le_max_of_le_left (le_sup _ i) · exact le_max_of_le_right (le_sup _ i) all_goals apply sup_le_of_range_subset.{_, max u v, w} rintro i ⟨a, rfl⟩ apply mem_range_self #align ordinal.sup_sum Ordinal.sup_sum theorem unbounded_range_of_sup_ge {α β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β → α) (h : type r ≤ sup.{u, u} (typein r ∘ f)) : Unbounded r (range f) := (not_bounded_iff _).1 fun ⟨x, hx⟩ => not_lt_of_le h <\| lt_of_le_of_lt (sup_le fun y => le_of_lt <\| (typein_lt_typein r).2 <\| hx _ <\| mem_range_self y) (typein_lt_type r x) #align ordinal.unbounded_range_of_sup_ge Ordinal.unbounded_range_of_sup_ge theorem le_sup_shrink_equiv {s : Set Ordinal.{u}} (hs : Small.{u} s) (a) (ha : a ∈ s) : a ≤ sup.{u, u} fun x => ((@equivShrink s hs).symm x).val := by convert le_sup.{u, u} (fun x => ((@equivShrink s hs).symm x).val) ((@equivShrink s hs) ⟨a, ha⟩) rw [symm_apply_apply] #align ordinal.le_sup_shrink_equiv Ordinal.le_sup_shrink_equiv instance small_Iio (o : Ordinal.{u}) : Small.{u} (Set.Iio o) := let f : o.out.α → Set.Iio o := fun x => ⟨typein ((· < ·) : o.out.α → o.out.α → Prop) x, typein_lt_self x⟩ let hf : Surjective f := fun b => ⟨enum (· < ·) b.val (by rw [type_lt] exact b.prop), Subtype.ext (typein_enum _ _)⟩ small_of_surjective hf #align ordinal.small_Iio Ordinal.small_Iio instance small_Iic (o : Ordinal.{u}) : Small.{u} (Set.Iic o) := by rw [← Iio_succ] infer_instance #align ordinal.small_Iic Ordinal.small_Iic theorem bddAbove_iff_small {s : Set Ordinal.{u}} : BddAbove s ↔ Small.{u} s := ⟨fun ⟨a, h⟩ => small_subset <\| show s ⊆ Iic a from fun _x hx => h hx, fun h => ⟨sup.{u, u} fun x => ((@equivShrink s h).symm x).val, le_sup_shrink_equiv h⟩⟩ #align ordinal.bdd_above_iff_small Ordinal.bddAbove_iff_small theorem bddAbove_of_small (s : Set Ordinal.{u}) [h : Small.{u} s] : BddAbove s := bddAbove_iff_small.2 h #align ordinal.bdd_above_of_small Ordinal.bddAbove_of_small theorem sup_eq_sSup {s : Set Ordinal.{u}} (hs : Small.{u} s) : (sup.{u, u} fun x => (@equivShrink s hs).symm x) = sSup s := let hs' := bddAbove_iff_small.2 hs ((csSup_le_iff' hs').2 (le_sup_shrink_equiv hs)).antisymm' (sup_le fun _x => le_csSup hs' (Subtype.mem _)) #align ordinal.sup_eq_Sup Ordinal.sup_eq_sSup theorem sSup_ord {s : Set Cardinal.{u}} (hs : BddAbove s) : (sSup s).ord = sSup (ord '' s) := eq_of_forall_ge_iff fun a => by rw [csSup_le_iff' (bddAbove_iff_small.2 (@small_image _ _ _ s (Cardinal.bddAbove_iff_small.1 hs))), ord_le, csSup_le_iff' hs] simp [ord_le] #align ordinal.Sup_ord Ordinal.sSup_ord theorem iSup_ord {ι} {f : ι → Cardinal} (hf : BddAbove (range f)) : (iSup f).ord = ⨆ i, (f i).ord := by unfold iSup convert sSup_ord hf -- Porting note: `change` is required. conv_lhs => change range (ord ∘ f) rw [range_comp] #align ordinal.supr_ord Ordinal.iSup_ord private theorem sup_le_sup {ι ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop) [IsWellOrder ι r] [IsWellOrder ι' r'] {o} (ho : type r = o) (ho' : type r' = o) (f : ∀ a < o, Ordinal.{max u v}) : sup.{_, v} (familyOfBFamily' r ho f) ≤ sup.{_, v} (familyOfBFamily' r' ho' f) := sup_le fun i => by cases' typein_surj r' (by rw [ho', ← ho] exact typein_lt_type r i) with j hj simp_rw [familyOfBFamily', ← hj] apply le_sup theorem sup_eq_sup {ι ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop) [IsWellOrder ι r] [IsWellOrder ι' r'] {o : Ordinal.{u}} (ho : type r = o) (ho' : type r' = o) (f : ∀ a < o, Ordinal.{max u v}) : sup.{_, v} (familyOfBFamily' r ho f) = sup.{_, v} (familyOfBFamily' r' ho' f) := sup_eq_of_range_eq.{u, u, v} (by simp) #align ordinal.sup_eq_sup Ordinal.sup_eq_sup /-- The supremum of a family of ordinals indexed by the set of ordinals less than some `o : Ordinal.{u}`. This is a special case of `sup` over the family provided by `familyOfBFamily`. -/ def bsup (o : Ordinal.{u}) (f : ∀ a < o, Ordinal.{max u v}) : Ordinal.{max u v} := sup.{_, v} (familyOfBFamily o f) #align ordinal.bsup Ordinal.bsup @[simp] theorem sup_eq_bsup {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) : sup.{_, v} (familyOfBFamily o f) = bsup.{_, v} o f := rfl #align ordinal.sup_eq_bsup Ordinal.sup_eq_bsup @[simp] theorem sup_eq_bsup' {o : Ordinal.{u}} {ι} (r : ι → ι → Prop) [IsWellOrder ι r] (ho : type r = o) (f : ∀ a < o, Ordinal.{max u v}) : sup.{_, v} (familyOfBFamily' r ho f) = bsup.{_, v} o f := sup_eq_sup r _ ho _ f #align ordinal.sup_eq_bsup' Ordinal.sup_eq_bsup' @[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this theorem sSup_eq_bsup {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) : sSup (brange o f) = bsup.{_, v} o f := by congr rw [range_familyOfBFamily] #align ordinal.Sup_eq_bsup Ordinal.sSup_eq_bsup @[simp] theorem bsup_eq_sup' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → Ordinal.{max u v}) : bsup.{_, v} _ (bfamilyOfFamily' r f) = sup.{_, v} f := by simp (config := { unfoldPartialApp := true }) only [← sup_eq_bsup' r, enum_typein, familyOfBFamily', bfamilyOfFamily'] #align ordinal.bsup_eq_sup' Ordinal.bsup_eq_sup' theorem bsup_eq_bsup {ι : Type u} (r r' : ι → ι → Prop) [IsWellOrder ι r] [IsWellOrder ι r'] (f : ι → Ordinal.{max u v}) : bsup.{_, v} _ (bfamilyOfFamily' r f) = bsup.{_, v} _ (bfamilyOfFamily' r' f) := by rw [bsup_eq_sup', bsup_eq_sup'] #align ordinal.bsup_eq_bsup Ordinal.bsup_eq_bsup @[simp] theorem bsup_eq_sup {ι : Type u} (f : ι → Ordinal.{max u v}) : bsup.{_, v} _ (bfamilyOfFamily f) = sup.{_, v} f := bsup_eq_sup' _ f #align ordinal.bsup_eq_sup Ordinal.bsup_eq_sup @[congr] theorem bsup_congr {o₁ o₂ : Ordinal.{u}} (f : ∀ a < o₁, Ordinal.{max u v}) (ho : o₁ = o₂) : bsup.{_, v} o₁ f = bsup.{_, v} o₂ fun a h => f a (h.trans_eq ho.symm) := by subst ho -- Porting note: `rfl` is required. rfl #align ordinal.bsup_congr Ordinal.bsup_congr theorem bsup_le_iff {o f a} : bsup.{u, v} o f ≤ a ↔ ∀ i h, f i h ≤ a := sup_le_iff.trans ⟨fun h i hi => by rw [← familyOfBFamily_enum o f] exact h _, fun h i => h _ _⟩ #align ordinal.bsup_le_iff Ordinal.bsup_le_iff theorem bsup_le {o : Ordinal} {f : ∀ b < o, Ordinal} {a} : (∀ i h, f i h ≤ a) → bsup.{u, v} o f ≤ a := bsup_le_iff.2 #align ordinal.bsup_le Ordinal.bsup_le theorem le_bsup {o} (f : ∀ a < o, Ordinal) (i h) : f i h ≤ bsup o f := bsup_le_iff.1 le_rfl _ _ #align ordinal.le_bsup Ordinal.le_bsup theorem lt_bsup {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) {a} : a < bsup.{_, v} o f ↔ ∃ i hi, a < f i hi := by simpa only [not_forall, not_le] using not_congr (@bsup_le_iff.{_, v} _ f a) #align ordinal.lt_bsup Ordinal.lt_bsup theorem IsNormal.bsup {f : Ordinal.{max u v} → Ordinal.{max u w}} (H : IsNormal f) {o : Ordinal.{u}} : ∀ (g : ∀ a < o, Ordinal), o ≠ 0 → f (bsup.{_, v} o g) = bsup.{_, w} o fun a h => f (g a h) := inductionOn o fun α r _ g h => by haveI := type_ne_zero_iff_nonempty.1 h rw [← sup_eq_bsup' r, IsNormal.sup.{_, v, w} H, ← sup_eq_bsup' r] <;> rfl #align ordinal.is_normal.bsup Ordinal.IsNormal.bsup theorem lt_bsup_of_ne_bsup {o : Ordinal.{u}} {f : ∀ a < o, Ordinal.{max u v}} : (∀ i h, f i h ≠ bsup.{_, v} o f) ↔ ∀ i h, f i h < bsup.{_, v} o f := ⟨fun hf _ _ => lt_of_le_of_ne (le_bsup _ _ _) (hf _ _), fun hf _ _ => ne_of_lt (hf _ _)⟩ #align ordinal.lt_bsup_of_ne_bsup Ordinal.lt_bsup_of_ne_bsup theorem bsup_not_succ_of_ne_bsup {o : Ordinal.{u}} {f : ∀ a < o, Ordinal.{max u v}} (hf : ∀ {i : Ordinal} (h : i < o), f i h ≠ bsup.{_, v} o f) (a) : a < bsup.{_, v} o f → succ a < bsup.{_, v} o f := by rw [← sup_eq_bsup] at * exact sup_not_succ_of_ne_sup fun i => hf _ #align ordinal.bsup_not_succ_of_ne_bsup Ordinal.bsup_not_succ_of_ne_bsup @[simp] theorem bsup_eq_zero_iff {o} {f : ∀ a < o, Ordinal} : bsup o f = 0 ↔ ∀ i hi, f i hi = 0 := by refine ⟨fun h i hi => ?_, fun h => le_antisymm (bsup_le fun i hi => Ordinal.le_zero.2 (h i hi)) (Ordinal.zero_le _)⟩ rw [← Ordinal.le_zero, ← h] exact le_bsup f i hi #align ordinal.bsup_eq_zero_iff Ordinal.bsup_eq_zero_iff theorem lt_bsup_of_limit {o : Ordinal} {f : ∀ a < o, Ordinal} (hf : ∀ {a a'} (ha : a < o) (ha' : a' < o), a < a' → f a ha < f a' ha') (ho : ∀ a < o, succ a < o) (i h) : f i h < bsup o f := (hf _ _ <\| lt_succ i).trans_le (le_bsup f (succ i) <\| ho _ h) #align ordinal.lt_bsup_of_limit Ordinal.lt_bsup_of_limit theorem bsup_succ_of_mono {o : Ordinal} {f : ∀ a < succ o, Ordinal} (hf : ∀ {i j} (hi hj), i ≤ j → f i hi ≤ f j hj) : bsup _ f = f o (lt_succ o) := le_antisymm (bsup_le fun _i hi => hf _ _ <\| le_of_lt_succ hi) (le_bsup _ _ _) #align ordinal.bsup_succ_of_mono Ordinal.bsup_succ_of_mono @[simp] theorem bsup_zero (f : ∀ a < (0 : Ordinal), Ordinal) : bsup 0 f = 0 := bsup_eq_zero_iff.2 fun i hi => (Ordinal.not_lt_zero i hi).elim #align ordinal.bsup_zero Ordinal.bsup_zero theorem bsup_const {o : Ordinal.{u}} (ho : o ≠ 0) (a : Ordinal.{max u v}) : (bsup.{_, v} o fun _ _ => a) = a := le_antisymm (bsup_le fun _ _ => le_rfl) (le_bsup _ 0 (Ordinal.pos_iff_ne_zero.2 ho)) #align ordinal.bsup_const Ordinal.bsup_const @[simp] theorem bsup_one (f : ∀ a < (1 : Ordinal), Ordinal) : bsup 1 f = f 0 zero_lt_one := by simp_rw [← sup_eq_bsup, sup_unique, familyOfBFamily, familyOfBFamily', typein_one_out] #align ordinal.bsup_one Ordinal.bsup_one theorem bsup_le_of_brange_subset {o o'} {f : ∀ a < o, Ordinal} {g : ∀ a < o', Ordinal} (h : brange o f ⊆ brange o' g) : bsup.{u, max v w} o f ≤ bsup.{v, max u w} o' g := bsup_le fun i hi => by obtain ⟨j, hj, hj'⟩ := h ⟨i, hi, rfl⟩ rw [← hj'] apply le_bsup #align ordinal.bsup_le_of_brange_subset Ordinal.bsup_le_of_brange_subset theorem bsup_eq_of_brange_eq {o o'} {f : ∀ a < o, Ordinal} {g : ∀ a < o', Ordinal} (h : brange o f = brange o' g) : bsup.{u, max v w} o f = bsup.{v, max u w} o' g := (bsup_le_of_brange_subset.{u, v, w} h.le).antisymm (bsup_le_of_brange_subset.{v, u, w} h.ge) #align ordinal.bsup_eq_of_brange_eq Ordinal.bsup_eq_of_brange_eq /-- The least strict upper bound of a family of ordinals. -/ def lsub {ι} (f : ι → Ordinal) : Ordinal := sup (succ ∘ f) #align ordinal.lsub Ordinal.lsub @[simp] theorem sup_eq_lsub {ι : Type u} (f : ι → Ordinal.{max u v}) : sup.{_, v} (succ ∘ f) = lsub.{_, v} f := rfl #align ordinal.sup_eq_lsub Ordinal.sup_eq_lsub theorem lsub_le_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : lsub.{_, v} f ≤ a ↔ ∀ i, f i < a := by convert sup_le_iff.{_, v} (f := succ ∘ f) (a := a) using 2 -- Porting note: `comp_apply` is required. simp only [comp_apply, succ_le_iff] #align ordinal.lsub_le_iff Ordinal.lsub_le_iff theorem lsub_le {ι} {f : ι → Ordinal} {a} : (∀ i, f i < a) → lsub f ≤ a := lsub_le_iff.2 #align ordinal.lsub_le Ordinal.lsub_le theorem lt_lsub {ι} (f : ι → Ordinal) (i) : f i < lsub f := succ_le_iff.1 (le_sup _ i) #align ordinal.lt_lsub Ordinal.lt_lsub	Mathlib/SetTheory/Ordinal/Arithmetic.lean	1,599	1,601	theorem lt_lsub_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : a < lsub.{_, v} f ↔ ∃ i, a ≤ f i := by	simpa only [not_forall, not_lt, not_le] using not_congr (@lsub_le_iff.{_, v} _ f a)
/- Copyright (c) 2022 Cuma Kökmen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Cuma Kökmen, Yury Kudryashov -/ import Mathlib.MeasureTheory.Constructions.Prod.Integral import Mathlib.MeasureTheory.Integral.CircleIntegral #align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" /-! # Integral over a torus in `ℂⁿ` In this file we define the integral of a function `f : ℂⁿ → E` over a torus `{z : ℂⁿ \| ∀ i, z i ∈ Metric.sphere (c i) (R i)}`. In order to do this, we define `torusMap (c : ℂⁿ) (R θ : ℝⁿ)` to be the point in `ℂⁿ` given by $z_k=c_k+R_ke^{θ_ki}$, where $i$ is the imaginary unit, then define `torusIntegral f c R` as the integral over the cube $[0, (fun _ ↦ 2π)] = \{θ\\|∀ k, 0 ≤ θ_k ≤ 2π\}$ of the Jacobian of the `torusMap` multiplied by `f (torusMap c R θ)`. We also define a predicate saying that `f ∘ torusMap c R` is integrable on the cube `[0, (fun _ ↦ 2π)]`. ## Main definitions * `torusMap c R`: the generalized multidimensional exponential map from `ℝⁿ` to `ℂⁿ` that sends $θ=(θ_0,…,θ_{n-1})$ to $z=(z_0,…,z_{n-1})$, where $z_k= c_k + R_ke^{θ_k i}$; * `TorusIntegrable f c R`: a function `f : ℂⁿ → E` is integrable over the generalized torus with center `c : ℂⁿ` and radius `R : ℝⁿ` if `f ∘ torusMap c R` is integrable on the closed cube `Icc (0 : ℝⁿ) (fun _ ↦ 2 * π)`; * `torusIntegral f c R`: the integral of a function `f : ℂⁿ → E` over a torus with center `c ∈ ℂⁿ` and radius `R ∈ ℝⁿ` defined as $\iiint_{[0, 2 * π]} (∏_{k = 1}^{n} i R_k e^{θ_k * i}) • f (c + Re^{θ_k i})\,dθ_0…dθ_{k-1}$. ## Main statements * `torusIntegral_dim0`, `torusIntegral_dim1`, `torusIntegral_succ`: formulas for `torusIntegral` in cases of dimension `0`, `1`, and `n + 1`. ## Notations - `ℝ⁰`, `ℝ¹`, `ℝⁿ`, `ℝⁿ⁺¹`: local notation for `Fin 0 → ℝ`, `Fin 1 → ℝ`, `Fin n → ℝ`, and `Fin (n + 1) → ℝ`, respectively; - `ℂ⁰`, `ℂ¹`, `ℂⁿ`, `ℂⁿ⁺¹`: local notation for `Fin 0 → ℂ`, `Fin 1 → ℂ`, `Fin n → ℂ`, and `Fin (n + 1) → ℂ`, respectively; - `∯ z in T(c, R), f z`: notation for `torusIntegral f c R`; - `∮ z in C(c, R), f z`: notation for `circleIntegral f c R`, defined elsewhere; - `∏ k, f k`: notation for `Finset.prod`, defined elsewhere; - `π`: notation for `Real.pi`, defined elsewhere. ## Tags integral, torus -/ variable {n : ℕ} variable {E : Type} [NormedAddCommGroup E] noncomputable section open Complex Set MeasureTheory Function Filter TopologicalSpace open scoped Real -- Porting note: notation copied from `./DivergenceTheorem` local macro:arg t:term:max noWs "ⁿ⁺¹" : term => `(Fin (n + 1) → $t) local macro:arg t:term:max noWs "ⁿ" : term => `(Fin n → $t) local macro:arg t:term:max noWs "⁰" : term => `(Fin 0 → $t) local macro:arg t:term:max noWs "¹" : term => `(Fin 1 → $t) /-! ### `torusMap`, a parametrization of a torus -/ /-- The n dimensional exponential map $θ_i ↦ c + R e^{θ_iI}, θ ∈ ℝⁿ$ representing a torus in `ℂⁿ` with center `c ∈ ℂⁿ` and generalized radius `R ∈ ℝⁿ`, so we can adjust it to every n axis. -/ def torusMap (c : ℂⁿ) (R : ℝⁿ) : ℝⁿ → ℂⁿ := fun θ i => c i + R i * exp (θ i * I) #align torus_map torusMap theorem torusMap_sub_center (c : ℂⁿ) (R : ℝⁿ) (θ : ℝⁿ) : torusMap c R θ - c = torusMap 0 R θ := by ext1 i; simp [torusMap] #align torus_map_sub_center torusMap_sub_center theorem torusMap_eq_center_iff {c : ℂⁿ} {R : ℝⁿ} {θ : ℝⁿ} : torusMap c R θ = c ↔ R = 0 := by simp [funext_iff, torusMap, exp_ne_zero] #align torus_map_eq_center_iff torusMap_eq_center_iff @[simp] theorem torusMap_zero_radius (c : ℂⁿ) : torusMap c 0 = const ℝⁿ c := funext fun _ ↦ torusMap_eq_center_iff.2 rfl #align torus_map_zero_radius torusMap_zero_radius /-! ### Integrability of a function on a generalized torus -/ /-- A function `f : ℂⁿ → E` is integrable on the generalized torus if the function `f ∘ torusMap c R θ` is integrable on `Icc (0 : ℝⁿ) (fun _ ↦ 2 * π)`. -/ def TorusIntegrable (f : ℂⁿ → E) (c : ℂⁿ) (R : ℝⁿ) : Prop := IntegrableOn (fun θ : ℝⁿ => f (torusMap c R θ)) (Icc (0 : ℝⁿ) fun _ => 2 * π) volume #align torus_integrable TorusIntegrable namespace TorusIntegrable -- Porting note (#11215): TODO: restore notation; `neg`, `add` etc fail if I use notation here variable {f g : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} /-- Constant functions are torus integrable -/ theorem torusIntegrable_const (a : E) (c : ℂⁿ) (R : ℝⁿ) : TorusIntegrable (fun _ => a) c R := by simp [TorusIntegrable, measure_Icc_lt_top] #align torus_integrable.torus_integrable_const TorusIntegrable.torusIntegrable_const /-- If `f` is torus integrable then `-f` is torus integrable. -/ protected nonrec theorem neg (hf : TorusIntegrable f c R) : TorusIntegrable (-f) c R := hf.neg #align torus_integrable.neg TorusIntegrable.neg /-- If `f` and `g` are two torus integrable functions, then so is `f + g`. -/ protected nonrec theorem add (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) : TorusIntegrable (f + g) c R := hf.add hg #align torus_integrable.add TorusIntegrable.add /-- If `f` and `g` are two torus integrable functions, then so is `f - g`. -/ protected nonrec theorem sub (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) : TorusIntegrable (f - g) c R := hf.sub hg #align torus_integrable.sub TorusIntegrable.sub theorem torusIntegrable_zero_radius {f : ℂⁿ → E} {c : ℂⁿ} : TorusIntegrable f c 0 := by rw [TorusIntegrable, torusMap_zero_radius] apply torusIntegrable_const (f c) c 0 #align torus_integrable.torus_integrable_zero_radius TorusIntegrable.torusIntegrable_zero_radius /-- The function given in the definition of `torusIntegral` is integrable. -/ theorem function_integrable [NormedSpace ℂ E] (hf : TorusIntegrable f c R) : IntegrableOn (fun θ : ℝⁿ => (∏ i, R i * exp (θ i * I) * I : ℂ) • f (torusMap c R θ)) (Icc (0 : ℝⁿ) fun _ => 2 * π) volume := by refine (hf.norm.const_mul (∏ i, \|R i\|)).mono' ?_ ?_ · refine (Continuous.aestronglyMeasurable ?_).smul hf.1; continuity simp [norm_smul, map_prod] #align torus_integrable.function_integrable TorusIntegrable.function_integrable end TorusIntegrable variable [NormedSpace ℂ E] [CompleteSpace E] {f g : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} /-- The integral over a generalized torus with center `c ∈ ℂⁿ` and radius `R ∈ ℝⁿ`, defined as the `•`-product of the derivative of `torusMap` and `f (torusMap c R θ)`-/ def torusIntegral (f : ℂⁿ → E) (c : ℂⁿ) (R : ℝⁿ) := ∫ θ : ℝⁿ in Icc (0 : ℝⁿ) fun _ => 2 * π, (∏ i, R i * exp (θ i * I) * I : ℂ) • f (torusMap c R θ) #align torus_integral torusIntegral @[inherit_doc torusIntegral] notation3"∯ "(...)" in ""T("c", "R")"", "r:(scoped f => torusIntegral f c R) => r theorem torusIntegral_radius_zero (hn : n ≠ 0) (f : ℂⁿ → E) (c : ℂⁿ) : (∯ x in T(c, 0), f x) = 0 := by simp only [torusIntegral, Pi.zero_apply, ofReal_zero, mul_zero, zero_mul, Fin.prod_const, zero_pow hn, zero_smul, integral_zero] #align torus_integral_radius_zero torusIntegral_radius_zero theorem torusIntegral_neg (f : ℂⁿ → E) (c : ℂⁿ) (R : ℝⁿ) : (∯ x in T(c, R), -f x) = -∯ x in T(c, R), f x := by simp [torusIntegral, integral_neg] #align torus_integral_neg torusIntegral_neg theorem torusIntegral_add (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) : (∯ x in T(c, R), f x + g x) = (∯ x in T(c, R), f x) + ∯ x in T(c, R), g x := by simpa only [torusIntegral, smul_add, Pi.add_apply] using integral_add hf.function_integrable hg.function_integrable #align torus_integral_add torusIntegral_add	Mathlib/MeasureTheory/Integral/TorusIntegral.lean	176	178	theorem torusIntegral_sub (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) : (∯ x in T(c, R), f x - g x) = (∯ x in T(c, R), f x) - ∯ x in T(c, R), g x := by	simpa only [sub_eq_add_neg, ← torusIntegral_neg] using torusIntegral_add hf hg.neg
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn -/ import Mathlib.Order.Hom.CompleteLattice import Mathlib.Topology.Bases import Mathlib.Topology.Homeomorph import Mathlib.Topology.ContinuousFunction.Basic import Mathlib.Order.CompactlyGenerated.Basic import Mathlib.Order.Copy #align_import topology.sets.opens from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" /-! # Open sets ## Summary We define the subtype of open sets in a topological space. ## Main Definitions ### Bundled open sets - `TopologicalSpace.Opens α` is the type of open subsets of a topological space `α`. - `TopologicalSpace.Opens.IsBasis` is a predicate saying that a set of `Opens`s form a topological basis. - `TopologicalSpace.Opens.comap`: preimage of an open set under a continuous map as a `FrameHom`. - `Homeomorph.opensCongr`: order-preserving equivalence between open sets in the domain and the codomain of a homeomorphism. ### Bundled open neighborhoods - `TopologicalSpace.OpenNhdsOf x` is the type of open subsets of a topological space `α` containing `x : α`. - `TopologicalSpace.OpenNhdsOf.comap f x U` is the preimage of open neighborhood `U` of `f x` under `f : C(α, β)`. ## Main results We define order structures on both `Opens α` (`CompleteLattice`, `Frame`) and `OpenNhdsOf x` (`OrderTop`, `DistribLattice`). ## TODO - Rename `TopologicalSpace.Opens` to `Open`? - Port the `auto_cases` tactic version (as a plugin if the ported `auto_cases` will allow plugins). -/ open Filter Function Order Set open Topology variable {ι α β γ : Type} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] namespace TopologicalSpace variable (α) /-- The type of open subsets of a topological space. -/ structure Opens where /-- The underlying set of a bundled `TopologicalSpace.Opens` object. -/ carrier : Set α /-- The `TopologicalSpace.Opens.carrier _` is an open set. -/ is_open' : IsOpen carrier #align topological_space.opens TopologicalSpace.Opens variable {α} namespace Opens instance : SetLike (Opens α) α where coe := Opens.carrier coe_injective' := fun ⟨_, _⟩ ⟨_, _⟩ _ => by congr instance : CanLift (Set α) (Opens α) (↑) IsOpen := ⟨fun s h => ⟨⟨s, h⟩, rfl⟩⟩ theorem «forall» {p : Opens α → Prop} : (∀ U, p U) ↔ ∀ (U : Set α) (hU : IsOpen U), p ⟨U, hU⟩ := ⟨fun h _ _ => h _, fun h _ => h _ _⟩ #align topological_space.opens.forall TopologicalSpace.Opens.forall @[simp] theorem carrier_eq_coe (U : Opens α) : U.1 = ↑U := rfl #align topological_space.opens.carrier_eq_coe TopologicalSpace.Opens.carrier_eq_coe /-- the coercion `Opens α → Set α` applied to a pair is the same as taking the first component -/ @[simp] theorem coe_mk {U : Set α} {hU : IsOpen U} : ↑(⟨U, hU⟩ : Opens α) = U := rfl #align topological_space.opens.coe_mk TopologicalSpace.Opens.coe_mk @[simp] theorem mem_mk {x : α} {U : Set α} {h : IsOpen U} : x ∈ mk U h ↔ x ∈ U := Iff.rfl #align topological_space.opens.mem_mk TopologicalSpace.Opens.mem_mk -- Porting note: removed @[simp] because LHS simplifies to `∃ x, x ∈ U` protected theorem nonempty_coeSort {U : Opens α} : Nonempty U ↔ (U : Set α).Nonempty := Set.nonempty_coe_sort #align topological_space.opens.nonempty_coe_sort TopologicalSpace.Opens.nonempty_coeSort -- Porting note (#10756): new lemma; todo: prove it for a `SetLike`? protected theorem nonempty_coe {U : Opens α} : (U : Set α).Nonempty ↔ ∃ x, x ∈ U := Iff.rfl @[ext] -- Porting note (#11215): TODO: replace with `∀ x, x ∈ U ↔ x ∈ V` theorem ext {U V : Opens α} (h : (U : Set α) = V) : U = V := SetLike.coe_injective h #align topological_space.opens.ext TopologicalSpace.Opens.ext -- Porting note: removed @[simp], simp can prove it theorem coe_inj {U V : Opens α} : (U : Set α) = V ↔ U = V := SetLike.ext'_iff.symm #align topological_space.opens.coe_inj TopologicalSpace.Opens.coe_inj protected theorem isOpen (U : Opens α) : IsOpen (U : Set α) := U.is_open' #align topological_space.opens.is_open TopologicalSpace.Opens.isOpen @[simp] theorem mk_coe (U : Opens α) : mk (↑U) U.isOpen = U := rfl #align topological_space.opens.mk_coe TopologicalSpace.Opens.mk_coe /-- See Note [custom simps projection]. -/ def Simps.coe (U : Opens α) : Set α := U #align topological_space.opens.simps.coe TopologicalSpace.Opens.Simps.coe initialize_simps_projections Opens (carrier → coe) /-- The interior of a set, as an element of `Opens`. -/ nonrec def interior (s : Set α) : Opens α := ⟨interior s, isOpen_interior⟩ #align topological_space.opens.interior TopologicalSpace.Opens.interior theorem gc : GaloisConnection ((↑) : Opens α → Set α) interior := fun U _ => ⟨fun h => interior_maximal h U.isOpen, fun h => le_trans h interior_subset⟩ #align topological_space.opens.gc TopologicalSpace.Opens.gc /-- The galois coinsertion between sets and opens. -/ def gi : GaloisCoinsertion (↑) (@interior α _) where choice s hs := ⟨s, interior_eq_iff_isOpen.mp <\| le_antisymm interior_subset hs⟩ gc := gc u_l_le _ := interior_subset choice_eq _s hs := le_antisymm hs interior_subset #align topological_space.opens.gi TopologicalSpace.Opens.gi instance : CompleteLattice (Opens α) := CompleteLattice.copy (GaloisCoinsertion.liftCompleteLattice gi) -- le (fun U V => (U : Set α) ⊆ V) rfl -- top ⟨univ, isOpen_univ⟩ (ext interior_univ.symm) -- bot ⟨∅, isOpen_empty⟩ rfl -- sup (fun U V => ⟨↑U ∪ ↑V, U.2.union V.2⟩) rfl -- inf (fun U V => ⟨↑U ∩ ↑V, U.2.inter V.2⟩) (funext₂ fun U V => ext (U.2.inter V.2).interior_eq.symm) -- sSup (fun S => ⟨⋃ s ∈ S, ↑s, isOpen_biUnion fun s _ => s.2⟩) (funext fun _ => ext sSup_image.symm) -- sInf _ rfl @[simp] theorem mk_inf_mk {U V : Set α} {hU : IsOpen U} {hV : IsOpen V} : (⟨U, hU⟩ ⊓ ⟨V, hV⟩ : Opens α) = ⟨U ⊓ V, IsOpen.inter hU hV⟩ := rfl #align topological_space.opens.mk_inf_mk TopologicalSpace.Opens.mk_inf_mk @[simp, norm_cast] theorem coe_inf (s t : Opens α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t := rfl #align topological_space.opens.coe_inf TopologicalSpace.Opens.coe_inf @[simp, norm_cast] theorem coe_sup (s t : Opens α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t := rfl #align topological_space.opens.coe_sup TopologicalSpace.Opens.coe_sup @[simp, norm_cast] theorem coe_bot : ((⊥ : Opens α) : Set α) = ∅ := rfl #align topological_space.opens.coe_bot TopologicalSpace.Opens.coe_bot @[simp] theorem mk_empty : (⟨∅, isOpen_empty⟩ : Opens α) = ⊥ := rfl -- Porting note (#10756): new lemma @[simp, norm_cast] theorem coe_eq_empty {U : Opens α} : (U : Set α) = ∅ ↔ U = ⊥ := SetLike.coe_injective.eq_iff' rfl @[simp, norm_cast] theorem coe_top : ((⊤ : Opens α) : Set α) = Set.univ := rfl #align topological_space.opens.coe_top TopologicalSpace.Opens.coe_top @[simp] theorem mk_univ : (⟨univ, isOpen_univ⟩ : Opens α) = ⊤ := rfl -- Porting note (#10756): new lemma @[simp, norm_cast] theorem coe_eq_univ {U : Opens α} : (U : Set α) = univ ↔ U = ⊤ := SetLike.coe_injective.eq_iff' rfl @[simp, norm_cast] theorem coe_sSup {S : Set (Opens α)} : (↑(sSup S) : Set α) = ⋃ i ∈ S, ↑i := rfl #align topological_space.opens.coe_Sup TopologicalSpace.Opens.coe_sSup @[simp, norm_cast] theorem coe_finset_sup (f : ι → Opens α) (s : Finset ι) : (↑(s.sup f) : Set α) = s.sup ((↑) ∘ f) := map_finset_sup (⟨⟨(↑), coe_sup⟩, coe_bot⟩ : SupBotHom (Opens α) (Set α)) _ _ #align topological_space.opens.coe_finset_sup TopologicalSpace.Opens.coe_finset_sup @[simp, norm_cast] theorem coe_finset_inf (f : ι → Opens α) (s : Finset ι) : (↑(s.inf f) : Set α) = s.inf ((↑) ∘ f) := map_finset_inf (⟨⟨(↑), coe_inf⟩, coe_top⟩ : InfTopHom (Opens α) (Set α)) _ _ #align topological_space.opens.coe_finset_inf TopologicalSpace.Opens.coe_finset_inf instance : Inhabited (Opens α) := ⟨⊥⟩ -- porting note (#10754): new instance instance [IsEmpty α] : Unique (Opens α) where uniq _ := ext <\| Subsingleton.elim _ _ -- porting note (#10754): new instance instance [Nonempty α] : Nontrivial (Opens α) where exists_pair_ne := ⟨⊥, ⊤, mt coe_inj.2 empty_ne_univ⟩ @[simp, norm_cast] theorem coe_iSup {ι} (s : ι → Opens α) : ((⨆ i, s i : Opens α) : Set α) = ⋃ i, s i := by simp [iSup] #align topological_space.opens.coe_supr TopologicalSpace.Opens.coe_iSup theorem iSup_def {ι} (s : ι → Opens α) : ⨆ i, s i = ⟨⋃ i, s i, isOpen_iUnion fun i => (s i).2⟩ := ext <\| coe_iSup s #align topological_space.opens.supr_def TopologicalSpace.Opens.iSup_def @[simp] theorem iSup_mk {ι} (s : ι → Set α) (h : ∀ i, IsOpen (s i)) : (⨆ i, ⟨s i, h i⟩ : Opens α) = ⟨⋃ i, s i, isOpen_iUnion h⟩ := iSup_def _ #align topological_space.opens.supr_mk TopologicalSpace.Opens.iSup_mk @[simp] theorem mem_iSup {ι} {x : α} {s : ι → Opens α} : x ∈ iSup s ↔ ∃ i, x ∈ s i := by rw [← SetLike.mem_coe] simp #align topological_space.opens.mem_supr TopologicalSpace.Opens.mem_iSup @[simp] theorem mem_sSup {Us : Set (Opens α)} {x : α} : x ∈ sSup Us ↔ ∃ u ∈ Us, x ∈ u := by simp_rw [sSup_eq_iSup, mem_iSup, exists_prop] #align topological_space.opens.mem_Sup TopologicalSpace.Opens.mem_sSup instance : Frame (Opens α) := { inferInstanceAs (CompleteLattice (Opens α)) with sSup := sSup inf_sSup_le_iSup_inf := fun a s => (ext <\| by simp only [coe_inf, coe_iSup, coe_sSup, Set.inter_iUnion₂]).le } theorem openEmbedding' (U : Opens α) : OpenEmbedding (Subtype.val : U → α) := U.isOpen.openEmbedding_subtype_val theorem openEmbedding_of_le {U V : Opens α} (i : U ≤ V) : OpenEmbedding (Set.inclusion <\| SetLike.coe_subset_coe.2 i) := { toEmbedding := embedding_inclusion i isOpen_range := by rw [Set.range_inclusion i] exact U.isOpen.preimage continuous_subtype_val } #align topological_space.opens.open_embedding_of_le TopologicalSpace.Opens.openEmbedding_of_le theorem not_nonempty_iff_eq_bot (U : Opens α) : ¬Set.Nonempty (U : Set α) ↔ U = ⊥ := by rw [← coe_inj, coe_bot, ← Set.not_nonempty_iff_eq_empty] #align topological_space.opens.not_nonempty_iff_eq_bot TopologicalSpace.Opens.not_nonempty_iff_eq_bot theorem ne_bot_iff_nonempty (U : Opens α) : U ≠ ⊥ ↔ Set.Nonempty (U : Set α) := by rw [Ne, ← not_nonempty_iff_eq_bot, not_not] #align topological_space.opens.ne_bot_iff_nonempty TopologicalSpace.Opens.ne_bot_iff_nonempty /-- An open set in the indiscrete topology is either empty or the whole space. -/ theorem eq_bot_or_top {α} [t : TopologicalSpace α] (h : t = ⊤) (U : Opens α) : U = ⊥ ∨ U = ⊤ := by subst h; letI : TopologicalSpace α := ⊤ rw [← coe_eq_empty, ← coe_eq_univ, ← isOpen_top_iff] exact U.2 #align topological_space.opens.eq_bot_or_top TopologicalSpace.Opens.eq_bot_or_top -- porting note (#10754): new instance instance [Nonempty α] [Subsingleton α] : IsSimpleOrder (Opens α) where eq_bot_or_eq_top := eq_bot_or_top <\| Subsingleton.elim _ _ /-- A set of `opens α` is a basis if the set of corresponding sets is a topological basis. -/ def IsBasis (B : Set (Opens α)) : Prop := IsTopologicalBasis (((↑) : _ → Set α) '' B) #align topological_space.opens.is_basis TopologicalSpace.Opens.IsBasis theorem isBasis_iff_nbhd {B : Set (Opens α)} : IsBasis B ↔ ∀ {U : Opens α} {x}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U := by constructor <;> intro h · rintro ⟨sU, hU⟩ x hx rcases h.mem_nhds_iff.mp (IsOpen.mem_nhds hU hx) with ⟨sV, ⟨⟨V, H₁, H₂⟩, hsV⟩⟩ refine ⟨V, H₁, ?_⟩ cases V dsimp at H₂ subst H₂ exact hsV · refine isTopologicalBasis_of_isOpen_of_nhds ?_ ?_ · rintro sU ⟨U, -, rfl⟩ exact U.2 · intro x sU hx hsU rcases @h ⟨sU, hsU⟩ x hx with ⟨V, hV, H⟩ exact ⟨V, ⟨V, hV, rfl⟩, H⟩ #align topological_space.opens.is_basis_iff_nbhd TopologicalSpace.Opens.isBasis_iff_nbhd theorem isBasis_iff_cover {B : Set (Opens α)} : IsBasis B ↔ ∀ U : Opens α, ∃ Us, Us ⊆ B ∧ U = sSup Us := by constructor · intro hB U refine ⟨{ V : Opens α \| V ∈ B ∧ V ≤ U }, fun U hU => hU.left, ext ?_⟩ rw [coe_sSup, hB.open_eq_sUnion' U.isOpen] simp_rw [sUnion_eq_biUnion, iUnion, mem_setOf_eq, iSup_and, iSup_image] rfl · intro h rw [isBasis_iff_nbhd] intro U x hx rcases h U with ⟨Us, hUs, rfl⟩ rcases mem_sSup.1 hx with ⟨U, Us, xU⟩ exact ⟨U, hUs Us, xU, le_sSup Us⟩ #align topological_space.opens.is_basis_iff_cover TopologicalSpace.Opens.isBasis_iff_cover /-- If `α` has a basis consisting of compact opens, then an open set in `α` is compact open iff it is a finite union of some elements in the basis -/ theorem IsBasis.isCompact_open_iff_eq_finite_iUnion {ι : Type} (b : ι → Opens α) (hb : IsBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i : Set α)) (U : Set α) : IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by apply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : ι => (b i).1 · convert (config := {transparency := .default}) hb ext simp · exact hb' #align topological_space.opens.is_basis.is_compact_open_iff_eq_finite_Union TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion @[simp]	Mathlib/Topology/Sets/Opens.lean	345	359	theorem isCompactElement_iff (s : Opens α) : CompleteLattice.IsCompactElement s ↔ IsCompact (s : Set α) := by	rw [isCompact_iff_finite_subcover, CompleteLattice.isCompactElement_iff] refine ⟨?_, fun H ι U hU => ?_⟩ · introv H hU hU' obtain ⟨t, ht⟩ := H ι (fun i => ⟨U i, hU i⟩) (by simpa) refine ⟨t, Set.Subset.trans ht ?_⟩ rw [coe_finset_sup, Finset.sup_eq_iSup] rfl · obtain ⟨t, ht⟩ := H (fun i => U i) (fun i => (U i).isOpen) (by simpa using show (s : Set α) ⊆ ↑(iSup U) from hU) refine ⟨t, Set.Subset.trans ht ?_⟩ simp only [Set.iUnion_subset_iff] show ∀ i ∈ t, U i ≤ t.sup U exact fun i => Finset.le_sup
/- Copyright (c) 2020 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis -/ import Mathlib.Algebra.Field.Basic import Mathlib.Algebra.Order.Field.Defs import Mathlib.Data.Tree.Basic import Mathlib.Logic.Basic import Mathlib.Tactic.NormNum.Core import Mathlib.Util.SynthesizeUsing import Mathlib.Util.Qq /-! # A tactic for canceling numeric denominators This file defines tactics that cancel numeric denominators from field Expressions. As an example, we want to transform a comparison `5(a/3 + b/4) < c/3` into the equivalent `5(4a + 3b) < 4c`. ## Implementation notes The tooling here was originally written for `linarith`, not intended as an interactive tactic. The interactive version has been split off because it is sometimes convenient to use on its own. There are likely some rough edges to it. Improving this tactic would be a good project for someone interested in learning tactic programming. -/ open Lean Parser Tactic Mathlib Meta NormNum Qq initialize registerTraceClass `CancelDenoms namespace CancelDenoms /-! ### Lemmas used in the procedure -/ theorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α} (h1 : n1 e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1, ← mul_assoc n2, mul_comm n2, mul_assoc, h2] #align cancel_factors.mul_subst CancelDenoms.mul_subst theorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α} (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul] #align cancel_factors.div_subst CancelDenoms.div_subst theorem cancel_factors_eq_div {α} [Field α] {n e e' : α} (h : n * e = e') (h2 : n ≠ 0) : e = e' / n := eq_div_of_mul_eq h2 <\| by rwa [mul_comm] at h #align cancel_factors.cancel_factors_eq_div CancelDenoms.cancel_factors_eq_div theorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) : n * (e1 + e2) = t1 + t2 := by simp [left_distrib, ] #align cancel_factors.add_subst CancelDenoms.add_subst theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n e1 = t1) (h2 : n * e2 = t2) : n * (e1 - e2) = t1 - t2 := by simp [left_distrib, , sub_eq_add_neg] #align cancel_factors.sub_subst CancelDenoms.sub_subst theorem neg_subst {α} [Ring α] {n e t : α} (h1 : n e = t) : n * -e = -t := by simp [] #align cancel_factors.neg_subst CancelDenoms.neg_subst theorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ} (h1 : n e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2 := by rw [← h2, ← h1, mul_pow, mul_assoc] theorem inv_subst {α} [Field α] {n k e : α} (h2 : e ≠ 0) (h3 : n * e = k) : k * (e ⁻¹) = n := by rw [← div_eq_mul_inv, ← h3, mul_div_cancel_right₀ _ h2] theorem cancel_factors_lt {α} [LinearOrderedField α] {a b ad bd a' b' gcd : α} (ha : ad * a = a') (hb : bd * b = b') (had : 0 < ad) (hbd : 0 < bd) (hgcd : 0 < gcd) : (a < b) = (1 / gcd * (bd * a') < 1 / gcd * (ad * b')) := by rw [mul_lt_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_lt_mul_left] · exact mul_pos had hbd · exact one_div_pos.2 hgcd #align cancel_factors.cancel_factors_lt CancelDenoms.cancel_factors_lt	Mathlib/Tactic/CancelDenoms/Core.lean	81	86	theorem cancel_factors_le {α} [LinearOrderedField α] {a b ad bd a' b' gcd : α} (ha : ad * a = a') (hb : bd * b = b') (had : 0 < ad) (hbd : 0 < bd) (hgcd : 0 < gcd) : (a ≤ b) = (1 / gcd * (bd * a') ≤ 1 / gcd * (ad * b')) := by	rw [mul_le_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_le_mul_left] · exact mul_pos had hbd · exact one_div_pos.2 hgcd
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn -/ import Mathlib.Data.Finsupp.Multiset import Mathlib.Order.Bounded import Mathlib.SetTheory.Cardinal.PartENat import Mathlib.SetTheory.Ordinal.Principal import Mathlib.Tactic.Linarith #align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f" /-! # Cardinals and ordinals Relationships between cardinals and ordinals, properties of cardinals that are proved using ordinals. ## Main definitions * The function `Cardinal.aleph'` gives the cardinals listed by their ordinal index, and is the inverse of `Cardinal.aleph/idx`. `aleph' n = n`, `aleph' ω = ℵ₀`, `aleph' (ω + 1) = succ ℵ₀`, etc. It is an order isomorphism between ordinals and cardinals. * The function `Cardinal.aleph` gives the infinite cardinals listed by their ordinal index. `aleph 0 = ℵ₀`, `aleph 1 = succ ℵ₀` is the first uncountable cardinal, and so on. The notation `ω_` combines the latter with `Cardinal.ord`, giving an enumeration of (infinite) initial ordinals. Thus `ω_ 0 = ω` and `ω₁ = ω_ 1` is the first uncountable ordinal. * The function `Cardinal.beth` enumerates the Beth cardinals. `beth 0 = ℵ₀`, `beth (succ o) = 2 ^ beth o`, and for a limit ordinal `o`, `beth o` is the supremum of `beth a` for `a < o`. ## Main Statements * `Cardinal.mul_eq_max` and `Cardinal.add_eq_max` state that the product (resp. sum) of two infinite cardinals is just their maximum. Several variations around this fact are also given. * `Cardinal.mk_list_eq_mk` : when `α` is infinite, `α` and `List α` have the same cardinality. * simp lemmas for inequalities between `bit0 a` and `bit1 b` are registered, making `simp` able to prove inequalities about numeral cardinals. ## Tags cardinal arithmetic (for infinite cardinals) -/ noncomputable section open Function Set Cardinal Equiv Order Ordinal open scoped Classical universe u v w namespace Cardinal section UsingOrdinals theorem ord_isLimit {c} (co : ℵ₀ ≤ c) : (ord c).IsLimit := by refine ⟨fun h => aleph0_ne_zero ?_, fun a => lt_imp_lt_of_le_imp_le fun h => ?_⟩ · rw [← Ordinal.le_zero, ord_le] at h simpa only [card_zero, nonpos_iff_eq_zero] using co.trans h · rw [ord_le] at h ⊢ rwa [← @add_one_of_aleph0_le (card a), ← card_succ] rw [← ord_le, ← le_succ_of_isLimit, ord_le] · exact co.trans h · rw [ord_aleph0] exact omega_isLimit #align cardinal.ord_is_limit Cardinal.ord_isLimit theorem noMaxOrder {c} (h : ℵ₀ ≤ c) : NoMaxOrder c.ord.out.α := Ordinal.out_no_max_of_succ_lt (ord_isLimit h).2 /-! ### Aleph cardinals -/ section aleph /-- The `aleph'` index function, which gives the ordinal index of a cardinal. (The `aleph'` part is because unlike `aleph` this counts also the finite stages. So `alephIdx n = n`, `alephIdx ω = ω`, `alephIdx ℵ₁ = ω + 1` and so on.) In this definition, we register additionally that this function is an initial segment, i.e., it is order preserving and its range is an initial segment of the ordinals. For the basic function version, see `alephIdx`. For an upgraded version stating that the range is everything, see `AlephIdx.rel_iso`. -/ def alephIdx.initialSeg : @InitialSeg Cardinal Ordinal (· < ·) (· < ·) := @RelEmbedding.collapse Cardinal Ordinal (· < ·) (· < ·) _ Cardinal.ord.orderEmbedding.ltEmbedding #align cardinal.aleph_idx.initial_seg Cardinal.alephIdx.initialSeg /-- The `aleph'` index function, which gives the ordinal index of a cardinal. (The `aleph'` part is because unlike `aleph` this counts also the finite stages. So `alephIdx n = n`, `alephIdx ω = ω`, `alephIdx ℵ₁ = ω + 1` and so on.) For an upgraded version stating that the range is everything, see `AlephIdx.rel_iso`. -/ def alephIdx : Cardinal → Ordinal := alephIdx.initialSeg #align cardinal.aleph_idx Cardinal.alephIdx @[simp] theorem alephIdx.initialSeg_coe : (alephIdx.initialSeg : Cardinal → Ordinal) = alephIdx := rfl #align cardinal.aleph_idx.initial_seg_coe Cardinal.alephIdx.initialSeg_coe @[simp] theorem alephIdx_lt {a b} : alephIdx a < alephIdx b ↔ a < b := alephIdx.initialSeg.toRelEmbedding.map_rel_iff #align cardinal.aleph_idx_lt Cardinal.alephIdx_lt @[simp] theorem alephIdx_le {a b} : alephIdx a ≤ alephIdx b ↔ a ≤ b := by rw [← not_lt, ← not_lt, alephIdx_lt] #align cardinal.aleph_idx_le Cardinal.alephIdx_le theorem alephIdx.init {a b} : b < alephIdx a → ∃ c, alephIdx c = b := alephIdx.initialSeg.init #align cardinal.aleph_idx.init Cardinal.alephIdx.init /-- The `aleph'` index function, which gives the ordinal index of a cardinal. (The `aleph'` part is because unlike `aleph` this counts also the finite stages. So `alephIdx n = n`, `alephIdx ℵ₀ = ω`, `alephIdx ℵ₁ = ω + 1` and so on.) In this version, we register additionally that this function is an order isomorphism between cardinals and ordinals. For the basic function version, see `alephIdx`. -/ def alephIdx.relIso : @RelIso Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) := @RelIso.ofSurjective Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) alephIdx.initialSeg.{u} <\| (InitialSeg.eq_or_principal alephIdx.initialSeg.{u}).resolve_right fun ⟨o, e⟩ => by have : ∀ c, alephIdx c < o := fun c => (e _).2 ⟨_, rfl⟩ refine Ordinal.inductionOn o ?_ this; intro α r _ h let s := ⨆ a, invFun alephIdx (Ordinal.typein r a) apply (lt_succ s).not_le have I : Injective.{u+2, u+2} alephIdx := alephIdx.initialSeg.toEmbedding.injective simpa only [typein_enum, leftInverse_invFun I (succ s)] using le_ciSup (Cardinal.bddAbove_range.{u, u} fun a : α => invFun alephIdx (Ordinal.typein r a)) (Ordinal.enum r _ (h (succ s))) #align cardinal.aleph_idx.rel_iso Cardinal.alephIdx.relIso @[simp] theorem alephIdx.relIso_coe : (alephIdx.relIso : Cardinal → Ordinal) = alephIdx := rfl #align cardinal.aleph_idx.rel_iso_coe Cardinal.alephIdx.relIso_coe @[simp] theorem type_cardinal : @type Cardinal (· < ·) _ = Ordinal.univ.{u, u + 1} := by rw [Ordinal.univ_id]; exact Quotient.sound ⟨alephIdx.relIso⟩ #align cardinal.type_cardinal Cardinal.type_cardinal @[simp] theorem mk_cardinal : #Cardinal = univ.{u, u + 1} := by simpa only [card_type, card_univ] using congr_arg card type_cardinal #align cardinal.mk_cardinal Cardinal.mk_cardinal /-- The `aleph'` function gives the cardinals listed by their ordinal index, and is the inverse of `aleph_idx`. `aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = succ ℵ₀`, etc. In this version, we register additionally that this function is an order isomorphism between ordinals and cardinals. For the basic function version, see `aleph'`. -/ def Aleph'.relIso := Cardinal.alephIdx.relIso.symm #align cardinal.aleph'.rel_iso Cardinal.Aleph'.relIso /-- The `aleph'` function gives the cardinals listed by their ordinal index, and is the inverse of `aleph_idx`. `aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = succ ℵ₀`, etc. -/ def aleph' : Ordinal → Cardinal := Aleph'.relIso #align cardinal.aleph' Cardinal.aleph' @[simp] theorem aleph'.relIso_coe : (Aleph'.relIso : Ordinal → Cardinal) = aleph' := rfl #align cardinal.aleph'.rel_iso_coe Cardinal.aleph'.relIso_coe @[simp] theorem aleph'_lt {o₁ o₂ : Ordinal} : aleph' o₁ < aleph' o₂ ↔ o₁ < o₂ := Aleph'.relIso.map_rel_iff #align cardinal.aleph'_lt Cardinal.aleph'_lt @[simp] theorem aleph'_le {o₁ o₂ : Ordinal} : aleph' o₁ ≤ aleph' o₂ ↔ o₁ ≤ o₂ := le_iff_le_iff_lt_iff_lt.2 aleph'_lt #align cardinal.aleph'_le Cardinal.aleph'_le @[simp] theorem aleph'_alephIdx (c : Cardinal) : aleph' c.alephIdx = c := Cardinal.alephIdx.relIso.toEquiv.symm_apply_apply c #align cardinal.aleph'_aleph_idx Cardinal.aleph'_alephIdx @[simp] theorem alephIdx_aleph' (o : Ordinal) : (aleph' o).alephIdx = o := Cardinal.alephIdx.relIso.toEquiv.apply_symm_apply o #align cardinal.aleph_idx_aleph' Cardinal.alephIdx_aleph' @[simp] theorem aleph'_zero : aleph' 0 = 0 := by rw [← nonpos_iff_eq_zero, ← aleph'_alephIdx 0, aleph'_le] apply Ordinal.zero_le #align cardinal.aleph'_zero Cardinal.aleph'_zero @[simp] theorem aleph'_succ {o : Ordinal} : aleph' (succ o) = succ (aleph' o) := by apply (succ_le_of_lt <\| aleph'_lt.2 <\| lt_succ o).antisymm' (Cardinal.alephIdx_le.1 <\| _) rw [alephIdx_aleph', succ_le_iff, ← aleph'_lt, aleph'_alephIdx] apply lt_succ #align cardinal.aleph'_succ Cardinal.aleph'_succ @[simp] theorem aleph'_nat : ∀ n : ℕ, aleph' n = n \| 0 => aleph'_zero \| n + 1 => show aleph' (succ n) = n.succ by rw [aleph'_succ, aleph'_nat n, nat_succ] #align cardinal.aleph'_nat Cardinal.aleph'_nat theorem aleph'_le_of_limit {o : Ordinal} (l : o.IsLimit) {c} : aleph' o ≤ c ↔ ∀ o' < o, aleph' o' ≤ c := ⟨fun h o' h' => (aleph'_le.2 <\| h'.le).trans h, fun h => by rw [← aleph'_alephIdx c, aleph'_le, limit_le l] intro x h' rw [← aleph'_le, aleph'_alephIdx] exact h _ h'⟩ #align cardinal.aleph'_le_of_limit Cardinal.aleph'_le_of_limit theorem aleph'_limit {o : Ordinal} (ho : o.IsLimit) : aleph' o = ⨆ a : Iio o, aleph' a := by refine le_antisymm ?_ (ciSup_le' fun i => aleph'_le.2 (le_of_lt i.2)) rw [aleph'_le_of_limit ho] exact fun a ha => le_ciSup (bddAbove_of_small _) (⟨a, ha⟩ : Iio o) #align cardinal.aleph'_limit Cardinal.aleph'_limit @[simp] theorem aleph'_omega : aleph' ω = ℵ₀ := eq_of_forall_ge_iff fun c => by simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le] exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat]) #align cardinal.aleph'_omega Cardinal.aleph'_omega /-- `aleph'` and `aleph_idx` form an equivalence between `Ordinal` and `Cardinal` -/ @[simp] def aleph'Equiv : Ordinal ≃ Cardinal := ⟨aleph', alephIdx, alephIdx_aleph', aleph'_alephIdx⟩ #align cardinal.aleph'_equiv Cardinal.aleph'Equiv /-- The `aleph` function gives the infinite cardinals listed by their ordinal index. `aleph 0 = ℵ₀`, `aleph 1 = succ ℵ₀` is the first uncountable cardinal, and so on. -/ def aleph (o : Ordinal) : Cardinal := aleph' (ω + o) #align cardinal.aleph Cardinal.aleph @[simp] theorem aleph_lt {o₁ o₂ : Ordinal} : aleph o₁ < aleph o₂ ↔ o₁ < o₂ := aleph'_lt.trans (add_lt_add_iff_left _) #align cardinal.aleph_lt Cardinal.aleph_lt @[simp] theorem aleph_le {o₁ o₂ : Ordinal} : aleph o₁ ≤ aleph o₂ ↔ o₁ ≤ o₂ := le_iff_le_iff_lt_iff_lt.2 aleph_lt #align cardinal.aleph_le Cardinal.aleph_le @[simp] theorem max_aleph_eq (o₁ o₂ : Ordinal) : max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) := by rcases le_total (aleph o₁) (aleph o₂) with h \| h · rw [max_eq_right h, max_eq_right (aleph_le.1 h)] · rw [max_eq_left h, max_eq_left (aleph_le.1 h)] #align cardinal.max_aleph_eq Cardinal.max_aleph_eq @[simp] theorem aleph_succ {o : Ordinal} : aleph (succ o) = succ (aleph o) := by rw [aleph, add_succ, aleph'_succ, aleph] #align cardinal.aleph_succ Cardinal.aleph_succ @[simp] theorem aleph_zero : aleph 0 = ℵ₀ := by rw [aleph, add_zero, aleph'_omega] #align cardinal.aleph_zero Cardinal.aleph_zero theorem aleph_limit {o : Ordinal} (ho : o.IsLimit) : aleph o = ⨆ a : Iio o, aleph a := by apply le_antisymm _ (ciSup_le' _) · rw [aleph, aleph'_limit (ho.add _)] refine ciSup_mono' (bddAbove_of_small _) ?_ rintro ⟨i, hi⟩ cases' lt_or_le i ω with h h · rcases lt_omega.1 h with ⟨n, rfl⟩ use ⟨0, ho.pos⟩ simpa using (nat_lt_aleph0 n).le · exact ⟨⟨_, (sub_lt_of_le h).2 hi⟩, aleph'_le.2 (le_add_sub _ _)⟩ · exact fun i => aleph_le.2 (le_of_lt i.2) #align cardinal.aleph_limit Cardinal.aleph_limit theorem aleph0_le_aleph' {o : Ordinal} : ℵ₀ ≤ aleph' o ↔ ω ≤ o := by rw [← aleph'_omega, aleph'_le] #align cardinal.aleph_0_le_aleph' Cardinal.aleph0_le_aleph' theorem aleph0_le_aleph (o : Ordinal) : ℵ₀ ≤ aleph o := by rw [aleph, aleph0_le_aleph'] apply Ordinal.le_add_right #align cardinal.aleph_0_le_aleph Cardinal.aleph0_le_aleph theorem aleph'_pos {o : Ordinal} (ho : 0 < o) : 0 < aleph' o := by rwa [← aleph'_zero, aleph'_lt] #align cardinal.aleph'_pos Cardinal.aleph'_pos theorem aleph_pos (o : Ordinal) : 0 < aleph o := aleph0_pos.trans_le (aleph0_le_aleph o) #align cardinal.aleph_pos Cardinal.aleph_pos @[simp] theorem aleph_toNat (o : Ordinal) : toNat (aleph o) = 0 := toNat_apply_of_aleph0_le <\| aleph0_le_aleph o #align cardinal.aleph_to_nat Cardinal.aleph_toNat @[simp] theorem aleph_toPartENat (o : Ordinal) : toPartENat (aleph o) = ⊤ := toPartENat_apply_of_aleph0_le <\| aleph0_le_aleph o #align cardinal.aleph_to_part_enat Cardinal.aleph_toPartENat instance nonempty_out_aleph (o : Ordinal) : Nonempty (aleph o).ord.out.α := by rw [out_nonempty_iff_ne_zero, ← ord_zero] exact fun h => (ord_injective h).not_gt (aleph_pos o) #align cardinal.nonempty_out_aleph Cardinal.nonempty_out_aleph theorem ord_aleph_isLimit (o : Ordinal) : (aleph o).ord.IsLimit := ord_isLimit <\| aleph0_le_aleph _ #align cardinal.ord_aleph_is_limit Cardinal.ord_aleph_isLimit instance (o : Ordinal) : NoMaxOrder (aleph o).ord.out.α := out_no_max_of_succ_lt (ord_aleph_isLimit o).2 theorem exists_aleph {c : Cardinal} : ℵ₀ ≤ c ↔ ∃ o, c = aleph o := ⟨fun h => ⟨alephIdx c - ω, by rw [aleph, Ordinal.add_sub_cancel_of_le, aleph'_alephIdx] rwa [← aleph0_le_aleph', aleph'_alephIdx]⟩, fun ⟨o, e⟩ => e.symm ▸ aleph0_le_aleph _⟩ #align cardinal.exists_aleph Cardinal.exists_aleph theorem aleph'_isNormal : IsNormal (ord ∘ aleph') := ⟨fun o => ord_lt_ord.2 <\| aleph'_lt.2 <\| lt_succ o, fun o l a => by simp [ord_le, aleph'_le_of_limit l]⟩ #align cardinal.aleph'_is_normal Cardinal.aleph'_isNormal theorem aleph_isNormal : IsNormal (ord ∘ aleph) := aleph'_isNormal.trans <\| add_isNormal ω #align cardinal.aleph_is_normal Cardinal.aleph_isNormal theorem succ_aleph0 : succ ℵ₀ = aleph 1 := by rw [← aleph_zero, ← aleph_succ, Ordinal.succ_zero] #align cardinal.succ_aleph_0 Cardinal.succ_aleph0 theorem aleph0_lt_aleph_one : ℵ₀ < aleph 1 := by rw [← succ_aleph0] apply lt_succ #align cardinal.aleph_0_lt_aleph_one Cardinal.aleph0_lt_aleph_one theorem countable_iff_lt_aleph_one {α : Type} (s : Set α) : s.Countable ↔ #s < aleph 1 := by rw [← succ_aleph0, lt_succ_iff, le_aleph0_iff_set_countable] #align cardinal.countable_iff_lt_aleph_one Cardinal.countable_iff_lt_aleph_one /-- Ordinals that are cardinals are unbounded. -/ theorem ord_card_unbounded : Unbounded (· < ·) { b : Ordinal \| b.card.ord = b } := unbounded_lt_iff.2 fun a => ⟨_, ⟨by dsimp rw [card_ord], (lt_ord_succ_card a).le⟩⟩ #align cardinal.ord_card_unbounded Cardinal.ord_card_unbounded theorem eq_aleph'_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) : ∃ a, (aleph' a).ord = o := ⟨Cardinal.alephIdx.relIso o.card, by simpa using ho⟩ #align cardinal.eq_aleph'_of_eq_card_ord Cardinal.eq_aleph'_of_eq_card_ord /-- `ord ∘ aleph'` enumerates the ordinals that are cardinals. -/ theorem ord_aleph'_eq_enum_card : ord ∘ aleph' = enumOrd { b : Ordinal \| b.card.ord = b } := by rw [← eq_enumOrd _ ord_card_unbounded, range_eq_iff] exact ⟨aleph'_isNormal.strictMono, ⟨fun a => by dsimp rw [card_ord], fun b hb => eq_aleph'_of_eq_card_ord hb⟩⟩ #align cardinal.ord_aleph'_eq_enum_card Cardinal.ord_aleph'_eq_enum_card /-- Infinite ordinals that are cardinals are unbounded. -/ theorem ord_card_unbounded' : Unbounded (· < ·) { b : Ordinal \| b.card.ord = b ∧ ω ≤ b } := (unbounded_lt_inter_le ω).2 ord_card_unbounded #align cardinal.ord_card_unbounded' Cardinal.ord_card_unbounded' theorem eq_aleph_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) (ho' : ω ≤ o) : ∃ a, (aleph a).ord = o := by cases' eq_aleph'_of_eq_card_ord ho with a ha use a - ω unfold aleph rwa [Ordinal.add_sub_cancel_of_le] rwa [← aleph0_le_aleph', ← ord_le_ord, ha, ord_aleph0] #align cardinal.eq_aleph_of_eq_card_ord Cardinal.eq_aleph_of_eq_card_ord /-- `ord ∘ aleph` enumerates the infinite ordinals that are cardinals. -/ theorem ord_aleph_eq_enum_card : ord ∘ aleph = enumOrd { b : Ordinal \| b.card.ord = b ∧ ω ≤ b } := by rw [← eq_enumOrd _ ord_card_unbounded'] use aleph_isNormal.strictMono rw [range_eq_iff] refine ⟨fun a => ⟨?_, ?_⟩, fun b hb => eq_aleph_of_eq_card_ord hb.1 hb.2⟩ · rw [Function.comp_apply, card_ord] · rw [← ord_aleph0, Function.comp_apply, ord_le_ord] exact aleph0_le_aleph _ #align cardinal.ord_aleph_eq_enum_card Cardinal.ord_aleph_eq_enum_card end aleph /-! ### Beth cardinals -/ section beth /-- Beth numbers are defined so that `beth 0 = ℵ₀`, `beth (succ o) = 2 ^ (beth o)`, and when `o` is a limit ordinal, `beth o` is the supremum of `beth o'` for `o' < o`. Assuming the generalized continuum hypothesis, which is undecidable in ZFC, `beth o = aleph o` for every `o`. -/ def beth (o : Ordinal.{u}) : Cardinal.{u} := limitRecOn o aleph0 (fun _ x => (2 : Cardinal) ^ x) fun a _ IH => ⨆ b : Iio a, IH b.1 b.2 #align cardinal.beth Cardinal.beth @[simp] theorem beth_zero : beth 0 = aleph0 := limitRecOn_zero _ _ _ #align cardinal.beth_zero Cardinal.beth_zero @[simp] theorem beth_succ (o : Ordinal) : beth (succ o) = 2 ^ beth o := limitRecOn_succ _ _ _ _ #align cardinal.beth_succ Cardinal.beth_succ theorem beth_limit {o : Ordinal} : o.IsLimit → beth o = ⨆ a : Iio o, beth a := limitRecOn_limit _ _ _ _ #align cardinal.beth_limit Cardinal.beth_limit theorem beth_strictMono : StrictMono beth := by intro a b induction' b using Ordinal.induction with b IH generalizing a intro h rcases zero_or_succ_or_limit b with (rfl \| ⟨c, rfl⟩ \| hb) · exact (Ordinal.not_lt_zero a h).elim · rw [lt_succ_iff] at h rw [beth_succ] apply lt_of_le_of_lt _ (cantor _) rcases eq_or_lt_of_le h with (rfl \| h) · rfl exact (IH c (lt_succ c) h).le · apply (cantor _).trans_le rw [beth_limit hb, ← beth_succ] exact le_ciSup (bddAbove_of_small _) (⟨_, hb.succ_lt h⟩ : Iio b) #align cardinal.beth_strict_mono Cardinal.beth_strictMono theorem beth_mono : Monotone beth := beth_strictMono.monotone #align cardinal.beth_mono Cardinal.beth_mono @[simp] theorem beth_lt {o₁ o₂ : Ordinal} : beth o₁ < beth o₂ ↔ o₁ < o₂ := beth_strictMono.lt_iff_lt #align cardinal.beth_lt Cardinal.beth_lt @[simp] theorem beth_le {o₁ o₂ : Ordinal} : beth o₁ ≤ beth o₂ ↔ o₁ ≤ o₂ := beth_strictMono.le_iff_le #align cardinal.beth_le Cardinal.beth_le theorem aleph_le_beth (o : Ordinal) : aleph o ≤ beth o := by induction o using limitRecOn with \| H₁ => simp \| H₂ o h => rw [aleph_succ, beth_succ, succ_le_iff] exact (cantor _).trans_le (power_le_power_left two_ne_zero h) \| H₃ o ho IH => rw [aleph_limit ho, beth_limit ho] exact ciSup_mono (bddAbove_of_small _) fun x => IH x.1 x.2 #align cardinal.aleph_le_beth Cardinal.aleph_le_beth theorem aleph0_le_beth (o : Ordinal) : ℵ₀ ≤ beth o := (aleph0_le_aleph o).trans <\| aleph_le_beth o #align cardinal.aleph_0_le_beth Cardinal.aleph0_le_beth theorem beth_pos (o : Ordinal) : 0 < beth o := aleph0_pos.trans_le <\| aleph0_le_beth o #align cardinal.beth_pos Cardinal.beth_pos theorem beth_ne_zero (o : Ordinal) : beth o ≠ 0 := (beth_pos o).ne' #align cardinal.beth_ne_zero Cardinal.beth_ne_zero theorem beth_normal : IsNormal.{u} fun o => (beth o).ord := (isNormal_iff_strictMono_limit _).2 ⟨ord_strictMono.comp beth_strictMono, fun o ho a ha => by rw [beth_limit ho, ord_le] exact ciSup_le' fun b => ord_le.1 (ha _ b.2)⟩ #align cardinal.beth_normal Cardinal.beth_normal end beth /-! ### Properties of `mul` -/ section mulOrdinals /-- If `α` is an infinite type, then `α × α` and `α` have the same cardinality. -/ theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c c = c := by refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c) -- the only nontrivial part is `c * c ≤ c`. We prove it inductively. refine Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => ?_) h -- consider the minimal well-order `r` on `α` (a type with cardinality `c`). rcases ord_eq α with ⟨r, wo, e⟩ letI := linearOrderOfSTO r haveI : IsWellOrder α (· < ·) := wo -- Define an order `s` on `α × α` by writing `(a, b) < (c, d)` if `max a b < max c d`, or -- the max are equal and `a < c`, or the max are equal and `a = c` and `b < d`. let g : α × α → α := fun p => max p.1 p.2 let f : α × α ↪ Ordinal × α × α := ⟨fun p : α × α => (typein (· < ·) (g p), p), fun p q => congr_arg Prod.snd⟩ let s := f ⁻¹'o Prod.Lex (· < ·) (Prod.Lex (· < ·) (· < ·)) -- this is a well order on `α × α`. haveI : IsWellOrder _ s := (RelEmbedding.preimage _ _).isWellOrder /- it suffices to show that this well order is smaller than `r` if it were larger, then `r` would be a strict prefix of `s`. It would be contained in `β × β` for some `β` of cardinality `< c`. By the inductive assumption, this set has the same cardinality as `β` (or it is finite if `β` is finite), so it is `< c`, which is a contradiction. -/ suffices type s ≤ type r by exact card_le_card this refine le_of_forall_lt fun o h => ?_ rcases typein_surj s h with ⟨p, rfl⟩ rw [← e, lt_ord] refine lt_of_le_of_lt (?_ : _ ≤ card (succ (typein (· < ·) (g p))) * card (succ (typein (· < ·) (g p)))) ?_ · have : { q \| s q p } ⊆ insert (g p) { x \| x < g p } ×ˢ insert (g p) { x \| x < g p } := by intro q h simp only [s, f, Preimage, ge_iff_le, Embedding.coeFn_mk, Prod.lex_def, typein_lt_typein, typein_inj, mem_setOf_eq] at h exact max_le_iff.1 (le_iff_lt_or_eq.2 <\| h.imp_right And.left) suffices H : (insert (g p) { x \| r x (g p) } : Set α) ≃ Sum { x \| r x (g p) } PUnit from ⟨(Set.embeddingOfSubset _ _ this).trans ((Equiv.Set.prod _ _).trans (H.prodCongr H)).toEmbedding⟩ refine (Equiv.Set.insert ?_).trans ((Equiv.refl _).sumCongr punitEquivPUnit) apply @irrefl _ r cases' lt_or_le (card (succ (typein (· < ·) (g p)))) ℵ₀ with qo qo · exact (mul_lt_aleph0 qo qo).trans_le ol · suffices (succ (typein LT.lt (g p))).card < ⟦α⟧ from (IH _ this qo).trans_lt this rw [← lt_ord] apply (ord_isLimit ol).2 rw [mk'_def, e] apply typein_lt_type #align cardinal.mul_eq_self Cardinal.mul_eq_self end mulOrdinals end UsingOrdinals /-! Properties of `mul`, not requiring ordinals -/ section mul /-- If `α` and `β` are infinite types, then the cardinality of `α × β` is the maximum of the cardinalities of `α` and `β`. -/ theorem mul_eq_max {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : ℵ₀ ≤ b) : a * b = max a b := le_antisymm (mul_eq_self (ha.trans (le_max_left a b)) ▸ mul_le_mul' (le_max_left _ _) (le_max_right _ _)) <\| max_le (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans hb) a) (by simpa only [one_mul] using mul_le_mul_right' (one_le_aleph0.trans ha) b) #align cardinal.mul_eq_max Cardinal.mul_eq_max @[simp] theorem mul_mk_eq_max {α β : Type u} [Infinite α] [Infinite β] : #α * #β = max #α #β := mul_eq_max (aleph0_le_mk α) (aleph0_le_mk β) #align cardinal.mul_mk_eq_max Cardinal.mul_mk_eq_max @[simp] theorem aleph_mul_aleph (o₁ o₂ : Ordinal) : aleph o₁ * aleph o₂ = aleph (max o₁ o₂) := by rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), max_aleph_eq] #align cardinal.aleph_mul_aleph Cardinal.aleph_mul_aleph @[simp] theorem aleph0_mul_eq {a : Cardinal} (ha : ℵ₀ ≤ a) : ℵ₀ * a = a := (mul_eq_max le_rfl ha).trans (max_eq_right ha) #align cardinal.aleph_0_mul_eq Cardinal.aleph0_mul_eq @[simp] theorem mul_aleph0_eq {a : Cardinal} (ha : ℵ₀ ≤ a) : a * ℵ₀ = a := (mul_eq_max ha le_rfl).trans (max_eq_left ha) #align cardinal.mul_aleph_0_eq Cardinal.mul_aleph0_eq -- Porting note (#10618): removed `simp`, `simp` can prove it theorem aleph0_mul_mk_eq {α : Type} [Infinite α] : ℵ₀ #α = #α := aleph0_mul_eq (aleph0_le_mk α) #align cardinal.aleph_0_mul_mk_eq Cardinal.aleph0_mul_mk_eq -- Porting note (#10618): removed `simp`, `simp` can prove it theorem mk_mul_aleph0_eq {α : Type} [Infinite α] : #α ℵ₀ = #α := mul_aleph0_eq (aleph0_le_mk α) #align cardinal.mk_mul_aleph_0_eq Cardinal.mk_mul_aleph0_eq @[simp] theorem aleph0_mul_aleph (o : Ordinal) : ℵ₀ * aleph o = aleph o := aleph0_mul_eq (aleph0_le_aleph o) #align cardinal.aleph_0_mul_aleph Cardinal.aleph0_mul_aleph @[simp] theorem aleph_mul_aleph0 (o : Ordinal) : aleph o * ℵ₀ = aleph o := mul_aleph0_eq (aleph0_le_aleph o) #align cardinal.aleph_mul_aleph_0 Cardinal.aleph_mul_aleph0 theorem mul_lt_of_lt {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a * b < c := (mul_le_mul' (le_max_left a b) (le_max_right a b)).trans_lt <\| (lt_or_le (max a b) ℵ₀).elim (fun h => (mul_lt_aleph0 h h).trans_le hc) fun h => by rw [mul_eq_self h] exact max_lt h1 h2 #align cardinal.mul_lt_of_lt Cardinal.mul_lt_of_lt theorem mul_le_max_of_aleph0_le_left {a b : Cardinal} (h : ℵ₀ ≤ a) : a * b ≤ max a b := by convert mul_le_mul' (le_max_left a b) (le_max_right a b) using 1 rw [mul_eq_self] exact h.trans (le_max_left a b) #align cardinal.mul_le_max_of_aleph_0_le_left Cardinal.mul_le_max_of_aleph0_le_left theorem mul_eq_max_of_aleph0_le_left {a b : Cardinal} (h : ℵ₀ ≤ a) (h' : b ≠ 0) : a * b = max a b := by rcases le_or_lt ℵ₀ b with hb \| hb · exact mul_eq_max h hb refine (mul_le_max_of_aleph0_le_left h).antisymm ?_ have : b ≤ a := hb.le.trans h rw [max_eq_left this] convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') a rw [mul_one] #align cardinal.mul_eq_max_of_aleph_0_le_left Cardinal.mul_eq_max_of_aleph0_le_left theorem mul_le_max_of_aleph0_le_right {a b : Cardinal} (h : ℵ₀ ≤ b) : a * b ≤ max a b := by simpa only [mul_comm b, max_comm b] using mul_le_max_of_aleph0_le_left h #align cardinal.mul_le_max_of_aleph_0_le_right Cardinal.mul_le_max_of_aleph0_le_right theorem mul_eq_max_of_aleph0_le_right {a b : Cardinal} (h' : a ≠ 0) (h : ℵ₀ ≤ b) : a * b = max a b := by rw [mul_comm, max_comm] exact mul_eq_max_of_aleph0_le_left h h' #align cardinal.mul_eq_max_of_aleph_0_le_right Cardinal.mul_eq_max_of_aleph0_le_right theorem mul_eq_max' {a b : Cardinal} (h : ℵ₀ ≤ a * b) : a * b = max a b := by rcases aleph0_le_mul_iff.mp h with ⟨ha, hb, ha' \| hb'⟩ · exact mul_eq_max_of_aleph0_le_left ha' hb · exact mul_eq_max_of_aleph0_le_right ha hb' #align cardinal.mul_eq_max' Cardinal.mul_eq_max' theorem mul_le_max (a b : Cardinal) : a * b ≤ max (max a b) ℵ₀ := by rcases eq_or_ne a 0 with (rfl \| ha0); · simp rcases eq_or_ne b 0 with (rfl \| hb0); · simp rcases le_or_lt ℵ₀ a with ha \| ha · rw [mul_eq_max_of_aleph0_le_left ha hb0] exact le_max_left _ _ · rcases le_or_lt ℵ₀ b with hb \| hb · rw [mul_comm, mul_eq_max_of_aleph0_le_left hb ha0, max_comm] exact le_max_left _ _ · exact le_max_of_le_right (mul_lt_aleph0 ha hb).le #align cardinal.mul_le_max Cardinal.mul_le_max theorem mul_eq_left {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) (hb' : b ≠ 0) : a * b = a := by rw [mul_eq_max_of_aleph0_le_left ha hb', max_eq_left hb] #align cardinal.mul_eq_left Cardinal.mul_eq_left theorem mul_eq_right {a b : Cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) (ha' : a ≠ 0) : a * b = b := by rw [mul_comm, mul_eq_left hb ha ha'] #align cardinal.mul_eq_right Cardinal.mul_eq_right theorem le_mul_left {a b : Cardinal} (h : b ≠ 0) : a ≤ b * a := by convert mul_le_mul_right' (one_le_iff_ne_zero.mpr h) a rw [one_mul] #align cardinal.le_mul_left Cardinal.le_mul_left theorem le_mul_right {a b : Cardinal} (h : b ≠ 0) : a ≤ a * b := by rw [mul_comm] exact le_mul_left h #align cardinal.le_mul_right Cardinal.le_mul_right theorem mul_eq_left_iff {a b : Cardinal} : a * b = a ↔ max ℵ₀ b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 := by rw [max_le_iff] refine ⟨fun h => ?_, ?_⟩ · rcases le_or_lt ℵ₀ a with ha \| ha · have : a ≠ 0 := by rintro rfl exact ha.not_lt aleph0_pos left rw [and_assoc] use ha constructor · rw [← not_lt] exact fun hb => ne_of_gt (hb.trans_le (le_mul_left this)) h · rintro rfl apply this rw [mul_zero] at h exact h.symm right by_cases h2a : a = 0 · exact Or.inr h2a have hb : b ≠ 0 := by rintro rfl apply h2a rw [mul_zero] at h exact h.symm left rw [← h, mul_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha rcases ha with (rfl \| rfl \| ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩) · contradiction · contradiction rw [← Ne] at h2a rw [← one_le_iff_ne_zero] at h2a hb norm_cast at h2a hb h ⊢ apply le_antisymm _ hb rw [← not_lt] apply fun h2b => ne_of_gt _ h conv_rhs => left; rw [← mul_one n] rw [mul_lt_mul_left] · exact id apply Nat.lt_of_succ_le h2a · rintro (⟨⟨ha, hab⟩, hb⟩ \| rfl \| rfl) · rw [mul_eq_max_of_aleph0_le_left ha hb, max_eq_left hab] all_goals simp #align cardinal.mul_eq_left_iff Cardinal.mul_eq_left_iff end mul /-! ### Properties of `add` -/ section add /-- If `α` is an infinite type, then `α ⊕ α` and `α` have the same cardinality. -/ theorem add_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c + c = c := le_antisymm (by convert mul_le_mul_right' ((nat_lt_aleph0 2).le.trans h) c using 1 <;> simp [two_mul, mul_eq_self h]) (self_le_add_left c c) #align cardinal.add_eq_self Cardinal.add_eq_self /-- If `α` is an infinite type, then the cardinality of `α ⊕ β` is the maximum of the cardinalities of `α` and `β`. -/ theorem add_eq_max {a b : Cardinal} (ha : ℵ₀ ≤ a) : a + b = max a b := le_antisymm (add_eq_self (ha.trans (le_max_left a b)) ▸ add_le_add (le_max_left _ _) (le_max_right _ _)) <\| max_le (self_le_add_right _ _) (self_le_add_left _ _) #align cardinal.add_eq_max Cardinal.add_eq_max theorem add_eq_max' {a b : Cardinal} (ha : ℵ₀ ≤ b) : a + b = max a b := by rw [add_comm, max_comm, add_eq_max ha] #align cardinal.add_eq_max' Cardinal.add_eq_max' @[simp] theorem add_mk_eq_max {α β : Type u} [Infinite α] : #α + #β = max #α #β := add_eq_max (aleph0_le_mk α) #align cardinal.add_mk_eq_max Cardinal.add_mk_eq_max @[simp] theorem add_mk_eq_max' {α β : Type u} [Infinite β] : #α + #β = max #α #β := add_eq_max' (aleph0_le_mk β) #align cardinal.add_mk_eq_max' Cardinal.add_mk_eq_max' theorem add_le_max (a b : Cardinal) : a + b ≤ max (max a b) ℵ₀ := by rcases le_or_lt ℵ₀ a with ha \| ha · rw [add_eq_max ha] exact le_max_left _ _ · rcases le_or_lt ℵ₀ b with hb \| hb · rw [add_comm, add_eq_max hb, max_comm] exact le_max_left _ _ · exact le_max_of_le_right (add_lt_aleph0 ha hb).le #align cardinal.add_le_max Cardinal.add_le_max theorem add_le_of_le {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a ≤ c) (h2 : b ≤ c) : a + b ≤ c := (add_le_add h1 h2).trans <\| le_of_eq <\| add_eq_self hc #align cardinal.add_le_of_le Cardinal.add_le_of_le theorem add_lt_of_lt {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a + b < c := (add_le_add (le_max_left a b) (le_max_right a b)).trans_lt <\| (lt_or_le (max a b) ℵ₀).elim (fun h => (add_lt_aleph0 h h).trans_le hc) fun h => by rw [add_eq_self h]; exact max_lt h1 h2 #align cardinal.add_lt_of_lt Cardinal.add_lt_of_lt theorem eq_of_add_eq_of_aleph0_le {a b c : Cardinal} (h : a + b = c) (ha : a < c) (hc : ℵ₀ ≤ c) : b = c := by apply le_antisymm · rw [← h] apply self_le_add_left rw [← not_lt]; intro hb have : a + b < c := add_lt_of_lt hc ha hb simp [h, lt_irrefl] at this #align cardinal.eq_of_add_eq_of_aleph_0_le Cardinal.eq_of_add_eq_of_aleph0_le theorem add_eq_left {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) : a + b = a := by rw [add_eq_max ha, max_eq_left hb] #align cardinal.add_eq_left Cardinal.add_eq_left theorem add_eq_right {a b : Cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) : a + b = b := by rw [add_comm, add_eq_left hb ha] #align cardinal.add_eq_right Cardinal.add_eq_right theorem add_eq_left_iff {a b : Cardinal} : a + b = a ↔ max ℵ₀ b ≤ a ∨ b = 0 := by rw [max_le_iff] refine ⟨fun h => ?_, ?_⟩ · rcases le_or_lt ℵ₀ a with ha \| ha · left use ha rw [← not_lt] apply fun hb => ne_of_gt _ h intro hb exact hb.trans_le (self_le_add_left b a) right rw [← h, add_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha rcases ha with ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩ norm_cast at h ⊢ rw [← add_right_inj, h, add_zero] · rintro (⟨h1, h2⟩ \| h3) · rw [add_eq_max h1, max_eq_left h2] · rw [h3, add_zero] #align cardinal.add_eq_left_iff Cardinal.add_eq_left_iff theorem add_eq_right_iff {a b : Cardinal} : a + b = b ↔ max ℵ₀ a ≤ b ∨ a = 0 := by rw [add_comm, add_eq_left_iff] #align cardinal.add_eq_right_iff Cardinal.add_eq_right_iff theorem add_nat_eq {a : Cardinal} (n : ℕ) (ha : ℵ₀ ≤ a) : a + n = a := add_eq_left ha ((nat_lt_aleph0 _).le.trans ha) #align cardinal.add_nat_eq Cardinal.add_nat_eq theorem nat_add_eq {a : Cardinal} (n : ℕ) (ha : ℵ₀ ≤ a) : n + a = a := by rw [add_comm, add_nat_eq n ha] theorem add_one_eq {a : Cardinal} (ha : ℵ₀ ≤ a) : a + 1 = a := add_one_of_aleph0_le ha #align cardinal.add_one_eq Cardinal.add_one_eq -- Porting note (#10618): removed `simp`, `simp` can prove it theorem mk_add_one_eq {α : Type} [Infinite α] : #α + 1 = #α := add_one_eq (aleph0_le_mk α) #align cardinal.mk_add_one_eq Cardinal.mk_add_one_eq protected theorem eq_of_add_eq_add_left {a b c : Cardinal} (h : a + b = a + c) (ha : a < ℵ₀) : b = c := by rcases le_or_lt ℵ₀ b with hb \| hb · have : a < b := ha.trans_le hb rw [add_eq_right hb this.le, eq_comm] at h rw [eq_of_add_eq_of_aleph0_le h this hb] · have hc : c < ℵ₀ := by rw [← not_le] intro hc apply lt_irrefl ℵ₀ apply (hc.trans (self_le_add_left _ a)).trans_lt rw [← h] apply add_lt_aleph0 ha hb rw [lt_aleph0] at rcases ha with ⟨n, rfl⟩ rcases hb with ⟨m, rfl⟩ rcases hc with ⟨k, rfl⟩ norm_cast at h ⊢ apply add_left_cancel h #align cardinal.eq_of_add_eq_add_left Cardinal.eq_of_add_eq_add_left protected theorem eq_of_add_eq_add_right {a b c : Cardinal} (h : a + b = c + b) (hb : b < ℵ₀) : a = c := by rw [add_comm a b, add_comm c b] at h exact Cardinal.eq_of_add_eq_add_left h hb #align cardinal.eq_of_add_eq_add_right Cardinal.eq_of_add_eq_add_right end add section ciSup variable {ι : Type u} {ι' : Type w} (f : ι → Cardinal.{v}) section add variable [Nonempty ι] [Nonempty ι'] (hf : BddAbove (range f)) protected theorem ciSup_add (c : Cardinal.{v}) : (⨆ i, f i) + c = ⨆ i, f i + c := by have : ∀ i, f i + c ≤ (⨆ i, f i) + c := fun i ↦ add_le_add_right (le_ciSup hf i) c refine le_antisymm ?_ (ciSup_le' this) have bdd : BddAbove (range (f · + c)) := ⟨_, forall_mem_range.mpr this⟩ obtain hs \| hs := lt_or_le (⨆ i, f i) ℵ₀ · obtain ⟨i, hi⟩ := exists_eq_of_iSup_eq_of_not_isLimit f hf _ (fun h ↦ hs.not_le h.aleph0_le) rfl exact hi ▸ le_ciSup bdd i rw [add_eq_max hs, max_le_iff] exact ⟨ciSup_mono bdd fun i ↦ self_le_add_right _ c, (self_le_add_left _ _).trans (le_ciSup bdd <\| Classical.arbitrary ι)⟩ protected theorem add_ciSup (c : Cardinal.{v}) : c + (⨆ i, f i) = ⨆ i, c + f i := by rw [add_comm, Cardinal.ciSup_add f hf]; simp_rw [add_comm] protected theorem ciSup_add_ciSup (g : ι' → Cardinal.{v}) (hg : BddAbove (range g)) : (⨆ i, f i) + (⨆ j, g j) = ⨆ (i) (j), f i + g j := by simp_rw [Cardinal.ciSup_add f hf, Cardinal.add_ciSup g hg] end add protected theorem ciSup_mul (c : Cardinal.{v}) : (⨆ i, f i) * c = ⨆ i, f i * c := by cases isEmpty_or_nonempty ι; · simp obtain rfl \| h0 := eq_or_ne c 0; · simp by_cases hf : BddAbove (range f); swap · have hfc : ¬ BddAbove (range (f · * c)) := fun bdd ↦ hf ⟨⨆ i, f i * c, forall_mem_range.mpr fun i ↦ (le_mul_right h0).trans (le_ciSup bdd i)⟩ simp [iSup, csSup_of_not_bddAbove, hf, hfc] have : ∀ i, f i * c ≤ (⨆ i, f i) * c := fun i ↦ mul_le_mul_right' (le_ciSup hf i) c refine le_antisymm ?_ (ciSup_le' this) have bdd : BddAbove (range (f · * c)) := ⟨_, forall_mem_range.mpr this⟩ obtain hs \| hs := lt_or_le (⨆ i, f i) ℵ₀ · obtain ⟨i, hi⟩ := exists_eq_of_iSup_eq_of_not_isLimit f hf _ (fun h ↦ hs.not_le h.aleph0_le) rfl exact hi ▸ le_ciSup bdd i rw [mul_eq_max_of_aleph0_le_left hs h0, max_le_iff] obtain ⟨i, hi⟩ := exists_lt_of_lt_ciSup' (one_lt_aleph0.trans_le hs) exact ⟨ciSup_mono bdd fun i ↦ le_mul_right h0, (le_mul_left (zero_lt_one.trans hi).ne').trans (le_ciSup bdd i)⟩ protected theorem mul_ciSup (c : Cardinal.{v}) : c * (⨆ i, f i) = ⨆ i, c * f i := by rw [mul_comm, Cardinal.ciSup_mul f]; simp_rw [mul_comm] protected theorem ciSup_mul_ciSup (g : ι' → Cardinal.{v}) : (⨆ i, f i) * (⨆ j, g j) = ⨆ (i) (j), f i * g j := by simp_rw [Cardinal.ciSup_mul f, Cardinal.mul_ciSup g] end ciSup @[simp] theorem aleph_add_aleph (o₁ o₂ : Ordinal) : aleph o₁ + aleph o₂ = aleph (max o₁ o₂) := by rw [Cardinal.add_eq_max (aleph0_le_aleph o₁), max_aleph_eq] #align cardinal.aleph_add_aleph Cardinal.aleph_add_aleph theorem principal_add_ord {c : Cardinal} (hc : ℵ₀ ≤ c) : Ordinal.Principal (· + ·) c.ord := fun a b ha hb => by rw [lt_ord, Ordinal.card_add] at * exact add_lt_of_lt hc ha hb #align cardinal.principal_add_ord Cardinal.principal_add_ord theorem principal_add_aleph (o : Ordinal) : Ordinal.Principal (· + ·) (aleph o).ord := principal_add_ord <\| aleph0_le_aleph o #align cardinal.principal_add_aleph Cardinal.principal_add_aleph theorem add_right_inj_of_lt_aleph0 {α β γ : Cardinal} (γ₀ : γ < aleph0) : α + γ = β + γ ↔ α = β := ⟨fun h => Cardinal.eq_of_add_eq_add_right h γ₀, fun h => congr_arg (· + γ) h⟩ #align cardinal.add_right_inj_of_lt_aleph_0 Cardinal.add_right_inj_of_lt_aleph0 @[simp] theorem add_nat_inj {α β : Cardinal} (n : ℕ) : α + n = β + n ↔ α = β := add_right_inj_of_lt_aleph0 (nat_lt_aleph0 _) #align cardinal.add_nat_inj Cardinal.add_nat_inj @[simp] theorem add_one_inj {α β : Cardinal} : α + 1 = β + 1 ↔ α = β := add_right_inj_of_lt_aleph0 one_lt_aleph0 #align cardinal.add_one_inj Cardinal.add_one_inj theorem add_le_add_iff_of_lt_aleph0 {α β γ : Cardinal} (γ₀ : γ < Cardinal.aleph0) : α + γ ≤ β + γ ↔ α ≤ β := by refine ⟨fun h => ?_, fun h => add_le_add_right h γ⟩ contrapose h rw [not_le, lt_iff_le_and_ne, Ne] at h ⊢ exact ⟨add_le_add_right h.1 γ, mt (add_right_inj_of_lt_aleph0 γ₀).1 h.2⟩ #align cardinal.add_le_add_iff_of_lt_aleph_0 Cardinal.add_le_add_iff_of_lt_aleph0 @[simp] theorem add_nat_le_add_nat_iff {α β : Cardinal} (n : ℕ) : α + n ≤ β + n ↔ α ≤ β := add_le_add_iff_of_lt_aleph0 (nat_lt_aleph0 n) #align cardinal.add_nat_le_add_nat_iff_of_lt_aleph_0 Cardinal.add_nat_le_add_nat_iff @[deprecated (since := "2024-02-12")] alias add_nat_le_add_nat_iff_of_lt_aleph_0 := add_nat_le_add_nat_iff @[simp] theorem add_one_le_add_one_iff {α β : Cardinal} : α + 1 ≤ β + 1 ↔ α ≤ β := add_le_add_iff_of_lt_aleph0 one_lt_aleph0 #align cardinal.add_one_le_add_one_iff_of_lt_aleph_0 Cardinal.add_one_le_add_one_iff @[deprecated (since := "2024-02-12")] alias add_one_le_add_one_iff_of_lt_aleph_0 := add_one_le_add_one_iff /-! ### Properties about power -/ section pow theorem pow_le {κ μ : Cardinal.{u}} (H1 : ℵ₀ ≤ κ) (H2 : μ < ℵ₀) : κ ^ μ ≤ κ := let ⟨n, H3⟩ := lt_aleph0.1 H2 H3.symm ▸ Quotient.inductionOn κ (fun α H1 => Nat.recOn n (lt_of_lt_of_le (by rw [Nat.cast_zero, power_zero] exact one_lt_aleph0) H1).le fun n ih => le_of_le_of_eq (by rw [Nat.cast_succ, power_add, power_one] exact mul_le_mul_right' ih _) (mul_eq_self H1)) H1 #align cardinal.pow_le Cardinal.pow_le theorem pow_eq {κ μ : Cardinal.{u}} (H1 : ℵ₀ ≤ κ) (H2 : 1 ≤ μ) (H3 : μ < ℵ₀) : κ ^ μ = κ := (pow_le H1 H3).antisymm <\| self_le_power κ H2 #align cardinal.pow_eq Cardinal.pow_eq theorem power_self_eq {c : Cardinal} (h : ℵ₀ ≤ c) : c ^ c = 2 ^ c := by apply ((power_le_power_right <\| (cantor c).le).trans _).antisymm · exact power_le_power_right ((nat_lt_aleph0 2).le.trans h) · rw [← power_mul, mul_eq_self h] #align cardinal.power_self_eq Cardinal.power_self_eq theorem prod_eq_two_power {ι : Type u} [Infinite ι] {c : ι → Cardinal.{v}} (h₁ : ∀ i, 2 ≤ c i) (h₂ : ∀ i, lift.{u} (c i) ≤ lift.{v} #ι) : prod c = 2 ^ lift.{v} #ι := by rw [← lift_id'.{u, v} (prod.{u, v} c), lift_prod, ← lift_two_power] apply le_antisymm · refine (prod_le_prod _ _ h₂).trans_eq ?_ rw [prod_const, lift_lift, ← lift_power, power_self_eq (aleph0_le_mk ι), lift_umax.{u, v}] · rw [← prod_const', lift_prod] refine prod_le_prod _ _ fun i => ?_ rw [lift_two, ← lift_two.{u, v}, lift_le] exact h₁ i #align cardinal.prod_eq_two_power Cardinal.prod_eq_two_power theorem power_eq_two_power {c₁ c₂ : Cardinal} (h₁ : ℵ₀ ≤ c₁) (h₂ : 2 ≤ c₂) (h₂' : c₂ ≤ c₁) : c₂ ^ c₁ = 2 ^ c₁ := le_antisymm (power_self_eq h₁ ▸ power_le_power_right h₂') (power_le_power_right h₂) #align cardinal.power_eq_two_power Cardinal.power_eq_two_power theorem nat_power_eq {c : Cardinal.{u}} (h : ℵ₀ ≤ c) {n : ℕ} (hn : 2 ≤ n) : (n : Cardinal.{u}) ^ c = 2 ^ c := power_eq_two_power h (by assumption_mod_cast) ((nat_lt_aleph0 n).le.trans h) #align cardinal.nat_power_eq Cardinal.nat_power_eq theorem power_nat_le {c : Cardinal.{u}} {n : ℕ} (h : ℵ₀ ≤ c) : c ^ n ≤ c := pow_le h (nat_lt_aleph0 n) #align cardinal.power_nat_le Cardinal.power_nat_le theorem power_nat_eq {c : Cardinal.{u}} {n : ℕ} (h1 : ℵ₀ ≤ c) (h2 : 1 ≤ n) : c ^ n = c := pow_eq h1 (mod_cast h2) (nat_lt_aleph0 n) #align cardinal.power_nat_eq Cardinal.power_nat_eq theorem power_nat_le_max {c : Cardinal.{u}} {n : ℕ} : c ^ (n : Cardinal.{u}) ≤ max c ℵ₀ := by rcases le_or_lt ℵ₀ c with hc \| hc · exact le_max_of_le_left (power_nat_le hc) · exact le_max_of_le_right (power_lt_aleph0 hc (nat_lt_aleph0 _)).le #align cardinal.power_nat_le_max Cardinal.power_nat_le_max theorem powerlt_aleph0 {c : Cardinal} (h : ℵ₀ ≤ c) : c ^< ℵ₀ = c := by apply le_antisymm · rw [powerlt_le] intro c' rw [lt_aleph0] rintro ⟨n, rfl⟩ apply power_nat_le h convert le_powerlt c one_lt_aleph0; rw [power_one] #align cardinal.powerlt_aleph_0 Cardinal.powerlt_aleph0 theorem powerlt_aleph0_le (c : Cardinal) : c ^< ℵ₀ ≤ max c ℵ₀ := by rcases le_or_lt ℵ₀ c with h \| h · rw [powerlt_aleph0 h] apply le_max_left rw [powerlt_le] exact fun c' hc' => (power_lt_aleph0 h hc').le.trans (le_max_right _ _) #align cardinal.powerlt_aleph_0_le Cardinal.powerlt_aleph0_le end pow /-! ### Computing cardinality of various types -/ section computing section Function variable {α β : Type u} {β' : Type v} theorem mk_equiv_eq_zero_iff_lift_ne : #(α ≃ β') = 0 ↔ lift.{v} #α ≠ lift.{u} #β' := by rw [mk_eq_zero_iff, ← not_nonempty_iff, ← lift_mk_eq'] theorem mk_equiv_eq_zero_iff_ne : #(α ≃ β) = 0 ↔ #α ≠ #β := by rw [mk_equiv_eq_zero_iff_lift_ne, lift_id, lift_id] /-- This lemma makes lemmas assuming `Infinite α` applicable to the situation where we have `Infinite β` instead. -/ theorem mk_equiv_comm : #(α ≃ β') = #(β' ≃ α) := (ofBijective _ symm_bijective).cardinal_eq theorem mk_embedding_eq_zero_iff_lift_lt : #(α ↪ β') = 0 ↔ lift.{u} #β' < lift.{v} #α := by rw [mk_eq_zero_iff, ← not_nonempty_iff, ← lift_mk_le', not_le] theorem mk_embedding_eq_zero_iff_lt : #(α ↪ β) = 0 ↔ #β < #α := by rw [mk_embedding_eq_zero_iff_lift_lt, lift_lt] theorem mk_arrow_eq_zero_iff : #(α → β') = 0 ↔ #α ≠ 0 ∧ #β' = 0 := by simp_rw [mk_eq_zero_iff, mk_ne_zero_iff, isEmpty_fun] theorem mk_surjective_eq_zero_iff_lift : #{f : α → β' \| Surjective f} = 0 ↔ lift.{v} #α < lift.{u} #β' ∨ (#α ≠ 0 ∧ #β' = 0) := by rw [← not_iff_not, not_or, not_lt, lift_mk_le', ← Ne, not_and_or, not_ne_iff, and_comm] simp_rw [mk_ne_zero_iff, mk_eq_zero_iff, nonempty_coe_sort, Set.Nonempty, mem_setOf, exists_surjective_iff, nonempty_fun] theorem mk_surjective_eq_zero_iff : #{f : α → β \| Surjective f} = 0 ↔ #α < #β ∨ (#α ≠ 0 ∧ #β = 0) := by rw [mk_surjective_eq_zero_iff_lift, lift_lt] variable (α β') theorem mk_equiv_le_embedding : #(α ≃ β') ≤ #(α ↪ β') := ⟨⟨_, Equiv.toEmbedding_injective⟩⟩ theorem mk_embedding_le_arrow : #(α ↪ β') ≤ #(α → β') := ⟨⟨_, DFunLike.coe_injective⟩⟩ variable [Infinite α] {α β'} theorem mk_perm_eq_self_power : #(Equiv.Perm α) = #α ^ #α := ((mk_equiv_le_embedding α α).trans (mk_embedding_le_arrow α α)).antisymm <\| by suffices Nonempty ((α → Bool) ↪ Equiv.Perm (α × Bool)) by obtain ⟨e⟩ : Nonempty (α ≃ α × Bool) := by erw [← Cardinal.eq, mk_prod, lift_uzero, mk_bool, lift_natCast, mul_two, add_eq_self (aleph0_le_mk α)] erw [← le_def, mk_arrow, lift_uzero, mk_bool, lift_natCast 2] at this rwa [← power_def, power_self_eq (aleph0_le_mk α), e.permCongr.cardinal_eq] refine ⟨⟨fun f ↦ Involutive.toPerm (fun x ↦ ⟨x.1, xor (f x.1) x.2⟩) fun x ↦ ?_, fun f g h ↦ ?_⟩⟩ · simp_rw [← Bool.xor_assoc, Bool.xor_self, Bool.false_xor] · ext a; rw [← (f a).xor_false, ← (g a).xor_false]; exact congr(($h ⟨a, false⟩).2) theorem mk_perm_eq_two_power : #(Equiv.Perm α) = 2 ^ #α := by rw [mk_perm_eq_self_power, power_self_eq (aleph0_le_mk α)] variable (leq : lift.{v} #α = lift.{u} #β') (eq : #α = #β) theorem mk_equiv_eq_arrow_of_lift_eq : #(α ≃ β') = #(α → β') := by obtain ⟨e⟩ := lift_mk_eq'.mp leq have e₁ := lift_mk_eq'.mpr ⟨.equivCongr (.refl α) e⟩ have e₂ := lift_mk_eq'.mpr ⟨.arrowCongr (.refl α) e⟩ rw [lift_id'.{u,v}] at e₁ e₂ rw [← e₁, ← e₂, lift_inj, mk_perm_eq_self_power, power_def] theorem mk_equiv_eq_arrow_of_eq : #(α ≃ β) = #(α → β) := mk_equiv_eq_arrow_of_lift_eq congr(lift $eq) theorem mk_equiv_of_lift_eq : #(α ≃ β') = 2 ^ lift.{v} #α := by erw [← (lift_mk_eq'.2 ⟨.equivCongr (.refl α) (lift_mk_eq'.1 leq).some⟩).trans (lift_id'.{u,v} _), lift_umax.{u,v}, mk_perm_eq_two_power, lift_power, lift_natCast]; rfl theorem mk_equiv_of_eq : #(α ≃ β) = 2 ^ #α := by rw [mk_equiv_of_lift_eq (lift_inj.mpr eq), lift_id] variable (lle : lift.{u} #β' ≤ lift.{v} #α) (le : #β ≤ #α) theorem mk_embedding_eq_arrow_of_lift_le : #(β' ↪ α) = #(β' → α) := (mk_embedding_le_arrow _ _).antisymm <\| by conv_rhs => rw [← (Equiv.embeddingCongr (.refl _) (Cardinal.eq.mp <\| mul_eq_self <\| aleph0_le_mk α).some).cardinal_eq] obtain ⟨e⟩ := lift_mk_le'.mp lle exact ⟨⟨fun f ↦ ⟨fun b ↦ ⟨e b, f b⟩, fun _ _ h ↦ e.injective congr(Prod.fst $h)⟩, fun f g h ↦ funext fun b ↦ congr(Prod.snd <\| $h b)⟩⟩ theorem mk_embedding_eq_arrow_of_le : #(β ↪ α) = #(β → α) := mk_embedding_eq_arrow_of_lift_le (lift_le.mpr le) theorem mk_surjective_eq_arrow_of_lift_le : #{f : α → β' \| Surjective f} = #(α → β') := (mk_set_le _).antisymm <\| have ⟨e⟩ : Nonempty (α ≃ α ⊕ β') := by simp_rw [← lift_mk_eq', mk_sum, lift_add, lift_lift]; rw [lift_umax.{u,v}, eq_comm] exact add_eq_left (aleph0_le_lift.mpr <\| aleph0_le_mk α) lle ⟨⟨fun f ↦ ⟨fun a ↦ (e a).elim f id, fun b ↦ ⟨e.symm (.inr b), congr_arg _ (e.right_inv _)⟩⟩, fun f g h ↦ funext fun a ↦ by simpa only [e.apply_symm_apply] using congr_fun (Subtype.ext_iff.mp h) (e.symm <\| .inl a)⟩⟩ theorem mk_surjective_eq_arrow_of_le : #{f : α → β \| Surjective f} = #(α → β) := mk_surjective_eq_arrow_of_lift_le (lift_le.mpr le) end Function @[simp] theorem mk_list_eq_mk (α : Type u) [Infinite α] : #(List α) = #α := have H1 : ℵ₀ ≤ #α := aleph0_le_mk α Eq.symm <\| le_antisymm ((le_def _ _).2 ⟨⟨fun a => [a], fun _ => by simp⟩⟩) <\| calc #(List α) = sum fun n : ℕ => #α ^ (n : Cardinal.{u}) := mk_list_eq_sum_pow α _ ≤ sum fun _ : ℕ => #α := sum_le_sum _ _ fun n => pow_le H1 <\| nat_lt_aleph0 n _ = #α := by simp [H1] #align cardinal.mk_list_eq_mk Cardinal.mk_list_eq_mk theorem mk_list_eq_aleph0 (α : Type u) [Countable α] [Nonempty α] : #(List α) = ℵ₀ := mk_le_aleph0.antisymm (aleph0_le_mk _) #align cardinal.mk_list_eq_aleph_0 Cardinal.mk_list_eq_aleph0 theorem mk_list_eq_max_mk_aleph0 (α : Type u) [Nonempty α] : #(List α) = max #α ℵ₀ := by cases finite_or_infinite α · rw [mk_list_eq_aleph0, eq_comm, max_eq_right] exact mk_le_aleph0 · rw [mk_list_eq_mk, eq_comm, max_eq_left] exact aleph0_le_mk α #align cardinal.mk_list_eq_max_mk_aleph_0 Cardinal.mk_list_eq_max_mk_aleph0 theorem mk_list_le_max (α : Type u) : #(List α) ≤ max ℵ₀ #α := by cases finite_or_infinite α · exact mk_le_aleph0.trans (le_max_left _ _) · rw [mk_list_eq_mk] apply le_max_right #align cardinal.mk_list_le_max Cardinal.mk_list_le_max @[simp] theorem mk_finset_of_infinite (α : Type u) [Infinite α] : #(Finset α) = #α := Eq.symm <\| le_antisymm (mk_le_of_injective fun _ _ => Finset.singleton_inj.1) <\| calc #(Finset α) ≤ #(List α) := mk_le_of_surjective List.toFinset_surjective _ = #α := mk_list_eq_mk α #align cardinal.mk_finset_of_infinite Cardinal.mk_finset_of_infinite @[simp] theorem mk_finsupp_lift_of_infinite (α : Type u) (β : Type v) [Infinite α] [Zero β] [Nontrivial β] : #(α →₀ β) = max (lift.{v} #α) (lift.{u} #β) := by apply le_antisymm · calc #(α →₀ β) ≤ #(Finset (α × β)) := mk_le_of_injective (Finsupp.graph_injective α β) _ = #(α × β) := mk_finset_of_infinite _ _ = max (lift.{v} #α) (lift.{u} #β) := by rw [mk_prod, mul_eq_max_of_aleph0_le_left] <;> simp · apply max_le <;> rw [← lift_id #(α →₀ β), ← lift_umax] · cases' exists_ne (0 : β) with b hb exact lift_mk_le.{v}.2 ⟨⟨_, Finsupp.single_left_injective hb⟩⟩ · inhabit α exact lift_mk_le.{u}.2 ⟨⟨_, Finsupp.single_injective default⟩⟩ #align cardinal.mk_finsupp_lift_of_infinite Cardinal.mk_finsupp_lift_of_infinite theorem mk_finsupp_of_infinite (α β : Type u) [Infinite α] [Zero β] [Nontrivial β] : #(α →₀ β) = max #α #β := by simp #align cardinal.mk_finsupp_of_infinite Cardinal.mk_finsupp_of_infinite @[simp] theorem mk_finsupp_lift_of_infinite' (α : Type u) (β : Type v) [Nonempty α] [Zero β] [Infinite β] : #(α →₀ β) = max (lift.{v} #α) (lift.{u} #β) := by cases fintypeOrInfinite α · rw [mk_finsupp_lift_of_fintype] have : ℵ₀ ≤ (#β).lift := aleph0_le_lift.2 (aleph0_le_mk β) rw [max_eq_right (le_trans _ this), power_nat_eq this] exacts [Fintype.card_pos, lift_le_aleph0.2 (lt_aleph0_of_finite _).le] · apply mk_finsupp_lift_of_infinite #align cardinal.mk_finsupp_lift_of_infinite' Cardinal.mk_finsupp_lift_of_infinite' theorem mk_finsupp_of_infinite' (α β : Type u) [Nonempty α] [Zero β] [Infinite β] : #(α →₀ β) = max #α #β := by simp #align cardinal.mk_finsupp_of_infinite' Cardinal.mk_finsupp_of_infinite' theorem mk_finsupp_nat (α : Type u) [Nonempty α] : #(α →₀ ℕ) = max #α ℵ₀ := by simp #align cardinal.mk_finsupp_nat Cardinal.mk_finsupp_nat @[simp] theorem mk_multiset_of_nonempty (α : Type u) [Nonempty α] : #(Multiset α) = max #α ℵ₀ := Multiset.toFinsupp.toEquiv.cardinal_eq.trans (mk_finsupp_nat α) #align cardinal.mk_multiset_of_nonempty Cardinal.mk_multiset_of_nonempty theorem mk_multiset_of_infinite (α : Type u) [Infinite α] : #(Multiset α) = #α := by simp #align cardinal.mk_multiset_of_infinite Cardinal.mk_multiset_of_infinite theorem mk_multiset_of_isEmpty (α : Type u) [IsEmpty α] : #(Multiset α) = 1 := Multiset.toFinsupp.toEquiv.cardinal_eq.trans (by simp) #align cardinal.mk_multiset_of_is_empty Cardinal.mk_multiset_of_isEmpty theorem mk_multiset_of_countable (α : Type u) [Countable α] [Nonempty α] : #(Multiset α) = ℵ₀ := Multiset.toFinsupp.toEquiv.cardinal_eq.trans (by simp) #align cardinal.mk_multiset_of_countable Cardinal.mk_multiset_of_countable theorem mk_bounded_set_le_of_infinite (α : Type u) [Infinite α] (c : Cardinal) : #{ t : Set α // #t ≤ c } ≤ #α ^ c := by refine le_trans ?_ (by rw [← add_one_eq (aleph0_le_mk α)]) induction' c using Cardinal.inductionOn with β fapply mk_le_of_surjective · intro f use Sum.inl ⁻¹' range f refine le_trans (mk_preimage_of_injective _ _ fun x y => Sum.inl.inj) ?_ apply mk_range_le rintro ⟨s, ⟨g⟩⟩ use fun y => if h : ∃ x : s, g x = y then Sum.inl (Classical.choose h).val else Sum.inr (ULift.up 0) apply Subtype.eq; ext x constructor · rintro ⟨y, h⟩ dsimp only at h by_cases h' : ∃ z : s, g z = y · rw [dif_pos h'] at h cases Sum.inl.inj h exact (Classical.choose h').2 · rw [dif_neg h'] at h cases h · intro h have : ∃ z : s, g z = g ⟨x, h⟩ := ⟨⟨x, h⟩, rfl⟩ use g ⟨x, h⟩ dsimp only rw [dif_pos this] congr suffices Classical.choose this = ⟨x, h⟩ from congr_arg Subtype.val this apply g.2 exact Classical.choose_spec this #align cardinal.mk_bounded_set_le_of_infinite Cardinal.mk_bounded_set_le_of_infinite theorem mk_bounded_set_le (α : Type u) (c : Cardinal) : #{ t : Set α // #t ≤ c } ≤ max #α ℵ₀ ^ c := by trans #{ t : Set (Sum (ULift.{u} ℕ) α) // #t ≤ c } · refine ⟨Embedding.subtypeMap ?_ ?_⟩ · apply Embedding.image use Sum.inr apply Sum.inr.inj intro s hs exact mk_image_le.trans hs apply (mk_bounded_set_le_of_infinite (Sum (ULift.{u} ℕ) α) c).trans rw [max_comm, ← add_eq_max] <;> rfl #align cardinal.mk_bounded_set_le Cardinal.mk_bounded_set_le theorem mk_bounded_subset_le {α : Type u} (s : Set α) (c : Cardinal.{u}) : #{ t : Set α // t ⊆ s ∧ #t ≤ c } ≤ max #s ℵ₀ ^ c := by refine le_trans ?_ (mk_bounded_set_le s c) refine ⟨Embedding.codRestrict _ ?_ ?_⟩ · use fun t => (↑) ⁻¹' t.1 rintro ⟨t, ht1, ht2⟩ ⟨t', h1t', h2t'⟩ h apply Subtype.eq dsimp only at h ⊢ refine (preimage_eq_preimage' ?_ ?_).1 h <;> rw [Subtype.range_coe] <;> assumption rintro ⟨t, _, h2t⟩; exact (mk_preimage_of_injective _ _ Subtype.val_injective).trans h2t #align cardinal.mk_bounded_subset_le Cardinal.mk_bounded_subset_le end computing /-! ### Properties of `compl` -/ section compl theorem mk_compl_of_infinite {α : Type} [Infinite α] (s : Set α) (h2 : #s < #α) : #(sᶜ : Set α) = #α := by refine eq_of_add_eq_of_aleph0_le ?_ h2 (aleph0_le_mk α) exact mk_sum_compl s #align cardinal.mk_compl_of_infinite Cardinal.mk_compl_of_infinite theorem mk_compl_finset_of_infinite {α : Type} [Infinite α] (s : Finset α) : #((↑s)ᶜ : Set α) = #α := by apply mk_compl_of_infinite exact (finset_card_lt_aleph0 s).trans_le (aleph0_le_mk α) #align cardinal.mk_compl_finset_of_infinite Cardinal.mk_compl_finset_of_infinite	Mathlib/SetTheory/Cardinal/Ordinal.lean	1,335	1,337	theorem mk_compl_eq_mk_compl_infinite {α : Type*} [Infinite α] {s t : Set α} (hs : #s < #α) (ht : #t < #α) : #(sᶜ : Set α) = #(tᶜ : Set α) := by	rw [mk_compl_of_infinite s hs, mk_compl_of_infinite t ht]
/- Copyright (c) 2021 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import Mathlib.MeasureTheory.Measure.Lebesgue.Complex import Mathlib.MeasureTheory.Integral.DivergenceTheorem import Mathlib.MeasureTheory.Integral.CircleIntegral import Mathlib.Analysis.Calculus.Dslope import Mathlib.Analysis.Analytic.Basic import Mathlib.Analysis.Complex.ReImTopology import Mathlib.Analysis.Calculus.DiffContOnCl import Mathlib.Analysis.Calculus.FDeriv.Analytic import Mathlib.Data.Real.Cardinality #align_import analysis.complex.cauchy_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" /-! # Cauchy integral formula In this file we prove the Cauchy-Goursat theorem and the Cauchy integral formula for integrals over circles. Most results are formulated for a function `f : ℂ → E` that takes values in a complex Banach space with second countable topology. ## Main statements In the following theorems, if the name ends with `off_countable`, then the actual theorem assumes differentiability at all but countably many points of the set mentioned below. * `Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable`: If a function `f : ℂ → E` is continuous on a closed rectangle and real differentiable on its interior, then its integral over the boundary of this rectangle is equal to the integral of `I • f' (x + y * I) 1 - f' (x + y * I) I` over the rectangle, where `f' z w : E` is the derivative of `f` at `z` in the direction `w` and `I = Complex.I` is the imaginary unit. * `Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable`: If a function `f : ℂ → E` is continuous on a closed rectangle and is complex differentiable on its interior, then its integral over the boundary of this rectangle is equal to zero. * `Complex.circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable`: If a function `f : ℂ → E` is continuous on a closed annulus `{z \| r ≤ \|z - c\| ≤ R}` and is complex differentiable on its interior `{z \| r < \|z - c\| < R}`, then the integrals of `(z - c)⁻¹ • f z` over the outer boundary and over the inner boundary are equal. * `Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto`, `Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable`: If a function `f : ℂ → E` is continuous on a punctured closed disc `{z \| \|z - c\| ≤ R ∧ z ≠ c}`, is complex differentiable on the corresponding punctured open disc, and tends to `y` as `z → c`, `z ≠ c`, then the integral of `(z - c)⁻¹ • f z` over the circle `\|z - c\| = R` is equal to `2πiy`. In particular, if `f` is continuous on the whole closed disc and is complex differentiable on the corresponding open disc, then this integral is equal to `2πif(c)`. * `Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`, `Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable` Cauchy integral formula: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is complex differentiable on the corresponding open disc, then for any `w` in the corresponding open disc the integral of `(z - w)⁻¹ • f z` over the boundary of the disc is equal to `2πif(w)`. Two versions of the lemma put the multiplier `2πi` at the different sides of the equality. * `Complex.hasFPowerSeriesOnBall_of_differentiable_off_countable`: If `f : ℂ → E` is continuous on a closed disc of positive radius and is complex differentiable on the corresponding open disc, then it is analytic on the corresponding open disc, and the coefficients of the power series are given by Cauchy integral formulas. * `DifferentiableOn.hasFPowerSeriesOnBall`: If `f : ℂ → E` is complex differentiable on a closed disc of positive radius, then it is analytic on the corresponding open disc, and the coefficients of the power series are given by Cauchy integral formulas. * `DifferentiableOn.analyticAt`, `Differentiable.analyticAt`: If `f : ℂ → E` is differentiable on a neighborhood of a point, then it is analytic at this point. In particular, if `f : ℂ → E` is differentiable on the whole `ℂ`, then it is analytic at every point `z : ℂ`. * `Differentiable.hasFPowerSeriesOnBall`: If `f : ℂ → E` is differentiable everywhere then the `cauchyPowerSeries f z R` is a formal power series representing `f` at `z` with infinite radius of convergence (this holds for any choice of `0 < R`). ## Implementation details The proof of the Cauchy integral formula in this file is based on a very general version of the divergence theorem, see `MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable` (a version for functions defined on `Fin (n + 1) → ℝ`), `MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le`, and `MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable` (versions for functions defined on `ℝ × ℝ`). Usually, the divergence theorem is formulated for a $C^1$ smooth function. The theorems formulated above deal with a function that is * continuous on a closed box/rectangle; * differentiable at all but countably many points of its interior; * have divergence integrable over the closed box/rectangle. First, we reformulate the theorem for a real-differentiable map `ℂ → E`, and relate the integral of `f` over the boundary of a rectangle in `ℂ` to the integral of the derivative $\frac{\partial f}{\partial \bar z}$ over the interior of this box. In particular, for a complex differentiable function, the latter derivative is zero, hence the integral over the boundary of a rectangle is zero. Thus we get the Cauchy-Goursat theorem for a rectangle in `ℂ`. Next, we apply this theorem to the function $F(z)=f(c+e^{z})$ on the rectangle $[\ln r, \ln R]\times [0, 2\pi]$ to prove that $$ \oint_{\|z-c\|=r}\frac{f(z)\,dz}{z-c}=\oint_{\|z-c\|=R}\frac{f(z)\,dz}{z-c} $$ provided that `f` is continuous on the closed annulus `r ≤ \|z - c\| ≤ R` and is complex differentiable on its interior `r < \|z - c\| < R` (possibly, at all but countably many points). Here and below, we write $\frac{f(z)}{z-c}$ in the documentation while the actual lemmas use `(z - c)⁻¹ • f z` because `f z` belongs to some Banach space over `ℂ` and `f z / (z - c)` is undefined. Taking the limit of this equality as `r` tends to `𝓝[>] 0`, we prove $$ \oint_{\|z-c\|=R}\frac{f(z)\,dz}{z-c}=2\pi if(c) $$ provided that `f` is continuous on the closed disc `\|z - c\| ≤ R` and is differentiable at all but countably many points of its interior. This is the Cauchy integral formula for the center of a circle. In particular, if we apply this function to `F z = (z - c) • f z`, then we get $$ \oint_{\|z-c\|=R} f(z)\,dz=0. $$ In order to deduce the Cauchy integral formula for any point `w`, `\|w - c\| < R`, we consider the slope function `g : ℂ → E` given by `g z = (z - w)⁻¹ • (f z - f w)` if `z ≠ w` and `g w = f' w`. This function satisfies assumptions of the previous theorem, so we have $$ \oint_{\|z-c\|=R} \frac{f(z)\,dz}{z-w}=\oint_{\|z-c\|=R} \frac{f(w)\,dz}{z-w}= \left(\oint_{\|z-c\|=R} \frac{dz}{z-w}\right)f(w). $$ The latter integral was computed in `circleIntegral.integral_sub_inv_of_mem_ball` and is equal to `2 * π * Complex.I`. There is one more step in the actual proof. Since we allow `f` to be non-differentiable on a countable set `s`, we cannot immediately claim that `g` is continuous at `w` if `w ∈ s`. So, we use the proof outlined in the previous paragraph for `w ∉ s` (see `Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux`), then use continuity of both sides of the formula and density of `sᶜ` to prove the formula for all points of the open ball, see `Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. Finally, we use the properties of the Cauchy integrals established elsewhere (see `hasFPowerSeriesOn_cauchy_integral`) and Cauchy integral formula to prove that the original function is analytic on the open ball. ## Tags Cauchy-Goursat theorem, Cauchy integral formula -/ open TopologicalSpace Set MeasureTheory intervalIntegral Metric Filter Function open scoped Interval Real NNReal ENNReal Topology noncomputable section universe u variable {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] namespace Complex /-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at `z w : ℂ`, is real differentiable at all but countably many points of the corresponding open rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the integral of `f` over the boundary of the rectangle is equal to the integral of $2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$ over the rectangle. -/ theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ) (s : Set ℂ) (hs : s.Countable) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x) (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) : (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) + I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) - I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) = ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdCLM.symm have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm have he₁ : e (1, 0) = 1 := rfl; have he₂ : e (0, 1) = I := rfl simp only [he] at * set F : ℝ × ℝ → E := f ∘ e set F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => (f' (e p)).comp (e : ℝ × ℝ →L[ℝ] ℂ) have hF' : ∀ p : ℝ × ℝ, (-(I • F' p)) (1, 0) + F' p (0, 1) = -(I • f' (e p) 1 - f' (e p) I) := by rintro ⟨x, y⟩ simp only [F', ContinuousLinearMap.neg_apply, ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe, he₁, he₂, neg_add_eq_sub, neg_sub] set R : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]] set t : Set (ℝ × ℝ) := e ⁻¹' s rw [uIcc_comm z.im] at Hc Hi; rw [min_comm z.im, max_comm z.im] at Hd have hR : e ⁻¹' ([[z.re, w.re]] ×ℂ [[w.im, z.im]]) = R := rfl have htc : ContinuousOn F R := Hc.comp e.continuousOn hR.ge have htd : ∀ p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \ t, HasFDerivAt F (F' p) p := fun p hp => (Hd (e p) hp).comp p e.hasFDerivAt simp_rw [← intervalIntegral.integral_smul, intervalIntegral.integral_symm w.im z.im, ← intervalIntegral.integral_neg, ← hF'] refine (integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (fun p => -(I • F p)) F (fun p => -(I • F' p)) F' z.re w.im w.re z.im t (hs.preimage e.injective) (htc.const_smul _).neg htc (fun p hp => ((htd p hp).const_smul I).neg) htd ?_).symm rw [← (volume_preserving_equiv_real_prod.symm _).integrableOn_comp_preimage (MeasurableEquiv.measurableEmbedding _)] at Hi simpa only [hF'] using Hi.neg #align complex.integral_boundary_rect_of_has_fderiv_at_real_off_countable Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable /-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at `z w : ℂ`, is real differentiable on the corresponding open rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the integral of `f` over the boundary of the rectangle is equal to the integral of $2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$ over the rectangle. -/ theorem integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im), HasFDerivAt f (f' x) x) (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) : (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) + I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) - I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := integral_boundary_rect_of_hasFDerivAt_real_off_countable f f' z w ∅ countable_empty Hc (fun x hx => Hd x hx.1) Hi #align complex.integral_boundary_rect_of_continuous_on_of_has_fderiv_at_real Complex.integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real /-- Suppose that a function `f : ℂ → E` is real differentiable on a closed rectangle with opposite corners at `z w : ℂ` and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the integral of `f` over the boundary of the rectangle is equal to the integral of $2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$ over the rectangle. -/ theorem integral_boundary_rect_of_differentiableOn_real (f : ℂ → E) (z w : ℂ) (Hd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hi : IntegrableOn (fun z => I • fderiv ℝ f z 1 - fderiv ℝ f z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) : (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) + I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) - I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • fderiv ℝ f (x + y * I) 1 - fderiv ℝ f (x + y * I) I := integral_boundary_rect_of_hasFDerivAt_real_off_countable f (fderiv ℝ f) z w ∅ countable_empty Hd.continuousOn (fun x hx => Hd.hasFDerivAt <\| by simpa only [← mem_interior_iff_mem_nhds, interior_reProdIm, uIcc, interior_Icc] using hx.1) Hi #align complex.integral_boundary_rect_of_differentiable_on_real Complex.integral_boundary_rect_of_differentiableOn_real /-- Cauchy-Goursat theorem for a rectangle: the integral of a complex differentiable function over the boundary of a rectangle equals zero. More precisely, if `f` is continuous on a closed rectangle and is complex differentiable at all but countably many points of the corresponding open rectangle, then its integral over the boundary of the rectangle equals zero. -/ theorem integral_boundary_rect_eq_zero_of_differentiable_on_off_countable (f : ℂ → E) (z w : ℂ) (s : Set ℂ) (hs : s.Countable) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, DifferentiableAt ℂ f x) : (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) + I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) - I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = 0 := by refine (integral_boundary_rect_of_hasFDerivAt_real_off_countable f (fun z => (fderiv ℂ f z).restrictScalars ℝ) z w s hs Hc (fun x hx => (Hd x hx).hasFDerivAt.restrictScalars ℝ) ?_).trans ?_ <;> simp [← ContinuousLinearMap.map_smul] #align complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable /-- Cauchy-Goursat theorem for a rectangle: the integral of a complex differentiable function over the boundary of a rectangle equals zero. More precisely, if `f` is continuous on a closed rectangle and is complex differentiable on the corresponding open rectangle, then its integral over the boundary of the rectangle equals zero. -/ theorem integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn (f : ℂ → E) (z w : ℂ) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd : DifferentiableOn ℂ f (Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im))) : (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) + I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) - I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = 0 := integral_boundary_rect_eq_zero_of_differentiable_on_off_countable f z w ∅ countable_empty Hc fun _x hx => Hd.differentiableAt <\| (isOpen_Ioo.reProdIm isOpen_Ioo).mem_nhds hx.1 #align complex.integral_boundary_rect_eq_zero_of_continuous_on_of_differentiable_on Complex.integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn /-- Cauchy-Goursat theorem for a rectangle: the integral of a complex differentiable function over the boundary of a rectangle equals zero. More precisely, if `f` is complex differentiable on a closed rectangle, then its integral over the boundary of the rectangle equals zero. -/ theorem integral_boundary_rect_eq_zero_of_differentiableOn (f : ℂ → E) (z w : ℂ) (H : DifferentiableOn ℂ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) : (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) + I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) - I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = 0 := integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn f z w H.continuousOn <\| H.mono <\| inter_subset_inter (preimage_mono Ioo_subset_Icc_self) (preimage_mono Ioo_subset_Icc_self) #align complex.integral_boundary_rect_eq_zero_of_differentiable_on Complex.integral_boundary_rect_eq_zero_of_differentiableOn /-- If `f : ℂ → E` is continuous on the closed annulus `r ≤ ‖z - c‖ ≤ R`, `0 < r ≤ R`, and is complex differentiable at all but countably many points of its interior, then the integrals of `f z / (z - c)` (formally, `(z - c)⁻¹ • f z`) over the circles `‖z - c‖ = r` and `‖z - c‖ = R` are equal to each other. -/	Mathlib/Analysis/Complex/CauchyIntegral.lean	295	324	theorem circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable {c : ℂ} {r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (closedBall c R \ ball c r)) (hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z) : (∮ z in C(c, R), (z - c)⁻¹ • f z) = ∮ z in C(c, r), (z - c)⁻¹ • f z := by	/- We apply the previous lemma to `fun z ↦ f (c + exp z)` on the rectangle `[log r, log R] × [0, 2 * π]`. -/ set A := closedBall c R \ ball c r obtain ⟨a, rfl⟩ : ∃ a, Real.exp a = r := ⟨Real.log r, Real.exp_log h0⟩ obtain ⟨b, rfl⟩ : ∃ b, Real.exp b = R := ⟨Real.log R, Real.exp_log (h0.trans_le hle)⟩ rw [Real.exp_le_exp] at hle -- Unfold definition of `circleIntegral` and cancel some terms. suffices (∫ θ in (0)..2 * π, I • f (circleMap c (Real.exp b) θ)) = ∫ θ in (0)..2 * π, I • f (circleMap c (Real.exp a) θ) by simpa only [circleIntegral, add_sub_cancel_left, ofReal_exp, ← exp_add, smul_smul, ← div_eq_mul_inv, mul_div_cancel_left₀ _ (circleMap_ne_center (Real.exp_pos _).ne'), circleMap_sub_center, deriv_circleMap] set R := [[a, b]] ×ℂ [[0, 2 * π]] set g : ℂ → ℂ := (c + exp ·) have hdg : Differentiable ℂ g := differentiable_exp.const_add _ replace hs : (g ⁻¹' s).Countable := (hs.preimage (add_right_injective c)).preimage_cexp have h_maps : MapsTo g R A := by rintro z ⟨h, -⟩; simpa [g, A, dist_eq, abs_exp, hle] using h.symm replace hc : ContinuousOn (f ∘ g) R := hc.comp hdg.continuous.continuousOn h_maps replace hd : ∀ z ∈ Ioo (min a b) (max a b) ×ℂ Ioo (min 0 (2 * π)) (max 0 (2 * π)) \ g ⁻¹' s, DifferentiableAt ℂ (f ∘ g) z := by refine fun z hz => (hd (g z) ⟨?_, hz.2⟩).comp z (hdg _) simpa [g, dist_eq, abs_exp, hle, and_comm] using hz.1.1 simpa [g, circleMap, exp_periodic _, sub_eq_zero, ← exp_add] using integral_boundary_rect_eq_zero_of_differentiable_on_off_countable _ ⟨a, 0⟩ ⟨b, 2 * π⟩ _ hs hc hd
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Yury Kudryashov -/ import Mathlib.MeasureTheory.OuterMeasure.Basic /-! # The “almost everywhere” filter of co-null sets. If `μ` is an outer measure or a measure on `α`, then `MeasureTheory.ae μ` is the filter of co-null sets: `s ∈ ae μ ↔ μ sᶜ = 0`. In this file we define the filter and prove some basic theorems about it. ## Notation - `∀ᵐ x ∂μ, p x`: the predicate `p` holds for `μ`-a.e. all `x`; - `∃ᶠ x ∂μ, p x`: the predicate `p` holds on a set of nonzero measure; - `f =ᵐ[μ] g`: `f x = g x` for `μ`-a.e. all `x`; - `f ≤ᵐ[μ] g`: `f x ≤ g x` for `μ`-a.e. all `x`. ## Implementation details All notation introduced in this file reducibly unfolds to the corresponding definitions about filters, so generic lemmas about `Filter.Eventually`, `Filter.EventuallyEq` etc apply. However, we restate some lemmas specifically for `ae`. ## Tags outer measure, measure, almost everywhere -/ open Filter Set open scoped ENNReal namespace MeasureTheory variable {α β F : Type} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α} /-- The “almost everywhere” filter of co-null sets. -/ def ae (μ : F) : Filter α := .ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fun _t ht _s hs ↦ measure_mono_null hs ht #align measure_theory.measure.ae MeasureTheory.ae /-- `∀ᵐ a ∂μ, p a` means that `p a` for a.e. `a`, i.e. `p` holds true away from a null set. This is notation for `Filter.Eventually p (MeasureTheory.ae μ)`. -/ notation3 "∀ᵐ "(...)" ∂"μ", "r:(scoped p => Filter.Eventually p <\| MeasureTheory.ae μ) => r /-- `∃ᵐ a ∂μ, p a` means that `p` holds `∂μ`-frequently, i.e. `p` holds on a set of positive measure. This is notation for `Filter.Frequently p (MeasureTheory.ae μ)`. -/ notation3 "∃ᵐ "(...)" ∂"μ", "r:(scoped P => Filter.Frequently P <\| MeasureTheory.ae μ) => r /-- `f =ᵐ[μ] g` means `f` and `g` are eventually equal along the a.e. filter, i.e. `f=g` away from a null set. This is notation for `Filter.EventuallyEq (MeasureTheory.ae μ) f g`. -/ notation:50 f " =ᵐ[" μ:50 "] " g:50 => Filter.EventuallyEq (MeasureTheory.ae μ) f g /-- `f ≤ᵐ[μ] g` means `f` is eventually less than `g` along the a.e. filter, i.e. `f ≤ g` away from a null set. This is notation for `Filter.EventuallyLE (MeasureTheory.ae μ) f g`. -/ notation:50 f " ≤ᵐ[" μ:50 "] " g:50 => Filter.EventuallyLE (MeasureTheory.ae μ) f g theorem mem_ae_iff {s : Set α} : s ∈ ae μ ↔ μ sᶜ = 0 := Iff.rfl #align measure_theory.mem_ae_iff MeasureTheory.mem_ae_iff theorem ae_iff {p : α → Prop} : (∀ᵐ a ∂μ, p a) ↔ μ { a \| ¬p a } = 0 := Iff.rfl #align measure_theory.ae_iff MeasureTheory.ae_iff theorem compl_mem_ae_iff {s : Set α} : sᶜ ∈ ae μ ↔ μ s = 0 := by simp only [mem_ae_iff, compl_compl] #align measure_theory.compl_mem_ae_iff MeasureTheory.compl_mem_ae_iff theorem frequently_ae_iff {p : α → Prop} : (∃ᵐ a ∂μ, p a) ↔ μ { a \| p a } ≠ 0 := not_congr compl_mem_ae_iff #align measure_theory.frequently_ae_iff MeasureTheory.frequently_ae_iff theorem frequently_ae_mem_iff {s : Set α} : (∃ᵐ a ∂μ, a ∈ s) ↔ μ s ≠ 0 := not_congr compl_mem_ae_iff #align measure_theory.frequently_ae_mem_iff MeasureTheory.frequently_ae_mem_iff theorem measure_zero_iff_ae_nmem {s : Set α} : μ s = 0 ↔ ∀ᵐ a ∂μ, a ∉ s := compl_mem_ae_iff.symm #align measure_theory.measure_zero_iff_ae_nmem MeasureTheory.measure_zero_iff_ae_nmem theorem ae_of_all {p : α → Prop} (μ : F) : (∀ a, p a) → ∀ᵐ a ∂μ, p a := eventually_of_forall #align measure_theory.ae_of_all MeasureTheory.ae_of_all instance instCountableInterFilter : CountableInterFilter (ae μ) := by unfold ae; infer_instance #align measure_theory.measure.ae.countable_Inter_filter MeasureTheory.instCountableInterFilter theorem ae_all_iff {ι : Sort} [Countable ι] {p : α → ι → Prop} : (∀ᵐ a ∂μ, ∀ i, p a i) ↔ ∀ i, ∀ᵐ a ∂μ, p a i := eventually_countable_forall #align measure_theory.ae_all_iff MeasureTheory.ae_all_iff theorem all_ae_of {ι : Sort} {p : α → ι → Prop} (hp : ∀ᵐ a ∂μ, ∀ i, p a i) (i : ι) : ∀ᵐ a ∂μ, p a i := by filter_upwards [hp] with a ha using ha i lemma ae_iff_of_countable [Countable α] {p : α → Prop} : (∀ᵐ x ∂μ, p x) ↔ ∀ x, μ {x} ≠ 0 → p x := by rw [ae_iff, measure_null_iff_singleton] exacts [forall_congr' fun _ ↦ not_imp_comm, Set.to_countable _] theorem ae_ball_iff {ι : Type} {S : Set ι} (hS : S.Countable) {p : α → ∀ i ∈ S, Prop} : (∀ᵐ x ∂μ, ∀ i (hi : i ∈ S), p x i hi) ↔ ∀ i (hi : i ∈ S), ∀ᵐ x ∂μ, p x i hi := eventually_countable_ball hS #align measure_theory.ae_ball_iff MeasureTheory.ae_ball_iff theorem ae_eq_refl (f : α → β) : f =ᵐ[μ] f := EventuallyEq.rfl #align measure_theory.ae_eq_refl MeasureTheory.ae_eq_refl theorem ae_eq_symm {f g : α → β} (h : f =ᵐ[μ] g) : g =ᵐ[μ] f := h.symm #align measure_theory.ae_eq_symm MeasureTheory.ae_eq_symm theorem ae_eq_trans {f g h : α → β} (h₁ : f =ᵐ[μ] g) (h₂ : g =ᵐ[μ] h) : f =ᵐ[μ] h := h₁.trans h₂ #align measure_theory.ae_eq_trans MeasureTheory.ae_eq_trans theorem ae_le_of_ae_lt {β : Type*} [Preorder β] {f g : α → β} (h : ∀ᵐ x ∂μ, f x < g x) : f ≤ᵐ[μ] g := h.mono fun _ ↦ le_of_lt #align measure_theory.ae_le_of_ae_lt MeasureTheory.ae_le_of_ae_lt @[simp] theorem ae_eq_empty : s =ᵐ[μ] (∅ : Set α) ↔ μ s = 0 := eventuallyEq_empty.trans <\| by simp only [ae_iff, Classical.not_not, setOf_mem_eq] #align measure_theory.ae_eq_empty MeasureTheory.ae_eq_empty -- Porting note: The priority should be higher than `eventuallyEq_univ`. @[simp high] theorem ae_eq_univ : s =ᵐ[μ] (univ : Set α) ↔ μ sᶜ = 0 := eventuallyEq_univ #align measure_theory.ae_eq_univ MeasureTheory.ae_eq_univ theorem ae_le_set : s ≤ᵐ[μ] t ↔ μ (s \ t) = 0 := calc s ≤ᵐ[μ] t ↔ ∀ᵐ x ∂μ, x ∈ s → x ∈ t := Iff.rfl _ ↔ μ (s \ t) = 0 := by simp [ae_iff]; rfl #align measure_theory.ae_le_set MeasureTheory.ae_le_set theorem ae_le_set_inter {s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t') : (s ∩ s' : Set α) ≤ᵐ[μ] (t ∩ t' : Set α) := h.inter h' #align measure_theory.ae_le_set_inter MeasureTheory.ae_le_set_inter theorem ae_le_set_union {s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t') : (s ∪ s' : Set α) ≤ᵐ[μ] (t ∪ t' : Set α) := h.union h' #align measure_theory.ae_le_set_union MeasureTheory.ae_le_set_union	Mathlib/MeasureTheory/OuterMeasure/AE.lean	164	166	theorem union_ae_eq_right : (s ∪ t : Set α) =ᵐ[μ] t ↔ μ (s \ t) = 0 := by	simp [eventuallyLE_antisymm_iff, ae_le_set, union_diff_right, diff_eq_empty.2 Set.subset_union_right]
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel, Rémy Degenne -/ import Mathlib.Analysis.SpecialFunctions.Pow.Continuity import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv import Mathlib.Analysis.Calculus.FDeriv.Extend import Mathlib.Analysis.Calculus.Deriv.Prod import Mathlib.Analysis.SpecialFunctions.Log.Deriv import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv #align_import analysis.special_functions.pow.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # Derivatives of power function on `ℂ`, `ℝ`, `ℝ≥0`, and `ℝ≥0∞` We also prove differentiability and provide derivatives for the power functions `x ^ y`. -/ noncomputable section open scoped Classical Real Topology NNReal ENNReal Filter open Filter namespace Complex theorem hasStrictFDerivAt_cpow {p : ℂ × ℂ} (hp : p.1 ∈ slitPlane) : HasStrictFDerivAt (fun x : ℂ × ℂ => x.1 ^ x.2) ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ + (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ) p := by have A : p.1 ≠ 0 := slitPlane_ne_zero hp have : (fun x : ℂ × ℂ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) := ((isOpen_ne.preimage continuous_fst).eventually_mem A).mono fun p hp => cpow_def_of_ne_zero hp _ rw [cpow_sub _ _ A, cpow_one, mul_div_left_comm, mul_smul, mul_smul] refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm simpa only [cpow_def_of_ne_zero A, div_eq_mul_inv, mul_smul, add_comm, smul_add] using ((hasStrictFDerivAt_fst.clog hp).mul hasStrictFDerivAt_snd).cexp #align complex.has_strict_fderiv_at_cpow Complex.hasStrictFDerivAt_cpow theorem hasStrictFDerivAt_cpow' {x y : ℂ} (hp : x ∈ slitPlane) : HasStrictFDerivAt (fun x : ℂ × ℂ => x.1 ^ x.2) ((y * x ^ (y - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ + (x ^ y * log x) • ContinuousLinearMap.snd ℂ ℂ ℂ) (x, y) := @hasStrictFDerivAt_cpow (x, y) hp #align complex.has_strict_fderiv_at_cpow' Complex.hasStrictFDerivAt_cpow' theorem hasStrictDerivAt_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) : HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y := by rcases em (x = 0) with (rfl \| hx) · replace h := h.neg_resolve_left rfl rw [log_zero, mul_zero] refine (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq ?_ exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm · simpa only [cpow_def_of_ne_zero hx, mul_one] using ((hasStrictDerivAt_id y).const_mul (log x)).cexp #align complex.has_strict_deriv_at_const_cpow Complex.hasStrictDerivAt_const_cpow theorem hasFDerivAt_cpow {p : ℂ × ℂ} (hp : p.1 ∈ slitPlane) : HasFDerivAt (fun x : ℂ × ℂ => x.1 ^ x.2) ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ + (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ) p := (hasStrictFDerivAt_cpow hp).hasFDerivAt #align complex.has_fderiv_at_cpow Complex.hasFDerivAt_cpow end Complex section fderiv open Complex variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : E → ℂ} {f' g' : E →L[ℂ] ℂ} {x : E} {s : Set E} {c : ℂ}	Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean	79	82	theorem HasStrictFDerivAt.cpow (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) (h0 : f x ∈ slitPlane) : HasStrictFDerivAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x := by	convert (@hasStrictFDerivAt_cpow ((fun x => (f x, g x)) x) h0).comp x (hf.prod hg)
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot -/ import Mathlib.Algebra.BigOperators.Group.Finset import Mathlib.Algebra.Order.Field.Defs import Mathlib.Algebra.Order.Group.Instances import Mathlib.Algebra.Order.Group.MinMax import Mathlib.Algebra.Order.Ring.Basic import Mathlib.Data.Finset.Preimage import Mathlib.Order.Interval.Set.Disjoint import Mathlib.Order.Interval.Set.OrderIso import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.Filter.Bases #align_import order.filter.at_top_bot from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58" /-! # `Filter.atTop` and `Filter.atBot` filters on preorders, monoids and groups. In this file we define the filters * `Filter.atTop`: corresponds to `n → +∞`; * `Filter.atBot`: corresponds to `n → -∞`. Then we prove many lemmas like “if `f → +∞`, then `f ± c → +∞`”. -/ set_option autoImplicit true variable {ι ι' α β γ : Type} open Set namespace Filter /-- `atTop` is the filter representing the limit `→ ∞` on an ordered set. It is generated by the collection of up-sets `{b \| a ≤ b}`. (The preorder need not have a top element for this to be well defined, and indeed is trivial when a top element exists.) -/ def atTop [Preorder α] : Filter α := ⨅ a, 𝓟 (Ici a) #align filter.at_top Filter.atTop /-- `atBot` is the filter representing the limit `→ -∞` on an ordered set. It is generated by the collection of down-sets `{b \| b ≤ a}`. (The preorder need not have a bottom element for this to be well defined, and indeed is trivial when a bottom element exists.) -/ def atBot [Preorder α] : Filter α := ⨅ a, 𝓟 (Iic a) #align filter.at_bot Filter.atBot theorem mem_atTop [Preorder α] (a : α) : { b : α \| a ≤ b } ∈ @atTop α _ := mem_iInf_of_mem a <\| Subset.refl _ #align filter.mem_at_top Filter.mem_atTop theorem Ici_mem_atTop [Preorder α] (a : α) : Ici a ∈ (atTop : Filter α) := mem_atTop a #align filter.Ici_mem_at_top Filter.Ici_mem_atTop theorem Ioi_mem_atTop [Preorder α] [NoMaxOrder α] (x : α) : Ioi x ∈ (atTop : Filter α) := let ⟨z, hz⟩ := exists_gt x mem_of_superset (mem_atTop z) fun _ h => lt_of_lt_of_le hz h #align filter.Ioi_mem_at_top Filter.Ioi_mem_atTop theorem mem_atBot [Preorder α] (a : α) : { b : α \| b ≤ a } ∈ @atBot α _ := mem_iInf_of_mem a <\| Subset.refl _ #align filter.mem_at_bot Filter.mem_atBot theorem Iic_mem_atBot [Preorder α] (a : α) : Iic a ∈ (atBot : Filter α) := mem_atBot a #align filter.Iic_mem_at_bot Filter.Iic_mem_atBot theorem Iio_mem_atBot [Preorder α] [NoMinOrder α] (x : α) : Iio x ∈ (atBot : Filter α) := let ⟨z, hz⟩ := exists_lt x mem_of_superset (mem_atBot z) fun _ h => lt_of_le_of_lt h hz #align filter.Iio_mem_at_bot Filter.Iio_mem_atBot theorem disjoint_atBot_principal_Ioi [Preorder α] (x : α) : Disjoint atBot (𝓟 (Ioi x)) := disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl) (Iic_mem_atBot x) (mem_principal_self _) #align filter.disjoint_at_bot_principal_Ioi Filter.disjoint_atBot_principal_Ioi theorem disjoint_atTop_principal_Iio [Preorder α] (x : α) : Disjoint atTop (𝓟 (Iio x)) := @disjoint_atBot_principal_Ioi αᵒᵈ _ _ #align filter.disjoint_at_top_principal_Iio Filter.disjoint_atTop_principal_Iio theorem disjoint_atTop_principal_Iic [Preorder α] [NoMaxOrder α] (x : α) : Disjoint atTop (𝓟 (Iic x)) := disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl).symm (Ioi_mem_atTop x) (mem_principal_self _) #align filter.disjoint_at_top_principal_Iic Filter.disjoint_atTop_principal_Iic theorem disjoint_atBot_principal_Ici [Preorder α] [NoMinOrder α] (x : α) : Disjoint atBot (𝓟 (Ici x)) := @disjoint_atTop_principal_Iic αᵒᵈ _ _ _ #align filter.disjoint_at_bot_principal_Ici Filter.disjoint_atBot_principal_Ici theorem disjoint_pure_atTop [Preorder α] [NoMaxOrder α] (x : α) : Disjoint (pure x) atTop := Disjoint.symm <\| (disjoint_atTop_principal_Iic x).mono_right <\| le_principal_iff.2 <\| mem_pure.2 right_mem_Iic #align filter.disjoint_pure_at_top Filter.disjoint_pure_atTop theorem disjoint_pure_atBot [Preorder α] [NoMinOrder α] (x : α) : Disjoint (pure x) atBot := @disjoint_pure_atTop αᵒᵈ _ _ _ #align filter.disjoint_pure_at_bot Filter.disjoint_pure_atBot theorem not_tendsto_const_atTop [Preorder α] [NoMaxOrder α] (x : α) (l : Filter β) [l.NeBot] : ¬Tendsto (fun _ => x) l atTop := tendsto_const_pure.not_tendsto (disjoint_pure_atTop x) #align filter.not_tendsto_const_at_top Filter.not_tendsto_const_atTop theorem not_tendsto_const_atBot [Preorder α] [NoMinOrder α] (x : α) (l : Filter β) [l.NeBot] : ¬Tendsto (fun _ => x) l atBot := tendsto_const_pure.not_tendsto (disjoint_pure_atBot x) #align filter.not_tendsto_const_at_bot Filter.not_tendsto_const_atBot theorem disjoint_atBot_atTop [PartialOrder α] [Nontrivial α] : Disjoint (atBot : Filter α) atTop := by rcases exists_pair_ne α with ⟨x, y, hne⟩ by_cases hle : x ≤ y · refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot x) (Ici_mem_atTop y) exact Iic_disjoint_Ici.2 (hle.lt_of_ne hne).not_le · refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot y) (Ici_mem_atTop x) exact Iic_disjoint_Ici.2 hle #align filter.disjoint_at_bot_at_top Filter.disjoint_atBot_atTop theorem disjoint_atTop_atBot [PartialOrder α] [Nontrivial α] : Disjoint (atTop : Filter α) atBot := disjoint_atBot_atTop.symm #align filter.disjoint_at_top_at_bot Filter.disjoint_atTop_atBot theorem hasAntitoneBasis_atTop [Nonempty α] [Preorder α] [IsDirected α (· ≤ ·)] : (@atTop α _).HasAntitoneBasis Ici := .iInf_principal fun _ _ ↦ Ici_subset_Ici.2 theorem atTop_basis [Nonempty α] [SemilatticeSup α] : (@atTop α _).HasBasis (fun _ => True) Ici := hasAntitoneBasis_atTop.1 #align filter.at_top_basis Filter.atTop_basis theorem atTop_eq_generate_Ici [SemilatticeSup α] : atTop = generate (range (Ici (α := α))) := by rcases isEmpty_or_nonempty α with hα\|hα · simp only [eq_iff_true_of_subsingleton] · simp [(atTop_basis (α := α)).eq_generate, range] theorem atTop_basis' [SemilatticeSup α] (a : α) : (@atTop α _).HasBasis (fun x => a ≤ x) Ici := ⟨fun _ => (@atTop_basis α ⟨a⟩ _).mem_iff.trans ⟨fun ⟨x, _, hx⟩ => ⟨x ⊔ a, le_sup_right, fun _y hy => hx (le_trans le_sup_left hy)⟩, fun ⟨x, _, hx⟩ => ⟨x, trivial, hx⟩⟩⟩ #align filter.at_top_basis' Filter.atTop_basis' theorem atBot_basis [Nonempty α] [SemilatticeInf α] : (@atBot α _).HasBasis (fun _ => True) Iic := @atTop_basis αᵒᵈ _ _ #align filter.at_bot_basis Filter.atBot_basis theorem atBot_basis' [SemilatticeInf α] (a : α) : (@atBot α _).HasBasis (fun x => x ≤ a) Iic := @atTop_basis' αᵒᵈ _ _ #align filter.at_bot_basis' Filter.atBot_basis' @[instance] theorem atTop_neBot [Nonempty α] [SemilatticeSup α] : NeBot (atTop : Filter α) := atTop_basis.neBot_iff.2 fun _ => nonempty_Ici #align filter.at_top_ne_bot Filter.atTop_neBot @[instance] theorem atBot_neBot [Nonempty α] [SemilatticeInf α] : NeBot (atBot : Filter α) := @atTop_neBot αᵒᵈ _ _ #align filter.at_bot_ne_bot Filter.atBot_neBot @[simp] theorem mem_atTop_sets [Nonempty α] [SemilatticeSup α] {s : Set α} : s ∈ (atTop : Filter α) ↔ ∃ a : α, ∀ b ≥ a, b ∈ s := atTop_basis.mem_iff.trans <\| exists_congr fun _ => true_and_iff _ #align filter.mem_at_top_sets Filter.mem_atTop_sets @[simp] theorem mem_atBot_sets [Nonempty α] [SemilatticeInf α] {s : Set α} : s ∈ (atBot : Filter α) ↔ ∃ a : α, ∀ b ≤ a, b ∈ s := @mem_atTop_sets αᵒᵈ _ _ _ #align filter.mem_at_bot_sets Filter.mem_atBot_sets @[simp] theorem eventually_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} : (∀ᶠ x in atTop, p x) ↔ ∃ a, ∀ b ≥ a, p b := mem_atTop_sets #align filter.eventually_at_top Filter.eventually_atTop @[simp] theorem eventually_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} : (∀ᶠ x in atBot, p x) ↔ ∃ a, ∀ b ≤ a, p b := mem_atBot_sets #align filter.eventually_at_bot Filter.eventually_atBot theorem eventually_ge_atTop [Preorder α] (a : α) : ∀ᶠ x in atTop, a ≤ x := mem_atTop a #align filter.eventually_ge_at_top Filter.eventually_ge_atTop theorem eventually_le_atBot [Preorder α] (a : α) : ∀ᶠ x in atBot, x ≤ a := mem_atBot a #align filter.eventually_le_at_bot Filter.eventually_le_atBot theorem eventually_gt_atTop [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ x in atTop, a < x := Ioi_mem_atTop a #align filter.eventually_gt_at_top Filter.eventually_gt_atTop theorem eventually_ne_atTop [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ x in atTop, x ≠ a := (eventually_gt_atTop a).mono fun _ => ne_of_gt #align filter.eventually_ne_at_top Filter.eventually_ne_atTop protected theorem Tendsto.eventually_gt_atTop [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, c < f x := hf.eventually (eventually_gt_atTop c) #align filter.tendsto.eventually_gt_at_top Filter.Tendsto.eventually_gt_atTop protected theorem Tendsto.eventually_ge_atTop [Preorder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, c ≤ f x := hf.eventually (eventually_ge_atTop c) #align filter.tendsto.eventually_ge_at_top Filter.Tendsto.eventually_ge_atTop protected theorem Tendsto.eventually_ne_atTop [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, f x ≠ c := hf.eventually (eventually_ne_atTop c) #align filter.tendsto.eventually_ne_at_top Filter.Tendsto.eventually_ne_atTop protected theorem Tendsto.eventually_ne_atTop' [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atTop) (c : α) : ∀ᶠ x in l, x ≠ c := (hf.eventually_ne_atTop (f c)).mono fun _ => ne_of_apply_ne f #align filter.tendsto.eventually_ne_at_top' Filter.Tendsto.eventually_ne_atTop' theorem eventually_lt_atBot [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ x in atBot, x < a := Iio_mem_atBot a #align filter.eventually_lt_at_bot Filter.eventually_lt_atBot theorem eventually_ne_atBot [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ x in atBot, x ≠ a := (eventually_lt_atBot a).mono fun _ => ne_of_lt #align filter.eventually_ne_at_bot Filter.eventually_ne_atBot protected theorem Tendsto.eventually_lt_atBot [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x < c := hf.eventually (eventually_lt_atBot c) #align filter.tendsto.eventually_lt_at_bot Filter.Tendsto.eventually_lt_atBot protected theorem Tendsto.eventually_le_atBot [Preorder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x ≤ c := hf.eventually (eventually_le_atBot c) #align filter.tendsto.eventually_le_at_bot Filter.Tendsto.eventually_le_atBot protected theorem Tendsto.eventually_ne_atBot [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} (hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x ≠ c := hf.eventually (eventually_ne_atBot c) #align filter.tendsto.eventually_ne_at_bot Filter.Tendsto.eventually_ne_atBot theorem eventually_forall_ge_atTop [Preorder α] {p : α → Prop} : (∀ᶠ x in atTop, ∀ y, x ≤ y → p y) ↔ ∀ᶠ x in atTop, p x := by refine ⟨fun h ↦ h.mono fun x hx ↦ hx x le_rfl, fun h ↦ ?_⟩ rcases (hasBasis_iInf_principal_finite _).eventually_iff.1 h with ⟨S, hSf, hS⟩ refine mem_iInf_of_iInter hSf (V := fun x ↦ Ici x.1) (fun _ ↦ Subset.rfl) fun x hx y hy ↦ ?_ simp only [mem_iInter] at hS hx exact hS fun z hz ↦ le_trans (hx ⟨z, hz⟩) hy theorem eventually_forall_le_atBot [Preorder α] {p : α → Prop} : (∀ᶠ x in atBot, ∀ y, y ≤ x → p y) ↔ ∀ᶠ x in atBot, p x := eventually_forall_ge_atTop (α := αᵒᵈ) theorem Tendsto.eventually_forall_ge_atTop {α β : Type} [Preorder β] {l : Filter α} {p : β → Prop} {f : α → β} (hf : Tendsto f l atTop) (h_evtl : ∀ᶠ x in atTop, p x) : ∀ᶠ x in l, ∀ y, f x ≤ y → p y := by rw [← Filter.eventually_forall_ge_atTop] at h_evtl; exact (h_evtl.comap f).filter_mono hf.le_comap theorem Tendsto.eventually_forall_le_atBot {α β : Type*} [Preorder β] {l : Filter α} {p : β → Prop} {f : α → β} (hf : Tendsto f l atBot) (h_evtl : ∀ᶠ x in atBot, p x) : ∀ᶠ x in l, ∀ y, y ≤ f x → p y := by rw [← Filter.eventually_forall_le_atBot] at h_evtl; exact (h_evtl.comap f).filter_mono hf.le_comap theorem atTop_basis_Ioi [Nonempty α] [SemilatticeSup α] [NoMaxOrder α] : (@atTop α _).HasBasis (fun _ => True) Ioi := atTop_basis.to_hasBasis (fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩) fun a ha => (exists_gt a).imp fun _b hb => ⟨ha, Ici_subset_Ioi.2 hb⟩ #align filter.at_top_basis_Ioi Filter.atTop_basis_Ioi lemma atTop_basis_Ioi' [SemilatticeSup α] [NoMaxOrder α] (a : α) : atTop.HasBasis (a < ·) Ioi := have : Nonempty α := ⟨a⟩ atTop_basis_Ioi.to_hasBasis (fun b _ ↦ let ⟨c, hc⟩ := exists_gt (a ⊔ b) ⟨c, le_sup_left.trans_lt hc, Ioi_subset_Ioi <\| le_sup_right.trans hc.le⟩) fun b _ ↦ ⟨b, trivial, Subset.rfl⟩ theorem atTop_countable_basis [Nonempty α] [SemilatticeSup α] [Countable α] : HasCountableBasis (atTop : Filter α) (fun _ => True) Ici := { atTop_basis with countable := to_countable _ } #align filter.at_top_countable_basis Filter.atTop_countable_basis theorem atBot_countable_basis [Nonempty α] [SemilatticeInf α] [Countable α] : HasCountableBasis (atBot : Filter α) (fun _ => True) Iic := { atBot_basis with countable := to_countable _ } #align filter.at_bot_countable_basis Filter.atBot_countable_basis instance (priority := 200) atTop.isCountablyGenerated [Preorder α] [Countable α] : (atTop : Filter <\| α).IsCountablyGenerated := isCountablyGenerated_seq _ #align filter.at_top.is_countably_generated Filter.atTop.isCountablyGenerated instance (priority := 200) atBot.isCountablyGenerated [Preorder α] [Countable α] : (atBot : Filter <\| α).IsCountablyGenerated := isCountablyGenerated_seq _ #align filter.at_bot.is_countably_generated Filter.atBot.isCountablyGenerated theorem _root_.IsTop.atTop_eq [Preorder α] {a : α} (ha : IsTop a) : atTop = 𝓟 (Ici a) := (iInf_le _ _).antisymm <\| le_iInf fun b ↦ principal_mono.2 <\| Ici_subset_Ici.2 <\| ha b theorem _root_.IsBot.atBot_eq [Preorder α] {a : α} (ha : IsBot a) : atBot = 𝓟 (Iic a) := ha.toDual.atTop_eq theorem OrderTop.atTop_eq (α) [PartialOrder α] [OrderTop α] : (atTop : Filter α) = pure ⊤ := by rw [isTop_top.atTop_eq, Ici_top, principal_singleton] #align filter.order_top.at_top_eq Filter.OrderTop.atTop_eq theorem OrderBot.atBot_eq (α) [PartialOrder α] [OrderBot α] : (atBot : Filter α) = pure ⊥ := @OrderTop.atTop_eq αᵒᵈ _ _ #align filter.order_bot.at_bot_eq Filter.OrderBot.atBot_eq @[nontriviality] theorem Subsingleton.atTop_eq (α) [Subsingleton α] [Preorder α] : (atTop : Filter α) = ⊤ := by refine top_unique fun s hs x => ?_ rw [atTop, ciInf_subsingleton x, mem_principal] at hs exact hs left_mem_Ici #align filter.subsingleton.at_top_eq Filter.Subsingleton.atTop_eq @[nontriviality] theorem Subsingleton.atBot_eq (α) [Subsingleton α] [Preorder α] : (atBot : Filter α) = ⊤ := @Subsingleton.atTop_eq αᵒᵈ _ _ #align filter.subsingleton.at_bot_eq Filter.Subsingleton.atBot_eq theorem tendsto_atTop_pure [PartialOrder α] [OrderTop α] (f : α → β) : Tendsto f atTop (pure <\| f ⊤) := (OrderTop.atTop_eq α).symm ▸ tendsto_pure_pure _ _ #align filter.tendsto_at_top_pure Filter.tendsto_atTop_pure theorem tendsto_atBot_pure [PartialOrder α] [OrderBot α] (f : α → β) : Tendsto f atBot (pure <\| f ⊥) := @tendsto_atTop_pure αᵒᵈ _ _ _ _ #align filter.tendsto_at_bot_pure Filter.tendsto_atBot_pure theorem Eventually.exists_forall_of_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} (h : ∀ᶠ x in atTop, p x) : ∃ a, ∀ b ≥ a, p b := eventually_atTop.mp h #align filter.eventually.exists_forall_of_at_top Filter.Eventually.exists_forall_of_atTop theorem Eventually.exists_forall_of_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} (h : ∀ᶠ x in atBot, p x) : ∃ a, ∀ b ≤ a, p b := eventually_atBot.mp h #align filter.eventually.exists_forall_of_at_bot Filter.Eventually.exists_forall_of_atBot lemma exists_eventually_atTop [SemilatticeSup α] [Nonempty α] {r : α → β → Prop} : (∃ b, ∀ᶠ a in atTop, r a b) ↔ ∀ᶠ a₀ in atTop, ∃ b, ∀ a ≥ a₀, r a b := by simp_rw [eventually_atTop, ← exists_swap (α := α)] exact exists_congr fun a ↦ .symm <\| forall_ge_iff <\| Monotone.exists fun _ _ _ hb H n hn ↦ H n (hb.trans hn) lemma exists_eventually_atBot [SemilatticeInf α] [Nonempty α] {r : α → β → Prop} : (∃ b, ∀ᶠ a in atBot, r a b) ↔ ∀ᶠ a₀ in atBot, ∃ b, ∀ a ≤ a₀, r a b := by simp_rw [eventually_atBot, ← exists_swap (α := α)] exact exists_congr fun a ↦ .symm <\| forall_le_iff <\| Antitone.exists fun _ _ _ hb H n hn ↦ H n (hn.trans hb) theorem frequently_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} : (∃ᶠ x in atTop, p x) ↔ ∀ a, ∃ b ≥ a, p b := atTop_basis.frequently_iff.trans <\| by simp #align filter.frequently_at_top Filter.frequently_atTop theorem frequently_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} : (∃ᶠ x in atBot, p x) ↔ ∀ a, ∃ b ≤ a, p b := @frequently_atTop αᵒᵈ _ _ _ #align filter.frequently_at_bot Filter.frequently_atBot theorem frequently_atTop' [SemilatticeSup α] [Nonempty α] [NoMaxOrder α] {p : α → Prop} : (∃ᶠ x in atTop, p x) ↔ ∀ a, ∃ b > a, p b := atTop_basis_Ioi.frequently_iff.trans <\| by simp #align filter.frequently_at_top' Filter.frequently_atTop' theorem frequently_atBot' [SemilatticeInf α] [Nonempty α] [NoMinOrder α] {p : α → Prop} : (∃ᶠ x in atBot, p x) ↔ ∀ a, ∃ b < a, p b := @frequently_atTop' αᵒᵈ _ _ _ _ #align filter.frequently_at_bot' Filter.frequently_atBot' theorem Frequently.forall_exists_of_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} (h : ∃ᶠ x in atTop, p x) : ∀ a, ∃ b ≥ a, p b := frequently_atTop.mp h #align filter.frequently.forall_exists_of_at_top Filter.Frequently.forall_exists_of_atTop theorem Frequently.forall_exists_of_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} (h : ∃ᶠ x in atBot, p x) : ∀ a, ∃ b ≤ a, p b := frequently_atBot.mp h #align filter.frequently.forall_exists_of_at_bot Filter.Frequently.forall_exists_of_atBot theorem map_atTop_eq [Nonempty α] [SemilatticeSup α] {f : α → β} : atTop.map f = ⨅ a, 𝓟 (f '' { a' \| a ≤ a' }) := (atTop_basis.map f).eq_iInf #align filter.map_at_top_eq Filter.map_atTop_eq theorem map_atBot_eq [Nonempty α] [SemilatticeInf α] {f : α → β} : atBot.map f = ⨅ a, 𝓟 (f '' { a' \| a' ≤ a }) := @map_atTop_eq αᵒᵈ _ _ _ _ #align filter.map_at_bot_eq Filter.map_atBot_eq theorem tendsto_atTop [Preorder β] {m : α → β} {f : Filter α} : Tendsto m f atTop ↔ ∀ b, ∀ᶠ a in f, b ≤ m a := by simp only [atTop, tendsto_iInf, tendsto_principal, mem_Ici] #align filter.tendsto_at_top Filter.tendsto_atTop theorem tendsto_atBot [Preorder β] {m : α → β} {f : Filter α} : Tendsto m f atBot ↔ ∀ b, ∀ᶠ a in f, m a ≤ b := @tendsto_atTop α βᵒᵈ _ m f #align filter.tendsto_at_bot Filter.tendsto_atBot theorem tendsto_atTop_mono' [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) (h₁ : Tendsto f₁ l atTop) : Tendsto f₂ l atTop := tendsto_atTop.2 fun b => by filter_upwards [tendsto_atTop.1 h₁ b, h] with x using le_trans #align filter.tendsto_at_top_mono' Filter.tendsto_atTop_mono' theorem tendsto_atBot_mono' [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) : Tendsto f₂ l atBot → Tendsto f₁ l atBot := @tendsto_atTop_mono' _ βᵒᵈ _ _ _ _ h #align filter.tendsto_at_bot_mono' Filter.tendsto_atBot_mono' theorem tendsto_atTop_mono [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ n, f n ≤ g n) : Tendsto f l atTop → Tendsto g l atTop := tendsto_atTop_mono' l <\| eventually_of_forall h #align filter.tendsto_at_top_mono Filter.tendsto_atTop_mono theorem tendsto_atBot_mono [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ n, f n ≤ g n) : Tendsto g l atBot → Tendsto f l atBot := @tendsto_atTop_mono _ βᵒᵈ _ _ _ _ h #align filter.tendsto_at_bot_mono Filter.tendsto_atBot_mono lemma atTop_eq_generate_of_forall_exists_le [LinearOrder α] {s : Set α} (hs : ∀ x, ∃ y ∈ s, x ≤ y) : (atTop : Filter α) = generate (Ici '' s) := by rw [atTop_eq_generate_Ici] apply le_antisymm · rw [le_generate_iff] rintro - ⟨y, -, rfl⟩ exact mem_generate_of_mem ⟨y, rfl⟩ · rw [le_generate_iff] rintro - ⟨x, -, -, rfl⟩ rcases hs x with ⟨y, ys, hy⟩ have A : Ici y ∈ generate (Ici '' s) := mem_generate_of_mem (mem_image_of_mem _ ys) have B : Ici y ⊆ Ici x := Ici_subset_Ici.2 hy exact sets_of_superset (generate (Ici '' s)) A B lemma atTop_eq_generate_of_not_bddAbove [LinearOrder α] {s : Set α} (hs : ¬ BddAbove s) : (atTop : Filter α) = generate (Ici '' s) := by refine atTop_eq_generate_of_forall_exists_le fun x ↦ ?_ obtain ⟨y, hy, hy'⟩ := not_bddAbove_iff.mp hs x exact ⟨y, hy, hy'.le⟩ end Filter namespace OrderIso open Filter variable [Preorder α] [Preorder β] @[simp] theorem comap_atTop (e : α ≃o β) : comap e atTop = atTop := by simp [atTop, ← e.surjective.iInf_comp] #align order_iso.comap_at_top OrderIso.comap_atTop @[simp] theorem comap_atBot (e : α ≃o β) : comap e atBot = atBot := e.dual.comap_atTop #align order_iso.comap_at_bot OrderIso.comap_atBot @[simp] theorem map_atTop (e : α ≃o β) : map (e : α → β) atTop = atTop := by rw [← e.comap_atTop, map_comap_of_surjective e.surjective] #align order_iso.map_at_top OrderIso.map_atTop @[simp] theorem map_atBot (e : α ≃o β) : map (e : α → β) atBot = atBot := e.dual.map_atTop #align order_iso.map_at_bot OrderIso.map_atBot theorem tendsto_atTop (e : α ≃o β) : Tendsto e atTop atTop := e.map_atTop.le #align order_iso.tendsto_at_top OrderIso.tendsto_atTop theorem tendsto_atBot (e : α ≃o β) : Tendsto e atBot atBot := e.map_atBot.le #align order_iso.tendsto_at_bot OrderIso.tendsto_atBot @[simp] theorem tendsto_atTop_iff {l : Filter γ} {f : γ → α} (e : α ≃o β) : Tendsto (fun x => e (f x)) l atTop ↔ Tendsto f l atTop := by rw [← e.comap_atTop, tendsto_comap_iff, Function.comp_def] #align order_iso.tendsto_at_top_iff OrderIso.tendsto_atTop_iff @[simp] theorem tendsto_atBot_iff {l : Filter γ} {f : γ → α} (e : α ≃o β) : Tendsto (fun x => e (f x)) l atBot ↔ Tendsto f l atBot := e.dual.tendsto_atTop_iff #align order_iso.tendsto_at_bot_iff OrderIso.tendsto_atBot_iff end OrderIso namespace Filter /-! ### Sequences -/ theorem inf_map_atTop_neBot_iff [SemilatticeSup α] [Nonempty α] {F : Filter β} {u : α → β} : NeBot (F ⊓ map u atTop) ↔ ∀ U ∈ F, ∀ N, ∃ n ≥ N, u n ∈ U := by simp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]; rfl #align filter.inf_map_at_top_ne_bot_iff Filter.inf_map_atTop_neBot_iff theorem inf_map_atBot_neBot_iff [SemilatticeInf α] [Nonempty α] {F : Filter β} {u : α → β} : NeBot (F ⊓ map u atBot) ↔ ∀ U ∈ F, ∀ N, ∃ n ≤ N, u n ∈ U := @inf_map_atTop_neBot_iff αᵒᵈ _ _ _ _ _ #align filter.inf_map_at_bot_ne_bot_iff Filter.inf_map_atBot_neBot_iff theorem extraction_of_frequently_atTop' {P : ℕ → Prop} (h : ∀ N, ∃ n > N, P n) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P (φ n) := by choose u hu hu' using h refine ⟨fun n => u^[n + 1] 0, strictMono_nat_of_lt_succ fun n => ?_, fun n => ?_⟩ · exact Trans.trans (hu _) (Function.iterate_succ_apply' _ _ _).symm · simpa only [Function.iterate_succ_apply'] using hu' _ #align filter.extraction_of_frequently_at_top' Filter.extraction_of_frequently_atTop'	Mathlib/Order/Filter/AtTopBot.lean	530	533	theorem extraction_of_frequently_atTop {P : ℕ → Prop} (h : ∃ᶠ n in atTop, P n) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P (φ n) := by	rw [frequently_atTop'] at h exact extraction_of_frequently_atTop' h
/- Copyright (c) 2020 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic import Mathlib.RingTheory.RootsOfUnity.Minpoly #align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f" /-! # Roots of cyclotomic polynomials. We gather results about roots of cyclotomic polynomials. In particular we show in `Polynomial.cyclotomic_eq_minpoly` that `cyclotomic n R` is the minimal polynomial of a primitive root of unity. ## Main results * `IsPrimitiveRoot.isRoot_cyclotomic` : Any `n`-th primitive root of unity is a root of `cyclotomic n R`. * `isRoot_cyclotomic_iff` : if `NeZero (n : R)`, then `μ` is a root of `cyclotomic n R` if and only if `μ` is a primitive root of unity. * `Polynomial.cyclotomic_eq_minpoly` : `cyclotomic n ℤ` is the minimal polynomial of a primitive `n`-th root of unity `μ`. * `Polynomial.cyclotomic.irreducible` : `cyclotomic n ℤ` is irreducible. ## Implementation details To prove `Polynomial.cyclotomic.irreducible`, the irreducibility of `cyclotomic n ℤ`, we show in `Polynomial.cyclotomic_eq_minpoly` that `cyclotomic n ℤ` is the minimal polynomial of any `n`-th primitive root of unity `μ : K`, where `K` is a field of characteristic `0`. -/ namespace Polynomial variable {R : Type} [CommRing R] {n : ℕ} theorem isRoot_of_unity_of_root_cyclotomic {ζ : R} {i : ℕ} (hi : i ∈ n.divisors) (h : (cyclotomic i R).IsRoot ζ) : ζ ^ n = 1 := by rcases n.eq_zero_or_pos with (rfl \| hn) · exact pow_zero _ have := congr_arg (eval ζ) (prod_cyclotomic_eq_X_pow_sub_one hn R).symm rw [eval_sub, eval_pow, eval_X, eval_one] at this convert eq_add_of_sub_eq' this convert (add_zero (M := R) _).symm apply eval_eq_zero_of_dvd_of_eval_eq_zero _ h exact Finset.dvd_prod_of_mem _ hi #align polynomial.is_root_of_unity_of_root_cyclotomic Polynomial.isRoot_of_unity_of_root_cyclotomic section IsDomain variable [IsDomain R] theorem _root_.isRoot_of_unity_iff (h : 0 < n) (R : Type) [CommRing R] [IsDomain R] {ζ : R} : ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).IsRoot ζ := by rw [← mem_nthRoots h, nthRoots, mem_roots <\| X_pow_sub_C_ne_zero h _, C_1, ← prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod] #align is_root_of_unity_iff isRoot_of_unity_iff /-- Any `n`-th primitive root of unity is a root of `cyclotomic n R`. -/ theorem _root_.IsPrimitiveRoot.isRoot_cyclotomic (hpos : 0 < n) {μ : R} (h : IsPrimitiveRoot μ n) : IsRoot (cyclotomic n R) μ := by rw [← mem_roots (cyclotomic_ne_zero n R), cyclotomic_eq_prod_X_sub_primitiveRoots h, roots_prod_X_sub_C, ← Finset.mem_def] rwa [← mem_primitiveRoots hpos] at h #align is_primitive_root.is_root_cyclotomic IsPrimitiveRoot.isRoot_cyclotomic private theorem isRoot_cyclotomic_iff' {n : ℕ} {K : Type} [Field K] {μ : K} [NeZero (n : K)] : IsRoot (cyclotomic n K) μ ↔ IsPrimitiveRoot μ n := by -- in this proof, `o` stands for `orderOf μ` have hnpos : 0 < n := (NeZero.of_neZero_natCast K).out.bot_lt refine ⟨fun hμ => ?_, IsPrimitiveRoot.isRoot_cyclotomic hnpos⟩ have hμn : μ ^ n = 1 := by rw [isRoot_of_unity_iff hnpos _] exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩ by_contra hnμ have ho : 0 < orderOf μ := (isOfFinOrder_iff_pow_eq_one.2 <\| ⟨n, hnpos, hμn⟩).orderOf_pos have := pow_orderOf_eq_one μ rw [isRoot_of_unity_iff ho] at this obtain ⟨i, hio, hiμ⟩ := this replace hio := Nat.dvd_of_mem_divisors hio rw [IsPrimitiveRoot.not_iff] at hnμ rw [← orderOf_dvd_iff_pow_eq_one] at hμn have key : i < n := (Nat.le_of_dvd ho hio).trans_lt ((Nat.le_of_dvd hnpos hμn).lt_of_ne hnμ) have key' : i ∣ n := hio.trans hμn rw [← Polynomial.dvd_iff_isRoot] at hμ hiμ have hni : {i, n} ⊆ n.divisors := by simpa [Finset.insert_subset_iff, key'] using hnpos.ne' obtain ⟨k, hk⟩ := hiμ obtain ⟨j, hj⟩ := hμ have := prod_cyclotomic_eq_X_pow_sub_one hnpos K rw [← Finset.prod_sdiff hni, Finset.prod_pair key.ne, hk, hj] at this have hn := (X_pow_sub_one_separable_iff.mpr <\| NeZero.natCast_ne n K).squarefree rw [← this, Squarefree] at hn specialize hn (X - C μ) ⟨(∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) k * j, by ring⟩ simp [Polynomial.isUnit_iff_degree_eq_zero] at hn theorem isRoot_cyclotomic_iff [NeZero (n : R)] {μ : R} : IsRoot (cyclotomic n R) μ ↔ IsPrimitiveRoot μ n := by have hf : Function.Injective _ := IsFractionRing.injective R (FractionRing R) haveI : NeZero (n : FractionRing R) := NeZero.nat_of_injective hf rw [← isRoot_map_iff hf, ← IsPrimitiveRoot.map_iff_of_injective hf, map_cyclotomic, ← isRoot_cyclotomic_iff'] #align polynomial.is_root_cyclotomic_iff Polynomial.isRoot_cyclotomic_iff theorem roots_cyclotomic_nodup [NeZero (n : R)] : (cyclotomic n R).roots.Nodup := by obtain h \| ⟨ζ, hζ⟩ := (cyclotomic n R).roots.empty_or_exists_mem · exact h.symm ▸ Multiset.nodup_zero rw [mem_roots <\| cyclotomic_ne_zero n R, isRoot_cyclotomic_iff] at hζ refine Multiset.nodup_of_le (roots.le_of_dvd (X_pow_sub_C_ne_zero (NeZero.pos_of_neZero_natCast R) 1) <\| cyclotomic.dvd_X_pow_sub_one n R) hζ.nthRoots_one_nodup #align polynomial.roots_cyclotomic_nodup Polynomial.roots_cyclotomic_nodup theorem cyclotomic.roots_to_finset_eq_primitiveRoots [NeZero (n : R)] : (⟨(cyclotomic n R).roots, roots_cyclotomic_nodup⟩ : Finset _) = primitiveRoots n R := by ext a -- Porting note: was -- `simp [cyclotomic_ne_zero n R, isRoot_cyclotomic_iff, mem_primitiveRoots,` -- ` NeZero.pos_of_neZero_natCast R]` simp only [mem_primitiveRoots, NeZero.pos_of_neZero_natCast R] convert isRoot_cyclotomic_iff (n := n) (μ := a) simp [cyclotomic_ne_zero n R] #align polynomial.cyclotomic.roots_to_finset_eq_primitive_roots Polynomial.cyclotomic.roots_to_finset_eq_primitiveRoots theorem cyclotomic.roots_eq_primitiveRoots_val [NeZero (n : R)] : (cyclotomic n R).roots = (primitiveRoots n R).val := by rw [← cyclotomic.roots_to_finset_eq_primitiveRoots] #align polynomial.cyclotomic.roots_eq_primitive_roots_val Polynomial.cyclotomic.roots_eq_primitiveRoots_val /-- If `R` is of characteristic zero, then `ζ` is a root of `cyclotomic n R` if and only if it is a primitive `n`-th root of unity. -/ theorem isRoot_cyclotomic_iff_charZero {n : ℕ} {R : Type*} [CommRing R] [IsDomain R] [CharZero R] {μ : R} (hn : 0 < n) : (Polynomial.cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n := letI := NeZero.of_gt hn isRoot_cyclotomic_iff #align polynomial.is_root_cyclotomic_iff_char_zero Polynomial.isRoot_cyclotomic_iff_charZero end IsDomain /-- Over a ring `R` of characteristic zero, `fun n => cyclotomic n R` is injective. -/ theorem cyclotomic_injective [CharZero R] : Function.Injective fun n => cyclotomic n R := by intro n m hnm simp only at hnm rcases eq_or_ne n 0 with (rfl \| hzero) · rw [cyclotomic_zero] at hnm replace hnm := congr_arg natDegree hnm rwa [natDegree_one, natDegree_cyclotomic, eq_comm, Nat.totient_eq_zero, eq_comm] at hnm · haveI := NeZero.mk hzero rw [← map_cyclotomic_int _ R, ← map_cyclotomic_int _ R] at hnm replace hnm := map_injective (Int.castRingHom R) Int.cast_injective hnm replace hnm := congr_arg (map (Int.castRingHom ℂ)) hnm rw [map_cyclotomic_int, map_cyclotomic_int] at hnm have hprim := Complex.isPrimitiveRoot_exp _ hzero have hroot := isRoot_cyclotomic_iff (R := ℂ).2 hprim rw [hnm] at hroot haveI hmzero : NeZero m := ⟨fun h => by simp [h] at hroot⟩ rw [isRoot_cyclotomic_iff (R := ℂ)] at hroot replace hprim := hprim.eq_orderOf rwa [← IsPrimitiveRoot.eq_orderOf hroot] at hprim #align polynomial.cyclotomic_injective Polynomial.cyclotomic_injective /-- The minimal polynomial of a primitive `n`-th root of unity `μ` divides `cyclotomic n ℤ`. -/	Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean	165	168	theorem _root_.IsPrimitiveRoot.minpoly_dvd_cyclotomic {n : ℕ} {K : Type*} [Field K] {μ : K} (h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K] : minpoly ℤ μ ∣ cyclotomic n ℤ := by	apply minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos) simpa [aeval_def, eval₂_eq_eval_map, IsRoot.def] using h.isRoot_cyclotomic hpos
/- Copyright (c) 2022 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Batteries.Tactic.SeqFocus /-! ## Ordering -/ namespace Ordering @[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl @[simp] theorem swap_inj {o₁ o₂ : Ordering} : o₁.swap = o₂.swap ↔ o₁ = o₂ := ⟨fun h => by simpa using congrArg swap h, congrArg _⟩ theorem swap_then (o₁ o₂ : Ordering) : (o₁.then o₂).swap = o₁.swap.then o₂.swap := by cases o₁ <;> rfl theorem then_eq_lt {o₁ o₂ : Ordering} : o₁.then o₂ = lt ↔ o₁ = lt ∨ o₁ = eq ∧ o₂ = lt := by cases o₁ <;> cases o₂ <;> decide theorem then_eq_eq {o₁ o₂ : Ordering} : o₁.then o₂ = eq ↔ o₁ = eq ∧ o₂ = eq := by cases o₁ <;> simp [«then»] theorem then_eq_gt {o₁ o₂ : Ordering} : o₁.then o₂ = gt ↔ o₁ = gt ∨ o₁ = eq ∧ o₂ = gt := by cases o₁ <;> cases o₂ <;> decide end Ordering namespace Batteries /-- `TotalBLE le` asserts that `le` has a total order, that is, `le a b ∨ le b a`. -/ class TotalBLE (le : α → α → Bool) : Prop where /-- `le` is total: either `le a b` or `le b a`. -/ total : le a b ∨ le b a /-- `OrientedCmp cmp` asserts that `cmp` is determined by the relation `cmp x y = .lt`. -/ class OrientedCmp (cmp : α → α → Ordering) : Prop where /-- The comparator operation is symmetric, in the sense that if `cmp x y` equals `.lt` then `cmp y x = .gt` and vice versa. -/ symm (x y) : (cmp x y).swap = cmp y x namespace OrientedCmp theorem cmp_eq_gt [OrientedCmp cmp] : cmp x y = .gt ↔ cmp y x = .lt := by rw [← Ordering.swap_inj, symm]; exact .rfl theorem cmp_ne_gt [OrientedCmp cmp] : cmp x y ≠ .gt ↔ cmp y x ≠ .lt := not_congr cmp_eq_gt theorem cmp_eq_eq_symm [OrientedCmp cmp] : cmp x y = .eq ↔ cmp y x = .eq := by rw [← Ordering.swap_inj, symm]; exact .rfl theorem cmp_refl [OrientedCmp cmp] : cmp x x = .eq := match e : cmp x x with \| .lt => nomatch e.symm.trans (cmp_eq_gt.2 e) \| .eq => rfl \| .gt => nomatch (cmp_eq_gt.1 e).symm.trans e end OrientedCmp /-- `TransCmp cmp` asserts that `cmp` induces a transitive relation. -/ class TransCmp (cmp : α → α → Ordering) extends OrientedCmp cmp : Prop where /-- The comparator operation is transitive. -/ le_trans : cmp x y ≠ .gt → cmp y z ≠ .gt → cmp x z ≠ .gt namespace TransCmp variable [TransCmp cmp] open OrientedCmp Decidable theorem ge_trans (h₁ : cmp x y ≠ .lt) (h₂ : cmp y z ≠ .lt) : cmp x z ≠ .lt := by have := @TransCmp.le_trans _ cmp _ z y x simp [cmp_eq_gt] at *; exact this h₂ h₁ theorem lt_asymm (h : cmp x y = .lt) : cmp y x ≠ .lt := fun h' => nomatch h.symm.trans (cmp_eq_gt.2 h') theorem gt_asymm (h : cmp x y = .gt) : cmp y x ≠ .gt := mt cmp_eq_gt.1 <\| lt_asymm <\| cmp_eq_gt.1 h theorem le_lt_trans (h₁ : cmp x y ≠ .gt) (h₂ : cmp y z = .lt) : cmp x z = .lt := byContradiction fun h₃ => ge_trans (mt cmp_eq_gt.2 h₁) h₃ h₂ theorem lt_le_trans (h₁ : cmp x y = .lt) (h₂ : cmp y z ≠ .gt) : cmp x z = .lt := byContradiction fun h₃ => ge_trans h₃ (mt cmp_eq_gt.2 h₂) h₁ theorem lt_trans (h₁ : cmp x y = .lt) (h₂ : cmp y z = .lt) : cmp x z = .lt := le_lt_trans (gt_asymm <\| cmp_eq_gt.2 h₁) h₂ theorem gt_trans (h₁ : cmp x y = .gt) (h₂ : cmp y z = .gt) : cmp x z = .gt := by rw [cmp_eq_gt] at h₁ h₂ ⊢; exact lt_trans h₂ h₁ theorem cmp_congr_left (xy : cmp x y = .eq) : cmp x z = cmp y z := match yz : cmp y z with \| .lt => byContradiction (ge_trans (nomatch ·.symm.trans (cmp_eq_eq_symm.1 xy)) · yz) \| .gt => byContradiction (le_trans (nomatch ·.symm.trans (cmp_eq_eq_symm.1 xy)) · yz) \| .eq => match xz : cmp x z with \| .lt => nomatch ge_trans (nomatch ·.symm.trans xy) (nomatch ·.symm.trans yz) xz \| .gt => nomatch le_trans (nomatch ·.symm.trans xy) (nomatch ·.symm.trans yz) xz \| .eq => rfl theorem cmp_congr_left' (xy : cmp x y = .eq) : cmp x = cmp y := funext fun _ => cmp_congr_left xy theorem cmp_congr_right [TransCmp cmp] (yz : cmp y z = .eq) : cmp x y = cmp x z := by rw [← Ordering.swap_inj, symm, symm, cmp_congr_left yz] end TransCmp instance [inst : OrientedCmp cmp] : OrientedCmp (flip cmp) where symm _ _ := inst.symm .. instance [inst : TransCmp cmp] : TransCmp (flip cmp) where le_trans h1 h2 := inst.le_trans h2 h1 /-- `BEqCmp cmp` asserts that `cmp x y = .eq` and `x == y` coincide. -/ class BEqCmp [BEq α] (cmp : α → α → Ordering) : Prop where /-- `cmp x y = .eq` holds iff `x == y` is true. -/ cmp_iff_beq : cmp x y = .eq ↔ x == y	.lake/packages/batteries/Batteries/Classes/Order.lean	121	122	theorem BEqCmp.cmp_iff_eq [BEq α] [LawfulBEq α] [BEqCmp (α := α) cmp] : cmp x y = .eq ↔ x = y := by	simp [BEqCmp.cmp_iff_beq]
/- Copyright (c) 2020 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov, Sébastien Gouëzel -/ import Mathlib.Analysis.NormedSpace.Banach import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace import Mathlib.Topology.PartialHomeomorph #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # Non-linear maps close to affine maps In this file we study a map `f` such that `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` on an open set `s`, where `f' : E →L[𝕜] F` is a continuous linear map and `c` is suitably small. Maps of this type behave like `f a + f' (x - a)` near each `a ∈ s`. When `f'` is onto, we show that `f` is locally onto. When `f'` is a continuous linear equiv, we show that `f` is a homeomorphism between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toPartialHomeomorph` to be a `PartialHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`. between `s` and `f '' s`. More precisely, we define `ApproximatesLinearOn.toPartialHomeomorph` to be a `PartialHomeomorph` with `toFun = f`, `source = s`, and `target = f '' s`. Maps of this type naturally appear in the proof of the inverse function theorem (see next section), and `ApproximatesLinearOn.toPartialHomeomorph` will imply that the locally inverse function and `ApproximatesLinearOn.toPartialHomeomorph` will imply that the locally inverse function exists. We define this auxiliary notion to split the proof of the inverse function theorem into small lemmas. This approach makes it possible - to prove a lower estimate on the size of the domain of the inverse function; - to reuse parts of the proofs in the case if a function is not strictly differentiable. E.g., for a function `f : E × F → G` with estimates on `f x y₁ - f x y₂` but not on `f x₁ y - f x₂ y`. ## Notations We introduce some `local notation` to make formulas shorter: * by `N` we denote `‖f'⁻¹‖`; * by `g` we denote the auxiliary contracting map `x ↦ x + f'.symm (y - f x)` used to prove that `{x \| f x = y}` is nonempty. -/ open Function Set Filter Metric open scoped Topology Classical NNReal noncomputable section variable {𝕜 : Type} [NontriviallyNormedField 𝕜] variable {E : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E] variable {F : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F] variable {G : Type} [NormedAddCommGroup G] [NormedSpace 𝕜 G] variable {G' : Type} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] variable {ε : ℝ} open Filter Metric Set open ContinuousLinearMap (id) /-- We say that `f` approximates a continuous linear map `f'` on `s` with constant `c`, if `‖f x - f y - f' (x - y)‖ ≤ c ‖x - y‖` whenever `x, y ∈ s`. This predicate is defined to facilitate the splitting of the inverse function theorem into small lemmas. Some of these lemmas can be useful, e.g., to prove that the inverse function is defined on a specific set. -/ def ApproximatesLinearOn (f : E → F) (f' : E →L[𝕜] F) (s : Set E) (c : ℝ≥0) : Prop := ∀ x ∈ s, ∀ y ∈ s, ‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖ #align approximates_linear_on ApproximatesLinearOn @[simp] theorem approximatesLinearOn_empty (f : E → F) (f' : E →L[𝕜] F) (c : ℝ≥0) : ApproximatesLinearOn f f' ∅ c := by simp [ApproximatesLinearOn] #align approximates_linear_on_empty approximatesLinearOn_empty namespace ApproximatesLinearOn variable [CompleteSpace E] {f : E → F} /-! First we prove some properties of a function that `ApproximatesLinearOn` a (not necessarily invertible) continuous linear map. -/ section variable {f' : E →L[𝕜] F} {s t : Set E} {c c' : ℝ≥0} theorem mono_num (hc : c ≤ c') (hf : ApproximatesLinearOn f f' s c) : ApproximatesLinearOn f f' s c' := fun x hx y hy => le_trans (hf x hx y hy) (mul_le_mul_of_nonneg_right hc <\| norm_nonneg _) #align approximates_linear_on.mono_num ApproximatesLinearOn.mono_num theorem mono_set (hst : s ⊆ t) (hf : ApproximatesLinearOn f f' t c) : ApproximatesLinearOn f f' s c := fun x hx y hy => hf x (hst hx) y (hst hy) #align approximates_linear_on.mono_set ApproximatesLinearOn.mono_set theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜] F} {s : Set E} {c : ℝ≥0} : ApproximatesLinearOn f f' s c ↔ LipschitzOnWith c (f - ⇑f') s := by have : ∀ x y, f x - f y - f' (x - y) = (f - f') x - (f - f') y := fun x y ↦ by simp only [map_sub, Pi.sub_apply]; abel simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn] #align approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith alias ⟨lipschitzOnWith, _root_.LipschitzOnWith.approximatesLinearOn⟩ := approximatesLinearOn_iff_lipschitzOnWith #align approximates_linear_on.lipschitz_on_with ApproximatesLinearOn.lipschitzOnWith #align lipschitz_on_with.approximates_linear_on LipschitzOnWith.approximatesLinearOn theorem lipschitz_sub (hf : ApproximatesLinearOn f f' s c) : LipschitzWith c fun x : s => f x - f' x := hf.lipschitzOnWith.to_restrict #align approximates_linear_on.lipschitz_sub ApproximatesLinearOn.lipschitz_sub protected theorem lipschitz (hf : ApproximatesLinearOn f f' s c) : LipschitzWith (‖f'‖₊ + c) (s.restrict f) := by simpa only [restrict_apply, add_sub_cancel] using (f'.lipschitz.restrict s).add hf.lipschitz_sub #align approximates_linear_on.lipschitz ApproximatesLinearOn.lipschitz protected theorem continuous (hf : ApproximatesLinearOn f f' s c) : Continuous (s.restrict f) := hf.lipschitz.continuous #align approximates_linear_on.continuous ApproximatesLinearOn.continuous protected theorem continuousOn (hf : ApproximatesLinearOn f f' s c) : ContinuousOn f s := continuousOn_iff_continuous_restrict.2 hf.continuous #align approximates_linear_on.continuous_on ApproximatesLinearOn.continuousOn end section LocallyOnto /-! We prove that a function which is linearly approximated by a continuous linear map with a nonlinear right inverse is locally onto. This will apply to the case where the approximating map is a linear equivalence, for the local inverse theorem, but also whenever the approximating map is onto, by Banach's open mapping theorem. -/ variable {s : Set E} {c : ℝ≥0} {f' : E →L[𝕜] F} /-- If a function is linearly approximated by a continuous linear map with a (possibly nonlinear) right inverse, then it is locally onto: a ball of an explicit radius is included in the image of the map. -/ theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) : SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) := by intro y hy rcases le_or_lt (f'symm.nnnorm : ℝ)⁻¹ c with hc \| hc · refine ⟨b, by simp [ε0], ?_⟩ have : dist y (f b) ≤ 0 := (mem_closedBall.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0) simp only [dist_le_zero] at this rw [this] have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If' have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith /- We have to show that `y` can be written as `f x` for some `x ∈ closedBall b ε`. The idea of the proof is to apply the Banach contraction principle to the map `g : x ↦ x + f'symm (y - f x)`, as a fixed point of this map satisfies `f x = y`. When `f'symm` is a genuine linear inverse, `g` is a contracting map. In our case, since `f'symm` is nonlinear, this map is not contracting (it is not even continuous), but still the proof of the contraction theorem holds: `uₙ = gⁿ b` is a Cauchy sequence, converging exponentially fast to the desired point `x`. Instead of appealing to general results, we check this by hand. The main point is that `f (u n)` becomes exponentially close to `y`, and therefore `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive bound on `dist (u n) b`, from which one checks that `u n` stays in the ball on which one has a control. Therefore, the bound can be checked at the next step, and so on inductively. -/ set g := fun x => x + f'symm (y - f x) with hg set u := fun n : ℕ => g^[n] b with hu have usucc : ∀ n, u (n + 1) = g (u n) := by simp [hu, ← iterate_succ_apply' g _ b] -- First bound: if `f z` is close to `y`, then `g z` is close to `z` (i.e., almost a fixed point). have A : ∀ z, dist (g z) z ≤ f'symm.nnnorm * dist (f z) y := by intro z rw [dist_eq_norm, hg, add_sub_cancel_left, dist_eq_norm'] exact f'symm.bound _ -- Second bound: if `z` and `g z` are in the set with good control, then `f (g z)` becomes closer -- to `y` than `f z` was (this uses the linear approximation property, and is the reason for the -- choice of the formula for `g`). have B : ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ c * f'symm.nnnorm * dist (f z) y := by intro z hz hgz set v := f'symm (y - f z) calc dist (f (g z)) y = ‖f (z + v) - y‖ := by rw [dist_eq_norm] _ = ‖f (z + v) - f z - f' v + f' v - (y - f z)‖ := by congr 1; abel _ = ‖f (z + v) - f z - f' (z + v - z)‖ := by simp only [v, ContinuousLinearMap.NonlinearRightInverse.right_inv, add_sub_cancel_left, sub_add_cancel] _ ≤ c * ‖z + v - z‖ := hf _ (hε hgz) _ (hε hz) _ ≤ c * (f'symm.nnnorm * dist (f z) y) := by gcongr simpa [dist_eq_norm'] using f'symm.bound (y - f z) _ = c * f'symm.nnnorm * dist (f z) y := by ring -- Third bound: a complicated bound on `dist w b` (that will show up in the induction) is enough -- to check that `w` is in the ball on which one has controls. Will be used to check that `u n` -- belongs to this ball for all `n`. have C : ∀ (n : ℕ) (w : E), dist w b ≤ f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) / (1 - c * f'symm.nnnorm) * dist (f b) y → w ∈ closedBall b ε := fun n w hw ↦ by apply hw.trans rw [div_mul_eq_mul_div, div_le_iff]; swap; · linarith calc (f'symm.nnnorm : ℝ) * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) * dist (f b) y = f'symm.nnnorm * dist (f b) y * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) := by ring _ ≤ f'symm.nnnorm * dist (f b) y * 1 := by gcongr rw [sub_le_self_iff] positivity _ ≤ f'symm.nnnorm * (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε) := by rw [mul_one] gcongr exact mem_closedBall'.1 hy _ = ε * (1 - c * f'symm.nnnorm) := by field_simp; ring /- Main inductive control: `f (u n)` becomes exponentially close to `y`, and therefore `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive bound on `dist (u n) b`, from which one checks that `u n` remains in the ball on which we have estimates. -/ have D : ∀ n : ℕ, dist (f (u n)) y ≤ ((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y ∧ dist (u n) b ≤ f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) / (1 - (c : ℝ) * f'symm.nnnorm) * dist (f b) y := fun n ↦ by induction' n with n IH; · simp [hu, le_refl] rw [usucc] have Ign : dist (g (u n)) b ≤ f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n.succ) / (1 - c * f'symm.nnnorm) * dist (f b) y := calc dist (g (u n)) b ≤ dist (g (u n)) (u n) + dist (u n) b := dist_triangle _ _ _ _ ≤ f'symm.nnnorm * dist (f (u n)) y + dist (u n) b := add_le_add (A _) le_rfl _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) + f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n) / (1 - c * f'symm.nnnorm) * dist (f b) y := by gcongr · exact IH.1 · exact IH.2 _ = f'symm.nnnorm * (1 - ((c : ℝ) * f'symm.nnnorm) ^ n.succ) / (1 - (c : ℝ) * f'symm.nnnorm) * dist (f b) y := by field_simp [Jcf', pow_succ]; ring refine ⟨?_, Ign⟩ calc dist (f (g (u n))) y ≤ c * f'symm.nnnorm * dist (f (u n)) y := B _ (C n _ IH.2) (C n.succ _ Ign) _ ≤ (c : ℝ) * f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) := by gcongr apply IH.1 _ = ((c : ℝ) * f'symm.nnnorm) ^ n.succ * dist (f b) y := by simp only [pow_succ']; ring -- Deduce from the inductive bound that `uₙ` is a Cauchy sequence, therefore converging. have : CauchySeq u := by refine cauchySeq_of_le_geometric _ (↑f'symm.nnnorm * dist (f b) y) Icf' fun n ↦ ?_ calc dist (u n) (u (n + 1)) = dist (g (u n)) (u n) := by rw [usucc, dist_comm] _ ≤ f'symm.nnnorm * dist (f (u n)) y := A _ _ ≤ f'symm.nnnorm * (((c : ℝ) * f'symm.nnnorm) ^ n * dist (f b) y) := by gcongr exact (D n).1 _ = f'symm.nnnorm * dist (f b) y * ((c : ℝ) * f'symm.nnnorm) ^ n := by ring obtain ⟨x, hx⟩ : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete this -- As all the `uₙ` belong to the ball `closedBall b ε`, so does their limit `x`. have xmem : x ∈ closedBall b ε := isClosed_ball.mem_of_tendsto hx (eventually_of_forall fun n => C n _ (D n).2) refine ⟨x, xmem, ?_⟩ -- It remains to check that `f x = y`. This follows from continuity of `f` on `closedBall b ε` -- and from the fact that `f uₙ` is converging to `y` by construction. have hx' : Tendsto u atTop (𝓝[closedBall b ε] x) := by simp only [nhdsWithin, tendsto_inf, hx, true_and_iff, ge_iff_le, tendsto_principal] exact eventually_of_forall fun n => C n _ (D n).2 have T1 : Tendsto (f ∘ u) atTop (𝓝 (f x)) := (hf.continuousOn.mono hε x xmem).tendsto.comp hx' have T2 : Tendsto (f ∘ u) atTop (𝓝 y) := by rw [tendsto_iff_dist_tendsto_zero] refine squeeze_zero (fun _ => dist_nonneg) (fun n => (D n).1) ?_ simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one (by positivity) Icf').mul tendsto_const_nhds exact tendsto_nhds_unique T1 T2 #align approximates_linear_on.surj_on_closed_ball_of_nonlinear_right_inverse ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) (hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) := by cases' hc with hE hc · exact isOpen_discrete _ simp only [isOpen_iff_mem_nhds, nhds_basis_closedBall.mem_iff, forall_mem_image] at hs ⊢ intro x hx rcases hs x hx with ⟨ε, ε0, hε⟩ refine ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, ?_⟩ exact (hf.surjOn_closedBall_of_nonlinearRightInverse f'symm (le_of_lt ε0) hε).mono hε Subset.rfl #align approximates_linear_on.open_image ApproximatesLinearOn.open_image theorem image_mem_nhds (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : f '' s ∈ 𝓝 (f x) := by obtain ⟨t, hts, ht, xt⟩ : ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := _root_.mem_nhds_iff.1 hs have := IsOpen.mem_nhds ((hf.mono_set hts).open_image f'symm ht hc) (mem_image_of_mem _ xt) exact mem_of_superset this (image_subset _ hts) #align approximates_linear_on.image_mem_nhds ApproximatesLinearOn.image_mem_nhds theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E} (hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : map f (𝓝 x) = 𝓝 (f x) := by refine le_antisymm ((hf.continuousOn x (mem_of_mem_nhds hs)).continuousAt hs) (le_map fun t ht => ?_) have : f '' (s ∩ t) ∈ 𝓝 (f x) := (hf.mono_set inter_subset_left).image_mem_nhds f'symm (inter_mem hs ht) hc exact mem_of_superset this (image_subset _ inter_subset_right) #align approximates_linear_on.map_nhds_eq ApproximatesLinearOn.map_nhds_eq end LocallyOnto /-! From now on we assume that `f` approximates an invertible continuous linear map `f : E ≃L[𝕜] F`. We also assume that either `E = {0}`, or `c < ‖f'⁻¹‖⁻¹`. We use `N` as an abbreviation for `‖f'⁻¹‖`. -/ variable {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} local notation "N" => ‖(f'.symm : F →L[𝕜] E)‖₊ protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) : AntilipschitzWith (N⁻¹ - c)⁻¹ (s.restrict f) := by cases' hc with hE hc · exact AntilipschitzWith.of_subsingleton convert (f'.antilipschitz.restrict s).add_lipschitzWith hf.lipschitz_sub hc simp [restrict] #align approximates_linear_on.antilipschitz ApproximatesLinearOn.antilipschitz protected theorem injective (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) : Injective (s.restrict f) := (hf.antilipschitz hc).injective #align approximates_linear_on.injective ApproximatesLinearOn.injective protected theorem injOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) : InjOn f s := injOn_iff_injective.2 <\| hf.injective hc #align approximates_linear_on.inj_on ApproximatesLinearOn.injOn protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c) (hc : Subsingleton E ∨ c < N⁻¹) : Surjective f := by cases' hc with hE hc · haveI : Subsingleton F := (Equiv.subsingleton_congr f'.toEquiv).1 hE exact surjective_to_subsingleton _ · apply forall_of_forall_mem_closedBall (fun y : F => ∃ a, f a = y) (f 0) _ have hc' : (0 : ℝ) < N⁻¹ - c := by rw [sub_pos]; exact hc let p : ℝ → Prop := fun R => closedBall (f 0) R ⊆ Set.range f have hp : ∀ᶠ r : ℝ in atTop, p ((N⁻¹ - c) * r) := by have hr : ∀ᶠ r : ℝ in atTop, 0 ≤ r := eventually_ge_atTop 0 refine hr.mono fun r hr => Subset.trans ?_ (image_subset_range f (closedBall 0 r)) refine hf.surjOn_closedBall_of_nonlinearRightInverse f'.toNonlinearRightInverse hr ?_ exact subset_univ _ refine ((tendsto_id.const_mul_atTop hc').frequently hp.frequently).mono ?_ exact fun R h y hy => h hy #align approximates_linear_on.surjective ApproximatesLinearOn.surjective /-- A map approximating a linear equivalence on a set defines a partial equivalence on this set. Should not be used outside of this file, because it is superseded by `toPartialHomeomorph` below. This is a first step towards the inverse function. -/ def toPartialEquiv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) : PartialEquiv E F := (hf.injOn hc).toPartialEquiv _ _ #align approximates_linear_on.to_local_equiv ApproximatesLinearOn.toPartialEquiv /-- The inverse function is continuous on `f '' s`. Use properties of `PartialHomeomorph` instead. -/ theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toPartialEquiv hc).symm (f '' s) := by apply continuousOn_iff_continuous_restrict.2 refine ((hf.antilipschitz hc).to_rightInvOn' ?_ (hf.toPartialEquiv hc).right_inv').continuous exact fun x hx => (hf.toPartialEquiv hc).map_target hx #align approximates_linear_on.inverse_continuous_on ApproximatesLinearOn.inverse_continuousOn /-- The inverse function is approximated linearly on `f '' s` by `f'.symm`. -/	Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean	377	398	theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) : ApproximatesLinearOn (hf.toPartialEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s) (N * (N⁻¹ - c)⁻¹ * c) := fun x hx y hy ↦ by set A := hf.toPartialEquiv hc have Af : ∀ z, A z = f z := fun z => rfl rcases (mem_image _ _ _).1 hx with ⟨x', x's, rfl⟩ rcases (mem_image _ _ _).1 hy with ⟨y', y's, rfl⟩ rw [← Af x', ← Af y', A.left_inv x's, A.left_inv y's] calc ‖x' - y' - f'.symm (A x' - A y')‖ ≤ N * ‖f' (x' - y' - f'.symm (A x' - A y'))‖ := (f' : E →L[𝕜] F).bound_of_antilipschitz f'.antilipschitz _ _ = N * ‖A y' - A x' - f' (y' - x')‖ := by	congr 2 simp only [ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.map_sub] abel _ ≤ N * (c * ‖y' - x'‖) := mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _) _ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) := by gcongr rw [← dist_eq_norm, ← dist_eq_norm] exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩ _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by simp only [norm_sub_rev, NNReal.coe_mul]; ring
/- Copyright (c) 2021 Jakob Scholbach. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jakob Scholbach -/ import Mathlib.Algebra.CharP.Basic import Mathlib.Algebra.CharP.Algebra import Mathlib.Data.Nat.Prime #align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" /-! # Exponential characteristic This file defines the exponential characteristic, which is defined to be 1 for a ring with characteristic 0 and the same as the ordinary characteristic, if the ordinary characteristic is prime. This concept is useful to simplify some theorem statements. This file establishes a few basic results relating it to the (ordinary characteristic). The definition is stated for a semiring, but the actual results are for nontrivial rings (as far as exponential characteristic one is concerned), respectively a ring without zero-divisors (for prime characteristic). ## Main results - `ExpChar`: the definition of exponential characteristic - `expChar_is_prime_or_one`: the exponential characteristic is a prime or one - `char_eq_expChar_iff`: the characteristic equals the exponential characteristic iff the characteristic is prime ## Tags exponential characteristic, characteristic -/ universe u variable (R : Type u) section Semiring variable [Semiring R] /-- The definition of the exponential characteristic of a semiring. -/ class inductive ExpChar (R : Type u) [Semiring R] : ℕ → Prop \| zero [CharZero R] : ExpChar R 1 \| prime {q : ℕ} (hprime : q.Prime) [hchar : CharP R q] : ExpChar R q #align exp_char ExpChar #align exp_char.prime ExpChar.prime instance expChar_prime (p) [CharP R p] [Fact p.Prime] : ExpChar R p := ExpChar.prime Fact.out instance expChar_zero [CharZero R] : ExpChar R 1 := ExpChar.zero instance (S : Type) [Semiring S] (p) [ExpChar R p] [ExpChar S p] : ExpChar (R × S) p := by obtain hp \| ⟨hp⟩ := ‹ExpChar R p› · have := Prod.charZero_of_left R S; exact .zero obtain _ \| _ := ‹ExpChar S p› · exact (Nat.not_prime_one hp).elim · have := Prod.charP R S p; exact .prime hp variable {R} in /-- The exponential characteristic is unique. -/ theorem ExpChar.eq {p q : ℕ} (hp : ExpChar R p) (hq : ExpChar R q) : p = q := by cases' hp with hp _ hp' hp · cases' hq with hq _ hq' hq exacts [rfl, False.elim (Nat.not_prime_zero (CharP.eq R hq (CharP.ofCharZero R) ▸ hq'))] · cases' hq with hq _ hq' hq exacts [False.elim (Nat.not_prime_zero (CharP.eq R hp (CharP.ofCharZero R) ▸ hp')), CharP.eq R hp hq] theorem ExpChar.congr {p : ℕ} (q : ℕ) [hq : ExpChar R q] (h : q = p) : ExpChar R p := h ▸ hq /-- Noncomputable function that outputs the unique exponential characteristic of a semiring. -/ noncomputable def ringExpChar (R : Type) [NonAssocSemiring R] : ℕ := max (ringChar R) 1 theorem ringExpChar.eq (q : ℕ) [h : ExpChar R q] : ringExpChar R = q := by cases' h with _ _ h _ · haveI := CharP.ofCharZero R rw [ringExpChar, ringChar.eq R 0]; rfl rw [ringExpChar, ringChar.eq R q] exact Nat.max_eq_left h.one_lt.le @[simp] theorem ringExpChar.eq_one (R : Type) [NonAssocSemiring R] [CharZero R] : ringExpChar R = 1 := by rw [ringExpChar, ringChar.eq_zero, max_eq_right zero_le_one] /-- The exponential characteristic is one if the characteristic is zero. -/ theorem expChar_one_of_char_zero (q : ℕ) [hp : CharP R 0] [hq : ExpChar R q] : q = 1 := by cases' hq with q hq_one hq_prime hq_hchar · rfl · exact False.elim <\| hq_prime.ne_zero <\| hq_hchar.eq R hp #align exp_char_one_of_char_zero expChar_one_of_char_zero /-- The characteristic equals the exponential characteristic iff the former is prime. -/ theorem char_eq_expChar_iff (p q : ℕ) [hp : CharP R p] [hq : ExpChar R q] : p = q ↔ p.Prime := by cases' hq with q hq_one hq_prime hq_hchar · rw [(CharP.eq R hp inferInstance : p = 0)] decide · exact ⟨fun hpq => hpq.symm ▸ hq_prime, fun _ => CharP.eq R hp hq_hchar⟩ #align char_eq_exp_char_iff char_eq_expChar_iff section Nontrivial variable [Nontrivial R] /-- The exponential characteristic is one if the characteristic is zero. -/ theorem char_zero_of_expChar_one (p : ℕ) [hp : CharP R p] [hq : ExpChar R 1] : p = 0 := by cases hq · exact CharP.eq R hp inferInstance · exact False.elim (CharP.char_ne_one R 1 rfl) #align char_zero_of_exp_char_one char_zero_of_expChar_one -- This could be an instance, but there are no `ExpChar R 1` instances in mathlib. /-- The characteristic is zero if the exponential characteristic is one. -/ theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by cases hq · assumption · exact False.elim (CharP.char_ne_one R 1 rfl) #align char_zero_of_exp_char_one' charZero_of_expChar_one' /-- The exponential characteristic is one iff the characteristic is zero. -/ theorem expChar_one_iff_char_zero (p q : ℕ) [CharP R p] [ExpChar R q] : q = 1 ↔ p = 0 := by constructor · rintro rfl exact char_zero_of_expChar_one R p · rintro rfl exact expChar_one_of_char_zero R q #align exp_char_one_iff_char_zero expChar_one_iff_char_zero section NoZeroDivisors variable [NoZeroDivisors R] /-- A helper lemma: the characteristic is prime if it is non-zero. -/ theorem char_prime_of_ne_zero {p : ℕ} [hp : CharP R p] (p_ne_zero : p ≠ 0) : Nat.Prime p := by cases' CharP.char_is_prime_or_zero R p with h h · exact h · contradiction #align char_prime_of_ne_zero char_prime_of_ne_zero /-- The exponential characteristic is a prime number or one. See also `CharP.char_is_prime_or_zero`. -/ theorem expChar_is_prime_or_one (q : ℕ) [hq : ExpChar R q] : Nat.Prime q ∨ q = 1 := by cases hq with \| zero => exact .inr rfl \| prime hp => exact .inl hp #align exp_char_is_prime_or_one expChar_is_prime_or_one /-- The exponential characteristic is positive. -/ theorem expChar_pos (q : ℕ) [ExpChar R q] : 0 < q := by rcases expChar_is_prime_or_one R q with h \| rfl exacts [Nat.Prime.pos h, Nat.one_pos] /-- Any power of the exponential characteristic is positive. -/ theorem expChar_pow_pos (q : ℕ) [ExpChar R q] (n : ℕ) : 0 < q ^ n := Nat.pos_pow_of_pos n (expChar_pos R q) end NoZeroDivisors end Nontrivial end Semiring theorem ExpChar.exists [Ring R] [IsDomain R] : ∃ q, ExpChar R q := by obtain _ \| ⟨p, ⟨hp⟩, _⟩ := CharP.exists' R exacts [⟨1, .zero⟩, ⟨p, .prime hp⟩] theorem ExpChar.exists_unique [Ring R] [IsDomain R] : ∃! q, ExpChar R q := let ⟨q, H⟩ := ExpChar.exists R ⟨q, H, fun _ H2 ↦ ExpChar.eq H2 H⟩ instance ringExpChar.expChar [Ring R] [IsDomain R] : ExpChar R (ringExpChar R) := by obtain ⟨q, _⟩ := ExpChar.exists R rwa [ringExpChar.eq R q] variable {R} in theorem ringExpChar.of_eq [Ring R] [IsDomain R] {q : ℕ} (h : ringExpChar R = q) : ExpChar R q := h ▸ ringExpChar.expChar R variable {R} in theorem ringExpChar.eq_iff [Ring R] [IsDomain R] {q : ℕ} : ringExpChar R = q ↔ ExpChar R q := ⟨ringExpChar.of_eq, fun _ ↦ ringExpChar.eq R q⟩ /-- If a ring homomorphism `R →+ A` is injective then `A` has the same exponential characteristic as `R`. -/ theorem expChar_of_injective_ringHom {R A : Type} [Semiring R] [Semiring A] {f : R →+ A} (h : Function.Injective f) (q : ℕ) [hR : ExpChar R q] : ExpChar A q := by cases' hR with _ _ hprime _ · haveI := charZero_of_injective_ringHom h; exact .zero haveI := charP_of_injective_ringHom h q; exact .prime hprime /-- If `R →+* A` is injective, and `A` is of exponential characteristic `p`, then `R` is also of exponential characteristic `p`. Similar to `RingHom.charZero`. -/ theorem RingHom.expChar {R A : Type} [Semiring R] [Semiring A] (f : R →+ A) (H : Function.Injective f) (p : ℕ) [ExpChar A p] : ExpChar R p := by cases ‹ExpChar A p› with \| zero => haveI := f.charZero; exact .zero \| prime hp => haveI := f.charP H p; exact .prime hp /-- If `R →+* A` is injective, then `R` is of exponential characteristic `p` if and only if `A` is also of exponential characteristic `p`. Similar to `RingHom.charZero_iff`. -/ theorem RingHom.expChar_iff {R A : Type} [Semiring R] [Semiring A] (f : R →+ A) (H : Function.Injective f) (p : ℕ) : ExpChar R p ↔ ExpChar A p := ⟨fun _ ↦ expChar_of_injective_ringHom H p, fun _ ↦ f.expChar H p⟩ /-- If the algebra map `R →+* A` is injective then `A` has the same exponential characteristic as `R`. -/ theorem expChar_of_injective_algebraMap {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (h : Function.Injective (algebraMap R A)) (q : ℕ) [ExpChar R q] : ExpChar A q := expChar_of_injective_ringHom h q theorem add_pow_expChar_of_commute [Semiring R] {q : ℕ} [hR : ExpChar R q] (x y : R) (h : Commute x y) : (x + y) ^ q = x ^ q + y ^ q := by cases' hR with _ _ hprime _ · simp only [pow_one] haveI := Fact.mk hprime; exact add_pow_char_of_commute R x y h theorem add_pow_expChar_pow_of_commute [Semiring R] {q : ℕ} [hR : ExpChar R q] {n : ℕ} (x y : R) (h : Commute x y) : (x + y) ^ q ^ n = x ^ q ^ n + y ^ q ^ n := by cases' hR with _ _ hprime _ · simp only [one_pow, pow_one] haveI := Fact.mk hprime; exact add_pow_char_pow_of_commute R x y h theorem sub_pow_expChar_of_commute [Ring R] {q : ℕ} [hR : ExpChar R q] (x y : R) (h : Commute x y) : (x - y) ^ q = x ^ q - y ^ q := by cases' hR with _ _ hprime _ · simp only [pow_one] haveI := Fact.mk hprime; exact sub_pow_char_of_commute R x y h theorem sub_pow_expChar_pow_of_commute [Ring R] {q : ℕ} [hR : ExpChar R q] {n : ℕ} (x y : R) (h : Commute x y) : (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n := by cases' hR with _ _ hprime _ · simp only [one_pow, pow_one] haveI := Fact.mk hprime; exact sub_pow_char_pow_of_commute R x y h theorem add_pow_expChar [CommSemiring R] {q : ℕ} [hR : ExpChar R q] (x y : R) : (x + y) ^ q = x ^ q + y ^ q := by cases' hR with _ _ hprime _ · simp only [pow_one] haveI := Fact.mk hprime; exact add_pow_char R x y theorem add_pow_expChar_pow [CommSemiring R] {q : ℕ} [hR : ExpChar R q] {n : ℕ} (x y : R) : (x + y) ^ q ^ n = x ^ q ^ n + y ^ q ^ n := by cases' hR with _ _ hprime _ · simp only [one_pow, pow_one] haveI := Fact.mk hprime; exact add_pow_char_pow R x y theorem sub_pow_expChar [CommRing R] {q : ℕ} [hR : ExpChar R q] (x y : R) : (x - y) ^ q = x ^ q - y ^ q := by cases' hR with _ _ hprime _ · simp only [pow_one] haveI := Fact.mk hprime; exact sub_pow_char R x y theorem sub_pow_expChar_pow [CommRing R] {q : ℕ} [hR : ExpChar R q] {n : ℕ} (x y : R) : (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n := by cases' hR with _ _ hprime _ · simp only [one_pow, pow_one] haveI := Fact.mk hprime; exact sub_pow_char_pow R x y theorem ExpChar.neg_one_pow_expChar [Ring R] (q : ℕ) [hR : ExpChar R q] : (-1 : R) ^ q = -1 := by cases' hR with _ _ hprime _ · simp only [pow_one] haveI := Fact.mk hprime; exact CharP.neg_one_pow_char R q	Mathlib/Algebra/CharP/ExpChar.lean	265	269	theorem ExpChar.neg_one_pow_expChar_pow [Ring R] (q n : ℕ) [hR : ExpChar R q] : (-1 : R) ^ q ^ n = -1 := by	cases' hR with _ _ hprime _ · simp only [one_pow, pow_one] haveI := Fact.mk hprime; exact CharP.neg_one_pow_char_pow R q n
/- Copyright (c) 2019 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Yakov Pechersky -/ import Mathlib.Data.List.Nodup import Mathlib.Data.List.Zip import Mathlib.Data.Nat.Defs import Mathlib.Data.List.Infix #align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e" /-! # List rotation This file proves basic results about `List.rotate`, the list rotation. ## Main declarations * `List.IsRotated l₁ l₂`: States that `l₁` is a rotated version of `l₂`. * `List.cyclicPermutations l`: The list of all cyclic permutants of `l`, up to the length of `l`. ## Tags rotated, rotation, permutation, cycle -/ universe u variable {α : Type u} open Nat Function namespace List theorem rotate_mod (l : List α) (n : ℕ) : l.rotate (n % l.length) = l.rotate n := by simp [rotate] #align list.rotate_mod List.rotate_mod @[simp] theorem rotate_nil (n : ℕ) : ([] : List α).rotate n = [] := by simp [rotate] #align list.rotate_nil List.rotate_nil @[simp] theorem rotate_zero (l : List α) : l.rotate 0 = l := by simp [rotate] #align list.rotate_zero List.rotate_zero -- Porting note: removing simp, simp can prove it theorem rotate'_nil (n : ℕ) : ([] : List α).rotate' n = [] := by cases n <;> rfl #align list.rotate'_nil List.rotate'_nil @[simp] theorem rotate'_zero (l : List α) : l.rotate' 0 = l := by cases l <;> rfl #align list.rotate'_zero List.rotate'_zero theorem rotate'_cons_succ (l : List α) (a : α) (n : ℕ) : (a :: l : List α).rotate' n.succ = (l ++ [a]).rotate' n := by simp [rotate'] #align list.rotate'_cons_succ List.rotate'_cons_succ @[simp] theorem length_rotate' : ∀ (l : List α) (n : ℕ), (l.rotate' n).length = l.length \| [], _ => by simp \| a :: l, 0 => rfl \| a :: l, n + 1 => by rw [List.rotate', length_rotate' (l ++ [a]) n]; simp #align list.length_rotate' List.length_rotate' theorem rotate'_eq_drop_append_take : ∀ {l : List α} {n : ℕ}, n ≤ l.length → l.rotate' n = l.drop n ++ l.take n \| [], n, h => by simp [drop_append_of_le_length h] \| l, 0, h => by simp [take_append_of_le_length h] \| a :: l, n + 1, h => by have hnl : n ≤ l.length := le_of_succ_le_succ h have hnl' : n ≤ (l ++ [a]).length := by rw [length_append, length_cons, List.length]; exact le_of_succ_le h rw [rotate'_cons_succ, rotate'_eq_drop_append_take hnl', drop, take, drop_append_of_le_length hnl, take_append_of_le_length hnl]; simp #align list.rotate'_eq_drop_append_take List.rotate'_eq_drop_append_take theorem rotate'_rotate' : ∀ (l : List α) (n m : ℕ), (l.rotate' n).rotate' m = l.rotate' (n + m) \| a :: l, 0, m => by simp \| [], n, m => by simp \| a :: l, n + 1, m => by rw [rotate'_cons_succ, rotate'_rotate' _ n, Nat.add_right_comm, ← rotate'_cons_succ, Nat.succ_eq_add_one] #align list.rotate'_rotate' List.rotate'_rotate' @[simp] theorem rotate'_length (l : List α) : rotate' l l.length = l := by rw [rotate'_eq_drop_append_take le_rfl]; simp #align list.rotate'_length List.rotate'_length @[simp] theorem rotate'_length_mul (l : List α) : ∀ n : ℕ, l.rotate' (l.length * n) = l \| 0 => by simp \| n + 1 => calc l.rotate' (l.length * (n + 1)) = (l.rotate' (l.length * n)).rotate' (l.rotate' (l.length * n)).length := by simp [-rotate'_length, Nat.mul_succ, rotate'_rotate'] _ = l := by rw [rotate'_length, rotate'_length_mul l n] #align list.rotate'_length_mul List.rotate'_length_mul theorem rotate'_mod (l : List α) (n : ℕ) : l.rotate' (n % l.length) = l.rotate' n := calc l.rotate' (n % l.length) = (l.rotate' (n % l.length)).rotate' ((l.rotate' (n % l.length)).length * (n / l.length)) := by rw [rotate'_length_mul] _ = l.rotate' n := by rw [rotate'_rotate', length_rotate', Nat.mod_add_div] #align list.rotate'_mod List.rotate'_mod theorem rotate_eq_rotate' (l : List α) (n : ℕ) : l.rotate n = l.rotate' n := if h : l.length = 0 then by simp_all [length_eq_zero] else by rw [← rotate'_mod, rotate'_eq_drop_append_take (le_of_lt (Nat.mod_lt _ (Nat.pos_of_ne_zero h)))]; simp [rotate] #align list.rotate_eq_rotate' List.rotate_eq_rotate' theorem rotate_cons_succ (l : List α) (a : α) (n : ℕ) : (a :: l : List α).rotate (n + 1) = (l ++ [a]).rotate n := by rw [rotate_eq_rotate', rotate_eq_rotate', rotate'_cons_succ] #align list.rotate_cons_succ List.rotate_cons_succ @[simp] theorem mem_rotate : ∀ {l : List α} {a : α} {n : ℕ}, a ∈ l.rotate n ↔ a ∈ l \| [], _, n => by simp \| a :: l, _, 0 => by simp \| a :: l, _, n + 1 => by simp [rotate_cons_succ, mem_rotate, or_comm] #align list.mem_rotate List.mem_rotate @[simp] theorem length_rotate (l : List α) (n : ℕ) : (l.rotate n).length = l.length := by rw [rotate_eq_rotate', length_rotate'] #align list.length_rotate List.length_rotate @[simp] theorem rotate_replicate (a : α) (n : ℕ) (k : ℕ) : (replicate n a).rotate k = replicate n a := eq_replicate.2 ⟨by rw [length_rotate, length_replicate], fun b hb => eq_of_mem_replicate <\| mem_rotate.1 hb⟩ #align list.rotate_replicate List.rotate_replicate theorem rotate_eq_drop_append_take {l : List α} {n : ℕ} : n ≤ l.length → l.rotate n = l.drop n ++ l.take n := by rw [rotate_eq_rotate']; exact rotate'_eq_drop_append_take #align list.rotate_eq_drop_append_take List.rotate_eq_drop_append_take theorem rotate_eq_drop_append_take_mod {l : List α} {n : ℕ} : l.rotate n = l.drop (n % l.length) ++ l.take (n % l.length) := by rcases l.length.zero_le.eq_or_lt with hl \| hl · simp [eq_nil_of_length_eq_zero hl.symm] rw [← rotate_eq_drop_append_take (n.mod_lt hl).le, rotate_mod] #align list.rotate_eq_drop_append_take_mod List.rotate_eq_drop_append_take_mod @[simp] theorem rotate_append_length_eq (l l' : List α) : (l ++ l').rotate l.length = l' ++ l := by rw [rotate_eq_rotate'] induction l generalizing l' · simp · simp_all [rotate'] #align list.rotate_append_length_eq List.rotate_append_length_eq theorem rotate_rotate (l : List α) (n m : ℕ) : (l.rotate n).rotate m = l.rotate (n + m) := by rw [rotate_eq_rotate', rotate_eq_rotate', rotate_eq_rotate', rotate'_rotate'] #align list.rotate_rotate List.rotate_rotate @[simp] theorem rotate_length (l : List α) : rotate l l.length = l := by rw [rotate_eq_rotate', rotate'_length] #align list.rotate_length List.rotate_length @[simp] theorem rotate_length_mul (l : List α) (n : ℕ) : l.rotate (l.length * n) = l := by rw [rotate_eq_rotate', rotate'_length_mul] #align list.rotate_length_mul List.rotate_length_mul theorem rotate_perm (l : List α) (n : ℕ) : l.rotate n ~ l := by rw [rotate_eq_rotate'] induction' n with n hn generalizing l · simp · cases' l with hd tl · simp · rw [rotate'_cons_succ] exact (hn _).trans (perm_append_singleton _ _) #align list.rotate_perm List.rotate_perm @[simp] theorem nodup_rotate {l : List α} {n : ℕ} : Nodup (l.rotate n) ↔ Nodup l := (rotate_perm l n).nodup_iff #align list.nodup_rotate List.nodup_rotate @[simp] theorem rotate_eq_nil_iff {l : List α} {n : ℕ} : l.rotate n = [] ↔ l = [] := by induction' n with n hn generalizing l · simp · cases' l with hd tl · simp · simp [rotate_cons_succ, hn] #align list.rotate_eq_nil_iff List.rotate_eq_nil_iff @[simp] theorem nil_eq_rotate_iff {l : List α} {n : ℕ} : [] = l.rotate n ↔ [] = l := by rw [eq_comm, rotate_eq_nil_iff, eq_comm] #align list.nil_eq_rotate_iff List.nil_eq_rotate_iff @[simp] theorem rotate_singleton (x : α) (n : ℕ) : [x].rotate n = [x] := rotate_replicate x 1 n #align list.rotate_singleton List.rotate_singleton theorem zipWith_rotate_distrib {β γ : Type} (f : α → β → γ) (l : List α) (l' : List β) (n : ℕ) (h : l.length = l'.length) : (zipWith f l l').rotate n = zipWith f (l.rotate n) (l'.rotate n) := by rw [rotate_eq_drop_append_take_mod, rotate_eq_drop_append_take_mod, rotate_eq_drop_append_take_mod, h, zipWith_append, ← zipWith_distrib_drop, ← zipWith_distrib_take, List.length_zipWith, h, min_self] rw [length_drop, length_drop, h] #align list.zip_with_rotate_distrib List.zipWith_rotate_distrib attribute [local simp] rotate_cons_succ -- Porting note: removing @[simp], simp can prove it theorem zipWith_rotate_one {β : Type} (f : α → α → β) (x y : α) (l : List α) : zipWith f (x :: y :: l) ((x :: y :: l).rotate 1) = f x y :: zipWith f (y :: l) (l ++ [x]) := by simp #align list.zip_with_rotate_one List.zipWith_rotate_one theorem get?_rotate {l : List α} {n m : ℕ} (hml : m < l.length) : (l.rotate n).get? m = l.get? ((m + n) % l.length) := by rw [rotate_eq_drop_append_take_mod] rcases lt_or_le m (l.drop (n % l.length)).length with hm \| hm · rw [get?_append hm, get?_drop, ← add_mod_mod] rw [length_drop, Nat.lt_sub_iff_add_lt] at hm rw [mod_eq_of_lt hm, Nat.add_comm] · have hlt : n % length l < length l := mod_lt _ (m.zero_le.trans_lt hml) rw [get?_append_right hm, get?_take, length_drop] · congr 1 rw [length_drop] at hm have hm' := Nat.sub_le_iff_le_add'.1 hm have : n % length l + m - length l < length l := by rw [Nat.sub_lt_iff_lt_add' hm'] exact Nat.add_lt_add hlt hml conv_rhs => rw [Nat.add_comm m, ← mod_add_mod, mod_eq_sub_mod hm', mod_eq_of_lt this] rw [← Nat.add_right_inj, ← Nat.add_sub_assoc, Nat.add_sub_sub_cancel, Nat.add_sub_cancel', Nat.add_comm] exacts [hm', hlt.le, hm] · rwa [Nat.sub_lt_iff_lt_add hm, length_drop, Nat.sub_add_cancel hlt.le] #align list.nth_rotate List.get?_rotate -- Porting note (#10756): new lemma theorem get_rotate (l : List α) (n : ℕ) (k : Fin (l.rotate n).length) : (l.rotate n).get k = l.get ⟨(k + n) % l.length, mod_lt _ (length_rotate l n ▸ k.1.zero_le.trans_lt k.2)⟩ := by rw [← Option.some_inj, ← get?_eq_get, ← get?_eq_get, get?_rotate] exact k.2.trans_eq (length_rotate _ _) theorem head?_rotate {l : List α} {n : ℕ} (h : n < l.length) : head? (l.rotate n) = l.get? n := by rw [← get?_zero, get?_rotate (n.zero_le.trans_lt h), Nat.zero_add, Nat.mod_eq_of_lt h] #align list.head'_rotate List.head?_rotate -- Porting note: moved down from its original location below `get_rotate` so that the -- non-deprecated lemma does not use the deprecated version set_option linter.deprecated false in @[deprecated get_rotate (since := "2023-01-13")] theorem nthLe_rotate (l : List α) (n k : ℕ) (hk : k < (l.rotate n).length) : (l.rotate n).nthLe k hk = l.nthLe ((k + n) % l.length) (mod_lt _ (length_rotate l n ▸ k.zero_le.trans_lt hk)) := get_rotate l n ⟨k, hk⟩ #align list.nth_le_rotate List.nthLe_rotate set_option linter.deprecated false in theorem nthLe_rotate_one (l : List α) (k : ℕ) (hk : k < (l.rotate 1).length) : (l.rotate 1).nthLe k hk = l.nthLe ((k + 1) % l.length) (mod_lt _ (length_rotate l 1 ▸ k.zero_le.trans_lt hk)) := nthLe_rotate l 1 k hk #align list.nth_le_rotate_one List.nthLe_rotate_one -- Porting note (#10756): new lemma /-- A version of `List.get_rotate` that represents `List.get l` in terms of `List.get (List.rotate l n)`, not vice versa. Can be used instead of rewriting `List.get_rotate` from right to left. -/ theorem get_eq_get_rotate (l : List α) (n : ℕ) (k : Fin l.length) : l.get k = (l.rotate n).get ⟨(l.length - n % l.length + k) % l.length, (Nat.mod_lt _ (k.1.zero_le.trans_lt k.2)).trans_eq (length_rotate _ _).symm⟩ := by rw [get_rotate] refine congr_arg l.get (Fin.eq_of_val_eq ?_) simp only [mod_add_mod] rw [← add_mod_mod, Nat.add_right_comm, Nat.sub_add_cancel, add_mod_left, mod_eq_of_lt] exacts [k.2, (mod_lt _ (k.1.zero_le.trans_lt k.2)).le] set_option linter.deprecated false in /-- A variant of `List.nthLe_rotate` useful for rewrites from right to left. -/ @[deprecated get_eq_get_rotate] theorem nthLe_rotate' (l : List α) (n k : ℕ) (hk : k < l.length) : (l.rotate n).nthLe ((l.length - n % l.length + k) % l.length) ((Nat.mod_lt _ (k.zero_le.trans_lt hk)).trans_le (length_rotate _ _).ge) = l.nthLe k hk := (get_eq_get_rotate l n ⟨k, hk⟩).symm #align list.nth_le_rotate' List.nthLe_rotate' theorem rotate_eq_self_iff_eq_replicate [hα : Nonempty α] : ∀ {l : List α}, (∀ n, l.rotate n = l) ↔ ∃ a, l = replicate l.length a \| [] => by simp \| a :: l => ⟨fun h => ⟨a, ext_get (length_replicate _ _).symm fun n h₁ h₂ => by rw [get_replicate, ← Option.some_inj, ← get?_eq_get, ← head?_rotate h₁, h, head?_cons]⟩, fun ⟨b, hb⟩ n => by rw [hb, rotate_replicate]⟩ #align list.rotate_eq_self_iff_eq_replicate List.rotate_eq_self_iff_eq_replicate theorem rotate_one_eq_self_iff_eq_replicate [Nonempty α] {l : List α} : l.rotate 1 = l ↔ ∃ a : α, l = List.replicate l.length a := ⟨fun h => rotate_eq_self_iff_eq_replicate.mp fun n => Nat.rec l.rotate_zero (fun n hn => by rwa [Nat.succ_eq_add_one, ← l.rotate_rotate, hn]) n, fun h => rotate_eq_self_iff_eq_replicate.mpr h 1⟩ #align list.rotate_one_eq_self_iff_eq_replicate List.rotate_one_eq_self_iff_eq_replicate theorem rotate_injective (n : ℕ) : Function.Injective fun l : List α => l.rotate n := by rintro l l' (h : l.rotate n = l'.rotate n) have hle : l.length = l'.length := (l.length_rotate n).symm.trans (h.symm ▸ l'.length_rotate n) rw [rotate_eq_drop_append_take_mod, rotate_eq_drop_append_take_mod] at h obtain ⟨hd, ht⟩ := append_inj h (by simp_all) rw [← take_append_drop _ l, ht, hd, take_append_drop] #align list.rotate_injective List.rotate_injective @[simp] theorem rotate_eq_rotate {l l' : List α} {n : ℕ} : l.rotate n = l'.rotate n ↔ l = l' := (rotate_injective n).eq_iff #align list.rotate_eq_rotate List.rotate_eq_rotate theorem rotate_eq_iff {l l' : List α} {n : ℕ} : l.rotate n = l' ↔ l = l'.rotate (l'.length - n % l'.length) := by rw [← @rotate_eq_rotate _ l _ n, rotate_rotate, ← rotate_mod l', add_mod] rcases l'.length.zero_le.eq_or_lt with hl \| hl · rw [eq_nil_of_length_eq_zero hl.symm, rotate_nil] · rcases (Nat.zero_le (n % l'.length)).eq_or_lt with hn \| hn · simp [← hn] · rw [mod_eq_of_lt (Nat.sub_lt hl hn), Nat.sub_add_cancel, mod_self, rotate_zero] exact (Nat.mod_lt _ hl).le #align list.rotate_eq_iff List.rotate_eq_iff @[simp] theorem rotate_eq_singleton_iff {l : List α} {n : ℕ} {x : α} : l.rotate n = [x] ↔ l = [x] := by rw [rotate_eq_iff, rotate_singleton] #align list.rotate_eq_singleton_iff List.rotate_eq_singleton_iff @[simp] theorem singleton_eq_rotate_iff {l : List α} {n : ℕ} {x : α} : [x] = l.rotate n ↔ [x] = l := by rw [eq_comm, rotate_eq_singleton_iff, eq_comm] #align list.singleton_eq_rotate_iff List.singleton_eq_rotate_iff theorem reverse_rotate (l : List α) (n : ℕ) : (l.rotate n).reverse = l.reverse.rotate (l.length - n % l.length) := by rw [← length_reverse l, ← rotate_eq_iff] induction' n with n hn generalizing l · simp · cases' l with hd tl · simp · rw [rotate_cons_succ, ← rotate_rotate, hn] simp #align list.reverse_rotate List.reverse_rotate theorem rotate_reverse (l : List α) (n : ℕ) : l.reverse.rotate n = (l.rotate (l.length - n % l.length)).reverse := by rw [← reverse_reverse l] simp_rw [reverse_rotate, reverse_reverse, rotate_eq_iff, rotate_rotate, length_rotate, length_reverse] rw [← length_reverse l] let k := n % l.reverse.length cases' hk' : k with k' · simp_all! [k, length_reverse, ← rotate_rotate] · cases' l with x l · simp · rw [Nat.mod_eq_of_lt, Nat.sub_add_cancel, rotate_length] · exact Nat.sub_le _ _ · exact Nat.sub_lt (by simp) (by simp_all! [k]) #align list.rotate_reverse List.rotate_reverse theorem map_rotate {β : Type} (f : α → β) (l : List α) (n : ℕ) : map f (l.rotate n) = (map f l).rotate n := by induction' n with n hn IH generalizing l · simp · cases' l with hd tl · simp · simp [hn] #align list.map_rotate List.map_rotate theorem Nodup.rotate_congr {l : List α} (hl : l.Nodup) (hn : l ≠ []) (i j : ℕ) (h : l.rotate i = l.rotate j) : i % l.length = j % l.length := by rw [← rotate_mod l i, ← rotate_mod l j] at h simpa only [head?_rotate, mod_lt, length_pos_of_ne_nil hn, get?_eq_get, Option.some_inj, hl.get_inj_iff, Fin.ext_iff] using congr_arg head? h #align list.nodup.rotate_congr List.Nodup.rotate_congr theorem Nodup.rotate_congr_iff {l : List α} (hl : l.Nodup) {i j : ℕ} : l.rotate i = l.rotate j ↔ i % l.length = j % l.length ∨ l = [] := by rcases eq_or_ne l [] with rfl \| hn · simp · simp only [hn, or_false] refine ⟨hl.rotate_congr hn _ _, fun h ↦ ?_⟩ rw [← rotate_mod, h, rotate_mod] theorem Nodup.rotate_eq_self_iff {l : List α} (hl : l.Nodup) {n : ℕ} : l.rotate n = l ↔ n % l.length = 0 ∨ l = [] := by rw [← zero_mod, ← hl.rotate_congr_iff, rotate_zero] #align list.nodup.rotate_eq_self_iff List.Nodup.rotate_eq_self_iff section IsRotated variable (l l' : List α) /-- `IsRotated l₁ l₂` or `l₁ ~r l₂` asserts that `l₁` and `l₂` are cyclic permutations of each other. This is defined by claiming that `∃ n, l.rotate n = l'`. -/ def IsRotated : Prop := ∃ n, l.rotate n = l' #align list.is_rotated List.IsRotated @[inherit_doc List.IsRotated] infixr:1000 " ~r " => IsRotated variable {l l'} @[refl] theorem IsRotated.refl (l : List α) : l ~r l := ⟨0, by simp⟩ #align list.is_rotated.refl List.IsRotated.refl @[symm] theorem IsRotated.symm (h : l ~r l') : l' ~r l := by obtain ⟨n, rfl⟩ := h cases' l with hd tl · exists 0 · use (hd :: tl).length n - n rw [rotate_rotate, Nat.add_sub_cancel', rotate_length_mul] exact Nat.le_mul_of_pos_left _ (by simp) #align list.is_rotated.symm List.IsRotated.symm theorem isRotated_comm : l ~r l' ↔ l' ~r l := ⟨IsRotated.symm, IsRotated.symm⟩ #align list.is_rotated_comm List.isRotated_comm @[simp] protected theorem IsRotated.forall (l : List α) (n : ℕ) : l.rotate n ~r l := IsRotated.symm ⟨n, rfl⟩ #align list.is_rotated.forall List.IsRotated.forall @[trans] theorem IsRotated.trans : ∀ {l l' l'' : List α}, l ~r l' → l' ~r l'' → l ~r l'' \| _, _, _, ⟨n, rfl⟩, ⟨m, rfl⟩ => ⟨n + m, by rw [rotate_rotate]⟩ #align list.is_rotated.trans List.IsRotated.trans theorem IsRotated.eqv : Equivalence (@IsRotated α) := Equivalence.mk IsRotated.refl IsRotated.symm IsRotated.trans #align list.is_rotated.eqv List.IsRotated.eqv /-- The relation `List.IsRotated l l'` forms a `Setoid` of cycles. -/ def IsRotated.setoid (α : Type) : Setoid (List α) where r := IsRotated iseqv := IsRotated.eqv #align list.is_rotated.setoid List.IsRotated.setoid theorem IsRotated.perm (h : l ~r l') : l ~ l' := Exists.elim h fun _ hl => hl ▸ (rotate_perm _ _).symm #align list.is_rotated.perm List.IsRotated.perm theorem IsRotated.nodup_iff (h : l ~r l') : Nodup l ↔ Nodup l' := h.perm.nodup_iff #align list.is_rotated.nodup_iff List.IsRotated.nodup_iff theorem IsRotated.mem_iff (h : l ~r l') {a : α} : a ∈ l ↔ a ∈ l' := h.perm.mem_iff #align list.is_rotated.mem_iff List.IsRotated.mem_iff @[simp] theorem isRotated_nil_iff : l ~r [] ↔ l = [] := ⟨fun ⟨n, hn⟩ => by simpa using hn, fun h => h ▸ by rfl⟩ #align list.is_rotated_nil_iff List.isRotated_nil_iff @[simp] theorem isRotated_nil_iff' : [] ~r l ↔ [] = l := by rw [isRotated_comm, isRotated_nil_iff, eq_comm] #align list.is_rotated_nil_iff' List.isRotated_nil_iff' @[simp] theorem isRotated_singleton_iff {x : α} : l ~r [x] ↔ l = [x] := ⟨fun ⟨n, hn⟩ => by simpa using hn, fun h => h ▸ by rfl⟩ #align list.is_rotated_singleton_iff List.isRotated_singleton_iff @[simp] theorem isRotated_singleton_iff' {x : α} : [x] ~r l ↔ [x] = l := by rw [isRotated_comm, isRotated_singleton_iff, eq_comm] #align list.is_rotated_singleton_iff' List.isRotated_singleton_iff' theorem isRotated_concat (hd : α) (tl : List α) : (tl ++ [hd]) ~r (hd :: tl) := IsRotated.symm ⟨1, by simp⟩ #align list.is_rotated_concat List.isRotated_concat theorem isRotated_append : (l ++ l') ~r (l' ++ l) := ⟨l.length, by simp⟩ #align list.is_rotated_append List.isRotated_append theorem IsRotated.reverse (h : l ~r l') : l.reverse ~r l'.reverse := by obtain ⟨n, rfl⟩ := h exact ⟨_, (reverse_rotate _ _).symm⟩ #align list.is_rotated.reverse List.IsRotated.reverse theorem isRotated_reverse_comm_iff : l.reverse ~r l' ↔ l ~r l'.reverse := by constructor <;> · intro h simpa using h.reverse #align list.is_rotated_reverse_comm_iff List.isRotated_reverse_comm_iff @[simp] theorem isRotated_reverse_iff : l.reverse ~r l'.reverse ↔ l ~r l' := by simp [isRotated_reverse_comm_iff] #align list.is_rotated_reverse_iff List.isRotated_reverse_iff theorem isRotated_iff_mod : l ~r l' ↔ ∃ n ≤ l.length, l.rotate n = l' := by refine ⟨fun h => ?_, fun ⟨n, _, h⟩ => ⟨n, h⟩⟩ obtain ⟨n, rfl⟩ := h cases' l with hd tl · simp · refine ⟨n % (hd :: tl).length, ?_, rotate_mod _ _⟩ refine (Nat.mod_lt _ ?_).le simp #align list.is_rotated_iff_mod List.isRotated_iff_mod theorem isRotated_iff_mem_map_range : l ~r l' ↔ l' ∈ (List.range (l.length + 1)).map l.rotate := by simp_rw [mem_map, mem_range, isRotated_iff_mod] exact ⟨fun ⟨n, hn, h⟩ => ⟨n, Nat.lt_succ_of_le hn, h⟩, fun ⟨n, hn, h⟩ => ⟨n, Nat.le_of_lt_succ hn, h⟩⟩ #align list.is_rotated_iff_mem_map_range List.isRotated_iff_mem_map_range -- Porting note: @[congr] only works for equality. -- @[congr] theorem IsRotated.map {β : Type} {l₁ l₂ : List α} (h : l₁ ~r l₂) (f : α → β) : map f l₁ ~r map f l₂ := by obtain ⟨n, rfl⟩ := h rw [map_rotate] use n #align list.is_rotated.map List.IsRotated.map /-- List of all cyclic permutations of `l`. The `cyclicPermutations` of a nonempty list `l` will always contain `List.length l` elements. This implies that under certain conditions, there are duplicates in `List.cyclicPermutations l`. The `n`th entry is equal to `l.rotate n`, proven in `List.get_cyclicPermutations`. The proof that every cyclic permutant of `l` is in the list is `List.mem_cyclicPermutations_iff`. cyclicPermutations [1, 2, 3, 2, 4] = [[1, 2, 3, 2, 4], [2, 3, 2, 4, 1], [3, 2, 4, 1, 2], [2, 4, 1, 2, 3], [4, 1, 2, 3, 2]] -/ def cyclicPermutations : List α → List (List α) \| [] => [[]] \| l@(_ :: _) => dropLast (zipWith (· ++ ·) (tails l) (inits l)) #align list.cyclic_permutations List.cyclicPermutations @[simp] theorem cyclicPermutations_nil : cyclicPermutations ([] : List α) = [[]] := rfl #align list.cyclic_permutations_nil List.cyclicPermutations_nil theorem cyclicPermutations_cons (x : α) (l : List α) : cyclicPermutations (x :: l) = dropLast (zipWith (· ++ ·) (tails (x :: l)) (inits (x :: l))) := rfl #align list.cyclic_permutations_cons List.cyclicPermutations_cons theorem cyclicPermutations_of_ne_nil (l : List α) (h : l ≠ []) : cyclicPermutations l = dropLast (zipWith (· ++ ·) (tails l) (inits l)) := by obtain ⟨hd, tl, rfl⟩ := exists_cons_of_ne_nil h exact cyclicPermutations_cons _ _ #align list.cyclic_permutations_of_ne_nil List.cyclicPermutations_of_ne_nil theorem length_cyclicPermutations_cons (x : α) (l : List α) : length (cyclicPermutations (x :: l)) = length l + 1 := by simp [cyclicPermutations_cons] #align list.length_cyclic_permutations_cons List.length_cyclicPermutations_cons @[simp] theorem length_cyclicPermutations_of_ne_nil (l : List α) (h : l ≠ []) : length (cyclicPermutations l) = length l := by simp [cyclicPermutations_of_ne_nil _ h] #align list.length_cyclic_permutations_of_ne_nil List.length_cyclicPermutations_of_ne_nil @[simp] theorem cyclicPermutations_ne_nil : ∀ l : List α, cyclicPermutations l ≠ [] \| a::l, h => by simpa using congr_arg length h @[simp] theorem get_cyclicPermutations (l : List α) (n : Fin (length (cyclicPermutations l))) : (cyclicPermutations l).get n = l.rotate n := by cases l with \| nil => simp \| cons a l => simp only [cyclicPermutations_cons, get_dropLast, get_zipWith, get_tails, get_inits] rw [rotate_eq_drop_append_take (by simpa using n.2.le)] #align list.nth_le_cyclic_permutations List.get_cyclicPermutations @[simp] theorem head_cyclicPermutations (l : List α) : (cyclicPermutations l).head (cyclicPermutations_ne_nil l) = l := by have h : 0 < length (cyclicPermutations l) := length_pos_of_ne_nil (cyclicPermutations_ne_nil _) rw [← get_mk_zero h, get_cyclicPermutations, Fin.val_mk, rotate_zero] @[simp] theorem head?_cyclicPermutations (l : List α) : (cyclicPermutations l).head? = l := by rw [head?_eq_head, head_cyclicPermutations] theorem cyclicPermutations_injective : Function.Injective (@cyclicPermutations α) := fun l l' h ↦ by simpa using congr_arg head? h @[simp] theorem cyclicPermutations_inj {l l' : List α} : cyclicPermutations l = cyclicPermutations l' ↔ l = l' := cyclicPermutations_injective.eq_iff theorem length_mem_cyclicPermutations (l : List α) (h : l' ∈ cyclicPermutations l) : length l' = length l := by obtain ⟨k, hk, rfl⟩ := get_of_mem h simp #align list.length_mem_cyclic_permutations List.length_mem_cyclicPermutations theorem mem_cyclicPermutations_self (l : List α) : l ∈ cyclicPermutations l := by simpa using head_mem (cyclicPermutations_ne_nil l) #align list.mem_cyclic_permutations_self List.mem_cyclicPermutations_self @[simp]	Mathlib/Data/List/Rotate.lean	624	632	theorem cyclicPermutations_rotate (l : List α) (k : ℕ) : (l.rotate k).cyclicPermutations = l.cyclicPermutations.rotate k := by	have : (l.rotate k).cyclicPermutations.length = length (l.cyclicPermutations.rotate k) := by cases l · simp · rw [length_cyclicPermutations_of_ne_nil] <;> simp refine ext_get this fun n hn hn' => ?_ rw [get_rotate, get_cyclicPermutations, rotate_rotate, ← rotate_mod, Nat.add_comm] cases l <;> simp
/- Copyright (c) 2021 Damiano Testa. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Damiano Testa -/ import Mathlib.Algebra.Regular.Basic import Mathlib.GroupTheory.GroupAction.Hom #align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11" /-! # Action of regular elements on a module We introduce `M`-regular elements, in the context of an `R`-module `M`. The corresponding predicate is called `IsSMulRegular`. There are very limited typeclass assumptions on `R` and `M`, but the "mathematical" case of interest is a commutative ring `R` acting on a module `M`. Since the properties are "multiplicative", there is no actual requirement of having an addition, but there is a zero in both `R` and `M`. SMultiplications involving `0` are, of course, all trivial. The defining property is that an element `a ∈ R` is `M`-regular if the smultiplication map `M → M`, defined by `m ↦ a • m`, is injective. This property is the direct generalization to modules of the property `IsLeftRegular` defined in `Algebra/Regular`. Lemma `isLeftRegular_iff` shows that indeed the two notions coincide. -/ variable {R S : Type} (M : Type) {a b : R} {s : S} /-- An `M`-regular element is an element `c` such that multiplication on the left by `c` is an injective map `M → M`. -/ def IsSMulRegular [SMul R M] (c : R) := Function.Injective ((c • ·) : M → M) #align is_smul_regular IsSMulRegular theorem IsLeftRegular.isSMulRegular [Mul R] {c : R} (h : IsLeftRegular c) : IsSMulRegular R c := h #align is_left_regular.is_smul_regular IsLeftRegular.isSMulRegular /-- Left-regular multiplication on `R` is equivalent to `R`-regularity of `R` itself. -/ theorem isLeftRegular_iff [Mul R] {a : R} : IsLeftRegular a ↔ IsSMulRegular R a := Iff.rfl #align is_left_regular_iff isLeftRegular_iff theorem IsRightRegular.isSMulRegular [Mul R] {c : R} (h : IsRightRegular c) : IsSMulRegular R (MulOpposite.op c) := h #align is_right_regular.is_smul_regular IsRightRegular.isSMulRegular /-- Right-regular multiplication on `R` is equivalent to `Rᵐᵒᵖ`-regularity of `R` itself. -/ theorem isRightRegular_iff [Mul R] {a : R} : IsRightRegular a ↔ IsSMulRegular R (MulOpposite.op a) := Iff.rfl #align is_right_regular_iff isRightRegular_iff namespace IsSMulRegular variable {M} section SMul variable [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] /-- The product of `M`-regular elements is `M`-regular. -/ theorem smul (ra : IsSMulRegular M a) (rs : IsSMulRegular M s) : IsSMulRegular M (a • s) := fun _ _ ab => rs (ra ((smul_assoc _ _ _).symm.trans (ab.trans (smul_assoc _ _ _)))) #align is_smul_regular.smul IsSMulRegular.smul /-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular element, then `b` is `M`-regular. -/ theorem of_smul (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s := @Function.Injective.of_comp _ _ _ (fun m : M => a • m) _ fun c d cd => by dsimp only [Function.comp_def] at cd rw [← smul_assoc, ← smul_assoc] at cd exact ab cd #align is_smul_regular.of_smul IsSMulRegular.of_smul /-- An element is `M`-regular if and only if multiplying it on the left by an `M`-regular element is `M`-regular. -/ @[simp] theorem smul_iff (b : S) (ha : IsSMulRegular M a) : IsSMulRegular M (a • b) ↔ IsSMulRegular M b := ⟨of_smul _, ha.smul⟩ #align is_smul_regular.smul_iff IsSMulRegular.smul_iff theorem isLeftRegular [Mul R] {a : R} (h : IsSMulRegular R a) : IsLeftRegular a := h #align is_smul_regular.is_left_regular IsSMulRegular.isLeftRegular theorem isRightRegular [Mul R] {a : R} (h : IsSMulRegular R (MulOpposite.op a)) : IsRightRegular a := h #align is_smul_regular.is_right_regular IsSMulRegular.isRightRegular theorem mul [Mul R] [IsScalarTower R R M] (ra : IsSMulRegular M a) (rb : IsSMulRegular M b) : IsSMulRegular M (a * b) := ra.smul rb #align is_smul_regular.mul IsSMulRegular.mul theorem of_mul [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) : IsSMulRegular M b := by rw [← smul_eq_mul] at ab exact ab.of_smul _ #align is_smul_regular.of_mul IsSMulRegular.of_mul @[simp] theorem mul_iff_right [Mul R] [IsScalarTower R R M] (ha : IsSMulRegular M a) : IsSMulRegular M (a * b) ↔ IsSMulRegular M b := ⟨of_mul, ha.mul⟩ #align is_smul_regular.mul_iff_right IsSMulRegular.mul_iff_right /-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a` are `M`-regular. -/ theorem mul_and_mul_iff [Mul R] [IsScalarTower R R M] : IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b := by refine ⟨?_, ?_⟩ · rintro ⟨ab, ba⟩ exact ⟨ba.of_mul, ab.of_mul⟩ · rintro ⟨ha, hb⟩ exact ⟨ha.mul hb, hb.mul ha⟩ #align is_smul_regular.mul_and_mul_iff IsSMulRegular.mul_and_mul_iff lemma of_injective {N F} [SMul R N] [FunLike F M N] [MulActionHomClass F R M N] (f : F) {r : R} (h1 : Function.Injective f) (h2 : IsSMulRegular N r) : IsSMulRegular M r := fun x y h3 => h1 <\| h2 <\| (map_smulₛₗ f r x).symm.trans ((congrArg f h3).trans (map_smulₛₗ f r y)) end SMul section Monoid variable [Monoid R] [MulAction R M] variable (M) /-- One is always `M`-regular. -/ @[simp] theorem one : IsSMulRegular M (1 : R) := fun a b ab => by dsimp only [Function.comp_def] at ab rw [one_smul, one_smul] at ab assumption #align is_smul_regular.one IsSMulRegular.one variable {M} /-- An element of `R` admitting a left inverse is `M`-regular. -/ theorem of_mul_eq_one (h : a * b = 1) : IsSMulRegular M b := of_mul (by rw [h] exact one M) #align is_smul_regular.of_mul_eq_one IsSMulRegular.of_mul_eq_one /-- Any power of an `M`-regular element is `M`-regular. -/ theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by induction' n with n hn · rw [pow_zero]; simp only [one] · rw [pow_succ'] exact (ra.smul_iff (a ^ n)).mpr hn #align is_smul_regular.pow IsSMulRegular.pow /-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/	Mathlib/Algebra/Regular/SMul.lean	164	167	theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a := by	refine ⟨?_, pow n⟩ rw [← Nat.succ_pred_eq_of_pos n0, pow_succ, ← smul_eq_mul] exact of_smul _
/- Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe -/ import Mathlib.Combinatorics.SimpleGraph.Init import Mathlib.Data.Rel import Mathlib.Data.Set.Finite import Mathlib.Data.Sym.Sym2 #align_import combinatorics.simple_graph.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe" /-! # Simple graphs This module defines simple graphs on a vertex type `V` as an irreflexive symmetric relation. ## Main definitions * `SimpleGraph` is a structure for symmetric, irreflexive relations * `SimpleGraph.neighborSet` is the `Set` of vertices adjacent to a given vertex * `SimpleGraph.commonNeighbors` is the intersection of the neighbor sets of two given vertices * `SimpleGraph.incidenceSet` is the `Set` of edges containing a given vertex * `CompleteAtomicBooleanAlgebra` instance: Under the subgraph relation, `SimpleGraph` forms a `CompleteAtomicBooleanAlgebra`. In other words, this is the complete lattice of spanning subgraphs of the complete graph. ## Todo * This is the simplest notion of an unoriented graph. This should eventually fit into a more complete combinatorics hierarchy which includes multigraphs and directed graphs. We begin with simple graphs in order to start learning what the combinatorics hierarchy should look like. -/ -- Porting note: using `aesop` for automation -- Porting note: These attributes are needed to use `aesop` as a replacement for `obviously` attribute [aesop norm unfold (rule_sets := [SimpleGraph])] Symmetric attribute [aesop norm unfold (rule_sets := [SimpleGraph])] Irreflexive -- Porting note: a thin wrapper around `aesop` for graph lemmas, modelled on `aesop_cat` /-- A variant of the `aesop` tactic for use in the graph library. Changes relative to standard `aesop`: - We use the `SimpleGraph` rule set in addition to the default rule sets. - We instruct Aesop's `intro` rule to unfold with `default` transparency. - We instruct Aesop to fail if it can't fully solve the goal. This allows us to use `aesop_graph` for auto-params. -/ macro (name := aesop_graph) "aesop_graph" c:Aesop.tactic_clause* : tactic => `(tactic\| aesop $c* (config := { introsTransparency? := some .default, terminal := true }) (rule_sets := [$(Lean.mkIdent `SimpleGraph):ident])) /-- Use `aesop_graph?` to pass along a `Try this` suggestion when using `aesop_graph` -/ macro (name := aesop_graph?) "aesop_graph?" c:Aesop.tactic_clause* : tactic => `(tactic\| aesop $c* (config := { introsTransparency? := some .default, terminal := true }) (rule_sets := [$(Lean.mkIdent `SimpleGraph):ident])) /-- A variant of `aesop_graph` which does not fail if it is unable to solve the goal. Use this only for exploration! Nonterminal Aesop is even worse than nonterminal `simp`. -/ macro (name := aesop_graph_nonterminal) "aesop_graph_nonterminal" c:Aesop.tactic_clause* : tactic => `(tactic\| aesop $c* (config := { introsTransparency? := some .default, warnOnNonterminal := false }) (rule_sets := [$(Lean.mkIdent `SimpleGraph):ident])) open Finset Function universe u v w /-- A simple graph is an irreflexive symmetric relation `Adj` on a vertex type `V`. The relation describes which pairs of vertices are adjacent. There is exactly one edge for every pair of adjacent vertices; see `SimpleGraph.edgeSet` for the corresponding edge set. -/ @[ext, aesop safe constructors (rule_sets := [SimpleGraph])] structure SimpleGraph (V : Type u) where /-- The adjacency relation of a simple graph. -/ Adj : V → V → Prop symm : Symmetric Adj := by aesop_graph loopless : Irreflexive Adj := by aesop_graph #align simple_graph SimpleGraph -- Porting note: changed `obviously` to `aesop` in the `structure` initialize_simps_projections SimpleGraph (Adj → adj) /-- Constructor for simple graphs using a symmetric irreflexive boolean function. -/ @[simps] def SimpleGraph.mk' {V : Type u} : {adj : V → V → Bool // (∀ x y, adj x y = adj y x) ∧ (∀ x, ¬ adj x x)} ↪ SimpleGraph V where toFun x := ⟨fun v w ↦ x.1 v w, fun v w ↦ by simp [x.2.1], fun v ↦ by simp [x.2.2]⟩ inj' := by rintro ⟨adj, _⟩ ⟨adj', _⟩ simp only [mk.injEq, Subtype.mk.injEq] intro h funext v w simpa [Bool.coe_iff_coe] using congr_fun₂ h v w /-- We can enumerate simple graphs by enumerating all functions `V → V → Bool` and filtering on whether they are symmetric and irreflexive. -/ instance {V : Type u} [Fintype V] [DecidableEq V] : Fintype (SimpleGraph V) where elems := Finset.univ.map SimpleGraph.mk' complete := by classical rintro ⟨Adj, hs, hi⟩ simp only [mem_map, mem_univ, true_and, Subtype.exists, Bool.not_eq_true] refine ⟨fun v w ↦ Adj v w, ⟨?_, ?_⟩, ?_⟩ · simp [hs.iff] · intro v; simp [hi v] · ext simp /-- Construct the simple graph induced by the given relation. It symmetrizes the relation and makes it irreflexive. -/ def SimpleGraph.fromRel {V : Type u} (r : V → V → Prop) : SimpleGraph V where Adj a b := a ≠ b ∧ (r a b ∨ r b a) symm := fun _ _ ⟨hn, hr⟩ => ⟨hn.symm, hr.symm⟩ loopless := fun _ ⟨hn, _⟩ => hn rfl #align simple_graph.from_rel SimpleGraph.fromRel @[simp] theorem SimpleGraph.fromRel_adj {V : Type u} (r : V → V → Prop) (v w : V) : (SimpleGraph.fromRel r).Adj v w ↔ v ≠ w ∧ (r v w ∨ r w v) := Iff.rfl #align simple_graph.from_rel_adj SimpleGraph.fromRel_adj -- Porting note: attributes needed for `completeGraph` attribute [aesop safe (rule_sets := [SimpleGraph])] Ne.symm attribute [aesop safe (rule_sets := [SimpleGraph])] Ne.irrefl /-- The complete graph on a type `V` is the simple graph with all pairs of distinct vertices adjacent. In `Mathlib`, this is usually referred to as `⊤`. -/ def completeGraph (V : Type u) : SimpleGraph V where Adj := Ne #align complete_graph completeGraph /-- The graph with no edges on a given vertex type `V`. `Mathlib` prefers the notation `⊥`. -/ def emptyGraph (V : Type u) : SimpleGraph V where Adj _ _ := False #align empty_graph emptyGraph /-- Two vertices are adjacent in the complete bipartite graph on two vertex types if and only if they are not from the same side. Any bipartite graph may be regarded as a subgraph of one of these. -/ @[simps] def completeBipartiteGraph (V W : Type) : SimpleGraph (Sum V W) where Adj v w := v.isLeft ∧ w.isRight ∨ v.isRight ∧ w.isLeft symm v w := by cases v <;> cases w <;> simp loopless v := by cases v <;> simp #align complete_bipartite_graph completeBipartiteGraph namespace SimpleGraph variable {ι : Sort} {V : Type u} (G : SimpleGraph V) {a b c u v w : V} {e : Sym2 V} @[simp] protected theorem irrefl {v : V} : ¬G.Adj v v := G.loopless v #align simple_graph.irrefl SimpleGraph.irrefl theorem adj_comm (u v : V) : G.Adj u v ↔ G.Adj v u := ⟨fun x => G.symm x, fun x => G.symm x⟩ #align simple_graph.adj_comm SimpleGraph.adj_comm @[symm] theorem adj_symm (h : G.Adj u v) : G.Adj v u := G.symm h #align simple_graph.adj_symm SimpleGraph.adj_symm theorem Adj.symm {G : SimpleGraph V} {u v : V} (h : G.Adj u v) : G.Adj v u := G.symm h #align simple_graph.adj.symm SimpleGraph.Adj.symm theorem ne_of_adj (h : G.Adj a b) : a ≠ b := by rintro rfl exact G.irrefl h #align simple_graph.ne_of_adj SimpleGraph.ne_of_adj protected theorem Adj.ne {G : SimpleGraph V} {a b : V} (h : G.Adj a b) : a ≠ b := G.ne_of_adj h #align simple_graph.adj.ne SimpleGraph.Adj.ne protected theorem Adj.ne' {G : SimpleGraph V} {a b : V} (h : G.Adj a b) : b ≠ a := h.ne.symm #align simple_graph.adj.ne' SimpleGraph.Adj.ne' theorem ne_of_adj_of_not_adj {v w x : V} (h : G.Adj v x) (hn : ¬G.Adj w x) : v ≠ w := fun h' => hn (h' ▸ h) #align simple_graph.ne_of_adj_of_not_adj SimpleGraph.ne_of_adj_of_not_adj theorem adj_injective : Injective (Adj : SimpleGraph V → V → V → Prop) := SimpleGraph.ext #align simple_graph.adj_injective SimpleGraph.adj_injective @[simp] theorem adj_inj {G H : SimpleGraph V} : G.Adj = H.Adj ↔ G = H := adj_injective.eq_iff #align simple_graph.adj_inj SimpleGraph.adj_inj section Order /-- The relation that one `SimpleGraph` is a subgraph of another. Note that this should be spelled `≤`. -/ def IsSubgraph (x y : SimpleGraph V) : Prop := ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w #align simple_graph.is_subgraph SimpleGraph.IsSubgraph instance : LE (SimpleGraph V) := ⟨IsSubgraph⟩ @[simp] theorem isSubgraph_eq_le : (IsSubgraph : SimpleGraph V → SimpleGraph V → Prop) = (· ≤ ·) := rfl #align simple_graph.is_subgraph_eq_le SimpleGraph.isSubgraph_eq_le /-- The supremum of two graphs `x ⊔ y` has edges where either `x` or `y` have edges. -/ instance : Sup (SimpleGraph V) where sup x y := { Adj := x.Adj ⊔ y.Adj symm := fun v w h => by rwa [Pi.sup_apply, Pi.sup_apply, x.adj_comm, y.adj_comm] } @[simp] theorem sup_adj (x y : SimpleGraph V) (v w : V) : (x ⊔ y).Adj v w ↔ x.Adj v w ∨ y.Adj v w := Iff.rfl #align simple_graph.sup_adj SimpleGraph.sup_adj /-- The infimum of two graphs `x ⊓ y` has edges where both `x` and `y` have edges. -/ instance : Inf (SimpleGraph V) where inf x y := { Adj := x.Adj ⊓ y.Adj symm := fun v w h => by rwa [Pi.inf_apply, Pi.inf_apply, x.adj_comm, y.adj_comm] } @[simp] theorem inf_adj (x y : SimpleGraph V) (v w : V) : (x ⊓ y).Adj v w ↔ x.Adj v w ∧ y.Adj v w := Iff.rfl #align simple_graph.inf_adj SimpleGraph.inf_adj /-- We define `Gᶜ` to be the `SimpleGraph V` such that no two adjacent vertices in `G` are adjacent in the complement, and every nonadjacent pair of vertices is adjacent (still ensuring that vertices are not adjacent to themselves). -/ instance hasCompl : HasCompl (SimpleGraph V) where compl G := { Adj := fun v w => v ≠ w ∧ ¬G.Adj v w symm := fun v w ⟨hne, _⟩ => ⟨hne.symm, by rwa [adj_comm]⟩ loopless := fun v ⟨hne, _⟩ => (hne rfl).elim } @[simp] theorem compl_adj (G : SimpleGraph V) (v w : V) : Gᶜ.Adj v w ↔ v ≠ w ∧ ¬G.Adj v w := Iff.rfl #align simple_graph.compl_adj SimpleGraph.compl_adj /-- The difference of two graphs `x \ y` has the edges of `x` with the edges of `y` removed. -/ instance sdiff : SDiff (SimpleGraph V) where sdiff x y := { Adj := x.Adj \ y.Adj symm := fun v w h => by change x.Adj w v ∧ ¬y.Adj w v; rwa [x.adj_comm, y.adj_comm] } @[simp] theorem sdiff_adj (x y : SimpleGraph V) (v w : V) : (x \ y).Adj v w ↔ x.Adj v w ∧ ¬y.Adj v w := Iff.rfl #align simple_graph.sdiff_adj SimpleGraph.sdiff_adj instance supSet : SupSet (SimpleGraph V) where sSup s := { Adj := fun a b => ∃ G ∈ s, Adj G a b symm := fun a b => Exists.imp fun _ => And.imp_right Adj.symm loopless := by rintro a ⟨G, _, ha⟩ exact ha.ne rfl } instance infSet : InfSet (SimpleGraph V) where sInf s := { Adj := fun a b => (∀ ⦃G⦄, G ∈ s → Adj G a b) ∧ a ≠ b symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) Ne.symm loopless := fun _ h => h.2 rfl } @[simp] theorem sSup_adj {s : Set (SimpleGraph V)} {a b : V} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.Sup_adj SimpleGraph.sSup_adj @[simp] theorem sInf_adj {s : Set (SimpleGraph V)} : (sInf s).Adj a b ↔ (∀ G ∈ s, Adj G a b) ∧ a ≠ b := Iff.rfl #align simple_graph.Inf_adj SimpleGraph.sInf_adj @[simp] theorem iSup_adj {f : ι → SimpleGraph V} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.supr_adj SimpleGraph.iSup_adj @[simp] theorem iInf_adj {f : ι → SimpleGraph V} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ a ≠ b := by simp [iInf] #align simple_graph.infi_adj SimpleGraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set (SimpleGraph V)} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G ∈ s, Adj G a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G, hG⟩ := hs exact fun h => (h _ hG).ne #align simple_graph.Inf_adj_of_nonempty SimpleGraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → SimpleGraph V} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _), Set.forall_mem_range] #align simple_graph.infi_adj_of_nonempty SimpleGraph.iInf_adj_of_nonempty /-- For graphs `G`, `H`, `G ≤ H` iff `∀ a b, G.Adj a b → H.Adj a b`. -/ instance distribLattice : DistribLattice (SimpleGraph V) := { show DistribLattice (SimpleGraph V) from adj_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun G H => ∀ ⦃a b⦄, G.Adj a b → H.Adj a b } instance completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (SimpleGraph V) := { SimpleGraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) compl := HasCompl.compl sdiff := (· \ ·) top := completeGraph V bot := emptyGraph V le_top := fun x v w h => x.ne_of_adj h bot_le := fun x v w h => h.elim sdiff_eq := fun x y => by ext v w refine ⟨fun h => ⟨h.1, ⟨?_, h.2⟩⟩, fun h => ⟨h.1, h.2.2⟩⟩ rintro rfl exact x.irrefl h.1 inf_compl_le_bot := fun G v w h => False.elim <\| h.2.2 h.1 top_le_sup_compl := fun G v w hvw => by by_cases h : G.Adj v w · exact Or.inl h · exact Or.inr ⟨hvw, h⟩ sSup := sSup le_sSup := fun s G hG a b hab => ⟨G, hG, hab⟩ sSup_le := fun s G hG a b => by rintro ⟨H, hH, hab⟩ exact hG _ hH hab sInf := sInf sInf_le := fun s G hG a b hab => hab.1 hG le_sInf := fun s G hG a b hab => ⟨fun H hH => hG _ hH hab, hab.ne⟩ iInf_iSup_eq := fun f => by ext; simp [Classical.skolem] } @[simp] theorem top_adj (v w : V) : (⊤ : SimpleGraph V).Adj v w ↔ v ≠ w := Iff.rfl #align simple_graph.top_adj SimpleGraph.top_adj @[simp] theorem bot_adj (v w : V) : (⊥ : SimpleGraph V).Adj v w ↔ False := Iff.rfl #align simple_graph.bot_adj SimpleGraph.bot_adj @[simp] theorem completeGraph_eq_top (V : Type u) : completeGraph V = ⊤ := rfl #align simple_graph.complete_graph_eq_top SimpleGraph.completeGraph_eq_top @[simp] theorem emptyGraph_eq_bot (V : Type u) : emptyGraph V = ⊥ := rfl #align simple_graph.empty_graph_eq_bot SimpleGraph.emptyGraph_eq_bot @[simps] instance (V : Type u) : Inhabited (SimpleGraph V) := ⟨⊥⟩ instance [Subsingleton V] : Unique (SimpleGraph V) where default := ⊥ uniq G := by ext a b; have := Subsingleton.elim a b; simp [this] instance [Nontrivial V] : Nontrivial (SimpleGraph V) := ⟨⟨⊥, ⊤, fun h ↦ not_subsingleton V ⟨by simpa only [← adj_inj, Function.funext_iff, bot_adj, top_adj, ne_eq, eq_iff_iff, false_iff, not_not] using h⟩⟩⟩ section Decidable variable (V) (H : SimpleGraph V) [DecidableRel G.Adj] [DecidableRel H.Adj] instance Bot.adjDecidable : DecidableRel (⊥ : SimpleGraph V).Adj := inferInstanceAs <\| DecidableRel fun _ _ => False #align simple_graph.bot.adj_decidable SimpleGraph.Bot.adjDecidable instance Sup.adjDecidable : DecidableRel (G ⊔ H).Adj := inferInstanceAs <\| DecidableRel fun v w => G.Adj v w ∨ H.Adj v w #align simple_graph.sup.adj_decidable SimpleGraph.Sup.adjDecidable instance Inf.adjDecidable : DecidableRel (G ⊓ H).Adj := inferInstanceAs <\| DecidableRel fun v w => G.Adj v w ∧ H.Adj v w #align simple_graph.inf.adj_decidable SimpleGraph.Inf.adjDecidable instance Sdiff.adjDecidable : DecidableRel (G \ H).Adj := inferInstanceAs <\| DecidableRel fun v w => G.Adj v w ∧ ¬H.Adj v w #align simple_graph.sdiff.adj_decidable SimpleGraph.Sdiff.adjDecidable variable [DecidableEq V] instance Top.adjDecidable : DecidableRel (⊤ : SimpleGraph V).Adj := inferInstanceAs <\| DecidableRel fun v w => v ≠ w #align simple_graph.top.adj_decidable SimpleGraph.Top.adjDecidable instance Compl.adjDecidable : DecidableRel (Gᶜ.Adj) := inferInstanceAs <\| DecidableRel fun v w => v ≠ w ∧ ¬G.Adj v w #align simple_graph.compl.adj_decidable SimpleGraph.Compl.adjDecidable end Decidable end Order /-- `G.support` is the set of vertices that form edges in `G`. -/ def support : Set V := Rel.dom G.Adj #align simple_graph.support SimpleGraph.support theorem mem_support {v : V} : v ∈ G.support ↔ ∃ w, G.Adj v w := Iff.rfl #align simple_graph.mem_support SimpleGraph.mem_support theorem support_mono {G G' : SimpleGraph V} (h : G ≤ G') : G.support ⊆ G'.support := Rel.dom_mono h #align simple_graph.support_mono SimpleGraph.support_mono /-- `G.neighborSet v` is the set of vertices adjacent to `v` in `G`. -/ def neighborSet (v : V) : Set V := {w \| G.Adj v w} #align simple_graph.neighbor_set SimpleGraph.neighborSet instance neighborSet.memDecidable (v : V) [DecidableRel G.Adj] : DecidablePred (· ∈ G.neighborSet v) := inferInstanceAs <\| DecidablePred (Adj G v) #align simple_graph.neighbor_set.mem_decidable SimpleGraph.neighborSet.memDecidable section EdgeSet variable {G₁ G₂ : SimpleGraph V} /-- The edges of G consist of the unordered pairs of vertices related by `G.Adj`. This is the order embedding; for the edge set of a particular graph, see `SimpleGraph.edgeSet`. The way `edgeSet` is defined is such that `mem_edgeSet` is proved by `Iff.rfl`. (That is, `s(v, w) ∈ G.edgeSet` is definitionally equal to `G.Adj v w`.) -/ -- Porting note: We need a separate definition so that dot notation works. def edgeSetEmbedding (V : Type*) : SimpleGraph V ↪o Set (Sym2 V) := OrderEmbedding.ofMapLEIff (fun G => Sym2.fromRel G.symm) fun _ _ => ⟨fun h a b => @h s(a, b), fun h e => Sym2.ind @h e⟩ /-- `G.edgeSet` is the edge set for `G`. This is an abbreviation for `edgeSetEmbedding G` that permits dot notation. -/ abbrev edgeSet (G : SimpleGraph V) : Set (Sym2 V) := edgeSetEmbedding V G #align simple_graph.edge_set SimpleGraph.edgeSetEmbedding @[simp] theorem mem_edgeSet : s(v, w) ∈ G.edgeSet ↔ G.Adj v w := Iff.rfl #align simple_graph.mem_edge_set SimpleGraph.mem_edgeSet theorem not_isDiag_of_mem_edgeSet : e ∈ edgeSet G → ¬e.IsDiag := Sym2.ind (fun _ _ => Adj.ne) e #align simple_graph.not_is_diag_of_mem_edge_set SimpleGraph.not_isDiag_of_mem_edgeSet theorem edgeSet_inj : G₁.edgeSet = G₂.edgeSet ↔ G₁ = G₂ := (edgeSetEmbedding V).eq_iff_eq #align simple_graph.edge_set_inj SimpleGraph.edgeSet_inj @[simp] theorem edgeSet_subset_edgeSet : edgeSet G₁ ⊆ edgeSet G₂ ↔ G₁ ≤ G₂ := (edgeSetEmbedding V).le_iff_le #align simple_graph.edge_set_subset_edge_set SimpleGraph.edgeSet_subset_edgeSet @[simp] theorem edgeSet_ssubset_edgeSet : edgeSet G₁ ⊂ edgeSet G₂ ↔ G₁ < G₂ := (edgeSetEmbedding V).lt_iff_lt #align simple_graph.edge_set_ssubset_edge_set SimpleGraph.edgeSet_ssubset_edgeSet theorem edgeSet_injective : Injective (edgeSet : SimpleGraph V → Set (Sym2 V)) := (edgeSetEmbedding V).injective #align simple_graph.edge_set_injective SimpleGraph.edgeSet_injective alias ⟨_, edgeSet_mono⟩ := edgeSet_subset_edgeSet #align simple_graph.edge_set_mono SimpleGraph.edgeSet_mono alias ⟨_, edgeSet_strict_mono⟩ := edgeSet_ssubset_edgeSet #align simple_graph.edge_set_strict_mono SimpleGraph.edgeSet_strict_mono attribute [mono] edgeSet_mono edgeSet_strict_mono variable (G₁ G₂) @[simp] theorem edgeSet_bot : (⊥ : SimpleGraph V).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_bot SimpleGraph.edgeSet_bot @[simp] theorem edgeSet_top : (⊤ : SimpleGraph V).edgeSet = {e \| ¬e.IsDiag} := Sym2.fromRel_ne @[simp] theorem edgeSet_subset_setOf_not_isDiag : G.edgeSet ⊆ {e \| ¬e.IsDiag} := fun _ h => (Sym2.fromRel_irreflexive (sym := G.symm)).mp G.loopless h @[simp] theorem edgeSet_sup : (G₁ ⊔ G₂).edgeSet = G₁.edgeSet ∪ G₂.edgeSet := by ext ⟨x, y⟩ rfl #align simple_graph.edge_set_sup SimpleGraph.edgeSet_sup @[simp] theorem edgeSet_inf : (G₁ ⊓ G₂).edgeSet = G₁.edgeSet ∩ G₂.edgeSet := by ext ⟨x, y⟩ rfl #align simple_graph.edge_set_inf SimpleGraph.edgeSet_inf @[simp] theorem edgeSet_sdiff : (G₁ \ G₂).edgeSet = G₁.edgeSet \ G₂.edgeSet := by ext ⟨x, y⟩ rfl #align simple_graph.edge_set_sdiff SimpleGraph.edgeSet_sdiff variable {G G₁ G₂} @[simp] lemma disjoint_edgeSet : Disjoint G₁.edgeSet G₂.edgeSet ↔ Disjoint G₁ G₂ := by rw [Set.disjoint_iff, disjoint_iff_inf_le, ← edgeSet_inf, ← edgeSet_bot, ← Set.le_iff_subset, OrderEmbedding.le_iff_le] #align simple_graph.disjoint_edge_set SimpleGraph.disjoint_edgeSet @[simp] lemma edgeSet_eq_empty : G.edgeSet = ∅ ↔ G = ⊥ := by rw [← edgeSet_bot, edgeSet_inj] #align simple_graph.edge_set_eq_empty SimpleGraph.edgeSet_eq_empty @[simp] lemma edgeSet_nonempty : G.edgeSet.Nonempty ↔ G ≠ ⊥ := by rw [Set.nonempty_iff_ne_empty, edgeSet_eq_empty.ne] #align simple_graph.edge_set_nonempty SimpleGraph.edgeSet_nonempty /-- This lemma, combined with `edgeSet_sdiff` and `edgeSet_from_edgeSet`, allows proving `(G \ from_edgeSet s).edge_set = G.edgeSet \ s` by `simp`. -/ @[simp] theorem edgeSet_sdiff_sdiff_isDiag (G : SimpleGraph V) (s : Set (Sym2 V)) : G.edgeSet \ (s \ { e \| e.IsDiag }) = G.edgeSet \ s := by ext e simp only [Set.mem_diff, Set.mem_setOf_eq, not_and, not_not, and_congr_right_iff] intro h simp only [G.not_isDiag_of_mem_edgeSet h, imp_false] #align simple_graph.edge_set_sdiff_sdiff_is_diag SimpleGraph.edgeSet_sdiff_sdiff_isDiag /-- Two vertices are adjacent iff there is an edge between them. The condition `v ≠ w` ensures they are different endpoints of the edge, which is necessary since when `v = w` the existential `∃ (e ∈ G.edgeSet), v ∈ e ∧ w ∈ e` is satisfied by every edge incident to `v`. -/ theorem adj_iff_exists_edge {v w : V} : G.Adj v w ↔ v ≠ w ∧ ∃ e ∈ G.edgeSet, v ∈ e ∧ w ∈ e := by refine ⟨fun _ => ⟨G.ne_of_adj ‹_›, s(v, w), by simpa⟩, ?_⟩ rintro ⟨hne, e, he, hv⟩ rw [Sym2.mem_and_mem_iff hne] at hv subst e rwa [mem_edgeSet] at he #align simple_graph.adj_iff_exists_edge SimpleGraph.adj_iff_exists_edge theorem adj_iff_exists_edge_coe : G.Adj a b ↔ ∃ e : G.edgeSet, e.val = s(a, b) := by simp only [mem_edgeSet, exists_prop, SetCoe.exists, exists_eq_right, Subtype.coe_mk] #align simple_graph.adj_iff_exists_edge_coe SimpleGraph.adj_iff_exists_edge_coe variable (G G₁ G₂) theorem edge_other_ne {e : Sym2 V} (he : e ∈ G.edgeSet) {v : V} (h : v ∈ e) : Sym2.Mem.other h ≠ v := by erw [← Sym2.other_spec h, Sym2.eq_swap] at he exact G.ne_of_adj he #align simple_graph.edge_other_ne SimpleGraph.edge_other_ne instance decidableMemEdgeSet [DecidableRel G.Adj] : DecidablePred (· ∈ G.edgeSet) := Sym2.fromRel.decidablePred G.symm #align simple_graph.decidable_mem_edge_set SimpleGraph.decidableMemEdgeSet instance fintypeEdgeSet [Fintype (Sym2 V)] [DecidableRel G.Adj] : Fintype G.edgeSet := Subtype.fintype _ #align simple_graph.fintype_edge_set SimpleGraph.fintypeEdgeSet instance fintypeEdgeSetBot : Fintype (⊥ : SimpleGraph V).edgeSet := by rw [edgeSet_bot] infer_instance #align simple_graph.fintype_edge_set_bot SimpleGraph.fintypeEdgeSetBot instance fintypeEdgeSetSup [DecidableEq V] [Fintype G₁.edgeSet] [Fintype G₂.edgeSet] : Fintype (G₁ ⊔ G₂).edgeSet := by rw [edgeSet_sup] infer_instance #align simple_graph.fintype_edge_set_sup SimpleGraph.fintypeEdgeSetSup instance fintypeEdgeSetInf [DecidableEq V] [Fintype G₁.edgeSet] [Fintype G₂.edgeSet] : Fintype (G₁ ⊓ G₂).edgeSet := by rw [edgeSet_inf] exact Set.fintypeInter _ _ #align simple_graph.fintype_edge_set_inf SimpleGraph.fintypeEdgeSetInf instance fintypeEdgeSetSdiff [DecidableEq V] [Fintype G₁.edgeSet] [Fintype G₂.edgeSet] : Fintype (G₁ \ G₂).edgeSet := by rw [edgeSet_sdiff] exact Set.fintypeDiff _ _ #align simple_graph.fintype_edge_set_sdiff SimpleGraph.fintypeEdgeSetSdiff end EdgeSet section FromEdgeSet variable (s : Set (Sym2 V)) /-- `fromEdgeSet` constructs a `SimpleGraph` from a set of edges, without loops. -/ def fromEdgeSet : SimpleGraph V where Adj := Sym2.ToRel s ⊓ Ne symm v w h := ⟨Sym2.toRel_symmetric s h.1, h.2.symm⟩ #align simple_graph.from_edge_set SimpleGraph.fromEdgeSet @[simp] theorem fromEdgeSet_adj : (fromEdgeSet s).Adj v w ↔ s(v, w) ∈ s ∧ v ≠ w := Iff.rfl #align simple_graph.from_edge_set_adj SimpleGraph.fromEdgeSet_adj -- Note: we need to make sure `fromEdgeSet_adj` and this lemma are confluent. -- In particular, both yield `s(u, v) ∈ (fromEdgeSet s).edgeSet` ==> `s(v, w) ∈ s ∧ v ≠ w`. @[simp] theorem edgeSet_fromEdgeSet : (fromEdgeSet s).edgeSet = s \ { e \| e.IsDiag } := by ext e exact Sym2.ind (by simp) e #align simple_graph.edge_set_from_edge_set SimpleGraph.edgeSet_fromEdgeSet @[simp]	Mathlib/Combinatorics/SimpleGraph/Basic.lean	644	646	theorem fromEdgeSet_edgeSet : fromEdgeSet G.edgeSet = G := by	ext v w exact ⟨fun h => h.1, fun h => ⟨h, G.ne_of_adj h⟩⟩
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne -/ import Mathlib.Order.MinMax import Mathlib.Data.Set.Subsingleton import Mathlib.Tactic.Says #align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c" /-! # Intervals In any preorder `α`, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. This file contains these definitions, and basic facts on inclusion, intersection, difference of intervals (where the precise statements may depend on the properties of the order, in particular for some statements it should be `LinearOrder` or `DenselyOrdered`). TODO: This is just the beginning; a lot of rules are missing -/ open Function open OrderDual (toDual ofDual) variable {α β : Type} namespace Set section Preorder variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α} /-- Left-open right-open interval -/ def Ioo (a b : α) := { x \| a < x ∧ x < b } #align set.Ioo Set.Ioo /-- Left-closed right-open interval -/ def Ico (a b : α) := { x \| a ≤ x ∧ x < b } #align set.Ico Set.Ico /-- Left-infinite right-open interval -/ def Iio (a : α) := { x \| x < a } #align set.Iio Set.Iio /-- Left-closed right-closed interval -/ def Icc (a b : α) := { x \| a ≤ x ∧ x ≤ b } #align set.Icc Set.Icc /-- Left-infinite right-closed interval -/ def Iic (b : α) := { x \| x ≤ b } #align set.Iic Set.Iic /-- Left-open right-closed interval -/ def Ioc (a b : α) := { x \| a < x ∧ x ≤ b } #align set.Ioc Set.Ioc /-- Left-closed right-infinite interval -/ def Ici (a : α) := { x \| a ≤ x } #align set.Ici Set.Ici /-- Left-open right-infinite interval -/ def Ioi (a : α) := { x \| a < x } #align set.Ioi Set.Ioi theorem Ioo_def (a b : α) : { x \| a < x ∧ x < b } = Ioo a b := rfl #align set.Ioo_def Set.Ioo_def theorem Ico_def (a b : α) : { x \| a ≤ x ∧ x < b } = Ico a b := rfl #align set.Ico_def Set.Ico_def theorem Iio_def (a : α) : { x \| x < a } = Iio a := rfl #align set.Iio_def Set.Iio_def theorem Icc_def (a b : α) : { x \| a ≤ x ∧ x ≤ b } = Icc a b := rfl #align set.Icc_def Set.Icc_def theorem Iic_def (b : α) : { x \| x ≤ b } = Iic b := rfl #align set.Iic_def Set.Iic_def theorem Ioc_def (a b : α) : { x \| a < x ∧ x ≤ b } = Ioc a b := rfl #align set.Ioc_def Set.Ioc_def theorem Ici_def (a : α) : { x \| a ≤ x } = Ici a := rfl #align set.Ici_def Set.Ici_def theorem Ioi_def (a : α) : { x \| a < x } = Ioi a := rfl #align set.Ioi_def Set.Ioi_def @[simp] theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b := Iff.rfl #align set.mem_Ioo Set.mem_Ioo @[simp] theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b := Iff.rfl #align set.mem_Ico Set.mem_Ico @[simp] theorem mem_Iio : x ∈ Iio b ↔ x < b := Iff.rfl #align set.mem_Iio Set.mem_Iio @[simp] theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b := Iff.rfl #align set.mem_Icc Set.mem_Icc @[simp] theorem mem_Iic : x ∈ Iic b ↔ x ≤ b := Iff.rfl #align set.mem_Iic Set.mem_Iic @[simp] theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b := Iff.rfl #align set.mem_Ioc Set.mem_Ioc @[simp] theorem mem_Ici : x ∈ Ici a ↔ a ≤ x := Iff.rfl #align set.mem_Ici Set.mem_Ici @[simp] theorem mem_Ioi : x ∈ Ioi a ↔ a < x := Iff.rfl #align set.mem_Ioi Set.mem_Ioi instance decidableMemIoo [Decidable (a < x ∧ x < b)] : Decidable (x ∈ Ioo a b) := by assumption #align set.decidable_mem_Ioo Set.decidableMemIoo instance decidableMemIco [Decidable (a ≤ x ∧ x < b)] : Decidable (x ∈ Ico a b) := by assumption #align set.decidable_mem_Ico Set.decidableMemIco instance decidableMemIio [Decidable (x < b)] : Decidable (x ∈ Iio b) := by assumption #align set.decidable_mem_Iio Set.decidableMemIio instance decidableMemIcc [Decidable (a ≤ x ∧ x ≤ b)] : Decidable (x ∈ Icc a b) := by assumption #align set.decidable_mem_Icc Set.decidableMemIcc instance decidableMemIic [Decidable (x ≤ b)] : Decidable (x ∈ Iic b) := by assumption #align set.decidable_mem_Iic Set.decidableMemIic instance decidableMemIoc [Decidable (a < x ∧ x ≤ b)] : Decidable (x ∈ Ioc a b) := by assumption #align set.decidable_mem_Ioc Set.decidableMemIoc instance decidableMemIci [Decidable (a ≤ x)] : Decidable (x ∈ Ici a) := by assumption #align set.decidable_mem_Ici Set.decidableMemIci instance decidableMemIoi [Decidable (a < x)] : Decidable (x ∈ Ioi a) := by assumption #align set.decidable_mem_Ioi Set.decidableMemIoi -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioo Set.left_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl] #align set.left_mem_Ico Set.left_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.left_mem_Icc Set.left_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioc Set.left_mem_Ioc theorem left_mem_Ici : a ∈ Ici a := by simp #align set.left_mem_Ici Set.left_mem_Ici -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioo : b ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ioo Set.right_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ico : b ∈ Ico a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ico Set.right_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.right_mem_Icc Set.right_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl] #align set.right_mem_Ioc Set.right_mem_Ioc theorem right_mem_Iic : a ∈ Iic a := by simp #align set.right_mem_Iic Set.right_mem_Iic @[simp] theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a := rfl #align set.dual_Ici Set.dual_Ici @[simp] theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a := rfl #align set.dual_Iic Set.dual_Iic @[simp] theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a := rfl #align set.dual_Ioi Set.dual_Ioi @[simp] theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a := rfl #align set.dual_Iio Set.dual_Iio @[simp] theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a := Set.ext fun _ => and_comm #align set.dual_Icc Set.dual_Icc @[simp] theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a := Set.ext fun _ => and_comm #align set.dual_Ioc Set.dual_Ioc @[simp] theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a := Set.ext fun _ => and_comm #align set.dual_Ico Set.dual_Ico @[simp] theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a := Set.ext fun _ => and_comm #align set.dual_Ioo Set.dual_Ioo @[simp] theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := ⟨fun ⟨_, hx⟩ => hx.1.trans hx.2, fun h => ⟨a, left_mem_Icc.2 h⟩⟩ #align set.nonempty_Icc Set.nonempty_Icc @[simp] theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_lt hx.2, fun h => ⟨a, left_mem_Ico.2 h⟩⟩ #align set.nonempty_Ico Set.nonempty_Ico @[simp] theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_le hx.2, fun h => ⟨b, right_mem_Ioc.2 h⟩⟩ #align set.nonempty_Ioc Set.nonempty_Ioc @[simp] theorem nonempty_Ici : (Ici a).Nonempty := ⟨a, left_mem_Ici⟩ #align set.nonempty_Ici Set.nonempty_Ici @[simp] theorem nonempty_Iic : (Iic a).Nonempty := ⟨a, right_mem_Iic⟩ #align set.nonempty_Iic Set.nonempty_Iic @[simp] theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := ⟨fun ⟨_, ha, hb⟩ => ha.trans hb, exists_between⟩ #align set.nonempty_Ioo Set.nonempty_Ioo @[simp] theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty := exists_gt a #align set.nonempty_Ioi Set.nonempty_Ioi @[simp] theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty := exists_lt a #align set.nonempty_Iio Set.nonempty_Iio theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) := Nonempty.to_subtype (nonempty_Icc.mpr h) #align set.nonempty_Icc_subtype Set.nonempty_Icc_subtype theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) := Nonempty.to_subtype (nonempty_Ico.mpr h) #align set.nonempty_Ico_subtype Set.nonempty_Ico_subtype theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) := Nonempty.to_subtype (nonempty_Ioc.mpr h) #align set.nonempty_Ioc_subtype Set.nonempty_Ioc_subtype /-- An interval `Ici a` is nonempty. -/ instance nonempty_Ici_subtype : Nonempty (Ici a) := Nonempty.to_subtype nonempty_Ici #align set.nonempty_Ici_subtype Set.nonempty_Ici_subtype /-- An interval `Iic a` is nonempty. -/ instance nonempty_Iic_subtype : Nonempty (Iic a) := Nonempty.to_subtype nonempty_Iic #align set.nonempty_Iic_subtype Set.nonempty_Iic_subtype theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) := Nonempty.to_subtype (nonempty_Ioo.mpr h) #align set.nonempty_Ioo_subtype Set.nonempty_Ioo_subtype /-- In an order without maximal elements, the intervals `Ioi` are nonempty. -/ instance nonempty_Ioi_subtype [NoMaxOrder α] : Nonempty (Ioi a) := Nonempty.to_subtype nonempty_Ioi #align set.nonempty_Ioi_subtype Set.nonempty_Ioi_subtype /-- In an order without minimal elements, the intervals `Iio` are nonempty. -/ instance nonempty_Iio_subtype [NoMinOrder α] : Nonempty (Iio a) := Nonempty.to_subtype nonempty_Iio #align set.nonempty_Iio_subtype Set.nonempty_Iio_subtype instance [NoMinOrder α] : NoMinOrder (Iio a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, lt_trans hb a.2⟩, hb⟩⟩ instance [NoMinOrder α] : NoMinOrder (Iic a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, hb.le.trans a.2⟩, hb⟩⟩ instance [NoMaxOrder α] : NoMaxOrder (Ioi a) := OrderDual.noMaxOrder (α := Iio (toDual a)) instance [NoMaxOrder α] : NoMaxOrder (Ici a) := OrderDual.noMaxOrder (α := Iic (toDual a)) @[simp] theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Icc_eq_empty Set.Icc_eq_empty @[simp] theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_lt hb) #align set.Ico_eq_empty Set.Ico_eq_empty @[simp] theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_le hb) #align set.Ioc_eq_empty Set.Ioc_eq_empty @[simp] theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Ioo_eq_empty Set.Ioo_eq_empty @[simp] theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ := Icc_eq_empty h.not_le #align set.Icc_eq_empty_of_lt Set.Icc_eq_empty_of_lt @[simp] theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ := Ico_eq_empty h.not_lt #align set.Ico_eq_empty_of_le Set.Ico_eq_empty_of_le @[simp] theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ := Ioc_eq_empty h.not_lt #align set.Ioc_eq_empty_of_le Set.Ioc_eq_empty_of_le @[simp] theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ := Ioo_eq_empty h.not_lt #align set.Ioo_eq_empty_of_le Set.Ioo_eq_empty_of_le -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ico_self (a : α) : Ico a a = ∅ := Ico_eq_empty <\| lt_irrefl _ #align set.Ico_self Set.Ico_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioc_self (a : α) : Ioc a a = ∅ := Ioc_eq_empty <\| lt_irrefl _ #align set.Ioc_self Set.Ioc_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioo_self (a : α) : Ioo a a = ∅ := Ioo_eq_empty <\| lt_irrefl _ #align set.Ioo_self Set.Ioo_self theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a := ⟨fun h => h <\| left_mem_Ici, fun h _ hx => h.trans hx⟩ #align set.Ici_subset_Ici Set.Ici_subset_Ici @[gcongr] alias ⟨_, _root_.GCongr.Ici_subset_Ici_of_le⟩ := Ici_subset_Ici theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b := @Ici_subset_Ici αᵒᵈ _ _ _ #align set.Iic_subset_Iic Set.Iic_subset_Iic @[gcongr] alias ⟨_, _root_.GCongr.Iic_subset_Iic_of_le⟩ := Iic_subset_Iic theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a := ⟨fun h => h left_mem_Ici, fun h _ hx => h.trans_le hx⟩ #align set.Ici_subset_Ioi Set.Ici_subset_Ioi theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b := ⟨fun h => h right_mem_Iic, fun h _ hx => lt_of_le_of_lt hx h⟩ #align set.Iic_subset_Iio Set.Iic_subset_Iio @[gcongr] theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans_le h₂⟩ #align set.Ioo_subset_Ioo Set.Ioo_subset_Ioo @[gcongr] theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b := Ioo_subset_Ioo h le_rfl #align set.Ioo_subset_Ioo_left Set.Ioo_subset_Ioo_left @[gcongr] theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ := Ioo_subset_Ioo le_rfl h #align set.Ioo_subset_Ioo_right Set.Ioo_subset_Ioo_right @[gcongr] theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, hx₂.trans_le h₂⟩ #align set.Ico_subset_Ico Set.Ico_subset_Ico @[gcongr] theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b := Ico_subset_Ico h le_rfl #align set.Ico_subset_Ico_left Set.Ico_subset_Ico_left @[gcongr] theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ := Ico_subset_Ico le_rfl h #align set.Ico_subset_Ico_right Set.Ico_subset_Ico_right @[gcongr] theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, le_trans hx₂ h₂⟩ #align set.Icc_subset_Icc Set.Icc_subset_Icc @[gcongr] theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b := Icc_subset_Icc h le_rfl #align set.Icc_subset_Icc_left Set.Icc_subset_Icc_left @[gcongr] theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ := Icc_subset_Icc le_rfl h #align set.Icc_subset_Icc_right Set.Icc_subset_Icc_right theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ hx => ⟨ha.trans_le hx.1, hx.2.trans_lt hb⟩ #align set.Icc_subset_Ioo Set.Icc_subset_Ioo theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := fun _ => And.left #align set.Icc_subset_Ici_self Set.Icc_subset_Ici_self theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := fun _ => And.right #align set.Icc_subset_Iic_self Set.Icc_subset_Iic_self theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := fun _ => And.right #align set.Ioc_subset_Iic_self Set.Ioc_subset_Iic_self @[gcongr] theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans h₂⟩ #align set.Ioc_subset_Ioc Set.Ioc_subset_Ioc @[gcongr] theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b := Ioc_subset_Ioc h le_rfl #align set.Ioc_subset_Ioc_left Set.Ioc_subset_Ioc_left @[gcongr] theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ := Ioc_subset_Ioc le_rfl h #align set.Ioc_subset_Ioc_right Set.Ioc_subset_Ioc_right theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := fun _ => And.imp_left h₁.trans_le #align set.Ico_subset_Ioo_left Set.Ico_subset_Ioo_left theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := fun _ => And.imp_right fun h' => h'.trans_lt h #align set.Ioc_subset_Ioo_right Set.Ioc_subset_Ioo_right theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := fun _ => And.imp_right fun h₂ => h₂.trans_lt h₁ #align set.Icc_subset_Ico_right Set.Icc_subset_Ico_right theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := fun _ => And.imp_left le_of_lt #align set.Ioo_subset_Ico_self Set.Ioo_subset_Ico_self theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := fun _ => And.imp_right le_of_lt #align set.Ioo_subset_Ioc_self Set.Ioo_subset_Ioc_self theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := fun _ => And.imp_right le_of_lt #align set.Ico_subset_Icc_self Set.Ico_subset_Icc_self theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := fun _ => And.imp_left le_of_lt #align set.Ioc_subset_Icc_self Set.Ioc_subset_Icc_self theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b := Subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self #align set.Ioo_subset_Icc_self Set.Ioo_subset_Icc_self theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := fun _ => And.right #align set.Ico_subset_Iio_self Set.Ico_subset_Iio_self theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := fun _ => And.right #align set.Ioo_subset_Iio_self Set.Ioo_subset_Iio_self theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := fun _ => And.left #align set.Ioc_subset_Ioi_self Set.Ioc_subset_Ioi_self theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := fun _ => And.left #align set.Ioo_subset_Ioi_self Set.Ioo_subset_Ioi_self theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a := fun _ hx => le_of_lt hx #align set.Ioi_subset_Ici_self Set.Ioi_subset_Ici_self theorem Iio_subset_Iic_self : Iio a ⊆ Iic a := fun _ hx => le_of_lt hx #align set.Iio_subset_Iic_self Set.Iio_subset_Iic_self theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := fun _ => And.left #align set.Ico_subset_Ici_self Set.Ico_subset_Ici_self theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a := ⟨Ioi_subset_Ici_self, fun h => lt_irrefl a (h le_rfl)⟩ #align set.Ioi_ssubset_Ici_self Set.Ioi_ssubset_Ici_self theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a := @Ioi_ssubset_Ici_self αᵒᵈ _ _ #align set.Iio_ssubset_Iic_self Set.Iio_ssubset_Iic_self theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Icc_iff Set.Icc_subset_Icc_iff theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ioo_iff Set.Icc_subset_Ioo_iff theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ico_iff Set.Icc_subset_Ico_iff theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Ioc_iff Set.Icc_subset_Ioc_iff theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans_lt h⟩ #align set.Icc_subset_Iio_iff Set.Icc_subset_Iio_iff theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans_le hx⟩ #align set.Icc_subset_Ioi_iff Set.Icc_subset_Ioi_iff theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans h⟩ #align set.Icc_subset_Iic_iff Set.Icc_subset_Iic_iff theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans hx⟩ #align set.Icc_subset_Ici_iff Set.Icc_subset_Ici_iff theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc (le_of_lt ha) hb)).mpr ⟨a₂, left_mem_Icc.mpr hI, not_and.mpr fun f _ => lt_irrefl a₂ (ha.trans_le f)⟩ #align set.Icc_ssubset_Icc_left Set.Icc_ssubset_Icc_left theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc ha (le_of_lt hb))).mpr ⟨b₂, right_mem_Icc.mpr hI, fun f => lt_irrefl b₁ (hb.trans_le f.2)⟩ #align set.Icc_ssubset_Icc_right Set.Icc_ssubset_Icc_right /-- If `a ≤ b`, then `(b, +∞) ⊆ (a, +∞)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Ioi_subset_Ioi_iff`. -/ @[gcongr] theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := fun _ hx => h.trans_lt hx #align set.Ioi_subset_Ioi Set.Ioi_subset_Ioi /-- If `a ≤ b`, then `(b, +∞) ⊆ [a, +∞)`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Ioi_subset_Ici_iff`. -/ theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a := Subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self #align set.Ioi_subset_Ici Set.Ioi_subset_Ici /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Iio_subset_Iio_iff`. -/ @[gcongr] theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := fun _ hx => lt_of_lt_of_le hx h #align set.Iio_subset_Iio Set.Iio_subset_Iio /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b]`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Iio_subset_Iic_iff`. -/ theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b := Subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self #align set.Iio_subset_Iic Set.Iio_subset_Iic theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b := rfl #align set.Ici_inter_Iic Set.Ici_inter_Iic theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b := rfl #align set.Ici_inter_Iio Set.Ici_inter_Iio theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b := rfl #align set.Ioi_inter_Iic Set.Ioi_inter_Iic theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b := rfl #align set.Ioi_inter_Iio Set.Ioi_inter_Iio theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a := inter_comm _ _ #align set.Iic_inter_Ici Set.Iic_inter_Ici theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a := inter_comm _ _ #align set.Iio_inter_Ici Set.Iio_inter_Ici theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a := inter_comm _ _ #align set.Iic_inter_Ioi Set.Iic_inter_Ioi theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a := inter_comm _ _ #align set.Iio_inter_Ioi Set.Iio_inter_Ioi theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b := Ioo_subset_Icc_self h #align set.mem_Icc_of_Ioo Set.mem_Icc_of_Ioo theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b := Ioo_subset_Ico_self h #align set.mem_Ico_of_Ioo Set.mem_Ico_of_Ioo theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b := Ioo_subset_Ioc_self h #align set.mem_Ioc_of_Ioo Set.mem_Ioc_of_Ioo theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b := Ico_subset_Icc_self h #align set.mem_Icc_of_Ico Set.mem_Icc_of_Ico theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b := Ioc_subset_Icc_self h #align set.mem_Icc_of_Ioc Set.mem_Icc_of_Ioc theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a := Ioi_subset_Ici_self h #align set.mem_Ici_of_Ioi Set.mem_Ici_of_Ioi theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a := Iio_subset_Iic_self h #align set.mem_Iic_of_Iio Set.mem_Iic_of_Iio theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc] #align set.Icc_eq_empty_iff Set.Icc_eq_empty_iff theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico] #align set.Ico_eq_empty_iff Set.Ico_eq_empty_iff theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioc] #align set.Ioc_eq_empty_iff Set.Ioc_eq_empty_iff theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo] #align set.Ioo_eq_empty_iff Set.Ioo_eq_empty_iff theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ := eq_univ_of_forall h #align is_top.Iic_eq IsTop.Iic_eq theorem _root_.IsBot.Ici_eq (h : IsBot a) : Ici a = univ := eq_univ_of_forall h #align is_bot.Ici_eq IsBot.Ici_eq theorem _root_.IsMax.Ioi_eq (h : IsMax a) : Ioi a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_max.Ioi_eq IsMax.Ioi_eq theorem _root_.IsMin.Iio_eq (h : IsMin a) : Iio a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_min.Iio_eq IsMin.Iio_eq theorem Iic_inter_Ioc_of_le (h : a ≤ c) : Iic a ∩ Ioc b c = Ioc b a := ext fun _ => ⟨fun H => ⟨H.2.1, H.1⟩, fun H => ⟨H.2, H.1, H.2.trans h⟩⟩ #align set.Iic_inter_Ioc_of_le Set.Iic_inter_Ioc_of_le theorem not_mem_Icc_of_lt (ha : c < a) : c ∉ Icc a b := fun h => ha.not_le h.1 #align set.not_mem_Icc_of_lt Set.not_mem_Icc_of_lt theorem not_mem_Icc_of_gt (hb : b < c) : c ∉ Icc a b := fun h => hb.not_le h.2 #align set.not_mem_Icc_of_gt Set.not_mem_Icc_of_gt theorem not_mem_Ico_of_lt (ha : c < a) : c ∉ Ico a b := fun h => ha.not_le h.1 #align set.not_mem_Ico_of_lt Set.not_mem_Ico_of_lt theorem not_mem_Ioc_of_gt (hb : b < c) : c ∉ Ioc a b := fun h => hb.not_le h.2 #align set.not_mem_Ioc_of_gt Set.not_mem_Ioc_of_gt -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Ioi_self : a ∉ Ioi a := lt_irrefl _ #align set.not_mem_Ioi_self Set.not_mem_Ioi_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Iio_self : b ∉ Iio b := lt_irrefl _ #align set.not_mem_Iio_self Set.not_mem_Iio_self theorem not_mem_Ioc_of_le (ha : c ≤ a) : c ∉ Ioc a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioc_of_le Set.not_mem_Ioc_of_le theorem not_mem_Ico_of_ge (hb : b ≤ c) : c ∉ Ico a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ico_of_ge Set.not_mem_Ico_of_ge theorem not_mem_Ioo_of_le (ha : c ≤ a) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioo_of_le Set.not_mem_Ioo_of_le theorem not_mem_Ioo_of_ge (hb : b ≤ c) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ioo_of_ge Set.not_mem_Ioo_of_ge end Preorder section PartialOrder variable [PartialOrder α] {a b c : α} @[simp] theorem Icc_self (a : α) : Icc a a = {a} := Set.ext <\| by simp [Icc, le_antisymm_iff, and_comm] #align set.Icc_self Set.Icc_self instance instIccUnique : Unique (Set.Icc a a) where default := ⟨a, by simp⟩ uniq y := Subtype.ext <\| by simpa using y.2 @[simp] theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by refine ⟨fun h => ?_, ?_⟩ · have hab : a ≤ b := nonempty_Icc.1 (h.symm.subst <\| singleton_nonempty c) exact ⟨eq_of_mem_singleton <\| h.subst <\| left_mem_Icc.2 hab, eq_of_mem_singleton <\| h.subst <\| right_mem_Icc.2 hab⟩ · rintro ⟨rfl, rfl⟩ exact Icc_self _ #align set.Icc_eq_singleton_iff Set.Icc_eq_singleton_iff lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) := fun _x ⟨hax, hxb⟩ _y ⟨hay, hyb⟩ ↦ le_antisymm (le_implies_le_of_le_of_le hxb hay hba) (le_implies_le_of_le_of_le hyb hax hba) #align set.subsingleton_Icc_of_ge Set.subsingleton_Icc_of_ge @[simp] lemma subsingleton_Icc_iff {α : Type} [LinearOrder α] {a b : α} : Set.Subsingleton (Icc a b) ↔ b ≤ a := by refine ⟨fun h ↦ ?_, subsingleton_Icc_of_ge⟩ contrapose! h simp only [ge_iff_le, gt_iff_lt, not_subsingleton_iff] exact ⟨a, ⟨le_refl _, h.le⟩, b, ⟨h.le, le_refl _⟩, h.ne⟩ @[simp] theorem Icc_diff_left : Icc a b \ {a} = Ioc a b := ext fun x => by simp [lt_iff_le_and_ne, eq_comm, and_right_comm] #align set.Icc_diff_left Set.Icc_diff_left @[simp] theorem Icc_diff_right : Icc a b \ {b} = Ico a b := ext fun x => by simp [lt_iff_le_and_ne, and_assoc] #align set.Icc_diff_right Set.Icc_diff_right @[simp] theorem Ico_diff_left : Ico a b \ {a} = Ioo a b := ext fun x => by simp [and_right_comm, ← lt_iff_le_and_ne, eq_comm] #align set.Ico_diff_left Set.Ico_diff_left @[simp] theorem Ioc_diff_right : Ioc a b \ {b} = Ioo a b := ext fun x => by simp [and_assoc, ← lt_iff_le_and_ne] #align set.Ioc_diff_right Set.Ioc_diff_right @[simp] theorem Icc_diff_both : Icc a b \ {a, b} = Ioo a b := by rw [insert_eq, ← diff_diff, Icc_diff_left, Ioc_diff_right] #align set.Icc_diff_both Set.Icc_diff_both @[simp] theorem Ici_diff_left : Ici a \ {a} = Ioi a := ext fun x => by simp [lt_iff_le_and_ne, eq_comm] #align set.Ici_diff_left Set.Ici_diff_left @[simp] theorem Iic_diff_right : Iic a \ {a} = Iio a := ext fun x => by simp [lt_iff_le_and_ne] #align set.Iic_diff_right Set.Iic_diff_right @[simp] theorem Ico_diff_Ioo_same (h : a < b) : Ico a b \ Ioo a b = {a} := by rw [← Ico_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Ico.2 h)] #align set.Ico_diff_Ioo_same Set.Ico_diff_Ioo_same @[simp] theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} := by rw [← Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Ioc.2 h)] #align set.Ioc_diff_Ioo_same Set.Ioc_diff_Ioo_same @[simp] theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Icc.2 h)] #align set.Icc_diff_Ico_same Set.Icc_diff_Ico_same @[simp] theorem Icc_diff_Ioc_same (h : a ≤ b) : Icc a b \ Ioc a b = {a} := by rw [← Icc_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Icc.2 h)] #align set.Icc_diff_Ioc_same Set.Icc_diff_Ioc_same @[simp] theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} := by rw [← Icc_diff_both, diff_diff_cancel_left] simp [insert_subset_iff, h] #align set.Icc_diff_Ioo_same Set.Icc_diff_Ioo_same @[simp] theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} := by rw [← Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)] #align set.Ici_diff_Ioi_same Set.Ici_diff_Ioi_same @[simp] theorem Iic_diff_Iio_same : Iic a \ Iio a = {a} := by rw [← Iic_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 right_mem_Iic)] #align set.Iic_diff_Iio_same Set.Iic_diff_Iio_same -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioi_union_left : Ioi a ∪ {a} = Ici a := ext fun x => by simp [eq_comm, le_iff_eq_or_lt] #align set.Ioi_union_left Set.Ioi_union_left -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Iio_union_right : Iio a ∪ {a} = Iic a := ext fun _ => le_iff_lt_or_eq.symm #align set.Iio_union_right Set.Iio_union_right theorem Ioo_union_left (hab : a < b) : Ioo a b ∪ {a} = Ico a b := by rw [← Ico_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Ico.2 hab)] #align set.Ioo_union_left Set.Ioo_union_left theorem Ioo_union_right (hab : a < b) : Ioo a b ∪ {b} = Ioc a b := by simpa only [dual_Ioo, dual_Ico] using Ioo_union_left hab.dual #align set.Ioo_union_right Set.Ioo_union_right theorem Ioo_union_both (h : a ≤ b) : Ioo a b ∪ {a, b} = Icc a b := by have : (Icc a b \ {a, b}) ∪ {a, b} = Icc a b := diff_union_of_subset fun \| x, .inl rfl => left_mem_Icc.mpr h \| x, .inr rfl => right_mem_Icc.mpr h rw [← this, Icc_diff_both] theorem Ioc_union_left (hab : a ≤ b) : Ioc a b ∪ {a} = Icc a b := by rw [← Icc_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Icc.2 hab)] #align set.Ioc_union_left Set.Ioc_union_left theorem Ico_union_right (hab : a ≤ b) : Ico a b ∪ {b} = Icc a b := by simpa only [dual_Ioc, dual_Icc] using Ioc_union_left hab.dual #align set.Ico_union_right Set.Ico_union_right @[simp] theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b := by rw [insert_eq, union_comm, Ico_union_right h] #align set.Ico_insert_right Set.Ico_insert_right @[simp] theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b := by rw [insert_eq, union_comm, Ioc_union_left h] #align set.Ioc_insert_left Set.Ioc_insert_left @[simp] theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b := by rw [insert_eq, union_comm, Ioo_union_left h] #align set.Ioo_insert_left Set.Ioo_insert_left @[simp] theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b := by rw [insert_eq, union_comm, Ioo_union_right h] #align set.Ioo_insert_right Set.Ioo_insert_right @[simp] theorem Iio_insert : insert a (Iio a) = Iic a := ext fun _ => le_iff_eq_or_lt.symm #align set.Iio_insert Set.Iio_insert @[simp] theorem Ioi_insert : insert a (Ioi a) = Ici a := ext fun _ => (or_congr_left eq_comm).trans le_iff_eq_or_lt.symm #align set.Ioi_insert Set.Ioi_insert theorem mem_Ici_Ioi_of_subset_of_subset {s : Set α} (ho : Ioi a ⊆ s) (hc : s ⊆ Ici a) : s ∈ ({Ici a, Ioi a} : Set (Set α)) := by_cases (fun h : a ∈ s => Or.inl <\| Subset.antisymm hc <\| by rw [← Ioi_union_left, union_subset_iff]; simp []) fun h => Or.inr <\| Subset.antisymm (fun x hx => lt_of_le_of_ne (hc hx) fun heq => h <\| heq.symm ▸ hx) ho #align set.mem_Ici_Ioi_of_subset_of_subset Set.mem_Ici_Ioi_of_subset_of_subset theorem mem_Iic_Iio_of_subset_of_subset {s : Set α} (ho : Iio a ⊆ s) (hc : s ⊆ Iic a) : s ∈ ({Iic a, Iio a} : Set (Set α)) := @mem_Ici_Ioi_of_subset_of_subset αᵒᵈ _ a s ho hc #align set.mem_Iic_Iio_of_subset_of_subset Set.mem_Iic_Iio_of_subset_of_subset theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) : s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α)) := by classical by_cases ha : a ∈ s <;> by_cases hb : b ∈ s · refine Or.inl (Subset.antisymm hc ?_) rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, ← Icc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_right] exact subset_diff_singleton hc hb · rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho · refine Or.inr <\| Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_left] exact subset_diff_singleton hc ha · rwa [← Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inr <\| Or.inr <\| Subset.antisymm ?_ ho rw [← Ico_diff_left, ← Icc_diff_right] apply_rules [subset_diff_singleton] #align set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset theorem eq_left_or_mem_Ioo_of_mem_Ico {x : α} (hmem : x ∈ Ico a b) : x = a ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => ⟨h, hmem.2⟩ #align set.eq_left_or_mem_Ioo_of_mem_Ico Set.eq_left_or_mem_Ioo_of_mem_Ico theorem eq_right_or_mem_Ioo_of_mem_Ioc {x : α} (hmem : x ∈ Ioc a b) : x = b ∨ x ∈ Ioo a b := hmem.2.eq_or_lt.imp_right <\| And.intro hmem.1 #align set.eq_right_or_mem_Ioo_of_mem_Ioc Set.eq_right_or_mem_Ioo_of_mem_Ioc theorem eq_endpoints_or_mem_Ioo_of_mem_Icc {x : α} (hmem : x ∈ Icc a b) : x = a ∨ x = b ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => eq_right_or_mem_Ioo_of_mem_Ioc ⟨h, hmem.2⟩ #align set.eq_endpoints_or_mem_Ioo_of_mem_Icc Set.eq_endpoints_or_mem_Ioo_of_mem_Icc theorem _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} := eq_singleton_iff_unique_mem.2 ⟨left_mem_Ici, fun _ => h.eq_of_ge⟩ #align is_max.Ici_eq IsMax.Ici_eq theorem _root_.IsMin.Iic_eq (h : IsMin a) : Iic a = {a} := h.toDual.Ici_eq #align is_min.Iic_eq IsMin.Iic_eq theorem Ici_injective : Injective (Ici : α → Set α) := fun _ _ => eq_of_forall_ge_iff ∘ Set.ext_iff.1 #align set.Ici_injective Set.Ici_injective theorem Iic_injective : Injective (Iic : α → Set α) := fun _ _ => eq_of_forall_le_iff ∘ Set.ext_iff.1 #align set.Iic_injective Set.Iic_injective theorem Ici_inj : Ici a = Ici b ↔ a = b := Ici_injective.eq_iff #align set.Ici_inj Set.Ici_inj theorem Iic_inj : Iic a = Iic b ↔ a = b := Iic_injective.eq_iff #align set.Iic_inj Set.Iic_inj end PartialOrder section OrderTop @[simp] theorem Ici_top [PartialOrder α] [OrderTop α] : Ici (⊤ : α) = {⊤} := isMax_top.Ici_eq #align set.Ici_top Set.Ici_top variable [Preorder α] [OrderTop α] {a : α} @[simp] theorem Ioi_top : Ioi (⊤ : α) = ∅ := isMax_top.Ioi_eq #align set.Ioi_top Set.Ioi_top @[simp] theorem Iic_top : Iic (⊤ : α) = univ := isTop_top.Iic_eq #align set.Iic_top Set.Iic_top @[simp] theorem Icc_top : Icc a ⊤ = Ici a := by simp [← Ici_inter_Iic] #align set.Icc_top Set.Icc_top @[simp] theorem Ioc_top : Ioc a ⊤ = Ioi a := by simp [← Ioi_inter_Iic] #align set.Ioc_top Set.Ioc_top end OrderTop section OrderBot @[simp] theorem Iic_bot [PartialOrder α] [OrderBot α] : Iic (⊥ : α) = {⊥} := isMin_bot.Iic_eq #align set.Iic_bot Set.Iic_bot variable [Preorder α] [OrderBot α] {a : α} @[simp] theorem Iio_bot : Iio (⊥ : α) = ∅ := isMin_bot.Iio_eq #align set.Iio_bot Set.Iio_bot @[simp] theorem Ici_bot : Ici (⊥ : α) = univ := isBot_bot.Ici_eq #align set.Ici_bot Set.Ici_bot @[simp] theorem Icc_bot : Icc ⊥ a = Iic a := by simp [← Ici_inter_Iic] #align set.Icc_bot Set.Icc_bot @[simp] theorem Ico_bot : Ico ⊥ a = Iio a := by simp [← Ici_inter_Iio] #align set.Ico_bot Set.Ico_bot end OrderBot theorem Icc_bot_top [PartialOrder α] [BoundedOrder α] : Icc (⊥ : α) ⊤ = univ := by simp #align set.Icc_bot_top Set.Icc_bot_top section LinearOrder variable [LinearOrder α] {a a₁ a₂ b b₁ b₂ c d : α} theorem not_mem_Ici : c ∉ Ici a ↔ c < a := not_le #align set.not_mem_Ici Set.not_mem_Ici theorem not_mem_Iic : c ∉ Iic b ↔ b < c := not_le #align set.not_mem_Iic Set.not_mem_Iic theorem not_mem_Ioi : c ∉ Ioi a ↔ c ≤ a := not_lt #align set.not_mem_Ioi Set.not_mem_Ioi theorem not_mem_Iio : c ∉ Iio b ↔ b ≤ c := not_lt #align set.not_mem_Iio Set.not_mem_Iio @[simp] theorem compl_Iic : (Iic a)ᶜ = Ioi a := ext fun _ => not_le #align set.compl_Iic Set.compl_Iic @[simp] theorem compl_Ici : (Ici a)ᶜ = Iio a := ext fun _ => not_le #align set.compl_Ici Set.compl_Ici @[simp] theorem compl_Iio : (Iio a)ᶜ = Ici a := ext fun _ => not_lt #align set.compl_Iio Set.compl_Iio @[simp] theorem compl_Ioi : (Ioi a)ᶜ = Iic a := ext fun _ => not_lt #align set.compl_Ioi Set.compl_Ioi @[simp] theorem Ici_diff_Ici : Ici a \ Ici b = Ico a b := by rw [diff_eq, compl_Ici, Ici_inter_Iio] #align set.Ici_diff_Ici Set.Ici_diff_Ici @[simp] theorem Ici_diff_Ioi : Ici a \ Ioi b = Icc a b := by rw [diff_eq, compl_Ioi, Ici_inter_Iic] #align set.Ici_diff_Ioi Set.Ici_diff_Ioi @[simp] theorem Ioi_diff_Ioi : Ioi a \ Ioi b = Ioc a b := by rw [diff_eq, compl_Ioi, Ioi_inter_Iic] #align set.Ioi_diff_Ioi Set.Ioi_diff_Ioi @[simp] theorem Ioi_diff_Ici : Ioi a \ Ici b = Ioo a b := by rw [diff_eq, compl_Ici, Ioi_inter_Iio] #align set.Ioi_diff_Ici Set.Ioi_diff_Ici @[simp] theorem Iic_diff_Iic : Iic b \ Iic a = Ioc a b := by rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iic] #align set.Iic_diff_Iic Set.Iic_diff_Iic @[simp] theorem Iio_diff_Iic : Iio b \ Iic a = Ioo a b := by rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iio] #align set.Iio_diff_Iic Set.Iio_diff_Iic @[simp] theorem Iic_diff_Iio : Iic b \ Iio a = Icc a b := by rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iic] #align set.Iic_diff_Iio Set.Iic_diff_Iio @[simp] theorem Iio_diff_Iio : Iio b \ Iio a = Ico a b := by rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iio] #align set.Iio_diff_Iio Set.Iio_diff_Iio theorem Ioi_injective : Injective (Ioi : α → Set α) := fun _ _ => eq_of_forall_gt_iff ∘ Set.ext_iff.1 #align set.Ioi_injective Set.Ioi_injective theorem Iio_injective : Injective (Iio : α → Set α) := fun _ _ => eq_of_forall_lt_iff ∘ Set.ext_iff.1 #align set.Iio_injective Set.Iio_injective theorem Ioi_inj : Ioi a = Ioi b ↔ a = b := Ioi_injective.eq_iff #align set.Ioi_inj Set.Ioi_inj theorem Iio_inj : Iio a = Iio b ↔ a = b := Iio_injective.eq_iff #align set.Iio_inj Set.Iio_inj theorem Ico_subset_Ico_iff (h₁ : a₁ < b₁) : Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => have : a₂ ≤ a₁ ∧ a₁ < b₂ := h ⟨le_rfl, h₁⟩ ⟨this.1, le_of_not_lt fun h' => lt_irrefl b₂ (h ⟨this.2.le, h'⟩).2⟩, fun ⟨h₁, h₂⟩ => Ico_subset_Ico h₁ h₂⟩ #align set.Ico_subset_Ico_iff Set.Ico_subset_Ico_iff theorem Ioc_subset_Ioc_iff (h₁ : a₁ < b₁) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ b₁ ≤ b₂ ∧ a₂ ≤ a₁ := by convert @Ico_subset_Ico_iff αᵒᵈ _ b₁ b₂ a₁ a₂ h₁ using 2 <;> exact (@dual_Ico α _ _ _).symm #align set.Ioc_subset_Ioc_iff Set.Ioc_subset_Ioc_iff theorem Ioo_subset_Ioo_iff [DenselyOrdered α] (h₁ : a₁ < b₁) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => by rcases exists_between h₁ with ⟨x, xa, xb⟩ constructor <;> refine le_of_not_lt fun h' => ?_ · have ab := (h ⟨xa, xb⟩).1.trans xb exact lt_irrefl _ (h ⟨h', ab⟩).1 · have ab := xa.trans (h ⟨xa, xb⟩).2 exact lt_irrefl _ (h ⟨ab, h'⟩).2, fun ⟨h₁, h₂⟩ => Ioo_subset_Ioo h₁ h₂⟩ #align set.Ioo_subset_Ioo_iff Set.Ioo_subset_Ioo_iff theorem Ico_eq_Ico_iff (h : a₁ < b₁ ∨ a₂ < b₂) : Ico a₁ b₁ = Ico a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ := ⟨fun e => by simp only [Subset.antisymm_iff] at e simp only [le_antisymm_iff] cases' h with h h <;> simp only [gt_iff_lt, not_lt, ge_iff_le, Ico_subset_Ico_iff h] at e <;> [ rcases e with ⟨⟨h₁, h₂⟩, e'⟩; rcases e with ⟨e', ⟨h₁, h₂⟩⟩ ] <;> -- Porting note: restore `tauto` have hab := (Ico_subset_Ico_iff <\| h₁.trans_lt <\| h.trans_le h₂).1 e' <;> [ exact ⟨⟨hab.left, h₁⟩, ⟨h₂, hab.right⟩⟩; exact ⟨⟨h₁, hab.left⟩, ⟨hab.right, h₂⟩⟩ ], fun ⟨h₁, h₂⟩ => by rw [h₁, h₂]⟩ #align set.Ico_eq_Ico_iff Set.Ico_eq_Ico_iff lemma Ici_eq_singleton_iff_isTop {x : α} : (Ici x = {x}) ↔ IsTop x := by refine ⟨fun h y ↦ ?_, fun h ↦ by ext y; simp [(h y).ge_iff_eq]⟩ by_contra! H have : y ∈ Ici x := H.le rw [h, mem_singleton_iff] at this exact lt_irrefl y (this.le.trans_lt H) open scoped Classical @[simp] theorem Ioi_subset_Ioi_iff : Ioi b ⊆ Ioi a ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Ioi_subset_Ioi h⟩ by_contra ba exact lt_irrefl _ (h (not_le.mp ba)) #align set.Ioi_subset_Ioi_iff Set.Ioi_subset_Ioi_iff @[simp] theorem Ioi_subset_Ici_iff [DenselyOrdered α] : Ioi b ⊆ Ici a ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Ioi_subset_Ici h⟩ by_contra ba obtain ⟨c, bc, ca⟩ : ∃ c, b < c ∧ c < a := exists_between (not_le.mp ba) exact lt_irrefl _ (ca.trans_le (h bc)) #align set.Ioi_subset_Ici_iff Set.Ioi_subset_Ici_iff @[simp] theorem Iio_subset_Iio_iff : Iio a ⊆ Iio b ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Iio_subset_Iio h⟩ by_contra ab exact lt_irrefl _ (h (not_le.mp ab)) #align set.Iio_subset_Iio_iff Set.Iio_subset_Iio_iff @[simp] theorem Iio_subset_Iic_iff [DenselyOrdered α] : Iio a ⊆ Iic b ↔ a ≤ b := by rw [← diff_eq_empty, Iio_diff_Iic, Ioo_eq_empty_iff, not_lt] #align set.Iio_subset_Iic_iff Set.Iio_subset_Iic_iff /-! ### Unions of adjacent intervals -/ /-! #### Two infinite intervals -/ theorem Iic_union_Ioi_of_le (h : a ≤ b) : Iic b ∪ Ioi a = univ := eq_univ_of_forall fun x => (h.lt_or_le x).symm #align set.Iic_union_Ioi_of_le Set.Iic_union_Ioi_of_le theorem Iio_union_Ici_of_le (h : a ≤ b) : Iio b ∪ Ici a = univ := eq_univ_of_forall fun x => (h.le_or_lt x).symm #align set.Iio_union_Ici_of_le Set.Iio_union_Ici_of_le theorem Iic_union_Ici_of_le (h : a ≤ b) : Iic b ∪ Ici a = univ := eq_univ_of_forall fun x => (h.le_or_le x).symm #align set.Iic_union_Ici_of_le Set.Iic_union_Ici_of_le theorem Iio_union_Ioi_of_lt (h : a < b) : Iio b ∪ Ioi a = univ := eq_univ_of_forall fun x => (h.lt_or_lt x).symm #align set.Iio_union_Ioi_of_lt Set.Iio_union_Ioi_of_lt @[simp] theorem Iic_union_Ici : Iic a ∪ Ici a = univ := Iic_union_Ici_of_le le_rfl #align set.Iic_union_Ici Set.Iic_union_Ici @[simp] theorem Iio_union_Ici : Iio a ∪ Ici a = univ := Iio_union_Ici_of_le le_rfl #align set.Iio_union_Ici Set.Iio_union_Ici @[simp] theorem Iic_union_Ioi : Iic a ∪ Ioi a = univ := Iic_union_Ioi_of_le le_rfl #align set.Iic_union_Ioi Set.Iic_union_Ioi @[simp] theorem Iio_union_Ioi : Iio a ∪ Ioi a = {a}ᶜ := ext fun _ => lt_or_lt_iff_ne #align set.Iio_union_Ioi Set.Iio_union_Ioi /-! #### A finite and an infinite interval -/ theorem Ioo_union_Ioi' (h₁ : c < b) : Ioo a b ∪ Ioi c = Ioi (min a c) := by ext1 x simp_rw [mem_union, mem_Ioo, mem_Ioi, min_lt_iff] by_cases hc : c < x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x < b := (le_of_not_gt hc).trans_lt h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ioo_union_Ioi' Set.Ioo_union_Ioi' theorem Ioo_union_Ioi (h : c < max a b) : Ioo a b ∪ Ioi c = Ioi (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ioo_union_Ioi' h · rw [min_comm] simp [, min_eq_left_of_lt] #align set.Ioo_union_Ioi Set.Ioo_union_Ioi theorem Ioi_subset_Ioo_union_Ici : Ioi a ⊆ Ioo a b ∪ Ici b := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ioi_subset_Ioo_union_Ici Set.Ioi_subset_Ioo_union_Ici @[simp] theorem Ioo_union_Ici_eq_Ioi (h : a < b) : Ioo a b ∪ Ici b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_le) Ioi_subset_Ioo_union_Ici #align set.Ioo_union_Ici_eq_Ioi Set.Ioo_union_Ici_eq_Ioi theorem Ici_subset_Ico_union_Ici : Ici a ⊆ Ico a b ∪ Ici b := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ici_subset_Ico_union_Ici Set.Ici_subset_Ico_union_Ici @[simp] theorem Ico_union_Ici_eq_Ici (h : a ≤ b) : Ico a b ∪ Ici b = Ici a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans) Ici_subset_Ico_union_Ici #align set.Ico_union_Ici_eq_Ici Set.Ico_union_Ici_eq_Ici theorem Ico_union_Ici' (h₁ : c ≤ b) : Ico a b ∪ Ici c = Ici (min a c) := by ext1 x simp_rw [mem_union, mem_Ico, mem_Ici, min_le_iff] by_cases hc : c ≤ x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ico_union_Ici' Set.Ico_union_Ici' theorem Ico_union_Ici (h : c ≤ max a b) : Ico a b ∪ Ici c = Ici (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ico_union_Ici' h · simp [] #align set.Ico_union_Ici Set.Ico_union_Ici theorem Ioi_subset_Ioc_union_Ioi : Ioi a ⊆ Ioc a b ∪ Ioi b := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ioi_subset_Ioc_union_Ioi Set.Ioi_subset_Ioc_union_Ioi @[simp] theorem Ioc_union_Ioi_eq_Ioi (h : a ≤ b) : Ioc a b ∪ Ioi b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_lt) Ioi_subset_Ioc_union_Ioi #align set.Ioc_union_Ioi_eq_Ioi Set.Ioc_union_Ioi_eq_Ioi theorem Ioc_union_Ioi' (h₁ : c ≤ b) : Ioc a b ∪ Ioi c = Ioi (min a c) := by ext1 x simp_rw [mem_union, mem_Ioc, mem_Ioi, min_lt_iff] by_cases hc : c < x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ioc_union_Ioi' Set.Ioc_union_Ioi' theorem Ioc_union_Ioi (h : c ≤ max a b) : Ioc a b ∪ Ioi c = Ioi (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ioc_union_Ioi' h · simp [] #align set.Ioc_union_Ioi Set.Ioc_union_Ioi theorem Ici_subset_Icc_union_Ioi : Ici a ⊆ Icc a b ∪ Ioi b := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ici_subset_Icc_union_Ioi Set.Ici_subset_Icc_union_Ioi @[simp] theorem Icc_union_Ioi_eq_Ici (h : a ≤ b) : Icc a b ∪ Ioi b = Ici a := Subset.antisymm (fun _ hx => (hx.elim And.left) fun hx' => h.trans <\| le_of_lt hx') Ici_subset_Icc_union_Ioi #align set.Icc_union_Ioi_eq_Ici Set.Icc_union_Ioi_eq_Ici theorem Ioi_subset_Ioc_union_Ici : Ioi a ⊆ Ioc a b ∪ Ici b := Subset.trans Ioi_subset_Ioo_union_Ici (union_subset_union_left _ Ioo_subset_Ioc_self) #align set.Ioi_subset_Ioc_union_Ici Set.Ioi_subset_Ioc_union_Ici @[simp] theorem Ioc_union_Ici_eq_Ioi (h : a < b) : Ioc a b ∪ Ici b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_le) Ioi_subset_Ioc_union_Ici #align set.Ioc_union_Ici_eq_Ioi Set.Ioc_union_Ici_eq_Ioi theorem Ici_subset_Icc_union_Ici : Ici a ⊆ Icc a b ∪ Ici b := Subset.trans Ici_subset_Ico_union_Ici (union_subset_union_left _ Ico_subset_Icc_self) #align set.Ici_subset_Icc_union_Ici Set.Ici_subset_Icc_union_Ici @[simp] theorem Icc_union_Ici_eq_Ici (h : a ≤ b) : Icc a b ∪ Ici b = Ici a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans) Ici_subset_Icc_union_Ici #align set.Icc_union_Ici_eq_Ici Set.Icc_union_Ici_eq_Ici theorem Icc_union_Ici' (h₁ : c ≤ b) : Icc a b ∪ Ici c = Ici (min a c) := by ext1 x simp_rw [mem_union, mem_Icc, mem_Ici, min_le_iff] by_cases hc : c ≤ x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Icc_union_Ici' Set.Icc_union_Ici' theorem Icc_union_Ici (h : c ≤ max a b) : Icc a b ∪ Ici c = Ici (min a c) := by rcases le_or_lt a b with hab \| hab <;> simp [hab] at h · exact Icc_union_Ici' h · cases' h with h h · simp [] · have hca : c ≤ a := h.trans hab.le simp [] #align set.Icc_union_Ici Set.Icc_union_Ici /-! #### An infinite and a finite interval -/ theorem Iic_subset_Iio_union_Icc : Iic b ⊆ Iio a ∪ Icc a b := fun x hx => (lt_or_le x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iic_subset_Iio_union_Icc Set.Iic_subset_Iio_union_Icc @[simp] theorem Iio_union_Icc_eq_Iic (h : a ≤ b) : Iio a ∪ Icc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx => (le_of_lt hx).trans h) And.right) Iic_subset_Iio_union_Icc #align set.Iio_union_Icc_eq_Iic Set.Iio_union_Icc_eq_Iic theorem Iio_subset_Iio_union_Ico : Iio b ⊆ Iio a ∪ Ico a b := fun x hx => (lt_or_le x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iio_subset_Iio_union_Ico Set.Iio_subset_Iio_union_Ico @[simp] theorem Iio_union_Ico_eq_Iio (h : a ≤ b) : Iio a ∪ Ico a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_lt_of_le hx' h) And.right) Iio_subset_Iio_union_Ico #align set.Iio_union_Ico_eq_Iio Set.Iio_union_Ico_eq_Iio theorem Iio_union_Ico' (h₁ : c ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by ext1 x simp_rw [mem_union, mem_Iio, mem_Ico, lt_max_iff] by_cases hc : c ≤ x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iio_union_Ico' Set.Iio_union_Ico' theorem Iio_union_Ico (h : min c d ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iio_union_Ico' h · simp [] #align set.Iio_union_Ico Set.Iio_union_Ico theorem Iic_subset_Iic_union_Ioc : Iic b ⊆ Iic a ∪ Ioc a b := fun x hx => (le_or_lt x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iic_subset_Iic_union_Ioc Set.Iic_subset_Iic_union_Ioc @[simp] theorem Iic_union_Ioc_eq_Iic (h : a ≤ b) : Iic a ∪ Ioc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => le_trans hx' h) And.right) Iic_subset_Iic_union_Ioc #align set.Iic_union_Ioc_eq_Iic Set.Iic_union_Ioc_eq_Iic theorem Iic_union_Ioc' (h₁ : c < b) : Iic b ∪ Ioc c d = Iic (max b d) := by ext1 x simp_rw [mem_union, mem_Iic, mem_Ioc, le_max_iff] by_cases hc : c < x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁.le simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iic_union_Ioc' Set.Iic_union_Ioc' theorem Iic_union_Ioc (h : min c d < b) : Iic b ∪ Ioc c d = Iic (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iic_union_Ioc' h · rw [max_comm] simp [, max_eq_right_of_lt h] #align set.Iic_union_Ioc Set.Iic_union_Ioc theorem Iio_subset_Iic_union_Ioo : Iio b ⊆ Iic a ∪ Ioo a b := fun x hx => (le_or_lt x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iio_subset_Iic_union_Ioo Set.Iio_subset_Iic_union_Ioo @[simp] theorem Iic_union_Ioo_eq_Iio (h : a < b) : Iic a ∪ Ioo a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_le_of_lt hx' h) And.right) Iio_subset_Iic_union_Ioo #align set.Iic_union_Ioo_eq_Iio Set.Iic_union_Ioo_eq_Iio theorem Iio_union_Ioo' (h₁ : c < b) : Iio b ∪ Ioo c d = Iio (max b d) := by ext x cases' lt_or_le x b with hba hba · simp [hba, h₁] · simp only [mem_Iio, mem_union, mem_Ioo, lt_max_iff] refine or_congr Iff.rfl ⟨And.right, ?_⟩ exact fun h₂ => ⟨h₁.trans_le hba, h₂⟩ #align set.Iio_union_Ioo' Set.Iio_union_Ioo' theorem Iio_union_Ioo (h : min c d < b) : Iio b ∪ Ioo c d = Iio (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iio_union_Ioo' h · rw [max_comm] simp [, max_eq_right_of_lt h] #align set.Iio_union_Ioo Set.Iio_union_Ioo theorem Iic_subset_Iic_union_Icc : Iic b ⊆ Iic a ∪ Icc a b := Subset.trans Iic_subset_Iic_union_Ioc (union_subset_union_right _ Ioc_subset_Icc_self) #align set.Iic_subset_Iic_union_Icc Set.Iic_subset_Iic_union_Icc @[simp] theorem Iic_union_Icc_eq_Iic (h : a ≤ b) : Iic a ∪ Icc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => le_trans hx' h) And.right) Iic_subset_Iic_union_Icc #align set.Iic_union_Icc_eq_Iic Set.Iic_union_Icc_eq_Iic theorem Iic_union_Icc' (h₁ : c ≤ b) : Iic b ∪ Icc c d = Iic (max b d) := by ext1 x simp_rw [mem_union, mem_Iic, mem_Icc, le_max_iff] by_cases hc : c ≤ x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iic_union_Icc' Set.Iic_union_Icc' theorem Iic_union_Icc (h : min c d ≤ b) : Iic b ∪ Icc c d = Iic (max b d) := by rcases le_or_lt c d with hcd \| hcd <;> simp [hcd] at h · exact Iic_union_Icc' h · cases' h with h h · have hdb : d ≤ b := hcd.le.trans h simp [] · simp [] #align set.Iic_union_Icc Set.Iic_union_Icc theorem Iio_subset_Iic_union_Ico : Iio b ⊆ Iic a ∪ Ico a b := Subset.trans Iio_subset_Iic_union_Ioo (union_subset_union_right _ Ioo_subset_Ico_self) #align set.Iio_subset_Iic_union_Ico Set.Iio_subset_Iic_union_Ico @[simp] theorem Iic_union_Ico_eq_Iio (h : a < b) : Iic a ∪ Ico a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_le_of_lt hx' h) And.right) Iio_subset_Iic_union_Ico #align set.Iic_union_Ico_eq_Iio Set.Iic_union_Ico_eq_Iio /-! #### Two finite intervals, `I?o` and `Ic?` -/ theorem Ioo_subset_Ioo_union_Ico : Ioo a c ⊆ Ioo a b ∪ Ico b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioo_subset_Ioo_union_Ico Set.Ioo_subset_Ioo_union_Ico @[simp] theorem Ioo_union_Ico_eq_Ioo (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Ico b c = Ioo a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_le h₂⟩) fun hx => ⟨h₁.trans_le hx.1, hx.2⟩) Ioo_subset_Ioo_union_Ico #align set.Ioo_union_Ico_eq_Ioo Set.Ioo_union_Ico_eq_Ioo theorem Ico_subset_Ico_union_Ico : Ico a c ⊆ Ico a b ∪ Ico b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ico_subset_Ico_union_Ico Set.Ico_subset_Ico_union_Ico @[simp] theorem Ico_union_Ico_eq_Ico (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Ico b c = Ico a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_le h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Ico_subset_Ico_union_Ico #align set.Ico_union_Ico_eq_Ico Set.Ico_union_Ico_eq_Ico theorem Ico_union_Ico' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ico a b ∪ Ico c d = Ico (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Ico, min_le_iff, lt_max_iff] by_cases hc : c ≤ x <;> by_cases hd : x < d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a ≤ x := h₂.trans (le_of_not_gt hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Ico_union_Ico' Set.Ico_union_Ico' theorem Ico_union_Ico (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b) : Ico a b ∪ Ico c d = Ico (min a c) (max b d) := by rcases le_total a b with hab \| hab <;> rcases le_total c d with hcd \| hcd <;> simp [] at h₁ h₂ · exact Ico_union_Ico' h₂ h₁ all_goals simp [] #align set.Ico_union_Ico Set.Ico_union_Ico theorem Icc_subset_Ico_union_Icc : Icc a c ⊆ Ico a b ∪ Icc b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Icc_subset_Ico_union_Icc Set.Icc_subset_Ico_union_Icc @[simp] theorem Ico_union_Icc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Icc b c = Icc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.le.trans h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Icc_subset_Ico_union_Icc #align set.Ico_union_Icc_eq_Icc Set.Ico_union_Icc_eq_Icc theorem Ioc_subset_Ioo_union_Icc : Ioc a c ⊆ Ioo a b ∪ Icc b c := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioc_subset_Ioo_union_Icc Set.Ioc_subset_Ioo_union_Icc @[simp] theorem Ioo_union_Icc_eq_Ioc (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Icc b c = Ioc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.le.trans h₂⟩) fun hx => ⟨h₁.trans_le hx.1, hx.2⟩) Ioc_subset_Ioo_union_Icc #align set.Ioo_union_Icc_eq_Ioc Set.Ioo_union_Icc_eq_Ioc /-! #### Two finite intervals, `I?c` and `Io?` -/ theorem Ioo_subset_Ioc_union_Ioo : Ioo a c ⊆ Ioc a b ∪ Ioo b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioo_subset_Ioc_union_Ioo Set.Ioo_subset_Ioc_union_Ioo @[simp] theorem Ioc_union_Ioo_eq_Ioo (h₁ : a ≤ b) (h₂ : b < c) : Ioc a b ∪ Ioo b c = Ioo a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans_lt hx.1, hx.2⟩) Ioo_subset_Ioc_union_Ioo #align set.Ioc_union_Ioo_eq_Ioo Set.Ioc_union_Ioo_eq_Ioo theorem Ico_subset_Icc_union_Ioo : Ico a c ⊆ Icc a b ∪ Ioo b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ico_subset_Icc_union_Ioo Set.Ico_subset_Icc_union_Ioo @[simp] theorem Icc_union_Ioo_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ioo b c = Ico a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans hx.1.le, hx.2⟩) Ico_subset_Icc_union_Ioo #align set.Icc_union_Ioo_eq_Ico Set.Icc_union_Ioo_eq_Ico theorem Icc_subset_Icc_union_Ioc : Icc a c ⊆ Icc a b ∪ Ioc b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Icc_subset_Icc_union_Ioc Set.Icc_subset_Icc_union_Ioc @[simp] theorem Icc_union_Ioc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Ioc b c = Icc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans hx.1.le, hx.2⟩) Icc_subset_Icc_union_Ioc #align set.Icc_union_Ioc_eq_Icc Set.Icc_union_Ioc_eq_Icc theorem Ioc_subset_Ioc_union_Ioc : Ioc a c ⊆ Ioc a b ∪ Ioc b c := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx.1, hxb⟩) fun hxb => Or.inr ⟨hxb, hx.2⟩ #align set.Ioc_subset_Ioc_union_Ioc Set.Ioc_subset_Ioc_union_Ioc @[simp] theorem Ioc_union_Ioc_eq_Ioc (h₁ : a ≤ b) (h₂ : b ≤ c) : Ioc a b ∪ Ioc b c = Ioc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans_lt hx.1, hx.2⟩) Ioc_subset_Ioc_union_Ioc #align set.Ioc_union_Ioc_eq_Ioc Set.Ioc_union_Ioc_eq_Ioc theorem Ioc_union_Ioc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Ioc, min_lt_iff, le_max_iff] by_cases hc : c < x <;> by_cases hd : x ≤ d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a < x := h₂.trans_lt (lt_of_not_ge hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Ioc_union_Ioc' Set.Ioc_union_Ioc' theorem Ioc_union_Ioc (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b) : Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) := by rcases le_total a b with hab \| hab <;> rcases le_total c d with hcd \| hcd <;> simp [] at h₁ h₂ · exact Ioc_union_Ioc' h₂ h₁ all_goals simp [] #align set.Ioc_union_Ioc Set.Ioc_union_Ioc /-! #### Two finite intervals with a common point -/ theorem Ioo_subset_Ioc_union_Ico : Ioo a c ⊆ Ioc a b ∪ Ico b c := Subset.trans Ioo_subset_Ioc_union_Ioo (union_subset_union_right _ Ioo_subset_Ico_self) #align set.Ioo_subset_Ioc_union_Ico Set.Ioo_subset_Ioc_union_Ico @[simp] theorem Ioc_union_Ico_eq_Ioo (h₁ : a < b) (h₂ : b < c) : Ioc a b ∪ Ico b c = Ioo a c := Subset.antisymm (fun _ hx => hx.elim (fun hx' => ⟨hx'.1, hx'.2.trans_lt h₂⟩) fun hx' => ⟨h₁.trans_le hx'.1, hx'.2⟩) Ioo_subset_Ioc_union_Ico #align set.Ioc_union_Ico_eq_Ioo Set.Ioc_union_Ico_eq_Ioo theorem Ico_subset_Icc_union_Ico : Ico a c ⊆ Icc a b ∪ Ico b c := Subset.trans Ico_subset_Icc_union_Ioo (union_subset_union_right _ Ioo_subset_Ico_self) #align set.Ico_subset_Icc_union_Ico Set.Ico_subset_Icc_union_Ico @[simp] theorem Icc_union_Ico_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ico b c = Ico a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans_lt h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Ico_subset_Icc_union_Ico #align set.Icc_union_Ico_eq_Ico Set.Icc_union_Ico_eq_Ico theorem Icc_subset_Icc_union_Icc : Icc a c ⊆ Icc a b ∪ Icc b c := Subset.trans Icc_subset_Icc_union_Ioc (union_subset_union_right _ Ioc_subset_Icc_self) #align set.Icc_subset_Icc_union_Icc Set.Icc_subset_Icc_union_Icc @[simp] theorem Icc_union_Icc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Icc b c = Icc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans hx.1, hx.2⟩) Icc_subset_Icc_union_Icc #align set.Icc_union_Icc_eq_Icc Set.Icc_union_Icc_eq_Icc theorem Icc_union_Icc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Icc a b ∪ Icc c d = Icc (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Icc, min_le_iff, le_max_iff] by_cases hc : c ≤ x <;> by_cases hd : x ≤ d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a ≤ x := h₂.trans (le_of_not_ge hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Icc_union_Icc' Set.Icc_union_Icc' /-- We cannot replace `<` by `≤` in the hypotheses. Otherwise for `b < a = d < c` the l.h.s. is `∅` and the r.h.s. is `{a}`. -/ theorem Icc_union_Icc (h₁ : min a b < max c d) (h₂ : min c d < max a b) : Icc a b ∪ Icc c d = Icc (min a c) (max b d) := by rcases le_or_lt a b with hab \| hab <;> rcases le_or_lt c d with hcd \| hcd <;> simp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, hab, hcd] at h₁ h₂ · exact Icc_union_Icc' h₂.le h₁.le all_goals simp [, min_eq_left_of_lt, max_eq_left_of_lt, min_eq_right_of_lt, max_eq_right_of_lt] #align set.Icc_union_Icc Set.Icc_union_Icc theorem Ioc_subset_Ioc_union_Icc : Ioc a c ⊆ Ioc a b ∪ Icc b c := Subset.trans Ioc_subset_Ioc_union_Ioc (union_subset_union_right _ Ioc_subset_Icc_self) #align set.Ioc_subset_Ioc_union_Icc Set.Ioc_subset_Ioc_union_Icc @[simp] theorem Ioc_union_Icc_eq_Ioc (h₁ : a < b) (h₂ : b ≤ c) : Ioc a b ∪ Icc b c = Ioc a c := Subset.antisymm (fun _ hx => hx.elim (fun hx => ⟨hx.1, hx.2.trans h₂⟩) fun hx => ⟨h₁.trans_le hx.1, hx.2⟩) Ioc_subset_Ioc_union_Icc #align set.Ioc_union_Icc_eq_Ioc Set.Ioc_union_Icc_eq_Ioc theorem Ioo_union_Ioo' (h₁ : c < b) (h₂ : a < d) : Ioo a b ∪ Ioo c d = Ioo (min a c) (max b d) := by ext1 x simp_rw [mem_union, mem_Ioo, min_lt_iff, lt_max_iff] by_cases hc : c < x <;> by_cases hd : x < d · simp only [hc, hd, and_self, or_true] -- Porting note: restore `tauto` · have hax : a < x := h₂.trans_le (le_of_not_lt hd) simp only [hax, true_and, hc, or_self] -- Porting note: restore `tauto` · have hxb : x < b := (le_of_not_lt hc).trans_lt h₁ simp only [hxb, and_true, hc, false_and, or_false, true_or] -- Porting note: restore `tauto` · simp only [hc, hd, and_self, or_false] -- Porting note: restore `tauto` #align set.Ioo_union_Ioo' Set.Ioo_union_Ioo' theorem Ioo_union_Ioo (h₁ : min a b < max c d) (h₂ : min c d < max a b) : Ioo a b ∪ Ioo c d = Ioo (min a c) (max b d) := by rcases le_total a b with hab \| hab <;> rcases le_total c d with hcd \| hcd <;> simp only [min_eq_left, min_eq_right, max_eq_left, max_eq_right, hab, hcd] at h₁ h₂ · exact Ioo_union_Ioo' h₂ h₁ all_goals simp [*, min_eq_left_of_lt, min_eq_right_of_lt, max_eq_left_of_lt, max_eq_right_of_lt, le_of_lt h₂, le_of_lt h₁] #align set.Ioo_union_Ioo Set.Ioo_union_Ioo end LinearOrder section Lattice section Inf variable [SemilatticeInf α] @[simp] theorem Iic_inter_Iic {a b : α} : Iic a ∩ Iic b = Iic (a ⊓ b) := by ext x simp [Iic] #align set.Iic_inter_Iic Set.Iic_inter_Iic @[simp] theorem Ioc_inter_Iic (a b c : α) : Ioc a b ∩ Iic c = Ioc a (b ⊓ c) := by rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, inter_assoc, Iic_inter_Iic] #align set.Ioc_inter_Iic Set.Ioc_inter_Iic end Inf section Sup variable [SemilatticeSup α] @[simp] theorem Ici_inter_Ici {a b : α} : Ici a ∩ Ici b = Ici (a ⊔ b) := by ext x simp [Ici] #align set.Ici_inter_Ici Set.Ici_inter_Ici @[simp] theorem Ico_inter_Ici (a b c : α) : Ico a b ∩ Ici c = Ico (a ⊔ c) b := by rw [← Ici_inter_Iio, ← Ici_inter_Iio, ← Ici_inter_Ici, inter_right_comm] #align set.Ico_inter_Ici Set.Ico_inter_Ici end Sup section Both variable [Lattice α] {a b c a₁ a₂ b₁ b₂ : α} theorem Icc_inter_Icc : Icc a₁ b₁ ∩ Icc a₂ b₂ = Icc (a₁ ⊔ a₂) (b₁ ⊓ b₂) := by simp only [Ici_inter_Iic.symm, Ici_inter_Ici.symm, Iic_inter_Iic.symm]; ac_rfl #align set.Icc_inter_Icc Set.Icc_inter_Icc @[simp] theorem Icc_inter_Icc_eq_singleton (hab : a ≤ b) (hbc : b ≤ c) : Icc a b ∩ Icc b c = {b} := by rw [Icc_inter_Icc, sup_of_le_right hab, inf_of_le_left hbc, Icc_self] #align set.Icc_inter_Icc_eq_singleton Set.Icc_inter_Icc_eq_singleton end Both end Lattice section LinearOrder variable [LinearOrder α] [LinearOrder β] {f : α → β} {a a₁ a₂ b b₁ b₂ c d : α} @[simp] theorem Ioi_inter_Ioi : Ioi a ∩ Ioi b = Ioi (a ⊔ b) := ext fun _ => sup_lt_iff.symm #align set.Ioi_inter_Ioi Set.Ioi_inter_Ioi @[simp] theorem Iio_inter_Iio : Iio a ∩ Iio b = Iio (a ⊓ b) := ext fun _ => lt_inf_iff.symm #align set.Iio_inter_Iio Set.Iio_inter_Iio theorem Ico_inter_Ico : Ico a₁ b₁ ∩ Ico a₂ b₂ = Ico (a₁ ⊔ a₂) (b₁ ⊓ b₂) := by simp only [Ici_inter_Iio.symm, Ici_inter_Ici.symm, Iio_inter_Iio.symm]; ac_rfl #align set.Ico_inter_Ico Set.Ico_inter_Ico theorem Ioc_inter_Ioc : Ioc a₁ b₁ ∩ Ioc a₂ b₂ = Ioc (a₁ ⊔ a₂) (b₁ ⊓ b₂) := by simp only [Ioi_inter_Iic.symm, Ioi_inter_Ioi.symm, Iic_inter_Iic.symm]; ac_rfl #align set.Ioc_inter_Ioc Set.Ioc_inter_Ioc theorem Ioo_inter_Ioo : Ioo a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (a₁ ⊔ a₂) (b₁ ⊓ b₂) := by simp only [Ioi_inter_Iio.symm, Ioi_inter_Ioi.symm, Iio_inter_Iio.symm]; ac_rfl #align set.Ioo_inter_Ioo Set.Ioo_inter_Ioo theorem Ioo_inter_Iio : Ioo a b ∩ Iio c = Ioo a (min b c) := by ext simp_rw [mem_inter_iff, mem_Ioo, mem_Iio, lt_min_iff, and_assoc] theorem Iio_inter_Ioo : Iio a ∩ Ioo b c = Ioo b (min a c) := by rw [Set.inter_comm, Set.Ioo_inter_Iio, min_comm] theorem Ioo_inter_Ioi : Ioo a b ∩ Ioi c = Ioo (max a c) b := by ext simp_rw [mem_inter_iff, mem_Ioo, mem_Ioi, max_lt_iff, and_assoc, and_comm] theorem Ioi_inter_Ioo : Set.Ioi a ∩ Set.Ioo b c = Set.Ioo (max a b) c := by rw [inter_comm, Ioo_inter_Ioi, max_comm] theorem Ioc_inter_Ioo_of_left_lt (h : b₁ < b₂) : Ioc a₁ b₁ ∩ Ioo a₂ b₂ = Ioc (max a₁ a₂) b₁ := ext fun x => by simp [and_assoc, @and_left_comm (x ≤ _), and_iff_left_iff_imp.2 fun h' => lt_of_le_of_lt h' h] #align set.Ioc_inter_Ioo_of_left_lt Set.Ioc_inter_Ioo_of_left_lt theorem Ioc_inter_Ioo_of_right_le (h : b₂ ≤ b₁) : Ioc a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (max a₁ a₂) b₂ := ext fun x => by simp [and_assoc, @and_left_comm (x ≤ _), and_iff_right_iff_imp.2 fun h' => (le_of_lt h').trans h] #align set.Ioc_inter_Ioo_of_right_le Set.Ioc_inter_Ioo_of_right_le theorem Ioo_inter_Ioc_of_left_le (h : b₁ ≤ b₂) : Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioo (max a₁ a₂) b₁ := by rw [inter_comm, Ioc_inter_Ioo_of_right_le h, max_comm] #align set.Ioo_inter_Ioc_of_left_le Set.Ioo_inter_Ioc_of_left_le theorem Ioo_inter_Ioc_of_right_lt (h : b₂ < b₁) : Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioc (max a₁ a₂) b₂ := by rw [inter_comm, Ioc_inter_Ioo_of_left_lt h, max_comm] #align set.Ioo_inter_Ioc_of_right_lt Set.Ioo_inter_Ioc_of_right_lt @[simp] theorem Ico_diff_Iio : Ico a b \ Iio c = Ico (max a c) b := by rw [diff_eq, compl_Iio, Ico_inter_Ici, sup_eq_max] #align set.Ico_diff_Iio Set.Ico_diff_Iio @[simp] theorem Ioc_diff_Ioi : Ioc a b \ Ioi c = Ioc a (min b c) := ext <\| by simp (config := { contextual := true }) [iff_def] #align set.Ioc_diff_Ioi Set.Ioc_diff_Ioi @[simp] theorem Ioc_inter_Ioi : Ioc a b ∩ Ioi c = Ioc (a ⊔ c) b := by rw [← Ioi_inter_Iic, inter_assoc, inter_comm, inter_assoc, Ioi_inter_Ioi, inter_comm, Ioi_inter_Iic, sup_comm] #align set.Ioc_inter_Ioi Set.Ioc_inter_Ioi @[simp] theorem Ico_inter_Iio : Ico a b ∩ Iio c = Ico a (min b c) := ext <\| by simp (config := { contextual := true }) [iff_def] #align set.Ico_inter_Iio Set.Ico_inter_Iio @[simp] theorem Ioc_diff_Iic : Ioc a b \ Iic c = Ioc (max a c) b := by rw [diff_eq, compl_Iic, Ioc_inter_Ioi, sup_eq_max] #align set.Ioc_diff_Iic Set.Ioc_diff_Iic @[simp] theorem Ioc_union_Ioc_right : Ioc a b ∪ Ioc a c = Ioc a (max b c) := by rw [Ioc_union_Ioc, min_self] <;> exact (min_le_left _ _).trans (le_max_left _ _) #align set.Ioc_union_Ioc_right Set.Ioc_union_Ioc_right @[simp]	Mathlib/Order/Interval/Set/Basic.lean	1,903	1,904	theorem Ioc_union_Ioc_left : Ioc a c ∪ Ioc b c = Ioc (min a b) c := by	rw [Ioc_union_Ioc, max_self] <;> exact (min_le_right _ _).trans (le_max_right _ _)
/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou -/ import Mathlib.MeasureTheory.Function.LpOrder #align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f" /-! # Integrable functions and `L¹` space In the first part of this file, the predicate `Integrable` is defined and basic properties of integrable functions are proved. Such a predicate is already available under the name `Memℒp 1`. We give a direct definition which is easier to use, and show that it is equivalent to `Memℒp 1` In the second part, we establish an API between `Integrable` and the space `L¹` of equivalence classes of integrable functions, already defined as a special case of `L^p` spaces for `p = 1`. ## Notation * `α →₁[μ] β` is the type of `L¹` space, where `α` is a `MeasureSpace` and `β` is a `NormedAddCommGroup` with a `SecondCountableTopology`. `f : α →ₘ β` is a "function" in `L¹`. In comments, `[f]` is also used to denote an `L¹` function. `₁` can be typed as `\1`. ## Main definitions * Let `f : α → β` be a function, where `α` is a `MeasureSpace` and `β` a `NormedAddCommGroup`. Then `HasFiniteIntegral f` means `(∫⁻ a, ‖f a‖₊) < ∞`. * If `β` is moreover a `MeasurableSpace` then `f` is called `Integrable` if `f` is `Measurable` and `HasFiniteIntegral f` holds. ## Implementation notes To prove something for an arbitrary integrable function, a useful theorem is `Integrable.induction` in the file `SetIntegral`. ## Tags integrable, function space, l1 -/ noncomputable section open scoped Classical open Topology ENNReal MeasureTheory NNReal open Set Filter TopologicalSpace ENNReal EMetric MeasureTheory variable {α β γ δ : Type} {m : MeasurableSpace α} {μ ν : Measure α} [MeasurableSpace δ] variable [NormedAddCommGroup β] variable [NormedAddCommGroup γ] namespace MeasureTheory /-! ### Some results about the Lebesgue integral involving a normed group -/ theorem lintegral_nnnorm_eq_lintegral_edist (f : α → β) : ∫⁻ a, ‖f a‖₊ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by simp only [edist_eq_coe_nnnorm] #align measure_theory.lintegral_nnnorm_eq_lintegral_edist MeasureTheory.lintegral_nnnorm_eq_lintegral_edist theorem lintegral_norm_eq_lintegral_edist (f : α → β) : ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by simp only [ofReal_norm_eq_coe_nnnorm, edist_eq_coe_nnnorm] #align measure_theory.lintegral_norm_eq_lintegral_edist MeasureTheory.lintegral_norm_eq_lintegral_edist theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ) (hh : AEStronglyMeasurable h μ) : (∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ := by rw [← lintegral_add_left' (hf.edist hh)] refine lintegral_mono fun a => ?_ apply edist_triangle_right #align measure_theory.lintegral_edist_triangle MeasureTheory.lintegral_edist_triangle theorem lintegral_nnnorm_zero : (∫⁻ _ : α, ‖(0 : β)‖₊ ∂μ) = 0 := by simp #align measure_theory.lintegral_nnnorm_zero MeasureTheory.lintegral_nnnorm_zero theorem lintegral_nnnorm_add_left {f : α → β} (hf : AEStronglyMeasurable f μ) (g : α → γ) : ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂μ := lintegral_add_left' hf.ennnorm _ #align measure_theory.lintegral_nnnorm_add_left MeasureTheory.lintegral_nnnorm_add_left theorem lintegral_nnnorm_add_right (f : α → β) {g : α → γ} (hg : AEStronglyMeasurable g μ) : ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂μ := lintegral_add_right' _ hg.ennnorm #align measure_theory.lintegral_nnnorm_add_right MeasureTheory.lintegral_nnnorm_add_right theorem lintegral_nnnorm_neg {f : α → β} : (∫⁻ a, ‖(-f) a‖₊ ∂μ) = ∫⁻ a, ‖f a‖₊ ∂μ := by simp only [Pi.neg_apply, nnnorm_neg] #align measure_theory.lintegral_nnnorm_neg MeasureTheory.lintegral_nnnorm_neg /-! ### The predicate `HasFiniteIntegral` -/ /-- `HasFiniteIntegral f μ` means that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite. `HasFiniteIntegral f` means `HasFiniteIntegral f volume`. -/ def HasFiniteIntegral {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop := (∫⁻ a, ‖f a‖₊ ∂μ) < ∞ #align measure_theory.has_finite_integral MeasureTheory.HasFiniteIntegral theorem hasFiniteIntegral_def {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : HasFiniteIntegral f μ ↔ ((∫⁻ a, ‖f a‖₊ ∂μ) < ∞) := Iff.rfl theorem hasFiniteIntegral_iff_norm (f : α → β) : HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) < ∞ := by simp only [HasFiniteIntegral, ofReal_norm_eq_coe_nnnorm] #align measure_theory.has_finite_integral_iff_norm MeasureTheory.hasFiniteIntegral_iff_norm theorem hasFiniteIntegral_iff_edist (f : α → β) : HasFiniteIntegral f μ ↔ (∫⁻ a, edist (f a) 0 ∂μ) < ∞ := by simp only [hasFiniteIntegral_iff_norm, edist_dist, dist_zero_right] #align measure_theory.has_finite_integral_iff_edist MeasureTheory.hasFiniteIntegral_iff_edist theorem hasFiniteIntegral_iff_ofReal {f : α → ℝ} (h : 0 ≤ᵐ[μ] f) : HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal (f a) ∂μ) < ∞ := by rw [HasFiniteIntegral, lintegral_nnnorm_eq_of_ae_nonneg h] #align measure_theory.has_finite_integral_iff_of_real MeasureTheory.hasFiniteIntegral_iff_ofReal theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} : HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by simp [hasFiniteIntegral_iff_norm] #align measure_theory.has_finite_integral_iff_of_nnreal MeasureTheory.hasFiniteIntegral_iff_ofNNReal theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral f μ := by simp only [hasFiniteIntegral_iff_norm] at calc (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ := lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h) _ < ∞ := hg #align measure_theory.has_finite_integral.mono MeasureTheory.HasFiniteIntegral.mono theorem HasFiniteIntegral.mono' {f : α → β} {g : α → ℝ} (hg : HasFiniteIntegral g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : HasFiniteIntegral f μ := hg.mono <\| h.mono fun _x hx => le_trans hx (le_abs_self _) #align measure_theory.has_finite_integral.mono' MeasureTheory.HasFiniteIntegral.mono' theorem HasFiniteIntegral.congr' {f : α → β} {g : α → γ} (hf : HasFiniteIntegral f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral g μ := hf.mono <\| EventuallyEq.le <\| EventuallyEq.symm h #align measure_theory.has_finite_integral.congr' MeasureTheory.HasFiniteIntegral.congr' theorem hasFiniteIntegral_congr' {f : α → β} {g : α → γ} (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ := ⟨fun hf => hf.congr' h, fun hg => hg.congr' <\| EventuallyEq.symm h⟩ #align measure_theory.has_finite_integral_congr' MeasureTheory.hasFiniteIntegral_congr' theorem HasFiniteIntegral.congr {f g : α → β} (hf : HasFiniteIntegral f μ) (h : f =ᵐ[μ] g) : HasFiniteIntegral g μ := hf.congr' <\| h.fun_comp norm #align measure_theory.has_finite_integral.congr MeasureTheory.HasFiniteIntegral.congr theorem hasFiniteIntegral_congr {f g : α → β} (h : f =ᵐ[μ] g) : HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ := hasFiniteIntegral_congr' <\| h.fun_comp norm #align measure_theory.has_finite_integral_congr MeasureTheory.hasFiniteIntegral_congr theorem hasFiniteIntegral_const_iff {c : β} : HasFiniteIntegral (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ := by simp [HasFiniteIntegral, lintegral_const, lt_top_iff_ne_top, ENNReal.mul_eq_top, or_iff_not_imp_left] #align measure_theory.has_finite_integral_const_iff MeasureTheory.hasFiniteIntegral_const_iff theorem hasFiniteIntegral_const [IsFiniteMeasure μ] (c : β) : HasFiniteIntegral (fun _ : α => c) μ := hasFiniteIntegral_const_iff.2 (Or.inr <\| measure_lt_top _ _) #align measure_theory.has_finite_integral_const MeasureTheory.hasFiniteIntegral_const theorem hasFiniteIntegral_of_bounded [IsFiniteMeasure μ] {f : α → β} {C : ℝ} (hC : ∀ᵐ a ∂μ, ‖f a‖ ≤ C) : HasFiniteIntegral f μ := (hasFiniteIntegral_const C).mono' hC #align measure_theory.has_finite_integral_of_bounded MeasureTheory.hasFiniteIntegral_of_bounded theorem HasFiniteIntegral.of_finite [Finite α] [IsFiniteMeasure μ] {f : α → β} : HasFiniteIntegral f μ := let ⟨_⟩ := nonempty_fintype α hasFiniteIntegral_of_bounded <\| ae_of_all μ <\| norm_le_pi_norm f @[deprecated (since := "2024-02-05")] alias hasFiniteIntegral_of_fintype := HasFiniteIntegral.of_finite theorem HasFiniteIntegral.mono_measure {f : α → β} (h : HasFiniteIntegral f ν) (hμ : μ ≤ ν) : HasFiniteIntegral f μ := lt_of_le_of_lt (lintegral_mono' hμ le_rfl) h #align measure_theory.has_finite_integral.mono_measure MeasureTheory.HasFiniteIntegral.mono_measure theorem HasFiniteIntegral.add_measure {f : α → β} (hμ : HasFiniteIntegral f μ) (hν : HasFiniteIntegral f ν) : HasFiniteIntegral f (μ + ν) := by simp only [HasFiniteIntegral, lintegral_add_measure] at * exact add_lt_top.2 ⟨hμ, hν⟩ #align measure_theory.has_finite_integral.add_measure MeasureTheory.HasFiniteIntegral.add_measure theorem HasFiniteIntegral.left_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) : HasFiniteIntegral f μ := h.mono_measure <\| Measure.le_add_right <\| le_rfl #align measure_theory.has_finite_integral.left_of_add_measure MeasureTheory.HasFiniteIntegral.left_of_add_measure theorem HasFiniteIntegral.right_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) : HasFiniteIntegral f ν := h.mono_measure <\| Measure.le_add_left <\| le_rfl #align measure_theory.has_finite_integral.right_of_add_measure MeasureTheory.HasFiniteIntegral.right_of_add_measure @[simp] theorem hasFiniteIntegral_add_measure {f : α → β} : HasFiniteIntegral f (μ + ν) ↔ HasFiniteIntegral f μ ∧ HasFiniteIntegral f ν := ⟨fun h => ⟨h.left_of_add_measure, h.right_of_add_measure⟩, fun h => h.1.add_measure h.2⟩ #align measure_theory.has_finite_integral_add_measure MeasureTheory.hasFiniteIntegral_add_measure theorem HasFiniteIntegral.smul_measure {f : α → β} (h : HasFiniteIntegral f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) : HasFiniteIntegral f (c • μ) := by simp only [HasFiniteIntegral, lintegral_smul_measure] at * exact mul_lt_top hc h.ne #align measure_theory.has_finite_integral.smul_measure MeasureTheory.HasFiniteIntegral.smul_measure @[simp] theorem hasFiniteIntegral_zero_measure {m : MeasurableSpace α} (f : α → β) : HasFiniteIntegral f (0 : Measure α) := by simp only [HasFiniteIntegral, lintegral_zero_measure, zero_lt_top] #align measure_theory.has_finite_integral_zero_measure MeasureTheory.hasFiniteIntegral_zero_measure variable (α β μ) @[simp] theorem hasFiniteIntegral_zero : HasFiniteIntegral (fun _ : α => (0 : β)) μ := by simp [HasFiniteIntegral] #align measure_theory.has_finite_integral_zero MeasureTheory.hasFiniteIntegral_zero variable {α β μ} theorem HasFiniteIntegral.neg {f : α → β} (hfi : HasFiniteIntegral f μ) : HasFiniteIntegral (-f) μ := by simpa [HasFiniteIntegral] using hfi #align measure_theory.has_finite_integral.neg MeasureTheory.HasFiniteIntegral.neg @[simp] theorem hasFiniteIntegral_neg_iff {f : α → β} : HasFiniteIntegral (-f) μ ↔ HasFiniteIntegral f μ := ⟨fun h => neg_neg f ▸ h.neg, HasFiniteIntegral.neg⟩ #align measure_theory.has_finite_integral_neg_iff MeasureTheory.hasFiniteIntegral_neg_iff theorem HasFiniteIntegral.norm {f : α → β} (hfi : HasFiniteIntegral f μ) : HasFiniteIntegral (fun a => ‖f a‖) μ := by have eq : (fun a => (nnnorm ‖f a‖ : ℝ≥0∞)) = fun a => (‖f a‖₊ : ℝ≥0∞) := by funext rw [nnnorm_norm] rwa [HasFiniteIntegral, eq] #align measure_theory.has_finite_integral.norm MeasureTheory.HasFiniteIntegral.norm theorem hasFiniteIntegral_norm_iff (f : α → β) : HasFiniteIntegral (fun a => ‖f a‖) μ ↔ HasFiniteIntegral f μ := hasFiniteIntegral_congr' <\| eventually_of_forall fun x => norm_norm (f x) #align measure_theory.has_finite_integral_norm_iff MeasureTheory.hasFiniteIntegral_norm_iff theorem hasFiniteIntegral_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hf : (∫⁻ x, f x ∂μ) ≠ ∞) : HasFiniteIntegral (fun x => (f x).toReal) μ := by have : ∀ x, (‖(f x).toReal‖₊ : ℝ≥0∞) = ENNReal.ofNNReal ⟨(f x).toReal, ENNReal.toReal_nonneg⟩ := by intro x rw [Real.nnnorm_of_nonneg] simp_rw [HasFiniteIntegral, this] refine lt_of_le_of_lt (lintegral_mono fun x => ?_) (lt_top_iff_ne_top.2 hf) by_cases hfx : f x = ∞ · simp [hfx] · lift f x to ℝ≥0 using hfx with fx h simp [← h, ← NNReal.coe_le_coe] #align measure_theory.has_finite_integral_to_real_of_lintegral_ne_top MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top theorem isFiniteMeasure_withDensity_ofReal {f : α → ℝ} (hfi : HasFiniteIntegral f μ) : IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal <\| f x) := by refine isFiniteMeasure_withDensity ((lintegral_mono fun x => ?_).trans_lt hfi).ne exact Real.ofReal_le_ennnorm (f x) #align measure_theory.is_finite_measure_with_density_of_real MeasureTheory.isFiniteMeasure_withDensity_ofReal section DominatedConvergence variable {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} theorem all_ae_ofReal_F_le_bound (h : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a) := fun n => (h n).mono fun _ h => ENNReal.ofReal_le_ofReal h set_option linter.uppercaseLean3 false in #align measure_theory.all_ae_of_real_F_le_bound MeasureTheory.all_ae_ofReal_F_le_bound theorem all_ae_tendsto_ofReal_norm (h : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop <\| 𝓝 <\| f a) : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop <\| 𝓝 <\| ENNReal.ofReal ‖f a‖ := h.mono fun _ h => tendsto_ofReal <\| Tendsto.comp (Continuous.tendsto continuous_norm _) h #align measure_theory.all_ae_tendsto_of_real_norm MeasureTheory.all_ae_tendsto_ofReal_norm theorem all_ae_ofReal_f_le_bound (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : ∀ᵐ a ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a) := by have F_le_bound := all_ae_ofReal_F_le_bound h_bound rw [← ae_all_iff] at F_le_bound apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _) intro a tendsto_norm F_le_bound exact le_of_tendsto' tendsto_norm F_le_bound #align measure_theory.all_ae_of_real_f_le_bound MeasureTheory.all_ae_ofReal_f_le_bound theorem hasFiniteIntegral_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (bound_hasFiniteIntegral : HasFiniteIntegral bound μ) (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : HasFiniteIntegral f μ := by /- `‖F n a‖ ≤ bound a` and `‖F n a‖ --> ‖f a‖` implies `‖f a‖ ≤ bound a`, and so `∫ ‖f‖ ≤ ∫ bound < ∞` since `bound` is has_finite_integral -/ rw [hasFiniteIntegral_iff_norm] calc (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := lintegral_mono_ae <\| all_ae_ofReal_f_le_bound h_bound h_lim _ < ∞ := by rw [← hasFiniteIntegral_iff_ofReal] · exact bound_hasFiniteIntegral exact (h_bound 0).mono fun a h => le_trans (norm_nonneg _) h #align measure_theory.has_finite_integral_of_dominated_convergence MeasureTheory.hasFiniteIntegral_of_dominated_convergence theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ} (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_hasFiniteIntegral : HasFiniteIntegral bound μ) (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0) := by have f_measurable : AEStronglyMeasurable f μ := aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim let b a := 2 * ENNReal.ofReal (bound a) /- `‖F n a‖ ≤ bound a` and `F n a --> f a` implies `‖f a‖ ≤ bound a`, and thus by the triangle inequality, have `‖F n a - f a‖ ≤ 2 * (bound a)`. -/ have hb : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a := by intro n filter_upwards [all_ae_ofReal_F_le_bound h_bound n, all_ae_ofReal_f_le_bound h_bound h_lim] with a h₁ h₂ calc ENNReal.ofReal ‖F n a - f a‖ ≤ ENNReal.ofReal ‖F n a‖ + ENNReal.ofReal ‖f a‖ := by rw [← ENNReal.ofReal_add] · apply ofReal_le_ofReal apply norm_sub_le · exact norm_nonneg _ · exact norm_nonneg _ _ ≤ ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) := add_le_add h₁ h₂ _ = b a := by rw [← two_mul] -- On the other hand, `F n a --> f a` implies that `‖F n a - f a‖ --> 0` have h : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0) := by rw [← ENNReal.ofReal_zero] refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_ rwa [← tendsto_iff_norm_sub_tendsto_zero] /- Therefore, by the dominated convergence theorem for nonnegative integration, have ` ∫ ‖f a - F n a‖ --> 0 ` -/ suffices Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 (∫⁻ _ : α, 0 ∂μ)) by rwa [lintegral_zero] at this -- Using the dominated convergence theorem. refine tendsto_lintegral_of_dominated_convergence' _ ?_ hb ?_ ?_ -- Show `fun a => ‖f a - F n a‖` is almost everywhere measurable for all `n` · exact fun n => measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable -- Show `2 * bound` `HasFiniteIntegral` · rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral · calc ∫⁻ a, b a ∂μ = 2 * ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := by rw [lintegral_const_mul'] exact coe_ne_top _ ≠ ∞ := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne filter_upwards [h_bound 0] with _ h using le_trans (norm_nonneg _) h -- Show `‖f a - F n a‖ --> 0` · exact h #align measure_theory.tendsto_lintegral_norm_of_dominated_convergence MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence end DominatedConvergence section PosPart /-! Lemmas used for defining the positive part of an `L¹` function -/ theorem HasFiniteIntegral.max_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) : HasFiniteIntegral (fun a => max (f a) 0) μ := hf.mono <\| eventually_of_forall fun x => by simp [abs_le, le_abs_self] #align measure_theory.has_finite_integral.max_zero MeasureTheory.HasFiniteIntegral.max_zero theorem HasFiniteIntegral.min_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) : HasFiniteIntegral (fun a => min (f a) 0) μ := hf.mono <\| eventually_of_forall fun x => by simpa [abs_le] using neg_abs_le _ #align measure_theory.has_finite_integral.min_zero MeasureTheory.HasFiniteIntegral.min_zero end PosPart section NormedSpace variable {𝕜 : Type} theorem HasFiniteIntegral.smul [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜) {f : α → β} : HasFiniteIntegral f μ → HasFiniteIntegral (c • f) μ := by simp only [HasFiniteIntegral]; intro hfi calc (∫⁻ a : α, ‖c • f a‖₊ ∂μ) ≤ ∫⁻ a : α, ‖c‖₊ ‖f a‖₊ ∂μ := by refine lintegral_mono ?_ intro i -- After leanprover/lean4#2734, we need to do beta reduction `exact mod_cast` beta_reduce exact mod_cast (nnnorm_smul_le c (f i)) _ < ∞ := by rw [lintegral_const_mul'] exacts [mul_lt_top coe_ne_top hfi.ne, coe_ne_top] #align measure_theory.has_finite_integral.smul MeasureTheory.HasFiniteIntegral.smul theorem hasFiniteIntegral_smul_iff [NormedRing 𝕜] [MulActionWithZero 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜} (hc : IsUnit c) (f : α → β) : HasFiniteIntegral (c • f) μ ↔ HasFiniteIntegral f μ := by obtain ⟨c, rfl⟩ := hc constructor · intro h simpa only [smul_smul, Units.inv_mul, one_smul] using h.smul ((c⁻¹ : 𝕜ˣ) : 𝕜) exact HasFiniteIntegral.smul _ #align measure_theory.has_finite_integral_smul_iff MeasureTheory.hasFiniteIntegral_smul_iff theorem HasFiniteIntegral.const_mul [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) : HasFiniteIntegral (fun x => c * f x) μ := h.smul c #align measure_theory.has_finite_integral.const_mul MeasureTheory.HasFiniteIntegral.const_mul theorem HasFiniteIntegral.mul_const [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) : HasFiniteIntegral (fun x => f x * c) μ := h.smul (MulOpposite.op c) #align measure_theory.has_finite_integral.mul_const MeasureTheory.HasFiniteIntegral.mul_const end NormedSpace /-! ### The predicate `Integrable` -/ -- variable [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] /-- `Integrable f μ` means that `f` is measurable and that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite. `Integrable f` means `Integrable f volume`. -/ def Integrable {α} {_ : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop := AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ #align measure_theory.integrable MeasureTheory.Integrable theorem memℒp_one_iff_integrable {f : α → β} : Memℒp f 1 μ ↔ Integrable f μ := by simp_rw [Integrable, HasFiniteIntegral, Memℒp, snorm_one_eq_lintegral_nnnorm] #align measure_theory.mem_ℒp_one_iff_integrable MeasureTheory.memℒp_one_iff_integrable theorem Integrable.aestronglyMeasurable {f : α → β} (hf : Integrable f μ) : AEStronglyMeasurable f μ := hf.1 #align measure_theory.integrable.ae_strongly_measurable MeasureTheory.Integrable.aestronglyMeasurable theorem Integrable.aemeasurable [MeasurableSpace β] [BorelSpace β] {f : α → β} (hf : Integrable f μ) : AEMeasurable f μ := hf.aestronglyMeasurable.aemeasurable #align measure_theory.integrable.ae_measurable MeasureTheory.Integrable.aemeasurable theorem Integrable.hasFiniteIntegral {f : α → β} (hf : Integrable f μ) : HasFiniteIntegral f μ := hf.2 #align measure_theory.integrable.has_finite_integral MeasureTheory.Integrable.hasFiniteIntegral theorem Integrable.mono {f : α → β} {g : α → γ} (hg : Integrable g μ) (hf : AEStronglyMeasurable f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : Integrable f μ := ⟨hf, hg.hasFiniteIntegral.mono h⟩ #align measure_theory.integrable.mono MeasureTheory.Integrable.mono theorem Integrable.mono' {f : α → β} {g : α → ℝ} (hg : Integrable g μ) (hf : AEStronglyMeasurable f μ) (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : Integrable f μ := ⟨hf, hg.hasFiniteIntegral.mono' h⟩ #align measure_theory.integrable.mono' MeasureTheory.Integrable.mono' theorem Integrable.congr' {f : α → β} {g : α → γ} (hf : Integrable f μ) (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Integrable g μ := ⟨hg, hf.hasFiniteIntegral.congr' h⟩ #align measure_theory.integrable.congr' MeasureTheory.Integrable.congr' theorem integrable_congr' {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Integrable f μ ↔ Integrable g μ := ⟨fun h2f => h2f.congr' hg h, fun h2g => h2g.congr' hf <\| EventuallyEq.symm h⟩ #align measure_theory.integrable_congr' MeasureTheory.integrable_congr' theorem Integrable.congr {f g : α → β} (hf : Integrable f μ) (h : f =ᵐ[μ] g) : Integrable g μ := ⟨hf.1.congr h, hf.2.congr h⟩ #align measure_theory.integrable.congr MeasureTheory.Integrable.congr theorem integrable_congr {f g : α → β} (h : f =ᵐ[μ] g) : Integrable f μ ↔ Integrable g μ := ⟨fun hf => hf.congr h, fun hg => hg.congr h.symm⟩ #align measure_theory.integrable_congr MeasureTheory.integrable_congr theorem integrable_const_iff {c : β} : Integrable (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ := by have : AEStronglyMeasurable (fun _ : α => c) μ := aestronglyMeasurable_const rw [Integrable, and_iff_right this, hasFiniteIntegral_const_iff] #align measure_theory.integrable_const_iff MeasureTheory.integrable_const_iff @[simp] theorem integrable_const [IsFiniteMeasure μ] (c : β) : Integrable (fun _ : α => c) μ := integrable_const_iff.2 <\| Or.inr <\| measure_lt_top _ _ #align measure_theory.integrable_const MeasureTheory.integrable_const @[simp] theorem Integrable.of_finite [Finite α] [MeasurableSpace α] [MeasurableSingletonClass α] (μ : Measure α) [IsFiniteMeasure μ] (f : α → β) : Integrable (fun a ↦ f a) μ := ⟨(StronglyMeasurable.of_finite f).aestronglyMeasurable, .of_finite⟩ @[deprecated (since := "2024-02-05")] alias integrable_of_fintype := Integrable.of_finite theorem Memℒp.integrable_norm_rpow {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by rw [← memℒp_one_iff_integrable] exact hf.norm_rpow hp_ne_zero hp_ne_top #align measure_theory.mem_ℒp.integrable_norm_rpow MeasureTheory.Memℒp.integrable_norm_rpow theorem Memℒp.integrable_norm_rpow' [IsFiniteMeasure μ] {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by by_cases h_zero : p = 0 · simp [h_zero, integrable_const] by_cases h_top : p = ∞ · simp [h_top, integrable_const] exact hf.integrable_norm_rpow h_zero h_top #align measure_theory.mem_ℒp.integrable_norm_rpow' MeasureTheory.Memℒp.integrable_norm_rpow' theorem Integrable.mono_measure {f : α → β} (h : Integrable f ν) (hμ : μ ≤ ν) : Integrable f μ := ⟨h.aestronglyMeasurable.mono_measure hμ, h.hasFiniteIntegral.mono_measure hμ⟩ #align measure_theory.integrable.mono_measure MeasureTheory.Integrable.mono_measure theorem Integrable.of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (hμ'_le : μ' ≤ c • μ) {f : α → β} (hf : Integrable f μ) : Integrable f μ' := by rw [← memℒp_one_iff_integrable] at hf ⊢ exact hf.of_measure_le_smul c hc hμ'_le #align measure_theory.integrable.of_measure_le_smul MeasureTheory.Integrable.of_measure_le_smul theorem Integrable.add_measure {f : α → β} (hμ : Integrable f μ) (hν : Integrable f ν) : Integrable f (μ + ν) := by simp_rw [← memℒp_one_iff_integrable] at hμ hν ⊢ refine ⟨hμ.aestronglyMeasurable.add_measure hν.aestronglyMeasurable, ?_⟩ rw [snorm_one_add_measure, ENNReal.add_lt_top] exact ⟨hμ.snorm_lt_top, hν.snorm_lt_top⟩ #align measure_theory.integrable.add_measure MeasureTheory.Integrable.add_measure theorem Integrable.left_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) : Integrable f μ := by rw [← memℒp_one_iff_integrable] at h ⊢ exact h.left_of_add_measure #align measure_theory.integrable.left_of_add_measure MeasureTheory.Integrable.left_of_add_measure theorem Integrable.right_of_add_measure {f : α → β} (h : Integrable f (μ + ν)) : Integrable f ν := by rw [← memℒp_one_iff_integrable] at h ⊢ exact h.right_of_add_measure #align measure_theory.integrable.right_of_add_measure MeasureTheory.Integrable.right_of_add_measure @[simp] theorem integrable_add_measure {f : α → β} : Integrable f (μ + ν) ↔ Integrable f μ ∧ Integrable f ν := ⟨fun h => ⟨h.left_of_add_measure, h.right_of_add_measure⟩, fun h => h.1.add_measure h.2⟩ #align measure_theory.integrable_add_measure MeasureTheory.integrable_add_measure @[simp] theorem integrable_zero_measure {_ : MeasurableSpace α} {f : α → β} : Integrable f (0 : Measure α) := ⟨aestronglyMeasurable_zero_measure f, hasFiniteIntegral_zero_measure f⟩ #align measure_theory.integrable_zero_measure MeasureTheory.integrable_zero_measure theorem integrable_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → β} {μ : ι → Measure α} {s : Finset ι} : Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i) := by induction s using Finset.induction_on <;> simp [*] #align measure_theory.integrable_finset_sum_measure MeasureTheory.integrable_finset_sum_measure theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) : Integrable f (c • μ) := by rw [← memℒp_one_iff_integrable] at h ⊢ exact h.smul_measure hc #align measure_theory.integrable.smul_measure MeasureTheory.Integrable.smul_measure theorem Integrable.smul_measure_nnreal {f : α → β} (h : Integrable f μ) {c : ℝ≥0} : Integrable f (c • μ) := by apply h.smul_measure simp theorem integrable_smul_measure {f : α → β} {c : ℝ≥0∞} (h₁ : c ≠ 0) (h₂ : c ≠ ∞) : Integrable f (c • μ) ↔ Integrable f μ := ⟨fun h => by simpa only [smul_smul, ENNReal.inv_mul_cancel h₁ h₂, one_smul] using h.smul_measure (ENNReal.inv_ne_top.2 h₁), fun h => h.smul_measure h₂⟩ #align measure_theory.integrable_smul_measure MeasureTheory.integrable_smul_measure theorem integrable_inv_smul_measure {f : α → β} {c : ℝ≥0∞} (h₁ : c ≠ 0) (h₂ : c ≠ ∞) : Integrable f (c⁻¹ • μ) ↔ Integrable f μ := integrable_smul_measure (by simpa using h₂) (by simpa using h₁) #align measure_theory.integrable_inv_smul_measure MeasureTheory.integrable_inv_smul_measure theorem Integrable.to_average {f : α → β} (h : Integrable f μ) : Integrable f ((μ univ)⁻¹ • μ) := by rcases eq_or_ne μ 0 with (rfl \| hne) · rwa [smul_zero] · apply h.smul_measure simpa #align measure_theory.integrable.to_average MeasureTheory.Integrable.to_average theorem integrable_average [IsFiniteMeasure μ] {f : α → β} : Integrable f ((μ univ)⁻¹ • μ) ↔ Integrable f μ := (eq_or_ne μ 0).by_cases (fun h => by simp [h]) fun h => integrable_smul_measure (ENNReal.inv_ne_zero.2 <\| measure_ne_top _ _) (ENNReal.inv_ne_top.2 <\| mt Measure.measure_univ_eq_zero.1 h) #align measure_theory.integrable_average MeasureTheory.integrable_average theorem integrable_map_measure {f : α → δ} {g : δ → β} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) : Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ := by simp_rw [← memℒp_one_iff_integrable] exact memℒp_map_measure_iff hg hf #align measure_theory.integrable_map_measure MeasureTheory.integrable_map_measure theorem Integrable.comp_aemeasurable {f : α → δ} {g : δ → β} (hg : Integrable g (Measure.map f μ)) (hf : AEMeasurable f μ) : Integrable (g ∘ f) μ := (integrable_map_measure hg.aestronglyMeasurable hf).mp hg #align measure_theory.integrable.comp_ae_measurable MeasureTheory.Integrable.comp_aemeasurable theorem Integrable.comp_measurable {f : α → δ} {g : δ → β} (hg : Integrable g (Measure.map f μ)) (hf : Measurable f) : Integrable (g ∘ f) μ := hg.comp_aemeasurable hf.aemeasurable #align measure_theory.integrable.comp_measurable MeasureTheory.Integrable.comp_measurable	Mathlib/MeasureTheory/Function/L1Space.lean	621	624	theorem _root_.MeasurableEmbedding.integrable_map_iff {f : α → δ} (hf : MeasurableEmbedding f) {g : δ → β} : Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ := by	simp_rw [← memℒp_one_iff_integrable] exact hf.memℒp_map_measure_iff
/- Copyright (c) 2019 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes -/ import Mathlib.NumberTheory.Zsqrtd.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Data.Complex.Basic import Mathlib.Data.Real.Archimedean #align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9" /-! # Gaussian integers The Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both integers. ## Main definitions The Euclidean domain structure on `ℤ[i]` is defined in this file. The homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file. ## See also See `NumberTheory.Zsqrtd.QuadraticReciprocity` for: * `prime_iff_mod_four_eq_three_of_nat_prime`: A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4` ## Notations This file uses the local notation `ℤ[i]` for `GaussianInt` ## Implementation notes Gaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers adjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties and definitions about `Zsqrtd` can easily be used. -/ open Zsqrtd Complex open scoped ComplexConjugate /-- The Gaussian integers, defined as `ℤ√(-1)`. -/ abbrev GaussianInt : Type := Zsqrtd (-1) #align gaussian_int GaussianInt local notation "ℤ[i]" => GaussianInt namespace GaussianInt instance : Repr ℤ[i] := ⟨fun x _ => "⟨" ++ repr x.re ++ ", " ++ repr x.im ++ "⟩"⟩ instance instCommRing : CommRing ℤ[i] := Zsqrtd.commRing #align gaussian_int.comm_ring GaussianInt.instCommRing section attribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily. /-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/ def toComplex : ℤ[i] →+* ℂ := Zsqrtd.lift ⟨I, by simp⟩ #align gaussian_int.to_complex GaussianInt.toComplex end instance : Coe ℤ[i] ℂ := ⟨toComplex⟩ theorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I := rfl #align gaussian_int.to_complex_def GaussianInt.toComplex_def theorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def] #align gaussian_int.to_complex_def' GaussianInt.toComplex_def' theorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by apply Complex.ext <;> simp [toComplex_def] #align gaussian_int.to_complex_def₂ GaussianInt.toComplex_def₂ @[simp] theorem to_real_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def] #align gaussian_int.to_real_re GaussianInt.to_real_re @[simp] theorem to_real_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def] #align gaussian_int.to_real_im GaussianInt.to_real_im @[simp] theorem toComplex_re (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def] #align gaussian_int.to_complex_re GaussianInt.toComplex_re @[simp] theorem toComplex_im (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def] #align gaussian_int.to_complex_im GaussianInt.toComplex_im -- Porting note (#10618): @[simp] can prove this theorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y := toComplex.map_add _ _ #align gaussian_int.to_complex_add GaussianInt.toComplex_add -- Porting note (#10618): @[simp] can prove this theorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y := toComplex.map_mul _ _ #align gaussian_int.to_complex_mul GaussianInt.toComplex_mul -- Porting note (#10618): @[simp] can prove this theorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 := toComplex.map_one #align gaussian_int.to_complex_one GaussianInt.toComplex_one -- Porting note (#10618): @[simp] can prove this theorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 := toComplex.map_zero #align gaussian_int.to_complex_zero GaussianInt.toComplex_zero -- Porting note (#10618): @[simp] can prove this theorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x := toComplex.map_neg _ #align gaussian_int.to_complex_neg GaussianInt.toComplex_neg -- Porting note (#10618): @[simp] can prove this theorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y := toComplex.map_sub _ _ #align gaussian_int.to_complex_sub GaussianInt.toComplex_sub @[simp] theorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by rw [toComplex_def₂, toComplex_def₂] exact congr_arg₂ _ rfl (Int.cast_neg _) #align gaussian_int.to_complex_star GaussianInt.toComplex_star @[simp] theorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by cases x; cases y; simp [toComplex_def₂] #align gaussian_int.to_complex_inj GaussianInt.toComplex_inj lemma toComplex_injective : Function.Injective GaussianInt.toComplex := fun ⦃_ _⦄ ↦ toComplex_inj.mp @[simp] theorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by rw [← toComplex_zero, toComplex_inj] #align gaussian_int.to_complex_eq_zero GaussianInt.toComplex_eq_zero @[simp] theorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by rw [Zsqrtd.norm, normSq]; simp #align gaussian_int.nat_cast_real_norm GaussianInt.intCast_real_norm @[deprecated (since := "2024-04-17")] alias int_cast_real_norm := intCast_real_norm @[simp] theorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) := by cases x; rw [Zsqrtd.norm, normSq]; simp #align gaussian_int.nat_cast_complex_norm GaussianInt.intCast_complex_norm @[deprecated (since := "2024-04-17")] alias int_cast_complex_norm := intCast_complex_norm theorem norm_nonneg (x : ℤ[i]) : 0 ≤ norm x := Zsqrtd.norm_nonneg (by norm_num) _ #align gaussian_int.norm_nonneg GaussianInt.norm_nonneg @[simp] theorem norm_eq_zero {x : ℤ[i]} : norm x = 0 ↔ x = 0 := by rw [← @Int.cast_inj ℝ _ _ _]; simp #align gaussian_int.norm_eq_zero GaussianInt.norm_eq_zero theorem norm_pos {x : ℤ[i]} : 0 < norm x ↔ x ≠ 0 := by rw [lt_iff_le_and_ne, Ne, eq_comm, norm_eq_zero]; simp [norm_nonneg] #align gaussian_int.norm_pos GaussianInt.norm_pos theorem abs_natCast_norm (x : ℤ[i]) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) #align gaussian_int.abs_coe_nat_norm GaussianInt.abs_natCast_norm -- 2024-04-05 @[deprecated] alias abs_coe_nat_norm := abs_natCast_norm @[simp] theorem natCast_natAbs_norm {α : Type} [Ring α] (x : ℤ[i]) : (x.norm.natAbs : α) = x.norm := by rw [← Int.cast_natCast, abs_natCast_norm] #align gaussian_int.nat_cast_nat_abs_norm GaussianInt.natCast_natAbs_norm @[deprecated (since := "2024-04-17")] alias nat_cast_natAbs_norm := natCast_natAbs_norm theorem natAbs_norm_eq (x : ℤ[i]) : x.norm.natAbs = x.re.natAbs x.re.natAbs + x.im.natAbs * x.im.natAbs := Int.ofNat.inj <\| by simp; simp [Zsqrtd.norm] #align gaussian_int.nat_abs_norm_eq GaussianInt.natAbs_norm_eq instance : Div ℤ[i] := ⟨fun x y => let n := (norm y : ℚ)⁻¹ let c := star y ⟨round ((x * c).re * n : ℚ), round ((x * c).im * n : ℚ)⟩⟩ theorem div_def (x y : ℤ[i]) : x / y = ⟨round ((x * star y).re / norm y : ℚ), round ((x * star y).im / norm y : ℚ)⟩ := show Zsqrtd.mk _ _ = _ by simp [div_eq_mul_inv] #align gaussian_int.div_def GaussianInt.div_def	Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean	212	214	theorem toComplex_div_re (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).re = round (x / y : ℂ).re := by	rw [div_def, ← @Rat.round_cast ℝ _ _] simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, mul_add, add_mul]
/- Copyright (c) 2020 Paul van Wamelen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Paul van Wamelen -/ import Mathlib.Algebra.Field.Basic import Mathlib.Algebra.Order.Group.Basic import Mathlib.Algebra.Order.Ring.Basic import Mathlib.RingTheory.Int.Basic import Mathlib.Tactic.Ring import Mathlib.Tactic.FieldSimp import Mathlib.Data.Int.NatPrime import Mathlib.Data.ZMod.Basic #align_import number_theory.pythagorean_triples from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" /-! # Pythagorean Triples The main result is the classification of Pythagorean triples. The final result is for general Pythagorean triples. It follows from the more interesting relatively prime case. We use the "rational parametrization of the circle" method for the proof. The parametrization maps the point `(x / z, y / z)` to the slope of the line through `(-1 , 0)` and `(x / z, y / z)`. This quickly shows that `(x / z, y / z) = (2 * m * n / (m ^ 2 + n ^ 2), (m ^ 2 - n ^ 2) / (m ^ 2 + n ^ 2))` where `m / n` is the slope. In order to identify numerators and denominators we now need results showing that these are coprime. This is easy except for the prime 2. In order to deal with that we have to analyze the parity of `x`, `y`, `m` and `n` and eliminate all the impossible cases. This takes up the bulk of the proof below. -/ theorem sq_ne_two_fin_zmod_four (z : ZMod 4) : z * z ≠ 2 := by change Fin 4 at z fin_cases z <;> decide #align sq_ne_two_fin_zmod_four sq_ne_two_fin_zmod_four theorem Int.sq_ne_two_mod_four (z : ℤ) : z * z % 4 ≠ 2 := by suffices ¬z * z % (4 : ℕ) = 2 % (4 : ℕ) by exact this rw [← ZMod.intCast_eq_intCast_iff'] simpa using sq_ne_two_fin_zmod_four _ #align int.sq_ne_two_mod_four Int.sq_ne_two_mod_four noncomputable section open scoped Classical /-- Three integers `x`, `y`, and `z` form a Pythagorean triple if `x * x + y * y = z * z`. -/ def PythagoreanTriple (x y z : ℤ) : Prop := x * x + y * y = z * z #align pythagorean_triple PythagoreanTriple /-- Pythagorean triples are interchangeable, i.e `x * x + y * y = y * y + x * x = z * z`. This comes from additive commutativity. -/ theorem pythagoreanTriple_comm {x y z : ℤ} : PythagoreanTriple x y z ↔ PythagoreanTriple y x z := by delta PythagoreanTriple rw [add_comm] #align pythagorean_triple_comm pythagoreanTriple_comm /-- The zeroth Pythagorean triple is all zeros. -/ theorem PythagoreanTriple.zero : PythagoreanTriple 0 0 0 := by simp only [PythagoreanTriple, zero_mul, zero_add] #align pythagorean_triple.zero PythagoreanTriple.zero namespace PythagoreanTriple variable {x y z : ℤ} (h : PythagoreanTriple x y z) theorem eq : x * x + y * y = z * z := h #align pythagorean_triple.eq PythagoreanTriple.eq @[symm] theorem symm : PythagoreanTriple y x z := by rwa [pythagoreanTriple_comm] #align pythagorean_triple.symm PythagoreanTriple.symm /-- A triple is still a triple if you multiply `x`, `y` and `z` by a constant `k`. -/ theorem mul (k : ℤ) : PythagoreanTriple (k * x) (k * y) (k * z) := calc k * x * (k * x) + k * y * (k * y) = k ^ 2 * (x * x + y * y) := by ring _ = k ^ 2 * (z * z) := by rw [h.eq] _ = k * z * (k * z) := by ring #align pythagorean_triple.mul PythagoreanTriple.mul /-- `(kx, ky, kz)` is a Pythagorean triple if and only if `(x, y, z)` is also a triple. -/ theorem mul_iff (k : ℤ) (hk : k ≠ 0) : PythagoreanTriple (k x) (k * y) (k * z) ↔ PythagoreanTriple x y z := by refine ⟨?_, fun h => h.mul k⟩ simp only [PythagoreanTriple] intro h rw [← mul_left_inj' (mul_ne_zero hk hk)] convert h using 1 <;> ring #align pythagorean_triple.mul_iff PythagoreanTriple.mul_iff /-- A Pythagorean triple `x, y, z` is “classified” if there exist integers `k, m, n` such that either * `x = k * (m ^ 2 - n ^ 2)` and `y = k * (2 * m * n)`, or * `x = k * (2 * m * n)` and `y = k * (m ^ 2 - n ^ 2)`. -/ @[nolint unusedArguments] def IsClassified (_ : PythagoreanTriple x y z) := ∃ k m n : ℤ, (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧ Int.gcd m n = 1 #align pythagorean_triple.is_classified PythagoreanTriple.IsClassified /-- A primitive Pythagorean triple `x, y, z` is a Pythagorean triple with `x` and `y` coprime. Such a triple is “primitively classified” if there exist coprime integers `m, n` such that either * `x = m ^ 2 - n ^ 2` and `y = 2 * m * n`, or * `x = 2 * m * n` and `y = m ^ 2 - n ^ 2`. -/ @[nolint unusedArguments] def IsPrimitiveClassified (_ : PythagoreanTriple x y z) := ∃ m n : ℤ, (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧ Int.gcd m n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) #align pythagorean_triple.is_primitive_classified PythagoreanTriple.IsPrimitiveClassified theorem mul_isClassified (k : ℤ) (hc : h.IsClassified) : (h.mul k).IsClassified := by obtain ⟨l, m, n, ⟨⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, co⟩⟩ := hc · use k * l, m, n apply And.intro _ co left constructor <;> ring · use k * l, m, n apply And.intro _ co right constructor <;> ring #align pythagorean_triple.mul_is_classified PythagoreanTriple.mul_isClassified theorem even_odd_of_coprime (hc : Int.gcd x y = 1) : x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0 := by cases' Int.emod_two_eq_zero_or_one x with hx hx <;> cases' Int.emod_two_eq_zero_or_one y with hy hy -- x even, y even · exfalso apply Nat.not_coprime_of_dvd_of_dvd (by decide : 1 < 2) _ _ hc · apply Int.natCast_dvd.1 apply Int.dvd_of_emod_eq_zero hx · apply Int.natCast_dvd.1 apply Int.dvd_of_emod_eq_zero hy -- x even, y odd · left exact ⟨hx, hy⟩ -- x odd, y even · right exact ⟨hx, hy⟩ -- x odd, y odd · exfalso obtain ⟨x0, y0, rfl, rfl⟩ : ∃ x0 y0, x = x0 * 2 + 1 ∧ y = y0 * 2 + 1 := by cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hx) with x0 hx2 cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hy) with y0 hy2 rw [sub_eq_iff_eq_add] at hx2 hy2 exact ⟨x0, y0, hx2, hy2⟩ apply Int.sq_ne_two_mod_four z rw [show z * z = 4 * (x0 * x0 + x0 + y0 * y0 + y0) + 2 by rw [← h.eq] ring] simp only [Int.add_emod, Int.mul_emod_right, zero_add] decide #align pythagorean_triple.even_odd_of_coprime PythagoreanTriple.even_odd_of_coprime theorem gcd_dvd : (Int.gcd x y : ℤ) ∣ z := by by_cases h0 : Int.gcd x y = 0 · have hx : x = 0 := by apply Int.natAbs_eq_zero.mp apply Nat.eq_zero_of_gcd_eq_zero_left h0 have hy : y = 0 := by apply Int.natAbs_eq_zero.mp apply Nat.eq_zero_of_gcd_eq_zero_right h0 have hz : z = 0 := by simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero, or_self_iff] using h simp only [hz, dvd_zero] obtain ⟨k, x0, y0, _, h2, rfl, rfl⟩ : ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k := Int.exists_gcd_one' (Nat.pos_of_ne_zero h0) rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul] rw [← Int.pow_dvd_pow_iff two_ne_zero, sq z, ← h.eq] rw [(by ring : x0 * k * (x0 * k) + y0 * k * (y0 * k) = (k : ℤ) ^ 2 * (x0 * x0 + y0 * y0))] exact dvd_mul_right _ _ #align pythagorean_triple.gcd_dvd PythagoreanTriple.gcd_dvd	Mathlib/NumberTheory/PythagoreanTriples.lean	185	207	theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by	by_cases h0 : Int.gcd x y = 0 · have hx : x = 0 := by apply Int.natAbs_eq_zero.mp apply Nat.eq_zero_of_gcd_eq_zero_left h0 have hy : y = 0 := by apply Int.natAbs_eq_zero.mp apply Nat.eq_zero_of_gcd_eq_zero_right h0 have hz : z = 0 := by simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero, or_self_iff] using h simp only [hx, hy, hz, Int.zero_div] exact zero rcases h.gcd_dvd with ⟨z0, rfl⟩ obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ : ∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k := Int.exists_gcd_one' (Nat.pos_of_ne_zero h0) have hk : (k : ℤ) ≠ 0 := by norm_cast rwa [pos_iff_ne_zero] at k0 rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul] at h ⊢ rw [mul_comm x0, mul_comm y0, mul_iff k hk] at h rwa [Int.mul_ediv_cancel _ hk, Int.mul_ediv_cancel _ hk, Int.mul_ediv_cancel_left _ hk]
/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ 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" /-! # Trigonometric functions ## Main definitions This file contains the definition of `π`. See also `Analysis.SpecialFunctions.Trigonometric.Inverse` and `Analysis.SpecialFunctions.Trigonometric.Arctan` for the inverse trigonometric functions. See also `Analysis.SpecialFunctions.Complex.Arg` and `Analysis.SpecialFunctions.Complex.Log` for the complex argument function and the complex logarithm. ## Main statements Many basic inequalities on the real trigonometric functions are established. The continuity of the usual trigonometric functions is proved. Several facts about the real trigonometric functions have the proofs deferred to `Analysis.SpecialFunctions.Trigonometric.Complex`, as they are most easily proved by appealing to the corresponding fact for complex trigonometric functions. See also `Analysis.SpecialFunctions.Trigonometric.Chebyshev` for the multiple angle formulas in terms of Chebyshev polynomials. ## Tags sin, cos, tan, angle -/ noncomputable section open scoped Classical open Topology Filter Set namespace Complex @[continuity, fun_prop] theorem continuous_sin : Continuous sin := by change Continuous fun z => (exp (-z * I) - exp (z * I)) * I / 2 continuity #align complex.continuous_sin Complex.continuous_sin @[fun_prop] theorem continuousOn_sin {s : Set ℂ} : ContinuousOn sin s := continuous_sin.continuousOn #align complex.continuous_on_sin Complex.continuousOn_sin @[continuity, fun_prop] theorem continuous_cos : Continuous cos := by change Continuous fun z => (exp (z * I) + exp (-z * I)) / 2 continuity #align complex.continuous_cos Complex.continuous_cos @[fun_prop] theorem continuousOn_cos {s : Set ℂ} : ContinuousOn cos s := continuous_cos.continuousOn #align complex.continuous_on_cos Complex.continuousOn_cos @[continuity, fun_prop] theorem continuous_sinh : Continuous sinh := by change Continuous fun z => (exp z - exp (-z)) / 2 continuity #align complex.continuous_sinh Complex.continuous_sinh @[continuity, fun_prop] theorem continuous_cosh : Continuous cosh := by change Continuous fun z => (exp z + exp (-z)) / 2 continuity #align complex.continuous_cosh Complex.continuous_cosh end Complex namespace Real variable {x y z : ℝ} @[continuity, fun_prop] theorem continuous_sin : Continuous sin := Complex.continuous_re.comp (Complex.continuous_sin.comp Complex.continuous_ofReal) #align real.continuous_sin Real.continuous_sin @[fun_prop] theorem continuousOn_sin {s} : ContinuousOn sin s := continuous_sin.continuousOn #align real.continuous_on_sin Real.continuousOn_sin @[continuity, fun_prop] theorem continuous_cos : Continuous cos := Complex.continuous_re.comp (Complex.continuous_cos.comp Complex.continuous_ofReal) #align real.continuous_cos Real.continuous_cos @[fun_prop] theorem continuousOn_cos {s} : ContinuousOn cos s := continuous_cos.continuousOn #align real.continuous_on_cos Real.continuousOn_cos @[continuity, fun_prop] theorem continuous_sinh : Continuous sinh := Complex.continuous_re.comp (Complex.continuous_sinh.comp Complex.continuous_ofReal) #align real.continuous_sinh Real.continuous_sinh @[continuity, fun_prop] theorem continuous_cosh : Continuous cosh := Complex.continuous_re.comp (Complex.continuous_cosh.comp Complex.continuous_ofReal) #align real.continuous_cosh Real.continuous_cosh end Real namespace Real theorem exists_cos_eq_zero : 0 ∈ cos '' Icc (1 : ℝ) 2 := intermediate_value_Icc' (by norm_num) continuousOn_cos ⟨le_of_lt cos_two_neg, le_of_lt cos_one_pos⟩ #align real.exists_cos_eq_zero Real.exists_cos_eq_zero /-- The number π = 3.14159265... Defined here using choice as twice a zero of cos in [1,2], from which one can derive all its properties. For explicit bounds on π, see `Data.Real.Pi.Bounds`. -/ protected noncomputable def pi : ℝ := 2 * Classical.choose exists_cos_eq_zero #align real.pi Real.pi @[inherit_doc] scoped notation "π" => Real.pi @[simp] theorem cos_pi_div_two : cos (π / 2) = 0 := by rw [Real.pi, mul_div_cancel_left₀ _ (two_ne_zero' ℝ)] exact (Classical.choose_spec exists_cos_eq_zero).2 #align real.cos_pi_div_two Real.cos_pi_div_two theorem one_le_pi_div_two : (1 : ℝ) ≤ π / 2 := by rw [Real.pi, mul_div_cancel_left₀ _ (two_ne_zero' ℝ)] exact (Classical.choose_spec exists_cos_eq_zero).1.1 #align real.one_le_pi_div_two Real.one_le_pi_div_two	Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean	152	154	theorem pi_div_two_le_two : π / 2 ≤ 2 := by	rw [Real.pi, mul_div_cancel_left₀ _ (two_ne_zero' ℝ)] exact (Classical.choose_spec exists_cos_eq_zero).1.2
/- Copyright (c) 2018 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Order.Bounds.Basic import Mathlib.Order.WellFounded import Mathlib.Data.Set.Image import Mathlib.Order.Interval.Set.Basic import Mathlib.Data.Set.Lattice #align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1" /-! # Theory of conditionally complete lattices. A conditionally complete lattice is a lattice in which every non-empty bounded subset `s` has a least upper bound and a greatest lower bound, denoted below by `sSup s` and `sInf s`. Typical examples are `ℝ`, `ℕ`, and `ℤ` with their usual orders. The theory is very comparable to the theory of complete lattices, except that suitable boundedness and nonemptiness assumptions have to be added to most statements. We introduce two predicates `BddAbove` and `BddBelow` to express this boundedness, prove their basic properties, and then go on to prove most useful properties of `sSup` and `sInf` in conditionally complete lattices. To differentiate the statements between complete lattices and conditionally complete lattices, we prefix `sInf` and `sSup` in the statements by `c`, giving `csInf` and `csSup`. For instance, `sInf_le` is a statement in complete lattices ensuring `sInf s ≤ x`, while `csInf_le` is the same statement in conditionally complete lattices with an additional assumption that `s` is bounded below. -/ open Function OrderDual Set variable {α β γ : Type} {ι : Sort} section /-! Extension of `sSup` and `sInf` from a preorder `α` to `WithTop α` and `WithBot α` -/ variable [Preorder α] open scoped Classical noncomputable instance WithTop.instSupSet [SupSet α] : SupSet (WithTop α) := ⟨fun S => if ⊤ ∈ S then ⊤ else if BddAbove ((fun (a : α) ↦ ↑a) ⁻¹' S : Set α) then ↑(sSup ((fun (a : α) ↦ (a : WithTop α)) ⁻¹' S : Set α)) else ⊤⟩ noncomputable instance WithTop.instInfSet [InfSet α] : InfSet (WithTop α) := ⟨fun S => if S ⊆ {⊤} ∨ ¬BddBelow S then ⊤ else ↑(sInf ((fun (a : α) ↦ ↑a) ⁻¹' S : Set α))⟩ noncomputable instance WithBot.instSupSet [SupSet α] : SupSet (WithBot α) := ⟨(WithTop.instInfSet (α := αᵒᵈ)).sInf⟩ noncomputable instance WithBot.instInfSet [InfSet α] : InfSet (WithBot α) := ⟨(WithTop.instSupSet (α := αᵒᵈ)).sSup⟩ theorem WithTop.sSup_eq [SupSet α] {s : Set (WithTop α)} (hs : ⊤ ∉ s) (hs' : BddAbove ((↑) ⁻¹' s : Set α)) : sSup s = ↑(sSup ((↑) ⁻¹' s) : α) := (if_neg hs).trans <\| if_pos hs' #align with_top.Sup_eq WithTop.sSup_eq theorem WithTop.sInf_eq [InfSet α] {s : Set (WithTop α)} (hs : ¬s ⊆ {⊤}) (h's : BddBelow s) : sInf s = ↑(sInf ((↑) ⁻¹' s) : α) := if_neg <\| by simp [hs, h's] #align with_top.Inf_eq WithTop.sInf_eq theorem WithBot.sInf_eq [InfSet α] {s : Set (WithBot α)} (hs : ⊥ ∉ s) (hs' : BddBelow ((↑) ⁻¹' s : Set α)) : sInf s = ↑(sInf ((↑) ⁻¹' s) : α) := (if_neg hs).trans <\| if_pos hs' #align with_bot.Inf_eq WithBot.sInf_eq theorem WithBot.sSup_eq [SupSet α] {s : Set (WithBot α)} (hs : ¬s ⊆ {⊥}) (h's : BddAbove s) : sSup s = ↑(sSup ((↑) ⁻¹' s) : α) := WithTop.sInf_eq (α := αᵒᵈ) hs h's #align with_bot.Sup_eq WithBot.sSup_eq @[simp] theorem WithTop.sInf_empty [InfSet α] : sInf (∅ : Set (WithTop α)) = ⊤ := if_pos <\| by simp #align with_top.cInf_empty WithTop.sInf_empty @[simp] theorem WithTop.iInf_empty [IsEmpty ι] [InfSet α] (f : ι → WithTop α) : ⨅ i, f i = ⊤ := by rw [iInf, range_eq_empty, WithTop.sInf_empty] #align with_top.cinfi_empty WithTop.iInf_empty theorem WithTop.coe_sInf' [InfSet α] {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) : ↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by obtain ⟨x, hx⟩ := hs change _ = ite _ _ _ split_ifs with h · rcases h with h1 \| h2 · cases h1 (mem_image_of_mem _ hx) · exact (h2 (Monotone.map_bddBelow coe_mono h's)).elim · rw [preimage_image_eq] exact Option.some_injective _ #align with_top.coe_Inf' WithTop.coe_sInf' -- Porting note: the mathlib3 proof uses `range_comp` in the opposite direction and -- does not need `rfl`. @[norm_cast] theorem WithTop.coe_iInf [Nonempty ι] [InfSet α] {f : ι → α} (hf : BddBelow (range f)) : ↑(⨅ i, f i) = (⨅ i, f i : WithTop α) := by rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp] rfl #align with_top.coe_infi WithTop.coe_iInf theorem WithTop.coe_sSup' [SupSet α] {s : Set α} (hs : BddAbove s) : ↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by change _ = ite _ _ _ rw [if_neg, preimage_image_eq, if_pos hs] · exact Option.some_injective _ · rintro ⟨x, _, ⟨⟩⟩ #align with_top.coe_Sup' WithTop.coe_sSup' -- Porting note: the mathlib3 proof uses `range_comp` in the opposite direction and -- does not need `rfl`. @[norm_cast]	Mathlib/Order/ConditionallyCompleteLattice/Basic.lean	127	129	theorem WithTop.coe_iSup [SupSet α] (f : ι → α) (h : BddAbove (Set.range f)) : ↑(⨆ i, f i) = (⨆ i, f i : WithTop α) := by	rw [iSup, iSup, WithTop.coe_sSup' h, ← range_comp]; rfl
/- Copyright (c) 2017 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import Mathlib.Data.List.Basic #align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2" /-! # Prefixes, suffixes, infixes This file proves properties about * `List.isPrefix`: `l₁` is a prefix of `l₂` if `l₂` starts with `l₁`. * `List.isSuffix`: `l₁` is a suffix of `l₂` if `l₂` ends with `l₁`. * `List.isInfix`: `l₁` is an infix of `l₂` if `l₁` is a prefix of some suffix of `l₂`. * `List.inits`: The list of prefixes of a list. * `List.tails`: The list of prefixes of a list. * `insert` on lists All those (except `insert`) are defined in `Mathlib.Data.List.Defs`. ## Notation * `l₁ <+: l₂`: `l₁` is a prefix of `l₂`. * `l₁ <:+ l₂`: `l₁` is a suffix of `l₂`. * `l₁ <:+: l₂`: `l₁` is an infix of `l₂`. -/ open Nat variable {α β : Type*} namespace List variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ} /-! ### prefix, suffix, infix -/ section Fix #align list.prefix_append List.prefix_append #align list.suffix_append List.suffix_append #align list.infix_append List.infix_append #align list.infix_append' List.infix_append' #align list.is_prefix.is_infix List.IsPrefix.isInfix #align list.is_suffix.is_infix List.IsSuffix.isInfix #align list.nil_prefix List.nil_prefix #align list.nil_suffix List.nil_suffix #align list.nil_infix List.nil_infix #align list.prefix_refl List.prefix_refl #align list.suffix_refl List.suffix_refl #align list.infix_refl List.infix_refl theorem prefix_rfl : l <+: l := prefix_refl _ #align list.prefix_rfl List.prefix_rfl theorem suffix_rfl : l <:+ l := suffix_refl _ #align list.suffix_rfl List.suffix_rfl theorem infix_rfl : l <:+: l := infix_refl _ #align list.infix_rfl List.infix_rfl #align list.suffix_cons List.suffix_cons theorem prefix_concat (a : α) (l) : l <+: concat l a := by simp #align list.prefix_concat List.prefix_concat	Mathlib/Data/List/Infix.lean	73	76	theorem prefix_concat_iff {l₁ l₂ : List α} {a : α} : l₁ <+: l₂ ++ [a] ↔ l₁ = l₂ ++ [a] ∨ l₁ <+: l₂ := by	simpa only [← reverse_concat', reverse_inj, reverse_suffix] using suffix_cons_iff (l₁ := l₁.reverse) (l₂ := l₂.reverse)
/- Copyright (c) 2019 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import Mathlib.Algebra.GroupWithZero.Divisibility import Mathlib.Algebra.Order.Group.Int import Mathlib.Algebra.Order.Ring.Nat import Mathlib.Algebra.Ring.Rat import Mathlib.Data.PNat.Defs #align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11" /-! # Further lemmas for the Rational Numbers -/ namespace Rat open Rat theorem num_dvd (a) {b : ℤ} (b0 : b ≠ 0) : (a /. b).num ∣ a := by cases' e : a /. b with n d h c rw [Rat.mk'_eq_divInt, divInt_eq_iff b0 (mod_cast h)] at e refine Int.natAbs_dvd.1 <\| Int.dvd_natAbs.1 <\| Int.natCast_dvd_natCast.2 <\| c.dvd_of_dvd_mul_right ?_ have := congr_arg Int.natAbs e simp only [Int.natAbs_mul, Int.natAbs_ofNat] at this; simp [this] #align rat.num_dvd Rat.num_dvd theorem den_dvd (a b : ℤ) : ((a /. b).den : ℤ) ∣ b := by by_cases b0 : b = 0; · simp [b0] cases' e : a /. b with n d h c rw [mk'_eq_divInt, divInt_eq_iff b0 (ne_of_gt (Int.natCast_pos.2 (Nat.pos_of_ne_zero h)))] at e refine Int.dvd_natAbs.1 <\| Int.natCast_dvd_natCast.2 <\| c.symm.dvd_of_dvd_mul_left ?_ rw [← Int.natAbs_mul, ← Int.natCast_dvd_natCast, Int.dvd_natAbs, ← e]; simp #align rat.denom_dvd Rat.den_dvd theorem num_den_mk {q : ℚ} {n d : ℤ} (hd : d ≠ 0) (qdf : q = n /. d) : ∃ c : ℤ, n = c * q.num ∧ d = c * q.den := by obtain rfl \| hn := eq_or_ne n 0 · simp [qdf] have : q.num * d = n * ↑q.den := by refine (divInt_eq_iff ?_ hd).mp ?_ · exact Int.natCast_ne_zero.mpr (Rat.den_nz _) · rwa [num_divInt_den] have hqdn : q.num ∣ n := by rw [qdf] exact Rat.num_dvd _ hd refine ⟨n / q.num, ?_, ?_⟩ · rw [Int.ediv_mul_cancel hqdn] · refine Int.eq_mul_div_of_mul_eq_mul_of_dvd_left ?_ hqdn this rw [qdf] exact Rat.num_ne_zero.2 ((divInt_ne_zero hd).mpr hn) #align rat.num_denom_mk Rat.num_den_mk #noalign rat.mk_pnat_num #noalign rat.mk_pnat_denom theorem num_mk (n d : ℤ) : (n /. d).num = d.sign * n / n.gcd d := by have (m : ℕ) : Int.natAbs (m + 1) = m + 1 := by rw [← Nat.cast_one, ← Nat.cast_add, Int.natAbs_cast] rcases d with ((_ \| _) \| _) <;> rw [← Int.div_eq_ediv_of_dvd] <;> simp [divInt, mkRat, Rat.normalize, Nat.succPNat, Int.sign, Int.gcd, Int.zero_ediv, Int.ofNat_dvd_left, Nat.gcd_dvd_left, this] #align rat.num_mk Rat.num_mk theorem den_mk (n d : ℤ) : (n /. d).den = if d = 0 then 1 else d.natAbs / n.gcd d := by have (m : ℕ) : Int.natAbs (m + 1) = m + 1 := by rw [← Nat.cast_one, ← Nat.cast_add, Int.natAbs_cast] rcases d with ((_ \| _) \| _) <;> simp [divInt, mkRat, Rat.normalize, Nat.succPNat, Int.sign, Int.gcd, if_neg (Nat.cast_add_one_ne_zero _), this] #align rat.denom_mk Rat.den_mk #noalign rat.mk_pnat_denom_dvd theorem add_den_dvd (q₁ q₂ : ℚ) : (q₁ + q₂).den ∣ q₁.den * q₂.den := by rw [add_def, normalize_eq] apply Nat.div_dvd_of_dvd apply Nat.gcd_dvd_right #align rat.add_denom_dvd Rat.add_den_dvd theorem mul_den_dvd (q₁ q₂ : ℚ) : (q₁ * q₂).den ∣ q₁.den * q₂.den := by rw [mul_def, normalize_eq] apply Nat.div_dvd_of_dvd apply Nat.gcd_dvd_right #align rat.mul_denom_dvd Rat.mul_den_dvd theorem mul_num (q₁ q₂ : ℚ) : (q₁ * q₂).num = q₁.num * q₂.num / Nat.gcd (q₁.num * q₂.num).natAbs (q₁.den * q₂.den) := by rw [mul_def, normalize_eq] #align rat.mul_num Rat.mul_num theorem mul_den (q₁ q₂ : ℚ) : (q₁ * q₂).den = q₁.den * q₂.den / Nat.gcd (q₁.num * q₂.num).natAbs (q₁.den * q₂.den) := by rw [mul_def, normalize_eq] #align rat.mul_denom Rat.mul_den theorem mul_self_num (q : ℚ) : (q * q).num = q.num * q.num := by rw [mul_num, Int.natAbs_mul, Nat.Coprime.gcd_eq_one, Int.ofNat_one, Int.ediv_one] exact (q.reduced.mul_right q.reduced).mul (q.reduced.mul_right q.reduced) #align rat.mul_self_num Rat.mul_self_num theorem mul_self_den (q : ℚ) : (q * q).den = q.den * q.den := by rw [Rat.mul_den, Int.natAbs_mul, Nat.Coprime.gcd_eq_one, Nat.div_one] exact (q.reduced.mul_right q.reduced).mul (q.reduced.mul_right q.reduced) #align rat.mul_self_denom Rat.mul_self_den theorem add_num_den (q r : ℚ) : q + r = (q.num * r.den + q.den * r.num : ℤ) /. (↑q.den * ↑r.den : ℤ) := by have hqd : (q.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 q.den_pos have hrd : (r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 r.den_pos conv_lhs => rw [← num_divInt_den q, ← num_divInt_den r, divInt_add_divInt _ _ hqd hrd] rw [mul_comm r.num q.den] #align rat.add_num_denom Rat.add_num_den section Casts theorem exists_eq_mul_div_num_and_eq_mul_div_den (n : ℤ) {d : ℤ} (d_ne_zero : d ≠ 0) : ∃ c : ℤ, n = c * ((n : ℚ) / d).num ∧ (d : ℤ) = c * ((n : ℚ) / d).den := haveI : (n : ℚ) / d = Rat.divInt n d := by rw [← Rat.divInt_eq_div] Rat.num_den_mk d_ne_zero this #align rat.exists_eq_mul_div_num_and_eq_mul_div_denom Rat.exists_eq_mul_div_num_and_eq_mul_div_den theorem mul_num_den' (q r : ℚ) : (q * r).num * q.den * r.den = q.num * r.num * (q * r).den := by let s := q.num * r.num /. (q.den * r.den : ℤ) have hs : (q.den * r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.mpr (mul_pos q.pos r.pos) obtain ⟨c, ⟨c_mul_num, c_mul_den⟩⟩ := exists_eq_mul_div_num_and_eq_mul_div_den (q.num * r.num) hs rw [c_mul_num, mul_assoc, mul_comm] nth_rw 1 [c_mul_den] rw [Int.mul_assoc, Int.mul_assoc, mul_eq_mul_left_iff, or_iff_not_imp_right] intro have h : _ = s := divInt_mul_divInt q.num r.num (mod_cast q.den_ne_zero) (mod_cast r.den_ne_zero) rw [num_divInt_den, num_divInt_den] at h rw [h, mul_comm, ← Rat.eq_iff_mul_eq_mul, ← divInt_eq_div] #align rat.mul_num_denom' Rat.mul_num_den' theorem add_num_den' (q r : ℚ) : (q + r).num * q.den * r.den = (q.num * r.den + r.num * q.den) * (q + r).den := by let s := divInt (q.num * r.den + r.num * q.den) (q.den * r.den : ℤ) have hs : (q.den * r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.mpr (mul_pos q.pos r.pos) obtain ⟨c, ⟨c_mul_num, c_mul_den⟩⟩ := exists_eq_mul_div_num_and_eq_mul_div_den (q.num * r.den + r.num * q.den) hs rw [c_mul_num, mul_assoc, mul_comm] nth_rw 1 [c_mul_den] repeat rw [Int.mul_assoc] apply mul_eq_mul_left_iff.2 rw [or_iff_not_imp_right] intro have h : _ = s := divInt_add_divInt q.num r.num (mod_cast q.den_ne_zero) (mod_cast r.den_ne_zero) rw [num_divInt_den, num_divInt_den] at h rw [h] rw [mul_comm] apply Rat.eq_iff_mul_eq_mul.mp rw [← divInt_eq_div] #align rat.add_num_denom' Rat.add_num_den' theorem substr_num_den' (q r : ℚ) : (q - r).num * q.den * r.den = (q.num * r.den - r.num * q.den) * (q - r).den := by rw [sub_eq_add_neg, sub_eq_add_neg, ← neg_mul, ← num_neg_eq_neg_num, ← den_neg_eq_den r, add_num_den' q (-r)] #align rat.substr_num_denom' Rat.substr_num_den' end Casts protected theorem inv_neg (q : ℚ) : (-q)⁻¹ = -q⁻¹ := by rw [← num_divInt_den q] simp only [Rat.neg_divInt, Rat.inv_divInt', eq_self_iff_true, Rat.divInt_neg] #align rat.inv_neg Rat.inv_neg theorem num_div_eq_of_coprime {a b : ℤ} (hb0 : 0 < b) (h : Nat.Coprime a.natAbs b.natAbs) : (a / b : ℚ).num = a := by -- Porting note: was `lift b to ℕ using le_of_lt hb0` rw [← Int.natAbs_of_nonneg hb0.le, ← Rat.divInt_eq_div, ← mk_eq_divInt _ _ (Int.natAbs_ne_zero.mpr hb0.ne') h] #align rat.num_div_eq_of_coprime Rat.num_div_eq_of_coprime theorem den_div_eq_of_coprime {a b : ℤ} (hb0 : 0 < b) (h : Nat.Coprime a.natAbs b.natAbs) : ((a / b : ℚ).den : ℤ) = b := by -- Porting note: was `lift b to ℕ using le_of_lt hb0` rw [← Int.natAbs_of_nonneg hb0.le, ← Rat.divInt_eq_div, ← mk_eq_divInt _ _ (Int.natAbs_ne_zero.mpr hb0.ne') h] #align rat.denom_div_eq_of_coprime Rat.den_div_eq_of_coprime theorem div_int_inj {a b c d : ℤ} (hb0 : 0 < b) (hd0 : 0 < d) (h1 : Nat.Coprime a.natAbs b.natAbs) (h2 : Nat.Coprime c.natAbs d.natAbs) (h : (a : ℚ) / b = (c : ℚ) / d) : a = c ∧ b = d := by apply And.intro · rw [← num_div_eq_of_coprime hb0 h1, h, num_div_eq_of_coprime hd0 h2] · rw [← den_div_eq_of_coprime hb0 h1, h, den_div_eq_of_coprime hd0 h2] #align rat.div_int_inj Rat.div_int_inj @[norm_cast] theorem intCast_div_self (n : ℤ) : ((n / n : ℤ) : ℚ) = n / n := by by_cases hn : n = 0 · subst hn simp only [Int.cast_zero, Int.zero_div, zero_div, Int.ediv_zero] · have : (n : ℚ) ≠ 0 := by rwa [← coe_int_inj] at hn simp only [Int.ediv_self hn, Int.cast_one, Ne, not_false_iff, div_self this] #align rat.coe_int_div_self Rat.intCast_div_self @[norm_cast] theorem natCast_div_self (n : ℕ) : ((n / n : ℕ) : ℚ) = n / n := intCast_div_self n #align rat.coe_nat_div_self Rat.natCast_div_self theorem intCast_div (a b : ℤ) (h : b ∣ a) : ((a / b : ℤ) : ℚ) = a / b := by rcases h with ⟨c, rfl⟩ rw [mul_comm b, Int.mul_ediv_assoc c (dvd_refl b), Int.cast_mul, intCast_div_self, Int.cast_mul, mul_div_assoc] #align rat.coe_int_div Rat.intCast_div theorem natCast_div (a b : ℕ) (h : b ∣ a) : ((a / b : ℕ) : ℚ) = a / b := intCast_div a b (Int.ofNat_dvd.mpr h) #align rat.coe_nat_div Rat.natCast_div	Mathlib/Data/Rat/Lemmas.lean	223	228	theorem den_div_intCast_eq_one_iff (m n : ℤ) (hn : n ≠ 0) : ((m : ℚ) / n).den = 1 ↔ n ∣ m := by	replace hn : (n : ℚ) ≠ 0 := num_ne_zero.mp hn constructor · rw [Rat.den_eq_one_iff, eq_div_iff hn] exact mod_cast (Dvd.intro_left _) · exact (intCast_div _ _ · ▸ rfl)
/- Copyright (c) 2019 Jan-David Salchow. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo -/ import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm import Mathlib.Analysis.NormedSpace.Span /-! # Operator norm for maps on normed spaces This file contains statements about operator norm for which it really matters that the underlying space has a norm (rather than just a seminorm). -/ suppress_compilation open Bornology open Filter hiding map_smul open scoped Classical NNReal Topology Uniformity -- the `ₗ` subscript variables are for special cases about linear (as opposed to semilinear) maps variable {𝕜 𝕜₂ 𝕜₃ E Eₗ F Fₗ G Gₗ 𝓕 : Type} section Normed variable [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] [NormedAddCommGroup Fₗ] open Metric ContinuousLinearMap section variable [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] [NormedSpace 𝕜 Fₗ] (c : 𝕜) {σ₁₂ : 𝕜 →+ 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} (f g : E →SL[σ₁₂] F) (x y z : E) namespace LinearMap	Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean	42	46	theorem bound_of_shell [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) : ‖f x‖ ≤ C * ‖x‖ := by	by_cases hx : x = 0; · simp [hx] exact SemilinearMapClass.bound_of_shell_semi_normed f ε_pos hc hf (norm_ne_zero_iff.2 hx)
/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis -/ import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.LinearAlgebra.SesquilinearForm #align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9" /-! # Orthogonal complements of submodules In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established. Some of the more subtle results about the orthogonal complement are delayed to `Analysis.InnerProductSpace.Projection`. See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form. ## Notation The orthogonal complement of a submodule `K` is denoted by `Kᗮ`. The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U ⟂ V`. Note this is not the same unicode symbol as `⊥` (`Bot`). -/ 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) /-- The subspace of vectors orthogonal to a given subspace. -/ 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 /-- When a vector is in `Kᗮ`. -/ theorem mem_orthogonal (v : E) : v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪u, v⟫ = 0 := Iff.rfl #align submodule.mem_orthogonal Submodule.mem_orthogonal /-- When a vector is in `Kᗮ`, with the inner product the other way round. -/ 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} /-- A vector in `K` is orthogonal to one in `Kᗮ`. -/ 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 /-- A vector in `Kᗮ` is orthogonal to one in `K`. -/ 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 /-- A vector is in `(𝕜 ∙ u)ᗮ` iff it is orthogonal to `u`. -/ 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 /-- A vector in `(𝕜 ∙ u)ᗮ` is orthogonal to `u`. -/ 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 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⟩ #align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : ∀ v : K, ⟪(v : E), x⟫ = ⟪(v : E), y⟫) : x - y ∈ Kᗮ := by intro u hu rw [inner_sub_right, sub_eq_zero] exact h ⟨u, hu⟩ #align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right variable (K) /-- `K` and `Kᗮ` have trivial intersection. -/	Mathlib/Analysis/InnerProductSpace/Orthogonal.lean	103	107	theorem inf_orthogonal_eq_bot : K ⊓ Kᗮ = ⊥ := by	rw [eq_bot_iff] intro x rw [mem_inf] exact fun ⟨hx, ho⟩ => inner_self_eq_zero.1 (ho x hx)
/- Copyright (c) 2022 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Probability.Variance import Mathlib.MeasureTheory.Function.UniformIntegrable #align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Identically distributed random variables Two random variables defined on two (possibly different) probability spaces but taking value in the same space are identically distributed if their distributions (i.e., the image probability measures on the target space) coincide. We define this concept and establish its basic properties in this file. ## Main definitions and results * `IdentDistrib f g μ ν` registers that the image of `μ` under `f` coincides with the image of `ν` under `g` (and that `f` and `g` are almost everywhere measurable, as otherwise the image measures don't make sense). The measures can be kept implicit as in `IdentDistrib f g` if the spaces are registered as measure spaces. * `IdentDistrib.comp`: being identically distributed is stable under composition with measurable maps. There are two main kinds of lemmas, under the assumption that `f` and `g` are identically distributed: lemmas saying that two quantities computed for `f` and `g` are the same, and lemmas saying that if `f` has some property then `g` also has it. The first kind is registered as `IdentDistrib.foo_fst`, the second one as `IdentDistrib.foo_snd` (in the latter case, to deduce a property of `f` from one of `g`, use `h.symm.foo_snd` where `h : IdentDistrib f g μ ν`). For instance: * `IdentDistrib.measure_mem_eq`: if `f` and `g` are identically distributed, then the probabilities that they belong to a given measurable set are the same. * `IdentDistrib.integral_eq`: if `f` and `g` are identically distributed, then their integrals are the same. * `IdentDistrib.variance_eq`: if `f` and `g` are identically distributed, then their variances are the same. * `IdentDistrib.aestronglyMeasurable_snd`: if `f` and `g` are identically distributed and `f` is almost everywhere strongly measurable, then so is `g`. * `IdentDistrib.memℒp_snd`: if `f` and `g` are identically distributed and `f` belongs to `ℒp`, then so does `g`. We also register several dot notation shortcuts for convenience. For instance, if `h : IdentDistrib f g μ ν`, then `h.sq` states that `f^2` and `g^2` are identically distributed, and `h.norm` states that `‖f‖` and `‖g‖` are identically distributed, and so on. -/ open MeasureTheory Filter Finset noncomputable section open scoped Topology MeasureTheory ENNReal NNReal variable {α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] namespace ProbabilityTheory /-- Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide. This only makes sense when the functions are ae measurable (as otherwise the image measures are not defined), so we require this as well in the definition. -/ structure IdentDistrib (f : α → γ) (g : β → γ) (μ : Measure α := by volume_tac) (ν : Measure β := by volume_tac) : Prop where aemeasurable_fst : AEMeasurable f μ aemeasurable_snd : AEMeasurable g ν map_eq : Measure.map f μ = Measure.map g ν #align probability_theory.ident_distrib ProbabilityTheory.IdentDistrib namespace IdentDistrib open TopologicalSpace variable {μ : Measure α} {ν : Measure β} {f : α → γ} {g : β → γ} protected theorem refl (hf : AEMeasurable f μ) : IdentDistrib f f μ μ := { aemeasurable_fst := hf aemeasurable_snd := hf map_eq := rfl } #align probability_theory.ident_distrib.refl ProbabilityTheory.IdentDistrib.refl protected theorem symm (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ := { aemeasurable_fst := h.aemeasurable_snd aemeasurable_snd := h.aemeasurable_fst map_eq := h.map_eq.symm } #align probability_theory.ident_distrib.symm ProbabilityTheory.IdentDistrib.symm protected theorem trans {ρ : Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν) (h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ := { aemeasurable_fst := h₁.aemeasurable_fst aemeasurable_snd := h₂.aemeasurable_snd map_eq := h₁.map_eq.trans h₂.map_eq } #align probability_theory.ident_distrib.trans ProbabilityTheory.IdentDistrib.trans protected theorem comp_of_aemeasurable {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν := { aemeasurable_fst := hu.comp_aemeasurable h.aemeasurable_fst aemeasurable_snd := by rw [h.map_eq] at hu; exact hu.comp_aemeasurable h.aemeasurable_snd map_eq := by rw [← AEMeasurable.map_map_of_aemeasurable hu h.aemeasurable_fst, ← AEMeasurable.map_map_of_aemeasurable _ h.aemeasurable_snd, h.map_eq] rwa [← h.map_eq] } #align probability_theory.ident_distrib.comp_of_ae_measurable ProbabilityTheory.IdentDistrib.comp_of_aemeasurable protected theorem comp {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) : IdentDistrib (u ∘ f) (u ∘ g) μ ν := h.comp_of_aemeasurable hu.aemeasurable #align probability_theory.ident_distrib.comp ProbabilityTheory.IdentDistrib.comp protected theorem of_ae_eq {g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) : IdentDistrib f g μ μ := { aemeasurable_fst := hf aemeasurable_snd := hf.congr heq map_eq := Measure.map_congr heq } #align probability_theory.ident_distrib.of_ae_eq ProbabilityTheory.IdentDistrib.of_ae_eq lemma _root_.MeasureTheory.AEMeasurable.identDistrib_mk (hf : AEMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ := IdentDistrib.of_ae_eq hf hf.ae_eq_mk lemma _root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ] (hf : AEStronglyMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ := IdentDistrib.of_ae_eq hf.aemeasurable hf.ae_eq_mk theorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) : μ (f ⁻¹' s) = ν (g ⁻¹' s) := by rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ← Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq] #align probability_theory.ident_distrib.measure_mem_eq ProbabilityTheory.IdentDistrib.measure_mem_eq alias measure_preimage_eq := measure_mem_eq #align probability_theory.ident_distrib.measure_preimage_eq ProbabilityTheory.IdentDistrib.measure_preimage_eq theorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x \| p x}) (hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by apply (ae_map_iff h.aemeasurable_snd pmeas).1 rw [← h.map_eq] exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp #align probability_theory.ident_distrib.ae_snd ProbabilityTheory.IdentDistrib.ae_snd theorem ae_mem_snd (h : IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t) (ht : ∀ᵐ x ∂μ, f x ∈ t) : ∀ᵐ x ∂ν, g x ∈ t := h.ae_snd tmeas ht #align probability_theory.ident_distrib.ae_mem_snd ProbabilityTheory.IdentDistrib.ae_mem_snd /-- In a second countable topology, the first function in an identically distributed pair is a.e. strongly measurable. So is the second function, but use `h.symm.aestronglyMeasurable_fst` as `h.aestronglyMeasurable_snd` has a different meaning. -/ theorem aestronglyMeasurable_fst [TopologicalSpace γ] [MetrizableSpace γ] [OpensMeasurableSpace γ] [SecondCountableTopology γ] (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ := h.aemeasurable_fst.aestronglyMeasurable #align probability_theory.ident_distrib.ae_strongly_measurable_fst ProbabilityTheory.IdentDistrib.aestronglyMeasurable_fst /-- If `f` and `g` are identically distributed and `f` is a.e. strongly measurable, so is `g`. -/ theorem aestronglyMeasurable_snd [TopologicalSpace γ] [MetrizableSpace γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν := by refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩ rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩ refine ⟨closure t, t_sep.closure, ?_⟩ apply h.ae_mem_snd isClosed_closure.measurableSet filter_upwards [ht] with x hx using subset_closure hx #align probability_theory.ident_distrib.ae_strongly_measurable_snd ProbabilityTheory.IdentDistrib.aestronglyMeasurable_snd theorem aestronglyMeasurable_iff [TopologicalSpace γ] [MetrizableSpace γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g ν := ⟨fun hf => h.aestronglyMeasurable_snd hf, fun hg => h.symm.aestronglyMeasurable_snd hg⟩ #align probability_theory.ident_distrib.ae_strongly_measurable_iff ProbabilityTheory.IdentDistrib.aestronglyMeasurable_iff theorem essSup_eq [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ] [OrderClosedTopology γ] (h : IdentDistrib f g μ ν) : essSup f μ = essSup g ν := by have I : ∀ a, μ {x : α \| a < f x} = ν {x : β \| a < g x} := fun a => h.measure_mem_eq measurableSet_Ioi simp_rw [essSup_eq_sInf, I] #align probability_theory.ident_distrib.ess_sup_eq ProbabilityTheory.IdentDistrib.essSup_eq theorem lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) : ∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν := by change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ← lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq] #align probability_theory.ident_distrib.lintegral_eq ProbabilityTheory.IdentDistrib.lintegral_eq theorem integral_eq [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) : ∫ x, f x ∂μ = ∫ x, g x ∂ν := by by_cases hf : AEStronglyMeasurable f μ · have A : AEStronglyMeasurable id (Measure.map f μ) := by rw [aestronglyMeasurable_iff_aemeasurable_separable] rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩ refine ⟨aemeasurable_id, ⟨closure t, t_sep.closure, ?_⟩⟩ rw [ae_map_iff h.aemeasurable_fst] · filter_upwards [ht] with x hx using subset_closure hx · exact isClosed_closure.measurableSet change ∫ x, id (f x) ∂μ = ∫ x, id (g x) ∂ν rw [← integral_map h.aemeasurable_fst A] rw [h.map_eq] at A rw [← integral_map h.aemeasurable_snd A, h.map_eq] · rw [integral_non_aestronglyMeasurable hf] rw [h.aestronglyMeasurable_iff] at hf rw [integral_non_aestronglyMeasurable hf] #align probability_theory.ident_distrib.integral_eq ProbabilityTheory.IdentDistrib.integral_eq	Mathlib/Probability/IdentDistrib.lean	209	221	theorem snorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν) (p : ℝ≥0∞) : snorm f p μ = snorm g p ν := by	by_cases h0 : p = 0 · simp [h0] by_cases h_top : p = ∞ · simp only [h_top, snorm, snormEssSup, ENNReal.top_ne_zero, eq_self_iff_true, if_true, if_false] apply essSup_eq exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm) simp only [snorm_eq_snorm' h0 h_top, snorm', one_div] congr 1 apply lintegral_eq exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm) p.toReal)
/- Copyright (c) 2020 Aaron Anderson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang -/ import Mathlib.Combinatorics.SimpleGraph.Basic import Mathlib.Combinatorics.SimpleGraph.Connectivity import Mathlib.LinearAlgebra.Matrix.Trace import Mathlib.LinearAlgebra.Matrix.Symmetric #align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1" /-! # Adjacency Matrices This module defines the adjacency matrix of a graph, and provides theorems connecting graph properties to computational properties of the matrix. ## Main definitions * `Matrix.IsAdjMatrix`: `A : Matrix V V α` is qualified as an "adjacency matrix" if (1) every entry of `A` is `0` or `1`, (2) `A` is symmetric, (3) every diagonal entry of `A` is `0`. * `Matrix.IsAdjMatrix.to_graph`: for `A : Matrix V V α` and `h : A.IsAdjMatrix`, `h.to_graph` is the simple graph induced by `A`. * `Matrix.compl`: for `A : Matrix V V α`, `A.compl` is supposed to be the adjacency matrix of the complement graph of the graph induced by `A`. * `SimpleGraph.adjMatrix`: the adjacency matrix of a `SimpleGraph`. * `SimpleGraph.adjMatrix_pow_apply_eq_card_walk`: each entry of the `n`th power of a graph's adjacency matrix counts the number of length-`n` walks between the corresponding pair of vertices. -/ open Matrix open Finset Matrix SimpleGraph variable {V α β : Type} namespace Matrix /-- `A : Matrix V V α` is qualified as an "adjacency matrix" if (1) every entry of `A` is `0` or `1`, (2) `A` is symmetric, (3) every diagonal entry of `A` is `0`. -/ structure IsAdjMatrix [Zero α] [One α] (A : Matrix V V α) : Prop where zero_or_one : ∀ i j, A i j = 0 ∨ A i j = 1 := by aesop symm : A.IsSymm := by aesop apply_diag : ∀ i, A i i = 0 := by aesop #align matrix.is_adj_matrix Matrix.IsAdjMatrix namespace IsAdjMatrix variable {A : Matrix V V α} @[simp] theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) : ¬A i i = 1 := by simp [h.apply_diag i] #align matrix.is_adj_matrix.apply_diag_ne Matrix.IsAdjMatrix.apply_diag_ne @[simp] theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) : ¬A i j = 1 ↔ A i j = 0 := by obtain h \| h := h.zero_or_one i j <;> simp [h] #align matrix.is_adj_matrix.apply_ne_one_iff Matrix.IsAdjMatrix.apply_ne_one_iff @[simp] theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) : ¬A i j = 0 ↔ A i j = 1 := by rw [← apply_ne_one_iff h, Classical.not_not] #align matrix.is_adj_matrix.apply_ne_zero_iff Matrix.IsAdjMatrix.apply_ne_zero_iff /-- For `A : Matrix V V α` and `h : IsAdjMatrix A`, `h.toGraph` is the simple graph whose adjacency matrix is `A`. -/ @[simps] def toGraph [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) : SimpleGraph V where Adj i j := A i j = 1 symm i j hij := by simp only; rwa [h.symm.apply i j] loopless i := by simp [h] #align matrix.is_adj_matrix.to_graph Matrix.IsAdjMatrix.toGraph instance [MulZeroOneClass α] [Nontrivial α] [DecidableEq α] (h : IsAdjMatrix A) : DecidableRel h.toGraph.Adj := by simp only [toGraph] infer_instance end IsAdjMatrix /-- For `A : Matrix V V α`, `A.compl` is supposed to be the adjacency matrix of the complement graph of the graph induced by `A.adjMatrix`. -/ def compl [Zero α] [One α] [DecidableEq α] [DecidableEq V] (A : Matrix V V α) : Matrix V V α := fun i j => ite (i = j) 0 (ite (A i j = 0) 1 0) #align matrix.compl Matrix.compl section Compl variable [DecidableEq α] [DecidableEq V] (A : Matrix V V α) @[simp] theorem compl_apply_diag [Zero α] [One α] (i : V) : A.compl i i = 0 := by simp [compl] #align matrix.compl_apply_diag Matrix.compl_apply_diag @[simp] theorem compl_apply [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1 := by unfold compl split_ifs <;> simp #align matrix.compl_apply Matrix.compl_apply @[simp] theorem isSymm_compl [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm := by ext simp [compl, h.apply, eq_comm] #align matrix.is_symm_compl Matrix.isSymm_compl @[simp] theorem isAdjMatrix_compl [Zero α] [One α] (h : A.IsSymm) : IsAdjMatrix A.compl := { symm := by simp [h] } #align matrix.is_adj_matrix_compl Matrix.isAdjMatrix_compl namespace IsAdjMatrix variable {A} @[simp] theorem compl [Zero α] [One α] (h : IsAdjMatrix A) : IsAdjMatrix A.compl := isAdjMatrix_compl A h.symm #align matrix.is_adj_matrix.compl Matrix.IsAdjMatrix.compl theorem toGraph_compl_eq [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) : h.compl.toGraph = h.toGraphᶜ := by ext v w cases' h.zero_or_one v w with h h <;> by_cases hvw : v = w <;> simp [Matrix.compl, h, hvw] #align matrix.is_adj_matrix.to_graph_compl_eq Matrix.IsAdjMatrix.toGraph_compl_eq end IsAdjMatrix end Compl end Matrix open Matrix namespace SimpleGraph variable (G : SimpleGraph V) [DecidableRel G.Adj] variable (α) /-- `adjMatrix G α` is the matrix `A` such that `A i j = (1 : α)` if `i` and `j` are adjacent in the simple graph `G`, and otherwise `A i j = 0`. -/ def adjMatrix [Zero α] [One α] : Matrix V V α := of fun i j => if G.Adj i j then (1 : α) else 0 #align simple_graph.adj_matrix SimpleGraph.adjMatrix variable {α} -- TODO: set as an equation lemma for `adjMatrix`, see mathlib4#3024 @[simp] theorem adjMatrix_apply (v w : V) [Zero α] [One α] : G.adjMatrix α v w = if G.Adj v w then 1 else 0 := rfl #align simple_graph.adj_matrix_apply SimpleGraph.adjMatrix_apply @[simp] theorem transpose_adjMatrix [Zero α] [One α] : (G.adjMatrix α)ᵀ = G.adjMatrix α := by ext simp [adj_comm] #align simple_graph.transpose_adj_matrix SimpleGraph.transpose_adjMatrix @[simp] theorem isSymm_adjMatrix [Zero α] [One α] : (G.adjMatrix α).IsSymm := transpose_adjMatrix G #align simple_graph.is_symm_adj_matrix SimpleGraph.isSymm_adjMatrix variable (α) /-- The adjacency matrix of `G` is an adjacency matrix. -/ @[simp] theorem isAdjMatrix_adjMatrix [Zero α] [One α] : (G.adjMatrix α).IsAdjMatrix := { zero_or_one := fun i j => by by_cases h : G.Adj i j <;> simp [h] } #align simple_graph.is_adj_matrix_adj_matrix SimpleGraph.isAdjMatrix_adjMatrix /-- The graph induced by the adjacency matrix of `G` is `G` itself. -/ theorem toGraph_adjMatrix_eq [MulZeroOneClass α] [Nontrivial α] : (G.isAdjMatrix_adjMatrix α).toGraph = G := by ext simp only [IsAdjMatrix.toGraph_adj, adjMatrix_apply, ite_eq_left_iff, zero_ne_one] apply Classical.not_not #align simple_graph.to_graph_adj_matrix_eq SimpleGraph.toGraph_adjMatrix_eq variable {α} [Fintype V] @[simp] theorem adjMatrix_dotProduct [NonAssocSemiring α] (v : V) (vec : V → α) : dotProduct (G.adjMatrix α v) vec = ∑ u ∈ G.neighborFinset v, vec u := by simp [neighborFinset_eq_filter, dotProduct, sum_filter] #align simple_graph.adj_matrix_dot_product SimpleGraph.adjMatrix_dotProduct @[simp] theorem dotProduct_adjMatrix [NonAssocSemiring α] (v : V) (vec : V → α) : dotProduct vec (G.adjMatrix α v) = ∑ u ∈ G.neighborFinset v, vec u := by simp [neighborFinset_eq_filter, dotProduct, sum_filter, Finset.sum_apply] #align simple_graph.dot_product_adj_matrix SimpleGraph.dotProduct_adjMatrix @[simp] theorem adjMatrix_mulVec_apply [NonAssocSemiring α] (v : V) (vec : V → α) : (G.adjMatrix α ᵥ vec) v = ∑ u ∈ G.neighborFinset v, vec u := by rw [mulVec, adjMatrix_dotProduct] #align simple_graph.adj_matrix_mul_vec_apply SimpleGraph.adjMatrix_mulVec_apply @[simp] theorem adjMatrix_vecMul_apply [NonAssocSemiring α] (v : V) (vec : V → α) : (vec ᵥ* G.adjMatrix α) v = ∑ u ∈ G.neighborFinset v, vec u := by simp only [← dotProduct_adjMatrix, vecMul] refine congr rfl ?_; ext x rw [← transpose_apply (adjMatrix α G) x v, transpose_adjMatrix] #align simple_graph.adj_matrix_vec_mul_apply SimpleGraph.adjMatrix_vecMul_apply @[simp] theorem adjMatrix_mul_apply [NonAssocSemiring α] (M : Matrix V V α) (v w : V) : (G.adjMatrix α * M) v w = ∑ u ∈ G.neighborFinset v, M u w := by simp [mul_apply, neighborFinset_eq_filter, sum_filter] #align simple_graph.adj_matrix_mul_apply SimpleGraph.adjMatrix_mul_apply @[simp] theorem mul_adjMatrix_apply [NonAssocSemiring α] (M : Matrix V V α) (v w : V) : (M * G.adjMatrix α) v w = ∑ u ∈ G.neighborFinset w, M v u := by simp [mul_apply, neighborFinset_eq_filter, sum_filter, adj_comm] #align simple_graph.mul_adj_matrix_apply SimpleGraph.mul_adjMatrix_apply variable (α) @[simp] theorem trace_adjMatrix [AddCommMonoid α] [One α] : Matrix.trace (G.adjMatrix α) = 0 := by simp [Matrix.trace] #align simple_graph.trace_adj_matrix SimpleGraph.trace_adjMatrix variable {α} theorem adjMatrix_mul_self_apply_self [NonAssocSemiring α] (i : V) : (G.adjMatrix α * G.adjMatrix α) i i = degree G i := by simp [filter_true_of_mem] #align simple_graph.adj_matrix_mul_self_apply_self SimpleGraph.adjMatrix_mul_self_apply_self variable {G} -- @[simp] -- Porting note (#10618): simp can prove this theorem adjMatrix_mulVec_const_apply [NonAssocSemiring α] {a : α} {v : V} : (G.adjMatrix α ᵥ Function.const _ a) v = G.degree v a := by simp #align simple_graph.adj_matrix_mul_vec_const_apply SimpleGraph.adjMatrix_mulVec_const_apply theorem adjMatrix_mulVec_const_apply_of_regular [NonAssocSemiring α] {d : ℕ} {a : α} (hd : G.IsRegularOfDegree d) {v : V} : (G.adjMatrix α ᵥ Function.const _ a) v = d a := by simp [hd v] #align simple_graph.adj_matrix_mul_vec_const_apply_of_regular SimpleGraph.adjMatrix_mulVec_const_apply_of_regular	Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean	260	277	theorem adjMatrix_pow_apply_eq_card_walk [DecidableEq V] [Semiring α] (n : ℕ) (u v : V) : (G.adjMatrix α ^ n) u v = Fintype.card { p : G.Walk u v \| p.length = n } := by	rw [card_set_walk_length_eq] induction' n with n ih generalizing u v · obtain rfl \| h := eq_or_ne u v <;> simp [finsetWalkLength, *] · simp only [pow_succ', finsetWalkLength, ih, adjMatrix_mul_apply] rw [Finset.card_biUnion] · norm_cast simp only [Nat.cast_sum, card_map, neighborFinset_def] apply Finset.sum_toFinset_eq_subtype -- Disjointness for card_bUnion · rintro ⟨x, hx⟩ - ⟨y, hy⟩ - hxy rw [disjoint_iff_inf_le] intro p hp simp only [inf_eq_inter, mem_inter, mem_map, Function.Embedding.coeFn_mk, exists_prop] at hp; obtain ⟨⟨px, _, rfl⟩, ⟨py, hpy, hp⟩⟩ := hp cases hp simp at hxy
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Chris Hughes -/ import Mathlib.Algebra.Algebra.Defs import Mathlib.Algebra.Polynomial.FieldDivision import Mathlib.FieldTheory.Minpoly.Basic import Mathlib.RingTheory.Adjoin.Basic import Mathlib.RingTheory.FinitePresentation import Mathlib.RingTheory.FiniteType import Mathlib.RingTheory.PowerBasis import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.RingTheory.QuotientNoetherian #align_import ring_theory.adjoin_root from "leanprover-community/mathlib"@"5c4b3d41a84bd2a1d79c7d9265e58a891e71be89" /-! # Adjoining roots of polynomials This file defines the commutative ring `AdjoinRoot f`, the ring R[X]/(f) obtained from a commutative ring `R` and a polynomial `f : R[X]`. If furthermore `R` is a field and `f` is irreducible, the field structure on `AdjoinRoot f` is constructed. We suggest stating results on `IsAdjoinRoot` instead of `AdjoinRoot` to achieve higher generality, since `IsAdjoinRoot` works for all different constructions of `R[α]` including `AdjoinRoot f = R[X]/(f)` itself. ## Main definitions and results The main definitions are in the `AdjoinRoot` namespace. * `mk f : R[X] →+* AdjoinRoot f`, the natural ring homomorphism. * `of f : R →+* AdjoinRoot f`, the natural ring homomorphism. * `root f : AdjoinRoot f`, the image of X in R[X]/(f). * `lift (i : R →+* S) (x : S) (h : f.eval₂ i x = 0) : (AdjoinRoot f) →+* S`, the ring homomorphism from R[X]/(f) to S extending `i : R →+* S` and sending `X` to `x`. * `lift_hom (x : S) (hfx : aeval x f = 0) : AdjoinRoot f →ₐ[R] S`, the algebra homomorphism from R[X]/(f) to S extending `algebraMap R S` and sending `X` to `x` * `equiv : (AdjoinRoot f →ₐ[F] E) ≃ {x // x ∈ f.aroots E}` a bijection between algebra homomorphisms from `AdjoinRoot` and roots of `f` in `S` -/ noncomputable section open scoped Classical open Polynomial universe u v w variable {R : Type u} {S : Type v} {K : Type w} open Polynomial Ideal /-- Adjoin a root of a polynomial `f` to a commutative ring `R`. We define the new ring as the quotient of `R[X]` by the principal ideal generated by `f`. -/ def AdjoinRoot [CommRing R] (f : R[X]) : Type u := Polynomial R ⧸ (span {f} : Ideal R[X]) #align adjoin_root AdjoinRoot namespace AdjoinRoot section CommRing variable [CommRing R] (f : R[X]) instance instCommRing : CommRing (AdjoinRoot f) := Ideal.Quotient.commRing _ #align adjoin_root.comm_ring AdjoinRoot.instCommRing instance : Inhabited (AdjoinRoot f) := ⟨0⟩ instance : DecidableEq (AdjoinRoot f) := Classical.decEq _ protected theorem nontrivial [IsDomain R] (h : degree f ≠ 0) : Nontrivial (AdjoinRoot f) := Ideal.Quotient.nontrivial (by simp_rw [Ne, span_singleton_eq_top, Polynomial.isUnit_iff, not_exists, not_and] rintro x hx rfl exact h (degree_C hx.ne_zero)) #align adjoin_root.nontrivial AdjoinRoot.nontrivial /-- Ring homomorphism from `R[x]` to `AdjoinRoot f` sending `X` to the `root`. -/ def mk : R[X] →+* AdjoinRoot f := Ideal.Quotient.mk _ #align adjoin_root.mk AdjoinRoot.mk @[elab_as_elim] theorem induction_on {C : AdjoinRoot f → Prop} (x : AdjoinRoot f) (ih : ∀ p : R[X], C (mk f p)) : C x := Quotient.inductionOn' x ih #align adjoin_root.induction_on AdjoinRoot.induction_on /-- Embedding of the original ring `R` into `AdjoinRoot f`. -/ def of : R →+* AdjoinRoot f := (mk f).comp C #align adjoin_root.of AdjoinRoot.of instance instSMulAdjoinRoot [DistribSMul S R] [IsScalarTower S R R] : SMul S (AdjoinRoot f) := Submodule.Quotient.instSMul' _ instance [DistribSMul S R] [IsScalarTower S R R] : DistribSMul S (AdjoinRoot f) := Submodule.Quotient.distribSMul' _ @[simp] theorem smul_mk [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R[X]) : a • mk f x = mk f (a • x) := rfl #align adjoin_root.smul_mk AdjoinRoot.smul_mk theorem smul_of [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R) : a • of f x = of f (a • x) := by rw [of, RingHom.comp_apply, RingHom.comp_apply, smul_mk, smul_C] #align adjoin_root.smul_of AdjoinRoot.smul_of instance (R₁ R₂ : Type) [SMul R₁ R₂] [DistribSMul R₁ R] [DistribSMul R₂ R] [IsScalarTower R₁ R R] [IsScalarTower R₂ R R] [IsScalarTower R₁ R₂ R] (f : R[X]) : IsScalarTower R₁ R₂ (AdjoinRoot f) := Submodule.Quotient.isScalarTower _ _ instance (R₁ R₂ : Type) [DistribSMul R₁ R] [DistribSMul R₂ R] [IsScalarTower R₁ R R] [IsScalarTower R₂ R R] [SMulCommClass R₁ R₂ R] (f : R[X]) : SMulCommClass R₁ R₂ (AdjoinRoot f) := Submodule.Quotient.smulCommClass _ _ instance isScalarTower_right [DistribSMul S R] [IsScalarTower S R R] : IsScalarTower S (AdjoinRoot f) (AdjoinRoot f) := Ideal.Quotient.isScalarTower_right #align adjoin_root.is_scalar_tower_right AdjoinRoot.isScalarTower_right instance [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (f : R[X]) : DistribMulAction S (AdjoinRoot f) := Submodule.Quotient.distribMulAction' _ instance [CommSemiring S] [Algebra S R] : Algebra S (AdjoinRoot f) := Ideal.Quotient.algebra S @[simp] theorem algebraMap_eq : algebraMap R (AdjoinRoot f) = of f := rfl #align adjoin_root.algebra_map_eq AdjoinRoot.algebraMap_eq variable (S) theorem algebraMap_eq' [CommSemiring S] [Algebra S R] : algebraMap S (AdjoinRoot f) = (of f).comp (algebraMap S R) := rfl #align adjoin_root.algebra_map_eq' AdjoinRoot.algebraMap_eq' variable {S} theorem finiteType : Algebra.FiniteType R (AdjoinRoot f) := (Algebra.FiniteType.polynomial R).of_surjective _ (Ideal.Quotient.mkₐ_surjective R _) #align adjoin_root.finite_type AdjoinRoot.finiteType theorem finitePresentation : Algebra.FinitePresentation R (AdjoinRoot f) := (Algebra.FinitePresentation.polynomial R).quotient (Submodule.fg_span_singleton f) #align adjoin_root.finite_presentation AdjoinRoot.finitePresentation /-- The adjoined root. -/ def root : AdjoinRoot f := mk f X #align adjoin_root.root AdjoinRoot.root variable {f} instance hasCoeT : CoeTC R (AdjoinRoot f) := ⟨of f⟩ #align adjoin_root.has_coe_t AdjoinRoot.hasCoeT /-- Two `R`-`AlgHom` from `AdjoinRoot f` to the same `R`-algebra are the same iff they agree on `root f`. -/ @[ext] theorem algHom_ext [Semiring S] [Algebra R S] {g₁ g₂ : AdjoinRoot f →ₐ[R] S} (h : g₁ (root f) = g₂ (root f)) : g₁ = g₂ := Ideal.Quotient.algHom_ext R <\| Polynomial.algHom_ext h #align adjoin_root.alg_hom_ext AdjoinRoot.algHom_ext @[simp] theorem mk_eq_mk {g h : R[X]} : mk f g = mk f h ↔ f ∣ g - h := Ideal.Quotient.eq.trans Ideal.mem_span_singleton #align adjoin_root.mk_eq_mk AdjoinRoot.mk_eq_mk @[simp] theorem mk_eq_zero {g : R[X]} : mk f g = 0 ↔ f ∣ g := mk_eq_mk.trans <\| by rw [sub_zero] #align adjoin_root.mk_eq_zero AdjoinRoot.mk_eq_zero @[simp] theorem mk_self : mk f f = 0 := Quotient.sound' <\| QuotientAddGroup.leftRel_apply.mpr (mem_span_singleton.2 <\| by simp) #align adjoin_root.mk_self AdjoinRoot.mk_self @[simp] theorem mk_C (x : R) : mk f (C x) = x := rfl set_option linter.uppercaseLean3 false in #align adjoin_root.mk_C AdjoinRoot.mk_C @[simp] theorem mk_X : mk f X = root f := rfl set_option linter.uppercaseLean3 false in #align adjoin_root.mk_X AdjoinRoot.mk_X theorem mk_ne_zero_of_degree_lt (hf : Monic f) {g : R[X]} (h0 : g ≠ 0) (hd : degree g < degree f) : mk f g ≠ 0 := mk_eq_zero.not.2 <\| hf.not_dvd_of_degree_lt h0 hd #align adjoin_root.mk_ne_zero_of_degree_lt AdjoinRoot.mk_ne_zero_of_degree_lt theorem mk_ne_zero_of_natDegree_lt (hf : Monic f) {g : R[X]} (h0 : g ≠ 0) (hd : natDegree g < natDegree f) : mk f g ≠ 0 := mk_eq_zero.not.2 <\| hf.not_dvd_of_natDegree_lt h0 hd #align adjoin_root.mk_ne_zero_of_nat_degree_lt AdjoinRoot.mk_ne_zero_of_natDegree_lt @[simp] theorem aeval_eq (p : R[X]) : aeval (root f) p = mk f p := Polynomial.induction_on p (fun x => by rw [aeval_C] rfl) (fun p q ihp ihq => by rw [AlgHom.map_add, RingHom.map_add, ihp, ihq]) fun n x _ => by rw [AlgHom.map_mul, aeval_C, AlgHom.map_pow, aeval_X, RingHom.map_mul, mk_C, RingHom.map_pow, mk_X] rfl #align adjoin_root.aeval_eq AdjoinRoot.aeval_eq -- Porting note: the following proof was partly in term-mode, but I was not able to fix it. theorem adjoinRoot_eq_top : Algebra.adjoin R ({root f} : Set (AdjoinRoot f)) = ⊤ := by refine Algebra.eq_top_iff.2 fun x => ?_ induction x using AdjoinRoot.induction_on with \| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩ #align adjoin_root.adjoin_root_eq_top AdjoinRoot.adjoinRoot_eq_top @[simp] theorem eval₂_root (f : R[X]) : f.eval₂ (of f) (root f) = 0 := by rw [← algebraMap_eq, ← aeval_def, aeval_eq, mk_self] #align adjoin_root.eval₂_root AdjoinRoot.eval₂_root theorem isRoot_root (f : R[X]) : IsRoot (f.map (of f)) (root f) := by rw [IsRoot, eval_map, eval₂_root] #align adjoin_root.is_root_root AdjoinRoot.isRoot_root theorem isAlgebraic_root (hf : f ≠ 0) : IsAlgebraic R (root f) := ⟨f, hf, eval₂_root f⟩ #align adjoin_root.is_algebraic_root AdjoinRoot.isAlgebraic_root theorem of.injective_of_degree_ne_zero [IsDomain R] (hf : f.degree ≠ 0) : Function.Injective (AdjoinRoot.of f) := by rw [injective_iff_map_eq_zero] intro p hp rw [AdjoinRoot.of, RingHom.comp_apply, AdjoinRoot.mk_eq_zero] at hp by_cases h : f = 0 · exact C_eq_zero.mp (eq_zero_of_zero_dvd (by rwa [h] at hp)) · contrapose! hf with h_contra rw [← degree_C h_contra] apply le_antisymm (degree_le_of_dvd hp (by rwa [Ne, C_eq_zero])) _ rwa [degree_C h_contra, zero_le_degree_iff] #align adjoin_root.of.injective_of_degree_ne_zero AdjoinRoot.of.injective_of_degree_ne_zero variable [CommRing S] /-- Lift a ring homomorphism `i : R →+* S` to `AdjoinRoot f →+* S`. -/ def lift (i : R →+* S) (x : S) (h : f.eval₂ i x = 0) : AdjoinRoot f →+* S := by apply Ideal.Quotient.lift _ (eval₂RingHom i x) intro g H rcases mem_span_singleton.1 H with ⟨y, hy⟩ rw [hy, RingHom.map_mul, coe_eval₂RingHom, h, zero_mul] #align adjoin_root.lift AdjoinRoot.lift variable {i : R →+* S} {a : S} (h : f.eval₂ i a = 0) @[simp] theorem lift_mk (g : R[X]) : lift i a h (mk f g) = g.eval₂ i a := Ideal.Quotient.lift_mk _ _ _ #align adjoin_root.lift_mk AdjoinRoot.lift_mk @[simp]	Mathlib/RingTheory/AdjoinRoot.lean	287	287	theorem lift_root : lift i a h (root f) = a := by	rw [root, lift_mk, eval₂_X]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn -/ import Mathlib.Data.Finsupp.Multiset import Mathlib.Order.Bounded import Mathlib.SetTheory.Cardinal.PartENat import Mathlib.SetTheory.Ordinal.Principal import Mathlib.Tactic.Linarith #align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f" /-! # Cardinals and ordinals Relationships between cardinals and ordinals, properties of cardinals that are proved using ordinals. ## Main definitions * The function `Cardinal.aleph'` gives the cardinals listed by their ordinal index, and is the inverse of `Cardinal.aleph/idx`. `aleph' n = n`, `aleph' ω = ℵ₀`, `aleph' (ω + 1) = succ ℵ₀`, etc. It is an order isomorphism between ordinals and cardinals. * The function `Cardinal.aleph` gives the infinite cardinals listed by their ordinal index. `aleph 0 = ℵ₀`, `aleph 1 = succ ℵ₀` is the first uncountable cardinal, and so on. The notation `ω_` combines the latter with `Cardinal.ord`, giving an enumeration of (infinite) initial ordinals. Thus `ω_ 0 = ω` and `ω₁ = ω_ 1` is the first uncountable ordinal. * The function `Cardinal.beth` enumerates the Beth cardinals. `beth 0 = ℵ₀`, `beth (succ o) = 2 ^ beth o`, and for a limit ordinal `o`, `beth o` is the supremum of `beth a` for `a < o`. ## Main Statements * `Cardinal.mul_eq_max` and `Cardinal.add_eq_max` state that the product (resp. sum) of two infinite cardinals is just their maximum. Several variations around this fact are also given. * `Cardinal.mk_list_eq_mk` : when `α` is infinite, `α` and `List α` have the same cardinality. * simp lemmas for inequalities between `bit0 a` and `bit1 b` are registered, making `simp` able to prove inequalities about numeral cardinals. ## Tags cardinal arithmetic (for infinite cardinals) -/ noncomputable section open Function Set Cardinal Equiv Order Ordinal open scoped Classical universe u v w namespace Cardinal section UsingOrdinals theorem ord_isLimit {c} (co : ℵ₀ ≤ c) : (ord c).IsLimit := by refine ⟨fun h => aleph0_ne_zero ?_, fun a => lt_imp_lt_of_le_imp_le fun h => ?_⟩ · rw [← Ordinal.le_zero, ord_le] at h simpa only [card_zero, nonpos_iff_eq_zero] using co.trans h · rw [ord_le] at h ⊢ rwa [← @add_one_of_aleph0_le (card a), ← card_succ] rw [← ord_le, ← le_succ_of_isLimit, ord_le] · exact co.trans h · rw [ord_aleph0] exact omega_isLimit #align cardinal.ord_is_limit Cardinal.ord_isLimit theorem noMaxOrder {c} (h : ℵ₀ ≤ c) : NoMaxOrder c.ord.out.α := Ordinal.out_no_max_of_succ_lt (ord_isLimit h).2 /-! ### Aleph cardinals -/ section aleph /-- The `aleph'` index function, which gives the ordinal index of a cardinal. (The `aleph'` part is because unlike `aleph` this counts also the finite stages. So `alephIdx n = n`, `alephIdx ω = ω`, `alephIdx ℵ₁ = ω + 1` and so on.) In this definition, we register additionally that this function is an initial segment, i.e., it is order preserving and its range is an initial segment of the ordinals. For the basic function version, see `alephIdx`. For an upgraded version stating that the range is everything, see `AlephIdx.rel_iso`. -/ def alephIdx.initialSeg : @InitialSeg Cardinal Ordinal (· < ·) (· < ·) := @RelEmbedding.collapse Cardinal Ordinal (· < ·) (· < ·) _ Cardinal.ord.orderEmbedding.ltEmbedding #align cardinal.aleph_idx.initial_seg Cardinal.alephIdx.initialSeg /-- The `aleph'` index function, which gives the ordinal index of a cardinal. (The `aleph'` part is because unlike `aleph` this counts also the finite stages. So `alephIdx n = n`, `alephIdx ω = ω`, `alephIdx ℵ₁ = ω + 1` and so on.) For an upgraded version stating that the range is everything, see `AlephIdx.rel_iso`. -/ def alephIdx : Cardinal → Ordinal := alephIdx.initialSeg #align cardinal.aleph_idx Cardinal.alephIdx @[simp] theorem alephIdx.initialSeg_coe : (alephIdx.initialSeg : Cardinal → Ordinal) = alephIdx := rfl #align cardinal.aleph_idx.initial_seg_coe Cardinal.alephIdx.initialSeg_coe @[simp] theorem alephIdx_lt {a b} : alephIdx a < alephIdx b ↔ a < b := alephIdx.initialSeg.toRelEmbedding.map_rel_iff #align cardinal.aleph_idx_lt Cardinal.alephIdx_lt @[simp] theorem alephIdx_le {a b} : alephIdx a ≤ alephIdx b ↔ a ≤ b := by rw [← not_lt, ← not_lt, alephIdx_lt] #align cardinal.aleph_idx_le Cardinal.alephIdx_le theorem alephIdx.init {a b} : b < alephIdx a → ∃ c, alephIdx c = b := alephIdx.initialSeg.init #align cardinal.aleph_idx.init Cardinal.alephIdx.init /-- The `aleph'` index function, which gives the ordinal index of a cardinal. (The `aleph'` part is because unlike `aleph` this counts also the finite stages. So `alephIdx n = n`, `alephIdx ℵ₀ = ω`, `alephIdx ℵ₁ = ω + 1` and so on.) In this version, we register additionally that this function is an order isomorphism between cardinals and ordinals. For the basic function version, see `alephIdx`. -/ def alephIdx.relIso : @RelIso Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) := @RelIso.ofSurjective Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) alephIdx.initialSeg.{u} <\| (InitialSeg.eq_or_principal alephIdx.initialSeg.{u}).resolve_right fun ⟨o, e⟩ => by have : ∀ c, alephIdx c < o := fun c => (e _).2 ⟨_, rfl⟩ refine Ordinal.inductionOn o ?_ this; intro α r _ h let s := ⨆ a, invFun alephIdx (Ordinal.typein r a) apply (lt_succ s).not_le have I : Injective.{u+2, u+2} alephIdx := alephIdx.initialSeg.toEmbedding.injective simpa only [typein_enum, leftInverse_invFun I (succ s)] using le_ciSup (Cardinal.bddAbove_range.{u, u} fun a : α => invFun alephIdx (Ordinal.typein r a)) (Ordinal.enum r _ (h (succ s))) #align cardinal.aleph_idx.rel_iso Cardinal.alephIdx.relIso @[simp] theorem alephIdx.relIso_coe : (alephIdx.relIso : Cardinal → Ordinal) = alephIdx := rfl #align cardinal.aleph_idx.rel_iso_coe Cardinal.alephIdx.relIso_coe @[simp] theorem type_cardinal : @type Cardinal (· < ·) _ = Ordinal.univ.{u, u + 1} := by rw [Ordinal.univ_id]; exact Quotient.sound ⟨alephIdx.relIso⟩ #align cardinal.type_cardinal Cardinal.type_cardinal @[simp] theorem mk_cardinal : #Cardinal = univ.{u, u + 1} := by simpa only [card_type, card_univ] using congr_arg card type_cardinal #align cardinal.mk_cardinal Cardinal.mk_cardinal /-- The `aleph'` function gives the cardinals listed by their ordinal index, and is the inverse of `aleph_idx`. `aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = succ ℵ₀`, etc. In this version, we register additionally that this function is an order isomorphism between ordinals and cardinals. For the basic function version, see `aleph'`. -/ def Aleph'.relIso := Cardinal.alephIdx.relIso.symm #align cardinal.aleph'.rel_iso Cardinal.Aleph'.relIso /-- The `aleph'` function gives the cardinals listed by their ordinal index, and is the inverse of `aleph_idx`. `aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = succ ℵ₀`, etc. -/ def aleph' : Ordinal → Cardinal := Aleph'.relIso #align cardinal.aleph' Cardinal.aleph' @[simp] theorem aleph'.relIso_coe : (Aleph'.relIso : Ordinal → Cardinal) = aleph' := rfl #align cardinal.aleph'.rel_iso_coe Cardinal.aleph'.relIso_coe @[simp] theorem aleph'_lt {o₁ o₂ : Ordinal} : aleph' o₁ < aleph' o₂ ↔ o₁ < o₂ := Aleph'.relIso.map_rel_iff #align cardinal.aleph'_lt Cardinal.aleph'_lt @[simp] theorem aleph'_le {o₁ o₂ : Ordinal} : aleph' o₁ ≤ aleph' o₂ ↔ o₁ ≤ o₂ := le_iff_le_iff_lt_iff_lt.2 aleph'_lt #align cardinal.aleph'_le Cardinal.aleph'_le @[simp] theorem aleph'_alephIdx (c : Cardinal) : aleph' c.alephIdx = c := Cardinal.alephIdx.relIso.toEquiv.symm_apply_apply c #align cardinal.aleph'_aleph_idx Cardinal.aleph'_alephIdx @[simp] theorem alephIdx_aleph' (o : Ordinal) : (aleph' o).alephIdx = o := Cardinal.alephIdx.relIso.toEquiv.apply_symm_apply o #align cardinal.aleph_idx_aleph' Cardinal.alephIdx_aleph' @[simp] theorem aleph'_zero : aleph' 0 = 0 := by rw [← nonpos_iff_eq_zero, ← aleph'_alephIdx 0, aleph'_le] apply Ordinal.zero_le #align cardinal.aleph'_zero Cardinal.aleph'_zero @[simp] theorem aleph'_succ {o : Ordinal} : aleph' (succ o) = succ (aleph' o) := by apply (succ_le_of_lt <\| aleph'_lt.2 <\| lt_succ o).antisymm' (Cardinal.alephIdx_le.1 <\| _) rw [alephIdx_aleph', succ_le_iff, ← aleph'_lt, aleph'_alephIdx] apply lt_succ #align cardinal.aleph'_succ Cardinal.aleph'_succ @[simp] theorem aleph'_nat : ∀ n : ℕ, aleph' n = n \| 0 => aleph'_zero \| n + 1 => show aleph' (succ n) = n.succ by rw [aleph'_succ, aleph'_nat n, nat_succ] #align cardinal.aleph'_nat Cardinal.aleph'_nat theorem aleph'_le_of_limit {o : Ordinal} (l : o.IsLimit) {c} : aleph' o ≤ c ↔ ∀ o' < o, aleph' o' ≤ c := ⟨fun h o' h' => (aleph'_le.2 <\| h'.le).trans h, fun h => by rw [← aleph'_alephIdx c, aleph'_le, limit_le l] intro x h' rw [← aleph'_le, aleph'_alephIdx] exact h _ h'⟩ #align cardinal.aleph'_le_of_limit Cardinal.aleph'_le_of_limit theorem aleph'_limit {o : Ordinal} (ho : o.IsLimit) : aleph' o = ⨆ a : Iio o, aleph' a := by refine le_antisymm ?_ (ciSup_le' fun i => aleph'_le.2 (le_of_lt i.2)) rw [aleph'_le_of_limit ho] exact fun a ha => le_ciSup (bddAbove_of_small _) (⟨a, ha⟩ : Iio o) #align cardinal.aleph'_limit Cardinal.aleph'_limit @[simp] theorem aleph'_omega : aleph' ω = ℵ₀ := eq_of_forall_ge_iff fun c => by simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le] exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat]) #align cardinal.aleph'_omega Cardinal.aleph'_omega /-- `aleph'` and `aleph_idx` form an equivalence between `Ordinal` and `Cardinal` -/ @[simp] def aleph'Equiv : Ordinal ≃ Cardinal := ⟨aleph', alephIdx, alephIdx_aleph', aleph'_alephIdx⟩ #align cardinal.aleph'_equiv Cardinal.aleph'Equiv /-- The `aleph` function gives the infinite cardinals listed by their ordinal index. `aleph 0 = ℵ₀`, `aleph 1 = succ ℵ₀` is the first uncountable cardinal, and so on. -/ def aleph (o : Ordinal) : Cardinal := aleph' (ω + o) #align cardinal.aleph Cardinal.aleph @[simp] theorem aleph_lt {o₁ o₂ : Ordinal} : aleph o₁ < aleph o₂ ↔ o₁ < o₂ := aleph'_lt.trans (add_lt_add_iff_left _) #align cardinal.aleph_lt Cardinal.aleph_lt @[simp] theorem aleph_le {o₁ o₂ : Ordinal} : aleph o₁ ≤ aleph o₂ ↔ o₁ ≤ o₂ := le_iff_le_iff_lt_iff_lt.2 aleph_lt #align cardinal.aleph_le Cardinal.aleph_le @[simp] theorem max_aleph_eq (o₁ o₂ : Ordinal) : max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) := by rcases le_total (aleph o₁) (aleph o₂) with h \| h · rw [max_eq_right h, max_eq_right (aleph_le.1 h)] · rw [max_eq_left h, max_eq_left (aleph_le.1 h)] #align cardinal.max_aleph_eq Cardinal.max_aleph_eq @[simp] theorem aleph_succ {o : Ordinal} : aleph (succ o) = succ (aleph o) := by rw [aleph, add_succ, aleph'_succ, aleph] #align cardinal.aleph_succ Cardinal.aleph_succ @[simp] theorem aleph_zero : aleph 0 = ℵ₀ := by rw [aleph, add_zero, aleph'_omega] #align cardinal.aleph_zero Cardinal.aleph_zero theorem aleph_limit {o : Ordinal} (ho : o.IsLimit) : aleph o = ⨆ a : Iio o, aleph a := by apply le_antisymm _ (ciSup_le' _) · rw [aleph, aleph'_limit (ho.add _)] refine ciSup_mono' (bddAbove_of_small _) ?_ rintro ⟨i, hi⟩ cases' lt_or_le i ω with h h · rcases lt_omega.1 h with ⟨n, rfl⟩ use ⟨0, ho.pos⟩ simpa using (nat_lt_aleph0 n).le · exact ⟨⟨_, (sub_lt_of_le h).2 hi⟩, aleph'_le.2 (le_add_sub _ _)⟩ · exact fun i => aleph_le.2 (le_of_lt i.2) #align cardinal.aleph_limit Cardinal.aleph_limit theorem aleph0_le_aleph' {o : Ordinal} : ℵ₀ ≤ aleph' o ↔ ω ≤ o := by rw [← aleph'_omega, aleph'_le] #align cardinal.aleph_0_le_aleph' Cardinal.aleph0_le_aleph' theorem aleph0_le_aleph (o : Ordinal) : ℵ₀ ≤ aleph o := by rw [aleph, aleph0_le_aleph'] apply Ordinal.le_add_right #align cardinal.aleph_0_le_aleph Cardinal.aleph0_le_aleph theorem aleph'_pos {o : Ordinal} (ho : 0 < o) : 0 < aleph' o := by rwa [← aleph'_zero, aleph'_lt] #align cardinal.aleph'_pos Cardinal.aleph'_pos theorem aleph_pos (o : Ordinal) : 0 < aleph o := aleph0_pos.trans_le (aleph0_le_aleph o) #align cardinal.aleph_pos Cardinal.aleph_pos @[simp] theorem aleph_toNat (o : Ordinal) : toNat (aleph o) = 0 := toNat_apply_of_aleph0_le <\| aleph0_le_aleph o #align cardinal.aleph_to_nat Cardinal.aleph_toNat @[simp] theorem aleph_toPartENat (o : Ordinal) : toPartENat (aleph o) = ⊤ := toPartENat_apply_of_aleph0_le <\| aleph0_le_aleph o #align cardinal.aleph_to_part_enat Cardinal.aleph_toPartENat instance nonempty_out_aleph (o : Ordinal) : Nonempty (aleph o).ord.out.α := by rw [out_nonempty_iff_ne_zero, ← ord_zero] exact fun h => (ord_injective h).not_gt (aleph_pos o) #align cardinal.nonempty_out_aleph Cardinal.nonempty_out_aleph theorem ord_aleph_isLimit (o : Ordinal) : (aleph o).ord.IsLimit := ord_isLimit <\| aleph0_le_aleph _ #align cardinal.ord_aleph_is_limit Cardinal.ord_aleph_isLimit instance (o : Ordinal) : NoMaxOrder (aleph o).ord.out.α := out_no_max_of_succ_lt (ord_aleph_isLimit o).2 theorem exists_aleph {c : Cardinal} : ℵ₀ ≤ c ↔ ∃ o, c = aleph o := ⟨fun h => ⟨alephIdx c - ω, by rw [aleph, Ordinal.add_sub_cancel_of_le, aleph'_alephIdx] rwa [← aleph0_le_aleph', aleph'_alephIdx]⟩, fun ⟨o, e⟩ => e.symm ▸ aleph0_le_aleph _⟩ #align cardinal.exists_aleph Cardinal.exists_aleph theorem aleph'_isNormal : IsNormal (ord ∘ aleph') := ⟨fun o => ord_lt_ord.2 <\| aleph'_lt.2 <\| lt_succ o, fun o l a => by simp [ord_le, aleph'_le_of_limit l]⟩ #align cardinal.aleph'_is_normal Cardinal.aleph'_isNormal theorem aleph_isNormal : IsNormal (ord ∘ aleph) := aleph'_isNormal.trans <\| add_isNormal ω #align cardinal.aleph_is_normal Cardinal.aleph_isNormal theorem succ_aleph0 : succ ℵ₀ = aleph 1 := by rw [← aleph_zero, ← aleph_succ, Ordinal.succ_zero] #align cardinal.succ_aleph_0 Cardinal.succ_aleph0 theorem aleph0_lt_aleph_one : ℵ₀ < aleph 1 := by rw [← succ_aleph0] apply lt_succ #align cardinal.aleph_0_lt_aleph_one Cardinal.aleph0_lt_aleph_one theorem countable_iff_lt_aleph_one {α : Type} (s : Set α) : s.Countable ↔ #s < aleph 1 := by rw [← succ_aleph0, lt_succ_iff, le_aleph0_iff_set_countable] #align cardinal.countable_iff_lt_aleph_one Cardinal.countable_iff_lt_aleph_one /-- Ordinals that are cardinals are unbounded. -/ theorem ord_card_unbounded : Unbounded (· < ·) { b : Ordinal \| b.card.ord = b } := unbounded_lt_iff.2 fun a => ⟨_, ⟨by dsimp rw [card_ord], (lt_ord_succ_card a).le⟩⟩ #align cardinal.ord_card_unbounded Cardinal.ord_card_unbounded theorem eq_aleph'_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) : ∃ a, (aleph' a).ord = o := ⟨Cardinal.alephIdx.relIso o.card, by simpa using ho⟩ #align cardinal.eq_aleph'_of_eq_card_ord Cardinal.eq_aleph'_of_eq_card_ord /-- `ord ∘ aleph'` enumerates the ordinals that are cardinals. -/ theorem ord_aleph'_eq_enum_card : ord ∘ aleph' = enumOrd { b : Ordinal \| b.card.ord = b } := by rw [← eq_enumOrd _ ord_card_unbounded, range_eq_iff] exact ⟨aleph'_isNormal.strictMono, ⟨fun a => by dsimp rw [card_ord], fun b hb => eq_aleph'_of_eq_card_ord hb⟩⟩ #align cardinal.ord_aleph'_eq_enum_card Cardinal.ord_aleph'_eq_enum_card /-- Infinite ordinals that are cardinals are unbounded. -/ theorem ord_card_unbounded' : Unbounded (· < ·) { b : Ordinal \| b.card.ord = b ∧ ω ≤ b } := (unbounded_lt_inter_le ω).2 ord_card_unbounded #align cardinal.ord_card_unbounded' Cardinal.ord_card_unbounded' theorem eq_aleph_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) (ho' : ω ≤ o) : ∃ a, (aleph a).ord = o := by cases' eq_aleph'_of_eq_card_ord ho with a ha use a - ω unfold aleph rwa [Ordinal.add_sub_cancel_of_le] rwa [← aleph0_le_aleph', ← ord_le_ord, ha, ord_aleph0] #align cardinal.eq_aleph_of_eq_card_ord Cardinal.eq_aleph_of_eq_card_ord /-- `ord ∘ aleph` enumerates the infinite ordinals that are cardinals. -/ theorem ord_aleph_eq_enum_card : ord ∘ aleph = enumOrd { b : Ordinal \| b.card.ord = b ∧ ω ≤ b } := by rw [← eq_enumOrd _ ord_card_unbounded'] use aleph_isNormal.strictMono rw [range_eq_iff] refine ⟨fun a => ⟨?_, ?_⟩, fun b hb => eq_aleph_of_eq_card_ord hb.1 hb.2⟩ · rw [Function.comp_apply, card_ord] · rw [← ord_aleph0, Function.comp_apply, ord_le_ord] exact aleph0_le_aleph _ #align cardinal.ord_aleph_eq_enum_card Cardinal.ord_aleph_eq_enum_card end aleph /-! ### Beth cardinals -/ section beth /-- Beth numbers are defined so that `beth 0 = ℵ₀`, `beth (succ o) = 2 ^ (beth o)`, and when `o` is a limit ordinal, `beth o` is the supremum of `beth o'` for `o' < o`. Assuming the generalized continuum hypothesis, which is undecidable in ZFC, `beth o = aleph o` for every `o`. -/ def beth (o : Ordinal.{u}) : Cardinal.{u} := limitRecOn o aleph0 (fun _ x => (2 : Cardinal) ^ x) fun a _ IH => ⨆ b : Iio a, IH b.1 b.2 #align cardinal.beth Cardinal.beth @[simp] theorem beth_zero : beth 0 = aleph0 := limitRecOn_zero _ _ _ #align cardinal.beth_zero Cardinal.beth_zero @[simp] theorem beth_succ (o : Ordinal) : beth (succ o) = 2 ^ beth o := limitRecOn_succ _ _ _ _ #align cardinal.beth_succ Cardinal.beth_succ theorem beth_limit {o : Ordinal} : o.IsLimit → beth o = ⨆ a : Iio o, beth a := limitRecOn_limit _ _ _ _ #align cardinal.beth_limit Cardinal.beth_limit theorem beth_strictMono : StrictMono beth := by intro a b induction' b using Ordinal.induction with b IH generalizing a intro h rcases zero_or_succ_or_limit b with (rfl \| ⟨c, rfl⟩ \| hb) · exact (Ordinal.not_lt_zero a h).elim · rw [lt_succ_iff] at h rw [beth_succ] apply lt_of_le_of_lt _ (cantor _) rcases eq_or_lt_of_le h with (rfl \| h) · rfl exact (IH c (lt_succ c) h).le · apply (cantor _).trans_le rw [beth_limit hb, ← beth_succ] exact le_ciSup (bddAbove_of_small _) (⟨_, hb.succ_lt h⟩ : Iio b) #align cardinal.beth_strict_mono Cardinal.beth_strictMono theorem beth_mono : Monotone beth := beth_strictMono.monotone #align cardinal.beth_mono Cardinal.beth_mono @[simp] theorem beth_lt {o₁ o₂ : Ordinal} : beth o₁ < beth o₂ ↔ o₁ < o₂ := beth_strictMono.lt_iff_lt #align cardinal.beth_lt Cardinal.beth_lt @[simp] theorem beth_le {o₁ o₂ : Ordinal} : beth o₁ ≤ beth o₂ ↔ o₁ ≤ o₂ := beth_strictMono.le_iff_le #align cardinal.beth_le Cardinal.beth_le theorem aleph_le_beth (o : Ordinal) : aleph o ≤ beth o := by induction o using limitRecOn with \| H₁ => simp \| H₂ o h => rw [aleph_succ, beth_succ, succ_le_iff] exact (cantor _).trans_le (power_le_power_left two_ne_zero h) \| H₃ o ho IH => rw [aleph_limit ho, beth_limit ho] exact ciSup_mono (bddAbove_of_small _) fun x => IH x.1 x.2 #align cardinal.aleph_le_beth Cardinal.aleph_le_beth theorem aleph0_le_beth (o : Ordinal) : ℵ₀ ≤ beth o := (aleph0_le_aleph o).trans <\| aleph_le_beth o #align cardinal.aleph_0_le_beth Cardinal.aleph0_le_beth theorem beth_pos (o : Ordinal) : 0 < beth o := aleph0_pos.trans_le <\| aleph0_le_beth o #align cardinal.beth_pos Cardinal.beth_pos theorem beth_ne_zero (o : Ordinal) : beth o ≠ 0 := (beth_pos o).ne' #align cardinal.beth_ne_zero Cardinal.beth_ne_zero theorem beth_normal : IsNormal.{u} fun o => (beth o).ord := (isNormal_iff_strictMono_limit _).2 ⟨ord_strictMono.comp beth_strictMono, fun o ho a ha => by rw [beth_limit ho, ord_le] exact ciSup_le' fun b => ord_le.1 (ha _ b.2)⟩ #align cardinal.beth_normal Cardinal.beth_normal end beth /-! ### Properties of `mul` -/ section mulOrdinals /-- If `α` is an infinite type, then `α × α` and `α` have the same cardinality. -/ theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c c = c := by refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c) -- the only nontrivial part is `c * c ≤ c`. We prove it inductively. refine Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => ?_) h -- consider the minimal well-order `r` on `α` (a type with cardinality `c`). rcases ord_eq α with ⟨r, wo, e⟩ letI := linearOrderOfSTO r haveI : IsWellOrder α (· < ·) := wo -- Define an order `s` on `α × α` by writing `(a, b) < (c, d)` if `max a b < max c d`, or -- the max are equal and `a < c`, or the max are equal and `a = c` and `b < d`. let g : α × α → α := fun p => max p.1 p.2 let f : α × α ↪ Ordinal × α × α := ⟨fun p : α × α => (typein (· < ·) (g p), p), fun p q => congr_arg Prod.snd⟩ let s := f ⁻¹'o Prod.Lex (· < ·) (Prod.Lex (· < ·) (· < ·)) -- this is a well order on `α × α`. haveI : IsWellOrder _ s := (RelEmbedding.preimage _ _).isWellOrder /- it suffices to show that this well order is smaller than `r` if it were larger, then `r` would be a strict prefix of `s`. It would be contained in `β × β` for some `β` of cardinality `< c`. By the inductive assumption, this set has the same cardinality as `β` (or it is finite if `β` is finite), so it is `< c`, which is a contradiction. -/ suffices type s ≤ type r by exact card_le_card this refine le_of_forall_lt fun o h => ?_ rcases typein_surj s h with ⟨p, rfl⟩ rw [← e, lt_ord] refine lt_of_le_of_lt (?_ : _ ≤ card (succ (typein (· < ·) (g p))) * card (succ (typein (· < ·) (g p)))) ?_ · have : { q \| s q p } ⊆ insert (g p) { x \| x < g p } ×ˢ insert (g p) { x \| x < g p } := by intro q h simp only [s, f, Preimage, ge_iff_le, Embedding.coeFn_mk, Prod.lex_def, typein_lt_typein, typein_inj, mem_setOf_eq] at h exact max_le_iff.1 (le_iff_lt_or_eq.2 <\| h.imp_right And.left) suffices H : (insert (g p) { x \| r x (g p) } : Set α) ≃ Sum { x \| r x (g p) } PUnit from ⟨(Set.embeddingOfSubset _ _ this).trans ((Equiv.Set.prod _ _).trans (H.prodCongr H)).toEmbedding⟩ refine (Equiv.Set.insert ?_).trans ((Equiv.refl _).sumCongr punitEquivPUnit) apply @irrefl _ r cases' lt_or_le (card (succ (typein (· < ·) (g p)))) ℵ₀ with qo qo · exact (mul_lt_aleph0 qo qo).trans_le ol · suffices (succ (typein LT.lt (g p))).card < ⟦α⟧ from (IH _ this qo).trans_lt this rw [← lt_ord] apply (ord_isLimit ol).2 rw [mk'_def, e] apply typein_lt_type #align cardinal.mul_eq_self Cardinal.mul_eq_self end mulOrdinals end UsingOrdinals /-! Properties of `mul`, not requiring ordinals -/ section mul /-- If `α` and `β` are infinite types, then the cardinality of `α × β` is the maximum of the cardinalities of `α` and `β`. -/ theorem mul_eq_max {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : ℵ₀ ≤ b) : a * b = max a b := le_antisymm (mul_eq_self (ha.trans (le_max_left a b)) ▸ mul_le_mul' (le_max_left _ _) (le_max_right _ _)) <\| max_le (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans hb) a) (by simpa only [one_mul] using mul_le_mul_right' (one_le_aleph0.trans ha) b) #align cardinal.mul_eq_max Cardinal.mul_eq_max @[simp] theorem mul_mk_eq_max {α β : Type u} [Infinite α] [Infinite β] : #α * #β = max #α #β := mul_eq_max (aleph0_le_mk α) (aleph0_le_mk β) #align cardinal.mul_mk_eq_max Cardinal.mul_mk_eq_max @[simp] theorem aleph_mul_aleph (o₁ o₂ : Ordinal) : aleph o₁ * aleph o₂ = aleph (max o₁ o₂) := by rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), max_aleph_eq] #align cardinal.aleph_mul_aleph Cardinal.aleph_mul_aleph @[simp] theorem aleph0_mul_eq {a : Cardinal} (ha : ℵ₀ ≤ a) : ℵ₀ * a = a := (mul_eq_max le_rfl ha).trans (max_eq_right ha) #align cardinal.aleph_0_mul_eq Cardinal.aleph0_mul_eq @[simp] theorem mul_aleph0_eq {a : Cardinal} (ha : ℵ₀ ≤ a) : a * ℵ₀ = a := (mul_eq_max ha le_rfl).trans (max_eq_left ha) #align cardinal.mul_aleph_0_eq Cardinal.mul_aleph0_eq -- Porting note (#10618): removed `simp`, `simp` can prove it theorem aleph0_mul_mk_eq {α : Type} [Infinite α] : ℵ₀ #α = #α := aleph0_mul_eq (aleph0_le_mk α) #align cardinal.aleph_0_mul_mk_eq Cardinal.aleph0_mul_mk_eq -- Porting note (#10618): removed `simp`, `simp` can prove it theorem mk_mul_aleph0_eq {α : Type} [Infinite α] : #α ℵ₀ = #α := mul_aleph0_eq (aleph0_le_mk α) #align cardinal.mk_mul_aleph_0_eq Cardinal.mk_mul_aleph0_eq @[simp] theorem aleph0_mul_aleph (o : Ordinal) : ℵ₀ * aleph o = aleph o := aleph0_mul_eq (aleph0_le_aleph o) #align cardinal.aleph_0_mul_aleph Cardinal.aleph0_mul_aleph @[simp] theorem aleph_mul_aleph0 (o : Ordinal) : aleph o * ℵ₀ = aleph o := mul_aleph0_eq (aleph0_le_aleph o) #align cardinal.aleph_mul_aleph_0 Cardinal.aleph_mul_aleph0 theorem mul_lt_of_lt {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a * b < c := (mul_le_mul' (le_max_left a b) (le_max_right a b)).trans_lt <\| (lt_or_le (max a b) ℵ₀).elim (fun h => (mul_lt_aleph0 h h).trans_le hc) fun h => by rw [mul_eq_self h] exact max_lt h1 h2 #align cardinal.mul_lt_of_lt Cardinal.mul_lt_of_lt theorem mul_le_max_of_aleph0_le_left {a b : Cardinal} (h : ℵ₀ ≤ a) : a * b ≤ max a b := by convert mul_le_mul' (le_max_left a b) (le_max_right a b) using 1 rw [mul_eq_self] exact h.trans (le_max_left a b) #align cardinal.mul_le_max_of_aleph_0_le_left Cardinal.mul_le_max_of_aleph0_le_left theorem mul_eq_max_of_aleph0_le_left {a b : Cardinal} (h : ℵ₀ ≤ a) (h' : b ≠ 0) : a * b = max a b := by rcases le_or_lt ℵ₀ b with hb \| hb · exact mul_eq_max h hb refine (mul_le_max_of_aleph0_le_left h).antisymm ?_ have : b ≤ a := hb.le.trans h rw [max_eq_left this] convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') a rw [mul_one] #align cardinal.mul_eq_max_of_aleph_0_le_left Cardinal.mul_eq_max_of_aleph0_le_left theorem mul_le_max_of_aleph0_le_right {a b : Cardinal} (h : ℵ₀ ≤ b) : a * b ≤ max a b := by simpa only [mul_comm b, max_comm b] using mul_le_max_of_aleph0_le_left h #align cardinal.mul_le_max_of_aleph_0_le_right Cardinal.mul_le_max_of_aleph0_le_right theorem mul_eq_max_of_aleph0_le_right {a b : Cardinal} (h' : a ≠ 0) (h : ℵ₀ ≤ b) : a * b = max a b := by rw [mul_comm, max_comm] exact mul_eq_max_of_aleph0_le_left h h' #align cardinal.mul_eq_max_of_aleph_0_le_right Cardinal.mul_eq_max_of_aleph0_le_right theorem mul_eq_max' {a b : Cardinal} (h : ℵ₀ ≤ a * b) : a * b = max a b := by rcases aleph0_le_mul_iff.mp h with ⟨ha, hb, ha' \| hb'⟩ · exact mul_eq_max_of_aleph0_le_left ha' hb · exact mul_eq_max_of_aleph0_le_right ha hb' #align cardinal.mul_eq_max' Cardinal.mul_eq_max' theorem mul_le_max (a b : Cardinal) : a * b ≤ max (max a b) ℵ₀ := by rcases eq_or_ne a 0 with (rfl \| ha0); · simp rcases eq_or_ne b 0 with (rfl \| hb0); · simp rcases le_or_lt ℵ₀ a with ha \| ha · rw [mul_eq_max_of_aleph0_le_left ha hb0] exact le_max_left _ _ · rcases le_or_lt ℵ₀ b with hb \| hb · rw [mul_comm, mul_eq_max_of_aleph0_le_left hb ha0, max_comm] exact le_max_left _ _ · exact le_max_of_le_right (mul_lt_aleph0 ha hb).le #align cardinal.mul_le_max Cardinal.mul_le_max theorem mul_eq_left {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) (hb' : b ≠ 0) : a * b = a := by rw [mul_eq_max_of_aleph0_le_left ha hb', max_eq_left hb] #align cardinal.mul_eq_left Cardinal.mul_eq_left theorem mul_eq_right {a b : Cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) (ha' : a ≠ 0) : a * b = b := by rw [mul_comm, mul_eq_left hb ha ha'] #align cardinal.mul_eq_right Cardinal.mul_eq_right theorem le_mul_left {a b : Cardinal} (h : b ≠ 0) : a ≤ b * a := by convert mul_le_mul_right' (one_le_iff_ne_zero.mpr h) a rw [one_mul] #align cardinal.le_mul_left Cardinal.le_mul_left theorem le_mul_right {a b : Cardinal} (h : b ≠ 0) : a ≤ a * b := by rw [mul_comm] exact le_mul_left h #align cardinal.le_mul_right Cardinal.le_mul_right theorem mul_eq_left_iff {a b : Cardinal} : a * b = a ↔ max ℵ₀ b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 := by rw [max_le_iff] refine ⟨fun h => ?_, ?_⟩ · rcases le_or_lt ℵ₀ a with ha \| ha · have : a ≠ 0 := by rintro rfl exact ha.not_lt aleph0_pos left rw [and_assoc] use ha constructor · rw [← not_lt] exact fun hb => ne_of_gt (hb.trans_le (le_mul_left this)) h · rintro rfl apply this rw [mul_zero] at h exact h.symm right by_cases h2a : a = 0 · exact Or.inr h2a have hb : b ≠ 0 := by rintro rfl apply h2a rw [mul_zero] at h exact h.symm left rw [← h, mul_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha rcases ha with (rfl \| rfl \| ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩) · contradiction · contradiction rw [← Ne] at h2a rw [← one_le_iff_ne_zero] at h2a hb norm_cast at h2a hb h ⊢ apply le_antisymm _ hb rw [← not_lt] apply fun h2b => ne_of_gt _ h conv_rhs => left; rw [← mul_one n] rw [mul_lt_mul_left] · exact id apply Nat.lt_of_succ_le h2a · rintro (⟨⟨ha, hab⟩, hb⟩ \| rfl \| rfl) · rw [mul_eq_max_of_aleph0_le_left ha hb, max_eq_left hab] all_goals simp #align cardinal.mul_eq_left_iff Cardinal.mul_eq_left_iff end mul /-! ### Properties of `add` -/ section add /-- If `α` is an infinite type, then `α ⊕ α` and `α` have the same cardinality. -/ theorem add_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c + c = c := le_antisymm (by convert mul_le_mul_right' ((nat_lt_aleph0 2).le.trans h) c using 1 <;> simp [two_mul, mul_eq_self h]) (self_le_add_left c c) #align cardinal.add_eq_self Cardinal.add_eq_self /-- If `α` is an infinite type, then the cardinality of `α ⊕ β` is the maximum of the cardinalities of `α` and `β`. -/ theorem add_eq_max {a b : Cardinal} (ha : ℵ₀ ≤ a) : a + b = max a b := le_antisymm (add_eq_self (ha.trans (le_max_left a b)) ▸ add_le_add (le_max_left _ _) (le_max_right _ _)) <\| max_le (self_le_add_right _ _) (self_le_add_left _ _) #align cardinal.add_eq_max Cardinal.add_eq_max theorem add_eq_max' {a b : Cardinal} (ha : ℵ₀ ≤ b) : a + b = max a b := by rw [add_comm, max_comm, add_eq_max ha] #align cardinal.add_eq_max' Cardinal.add_eq_max' @[simp] theorem add_mk_eq_max {α β : Type u} [Infinite α] : #α + #β = max #α #β := add_eq_max (aleph0_le_mk α) #align cardinal.add_mk_eq_max Cardinal.add_mk_eq_max @[simp] theorem add_mk_eq_max' {α β : Type u} [Infinite β] : #α + #β = max #α #β := add_eq_max' (aleph0_le_mk β) #align cardinal.add_mk_eq_max' Cardinal.add_mk_eq_max' theorem add_le_max (a b : Cardinal) : a + b ≤ max (max a b) ℵ₀ := by rcases le_or_lt ℵ₀ a with ha \| ha · rw [add_eq_max ha] exact le_max_left _ _ · rcases le_or_lt ℵ₀ b with hb \| hb · rw [add_comm, add_eq_max hb, max_comm] exact le_max_left _ _ · exact le_max_of_le_right (add_lt_aleph0 ha hb).le #align cardinal.add_le_max Cardinal.add_le_max theorem add_le_of_le {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a ≤ c) (h2 : b ≤ c) : a + b ≤ c := (add_le_add h1 h2).trans <\| le_of_eq <\| add_eq_self hc #align cardinal.add_le_of_le Cardinal.add_le_of_le theorem add_lt_of_lt {a b c : Cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a + b < c := (add_le_add (le_max_left a b) (le_max_right a b)).trans_lt <\| (lt_or_le (max a b) ℵ₀).elim (fun h => (add_lt_aleph0 h h).trans_le hc) fun h => by rw [add_eq_self h]; exact max_lt h1 h2 #align cardinal.add_lt_of_lt Cardinal.add_lt_of_lt theorem eq_of_add_eq_of_aleph0_le {a b c : Cardinal} (h : a + b = c) (ha : a < c) (hc : ℵ₀ ≤ c) : b = c := by apply le_antisymm · rw [← h] apply self_le_add_left rw [← not_lt]; intro hb have : a + b < c := add_lt_of_lt hc ha hb simp [h, lt_irrefl] at this #align cardinal.eq_of_add_eq_of_aleph_0_le Cardinal.eq_of_add_eq_of_aleph0_le	Mathlib/SetTheory/Cardinal/Ordinal.lean	785	786	theorem add_eq_left {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) : a + b = a := by	rw [add_eq_max ha, max_eq_left hb]
/- Copyright (c) 2018 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import Mathlib.Logic.Function.Basic import Mathlib.Logic.Relator import Mathlib.Init.Data.Quot import Mathlib.Tactic.Cases import Mathlib.Tactic.Use import Mathlib.Tactic.MkIffOfInductiveProp import Mathlib.Tactic.SimpRw #align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe" /-! # Relation closures This file defines the reflexive, transitive, and reflexive transitive closures of relations. It also proves some basic results on definitions such as `EqvGen`. Note that this is about unbundled relations, that is terms of types of the form `α → β → Prop`. For the bundled version, see `Rel`. ## Definitions * `Relation.ReflGen`: Reflexive closure. `ReflGen r` relates everything `r` related, plus for all `a` it relates `a` with itself. So `ReflGen r a b ↔ r a b ∨ a = b`. * `Relation.TransGen`: Transitive closure. `TransGen r` relates everything `r` related transitively. So `TransGen r a b ↔ ∃ x₀ ... xₙ, r a x₀ ∧ r x₀ x₁ ∧ ... ∧ r xₙ b`. * `Relation.ReflTransGen`: Reflexive transitive closure. `ReflTransGen r` relates everything `r` related transitively, plus for all `a` it relates `a` with itself. So `ReflTransGen r a b ↔ (∃ x₀ ... xₙ, r a x₀ ∧ r x₀ x₁ ∧ ... ∧ r xₙ b) ∨ a = b`. It is the same as the reflexive closure of the transitive closure, or the transitive closure of the reflexive closure. In terms of rewriting systems, this means that `a` can be rewritten to `b` in a number of rewrites. * `Relation.Comp`: Relation composition. We provide notation `∘r`. For `r : α → β → Prop` and `s : β → γ → Prop`, `r ∘r s`relates `a : α` and `c : γ` iff there exists `b : β` that's related to both. * `Relation.Map`: Image of a relation under a pair of maps. For `r : α → β → Prop`, `f : α → γ`, `g : β → δ`, `Map r f g` is the relation `γ → δ → Prop` relating `f a` and `g b` for all `a`, `b` related by `r`. * `Relation.Join`: Join of a relation. For `r : α → α → Prop`, `Join r a b ↔ ∃ c, r a c ∧ r b c`. In terms of rewriting systems, this means that `a` and `b` can be rewritten to the same term. -/ open Function variable {α β γ δ ε ζ : Type*} section NeImp variable {r : α → α → Prop} theorem IsRefl.reflexive [IsRefl α r] : Reflexive r := fun x ↦ IsRefl.refl x #align is_refl.reflexive IsRefl.reflexive /-- To show a reflexive relation `r : α → α → Prop` holds over `x y : α`, it suffices to show it holds when `x ≠ y`. -/ theorem Reflexive.rel_of_ne_imp (h : Reflexive r) {x y : α} (hr : x ≠ y → r x y) : r x y := by by_cases hxy : x = y · exact hxy ▸ h x · exact hr hxy #align reflexive.rel_of_ne_imp Reflexive.rel_of_ne_imp /-- If a reflexive relation `r : α → α → Prop` holds over `x y : α`, then it holds whether or not `x ≠ y`. -/ theorem Reflexive.ne_imp_iff (h : Reflexive r) {x y : α} : x ≠ y → r x y ↔ r x y := ⟨h.rel_of_ne_imp, fun hr _ ↦ hr⟩ #align reflexive.ne_imp_iff Reflexive.ne_imp_iff /-- If a reflexive relation `r : α → α → Prop` holds over `x y : α`, then it holds whether or not `x ≠ y`. Unlike `Reflexive.ne_imp_iff`, this uses `[IsRefl α r]`. -/ theorem reflexive_ne_imp_iff [IsRefl α r] {x y : α} : x ≠ y → r x y ↔ r x y := IsRefl.reflexive.ne_imp_iff #align reflexive_ne_imp_iff reflexive_ne_imp_iff protected theorem Symmetric.iff (H : Symmetric r) (x y : α) : r x y ↔ r y x := ⟨fun h ↦ H h, fun h ↦ H h⟩ #align symmetric.iff Symmetric.iff theorem Symmetric.flip_eq (h : Symmetric r) : flip r = r := funext₂ fun _ _ ↦ propext <\| h.iff _ _ #align symmetric.flip_eq Symmetric.flip_eq theorem Symmetric.swap_eq : Symmetric r → swap r = r := Symmetric.flip_eq #align symmetric.swap_eq Symmetric.swap_eq theorem flip_eq_iff : flip r = r ↔ Symmetric r := ⟨fun h _ _ ↦ (congr_fun₂ h _ _).mp, Symmetric.flip_eq⟩ #align flip_eq_iff flip_eq_iff theorem swap_eq_iff : swap r = r ↔ Symmetric r := flip_eq_iff #align swap_eq_iff swap_eq_iff end NeImp section Comap variable {r : β → β → Prop} theorem Reflexive.comap (h : Reflexive r) (f : α → β) : Reflexive (r on f) := fun a ↦ h (f a) #align reflexive.comap Reflexive.comap theorem Symmetric.comap (h : Symmetric r) (f : α → β) : Symmetric (r on f) := fun _ _ hab ↦ h hab #align symmetric.comap Symmetric.comap theorem Transitive.comap (h : Transitive r) (f : α → β) : Transitive (r on f) := fun _ _ _ hab hbc ↦ h hab hbc #align transitive.comap Transitive.comap theorem Equivalence.comap (h : Equivalence r) (f : α → β) : Equivalence (r on f) := ⟨h.reflexive.comap f, @(h.symmetric.comap f), @(h.transitive.comap f)⟩ #align equivalence.comap Equivalence.comap end Comap namespace Relation section Comp variable {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop} /-- The composition of two relations, yielding a new relation. The result relates a term of `α` and a term of `γ` if there is an intermediate term of `β` related to both. -/ def Comp (r : α → β → Prop) (p : β → γ → Prop) (a : α) (c : γ) : Prop := ∃ b, r a b ∧ p b c #align relation.comp Relation.Comp @[inherit_doc] local infixr:80 " ∘r " => Relation.Comp theorem comp_eq : r ∘r (· = ·) = r := funext fun _ ↦ funext fun b ↦ propext <\| Iff.intro (fun ⟨_, h, Eq⟩ ↦ Eq ▸ h) fun h ↦ ⟨b, h, rfl⟩ #align relation.comp_eq Relation.comp_eq theorem eq_comp : (· = ·) ∘r r = r := funext fun a ↦ funext fun _ ↦ propext <\| Iff.intro (fun ⟨_, Eq, h⟩ ↦ Eq.symm ▸ h) fun h ↦ ⟨a, rfl, h⟩ #align relation.eq_comp Relation.eq_comp theorem iff_comp {r : Prop → α → Prop} : (· ↔ ·) ∘r r = r := by have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq rw [this, eq_comp] #align relation.iff_comp Relation.iff_comp theorem comp_iff {r : α → Prop → Prop} : r ∘r (· ↔ ·) = r := by have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq rw [this, comp_eq] #align relation.comp_iff Relation.comp_iff theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by funext a d apply propext constructor · exact fun ⟨c, ⟨b, hab, hbc⟩, hcd⟩ ↦ ⟨b, hab, c, hbc, hcd⟩ · exact fun ⟨b, hab, c, hbc, hcd⟩ ↦ ⟨c, ⟨b, hab, hbc⟩, hcd⟩ #align relation.comp_assoc Relation.comp_assoc theorem flip_comp : flip (r ∘r p) = flip p ∘r flip r := by funext c a apply propext constructor · exact fun ⟨b, hab, hbc⟩ ↦ ⟨b, hbc, hab⟩ · exact fun ⟨b, hbc, hab⟩ ↦ ⟨b, hab, hbc⟩ #align relation.flip_comp Relation.flip_comp end Comp section Fibration variable (rα : α → α → Prop) (rβ : β → β → Prop) (f : α → β) /-- A function `f : α → β` is a fibration between the relation `rα` and `rβ` if for all `a : α` and `b : β`, whenever `b : β` and `f a` are related by `rβ`, `b` is the image of some `a' : α` under `f`, and `a'` and `a` are related by `rα`. -/ def Fibration := ∀ ⦃a b⦄, rβ b (f a) → ∃ a', rα a' a ∧ f a' = b #align relation.fibration Relation.Fibration variable {rα rβ} /-- If `f : α → β` is a fibration between relations `rα` and `rβ`, and `a : α` is accessible under `rα`, then `f a` is accessible under `rβ`. -/ theorem _root_.Acc.of_fibration (fib : Fibration rα rβ f) {a} (ha : Acc rα a) : Acc rβ (f a) := by induction' ha with a _ ih refine Acc.intro (f a) fun b hr ↦ ?_ obtain ⟨a', hr', rfl⟩ := fib hr exact ih a' hr' #align acc.of_fibration Acc.of_fibration theorem _root_.Acc.of_downward_closed (dc : ∀ {a b}, rβ b (f a) → ∃ c, f c = b) (a : α) (ha : Acc (InvImage rβ f) a) : Acc rβ (f a) := ha.of_fibration f fun a _ h ↦ let ⟨a', he⟩ := dc h -- Porting note: Lean 3 did not need the motive ⟨a', he.substr (p := fun x ↦ rβ x (f a)) h, he⟩ #align acc.of_downward_closed Acc.of_downward_closed end Fibration section Map variable {r : α → β → Prop} {f : α → γ} {g : β → δ} {c : γ} {d : δ} /-- The map of a relation `r` through a pair of functions pushes the relation to the codomains of the functions. The resulting relation is defined by having pairs of terms related if they have preimages related by `r`. -/ protected def Map (r : α → β → Prop) (f : α → γ) (g : β → δ) : γ → δ → Prop := fun c d ↦ ∃ a b, r a b ∧ f a = c ∧ g b = d #align relation.map Relation.Map lemma map_apply : Relation.Map r f g c d ↔ ∃ a b, r a b ∧ f a = c ∧ g b = d := Iff.rfl #align relation.map_apply Relation.map_apply @[simp] lemma map_map (r : α → β → Prop) (f₁ : α → γ) (g₁ : β → δ) (f₂ : γ → ε) (g₂ : δ → ζ) : Relation.Map (Relation.Map r f₁ g₁) f₂ g₂ = Relation.Map r (f₂ ∘ f₁) (g₂ ∘ g₁) := by ext a b simp_rw [Relation.Map, Function.comp_apply, ← exists_and_right, @exists_comm γ, @exists_comm δ] refine exists₂_congr fun a b ↦ ⟨?_, fun h ↦ ⟨_, _, ⟨⟨h.1, rfl, rfl⟩, h.2⟩⟩⟩ rintro ⟨_, _, ⟨hab, rfl, rfl⟩, h⟩ exact ⟨hab, h⟩ #align relation.map_map Relation.map_map @[simp] lemma map_apply_apply (hf : Injective f) (hg : Injective g) (r : α → β → Prop) (a : α) (b : β) : Relation.Map r f g (f a) (g b) ↔ r a b := by simp [Relation.Map, hf.eq_iff, hg.eq_iff] @[simp] lemma map_id_id (r : α → β → Prop) : Relation.Map r id id = r := by ext; simp [Relation.Map] #align relation.map_id_id Relation.map_id_id instance [Decidable (∃ a b, r a b ∧ f a = c ∧ g b = d)] : Decidable (Relation.Map r f g c d) := ‹Decidable _› end Map variable {r : α → α → Prop} {a b c d : α} /-- `ReflTransGen r`: reflexive transitive closure of `r` -/ @[mk_iff ReflTransGen.cases_tail_iff] inductive ReflTransGen (r : α → α → Prop) (a : α) : α → Prop \| refl : ReflTransGen r a a \| tail {b c} : ReflTransGen r a b → r b c → ReflTransGen r a c #align relation.refl_trans_gen Relation.ReflTransGen #align relation.refl_trans_gen.cases_tail_iff Relation.ReflTransGen.cases_tail_iff attribute [refl] ReflTransGen.refl /-- `ReflGen r`: reflexive closure of `r` -/ @[mk_iff] inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop \| refl : ReflGen r a a \| single {b} : r a b → ReflGen r a b #align relation.refl_gen Relation.ReflGen #align relation.refl_gen_iff Relation.reflGen_iff /-- `TransGen r`: transitive closure of `r` -/ @[mk_iff] inductive TransGen (r : α → α → Prop) (a : α) : α → Prop \| single {b} : r a b → TransGen r a b \| tail {b c} : TransGen r a b → r b c → TransGen r a c #align relation.trans_gen Relation.TransGen #align relation.trans_gen_iff Relation.transGen_iff attribute [refl] ReflGen.refl namespace ReflGen theorem to_reflTransGen : ∀ {a b}, ReflGen r a b → ReflTransGen r a b \| a, _, refl => by rfl \| a, b, single h => ReflTransGen.tail ReflTransGen.refl h #align relation.refl_gen.to_refl_trans_gen Relation.ReflGen.to_reflTransGen theorem mono {p : α → α → Prop} (hp : ∀ a b, r a b → p a b) : ∀ {a b}, ReflGen r a b → ReflGen p a b \| a, _, ReflGen.refl => by rfl \| a, b, single h => single (hp a b h) #align relation.refl_gen.mono Relation.ReflGen.mono instance : IsRefl α (ReflGen r) := ⟨@refl α r⟩ end ReflGen namespace ReflTransGen @[trans]	Mathlib/Logic/Relation.lean	296	299	theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by	induction hbc with \| refl => assumption \| tail _ hcd hac => exact hac.tail hcd
/- Copyright (c) 2021 Anatole Dedecker. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anatole Dedecker, Bhavik Mehta -/ import Mathlib.Analysis.Calculus.Deriv.Support import Mathlib.Analysis.SpecialFunctions.Pow.Deriv import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Order.Filter.AtTopBot import Mathlib.MeasureTheory.Function.Jacobian import Mathlib.MeasureTheory.Measure.Haar.NormedSpace import Mathlib.MeasureTheory.Measure.Haar.Unique #align_import measure_theory.integral.integral_eq_improper from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5" /-! # Links between an integral and its "improper" version In its current state, mathlib only knows how to talk about definite ("proper") integrals, in the sense that it treats integrals over `[x, +∞)` the same as it treats integrals over `[y, z]`. For example, the integral over `[1, +∞)` is not defined to be the limit of the integral over `[1, x]` as `x` tends to `+∞`, which is known as an improper integral. Indeed, the "proper" definition is stronger than the "improper" one. The usual counterexample is `x ↦ sin(x)/x`, which has an improper integral over `[1, +∞)` but no definite integral. Although definite integrals have better properties, they are hardly usable when it comes to computing integrals on unbounded sets, which is much easier using limits. Thus, in this file, we prove various ways of studying the proper integral by studying the improper one. ## Definitions The main definition of this file is `MeasureTheory.AECover`. It is a rather technical definition whose sole purpose is generalizing and factoring proofs. Given an index type `ι`, a countably generated filter `l` over `ι`, and an `ι`-indexed family `φ` of subsets of a measurable space `α` equipped with a measure `μ`, one should think of a hypothesis `hφ : MeasureTheory.AECover μ l φ` as a sufficient condition for being able to interpret `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ i, f x ∂μ` as `i` tends to `l`. When using this definition with a measure restricted to a set `s`, which happens fairly often, one should not try too hard to use a `MeasureTheory.AECover` of subsets of `s`, as it often makes proofs more complicated than necessary. See for example the proof of `MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(fun x ↦ oi x)` as a `MeasureTheory.AECover` w.r.t. `μ.restrict (Iic b)`, instead of using `(fun x ↦ Ioc x b)`. ## Main statements - `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated` : if `φ` is a `MeasureTheory.AECover μ l`, where `l` is a countably generated filter, and if `f` is a measurable `ENNReal`-valued function, then `∫⁻ x in φ n, f x ∂μ` tends to `∫⁻ x, f x ∂μ` as `n` tends to `l` - `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` : if `φ` is a `MeasureTheory.AECover μ l`, where `l` is a countably generated filter, if `f` is measurable and integrable on each `φ n`, and if `∫ x in φ n, ‖f x‖ ∂μ` tends to some `I : ℝ` as n tends to `l`, then `f` is integrable - `MeasureTheory.AECover.integral_tendsto_of_countably_generated` : if `φ` is a `MeasureTheory.AECover μ l`, where `l` is a countably generated filter, and if `f` is measurable and integrable (globally), then `∫ x in φ n, f x ∂μ` tends to `∫ x, f x ∂μ` as `n` tends to `+∞`. We then specialize these lemmas to various use cases involving intervals, which are frequent in analysis. In particular, - `MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto` is a version of FTC-2 on the interval `(a, +∞)`, giving the formula `∫ x in (a, +∞), g' x = l - g a` if `g'` is integrable and `g` tends to `l` at `+∞`. - `MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg` gives the same result assuming that `g'` is nonnegative instead of integrable. Its automatic integrability in this context is proved in `MeasureTheory.integrableOn_Ioi_deriv_of_nonneg`. - `MeasureTheory.integral_comp_smul_deriv_Ioi` is a version of the change of variables formula on semi-infinite intervals. - `MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi` shows that a function whose derivative is integrable on `(a, +∞)` has a limit at `+∞`. - `MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi` shows that an integrable function whose derivative is integrable on `(a, +∞)` tends to `0` at `+∞`. Versions of these results are also given on the intervals `(-∞, a]` and `(-∞, +∞)`, as well as the corresponding versions of integration by parts. -/ open MeasureTheory Filter Set TopologicalSpace open scoped ENNReal NNReal Topology namespace MeasureTheory section AECover variable {α ι : Type} [MeasurableSpace α] (μ : Measure α) (l : Filter ι) /-- A sequence `φ` of subsets of `α` is a `MeasureTheory.AECover` w.r.t. a measure `μ` and a filter `l` if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if each `φ n` is measurable. This definition is a technical way to avoid duplicating a lot of proofs. It should be thought of as a sufficient condition for being able to interpret `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ n, f x ∂μ` as `n` tends to `l`. See for example `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated`, `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` and `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/ structure AECover (φ : ι → Set α) : Prop where ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i protected measurableSet : ∀ i, MeasurableSet <\| φ i #align measure_theory.ae_cover MeasureTheory.AECover #align measure_theory.ae_cover.ae_eventually_mem MeasureTheory.AECover.ae_eventually_mem #align measure_theory.ae_cover.measurable MeasureTheory.AECover.measurableSet variable {μ} {l} namespace AECover /-! ## Operations on `AECover`s Porting note: this is a new section. -/ /-- Elementwise intersection of two `AECover`s is an `AECover`. -/ theorem inter {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hψ : AECover μ l ψ) : AECover μ l (fun i ↦ φ i ∩ ψ i) where ae_eventually_mem := hψ.1.mp <\| hφ.1.mono fun _ ↦ Eventually.and measurableSet _ := (hφ.2 _).inter (hψ.2 _) theorem superset {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hsub : ∀ i, φ i ⊆ ψ i) (hmeas : ∀ i, MeasurableSet (ψ i)) : AECover μ l ψ := ⟨hφ.1.mono fun _x hx ↦ hx.mono fun i hi ↦ hsub i hi, hmeas⟩ theorem mono_ac {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≪ μ) : AECover ν l φ := ⟨hle hφ.1, hφ.2⟩ theorem mono {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≤ μ) : AECover ν l φ := hφ.mono_ac hle.absolutelyContinuous end AECover section MetricSpace variable [PseudoMetricSpace α] [OpensMeasurableSpace α] theorem aecover_ball {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) : AECover μ l (fun i ↦ Metric.ball x (r i)) where measurableSet _ := Metric.isOpen_ball.measurableSet ae_eventually_mem := by filter_upwards with y filter_upwards [hr (Ioi_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) : AECover μ l (fun i ↦ Metric.closedBall x (r i)) where measurableSet _ := Metric.isClosed_ball.measurableSet ae_eventually_mem := by filter_upwards with y filter_upwards [hr (Ici_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha end MetricSpace section Preorderα variable [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) theorem aecover_Ici : AECover μ l fun i => Ici (a i) where ae_eventually_mem := ae_of_all μ ha.eventually_le_atBot measurableSet _ := measurableSet_Ici #align measure_theory.ae_cover_Ici MeasureTheory.aecover_Ici theorem aecover_Iic : AECover μ l fun i => Iic <\| b i := aecover_Ici (α := αᵒᵈ) hb #align measure_theory.ae_cover_Iic MeasureTheory.aecover_Iic theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) := (aecover_Ici ha).inter (aecover_Iic hb) #align measure_theory.ae_cover_Icc MeasureTheory.aecover_Icc end Preorderα section LinearOrderα variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) theorem aecover_Ioi [NoMinOrder α] : AECover μ l fun i => Ioi (a i) where ae_eventually_mem := ae_of_all μ ha.eventually_lt_atBot measurableSet _ := measurableSet_Ioi #align measure_theory.ae_cover_Ioi MeasureTheory.aecover_Ioi theorem aecover_Iio [NoMaxOrder α] : AECover μ l fun i => Iio (b i) := aecover_Ioi (α := αᵒᵈ) hb #align measure_theory.ae_cover_Iio MeasureTheory.aecover_Iio theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo (a i) (b i) := (aecover_Ioi ha).inter (aecover_Iio hb) #align measure_theory.ae_cover_Ioo MeasureTheory.aecover_Ioo theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) := (aecover_Ioi ha).inter (aecover_Iic hb) #align measure_theory.ae_cover_Ioc MeasureTheory.aecover_Ioc theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) := (aecover_Ici ha).inter (aecover_Iio hb) #align measure_theory.ae_cover_Ico MeasureTheory.aecover_Ico end LinearOrderα section FiniteIntervals variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) -- Porting note (#10756): new lemma theorem aecover_Ioi_of_Ioi : AECover (μ.restrict (Ioi A)) l fun i ↦ Ioi (a i) where ae_eventually_mem := (ae_restrict_mem measurableSet_Ioi).mono fun _x hx ↦ ha.eventually <\| eventually_lt_nhds hx measurableSet _ := measurableSet_Ioi -- Porting note (#10756): new lemma theorem aecover_Iio_of_Iio : AECover (μ.restrict (Iio B)) l fun i ↦ Iio (b i) := aecover_Ioi_of_Ioi (α := αᵒᵈ) hb -- Porting note (#10756): new lemma theorem aecover_Ioi_of_Ici : AECover (μ.restrict (Ioi A)) l fun i ↦ Ici (a i) := (aecover_Ioi_of_Ioi ha).superset (fun _ ↦ Ioi_subset_Ici_self) fun _ ↦ measurableSet_Ici -- Porting note (#10756): new lemma theorem aecover_Iio_of_Iic : AECover (μ.restrict (Iio B)) l fun i ↦ Iic (b i) := aecover_Ioi_of_Ici (α := αᵒᵈ) hb theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <\| Ioo A B) l fun i => Ioo (a i) (b i) := ((aecover_Ioi_of_Ioi ha).mono <\| Measure.restrict_mono Ioo_subset_Ioi_self le_rfl).inter ((aecover_Iio_of_Iio hb).mono <\| Measure.restrict_mono Ioo_subset_Iio_self le_rfl) #align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aecover_Ioo_of_Ioo theorem aecover_Ioo_of_Icc : AECover (μ.restrict <\| Ioo A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).superset (fun _ ↦ Ioo_subset_Icc_self) fun _ ↦ measurableSet_Icc #align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aecover_Ioo_of_Icc theorem aecover_Ioo_of_Ico : AECover (μ.restrict <\| Ioo A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).superset (fun _ ↦ Ioo_subset_Ico_self) fun _ ↦ measurableSet_Ico #align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aecover_Ioo_of_Ico theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <\| Ioo A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).superset (fun _ ↦ Ioo_subset_Ioc_self) fun _ ↦ measurableSet_Ioc #align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aecover_Ioo_of_Ioc variable [NoAtoms μ] theorem aecover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Icc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Icc MeasureTheory.aecover_Ioc_of_Icc theorem aecover_Ioc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ico ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Ico MeasureTheory.aecover_Ioc_of_Ico theorem aecover_Ioc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Ioc MeasureTheory.aecover_Ioc_of_Ioc theorem aecover_Ioc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Ioo (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Ioo MeasureTheory.aecover_Ioc_of_Ioo theorem aecover_Ico_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Icc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Icc MeasureTheory.aecover_Ico_of_Icc theorem aecover_Ico_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ico ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Ico MeasureTheory.aecover_Ico_of_Ico theorem aecover_Ico_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Ioc MeasureTheory.aecover_Ico_of_Ioc theorem aecover_Ico_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Ioo (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Ioo MeasureTheory.aecover_Ico_of_Ioo theorem aecover_Icc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Icc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Icc MeasureTheory.aecover_Icc_of_Icc theorem aecover_Icc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ico ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Ico MeasureTheory.aecover_Icc_of_Ico theorem aecover_Icc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Ioc MeasureTheory.aecover_Icc_of_Ioc theorem aecover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Ioo (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Ioo MeasureTheory.aecover_Icc_of_Ioo end FiniteIntervals protected theorem AECover.restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} : AECover (μ.restrict s) l φ := hφ.mono Measure.restrict_le_self #align measure_theory.ae_cover.restrict MeasureTheory.AECover.restrict theorem aecover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s) (ae_eventually_mem : ∀ᵐ x ∂μ, x ∈ s → ∀ᶠ n in l, x ∈ φ n) (measurable : ∀ n, MeasurableSet <\| φ n) : AECover (μ.restrict s) l φ where ae_eventually_mem := by rwa [ae_restrict_iff' hs] measurableSet := measurable #align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aecover_restrict_of_ae_imp theorem AECover.inter_restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} (hs : MeasurableSet s) : AECover (μ.restrict s) l fun i => φ i ∩ s := aecover_restrict_of_ae_imp hs (hφ.ae_eventually_mem.mono fun _x hx hxs => hx.mono fun _i hi => ⟨hi, hxs⟩) fun i => (hφ.measurableSet i).inter hs #align measure_theory.ae_cover.inter_restrict MeasureTheory.AECover.inter_restrict theorem AECover.ae_tendsto_indicator {β : Type} [Zero β] [TopologicalSpace β] (f : α → β) {φ : ι → Set α} (hφ : AECover μ l φ) : ∀ᵐ x ∂μ, Tendsto (fun i => (φ i).indicator f x) l (𝓝 <\| f x) := hφ.ae_eventually_mem.mono fun _x hx => tendsto_const_nhds.congr' <\| hx.mono fun _n hn => (indicator_of_mem hn _).symm #align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AECover.ae_tendsto_indicator theorem AECover.aemeasurable {β : Type} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ) (hfm : ∀ i, AEMeasurable f (μ.restrict <\| φ i)) : AEMeasurable f μ := by obtain ⟨u, hu⟩ := l.exists_seq_tendsto have := aemeasurable_iUnion_iff.mpr fun n : ℕ => hfm (u n) rwa [Measure.restrict_eq_self_of_ae_mem] at this filter_upwards [hφ.ae_eventually_mem] with x hx using mem_iUnion.mpr (hu.eventually hx).exists #align measure_theory.ae_cover.ae_measurable MeasureTheory.AECover.aemeasurable theorem AECover.aestronglyMeasurable {β : Type} [TopologicalSpace β] [PseudoMetrizableSpace β] [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ) (hfm : ∀ i, AEStronglyMeasurable f (μ.restrict <\| φ i)) : AEStronglyMeasurable f μ := by obtain ⟨u, hu⟩ := l.exists_seq_tendsto have := aestronglyMeasurable_iUnion_iff.mpr fun n : ℕ => hfm (u n) rwa [Measure.restrict_eq_self_of_ae_mem] at this filter_upwards [hφ.ae_eventually_mem] with x hx using mem_iUnion.mpr (hu.eventually hx).exists #align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AECover.aestronglyMeasurable end AECover theorem AECover.comp_tendsto {α ι ι' : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} {l' : Filter ι'} {φ : ι → Set α} (hφ : AECover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) : AECover μ l' (φ ∘ u) where ae_eventually_mem := hφ.ae_eventually_mem.mono fun _x hx => hu.eventually hx measurableSet i := hφ.measurableSet (u i) #align measure_theory.ae_cover.comp_tendsto MeasureTheory.AECover.comp_tendsto section AECoverUnionInterCountable variable {α ι : Type} [Countable ι] [MeasurableSpace α] {μ : Measure α} theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋃ (k) (_h : k ∈ Iic n), φ k := hφ.superset (fun _ ↦ subset_biUnion_of_mem right_mem_Iic) fun _ ↦ .biUnion (to_countable _) fun _ _ ↦ (hφ.2 _) #align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AECover.biUnion_Iic_aecover -- Porting note: generalized from `[SemilatticeSup ι] [Nonempty ι]` to `[Preorder ι]` theorem AECover.biInter_Ici_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (_h : k ∈ Ici n), φ k where ae_eventually_mem := hφ.ae_eventually_mem.mono fun x h ↦ by simpa only [mem_iInter, mem_Ici, eventually_forall_ge_atTop] measurableSet i := .biInter (to_countable _) fun n _ => hφ.measurableSet n #align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AECover.biInter_Ici_aecover end AECoverUnionInterCountable section Lintegral variable {α ι : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <\| ∫⁻ x, f x ∂μ) := let F n := (φ n).indicator f have key₁ : ∀ n, AEMeasurable (F n) μ := fun n => hfm.indicator (hφ.measurableSet n) have key₂ : ∀ᵐ x : α ∂μ, Monotone fun n => F n x := ae_of_all _ fun x _i _j hij => indicator_le_indicator_of_subset (hmono hij) (fun x => zero_le <\| f x) x have key₃ : ∀ᵐ x : α ∂μ, Tendsto (fun n => F n x) atTop (𝓝 (f x)) := hφ.ae_tendsto_indicator f (lintegral_tendsto_of_tendsto_of_monotone key₁ key₂ key₃).congr fun n => lintegral_indicator f (hφ.measurableSet n) theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : Tendsto (∫⁻ x in φ ·, f x ∂μ) atTop (𝓝 <\| ∫⁻ x, f x ∂μ) := by have lim₁ := lintegral_tendsto_of_monotone_of_nat hφ.biInter_Ici_aecover (fun i j hij => biInter_subset_biInter_left (Ici_subset_Ici.mpr hij)) hfm have lim₂ := lintegral_tendsto_of_monotone_of_nat hφ.biUnion_Iic_aecover (fun i j hij => biUnion_subset_biUnion_left (Iic_subset_Iic.mpr hij)) hfm refine tendsto_of_tendsto_of_tendsto_of_le_of_le lim₁ lim₂ (fun n ↦ ?_) fun n ↦ ?_ exacts [lintegral_mono_set (biInter_subset_of_mem left_mem_Ici), lintegral_mono_set (subset_biUnion_of_mem right_mem_Iic)] #align measure_theory.ae_cover.lintegral_tendsto_of_nat MeasureTheory.AECover.lintegral_tendsto_of_nat theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 <\| ∫⁻ x, f x ∂μ) := tendsto_of_seq_tendsto fun _u hu => (hφ.comp_tendsto hu).lintegral_tendsto_of_nat hfm #align measure_theory.ae_cover.lintegral_tendsto_of_countably_generated MeasureTheory.AECover.lintegral_tendsto_of_countably_generated theorem AECover.lintegral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ) (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : ∫⁻ x, f x ∂μ = I := tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AECover.lintegral_eq_of_tendsto theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : ⨆ i : ι, ∫⁻ x in φ i, f x ∂μ = ∫⁻ x, f x ∂μ := by have := hφ.lintegral_tendsto_of_countably_generated hfm refine ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun i => lintegral_mono' Measure.restrict_le_self le_rfl) fun w hw => ?_ rcases exists_between hw with ⟨m, hm₁, hm₂⟩ rcases (eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩ exact ⟨i, lt_of_lt_of_le hm₁ hi⟩ #align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated end Lintegral section Integrable variable {α ι E : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E] theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ) (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ := by refine ⟨hfm, (le_of_tendsto ?_ hbounded).trans_lt ENNReal.ofReal_lt_top⟩ exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ) (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <\| ENNReal.ofReal I)) : Integrable f μ := by refine hφ.integrable_of_lintegral_nnnorm_bounded (max 1 (I + 1)) hfm ?_ refine htendsto.eventually (ge_mem_nhds ?_) refine (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 ?_ exact lt_max_of_lt_right (lt_add_one I) #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ) (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ := hφ.integrable_of_lintegral_nnnorm_bounded I hfm (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded) #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded' theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ) (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ := hφ.integrable_of_lintegral_nnnorm_tendsto I hfm (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto) #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto' theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ := by have hfm : AEStronglyMeasurable f μ := hφ.aestronglyMeasurable fun i => (hfi i).aestronglyMeasurable refine hφ.integrable_of_lintegral_nnnorm_bounded I hfm ?_ conv at hbounded in integral _ _ => rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x)) hfm.norm.restrict] conv at hbounded in ENNReal.ofReal _ => rw [← coe_nnnorm] rw [ENNReal.ofReal_coe_nnreal] refine hbounded.mono fun i hi => ?_ rw [← ENNReal.ofReal_toReal (ne_top_of_lt (hfi i).2)] apply ENNReal.ofReal_le_ofReal hi #align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AECover.integrable_of_integral_norm_bounded theorem AECover.integrable_of_integral_norm_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ := let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le hφ.integrable_of_integral_norm_bounded I' hfi hI' #align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AECover.integrable_of_integral_norm_tendsto theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ := hφ.integrable_of_integral_norm_bounded I hfi <\| hbounded.mono fun _i hi => (integral_congr_ae <\| ae_restrict_of_ae <\| hnng.mono fun _ => Real.norm_of_nonneg).le.trans hi #align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) : Integrable f μ := let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le hφ.integrable_of_integral_bounded_of_nonneg_ae I' hfi hnng hI' #align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae end Integrable section Integral variable {α ι E : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (hfi : Integrable f μ) : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <\| ∫ x, f x ∂μ) := suffices h : Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ)) from by convert h using 2; rw [integral_indicator (hφ.measurableSet _)] tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖) (eventually_of_forall fun i => hfi.aestronglyMeasurable.indicator <\| hφ.measurableSet i) (eventually_of_forall fun i => ae_of_all _ fun x => norm_indicator_le_norm_self _ _) hfi.norm (hφ.ae_tendsto_indicator f) #align measure_theory.ae_cover.integral_tendsto_of_countably_generated MeasureTheory.AECover.integral_tendsto_of_countably_generated /-- Slight reformulation of `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/ theorem AECover.integral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ) (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I := tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h #align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AECover.integral_eq_of_tendsto theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f) (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I := have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto hφ.integral_eq_of_tendsto I hfi' htendsto #align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae end Integral section IntegrableOfIntervalIntegral variable {ι E : Type} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [IsCountablyGenerated l] [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E} theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ := by have hφ : AECover μ l _ := aecover_Ioc ha hb refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_) filter_upwards [ha.eventually (eventually_le_atBot 0), hb.eventually (eventually_ge_atTop 0)] with i hai hbi ht rwa [← intervalIntegral.integral_of_le (hai.trans hbi)] #align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrable_of_intervalIntegral_norm_bounded /-- If `f` is integrable on intervals `Ioc (a i) (b i)`, where `a i` tends to -∞ and `b i` tends to ∞, and `∫ x in a i .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`, then `f` is integrable on the interval (-∞, ∞) -/ theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a i..b i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ := let ⟨I', hI'⟩ := h.isBoundedUnder_le integrable_of_intervalIntegral_norm_bounded I' hfi ha hb hI' #align measure_theory.integrable_of_interval_integral_norm_tendsto MeasureTheory.integrable_of_intervalIntegral_norm_tendsto theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot) (h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ := by have hφ : AECover (μ.restrict <\| Iic b) l _ := aecover_Ioi ha have hfi : ∀ i, IntegrableOn f (Ioi (a i)) (μ.restrict <\| Iic b) := by intro i rw [IntegrableOn, Measure.restrict_restrict (hφ.measurableSet i)] exact hfi i refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_) filter_upwards [ha.eventually (eventually_le_atBot b)] with i hai rw [intervalIntegral.integral_of_le hai, Measure.restrict_restrict (hφ.measurableSet i)] exact id #align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded /-- If `f` is integrable on intervals `Ioc (a i) b`, where `a i` tends to -∞, and `∫ x in a i .. b, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`, then `f` is integrable on the interval (-∞, b) -/ theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot) (h : Tendsto (fun i => ∫ x in a i..b, ‖f x‖ ∂μ) l (𝓝 I)) : IntegrableOn f (Iic b) μ := let ⟨I', hI'⟩ := h.isBoundedUnder_le integrableOn_Iic_of_intervalIntegral_norm_bounded I' b hfi ha hI' #align measure_theory.integrable_on_Iic_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a..b i, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Ioi a) μ := by have hφ : AECover (μ.restrict <\| Ioi a) l _ := aecover_Iic hb have hfi : ∀ i, IntegrableOn f (Iic (b i)) (μ.restrict <\| Ioi a) := by intro i rw [IntegrableOn, Measure.restrict_restrict (hφ.measurableSet i), inter_comm] exact hfi i refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_) filter_upwards [hb.eventually (eventually_ge_atTop a)] with i hbi rw [intervalIntegral.integral_of_le hbi, Measure.restrict_restrict (hφ.measurableSet i), inter_comm] exact id #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded /-- If `f` is integrable on intervals `Ioc a (b i)`, where `b i` tends to ∞, and `∫ x in a .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`, then `f` is integrable on the interval (a, ∞) -/ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a..b i, ‖f x‖ ∂μ) l (𝓝 <\| I)) : IntegrableOn f (Ioi a) μ := let ⟨I', hI'⟩ := h.isBoundedUnder_le integrableOn_Ioi_of_intervalIntegral_norm_bounded I' a hfi hb hI' #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ} (hfi : ∀ i, IntegrableOn f <\| Ioc (a i) (b i)) (ha : Tendsto a l <\| 𝓝 a₀) (hb : Tendsto b l <\| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b₀) := by refine (aecover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I (fun i => (hfi i).restrict measurableSet_Ioc) (h.mono fun i hi ↦ ?_) rw [Measure.restrict_restrict measurableSet_Ioc] refine le_trans (setIntegral_mono_set (hfi i).norm ?_ ?_) hi <;> apply ae_of_all · simp only [Pi.zero_apply, norm_nonneg, forall_const] · intro c hc; exact hc.1 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ} (hfi : ∀ i, IntegrableOn f <\| Ioc (a i) b) (ha : Tendsto a l <\| 𝓝 a₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) := integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi ha tendsto_const_nhds h #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_right {I a b₀ : ℝ} (hfi : ∀ i, IntegrableOn f <\| Ioc a (b i)) (hb : Tendsto b l <\| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) := integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi tendsto_const_nhds hb h #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right @[deprecated (since := "2024-04-06")] alias integrableOn_Ioc_of_interval_integral_norm_bounded := integrableOn_Ioc_of_intervalIntegral_norm_bounded @[deprecated (since := "2024-04-06")] alias integrableOn_Ioc_of_interval_integral_norm_bounded_left := integrableOn_Ioc_of_intervalIntegral_norm_bounded_left @[deprecated (since := "2024-04-06")] alias integrableOn_Ioc_of_interval_integral_norm_bounded_right := integrableOn_Ioc_of_intervalIntegral_norm_bounded_right end IntegrableOfIntervalIntegral section IntegralOfIntervalIntegral variable {ι E : Type} {μ : Measure ℝ} {l : Filter ι} [IsCountablyGenerated l] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {a b : ι → ℝ} {f : ℝ → E} theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) : Tendsto (fun i => ∫ x in a i..b i, f x ∂μ) l (𝓝 <\| ∫ x, f x ∂μ) := by let φ i := Ioc (a i) (b i) have hφ : AECover μ l φ := aecover_Ioc ha hb refine (hφ.integral_tendsto_of_countably_generated hfi).congr' ?_ filter_upwards [ha.eventually (eventually_le_atBot 0), hb.eventually (eventually_ge_atTop 0)] with i hai hbi exact (intervalIntegral.integral_of_le (hai.trans hbi)).symm #align measure_theory.interval_integral_tendsto_integral MeasureTheory.intervalIntegral_tendsto_integral theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ) (ha : Tendsto a l atBot) : Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <\| ∫ x in Iic b, f x ∂μ) := by let φ i := Ioi (a i) have hφ : AECover (μ.restrict <\| Iic b) l φ := aecover_Ioi ha refine (hφ.integral_tendsto_of_countably_generated hfi).congr' ?_ filter_upwards [ha.eventually (eventually_le_atBot <\| b)] with i hai rw [intervalIntegral.integral_of_le hai, Measure.restrict_restrict (hφ.measurableSet i)] rfl #align measure_theory.interval_integral_tendsto_integral_Iic MeasureTheory.intervalIntegral_tendsto_integral_Iic theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (Ioi a) μ) (hb : Tendsto b l atTop) : Tendsto (fun i => ∫ x in a..b i, f x ∂μ) l (𝓝 <\| ∫ x in Ioi a, f x ∂μ) := by let φ i := Iic (b i) have hφ : AECover (μ.restrict <\| Ioi a) l φ := aecover_Iic hb refine (hφ.integral_tendsto_of_countably_generated hfi).congr' ?_ filter_upwards [hb.eventually (eventually_ge_atTop <\| a)] with i hbi rw [intervalIntegral.integral_of_le hbi, Measure.restrict_restrict (hφ.measurableSet i), inter_comm] rfl #align measure_theory.interval_integral_tendsto_integral_Ioi MeasureTheory.intervalIntegral_tendsto_integral_Ioi end IntegralOfIntervalIntegral open Real open scoped Interval section IoiFTC variable {E : Type} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E] [NormedSpace ℝ E] /-- If the derivative of a function defined on the real line is integrable close to `+∞`, then the function has a limit at `+∞`. -/ theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E] (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) : Tendsto f atTop (𝓝 (limUnder atTop f)) := by suffices ∃ a, Tendsto f atTop (𝓝 a) from tendsto_nhds_limUnder this suffices CauchySeq f from cauchySeq_tendsto_of_complete this apply Metric.cauchySeq_iff'.2 (fun ε εpos ↦ ?_) have A : ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε := by have L : Tendsto (fun (n : ℕ) ↦ ∫ x in Ici (n : ℝ), ‖f' x‖) atTop (𝓝 (∫ x in ⋂ (n : ℕ), Ici (n : ℝ), ‖f' x‖)) := by apply tendsto_setIntegral_of_antitone (fun n ↦ measurableSet_Ici) · intro m n hmn exact Ici_subset_Ici.2 (Nat.cast_le.mpr hmn) · rcases exists_nat_gt a with ⟨n, hn⟩ exact ⟨n, IntegrableOn.mono_set f'int.norm (Ici_subset_Ioi.2 hn)⟩ have B : ⋂ (n : ℕ), Ici (n : ℝ) = ∅ := by apply eq_empty_of_forall_not_mem (fun x ↦ ?_) simpa only [mem_iInter, mem_Ici, not_forall, not_le] using exists_nat_gt x simp only [B, Measure.restrict_empty, integral_zero_measure] at L exact (tendsto_order.1 L).2 _ εpos have B : ∀ᶠ (n : ℕ) in atTop, a < n := by rcases exists_nat_gt a with ⟨n, hn⟩ filter_upwards [Ioi_mem_atTop n] with m (hm : n < m) using hn.trans (Nat.cast_lt.mpr hm) rcases (A.and B).exists with ⟨N, hN, h'N⟩ refine ⟨N, fun x hx ↦ ?_⟩ calc dist (f x) (f ↑N) = ‖f x - f N‖ := dist_eq_norm _ _ _ = ‖∫ t in Ioc ↑N x, f' t‖ := by rw [← intervalIntegral.integral_of_le hx, intervalIntegral.integral_eq_sub_of_hasDerivAt] · intro y hy simp only [hx, uIcc_of_le, mem_Icc] at hy exact hderiv _ (h'N.trans_le hy.1) · rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hx] exact f'int.mono_set (Ioc_subset_Ioi_self.trans (Ioi_subset_Ioi h'N.le)) _ ≤ ∫ t in Ioc ↑N x, ‖f' t‖ := norm_integral_le_integral_norm fun a ↦ f' a _ ≤ ∫ t in Ici ↑N, ‖f' t‖ := by apply setIntegral_mono_set · apply IntegrableOn.mono_set f'int.norm (Ici_subset_Ioi.2 h'N) · filter_upwards with x using norm_nonneg _ · have : Ioc (↑N) x ⊆ Ici ↑N := Ioc_subset_Ioi_self.trans Ioi_subset_Ici_self exact this.eventuallyLE _ < ε := hN open UniformSpace in /-- If a function and its derivative are integrable on `(a, +∞)`, then the function tends to zero at `+∞`. -/ theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) (fint : IntegrableOn f (Ioi a)) : Tendsto f atTop (𝓝 0) := by let F : E →L[ℝ] Completion E := Completion.toComplL have Fderiv : ∀ x ∈ Ioi a, HasDerivAt (F ∘ f) (F (f' x)) x := fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx) have Fint : IntegrableOn (F ∘ f) (Ioi a) := by apply F.integrable_comp fint have F'int : IntegrableOn (F ∘ f') (Ioi a) := by apply F.integrable_comp f'int have A : Tendsto (F ∘ f) atTop (𝓝 (limUnder atTop (F ∘ f))) := by apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi Fderiv F'int have B : limUnder atTop (F ∘ f) = F 0 := by have : IntegrableAtFilter (F ∘ f) atTop := by exact ⟨Ioi a, Ioi_mem_atTop _, Fint⟩ apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A intro s hs rcases mem_atTop_sets.1 hs with ⟨b, hb⟩ rw [← top_le_iff, ← volume_Ici (a := b)] exact measure_mono hb rwa [B, ← Embedding.tendsto_nhds_iff] at A exact (Completion.uniformEmbedding_coe E).embedding variable [CompleteSpace E] /-- Fundamental theorem of calculus-2, on semi-infinite intervals `(a, +∞)`. When a function has a limit at infinity `m`, and its derivative is integrable, then the integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability on `(a, +∞)` and continuity at `a⁺`. Note that such a function always has a limit at infinity, see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/ theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici a) a) (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a := by have hcont : ContinuousOn f (Ici a) := by intro x hx rcases hx.out.eq_or_lt with rfl\|hx · exact hcont · exact (hderiv x hx).continuousAt.continuousWithinAt refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Ioi a f'int tendsto_id) ?_ apply Tendsto.congr' _ (hf.sub_const _) filter_upwards [Ioi_mem_atTop a] with x hx have h'x : a ≤ id x := le_of_lt hx symm apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1 rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x] exact f'int.mono (fun y hy => hy.1) le_rfl #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto /-- Fundamental theorem of calculus-2, on semi-infinite intervals `(a, +∞)`. When a function has a limit at infinity `m`, and its derivative is integrable, then the integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability on `[a, +∞)`. Note that such a function always has a limit at infinity, see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/ theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a := by refine integral_Ioi_of_hasDerivAt_of_tendsto ?_ (fun x hx => hderiv x hx.out.le) f'int hf exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt #align measure_theory.integral_Ioi_of_has_deriv_at_of_tendsto' MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto' /-- A special case of `integral_Ioi_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with compact support. -/ theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b := by have := fun x (_ : x ∈ Ioi b) ↦ hf.differentiable le_rfl x \|>.hasDerivAt rw [integral_Ioi_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, zero_sub] · refine hf.continuous_deriv le_rfl \|>.integrable_of_hasCompactSupport h2f.deriv \|>.integrableOn rw [hasCompactSupport_iff_eventuallyEq, Filter.coclosedCompact_eq_cocompact] at h2f exact h2f.filter_mono _root_.atTop_le_cocompact \|>.tendsto /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative is automatically integrable on `(a, +∞)`. Version assuming differentiability on `(a, +∞)` and continuity at `a⁺`. -/ theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a) (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by have hcont : ContinuousOn g (Ici a) := by intro x hx rcases hx.out.eq_or_lt with rfl\|hx · exact hcont · exact (hderiv x hx).continuousAt.continuousWithinAt refine integrableOn_Ioi_of_intervalIntegral_norm_tendsto (l - g a) a (fun x => ?_) tendsto_id ?_ · exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self) (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1 apply Tendsto.congr' _ (hg.sub_const _) filter_upwards [Ioi_mem_atTop a] with x hx have h'x : a ≤ id x := le_of_lt hx calc g x - g a = ∫ y in a..id x, g' y := by symm apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le h'x (hcont.mono Icc_subset_Ici_self) fun y hy => hderiv y hy.1 rw [intervalIntegrable_iff_integrableOn_Ioc_of_le h'x] exact intervalIntegral.integrableOn_deriv_of_nonneg (hcont.mono Icc_subset_Ici_self) (fun y hy => hderiv y hy.1) fun y hy => g'pos y hy.1 _ = ∫ y in a..id x, ‖g' y‖ := by simp_rw [intervalIntegral.integral_of_le h'x] refine setIntegral_congr measurableSet_Ioc fun y hy => ?_ dsimp rw [abs_of_nonneg] exact g'pos _ hy.1 #align measure_theory.integrable_on_Ioi_deriv_of_nonneg MeasureTheory.integrableOn_Ioi_deriv_of_nonneg /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative is automatically integrable on `(a, +∞)`. Version assuming differentiability on `[a, +∞)`. -/ theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by refine integrableOn_Ioi_deriv_of_nonneg ?_ (fun x hx => hderiv x hx.out.le) g'pos hg exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt #align measure_theory.integrable_on_Ioi_deriv_of_nonneg' MeasureTheory.integrableOn_Ioi_deriv_of_nonneg' /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see `integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and continuity at `a⁺`. -/ theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a) (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a := integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv (integrableOn_Ioi_deriv_of_nonneg hcont hderiv g'pos hg) hg #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/ theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a := integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonneg' hderiv g'pos hg) hg #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonneg' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg' /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative is automatically integrable on `(a, +∞)`. Version assuming differentiability on `(a, +∞)` and continuity at `a⁺`. -/ theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a) (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by apply integrable_neg_iff.1 exact integrableOn_Ioi_deriv_of_nonneg hcont.neg (fun x hx => (hderiv x hx).neg) (fun x hx => neg_nonneg_of_nonpos (g'neg x hx)) hg.neg #align measure_theory.integrable_on_Ioi_deriv_of_nonpos MeasureTheory.integrableOn_Ioi_deriv_of_nonpos /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative is automatically integrable on `(a, +∞)`. Version assuming differentiability on `[a, +∞)`. -/ theorem integrableOn_Ioi_deriv_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by refine integrableOn_Ioi_deriv_of_nonpos ?_ (fun x hx ↦ hderiv x hx.out.le) g'neg hg exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt #align measure_theory.integrable_on_Ioi_deriv_of_nonpos' MeasureTheory.integrableOn_Ioi_deriv_of_nonpos' /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see `integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and continuity at `a⁺`. -/ theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a) (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a := integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv (integrableOn_Ioi_deriv_of_nonpos hcont hderiv g'neg hg) hg #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see `integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/ theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a := integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (integrableOn_Ioi_deriv_of_nonpos' hderiv g'neg hg) hg #align measure_theory.integral_Ioi_of_has_deriv_at_of_nonpos' MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos' end IoiFTC section IicFTC variable {E : Type} {f f' : ℝ → E} {g g' : ℝ → ℝ} {a b l : ℝ} {m : E} [NormedAddCommGroup E] [NormedSpace ℝ E] /-- If the derivative of a function defined on the real line is integrable close to `-∞`, then the function has a limit at `-∞`. -/ theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic [CompleteSpace E] (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) : Tendsto f atBot (𝓝 (limUnder atBot f)) := by suffices ∃ a, Tendsto f atBot (𝓝 a) from tendsto_nhds_limUnder this let g := f ∘ (fun x ↦ -x) have hdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x := by intro x hx have : -x ∈ Iic a := by simp only [mem_Iic, mem_Ioi, neg_le] at ; exact hx.le simpa using HasDerivAt.scomp x (hderiv (-x) this) (hasDerivAt_neg' x) have L : Tendsto g atTop (𝓝 (limUnder atTop g)) := by apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi hdg exact ((MeasurePreserving.integrableOn_comp_preimage (Measure.measurePreserving_neg _) (Homeomorph.neg ℝ).measurableEmbedding).2 f'int.neg).mono_set (by simp) refine ⟨limUnder atTop g, ?_⟩ have : Tendsto (fun x ↦ g (-x)) atBot (𝓝 (limUnder atTop g)) := L.comp tendsto_neg_atBot_atTop simpa [g] using this open UniformSpace in /-- If a function and its derivative are integrable on `(-∞, a]`, then the function tends to zero at `-∞`. -/ theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Iic (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) (fint : IntegrableOn f (Iic a)) : Tendsto f atBot (𝓝 0) := by let F : E →L[ℝ] Completion E := Completion.toComplL have Fderiv : ∀ x ∈ Iic a, HasDerivAt (F ∘ f) (F (f' x)) x := fun x hx ↦ F.hasFDerivAt.comp_hasDerivAt _ (hderiv x hx) have Fint : IntegrableOn (F ∘ f) (Iic a) := by apply F.integrable_comp fint have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.integrable_comp f'int have A : Tendsto (F ∘ f) atBot (𝓝 (limUnder atBot (F ∘ f))) := by apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic Fderiv F'int have B : limUnder atBot (F ∘ f) = F 0 := by have : IntegrableAtFilter (F ∘ f) atBot := by exact ⟨Iic a, Iic_mem_atBot _, Fint⟩ apply IntegrableAtFilter.eq_zero_of_tendsto this ?_ A intro s hs rcases mem_atBot_sets.1 hs with ⟨b, hb⟩ apply le_antisymm (le_top) rw [← volume_Iic (a := b)] exact measure_mono hb rwa [B, ← Embedding.tendsto_nhds_iff] at A exact (Completion.uniformEmbedding_coe E).embedding variable [CompleteSpace E] /-- Fundamental theorem of calculus-2, on semi-infinite intervals `(-∞, a)`. When a function has a limit `m` at `-∞`, and its derivative is integrable, then the integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability on `(-∞, a)` and continuity at `a⁻`. Note that such a function always has a limit at minus infinity, see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic`. -/ theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a) (hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by have hcont : ContinuousOn f (Iic a) := by intro x hx rcases hx.out.eq_or_lt with rfl\|hx · exact hcont · exact (hderiv x hx).continuousAt.continuousWithinAt refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Iic a f'int tendsto_id) ?_ apply Tendsto.congr' _ (hf.const_sub _) filter_upwards [Iic_mem_atBot a] with x hx symm apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hx (hcont.mono Icc_subset_Iic_self) fun y hy => hderiv y hy.2 rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hx] exact f'int.mono (fun y hy => hy.2) le_rfl /-- Fundamental theorem of calculus-2, on semi-infinite intervals `(-∞, a)`. When a function has a limit `m` at `-∞`, and its derivative is integrable, then the integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability on `(-∞, a]`. Note that such a function always has a limit at minus infinity, see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic`. -/	Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean	1,012	1,017	theorem integral_Iic_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by	refine integral_Iic_of_hasDerivAt_of_tendsto ?_ (fun x hx => hderiv x hx.out.le) f'int hf exact (hderiv a right_mem_Iic).continuousAt.continuousWithinAt
/- Copyright (c) 2022 Vincent Beffara. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Vincent Beffara -/ import Mathlib.Analysis.Complex.RemovableSingularity import Mathlib.Analysis.Calculus.UniformLimitsDeriv import Mathlib.Analysis.NormedSpace.FunctionSeries #align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081" /-! # Locally uniform limits of holomorphic functions This file gathers some results about locally uniform limits of holomorphic functions on an open subset of the complex plane. ## Main results * `TendstoLocallyUniformlyOn.differentiableOn`: A locally uniform limit of holomorphic functions is holomorphic. * `TendstoLocallyUniformlyOn.deriv`: Locally uniform convergence implies locally uniform convergence of the derivatives to the derivative of the limit. -/ open Set Metric MeasureTheory Filter Complex intervalIntegral open scoped Real Topology variable {E ι : Type} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {U K : Set ℂ} {z : ℂ} {M r δ : ℝ} {φ : Filter ι} {F : ι → ℂ → E} {f g : ℂ → E} namespace Complex section Cderiv /-- A circle integral which coincides with `deriv f z` whenever one can apply the Cauchy formula for the derivative. It is useful in the proof that locally uniform limits of holomorphic functions are holomorphic, because it depends continuously on `f` for the uniform topology. -/ noncomputable def cderiv (r : ℝ) (f : ℂ → E) (z : ℂ) : E := (2 π * I : ℂ)⁻¹ • ∮ w in C(z, r), ((w - z) ^ 2)⁻¹ • f w #align complex.cderiv Complex.cderiv theorem cderiv_eq_deriv (hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (hr : 0 < r) (hzr : closedBall z r ⊆ U) : cderiv r f z = deriv f z := two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable hU hzr hf (mem_ball_self hr) #align complex.cderiv_eq_deriv Complex.cderiv_eq_deriv	Mathlib/Analysis/Complex/LocallyUniformLimit.lean	50	64	theorem norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) : ‖cderiv r f z‖ ≤ M / r := by	have hM : 0 ≤ M := by obtain ⟨w, hw⟩ : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le exact (norm_nonneg _).trans (hf w hw) have h1 : ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2 := by intro w hw simp only [mem_sphere_iff_norm, norm_eq_abs] at hw simp only [norm_smul, inv_mul_eq_div, hw, norm_eq_abs, map_inv₀, Complex.abs_pow] exact div_le_div hM (hf w hw) (sq_pos_of_pos hr) le_rfl have h2 := circleIntegral.norm_integral_le_of_norm_le_const hr.le h1 simp only [cderiv, norm_smul] refine (mul_le_mul le_rfl h2 (norm_nonneg _) (norm_nonneg _)).trans (le_of_eq ?_) field_simp [_root_.abs_of_nonneg Real.pi_pos.le] ring
/- Copyright (c) 2022 Andrew Yang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang -/ import Mathlib.CategoryTheory.Extensive import Mathlib.CategoryTheory.Limits.Shapes.KernelPair #align_import category_theory.adhesive from "leanprover-community/mathlib"@"afff1f24a6b68d0077c9d63782a1d093e337758c" /-! # Adhesive categories ## Main definitions - `CategoryTheory.IsPushout.IsVanKampen`: A convenience formulation for a pushout being a van Kampen colimit. - `CategoryTheory.Adhesive`: A category is adhesive if it has pushouts and pullbacks along monomorphisms, and such pushouts are van Kampen. ## Main Results - `CategoryTheory.Type.adhesive`: The category of `Type` is adhesive. - `CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_left`: In adhesive categories, pushouts along monomorphisms are pullbacks. - `CategoryTheory.Adhesive.mono_of_isPushout_of_mono_left`: In adhesive categories, monomorphisms are stable under pushouts. - `CategoryTheory.Adhesive.toRegularMonoCategory`: Monomorphisms in adhesive categories are regular (this implies that adhesive categories are balanced). - `CategoryTheory.adhesive_functor`: The category `C ⥤ D` is adhesive if `D` has all pullbacks and all pushouts and is adhesive ## References - https://ncatlab.org/nlab/show/adhesive+category - [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004] -/ namespace CategoryTheory open Limits universe v' u' v u variable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C] variable {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} -- This only makes sense when the original diagram is a pushout. /-- A convenience formulation for a pushout being a van Kampen colimit. See `IsPushout.isVanKampen_iff` below. -/ @[nolint unusedArguments] def IsPushout.IsVanKampen (_ : IsPushout f g h i) : Prop := ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αW : W' ⟶ W) (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z) (_ : IsPullback f' αW αX f) (_ : IsPullback g' αW αY g) (_ : CommSq h' αX αZ h) (_ : CommSq i' αY αZ i) (_ : CommSq f' g' h' i'), IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i #align category_theory.is_pushout.is_van_kampen CategoryTheory.IsPushout.IsVanKampen	Mathlib/CategoryTheory/Adhesive.lean	59	63	theorem IsPushout.IsVanKampen.flip {H : IsPushout f g h i} (H' : H.IsVanKampen) : H.flip.IsVanKampen := by	introv W' hf hg hh hi w simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip
/- Copyright (c) 2020 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov -/ import Mathlib.LinearAlgebra.AffineSpace.AffineMap import Mathlib.Tactic.FieldSimp #align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" /-! # Slope of a function In this file we define the slope of a function `f : k → PE` taking values in an affine space over `k` and prove some basic theorems about `slope`. The `slope` function naturally appears in the Mean Value Theorem, and in the proof of the fact that a function with nonnegative second derivative on an interval is convex on this interval. ## Tags affine space, slope -/ open AffineMap variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE] /-- `slope f a b = (b - a)⁻¹ • (f b -ᵥ f a)` is the slope of a function `f` on the interval `[a, b]`. Note that `slope f a a = 0`, not the derivative of `f` at `a`. -/ def slope (f : k → PE) (a b : k) : E := (b - a)⁻¹ • (f b -ᵥ f a) #align slope slope theorem slope_fun_def (f : k → PE) : slope f = fun a b => (b - a)⁻¹ • (f b -ᵥ f a) := rfl #align slope_fun_def slope_fun_def theorem slope_def_field (f : k → k) (a b : k) : slope f a b = (f b - f a) / (b - a) := (div_eq_inv_mul _ _).symm #align slope_def_field slope_def_field theorem slope_fun_def_field (f : k → k) (a : k) : slope f a = fun b => (f b - f a) / (b - a) := (div_eq_inv_mul _ _).symm #align slope_fun_def_field slope_fun_def_field @[simp] theorem slope_same (f : k → PE) (a : k) : (slope f a a : E) = 0 := by rw [slope, sub_self, inv_zero, zero_smul] #align slope_same slope_same theorem slope_def_module (f : k → E) (a b : k) : slope f a b = (b - a)⁻¹ • (f b - f a) := rfl #align slope_def_module slope_def_module @[simp] theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a := by rcases eq_or_ne a b with (rfl \| hne) · rw [sub_self, zero_smul, vsub_self] · rw [slope, smul_inv_smul₀ (sub_ne_zero.2 hne.symm)] #align sub_smul_slope sub_smul_slope theorem sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b := by rw [sub_smul_slope, vsub_vadd] #align sub_smul_slope_vadd sub_smul_slope_vadd @[simp] theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f := by ext a b simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub] #align slope_vadd_const slope_vadd_const @[simp] theorem slope_sub_smul (f : k → E) {a b : k} (h : a ≠ b) : slope (fun x => (x - a) • f x) a b = f b := by simp [slope, inv_smul_smul₀ (sub_ne_zero.2 h.symm)] #align slope_sub_smul slope_sub_smul	Mathlib/LinearAlgebra/AffineSpace/Slope.lean	78	79	theorem eq_of_slope_eq_zero {f : k → PE} {a b : k} (h : slope f a b = (0 : E)) : f a = f b := by	rw [← sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd]
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Aurélien Saue, Anne Baanen -/ import Mathlib.Algebra.Order.Ring.Rat import Mathlib.Tactic.NormNum.Inv import Mathlib.Tactic.NormNum.Pow import Mathlib.Util.AtomM /-! # `ring` tactic A tactic for solving equations in commutative (semi)rings, where the exponents can also contain variables. Based on <http://www.cs.ru.nl/~freek/courses/tt-2014/read/10.1.1.61.3041.pdf> . More precisely, expressions of the following form are supported: - constants (non-negative integers) - variables - coefficients (any rational number, embedded into the (semi)ring) - addition of expressions - multiplication of expressions (`a * b`) - scalar multiplication of expressions (`n • a`; the multiplier must have type `ℕ`) - exponentiation of expressions (the exponent must have type `ℕ`) - subtraction and negation of expressions (if the base is a full ring) The extension to exponents means that something like `2 * 2^n * b = b * 2^(n+1)` can be proved, even though it is not strictly speaking an equation in the language of commutative rings. ## Implementation notes The basic approach to prove equalities is to normalise both sides and check for equality. The normalisation is guided by building a value in the type `ExSum` at the meta level, together with a proof (at the base level) that the original value is equal to the normalised version. The outline of the file: - Define a mutual inductive family of types `ExSum`, `ExProd`, `ExBase`, which can represent expressions with `+`, ``, `^` and rational numerals. The mutual induction ensures that associativity and distributivity are applied, by restricting which kinds of subexpressions appear as arguments to the various operators. - Represent addition, multiplication and exponentiation in the `ExSum` type, thus allowing us to map expressions to `ExSum` (the `eval` function drives this). We apply associativity and distributivity of the operators here (helped by `Ex` types) and commutativity as well (by sorting the subterms; unfortunately not helped by anything). Any expression not of the above formats is treated as an atom (the same as a variable). There are some details we glossed over which make the plan more complicated: - The order on atoms is not initially obvious. We construct a list containing them in order of initial appearance in the expression, then use the index into the list as a key to order on. - For `pow`, the exponent must be a natural number, while the base can be any semiring `α`. We swap out operations for the base ring `α` with those for the exponent ring `ℕ` as soon as we deal with exponents. ## Caveats and future work The normalized form of an expression is the one that is useful for the tactic, but not as nice to read. To remedy this, the user-facing normalization calls `ringNFCore`. Subtraction cancels out identical terms, but division does not. That is: `a - a = 0 := by ring` solves the goal, but `a / a := 1 by ring` doesn't. Note that `0 / 0` is generally defined to be `0`, so division cancelling out is not true in general. Multiplication of powers can be simplified a little bit further: `2 ^ n * 2 ^ n = 4 ^ n := by ring` could be implemented in a similar way that `2 * a + 2 * a = 4 * a := by ring` already works. This feature wasn't needed yet, so it's not implemented yet. ## Tags ring, semiring, exponent, power -/ set_option autoImplicit true namespace Mathlib.Tactic namespace Ring open Mathlib.Meta Qq NormNum Lean.Meta AtomM open Lean (MetaM Expr mkRawNatLit) /-- A shortcut instance for `CommSemiring ℕ` used by ring. -/ def instCommSemiringNat : CommSemiring ℕ := inferInstance /-- A typed expression of type `CommSemiring ℕ` used when we are working on ring subexpressions of type `ℕ`. -/ def sℕ : Q(CommSemiring ℕ) := q(instCommSemiringNat) -- In this file, we would like to use multi-character auto-implicits. set_option relaxedAutoImplicit true mutual /-- The base `e` of a normalized exponent expression. -/ inductive ExBase : ∀ {α : Q(Type u)}, Q(CommSemiring $α) → (e : Q($α)) → Type /-- An atomic expression `e` with id `id`. Atomic expressions are those which `ring` cannot parse any further. For instance, `a + (a % b)` has `a` and `(a % b)` as atoms. The `ring1` tactic does not normalize the subexpressions in atoms, but `ring_nf` does. Atoms in fact represent equivalence classes of expressions, modulo definitional equality. The field `index : ℕ` should be a unique number for each class, while `value : expr` contains a representative of this class. The function `resolve_atom` determines the appropriate atom for a given expression. -/ \| atom (id : ℕ) : ExBase sα e /-- A sum of monomials. -/ \| sum (_ : ExSum sα e) : ExBase sα e /-- A monomial, which is a product of powers of `ExBase` expressions, terminated by a (nonzero) constant coefficient. -/ inductive ExProd : ∀ {α : Q(Type u)}, Q(CommSemiring $α) → (e : Q($α)) → Type /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast. If `value` is not an integer, then `hyp` should be a proof of `(value.den : α) ≠ 0`. -/ \| const (value : ℚ) (hyp : Option Expr := none) : ExProd sα e /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase` and `e` is an `ExProd` representing a monomial expression in `ℕ` (it is a monomial instead of a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/ \| mul {α : Q(Type u)} {sα : Q(CommSemiring $α)} {x : Q($α)} {e : Q(ℕ)} {b : Q($α)} : ExBase sα x → ExProd sℕ e → ExProd sα b → ExProd sα q($x ^ $e * $b) /-- A polynomial expression, which is a sum of monomials. -/ inductive ExSum : ∀ {α : Q(Type u)}, Q(CommSemiring $α) → (e : Q($α)) → Type /-- Zero is a polynomial. `e` is the expression `0`. -/ \| zero {α : Q(Type u)} {sα : Q(CommSemiring $α)} : ExSum sα q(0 : $α) /-- A sum `a + b` is a polynomial if `a` is a monomial and `b` is another polynomial. -/ \| add {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} : ExProd sα a → ExSum sα b → ExSum sα q($a + $b) end mutual -- partial only to speed up compilation /-- Equality test for expressions. This is not a `BEq` instance because it is heterogeneous. -/ partial def ExBase.eq : ExBase sα a → ExBase sα b → Bool \| .atom i, .atom j => i == j \| .sum a, .sum b => a.eq b \| _, _ => false @[inherit_doc ExBase.eq] partial def ExProd.eq : ExProd sα a → ExProd sα b → Bool \| .const i _, .const j _ => i == j \| .mul a₁ a₂ a₃, .mul b₁ b₂ b₃ => a₁.eq b₁ && a₂.eq b₂ && a₃.eq b₃ \| _, _ => false @[inherit_doc ExBase.eq] partial def ExSum.eq : ExSum sα a → ExSum sα b → Bool \| .zero, .zero => true \| .add a₁ a₂, .add b₁ b₂ => a₁.eq b₁ && a₂.eq b₂ \| _, _ => false end mutual -- partial only to speed up compilation /-- A total order on normalized expressions. This is not an `Ord` instance because it is heterogeneous. -/ partial def ExBase.cmp : ExBase sα a → ExBase sα b → Ordering \| .atom i, .atom j => compare i j \| .sum a, .sum b => a.cmp b \| .atom .., .sum .. => .lt \| .sum .., .atom .. => .gt @[inherit_doc ExBase.cmp] partial def ExProd.cmp : ExProd sα a → ExProd sα b → Ordering \| .const i _, .const j _ => compare i j \| .mul a₁ a₂ a₃, .mul b₁ b₂ b₃ => (a₁.cmp b₁).then (a₂.cmp b₂) \|>.then (a₃.cmp b₃) \| .const _ _, .mul .. => .lt \| .mul .., .const _ _ => .gt @[inherit_doc ExBase.cmp] partial def ExSum.cmp : ExSum sα a → ExSum sα b → Ordering \| .zero, .zero => .eq \| .add a₁ a₂, .add b₁ b₂ => (a₁.cmp b₁).then (a₂.cmp b₂) \| .zero, .add .. => .lt \| .add .., .zero => .gt end instance : Inhabited (Σ e, (ExBase sα) e) := ⟨default, .atom 0⟩ instance : Inhabited (Σ e, (ExSum sα) e) := ⟨_, .zero⟩ instance : Inhabited (Σ e, (ExProd sα) e) := ⟨default, .const 0 none⟩ mutual /-- Converts `ExBase sα` to `ExBase sβ`, assuming `sα` and `sβ` are defeq. -/ partial def ExBase.cast : ExBase sα a → Σ a, ExBase sβ a \| .atom i => ⟨a, .atom i⟩ \| .sum a => let ⟨_, vb⟩ := a.cast; ⟨_, .sum vb⟩ /-- Converts `ExProd sα` to `ExProd sβ`, assuming `sα` and `sβ` are defeq. -/ partial def ExProd.cast : ExProd sα a → Σ a, ExProd sβ a \| .const i h => ⟨a, .const i h⟩ \| .mul a₁ a₂ a₃ => ⟨_, .mul a₁.cast.2 a₂ a₃.cast.2⟩ /-- Converts `ExSum sα` to `ExSum sβ`, assuming `sα` and `sβ` are defeq. -/ partial def ExSum.cast : ExSum sα a → Σ a, ExSum sβ a \| .zero => ⟨_, .zero⟩ \| .add a₁ a₂ => ⟨_, .add a₁.cast.2 a₂.cast.2⟩ end /-- The result of evaluating an (unnormalized) expression `e` into the type family `E` (one of `ExSum`, `ExProd`, `ExBase`) is a (normalized) element `e'` and a representation `E e'` for it, and a proof of `e = e'`. -/ structure Result {α : Q(Type u)} (E : Q($α) → Type) (e : Q($α)) where /-- The normalized result. -/ expr : Q($α) /-- The data associated to the normalization. -/ val : E expr /-- A proof that the original expression is equal to the normalized result. -/ proof : Q($e = $expr) instance [Inhabited (Σ e, E e)] : Inhabited (Result E e) := let ⟨e', v⟩ : Σ e, E e := default; ⟨e', v, default⟩ variable {α : Q(Type u)} (sα : Q(CommSemiring $α)) [CommSemiring R] /-- Constructs the expression corresponding to `.const n`. (The `.const` constructor does not check that the expression is correct.) -/ def ExProd.mkNat (n : ℕ) : (e : Q($α)) × ExProd sα e := let lit : Q(ℕ) := mkRawNatLit n ⟨q(($lit).rawCast : $α), .const n none⟩ /-- Constructs the expression corresponding to `.const (-n)`. (The `.const` constructor does not check that the expression is correct.) -/ def ExProd.mkNegNat (_ : Q(Ring $α)) (n : ℕ) : (e : Q($α)) × ExProd sα e := let lit : Q(ℕ) := mkRawNatLit n ⟨q((Int.negOfNat $lit).rawCast : $α), .const (-n) none⟩ /-- Constructs the expression corresponding to `.const (-n)`. (The `.const` constructor does not check that the expression is correct.) -/ def ExProd.mkRat (_ : Q(DivisionRing $α)) (q : ℚ) (n : Q(ℤ)) (d : Q(ℕ)) (h : Expr) : (e : Q($α)) × ExProd sα e := ⟨q(Rat.rawCast $n $d : $α), .const q h⟩ section variable {sα} /-- Embed an exponent (an `ExBase, ExProd` pair) as an `ExProd` by multiplying by 1. -/ def ExBase.toProd (va : ExBase sα a) (vb : ExProd sℕ b) : ExProd sα q($a ^ $b * (nat_lit 1).rawCast) := .mul va vb (.const 1 none) /-- Embed `ExProd` in `ExSum` by adding 0. -/ def ExProd.toSum (v : ExProd sα e) : ExSum sα q($e + 0) := .add v .zero /-- Get the leading coefficient of an `ExProd`. -/ def ExProd.coeff : ExProd sα e → ℚ \| .const q _ => q \| .mul _ _ v => v.coeff end /-- Two monomials are said to "overlap" if they differ by a constant factor, in which case the constants just add. When this happens, the constant may be either zero (if the monomials cancel) or nonzero (if they add up); the zero case is handled specially. -/ inductive Overlap (e : Q($α)) where /-- The expression `e` (the sum of monomials) is equal to `0`. -/ \| zero (_ : Q(IsNat $e (nat_lit 0))) /-- The expression `e` (the sum of monomials) is equal to another monomial (with nonzero leading coefficient). -/ \| nonzero (_ : Result (ExProd sα) e) theorem add_overlap_pf (x : R) (e) (pq_pf : a + b = c) : x ^ e * a + x ^ e * b = x ^ e * c := by subst_vars; simp [mul_add] theorem add_overlap_pf_zero (x : R) (e) : IsNat (a + b) (nat_lit 0) → IsNat (x ^ e * a + x ^ e * b) (nat_lit 0) \| ⟨h⟩ => ⟨by simp [h, ← mul_add]⟩ /-- Given monomials `va, vb`, attempts to add them together to get another monomial. If the monomials are not compatible, returns `none`. For example, `xy + 2xy = 3xy` is a `.nonzero` overlap, while `xy + xz` returns `none` and `xy + -xy = 0` is a `.zero` overlap. -/ def evalAddOverlap (va : ExProd sα a) (vb : ExProd sα b) : Option (Overlap sα q($a + $b)) := match va, vb with \| .const za ha, .const zb hb => do let ra := Result.ofRawRat za a ha; let rb := Result.ofRawRat zb b hb let res ← NormNum.evalAdd.core q($a + $b) q(HAdd.hAdd) a b ra rb match res with \| .isNat _ (.lit (.natVal 0)) p => pure <\| .zero p \| rc => let ⟨zc, hc⟩ ← rc.toRatNZ let ⟨c, pc⟩ := rc.toRawEq pure <\| .nonzero ⟨c, .const zc hc, pc⟩ \| .mul (x := a₁) (e := a₂) va₁ va₂ va₃, .mul vb₁ vb₂ vb₃ => do guard (va₁.eq vb₁ && va₂.eq vb₂) match ← evalAddOverlap va₃ vb₃ with \| .zero p => pure <\| .zero (q(add_overlap_pf_zero $a₁ $a₂ $p) : Expr) \| .nonzero ⟨_, vc, p⟩ => pure <\| .nonzero ⟨_, .mul va₁ va₂ vc, (q(add_overlap_pf $a₁ $a₂ $p) : Expr)⟩ \| _, _ => none theorem add_pf_zero_add (b : R) : 0 + b = b := by simp theorem add_pf_add_zero (a : R) : a + 0 = a := by simp theorem add_pf_add_overlap (_ : a₁ + b₁ = c₁) (_ : a₂ + b₂ = c₂) : (a₁ + a₂ : R) + (b₁ + b₂) = c₁ + c₂ := by subst_vars; simp [add_assoc, add_left_comm] theorem add_pf_add_overlap_zero (h : IsNat (a₁ + b₁) (nat_lit 0)) (h₄ : a₂ + b₂ = c) : (a₁ + a₂ : R) + (b₁ + b₂) = c := by subst_vars; rw [add_add_add_comm, h.1, Nat.cast_zero, add_pf_zero_add] theorem add_pf_add_lt (a₁ : R) (_ : a₂ + b = c) : (a₁ + a₂) + b = a₁ + c := by simp [, add_assoc] theorem add_pf_add_gt (b₁ : R) (_ : a + b₂ = c) : a + (b₁ + b₂) = b₁ + c := by subst_vars; simp [add_left_comm] /-- Adds two polynomials `va, vb` together to get a normalized result polynomial. `0 + b = b` * `a + 0 = a` * `a * x + a * y = a * (x + y)` (for `x`, `y` coefficients; uses `evalAddOverlap`) * `(a₁ + a₂) + (b₁ + b₂) = a₁ + (a₂ + (b₁ + b₂))` (if `a₁.lt b₁`) * `(a₁ + a₂) + (b₁ + b₂) = b₁ + ((a₁ + a₂) + b₂)` (if not `a₁.lt b₁`) -/ partial def evalAdd (va : ExSum sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a + $b) := match va, vb with \| .zero, vb => ⟨b, vb, q(add_pf_zero_add $b)⟩ \| va, .zero => ⟨a, va, q(add_pf_add_zero $a)⟩ \| .add (a := a₁) (b := _a₂) va₁ va₂, .add (a := b₁) (b := _b₂) vb₁ vb₂ => match evalAddOverlap sα va₁ vb₁ with \| some (.nonzero ⟨_, vc₁, pc₁⟩) => let ⟨_, vc₂, pc₂⟩ := evalAdd va₂ vb₂ ⟨_, .add vc₁ vc₂, q(add_pf_add_overlap $pc₁ $pc₂)⟩ \| some (.zero pc₁) => let ⟨c₂, vc₂, pc₂⟩ := evalAdd va₂ vb₂ ⟨c₂, vc₂, q(add_pf_add_overlap_zero $pc₁ $pc₂)⟩ \| none => if let .lt := va₁.cmp vb₁ then let ⟨_c, vc, (pc : Q($_a₂ + ($b₁ + $_b₂) = $_c))⟩ := evalAdd va₂ vb ⟨_, .add va₁ vc, q(add_pf_add_lt $a₁ $pc)⟩ else let ⟨_c, vc, (pc : Q($a₁ + $_a₂ + $_b₂ = $_c))⟩ := evalAdd va vb₂ ⟨_, .add vb₁ vc, q(add_pf_add_gt $b₁ $pc)⟩ theorem one_mul (a : R) : (nat_lit 1).rawCast * a = a := by simp [Nat.rawCast] theorem mul_one (a : R) : a * (nat_lit 1).rawCast = a := by simp [Nat.rawCast] theorem mul_pf_left (a₁ : R) (a₂) (_ : a₃ * b = c) : (a₁ ^ a₂ * a₃ : R) * b = a₁ ^ a₂ * c := by subst_vars; rw [mul_assoc] theorem mul_pf_right (b₁ : R) (b₂) (_ : a * b₃ = c) : a * (b₁ ^ b₂ * b₃) = b₁ ^ b₂ * c := by subst_vars; rw [mul_left_comm] theorem mul_pp_pf_overlap (x : R) (_ : ea + eb = e) (_ : a₂ * b₂ = c) : (x ^ ea * a₂ : R) * (x ^ eb * b₂) = x ^ e * c := by subst_vars; simp [pow_add, mul_mul_mul_comm] /-- Multiplies two monomials `va, vb` together to get a normalized result monomial. * `x * y = (x * y)` (for `x`, `y` coefficients) * `x * (b₁ * b₂) = b₁ * (b₂ * x)` (for `x` coefficient) * `(a₁ * a₂) * y = a₁ * (a₂ * y)` (for `y` coefficient) * `(x ^ ea * a₂) * (x ^ eb * b₂) = x ^ (ea + eb) * (a₂ * b₂)` (if `ea` and `eb` are identical except coefficient) * `(a₁ * a₂) * (b₁ * b₂) = a₁ * (a₂ * (b₁ * b₂))` (if `a₁.lt b₁`) * `(a₁ * a₂) * (b₁ * b₂) = b₁ * ((a₁ * a₂) * b₂)` (if not `a₁.lt b₁`) -/ partial def evalMulProd (va : ExProd sα a) (vb : ExProd sα b) : Result (ExProd sα) q($a * $b) := match va, vb with \| .const za ha, .const zb hb => if za = 1 then ⟨b, .const zb hb, (q(one_mul $b) : Expr)⟩ else if zb = 1 then ⟨a, .const za ha, (q(mul_one $a) : Expr)⟩ else let ra := Result.ofRawRat za a ha; let rb := Result.ofRawRat zb b hb let rc := (NormNum.evalMul.core q($a * $b) q(HMul.hMul) _ _ q(CommSemiring.toSemiring) ra rb).get! let ⟨zc, hc⟩ := rc.toRatNZ.get! let ⟨c, pc⟩ := rc.toRawEq ⟨c, .const zc hc, pc⟩ \| .mul (x := a₁) (e := a₂) va₁ va₂ va₃, .const _ _ => let ⟨_, vc, pc⟩ := evalMulProd va₃ vb ⟨_, .mul va₁ va₂ vc, (q(mul_pf_left $a₁ $a₂ $pc) : Expr)⟩ \| .const _ _, .mul (x := b₁) (e := b₂) vb₁ vb₂ vb₃ => let ⟨_, vc, pc⟩ := evalMulProd va vb₃ ⟨_, .mul vb₁ vb₂ vc, (q(mul_pf_right $b₁ $b₂ $pc) : Expr)⟩ \| .mul (x := xa) (e := ea) vxa vea va₂, .mul (x := xb) (e := eb) vxb veb vb₂ => Id.run do if vxa.eq vxb then if let some (.nonzero ⟨_, ve, pe⟩) := evalAddOverlap sℕ vea veb then let ⟨_, vc, pc⟩ := evalMulProd va₂ vb₂ return ⟨_, .mul vxa ve vc, (q(mul_pp_pf_overlap $xa $pe $pc) : Expr)⟩ if let .lt := (vxa.cmp vxb).then (vea.cmp veb) then let ⟨_, vc, pc⟩ := evalMulProd va₂ vb ⟨_, .mul vxa vea vc, (q(mul_pf_left $xa $ea $pc) : Expr)⟩ else let ⟨_, vc, pc⟩ := evalMulProd va vb₂ ⟨_, .mul vxb veb vc, (q(mul_pf_right $xb $eb $pc) : Expr)⟩ theorem mul_zero (a : R) : a * 0 = 0 := by simp theorem mul_add (_ : (a : R) * b₁ = c₁) (_ : a * b₂ = c₂) (_ : c₁ + 0 + c₂ = d) : a * (b₁ + b₂) = d := by subst_vars; simp [_root_.mul_add] /-- Multiplies a monomial `va` to a polynomial `vb` to get a normalized result polynomial. * `a * 0 = 0` * `a * (b₁ + b₂) = (a * b₁) + (a * b₂)` -/ def evalMul₁ (va : ExProd sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a * $b) := match vb with \| .zero => ⟨_, .zero, q(mul_zero $a)⟩ \| .add vb₁ vb₂ => let ⟨_, vc₁, pc₁⟩ := evalMulProd sα va vb₁ let ⟨_, vc₂, pc₂⟩ := evalMul₁ va vb₂ let ⟨_, vd, pd⟩ := evalAdd sα vc₁.toSum vc₂ ⟨_, vd, q(mul_add $pc₁ $pc₂ $pd)⟩ theorem zero_mul (b : R) : 0 * b = 0 := by simp theorem add_mul (_ : (a₁ : R) * b = c₁) (_ : a₂ * b = c₂) (_ : c₁ + c₂ = d) : (a₁ + a₂) * b = d := by subst_vars; simp [_root_.add_mul] /-- Multiplies two polynomials `va, vb` together to get a normalized result polynomial. * `0 * b = 0` * `(a₁ + a₂) * b = (a₁ * b) + (a₂ * b)` -/ def evalMul (va : ExSum sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a * $b) := match va with \| .zero => ⟨_, .zero, q(zero_mul $b)⟩ \| .add va₁ va₂ => let ⟨_, vc₁, pc₁⟩ := evalMul₁ sα va₁ vb let ⟨_, vc₂, pc₂⟩ := evalMul va₂ vb let ⟨_, vd, pd⟩ := evalAdd sα vc₁ vc₂ ⟨_, vd, q(add_mul $pc₁ $pc₂ $pd)⟩ theorem natCast_nat (n) : ((Nat.rawCast n : ℕ) : R) = Nat.rawCast n := by simp theorem natCast_mul (a₂) (_ : ((a₁ : ℕ) : R) = b₁) (_ : ((a₃ : ℕ) : R) = b₃) : ((a₁ ^ a₂ * a₃ : ℕ) : R) = b₁ ^ a₂ * b₃ := by subst_vars; simp theorem natCast_zero : ((0 : ℕ) : R) = 0 := Nat.cast_zero theorem natCast_add (_ : ((a₁ : ℕ) : R) = b₁) (_ : ((a₂ : ℕ) : R) = b₂) : ((a₁ + a₂ : ℕ) : R) = b₁ + b₂ := by subst_vars; simp mutual /-- Applies `Nat.cast` to a nat polynomial to produce a polynomial in `α`. * An atom `e` causes `↑e` to be allocated as a new atom. * A sum delegates to `ExSum.evalNatCast`. -/ partial def ExBase.evalNatCast (va : ExBase sℕ a) : AtomM (Result (ExBase sα) q($a)) := match va with \| .atom _ => do let a' : Q($α) := q($a) let i ← addAtom a' pure ⟨a', ExBase.atom i, (q(Eq.refl $a') : Expr)⟩ \| .sum va => do let ⟨_, vc, p⟩ ← va.evalNatCast pure ⟨_, .sum vc, p⟩ /-- Applies `Nat.cast` to a nat monomial to produce a monomial in `α`. * `↑c = c` if `c` is a numeric literal * `↑(a ^ n * b) = ↑a ^ n * ↑b` -/ partial def ExProd.evalNatCast (va : ExProd sℕ a) : AtomM (Result (ExProd sα) q($a)) := match va with \| .const c hc => have n : Q(ℕ) := a.appArg! pure ⟨q(Nat.rawCast $n), .const c hc, (q(natCast_nat (R := $α) $n) : Expr)⟩ \| .mul (e := a₂) va₁ va₂ va₃ => do let ⟨_, vb₁, pb₁⟩ ← va₁.evalNatCast let ⟨_, vb₃, pb₃⟩ ← va₃.evalNatCast pure ⟨_, .mul vb₁ va₂ vb₃, q(natCast_mul $a₂ $pb₁ $pb₃)⟩ /-- Applies `Nat.cast` to a nat polynomial to produce a polynomial in `α`. * `↑0 = 0` * `↑(a + b) = ↑a + ↑b` -/ partial def ExSum.evalNatCast (va : ExSum sℕ a) : AtomM (Result (ExSum sα) q($a)) := match va with \| .zero => pure ⟨_, .zero, q(natCast_zero (R := $α))⟩ \| .add va₁ va₂ => do let ⟨_, vb₁, pb₁⟩ ← va₁.evalNatCast let ⟨_, vb₂, pb₂⟩ ← va₂.evalNatCast pure ⟨_, .add vb₁ vb₂, q(natCast_add $pb₁ $pb₂)⟩ end theorem smul_nat (_ : (a * b : ℕ) = c) : a • b = c := by subst_vars; simp theorem smul_eq_cast (_ : ((a : ℕ) : R) = a') (_ : a' * b = c) : a • b = c := by subst_vars; simp /-- Constructs the scalar multiplication `n • a`, where both `n : ℕ` and `a : α` are normalized polynomial expressions. * `a • b = a * b` if `α = ℕ` * `a • b = ↑a * b` otherwise -/ def evalNSMul (va : ExSum sℕ a) (vb : ExSum sα b) : AtomM (Result (ExSum sα) q($a • $b)) := do if ← isDefEq sα sℕ then let ⟨_, va'⟩ := va.cast have _b : Q(ℕ) := b let ⟨(_c : Q(ℕ)), vc, (pc : Q($a * $_b = $_c))⟩ := evalMul sα va' vb pure ⟨_, vc, (q(smul_nat $pc) : Expr)⟩ else let ⟨_, va', pa'⟩ ← va.evalNatCast sα let ⟨_, vc, pc⟩ := evalMul sα va' vb pure ⟨_, vc, (q(smul_eq_cast $pa' $pc) : Expr)⟩ theorem neg_one_mul {R} [Ring R] {a b : R} (_ : (Int.negOfNat (nat_lit 1)).rawCast * a = b) : -a = b := by subst_vars; simp [Int.negOfNat] theorem neg_mul {R} [Ring R] (a₁ : R) (a₂) {a₃ b : R} (_ : -a₃ = b) : -(a₁ ^ a₂ * a₃) = a₁ ^ a₂ * b := by subst_vars; simp /-- Negates a monomial `va` to get another monomial. * `-c = (-c)` (for `c` coefficient) * `-(a₁ * a₂) = a₁ * -a₂` -/ def evalNegProd (rα : Q(Ring $α)) (va : ExProd sα a) : Result (ExProd sα) q(-$a) := match va with \| .const za ha => let lit : Q(ℕ) := mkRawNatLit 1 let ⟨m1, _⟩ := ExProd.mkNegNat sα rα 1 let rm := Result.isNegNat rα lit (q(IsInt.of_raw $α (.negOfNat $lit)) : Expr) let ra := Result.ofRawRat za a ha let rb := (NormNum.evalMul.core q($m1 * $a) q(HMul.hMul) _ _ q(CommSemiring.toSemiring) rm ra).get! let ⟨zb, hb⟩ := rb.toRatNZ.get! let ⟨b, (pb : Q((Int.negOfNat (nat_lit 1)).rawCast * $a = $b))⟩ := rb.toRawEq ⟨b, .const zb hb, (q(neg_one_mul (R := $α) $pb) : Expr)⟩ \| .mul (x := a₁) (e := a₂) va₁ va₂ va₃ => let ⟨_, vb, pb⟩ := evalNegProd rα va₃ ⟨_, .mul va₁ va₂ vb, (q(neg_mul $a₁ $a₂ $pb) : Expr)⟩ theorem neg_zero {R} [Ring R] : -(0 : R) = 0 := by simp theorem neg_add {R} [Ring R] {a₁ a₂ b₁ b₂ : R} (_ : -a₁ = b₁) (_ : -a₂ = b₂) : -(a₁ + a₂) = b₁ + b₂ := by subst_vars; simp [add_comm] /-- Negates a polynomial `va` to get another polynomial. * `-0 = 0` (for `c` coefficient) * `-(a₁ + a₂) = -a₁ + -a₂` -/ def evalNeg (rα : Q(Ring $α)) (va : ExSum sα a) : Result (ExSum sα) q(-$a) := match va with \| .zero => ⟨_, .zero, (q(neg_zero (R := $α)) : Expr)⟩ \| .add va₁ va₂ => let ⟨_, vb₁, pb₁⟩ := evalNegProd sα rα va₁ let ⟨_, vb₂, pb₂⟩ := evalNeg rα va₂ ⟨_, .add vb₁ vb₂, (q(neg_add $pb₁ $pb₂) : Expr)⟩ theorem sub_pf {R} [Ring R] {a b c d : R} (_ : -b = c) (_ : a + c = d) : a - b = d := by subst_vars; simp [sub_eq_add_neg] /-- Subtracts two polynomials `va, vb` to get a normalized result polynomial. * `a - b = a + -b` -/ def evalSub (rα : Q(Ring $α)) (va : ExSum sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a - $b) := let ⟨_c, vc, pc⟩ := evalNeg sα rα vb let ⟨d, vd, (pd : Q($a + $_c = $d))⟩ := evalAdd sα va vc ⟨d, vd, (q(sub_pf $pc $pd) : Expr)⟩ theorem pow_prod_atom (a : R) (b) : a ^ b = (a + 0) ^ b * (nat_lit 1).rawCast := by simp /-- The fallback case for exponentiating polynomials is to use `ExBase.toProd` to just build an exponent expression. (This has a slightly different normalization than `evalPowAtom` because the input types are different.) * `x ^ e = (x + 0) ^ e * 1` -/ def evalPowProdAtom (va : ExProd sα a) (vb : ExProd sℕ b) : Result (ExProd sα) q($a ^ $b) := ⟨_, (ExBase.sum va.toSum).toProd vb, q(pow_prod_atom $a $b)⟩	Mathlib/Tactic/Ring/Basic.lean	598	598	theorem pow_atom (a : R) (b) : a ^ b = a ^ b * (nat_lit 1).rawCast + 0 := by	simp
/- Copyright (c) 2021 Bolton Bailey. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bolton Bailey -/ import Mathlib.Algebra.Periodic import Mathlib.Data.Nat.Count import Mathlib.Data.Nat.GCD.Basic import Mathlib.Order.Interval.Finset.Nat #align_import data.nat.periodic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" /-! # Periodic Functions on ℕ This file identifies a few functions on `ℕ` which are periodic, and also proves a lemma about periodic predicates which helps determine their cardinality when filtering intervals over them. -/ namespace Nat open Nat Function theorem periodic_gcd (a : ℕ) : Periodic (gcd a) a := by simp only [forall_const, gcd_add_self_right, eq_self_iff_true, Periodic] #align nat.periodic_gcd Nat.periodic_gcd theorem periodic_coprime (a : ℕ) : Periodic (Coprime a) a := by simp only [coprime_add_self_right, forall_const, iff_self_iff, eq_iff_iff, Periodic] #align nat.periodic_coprime Nat.periodic_coprime theorem periodic_mod (a : ℕ) : Periodic (fun n => n % a) a := by simp only [forall_const, eq_self_iff_true, add_mod_right, Periodic] #align nat.periodic_mod Nat.periodic_mod theorem _root_.Function.Periodic.map_mod_nat {α : Type*} {f : ℕ → α} {a : ℕ} (hf : Periodic f a) : ∀ n, f (n % a) = f n := fun n => by conv_rhs => rw [← Nat.mod_add_div n a, mul_comm, ← Nat.nsmul_eq_mul, hf.nsmul] #align function.periodic.map_mod_nat Function.Periodic.map_mod_nat section Multiset open Multiset /-- An interval of length `a` filtered over a periodic predicate of period `a` has cardinality equal to the number naturals below `a` for which `p a` is true. -/	Mathlib/Data/Nat/Periodic.lean	48	54	theorem filter_multiset_Ico_card_eq_of_periodic (n a : ℕ) (p : ℕ → Prop) [DecidablePred p] (pp : Periodic p a) : card (filter p (Ico n (n + a))) = a.count p := by	rw [count_eq_card_filter_range, Finset.card, Finset.filter_val, Finset.range_val, ← multiset_Ico_map_mod n, ← map_count_True_eq_filter_card, ← map_count_True_eq_filter_card, map_map] congr; funext n exact (Function.Periodic.map_mod_nat pp n).symm
/- Copyright (c) 2020 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Johan Commelin -/ import Mathlib.Analysis.Analytic.Basic import Mathlib.Combinatorics.Enumerative.Composition #align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a" /-! # Composition of analytic functions In this file we prove that the composition of analytic functions is analytic. The argument is the following. Assume `g z = ∑' qₙ (z, ..., z)` and `f y = ∑' pₖ (y, ..., y)`. Then `g (f y) = ∑' qₙ (∑' pₖ (y, ..., y), ..., ∑' pₖ (y, ..., y)) = ∑' qₙ (p_{i₁} (y, ..., y), ..., p_{iₙ} (y, ..., y))`. For each `n` and `i₁, ..., iₙ`, define a `i₁ + ... + iₙ` multilinear function mapping `(y₀, ..., y_{i₁ + ... + iₙ - 1})` to `qₙ (p_{i₁} (y₀, ..., y_{i₁-1}), p_{i₂} (y_{i₁}, ..., y_{i₁ + i₂ - 1}), ..., p_{iₙ} (....)))`. Then `g ∘ f` is obtained by summing all these multilinear functions. To formalize this, we use compositions of an integer `N`, i.e., its decompositions into a sum `i₁ + ... + iₙ` of positive integers. Given such a composition `c` and two formal multilinear series `q` and `p`, let `q.comp_along_composition p c` be the above multilinear function. Then the `N`-th coefficient in the power series expansion of `g ∘ f` is the sum of these terms over all `c : composition N`. To complete the proof, we need to show that this power series has a positive radius of convergence. This follows from the fact that `composition N` has cardinality `2^(N-1)` and estimates on the norm of `qₙ` and `pₖ`, which give summability. We also need to show that it indeed converges to `g ∘ f`. For this, we note that the composition of partial sums converges to `g ∘ f`, and that it corresponds to a part of the whole sum, on a subset that increases to the whole space. By summability of the norms, this implies the overall convergence. ## Main results * `q.comp p` is the formal composition of the formal multilinear series `q` and `p`. * `HasFPowerSeriesAt.comp` states that if two functions `g` and `f` admit power series expansions `q` and `p`, then `g ∘ f` admits a power series expansion given by `q.comp p`. * `AnalyticAt.comp` states that the composition of analytic functions is analytic. * `FormalMultilinearSeries.comp_assoc` states that composition is associative on formal multilinear series. ## Implementation details The main technical difficulty is to write down things. In particular, we need to define precisely `q.comp_along_composition p c` and to show that it is indeed a continuous multilinear function. This requires a whole interface built on the class `Composition`. Once this is set, the main difficulty is to reorder the sums, writing the composition of the partial sums as a sum over some subset of `Σ n, composition n`. We need to check that the reordering is a bijection, running over difficulties due to the dependent nature of the types under consideration, that are controlled thanks to the interface for `Composition`. The associativity of composition on formal multilinear series is a nontrivial result: it does not follow from the associativity of composition of analytic functions, as there is no uniqueness for the formal multilinear series representing a function (and also, it holds even when the radius of convergence of the series is `0`). Instead, we give a direct proof, which amounts to reordering double sums in a careful way. The change of variables is a canonical (combinatorial) bijection `Composition.sigmaEquivSigmaPi` between `(Σ (a : composition n), composition a.length)` and `(Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i))`, and is described in more details below in the paragraph on associativity. -/ noncomputable section variable {𝕜 : Type} {E F G H : Type} open Filter List open scoped Topology Classical NNReal ENNReal section Topological variable [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] variable [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] variable [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] /-! ### Composing formal multilinear series -/ namespace FormalMultilinearSeries variable [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] variable [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] variable [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] /-! In this paragraph, we define the composition of formal multilinear series, by summing over all possible compositions of `n`. -/ /-- Given a formal multilinear series `p`, a composition `c` of `n` and the index `i` of a block of `c`, we may define a function on `fin n → E` by picking the variables in the `i`-th block of `n`, and applying the corresponding coefficient of `p` to these variables. This function is called `p.apply_composition c v i` for `v : fin n → E` and `i : fin c.length`. -/ def applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n) : (Fin n → E) → Fin c.length → F := fun v i => p (c.blocksFun i) (v ∘ c.embedding i) #align formal_multilinear_series.apply_composition FormalMultilinearSeries.applyComposition theorem applyComposition_ones (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : p.applyComposition (Composition.ones n) = fun v i => p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) := by funext v i apply p.congr (Composition.ones_blocksFun _ _) intro j hjn hj1 obtain rfl : j = 0 := by omega refine congr_arg v ?_ rw [Fin.ext_iff, Fin.coe_castLE, Composition.ones_embedding, Fin.val_mk] #align formal_multilinear_series.apply_composition_ones FormalMultilinearSeries.applyComposition_ones theorem applyComposition_single (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n) (v : Fin n → E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v := by ext j refine p.congr (by simp) fun i hi1 hi2 => ?_ dsimp congr 1 convert Composition.single_embedding hn ⟨i, hi2⟩ using 1 cases' j with j_val j_property have : j_val = 0 := le_bot_iff.1 (Nat.lt_succ_iff.1 j_property) congr! simp #align formal_multilinear_series.apply_composition_single FormalMultilinearSeries.applyComposition_single @[simp] theorem removeZero_applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by ext v i simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos] #align formal_multilinear_series.remove_zero_apply_composition FormalMultilinearSeries.removeZero_applyComposition /-- Technical lemma stating how `p.apply_composition` commutes with updating variables. This will be the key point to show that functions constructed from `apply_composition` retain multilinearity. -/ theorem applyComposition_update (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n) (j : Fin n) (v : Fin n → E) (z : E) : p.applyComposition c (Function.update v j z) = Function.update (p.applyComposition c v) (c.index j) (p (c.blocksFun (c.index j)) (Function.update (v ∘ c.embedding (c.index j)) (c.invEmbedding j) z)) := by ext k by_cases h : k = c.index j · rw [h] let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j) simp only [Function.update_same] change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _ let j' := c.invEmbedding j suffices B : Function.update v j z ∘ r = Function.update (v ∘ r) j' z by rw [B] suffices C : Function.update v (r j') z ∘ r = Function.update (v ∘ r) j' z by convert C; exact (c.embedding_comp_inv j).symm exact Function.update_comp_eq_of_injective _ (c.embedding _).injective _ _ · simp only [h, Function.update_eq_self, Function.update_noteq, Ne, not_false_iff] let r : Fin (c.blocksFun k) → Fin n := c.embedding k change p (c.blocksFun k) (Function.update v j z ∘ r) = p (c.blocksFun k) (v ∘ r) suffices B : Function.update v j z ∘ r = v ∘ r by rw [B] apply Function.update_comp_eq_of_not_mem_range rwa [c.mem_range_embedding_iff'] #align formal_multilinear_series.apply_composition_update FormalMultilinearSeries.applyComposition_update @[simp] theorem compContinuousLinearMap_applyComposition {n : ℕ} (p : FormalMultilinearSeries 𝕜 F G) (f : E →L[𝕜] F) (c : Composition n) (v : Fin n → E) : (p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (f ∘ v) := by simp (config := {unfoldPartialApp := true}) [applyComposition]; rfl #align formal_multilinear_series.comp_continuous_linear_map_apply_composition FormalMultilinearSeries.compContinuousLinearMap_applyComposition end FormalMultilinearSeries namespace ContinuousMultilinearMap open FormalMultilinearSeries variable [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] variable [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] /-- Given a formal multilinear series `p`, a composition `c` of `n` and a continuous multilinear map `f` in `c.length` variables, one may form a continuous multilinear map in `n` variables by applying the right coefficient of `p` to each block of the composition, and then applying `f` to the resulting vector. It is called `f.comp_along_composition p c`. -/ def compAlongComposition {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (f : ContinuousMultilinearMap 𝕜 (fun _i : Fin c.length => F) G) : ContinuousMultilinearMap 𝕜 (fun _i : Fin n => E) G where toFun v := f (p.applyComposition c v) map_add' v i x y := by cases Subsingleton.elim ‹_› (instDecidableEqFin _) simp only [applyComposition_update, ContinuousMultilinearMap.map_add] map_smul' v i c x := by cases Subsingleton.elim ‹_› (instDecidableEqFin _) simp only [applyComposition_update, ContinuousMultilinearMap.map_smul] cont := f.cont.comp <\| continuous_pi fun i => (coe_continuous _).comp <\| continuous_pi fun j => continuous_apply _ #align continuous_multilinear_map.comp_along_composition ContinuousMultilinearMap.compAlongComposition @[simp] theorem compAlongComposition_apply {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (f : ContinuousMultilinearMap 𝕜 (fun _i : Fin c.length => F) G) (v : Fin n → E) : (f.compAlongComposition p c) v = f (p.applyComposition c v) := rfl #align continuous_multilinear_map.comp_along_composition_apply ContinuousMultilinearMap.compAlongComposition_apply end ContinuousMultilinearMap namespace FormalMultilinearSeries variable [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] variable [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] variable [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] /-- Given two formal multilinear series `q` and `p` and a composition `c` of `n`, one may form a continuous multilinear map in `n` variables by applying the right coefficient of `p` to each block of the composition, and then applying `q c.length` to the resulting vector. It is called `q.comp_along_composition p c`. -/ def compAlongComposition {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) : ContinuousMultilinearMap 𝕜 (fun _i : Fin n => E) G := (q c.length).compAlongComposition p c #align formal_multilinear_series.comp_along_composition FormalMultilinearSeries.compAlongComposition @[simp] theorem compAlongComposition_apply {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (v : Fin n → E) : (q.compAlongComposition p c) v = q c.length (p.applyComposition c v) := rfl #align formal_multilinear_series.comp_along_composition_apply FormalMultilinearSeries.compAlongComposition_apply /-- Formal composition of two formal multilinear series. The `n`-th coefficient in the composition is defined to be the sum of `q.comp_along_composition p c` over all compositions of `n`. In other words, this term (as a multilinear function applied to `v_0, ..., v_{n-1}`) is `∑'_{k} ∑'_{i₁ + ... + iₖ = n} qₖ (p_{i_1} (...), ..., p_{i_k} (...))`, where one puts all variables `v_0, ..., v_{n-1}` in increasing order in the dots. In general, the composition `q ∘ p` only makes sense when the constant coefficient of `p` vanishes. We give a general formula but which ignores the value of `p 0` instead. -/ protected def comp (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) : FormalMultilinearSeries 𝕜 E G := fun n => ∑ c : Composition n, q.compAlongComposition p c #align formal_multilinear_series.comp FormalMultilinearSeries.comp /-- The `0`-th coefficient of `q.comp p` is `q 0`. Since these maps are multilinear maps in zero variables, but on different spaces, we can not state this directly, so we state it when applied to arbitrary vectors (which have to be the zero vector). -/ theorem comp_coeff_zero (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (v : Fin 0 → E) (v' : Fin 0 → F) : (q.comp p) 0 v = q 0 v' := by let c : Composition 0 := Composition.ones 0 dsimp [FormalMultilinearSeries.comp] have : {c} = (Finset.univ : Finset (Composition 0)) := by apply Finset.eq_of_subset_of_card_le <;> simp [Finset.card_univ, composition_card 0] rw [← this, Finset.sum_singleton, compAlongComposition_apply] symm; congr! -- Porting note: needed the stronger `congr!`! #align formal_multilinear_series.comp_coeff_zero FormalMultilinearSeries.comp_coeff_zero @[simp] theorem comp_coeff_zero' (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (v : Fin 0 → E) : (q.comp p) 0 v = q 0 fun _i => 0 := q.comp_coeff_zero p v _ #align formal_multilinear_series.comp_coeff_zero' FormalMultilinearSeries.comp_coeff_zero' /-- The `0`-th coefficient of `q.comp p` is `q 0`. When `p` goes from `E` to `E`, this can be expressed as a direct equality -/	Mathlib/Analysis/Analytic/Composition.lean	266	267	theorem comp_coeff_zero'' (q : FormalMultilinearSeries 𝕜 E F) (p : FormalMultilinearSeries 𝕜 E E) : (q.comp p) 0 = q 0 := by	ext v; exact q.comp_coeff_zero p _ _
/- Copyright (c) 2021 Kalle Kytölä. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kalle Kytölä -/ import Mathlib.Topology.Algebra.Module.WeakDual import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed #align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Finite measures This file defines the type of finite measures on a given measurable space. When the underlying space has a topology and the measurable space structure (sigma algebra) is finer than the Borel sigma algebra, then the type of finite measures is equipped with the topology of weak convergence of measures. The topology of weak convergence is the coarsest topology w.r.t. which for every bounded continuous `ℝ≥0`-valued function `f`, the integration of `f` against the measure is continuous. ## Main definitions The main definitions are * `MeasureTheory.FiniteMeasure Ω`: The type of finite measures on `Ω` with the topology of weak convergence of measures. * `MeasureTheory.FiniteMeasure.toWeakDualBCNN : FiniteMeasure Ω → (WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0))`: Interpret a finite measure as a continuous linear functional on the space of bounded continuous nonnegative functions on `Ω`. This is used for the definition of the topology of weak convergence. * `MeasureTheory.FiniteMeasure.map`: The push-forward `f* μ` of a finite measure `μ` on `Ω` along a measurable function `f : Ω → Ω'`. * `MeasureTheory.FiniteMeasure.mapCLM`: The push-forward along a given continuous `f : Ω → Ω'` as a continuous linear map `f* : FiniteMeasure Ω →L[ℝ≥0] FiniteMeasure Ω'`. ## Main results * Finite measures `μ` on `Ω` give rise to continuous linear functionals on the space of bounded continuous nonnegative functions on `Ω` via integration: `MeasureTheory.FiniteMeasure.toWeakDualBCNN : FiniteMeasure Ω → (WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0))` * `MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto`: Convergence of finite measures is characterized by the convergence of integrals of all bounded continuous functions. This shows that the chosen definition of topology coincides with the common textbook definition of weak convergence of measures. A similar characterization by the convergence of integrals (in the `MeasureTheory.lintegral` sense) of all bounded continuous nonnegative functions is `MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto`. * `MeasureTheory.FiniteMeasure.continuous_map`: For a continuous function `f : Ω → Ω'`, the push-forward of finite measures `f* : FiniteMeasure Ω → FiniteMeasure Ω'` is continuous. * `MeasureTheory.FiniteMeasure.t2Space`: The topology of weak convergence of finite Borel measures is Hausdorff on spaces where indicators of closed sets have continuous decreasing approximating sequences (in particular on any pseudo-metrizable spaces). ## Implementation notes The topology of weak convergence of finite Borel measures is defined using a mapping from `MeasureTheory.FiniteMeasure Ω` to `WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0)`, inheriting the topology from the latter. The implementation of `MeasureTheory.FiniteMeasure Ω` and is directly as a subtype of `MeasureTheory.Measure Ω`, and the coercion to a function is the composition `ENNReal.toNNReal` and the coercion to function of `MeasureTheory.Measure Ω`. Another alternative would have been to use a bijection with `MeasureTheory.VectorMeasure Ω ℝ≥0` as an intermediate step. Some considerations: * Potential advantages of using the `NNReal`-valued vector measure alternative: * The coercion to function would avoid need to compose with `ENNReal.toNNReal`, the `NNReal`-valued API could be more directly available. * Potential drawbacks of the vector measure alternative: * The coercion to function would lose monotonicity, as non-measurable sets would be defined to have measure 0. * No integration theory directly. E.g., the topology definition requires `MeasureTheory.lintegral` w.r.t. a coercion to `MeasureTheory.Measure Ω` in any case. ## References * [Billingsley, Convergence of probability measures][billingsley1999] ## Tags weak convergence of measures, finite measure -/ noncomputable section open MeasureTheory open Set open Filter open BoundedContinuousFunction open scoped Topology ENNReal NNReal BoundedContinuousFunction namespace MeasureTheory namespace FiniteMeasure section FiniteMeasure /-! ### Finite measures In this section we define the `Type` of `MeasureTheory.FiniteMeasure Ω`, when `Ω` is a measurable space. Finite measures on `Ω` are a module over `ℝ≥0`. If `Ω` is moreover a topological space and the sigma algebra on `Ω` is finer than the Borel sigma algebra (i.e. `[OpensMeasurableSpace Ω]`), then `MeasureTheory.FiniteMeasure Ω` is equipped with the topology of weak convergence of measures. This is implemented by defining a pairing of finite measures `μ` on `Ω` with continuous bounded nonnegative functions `f : Ω →ᵇ ℝ≥0` via integration, and using the associated weak topology (essentially the weak-star topology on the dual of `Ω →ᵇ ℝ≥0`). -/ variable {Ω : Type} [MeasurableSpace Ω] /-- Finite measures are defined as the subtype of measures that have the property of being finite measures (i.e., their total mass is finite). -/ def _root_.MeasureTheory.FiniteMeasure (Ω : Type) [MeasurableSpace Ω] : Type _ := { μ : Measure Ω // IsFiniteMeasure μ } #align measure_theory.finite_measure MeasureTheory.FiniteMeasure -- Porting note: as with other subtype synonyms (e.g., `ℝ≥0`, we need a new function for the -- coercion instead of relying on `Subtype.val`. /-- Coercion from `MeasureTheory.FiniteMeasure Ω` to `MeasureTheory.Measure Ω`. -/ @[coe] def toMeasure : FiniteMeasure Ω → Measure Ω := Subtype.val /-- A finite measure can be interpreted as a measure. -/ instance instCoe : Coe (FiniteMeasure Ω) (MeasureTheory.Measure Ω) where coe := toMeasure instance isFiniteMeasure (μ : FiniteMeasure Ω) : IsFiniteMeasure (μ : Measure Ω) := μ.prop #align measure_theory.finite_measure.is_finite_measure MeasureTheory.FiniteMeasure.isFiniteMeasure @[simp] theorem val_eq_toMeasure (ν : FiniteMeasure Ω) : ν.val = (ν : Measure Ω) := rfl #align measure_theory.finite_measure.val_eq_to_measure MeasureTheory.FiniteMeasure.val_eq_toMeasure theorem toMeasure_injective : Function.Injective ((↑) : FiniteMeasure Ω → Measure Ω) := Subtype.coe_injective #align measure_theory.finite_measure.coe_injective MeasureTheory.FiniteMeasure.toMeasure_injective instance instFunLike : FunLike (FiniteMeasure Ω) (Set Ω) ℝ≥0 where coe μ s := ((μ : Measure Ω) s).toNNReal coe_injective' μ ν h := toMeasure_injective $ Measure.ext fun s _ ↦ by simpa [ENNReal.toNNReal_eq_toNNReal_iff, measure_ne_top] using congr_fun h s lemma coeFn_def (μ : FiniteMeasure Ω) : μ = fun s ↦ ((μ : Measure Ω) s).toNNReal := rfl #align measure_theory.finite_measure.coe_fn_eq_to_nnreal_coe_fn_to_measure MeasureTheory.FiniteMeasure.coeFn_def lemma coeFn_mk (μ : Measure Ω) (hμ) : DFunLike.coe (F := FiniteMeasure Ω) ⟨μ, hμ⟩ = fun s ↦ (μ s).toNNReal := rfl @[simp, norm_cast] lemma mk_apply (μ : Measure Ω) (hμ) (s : Set Ω) : DFunLike.coe (F := FiniteMeasure Ω) ⟨μ, hμ⟩ s = (μ s).toNNReal := rfl @[simp] theorem ennreal_coeFn_eq_coeFn_toMeasure (ν : FiniteMeasure Ω) (s : Set Ω) : (ν s : ℝ≥0∞) = (ν : Measure Ω) s := ENNReal.coe_toNNReal (measure_lt_top (↑ν) s).ne #align measure_theory.finite_measure.ennreal_coe_fn_eq_coe_fn_to_measure MeasureTheory.FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure theorem apply_mono (μ : FiniteMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ := by change ((μ : Measure Ω) s₁).toNNReal ≤ ((μ : Measure Ω) s₂).toNNReal have key : (μ : Measure Ω) s₁ ≤ (μ : Measure Ω) s₂ := (μ : Measure Ω).mono h apply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ s₁) (measure_ne_top _ s₂)).mpr key #align measure_theory.finite_measure.apply_mono MeasureTheory.FiniteMeasure.apply_mono /-- The (total) mass of a finite measure `μ` is `μ univ`, i.e., the cast to `NNReal` of `(μ : measure Ω) univ`. -/ def mass (μ : FiniteMeasure Ω) : ℝ≥0 := μ univ #align measure_theory.finite_measure.mass MeasureTheory.FiniteMeasure.mass @[simp] theorem apply_le_mass (μ : FiniteMeasure Ω) (s : Set Ω) : μ s ≤ μ.mass := by simpa using apply_mono μ (subset_univ s) @[simp] theorem ennreal_mass {μ : FiniteMeasure Ω} : (μ.mass : ℝ≥0∞) = (μ : Measure Ω) univ := ennreal_coeFn_eq_coeFn_toMeasure μ Set.univ #align measure_theory.finite_measure.ennreal_mass MeasureTheory.FiniteMeasure.ennreal_mass instance instZero : Zero (FiniteMeasure Ω) where zero := ⟨0, MeasureTheory.isFiniteMeasureZero⟩ #align measure_theory.finite_measure.has_zero MeasureTheory.FiniteMeasure.instZero @[simp, norm_cast] lemma coeFn_zero : ⇑(0 : FiniteMeasure Ω) = 0 := rfl #align measure_theory.finite_measure.coe_fn_zero MeasureTheory.FiniteMeasure.coeFn_zero @[simp] theorem zero_mass : (0 : FiniteMeasure Ω).mass = 0 := rfl #align measure_theory.finite_measure.zero.mass MeasureTheory.FiniteMeasure.zero_mass @[simp] theorem mass_zero_iff (μ : FiniteMeasure Ω) : μ.mass = 0 ↔ μ = 0 := by refine ⟨fun μ_mass => ?_, fun hμ => by simp only [hμ, zero_mass]⟩ apply toMeasure_injective apply Measure.measure_univ_eq_zero.mp rwa [← ennreal_mass, ENNReal.coe_eq_zero] #align measure_theory.finite_measure.mass_zero_iff MeasureTheory.FiniteMeasure.mass_zero_iff theorem mass_nonzero_iff (μ : FiniteMeasure Ω) : μ.mass ≠ 0 ↔ μ ≠ 0 := by rw [not_iff_not] exact FiniteMeasure.mass_zero_iff μ #align measure_theory.finite_measure.mass_nonzero_iff MeasureTheory.FiniteMeasure.mass_nonzero_iff @[ext]	Mathlib/MeasureTheory/Measure/FiniteMeasure.lean	213	217	theorem eq_of_forall_toMeasure_apply_eq (μ ν : FiniteMeasure Ω) (h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by	apply Subtype.ext ext1 s s_mble exact h s s_mble
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne -/ import Mathlib.Order.MinMax import Mathlib.Data.Set.Subsingleton import Mathlib.Tactic.Says #align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c" /-! # Intervals In any preorder `α`, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. This file contains these definitions, and basic facts on inclusion, intersection, difference of intervals (where the precise statements may depend on the properties of the order, in particular for some statements it should be `LinearOrder` or `DenselyOrdered`). TODO: This is just the beginning; a lot of rules are missing -/ open Function open OrderDual (toDual ofDual) variable {α β : Type} namespace Set section Preorder variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α} /-- Left-open right-open interval -/ def Ioo (a b : α) := { x \| a < x ∧ x < b } #align set.Ioo Set.Ioo /-- Left-closed right-open interval -/ def Ico (a b : α) := { x \| a ≤ x ∧ x < b } #align set.Ico Set.Ico /-- Left-infinite right-open interval -/ def Iio (a : α) := { x \| x < a } #align set.Iio Set.Iio /-- Left-closed right-closed interval -/ def Icc (a b : α) := { x \| a ≤ x ∧ x ≤ b } #align set.Icc Set.Icc /-- Left-infinite right-closed interval -/ def Iic (b : α) := { x \| x ≤ b } #align set.Iic Set.Iic /-- Left-open right-closed interval -/ def Ioc (a b : α) := { x \| a < x ∧ x ≤ b } #align set.Ioc Set.Ioc /-- Left-closed right-infinite interval -/ def Ici (a : α) := { x \| a ≤ x } #align set.Ici Set.Ici /-- Left-open right-infinite interval -/ def Ioi (a : α) := { x \| a < x } #align set.Ioi Set.Ioi theorem Ioo_def (a b : α) : { x \| a < x ∧ x < b } = Ioo a b := rfl #align set.Ioo_def Set.Ioo_def theorem Ico_def (a b : α) : { x \| a ≤ x ∧ x < b } = Ico a b := rfl #align set.Ico_def Set.Ico_def theorem Iio_def (a : α) : { x \| x < a } = Iio a := rfl #align set.Iio_def Set.Iio_def theorem Icc_def (a b : α) : { x \| a ≤ x ∧ x ≤ b } = Icc a b := rfl #align set.Icc_def Set.Icc_def theorem Iic_def (b : α) : { x \| x ≤ b } = Iic b := rfl #align set.Iic_def Set.Iic_def theorem Ioc_def (a b : α) : { x \| a < x ∧ x ≤ b } = Ioc a b := rfl #align set.Ioc_def Set.Ioc_def theorem Ici_def (a : α) : { x \| a ≤ x } = Ici a := rfl #align set.Ici_def Set.Ici_def theorem Ioi_def (a : α) : { x \| a < x } = Ioi a := rfl #align set.Ioi_def Set.Ioi_def @[simp] theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b := Iff.rfl #align set.mem_Ioo Set.mem_Ioo @[simp] theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b := Iff.rfl #align set.mem_Ico Set.mem_Ico @[simp] theorem mem_Iio : x ∈ Iio b ↔ x < b := Iff.rfl #align set.mem_Iio Set.mem_Iio @[simp] theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b := Iff.rfl #align set.mem_Icc Set.mem_Icc @[simp] theorem mem_Iic : x ∈ Iic b ↔ x ≤ b := Iff.rfl #align set.mem_Iic Set.mem_Iic @[simp] theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b := Iff.rfl #align set.mem_Ioc Set.mem_Ioc @[simp] theorem mem_Ici : x ∈ Ici a ↔ a ≤ x := Iff.rfl #align set.mem_Ici Set.mem_Ici @[simp] theorem mem_Ioi : x ∈ Ioi a ↔ a < x := Iff.rfl #align set.mem_Ioi Set.mem_Ioi instance decidableMemIoo [Decidable (a < x ∧ x < b)] : Decidable (x ∈ Ioo a b) := by assumption #align set.decidable_mem_Ioo Set.decidableMemIoo instance decidableMemIco [Decidable (a ≤ x ∧ x < b)] : Decidable (x ∈ Ico a b) := by assumption #align set.decidable_mem_Ico Set.decidableMemIco instance decidableMemIio [Decidable (x < b)] : Decidable (x ∈ Iio b) := by assumption #align set.decidable_mem_Iio Set.decidableMemIio instance decidableMemIcc [Decidable (a ≤ x ∧ x ≤ b)] : Decidable (x ∈ Icc a b) := by assumption #align set.decidable_mem_Icc Set.decidableMemIcc instance decidableMemIic [Decidable (x ≤ b)] : Decidable (x ∈ Iic b) := by assumption #align set.decidable_mem_Iic Set.decidableMemIic instance decidableMemIoc [Decidable (a < x ∧ x ≤ b)] : Decidable (x ∈ Ioc a b) := by assumption #align set.decidable_mem_Ioc Set.decidableMemIoc instance decidableMemIci [Decidable (a ≤ x)] : Decidable (x ∈ Ici a) := by assumption #align set.decidable_mem_Ici Set.decidableMemIci instance decidableMemIoi [Decidable (a < x)] : Decidable (x ∈ Ioi a) := by assumption #align set.decidable_mem_Ioi Set.decidableMemIoi -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioo Set.left_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl] #align set.left_mem_Ico Set.left_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.left_mem_Icc Set.left_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioc Set.left_mem_Ioc theorem left_mem_Ici : a ∈ Ici a := by simp #align set.left_mem_Ici Set.left_mem_Ici -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioo : b ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ioo Set.right_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ico : b ∈ Ico a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ico Set.right_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.right_mem_Icc Set.right_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl] #align set.right_mem_Ioc Set.right_mem_Ioc theorem right_mem_Iic : a ∈ Iic a := by simp #align set.right_mem_Iic Set.right_mem_Iic @[simp] theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a := rfl #align set.dual_Ici Set.dual_Ici @[simp] theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a := rfl #align set.dual_Iic Set.dual_Iic @[simp] theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a := rfl #align set.dual_Ioi Set.dual_Ioi @[simp] theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a := rfl #align set.dual_Iio Set.dual_Iio @[simp] theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a := Set.ext fun _ => and_comm #align set.dual_Icc Set.dual_Icc @[simp] theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a := Set.ext fun _ => and_comm #align set.dual_Ioc Set.dual_Ioc @[simp] theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a := Set.ext fun _ => and_comm #align set.dual_Ico Set.dual_Ico @[simp] theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a := Set.ext fun _ => and_comm #align set.dual_Ioo Set.dual_Ioo @[simp] theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := ⟨fun ⟨_, hx⟩ => hx.1.trans hx.2, fun h => ⟨a, left_mem_Icc.2 h⟩⟩ #align set.nonempty_Icc Set.nonempty_Icc @[simp] theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_lt hx.2, fun h => ⟨a, left_mem_Ico.2 h⟩⟩ #align set.nonempty_Ico Set.nonempty_Ico @[simp] theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_le hx.2, fun h => ⟨b, right_mem_Ioc.2 h⟩⟩ #align set.nonempty_Ioc Set.nonempty_Ioc @[simp] theorem nonempty_Ici : (Ici a).Nonempty := ⟨a, left_mem_Ici⟩ #align set.nonempty_Ici Set.nonempty_Ici @[simp] theorem nonempty_Iic : (Iic a).Nonempty := ⟨a, right_mem_Iic⟩ #align set.nonempty_Iic Set.nonempty_Iic @[simp] theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := ⟨fun ⟨_, ha, hb⟩ => ha.trans hb, exists_between⟩ #align set.nonempty_Ioo Set.nonempty_Ioo @[simp] theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty := exists_gt a #align set.nonempty_Ioi Set.nonempty_Ioi @[simp] theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty := exists_lt a #align set.nonempty_Iio Set.nonempty_Iio theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) := Nonempty.to_subtype (nonempty_Icc.mpr h) #align set.nonempty_Icc_subtype Set.nonempty_Icc_subtype theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) := Nonempty.to_subtype (nonempty_Ico.mpr h) #align set.nonempty_Ico_subtype Set.nonempty_Ico_subtype theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) := Nonempty.to_subtype (nonempty_Ioc.mpr h) #align set.nonempty_Ioc_subtype Set.nonempty_Ioc_subtype /-- An interval `Ici a` is nonempty. -/ instance nonempty_Ici_subtype : Nonempty (Ici a) := Nonempty.to_subtype nonempty_Ici #align set.nonempty_Ici_subtype Set.nonempty_Ici_subtype /-- An interval `Iic a` is nonempty. -/ instance nonempty_Iic_subtype : Nonempty (Iic a) := Nonempty.to_subtype nonempty_Iic #align set.nonempty_Iic_subtype Set.nonempty_Iic_subtype theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) := Nonempty.to_subtype (nonempty_Ioo.mpr h) #align set.nonempty_Ioo_subtype Set.nonempty_Ioo_subtype /-- In an order without maximal elements, the intervals `Ioi` are nonempty. -/ instance nonempty_Ioi_subtype [NoMaxOrder α] : Nonempty (Ioi a) := Nonempty.to_subtype nonempty_Ioi #align set.nonempty_Ioi_subtype Set.nonempty_Ioi_subtype /-- In an order without minimal elements, the intervals `Iio` are nonempty. -/ instance nonempty_Iio_subtype [NoMinOrder α] : Nonempty (Iio a) := Nonempty.to_subtype nonempty_Iio #align set.nonempty_Iio_subtype Set.nonempty_Iio_subtype instance [NoMinOrder α] : NoMinOrder (Iio a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, lt_trans hb a.2⟩, hb⟩⟩ instance [NoMinOrder α] : NoMinOrder (Iic a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, hb.le.trans a.2⟩, hb⟩⟩ instance [NoMaxOrder α] : NoMaxOrder (Ioi a) := OrderDual.noMaxOrder (α := Iio (toDual a)) instance [NoMaxOrder α] : NoMaxOrder (Ici a) := OrderDual.noMaxOrder (α := Iic (toDual a)) @[simp] theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Icc_eq_empty Set.Icc_eq_empty @[simp] theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_lt hb) #align set.Ico_eq_empty Set.Ico_eq_empty @[simp] theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_le hb) #align set.Ioc_eq_empty Set.Ioc_eq_empty @[simp] theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Ioo_eq_empty Set.Ioo_eq_empty @[simp] theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ := Icc_eq_empty h.not_le #align set.Icc_eq_empty_of_lt Set.Icc_eq_empty_of_lt @[simp] theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ := Ico_eq_empty h.not_lt #align set.Ico_eq_empty_of_le Set.Ico_eq_empty_of_le @[simp] theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ := Ioc_eq_empty h.not_lt #align set.Ioc_eq_empty_of_le Set.Ioc_eq_empty_of_le @[simp] theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ := Ioo_eq_empty h.not_lt #align set.Ioo_eq_empty_of_le Set.Ioo_eq_empty_of_le -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ico_self (a : α) : Ico a a = ∅ := Ico_eq_empty <\| lt_irrefl _ #align set.Ico_self Set.Ico_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioc_self (a : α) : Ioc a a = ∅ := Ioc_eq_empty <\| lt_irrefl _ #align set.Ioc_self Set.Ioc_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioo_self (a : α) : Ioo a a = ∅ := Ioo_eq_empty <\| lt_irrefl _ #align set.Ioo_self Set.Ioo_self theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a := ⟨fun h => h <\| left_mem_Ici, fun h _ hx => h.trans hx⟩ #align set.Ici_subset_Ici Set.Ici_subset_Ici @[gcongr] alias ⟨_, _root_.GCongr.Ici_subset_Ici_of_le⟩ := Ici_subset_Ici theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b := @Ici_subset_Ici αᵒᵈ _ _ _ #align set.Iic_subset_Iic Set.Iic_subset_Iic @[gcongr] alias ⟨_, _root_.GCongr.Iic_subset_Iic_of_le⟩ := Iic_subset_Iic theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a := ⟨fun h => h left_mem_Ici, fun h _ hx => h.trans_le hx⟩ #align set.Ici_subset_Ioi Set.Ici_subset_Ioi theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b := ⟨fun h => h right_mem_Iic, fun h _ hx => lt_of_le_of_lt hx h⟩ #align set.Iic_subset_Iio Set.Iic_subset_Iio @[gcongr] theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans_le h₂⟩ #align set.Ioo_subset_Ioo Set.Ioo_subset_Ioo @[gcongr] theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b := Ioo_subset_Ioo h le_rfl #align set.Ioo_subset_Ioo_left Set.Ioo_subset_Ioo_left @[gcongr] theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ := Ioo_subset_Ioo le_rfl h #align set.Ioo_subset_Ioo_right Set.Ioo_subset_Ioo_right @[gcongr] theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, hx₂.trans_le h₂⟩ #align set.Ico_subset_Ico Set.Ico_subset_Ico @[gcongr] theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b := Ico_subset_Ico h le_rfl #align set.Ico_subset_Ico_left Set.Ico_subset_Ico_left @[gcongr] theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ := Ico_subset_Ico le_rfl h #align set.Ico_subset_Ico_right Set.Ico_subset_Ico_right @[gcongr] theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, le_trans hx₂ h₂⟩ #align set.Icc_subset_Icc Set.Icc_subset_Icc @[gcongr] theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b := Icc_subset_Icc h le_rfl #align set.Icc_subset_Icc_left Set.Icc_subset_Icc_left @[gcongr] theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ := Icc_subset_Icc le_rfl h #align set.Icc_subset_Icc_right Set.Icc_subset_Icc_right theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ hx => ⟨ha.trans_le hx.1, hx.2.trans_lt hb⟩ #align set.Icc_subset_Ioo Set.Icc_subset_Ioo theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := fun _ => And.left #align set.Icc_subset_Ici_self Set.Icc_subset_Ici_self theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := fun _ => And.right #align set.Icc_subset_Iic_self Set.Icc_subset_Iic_self theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := fun _ => And.right #align set.Ioc_subset_Iic_self Set.Ioc_subset_Iic_self @[gcongr] theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans h₂⟩ #align set.Ioc_subset_Ioc Set.Ioc_subset_Ioc @[gcongr] theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b := Ioc_subset_Ioc h le_rfl #align set.Ioc_subset_Ioc_left Set.Ioc_subset_Ioc_left @[gcongr] theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ := Ioc_subset_Ioc le_rfl h #align set.Ioc_subset_Ioc_right Set.Ioc_subset_Ioc_right theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := fun _ => And.imp_left h₁.trans_le #align set.Ico_subset_Ioo_left Set.Ico_subset_Ioo_left theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := fun _ => And.imp_right fun h' => h'.trans_lt h #align set.Ioc_subset_Ioo_right Set.Ioc_subset_Ioo_right theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := fun _ => And.imp_right fun h₂ => h₂.trans_lt h₁ #align set.Icc_subset_Ico_right Set.Icc_subset_Ico_right theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := fun _ => And.imp_left le_of_lt #align set.Ioo_subset_Ico_self Set.Ioo_subset_Ico_self theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := fun _ => And.imp_right le_of_lt #align set.Ioo_subset_Ioc_self Set.Ioo_subset_Ioc_self theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := fun _ => And.imp_right le_of_lt #align set.Ico_subset_Icc_self Set.Ico_subset_Icc_self theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := fun _ => And.imp_left le_of_lt #align set.Ioc_subset_Icc_self Set.Ioc_subset_Icc_self theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b := Subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self #align set.Ioo_subset_Icc_self Set.Ioo_subset_Icc_self theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := fun _ => And.right #align set.Ico_subset_Iio_self Set.Ico_subset_Iio_self theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := fun _ => And.right #align set.Ioo_subset_Iio_self Set.Ioo_subset_Iio_self theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := fun _ => And.left #align set.Ioc_subset_Ioi_self Set.Ioc_subset_Ioi_self theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := fun _ => And.left #align set.Ioo_subset_Ioi_self Set.Ioo_subset_Ioi_self theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a := fun _ hx => le_of_lt hx #align set.Ioi_subset_Ici_self Set.Ioi_subset_Ici_self theorem Iio_subset_Iic_self : Iio a ⊆ Iic a := fun _ hx => le_of_lt hx #align set.Iio_subset_Iic_self Set.Iio_subset_Iic_self theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := fun _ => And.left #align set.Ico_subset_Ici_self Set.Ico_subset_Ici_self theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a := ⟨Ioi_subset_Ici_self, fun h => lt_irrefl a (h le_rfl)⟩ #align set.Ioi_ssubset_Ici_self Set.Ioi_ssubset_Ici_self theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a := @Ioi_ssubset_Ici_self αᵒᵈ _ _ #align set.Iio_ssubset_Iic_self Set.Iio_ssubset_Iic_self theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Icc_iff Set.Icc_subset_Icc_iff theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ioo_iff Set.Icc_subset_Ioo_iff theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ico_iff Set.Icc_subset_Ico_iff theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Ioc_iff Set.Icc_subset_Ioc_iff theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans_lt h⟩ #align set.Icc_subset_Iio_iff Set.Icc_subset_Iio_iff theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans_le hx⟩ #align set.Icc_subset_Ioi_iff Set.Icc_subset_Ioi_iff theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans h⟩ #align set.Icc_subset_Iic_iff Set.Icc_subset_Iic_iff theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans hx⟩ #align set.Icc_subset_Ici_iff Set.Icc_subset_Ici_iff theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc (le_of_lt ha) hb)).mpr ⟨a₂, left_mem_Icc.mpr hI, not_and.mpr fun f _ => lt_irrefl a₂ (ha.trans_le f)⟩ #align set.Icc_ssubset_Icc_left Set.Icc_ssubset_Icc_left theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc ha (le_of_lt hb))).mpr ⟨b₂, right_mem_Icc.mpr hI, fun f => lt_irrefl b₁ (hb.trans_le f.2)⟩ #align set.Icc_ssubset_Icc_right Set.Icc_ssubset_Icc_right /-- If `a ≤ b`, then `(b, +∞) ⊆ (a, +∞)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Ioi_subset_Ioi_iff`. -/ @[gcongr] theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := fun _ hx => h.trans_lt hx #align set.Ioi_subset_Ioi Set.Ioi_subset_Ioi /-- If `a ≤ b`, then `(b, +∞) ⊆ [a, +∞)`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Ioi_subset_Ici_iff`. -/ theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a := Subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self #align set.Ioi_subset_Ici Set.Ioi_subset_Ici /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Iio_subset_Iio_iff`. -/ @[gcongr] theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := fun _ hx => lt_of_lt_of_le hx h #align set.Iio_subset_Iio Set.Iio_subset_Iio /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b]`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Iio_subset_Iic_iff`. -/ theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b := Subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self #align set.Iio_subset_Iic Set.Iio_subset_Iic theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b := rfl #align set.Ici_inter_Iic Set.Ici_inter_Iic theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b := rfl #align set.Ici_inter_Iio Set.Ici_inter_Iio theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b := rfl #align set.Ioi_inter_Iic Set.Ioi_inter_Iic theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b := rfl #align set.Ioi_inter_Iio Set.Ioi_inter_Iio theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a := inter_comm _ _ #align set.Iic_inter_Ici Set.Iic_inter_Ici theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a := inter_comm _ _ #align set.Iio_inter_Ici Set.Iio_inter_Ici theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a := inter_comm _ _ #align set.Iic_inter_Ioi Set.Iic_inter_Ioi theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a := inter_comm _ _ #align set.Iio_inter_Ioi Set.Iio_inter_Ioi theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b := Ioo_subset_Icc_self h #align set.mem_Icc_of_Ioo Set.mem_Icc_of_Ioo theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b := Ioo_subset_Ico_self h #align set.mem_Ico_of_Ioo Set.mem_Ico_of_Ioo theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b := Ioo_subset_Ioc_self h #align set.mem_Ioc_of_Ioo Set.mem_Ioc_of_Ioo theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b := Ico_subset_Icc_self h #align set.mem_Icc_of_Ico Set.mem_Icc_of_Ico theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b := Ioc_subset_Icc_self h #align set.mem_Icc_of_Ioc Set.mem_Icc_of_Ioc theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a := Ioi_subset_Ici_self h #align set.mem_Ici_of_Ioi Set.mem_Ici_of_Ioi theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a := Iio_subset_Iic_self h #align set.mem_Iic_of_Iio Set.mem_Iic_of_Iio theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc] #align set.Icc_eq_empty_iff Set.Icc_eq_empty_iff theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico] #align set.Ico_eq_empty_iff Set.Ico_eq_empty_iff theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioc] #align set.Ioc_eq_empty_iff Set.Ioc_eq_empty_iff theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo] #align set.Ioo_eq_empty_iff Set.Ioo_eq_empty_iff theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ := eq_univ_of_forall h #align is_top.Iic_eq IsTop.Iic_eq theorem _root_.IsBot.Ici_eq (h : IsBot a) : Ici a = univ := eq_univ_of_forall h #align is_bot.Ici_eq IsBot.Ici_eq theorem _root_.IsMax.Ioi_eq (h : IsMax a) : Ioi a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_max.Ioi_eq IsMax.Ioi_eq theorem _root_.IsMin.Iio_eq (h : IsMin a) : Iio a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_min.Iio_eq IsMin.Iio_eq theorem Iic_inter_Ioc_of_le (h : a ≤ c) : Iic a ∩ Ioc b c = Ioc b a := ext fun _ => ⟨fun H => ⟨H.2.1, H.1⟩, fun H => ⟨H.2, H.1, H.2.trans h⟩⟩ #align set.Iic_inter_Ioc_of_le Set.Iic_inter_Ioc_of_le theorem not_mem_Icc_of_lt (ha : c < a) : c ∉ Icc a b := fun h => ha.not_le h.1 #align set.not_mem_Icc_of_lt Set.not_mem_Icc_of_lt theorem not_mem_Icc_of_gt (hb : b < c) : c ∉ Icc a b := fun h => hb.not_le h.2 #align set.not_mem_Icc_of_gt Set.not_mem_Icc_of_gt theorem not_mem_Ico_of_lt (ha : c < a) : c ∉ Ico a b := fun h => ha.not_le h.1 #align set.not_mem_Ico_of_lt Set.not_mem_Ico_of_lt theorem not_mem_Ioc_of_gt (hb : b < c) : c ∉ Ioc a b := fun h => hb.not_le h.2 #align set.not_mem_Ioc_of_gt Set.not_mem_Ioc_of_gt -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Ioi_self : a ∉ Ioi a := lt_irrefl _ #align set.not_mem_Ioi_self Set.not_mem_Ioi_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Iio_self : b ∉ Iio b := lt_irrefl _ #align set.not_mem_Iio_self Set.not_mem_Iio_self theorem not_mem_Ioc_of_le (ha : c ≤ a) : c ∉ Ioc a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioc_of_le Set.not_mem_Ioc_of_le theorem not_mem_Ico_of_ge (hb : b ≤ c) : c ∉ Ico a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ico_of_ge Set.not_mem_Ico_of_ge theorem not_mem_Ioo_of_le (ha : c ≤ a) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioo_of_le Set.not_mem_Ioo_of_le theorem not_mem_Ioo_of_ge (hb : b ≤ c) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ioo_of_ge Set.not_mem_Ioo_of_ge end Preorder section PartialOrder variable [PartialOrder α] {a b c : α} @[simp] theorem Icc_self (a : α) : Icc a a = {a} := Set.ext <\| by simp [Icc, le_antisymm_iff, and_comm] #align set.Icc_self Set.Icc_self instance instIccUnique : Unique (Set.Icc a a) where default := ⟨a, by simp⟩ uniq y := Subtype.ext <\| by simpa using y.2 @[simp] theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by refine ⟨fun h => ?_, ?_⟩ · have hab : a ≤ b := nonempty_Icc.1 (h.symm.subst <\| singleton_nonempty c) exact ⟨eq_of_mem_singleton <\| h.subst <\| left_mem_Icc.2 hab, eq_of_mem_singleton <\| h.subst <\| right_mem_Icc.2 hab⟩ · rintro ⟨rfl, rfl⟩ exact Icc_self _ #align set.Icc_eq_singleton_iff Set.Icc_eq_singleton_iff lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) := fun _x ⟨hax, hxb⟩ _y ⟨hay, hyb⟩ ↦ le_antisymm (le_implies_le_of_le_of_le hxb hay hba) (le_implies_le_of_le_of_le hyb hax hba) #align set.subsingleton_Icc_of_ge Set.subsingleton_Icc_of_ge @[simp] lemma subsingleton_Icc_iff {α : Type} [LinearOrder α] {a b : α} : Set.Subsingleton (Icc a b) ↔ b ≤ a := by refine ⟨fun h ↦ ?_, subsingleton_Icc_of_ge⟩ contrapose! h simp only [ge_iff_le, gt_iff_lt, not_subsingleton_iff] exact ⟨a, ⟨le_refl _, h.le⟩, b, ⟨h.le, le_refl _⟩, h.ne⟩ @[simp] theorem Icc_diff_left : Icc a b \ {a} = Ioc a b := ext fun x => by simp [lt_iff_le_and_ne, eq_comm, and_right_comm] #align set.Icc_diff_left Set.Icc_diff_left @[simp] theorem Icc_diff_right : Icc a b \ {b} = Ico a b := ext fun x => by simp [lt_iff_le_and_ne, and_assoc] #align set.Icc_diff_right Set.Icc_diff_right @[simp] theorem Ico_diff_left : Ico a b \ {a} = Ioo a b := ext fun x => by simp [and_right_comm, ← lt_iff_le_and_ne, eq_comm] #align set.Ico_diff_left Set.Ico_diff_left @[simp] theorem Ioc_diff_right : Ioc a b \ {b} = Ioo a b := ext fun x => by simp [and_assoc, ← lt_iff_le_and_ne] #align set.Ioc_diff_right Set.Ioc_diff_right @[simp] theorem Icc_diff_both : Icc a b \ {a, b} = Ioo a b := by rw [insert_eq, ← diff_diff, Icc_diff_left, Ioc_diff_right] #align set.Icc_diff_both Set.Icc_diff_both @[simp] theorem Ici_diff_left : Ici a \ {a} = Ioi a := ext fun x => by simp [lt_iff_le_and_ne, eq_comm] #align set.Ici_diff_left Set.Ici_diff_left @[simp] theorem Iic_diff_right : Iic a \ {a} = Iio a := ext fun x => by simp [lt_iff_le_and_ne] #align set.Iic_diff_right Set.Iic_diff_right @[simp] theorem Ico_diff_Ioo_same (h : a < b) : Ico a b \ Ioo a b = {a} := by rw [← Ico_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Ico.2 h)] #align set.Ico_diff_Ioo_same Set.Ico_diff_Ioo_same @[simp] theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} := by rw [← Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Ioc.2 h)] #align set.Ioc_diff_Ioo_same Set.Ioc_diff_Ioo_same @[simp] theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Icc.2 h)] #align set.Icc_diff_Ico_same Set.Icc_diff_Ico_same @[simp] theorem Icc_diff_Ioc_same (h : a ≤ b) : Icc a b \ Ioc a b = {a} := by rw [← Icc_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Icc.2 h)] #align set.Icc_diff_Ioc_same Set.Icc_diff_Ioc_same @[simp] theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} := by rw [← Icc_diff_both, diff_diff_cancel_left] simp [insert_subset_iff, h] #align set.Icc_diff_Ioo_same Set.Icc_diff_Ioo_same @[simp] theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} := by rw [← Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)] #align set.Ici_diff_Ioi_same Set.Ici_diff_Ioi_same @[simp] theorem Iic_diff_Iio_same : Iic a \ Iio a = {a} := by rw [← Iic_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 right_mem_Iic)] #align set.Iic_diff_Iio_same Set.Iic_diff_Iio_same -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioi_union_left : Ioi a ∪ {a} = Ici a := ext fun x => by simp [eq_comm, le_iff_eq_or_lt] #align set.Ioi_union_left Set.Ioi_union_left -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Iio_union_right : Iio a ∪ {a} = Iic a := ext fun _ => le_iff_lt_or_eq.symm #align set.Iio_union_right Set.Iio_union_right theorem Ioo_union_left (hab : a < b) : Ioo a b ∪ {a} = Ico a b := by rw [← Ico_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Ico.2 hab)] #align set.Ioo_union_left Set.Ioo_union_left theorem Ioo_union_right (hab : a < b) : Ioo a b ∪ {b} = Ioc a b := by simpa only [dual_Ioo, dual_Ico] using Ioo_union_left hab.dual #align set.Ioo_union_right Set.Ioo_union_right theorem Ioo_union_both (h : a ≤ b) : Ioo a b ∪ {a, b} = Icc a b := by have : (Icc a b \ {a, b}) ∪ {a, b} = Icc a b := diff_union_of_subset fun \| x, .inl rfl => left_mem_Icc.mpr h \| x, .inr rfl => right_mem_Icc.mpr h rw [← this, Icc_diff_both] theorem Ioc_union_left (hab : a ≤ b) : Ioc a b ∪ {a} = Icc a b := by rw [← Icc_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Icc.2 hab)] #align set.Ioc_union_left Set.Ioc_union_left theorem Ico_union_right (hab : a ≤ b) : Ico a b ∪ {b} = Icc a b := by simpa only [dual_Ioc, dual_Icc] using Ioc_union_left hab.dual #align set.Ico_union_right Set.Ico_union_right @[simp] theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b := by rw [insert_eq, union_comm, Ico_union_right h] #align set.Ico_insert_right Set.Ico_insert_right @[simp] theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b := by rw [insert_eq, union_comm, Ioc_union_left h] #align set.Ioc_insert_left Set.Ioc_insert_left @[simp] theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b := by rw [insert_eq, union_comm, Ioo_union_left h] #align set.Ioo_insert_left Set.Ioo_insert_left @[simp] theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b := by rw [insert_eq, union_comm, Ioo_union_right h] #align set.Ioo_insert_right Set.Ioo_insert_right @[simp] theorem Iio_insert : insert a (Iio a) = Iic a := ext fun _ => le_iff_eq_or_lt.symm #align set.Iio_insert Set.Iio_insert @[simp] theorem Ioi_insert : insert a (Ioi a) = Ici a := ext fun _ => (or_congr_left eq_comm).trans le_iff_eq_or_lt.symm #align set.Ioi_insert Set.Ioi_insert theorem mem_Ici_Ioi_of_subset_of_subset {s : Set α} (ho : Ioi a ⊆ s) (hc : s ⊆ Ici a) : s ∈ ({Ici a, Ioi a} : Set (Set α)) := by_cases (fun h : a ∈ s => Or.inl <\| Subset.antisymm hc <\| by rw [← Ioi_union_left, union_subset_iff]; simp [*]) fun h => Or.inr <\| Subset.antisymm (fun x hx => lt_of_le_of_ne (hc hx) fun heq => h <\| heq.symm ▸ hx) ho #align set.mem_Ici_Ioi_of_subset_of_subset Set.mem_Ici_Ioi_of_subset_of_subset theorem mem_Iic_Iio_of_subset_of_subset {s : Set α} (ho : Iio a ⊆ s) (hc : s ⊆ Iic a) : s ∈ ({Iic a, Iio a} : Set (Set α)) := @mem_Ici_Ioi_of_subset_of_subset αᵒᵈ _ a s ho hc #align set.mem_Iic_Iio_of_subset_of_subset Set.mem_Iic_Iio_of_subset_of_subset theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) : s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α)) := by classical by_cases ha : a ∈ s <;> by_cases hb : b ∈ s · refine Or.inl (Subset.antisymm hc ?_) rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, ← Icc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_right] exact subset_diff_singleton hc hb · rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho · refine Or.inr <\| Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_left] exact subset_diff_singleton hc ha · rwa [← Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inr <\| Or.inr <\| Subset.antisymm ?_ ho rw [← Ico_diff_left, ← Icc_diff_right] apply_rules [subset_diff_singleton] #align set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset theorem eq_left_or_mem_Ioo_of_mem_Ico {x : α} (hmem : x ∈ Ico a b) : x = a ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => ⟨h, hmem.2⟩ #align set.eq_left_or_mem_Ioo_of_mem_Ico Set.eq_left_or_mem_Ioo_of_mem_Ico theorem eq_right_or_mem_Ioo_of_mem_Ioc {x : α} (hmem : x ∈ Ioc a b) : x = b ∨ x ∈ Ioo a b := hmem.2.eq_or_lt.imp_right <\| And.intro hmem.1 #align set.eq_right_or_mem_Ioo_of_mem_Ioc Set.eq_right_or_mem_Ioo_of_mem_Ioc theorem eq_endpoints_or_mem_Ioo_of_mem_Icc {x : α} (hmem : x ∈ Icc a b) : x = a ∨ x = b ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => eq_right_or_mem_Ioo_of_mem_Ioc ⟨h, hmem.2⟩ #align set.eq_endpoints_or_mem_Ioo_of_mem_Icc Set.eq_endpoints_or_mem_Ioo_of_mem_Icc theorem _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} := eq_singleton_iff_unique_mem.2 ⟨left_mem_Ici, fun _ => h.eq_of_ge⟩ #align is_max.Ici_eq IsMax.Ici_eq theorem _root_.IsMin.Iic_eq (h : IsMin a) : Iic a = {a} := h.toDual.Ici_eq #align is_min.Iic_eq IsMin.Iic_eq theorem Ici_injective : Injective (Ici : α → Set α) := fun _ _ => eq_of_forall_ge_iff ∘ Set.ext_iff.1 #align set.Ici_injective Set.Ici_injective theorem Iic_injective : Injective (Iic : α → Set α) := fun _ _ => eq_of_forall_le_iff ∘ Set.ext_iff.1 #align set.Iic_injective Set.Iic_injective theorem Ici_inj : Ici a = Ici b ↔ a = b := Ici_injective.eq_iff #align set.Ici_inj Set.Ici_inj theorem Iic_inj : Iic a = Iic b ↔ a = b := Iic_injective.eq_iff #align set.Iic_inj Set.Iic_inj end PartialOrder section OrderTop @[simp] theorem Ici_top [PartialOrder α] [OrderTop α] : Ici (⊤ : α) = {⊤} := isMax_top.Ici_eq #align set.Ici_top Set.Ici_top variable [Preorder α] [OrderTop α] {a : α} @[simp] theorem Ioi_top : Ioi (⊤ : α) = ∅ := isMax_top.Ioi_eq #align set.Ioi_top Set.Ioi_top @[simp] theorem Iic_top : Iic (⊤ : α) = univ := isTop_top.Iic_eq #align set.Iic_top Set.Iic_top @[simp] theorem Icc_top : Icc a ⊤ = Ici a := by simp [← Ici_inter_Iic] #align set.Icc_top Set.Icc_top @[simp] theorem Ioc_top : Ioc a ⊤ = Ioi a := by simp [← Ioi_inter_Iic] #align set.Ioc_top Set.Ioc_top end OrderTop section OrderBot @[simp] theorem Iic_bot [PartialOrder α] [OrderBot α] : Iic (⊥ : α) = {⊥} := isMin_bot.Iic_eq #align set.Iic_bot Set.Iic_bot variable [Preorder α] [OrderBot α] {a : α} @[simp] theorem Iio_bot : Iio (⊥ : α) = ∅ := isMin_bot.Iio_eq #align set.Iio_bot Set.Iio_bot @[simp] theorem Ici_bot : Ici (⊥ : α) = univ := isBot_bot.Ici_eq #align set.Ici_bot Set.Ici_bot @[simp]	Mathlib/Order/Interval/Set/Basic.lean	1,063	1,063	theorem Icc_bot : Icc ⊥ a = Iic a := by	simp [← Ici_inter_Iic]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import Mathlib.MeasureTheory.Measure.NullMeasurable import Mathlib.MeasureTheory.MeasurableSpace.Basic import Mathlib.Topology.Algebra.Order.LiminfLimsup #align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55" /-! # Measure spaces The definition of a measure and a measure space are in `MeasureTheory.MeasureSpaceDef`, with only a few basic properties. This file provides many more properties of these objects. This separation allows the measurability tactic to import only the file `MeasureSpaceDef`, and to be available in `MeasureSpace` (through `MeasurableSpace`). Given a measurable space `α`, a measure on `α` is a function that sends measurable sets to the extended nonnegative reals that satisfies the following conditions: 1. `μ ∅ = 0`; 2. `μ` is countably additive. This means that the measure of a countable union of pairwise disjoint sets is equal to the measure of the individual sets. Every measure can be canonically extended to an outer measure, so that it assigns values to all subsets, not just the measurable subsets. On the other hand, a measure that is countably additive on measurable sets can be restricted to measurable sets to obtain a measure. In this file a measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. Measures on `α` form a complete lattice, and are closed under scalar multiplication with `ℝ≥0∞`. Given a measure, the null sets are the sets where `μ s = 0`, where `μ` denotes the corresponding outer measure (so `s` might not be measurable). We can then define the completion of `μ` as the measure on the least `σ`-algebra that also contains all null sets, by defining the measure to be `0` on the null sets. ## Main statements * `completion` is the completion of a measure to all null measurable sets. * `Measure.ofMeasurable` and `OuterMeasure.toMeasure` are two important ways to define a measure. ## Implementation notes Given `μ : Measure α`, `μ s` is the value of the outer measure applied to `s`. This conveniently allows us to apply the measure to sets without proving that they are measurable. We get countable subadditivity for all sets, but only countable additivity for measurable sets. You often don't want to define a measure via its constructor. Two ways that are sometimes more convenient: * `Measure.ofMeasurable` is a way to define a measure by only giving its value on measurable sets and proving the properties (1) and (2) mentioned above. * `OuterMeasure.toMeasure` is a way of obtaining a measure from an outer measure by showing that all measurable sets in the measurable space are Carathéodory measurable. To prove that two measures are equal, there are multiple options: * `ext`: two measures are equal if they are equal on all measurable sets. * `ext_of_generateFrom_of_iUnion`: two measures are equal if they are equal on a π-system generating the measurable sets, if the π-system contains a spanning increasing sequence of sets where the measures take finite value (in particular the measures are σ-finite). This is a special case of the more general `ext_of_generateFrom_of_cover` * `ext_of_generate_finite`: two finite measures are equal if they are equal on a π-system generating the measurable sets. This is a special case of `ext_of_generateFrom_of_iUnion` using `C ∪ {univ}`, but is easier to work with. A `MeasureSpace` is a class that is a measurable space with a canonical measure. The measure is denoted `volume`. ## References * <https://en.wikipedia.org/wiki/Measure_(mathematics)> * <https://en.wikipedia.org/wiki/Complete_measure> * <https://en.wikipedia.org/wiki/Almost_everywhere> ## Tags measure, almost everywhere, measure space, completion, null set, null measurable set -/ noncomputable section open Set open Filter hiding map open Function MeasurableSpace open scoped Classical symmDiff open Topology Filter ENNReal NNReal Interval MeasureTheory variable {α β γ δ ι R R' : Type} namespace MeasureTheory section variable {m : MeasurableSpace α} {μ μ₁ μ₂ : Measure α} {s s₁ s₂ t : Set α} instance ae_isMeasurablyGenerated : IsMeasurablyGenerated (ae μ) := ⟨fun _s hs => let ⟨t, hst, htm, htμ⟩ := exists_measurable_superset_of_null hs ⟨tᶜ, compl_mem_ae_iff.2 htμ, htm.compl, compl_subset_comm.1 hst⟩⟩ #align measure_theory.ae_is_measurably_generated MeasureTheory.ae_isMeasurablyGenerated /-- See also `MeasureTheory.ae_restrict_uIoc_iff`. -/ theorem ae_uIoc_iff [LinearOrder α] {a b : α} {P : α → Prop} : (∀ᵐ x ∂μ, x ∈ Ι a b → P x) ↔ (∀ᵐ x ∂μ, x ∈ Ioc a b → P x) ∧ ∀ᵐ x ∂μ, x ∈ Ioc b a → P x := by simp only [uIoc_eq_union, mem_union, or_imp, eventually_and] #align measure_theory.ae_uIoc_iff MeasureTheory.ae_uIoc_iff theorem measure_union (hd : Disjoint s₁ s₂) (h : MeasurableSet s₂) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂ := measure_union₀ h.nullMeasurableSet hd.aedisjoint #align measure_theory.measure_union MeasureTheory.measure_union theorem measure_union' (hd : Disjoint s₁ s₂) (h : MeasurableSet s₁) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂ := measure_union₀' h.nullMeasurableSet hd.aedisjoint #align measure_theory.measure_union' MeasureTheory.measure_union' theorem measure_inter_add_diff (s : Set α) (ht : MeasurableSet t) : μ (s ∩ t) + μ (s \ t) = μ s := measure_inter_add_diff₀ _ ht.nullMeasurableSet #align measure_theory.measure_inter_add_diff MeasureTheory.measure_inter_add_diff theorem measure_diff_add_inter (s : Set α) (ht : MeasurableSet t) : μ (s \ t) + μ (s ∩ t) = μ s := (add_comm _ _).trans (measure_inter_add_diff s ht) #align measure_theory.measure_diff_add_inter MeasureTheory.measure_diff_add_inter theorem measure_union_add_inter (s : Set α) (ht : MeasurableSet t) : μ (s ∪ t) + μ (s ∩ t) = μ s + μ t := by rw [← measure_inter_add_diff (s ∪ t) ht, Set.union_inter_cancel_right, union_diff_right, ← measure_inter_add_diff s ht] ac_rfl #align measure_theory.measure_union_add_inter MeasureTheory.measure_union_add_inter theorem measure_union_add_inter' (hs : MeasurableSet s) (t : Set α) : μ (s ∪ t) + μ (s ∩ t) = μ s + μ t := by rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm] #align measure_theory.measure_union_add_inter' MeasureTheory.measure_union_add_inter' lemma measure_symmDiff_eq (hs : MeasurableSet s) (ht : MeasurableSet t) : μ (s ∆ t) = μ (s \ t) + μ (t \ s) := by simpa only [symmDiff_def, sup_eq_union] using measure_union disjoint_sdiff_sdiff (ht.diff hs) lemma measure_symmDiff_le (s t u : Set α) : μ (s ∆ u) ≤ μ (s ∆ t) + μ (t ∆ u) := le_trans (μ.mono <\| symmDiff_triangle s t u) (measure_union_le (s ∆ t) (t ∆ u)) theorem measure_add_measure_compl (h : MeasurableSet s) : μ s + μ sᶜ = μ univ := measure_add_measure_compl₀ h.nullMeasurableSet #align measure_theory.measure_add_measure_compl MeasureTheory.measure_add_measure_compl theorem measure_biUnion₀ {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.Pairwise (AEDisjoint μ on f)) (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : μ (⋃ b ∈ s, f b) = ∑' p : s, μ (f p) := by haveI := hs.toEncodable rw [biUnion_eq_iUnion] exact measure_iUnion₀ (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2 #align measure_theory.measure_bUnion₀ MeasureTheory.measure_biUnion₀ theorem measure_biUnion {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.PairwiseDisjoint f) (h : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑' p : s, μ (f p) := measure_biUnion₀ hs hd.aedisjoint fun b hb => (h b hb).nullMeasurableSet #align measure_theory.measure_bUnion MeasureTheory.measure_biUnion theorem measure_sUnion₀ {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise (AEDisjoint μ)) (h : ∀ s ∈ S, NullMeasurableSet s μ) : μ (⋃₀ S) = ∑' s : S, μ s := by rw [sUnion_eq_biUnion, measure_biUnion₀ hs hd h] #align measure_theory.measure_sUnion₀ MeasureTheory.measure_sUnion₀ theorem measure_sUnion {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise Disjoint) (h : ∀ s ∈ S, MeasurableSet s) : μ (⋃₀ S) = ∑' s : S, μ s := by rw [sUnion_eq_biUnion, measure_biUnion hs hd h] #align measure_theory.measure_sUnion MeasureTheory.measure_sUnion theorem measure_biUnion_finset₀ {s : Finset ι} {f : ι → Set α} (hd : Set.Pairwise (↑s) (AEDisjoint μ on f)) (hm : ∀ b ∈ s, NullMeasurableSet (f b) μ) : μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p) := by rw [← Finset.sum_attach, Finset.attach_eq_univ, ← tsum_fintype] exact measure_biUnion₀ s.countable_toSet hd hm #align measure_theory.measure_bUnion_finset₀ MeasureTheory.measure_biUnion_finset₀ theorem measure_biUnion_finset {s : Finset ι} {f : ι → Set α} (hd : PairwiseDisjoint (↑s) f) (hm : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p) := measure_biUnion_finset₀ hd.aedisjoint fun b hb => (hm b hb).nullMeasurableSet #align measure_theory.measure_bUnion_finset MeasureTheory.measure_biUnion_finset /-- The measure of an a.e. disjoint union (even uncountable) of null-measurable sets is at least the sum of the measures of the sets. -/ theorem tsum_meas_le_meas_iUnion_of_disjoint₀ {ι : Type} [MeasurableSpace α] (μ : Measure α) {As : ι → Set α} (As_mble : ∀ i : ι, NullMeasurableSet (As i) μ) (As_disj : Pairwise (AEDisjoint μ on As)) : (∑' i, μ (As i)) ≤ μ (⋃ i, As i) := by rw [ENNReal.tsum_eq_iSup_sum, iSup_le_iff] intro s simp only [← measure_biUnion_finset₀ (fun _i _hi _j _hj hij => As_disj hij) fun i _ => As_mble i] gcongr exact iUnion_subset fun _ ↦ Subset.rfl /-- The measure of a disjoint union (even uncountable) of measurable sets is at least the sum of the measures of the sets. -/ theorem tsum_meas_le_meas_iUnion_of_disjoint {ι : Type} [MeasurableSpace α] (μ : Measure α) {As : ι → Set α} (As_mble : ∀ i : ι, MeasurableSet (As i)) (As_disj : Pairwise (Disjoint on As)) : (∑' i, μ (As i)) ≤ μ (⋃ i, As i) := tsum_meas_le_meas_iUnion_of_disjoint₀ μ (fun i ↦ (As_mble i).nullMeasurableSet) (fun _ _ h ↦ Disjoint.aedisjoint (As_disj h)) #align measure_theory.tsum_meas_le_meas_Union_of_disjoint MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint /-- If `s` is a countable set, then the measure of its preimage can be found as the sum of measures of the fibers `f ⁻¹' {y}`. -/ theorem tsum_measure_preimage_singleton {s : Set β} (hs : s.Countable) {f : α → β} (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : (∑' b : s, μ (f ⁻¹' {↑b})) = μ (f ⁻¹' s) := by rw [← Set.biUnion_preimage_singleton, measure_biUnion hs (pairwiseDisjoint_fiber f s) hf] #align measure_theory.tsum_measure_preimage_singleton MeasureTheory.tsum_measure_preimage_singleton lemma measure_preimage_eq_zero_iff_of_countable {s : Set β} {f : α → β} (hs : s.Countable) : μ (f ⁻¹' s) = 0 ↔ ∀ x ∈ s, μ (f ⁻¹' {x}) = 0 := by rw [← biUnion_preimage_singleton, measure_biUnion_null_iff hs] /-- If `s` is a `Finset`, then the measure of its preimage can be found as the sum of measures of the fibers `f ⁻¹' {y}`. -/ theorem sum_measure_preimage_singleton (s : Finset β) {f : α → β} (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : (∑ b ∈ s, μ (f ⁻¹' {b})) = μ (f ⁻¹' ↑s) := by simp only [← measure_biUnion_finset (pairwiseDisjoint_fiber f s) hf, Finset.set_biUnion_preimage_singleton] #align measure_theory.sum_measure_preimage_singleton MeasureTheory.sum_measure_preimage_singleton theorem measure_diff_null' (h : μ (s₁ ∩ s₂) = 0) : μ (s₁ \ s₂) = μ s₁ := measure_congr <\| diff_ae_eq_self.2 h #align measure_theory.measure_diff_null' MeasureTheory.measure_diff_null' theorem measure_add_diff (hs : MeasurableSet s) (t : Set α) : μ s + μ (t \ s) = μ (s ∪ t) := by rw [← measure_union' disjoint_sdiff_right hs, union_diff_self] #align measure_theory.measure_add_diff MeasureTheory.measure_add_diff theorem measure_diff' (s : Set α) (hm : MeasurableSet t) (h_fin : μ t ≠ ∞) : μ (s \ t) = μ (s ∪ t) - μ t := Eq.symm <\| ENNReal.sub_eq_of_add_eq h_fin <\| by rw [add_comm, measure_add_diff hm, union_comm] #align measure_theory.measure_diff' MeasureTheory.measure_diff' theorem measure_diff (h : s₂ ⊆ s₁) (h₂ : MeasurableSet s₂) (h_fin : μ s₂ ≠ ∞) : μ (s₁ \ s₂) = μ s₁ - μ s₂ := by rw [measure_diff' _ h₂ h_fin, union_eq_self_of_subset_right h] #align measure_theory.measure_diff MeasureTheory.measure_diff theorem le_measure_diff : μ s₁ - μ s₂ ≤ μ (s₁ \ s₂) := tsub_le_iff_left.2 <\| (measure_le_inter_add_diff μ s₁ s₂).trans <\| by gcongr; apply inter_subset_right #align measure_theory.le_measure_diff MeasureTheory.le_measure_diff /-- If the measure of the symmetric difference of two sets is finite, then one has infinite measure if and only if the other one does. -/ theorem measure_eq_top_iff_of_symmDiff (hμst : μ (s ∆ t) ≠ ∞) : μ s = ∞ ↔ μ t = ∞ := by suffices h : ∀ u v, μ (u ∆ v) ≠ ∞ → μ u = ∞ → μ v = ∞ from ⟨h s t hμst, h t s (symmDiff_comm s t ▸ hμst)⟩ intro u v hμuv hμu by_contra! hμv apply hμuv rw [Set.symmDiff_def, eq_top_iff] calc ∞ = μ u - μ v := (WithTop.sub_eq_top_iff.2 ⟨hμu, hμv⟩).symm _ ≤ μ (u \ v) := le_measure_diff _ ≤ μ (u \ v ∪ v \ u) := measure_mono subset_union_left /-- If the measure of the symmetric difference of two sets is finite, then one has finite measure if and only if the other one does. -/ theorem measure_ne_top_iff_of_symmDiff (hμst : μ (s ∆ t) ≠ ∞) : μ s ≠ ∞ ↔ μ t ≠ ∞ := (measure_eq_top_iff_of_symmDiff hμst).ne theorem measure_diff_lt_of_lt_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞} (h : μ t < μ s + ε) : μ (t \ s) < ε := by rw [measure_diff hst hs hs']; rw [add_comm] at h exact ENNReal.sub_lt_of_lt_add (measure_mono hst) h #align measure_theory.measure_diff_lt_of_lt_add MeasureTheory.measure_diff_lt_of_lt_add theorem measure_diff_le_iff_le_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞} : μ (t \ s) ≤ ε ↔ μ t ≤ μ s + ε := by rw [measure_diff hst hs hs', tsub_le_iff_left] #align measure_theory.measure_diff_le_iff_le_add MeasureTheory.measure_diff_le_iff_le_add theorem measure_eq_measure_of_null_diff {s t : Set α} (hst : s ⊆ t) (h_nulldiff : μ (t \ s) = 0) : μ s = μ t := measure_congr <\| EventuallyLE.antisymm (HasSubset.Subset.eventuallyLE hst) (ae_le_set.mpr h_nulldiff) #align measure_theory.measure_eq_measure_of_null_diff MeasureTheory.measure_eq_measure_of_null_diff theorem measure_eq_measure_of_between_null_diff {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₁ = μ s₂ ∧ μ s₂ = μ s₃ := by have le12 : μ s₁ ≤ μ s₂ := measure_mono h12 have le23 : μ s₂ ≤ μ s₃ := measure_mono h23 have key : μ s₃ ≤ μ s₁ := calc μ s₃ = μ (s₃ \ s₁ ∪ s₁) := by rw [diff_union_of_subset (h12.trans h23)] _ ≤ μ (s₃ \ s₁) + μ s₁ := measure_union_le _ _ _ = μ s₁ := by simp only [h_nulldiff, zero_add] exact ⟨le12.antisymm (le23.trans key), le23.antisymm (key.trans le12)⟩ #align measure_theory.measure_eq_measure_of_between_null_diff MeasureTheory.measure_eq_measure_of_between_null_diff theorem measure_eq_measure_smaller_of_between_null_diff {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₁ = μ s₂ := (measure_eq_measure_of_between_null_diff h12 h23 h_nulldiff).1 #align measure_theory.measure_eq_measure_smaller_of_between_null_diff MeasureTheory.measure_eq_measure_smaller_of_between_null_diff theorem measure_eq_measure_larger_of_between_null_diff {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₂ = μ s₃ := (measure_eq_measure_of_between_null_diff h12 h23 h_nulldiff).2 #align measure_theory.measure_eq_measure_larger_of_between_null_diff MeasureTheory.measure_eq_measure_larger_of_between_null_diff lemma measure_compl₀ (h : NullMeasurableSet s μ) (hs : μ s ≠ ∞) : μ sᶜ = μ Set.univ - μ s := by rw [← measure_add_measure_compl₀ h, ENNReal.add_sub_cancel_left hs] theorem measure_compl (h₁ : MeasurableSet s) (h_fin : μ s ≠ ∞) : μ sᶜ = μ univ - μ s := measure_compl₀ h₁.nullMeasurableSet h_fin #align measure_theory.measure_compl MeasureTheory.measure_compl lemma measure_inter_conull' (ht : μ (s \ t) = 0) : μ (s ∩ t) = μ s := by rw [← diff_compl, measure_diff_null']; rwa [← diff_eq] lemma measure_inter_conull (ht : μ tᶜ = 0) : μ (s ∩ t) = μ s := by rw [← diff_compl, measure_diff_null ht] @[simp] theorem union_ae_eq_left_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] s ↔ t ≤ᵐ[μ] s := by rw [ae_le_set] refine ⟨fun h => by simpa only [union_diff_left] using (ae_eq_set.mp h).1, fun h => eventuallyLE_antisymm_iff.mpr ⟨by rwa [ae_le_set, union_diff_left], HasSubset.Subset.eventuallyLE subset_union_left⟩⟩ #align measure_theory.union_ae_eq_left_iff_ae_subset MeasureTheory.union_ae_eq_left_iff_ae_subset @[simp] theorem union_ae_eq_right_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] t ↔ s ≤ᵐ[μ] t := by rw [union_comm, union_ae_eq_left_iff_ae_subset] #align measure_theory.union_ae_eq_right_iff_ae_subset MeasureTheory.union_ae_eq_right_iff_ae_subset theorem ae_eq_of_ae_subset_of_measure_ge (h₁ : s ≤ᵐ[μ] t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s) (ht : μ t ≠ ∞) : s =ᵐ[μ] t := by refine eventuallyLE_antisymm_iff.mpr ⟨h₁, ae_le_set.mpr ?_⟩ replace h₂ : μ t = μ s := h₂.antisymm (measure_mono_ae h₁) replace ht : μ s ≠ ∞ := h₂ ▸ ht rw [measure_diff' t hsm ht, measure_congr (union_ae_eq_left_iff_ae_subset.mpr h₁), h₂, tsub_self] #align measure_theory.ae_eq_of_ae_subset_of_measure_ge MeasureTheory.ae_eq_of_ae_subset_of_measure_ge /-- If `s ⊆ t`, `μ t ≤ μ s`, `μ t ≠ ∞`, and `s` is measurable, then `s =ᵐ[μ] t`. -/ theorem ae_eq_of_subset_of_measure_ge (h₁ : s ⊆ t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s) (ht : μ t ≠ ∞) : s =ᵐ[μ] t := ae_eq_of_ae_subset_of_measure_ge (HasSubset.Subset.eventuallyLE h₁) h₂ hsm ht #align measure_theory.ae_eq_of_subset_of_measure_ge MeasureTheory.ae_eq_of_subset_of_measure_ge theorem measure_iUnion_congr_of_subset [Countable β] {s : β → Set α} {t : β → Set α} (hsub : ∀ b, s b ⊆ t b) (h_le : ∀ b, μ (t b) ≤ μ (s b)) : μ (⋃ b, s b) = μ (⋃ b, t b) := by rcases Classical.em (∃ b, μ (t b) = ∞) with (⟨b, hb⟩ \| htop) · calc μ (⋃ b, s b) = ∞ := top_unique (hb ▸ (h_le b).trans <\| measure_mono <\| subset_iUnion _ _) _ = μ (⋃ b, t b) := Eq.symm <\| top_unique <\| hb ▸ measure_mono (subset_iUnion _ _) push_neg at htop refine le_antisymm (measure_mono (iUnion_mono hsub)) ?_ set M := toMeasurable μ have H : ∀ b, (M (t b) ∩ M (⋃ b, s b) : Set α) =ᵐ[μ] M (t b) := by refine fun b => ae_eq_of_subset_of_measure_ge inter_subset_left ?_ ?_ ?_ · calc μ (M (t b)) = μ (t b) := measure_toMeasurable _ _ ≤ μ (s b) := h_le b _ ≤ μ (M (t b) ∩ M (⋃ b, s b)) := measure_mono <\| subset_inter ((hsub b).trans <\| subset_toMeasurable _ _) ((subset_iUnion _ _).trans <\| subset_toMeasurable _ _) · exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _) · rw [measure_toMeasurable] exact htop b calc μ (⋃ b, t b) ≤ μ (⋃ b, M (t b)) := measure_mono (iUnion_mono fun b => subset_toMeasurable _ _) _ = μ (⋃ b, M (t b) ∩ M (⋃ b, s b)) := measure_congr (EventuallyEq.countable_iUnion H).symm _ ≤ μ (M (⋃ b, s b)) := measure_mono (iUnion_subset fun b => inter_subset_right) _ = μ (⋃ b, s b) := measure_toMeasurable _ #align measure_theory.measure_Union_congr_of_subset MeasureTheory.measure_iUnion_congr_of_subset theorem measure_union_congr_of_subset {t₁ t₂ : Set α} (hs : s₁ ⊆ s₂) (hsμ : μ s₂ ≤ μ s₁) (ht : t₁ ⊆ t₂) (htμ : μ t₂ ≤ μ t₁) : μ (s₁ ∪ t₁) = μ (s₂ ∪ t₂) := by rw [union_eq_iUnion, union_eq_iUnion] exact measure_iUnion_congr_of_subset (Bool.forall_bool.2 ⟨ht, hs⟩) (Bool.forall_bool.2 ⟨htμ, hsμ⟩) #align measure_theory.measure_union_congr_of_subset MeasureTheory.measure_union_congr_of_subset @[simp] theorem measure_iUnion_toMeasurable [Countable β] (s : β → Set α) : μ (⋃ b, toMeasurable μ (s b)) = μ (⋃ b, s b) := Eq.symm <\| measure_iUnion_congr_of_subset (fun _b => subset_toMeasurable _ _) fun _b => (measure_toMeasurable _).le #align measure_theory.measure_Union_to_measurable MeasureTheory.measure_iUnion_toMeasurable theorem measure_biUnion_toMeasurable {I : Set β} (hc : I.Countable) (s : β → Set α) : μ (⋃ b ∈ I, toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b) := by haveI := hc.toEncodable simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable] #align measure_theory.measure_bUnion_to_measurable MeasureTheory.measure_biUnion_toMeasurable @[simp] theorem measure_toMeasurable_union : μ (toMeasurable μ s ∪ t) = μ (s ∪ t) := Eq.symm <\| measure_union_congr_of_subset (subset_toMeasurable _ _) (measure_toMeasurable _).le Subset.rfl le_rfl #align measure_theory.measure_to_measurable_union MeasureTheory.measure_toMeasurable_union @[simp] theorem measure_union_toMeasurable : μ (s ∪ toMeasurable μ t) = μ (s ∪ t) := Eq.symm <\| measure_union_congr_of_subset Subset.rfl le_rfl (subset_toMeasurable _ _) (measure_toMeasurable _).le #align measure_theory.measure_union_to_measurable MeasureTheory.measure_union_toMeasurable theorem sum_measure_le_measure_univ {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasurableSet (t i)) (H : Set.PairwiseDisjoint (↑s) t) : (∑ i ∈ s, μ (t i)) ≤ μ (univ : Set α) := by rw [← measure_biUnion_finset H h] exact measure_mono (subset_univ _) #align measure_theory.sum_measure_le_measure_univ MeasureTheory.sum_measure_le_measure_univ theorem tsum_measure_le_measure_univ {s : ι → Set α} (hs : ∀ i, MeasurableSet (s i)) (H : Pairwise (Disjoint on s)) : (∑' i, μ (s i)) ≤ μ (univ : Set α) := by rw [ENNReal.tsum_eq_iSup_sum] exact iSup_le fun s => sum_measure_le_measure_univ (fun i _hi => hs i) fun i _hi j _hj hij => H hij #align measure_theory.tsum_measure_le_measure_univ MeasureTheory.tsum_measure_le_measure_univ /-- Pigeonhole principle for measure spaces: if `∑' i, μ (s i) > μ univ`, then one of the intersections `s i ∩ s j` is not empty. -/ theorem exists_nonempty_inter_of_measure_univ_lt_tsum_measure {m : MeasurableSpace α} (μ : Measure α) {s : ι → Set α} (hs : ∀ i, MeasurableSet (s i)) (H : μ (univ : Set α) < ∑' i, μ (s i)) : ∃ i j, i ≠ j ∧ (s i ∩ s j).Nonempty := by contrapose! H apply tsum_measure_le_measure_univ hs intro i j hij exact disjoint_iff_inter_eq_empty.mpr (H i j hij) #align measure_theory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure /-- Pigeonhole principle for measure spaces: if `s` is a `Finset` and `∑ i ∈ s, μ (t i) > μ univ`, then one of the intersections `t i ∩ t j` is not empty. -/ theorem exists_nonempty_inter_of_measure_univ_lt_sum_measure {m : MeasurableSpace α} (μ : Measure α) {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasurableSet (t i)) (H : μ (univ : Set α) < ∑ i ∈ s, μ (t i)) : ∃ i ∈ s, ∃ j ∈ s, ∃ _h : i ≠ j, (t i ∩ t j).Nonempty := by contrapose! H apply sum_measure_le_measure_univ h intro i hi j hj hij exact disjoint_iff_inter_eq_empty.mpr (H i hi j hj hij) #align measure_theory.exists_nonempty_inter_of_measure_univ_lt_sum_measure MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_sum_measure /-- If two sets `s` and `t` are included in a set `u`, and `μ s + μ t > μ u`, then `s` intersects `t`. Version assuming that `t` is measurable. -/ theorem nonempty_inter_of_measure_lt_add {m : MeasurableSpace α} (μ : Measure α) {s t u : Set α} (ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) : (s ∩ t).Nonempty := by rw [← Set.not_disjoint_iff_nonempty_inter] contrapose! h calc μ s + μ t = μ (s ∪ t) := (measure_union h ht).symm _ ≤ μ u := measure_mono (union_subset h's h't) #align measure_theory.nonempty_inter_of_measure_lt_add MeasureTheory.nonempty_inter_of_measure_lt_add /-- If two sets `s` and `t` are included in a set `u`, and `μ s + μ t > μ u`, then `s` intersects `t`. Version assuming that `s` is measurable. -/ theorem nonempty_inter_of_measure_lt_add' {m : MeasurableSpace α} (μ : Measure α) {s t u : Set α} (hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) : (s ∩ t).Nonempty := by rw [add_comm] at h rw [inter_comm] exact nonempty_inter_of_measure_lt_add μ hs h't h's h #align measure_theory.nonempty_inter_of_measure_lt_add' MeasureTheory.nonempty_inter_of_measure_lt_add' /-- Continuity from below: the measure of the union of a directed sequence of (not necessarily -measurable) sets is the supremum of the measures. -/ theorem measure_iUnion_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s) : μ (⋃ i, s i) = ⨆ i, μ (s i) := by cases nonempty_encodable ι -- WLOG, `ι = ℕ` generalize ht : Function.extend Encodable.encode s ⊥ = t replace hd : Directed (· ⊆ ·) t := ht ▸ hd.extend_bot Encodable.encode_injective suffices μ (⋃ n, t n) = ⨆ n, μ (t n) by simp only [← ht, Function.apply_extend μ, ← iSup_eq_iUnion, iSup_extend_bot Encodable.encode_injective, (· ∘ ·), Pi.bot_apply, bot_eq_empty, measure_empty] at this exact this.trans (iSup_extend_bot Encodable.encode_injective _) clear! ι -- The `≥` inequality is trivial refine le_antisymm ?_ (iSup_le fun i => measure_mono <\| subset_iUnion _ _) -- Choose `T n ⊇ t n` of the same measure, put `Td n = disjointed T` set T : ℕ → Set α := fun n => toMeasurable μ (t n) set Td : ℕ → Set α := disjointed T have hm : ∀ n, MeasurableSet (Td n) := MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _ calc μ (⋃ n, t n) ≤ μ (⋃ n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _) _ = μ (⋃ n, Td n) := by rw [iUnion_disjointed] _ ≤ ∑' n, μ (Td n) := measure_iUnion_le _ _ = ⨆ I : Finset ℕ, ∑ n ∈ I, μ (Td n) := ENNReal.tsum_eq_iSup_sum _ ≤ ⨆ n, μ (t n) := iSup_le fun I => by rcases hd.finset_le I with ⟨N, hN⟩ calc (∑ n ∈ I, μ (Td n)) = μ (⋃ n ∈ I, Td n) := (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm _ ≤ μ (⋃ n ∈ I, T n) := measure_mono (iUnion₂_mono fun n _hn => disjointed_subset _ _) _ = μ (⋃ n ∈ I, t n) := measure_biUnion_toMeasurable I.countable_toSet _ _ ≤ μ (t N) := measure_mono (iUnion₂_subset hN) _ ≤ ⨆ n, μ (t n) := le_iSup (μ ∘ t) N #align measure_theory.measure_Union_eq_supr MeasureTheory.measure_iUnion_eq_iSup /-- Continuity from below: the measure of the union of a sequence of (not necessarily measurable) sets is the supremum of the measures of the partial unions. -/ theorem measure_iUnion_eq_iSup' {α ι : Type} [MeasurableSpace α] {μ : Measure α} [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} : μ (⋃ i, f i) = ⨆ i, μ (Accumulate f i) := by have hd : Directed (· ⊆ ·) (Accumulate f) := by intro i j rcases directed_of (· ≤ ·) i j with ⟨k, rik, rjk⟩ exact ⟨k, biUnion_subset_biUnion_left fun l rli ↦ le_trans rli rik, biUnion_subset_biUnion_left fun l rlj ↦ le_trans rlj rjk⟩ rw [← iUnion_accumulate] exact measure_iUnion_eq_iSup hd theorem measure_biUnion_eq_iSup {s : ι → Set α} {t : Set ι} (ht : t.Countable) (hd : DirectedOn ((· ⊆ ·) on s) t) : μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i) := by haveI := ht.toEncodable rw [biUnion_eq_iUnion, measure_iUnion_eq_iSup hd.directed_val, ← iSup_subtype''] #align measure_theory.measure_bUnion_eq_supr MeasureTheory.measure_biUnion_eq_iSup /-- Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the infimum of the measures. -/ theorem measure_iInter_eq_iInf [Countable ι] {s : ι → Set α} (h : ∀ i, MeasurableSet (s i)) (hd : Directed (· ⊇ ·) s) (hfin : ∃ i, μ (s i) ≠ ∞) : μ (⋂ i, s i) = ⨅ i, μ (s i) := by rcases hfin with ⟨k, hk⟩ have : ∀ t ⊆ s k, μ t ≠ ∞ := fun t ht => ne_top_of_le_ne_top hk (measure_mono ht) rw [← ENNReal.sub_sub_cancel hk (iInf_le _ k), ENNReal.sub_iInf, ← ENNReal.sub_sub_cancel hk (measure_mono (iInter_subset _ k)), ← measure_diff (iInter_subset _ k) (MeasurableSet.iInter h) (this _ (iInter_subset _ k)), diff_iInter, measure_iUnion_eq_iSup] · congr 1 refine le_antisymm (iSup_mono' fun i => ?_) (iSup_mono fun i => ?_) · rcases hd i k with ⟨j, hji, hjk⟩ use j rw [← measure_diff hjk (h _) (this _ hjk)] gcongr · rw [tsub_le_iff_right, ← measure_union, Set.union_comm] · exact measure_mono (diff_subset_iff.1 Subset.rfl) · apply disjoint_sdiff_left · apply h i · exact hd.mono_comp _ fun _ _ => diff_subset_diff_right #align measure_theory.measure_Inter_eq_infi MeasureTheory.measure_iInter_eq_iInf /-- Continuity from above: the measure of the intersection of a sequence of measurable sets is the infimum of the measures of the partial intersections. -/ theorem measure_iInter_eq_iInf' {α ι : Type} [MeasurableSpace α] {μ : Measure α} [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} (h : ∀ i, MeasurableSet (f i)) (hfin : ∃ i, μ (f i) ≠ ∞) : μ (⋂ i, f i) = ⨅ i, μ (⋂ j ≤ i, f j) := by let s := fun i ↦ ⋂ j ≤ i, f j have iInter_eq : ⋂ i, f i = ⋂ i, s i := by ext x; simp [s]; constructor · exact fun h _ j _ ↦ h j · intro h i rcases directed_of (· ≤ ·) i i with ⟨j, rij, -⟩ exact h j i rij have ms : ∀ i, MeasurableSet (s i) := fun i ↦ MeasurableSet.biInter (countable_univ.mono <\| subset_univ _) fun i _ ↦ h i have hd : Directed (· ⊇ ·) s := by intro i j rcases directed_of (· ≤ ·) i j with ⟨k, rik, rjk⟩ exact ⟨k, biInter_subset_biInter_left fun j rji ↦ le_trans rji rik, biInter_subset_biInter_left fun i rij ↦ le_trans rij rjk⟩ have hfin' : ∃ i, μ (s i) ≠ ∞ := by rcases hfin with ⟨i, hi⟩ rcases directed_of (· ≤ ·) i i with ⟨j, rij, -⟩ exact ⟨j, ne_top_of_le_ne_top hi <\| measure_mono <\| biInter_subset_of_mem rij⟩ exact iInter_eq ▸ measure_iInter_eq_iInf ms hd hfin' /-- Continuity from below: the measure of the union of an increasing sequence of (not necessarily measurable) sets is the limit of the measures. -/ theorem tendsto_measure_iUnion [Preorder ι] [IsDirected ι (· ≤ ·)] [Countable ι] {s : ι → Set α} (hm : Monotone s) : Tendsto (μ ∘ s) atTop (𝓝 (μ (⋃ n, s n))) := by rw [measure_iUnion_eq_iSup hm.directed_le] exact tendsto_atTop_iSup fun n m hnm => measure_mono <\| hm hnm #align measure_theory.tendsto_measure_Union MeasureTheory.tendsto_measure_iUnion /-- Continuity from below: the measure of the union of a sequence of (not necessarily measurable) sets is the limit of the measures of the partial unions. -/ theorem tendsto_measure_iUnion' {α ι : Type} [MeasurableSpace α] {μ : Measure α} [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} : Tendsto (fun i ↦ μ (Accumulate f i)) atTop (𝓝 (μ (⋃ i, f i))) := by rw [measure_iUnion_eq_iSup'] exact tendsto_atTop_iSup fun i j hij ↦ by gcongr /-- Continuity from above: the measure of the intersection of a decreasing sequence of measurable sets is the limit of the measures. -/ theorem tendsto_measure_iInter [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {s : ι → Set α} (hs : ∀ n, MeasurableSet (s n)) (hm : Antitone s) (hf : ∃ i, μ (s i) ≠ ∞) : Tendsto (μ ∘ s) atTop (𝓝 (μ (⋂ n, s n))) := by rw [measure_iInter_eq_iInf hs hm.directed_ge hf] exact tendsto_atTop_iInf fun n m hnm => measure_mono <\| hm hnm #align measure_theory.tendsto_measure_Inter MeasureTheory.tendsto_measure_iInter /-- Continuity from above: the measure of the intersection of a sequence of measurable sets such that one has finite measure is the limit of the measures of the partial intersections. -/ theorem tendsto_measure_iInter' {α ι : Type} [MeasurableSpace α] {μ : Measure α} [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} (hm : ∀ i, MeasurableSet (f i)) (hf : ∃ i, μ (f i) ≠ ∞) : Tendsto (fun i ↦ μ (⋂ j ∈ {j \| j ≤ i}, f j)) atTop (𝓝 (μ (⋂ i, f i))) := by rw [measure_iInter_eq_iInf' hm hf] exact tendsto_atTop_iInf fun i j hij ↦ measure_mono <\| biInter_subset_biInter_left fun k hki ↦ le_trans hki hij /-- The measure of the intersection of a decreasing sequence of measurable sets indexed by a linear order with first countable topology is the limit of the measures. -/ theorem tendsto_measure_biInter_gt {ι : Type} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α} {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j) (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by refine tendsto_order.2 ⟨fun l hl => ?_, fun L hL => ?_⟩ · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le (measure_mono (biInter_subset_of_mem hr)) obtain ⟨u, u_anti, u_pos, u_lim⟩ : ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by rcases hf with ⟨r, ar, _⟩ rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩ exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩ have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by refine tendsto_measure_iInter (fun n => hs _ (u_pos n)) ?_ ?_ · intro m n hmn exact hm _ _ (u_pos n) (u_anti.antitone hmn) · rcases hf with ⟨r, rpos, hr⟩ obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists refine ⟨n, ne_of_lt (lt_of_le_of_lt ?_ hr.lt_top)⟩ exact measure_mono (hm _ _ (u_pos n) hn.le) have B : ⋂ n, s (u n) = ⋂ r > a, s r := by apply Subset.antisymm · simp only [subset_iInter_iff, gt_iff_lt] intro r rpos obtain ⟨n, hn⟩ : ∃ n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists exact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le) · simp only [subset_iInter_iff, gt_iff_lt] intro n apply biInter_subset_of_mem exact u_pos n rw [B] at A obtain ⟨n, hn⟩ : ∃ n, μ (s (u n)) < L := ((tendsto_order.1 A).2 _ hL).exists have : Ioc a (u n) ∈ 𝓝[>] a := Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, u_pos n⟩ filter_upwards [this] with r hr using lt_of_le_of_lt (measure_mono (hm _ _ hr.1 hr.2)) hn #align measure_theory.tendsto_measure_bInter_gt MeasureTheory.tendsto_measure_biInter_gt /-- One direction of the Borel-Cantelli lemma (sometimes called the "first Borel-Cantelli lemma"): if (sᵢ) is a sequence of sets such that `∑ μ sᵢ` is finite, then the limit superior of the `sᵢ` is a null set. Note: for the second Borel-Cantelli lemma (applying to independent sets in a probability space), see `ProbabilityTheory.measure_limsup_eq_one`. -/ theorem measure_limsup_eq_zero {s : ℕ → Set α} (hs : (∑' i, μ (s i)) ≠ ∞) : μ (limsup s atTop) = 0 := by -- First we replace the sequence `sₙ` with a sequence of measurable sets `tₙ ⊇ sₙ` of the same -- measure. set t : ℕ → Set α := fun n => toMeasurable μ (s n) have ht : (∑' i, μ (t i)) ≠ ∞ := by simpa only [t, measure_toMeasurable] using hs suffices μ (limsup t atTop) = 0 by have A : s ≤ t := fun n => subset_toMeasurable μ (s n) -- TODO default args fail exact measure_mono_null (limsup_le_limsup (eventually_of_forall (Pi.le_def.mp A))) this -- Next we unfold `limsup` for sets and replace equality with an inequality simp only [limsup_eq_iInf_iSup_of_nat', Set.iInf_eq_iInter, Set.iSup_eq_iUnion, ← nonpos_iff_eq_zero] -- Finally, we estimate `μ (⋃ i, t (i + n))` by `∑ i', μ (t (i + n))` refine le_of_tendsto_of_tendsto' (tendsto_measure_iInter (fun i => MeasurableSet.iUnion fun b => measurableSet_toMeasurable _ _) ?_ ⟨0, ne_top_of_le_ne_top ht (measure_iUnion_le t)⟩) (ENNReal.tendsto_sum_nat_add (μ ∘ t) ht) fun n => measure_iUnion_le _ intro n m hnm x simp only [Set.mem_iUnion] exact fun ⟨i, hi⟩ => ⟨i + (m - n), by simpa only [add_assoc, tsub_add_cancel_of_le hnm] using hi⟩ #align measure_theory.measure_limsup_eq_zero MeasureTheory.measure_limsup_eq_zero theorem measure_liminf_eq_zero {s : ℕ → Set α} (h : (∑' i, μ (s i)) ≠ ∞) : μ (liminf s atTop) = 0 := by rw [← le_zero_iff] have : liminf s atTop ≤ limsup s atTop := liminf_le_limsup exact (μ.mono this).trans (by simp [measure_limsup_eq_zero h]) #align measure_theory.measure_liminf_eq_zero MeasureTheory.measure_liminf_eq_zero -- Need to specify `α := Set α` below because of diamond; see #19041 theorem limsup_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α} (h : ∀ n, s n =ᵐ[μ] t) : limsup (α := Set α) s atTop =ᵐ[μ] t := by simp_rw [ae_eq_set] at h ⊢ constructor · rw [atTop.limsup_sdiff s t] apply measure_limsup_eq_zero simp [h] · rw [atTop.sdiff_limsup s t] apply measure_liminf_eq_zero simp [h] #align measure_theory.limsup_ae_eq_of_forall_ae_eq MeasureTheory.limsup_ae_eq_of_forall_ae_eq -- Need to specify `α := Set α` above because of diamond; see #19041 theorem liminf_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α} (h : ∀ n, s n =ᵐ[μ] t) : liminf (α := Set α) s atTop =ᵐ[μ] t := by simp_rw [ae_eq_set] at h ⊢ constructor · rw [atTop.liminf_sdiff s t] apply measure_liminf_eq_zero simp [h] · rw [atTop.sdiff_liminf s t] apply measure_limsup_eq_zero simp [h] #align measure_theory.liminf_ae_eq_of_forall_ae_eq MeasureTheory.liminf_ae_eq_of_forall_ae_eq theorem measure_if {x : β} {t : Set β} {s : Set α} : μ (if x ∈ t then s else ∅) = indicator t (fun _ => μ s) x := by split_ifs with h <;> simp [h] #align measure_theory.measure_if MeasureTheory.measure_if end section OuterMeasure variable [ms : MeasurableSpace α] {s t : Set α} /-- Obtain a measure by giving an outer measure where all sets in the σ-algebra are Carathéodory measurable. -/ def OuterMeasure.toMeasure (m : OuterMeasure α) (h : ms ≤ m.caratheodory) : Measure α := Measure.ofMeasurable (fun s _ => m s) m.empty fun _f hf hd => m.iUnion_eq_of_caratheodory (fun i => h _ (hf i)) hd #align measure_theory.outer_measure.to_measure MeasureTheory.OuterMeasure.toMeasure theorem le_toOuterMeasure_caratheodory (μ : Measure α) : ms ≤ μ.toOuterMeasure.caratheodory := fun _s hs _t => (measure_inter_add_diff _ hs).symm #align measure_theory.le_to_outer_measure_caratheodory MeasureTheory.le_toOuterMeasure_caratheodory @[simp] theorem toMeasure_toOuterMeasure (m : OuterMeasure α) (h : ms ≤ m.caratheodory) : (m.toMeasure h).toOuterMeasure = m.trim := rfl #align measure_theory.to_measure_to_outer_measure MeasureTheory.toMeasure_toOuterMeasure @[simp] theorem toMeasure_apply (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α} (hs : MeasurableSet s) : m.toMeasure h s = m s := m.trim_eq hs #align measure_theory.to_measure_apply MeasureTheory.toMeasure_apply theorem le_toMeasure_apply (m : OuterMeasure α) (h : ms ≤ m.caratheodory) (s : Set α) : m s ≤ m.toMeasure h s := m.le_trim s #align measure_theory.le_to_measure_apply MeasureTheory.le_toMeasure_apply theorem toMeasure_apply₀ (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α} (hs : NullMeasurableSet s (m.toMeasure h)) : m.toMeasure h s = m s := by refine le_antisymm ?_ (le_toMeasure_apply _ _ _) rcases hs.exists_measurable_subset_ae_eq with ⟨t, hts, htm, heq⟩ calc m.toMeasure h s = m.toMeasure h t := measure_congr heq.symm _ = m t := toMeasure_apply m h htm _ ≤ m s := m.mono hts #align measure_theory.to_measure_apply₀ MeasureTheory.toMeasure_apply₀ @[simp] theorem toOuterMeasure_toMeasure {μ : Measure α} : μ.toOuterMeasure.toMeasure (le_toOuterMeasure_caratheodory _) = μ := Measure.ext fun _s => μ.toOuterMeasure.trim_eq #align measure_theory.to_outer_measure_to_measure MeasureTheory.toOuterMeasure_toMeasure @[simp] theorem boundedBy_measure (μ : Measure α) : OuterMeasure.boundedBy μ = μ.toOuterMeasure := μ.toOuterMeasure.boundedBy_eq_self #align measure_theory.bounded_by_measure MeasureTheory.boundedBy_measure end OuterMeasure section /- Porting note: These variables are wrapped by an anonymous section because they interrupt synthesizing instances in `MeasureSpace` section. -/ variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] variable {μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α} {s s' t : Set α} namespace Measure /-- If `u` is a superset of `t` with the same (finite) measure (both sets possibly non-measurable), then for any measurable set `s` one also has `μ (t ∩ s) = μ (u ∩ s)`. -/ theorem measure_inter_eq_of_measure_eq {s t u : Set α} (hs : MeasurableSet s) (h : μ t = μ u) (htu : t ⊆ u) (ht_ne_top : μ t ≠ ∞) : μ (t ∩ s) = μ (u ∩ s) := by rw [h] at ht_ne_top refine le_antisymm (by gcongr) ?_ have A : μ (u ∩ s) + μ (u \ s) ≤ μ (t ∩ s) + μ (u \ s) := calc μ (u ∩ s) + μ (u \ s) = μ u := measure_inter_add_diff _ hs _ = μ t := h.symm _ = μ (t ∩ s) + μ (t \ s) := (measure_inter_add_diff _ hs).symm _ ≤ μ (t ∩ s) + μ (u \ s) := by gcongr have B : μ (u \ s) ≠ ∞ := (lt_of_le_of_lt (measure_mono diff_subset) ht_ne_top.lt_top).ne exact ENNReal.le_of_add_le_add_right B A #align measure_theory.measure.measure_inter_eq_of_measure_eq MeasureTheory.Measure.measure_inter_eq_of_measure_eq /-- The measurable superset `toMeasurable μ t` of `t` (which has the same measure as `t`) satisfies, for any measurable set `s`, the equality `μ (toMeasurable μ t ∩ s) = μ (u ∩ s)`. Here, we require that the measure of `t` is finite. The conclusion holds without this assumption when the measure is s-finite (for example when it is σ-finite), see `measure_toMeasurable_inter_of_sFinite`. -/ theorem measure_toMeasurable_inter {s t : Set α} (hs : MeasurableSet s) (ht : μ t ≠ ∞) : μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := (measure_inter_eq_of_measure_eq hs (measure_toMeasurable t).symm (subset_toMeasurable μ t) ht).symm #align measure_theory.measure.measure_to_measurable_inter MeasureTheory.Measure.measure_toMeasurable_inter /-! ### The `ℝ≥0∞`-module of measures -/ instance instZero [MeasurableSpace α] : Zero (Measure α) := ⟨{ toOuterMeasure := 0 m_iUnion := fun _f _hf _hd => tsum_zero.symm trim_le := OuterMeasure.trim_zero.le }⟩ #align measure_theory.measure.has_zero MeasureTheory.Measure.instZero @[simp] theorem zero_toOuterMeasure {_m : MeasurableSpace α} : (0 : Measure α).toOuterMeasure = 0 := rfl #align measure_theory.measure.zero_to_outer_measure MeasureTheory.Measure.zero_toOuterMeasure @[simp, norm_cast] theorem coe_zero {_m : MeasurableSpace α} : ⇑(0 : Measure α) = 0 := rfl #align measure_theory.measure.coe_zero MeasureTheory.Measure.coe_zero @[nontriviality] lemma apply_eq_zero_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : μ s = 0 := by rw [eq_empty_of_isEmpty s, measure_empty] instance instSubsingleton [IsEmpty α] {m : MeasurableSpace α} : Subsingleton (Measure α) := ⟨fun μ ν => by ext1 s _; rw [apply_eq_zero_of_isEmpty, apply_eq_zero_of_isEmpty]⟩ #align measure_theory.measure.subsingleton MeasureTheory.Measure.instSubsingleton theorem eq_zero_of_isEmpty [IsEmpty α] {_m : MeasurableSpace α} (μ : Measure α) : μ = 0 := Subsingleton.elim μ 0 #align measure_theory.measure.eq_zero_of_is_empty MeasureTheory.Measure.eq_zero_of_isEmpty instance instInhabited [MeasurableSpace α] : Inhabited (Measure α) := ⟨0⟩ #align measure_theory.measure.inhabited MeasureTheory.Measure.instInhabited instance instAdd [MeasurableSpace α] : Add (Measure α) := ⟨fun μ₁ μ₂ => { toOuterMeasure := μ₁.toOuterMeasure + μ₂.toOuterMeasure m_iUnion := fun s hs hd => show μ₁ (⋃ i, s i) + μ₂ (⋃ i, s i) = ∑' i, (μ₁ (s i) + μ₂ (s i)) by rw [ENNReal.tsum_add, measure_iUnion hd hs, measure_iUnion hd hs] trim_le := by rw [OuterMeasure.trim_add, μ₁.trimmed, μ₂.trimmed] }⟩ #align measure_theory.measure.has_add MeasureTheory.Measure.instAdd @[simp] theorem add_toOuterMeasure {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) : (μ₁ + μ₂).toOuterMeasure = μ₁.toOuterMeasure + μ₂.toOuterMeasure := rfl #align measure_theory.measure.add_to_outer_measure MeasureTheory.Measure.add_toOuterMeasure @[simp, norm_cast] theorem coe_add {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) : ⇑(μ₁ + μ₂) = μ₁ + μ₂ := rfl #align measure_theory.measure.coe_add MeasureTheory.Measure.coe_add theorem add_apply {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) (s : Set α) : (μ₁ + μ₂) s = μ₁ s + μ₂ s := rfl #align measure_theory.measure.add_apply MeasureTheory.Measure.add_apply section SMul variable [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] variable [SMul R' ℝ≥0∞] [IsScalarTower R' ℝ≥0∞ ℝ≥0∞] instance instSMul [MeasurableSpace α] : SMul R (Measure α) := ⟨fun c μ => { toOuterMeasure := c • μ.toOuterMeasure m_iUnion := fun s hs hd => by simp only [OuterMeasure.smul_apply, coe_toOuterMeasure, ENNReal.tsum_const_smul, measure_iUnion hd hs] trim_le := by rw [OuterMeasure.trim_smul, μ.trimmed] }⟩ #align measure_theory.measure.has_smul MeasureTheory.Measure.instSMul @[simp] theorem smul_toOuterMeasure {_m : MeasurableSpace α} (c : R) (μ : Measure α) : (c • μ).toOuterMeasure = c • μ.toOuterMeasure := rfl #align measure_theory.measure.smul_to_outer_measure MeasureTheory.Measure.smul_toOuterMeasure @[simp, norm_cast] theorem coe_smul {_m : MeasurableSpace α} (c : R) (μ : Measure α) : ⇑(c • μ) = c • ⇑μ := rfl #align measure_theory.measure.coe_smul MeasureTheory.Measure.coe_smul @[simp] theorem smul_apply {_m : MeasurableSpace α} (c : R) (μ : Measure α) (s : Set α) : (c • μ) s = c • μ s := rfl #align measure_theory.measure.smul_apply MeasureTheory.Measure.smul_apply instance instSMulCommClass [SMulCommClass R R' ℝ≥0∞] [MeasurableSpace α] : SMulCommClass R R' (Measure α) := ⟨fun _ _ _ => ext fun _ _ => smul_comm _ _ _⟩ #align measure_theory.measure.smul_comm_class MeasureTheory.Measure.instSMulCommClass instance instIsScalarTower [SMul R R'] [IsScalarTower R R' ℝ≥0∞] [MeasurableSpace α] : IsScalarTower R R' (Measure α) := ⟨fun _ _ _ => ext fun _ _ => smul_assoc _ _ _⟩ #align measure_theory.measure.is_scalar_tower MeasureTheory.Measure.instIsScalarTower instance instIsCentralScalar [SMul Rᵐᵒᵖ ℝ≥0∞] [IsCentralScalar R ℝ≥0∞] [MeasurableSpace α] : IsCentralScalar R (Measure α) := ⟨fun _ _ => ext fun _ _ => op_smul_eq_smul _ _⟩ #align measure_theory.measure.is_central_scalar MeasureTheory.Measure.instIsCentralScalar end SMul instance instNoZeroSMulDivisors [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [NoZeroSMulDivisors R ℝ≥0∞] : NoZeroSMulDivisors R (Measure α) where eq_zero_or_eq_zero_of_smul_eq_zero h := by simpa [Ne, ext_iff', forall_or_left] using h instance instMulAction [Monoid R] [MulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [MeasurableSpace α] : MulAction R (Measure α) := Injective.mulAction _ toOuterMeasure_injective smul_toOuterMeasure #align measure_theory.measure.mul_action MeasureTheory.Measure.instMulAction instance instAddCommMonoid [MeasurableSpace α] : AddCommMonoid (Measure α) := toOuterMeasure_injective.addCommMonoid toOuterMeasure zero_toOuterMeasure add_toOuterMeasure fun _ _ => smul_toOuterMeasure _ _ #align measure_theory.measure.add_comm_monoid MeasureTheory.Measure.instAddCommMonoid /-- Coercion to function as an additive monoid homomorphism. -/ def coeAddHom {_ : MeasurableSpace α} : Measure α →+ Set α → ℝ≥0∞ where toFun := (⇑) map_zero' := coe_zero map_add' := coe_add #align measure_theory.measure.coe_add_hom MeasureTheory.Measure.coeAddHom @[simp] theorem coe_finset_sum {_m : MeasurableSpace α} (I : Finset ι) (μ : ι → Measure α) : ⇑(∑ i ∈ I, μ i) = ∑ i ∈ I, ⇑(μ i) := map_sum coeAddHom μ I #align measure_theory.measure.coe_finset_sum MeasureTheory.Measure.coe_finset_sum theorem finset_sum_apply {m : MeasurableSpace α} (I : Finset ι) (μ : ι → Measure α) (s : Set α) : (∑ i ∈ I, μ i) s = ∑ i ∈ I, μ i s := by rw [coe_finset_sum, Finset.sum_apply] #align measure_theory.measure.finset_sum_apply MeasureTheory.Measure.finset_sum_apply instance instDistribMulAction [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [MeasurableSpace α] : DistribMulAction R (Measure α) := Injective.distribMulAction ⟨⟨toOuterMeasure, zero_toOuterMeasure⟩, add_toOuterMeasure⟩ toOuterMeasure_injective smul_toOuterMeasure #align measure_theory.measure.distrib_mul_action MeasureTheory.Measure.instDistribMulAction instance instModule [Semiring R] [Module R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [MeasurableSpace α] : Module R (Measure α) := Injective.module R ⟨⟨toOuterMeasure, zero_toOuterMeasure⟩, add_toOuterMeasure⟩ toOuterMeasure_injective smul_toOuterMeasure #align measure_theory.measure.module MeasureTheory.Measure.instModule @[simp] theorem coe_nnreal_smul_apply {_m : MeasurableSpace α} (c : ℝ≥0) (μ : Measure α) (s : Set α) : (c • μ) s = c * μ s := rfl #align measure_theory.measure.coe_nnreal_smul_apply MeasureTheory.Measure.coe_nnreal_smul_apply @[simp] theorem nnreal_smul_coe_apply {_m : MeasurableSpace α} (c : ℝ≥0) (μ : Measure α) (s : Set α) : c • μ s = c * μ s := by rfl theorem ae_smul_measure_iff {p : α → Prop} {c : ℝ≥0∞} (hc : c ≠ 0) : (∀ᵐ x ∂c • μ, p x) ↔ ∀ᵐ x ∂μ, p x := by simp only [ae_iff, Algebra.id.smul_eq_mul, smul_apply, or_iff_right_iff_imp, mul_eq_zero] simp only [IsEmpty.forall_iff, hc] #align measure_theory.measure.ae_smul_measure_iff MeasureTheory.Measure.ae_smul_measure_iff theorem measure_eq_left_of_subset_of_measure_add_eq {s t : Set α} (h : (μ + ν) t ≠ ∞) (h' : s ⊆ t) (h'' : (μ + ν) s = (μ + ν) t) : μ s = μ t := by refine le_antisymm (measure_mono h') ?_ have : μ t + ν t ≤ μ s + ν t := calc μ t + ν t = μ s + ν s := h''.symm _ ≤ μ s + ν t := by gcongr apply ENNReal.le_of_add_le_add_right _ this exact ne_top_of_le_ne_top h (le_add_left le_rfl) #align measure_theory.measure.measure_eq_left_of_subset_of_measure_add_eq MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq theorem measure_eq_right_of_subset_of_measure_add_eq {s t : Set α} (h : (μ + ν) t ≠ ∞) (h' : s ⊆ t) (h'' : (μ + ν) s = (μ + ν) t) : ν s = ν t := by rw [add_comm] at h'' h exact measure_eq_left_of_subset_of_measure_add_eq h h' h'' #align measure_theory.measure.measure_eq_right_of_subset_of_measure_add_eq MeasureTheory.Measure.measure_eq_right_of_subset_of_measure_add_eq theorem measure_toMeasurable_add_inter_left {s t : Set α} (hs : MeasurableSet s) (ht : (μ + ν) t ≠ ∞) : μ (toMeasurable (μ + ν) t ∩ s) = μ (t ∩ s) := by refine (measure_inter_eq_of_measure_eq hs ?_ (subset_toMeasurable _ _) ?_).symm · refine measure_eq_left_of_subset_of_measure_add_eq ?_ (subset_toMeasurable _ _) (measure_toMeasurable t).symm rwa [measure_toMeasurable t] · simp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne, coe_add] at ht exact ht.1 #align measure_theory.measure.measure_to_measurable_add_inter_left MeasureTheory.Measure.measure_toMeasurable_add_inter_left theorem measure_toMeasurable_add_inter_right {s t : Set α} (hs : MeasurableSet s) (ht : (μ + ν) t ≠ ∞) : ν (toMeasurable (μ + ν) t ∩ s) = ν (t ∩ s) := by rw [add_comm] at ht ⊢ exact measure_toMeasurable_add_inter_left hs ht #align measure_theory.measure.measure_to_measurable_add_inter_right MeasureTheory.Measure.measure_toMeasurable_add_inter_right /-! ### The complete lattice of measures -/ /-- Measures are partially ordered. -/ instance instPartialOrder [MeasurableSpace α] : PartialOrder (Measure α) where le m₁ m₂ := ∀ s, m₁ s ≤ m₂ s le_refl m s := le_rfl le_trans m₁ m₂ m₃ h₁ h₂ s := le_trans (h₁ s) (h₂ s) le_antisymm m₁ m₂ h₁ h₂ := ext fun s _ => le_antisymm (h₁ s) (h₂ s) #align measure_theory.measure.partial_order MeasureTheory.Measure.instPartialOrder theorem toOuterMeasure_le : μ₁.toOuterMeasure ≤ μ₂.toOuterMeasure ↔ μ₁ ≤ μ₂ := .rfl #align measure_theory.measure.to_outer_measure_le MeasureTheory.Measure.toOuterMeasure_le theorem le_iff : μ₁ ≤ μ₂ ↔ ∀ s, MeasurableSet s → μ₁ s ≤ μ₂ s := outerMeasure_le_iff #align measure_theory.measure.le_iff MeasureTheory.Measure.le_iff theorem le_intro (h : ∀ s, MeasurableSet s → s.Nonempty → μ₁ s ≤ μ₂ s) : μ₁ ≤ μ₂ := le_iff.2 fun s hs ↦ s.eq_empty_or_nonempty.elim (by rintro rfl; simp) (h s hs) theorem le_iff' : μ₁ ≤ μ₂ ↔ ∀ s, μ₁ s ≤ μ₂ s := .rfl #align measure_theory.measure.le_iff' MeasureTheory.Measure.le_iff' theorem lt_iff : μ < ν ↔ μ ≤ ν ∧ ∃ s, MeasurableSet s ∧ μ s < ν s := lt_iff_le_not_le.trans <\| and_congr Iff.rfl <\| by simp only [le_iff, not_forall, not_le, exists_prop] #align measure_theory.measure.lt_iff MeasureTheory.Measure.lt_iff theorem lt_iff' : μ < ν ↔ μ ≤ ν ∧ ∃ s, μ s < ν s := lt_iff_le_not_le.trans <\| and_congr Iff.rfl <\| by simp only [le_iff', not_forall, not_le] #align measure_theory.measure.lt_iff' MeasureTheory.Measure.lt_iff' instance covariantAddLE [MeasurableSpace α] : CovariantClass (Measure α) (Measure α) (· + ·) (· ≤ ·) := ⟨fun _ν _μ₁ _μ₂ hμ s => add_le_add_left (hμ s) _⟩ #align measure_theory.measure.covariant_add_le MeasureTheory.Measure.covariantAddLE protected theorem le_add_left (h : μ ≤ ν) : μ ≤ ν' + ν := fun s => le_add_left (h s) #align measure_theory.measure.le_add_left MeasureTheory.Measure.le_add_left protected theorem le_add_right (h : μ ≤ ν) : μ ≤ ν + ν' := fun s => le_add_right (h s) #align measure_theory.measure.le_add_right MeasureTheory.Measure.le_add_right section sInf variable {m : Set (Measure α)} theorem sInf_caratheodory (s : Set α) (hs : MeasurableSet s) : MeasurableSet[(sInf (toOuterMeasure '' m)).caratheodory] s := by rw [OuterMeasure.sInf_eq_boundedBy_sInfGen] refine OuterMeasure.boundedBy_caratheodory fun t => ?_ simp only [OuterMeasure.sInfGen, le_iInf_iff, forall_mem_image, measure_eq_iInf t, coe_toOuterMeasure] intro μ hμ u htu _hu have hm : ∀ {s t}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t := by intro s t hst rw [OuterMeasure.sInfGen_def, iInf_image] exact iInf₂_le_of_le μ hμ <\| measure_mono hst rw [← measure_inter_add_diff u hs] exact add_le_add (hm <\| inter_subset_inter_left _ htu) (hm <\| diff_subset_diff_left htu) #align measure_theory.measure.Inf_caratheodory MeasureTheory.Measure.sInf_caratheodory instance [MeasurableSpace α] : InfSet (Measure α) := ⟨fun m => (sInf (toOuterMeasure '' m)).toMeasure <\| sInf_caratheodory⟩ theorem sInf_apply (hs : MeasurableSet s) : sInf m s = sInf (toOuterMeasure '' m) s := toMeasure_apply _ _ hs #align measure_theory.measure.Inf_apply MeasureTheory.Measure.sInf_apply private theorem measure_sInf_le (h : μ ∈ m) : sInf m ≤ μ := have : sInf (toOuterMeasure '' m) ≤ μ.toOuterMeasure := sInf_le (mem_image_of_mem _ h) le_iff.2 fun s hs => by rw [sInf_apply hs]; exact this s private theorem measure_le_sInf (h : ∀ μ' ∈ m, μ ≤ μ') : μ ≤ sInf m := have : μ.toOuterMeasure ≤ sInf (toOuterMeasure '' m) := le_sInf <\| forall_mem_image.2 fun μ hμ ↦ toOuterMeasure_le.2 <\| h _ hμ le_iff.2 fun s hs => by rw [sInf_apply hs]; exact this s instance instCompleteSemilatticeInf [MeasurableSpace α] : CompleteSemilatticeInf (Measure α) := { (by infer_instance : PartialOrder (Measure α)), (by infer_instance : InfSet (Measure α)) with sInf_le := fun _s _a => measure_sInf_le le_sInf := fun _s _a => measure_le_sInf } #align measure_theory.measure.complete_semilattice_Inf MeasureTheory.Measure.instCompleteSemilatticeInf instance instCompleteLattice [MeasurableSpace α] : CompleteLattice (Measure α) := { completeLatticeOfCompleteSemilatticeInf (Measure α) with top := { toOuterMeasure := ⊤, m_iUnion := by intro f _ _ refine (measure_iUnion_le _).antisymm ?_ if hne : (⋃ i, f i).Nonempty then rw [OuterMeasure.top_apply hne] exact le_top else simp_all [Set.not_nonempty_iff_eq_empty] trim_le := le_top }, le_top := fun μ => toOuterMeasure_le.mp le_top bot := 0 bot_le := fun _a _s => bot_le } #align measure_theory.measure.complete_lattice MeasureTheory.Measure.instCompleteLattice end sInf @[simp] theorem _root_.MeasureTheory.OuterMeasure.toMeasure_top : (⊤ : OuterMeasure α).toMeasure (by rw [OuterMeasure.top_caratheodory]; exact le_top) = (⊤ : Measure α) := toOuterMeasure_toMeasure (μ := ⊤) #align measure_theory.outer_measure.to_measure_top MeasureTheory.OuterMeasure.toMeasure_top @[simp] theorem toOuterMeasure_top [MeasurableSpace α] : (⊤ : Measure α).toOuterMeasure = (⊤ : OuterMeasure α) := rfl #align measure_theory.measure.to_outer_measure_top MeasureTheory.Measure.toOuterMeasure_top @[simp] theorem top_add : ⊤ + μ = ⊤ := top_unique <\| Measure.le_add_right le_rfl #align measure_theory.measure.top_add MeasureTheory.Measure.top_add @[simp] theorem add_top : μ + ⊤ = ⊤ := top_unique <\| Measure.le_add_left le_rfl #align measure_theory.measure.add_top MeasureTheory.Measure.add_top protected theorem zero_le {_m0 : MeasurableSpace α} (μ : Measure α) : 0 ≤ μ := bot_le #align measure_theory.measure.zero_le MeasureTheory.Measure.zero_le theorem nonpos_iff_eq_zero' : μ ≤ 0 ↔ μ = 0 := μ.zero_le.le_iff_eq #align measure_theory.measure.nonpos_iff_eq_zero' MeasureTheory.Measure.nonpos_iff_eq_zero' @[simp] theorem measure_univ_eq_zero : μ univ = 0 ↔ μ = 0 := ⟨fun h => bot_unique fun s => (h ▸ measure_mono (subset_univ s) : μ s ≤ 0), fun h => h.symm ▸ rfl⟩ #align measure_theory.measure.measure_univ_eq_zero MeasureTheory.Measure.measure_univ_eq_zero theorem measure_univ_ne_zero : μ univ ≠ 0 ↔ μ ≠ 0 := measure_univ_eq_zero.not #align measure_theory.measure.measure_univ_ne_zero MeasureTheory.Measure.measure_univ_ne_zero instance [NeZero μ] : NeZero (μ univ) := ⟨measure_univ_ne_zero.2 <\| NeZero.ne μ⟩ @[simp] theorem measure_univ_pos : 0 < μ univ ↔ μ ≠ 0 := pos_iff_ne_zero.trans measure_univ_ne_zero #align measure_theory.measure.measure_univ_pos MeasureTheory.Measure.measure_univ_pos /-! ### Pushforward and pullback -/ /-- Lift a linear map between `OuterMeasure` spaces such that for each measure `μ` every measurable set is caratheodory-measurable w.r.t. `f μ` to a linear map between `Measure` spaces. -/ def liftLinear {m0 : MeasurableSpace α} (f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β) (hf : ∀ μ : Measure α, ‹_› ≤ (f μ.toOuterMeasure).caratheodory) : Measure α →ₗ[ℝ≥0∞] Measure β where toFun μ := (f μ.toOuterMeasure).toMeasure (hf μ) map_add' μ₁ μ₂ := ext fun s hs => by simp only [map_add, coe_add, Pi.add_apply, toMeasure_apply, add_toOuterMeasure, OuterMeasure.coe_add, hs] map_smul' c μ := ext fun s hs => by simp only [LinearMap.map_smulₛₗ, coe_smul, Pi.smul_apply, toMeasure_apply, smul_toOuterMeasure (R := ℝ≥0∞), OuterMeasure.coe_smul (R := ℝ≥0∞), smul_apply, hs] #align measure_theory.measure.lift_linear MeasureTheory.Measure.liftLinear lemma liftLinear_apply₀ {f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β} (hf) {s : Set β} (hs : NullMeasurableSet s (liftLinear f hf μ)) : liftLinear f hf μ s = f μ.toOuterMeasure s := toMeasure_apply₀ _ (hf μ) hs @[simp] theorem liftLinear_apply {f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β} (hf) {s : Set β} (hs : MeasurableSet s) : liftLinear f hf μ s = f μ.toOuterMeasure s := toMeasure_apply _ (hf μ) hs #align measure_theory.measure.lift_linear_apply MeasureTheory.Measure.liftLinear_apply theorem le_liftLinear_apply {f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β} (hf) (s : Set β) : f μ.toOuterMeasure s ≤ liftLinear f hf μ s := le_toMeasure_apply _ (hf μ) s #align measure_theory.measure.le_lift_linear_apply MeasureTheory.Measure.le_liftLinear_apply /-- The pushforward of a measure as a linear map. It is defined to be `0` if `f` is not a measurable function. -/ def mapₗ [MeasurableSpace α] (f : α → β) : Measure α →ₗ[ℝ≥0∞] Measure β := if hf : Measurable f then liftLinear (OuterMeasure.map f) fun μ _s hs t => le_toOuterMeasure_caratheodory μ _ (hf hs) (f ⁻¹' t) else 0 #align measure_theory.measure.mapₗ MeasureTheory.Measure.mapₗ theorem mapₗ_congr {f g : α → β} (hf : Measurable f) (hg : Measurable g) (h : f =ᵐ[μ] g) : mapₗ f μ = mapₗ g μ := by ext1 s hs simpa only [mapₗ, hf, hg, hs, dif_pos, liftLinear_apply, OuterMeasure.map_apply] using measure_congr (h.preimage s) #align measure_theory.measure.mapₗ_congr MeasureTheory.Measure.mapₗ_congr /-- The pushforward of a measure. It is defined to be `0` if `f` is not an almost everywhere measurable function. -/ irreducible_def map [MeasurableSpace α] (f : α → β) (μ : Measure α) : Measure β := if hf : AEMeasurable f μ then mapₗ (hf.mk f) μ else 0 #align measure_theory.measure.map MeasureTheory.Measure.map theorem mapₗ_mk_apply_of_aemeasurable {f : α → β} (hf : AEMeasurable f μ) : mapₗ (hf.mk f) μ = map f μ := by simp [map, hf] #align measure_theory.measure.mapₗ_mk_apply_of_ae_measurable MeasureTheory.Measure.mapₗ_mk_apply_of_aemeasurable theorem mapₗ_apply_of_measurable {f : α → β} (hf : Measurable f) (μ : Measure α) : mapₗ f μ = map f μ := by simp only [← mapₗ_mk_apply_of_aemeasurable hf.aemeasurable] exact mapₗ_congr hf hf.aemeasurable.measurable_mk hf.aemeasurable.ae_eq_mk #align measure_theory.measure.mapₗ_apply_of_measurable MeasureTheory.Measure.mapₗ_apply_of_measurable @[simp] theorem map_add (μ ν : Measure α) {f : α → β} (hf : Measurable f) : (μ + ν).map f = μ.map f + ν.map f := by simp [← mapₗ_apply_of_measurable hf] #align measure_theory.measure.map_add MeasureTheory.Measure.map_add @[simp] theorem map_zero (f : α → β) : (0 : Measure α).map f = 0 := by by_cases hf : AEMeasurable f (0 : Measure α) <;> simp [map, hf] #align measure_theory.measure.map_zero MeasureTheory.Measure.map_zero @[simp] theorem map_of_not_aemeasurable {f : α → β} {μ : Measure α} (hf : ¬AEMeasurable f μ) : μ.map f = 0 := by simp [map, hf] #align measure_theory.measure.map_of_not_ae_measurable MeasureTheory.Measure.map_of_not_aemeasurable theorem map_congr {f g : α → β} (h : f =ᵐ[μ] g) : Measure.map f μ = Measure.map g μ := by by_cases hf : AEMeasurable f μ · have hg : AEMeasurable g μ := hf.congr h simp only [← mapₗ_mk_apply_of_aemeasurable hf, ← mapₗ_mk_apply_of_aemeasurable hg] exact mapₗ_congr hf.measurable_mk hg.measurable_mk (hf.ae_eq_mk.symm.trans (h.trans hg.ae_eq_mk)) · have hg : ¬AEMeasurable g μ := by simpa [← aemeasurable_congr h] using hf simp [map_of_not_aemeasurable, hf, hg] #align measure_theory.measure.map_congr MeasureTheory.Measure.map_congr @[simp] protected theorem map_smul (c : ℝ≥0∞) (μ : Measure α) (f : α → β) : (c • μ).map f = c • μ.map f := by rcases eq_or_ne c 0 with (rfl \| hc); · simp by_cases hf : AEMeasurable f μ · have hfc : AEMeasurable f (c • μ) := ⟨hf.mk f, hf.measurable_mk, (ae_smul_measure_iff hc).2 hf.ae_eq_mk⟩ simp only [← mapₗ_mk_apply_of_aemeasurable hf, ← mapₗ_mk_apply_of_aemeasurable hfc, LinearMap.map_smulₛₗ, RingHom.id_apply] congr 1 apply mapₗ_congr hfc.measurable_mk hf.measurable_mk exact EventuallyEq.trans ((ae_smul_measure_iff hc).1 hfc.ae_eq_mk.symm) hf.ae_eq_mk · have hfc : ¬AEMeasurable f (c • μ) := by intro hfc exact hf ⟨hfc.mk f, hfc.measurable_mk, (ae_smul_measure_iff hc).1 hfc.ae_eq_mk⟩ simp [map_of_not_aemeasurable hf, map_of_not_aemeasurable hfc] #align measure_theory.measure.map_smul MeasureTheory.Measure.map_smul @[simp] protected theorem map_smul_nnreal (c : ℝ≥0) (μ : Measure α) (f : α → β) : (c • μ).map f = c • μ.map f := μ.map_smul (c : ℝ≥0∞) f #align measure_theory.measure.map_smul_nnreal MeasureTheory.Measure.map_smul_nnreal variable {f : α → β} lemma map_apply₀ {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : NullMeasurableSet s (map f μ)) : μ.map f s = μ (f ⁻¹' s) := by rw [map, dif_pos hf, mapₗ, dif_pos hf.measurable_mk] at hs ⊢ rw [liftLinear_apply₀ _ hs, measure_congr (hf.ae_eq_mk.preimage s)] rfl /-- We can evaluate the pushforward on measurable sets. For non-measurable sets, see `MeasureTheory.Measure.le_map_apply` and `MeasurableEquiv.map_apply`. -/ @[simp] theorem map_apply_of_aemeasurable (hf : AEMeasurable f μ) {s : Set β} (hs : MeasurableSet s) : μ.map f s = μ (f ⁻¹' s) := map_apply₀ hf hs.nullMeasurableSet #align measure_theory.measure.map_apply_of_ae_measurable MeasureTheory.Measure.map_apply_of_aemeasurable @[simp] theorem map_apply (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) : μ.map f s = μ (f ⁻¹' s) := map_apply_of_aemeasurable hf.aemeasurable hs #align measure_theory.measure.map_apply MeasureTheory.Measure.map_apply theorem map_toOuterMeasure (hf : AEMeasurable f μ) : (μ.map f).toOuterMeasure = (OuterMeasure.map f μ.toOuterMeasure).trim := by rw [← trimmed, OuterMeasure.trim_eq_trim_iff] intro s hs simp [hf, hs] #align measure_theory.measure.map_to_outer_measure MeasureTheory.Measure.map_toOuterMeasure @[simp] lemma map_eq_zero_iff (hf : AEMeasurable f μ) : μ.map f = 0 ↔ μ = 0 := by simp_rw [← measure_univ_eq_zero, map_apply_of_aemeasurable hf .univ, preimage_univ] @[simp] lemma mapₗ_eq_zero_iff (hf : Measurable f) : Measure.mapₗ f μ = 0 ↔ μ = 0 := by rw [mapₗ_apply_of_measurable hf, map_eq_zero_iff hf.aemeasurable] lemma map_ne_zero_iff (hf : AEMeasurable f μ) : μ.map f ≠ 0 ↔ μ ≠ 0 := (map_eq_zero_iff hf).not lemma mapₗ_ne_zero_iff (hf : Measurable f) : Measure.mapₗ f μ ≠ 0 ↔ μ ≠ 0 := (mapₗ_eq_zero_iff hf).not @[simp] theorem map_id : map id μ = μ := ext fun _ => map_apply measurable_id #align measure_theory.measure.map_id MeasureTheory.Measure.map_id @[simp] theorem map_id' : map (fun x => x) μ = μ := map_id #align measure_theory.measure.map_id' MeasureTheory.Measure.map_id' theorem map_map {g : β → γ} {f : α → β} (hg : Measurable g) (hf : Measurable f) : (μ.map f).map g = μ.map (g ∘ f) := ext fun s hs => by simp [hf, hg, hs, hg hs, hg.comp hf, ← preimage_comp] #align measure_theory.measure.map_map MeasureTheory.Measure.map_map @[mono] theorem map_mono {f : α → β} (h : μ ≤ ν) (hf : Measurable f) : μ.map f ≤ ν.map f := le_iff.2 fun s hs ↦ by simp [hf.aemeasurable, hs, h _] #align measure_theory.measure.map_mono MeasureTheory.Measure.map_mono /-- Even if `s` is not measurable, we can bound `map f μ s` from below. See also `MeasurableEquiv.map_apply`. -/ theorem le_map_apply {f : α → β} (hf : AEMeasurable f μ) (s : Set β) : μ (f ⁻¹' s) ≤ μ.map f s := calc μ (f ⁻¹' s) ≤ μ (f ⁻¹' toMeasurable (μ.map f) s) := by gcongr; apply subset_toMeasurable _ = μ.map f (toMeasurable (μ.map f) s) := (map_apply_of_aemeasurable hf <\| measurableSet_toMeasurable _ _).symm _ = μ.map f s := measure_toMeasurable _ #align measure_theory.measure.le_map_apply MeasureTheory.Measure.le_map_apply theorem le_map_apply_image {f : α → β} (hf : AEMeasurable f μ) (s : Set α) : μ s ≤ μ.map f (f '' s) := (measure_mono (subset_preimage_image f s)).trans (le_map_apply hf _) /-- Even if `s` is not measurable, `map f μ s = 0` implies that `μ (f ⁻¹' s) = 0`. -/ theorem preimage_null_of_map_null {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : μ.map f s = 0) : μ (f ⁻¹' s) = 0 := nonpos_iff_eq_zero.mp <\| (le_map_apply hf s).trans_eq hs #align measure_theory.measure.preimage_null_of_map_null MeasureTheory.Measure.preimage_null_of_map_null theorem tendsto_ae_map {f : α → β} (hf : AEMeasurable f μ) : Tendsto f (ae μ) (ae (μ.map f)) := fun _ hs => preimage_null_of_map_null hf hs #align measure_theory.measure.tendsto_ae_map MeasureTheory.Measure.tendsto_ae_map /-- Pullback of a `Measure` as a linear map. If `f` sends each measurable set to a measurable set, then for each measurable set `s` we have `comapₗ f μ s = μ (f '' s)`. If the linearity is not needed, please use `comap` instead, which works for a larger class of functions. -/ def comapₗ [MeasurableSpace α] (f : α → β) : Measure β →ₗ[ℝ≥0∞] Measure α := if hf : Injective f ∧ ∀ s, MeasurableSet s → MeasurableSet (f '' s) then liftLinear (OuterMeasure.comap f) fun μ s hs t => by simp only [OuterMeasure.comap_apply, image_inter hf.1, image_diff hf.1] apply le_toOuterMeasure_caratheodory exact hf.2 s hs else 0 #align measure_theory.measure.comapₗ MeasureTheory.Measure.comapₗ theorem comapₗ_apply {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β) (hs : MeasurableSet s) : comapₗ f μ s = μ (f '' s) := by rw [comapₗ, dif_pos, liftLinear_apply _ hs, OuterMeasure.comap_apply, coe_toOuterMeasure] exact ⟨hfi, hf⟩ #align measure_theory.measure.comapₗ_apply MeasureTheory.Measure.comapₗ_apply /-- Pullback of a `Measure`. If `f` sends each measurable set to a null-measurable set, then for each measurable set `s` we have `comap f μ s = μ (f '' s)`. -/ def comap [MeasurableSpace α] (f : α → β) (μ : Measure β) : Measure α := if hf : Injective f ∧ ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ then (OuterMeasure.comap f μ.toOuterMeasure).toMeasure fun s hs t => by simp only [OuterMeasure.comap_apply, image_inter hf.1, image_diff hf.1] exact (measure_inter_add_diff₀ _ (hf.2 s hs)).symm else 0 #align measure_theory.measure.comap MeasureTheory.Measure.comap theorem comap_apply₀ [MeasurableSpace α] (f : α → β) (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) (hs : NullMeasurableSet s (comap f μ)) : comap f μ s = μ (f '' s) := by rw [comap, dif_pos (And.intro hfi hf)] at hs ⊢ rw [toMeasure_apply₀ _ _ hs, OuterMeasure.comap_apply, coe_toOuterMeasure] #align measure_theory.measure.comap_apply₀ MeasureTheory.Measure.comap_apply₀ theorem le_comap_apply {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) (s : Set α) : μ (f '' s) ≤ comap f μ s := by rw [comap, dif_pos (And.intro hfi hf)] exact le_toMeasure_apply _ _ _ #align measure_theory.measure.le_comap_apply MeasureTheory.Measure.le_comap_apply theorem comap_apply {β} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β) (hs : MeasurableSet s) : comap f μ s = μ (f '' s) := comap_apply₀ f μ hfi (fun s hs => (hf s hs).nullMeasurableSet) hs.nullMeasurableSet #align measure_theory.measure.comap_apply MeasureTheory.Measure.comap_apply theorem comapₗ_eq_comap {β} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β) (hs : MeasurableSet s) : comapₗ f μ s = comap f μ s := (comapₗ_apply f hfi hf μ hs).trans (comap_apply f hfi hf μ hs).symm #align measure_theory.measure.comapₗ_eq_comap MeasureTheory.Measure.comapₗ_eq_comap theorem measure_image_eq_zero_of_comap_eq_zero {β} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β) (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) {s : Set α} (hs : comap f μ s = 0) : μ (f '' s) = 0 := le_antisymm ((le_comap_apply f μ hfi hf s).trans hs.le) (zero_le _) #align measure_theory.measure.measure_image_eq_zero_of_comap_eq_zero MeasureTheory.Measure.measure_image_eq_zero_of_comap_eq_zero theorem ae_eq_image_of_ae_eq_comap {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) {s t : Set α} (hst : s =ᵐ[comap f μ] t) : f '' s =ᵐ[μ] f '' t := by rw [EventuallyEq, ae_iff] at hst ⊢ have h_eq_α : { a : α \| ¬s a = t a } = s \ t ∪ t \ s := by ext1 x simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff] tauto have h_eq_β : { a : β \| ¬(f '' s) a = (f '' t) a } = f '' s \ f '' t ∪ f '' t \ f '' s := by ext1 x simp only [eq_iff_iff, mem_setOf_eq, mem_union, mem_diff] tauto rw [← Set.image_diff hfi, ← Set.image_diff hfi, ← Set.image_union] at h_eq_β rw [h_eq_β] rw [h_eq_α] at hst exact measure_image_eq_zero_of_comap_eq_zero f μ hfi hf hst #align measure_theory.measure.ae_eq_image_of_ae_eq_comap MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap theorem NullMeasurableSet.image {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) {s : Set α} (hs : NullMeasurableSet s (μ.comap f)) : NullMeasurableSet (f '' s) μ := by refine ⟨toMeasurable μ (f '' toMeasurable (μ.comap f) s), measurableSet_toMeasurable _ _, ?_⟩ refine EventuallyEq.trans ?_ (NullMeasurableSet.toMeasurable_ae_eq ?_).symm swap · exact hf _ (measurableSet_toMeasurable _ _) have h : toMeasurable (comap f μ) s =ᵐ[comap f μ] s := NullMeasurableSet.toMeasurable_ae_eq hs exact ae_eq_image_of_ae_eq_comap f μ hfi hf h.symm #align measure_theory.measure.null_measurable_set.image MeasureTheory.Measure.NullMeasurableSet.image theorem comap_preimage {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : Measure β) {s : Set β} (hf : Injective f) (hf' : Measurable f) (h : ∀ t, MeasurableSet t → NullMeasurableSet (f '' t) μ) (hs : MeasurableSet s) : μ.comap f (f ⁻¹' s) = μ (s ∩ range f) := by rw [comap_apply₀ _ _ hf h (hf' hs).nullMeasurableSet, image_preimage_eq_inter_range] #align measure_theory.measure.comap_preimage MeasureTheory.Measure.comap_preimage section Sum /-- Sum of an indexed family of measures. -/ noncomputable def sum (f : ι → Measure α) : Measure α := (OuterMeasure.sum fun i => (f i).toOuterMeasure).toMeasure <\| le_trans (le_iInf fun _ => le_toOuterMeasure_caratheodory _) (OuterMeasure.le_sum_caratheodory _) #align measure_theory.measure.sum MeasureTheory.Measure.sum theorem le_sum_apply (f : ι → Measure α) (s : Set α) : ∑' i, f i s ≤ sum f s := le_toMeasure_apply _ _ _ #align measure_theory.measure.le_sum_apply MeasureTheory.Measure.le_sum_apply @[simp] theorem sum_apply (f : ι → Measure α) {s : Set α} (hs : MeasurableSet s) : sum f s = ∑' i, f i s := toMeasure_apply _ _ hs #align measure_theory.measure.sum_apply MeasureTheory.Measure.sum_apply theorem sum_apply₀ (f : ι → Measure α) {s : Set α} (hs : NullMeasurableSet s (sum f)) : sum f s = ∑' i, f i s := by apply le_antisymm ?_ (le_sum_apply _ _) rcases hs.exists_measurable_subset_ae_eq with ⟨t, ts, t_meas, ht⟩ calc sum f s = sum f t := measure_congr ht.symm _ = ∑' i, f i t := sum_apply _ t_meas _ ≤ ∑' i, f i s := ENNReal.tsum_le_tsum fun i ↦ measure_mono ts /-! For the next theorem, the countability assumption is necessary. For a counterexample, consider an uncountable space, with a distinguished point `x₀`, and the sigma-algebra made of countable sets not containing `x₀`, and their complements. All points but `x₀` are measurable. Consider the sum of the Dirac masses at points different from `x₀`, and `s = x₀`. For any Dirac mass `δ_x`, we have `δ_x (x₀) = 0`, so `∑' x, δ_x (x₀) = 0`. On the other hand, the measure `sum δ_x` gives mass one to each point different from `x₀`, so it gives infinite mass to any measurable set containing `x₀` (as such a set is uncountable), and by outer regularity one get `sum δ_x {x₀} = ∞`. -/ theorem sum_apply_of_countable [Countable ι] (f : ι → Measure α) (s : Set α) : sum f s = ∑' i, f i s := by apply le_antisymm ?_ (le_sum_apply _ _) rcases exists_measurable_superset_forall_eq f s with ⟨t, hst, htm, ht⟩ calc sum f s ≤ sum f t := measure_mono hst _ = ∑' i, f i t := sum_apply _ htm _ = ∑' i, f i s := by simp [ht] theorem le_sum (μ : ι → Measure α) (i : ι) : μ i ≤ sum μ := le_iff.2 fun s hs ↦ by simpa only [sum_apply μ hs] using ENNReal.le_tsum i #align measure_theory.measure.le_sum MeasureTheory.Measure.le_sum @[simp] theorem sum_apply_eq_zero [Countable ι] {μ : ι → Measure α} {s : Set α} : sum μ s = 0 ↔ ∀ i, μ i s = 0 := by simp [sum_apply_of_countable] #align measure_theory.measure.sum_apply_eq_zero MeasureTheory.Measure.sum_apply_eq_zero theorem sum_apply_eq_zero' {μ : ι → Measure α} {s : Set α} (hs : MeasurableSet s) : sum μ s = 0 ↔ ∀ i, μ i s = 0 := by simp [hs] #align measure_theory.measure.sum_apply_eq_zero' MeasureTheory.Measure.sum_apply_eq_zero' @[simp] lemma sum_zero : Measure.sum (fun (_ : ι) ↦ (0 : Measure α)) = 0 := by ext s hs simp [Measure.sum_apply _ hs] theorem sum_sum {ι' : Type} (μ : ι → ι' → Measure α) : (sum fun n => sum (μ n)) = sum (fun (p : ι × ι') ↦ μ p.1 p.2) := by ext1 s hs simp [sum_apply _ hs, ENNReal.tsum_prod'] theorem sum_comm {ι' : Type} (μ : ι → ι' → Measure α) : (sum fun n => sum (μ n)) = sum fun m => sum fun n => μ n m := by ext1 s hs simp_rw [sum_apply _ hs] rw [ENNReal.tsum_comm] #align measure_theory.measure.sum_comm MeasureTheory.Measure.sum_comm theorem ae_sum_iff [Countable ι] {μ : ι → Measure α} {p : α → Prop} : (∀ᵐ x ∂sum μ, p x) ↔ ∀ i, ∀ᵐ x ∂μ i, p x := sum_apply_eq_zero #align measure_theory.measure.ae_sum_iff MeasureTheory.Measure.ae_sum_iff theorem ae_sum_iff' {μ : ι → Measure α} {p : α → Prop} (h : MeasurableSet { x \| p x }) : (∀ᵐ x ∂sum μ, p x) ↔ ∀ i, ∀ᵐ x ∂μ i, p x := sum_apply_eq_zero' h.compl #align measure_theory.measure.ae_sum_iff' MeasureTheory.Measure.ae_sum_iff' @[simp] theorem sum_fintype [Fintype ι] (μ : ι → Measure α) : sum μ = ∑ i, μ i := by ext1 s hs simp only [sum_apply, finset_sum_apply, hs, tsum_fintype] #align measure_theory.measure.sum_fintype MeasureTheory.Measure.sum_fintype theorem sum_coe_finset (s : Finset ι) (μ : ι → Measure α) : (sum fun i : s => μ i) = ∑ i ∈ s, μ i := by rw [sum_fintype, Finset.sum_coe_sort s μ] #align measure_theory.measure.sum_coe_finset MeasureTheory.Measure.sum_coe_finset @[simp] theorem ae_sum_eq [Countable ι] (μ : ι → Measure α) : ae (sum μ) = ⨆ i, ae (μ i) := Filter.ext fun _ => ae_sum_iff.trans mem_iSup.symm #align measure_theory.measure.ae_sum_eq MeasureTheory.Measure.ae_sum_eq theorem sum_bool (f : Bool → Measure α) : sum f = f true + f false := by rw [sum_fintype, Fintype.sum_bool] #align measure_theory.measure.sum_bool MeasureTheory.Measure.sum_bool theorem sum_cond (μ ν : Measure α) : (sum fun b => cond b μ ν) = μ + ν := sum_bool _ #align measure_theory.measure.sum_cond MeasureTheory.Measure.sum_cond @[simp] theorem sum_of_empty [IsEmpty ι] (μ : ι → Measure α) : sum μ = 0 := by rw [← measure_univ_eq_zero, sum_apply _ MeasurableSet.univ, tsum_empty] #align measure_theory.measure.sum_of_empty MeasureTheory.Measure.sum_of_empty theorem sum_add_sum_compl (s : Set ι) (μ : ι → Measure α) : ((sum fun i : s => μ i) + sum fun i : ↥sᶜ => μ i) = sum μ := by ext1 t ht simp only [add_apply, sum_apply _ ht] exact tsum_add_tsum_compl (f := fun i => μ i t) ENNReal.summable ENNReal.summable #align measure_theory.measure.sum_add_sum_compl MeasureTheory.Measure.sum_add_sum_compl theorem sum_congr {μ ν : ℕ → Measure α} (h : ∀ n, μ n = ν n) : sum μ = sum ν := congr_arg sum (funext h) #align measure_theory.measure.sum_congr MeasureTheory.Measure.sum_congr theorem sum_add_sum {ι : Type} (μ ν : ι → Measure α) : sum μ + sum ν = sum fun n => μ n + ν n := by ext1 s hs simp only [add_apply, sum_apply _ hs, Pi.add_apply, coe_add, tsum_add ENNReal.summable ENNReal.summable] #align measure_theory.measure.sum_add_sum MeasureTheory.Measure.sum_add_sum @[simp] lemma sum_comp_equiv {ι ι' : Type} (e : ι' ≃ ι) (m : ι → Measure α) : sum (m ∘ e) = sum m := by ext s hs simpa [hs, sum_apply] using e.tsum_eq (fun n ↦ m n s) @[simp] lemma sum_extend_zero {ι ι' : Type} {f : ι → ι'} (hf : Injective f) (m : ι → Measure α) : sum (Function.extend f m 0) = sum m := by ext s hs simp [, Function.apply_extend (fun μ : Measure α ↦ μ s)] end Sum /-! ### Absolute continuity -/ /-- We say that `μ` is absolutely continuous with respect to `ν`, or that `μ` is dominated by `ν`, if `ν(A) = 0` implies that `μ(A) = 0`. -/ def AbsolutelyContinuous {_m0 : MeasurableSpace α} (μ ν : Measure α) : Prop := ∀ ⦃s : Set α⦄, ν s = 0 → μ s = 0 #align measure_theory.measure.absolutely_continuous MeasureTheory.Measure.AbsolutelyContinuous @[inherit_doc MeasureTheory.Measure.AbsolutelyContinuous] scoped[MeasureTheory] infixl:50 " ≪ " => MeasureTheory.Measure.AbsolutelyContinuous theorem absolutelyContinuous_of_le (h : μ ≤ ν) : μ ≪ ν := fun s hs => nonpos_iff_eq_zero.1 <\| hs ▸ le_iff'.1 h s #align measure_theory.measure.absolutely_continuous_of_le MeasureTheory.Measure.absolutelyContinuous_of_le alias _root_.LE.le.absolutelyContinuous := absolutelyContinuous_of_le #align has_le.le.absolutely_continuous LE.le.absolutelyContinuous theorem absolutelyContinuous_of_eq (h : μ = ν) : μ ≪ ν := h.le.absolutelyContinuous #align measure_theory.measure.absolutely_continuous_of_eq MeasureTheory.Measure.absolutelyContinuous_of_eq alias _root_.Eq.absolutelyContinuous := absolutelyContinuous_of_eq #align eq.absolutely_continuous Eq.absolutelyContinuous namespace AbsolutelyContinuous theorem mk (h : ∀ ⦃s : Set α⦄, MeasurableSet s → ν s = 0 → μ s = 0) : μ ≪ ν := by intro s hs rcases exists_measurable_superset_of_null hs with ⟨t, h1t, h2t, h3t⟩ exact measure_mono_null h1t (h h2t h3t) #align measure_theory.measure.absolutely_continuous.mk MeasureTheory.Measure.AbsolutelyContinuous.mk @[refl] protected theorem refl {_m0 : MeasurableSpace α} (μ : Measure α) : μ ≪ μ := rfl.absolutelyContinuous #align measure_theory.measure.absolutely_continuous.refl MeasureTheory.Measure.AbsolutelyContinuous.refl protected theorem rfl : μ ≪ μ := fun _s hs => hs #align measure_theory.measure.absolutely_continuous.rfl MeasureTheory.Measure.AbsolutelyContinuous.rfl instance instIsRefl [MeasurableSpace α] : IsRefl (Measure α) (· ≪ ·) := ⟨fun _ => AbsolutelyContinuous.rfl⟩ #align measure_theory.measure.absolutely_continuous.is_refl MeasureTheory.Measure.AbsolutelyContinuous.instIsRefl @[simp] protected lemma zero (μ : Measure α) : 0 ≪ μ := fun s _ ↦ by simp @[trans] protected theorem trans (h1 : μ₁ ≪ μ₂) (h2 : μ₂ ≪ μ₃) : μ₁ ≪ μ₃ := fun _s hs => h1 <\| h2 hs #align measure_theory.measure.absolutely_continuous.trans MeasureTheory.Measure.AbsolutelyContinuous.trans @[mono] protected theorem map (h : μ ≪ ν) {f : α → β} (hf : Measurable f) : μ.map f ≪ ν.map f := AbsolutelyContinuous.mk fun s hs => by simpa [hf, hs] using @h _ #align measure_theory.measure.absolutely_continuous.map MeasureTheory.Measure.AbsolutelyContinuous.map protected theorem smul [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (h : μ ≪ ν) (c : R) : c • μ ≪ ν := fun s hνs => by simp only [h hνs, smul_eq_mul, smul_apply, smul_zero] #align measure_theory.measure.absolutely_continuous.smul MeasureTheory.Measure.AbsolutelyContinuous.smul protected lemma add (h1 : μ₁ ≪ ν) (h2 : μ₂ ≪ ν') : μ₁ + μ₂ ≪ ν + ν' := by intro s hs simp only [coe_add, Pi.add_apply, add_eq_zero] at hs ⊢ exact ⟨h1 hs.1, h2 hs.2⟩ lemma add_left_iff {μ₁ μ₂ ν : Measure α} : μ₁ + μ₂ ≪ ν ↔ μ₁ ≪ ν ∧ μ₂ ≪ ν := by refine ⟨fun h ↦ ?_, fun h ↦ (h.1.add h.2).trans ?_⟩ · have : ∀ s, ν s = 0 → μ₁ s = 0 ∧ μ₂ s = 0 := by intro s hs0; simpa using h hs0 exact ⟨fun s hs0 ↦ (this s hs0).1, fun s hs0 ↦ (this s hs0).2⟩ · have : ν + ν = 2 • ν := by ext; simp [two_mul] rw [this] exact AbsolutelyContinuous.rfl.smul 2 lemma add_right (h1 : μ ≪ ν) (ν' : Measure α) : μ ≪ ν + ν' := by intro s hs simp only [coe_add, Pi.add_apply, add_eq_zero] at hs ⊢ exact h1 hs.1 end AbsolutelyContinuous @[simp] lemma absolutelyContinuous_zero_iff : μ ≪ 0 ↔ μ = 0 := ⟨fun h ↦ measure_univ_eq_zero.mp (h rfl), fun h ↦ h.symm ▸ AbsolutelyContinuous.zero _⟩ alias absolutelyContinuous_refl := AbsolutelyContinuous.refl alias absolutelyContinuous_rfl := AbsolutelyContinuous.rfl lemma absolutelyContinuous_sum_left {μs : ι → Measure α} (hμs : ∀ i, μs i ≪ ν) : Measure.sum μs ≪ ν := AbsolutelyContinuous.mk fun s hs hs0 ↦ by simp [sum_apply _ hs, fun i ↦ hμs i hs0] lemma absolutelyContinuous_sum_right {μs : ι → Measure α} (i : ι) (hνμ : ν ≪ μs i) : ν ≪ Measure.sum μs := by refine AbsolutelyContinuous.mk fun s hs hs0 ↦ ?_ simp only [sum_apply _ hs, ENNReal.tsum_eq_zero] at hs0 exact hνμ (hs0 i) theorem absolutelyContinuous_of_le_smul {μ' : Measure α} {c : ℝ≥0∞} (hμ'_le : μ' ≤ c • μ) : μ' ≪ μ := (Measure.absolutelyContinuous_of_le hμ'_le).trans (Measure.AbsolutelyContinuous.rfl.smul c) #align measure_theory.measure.absolutely_continuous_of_le_smul MeasureTheory.Measure.absolutelyContinuous_of_le_smul lemma smul_absolutelyContinuous {c : ℝ≥0∞} : c • μ ≪ μ := absolutelyContinuous_of_le_smul le_rfl lemma absolutelyContinuous_smul {c : ℝ≥0∞} (hc : c ≠ 0) : μ ≪ c • μ := by simp [AbsolutelyContinuous, hc] theorem ae_le_iff_absolutelyContinuous : ae μ ≤ ae ν ↔ μ ≪ ν := ⟨fun h s => by rw [measure_zero_iff_ae_nmem, measure_zero_iff_ae_nmem] exact fun hs => h hs, fun h s hs => h hs⟩ #align measure_theory.measure.ae_le_iff_absolutely_continuous MeasureTheory.Measure.ae_le_iff_absolutelyContinuous alias ⟨_root_.LE.le.absolutelyContinuous_of_ae, AbsolutelyContinuous.ae_le⟩ := ae_le_iff_absolutelyContinuous #align has_le.le.absolutely_continuous_of_ae LE.le.absolutelyContinuous_of_ae #align measure_theory.measure.absolutely_continuous.ae_le MeasureTheory.Measure.AbsolutelyContinuous.ae_le alias ae_mono' := AbsolutelyContinuous.ae_le #align measure_theory.measure.ae_mono' MeasureTheory.Measure.ae_mono' theorem AbsolutelyContinuous.ae_eq (h : μ ≪ ν) {f g : α → δ} (h' : f =ᵐ[ν] g) : f =ᵐ[μ] g := h.ae_le h' #align measure_theory.measure.absolutely_continuous.ae_eq MeasureTheory.Measure.AbsolutelyContinuous.ae_eq protected theorem _root_.MeasureTheory.AEDisjoint.of_absolutelyContinuous (h : AEDisjoint μ s t) {ν : Measure α} (h' : ν ≪ μ) : AEDisjoint ν s t := h' h protected theorem _root_.MeasureTheory.AEDisjoint.of_le (h : AEDisjoint μ s t) {ν : Measure α} (h' : ν ≤ μ) : AEDisjoint ν s t := h.of_absolutelyContinuous (Measure.absolutelyContinuous_of_le h') /-! ### Quasi measure preserving maps (a.k.a. non-singular maps) -/ /-- A map `f : α → β` is said to be quasi measure preserving (a.k.a. non-singular) w.r.t. measures `μa` and `μb` if it is measurable and `μb s = 0` implies `μa (f ⁻¹' s) = 0`. -/ structure QuasiMeasurePreserving {m0 : MeasurableSpace α} (f : α → β) (μa : Measure α := by volume_tac) (μb : Measure β := by volume_tac) : Prop where protected measurable : Measurable f protected absolutelyContinuous : μa.map f ≪ μb #align measure_theory.measure.quasi_measure_preserving MeasureTheory.Measure.QuasiMeasurePreserving #align measure_theory.measure.quasi_measure_preserving.measurable MeasureTheory.Measure.QuasiMeasurePreserving.measurable #align measure_theory.measure.quasi_measure_preserving.absolutely_continuous MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous namespace QuasiMeasurePreserving protected theorem id {_m0 : MeasurableSpace α} (μ : Measure α) : QuasiMeasurePreserving id μ μ := ⟨measurable_id, map_id.absolutelyContinuous⟩ #align measure_theory.measure.quasi_measure_preserving.id MeasureTheory.Measure.QuasiMeasurePreserving.id variable {μa μa' : Measure α} {μb μb' : Measure β} {μc : Measure γ} {f : α → β} protected theorem _root_.Measurable.quasiMeasurePreserving {_m0 : MeasurableSpace α} (hf : Measurable f) (μ : Measure α) : QuasiMeasurePreserving f μ (μ.map f) := ⟨hf, AbsolutelyContinuous.rfl⟩ #align measurable.quasi_measure_preserving Measurable.quasiMeasurePreserving theorem mono_left (h : QuasiMeasurePreserving f μa μb) (ha : μa' ≪ μa) : QuasiMeasurePreserving f μa' μb := ⟨h.1, (ha.map h.1).trans h.2⟩ #align measure_theory.measure.quasi_measure_preserving.mono_left MeasureTheory.Measure.QuasiMeasurePreserving.mono_left theorem mono_right (h : QuasiMeasurePreserving f μa μb) (ha : μb ≪ μb') : QuasiMeasurePreserving f μa μb' := ⟨h.1, h.2.trans ha⟩ #align measure_theory.measure.quasi_measure_preserving.mono_right MeasureTheory.Measure.QuasiMeasurePreserving.mono_right @[mono] theorem mono (ha : μa' ≪ μa) (hb : μb ≪ μb') (h : QuasiMeasurePreserving f μa μb) : QuasiMeasurePreserving f μa' μb' := (h.mono_left ha).mono_right hb #align measure_theory.measure.quasi_measure_preserving.mono MeasureTheory.Measure.QuasiMeasurePreserving.mono protected theorem comp {g : β → γ} {f : α → β} (hg : QuasiMeasurePreserving g μb μc) (hf : QuasiMeasurePreserving f μa μb) : QuasiMeasurePreserving (g ∘ f) μa μc := ⟨hg.measurable.comp hf.measurable, by rw [← map_map hg.1 hf.1] exact (hf.2.map hg.1).trans hg.2⟩ #align measure_theory.measure.quasi_measure_preserving.comp MeasureTheory.Measure.QuasiMeasurePreserving.comp protected theorem iterate {f : α → α} (hf : QuasiMeasurePreserving f μa μa) : ∀ n, QuasiMeasurePreserving f^[n] μa μa \| 0 => QuasiMeasurePreserving.id μa \| n + 1 => (hf.iterate n).comp hf #align measure_theory.measure.quasi_measure_preserving.iterate MeasureTheory.Measure.QuasiMeasurePreserving.iterate protected theorem aemeasurable (hf : QuasiMeasurePreserving f μa μb) : AEMeasurable f μa := hf.1.aemeasurable #align measure_theory.measure.quasi_measure_preserving.ae_measurable MeasureTheory.Measure.QuasiMeasurePreserving.aemeasurable theorem ae_map_le (h : QuasiMeasurePreserving f μa μb) : ae (μa.map f) ≤ ae μb := h.2.ae_le #align measure_theory.measure.quasi_measure_preserving.ae_map_le MeasureTheory.Measure.QuasiMeasurePreserving.ae_map_le theorem tendsto_ae (h : QuasiMeasurePreserving f μa μb) : Tendsto f (ae μa) (ae μb) := (tendsto_ae_map h.aemeasurable).mono_right h.ae_map_le #align measure_theory.measure.quasi_measure_preserving.tendsto_ae MeasureTheory.Measure.QuasiMeasurePreserving.tendsto_ae theorem ae (h : QuasiMeasurePreserving f μa μb) {p : β → Prop} (hg : ∀ᵐ x ∂μb, p x) : ∀ᵐ x ∂μa, p (f x) := h.tendsto_ae hg #align measure_theory.measure.quasi_measure_preserving.ae MeasureTheory.Measure.QuasiMeasurePreserving.ae theorem ae_eq (h : QuasiMeasurePreserving f μa μb) {g₁ g₂ : β → δ} (hg : g₁ =ᵐ[μb] g₂) : g₁ ∘ f =ᵐ[μa] g₂ ∘ f := h.ae hg #align measure_theory.measure.quasi_measure_preserving.ae_eq MeasureTheory.Measure.QuasiMeasurePreserving.ae_eq theorem preimage_null (h : QuasiMeasurePreserving f μa μb) {s : Set β} (hs : μb s = 0) : μa (f ⁻¹' s) = 0 := preimage_null_of_map_null h.aemeasurable (h.2 hs) #align measure_theory.measure.quasi_measure_preserving.preimage_null MeasureTheory.Measure.QuasiMeasurePreserving.preimage_null theorem preimage_mono_ae {s t : Set β} (hf : QuasiMeasurePreserving f μa μb) (h : s ≤ᵐ[μb] t) : f ⁻¹' s ≤ᵐ[μa] f ⁻¹' t := eventually_map.mp <\| Eventually.filter_mono (tendsto_ae_map hf.aemeasurable) (Eventually.filter_mono hf.ae_map_le h) #align measure_theory.measure.quasi_measure_preserving.preimage_mono_ae MeasureTheory.Measure.QuasiMeasurePreserving.preimage_mono_ae theorem preimage_ae_eq {s t : Set β} (hf : QuasiMeasurePreserving f μa μb) (h : s =ᵐ[μb] t) : f ⁻¹' s =ᵐ[μa] f ⁻¹' t := EventuallyLE.antisymm (hf.preimage_mono_ae h.le) (hf.preimage_mono_ae h.symm.le) #align measure_theory.measure.quasi_measure_preserving.preimage_ae_eq MeasureTheory.Measure.QuasiMeasurePreserving.preimage_ae_eq theorem preimage_iterate_ae_eq {s : Set α} {f : α → α} (hf : QuasiMeasurePreserving f μ μ) (k : ℕ) (hs : f ⁻¹' s =ᵐ[μ] s) : f^[k] ⁻¹' s =ᵐ[μ] s := by induction' k with k ih; · rfl rw [iterate_succ, preimage_comp] exact EventuallyEq.trans (hf.preimage_ae_eq ih) hs #align measure_theory.measure.quasi_measure_preserving.preimage_iterate_ae_eq MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq theorem image_zpow_ae_eq {s : Set α} {e : α ≃ α} (he : QuasiMeasurePreserving e μ μ) (he' : QuasiMeasurePreserving e.symm μ μ) (k : ℤ) (hs : e '' s =ᵐ[μ] s) : (⇑(e ^ k)) '' s =ᵐ[μ] s := by rw [Equiv.image_eq_preimage] obtain ⟨k, rfl \| rfl⟩ := k.eq_nat_or_neg · replace hs : (⇑e⁻¹) ⁻¹' s =ᵐ[μ] s := by rwa [Equiv.image_eq_preimage] at hs replace he' : (⇑e⁻¹)^[k] ⁻¹' s =ᵐ[μ] s := he'.preimage_iterate_ae_eq k hs rwa [Equiv.Perm.iterate_eq_pow e⁻¹ k, inv_pow e k] at he' · rw [zpow_neg, zpow_natCast] replace hs : e ⁻¹' s =ᵐ[μ] s := by convert he.preimage_ae_eq hs.symm rw [Equiv.preimage_image] replace he : (⇑e)^[k] ⁻¹' s =ᵐ[μ] s := he.preimage_iterate_ae_eq k hs rwa [Equiv.Perm.iterate_eq_pow e k] at he #align measure_theory.measure.quasi_measure_preserving.image_zpow_ae_eq MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq -- Need to specify `α := Set α` below because of diamond; see #19041	Mathlib/MeasureTheory/Measure/MeasureSpace.lean	1,856	1,863	theorem limsup_preimage_iterate_ae_eq {f : α → α} (hf : QuasiMeasurePreserving f μ μ) (hs : f ⁻¹' s =ᵐ[μ] s) : limsup (α := Set α) (fun n => (preimage f)^[n] s) atTop =ᵐ[μ] s := haveI : ∀ n, (preimage f)^[n] s =ᵐ[μ] s := by	intro n induction' n with n ih · rfl simpa only [iterate_succ', comp_apply] using ae_eq_trans (hf.ae_eq ih) hs (limsup_ae_eq_of_forall_ae_eq (fun n => (preimage f)^[n] s) this).trans (ae_eq_refl _)
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import Mathlib.Data.Int.Interval import Mathlib.Data.Int.SuccPred import Mathlib.Data.Int.ConditionallyCompleteOrder import Mathlib.Topology.Instances.Discrete import Mathlib.Topology.MetricSpace.Bounded import Mathlib.Order.Filter.Archimedean #align_import topology.instances.int from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" /-! # Topology on the integers The structure of a metric space on `ℤ` is introduced in this file, induced from `ℝ`. -/ noncomputable section open Metric Set Filter namespace Int instance : Dist ℤ := ⟨fun x y => dist (x : ℝ) y⟩ theorem dist_eq (x y : ℤ) : dist x y = \|(x : ℝ) - y\| := rfl #align int.dist_eq Int.dist_eq theorem dist_eq' (m n : ℤ) : dist m n = \|m - n\| := by rw [dist_eq]; norm_cast @[norm_cast, simp] theorem dist_cast_real (x y : ℤ) : dist (x : ℝ) y = dist x y := rfl #align int.dist_cast_real Int.dist_cast_real	Mathlib/Topology/Instances/Int.lean	41	43	theorem pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n := by	intro m n hne rw [dist_eq]; norm_cast; rwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero]
/- Copyright (c) 2021 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Algebra.GroupWithZero.Indicator import Mathlib.Topology.ContinuousOn import Mathlib.Topology.Instances.ENNReal #align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Semicontinuous maps A function `f` from a topological space `α` to an ordered space `β` is lower semicontinuous at a point `x` if, for any `y < f x`, for any `x'` close enough to `x`, one has `f x' > y`. In other words, `f` can jump up, but it can not jump down. Upper semicontinuous functions are defined similarly. This file introduces these notions, and a basic API around them mimicking the API for continuous functions. ## Main definitions and results We introduce 4 definitions related to lower semicontinuity: * `LowerSemicontinuousWithinAt f s x` * `LowerSemicontinuousAt f x` * `LowerSemicontinuousOn f s` * `LowerSemicontinuous f` We build a basic API using dot notation around these notions, and we prove that * constant functions are lower semicontinuous; * `indicator s (fun _ ↦ y)` is lower semicontinuous when `s` is open and `0 ≤ y`, or when `s` is closed and `y ≤ 0`; * continuous functions are lower semicontinuous; * left composition with a continuous monotone functions maps lower semicontinuous functions to lower semicontinuous functions. If the function is anti-monotone, it instead maps lower semicontinuous functions to upper semicontinuous functions; * right composition with continuous functions preserves lower and upper semicontinuity; * a sum of two (or finitely many) lower semicontinuous functions is lower semicontinuous; * a supremum of a family of lower semicontinuous functions is lower semicontinuous; * An infinite sum of `ℝ≥0∞`-valued lower semicontinuous functions is lower semicontinuous. Similar results are stated and proved for upper semicontinuity. We also prove that a function is continuous if and only if it is both lower and upper semicontinuous. We have some equivalent definitions of lower- and upper-semicontinuity (under certain restrictions on the order on the codomain): * `lowerSemicontinuous_iff_isOpen_preimage` in a linear order; * `lowerSemicontinuous_iff_isClosed_preimage` in a linear order; * `lowerSemicontinuousAt_iff_le_liminf` in a dense complete linear order; * `lowerSemicontinuous_iff_isClosed_epigraph` in a dense complete linear order with the order topology. ## Implementation details All the nontrivial results for upper semicontinuous functions are deduced from the corresponding ones for lower semicontinuous functions using `OrderDual`. ## References * <https://en.wikipedia.org/wiki/Closed_convex_function> * <https://en.wikipedia.org/wiki/Semi-continuity> -/ open Topology ENNReal open Set Function Filter variable {α : Type} [TopologicalSpace α] {β : Type} [Preorder β] {f g : α → β} {x : α} {s t : Set α} {y z : β} /-! ### Main definitions -/ /-- A real function `f` is lower semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) := ∀ y < f x, ∀ᶠ x' in 𝓝[s] x, y < f x' #align lower_semicontinuous_within_at LowerSemicontinuousWithinAt /-- A real function `f` is lower semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`, for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuousOn (f : α → β) (s : Set α) := ∀ x ∈ s, LowerSemicontinuousWithinAt f s x #align lower_semicontinuous_on LowerSemicontinuousOn /-- A real function `f` is lower semicontinuous at `x` if, for any `ε > 0`, for all `x'` close enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuousAt (f : α → β) (x : α) := ∀ y < f x, ∀ᶠ x' in 𝓝 x, y < f x' #align lower_semicontinuous_at LowerSemicontinuousAt /-- A real function `f` is lower semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuous (f : α → β) := ∀ x, LowerSemicontinuousAt f x #align lower_semicontinuous LowerSemicontinuous /-- A real function `f` is upper semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) := ∀ y, f x < y → ∀ᶠ x' in 𝓝[s] x, f x' < y #align upper_semicontinuous_within_at UpperSemicontinuousWithinAt /-- A real function `f` is upper semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`, for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuousOn (f : α → β) (s : Set α) := ∀ x ∈ s, UpperSemicontinuousWithinAt f s x #align upper_semicontinuous_on UpperSemicontinuousOn /-- A real function `f` is upper semicontinuous at `x` if, for any `ε > 0`, for all `x'` close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuousAt (f : α → β) (x : α) := ∀ y, f x < y → ∀ᶠ x' in 𝓝 x, f x' < y #align upper_semicontinuous_at UpperSemicontinuousAt /-- A real function `f` is upper semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuous (f : α → β) := ∀ x, UpperSemicontinuousAt f x #align upper_semicontinuous UpperSemicontinuous /-! ### Lower semicontinuous functions -/ /-! #### Basic dot notation interface for lower semicontinuity -/ theorem LowerSemicontinuousWithinAt.mono (h : LowerSemicontinuousWithinAt f s x) (hst : t ⊆ s) : LowerSemicontinuousWithinAt f t x := fun y hy => Filter.Eventually.filter_mono (nhdsWithin_mono _ hst) (h y hy) #align lower_semicontinuous_within_at.mono LowerSemicontinuousWithinAt.mono theorem lowerSemicontinuousWithinAt_univ_iff : LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x := by simp [LowerSemicontinuousWithinAt, LowerSemicontinuousAt, nhdsWithin_univ] #align lower_semicontinuous_within_at_univ_iff lowerSemicontinuousWithinAt_univ_iff theorem LowerSemicontinuousAt.lowerSemicontinuousWithinAt (s : Set α) (h : LowerSemicontinuousAt f x) : LowerSemicontinuousWithinAt f s x := fun y hy => Filter.Eventually.filter_mono nhdsWithin_le_nhds (h y hy) #align lower_semicontinuous_at.lower_semicontinuous_within_at LowerSemicontinuousAt.lowerSemicontinuousWithinAt theorem LowerSemicontinuousOn.lowerSemicontinuousWithinAt (h : LowerSemicontinuousOn f s) (hx : x ∈ s) : LowerSemicontinuousWithinAt f s x := h x hx #align lower_semicontinuous_on.lower_semicontinuous_within_at LowerSemicontinuousOn.lowerSemicontinuousWithinAt theorem LowerSemicontinuousOn.mono (h : LowerSemicontinuousOn f s) (hst : t ⊆ s) : LowerSemicontinuousOn f t := fun x hx => (h x (hst hx)).mono hst #align lower_semicontinuous_on.mono LowerSemicontinuousOn.mono theorem lowerSemicontinuousOn_univ_iff : LowerSemicontinuousOn f univ ↔ LowerSemicontinuous f := by simp [LowerSemicontinuousOn, LowerSemicontinuous, lowerSemicontinuousWithinAt_univ_iff] #align lower_semicontinuous_on_univ_iff lowerSemicontinuousOn_univ_iff theorem LowerSemicontinuous.lowerSemicontinuousAt (h : LowerSemicontinuous f) (x : α) : LowerSemicontinuousAt f x := h x #align lower_semicontinuous.lower_semicontinuous_at LowerSemicontinuous.lowerSemicontinuousAt theorem LowerSemicontinuous.lowerSemicontinuousWithinAt (h : LowerSemicontinuous f) (s : Set α) (x : α) : LowerSemicontinuousWithinAt f s x := (h x).lowerSemicontinuousWithinAt s #align lower_semicontinuous.lower_semicontinuous_within_at LowerSemicontinuous.lowerSemicontinuousWithinAt theorem LowerSemicontinuous.lowerSemicontinuousOn (h : LowerSemicontinuous f) (s : Set α) : LowerSemicontinuousOn f s := fun x _hx => h.lowerSemicontinuousWithinAt s x #align lower_semicontinuous.lower_semicontinuous_on LowerSemicontinuous.lowerSemicontinuousOn /-! #### Constants -/ theorem lowerSemicontinuousWithinAt_const : LowerSemicontinuousWithinAt (fun _x => z) s x := fun _y hy => Filter.eventually_of_forall fun _x => hy #align lower_semicontinuous_within_at_const lowerSemicontinuousWithinAt_const theorem lowerSemicontinuousAt_const : LowerSemicontinuousAt (fun _x => z) x := fun _y hy => Filter.eventually_of_forall fun _x => hy #align lower_semicontinuous_at_const lowerSemicontinuousAt_const theorem lowerSemicontinuousOn_const : LowerSemicontinuousOn (fun _x => z) s := fun _x _hx => lowerSemicontinuousWithinAt_const #align lower_semicontinuous_on_const lowerSemicontinuousOn_const theorem lowerSemicontinuous_const : LowerSemicontinuous fun _x : α => z := fun _x => lowerSemicontinuousAt_const #align lower_semicontinuous_const lowerSemicontinuous_const /-! #### Indicators -/ section variable [Zero β] theorem IsOpen.lowerSemicontinuous_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuous (indicator s fun _x => y) := by intro x z hz by_cases h : x ∈ s <;> simp [h] at hz · filter_upwards [hs.mem_nhds h] simp (config := { contextual := true }) [hz] · refine Filter.eventually_of_forall fun x' => ?_ by_cases h' : x' ∈ s <;> simp [h', hz.trans_le hy, hz] #align is_open.lower_semicontinuous_indicator IsOpen.lowerSemicontinuous_indicator theorem IsOpen.lowerSemicontinuousOn_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuousOn (indicator s fun _x => y) t := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousOn t #align is_open.lower_semicontinuous_on_indicator IsOpen.lowerSemicontinuousOn_indicator theorem IsOpen.lowerSemicontinuousAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuousAt (indicator s fun _x => y) x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x #align is_open.lower_semicontinuous_at_indicator IsOpen.lowerSemicontinuousAt_indicator theorem IsOpen.lowerSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuousWithinAt (indicator s fun _x => y) t x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousWithinAt t x #align is_open.lower_semicontinuous_within_at_indicator IsOpen.lowerSemicontinuousWithinAt_indicator theorem IsClosed.lowerSemicontinuous_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuous (indicator s fun _x => y) := by intro x z hz by_cases h : x ∈ s <;> simp [h] at hz · refine Filter.eventually_of_forall fun x' => ?_ by_cases h' : x' ∈ s <;> simp [h', hz, hz.trans_le hy] · filter_upwards [hs.isOpen_compl.mem_nhds h] simp (config := { contextual := true }) [hz] #align is_closed.lower_semicontinuous_indicator IsClosed.lowerSemicontinuous_indicator theorem IsClosed.lowerSemicontinuousOn_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuousOn (indicator s fun _x => y) t := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousOn t #align is_closed.lower_semicontinuous_on_indicator IsClosed.lowerSemicontinuousOn_indicator theorem IsClosed.lowerSemicontinuousAt_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuousAt (indicator s fun _x => y) x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x #align is_closed.lower_semicontinuous_at_indicator IsClosed.lowerSemicontinuousAt_indicator theorem IsClosed.lowerSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuousWithinAt (indicator s fun _x => y) t x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousWithinAt t x #align is_closed.lower_semicontinuous_within_at_indicator IsClosed.lowerSemicontinuousWithinAt_indicator end /-! #### Relationship with continuity -/ theorem lowerSemicontinuous_iff_isOpen_preimage : LowerSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Ioi y) := ⟨fun H y => isOpen_iff_mem_nhds.2 fun x hx => H x y hx, fun H _x y y_lt => IsOpen.mem_nhds (H y) y_lt⟩ #align lower_semicontinuous_iff_is_open_preimage lowerSemicontinuous_iff_isOpen_preimage theorem LowerSemicontinuous.isOpen_preimage (hf : LowerSemicontinuous f) (y : β) : IsOpen (f ⁻¹' Ioi y) := lowerSemicontinuous_iff_isOpen_preimage.1 hf y #align lower_semicontinuous.is_open_preimage LowerSemicontinuous.isOpen_preimage section variable {γ : Type} [LinearOrder γ] theorem lowerSemicontinuous_iff_isClosed_preimage {f : α → γ} : LowerSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Iic y) := by rw [lowerSemicontinuous_iff_isOpen_preimage] simp only [← isOpen_compl_iff, ← preimage_compl, compl_Iic] #align lower_semicontinuous_iff_is_closed_preimage lowerSemicontinuous_iff_isClosed_preimage theorem LowerSemicontinuous.isClosed_preimage {f : α → γ} (hf : LowerSemicontinuous f) (y : γ) : IsClosed (f ⁻¹' Iic y) := lowerSemicontinuous_iff_isClosed_preimage.1 hf y #align lower_semicontinuous.is_closed_preimage LowerSemicontinuous.isClosed_preimage variable [TopologicalSpace γ] [OrderTopology γ] theorem ContinuousWithinAt.lowerSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) : LowerSemicontinuousWithinAt f s x := fun _y hy => h (Ioi_mem_nhds hy) #align continuous_within_at.lower_semicontinuous_within_at ContinuousWithinAt.lowerSemicontinuousWithinAt theorem ContinuousAt.lowerSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) : LowerSemicontinuousAt f x := fun _y hy => h (Ioi_mem_nhds hy) #align continuous_at.lower_semicontinuous_at ContinuousAt.lowerSemicontinuousAt theorem ContinuousOn.lowerSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) : LowerSemicontinuousOn f s := fun x hx => (h x hx).lowerSemicontinuousWithinAt #align continuous_on.lower_semicontinuous_on ContinuousOn.lowerSemicontinuousOn theorem Continuous.lowerSemicontinuous {f : α → γ} (h : Continuous f) : LowerSemicontinuous f := fun _x => h.continuousAt.lowerSemicontinuousAt #align continuous.lower_semicontinuous Continuous.lowerSemicontinuous end /-! #### Equivalent definitions -/ section variable {γ : Type} [CompleteLinearOrder γ] [DenselyOrdered γ] theorem lowerSemicontinuousWithinAt_iff_le_liminf {f : α → γ} : LowerSemicontinuousWithinAt f s x ↔ f x ≤ liminf f (𝓝[s] x) := by constructor · intro hf; unfold LowerSemicontinuousWithinAt at hf contrapose! hf obtain ⟨y, lty, ylt⟩ := exists_between hf; use y exact ⟨ylt, fun h => lty.not_le (le_liminf_of_le (by isBoundedDefault) (h.mono fun _ hx => le_of_lt hx))⟩ exact fun hf y ylt => eventually_lt_of_lt_liminf (ylt.trans_le hf) alias ⟨LowerSemicontinuousWithinAt.le_liminf, _⟩ := lowerSemicontinuousWithinAt_iff_le_liminf theorem lowerSemicontinuousAt_iff_le_liminf {f : α → γ} : LowerSemicontinuousAt f x ↔ f x ≤ liminf f (𝓝 x) := by rw [← lowerSemicontinuousWithinAt_univ_iff, lowerSemicontinuousWithinAt_iff_le_liminf, ← nhdsWithin_univ] alias ⟨LowerSemicontinuousAt.le_liminf, _⟩ := lowerSemicontinuousAt_iff_le_liminf theorem lowerSemicontinuous_iff_le_liminf {f : α → γ} : LowerSemicontinuous f ↔ ∀ x, f x ≤ liminf f (𝓝 x) := by simp only [← lowerSemicontinuousAt_iff_le_liminf, LowerSemicontinuous] alias ⟨LowerSemicontinuous.le_liminf, _⟩ := lowerSemicontinuous_iff_le_liminf theorem lowerSemicontinuousOn_iff_le_liminf {f : α → γ} : LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ liminf f (𝓝[s] x) := by simp only [← lowerSemicontinuousWithinAt_iff_le_liminf, LowerSemicontinuousOn] alias ⟨LowerSemicontinuousOn.le_liminf, _⟩ := lowerSemicontinuousOn_iff_le_liminf variable [TopologicalSpace γ] [OrderTopology γ] theorem lowerSemicontinuous_iff_isClosed_epigraph {f : α → γ} : LowerSemicontinuous f ↔ IsClosed {p : α × γ \| f p.1 ≤ p.2} := by constructor · rw [lowerSemicontinuous_iff_le_liminf, isClosed_iff_forall_filter] rintro hf ⟨x, y⟩ F F_ne h h' rw [nhds_prod_eq, le_prod] at h' calc f x ≤ liminf f (𝓝 x) := hf x _ ≤ liminf f (map Prod.fst F) := liminf_le_liminf_of_le h'.1 _ = liminf (f ∘ Prod.fst) F := (Filter.liminf_comp _ _ _).symm _ ≤ liminf Prod.snd F := liminf_le_liminf <\| by simpa using (eventually_principal.2 fun (_ : α × γ) ↦ id).filter_mono h _ = y := h'.2.liminf_eq · rw [lowerSemicontinuous_iff_isClosed_preimage] exact fun hf y ↦ hf.preimage (Continuous.Prod.mk_left y) @[deprecated (since := "2024-03-02")] alias lowerSemicontinuous_iff_IsClosed_epigraph := lowerSemicontinuous_iff_isClosed_epigraph alias ⟨LowerSemicontinuous.isClosed_epigraph, _⟩ := lowerSemicontinuous_iff_isClosed_epigraph @[deprecated (since := "2024-03-02")] alias LowerSemicontinuous.IsClosed_epigraph := LowerSemicontinuous.isClosed_epigraph end /-! ### Composition -/ section variable {γ : Type} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] variable {δ : Type} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] variable {ι : Type*} [TopologicalSpace ι] theorem ContinuousAt.comp_lowerSemicontinuousWithinAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Monotone g) : LowerSemicontinuousWithinAt (g ∘ f) s x := by intro y hy by_cases h : ∃ l, l < f x · obtain ⟨z, zlt, hz⟩ : ∃ z < f x, Ioc z (f x) ⊆ g ⁻¹' Ioi y := exists_Ioc_subset_of_mem_nhds (hg (Ioi_mem_nhds hy)) h filter_upwards [hf z zlt] with a ha calc y < g (min (f x) (f a)) := hz (by simp [zlt, ha, le_refl]) _ ≤ g (f a) := gmon (min_le_right _ _) · simp only [not_exists, not_lt] at h exact Filter.eventually_of_forall fun a => hy.trans_le (gmon (h (f a))) #align continuous_at.comp_lower_semicontinuous_within_at ContinuousAt.comp_lowerSemicontinuousWithinAt	Mathlib/Topology/Semicontinuous.lean	403	406	theorem ContinuousAt.comp_lowerSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousAt f x) (gmon : Monotone g) : LowerSemicontinuousAt (g ∘ f) x := by	simp only [← lowerSemicontinuousWithinAt_univ_iff] at hf ⊢ exact hg.comp_lowerSemicontinuousWithinAt hf gmon
/- Copyright (c) 2020 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import Mathlib.CategoryTheory.Category.Cat import Mathlib.CategoryTheory.Elements #align_import category_theory.grothendieck from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7" /-! # The Grothendieck construction Given a functor `F : C ⥤ Cat`, the objects of `Grothendieck F` consist of dependent pairs `(b, f)`, where `b : C` and `f : F.obj c`, and a morphism `(b, f) ⟶ (b', f')` is a pair `β : b ⟶ b'` in `C`, and `φ : (F.map β).obj f ⟶ f'` Categories such as `PresheafedSpace` are in fact examples of this construction, and it may be interesting to try to generalize some of the development there. ## Implementation notes Really we should treat `Cat` as a 2-category, and allow `F` to be a 2-functor. There is also a closely related construction starting with `G : Cᵒᵖ ⥤ Cat`, where morphisms consists again of `β : b ⟶ b'` and `φ : f ⟶ (F.map (op β)).obj f'`. ## References See also `CategoryTheory.Functor.Elements` for the category of elements of functor `F : C ⥤ Type`. * https://stacks.math.columbia.edu/tag/02XV * https://ncatlab.org/nlab/show/Grothendieck+construction -/ universe u namespace CategoryTheory variable {C D : Type} [Category C] [Category D] variable (F : C ⥤ Cat) /-- The Grothendieck construction (often written as `∫ F` in mathematics) for a functor `F : C ⥤ Cat` gives a category whose objects `X` consist of `X.base : C` and `X.fiber : F.obj base` * morphisms `f : X ⟶ Y` consist of `base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber` -/ -- Porting note(#5171): no such linter yet -- @[nolint has_nonempty_instance] structure Grothendieck where /-- The underlying object in `C` -/ base : C /-- The object in the fiber of the base object. -/ fiber : F.obj base #align category_theory.grothendieck CategoryTheory.Grothendieck namespace Grothendieck variable {F} /-- A morphism in the Grothendieck category `F : C ⥤ Cat` consists of `base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`. -/ structure Hom (X Y : Grothendieck F) where /-- The morphism between base objects. -/ base : X.base ⟶ Y.base /-- The morphism from the pushforward to the source fiber object to the target fiber object. -/ fiber : (F.map base).obj X.fiber ⟶ Y.fiber #align category_theory.grothendieck.hom CategoryTheory.Grothendieck.Hom @[ext] theorem ext {X Y : Grothendieck F} (f g : Hom X Y) (w_base : f.base = g.base) (w_fiber : eqToHom (by rw [w_base]) ≫ f.fiber = g.fiber) : f = g := by cases f; cases g congr dsimp at w_base aesop_cat #align category_theory.grothendieck.ext CategoryTheory.Grothendieck.ext /-- The identity morphism in the Grothendieck category. -/ @[simps] def id (X : Grothendieck F) : Hom X X where base := 𝟙 X.base fiber := eqToHom (by erw [CategoryTheory.Functor.map_id, Functor.id_obj X.fiber]) #align category_theory.grothendieck.id CategoryTheory.Grothendieck.id instance (X : Grothendieck F) : Inhabited (Hom X X) := ⟨id X⟩ /-- Composition of morphisms in the Grothendieck category. -/ @[simps] def comp {X Y Z : Grothendieck F} (f : Hom X Y) (g : Hom Y Z) : Hom X Z where base := f.base ≫ g.base fiber := eqToHom (by erw [Functor.map_comp, Functor.comp_obj]) ≫ (F.map g.base).map f.fiber ≫ g.fiber #align category_theory.grothendieck.comp CategoryTheory.Grothendieck.comp attribute [local simp] eqToHom_map instance : Category (Grothendieck F) where Hom X Y := Grothendieck.Hom X Y id X := Grothendieck.id X comp := @fun X Y Z f g => Grothendieck.comp f g comp_id := @fun X Y f => by dsimp; ext · simp · dsimp rw [← NatIso.naturality_2 (eqToIso (F.map_id Y.base)) f.fiber] simp id_comp := @fun X Y f => by dsimp; ext <;> simp assoc := @fun W X Y Z f g h => by dsimp; ext · simp · dsimp rw [← NatIso.naturality_2 (eqToIso (F.map_comp _ _)) f.fiber] simp @[simp] theorem id_fiber' (X : Grothendieck F) : Hom.fiber (𝟙 X) = eqToHom (by erw [CategoryTheory.Functor.map_id, Functor.id_obj X.fiber]) := id_fiber X #align category_theory.grothendieck.id_fiber' CategoryTheory.Grothendieck.id_fiber'	Mathlib/CategoryTheory/Grothendieck.lean	132	136	theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) : f.fiber = eqToHom (by subst h; rfl) ≫ g.fiber := by	subst h dsimp simp
/- Copyright (c) 2022 Anatole Dedecker. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anatole Dedecker -/ import Mathlib.Topology.UniformSpace.UniformConvergenceTopology #align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Equicontinuity of a family of functions Let `X` be a topological space and `α` a `UniformSpace`. A family of functions `F : ι → X → α` is said to be equicontinuous at a point `x₀ : X` when, for any entourage `U` in `α`, there is a neighborhood `V` of `x₀` such that, for all `x ∈ V`, and for all `i`, `F i x` is `U`-close to `F i x₀`. In other words, one has `∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U`. For maps between metric spaces, this corresponds to `∀ ε > 0, ∃ δ > 0, ∀ x, ∀ i, dist x₀ x < δ → dist (F i x₀) (F i x) < ε`. `F` is said to be equicontinuous if it is equicontinuous at each point. A closely related concept is that of *uniform* equicontinuity of a family of functions `F : ι → β → α` between uniform spaces, which means that, for any entourage `U` in `α`, there is an entourage `V` in `β` such that, if `x` and `y` are `V`-close, then for all `i`, `F i x` and `F i y` are `U`-close. In other words, one has `∀ U ∈ 𝓤 α, ∀ᶠ xy in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U`. For maps between metric spaces, this corresponds to `∀ ε > 0, ∃ δ > 0, ∀ x y, ∀ i, dist x y < δ → dist (F i x₀) (F i x) < ε`. ## Main definitions * `EquicontinuousAt`: equicontinuity of a family of functions at a point * `Equicontinuous`: equicontinuity of a family of functions on the whole domain * `UniformEquicontinuous`: uniform equicontinuity of a family of functions on the whole domain We also introduce relative versions, namely `EquicontinuousWithinAt`, `EquicontinuousOn` and `UniformEquicontinuousOn`, akin to `ContinuousWithinAt`, `ContinuousOn` and `UniformContinuousOn` respectively. ## Main statements * `equicontinuous_iff_continuous`: equicontinuity can be expressed as a simple continuity condition between well-chosen function spaces. This is really useful for building up the theory. * `Equicontinuous.closure`: if a set of functions is equicontinuous, its closure for the topology of pointwise convergence is also equicontinuous. ## Notations Throughout this file, we use : - `ι`, `κ` for indexing types - `X`, `Y`, `Z` for topological spaces - `α`, `β`, `γ` for uniform spaces ## Implementation details We choose to express equicontinuity as a properties of indexed families of functions rather than sets of functions for the following reasons: - it is really easy to express equicontinuity of `H : Set (X → α)` using our setup: it is just equicontinuity of the family `(↑) : ↥H → (X → α)`. On the other hand, going the other way around would require working with the range of the family, which is always annoying because it introduces useless existentials. - in most applications, one doesn't work with bare functions but with a more specific hom type `hom`. Equicontinuity of a set `H : Set hom` would then have to be expressed as equicontinuity of `coe_fn '' H`, which is super annoying to work with. This is much simpler with families, because equicontinuity of a family `𝓕 : ι → hom` would simply be expressed as equicontinuity of `coe_fn ∘ 𝓕`, which doesn't introduce any nasty existentials. To simplify statements, we do provide abbreviations `Set.EquicontinuousAt`, `Set.Equicontinuous` and `Set.UniformEquicontinuous` asserting the corresponding fact about the family `(↑) : ↥H → (X → α)` where `H : Set (X → α)`. Note however that these won't work for sets of hom types, and in that case one should go back to the family definition rather than using `Set.image`. ## References * [N. Bourbaki, General Topology, Chapter X][bourbaki1966] ## Tags equicontinuity, uniform convergence, ascoli -/ section open UniformSpace Filter Set Uniformity Topology UniformConvergence Function variable {ι κ X X' Y Z α α' β β' γ 𝓕 : Type} [tX : TopologicalSpace X] [tY : TopologicalSpace Y] [tZ : TopologicalSpace Z] [uα : UniformSpace α] [uβ : UniformSpace β] [uγ : UniformSpace γ] /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is equicontinuous at `x₀ : X`* if, for all entourages `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀` such that, for all `x ∈ V` and for all `i : ι`, `F i x` is `U`-close to `F i x₀`. -/ def EquicontinuousAt (F : ι → X → α) (x₀ : X) : Prop := ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U #align equicontinuous_at EquicontinuousAt /-- We say that a set `H : Set (X → α)` of functions is equicontinuous at a point if the family `(↑) : ↥H → (X → α)` is equicontinuous at that point. -/ protected abbrev Set.EquicontinuousAt (H : Set <\| X → α) (x₀ : X) : Prop := EquicontinuousAt ((↑) : H → X → α) x₀ #align set.equicontinuous_at Set.EquicontinuousAt /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is equicontinuous at `x₀ : X` within `S : Set X` if, for all entourages `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀` within `S` such that, for all `x ∈ V` and for all `i : ι`, `F i x` is `U`-close to `F i x₀`. -/ def EquicontinuousWithinAt (F : ι → X → α) (S : Set X) (x₀ : X) : Prop := ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ U /-- We say that a set `H : Set (X → α)` of functions is equicontinuous at a point within a subset if the family `(↑) : ↥H → (X → α)` is equicontinuous at that point within that same subset. -/ protected abbrev Set.EquicontinuousWithinAt (H : Set <\| X → α) (S : Set X) (x₀ : X) : Prop := EquicontinuousWithinAt ((↑) : H → X → α) S x₀ /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is equicontinuous on all of `X` if it is equicontinuous at each point of `X`. -/ def Equicontinuous (F : ι → X → α) : Prop := ∀ x₀, EquicontinuousAt F x₀ #align equicontinuous Equicontinuous /-- We say that a set `H : Set (X → α)` of functions is equicontinuous if the family `(↑) : ↥H → (X → α)` is equicontinuous. -/ protected abbrev Set.Equicontinuous (H : Set <\| X → α) : Prop := Equicontinuous ((↑) : H → X → α) #align set.equicontinuous Set.Equicontinuous /-- A family `F : ι → X → α` of functions from a topological space to a uniform space is equicontinuous on `S : Set X` if it is equicontinuous within `S` at each point of `S`. -/ def EquicontinuousOn (F : ι → X → α) (S : Set X) : Prop := ∀ x₀ ∈ S, EquicontinuousWithinAt F S x₀ /-- We say that a set `H : Set (X → α)` of functions is equicontinuous on a subset if the family `(↑) : ↥H → (X → α)` is equicontinuous on that subset. -/ protected abbrev Set.EquicontinuousOn (H : Set <\| X → α) (S : Set X) : Prop := EquicontinuousOn ((↑) : H → X → α) S /-- A family `F : ι → β → α` of functions between uniform spaces is uniformly equicontinuous if, for all entourages `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are `V`-close, we have that, for all `i : ι`, `F i x` is `U`-close to `F i y`. -/ def UniformEquicontinuous (F : ι → β → α) : Prop := ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U #align uniform_equicontinuous UniformEquicontinuous /-- We say that a set `H : Set (X → α)` of functions is uniformly equicontinuous if the family `(↑) : ↥H → (X → α)` is uniformly equicontinuous. -/ protected abbrev Set.UniformEquicontinuous (H : Set <\| β → α) : Prop := UniformEquicontinuous ((↑) : H → β → α) #align set.uniform_equicontinuous Set.UniformEquicontinuous /-- A family `F : ι → β → α` of functions between uniform spaces is uniformly equicontinuous on `S : Set β` if, for all entourages `U ∈ 𝓤 α`, there is a relative entourage `V ∈ 𝓤 β ⊓ 𝓟 (S ×ˢ S)` such that, whenever `x` and `y` are `V`-close, we have that, for all `i : ι`, `F i x` is `U`-close to `F i y`. -/ def UniformEquicontinuousOn (F : ι → β → α) (S : Set β) : Prop := ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ U /-- We say that a set `H : Set (X → α)` of functions is uniformly equicontinuous on a subset if the family `(↑) : ↥H → (X → α)` is uniformly equicontinuous on that subset. -/ protected abbrev Set.UniformEquicontinuousOn (H : Set <\| β → α) (S : Set β) : Prop := UniformEquicontinuousOn ((↑) : H → β → α) S lemma EquicontinuousAt.equicontinuousWithinAt {F : ι → X → α} {x₀ : X} (H : EquicontinuousAt F x₀) (S : Set X) : EquicontinuousWithinAt F S x₀ := fun U hU ↦ (H U hU).filter_mono inf_le_left lemma EquicontinuousWithinAt.mono {F : ι → X → α} {x₀ : X} {S T : Set X} (H : EquicontinuousWithinAt F T x₀) (hST : S ⊆ T) : EquicontinuousWithinAt F S x₀ := fun U hU ↦ (H U hU).filter_mono <\| nhdsWithin_mono x₀ hST @[simp] lemma equicontinuousWithinAt_univ (F : ι → X → α) (x₀ : X) : EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ := by rw [EquicontinuousWithinAt, EquicontinuousAt, nhdsWithin_univ] lemma equicontinuousAt_restrict_iff (F : ι → X → α) {S : Set X} (x₀ : S) : EquicontinuousAt (S.restrict ∘ F) x₀ ↔ EquicontinuousWithinAt F S x₀ := by simp [EquicontinuousWithinAt, EquicontinuousAt, ← eventually_nhds_subtype_iff] lemma Equicontinuous.equicontinuousOn {F : ι → X → α} (H : Equicontinuous F) (S : Set X) : EquicontinuousOn F S := fun x _ ↦ (H x).equicontinuousWithinAt S lemma EquicontinuousOn.mono {F : ι → X → α} {S T : Set X} (H : EquicontinuousOn F T) (hST : S ⊆ T) : EquicontinuousOn F S := fun x hx ↦ (H x (hST hx)).mono hST lemma equicontinuousOn_univ (F : ι → X → α) : EquicontinuousOn F univ ↔ Equicontinuous F := by simp [EquicontinuousOn, Equicontinuous] lemma equicontinuous_restrict_iff (F : ι → X → α) {S : Set X} : Equicontinuous (S.restrict ∘ F) ↔ EquicontinuousOn F S := by simp [Equicontinuous, EquicontinuousOn, equicontinuousAt_restrict_iff] lemma UniformEquicontinuous.uniformEquicontinuousOn {F : ι → β → α} (H : UniformEquicontinuous F) (S : Set β) : UniformEquicontinuousOn F S := fun U hU ↦ (H U hU).filter_mono inf_le_left lemma UniformEquicontinuousOn.mono {F : ι → β → α} {S T : Set β} (H : UniformEquicontinuousOn F T) (hST : S ⊆ T) : UniformEquicontinuousOn F S := fun U hU ↦ (H U hU).filter_mono <\| by gcongr lemma uniformEquicontinuousOn_univ (F : ι → β → α) : UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F := by simp [UniformEquicontinuousOn, UniformEquicontinuous] lemma uniformEquicontinuous_restrict_iff (F : ι → β → α) {S : Set β} : UniformEquicontinuous (S.restrict ∘ F) ↔ UniformEquicontinuousOn F S := by rw [UniformEquicontinuous, UniformEquicontinuousOn] conv in _ ⊓ _ => rw [← Subtype.range_val (s := S), ← range_prod_map, ← map_comap] rfl /-! ### Empty index type -/ @[simp] lemma equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) : EquicontinuousAt F x₀ := fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim) @[simp] lemma equicontinuousWithinAt_empty [h : IsEmpty ι] (F : ι → X → α) (S : Set X) (x₀ : X) : EquicontinuousWithinAt F S x₀ := fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim) @[simp] lemma equicontinuous_empty [IsEmpty ι] (F : ι → X → α) : Equicontinuous F := equicontinuousAt_empty F @[simp] lemma equicontinuousOn_empty [IsEmpty ι] (F : ι → X → α) (S : Set X) : EquicontinuousOn F S := fun x₀ _ ↦ equicontinuousWithinAt_empty F S x₀ @[simp] lemma uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) : UniformEquicontinuous F := fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim) @[simp] lemma uniformEquicontinuousOn_empty [h : IsEmpty ι] (F : ι → β → α) (S : Set β) : UniformEquicontinuousOn F S := fun _ _ ↦ eventually_of_forall (fun _ ↦ h.elim) /-! ### Finite index type -/ theorem equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} : EquicontinuousAt F x₀ ↔ ∀ i, ContinuousAt (F i) x₀ := by simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff, UniformSpace.ball, @forall_swap _ ι] theorem equicontinuousWithinAt_finite [Finite ι] {F : ι → X → α} {S : Set X} {x₀ : X} : EquicontinuousWithinAt F S x₀ ↔ ∀ i, ContinuousWithinAt (F i) S x₀ := by simp [EquicontinuousWithinAt, ContinuousWithinAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff, UniformSpace.ball, @forall_swap _ ι] theorem equicontinuous_finite [Finite ι] {F : ι → X → α} : Equicontinuous F ↔ ∀ i, Continuous (F i) := by simp only [Equicontinuous, equicontinuousAt_finite, continuous_iff_continuousAt, @forall_swap ι] theorem equicontinuousOn_finite [Finite ι] {F : ι → X → α} {S : Set X} : EquicontinuousOn F S ↔ ∀ i, ContinuousOn (F i) S := by simp only [EquicontinuousOn, equicontinuousWithinAt_finite, ContinuousOn, @forall_swap ι] theorem uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} : UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) := by simp only [UniformEquicontinuous, eventually_all, @forall_swap _ ι]; rfl theorem uniformEquicontinuousOn_finite [Finite ι] {F : ι → β → α} {S : Set β} : UniformEquicontinuousOn F S ↔ ∀ i, UniformContinuousOn (F i) S := by simp only [UniformEquicontinuousOn, eventually_all, @forall_swap _ ι]; rfl /-! ### Index type with a unique element -/ theorem equicontinuousAt_unique [Unique ι] {F : ι → X → α} {x : X} : EquicontinuousAt F x ↔ ContinuousAt (F default) x := equicontinuousAt_finite.trans Unique.forall_iff theorem equicontinuousWithinAt_unique [Unique ι] {F : ι → X → α} {S : Set X} {x : X} : EquicontinuousWithinAt F S x ↔ ContinuousWithinAt (F default) S x := equicontinuousWithinAt_finite.trans Unique.forall_iff theorem equicontinuous_unique [Unique ι] {F : ι → X → α} : Equicontinuous F ↔ Continuous (F default) := equicontinuous_finite.trans Unique.forall_iff theorem equicontinuousOn_unique [Unique ι] {F : ι → X → α} {S : Set X} : EquicontinuousOn F S ↔ ContinuousOn (F default) S := equicontinuousOn_finite.trans Unique.forall_iff theorem uniformEquicontinuous_unique [Unique ι] {F : ι → β → α} : UniformEquicontinuous F ↔ UniformContinuous (F default) := uniformEquicontinuous_finite.trans Unique.forall_iff theorem uniformEquicontinuousOn_unique [Unique ι] {F : ι → β → α} {S : Set β} : UniformEquicontinuousOn F S ↔ UniformContinuousOn (F default) S := uniformEquicontinuousOn_finite.trans Unique.forall_iff /-- Reformulation of equicontinuity at `x₀` within a set `S`, comparing two variables near `x₀` instead of comparing only one with `x₀`. -/ theorem equicontinuousWithinAt_iff_pair {F : ι → X → α} {S : Set X} {x₀ : X} (hx₀ : x₀ ∈ S) : EquicontinuousWithinAt F S x₀ ↔ ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by constructor <;> intro H U hU · rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩ refine ⟨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel ?_ (hy i))⟩ exact hVsymm.mk_mem_comm.mp (hx i) · rcases H U hU with ⟨V, hV, hVU⟩ filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhdsWithin hx₀ hV) x hx i /-- Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing only one with `x₀`. -/ theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} : EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by simp_rw [← equicontinuousWithinAt_univ, equicontinuousWithinAt_iff_pair (mem_univ x₀), nhdsWithin_univ] #align equicontinuous_at_iff_pair equicontinuousAt_iff_pair /-- Uniform equicontinuity implies equicontinuity. -/ theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) : Equicontinuous F := fun x₀ U hU ↦ mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun _ hx i ↦ hx i #align uniform_equicontinuous.equicontinuous UniformEquicontinuous.equicontinuous /-- Uniform equicontinuity on a subset implies equicontinuity on that subset. -/ theorem UniformEquicontinuousOn.equicontinuousOn {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S) : EquicontinuousOn F S := fun _ hx₀ U hU ↦ mem_of_superset (ball_mem_nhdsWithin hx₀ (h U hU)) fun _ hx i ↦ hx i /-- Each function of a family equicontinuous at `x₀` is continuous at `x₀`. -/ theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) : ContinuousAt (F i) x₀ := (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i #align equicontinuous_at.continuous_at EquicontinuousAt.continuousAt /-- Each function of a family equicontinuous at `x₀` within `S` is continuous at `x₀` within `S`. -/ theorem EquicontinuousWithinAt.continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} (h : EquicontinuousWithinAt F S x₀) (i : ι) : ContinuousWithinAt (F i) S x₀ := (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <\| X → α} {x₀ : X} (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ := h.continuousAt ⟨f, hf⟩ #align set.equicontinuous_at.continuous_at_of_mem Set.EquicontinuousAt.continuousAt_of_mem protected theorem Set.EquicontinuousWithinAt.continuousWithinAt_of_mem {H : Set <\| X → α} {S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) {f : X → α} (hf : f ∈ H) : ContinuousWithinAt f S x₀ := h.continuousWithinAt ⟨f, hf⟩ /-- Each function of an equicontinuous family is continuous. -/ theorem Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) : Continuous (F i) := continuous_iff_continuousAt.mpr fun x => (h x).continuousAt i #align equicontinuous.continuous Equicontinuous.continuous /-- Each function of a family equicontinuous on `S` is continuous on `S`. -/ theorem EquicontinuousOn.continuousOn {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (i : ι) : ContinuousOn (F i) S := fun x hx ↦ (h x hx).continuousWithinAt i protected theorem Set.Equicontinuous.continuous_of_mem {H : Set <\| X → α} (h : H.Equicontinuous) {f : X → α} (hf : f ∈ H) : Continuous f := h.continuous ⟨f, hf⟩ #align set.equicontinuous.continuous_of_mem Set.Equicontinuous.continuous_of_mem protected theorem Set.EquicontinuousOn.continuousOn_of_mem {H : Set <\| X → α} {S : Set X} (h : H.EquicontinuousOn S) {f : X → α} (hf : f ∈ H) : ContinuousOn f S := h.continuousOn ⟨f, hf⟩ /-- Each function of a uniformly equicontinuous family is uniformly continuous. -/ theorem UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F) (i : ι) : UniformContinuous (F i) := fun U hU => mem_map.mpr (mem_of_superset (h U hU) fun _ hxy => hxy i) #align uniform_equicontinuous.uniform_continuous UniformEquicontinuous.uniformContinuous /-- Each function of a family uniformly equicontinuous on `S` is uniformly continuous on `S`. -/ theorem UniformEquicontinuousOn.uniformContinuousOn {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S) (i : ι) : UniformContinuousOn (F i) S := fun U hU => mem_map.mpr (mem_of_superset (h U hU) fun _ hxy => hxy i) protected theorem Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <\| β → α} (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f := h.uniformContinuous ⟨f, hf⟩ #align set.uniform_equicontinuous.uniform_continuous_of_mem Set.UniformEquicontinuous.uniformContinuous_of_mem protected theorem Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem {H : Set <\| β → α} {S : Set β} (h : H.UniformEquicontinuousOn S) {f : β → α} (hf : f ∈ H) : UniformContinuousOn f S := h.uniformContinuousOn ⟨f, hf⟩ /-- Taking sub-families preserves equicontinuity at a point. -/ theorem EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) : EquicontinuousAt (F ∘ u) x₀ := fun U hU => (h U hU).mono fun _ H k => H (u k) #align equicontinuous_at.comp EquicontinuousAt.comp /-- Taking sub-families preserves equicontinuity at a point within a subset. -/ theorem EquicontinuousWithinAt.comp {F : ι → X → α} {S : Set X} {x₀ : X} (h : EquicontinuousWithinAt F S x₀) (u : κ → ι) : EquicontinuousWithinAt (F ∘ u) S x₀ := fun U hU ↦ (h U hU).mono fun _ H k => H (u k) protected theorem Set.EquicontinuousAt.mono {H H' : Set <\| X → α} {x₀ : X} (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ := h.comp (inclusion hH) #align set.equicontinuous_at.mono Set.EquicontinuousAt.mono protected theorem Set.EquicontinuousWithinAt.mono {H H' : Set <\| X → α} {S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) (hH : H' ⊆ H) : H'.EquicontinuousWithinAt S x₀ := h.comp (inclusion hH) /-- Taking sub-families preserves equicontinuity. -/ theorem Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) : Equicontinuous (F ∘ u) := fun x => (h x).comp u #align equicontinuous.comp Equicontinuous.comp /-- Taking sub-families preserves equicontinuity on a subset. -/ theorem EquicontinuousOn.comp {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (u : κ → ι) : EquicontinuousOn (F ∘ u) S := fun x hx ↦ (h x hx).comp u protected theorem Set.Equicontinuous.mono {H H' : Set <\| X → α} (h : H.Equicontinuous) (hH : H' ⊆ H) : H'.Equicontinuous := h.comp (inclusion hH) #align set.equicontinuous.mono Set.Equicontinuous.mono protected theorem Set.EquicontinuousOn.mono {H H' : Set <\| X → α} {S : Set X} (h : H.EquicontinuousOn S) (hH : H' ⊆ H) : H'.EquicontinuousOn S := h.comp (inclusion hH) /-- Taking sub-families preserves uniform equicontinuity. -/ theorem UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) : UniformEquicontinuous (F ∘ u) := fun U hU => (h U hU).mono fun _ H k => H (u k) #align uniform_equicontinuous.comp UniformEquicontinuous.comp /-- Taking sub-families preserves uniform equicontinuity on a subset. -/ theorem UniformEquicontinuousOn.comp {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S) (u : κ → ι) : UniformEquicontinuousOn (F ∘ u) S := fun U hU ↦ (h U hU).mono fun _ H k => H (u k) protected theorem Set.UniformEquicontinuous.mono {H H' : Set <\| β → α} (h : H.UniformEquicontinuous) (hH : H' ⊆ H) : H'.UniformEquicontinuous := h.comp (inclusion hH) #align set.uniform_equicontinuous.mono Set.UniformEquicontinuous.mono protected theorem Set.UniformEquicontinuousOn.mono {H H' : Set <\| β → α} {S : Set β} (h : H.UniformEquicontinuousOn S) (hH : H' ⊆ H) : H'.UniformEquicontinuousOn S := h.comp (inclusion hH) /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`, i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀`. -/ theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ := by simp only [EquicontinuousAt, forall_subtype_range_iff] #align equicontinuous_at_iff_range equicontinuousAt_iff_range /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff `range 𝓕` is equicontinuous at `x₀` within `S`, i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀` within `S`. -/ theorem equicontinuousWithinAt_iff_range {F : ι → X → α} {S : Set X} {x₀ : X} : EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((↑) : range F → X → α) S x₀ := by simp only [EquicontinuousWithinAt, forall_subtype_range_iff] /-- A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous, i.e the family `(↑) : range F → X → α` is equicontinuous. -/ theorem equicontinuous_iff_range {F : ι → X → α} : Equicontinuous F ↔ Equicontinuous ((↑) : range F → X → α) := forall_congr' fun _ => equicontinuousAt_iff_range #align equicontinuous_iff_range equicontinuous_iff_range /-- A family `𝓕 : ι → X → α` is equicontinuous on `S` iff `range 𝓕` is equicontinuous on `S`, i.e the family `(↑) : range F → X → α` is equicontinuous on `S`. -/ theorem equicontinuousOn_iff_range {F : ι → X → α} {S : Set X} : EquicontinuousOn F S ↔ EquicontinuousOn ((↑) : range F → X → α) S := forall_congr' fun _ ↦ forall_congr' fun _ ↦ equicontinuousWithinAt_iff_range /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous, i.e the family `(↑) : range F → β → α` is uniformly equicontinuous. -/ theorem uniformEquicontinuous_iff_range {F : ι → β → α} : UniformEquicontinuous F ↔ UniformEquicontinuous ((↑) : range F → β → α) := ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h => h.comp (rangeFactorization F)⟩ #align uniform_equicontinuous_at_iff_range uniformEquicontinuous_iff_range /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous on `S` iff `range 𝓕` is uniformly equicontinuous on `S`, i.e the family `(↑) : range F → β → α` is uniformly equicontinuous on `S`. -/ theorem uniformEquicontinuousOn_iff_range {F : ι → β → α} {S : Set β} : UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((↑) : range F → β → α) S := ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h => h.comp (rangeFactorization F)⟩ section open UniformFun /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is continuous at `x₀` when `ι → α` is equipped with the topology of uniform convergence. This is very useful for developping the equicontinuity API, but it should not be used directly for other purposes. -/ theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} : EquicontinuousAt F x₀ ↔ ContinuousAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) x₀ := by rw [ContinuousAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff] rfl #align equicontinuous_at_iff_continuous_at equicontinuousAt_iff_continuousAt /-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff the function `swap 𝓕 : X → ι → α` is continuous at `x₀` within `S` when `ι → α` is equipped with the topology of uniform convergence. This is very useful for developping the equicontinuity API, but it should not be used directly for other purposes. -/ theorem equicontinuousWithinAt_iff_continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} : EquicontinuousWithinAt F S x₀ ↔ ContinuousWithinAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) S x₀ := by rw [ContinuousWithinAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff] rfl /-- A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is continuous when `ι → α` is equipped with the topology of uniform convergence. This is very useful for developping the equicontinuity API, but it should not be used directly for other purposes. -/ theorem equicontinuous_iff_continuous {F : ι → X → α} : Equicontinuous F ↔ Continuous (ofFun ∘ Function.swap F : X → ι →ᵤ α) := by simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt] #align equicontinuous_iff_continuous equicontinuous_iff_continuous /-- A family `𝓕 : ι → X → α` is equicontinuous on `S` iff the function `swap 𝓕 : X → ι → α` is continuous on `S` when `ι → α` is equipped with the topology of uniform convergence. This is very useful for developping the equicontinuity API, but it should not be used directly for other purposes. -/ theorem equicontinuousOn_iff_continuousOn {F : ι → X → α} {S : Set X} : EquicontinuousOn F S ↔ ContinuousOn (ofFun ∘ Function.swap F : X → ι →ᵤ α) S := by simp_rw [EquicontinuousOn, ContinuousOn, equicontinuousWithinAt_iff_continuousWithinAt] /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is uniformly continuous when `ι → α` is equipped with the uniform structure of uniform convergence. This is very useful for developping the equicontinuity API, but it should not be used directly for other purposes. -/ theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} : UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) := by rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff] rfl #align uniform_equicontinuous_iff_uniform_continuous uniformEquicontinuous_iff_uniformContinuous /-- A family `𝓕 : ι → β → α` is uniformly equicontinuous on `S` iff the function `swap 𝓕 : β → ι → α` is uniformly continuous on `S` when `ι → α` is equipped with the uniform structure of uniform convergence. This is very useful for developping the equicontinuity API, but it should not be used directly for other purposes. -/ theorem uniformEquicontinuousOn_iff_uniformContinuousOn {F : ι → β → α} {S : Set β} : UniformEquicontinuousOn F S ↔ UniformContinuousOn (ofFun ∘ Function.swap F : β → ι →ᵤ α) S := by rw [UniformContinuousOn, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff] rfl theorem equicontinuousWithinAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} {S : Set X} {x₀ : X} : EquicontinuousWithinAt (uα := ⨅ k, u k) F S x₀ ↔ ∀ k, EquicontinuousWithinAt (uα := u k) F S x₀ := by simp only [equicontinuousWithinAt_iff_continuousWithinAt (uα := _), topologicalSpace] unfold ContinuousWithinAt rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf] theorem equicontinuousAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} {x₀ : X} : EquicontinuousAt (uα := ⨅ k, u k) F x₀ ↔ ∀ k, EquicontinuousAt (uα := u k) F x₀ := by simp only [← equicontinuousWithinAt_univ (uα := _), equicontinuousWithinAt_iInf_rng] theorem equicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} : Equicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, Equicontinuous (uα := u k) F := by simp_rw [equicontinuous_iff_continuous (uα := _), UniformFun.topologicalSpace] rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng] theorem equicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} {S : Set X} : EquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, EquicontinuousOn (uα := u k) F S := by simp_rw [EquicontinuousOn, equicontinuousWithinAt_iInf_rng, @forall_swap _ κ] theorem uniformEquicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'} : UniformEquicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, UniformEquicontinuous (uα := u k) F := by simp_rw [uniformEquicontinuous_iff_uniformContinuous (uα := _)] rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng] theorem uniformEquicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'} {S : Set β} : UniformEquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, UniformEquicontinuousOn (uα := u k) F S := by simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uα := _)] unfold UniformContinuousOn rw [UniformFun.iInf_eq, iInf_uniformity, tendsto_iInf] theorem equicontinuousWithinAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α} {S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt (tX := t k) F S x₀) : EquicontinuousWithinAt (tX := ⨅ k, t k) F S x₀ := by simp [equicontinuousWithinAt_iff_continuousWithinAt (tX := _)] at hk ⊢ unfold ContinuousWithinAt nhdsWithin at hk ⊢ rw [nhds_iInf] exact hk.mono_left <\| inf_le_inf_right _ <\| iInf_le _ k theorem equicontinuousAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α} {x₀ : X'} {k : κ} (hk : EquicontinuousAt (tX := t k) F x₀) : EquicontinuousAt (tX := ⨅ k, t k) F x₀ := by rw [← equicontinuousWithinAt_univ (tX := _)] at hk ⊢ exact equicontinuousWithinAt_iInf_dom hk theorem equicontinuous_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α} {k : κ} (hk : Equicontinuous (tX := t k) F) : Equicontinuous (tX := ⨅ k, t k) F := fun x ↦ equicontinuousAt_iInf_dom (hk x) theorem equicontinuousOn_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α} {S : Set X'} {k : κ} (hk : EquicontinuousOn (tX := t k) F S) : EquicontinuousOn (tX := ⨅ k, t k) F S := fun x hx ↦ equicontinuousWithinAt_iInf_dom (hk x hx) theorem uniformEquicontinuous_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α} {k : κ} (hk : UniformEquicontinuous (uβ := u k) F) : UniformEquicontinuous (uβ := ⨅ k, u k) F := by simp_rw [uniformEquicontinuous_iff_uniformContinuous (uβ := _)] at hk ⊢ exact uniformContinuous_iInf_dom hk theorem uniformEquicontinuousOn_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α} {S : Set β'} {k : κ} (hk : UniformEquicontinuousOn (uβ := u k) F S) : UniformEquicontinuousOn (uβ := ⨅ k, u k) F S := by simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uβ := _)] at hk ⊢ unfold UniformContinuousOn rw [iInf_uniformity] exact hk.mono_left <\| inf_le_inf_right _ <\| iInf_le _ k theorem Filter.HasBasis.equicontinuousAt_iff_left {p : κ → Prop} {s : κ → Set X} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) : EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by rw [equicontinuousAt_iff_continuousAt, ContinuousAt, hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)] rfl #align filter.has_basis.equicontinuous_at_iff_left Filter.HasBasis.equicontinuousAt_iff_left theorem Filter.HasBasis.equicontinuousWithinAt_iff_left {p : κ → Prop} {s : κ → Set X} {F : ι → X → α} {S : Set X} {x₀ : X} (hX : (𝓝[S] x₀).HasBasis p s) : EquicontinuousWithinAt F S x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U := by rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt, hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)] rfl theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) : EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k := by rw [equicontinuousAt_iff_continuousAt, ContinuousAt, (UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff] rfl #align filter.has_basis.equicontinuous_at_iff_right Filter.HasBasis.equicontinuousAt_iff_right theorem Filter.HasBasis.equicontinuousWithinAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (𝓤 α).HasBasis p s) : EquicontinuousWithinAt F S x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ s k := by rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt, (UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff] rfl theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) : EquicontinuousAt F x₀ ↔ ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by rw [equicontinuousAt_iff_continuousAt, ContinuousAt, hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)] rfl #align filter.has_basis.equicontinuous_at_iff Filter.HasBasis.equicontinuousAt_iff theorem Filter.HasBasis.equicontinuousWithinAt_iff {κ₁ κ₂ : Type} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hX : (𝓝[S] x₀).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) : EquicontinuousWithinAt F S x₀ ↔ ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt, hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)] rfl theorem Filter.HasBasis.uniformEquicontinuous_iff_left {p : κ → Prop} {s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) : UniformEquicontinuous F ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous, hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)] simp only [Prod.forall] rfl #align filter.has_basis.uniform_equicontinuous_iff_left Filter.HasBasis.uniformEquicontinuous_iff_left theorem Filter.HasBasis.uniformEquicontinuousOn_iff_left {p : κ → Prop} {s : κ → Set (β × β)} {F : ι → β → α} {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p s) : UniformEquicontinuousOn F S ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn, hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)] simp only [Prod.forall] rfl theorem Filter.HasBasis.uniformEquicontinuous_iff_right {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) : UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k := by rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous, (UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff] rfl #align filter.has_basis.uniform_equicontinuous_iff_right Filter.HasBasis.uniformEquicontinuous_iff_right theorem Filter.HasBasis.uniformEquicontinuousOn_iff_right {p : κ → Prop} {s : κ → Set (α × α)} {F : ι → β → α} {S : Set β} (hα : (𝓤 α).HasBasis p s) : UniformEquicontinuousOn F S ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ s k := by rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn, (UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff] rfl theorem Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) : UniformEquicontinuous F ↔ ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous, hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)] simp only [Prod.forall] rfl #align filter.has_basis.uniform_equicontinuous_iff Filter.HasBasis.uniformEquicontinuous_iff theorem Filter.HasBasis.uniformEquicontinuousOn_iff {κ₁ κ₂ : Type} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α} {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) : UniformEquicontinuousOn F S ↔ ∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn, hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)] simp only [Prod.forall] rfl /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point `x₀ : X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous at `x₀`. -/ theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β} (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ := by have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff] rfl #align uniform_inducing.equicontinuous_at_iff UniformInducing.equicontinuousAt_iff /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point `x₀ : X` within a subset `S : Set X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous at `x₀` within `S`. -/ theorem UniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β} (hu : UniformInducing u) : EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ := by have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing simp only [equicontinuousWithinAt_iff_continuousWithinAt, this.continuousWithinAt_iff] rfl /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous. -/ theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) : Equicontinuous F ↔ Equicontinuous ((u ∘ ·) ∘ F) := by congrm ∀ x, ?_ rw [hu.equicontinuousAt_iff] #align uniform_inducing.equicontinuous_iff UniformInducing.equicontinuous_iff /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous on a subset `S : Set X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous on `S`. -/ theorem UniformInducing.equicontinuousOn_iff {F : ι → X → α} {S : Set X} {u : α → β} (hu : UniformInducing u) : EquicontinuousOn F S ↔ EquicontinuousOn ((u ∘ ·) ∘ F) S := by congrm ∀ x ∈ S, ?_ rw [hu.equicontinuousWithinAt_iff] /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is uniformly equicontinuous. -/ theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ} (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) := by have := UniformFun.postcomp_uniformInducing (α := ι) hu simp only [uniformEquicontinuous_iff_uniformContinuous, this.uniformContinuous_iff] rfl #align uniform_inducing.uniform_equicontinuous_iff UniformInducing.uniformEquicontinuous_iff /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous on a subset `S : Set β` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is uniformly equicontinuous on `S`. -/	Mathlib/Topology/UniformSpace/Equicontinuity.lean	787	792	theorem UniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ} (hu : UniformInducing u) : UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S := by	have := UniformFun.postcomp_uniformInducing (α := ι) hu simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff] rfl
/- Copyright (c) 2015 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro -/ import Mathlib.Algebra.Group.Embedding import Mathlib.Data.Fin.Basic import Mathlib.Data.Finset.Union #align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83" /-! # Image and map operations on finite sets This file provides the finite analog of `Set.image`, along with some other similar functions. Note there are two ways to take the image over a finset; via `Finset.image` which applies the function then removes duplicates (requiring `DecidableEq`), or via `Finset.map` which exploits injectivity of the function to avoid needing to deduplicate. Choosing between these is similar to choosing between `insert` and `Finset.cons`, or between `Finset.union` and `Finset.disjUnion`. ## Main definitions * `Finset.image`: Given a function `f : α → β`, `s.image f` is the image finset in `β`. * `Finset.map`: Given an embedding `f : α ↪ β`, `s.map f` is the image finset in `β`. * `Finset.filterMap` Given a function `f : α → Option β`, `s.filterMap f` is the image finset in `β`, filtering out `none`s. * `Finset.subtype`: `s.subtype p` is the finset of `Subtype p` whose elements belong to `s`. * `Finset.fin`:`s.fin n` is the finset of all elements of `s` less than `n`. ## TODO Move the material about `Finset.range` so that the `Mathlib.Algebra.Group.Embedding` import can be removed. -/ -- TODO -- assert_not_exists OrderedCommMonoid assert_not_exists MonoidWithZero assert_not_exists MulAction variable {α β γ : Type*} open Multiset open Function namespace Finset /-! ### map -/ section Map open Function /-- When `f` is an embedding of `α` in `β` and `s` is a finset in `α`, then `s.map f` is the image finset in `β`. The embedding condition guarantees that there are no duplicates in the image. -/ def map (f : α ↪ β) (s : Finset α) : Finset β := ⟨s.1.map f, s.2.map f.2⟩ #align finset.map Finset.map @[simp] theorem map_val (f : α ↪ β) (s : Finset α) : (map f s).1 = s.1.map f := rfl #align finset.map_val Finset.map_val @[simp] theorem map_empty (f : α ↪ β) : (∅ : Finset α).map f = ∅ := rfl #align finset.map_empty Finset.map_empty variable {f : α ↪ β} {s : Finset α} @[simp] theorem mem_map {b : β} : b ∈ s.map f ↔ ∃ a ∈ s, f a = b := Multiset.mem_map #align finset.mem_map Finset.mem_map -- Porting note: Higher priority to apply before `mem_map`. @[simp 1100] theorem mem_map_equiv {f : α ≃ β} {b : β} : b ∈ s.map f.toEmbedding ↔ f.symm b ∈ s := by rw [mem_map] exact ⟨by rintro ⟨a, H, rfl⟩ simpa, fun h => ⟨_, h, by simp⟩⟩ #align finset.mem_map_equiv Finset.mem_map_equiv -- The simpNF linter says that the LHS can be simplified via `Finset.mem_map`. -- However this is a higher priority lemma. -- https://github.com/leanprover/std4/issues/207 @[simp 1100, nolint simpNF] theorem mem_map' (f : α ↪ β) {a} {s : Finset α} : f a ∈ s.map f ↔ a ∈ s := mem_map_of_injective f.2 #align finset.mem_map' Finset.mem_map' theorem mem_map_of_mem (f : α ↪ β) {a} {s : Finset α} : a ∈ s → f a ∈ s.map f := (mem_map' _).2 #align finset.mem_map_of_mem Finset.mem_map_of_mem theorem forall_mem_map {f : α ↪ β} {s : Finset α} {p : ∀ a, a ∈ s.map f → Prop} : (∀ y (H : y ∈ s.map f), p y H) ↔ ∀ x (H : x ∈ s), p (f x) (mem_map_of_mem _ H) := ⟨fun h y hy => h (f y) (mem_map_of_mem _ hy), fun h x hx => by obtain ⟨y, hy, rfl⟩ := mem_map.1 hx exact h _ hy⟩ #align finset.forall_mem_map Finset.forall_mem_map theorem apply_coe_mem_map (f : α ↪ β) (s : Finset α) (x : s) : f x ∈ s.map f := mem_map_of_mem f x.prop #align finset.apply_coe_mem_map Finset.apply_coe_mem_map @[simp, norm_cast] theorem coe_map (f : α ↪ β) (s : Finset α) : (s.map f : Set β) = f '' s := Set.ext (by simp only [mem_coe, mem_map, Set.mem_image, implies_true]) #align finset.coe_map Finset.coe_map theorem coe_map_subset_range (f : α ↪ β) (s : Finset α) : (s.map f : Set β) ⊆ Set.range f := calc ↑(s.map f) = f '' s := coe_map f s _ ⊆ Set.range f := Set.image_subset_range f ↑s #align finset.coe_map_subset_range Finset.coe_map_subset_range /-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect. -/ theorem map_perm {σ : Equiv.Perm α} (hs : { a \| σ a ≠ a } ⊆ s) : s.map (σ : α ↪ α) = s := coe_injective <\| (coe_map _ _).trans <\| Set.image_perm hs #align finset.map_perm Finset.map_perm theorem map_toFinset [DecidableEq α] [DecidableEq β] {s : Multiset α} : s.toFinset.map f = (s.map f).toFinset := ext fun _ => by simp only [mem_map, Multiset.mem_map, exists_prop, Multiset.mem_toFinset] #align finset.map_to_finset Finset.map_toFinset @[simp] theorem map_refl : s.map (Embedding.refl _) = s := ext fun _ => by simpa only [mem_map, exists_prop] using exists_eq_right #align finset.map_refl Finset.map_refl @[simp] theorem map_cast_heq {α β} (h : α = β) (s : Finset α) : HEq (s.map (Equiv.cast h).toEmbedding) s := by subst h simp #align finset.map_cast_heq Finset.map_cast_heq theorem map_map (f : α ↪ β) (g : β ↪ γ) (s : Finset α) : (s.map f).map g = s.map (f.trans g) := eq_of_veq <\| by simp only [map_val, Multiset.map_map]; rfl #align finset.map_map Finset.map_map theorem map_comm {β'} {f : β ↪ γ} {g : α ↪ β} {f' : α ↪ β'} {g' : β' ↪ γ} (h_comm : ∀ a, f (g a) = g' (f' a)) : (s.map g).map f = (s.map f').map g' := by simp_rw [map_map, Embedding.trans, Function.comp, h_comm] #align finset.map_comm Finset.map_comm theorem _root_.Function.Semiconj.finset_map {f : α ↪ β} {ga : α ↪ α} {gb : β ↪ β} (h : Function.Semiconj f ga gb) : Function.Semiconj (map f) (map ga) (map gb) := fun _ => map_comm h #align function.semiconj.finset_map Function.Semiconj.finset_map theorem _root_.Function.Commute.finset_map {f g : α ↪ α} (h : Function.Commute f g) : Function.Commute (map f) (map g) := Function.Semiconj.finset_map h #align function.commute.finset_map Function.Commute.finset_map @[simp] theorem map_subset_map {s₁ s₂ : Finset α} : s₁.map f ⊆ s₂.map f ↔ s₁ ⊆ s₂ := ⟨fun h x xs => (mem_map' _).1 <\| h <\| (mem_map' f).2 xs, fun h => by simp [subset_def, Multiset.map_subset_map h]⟩ #align finset.map_subset_map Finset.map_subset_map @[gcongr] alias ⟨_, _root_.GCongr.finsetMap_subset⟩ := map_subset_map /-- The `Finset` version of `Equiv.subset_symm_image`. -/ theorem subset_map_symm {t : Finset β} {f : α ≃ β} : s ⊆ t.map f.symm ↔ s.map f ⊆ t := by constructor <;> intro h x hx · simp only [mem_map_equiv, Equiv.symm_symm] at hx simpa using h hx · simp only [mem_map_equiv] exact h (by simp [hx]) /-- The `Finset` version of `Equiv.symm_image_subset`. -/ theorem map_symm_subset {t : Finset β} {f : α ≃ β} : t.map f.symm ⊆ s ↔ t ⊆ s.map f := by simp only [← subset_map_symm, Equiv.symm_symm] /-- Associate to an embedding `f` from `α` to `β` the order embedding that maps a finset to its image under `f`. -/ def mapEmbedding (f : α ↪ β) : Finset α ↪o Finset β := OrderEmbedding.ofMapLEIff (map f) fun _ _ => map_subset_map #align finset.map_embedding Finset.mapEmbedding @[simp] theorem map_inj {s₁ s₂ : Finset α} : s₁.map f = s₂.map f ↔ s₁ = s₂ := (mapEmbedding f).injective.eq_iff #align finset.map_inj Finset.map_inj theorem map_injective (f : α ↪ β) : Injective (map f) := (mapEmbedding f).injective #align finset.map_injective Finset.map_injective @[simp] theorem map_ssubset_map {s t : Finset α} : s.map f ⊂ t.map f ↔ s ⊂ t := (mapEmbedding f).lt_iff_lt @[gcongr] alias ⟨_, _root_.GCongr.finsetMap_ssubset⟩ := map_ssubset_map @[simp] theorem mapEmbedding_apply : mapEmbedding f s = map f s := rfl #align finset.map_embedding_apply Finset.mapEmbedding_apply theorem filter_map {p : β → Prop} [DecidablePred p] : (s.map f).filter p = (s.filter (p ∘ f)).map f := eq_of_veq (map_filter _ _ _) #align finset.filter_map Finset.filter_map lemma map_filter' (p : α → Prop) [DecidablePred p] (f : α ↪ β) (s : Finset α) [DecidablePred (∃ a, p a ∧ f a = ·)] : (s.filter p).map f = (s.map f).filter fun b => ∃ a, p a ∧ f a = b := by simp [(· ∘ ·), filter_map, f.injective.eq_iff] #align finset.map_filter' Finset.map_filter' lemma filter_attach' [DecidableEq α] (s : Finset α) (p : s → Prop) [DecidablePred p] : s.attach.filter p = (s.filter fun x => ∃ h, p ⟨x, h⟩).attach.map ⟨Subtype.map id <\| filter_subset _ _, Subtype.map_injective _ injective_id⟩ := eq_of_veq <\| Multiset.filter_attach' _ _ #align finset.filter_attach' Finset.filter_attach' lemma filter_attach (p : α → Prop) [DecidablePred p] (s : Finset α) : s.attach.filter (fun a : s ↦ p a) = (s.filter p).attach.map ((Embedding.refl _).subtypeMap mem_of_mem_filter) := eq_of_veq <\| Multiset.filter_attach _ _ #align finset.filter_attach Finset.filter_attach theorem map_filter {f : α ≃ β} {p : α → Prop} [DecidablePred p] : (s.filter p).map f.toEmbedding = (s.map f.toEmbedding).filter (p ∘ f.symm) := by simp only [filter_map, Function.comp, Equiv.toEmbedding_apply, Equiv.symm_apply_apply] #align finset.map_filter Finset.map_filter @[simp] theorem disjoint_map {s t : Finset α} (f : α ↪ β) : Disjoint (s.map f) (t.map f) ↔ Disjoint s t := mod_cast Set.disjoint_image_iff f.injective (s := s) (t := t) #align finset.disjoint_map Finset.disjoint_map theorem map_disjUnion {f : α ↪ β} (s₁ s₂ : Finset α) (h) (h' := (disjoint_map _).mpr h) : (s₁.disjUnion s₂ h).map f = (s₁.map f).disjUnion (s₂.map f) h' := eq_of_veq <\| Multiset.map_add _ _ _ #align finset.map_disj_union Finset.map_disjUnion /-- A version of `Finset.map_disjUnion` for writing in the other direction. -/ theorem map_disjUnion' {f : α ↪ β} (s₁ s₂ : Finset α) (h') (h := (disjoint_map _).mp h') : (s₁.disjUnion s₂ h).map f = (s₁.map f).disjUnion (s₂.map f) h' := map_disjUnion _ _ _ #align finset.map_disj_union' Finset.map_disjUnion' theorem map_union [DecidableEq α] [DecidableEq β] {f : α ↪ β} (s₁ s₂ : Finset α) : (s₁ ∪ s₂).map f = s₁.map f ∪ s₂.map f := mod_cast Set.image_union f s₁ s₂ #align finset.map_union Finset.map_union theorem map_inter [DecidableEq α] [DecidableEq β] {f : α ↪ β} (s₁ s₂ : Finset α) : (s₁ ∩ s₂).map f = s₁.map f ∩ s₂.map f := mod_cast Set.image_inter f.injective (s := s₁) (t := s₂) #align finset.map_inter Finset.map_inter @[simp] theorem map_singleton (f : α ↪ β) (a : α) : map f {a} = {f a} := coe_injective <\| by simp only [coe_map, coe_singleton, Set.image_singleton] #align finset.map_singleton Finset.map_singleton @[simp] theorem map_insert [DecidableEq α] [DecidableEq β] (f : α ↪ β) (a : α) (s : Finset α) : (insert a s).map f = insert (f a) (s.map f) := by simp only [insert_eq, map_union, map_singleton] #align finset.map_insert Finset.map_insert @[simp] theorem map_cons (f : α ↪ β) (a : α) (s : Finset α) (ha : a ∉ s) : (cons a s ha).map f = cons (f a) (s.map f) (by simpa using ha) := eq_of_veq <\| Multiset.map_cons f a s.val #align finset.map_cons Finset.map_cons @[simp] theorem map_eq_empty : s.map f = ∅ ↔ s = ∅ := (map_injective f).eq_iff' (map_empty f) #align finset.map_eq_empty Finset.map_eq_empty @[simp, aesop safe apply (rule_sets := [finsetNonempty])] theorem map_nonempty : (s.map f).Nonempty ↔ s.Nonempty := mod_cast Set.image_nonempty (f := f) (s := s) #align finset.map_nonempty Finset.map_nonempty protected alias ⟨_, Nonempty.map⟩ := map_nonempty #align finset.nonempty.map Finset.Nonempty.map @[simp] theorem map_nontrivial : (s.map f).Nontrivial ↔ s.Nontrivial := mod_cast Set.image_nontrivial f.injective (s := s) theorem attach_map_val {s : Finset α} : s.attach.map (Embedding.subtype _) = s := eq_of_veq <\| by rw [map_val, attach_val]; exact Multiset.attach_map_val _ #align finset.attach_map_val Finset.attach_map_val theorem disjoint_range_addLeftEmbedding (a b : ℕ) : Disjoint (range a) (map (addLeftEmbedding a) (range b)) := by simp [disjoint_left]; omega #align finset.disjoint_range_add_left_embedding Finset.disjoint_range_addLeftEmbedding theorem disjoint_range_addRightEmbedding (a b : ℕ) : Disjoint (range a) (map (addRightEmbedding a) (range b)) := by simp [disjoint_left]; omega #align finset.disjoint_range_add_right_embedding Finset.disjoint_range_addRightEmbedding theorem map_disjiUnion {f : α ↪ β} {s : Finset α} {t : β → Finset γ} {h} : (s.map f).disjiUnion t h = s.disjiUnion (fun a => t (f a)) fun _ ha _ hb hab => h (mem_map_of_mem _ ha) (mem_map_of_mem _ hb) (f.injective.ne hab) := eq_of_veq <\| Multiset.bind_map _ _ _ #align finset.map_disj_Union Finset.map_disjiUnion theorem disjiUnion_map {s : Finset α} {t : α → Finset β} {f : β ↪ γ} {h} : (s.disjiUnion t h).map f = s.disjiUnion (fun a => (t a).map f) (h.mono' fun _ _ ↦ (disjoint_map _).2) := eq_of_veq <\| Multiset.map_bind _ _ _ #align finset.disj_Union_map Finset.disjiUnion_map end Map theorem range_add_one' (n : ℕ) : range (n + 1) = insert 0 ((range n).map ⟨fun i => i + 1, fun i j => by simp⟩) := by ext (⟨⟩ \| ⟨n⟩) <;> simp [Nat.succ_eq_add_one, Nat.zero_lt_succ n] #align finset.range_add_one' Finset.range_add_one' /-! ### image -/ section Image variable [DecidableEq β] /-- `image f s` is the forward image of `s` under `f`. -/ def image (f : α → β) (s : Finset α) : Finset β := (s.1.map f).toFinset #align finset.image Finset.image @[simp] theorem image_val (f : α → β) (s : Finset α) : (image f s).1 = (s.1.map f).dedup := rfl #align finset.image_val Finset.image_val @[simp] theorem image_empty (f : α → β) : (∅ : Finset α).image f = ∅ := rfl #align finset.image_empty Finset.image_empty variable {f g : α → β} {s : Finset α} {t : Finset β} {a : α} {b c : β} @[simp] theorem mem_image : b ∈ s.image f ↔ ∃ a ∈ s, f a = b := by simp only [mem_def, image_val, mem_dedup, Multiset.mem_map, exists_prop] #align finset.mem_image Finset.mem_image theorem mem_image_of_mem (f : α → β) {a} (h : a ∈ s) : f a ∈ s.image f := mem_image.2 ⟨_, h, rfl⟩ #align finset.mem_image_of_mem Finset.mem_image_of_mem theorem forall_image {p : β → Prop} : (∀ b ∈ s.image f, p b) ↔ ∀ a ∈ s, p (f a) := by simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] #align finset.forall_image Finset.forall_image theorem map_eq_image (f : α ↪ β) (s : Finset α) : s.map f = s.image f := eq_of_veq (s.map f).2.dedup.symm #align finset.map_eq_image Finset.map_eq_image --@[simp] Porting note: removing simp, `simp` [Nonempty] can prove it theorem mem_image_const : c ∈ s.image (const α b) ↔ s.Nonempty ∧ b = c := by rw [mem_image] simp only [exists_prop, const_apply, exists_and_right] rfl #align finset.mem_image_const Finset.mem_image_const theorem mem_image_const_self : b ∈ s.image (const α b) ↔ s.Nonempty := mem_image_const.trans <\| and_iff_left rfl #align finset.mem_image_const_self Finset.mem_image_const_self instance canLift (c) (p) [CanLift β α c p] : CanLift (Finset β) (Finset α) (image c) fun s => ∀ x ∈ s, p x where prf := by rintro ⟨⟨l⟩, hd : l.Nodup⟩ hl lift l to List α using hl exact ⟨⟨l, hd.of_map _⟩, ext fun a => by simp⟩ #align finset.can_lift Finset.canLift theorem image_congr (h : (s : Set α).EqOn f g) : Finset.image f s = Finset.image g s := by ext simp_rw [mem_image, ← bex_def] exact exists₂_congr fun x hx => by rw [h hx] #align finset.image_congr Finset.image_congr	Mathlib/Data/Finset/Image.lean	398	402	theorem _root_.Function.Injective.mem_finset_image (hf : Injective f) : f a ∈ s.image f ↔ a ∈ s := by	refine ⟨fun h => ?_, Finset.mem_image_of_mem f⟩ obtain ⟨y, hy, heq⟩ := mem_image.1 h exact hf heq ▸ hy
/- Copyright (c) 2022 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Analysis.Calculus.FDeriv.Add import Mathlib.Analysis.Calculus.FDeriv.Equiv import Mathlib.Analysis.Calculus.FDeriv.Prod import Mathlib.Analysis.Calculus.Monotone import Mathlib.Data.Set.Function import Mathlib.Algebra.Group.Basic import Mathlib.Tactic.WLOG #align_import analysis.bounded_variation from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # Functions of bounded variation We study functions of bounded variation. In particular, we show that a bounded variation function is a difference of monotone functions, and differentiable almost everywhere. This implies that Lipschitz functions from the real line into finite-dimensional vector space are also differentiable almost everywhere. ## Main definitions and results * `eVariationOn f s` is the total variation of the function `f` on the set `s`, in `ℝ≥0∞`. * `BoundedVariationOn f s` registers that the variation of `f` on `s` is finite. * `LocallyBoundedVariationOn f s` registers that `f` has finite variation on any compact subinterval of `s`. * `variationOnFromTo f s a b` is the signed variation of `f` on `s ∩ Icc a b`, converted to `ℝ`. * `eVariationOn.Icc_add_Icc` states that the variation of `f` on `[a, c]` is the sum of its variations on `[a, b]` and `[b, c]`. * `LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn` proves that a function with locally bounded variation is the difference of two monotone functions. * `LipschitzWith.locallyBoundedVariationOn` shows that a Lipschitz function has locally bounded variation. * `LocallyBoundedVariationOn.ae_differentiableWithinAt` shows that a bounded variation function into a finite dimensional real vector space is differentiable almost everywhere. * `LipschitzOnWith.ae_differentiableWithinAt` is the same result for Lipschitz functions. We also give several variations around these results. ## Implementation We define the variation as an extended nonnegative real, to allow for infinite variation. This makes it possible to use the complete linear order structure of `ℝ≥0∞`. The proofs would be much more tedious with an `ℝ`-valued or `ℝ≥0`-valued variation, since one would always need to check that the sets one uses are nonempty and bounded above as these are only conditionally complete. -/ open scoped NNReal ENNReal Topology UniformConvergence open Set MeasureTheory Filter -- Porting note: sectioned variables because a `wlog` was broken due to extra variables in context variable {α : Type} [LinearOrder α] {E : Type} [PseudoEMetricSpace E] /-- The (extended real valued) variation of a function `f` on a set `s` inside a linear order is the supremum of the sum of `edist (f (u (i+1))) (f (u i))` over all finite increasing sequences `u` in `s`. -/ noncomputable def eVariationOn (f : α → E) (s : Set α) : ℝ≥0∞ := ⨆ p : ℕ × { u : ℕ → α // Monotone u ∧ ∀ i, u i ∈ s }, ∑ i ∈ Finset.range p.1, edist (f (p.2.1 (i + 1))) (f (p.2.1 i)) #align evariation_on eVariationOn /-- A function has bounded variation on a set `s` if its total variation there is finite. -/ def BoundedVariationOn (f : α → E) (s : Set α) := eVariationOn f s ≠ ∞ #align has_bounded_variation_on BoundedVariationOn /-- A function has locally bounded variation on a set `s` if, given any interval `[a, b]` with endpoints in `s`, then the function has finite variation on `s ∩ [a, b]`. -/ def LocallyBoundedVariationOn (f : α → E) (s : Set α) := ∀ a b, a ∈ s → b ∈ s → BoundedVariationOn f (s ∩ Icc a b) #align has_locally_bounded_variation_on LocallyBoundedVariationOn /-! ## Basic computations of variation -/ namespace eVariationOn theorem nonempty_monotone_mem {s : Set α} (hs : s.Nonempty) : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } := by obtain ⟨x, hx⟩ := hs exact ⟨⟨fun _ => x, fun i j _ => le_rfl, fun _ => hx⟩⟩ #align evariation_on.nonempty_monotone_mem eVariationOn.nonempty_monotone_mem theorem eq_of_edist_zero_on {f f' : α → E} {s : Set α} (h : ∀ ⦃x⦄, x ∈ s → edist (f x) (f' x) = 0) : eVariationOn f s = eVariationOn f' s := by dsimp only [eVariationOn] congr 1 with p : 1 congr 1 with i : 1 rw [edist_congr_right (h <\| p.snd.prop.2 (i + 1)), edist_congr_left (h <\| p.snd.prop.2 i)] #align evariation_on.eq_of_edist_zero_on eVariationOn.eq_of_edist_zero_on theorem eq_of_eqOn {f f' : α → E} {s : Set α} (h : EqOn f f' s) : eVariationOn f s = eVariationOn f' s := eq_of_edist_zero_on fun x xs => by rw [h xs, edist_self] #align evariation_on.eq_of_eq_on eVariationOn.eq_of_eqOn theorem sum_le (f : α → E) {s : Set α} (n : ℕ) {u : ℕ → α} (hu : Monotone u) (us : ∀ i, u i ∈ s) : (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := le_iSup_of_le ⟨n, u, hu, us⟩ le_rfl #align evariation_on.sum_le eVariationOn.sum_le theorem sum_le_of_monotoneOn_Icc (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α} (hu : MonotoneOn u (Icc m n)) (us : ∀ i ∈ Icc m n, u i ∈ s) : (∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := by rcases le_total n m with hnm \| hmn · simp [Finset.Ico_eq_empty_of_le hnm] let π := projIcc m n hmn let v i := u (π i) calc ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) = ∑ i ∈ Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) := Finset.sum_congr rfl fun i hi ↦ by rw [Finset.mem_Ico] at hi simp only [v, π, projIcc_of_mem hmn ⟨hi.1, hi.2.le⟩, projIcc_of_mem hmn ⟨hi.1.trans i.le_succ, hi.2⟩] _ ≤ ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i)) := Finset.sum_mono_set _ (Nat.Iio_eq_range ▸ Finset.Ico_subset_Iio_self) _ ≤ eVariationOn f s := sum_le _ _ (fun i j h ↦ hu (π i).2 (π j).2 (monotone_projIcc hmn h)) fun i ↦ us _ (π i).2 #align evariation_on.sum_le_of_monotone_on_Icc eVariationOn.sum_le_of_monotoneOn_Icc theorem sum_le_of_monotoneOn_Iic (f : α → E) {s : Set α} {n : ℕ} {u : ℕ → α} (hu : MonotoneOn u (Iic n)) (us : ∀ i ≤ n, u i ∈ s) : (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := by simpa using sum_le_of_monotoneOn_Icc f (m := 0) (hu.mono Icc_subset_Iic_self) fun i hi ↦ us i hi.2 #align evariation_on.sum_le_of_monotone_on_Iic eVariationOn.sum_le_of_monotoneOn_Iic theorem mono (f : α → E) {s t : Set α} (hst : t ⊆ s) : eVariationOn f t ≤ eVariationOn f s := by apply iSup_le _ rintro ⟨n, ⟨u, hu, ut⟩⟩ exact sum_le f n hu fun i => hst (ut i) #align evariation_on.mono eVariationOn.mono theorem _root_.BoundedVariationOn.mono {f : α → E} {s : Set α} (h : BoundedVariationOn f s) {t : Set α} (ht : t ⊆ s) : BoundedVariationOn f t := ne_top_of_le_ne_top h (eVariationOn.mono f ht) #align has_bounded_variation_on.mono BoundedVariationOn.mono theorem _root_.BoundedVariationOn.locallyBoundedVariationOn {f : α → E} {s : Set α} (h : BoundedVariationOn f s) : LocallyBoundedVariationOn f s := fun _ _ _ _ => h.mono inter_subset_left #align has_bounded_variation_on.has_locally_bounded_variation_on BoundedVariationOn.locallyBoundedVariationOn	Mathlib/Analysis/BoundedVariation.lean	149	161	theorem edist_le (f : α → E) {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) : edist (f x) (f y) ≤ eVariationOn f s := by	wlog hxy : y ≤ x generalizing x y · rw [edist_comm] exact this hy hx (le_of_not_le hxy) let u : ℕ → α := fun n => if n = 0 then y else x have hu : Monotone u := monotone_nat_of_le_succ fun \| 0 => hxy \| (_ + 1) => le_rfl have us : ∀ i, u i ∈ s := fun \| 0 => hy \| (_ + 1) => hx simpa only [Finset.sum_range_one] using sum_le f 1 hu us
/- Copyright (c) 2016 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro -/ import Batteries.Tactic.Alias import Batteries.Data.Nat.Basic /-! # Basic lemmas about natural numbers The primary purpose of the lemmas in this file is to assist with reasoning about sizes of objects, array indices and such. For a more thorough development of the theory of natural numbers, we recommend using Mathlib. -/ namespace Nat /-! ### rec/cases -/ @[simp] theorem recAux_zero {motive : Nat → Sort _} (zero : motive 0) (succ : ∀ n, motive n → motive (n+1)) : Nat.recAux zero succ 0 = zero := rfl theorem recAux_succ {motive : Nat → Sort _} (zero : motive 0) (succ : ∀ n, motive n → motive (n+1)) (n) : Nat.recAux zero succ (n+1) = succ n (Nat.recAux zero succ n) := rfl @[simp] theorem recAuxOn_zero {motive : Nat → Sort _} (zero : motive 0) (succ : ∀ n, motive n → motive (n+1)) : Nat.recAuxOn 0 zero succ = zero := rfl theorem recAuxOn_succ {motive : Nat → Sort _} (zero : motive 0) (succ : ∀ n, motive n → motive (n+1)) (n) : Nat.recAuxOn (n+1) zero succ = succ n (Nat.recAuxOn n zero succ) := rfl @[simp] theorem casesAuxOn_zero {motive : Nat → Sort _} (zero : motive 0) (succ : ∀ n, motive (n+1)) : Nat.casesAuxOn 0 zero succ = zero := rfl theorem casesAuxOn_succ {motive : Nat → Sort _} (zero : motive 0) (succ : ∀ n, motive (n+1)) (n) : Nat.casesAuxOn (n+1) zero succ = succ n := rfl theorem strongRec_eq {motive : Nat → Sort _} (ind : ∀ n, (∀ m, m < n → motive m) → motive n) (t : Nat) : Nat.strongRec ind t = ind t fun m _ => Nat.strongRec ind m := by conv => lhs; unfold Nat.strongRec theorem strongRecOn_eq {motive : Nat → Sort _} (ind : ∀ n, (∀ m, m < n → motive m) → motive n) (t : Nat) : Nat.strongRecOn t ind = ind t fun m _ => Nat.strongRecOn m ind := Nat.strongRec_eq .. @[simp] theorem recDiagAux_zero_left {motive : Nat → Nat → Sort _} (zero_left : ∀ n, motive 0 n) (zero_right : ∀ m, motive m 0) (succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (n) : Nat.recDiagAux zero_left zero_right succ_succ 0 n = zero_left n := by cases n <;> rfl @[simp] theorem recDiagAux_zero_right {motive : Nat → Nat → Sort _} (zero_left : ∀ n, motive 0 n) (zero_right : ∀ m, motive m 0) (succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (m) (h : zero_left 0 = zero_right 0 := by first \| assumption \| trivial) : Nat.recDiagAux zero_left zero_right succ_succ m 0 = zero_right m := by cases m; exact h; rfl theorem recDiagAux_succ_succ {motive : Nat → Nat → Sort _} (zero_left : ∀ n, motive 0 n) (zero_right : ∀ m, motive m 0) (succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (m n) : Nat.recDiagAux zero_left zero_right succ_succ (m+1) (n+1) = succ_succ m n (Nat.recDiagAux zero_left zero_right succ_succ m n) := rfl @[simp] theorem recDiag_zero_zero {motive : Nat → Nat → Sort _} (zero_zero : motive 0 0) (zero_succ : ∀ n, motive 0 n → motive 0 (n+1)) (succ_zero : ∀ m, motive m 0 → motive (m+1) 0) (succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) : Nat.recDiag (motive:=motive) zero_zero zero_succ succ_zero succ_succ 0 0 = zero_zero := rfl	.lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean	74	79	theorem recDiag_zero_succ {motive : Nat → Nat → Sort _} (zero_zero : motive 0 0) (zero_succ : ∀ n, motive 0 n → motive 0 (n+1)) (succ_zero : ∀ m, motive m 0 → motive (m+1) 0) (succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (n) : Nat.recDiag zero_zero zero_succ succ_zero succ_succ 0 (n+1) = zero_succ n (Nat.recDiag zero_zero zero_succ succ_zero succ_succ 0 n) := by	simp [Nat.recDiag]; rfl
/- Copyright (c) 2020 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen -/ import Mathlib.Algebra.BigOperators.Fin import Mathlib.LinearAlgebra.Finsupp import Mathlib.LinearAlgebra.Prod import Mathlib.SetTheory.Cardinal.Basic import Mathlib.Tactic.FinCases import Mathlib.Tactic.LinearCombination import Mathlib.Lean.Expr.ExtraRecognizers import Mathlib.Data.Set.Subsingleton #align_import linear_algebra.linear_independent from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb" /-! # Linear independence This file defines linear independence in a module or vector space. It is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light. We define `LinearIndependent R v` as `ker (Finsupp.total ι M R v) = ⊥`. Here `Finsupp.total` is the linear map sending a function `f : ι →₀ R` with finite support to the linear combination of vectors from `v` with these coefficients. Then we prove that several other statements are equivalent to this one, including injectivity of `Finsupp.total ι M R v` and some versions with explicitly written linear combinations. ## Main definitions All definitions are given for families of vectors, i.e. `v : ι → M` where `M` is the module or vector space and `ι : Type` is an arbitrary indexing type. `LinearIndependent R v` states that the elements of the family `v` are linearly independent. * `LinearIndependent.repr hv x` returns the linear combination representing `x : span R (range v)` on the linearly independent vectors `v`, given `hv : LinearIndependent R v` (using classical choice). `LinearIndependent.repr hv` is provided as a linear map. ## Main statements We prove several specialized tests for linear independence of families of vectors and of sets of vectors. * `Fintype.linearIndependent_iff`: if `ι` is a finite type, then any function `f : ι → R` has finite support, so we can reformulate the statement using `∑ i : ι, f i • v i` instead of a sum over an auxiliary `s : Finset ι`; * `linearIndependent_empty_type`: a family indexed by an empty type is linearly independent; * `linearIndependent_unique_iff`: if `ι` is a singleton, then `LinearIndependent K v` is equivalent to `v default ≠ 0`; * `linearIndependent_option`, `linearIndependent_sum`, `linearIndependent_fin_cons`, `linearIndependent_fin_succ`: type-specific tests for linear independence of families of vector fields; * `linearIndependent_insert`, `linearIndependent_union`, `linearIndependent_pair`, `linearIndependent_singleton`: linear independence tests for set operations. In many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to make the linear independence tests usable as `hv.insert ha` etc. We also prove that, when working over a division ring, any family of vectors includes a linear independent subfamily spanning the same subspace. ## Implementation notes We use families instead of sets because it allows us to say that two identical vectors are linearly dependent. If you want to use sets, use the family `(fun x ↦ x : s → M)` given a set `s : Set M`. The lemmas `LinearIndependent.to_subtype_range` and `LinearIndependent.of_subtype_range` connect those two worlds. ## Tags linearly dependent, linear dependence, linearly independent, linear independence -/ noncomputable section open Function Set Submodule open Cardinal universe u' u variable {ι : Type u'} {ι' : Type} {R : Type} {K : Type} variable {M : Type} {M' M'' : Type} {V : Type u} {V' : Type} section Module variable {v : ι → M} variable [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid M''] variable [Module R M] [Module R M'] [Module R M''] variable {a b : R} {x y : M} variable (R) (v) /-- `LinearIndependent R v` states the family of vectors `v` is linearly independent over `R`. -/ def LinearIndependent : Prop := LinearMap.ker (Finsupp.total ι M R v) = ⊥ #align linear_independent LinearIndependent open Lean PrettyPrinter.Delaborator SubExpr in /-- Delaborator for `LinearIndependent` that suggests pretty printing with type hints in case the family of vectors is over a `Set`. Type hints look like `LinearIndependent fun (v : ↑s) => ↑v` or `LinearIndependent (ι := ↑s) f`, depending on whether the family is a lambda expression or not. -/ @[delab app.LinearIndependent] def delabLinearIndependent : Delab := whenPPOption getPPNotation <\| whenNotPPOption getPPAnalysisSkip <\| withOptionAtCurrPos `pp.analysis.skip true do let e ← getExpr guard <\| e.isAppOfArity ``LinearIndependent 7 let some _ := (e.getArg! 0).coeTypeSet? \| failure let optionsPerPos ← if (e.getArg! 3).isLambda then withNaryArg 3 do return (← read).optionsPerPos.setBool (← getPos) pp.funBinderTypes.name true else withNaryArg 0 do return (← read).optionsPerPos.setBool (← getPos) `pp.analysis.namedArg true withTheReader Context ({· with optionsPerPos}) delab variable {R} {v} theorem linearIndependent_iff : LinearIndependent R v ↔ ∀ l, Finsupp.total ι M R v l = 0 → l = 0 := by simp [LinearIndependent, LinearMap.ker_eq_bot'] #align linear_independent_iff linearIndependent_iff theorem linearIndependent_iff' : LinearIndependent R v ↔ ∀ s : Finset ι, ∀ g : ι → R, ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0 := linearIndependent_iff.trans ⟨fun hf s g hg i his => have h := hf (∑ i ∈ s, Finsupp.single i (g i)) <\| by simpa only [map_sum, Finsupp.total_single] using hg calc g i = (Finsupp.lapply i : (ι →₀ R) →ₗ[R] R) (Finsupp.single i (g i)) := by { rw [Finsupp.lapply_apply, Finsupp.single_eq_same] } _ = ∑ j ∈ s, (Finsupp.lapply i : (ι →₀ R) →ₗ[R] R) (Finsupp.single j (g j)) := Eq.symm <\| Finset.sum_eq_single i (fun j _hjs hji => by rw [Finsupp.lapply_apply, Finsupp.single_eq_of_ne hji]) fun hnis => hnis.elim his _ = (∑ j ∈ s, Finsupp.single j (g j)) i := (map_sum ..).symm _ = 0 := DFunLike.ext_iff.1 h i, fun hf l hl => Finsupp.ext fun i => _root_.by_contradiction fun hni => hni <\| hf _ _ hl _ <\| Finsupp.mem_support_iff.2 hni⟩ #align linear_independent_iff' linearIndependent_iff' theorem linearIndependent_iff'' : LinearIndependent R v ↔ ∀ (s : Finset ι) (g : ι → R), (∀ i ∉ s, g i = 0) → ∑ i ∈ s, g i • v i = 0 → ∀ i, g i = 0 := by classical exact linearIndependent_iff'.trans ⟨fun H s g hg hv i => if his : i ∈ s then H s g hv i his else hg i his, fun H s g hg i hi => by convert H s (fun j => if j ∈ s then g j else 0) (fun j hj => if_neg hj) (by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]) i exact (if_pos hi).symm⟩ #align linear_independent_iff'' linearIndependent_iff'' theorem not_linearIndependent_iff : ¬LinearIndependent R v ↔ ∃ s : Finset ι, ∃ g : ι → R, ∑ i ∈ s, g i • v i = 0 ∧ ∃ i ∈ s, g i ≠ 0 := by rw [linearIndependent_iff'] simp only [exists_prop, not_forall] #align not_linear_independent_iff not_linearIndependent_iff theorem Fintype.linearIndependent_iff [Fintype ι] : LinearIndependent R v ↔ ∀ g : ι → R, ∑ i, g i • v i = 0 → ∀ i, g i = 0 := by refine ⟨fun H g => by simpa using linearIndependent_iff'.1 H Finset.univ g, fun H => linearIndependent_iff''.2 fun s g hg hs i => H _ ?_ _⟩ rw [← hs] refine (Finset.sum_subset (Finset.subset_univ _) fun i _ hi => ?_).symm rw [hg i hi, zero_smul] #align fintype.linear_independent_iff Fintype.linearIndependent_iff /-- A finite family of vectors `v i` is linear independent iff the linear map that sends `c : ι → R` to `∑ i, c i • v i` has the trivial kernel. -/ theorem Fintype.linearIndependent_iff' [Fintype ι] [DecidableEq ι] : LinearIndependent R v ↔ LinearMap.ker (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) = ⊥ := by simp [Fintype.linearIndependent_iff, LinearMap.ker_eq_bot', funext_iff] #align fintype.linear_independent_iff' Fintype.linearIndependent_iff' theorem Fintype.not_linearIndependent_iff [Fintype ι] : ¬LinearIndependent R v ↔ ∃ g : ι → R, ∑ i, g i • v i = 0 ∧ ∃ i, g i ≠ 0 := by simpa using not_iff_not.2 Fintype.linearIndependent_iff #align fintype.not_linear_independent_iff Fintype.not_linearIndependent_iff theorem linearIndependent_empty_type [IsEmpty ι] : LinearIndependent R v := linearIndependent_iff.mpr fun v _hv => Subsingleton.elim v 0 #align linear_independent_empty_type linearIndependent_empty_type theorem LinearIndependent.ne_zero [Nontrivial R] (i : ι) (hv : LinearIndependent R v) : v i ≠ 0 := fun h => zero_ne_one' R <\| Eq.symm (by suffices (Finsupp.single i 1 : ι →₀ R) i = 0 by simpa rw [linearIndependent_iff.1 hv (Finsupp.single i 1)] · simp · simp [h]) #align linear_independent.ne_zero LinearIndependent.ne_zero lemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y]) {s t : R} (h' : s • x + t • y = 0) : s = 0 ∧ t = 0 := by have := linearIndependent_iff'.1 h Finset.univ ![s, t] simp only [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons, h', Finset.mem_univ, forall_true_left] at this exact ⟨this 0, this 1⟩ /-- Also see `LinearIndependent.pair_iff'` for a simpler version over fields. -/ lemma LinearIndependent.pair_iff {x y : M} : LinearIndependent R ![x, y] ↔ ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0 := by refine ⟨fun h s t hst ↦ h.eq_zero_of_pair hst, fun h ↦ ?_⟩ apply Fintype.linearIndependent_iff.2 intro g hg simp only [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons] at hg intro i fin_cases i exacts [(h _ _ hg).1, (h _ _ hg).2] /-- A subfamily of a linearly independent family (i.e., a composition with an injective map) is a linearly independent family. -/ theorem LinearIndependent.comp (h : LinearIndependent R v) (f : ι' → ι) (hf : Injective f) : LinearIndependent R (v ∘ f) := by rw [linearIndependent_iff, Finsupp.total_comp] intro l hl have h_map_domain : ∀ x, (Finsupp.mapDomain f l) (f x) = 0 := by rw [linearIndependent_iff.1 h (Finsupp.mapDomain f l) hl]; simp ext x convert h_map_domain x rw [Finsupp.mapDomain_apply hf] #align linear_independent.comp LinearIndependent.comp /-- A family is linearly independent if and only if all of its finite subfamily is linearly independent. -/ theorem linearIndependent_iff_finset_linearIndependent : LinearIndependent R v ↔ ∀ (s : Finset ι), LinearIndependent R (v ∘ (Subtype.val : s → ι)) := ⟨fun H _ ↦ H.comp _ Subtype.val_injective, fun H ↦ linearIndependent_iff'.2 fun s g hg i hi ↦ Fintype.linearIndependent_iff.1 (H s) (g ∘ Subtype.val) (hg ▸ Finset.sum_attach s fun j ↦ g j • v j) ⟨i, hi⟩⟩ theorem LinearIndependent.coe_range (i : LinearIndependent R v) : LinearIndependent R ((↑) : range v → M) := by simpa using i.comp _ (rangeSplitting_injective v) #align linear_independent.coe_range LinearIndependent.coe_range /-- If `v` is a linearly independent family of vectors and the kernel of a linear map `f` is disjoint with the submodule spanned by the vectors of `v`, then `f ∘ v` is a linearly independent family of vectors. See also `LinearIndependent.map'` for a special case assuming `ker f = ⊥`. -/ theorem LinearIndependent.map (hv : LinearIndependent R v) {f : M →ₗ[R] M'} (hf_inj : Disjoint (span R (range v)) (LinearMap.ker f)) : LinearIndependent R (f ∘ v) := by rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_total, map_inf_eq_map_inf_comap, map_le_iff_le_comap, comap_bot, Finsupp.supported_univ, top_inf_eq] at hf_inj unfold LinearIndependent at hv ⊢ rw [hv, le_bot_iff] at hf_inj haveI : Inhabited M := ⟨0⟩ rw [Finsupp.total_comp, Finsupp.lmapDomain_total _ _ f, LinearMap.ker_comp, hf_inj] exact fun _ => rfl #align linear_independent.map LinearIndependent.map /-- If `v` is an injective family of vectors such that `f ∘ v` is linearly independent, then `v` spans a submodule disjoint from the kernel of `f` -/ theorem Submodule.range_ker_disjoint {f : M →ₗ[R] M'} (hv : LinearIndependent R (f ∘ v)) : Disjoint (span R (range v)) (LinearMap.ker f) := by rw [LinearIndependent, Finsupp.total_comp, Finsupp.lmapDomain_total R _ f (fun _ ↦ rfl), LinearMap.ker_comp] at hv rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_total, map_inf_eq_map_inf_comap, hv, inf_bot_eq, map_bot] /-- An injective linear map sends linearly independent families of vectors to linearly independent families of vectors. See also `LinearIndependent.map` for a more general statement. -/ theorem LinearIndependent.map' (hv : LinearIndependent R v) (f : M →ₗ[R] M') (hf_inj : LinearMap.ker f = ⊥) : LinearIndependent R (f ∘ v) := hv.map <\| by simp [hf_inj] #align linear_independent.map' LinearIndependent.map' /-- If `M / R` and `M' / R'` are modules, `i : R' → R` is a map, `j : M →+ M'` is a monoid map, such that they send non-zero elements to non-zero elements, and compatible with the scalar multiplications on `M` and `M'`, then `j` sends linearly independent families of vectors to linearly independent families of vectors. As a special case, taking `R = R'` it is `LinearIndependent.map'`. -/ theorem LinearIndependent.map_of_injective_injective {R' : Type} {M' : Type} [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v) (i : R' → R) (j : M →+ M') (hi : ∀ r, i r = 0 → r = 0) (hj : ∀ m, j m = 0 → m = 0) (hc : ∀ (r : R') (m : M), j (i r • m) = r • j m) : LinearIndependent R' (j ∘ v) := by rw [linearIndependent_iff'] at hv ⊢ intro S r' H s hs simp_rw [comp_apply, ← hc, ← map_sum] at H exact hi _ <\| hv _ _ (hj _ H) s hs /-- If `M / R` and `M' / R'` are modules, `i : R → R'` is a surjective map which maps zero to zero, `j : M →+ M'` is a monoid map which sends non-zero elements to non-zero elements, such that the scalar multiplications on `M` and `M'` are compatible, then `j` sends linearly independent families of vectors to linearly independent families of vectors. As a special case, taking `R = R'` it is `LinearIndependent.map'`. -/ theorem LinearIndependent.map_of_surjective_injective {R' : Type} {M' : Type} [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v) (i : ZeroHom R R') (j : M →+ M') (hi : Surjective i) (hj : ∀ m, j m = 0 → m = 0) (hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) : LinearIndependent R' (j ∘ v) := by obtain ⟨i', hi'⟩ := hi.hasRightInverse refine hv.map_of_injective_injective i' j (fun _ h ↦ ?_) hj fun r m ↦ ?_ · apply_fun i at h rwa [hi', i.map_zero] at h rw [hc (i' r) m, hi'] /-- If the image of a family of vectors under a linear map is linearly independent, then so is the original family. -/ theorem LinearIndependent.of_comp (f : M →ₗ[R] M') (hfv : LinearIndependent R (f ∘ v)) : LinearIndependent R v := linearIndependent_iff'.2 fun s g hg i his => have : (∑ i ∈ s, g i • f (v i)) = 0 := by simp_rw [← map_smul, ← map_sum, hg, f.map_zero] linearIndependent_iff'.1 hfv s g this i his #align linear_independent.of_comp LinearIndependent.of_comp /-- If `f` is an injective linear map, then the family `f ∘ v` is linearly independent if and only if the family `v` is linearly independent. -/ protected theorem LinearMap.linearIndependent_iff (f : M →ₗ[R] M') (hf_inj : LinearMap.ker f = ⊥) : LinearIndependent R (f ∘ v) ↔ LinearIndependent R v := ⟨fun h => h.of_comp f, fun h => h.map <\| by simp only [hf_inj, disjoint_bot_right]⟩ #align linear_map.linear_independent_iff LinearMap.linearIndependent_iff @[nontriviality] theorem linearIndependent_of_subsingleton [Subsingleton R] : LinearIndependent R v := linearIndependent_iff.2 fun _l _hl => Subsingleton.elim _ _ #align linear_independent_of_subsingleton linearIndependent_of_subsingleton theorem linearIndependent_equiv (e : ι ≃ ι') {f : ι' → M} : LinearIndependent R (f ∘ e) ↔ LinearIndependent R f := ⟨fun h => Function.comp_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective, fun h => h.comp _ e.injective⟩ #align linear_independent_equiv linearIndependent_equiv theorem linearIndependent_equiv' (e : ι ≃ ι') {f : ι' → M} {g : ι → M} (h : f ∘ e = g) : LinearIndependent R g ↔ LinearIndependent R f := h ▸ linearIndependent_equiv e #align linear_independent_equiv' linearIndependent_equiv' theorem linearIndependent_subtype_range {ι} {f : ι → M} (hf : Injective f) : LinearIndependent R ((↑) : range f → M) ↔ LinearIndependent R f := Iff.symm <\| linearIndependent_equiv' (Equiv.ofInjective f hf) rfl #align linear_independent_subtype_range linearIndependent_subtype_range alias ⟨LinearIndependent.of_subtype_range, _⟩ := linearIndependent_subtype_range #align linear_independent.of_subtype_range LinearIndependent.of_subtype_range theorem linearIndependent_image {ι} {s : Set ι} {f : ι → M} (hf : Set.InjOn f s) : (LinearIndependent R fun x : s => f x) ↔ LinearIndependent R fun x : f '' s => (x : M) := linearIndependent_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl #align linear_independent_image linearIndependent_image theorem linearIndependent_span (hs : LinearIndependent R v) : LinearIndependent R (M := span R (range v)) (fun i : ι => ⟨v i, subset_span (mem_range_self i)⟩) := LinearIndependent.of_comp (span R (range v)).subtype hs #align linear_independent_span linearIndependent_span /-- See `LinearIndependent.fin_cons` for a family of elements in a vector space. -/ theorem LinearIndependent.fin_cons' {m : ℕ} (x : M) (v : Fin m → M) (hli : LinearIndependent R v) (x_ortho : ∀ (c : R) (y : Submodule.span R (Set.range v)), c • x + y = (0 : M) → c = 0) : LinearIndependent R (Fin.cons x v : Fin m.succ → M) := by rw [Fintype.linearIndependent_iff] at hli ⊢ rintro g total_eq j simp_rw [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ] at total_eq have : g 0 = 0 := by refine x_ortho (g 0) ⟨∑ i : Fin m, g i.succ • v i, ?_⟩ total_eq exact sum_mem fun i _ => smul_mem _ _ (subset_span ⟨i, rfl⟩) rw [this, zero_smul, zero_add] at total_eq exact Fin.cases this (hli _ total_eq) j #align linear_independent.fin_cons' LinearIndependent.fin_cons' /-- A set of linearly independent vectors in a module `M` over a semiring `K` is also linearly independent over a subring `R` of `K`. The implementation uses minimal assumptions about the relationship between `R`, `K` and `M`. The version where `K` is an `R`-algebra is `LinearIndependent.restrict_scalars_algebras`. -/ theorem LinearIndependent.restrict_scalars [Semiring K] [SMulWithZero R K] [Module K M] [IsScalarTower R K M] (hinj : Function.Injective fun r : R => r • (1 : K)) (li : LinearIndependent K v) : LinearIndependent R v := by refine linearIndependent_iff'.mpr fun s g hg i hi => hinj ?_ dsimp only; rw [zero_smul] refine (linearIndependent_iff'.mp li : _) _ (g · • (1:K)) ?_ i hi simp_rw [smul_assoc, one_smul] exact hg #align linear_independent.restrict_scalars LinearIndependent.restrict_scalars /-- Every finite subset of a linearly independent set is linearly independent. -/ theorem linearIndependent_finset_map_embedding_subtype (s : Set M) (li : LinearIndependent R ((↑) : s → M)) (t : Finset s) : LinearIndependent R ((↑) : Finset.map (Embedding.subtype s) t → M) := by let f : t.map (Embedding.subtype s) → s := fun x => ⟨x.1, by obtain ⟨x, h⟩ := x rw [Finset.mem_map] at h obtain ⟨a, _ha, rfl⟩ := h simp only [Subtype.coe_prop, Embedding.coe_subtype]⟩ convert LinearIndependent.comp li f ?_ rintro ⟨x, hx⟩ ⟨y, hy⟩ rw [Finset.mem_map] at hx hy obtain ⟨a, _ha, rfl⟩ := hx obtain ⟨b, _hb, rfl⟩ := hy simp only [f, imp_self, Subtype.mk_eq_mk] #align linear_independent_finset_map_embedding_subtype linearIndependent_finset_map_embedding_subtype /-- If every finite set of linearly independent vectors has cardinality at most `n`, then the same is true for arbitrary sets of linearly independent vectors. -/ theorem linearIndependent_bounded_of_finset_linearIndependent_bounded {n : ℕ} (H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) : ∀ s : Set M, LinearIndependent R ((↑) : s → M) → #s ≤ n := by intro s li apply Cardinal.card_le_of intro t rw [← Finset.card_map (Embedding.subtype s)] apply H apply linearIndependent_finset_map_embedding_subtype _ li #align linear_independent_bounded_of_finset_linear_independent_bounded linearIndependent_bounded_of_finset_linearIndependent_bounded section Subtype /-! The following lemmas use the subtype defined by a set in `M` as the index set `ι`. -/ theorem linearIndependent_comp_subtype {s : Set ι} : LinearIndependent R (v ∘ (↑) : s → M) ↔ ∀ l ∈ Finsupp.supported R R s, (Finsupp.total ι M R v) l = 0 → l = 0 := by simp only [linearIndependent_iff, (· ∘ ·), Finsupp.mem_supported, Finsupp.total_apply, Set.subset_def, Finset.mem_coe] constructor · intro h l hl₁ hl₂ have := h (l.subtypeDomain s) ((Finsupp.sum_subtypeDomain_index hl₁).trans hl₂) exact (Finsupp.subtypeDomain_eq_zero_iff hl₁).1 this · intro h l hl refine Finsupp.embDomain_eq_zero.1 (h (l.embDomain <\| Function.Embedding.subtype s) ?_ ?_) · suffices ∀ i hi, ¬l ⟨i, hi⟩ = 0 → i ∈ s by simpa intros assumption · rwa [Finsupp.embDomain_eq_mapDomain, Finsupp.sum_mapDomain_index] exacts [fun _ => zero_smul _ _, fun _ _ _ => add_smul _ _ _] #align linear_independent_comp_subtype linearIndependent_comp_subtype theorem linearDependent_comp_subtype' {s : Set ι} : ¬LinearIndependent R (v ∘ (↑) : s → M) ↔ ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ Finsupp.total ι M R v f = 0 ∧ f ≠ 0 := by simp [linearIndependent_comp_subtype, and_left_comm] #align linear_dependent_comp_subtype' linearDependent_comp_subtype' /-- A version of `linearDependent_comp_subtype'` with `Finsupp.total` unfolded. -/ theorem linearDependent_comp_subtype {s : Set ι} : ¬LinearIndependent R (v ∘ (↑) : s → M) ↔ ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ ∑ i ∈ f.support, f i • v i = 0 ∧ f ≠ 0 := linearDependent_comp_subtype' #align linear_dependent_comp_subtype linearDependent_comp_subtype theorem linearIndependent_subtype {s : Set M} : LinearIndependent R (fun x => x : s → M) ↔ ∀ l ∈ Finsupp.supported R R s, (Finsupp.total M M R id) l = 0 → l = 0 := by apply linearIndependent_comp_subtype (v := id) #align linear_independent_subtype linearIndependent_subtype theorem linearIndependent_comp_subtype_disjoint {s : Set ι} : LinearIndependent R (v ∘ (↑) : s → M) ↔ Disjoint (Finsupp.supported R R s) (LinearMap.ker <\| Finsupp.total ι M R v) := by rw [linearIndependent_comp_subtype, LinearMap.disjoint_ker] #align linear_independent_comp_subtype_disjoint linearIndependent_comp_subtype_disjoint theorem linearIndependent_subtype_disjoint {s : Set M} : LinearIndependent R (fun x => x : s → M) ↔ Disjoint (Finsupp.supported R R s) (LinearMap.ker <\| Finsupp.total M M R id) := by apply linearIndependent_comp_subtype_disjoint (v := id) #align linear_independent_subtype_disjoint linearIndependent_subtype_disjoint theorem linearIndependent_iff_totalOn {s : Set M} : LinearIndependent R (fun x => x : s → M) ↔ (LinearMap.ker <\| Finsupp.totalOn M M R id s) = ⊥ := by rw [Finsupp.totalOn, LinearMap.ker, LinearMap.comap_codRestrict, Submodule.map_bot, comap_bot, LinearMap.ker_comp, linearIndependent_subtype_disjoint, disjoint_iff_inf_le, ← map_comap_subtype, map_le_iff_le_comap, comap_bot, ker_subtype, le_bot_iff] #align linear_independent_iff_total_on linearIndependent_iff_totalOn theorem LinearIndependent.restrict_of_comp_subtype {s : Set ι} (hs : LinearIndependent R (v ∘ (↑) : s → M)) : LinearIndependent R (s.restrict v) := hs #align linear_independent.restrict_of_comp_subtype LinearIndependent.restrict_of_comp_subtype variable (R M) theorem linearIndependent_empty : LinearIndependent R (fun x => x : (∅ : Set M) → M) := by simp [linearIndependent_subtype_disjoint] #align linear_independent_empty linearIndependent_empty variable {R M} theorem LinearIndependent.mono {t s : Set M} (h : t ⊆ s) : LinearIndependent R (fun x => x : s → M) → LinearIndependent R (fun x => x : t → M) := by simp only [linearIndependent_subtype_disjoint] exact Disjoint.mono_left (Finsupp.supported_mono h) #align linear_independent.mono LinearIndependent.mono theorem linearIndependent_of_finite (s : Set M) (H : ∀ t ⊆ s, Set.Finite t → LinearIndependent R (fun x => x : t → M)) : LinearIndependent R (fun x => x : s → M) := linearIndependent_subtype.2 fun l hl => linearIndependent_subtype.1 (H _ hl (Finset.finite_toSet _)) l (Subset.refl _) #align linear_independent_of_finite linearIndependent_of_finite theorem linearIndependent_iUnion_of_directed {η : Type} {s : η → Set M} (hs : Directed (· ⊆ ·) s) (h : ∀ i, LinearIndependent R (fun x => x : s i → M)) : LinearIndependent R (fun x => x : (⋃ i, s i) → M) := by by_cases hη : Nonempty η · refine linearIndependent_of_finite (⋃ i, s i) fun t ht ft => ?_ rcases finite_subset_iUnion ft ht with ⟨I, fi, hI⟩ rcases hs.finset_le fi.toFinset with ⟨i, hi⟩ exact (h i).mono (Subset.trans hI <\| iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj)) · refine (linearIndependent_empty R M).mono (t := iUnion (s ·)) ?_ rintro _ ⟨_, ⟨i, _⟩, _⟩ exact hη ⟨i⟩ #align linear_independent_Union_of_directed linearIndependent_iUnion_of_directed theorem linearIndependent_sUnion_of_directed {s : Set (Set M)} (hs : DirectedOn (· ⊆ ·) s) (h : ∀ a ∈ s, LinearIndependent R ((↑) : ((a : Set M) : Type _) → M)) : LinearIndependent R (fun x => x : ⋃₀ s → M) := by rw [sUnion_eq_iUnion]; exact linearIndependent_iUnion_of_directed hs.directed_val (by simpa using h) #align linear_independent_sUnion_of_directed linearIndependent_sUnion_of_directed theorem linearIndependent_biUnion_of_directed {η} {s : Set η} {t : η → Set M} (hs : DirectedOn (t ⁻¹'o (· ⊆ ·)) s) (h : ∀ a ∈ s, LinearIndependent R (fun x => x : t a → M)) : LinearIndependent R (fun x => x : (⋃ a ∈ s, t a) → M) := by rw [biUnion_eq_iUnion] exact linearIndependent_iUnion_of_directed (directed_comp.2 <\| hs.directed_val) (by simpa using h) #align linear_independent_bUnion_of_directed linearIndependent_biUnion_of_directed end Subtype end Module /-! ### Properties which require `Ring R` -/ section Module variable {v : ι → M} variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M''] variable [Module R M] [Module R M'] [Module R M''] variable {a b : R} {x y : M} theorem linearIndependent_iff_injective_total : LinearIndependent R v ↔ Function.Injective (Finsupp.total ι M R v) := linearIndependent_iff.trans (injective_iff_map_eq_zero (Finsupp.total ι M R v).toAddMonoidHom).symm #align linear_independent_iff_injective_total linearIndependent_iff_injective_total alias ⟨LinearIndependent.injective_total, _⟩ := linearIndependent_iff_injective_total #align linear_independent.injective_total LinearIndependent.injective_total theorem LinearIndependent.injective [Nontrivial R] (hv : LinearIndependent R v) : Injective v := by intro i j hij let l : ι →₀ R := Finsupp.single i (1 : R) - Finsupp.single j 1 have h_total : Finsupp.total ι M R v l = 0 := by simp_rw [l, LinearMap.map_sub, Finsupp.total_apply] simp [hij] have h_single_eq : Finsupp.single i (1 : R) = Finsupp.single j 1 := by rw [linearIndependent_iff] at hv simp [eq_add_of_sub_eq' (hv l h_total)] simpa [Finsupp.single_eq_single_iff] using h_single_eq #align linear_independent.injective LinearIndependent.injective theorem LinearIndependent.to_subtype_range {ι} {f : ι → M} (hf : LinearIndependent R f) : LinearIndependent R ((↑) : range f → M) := by nontriviality R exact (linearIndependent_subtype_range hf.injective).2 hf #align linear_independent.to_subtype_range LinearIndependent.to_subtype_range theorem LinearIndependent.to_subtype_range' {ι} {f : ι → M} (hf : LinearIndependent R f) {t} (ht : range f = t) : LinearIndependent R ((↑) : t → M) := ht ▸ hf.to_subtype_range #align linear_independent.to_subtype_range' LinearIndependent.to_subtype_range' theorem LinearIndependent.image_of_comp {ι ι'} (s : Set ι) (f : ι → ι') (g : ι' → M) (hs : LinearIndependent R fun x : s => g (f x)) : LinearIndependent R fun x : f '' s => g x := by nontriviality R have : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp exact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs #align linear_independent.image_of_comp LinearIndependent.image_of_comp theorem LinearIndependent.image {ι} {s : Set ι} {f : ι → M} (hs : LinearIndependent R fun x : s => f x) : LinearIndependent R fun x : f '' s => (x : M) := by convert LinearIndependent.image_of_comp s f id hs #align linear_independent.image LinearIndependent.image theorem LinearIndependent.group_smul {G : Type} [hG : Group G] [DistribMulAction G R] [DistribMulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : ι → M} (hv : LinearIndependent R v) (w : ι → G) : LinearIndependent R (w • v) := by rw [linearIndependent_iff''] at hv ⊢ intro s g hgs hsum i refine (smul_eq_zero_iff_eq (w i)).1 ?_ refine hv s (fun i => w i • g i) (fun i hi => ?_) ?_ i · dsimp only exact (hgs i hi).symm ▸ smul_zero _ · rw [← hsum, Finset.sum_congr rfl _] intros dsimp rw [smul_assoc, smul_comm] #align linear_independent.group_smul LinearIndependent.group_smul -- This lemma cannot be proved with `LinearIndependent.group_smul` since the action of -- `Rˣ` on `R` is not commutative. theorem LinearIndependent.units_smul {v : ι → M} (hv : LinearIndependent R v) (w : ι → Rˣ) : LinearIndependent R (w • v) := by rw [linearIndependent_iff''] at hv ⊢ intro s g hgs hsum i rw [← (w i).mul_left_eq_zero] refine hv s (fun i => g i • (w i : R)) (fun i hi => ?_) ?_ i · dsimp only exact (hgs i hi).symm ▸ zero_smul _ _ · rw [← hsum, Finset.sum_congr rfl _] intros erw [Pi.smul_apply, smul_assoc] rfl #align linear_independent.units_smul LinearIndependent.units_smul lemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y]) {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' := by have : (s - s') • x + (t - t') • y = 0 := by rw [← sub_eq_zero_of_eq h', ← sub_eq_zero] simp only [sub_smul] abel simpa [sub_eq_zero] using h.eq_zero_of_pair this lemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y]) {s t : R} (h' : s • x = t • y) : s = 0 ∧ t = 0 := by suffices H : s = 0 ∧ 0 = t from ⟨H.1, H.2.symm⟩ exact h.eq_of_pair (by simpa using h') /-- If two vectors `x` and `y` are linearly independent, so are their linear combinations `a x + b y` and `c x + d y` provided the determinant `a * d - b * c` is nonzero. -/ lemma LinearIndependent.linear_combination_pair_of_det_ne_zero {R M : Type} [CommRing R] [NoZeroDivisors R] [AddCommGroup M] [Module R M] {x y : M} (h : LinearIndependent R ![x, y]) {a b c d : R} (h' : a d - b * c ≠ 0) : LinearIndependent R ![a • x + b • y, c • x + d • y] := by apply LinearIndependent.pair_iff.2 (fun s t hst ↦ ?_) have H : (s * a + t * c) • x + (s * b + t * d) • y = 0 := by convert hst using 1 simp only [_root_.add_smul, smul_add, smul_smul] abel have I1 : s * a + t * c = 0 := (h.eq_zero_of_pair H).1 have I2 : s * b + t * d = 0 := (h.eq_zero_of_pair H).2 have J1 : (a * d - b * c) * s = 0 := by linear_combination d * I1 - c * I2 have J2 : (a * d - b * c) * t = 0 := by linear_combination -b * I1 + a * I2 exact ⟨by simpa [h'] using mul_eq_zero.1 J1, by simpa [h'] using mul_eq_zero.1 J2⟩ section Maximal universe v w /-- A linearly independent family is maximal if there is no strictly larger linearly independent family. -/ @[nolint unusedArguments] def LinearIndependent.Maximal {ι : Type w} {R : Type u} [Semiring R] {M : Type v} [AddCommMonoid M] [Module R M] {v : ι → M} (_i : LinearIndependent R v) : Prop := ∀ (s : Set M) (_i' : LinearIndependent R ((↑) : s → M)) (_h : range v ≤ s), range v = s #align linear_independent.maximal LinearIndependent.Maximal /-- An alternative characterization of a maximal linearly independent family, quantifying over types (in the same universe as `M`) into which the indexing family injects. -/ theorem LinearIndependent.maximal_iff {ι : Type w} {R : Type u} [Ring R] [Nontrivial R] {M : Type v} [AddCommGroup M] [Module R M] {v : ι → M} (i : LinearIndependent R v) : i.Maximal ↔ ∀ (κ : Type v) (w : κ → M) (_i' : LinearIndependent R w) (j : ι → κ) (_h : w ∘ j = v), Surjective j := by constructor · rintro p κ w i' j rfl specialize p (range w) i'.coe_range (range_comp_subset_range _ _) rw [range_comp, ← image_univ (f := w)] at p exact range_iff_surjective.mp (image_injective.mpr i'.injective p) · intro p w i' h specialize p w ((↑) : w → M) i' (fun i => ⟨v i, range_subset_iff.mp h i⟩) (by ext simp) have q := congr_arg (fun s => ((↑) : w → M) '' s) p.range_eq dsimp at q rw [← image_univ, image_image] at q simpa using q #align linear_independent.maximal_iff LinearIndependent.maximal_iff end Maximal /-- Linear independent families are injective, even if you multiply either side. -/ theorem LinearIndependent.eq_of_smul_apply_eq_smul_apply {M : Type} [AddCommGroup M] [Module R M] {v : ι → M} (li : LinearIndependent R v) (c d : R) (i j : ι) (hc : c ≠ 0) (h : c • v i = d • v j) : i = j := by let l : ι →₀ R := Finsupp.single i c - Finsupp.single j d have h_total : Finsupp.total ι M R v l = 0 := by simp_rw [l, LinearMap.map_sub, Finsupp.total_apply] simp [h] have h_single_eq : Finsupp.single i c = Finsupp.single j d := by rw [linearIndependent_iff] at li simp [eq_add_of_sub_eq' (li l h_total)] rcases (Finsupp.single_eq_single_iff ..).mp h_single_eq with (⟨H, _⟩ \| ⟨hc, _⟩) · exact H · contradiction #align linear_independent.eq_of_smul_apply_eq_smul_apply LinearIndependent.eq_of_smul_apply_eq_smul_apply section Subtype /-! The following lemmas use the subtype defined by a set in `M` as the index set `ι`. -/ theorem LinearIndependent.disjoint_span_image (hv : LinearIndependent R v) {s t : Set ι} (hs : Disjoint s t) : Disjoint (Submodule.span R <\| v '' s) (Submodule.span R <\| v '' t) := by simp only [disjoint_def, Finsupp.mem_span_image_iff_total] rintro _ ⟨l₁, hl₁, rfl⟩ ⟨l₂, hl₂, H⟩ rw [hv.injective_total.eq_iff] at H; subst l₂ have : l₁ = 0 := Submodule.disjoint_def.mp (Finsupp.disjoint_supported_supported hs) _ hl₁ hl₂ simp [this] #align linear_independent.disjoint_span_image LinearIndependent.disjoint_span_image theorem LinearIndependent.not_mem_span_image [Nontrivial R] (hv : LinearIndependent R v) {s : Set ι} {x : ι} (h : x ∉ s) : v x ∉ Submodule.span R (v '' s) := by have h' : v x ∈ Submodule.span R (v '' {x}) := by rw [Set.image_singleton] exact mem_span_singleton_self (v x) intro w apply LinearIndependent.ne_zero x hv refine disjoint_def.1 (hv.disjoint_span_image ?_) (v x) h' w simpa using h #align linear_independent.not_mem_span_image LinearIndependent.not_mem_span_image theorem LinearIndependent.total_ne_of_not_mem_support [Nontrivial R] (hv : LinearIndependent R v) {x : ι} (f : ι →₀ R) (h : x ∉ f.support) : Finsupp.total ι M R v f ≠ v x := by replace h : x ∉ (f.support : Set ι) := h have p := hv.not_mem_span_image h intro w rw [← w] at p rw [Finsupp.span_image_eq_map_total] at p simp only [not_exists, not_and, mem_map] at p -- Porting note: `mem_map` isn't currently triggered exact p f (f.mem_supported_support R) rfl #align linear_independent.total_ne_of_not_mem_support LinearIndependent.total_ne_of_not_mem_support theorem linearIndependent_sum {v : Sum ι ι' → M} : LinearIndependent R v ↔ LinearIndependent R (v ∘ Sum.inl) ∧ LinearIndependent R (v ∘ Sum.inr) ∧ Disjoint (Submodule.span R (range (v ∘ Sum.inl))) (Submodule.span R (range (v ∘ Sum.inr))) := by classical rw [range_comp v, range_comp v] refine ⟨?_, ?_⟩ · intro h refine ⟨h.comp _ Sum.inl_injective, h.comp _ Sum.inr_injective, ?_⟩ refine h.disjoint_span_image ?_ -- Porting note: `isCompl_range_inl_range_inr.1` timeouts. exact IsCompl.disjoint isCompl_range_inl_range_inr rintro ⟨hl, hr, hlr⟩ rw [linearIndependent_iff'] at intro s g hg i hi have : ((∑ i ∈ s.preimage Sum.inl Sum.inl_injective.injOn, (fun x => g x • v x) (Sum.inl i)) + ∑ i ∈ s.preimage Sum.inr Sum.inr_injective.injOn, (fun x => g x • v x) (Sum.inr i)) = 0 := by -- Porting note: `g` must be specified. rw [Finset.sum_preimage' (g := fun x => g x • v x), Finset.sum_preimage' (g := fun x => g x • v x), ← Finset.sum_union, ← Finset.filter_or] · simpa only [← mem_union, range_inl_union_range_inr, mem_univ, Finset.filter_True] · -- Porting note: Here was one `exact`, but timeouted. refine Finset.disjoint_filter.2 fun x _ hx => disjoint_left.1 ?_ hx exact IsCompl.disjoint isCompl_range_inl_range_inr rw [← eq_neg_iff_add_eq_zero] at this rw [disjoint_def'] at hlr have A := by refine hlr _ (sum_mem fun i _ => ?_) _ (neg_mem <\| sum_mem fun i _ => ?_) this · exact smul_mem _ _ (subset_span ⟨Sum.inl i, mem_range_self _, rfl⟩) · exact smul_mem _ _ (subset_span ⟨Sum.inr i, mem_range_self _, rfl⟩) cases' i with i i · exact hl _ _ A i (Finset.mem_preimage.2 hi) · rw [this, neg_eq_zero] at A exact hr _ _ A i (Finset.mem_preimage.2 hi) #align linear_independent_sum linearIndependent_sum theorem LinearIndependent.sum_type {v' : ι' → M} (hv : LinearIndependent R v) (hv' : LinearIndependent R v') (h : Disjoint (Submodule.span R (range v)) (Submodule.span R (range v'))) : LinearIndependent R (Sum.elim v v') := linearIndependent_sum.2 ⟨hv, hv', h⟩ #align linear_independent.sum_type LinearIndependent.sum_type theorem LinearIndependent.union {s t : Set M} (hs : LinearIndependent R (fun x => x : s → M)) (ht : LinearIndependent R (fun x => x : t → M)) (hst : Disjoint (span R s) (span R t)) : LinearIndependent R (fun x => x : ↥(s ∪ t) → M) := (hs.sum_type ht <\| by simpa).to_subtype_range' <\| by simp #align linear_independent.union LinearIndependent.union theorem linearIndependent_iUnion_finite_subtype {ι : Type} {f : ι → Set M} (hl : ∀ i, LinearIndependent R (fun x => x : f i → M)) (hd : ∀ i, ∀ t : Set ι, t.Finite → i ∉ t → Disjoint (span R (f i)) (⨆ i ∈ t, span R (f i))) : LinearIndependent R (fun x => x : (⋃ i, f i) → M) := by classical rw [iUnion_eq_iUnion_finset f] apply linearIndependent_iUnion_of_directed · apply directed_of_isDirected_le exact fun t₁ t₂ ht => iUnion_mono fun i => iUnion_subset_iUnion_const fun h => ht h intro t induction' t using Finset.induction_on with i s his ih · refine (linearIndependent_empty R M).mono ?_ simp · rw [Finset.set_biUnion_insert] refine (hl _).union ih ?_ rw [span_iUnion₂] exact hd i s s.finite_toSet his #align linear_independent_Union_finite_subtype linearIndependent_iUnion_finite_subtype theorem linearIndependent_iUnion_finite {η : Type} {ιs : η → Type*} {f : ∀ j : η, ιs j → M} (hindep : ∀ j, LinearIndependent R (f j)) (hd : ∀ i, ∀ t : Set η, t.Finite → i ∉ t → Disjoint (span R (range (f i))) (⨆ i ∈ t, span R (range (f i)))) : LinearIndependent R fun ji : Σ j, ιs j => f ji.1 ji.2 := by nontriviality R apply LinearIndependent.of_subtype_range · rintro ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ hxy by_cases h_cases : x₁ = y₁ · subst h_cases refine Sigma.eq rfl ?_ rw [LinearIndependent.injective (hindep _) hxy] · have h0 : f x₁ x₂ = 0 := by apply disjoint_def.1 (hd x₁ {y₁} (finite_singleton y₁) fun h => h_cases (eq_of_mem_singleton h)) (f x₁ x₂) (subset_span (mem_range_self _)) rw [iSup_singleton] simp only at hxy rw [hxy] exact subset_span (mem_range_self y₂) exact False.elim ((hindep x₁).ne_zero _ h0) rw [range_sigma_eq_iUnion_range] apply linearIndependent_iUnion_finite_subtype (fun j => (hindep j).to_subtype_range) hd #align linear_independent_Union_finite linearIndependent_iUnion_finite end Subtype section repr variable (hv : LinearIndependent R v) /-- Canonical isomorphism between linear combinations and the span of linearly independent vectors. -/ @[simps (config := { rhsMd := default }) symm_apply] def LinearIndependent.totalEquiv (hv : LinearIndependent R v) : (ι →₀ R) ≃ₗ[R] span R (range v) := by apply LinearEquiv.ofBijective (LinearMap.codRestrict (span R (range v)) (Finsupp.total ι M R v) _) constructor · rw [← LinearMap.ker_eq_bot, LinearMap.ker_codRestrict] · apply hv · intro l rw [← Finsupp.range_total] rw [LinearMap.mem_range] apply mem_range_self l · rw [← LinearMap.range_eq_top, LinearMap.range_eq_map, LinearMap.map_codRestrict, ← LinearMap.range_le_iff_comap, range_subtype, Submodule.map_top] rw [Finsupp.range_total] #align linear_independent.total_equiv LinearIndependent.totalEquiv #align linear_independent.total_equiv_symm_apply LinearIndependent.totalEquiv_symm_apply -- Porting note: The original theorem generated by `simps` was -- different from the theorem on Lean 3, and not simp-normal form. @[simp] theorem LinearIndependent.totalEquiv_apply_coe (hv : LinearIndependent R v) (l : ι →₀ R) : hv.totalEquiv l = Finsupp.total ι M R v l := rfl #align linear_independent.total_equiv_apply_coe LinearIndependent.totalEquiv_apply_coe /-- Linear combination representing a vector in the span of linearly independent vectors. Given a family of linearly independent vectors, we can represent any vector in their span as a linear combination of these vectors. These are provided by this linear map. It is simply one direction of `LinearIndependent.total_equiv`. -/ def LinearIndependent.repr (hv : LinearIndependent R v) : span R (range v) →ₗ[R] ι →₀ R := hv.totalEquiv.symm #align linear_independent.repr LinearIndependent.repr @[simp] theorem LinearIndependent.total_repr (x) : Finsupp.total ι M R v (hv.repr x) = x := Subtype.ext_iff.1 (LinearEquiv.apply_symm_apply hv.totalEquiv x) #align linear_independent.total_repr LinearIndependent.total_repr theorem LinearIndependent.total_comp_repr : (Finsupp.total ι M R v).comp hv.repr = Submodule.subtype _ := LinearMap.ext <\| hv.total_repr #align linear_independent.total_comp_repr LinearIndependent.total_comp_repr theorem LinearIndependent.repr_ker : LinearMap.ker hv.repr = ⊥ := by rw [LinearIndependent.repr, LinearEquiv.ker] #align linear_independent.repr_ker LinearIndependent.repr_ker theorem LinearIndependent.repr_range : LinearMap.range hv.repr = ⊤ := by rw [LinearIndependent.repr, LinearEquiv.range] #align linear_independent.repr_range LinearIndependent.repr_range theorem LinearIndependent.repr_eq {l : ι →₀ R} {x : span R (range v)} (eq : Finsupp.total ι M R v l = ↑x) : hv.repr x = l := by have : ↑((LinearIndependent.totalEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l) = Finsupp.total ι M R v l := rfl have : (LinearIndependent.totalEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l = x := by rw [eq] at this exact Subtype.ext_iff.2 this rw [← LinearEquiv.symm_apply_apply hv.totalEquiv l] rw [← this] rfl #align linear_independent.repr_eq LinearIndependent.repr_eq theorem LinearIndependent.repr_eq_single (i) (x : span R (range v)) (hx : ↑x = v i) : hv.repr x = Finsupp.single i 1 := by apply hv.repr_eq simp [Finsupp.total_single, hx] #align linear_independent.repr_eq_single LinearIndependent.repr_eq_single theorem LinearIndependent.span_repr_eq [Nontrivial R] (x) : Span.repr R (Set.range v) x = (hv.repr x).equivMapDomain (Equiv.ofInjective _ hv.injective) := by have p : (Span.repr R (Set.range v) x).equivMapDomain (Equiv.ofInjective _ hv.injective).symm = hv.repr x := by apply (LinearIndependent.totalEquiv hv).injective ext simp only [LinearIndependent.totalEquiv_apply_coe, Equiv.self_comp_ofInjective_symm, LinearIndependent.total_repr, Finsupp.total_equivMapDomain, Span.finsupp_total_repr] ext ⟨_, ⟨i, rfl⟩⟩ simp [← p] #align linear_independent.span_repr_eq LinearIndependent.span_repr_eq theorem linearIndependent_iff_not_smul_mem_span : LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0 := ⟨fun hv i a ha => by rw [Finsupp.span_image_eq_map_total, mem_map] at ha rcases ha with ⟨l, hl, e⟩ rw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl by_contra hn exact (not_mem_of_mem_diff (hl <\| by simp [hn])) (mem_singleton _), fun H => linearIndependent_iff.2 fun l hl => by ext i; simp only [Finsupp.zero_apply] by_contra hn refine hn (H i _ ?_) refine (Finsupp.mem_span_image_iff_total R).2 ⟨Finsupp.single i (l i) - l, ?_, ?_⟩ · rw [Finsupp.mem_supported'] intro j hj have hij : j = i := Classical.not_not.1 fun hij : j ≠ i => hj ((mem_diff _).2 ⟨mem_univ _, fun h => hij (eq_of_mem_singleton h)⟩) simp [hij] · simp [hl]⟩ #align linear_independent_iff_not_smul_mem_span linearIndependent_iff_not_smul_mem_span /-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/ theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) : CompleteLattice.Independent fun i => R ∙ v i := by refine CompleteLattice.independent_def.mp fun i => ?_ rw [disjoint_iff_inf_le] intro m hm simp only [mem_inf, mem_span_singleton, iSup_subtype'] at hm rw [← span_range_eq_iSup] at hm obtain ⟨⟨r, rfl⟩, hm⟩ := hm suffices r = 0 by simp [this] apply linearIndependent_iff_not_smul_mem_span.mp hv i -- Porting note: The original proof was using `convert hm`. suffices v '' (univ \ {i}) = range fun j : { j // j ≠ i } => v j by rwa [this] ext simp #align linear_independent.independent_span_singleton LinearIndependent.independent_span_singleton variable (R) theorem exists_maximal_independent' (s : ι → M) : ∃ I : Set ι, (LinearIndependent R fun x : I => s x) ∧ ∀ J : Set ι, I ⊆ J → (LinearIndependent R fun x : J => s x) → I = J := by let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M) let X := { I : Set ι // indep I } let r : X → X → Prop := fun I J => I.1 ⊆ J.1 have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by intro c hc dsimp [indep] rw [linearIndependent_comp_subtype] intro f hsupport hsum rcases eq_empty_or_nonempty c with (rfl \| hn) · simpa using hsupport haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩ obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I := hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport exact linearIndependent_comp_subtype.mp I.2 f hI hsum have trans : Transitive r := fun I J K => Set.Subset.trans obtain ⟨⟨I, hli : indep I⟩, hmax : ∀ a, r ⟨I, hli⟩ a → r a ⟨I, hli⟩⟩ := exists_maximal_of_chains_bounded (fun c hc => ⟨⟨⋃ I ∈ c, (I : Set ι), key c hc⟩, fun I => Set.subset_biUnion_of_mem⟩) @trans exact ⟨I, hli, fun J hsub hli => Set.Subset.antisymm hsub (hmax ⟨J, hli⟩ hsub)⟩ #align exists_maximal_independent' exists_maximal_independent'	Mathlib/LinearAlgebra/LinearIndependent.lean	1,017	1,049	theorem exists_maximal_independent (s : ι → M) : ∃ I : Set ι, (LinearIndependent R fun x : I => s x) ∧ ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by	classical rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩ use I, hIlinind intro i hi specialize hImaximal (I ∪ {i}) (by simp) set J := I ∪ {i} with hJ have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ] have hiJ : i ∈ J := by simp [J] have h := by refine mt hImaximal ?_ · intro h2 rw [h2] at hi exact absurd hiJ hi obtain ⟨f, supp_f, sum_f, f_ne⟩ := linearDependent_comp_subtype.mp h have hfi : f i ≠ 0 := by contrapose hIlinind refine linearDependent_comp_subtype.mpr ⟨f, ?_, sum_f, f_ne⟩ simp only [Finsupp.mem_supported, hJ] at supp_f ⊢ rintro x hx refine (memJ.mp (supp_f hx)).resolve_left ?_ rintro rfl exact hIlinind (Finsupp.mem_support_iff.mp hx) use f i, hfi have hfi' : i ∈ f.support := Finsupp.mem_support_iff.mpr hfi rw [← Finset.insert_erase hfi', Finset.sum_insert (Finset.not_mem_erase _ _), add_eq_zero_iff_eq_neg] at sum_f rw [sum_f] refine neg_mem (sum_mem fun c hc => smul_mem _ _ (subset_span ⟨c, ?_, rfl⟩)) exact (memJ.mp (supp_f (Finset.erase_subset _ _ hc))).resolve_left (Finset.ne_of_mem_erase hc)
/- Copyright (c) 2019 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import Mathlib.MeasureTheory.Integral.Lebesgue #align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625" /-! # The Giry monad Let X be a measurable space. The collection of all measures on X again forms a measurable space. This construction forms a monad on measurable spaces and measurable functions, called the Giry monad. Note that most sources use the term "Giry monad" for the restriction to probability measures. Here we include all measures on X. See also `MeasureTheory/Category/MeasCat.lean`, containing an upgrade of the type-level monad to an honest monad of the functor `measure : MeasCat ⥤ MeasCat`. ## References * <https://ncatlab.org/nlab/show/Giry+monad> ## Tags giry monad -/ noncomputable section open scoped Classical open ENNReal open scoped Classical open Set Filter variable {α β : Type} namespace MeasureTheory namespace Measure variable [MeasurableSpace α] [MeasurableSpace β] /-- Measurability structure on `Measure`: Measures are measurable w.r.t. all projections -/ instance instMeasurableSpace : MeasurableSpace (Measure α) := ⨆ (s : Set α) (_ : MeasurableSet s), (borel ℝ≥0∞).comap fun μ => μ s #align measure_theory.measure.measurable_space MeasureTheory.Measure.instMeasurableSpace theorem measurable_coe {s : Set α} (hs : MeasurableSet s) : Measurable fun μ : Measure α => μ s := Measurable.of_comap_le <\| le_iSup_of_le s <\| le_iSup_of_le hs <\| le_rfl #align measure_theory.measure.measurable_coe MeasureTheory.Measure.measurable_coe theorem measurable_of_measurable_coe (f : β → Measure α) (h : ∀ (s : Set α), MeasurableSet s → Measurable fun b => f b s) : Measurable f := Measurable.of_le_map <\| iSup₂_le fun s hs => MeasurableSpace.comap_le_iff_le_map.2 <\| by rw [MeasurableSpace.map_comp]; exact h s hs #align measure_theory.measure.measurable_of_measurable_coe MeasureTheory.Measure.measurable_of_measurable_coe instance instMeasurableAdd₂ {α : Type} {m : MeasurableSpace α} : MeasurableAdd₂ (Measure α) := by refine ⟨Measure.measurable_of_measurable_coe _ fun s hs => ?_⟩ simp_rw [Measure.coe_add, Pi.add_apply] refine Measurable.add ?_ ?_ · exact (Measure.measurable_coe hs).comp measurable_fst · exact (Measure.measurable_coe hs).comp measurable_snd #align measure_theory.measure.has_measurable_add₂ MeasureTheory.Measure.instMeasurableAdd₂ theorem measurable_measure {μ : α → Measure β} : Measurable μ ↔ ∀ (s : Set β), MeasurableSet s → Measurable fun b => μ b s := ⟨fun hμ _s hs => (measurable_coe hs).comp hμ, measurable_of_measurable_coe μ⟩ #align measure_theory.measure.measurable_measure MeasureTheory.Measure.measurable_measure theorem measurable_map (f : α → β) (hf : Measurable f) : Measurable fun μ : Measure α => map f μ := by refine measurable_of_measurable_coe _ fun s hs => ?_ simp_rw [map_apply hf hs] exact measurable_coe (hf hs) #align measure_theory.measure.measurable_map MeasureTheory.Measure.measurable_map theorem measurable_dirac : Measurable (Measure.dirac : α → Measure α) := by refine measurable_of_measurable_coe _ fun s hs => ?_ simp_rw [dirac_apply' _ hs] exact measurable_one.indicator hs #align measure_theory.measure.measurable_dirac MeasureTheory.Measure.measurable_dirac theorem measurable_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) : Measurable fun μ : Measure α => ∫⁻ x, f x ∂μ := by simp only [lintegral_eq_iSup_eapprox_lintegral, hf, SimpleFunc.lintegral] refine measurable_iSup fun n => Finset.measurable_sum _ fun i _ => ?_ refine Measurable.const_mul ?_ _ exact measurable_coe ((SimpleFunc.eapprox f n).measurableSet_preimage _) #align measure_theory.measure.measurable_lintegral MeasureTheory.Measure.measurable_lintegral /-- Monadic join on `Measure` in the category of measurable spaces and measurable functions. -/ def join (m : Measure (Measure α)) : Measure α := Measure.ofMeasurable (fun s _ => ∫⁻ μ, μ s ∂m) (by simp only [measure_empty, lintegral_const, zero_mul]) (by intro f hf h simp_rw [measure_iUnion h hf] apply lintegral_tsum intro i; exact (measurable_coe (hf i)).aemeasurable) #align measure_theory.measure.join MeasureTheory.Measure.join @[simp] theorem join_apply {m : Measure (Measure α)} {s : Set α} (hs : MeasurableSet s) : join m s = ∫⁻ μ, μ s ∂m := Measure.ofMeasurable_apply s hs #align measure_theory.measure.join_apply MeasureTheory.Measure.join_apply @[simp] theorem join_zero : (0 : Measure (Measure α)).join = 0 := by ext1 s hs simp only [hs, join_apply, lintegral_zero_measure, coe_zero, Pi.zero_apply] #align measure_theory.measure.join_zero MeasureTheory.Measure.join_zero theorem measurable_join : Measurable (join : Measure (Measure α) → Measure α) := measurable_of_measurable_coe _ fun s hs => by simp only [join_apply hs]; exact measurable_lintegral (measurable_coe hs) #align measure_theory.measure.measurable_join MeasureTheory.Measure.measurable_join	Mathlib/MeasureTheory/Measure/GiryMonad.lean	128	149	theorem lintegral_join {m : Measure (Measure α)} {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ x, f x ∂join m = ∫⁻ μ, ∫⁻ x, f x ∂μ ∂m := by	simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral, join_apply (SimpleFunc.measurableSet_preimage _ _)] suffices ∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ n r, Measurable (f n r)) → Monotone (fun n μ => ∑ r ∈ s n, r * f n r μ) → ⨆ n, ∑ r ∈ s n, r * ∫⁻ μ, f n r μ ∂m = ∫⁻ μ, ⨆ n, ∑ r ∈ s n, r * f n r μ ∂m by refine this (fun n => SimpleFunc.range (SimpleFunc.eapprox f n)) (fun n r μ => μ (SimpleFunc.eapprox f n ⁻¹' {r})) ?_ ?_ · exact fun n r => measurable_coe (SimpleFunc.measurableSet_preimage _ _) · exact fun n m h μ => SimpleFunc.lintegral_mono (SimpleFunc.monotone_eapprox _ h) le_rfl intro s f hf hm rw [lintegral_iSup _ hm] swap · exact fun n => Finset.measurable_sum _ fun r _ => (hf _ _).const_mul _ congr funext n rw [lintegral_finset_sum (s n)] · simp_rw [lintegral_const_mul _ (hf _ _)] · exact fun r _ => (hf _ _).const_mul _
/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yourong Zang -/ import Mathlib.Analysis.Calculus.ContDiff.Basic import Mathlib.Analysis.Calculus.Deriv.Linear import Mathlib.Analysis.Complex.Conformal import Mathlib.Analysis.Calculus.Conformal.NormedSpace #align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # Real differentiability of complex-differentiable functions `HasDerivAt.real_of_complex` expresses that, if a function on `ℂ` is differentiable (over `ℂ`), then its restriction to `ℝ` is differentiable over `ℝ`, with derivative the real part of the complex derivative. `DifferentiableAt.conformalAt` states that a real-differentiable function with a nonvanishing differential from the complex plane into an arbitrary complex-normed space is conformal at a point if it's holomorphic at that point. This is a version of Cauchy-Riemann equations. `conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj` proves that a real-differential function with a nonvanishing differential between the complex plane is conformal at a point if and only if it's holomorphic or antiholomorphic at that point. ## TODO * The classical form of Cauchy-Riemann equations * On a connected open set `u`, a function which is `ConformalAt` each point is either holomorphic throughout or antiholomorphic throughout. ## Warning We do NOT require conformal functions to be orientation-preserving in this file. -/ section RealDerivOfComplex /-! ### Differentiability of the restriction to `ℝ` of complex functions -/ open Complex variable {e : ℂ → ℂ} {e' : ℂ} {z : ℝ} /-- If a complex function is differentiable at a real point, then the induced real function is also differentiable at this point, with a derivative equal to the real part of the complex derivative. -/ theorem HasStrictDerivAt.real_of_complex (h : HasStrictDerivAt e e' z) : HasStrictDerivAt (fun x : ℝ => (e x).re) e'.re z := by have A : HasStrictFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasStrictFDerivAt have B : HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ) (ofRealCLM z) := h.hasStrictFDerivAt.restrictScalars ℝ have C : HasStrictFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasStrictFDerivAt -- Porting note: this should be by: -- simpa using (C.comp z (B.comp z A)).hasStrictDerivAt -- but for some reason simp can not use `ContinuousLinearMap.comp_apply` convert (C.comp z (B.comp z A)).hasStrictDerivAt rw [ContinuousLinearMap.comp_apply, ContinuousLinearMap.comp_apply] simp #align has_strict_deriv_at.real_of_complex HasStrictDerivAt.real_of_complex /-- If a complex function `e` is differentiable at a real point, then the function `ℝ → ℝ` given by the real part of `e` is also differentiable at this point, with a derivative equal to the real part of the complex derivative. -/ theorem HasDerivAt.real_of_complex (h : HasDerivAt e e' z) : HasDerivAt (fun x : ℝ => (e x).re) e'.re z := by have A : HasFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasFDerivAt have B : HasFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ) (ofRealCLM z) := h.hasFDerivAt.restrictScalars ℝ have C : HasFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasFDerivAt -- Porting note: this should be by: -- simpa using (C.comp z (B.comp z A)).hasStrictDerivAt -- but for some reason simp can not use `ContinuousLinearMap.comp_apply` convert (C.comp z (B.comp z A)).hasDerivAt rw [ContinuousLinearMap.comp_apply, ContinuousLinearMap.comp_apply] simp #align has_deriv_at.real_of_complex HasDerivAt.real_of_complex theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) : ContDiffAt ℝ n (fun x : ℝ => (e x).re) z := by have A : ContDiffAt ℝ n ((↑) : ℝ → ℂ) z := ofRealCLM.contDiff.contDiffAt have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ have C : ContDiffAt ℝ n re (e z) := reCLM.contDiff.contDiffAt exact C.comp z (B.comp z A) #align cont_diff_at.real_of_complex ContDiffAt.real_of_complex theorem ContDiff.real_of_complex {n : ℕ∞} (h : ContDiff ℂ n e) : ContDiff ℝ n fun x : ℝ => (e x).re := contDiff_iff_contDiffAt.2 fun _ => h.contDiffAt.real_of_complex #align cont_diff.real_of_complex ContDiff.real_of_complex variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] theorem HasStrictDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasStrictDerivAt f f' x) : HasStrictFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by simpa only [Complex.restrictScalars_one_smulRight'] using h.hasStrictFDerivAt.restrictScalars ℝ #align has_strict_deriv_at.complex_to_real_fderiv' HasStrictDerivAt.complexToReal_fderiv' theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) : HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivAt.restrictScalars ℝ #align has_deriv_at.complex_to_real_fderiv' HasDerivAt.complexToReal_fderiv' theorem HasDerivWithinAt.complexToReal_fderiv' {f : ℂ → E} {s : Set ℂ} {x : ℂ} {f' : E} (h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (reCLM.smulRight f' + I • imCLM.smulRight f') s x := by simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivWithinAt.restrictScalars ℝ #align has_deriv_within_at.complex_to_real_fderiv' HasDerivWithinAt.complexToReal_fderiv'	Mathlib/Analysis/Complex/RealDeriv.lean	118	120	theorem HasStrictDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasStrictDerivAt f f' x) : HasStrictFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x := by	simpa only [Complex.restrictScalars_one_smulRight] using h.hasStrictFDerivAt.restrictScalars ℝ
/- Copyright (c) 2021 Kexing Ying. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kexing Ying, Rémy Degenne -/ import Mathlib.Probability.Process.Adapted import Mathlib.MeasureTheory.Constructions.BorelSpace.Order #align_import probability.process.stopping from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca" /-! # Stopping times, stopped processes and stopped values Definition and properties of stopping times. ## Main definitions * `MeasureTheory.IsStoppingTime`: a stopping time with respect to some filtration `f` is a function `τ` such that for all `i`, the preimage of `{j \| j ≤ i}` along `τ` is `f i`-measurable * `MeasureTheory.IsStoppingTime.measurableSpace`: the σ-algebra associated with a stopping time ## Main results * `ProgMeasurable.stoppedProcess`: the stopped process of a progressively measurable process is progressively measurable. * `memℒp_stoppedProcess`: if a process belongs to `ℒp` at every time in `ℕ`, then its stopped process belongs to `ℒp` as well. ## Tags stopping time, stochastic process -/ open Filter Order TopologicalSpace open scoped Classical MeasureTheory NNReal ENNReal Topology namespace MeasureTheory variable {Ω β ι : Type*} {m : MeasurableSpace Ω} /-! ### Stopping times -/ /-- A stopping time with respect to some filtration `f` is a function `τ` such that for all `i`, the preimage of `{j \| j ≤ i}` along `τ` is measurable with respect to `f i`. Intuitively, the stopping time `τ` describes some stopping rule such that at time `i`, we may determine it with the information we have at time `i`. -/ def IsStoppingTime [Preorder ι] (f : Filtration ι m) (τ : Ω → ι) := ∀ i : ι, MeasurableSet[f i] <\| {ω \| τ ω ≤ i} #align measure_theory.is_stopping_time MeasureTheory.IsStoppingTime theorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) : IsStoppingTime f fun _ => i := fun j => by simp only [MeasurableSet.const] #align measure_theory.is_stopping_time_const MeasureTheory.isStoppingTime_const section MeasurableSet section Preorder variable [Preorder ι] {f : Filtration ι m} {τ : Ω → ι} protected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω ≤ i} := hτ i #align measure_theory.is_stopping_time.measurable_set_le MeasureTheory.IsStoppingTime.measurableSet_le theorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := by by_cases hi_min : IsMin i · suffices {ω : Ω \| τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i) ext1 ω simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] rw [isMin_iff_forall_not_lt] at hi_min exact hi_min (τ ω) have : {ω : Ω \| τ ω < i} = τ ⁻¹' Set.Iic (pred i) := by ext; simp [Iic_pred_of_not_isMin hi_min] rw [this] exact f.mono (pred_le i) _ (hτ.measurableSet_le <\| pred i) #align measure_theory.is_stopping_time.measurable_set_lt_of_pred MeasureTheory.IsStoppingTime.measurableSet_lt_of_pred end Preorder section CountableStoppingTime namespace IsStoppingTime variable [PartialOrder ι] {τ : Ω → ι} {f : Filtration ι m} protected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω \| τ ω = i} := by have : {ω \| τ ω = i} = {ω \| τ ω ≤ i} \ ⋃ (j ∈ Set.range τ) (_ : j < i), {ω \| τ ω ≤ j} := by ext1 a simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq', Set.mem_diff, Set.mem_iUnion, exists_prop, not_exists, not_and, not_le] constructor <;> intro h · simp only [h, lt_iff_le_not_le, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff] · exact h.1.eq_or_lt.resolve_right fun h_lt => h.2 a h_lt le_rfl rw [this] refine (hτ.measurableSet_le i).diff ?_ refine MeasurableSet.biUnion h_countable fun j _ => ?_ rw [Set.iUnion_eq_if] split_ifs with hji · exact f.mono hji.le _ (hτ.measurableSet_le j) · exact @MeasurableSet.empty _ (f i) #align measure_theory.is_stopping_time.measurable_set_eq_of_countable_range MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range protected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω = i} := hτ.measurableSet_eq_of_countable_range (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_eq_of_countable MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable protected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := by have : {ω \| τ ω < i} = {ω \| τ ω ≤ i} \ {ω \| τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne] rw [this] exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i) #align measure_theory.is_stopping_time.measurable_set_lt_of_countable_range MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range protected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := hτ.measurableSet_lt_of_countable_range (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_lt_of_countable MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable protected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → ι} {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω < i}ᶜ := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] rw [this] exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl #align measure_theory.is_stopping_time.measurable_set_ge_of_countable_range MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range protected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → ι} {f : Filtration ι m} [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| i ≤ τ ω} := hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_ge_of_countable MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable end IsStoppingTime end CountableStoppingTime section LinearOrder variable [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι} theorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| i < τ ω} := by have : {ω \| i < τ ω} = {ω \| τ ω ≤ i}ᶜ := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le] rw [this] exact (hτ.measurableSet_le i).compl #align measure_theory.is_stopping_time.measurable_set_gt MeasureTheory.IsStoppingTime.measurableSet_gt section TopologicalSpace variable [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] /-- Auxiliary lemma for `MeasureTheory.IsStoppingTime.measurableSet_lt`. -/ theorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι) (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω \| τ ω < i} := by by_cases hi_min : IsMin i · suffices {ω \| τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i) ext1 ω simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] exact isMin_iff_forall_not_lt.mp hi_min (τ ω) obtain ⟨seq, -, -, h_tendsto, h_bound⟩ : ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i := h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min) have h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k \| k ≤ seq j} := by ext1 k simp only [Set.mem_Iio, Set.mem_iUnion, Set.mem_setOf_eq] refine ⟨fun hk_lt_i => ?_, fun h_exists_k_le_seq => ?_⟩ · rw [tendsto_atTop'] at h_tendsto have h_nhds : Set.Ici k ∈ 𝓝 i := mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩ obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds exact ⟨a, ha a le_rfl⟩ · obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq exact hk_seq_j.trans_lt (h_bound j) have h_lt_eq_preimage : {ω \| τ ω < i} = τ ⁻¹' Set.Iio i := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iio] rw [h_lt_eq_preimage, h_Ioi_eq_Union] simp only [Set.preimage_iUnion, Set.preimage_setOf_eq] exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n)) #align measure_theory.is_stopping_time.measurable_set_lt_of_is_lub MeasureTheory.IsStoppingTime.measurableSet_lt_of_isLUB theorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := by obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i cases' lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i h_Iio_eq_Iic · rw [← hi'_eq_i] at hi'_lub ⊢ exact hτ.measurableSet_lt_of_isLUB i' hi'_lub · have h_lt_eq_preimage : {ω : Ω \| τ ω < i} = τ ⁻¹' Set.Iio i := rfl rw [h_lt_eq_preimage, h_Iio_eq_Iic] exact f.mono (lub_Iio_le i hi'_lub) _ (hτ.measurableSet_le i') #align measure_theory.is_stopping_time.measurable_set_lt MeasureTheory.IsStoppingTime.measurableSet_lt theorem IsStoppingTime.measurableSet_ge (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω < i}ᶜ := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] rw [this] exact (hτ.measurableSet_lt i).compl #align measure_theory.is_stopping_time.measurable_set_ge MeasureTheory.IsStoppingTime.measurableSet_ge theorem IsStoppingTime.measurableSet_eq (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω = i} := by have : {ω \| τ ω = i} = {ω \| τ ω ≤ i} ∩ {ω \| τ ω ≥ i} := by ext1 ω; simp only [Set.mem_setOf_eq, ge_iff_le, Set.mem_inter_iff, le_antisymm_iff] rw [this] exact (hτ.measurableSet_le i).inter (hτ.measurableSet_ge i) #align measure_theory.is_stopping_time.measurable_set_eq MeasureTheory.IsStoppingTime.measurableSet_eq theorem IsStoppingTime.measurableSet_eq_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) : MeasurableSet[f j] {ω \| τ ω = i} := f.mono hle _ <\| hτ.measurableSet_eq i #align measure_theory.is_stopping_time.measurable_set_eq_le MeasureTheory.IsStoppingTime.measurableSet_eq_le theorem IsStoppingTime.measurableSet_lt_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) : MeasurableSet[f j] {ω \| τ ω < i} := f.mono hle _ <\| hτ.measurableSet_lt i #align measure_theory.is_stopping_time.measurable_set_lt_le MeasureTheory.IsStoppingTime.measurableSet_lt_le end TopologicalSpace end LinearOrder section Countable theorem isStoppingTime_of_measurableSet_eq [Preorder ι] [Countable ι] {f : Filtration ι m} {τ : Ω → ι} (hτ : ∀ i, MeasurableSet[f i] {ω \| τ ω = i}) : IsStoppingTime f τ := by intro i rw [show {ω \| τ ω ≤ i} = ⋃ k ≤ i, {ω \| τ ω = k} by ext; simp] refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_ exact f.mono hk _ (hτ k) #align measure_theory.is_stopping_time_of_measurable_set_eq MeasureTheory.isStoppingTime_of_measurableSet_eq end Countable end MeasurableSet namespace IsStoppingTime protected theorem max [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : IsStoppingTime f fun ω => max (τ ω) (π ω) := by intro i simp_rw [max_le_iff, Set.setOf_and] exact (hτ i).inter (hπ i) #align measure_theory.is_stopping_time.max MeasureTheory.IsStoppingTime.max protected theorem max_const [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ) (i : ι) : IsStoppingTime f fun ω => max (τ ω) i := hτ.max (isStoppingTime_const f i) #align measure_theory.is_stopping_time.max_const MeasureTheory.IsStoppingTime.max_const protected theorem min [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : IsStoppingTime f fun ω => min (τ ω) (π ω) := by intro i simp_rw [min_le_iff, Set.setOf_or] exact (hτ i).union (hπ i) #align measure_theory.is_stopping_time.min MeasureTheory.IsStoppingTime.min protected theorem min_const [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ) (i : ι) : IsStoppingTime f fun ω => min (τ ω) i := hτ.min (isStoppingTime_const f i) #align measure_theory.is_stopping_time.min_const MeasureTheory.IsStoppingTime.min_const theorem add_const [AddGroup ι] [Preorder ι] [CovariantClass ι ι (Function.swap (· + ·)) (· ≤ ·)] [CovariantClass ι ι (· + ·) (· ≤ ·)] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ) {i : ι} (hi : 0 ≤ i) : IsStoppingTime f fun ω => τ ω + i := by intro j simp_rw [← le_sub_iff_add_le] exact f.mono (sub_le_self j hi) _ (hτ (j - i)) #align measure_theory.is_stopping_time.add_const MeasureTheory.IsStoppingTime.add_const theorem add_const_nat {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) {i : ℕ} : IsStoppingTime f fun ω => τ ω + i := by refine isStoppingTime_of_measurableSet_eq fun j => ?_ by_cases hij : i ≤ j · simp_rw [eq_comm, ← Nat.sub_eq_iff_eq_add hij, eq_comm] exact f.mono (j.sub_le i) _ (hτ.measurableSet_eq (j - i)) · rw [not_le] at hij convert @MeasurableSet.empty _ (f.1 j) ext ω simp only [Set.mem_empty_iff_false, iff_false_iff, Set.mem_setOf] omega #align measure_theory.is_stopping_time.add_const_nat MeasureTheory.IsStoppingTime.add_const_nat -- generalize to certain countable type? theorem add {f : Filtration ℕ m} {τ π : Ω → ℕ} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : IsStoppingTime f (τ + π) := by intro i rw [(_ : {ω \| (τ + π) ω ≤ i} = ⋃ k ≤ i, {ω \| π ω = k} ∩ {ω \| τ ω + k ≤ i})] · exact MeasurableSet.iUnion fun k => MeasurableSet.iUnion fun hk => (hπ.measurableSet_eq_le hk).inter (hτ.add_const_nat i) ext ω simp only [Pi.add_apply, Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_prop] refine ⟨fun h => ⟨π ω, by omega, rfl, h⟩, ?_⟩ rintro ⟨j, hj, rfl, h⟩ assumption #align measure_theory.is_stopping_time.add MeasureTheory.IsStoppingTime.add section Preorder variable [Preorder ι] {f : Filtration ι m} {τ π : Ω → ι} /-- The associated σ-algebra with a stopping time. -/ protected def measurableSpace (hτ : IsStoppingTime f τ) : MeasurableSpace Ω where MeasurableSet' s := ∀ i : ι, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) measurableSet_empty i := (Set.empty_inter {ω \| τ ω ≤ i}).symm ▸ @MeasurableSet.empty _ (f i) measurableSet_compl s hs i := by rw [(_ : sᶜ ∩ {ω \| τ ω ≤ i} = (sᶜ ∪ {ω \| τ ω ≤ i}ᶜ) ∩ {ω \| τ ω ≤ i})] · refine MeasurableSet.inter ?_ ?_ · rw [← Set.compl_inter] exact (hs i).compl · exact hτ i · rw [Set.union_inter_distrib_right] simp only [Set.compl_inter_self, Set.union_empty] measurableSet_iUnion s hs i := by rw [forall_swap] at hs rw [Set.iUnion_inter] exact MeasurableSet.iUnion (hs i) #align measure_theory.is_stopping_time.measurable_space MeasureTheory.IsStoppingTime.measurableSpace protected theorem measurableSet (hτ : IsStoppingTime f τ) (s : Set Ω) : MeasurableSet[hτ.measurableSpace] s ↔ ∀ i : ι, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) := Iff.rfl #align measure_theory.is_stopping_time.measurable_set MeasureTheory.IsStoppingTime.measurableSet theorem measurableSpace_mono (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) (hle : τ ≤ π) : hτ.measurableSpace ≤ hπ.measurableSpace := by intro s hs i rw [(_ : s ∩ {ω \| π ω ≤ i} = s ∩ {ω \| τ ω ≤ i} ∩ {ω \| π ω ≤ i})] · exact (hs i).inter (hπ i) · ext simp only [Set.mem_inter_iff, iff_self_and, and_congr_left_iff, Set.mem_setOf_eq] intro hle' _ exact le_trans (hle _) hle' #align measure_theory.is_stopping_time.measurable_space_mono MeasureTheory.IsStoppingTime.measurableSpace_mono theorem measurableSpace_le_of_countable [Countable ι] (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := by intro s hs change ∀ i, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) at hs rw [(_ : s = ⋃ i, s ∩ {ω \| τ ω ≤ i})] · exact MeasurableSet.iUnion fun i => f.le i _ (hs i) · ext ω; constructor <;> rw [Set.mem_iUnion] · exact fun hx => ⟨τ ω, hx, le_rfl⟩ · rintro ⟨_, hx, _⟩ exact hx #align measure_theory.is_stopping_time.measurable_space_le_of_countable MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable theorem measurableSpace_le' [IsCountablyGenerated (atTop : Filter ι)] [(atTop : Filter ι).NeBot] (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := by intro s hs change ∀ i, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) at hs obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := (atTop : Filter ι).exists_seq_tendsto rw [(_ : s = ⋃ n, s ∩ {ω \| τ ω ≤ seq n})] · exact MeasurableSet.iUnion fun i => f.le (seq i) _ (hs (seq i)) · ext ω; constructor <;> rw [Set.mem_iUnion] · intro hx suffices ∃ i, τ ω ≤ seq i from ⟨this.choose, hx, this.choose_spec⟩ rw [tendsto_atTop] at h_seq_tendsto exact (h_seq_tendsto (τ ω)).exists · rintro ⟨_, hx, _⟩ exact hx #align measure_theory.is_stopping_time.measurable_space_le' MeasureTheory.IsStoppingTime.measurableSpace_le' theorem measurableSpace_le {ι} [SemilatticeSup ι] {f : Filtration ι m} {τ : Ω → ι} [IsCountablyGenerated (atTop : Filter ι)] (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := by cases isEmpty_or_nonempty ι · haveI : IsEmpty Ω := ⟨fun ω => IsEmpty.false (τ ω)⟩ intro s _ suffices hs : s = ∅ by rw [hs]; exact MeasurableSet.empty haveI : Unique (Set Ω) := Set.uniqueEmpty rw [Unique.eq_default s, Unique.eq_default ∅] exact measurableSpace_le' hτ #align measure_theory.is_stopping_time.measurable_space_le MeasureTheory.IsStoppingTime.measurableSpace_le example {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := hτ.measurableSpace_le example {f : Filtration ℝ m} {τ : Ω → ℝ} (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := hτ.measurableSpace_le @[simp] theorem measurableSpace_const (f : Filtration ι m) (i : ι) : (isStoppingTime_const f i).measurableSpace = f i := by ext1 s change MeasurableSet[(isStoppingTime_const f i).measurableSpace] s ↔ MeasurableSet[f i] s rw [IsStoppingTime.measurableSet] constructor <;> intro h · specialize h i simpa only [le_refl, Set.setOf_true, Set.inter_univ] using h · intro j by_cases hij : i ≤ j · simp only [hij, Set.setOf_true, Set.inter_univ] exact f.mono hij _ h · simp only [hij, Set.setOf_false, Set.inter_empty, @MeasurableSet.empty _ (f.1 j)] #align measure_theory.is_stopping_time.measurable_space_const MeasureTheory.IsStoppingTime.measurableSpace_const theorem measurableSet_inter_eq_iff (hτ : IsStoppingTime f τ) (s : Set Ω) (i : ι) : MeasurableSet[hτ.measurableSpace] (s ∩ {ω \| τ ω = i}) ↔ MeasurableSet[f i] (s ∩ {ω \| τ ω = i}) := by have : ∀ j, {ω : Ω \| τ ω = i} ∩ {ω : Ω \| τ ω ≤ j} = {ω : Ω \| τ ω = i} ∩ {_ω \| i ≤ j} := by intro j ext1 ω simp only [Set.mem_inter_iff, Set.mem_setOf_eq, and_congr_right_iff] intro hxi rw [hxi] constructor <;> intro h · specialize h i simpa only [Set.inter_assoc, this, le_refl, Set.setOf_true, Set.inter_univ] using h · intro j rw [Set.inter_assoc, this] by_cases hij : i ≤ j · simp only [hij, Set.setOf_true, Set.inter_univ] exact f.mono hij _ h · set_option tactic.skipAssignedInstances false in simp [hij] convert @MeasurableSet.empty _ (Filtration.seq f j) #align measure_theory.is_stopping_time.measurable_set_inter_eq_iff MeasureTheory.IsStoppingTime.measurableSet_inter_eq_iff theorem measurableSpace_le_of_le_const (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) : hτ.measurableSpace ≤ f i := (measurableSpace_mono hτ _ hτ_le).trans (measurableSpace_const _ _).le #align measure_theory.is_stopping_time.measurable_space_le_of_le_const MeasureTheory.IsStoppingTime.measurableSpace_le_of_le_const theorem measurableSpace_le_of_le (hτ : IsStoppingTime f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) : hτ.measurableSpace ≤ m := (hτ.measurableSpace_le_of_le_const hτ_le).trans (f.le n) #align measure_theory.is_stopping_time.measurable_space_le_of_le MeasureTheory.IsStoppingTime.measurableSpace_le_of_le theorem le_measurableSpace_of_const_le (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, i ≤ τ ω) : f i ≤ hτ.measurableSpace := (measurableSpace_const _ _).symm.le.trans (measurableSpace_mono _ hτ hτ_le) #align measure_theory.is_stopping_time.le_measurable_space_of_const_le MeasureTheory.IsStoppingTime.le_measurableSpace_of_const_le end Preorder instance sigmaFinite_stopping_time {ι} [SemilatticeSup ι] [OrderBot ι] [(Filter.atTop : Filter ι).IsCountablyGenerated] {μ : Measure Ω} {f : Filtration ι m} {τ : Ω → ι} [SigmaFiniteFiltration μ f] (hτ : IsStoppingTime f τ) : SigmaFinite (μ.trim hτ.measurableSpace_le) := by refine @sigmaFiniteTrim_mono _ _ ?_ _ _ _ ?_ ?_ · exact f ⊥ · exact hτ.le_measurableSpace_of_const_le fun _ => bot_le · infer_instance #align measure_theory.is_stopping_time.sigma_finite_stopping_time MeasureTheory.IsStoppingTime.sigmaFinite_stopping_time instance sigmaFinite_stopping_time_of_le {ι} [SemilatticeSup ι] [OrderBot ι] {μ : Measure Ω} {f : Filtration ι m} {τ : Ω → ι} [SigmaFiniteFiltration μ f] (hτ : IsStoppingTime f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) : SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le)) := by refine @sigmaFiniteTrim_mono _ _ ?_ _ _ _ ?_ ?_ · exact f ⊥ · exact hτ.le_measurableSpace_of_const_le fun _ => bot_le · infer_instance #align measure_theory.is_stopping_time.sigma_finite_stopping_time_of_le MeasureTheory.IsStoppingTime.sigmaFinite_stopping_time_of_le section LinearOrder variable [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} protected theorem measurableSet_le' (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω ≤ i} := by intro j have : {ω : Ω \| τ ω ≤ i} ∩ {ω : Ω \| τ ω ≤ j} = {ω : Ω \| τ ω ≤ min i j} := by ext1 ω; simp only [Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff] rw [this] exact f.mono (min_le_right i j) _ (hτ _) #align measure_theory.is_stopping_time.measurable_set_le' MeasureTheory.IsStoppingTime.measurableSet_le' protected theorem measurableSet_gt' (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i < τ ω} := by have : {ω : Ω \| i < τ ω} = {ω : Ω \| τ ω ≤ i}ᶜ := by ext1 ω; simp rw [this] exact (hτ.measurableSet_le' i).compl #align measure_theory.is_stopping_time.measurable_set_gt' MeasureTheory.IsStoppingTime.measurableSet_gt' protected theorem measurableSet_eq' [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω = i} := by rw [← Set.univ_inter {ω \| τ ω = i}, measurableSet_inter_eq_iff, Set.univ_inter] exact hτ.measurableSet_eq i #align measure_theory.is_stopping_time.measurable_set_eq' MeasureTheory.IsStoppingTime.measurableSet_eq' protected theorem measurableSet_ge' [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω = i} ∪ {ω \| i < τ ω} := by ext1 ω simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union] rw [@eq_comm _ i, or_comm] rw [this] exact (hτ.measurableSet_eq' i).union (hτ.measurableSet_gt' i) #align measure_theory.is_stopping_time.measurable_set_ge' MeasureTheory.IsStoppingTime.measurableSet_ge' protected theorem measurableSet_lt' [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω < i} := by have : {ω \| τ ω < i} = {ω \| τ ω ≤ i} \ {ω \| τ ω = i} := by ext1 ω simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] rw [this] exact (hτ.measurableSet_le' i).diff (hτ.measurableSet_eq' i) #align measure_theory.is_stopping_time.measurable_set_lt' MeasureTheory.IsStoppingTime.measurableSet_lt' section Countable protected theorem measurableSet_eq_of_countable_range' (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω = i} := by rw [← Set.univ_inter {ω \| τ ω = i}, measurableSet_inter_eq_iff, Set.univ_inter] exact hτ.measurableSet_eq_of_countable_range h_countable i #align measure_theory.is_stopping_time.measurable_set_eq_of_countable_range' MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range' protected theorem measurableSet_eq_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω = i} := hτ.measurableSet_eq_of_countable_range' (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_eq_of_countable' MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable' protected theorem measurableSet_ge_of_countable_range' (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω = i} ∪ {ω \| i < τ ω} := by ext1 ω simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union] rw [@eq_comm _ i, or_comm] rw [this] exact (hτ.measurableSet_eq_of_countable_range' h_countable i).union (hτ.measurableSet_gt' i) #align measure_theory.is_stopping_time.measurable_set_ge_of_countable_range' MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range' protected theorem measurableSet_ge_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i ≤ τ ω} := hτ.measurableSet_ge_of_countable_range' (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_ge_of_countable' MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable' protected theorem measurableSet_lt_of_countable_range' (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω < i} := by have : {ω \| τ ω < i} = {ω \| τ ω ≤ i} \ {ω \| τ ω = i} := by ext1 ω simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] rw [this] exact (hτ.measurableSet_le' i).diff (hτ.measurableSet_eq_of_countable_range' h_countable i) #align measure_theory.is_stopping_time.measurable_set_lt_of_countable_range' MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range' protected theorem measurableSet_lt_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω < i} := hτ.measurableSet_lt_of_countable_range' (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_lt_of_countable' MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable' protected theorem measurableSpace_le_of_countable_range (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) : hτ.measurableSpace ≤ m := by intro s hs change ∀ i, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) at hs rw [(_ : s = ⋃ i ∈ Set.range τ, s ∩ {ω \| τ ω ≤ i})] · exact MeasurableSet.biUnion h_countable fun i _ => f.le i _ (hs i) · ext ω constructor <;> rw [Set.mem_iUnion] · exact fun hx => ⟨τ ω, by simpa using hx⟩ · rintro ⟨i, hx⟩ simp only [Set.mem_range, Set.iUnion_exists, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_setOf_eq, exists_prop, exists_and_right] at hx exact hx.2.1 #align measure_theory.is_stopping_time.measurable_space_le_of_countable_range MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable_range end Countable protected theorem measurable [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) : Measurable[hτ.measurableSpace] τ := @measurable_of_Iic ι Ω _ _ _ hτ.measurableSpace _ _ _ _ fun i => hτ.measurableSet_le' i #align measure_theory.is_stopping_time.measurable MeasureTheory.IsStoppingTime.measurable protected theorem measurable_of_le [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) : Measurable[f i] τ := hτ.measurable.mono (measurableSpace_le_of_le_const _ hτ_le) le_rfl #align measure_theory.is_stopping_time.measurable_of_le MeasureTheory.IsStoppingTime.measurable_of_le theorem measurableSpace_min (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : (hτ.min hπ).measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace := by refine le_antisymm ?_ ?_ · exact le_inf (measurableSpace_mono _ hτ fun _ => min_le_left _ _) (measurableSpace_mono _ hπ fun _ => min_le_right _ _) · intro s change MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s → MeasurableSet[(hτ.min hπ).measurableSpace] s simp_rw [IsStoppingTime.measurableSet] have : ∀ i, {ω \| min (τ ω) (π ω) ≤ i} = {ω \| τ ω ≤ i} ∪ {ω \| π ω ≤ i} := by intro i; ext1 ω; simp simp_rw [this, Set.inter_union_distrib_left] exact fun h i => (h.left i).union (h.right i) #align measure_theory.is_stopping_time.measurable_space_min MeasureTheory.IsStoppingTime.measurableSpace_min theorem measurableSet_min_iff (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) (s : Set Ω) : MeasurableSet[(hτ.min hπ).measurableSpace] s ↔ MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s := by rw [measurableSpace_min hτ hπ]; rfl #align measure_theory.is_stopping_time.measurable_set_min_iff MeasureTheory.IsStoppingTime.measurableSet_min_iff theorem measurableSpace_min_const (hτ : IsStoppingTime f τ) {i : ι} : (hτ.min_const i).measurableSpace = hτ.measurableSpace ⊓ f i := by rw [hτ.measurableSpace_min (isStoppingTime_const _ i), measurableSpace_const] #align measure_theory.is_stopping_time.measurable_space_min_const MeasureTheory.IsStoppingTime.measurableSpace_min_const	Mathlib/Probability/Process/Stopping.lean	612	615	theorem measurableSet_min_const_iff (hτ : IsStoppingTime f τ) (s : Set Ω) {i : ι} : MeasurableSet[(hτ.min_const i).measurableSpace] s ↔ MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[f i] s := by	rw [measurableSpace_min_const hτ]; apply MeasurableSpace.measurableSet_inf
/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.LinearAlgebra.Quotient import Mathlib.RingTheory.Ideal.Operations /-! # The colon ideal This file defines `Submodule.colon N P` as the ideal of all elements `r : R` such that `r • P ⊆ N`. The normal notation for this would be `N : P` which has already been taken by type theory. -/ namespace Submodule open Pointwise variable {R M M' F G : Type} [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ ((Quotient.mk_eq_zero N).2 <\| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' @[simp] theorem colon_top {I : Ideal R} : I.colon ⊤ = I := by simp_rw [SetLike.ext_iff, mem_colon, smul_eq_mul] exact fun x ↦ ⟨fun h ↦ mul_one x ▸ h 1 trivial, fun h _ _ ↦ I.mul_mem_right _ h⟩ @[simp] theorem colon_bot : colon ⊥ N = N.annihilator := by simp_rw [SetLike.ext_iff, mem_colon, mem_annihilator, mem_bot, forall_const] theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <\| mem_colon.1 hrnp p₁ <\| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort) (f : ι₁ → Submodule R M) (ι₂ : Sort*) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <\| iSup_le fun j => map_le_iff_le_comap.1 <\| le_iInf fun i => map_le_iff_le_comap.2 <\| mem_colon'.1 <\| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp]	Mathlib/RingTheory/Ideal/Colon.lean	76	78	theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by	simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul]
/- Copyright (c) 2021 Thomas Browning. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Thomas Browning -/ import Mathlib.Algebra.BigOperators.GroupWithZero.Finset import Mathlib.Data.Finite.Card import Mathlib.GroupTheory.Finiteness import Mathlib.GroupTheory.GroupAction.Quotient #align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" /-! # Index of a Subgroup In this file we define the index of a subgroup, and prove several divisibility properties. Several theorems proved in this file are known as Lagrange's theorem. ## Main definitions - `H.index` : the index of `H : Subgroup G` as a natural number, and returns 0 if the index is infinite. - `H.relindex K` : the relative index of `H : Subgroup G` in `K : Subgroup G` as a natural number, and returns 0 if the relative index is infinite. # Main results - `card_mul_index` : `Nat.card H * H.index = Nat.card G` - `index_mul_card` : `H.index * Fintype.card H = Fintype.card G` - `index_dvd_card` : `H.index ∣ Fintype.card G` - `relindex_mul_index` : If `H ≤ K`, then `H.relindex K * K.index = H.index` - `index_dvd_of_le` : If `H ≤ K`, then `K.index ∣ H.index` - `relindex_mul_relindex` : `relindex` is multiplicative in towers -/ namespace Subgroup open Cardinal variable {G : Type} [Group G] (H K L : Subgroup G) /-- The index of a subgroup as a natural number, and returns 0 if the index is infinite. -/ @[to_additive "The index of a subgroup as a natural number, and returns 0 if the index is infinite."] noncomputable def index : ℕ := Nat.card (G ⧸ H) #align subgroup.index Subgroup.index #align add_subgroup.index AddSubgroup.index /-- The relative index of a subgroup as a natural number, and returns 0 if the relative index is infinite. -/ @[to_additive "The relative index of a subgroup as a natural number, and returns 0 if the relative index is infinite."] noncomputable def relindex : ℕ := (H.subgroupOf K).index #align subgroup.relindex Subgroup.relindex #align add_subgroup.relindex AddSubgroup.relindex @[to_additive] theorem index_comap_of_surjective {G' : Type} [Group G'] {f : G' →* G} (hf : Function.Surjective f) : (H.comap f).index = H.index := by letI := QuotientGroup.leftRel H letI := QuotientGroup.leftRel (H.comap f) have key : ∀ x y : G', Setoid.r x y ↔ Setoid.r (f x) (f y) := by simp only [QuotientGroup.leftRel_apply] exact fun x y => iff_of_eq (congr_arg (· ∈ H) (by rw [f.map_mul, f.map_inv])) refine Cardinal.toNat_congr (Equiv.ofBijective (Quotient.map' f fun x y => (key x y).mp) ⟨?_, ?_⟩) · simp_rw [← Quotient.eq''] at key refine Quotient.ind' fun x => ?_ refine Quotient.ind' fun y => ?_ exact (key x y).mpr · refine Quotient.ind' fun x => ?_ obtain ⟨y, hy⟩ := hf x exact ⟨y, (Quotient.map'_mk'' f _ y).trans (congr_arg Quotient.mk'' hy)⟩ #align subgroup.index_comap_of_surjective Subgroup.index_comap_of_surjective #align add_subgroup.index_comap_of_surjective AddSubgroup.index_comap_of_surjective @[to_additive] theorem index_comap {G' : Type} [Group G'] (f : G' → G) : (H.comap f).index = H.relindex f.range := Eq.trans (congr_arg index (by rfl)) ((H.subgroupOf f.range).index_comap_of_surjective f.rangeRestrict_surjective) #align subgroup.index_comap Subgroup.index_comap #align add_subgroup.index_comap AddSubgroup.index_comap @[to_additive] theorem relindex_comap {G' : Type} [Group G'] (f : G' → G) (K : Subgroup G') : relindex (comap f H) K = relindex H (map f K) := by rw [relindex, subgroupOf, comap_comap, index_comap, ← f.map_range, K.subtype_range] #align subgroup.relindex_comap Subgroup.relindex_comap #align add_subgroup.relindex_comap AddSubgroup.relindex_comap variable {H K L} @[to_additive relindex_mul_index] theorem relindex_mul_index (h : H ≤ K) : H.relindex K * K.index = H.index := ((mul_comm _ _).trans (Cardinal.toNat_mul _ _).symm).trans (congr_arg Cardinal.toNat (Equiv.cardinal_eq (quotientEquivProdOfLE h))).symm #align subgroup.relindex_mul_index Subgroup.relindex_mul_index #align add_subgroup.relindex_mul_index AddSubgroup.relindex_mul_index @[to_additive] theorem index_dvd_of_le (h : H ≤ K) : K.index ∣ H.index := dvd_of_mul_left_eq (H.relindex K) (relindex_mul_index h) #align subgroup.index_dvd_of_le Subgroup.index_dvd_of_le #align add_subgroup.index_dvd_of_le AddSubgroup.index_dvd_of_le @[to_additive] theorem relindex_dvd_index_of_le (h : H ≤ K) : H.relindex K ∣ H.index := dvd_of_mul_right_eq K.index (relindex_mul_index h) #align subgroup.relindex_dvd_index_of_le Subgroup.relindex_dvd_index_of_le #align add_subgroup.relindex_dvd_index_of_le AddSubgroup.relindex_dvd_index_of_le @[to_additive] theorem relindex_subgroupOf (hKL : K ≤ L) : (H.subgroupOf L).relindex (K.subgroupOf L) = H.relindex K := ((index_comap (H.subgroupOf L) (inclusion hKL)).trans (congr_arg _ (inclusion_range hKL))).symm #align subgroup.relindex_subgroup_of Subgroup.relindex_subgroupOf #align add_subgroup.relindex_add_subgroup_of AddSubgroup.relindex_addSubgroupOf variable (H K L) @[to_additive relindex_mul_relindex] theorem relindex_mul_relindex (hHK : H ≤ K) (hKL : K ≤ L) : H.relindex K * K.relindex L = H.relindex L := by rw [← relindex_subgroupOf hKL] exact relindex_mul_index fun x hx => hHK hx #align subgroup.relindex_mul_relindex Subgroup.relindex_mul_relindex #align add_subgroup.relindex_mul_relindex AddSubgroup.relindex_mul_relindex @[to_additive] theorem inf_relindex_right : (H ⊓ K).relindex K = H.relindex K := by rw [relindex, relindex, inf_subgroupOf_right] #align subgroup.inf_relindex_right Subgroup.inf_relindex_right #align add_subgroup.inf_relindex_right AddSubgroup.inf_relindex_right @[to_additive]	Mathlib/GroupTheory/Index.lean	140	141	theorem inf_relindex_left : (H ⊓ K).relindex H = K.relindex H := by	rw [inf_comm, inf_relindex_right]
/- Copyright (c) 2024 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Normed.Group.Basic import Mathlib.Topology.ContinuousFunction.CocompactMap /-! # Cocompact maps in normed groups This file gives a characterization of cocompact maps in terms of norm estimates. ## Main statements * `CocompactMapClass.norm_le`: Every cocompact map satisfies a norm estimate * `ContinuousMapClass.toCocompactMapClass_of_norm`: Conversely, this norm estimate implies that a map is cocompact. -/ open Filter Metric variable {𝕜 E F 𝓕 : Type*} variable [NormedAddCommGroup E] [NormedAddCommGroup F] [ProperSpace E] [ProperSpace F] variable {f : 𝓕}	Mathlib/Analysis/Normed/Group/CocompactMap.lean	29	39	theorem CocompactMapClass.norm_le [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F] (ε : ℝ) : ∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖ := by	have h := cocompact_tendsto f rw [tendsto_def] at h specialize h (Metric.closedBall 0 ε)ᶜ (mem_cocompact_of_closedBall_compl_subset 0 ⟨ε, rfl.subset⟩) rcases closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩ use r intro x hx suffices x ∈ f⁻¹' (Metric.closedBall 0 ε)ᶜ by aesop apply hr simp [hx]
/- Copyright (c) 2021 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import Mathlib.Analysis.Convex.Between import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic import Mathlib.MeasureTheory.Measure.Lebesgue.Basic import Mathlib.Topology.MetricSpace.Holder import Mathlib.Topology.MetricSpace.MetricSeparated #align_import measure_theory.measure.hausdorff from "leanprover-community/mathlib"@"3d5c4a7a5fb0d982f97ed953161264f1dbd90ead" /-! # Hausdorff measure and metric (outer) measures In this file we define the `d`-dimensional Hausdorff measure on an (extended) metric space `X` and the Hausdorff dimension of a set in an (extended) metric space. Let `μ d δ` be the maximal outer measure such that `μ d δ s ≤ (EMetric.diam s) ^ d` for every set of diameter less than `δ`. Then the Hausdorff measure `μH[d] s` of `s` is defined as `⨆ δ > 0, μ d δ s`. By Caratheodory theorem `MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory`, this is a Borel measure on `X`. The value of `μH[d]`, `d > 0`, on a set `s` (measurable or not) is given by ``` μH[d] s = ⨆ (r : ℝ≥0∞) (hr : 0 < r), ⨅ (t : ℕ → Set X) (hts : s ⊆ ⋃ n, t n) (ht : ∀ n, EMetric.diam (t n) ≤ r), ∑' n, EMetric.diam (t n) ^ d ``` For every set `s` for any `d < d'` we have either `μH[d] s = ∞` or `μH[d'] s = 0`, see `MeasureTheory.Measure.hausdorffMeasure_zero_or_top`. In `Mathlib.Topology.MetricSpace.HausdorffDimension` we use this fact to define the Hausdorff dimension `dimH` of a set in an (extended) metric space. We also define two generalizations of the Hausdorff measure. In one generalization (see `MeasureTheory.Measure.mkMetric`) we take any function `m (diam s)` instead of `(diam s) ^ d`. In an even more general definition (see `MeasureTheory.Measure.mkMetric'`) we use any function of `m : Set X → ℝ≥0∞`. Some authors start with a partial function `m` defined only on some sets `s : Set X` (e.g., only on balls or only on measurable sets). This is equivalent to our definition applied to `MeasureTheory.extend m`. We also define a predicate `MeasureTheory.OuterMeasure.IsMetric` which says that an outer measure is additive on metric separated pairs of sets: `μ (s ∪ t) = μ s + μ t` provided that `⨅ (x ∈ s) (y ∈ t), edist x y ≠ 0`. This is the property required for the Caratheodory theorem `MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory`, so we prove this theorem for any metric outer measure, then prove that outer measures constructed using `mkMetric'` are metric outer measures. ## Main definitions * `MeasureTheory.OuterMeasure.IsMetric`: an outer measure `μ` is called metric if `μ (s ∪ t) = μ s + μ t` for any two metric separated sets `s` and `t`. A metric outer measure in a Borel extended metric space is guaranteed to satisfy the Caratheodory condition, see `MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory`. * `MeasureTheory.OuterMeasure.mkMetric'` and its particular case `MeasureTheory.OuterMeasure.mkMetric`: a construction of an outer measure that is guaranteed to be metric. Both constructions are generalizations of the Hausdorff measure. The same measures interpreted as Borel measures are called `MeasureTheory.Measure.mkMetric'` and `MeasureTheory.Measure.mkMetric`. * `MeasureTheory.Measure.hausdorffMeasure` a.k.a. `μH[d]`: the `d`-dimensional Hausdorff measure. There are many definitions of the Hausdorff measure that differ from each other by a multiplicative constant. We put `μH[d] s = ⨆ r > 0, ⨅ (t : ℕ → Set X) (hts : s ⊆ ⋃ n, t n) (ht : ∀ n, EMetric.diam (t n) ≤ r), ∑' n, ⨆ (ht : ¬Set.Subsingleton (t n)), (EMetric.diam (t n)) ^ d`, see `MeasureTheory.Measure.hausdorffMeasure_apply`. In the most interesting case `0 < d` one can omit the `⨆ (ht : ¬Set.Subsingleton (t n))` part. ## Main statements ### Basic properties * `MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory`: if `μ` is a metric outer measure on an extended metric space `X` (that is, it is additive on pairs of metric separated sets), then every Borel set is Caratheodory measurable (hence, `μ` defines an actual `MeasureTheory.Measure`). See also `MeasureTheory.Measure.mkMetric`. * `MeasureTheory.Measure.hausdorffMeasure_mono`: `μH[d] s` is an antitone function of `d`. * `MeasureTheory.Measure.hausdorffMeasure_zero_or_top`: if `d₁ < d₂`, then for any `s`, either `μH[d₂] s = 0` or `μH[d₁] s = ∞`. Together with the previous lemma, this means that `μH[d] s` is equal to infinity on some ray `(-∞, D)` and is equal to zero on `(D, +∞)`, where `D` is a possibly infinite number called the Hausdorff dimension of `s`; `μH[D] s` can be zero, infinity, or anything in between. * `MeasureTheory.Measure.noAtoms_hausdorff`: Hausdorff measure has no atoms. ### Hausdorff measure in `ℝⁿ` * `MeasureTheory.hausdorffMeasure_pi_real`: for a nonempty `ι`, `μH[card ι]` on `ι → ℝ` equals Lebesgue measure. ## Notations We use the following notation localized in `MeasureTheory`. - `μH[d]` : `MeasureTheory.Measure.hausdorffMeasure d` ## Implementation notes There are a few similar constructions called the `d`-dimensional Hausdorff measure. E.g., some sources only allow coverings by balls and use `r ^ d` instead of `(diam s) ^ d`. While these construction lead to different Hausdorff measures, they lead to the same notion of the Hausdorff dimension. ## References * [Herbert Federer, Geometric Measure Theory, Chapter 2.10][Federer1996] ## Tags Hausdorff measure, measure, metric measure -/ open scoped NNReal ENNReal Topology open EMetric Set Function Filter Encodable FiniteDimensional TopologicalSpace noncomputable section variable {ι X Y : Type} [EMetricSpace X] [EMetricSpace Y] namespace MeasureTheory namespace OuterMeasure /-! ### Metric outer measures In this section we define metric outer measures and prove Caratheodory theorem: a metric outer measure has the Caratheodory property. -/ /-- We say that an outer measure `μ` in an (e)metric space is metric* if `μ (s ∪ t) = μ s + μ t` for any two metric separated sets `s`, `t`. -/ def IsMetric (μ : OuterMeasure X) : Prop := ∀ s t : Set X, IsMetricSeparated s t → μ (s ∪ t) = μ s + μ t #align measure_theory.outer_measure.is_metric MeasureTheory.OuterMeasure.IsMetric namespace IsMetric variable {μ : OuterMeasure X} /-- A metric outer measure is additive on a finite set of pairwise metric separated sets. -/ theorem finset_iUnion_of_pairwise_separated (hm : IsMetric μ) {I : Finset ι} {s : ι → Set X} (hI : ∀ i ∈ I, ∀ j ∈ I, i ≠ j → IsMetricSeparated (s i) (s j)) : μ (⋃ i ∈ I, s i) = ∑ i ∈ I, μ (s i) := by classical induction' I using Finset.induction_on with i I hiI ihI hI · simp simp only [Finset.mem_insert] at hI rw [Finset.set_biUnion_insert, hm, ihI, Finset.sum_insert hiI] exacts [fun i hi j hj hij => hI i (Or.inr hi) j (Or.inr hj) hij, IsMetricSeparated.finset_iUnion_right fun j hj => hI i (Or.inl rfl) j (Or.inr hj) (ne_of_mem_of_not_mem hj hiI).symm] #align measure_theory.outer_measure.is_metric.finset_Union_of_pairwise_separated MeasureTheory.OuterMeasure.IsMetric.finset_iUnion_of_pairwise_separated /-- Caratheodory theorem. If `m` is a metric outer measure, then every Borel measurable set `t` is Caratheodory measurable: for any (not necessarily measurable) set `s` we have `μ (s ∩ t) + μ (s \ t) = μ s`. -/ theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory := by rw [borel_eq_generateFrom_isClosed] refine MeasurableSpace.generateFrom_le fun t ht => μ.isCaratheodory_iff_le.2 fun s => ?_ set S : ℕ → Set X := fun n => {x ∈ s \| (↑n)⁻¹ ≤ infEdist x t} have Ssep (n) : IsMetricSeparated (S n) t := ⟨n⁻¹, ENNReal.inv_ne_zero.2 (ENNReal.natCast_ne_top _), fun x hx y hy ↦ hx.2.trans <\| infEdist_le_edist_of_mem hy⟩ have Ssep' : ∀ n, IsMetricSeparated (S n) (s ∩ t) := fun n => (Ssep n).mono Subset.rfl inter_subset_right have S_sub : ∀ n, S n ⊆ s \ t := fun n => subset_inter inter_subset_left (Ssep n).subset_compl_right have hSs : ∀ n, μ (s ∩ t) + μ (S n) ≤ μ s := fun n => calc μ (s ∩ t) + μ (S n) = μ (s ∩ t ∪ S n) := Eq.symm <\| hm _ _ <\| (Ssep' n).symm _ ≤ μ (s ∩ t ∪ s \ t) := μ.mono <\| union_subset_union_right _ <\| S_sub n _ = μ s := by rw [inter_union_diff] have iUnion_S : ⋃ n, S n = s \ t := by refine Subset.antisymm (iUnion_subset S_sub) ?_ rintro x ⟨hxs, hxt⟩ rw [mem_iff_infEdist_zero_of_closed ht] at hxt rcases ENNReal.exists_inv_nat_lt hxt with ⟨n, hn⟩ exact mem_iUnion.2 ⟨n, hxs, hn.le⟩ /- Now we have `∀ n, μ (s ∩ t) + μ (S n) ≤ μ s` and we need to prove `μ (s ∩ t) + μ (⋃ n, S n) ≤ μ s`. We can't pass to the limit because `μ` is only an outer measure. -/ by_cases htop : μ (s \ t) = ∞ · rw [htop, add_top, ← htop] exact μ.mono diff_subset suffices μ (⋃ n, S n) ≤ ⨆ n, μ (S n) by calc μ (s ∩ t) + μ (s \ t) = μ (s ∩ t) + μ (⋃ n, S n) := by rw [iUnion_S] _ ≤ μ (s ∩ t) + ⨆ n, μ (S n) := by gcongr _ = ⨆ n, μ (s ∩ t) + μ (S n) := ENNReal.add_iSup _ ≤ μ s := iSup_le hSs /- It suffices to show that `∑' k, μ (S (k + 1) \ S k) ≠ ∞`. Indeed, if we have this, then for all `N` we have `μ (⋃ n, S n) ≤ μ (S N) + ∑' k, m (S (N + k + 1) \ S (N + k))` and the second term tends to zero, see `OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top` for details. -/ have : ∀ n, S n ⊆ S (n + 1) := fun n x hx => ⟨hx.1, le_trans (ENNReal.inv_le_inv.2 <\| Nat.cast_le.2 n.le_succ) hx.2⟩ classical -- Porting note: Added this to get the next tactic to work refine (μ.iUnion_nat_of_monotone_of_tsum_ne_top this ?_).le; clear this /- While the sets `S (k + 1) \ S k` are not pairwise metric separated, the sets in each subsequence `S (2 * k + 1) \ S (2 * k)` and `S (2 * k + 2) \ S (2 * k)` are metric separated, so `m` is additive on each of those sequences. -/ rw [← tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top] suffices ∀ a, (∑' k : ℕ, μ (S (2 * k + 1 + a) \ S (2 * k + a))) ≠ ∞ from ⟨by simpa using this 0, by simpa using this 1⟩ refine fun r => ne_top_of_le_ne_top htop ?_ rw [← iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff] intro n rw [← hm.finset_iUnion_of_pairwise_separated] · exact μ.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 ⟨_, hx.1⟩) suffices ∀ i j, i < j → IsMetricSeparated (S (2 * i + 1 + r)) (s \ S (2 * j + r)) from fun i _ j _ hij => hij.lt_or_lt.elim (fun h => (this i j h).mono inter_subset_left fun x hx => by exact ⟨hx.1.1, hx.2⟩) fun h => (this j i h).symm.mono (fun x hx => by exact ⟨hx.1.1, hx.2⟩) inter_subset_left intro i j hj have A : ((↑(2 * j + r))⁻¹ : ℝ≥0∞) < (↑(2 * i + 1 + r))⁻¹ := by rw [ENNReal.inv_lt_inv, Nat.cast_lt]; omega refine ⟨(↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹, by simpa [tsub_eq_zero_iff_le] using A, fun x hx y hy => ?_⟩ have : infEdist y t < (↑(2 * j + r))⁻¹ := not_le.1 fun hle => hy.2 ⟨hy.1, hle⟩ rcases infEdist_lt_iff.mp this with ⟨z, hzt, hyz⟩ have hxz : (↑(2 * i + 1 + r))⁻¹ ≤ edist x z := le_infEdist.1 hx.2 _ hzt apply ENNReal.le_of_add_le_add_right hyz.ne_top refine le_trans ?_ (edist_triangle _ _ _) refine (add_le_add le_rfl hyz.le).trans (Eq.trans_le ?_ hxz) rw [tsub_add_cancel_of_le A.le] #align measure_theory.outer_measure.is_metric.borel_le_caratheodory MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory theorem le_caratheodory [MeasurableSpace X] [BorelSpace X] (hm : IsMetric μ) : ‹MeasurableSpace X› ≤ μ.caratheodory := by rw [BorelSpace.measurable_eq (α := X)] exact hm.borel_le_caratheodory #align measure_theory.outer_measure.is_metric.le_caratheodory MeasureTheory.OuterMeasure.IsMetric.le_caratheodory end IsMetric /-! ### Constructors of metric outer measures In this section we provide constructors `MeasureTheory.OuterMeasure.mkMetric'` and `MeasureTheory.OuterMeasure.mkMetric` and prove that these outer measures are metric outer measures. We also prove basic lemmas about `map`/`comap` of these measures. -/ /-- Auxiliary definition for `OuterMeasure.mkMetric'`: given a function on sets `m : Set X → ℝ≥0∞`, returns the maximal outer measure `μ` such that `μ s ≤ m s` for any set `s` of diameter at most `r`. -/ def mkMetric'.pre (m : Set X → ℝ≥0∞) (r : ℝ≥0∞) : OuterMeasure X := boundedBy <\| extend fun s (_ : diam s ≤ r) => m s #align measure_theory.outer_measure.mk_metric'.pre MeasureTheory.OuterMeasure.mkMetric'.pre /-- Given a function `m : Set X → ℝ≥0∞`, `mkMetric' m` is the supremum of `mkMetric'.pre m r` over `r > 0`. Equivalently, it is the limit of `mkMetric'.pre m r` as `r` tends to zero from the right. -/ def mkMetric' (m : Set X → ℝ≥0∞) : OuterMeasure X := ⨆ r > 0, mkMetric'.pre m r #align measure_theory.outer_measure.mk_metric' MeasureTheory.OuterMeasure.mkMetric' /-- Given a function `m : ℝ≥0∞ → ℝ≥0∞` and `r > 0`, let `μ r` be the maximal outer measure such that `μ s ≤ m (EMetric.diam s)` whenever `EMetric.diam s < r`. Then `mkMetric m = ⨆ r > 0, μ r`. -/ def mkMetric (m : ℝ≥0∞ → ℝ≥0∞) : OuterMeasure X := mkMetric' fun s => m (diam s) #align measure_theory.outer_measure.mk_metric MeasureTheory.OuterMeasure.mkMetric namespace mkMetric' variable {m : Set X → ℝ≥0∞} {r : ℝ≥0∞} {μ : OuterMeasure X} {s : Set X} theorem le_pre : μ ≤ pre m r ↔ ∀ s : Set X, diam s ≤ r → μ s ≤ m s := by simp only [pre, le_boundedBy, extend, le_iInf_iff] #align measure_theory.outer_measure.mk_metric'.le_pre MeasureTheory.OuterMeasure.mkMetric'.le_pre theorem pre_le (hs : diam s ≤ r) : pre m r s ≤ m s := (boundedBy_le _).trans <\| iInf_le _ hs #align measure_theory.outer_measure.mk_metric'.pre_le MeasureTheory.OuterMeasure.mkMetric'.pre_le theorem mono_pre (m : Set X → ℝ≥0∞) {r r' : ℝ≥0∞} (h : r ≤ r') : pre m r' ≤ pre m r := le_pre.2 fun _ hs => pre_le (hs.trans h) #align measure_theory.outer_measure.mk_metric'.mono_pre MeasureTheory.OuterMeasure.mkMetric'.mono_pre theorem mono_pre_nat (m : Set X → ℝ≥0∞) : Monotone fun k : ℕ => pre m k⁻¹ := fun k l h => le_pre.2 fun s hs => pre_le (hs.trans <\| by simpa) #align measure_theory.outer_measure.mk_metric'.mono_pre_nat MeasureTheory.OuterMeasure.mkMetric'.mono_pre_nat theorem tendsto_pre (m : Set X → ℝ≥0∞) (s : Set X) : Tendsto (fun r => pre m r s) (𝓝[>] 0) (𝓝 <\| mkMetric' m s) := by rw [← map_coe_Ioi_atBot, tendsto_map'_iff] simp only [mkMetric', OuterMeasure.iSup_apply, iSup_subtype'] exact tendsto_atBot_iSup fun r r' hr => mono_pre _ hr _ #align measure_theory.outer_measure.mk_metric'.tendsto_pre MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre theorem tendsto_pre_nat (m : Set X → ℝ≥0∞) (s : Set X) : Tendsto (fun n : ℕ => pre m n⁻¹ s) atTop (𝓝 <\| mkMetric' m s) := by refine (tendsto_pre m s).comp (tendsto_inf.2 ⟨ENNReal.tendsto_inv_nat_nhds_zero, ?_⟩) refine tendsto_principal.2 (eventually_of_forall fun n => ?_) simp #align measure_theory.outer_measure.mk_metric'.tendsto_pre_nat MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre_nat theorem eq_iSup_nat (m : Set X → ℝ≥0∞) : mkMetric' m = ⨆ n : ℕ, mkMetric'.pre m n⁻¹ := by ext1 s rw [iSup_apply] refine tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s) (tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s) #align measure_theory.outer_measure.mk_metric'.eq_supr_nat MeasureTheory.OuterMeasure.mkMetric'.eq_iSup_nat /-- `MeasureTheory.OuterMeasure.mkMetric'.pre m r` is a trimmed measure provided that `m (closure s) = m s` for any set `s`. -/ theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞) (hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r := by refine le_antisymm (le_pre.2 fun s hs => ?_) (le_trim _) rw [trim_eq_iInf] refine iInf_le_of_le (closure s) <\| iInf_le_of_le subset_closure <\| iInf_le_of_le measurableSet_closure ((pre_le ?_).trans_eq (hcl _)) rwa [diam_closure] #align measure_theory.outer_measure.mk_metric'.trim_pre MeasureTheory.OuterMeasure.mkMetric'.trim_pre end mkMetric' /-- An outer measure constructed using `OuterMeasure.mkMetric'` is a metric outer measure. -/ theorem mkMetric'_isMetric (m : Set X → ℝ≥0∞) : (mkMetric' m).IsMetric := by rintro s t ⟨r, r0, hr⟩ refine tendsto_nhds_unique_of_eventuallyEq (mkMetric'.tendsto_pre _ _) ((mkMetric'.tendsto_pre _ _).add (mkMetric'.tendsto_pre _ _)) ?_ rw [← pos_iff_ne_zero] at r0 filter_upwards [Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, r0⟩] rintro ε ⟨_, εr⟩ refine boundedBy_union_of_top_of_nonempty_inter ?_ rintro u ⟨x, hxs, hxu⟩ ⟨y, hyt, hyu⟩ have : ε < diam u := εr.trans_le ((hr x hxs y hyt).trans <\| edist_le_diam_of_mem hxu hyu) exact iInf_eq_top.2 fun h => (this.not_le h).elim #align measure_theory.outer_measure.mk_metric'_is_metric MeasureTheory.OuterMeasure.mkMetric'_isMetric /-- If `c ∉ {0, ∞}` and `m₁ d ≤ c * m₂ d` for `d < ε` for some `ε > 0` (we use `≤ᶠ[𝓝[≥] 0]` to state this), then `mkMetric m₁ hm₁ ≤ c • mkMetric m₂ hm₂`. -/	Mathlib/MeasureTheory/Measure/Hausdorff.lean	336	351	theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0) (hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) : (mkMetric m₁ : OuterMeasure X) ≤ c • mkMetric m₂ := by	classical rcases (mem_nhdsWithin_Ici_iff_exists_Ico_subset' zero_lt_one).1 hle with ⟨r, hr0, hr⟩ refine fun s => le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s) (ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc)) (mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, hr0⟩) fun r' hr' => ?_) simp only [mem_setOf_eq, mkMetric'.pre, RingHom.id_apply] rw [← smul_eq_mul, ← smul_apply, smul_boundedBy hc] refine le_boundedBy.2 (fun t => (boundedBy_le _).trans ?_) _ simp only [smul_eq_mul, Pi.smul_apply, extend, iInf_eq_if] split_ifs with ht · apply hr exact ⟨zero_le _, ht.trans_lt hr'.2⟩ · simp [h0]
/- Copyright (c) 2021 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Algebra.GroupWithZero.Indicator import Mathlib.Topology.ContinuousOn import Mathlib.Topology.Instances.ENNReal #align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" /-! # Semicontinuous maps A function `f` from a topological space `α` to an ordered space `β` is lower semicontinuous at a point `x` if, for any `y < f x`, for any `x'` close enough to `x`, one has `f x' > y`. In other words, `f` can jump up, but it can not jump down. Upper semicontinuous functions are defined similarly. This file introduces these notions, and a basic API around them mimicking the API for continuous functions. ## Main definitions and results We introduce 4 definitions related to lower semicontinuity: * `LowerSemicontinuousWithinAt f s x` * `LowerSemicontinuousAt f x` * `LowerSemicontinuousOn f s` * `LowerSemicontinuous f` We build a basic API using dot notation around these notions, and we prove that * constant functions are lower semicontinuous; * `indicator s (fun _ ↦ y)` is lower semicontinuous when `s` is open and `0 ≤ y`, or when `s` is closed and `y ≤ 0`; * continuous functions are lower semicontinuous; * left composition with a continuous monotone functions maps lower semicontinuous functions to lower semicontinuous functions. If the function is anti-monotone, it instead maps lower semicontinuous functions to upper semicontinuous functions; * right composition with continuous functions preserves lower and upper semicontinuity; * a sum of two (or finitely many) lower semicontinuous functions is lower semicontinuous; * a supremum of a family of lower semicontinuous functions is lower semicontinuous; * An infinite sum of `ℝ≥0∞`-valued lower semicontinuous functions is lower semicontinuous. Similar results are stated and proved for upper semicontinuity. We also prove that a function is continuous if and only if it is both lower and upper semicontinuous. We have some equivalent definitions of lower- and upper-semicontinuity (under certain restrictions on the order on the codomain): * `lowerSemicontinuous_iff_isOpen_preimage` in a linear order; * `lowerSemicontinuous_iff_isClosed_preimage` in a linear order; * `lowerSemicontinuousAt_iff_le_liminf` in a dense complete linear order; * `lowerSemicontinuous_iff_isClosed_epigraph` in a dense complete linear order with the order topology. ## Implementation details All the nontrivial results for upper semicontinuous functions are deduced from the corresponding ones for lower semicontinuous functions using `OrderDual`. ## References * <https://en.wikipedia.org/wiki/Closed_convex_function> * <https://en.wikipedia.org/wiki/Semi-continuity> -/ open Topology ENNReal open Set Function Filter variable {α : Type} [TopologicalSpace α] {β : Type} [Preorder β] {f g : α → β} {x : α} {s t : Set α} {y z : β} /-! ### Main definitions -/ /-- A real function `f` is lower semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) := ∀ y < f x, ∀ᶠ x' in 𝓝[s] x, y < f x' #align lower_semicontinuous_within_at LowerSemicontinuousWithinAt /-- A real function `f` is lower semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`, for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuousOn (f : α → β) (s : Set α) := ∀ x ∈ s, LowerSemicontinuousWithinAt f s x #align lower_semicontinuous_on LowerSemicontinuousOn /-- A real function `f` is lower semicontinuous at `x` if, for any `ε > 0`, for all `x'` close enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuousAt (f : α → β) (x : α) := ∀ y < f x, ∀ᶠ x' in 𝓝 x, y < f x' #align lower_semicontinuous_at LowerSemicontinuousAt /-- A real function `f` is lower semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ def LowerSemicontinuous (f : α → β) := ∀ x, LowerSemicontinuousAt f x #align lower_semicontinuous LowerSemicontinuous /-- A real function `f` is upper semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuousWithinAt (f : α → β) (s : Set α) (x : α) := ∀ y, f x < y → ∀ᶠ x' in 𝓝[s] x, f x' < y #align upper_semicontinuous_within_at UpperSemicontinuousWithinAt /-- A real function `f` is upper semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`, for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuousOn (f : α → β) (s : Set α) := ∀ x ∈ s, UpperSemicontinuousWithinAt f s x #align upper_semicontinuous_on UpperSemicontinuousOn /-- A real function `f` is upper semicontinuous at `x` if, for any `ε > 0`, for all `x'` close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuousAt (f : α → β) (x : α) := ∀ y, f x < y → ∀ᶠ x' in 𝓝 x, f x' < y #align upper_semicontinuous_at UpperSemicontinuousAt /-- A real function `f` is upper semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ def UpperSemicontinuous (f : α → β) := ∀ x, UpperSemicontinuousAt f x #align upper_semicontinuous UpperSemicontinuous /-! ### Lower semicontinuous functions -/ /-! #### Basic dot notation interface for lower semicontinuity -/ theorem LowerSemicontinuousWithinAt.mono (h : LowerSemicontinuousWithinAt f s x) (hst : t ⊆ s) : LowerSemicontinuousWithinAt f t x := fun y hy => Filter.Eventually.filter_mono (nhdsWithin_mono _ hst) (h y hy) #align lower_semicontinuous_within_at.mono LowerSemicontinuousWithinAt.mono theorem lowerSemicontinuousWithinAt_univ_iff : LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x := by simp [LowerSemicontinuousWithinAt, LowerSemicontinuousAt, nhdsWithin_univ] #align lower_semicontinuous_within_at_univ_iff lowerSemicontinuousWithinAt_univ_iff theorem LowerSemicontinuousAt.lowerSemicontinuousWithinAt (s : Set α) (h : LowerSemicontinuousAt f x) : LowerSemicontinuousWithinAt f s x := fun y hy => Filter.Eventually.filter_mono nhdsWithin_le_nhds (h y hy) #align lower_semicontinuous_at.lower_semicontinuous_within_at LowerSemicontinuousAt.lowerSemicontinuousWithinAt theorem LowerSemicontinuousOn.lowerSemicontinuousWithinAt (h : LowerSemicontinuousOn f s) (hx : x ∈ s) : LowerSemicontinuousWithinAt f s x := h x hx #align lower_semicontinuous_on.lower_semicontinuous_within_at LowerSemicontinuousOn.lowerSemicontinuousWithinAt theorem LowerSemicontinuousOn.mono (h : LowerSemicontinuousOn f s) (hst : t ⊆ s) : LowerSemicontinuousOn f t := fun x hx => (h x (hst hx)).mono hst #align lower_semicontinuous_on.mono LowerSemicontinuousOn.mono theorem lowerSemicontinuousOn_univ_iff : LowerSemicontinuousOn f univ ↔ LowerSemicontinuous f := by simp [LowerSemicontinuousOn, LowerSemicontinuous, lowerSemicontinuousWithinAt_univ_iff] #align lower_semicontinuous_on_univ_iff lowerSemicontinuousOn_univ_iff theorem LowerSemicontinuous.lowerSemicontinuousAt (h : LowerSemicontinuous f) (x : α) : LowerSemicontinuousAt f x := h x #align lower_semicontinuous.lower_semicontinuous_at LowerSemicontinuous.lowerSemicontinuousAt theorem LowerSemicontinuous.lowerSemicontinuousWithinAt (h : LowerSemicontinuous f) (s : Set α) (x : α) : LowerSemicontinuousWithinAt f s x := (h x).lowerSemicontinuousWithinAt s #align lower_semicontinuous.lower_semicontinuous_within_at LowerSemicontinuous.lowerSemicontinuousWithinAt theorem LowerSemicontinuous.lowerSemicontinuousOn (h : LowerSemicontinuous f) (s : Set α) : LowerSemicontinuousOn f s := fun x _hx => h.lowerSemicontinuousWithinAt s x #align lower_semicontinuous.lower_semicontinuous_on LowerSemicontinuous.lowerSemicontinuousOn /-! #### Constants -/ theorem lowerSemicontinuousWithinAt_const : LowerSemicontinuousWithinAt (fun _x => z) s x := fun _y hy => Filter.eventually_of_forall fun _x => hy #align lower_semicontinuous_within_at_const lowerSemicontinuousWithinAt_const theorem lowerSemicontinuousAt_const : LowerSemicontinuousAt (fun _x => z) x := fun _y hy => Filter.eventually_of_forall fun _x => hy #align lower_semicontinuous_at_const lowerSemicontinuousAt_const theorem lowerSemicontinuousOn_const : LowerSemicontinuousOn (fun _x => z) s := fun _x _hx => lowerSemicontinuousWithinAt_const #align lower_semicontinuous_on_const lowerSemicontinuousOn_const theorem lowerSemicontinuous_const : LowerSemicontinuous fun _x : α => z := fun _x => lowerSemicontinuousAt_const #align lower_semicontinuous_const lowerSemicontinuous_const /-! #### Indicators -/ section variable [Zero β] theorem IsOpen.lowerSemicontinuous_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuous (indicator s fun _x => y) := by intro x z hz by_cases h : x ∈ s <;> simp [h] at hz · filter_upwards [hs.mem_nhds h] simp (config := { contextual := true }) [hz] · refine Filter.eventually_of_forall fun x' => ?_ by_cases h' : x' ∈ s <;> simp [h', hz.trans_le hy, hz] #align is_open.lower_semicontinuous_indicator IsOpen.lowerSemicontinuous_indicator theorem IsOpen.lowerSemicontinuousOn_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuousOn (indicator s fun _x => y) t := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousOn t #align is_open.lower_semicontinuous_on_indicator IsOpen.lowerSemicontinuousOn_indicator theorem IsOpen.lowerSemicontinuousAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuousAt (indicator s fun _x => y) x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x #align is_open.lower_semicontinuous_at_indicator IsOpen.lowerSemicontinuousAt_indicator theorem IsOpen.lowerSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) : LowerSemicontinuousWithinAt (indicator s fun _x => y) t x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousWithinAt t x #align is_open.lower_semicontinuous_within_at_indicator IsOpen.lowerSemicontinuousWithinAt_indicator theorem IsClosed.lowerSemicontinuous_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuous (indicator s fun _x => y) := by intro x z hz by_cases h : x ∈ s <;> simp [h] at hz · refine Filter.eventually_of_forall fun x' => ?_ by_cases h' : x' ∈ s <;> simp [h', hz, hz.trans_le hy] · filter_upwards [hs.isOpen_compl.mem_nhds h] simp (config := { contextual := true }) [hz] #align is_closed.lower_semicontinuous_indicator IsClosed.lowerSemicontinuous_indicator theorem IsClosed.lowerSemicontinuousOn_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuousOn (indicator s fun _x => y) t := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousOn t #align is_closed.lower_semicontinuous_on_indicator IsClosed.lowerSemicontinuousOn_indicator theorem IsClosed.lowerSemicontinuousAt_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuousAt (indicator s fun _x => y) x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousAt x #align is_closed.lower_semicontinuous_at_indicator IsClosed.lowerSemicontinuousAt_indicator theorem IsClosed.lowerSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuousWithinAt (indicator s fun _x => y) t x := (hs.lowerSemicontinuous_indicator hy).lowerSemicontinuousWithinAt t x #align is_closed.lower_semicontinuous_within_at_indicator IsClosed.lowerSemicontinuousWithinAt_indicator end /-! #### Relationship with continuity -/ theorem lowerSemicontinuous_iff_isOpen_preimage : LowerSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Ioi y) := ⟨fun H y => isOpen_iff_mem_nhds.2 fun x hx => H x y hx, fun H _x y y_lt => IsOpen.mem_nhds (H y) y_lt⟩ #align lower_semicontinuous_iff_is_open_preimage lowerSemicontinuous_iff_isOpen_preimage theorem LowerSemicontinuous.isOpen_preimage (hf : LowerSemicontinuous f) (y : β) : IsOpen (f ⁻¹' Ioi y) := lowerSemicontinuous_iff_isOpen_preimage.1 hf y #align lower_semicontinuous.is_open_preimage LowerSemicontinuous.isOpen_preimage section variable {γ : Type} [LinearOrder γ] theorem lowerSemicontinuous_iff_isClosed_preimage {f : α → γ} : LowerSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Iic y) := by rw [lowerSemicontinuous_iff_isOpen_preimage] simp only [← isOpen_compl_iff, ← preimage_compl, compl_Iic] #align lower_semicontinuous_iff_is_closed_preimage lowerSemicontinuous_iff_isClosed_preimage theorem LowerSemicontinuous.isClosed_preimage {f : α → γ} (hf : LowerSemicontinuous f) (y : γ) : IsClosed (f ⁻¹' Iic y) := lowerSemicontinuous_iff_isClosed_preimage.1 hf y #align lower_semicontinuous.is_closed_preimage LowerSemicontinuous.isClosed_preimage variable [TopologicalSpace γ] [OrderTopology γ] theorem ContinuousWithinAt.lowerSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) : LowerSemicontinuousWithinAt f s x := fun _y hy => h (Ioi_mem_nhds hy) #align continuous_within_at.lower_semicontinuous_within_at ContinuousWithinAt.lowerSemicontinuousWithinAt theorem ContinuousAt.lowerSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) : LowerSemicontinuousAt f x := fun _y hy => h (Ioi_mem_nhds hy) #align continuous_at.lower_semicontinuous_at ContinuousAt.lowerSemicontinuousAt theorem ContinuousOn.lowerSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) : LowerSemicontinuousOn f s := fun x hx => (h x hx).lowerSemicontinuousWithinAt #align continuous_on.lower_semicontinuous_on ContinuousOn.lowerSemicontinuousOn theorem Continuous.lowerSemicontinuous {f : α → γ} (h : Continuous f) : LowerSemicontinuous f := fun _x => h.continuousAt.lowerSemicontinuousAt #align continuous.lower_semicontinuous Continuous.lowerSemicontinuous end /-! #### Equivalent definitions -/ section variable {γ : Type} [CompleteLinearOrder γ] [DenselyOrdered γ] theorem lowerSemicontinuousWithinAt_iff_le_liminf {f : α → γ} : LowerSemicontinuousWithinAt f s x ↔ f x ≤ liminf f (𝓝[s] x) := by constructor · intro hf; unfold LowerSemicontinuousWithinAt at hf contrapose! hf obtain ⟨y, lty, ylt⟩ := exists_between hf; use y exact ⟨ylt, fun h => lty.not_le (le_liminf_of_le (by isBoundedDefault) (h.mono fun _ hx => le_of_lt hx))⟩ exact fun hf y ylt => eventually_lt_of_lt_liminf (ylt.trans_le hf) alias ⟨LowerSemicontinuousWithinAt.le_liminf, _⟩ := lowerSemicontinuousWithinAt_iff_le_liminf theorem lowerSemicontinuousAt_iff_le_liminf {f : α → γ} : LowerSemicontinuousAt f x ↔ f x ≤ liminf f (𝓝 x) := by rw [← lowerSemicontinuousWithinAt_univ_iff, lowerSemicontinuousWithinAt_iff_le_liminf, ← nhdsWithin_univ] alias ⟨LowerSemicontinuousAt.le_liminf, _⟩ := lowerSemicontinuousAt_iff_le_liminf theorem lowerSemicontinuous_iff_le_liminf {f : α → γ} : LowerSemicontinuous f ↔ ∀ x, f x ≤ liminf f (𝓝 x) := by simp only [← lowerSemicontinuousAt_iff_le_liminf, LowerSemicontinuous] alias ⟨LowerSemicontinuous.le_liminf, _⟩ := lowerSemicontinuous_iff_le_liminf theorem lowerSemicontinuousOn_iff_le_liminf {f : α → γ} : LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ liminf f (𝓝[s] x) := by simp only [← lowerSemicontinuousWithinAt_iff_le_liminf, LowerSemicontinuousOn] alias ⟨LowerSemicontinuousOn.le_liminf, _⟩ := lowerSemicontinuousOn_iff_le_liminf variable [TopologicalSpace γ] [OrderTopology γ] theorem lowerSemicontinuous_iff_isClosed_epigraph {f : α → γ} : LowerSemicontinuous f ↔ IsClosed {p : α × γ \| f p.1 ≤ p.2} := by constructor · rw [lowerSemicontinuous_iff_le_liminf, isClosed_iff_forall_filter] rintro hf ⟨x, y⟩ F F_ne h h' rw [nhds_prod_eq, le_prod] at h' calc f x ≤ liminf f (𝓝 x) := hf x _ ≤ liminf f (map Prod.fst F) := liminf_le_liminf_of_le h'.1 _ = liminf (f ∘ Prod.fst) F := (Filter.liminf_comp _ _ _).symm _ ≤ liminf Prod.snd F := liminf_le_liminf <\| by simpa using (eventually_principal.2 fun (_ : α × γ) ↦ id).filter_mono h _ = y := h'.2.liminf_eq · rw [lowerSemicontinuous_iff_isClosed_preimage] exact fun hf y ↦ hf.preimage (Continuous.Prod.mk_left y) @[deprecated (since := "2024-03-02")] alias lowerSemicontinuous_iff_IsClosed_epigraph := lowerSemicontinuous_iff_isClosed_epigraph alias ⟨LowerSemicontinuous.isClosed_epigraph, _⟩ := lowerSemicontinuous_iff_isClosed_epigraph @[deprecated (since := "2024-03-02")] alias LowerSemicontinuous.IsClosed_epigraph := LowerSemicontinuous.isClosed_epigraph end /-! ### Composition -/ section variable {γ : Type} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] variable {δ : Type} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] variable {ι : Type} [TopologicalSpace ι] theorem ContinuousAt.comp_lowerSemicontinuousWithinAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Monotone g) : LowerSemicontinuousWithinAt (g ∘ f) s x := by intro y hy by_cases h : ∃ l, l < f x · obtain ⟨z, zlt, hz⟩ : ∃ z < f x, Ioc z (f x) ⊆ g ⁻¹' Ioi y := exists_Ioc_subset_of_mem_nhds (hg (Ioi_mem_nhds hy)) h filter_upwards [hf z zlt] with a ha calc y < g (min (f x) (f a)) := hz (by simp [zlt, ha, le_refl]) _ ≤ g (f a) := gmon (min_le_right _ _) · simp only [not_exists, not_lt] at h exact Filter.eventually_of_forall fun a => hy.trans_le (gmon (h (f a))) #align continuous_at.comp_lower_semicontinuous_within_at ContinuousAt.comp_lowerSemicontinuousWithinAt theorem ContinuousAt.comp_lowerSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousAt f x) (gmon : Monotone g) : LowerSemicontinuousAt (g ∘ f) x := by simp only [← lowerSemicontinuousWithinAt_univ_iff] at hf ⊢ exact hg.comp_lowerSemicontinuousWithinAt hf gmon #align continuous_at.comp_lower_semicontinuous_at ContinuousAt.comp_lowerSemicontinuousAt theorem Continuous.comp_lowerSemicontinuousOn {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuousOn f s) (gmon : Monotone g) : LowerSemicontinuousOn (g ∘ f) s := fun x hx => hg.continuousAt.comp_lowerSemicontinuousWithinAt (hf x hx) gmon #align continuous.comp_lower_semicontinuous_on Continuous.comp_lowerSemicontinuousOn theorem Continuous.comp_lowerSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuous f) (gmon : Monotone g) : LowerSemicontinuous (g ∘ f) := fun x => hg.continuousAt.comp_lowerSemicontinuousAt (hf x) gmon #align continuous.comp_lower_semicontinuous Continuous.comp_lowerSemicontinuous theorem ContinuousAt.comp_lowerSemicontinuousWithinAt_antitone {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Antitone g) : UpperSemicontinuousWithinAt (g ∘ f) s x := @ContinuousAt.comp_lowerSemicontinuousWithinAt α _ x s γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon #align continuous_at.comp_lower_semicontinuous_within_at_antitone ContinuousAt.comp_lowerSemicontinuousWithinAt_antitone theorem ContinuousAt.comp_lowerSemicontinuousAt_antitone {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousAt f x) (gmon : Antitone g) : UpperSemicontinuousAt (g ∘ f) x := @ContinuousAt.comp_lowerSemicontinuousAt α _ x γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon #align continuous_at.comp_lower_semicontinuous_at_antitone ContinuousAt.comp_lowerSemicontinuousAt_antitone theorem Continuous.comp_lowerSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuousOn f s) (gmon : Antitone g) : UpperSemicontinuousOn (g ∘ f) s := fun x hx => hg.continuousAt.comp_lowerSemicontinuousWithinAt_antitone (hf x hx) gmon #align continuous.comp_lower_semicontinuous_on_antitone Continuous.comp_lowerSemicontinuousOn_antitone theorem Continuous.comp_lowerSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuous f) (gmon : Antitone g) : UpperSemicontinuous (g ∘ f) := fun x => hg.continuousAt.comp_lowerSemicontinuousAt_antitone (hf x) gmon #align continuous.comp_lower_semicontinuous_antitone Continuous.comp_lowerSemicontinuous_antitone theorem LowerSemicontinuousAt.comp_continuousAt {f : α → β} {g : ι → α} {x : ι} (hf : LowerSemicontinuousAt f (g x)) (hg : ContinuousAt g x) : LowerSemicontinuousAt (fun x ↦ f (g x)) x := fun _ lt ↦ hg.eventually (hf _ lt) theorem LowerSemicontinuousAt.comp_continuousAt_of_eq {f : α → β} {g : ι → α} {y : α} {x : ι} (hf : LowerSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y) : LowerSemicontinuousAt (fun x ↦ f (g x)) x := by rw [← hy] at hf exact comp_continuousAt hf hg theorem LowerSemicontinuous.comp_continuous {f : α → β} {g : ι → α} (hf : LowerSemicontinuous f) (hg : Continuous g) : LowerSemicontinuous fun x ↦ f (g x) := fun x ↦ (hf (g x)).comp_continuousAt hg.continuousAt end /-! #### Addition -/ section variable {ι : Type} {γ : Type} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem LowerSemicontinuousWithinAt.add' {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x) (hg : LowerSemicontinuousWithinAt g s x) (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : LowerSemicontinuousWithinAt (fun z => f z + g z) s x := by intro y hy obtain ⟨u, v, u_open, xu, v_open, xv, h⟩ : ∃ u v : Set γ, IsOpen u ∧ f x ∈ u ∧ IsOpen v ∧ g x ∈ v ∧ u ×ˢ v ⊆ { p : γ × γ \| y < p.fst + p.snd } := mem_nhds_prod_iff'.1 (hcont (isOpen_Ioi.mem_nhds hy)) by_cases hx₁ : ∃ l, l < f x · obtain ⟨z₁, z₁lt, h₁⟩ : ∃ z₁ < f x, Ioc z₁ (f x) ⊆ u := exists_Ioc_subset_of_mem_nhds (u_open.mem_nhds xu) hx₁ by_cases hx₂ : ∃ l, l < g x · obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v := exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂ filter_upwards [hf z₁ z₁lt, hg z₂ z₂lt] with z h₁z h₂z have A1 : min (f z) (f x) ∈ u := by by_cases H : f z ≤ f x · simp [H] exact h₁ ⟨h₁z, H⟩ · simp [le_of_not_le H] exact h₁ ⟨z₁lt, le_rfl⟩ have A2 : min (g z) (g x) ∈ v := by by_cases H : g z ≤ g x · simp [H] exact h₂ ⟨h₂z, H⟩ · simp [le_of_not_le H] exact h₂ ⟨z₂lt, le_rfl⟩ have : (min (f z) (f x), min (g z) (g x)) ∈ u ×ˢ v := ⟨A1, A2⟩ calc y < min (f z) (f x) + min (g z) (g x) := h this _ ≤ f z + g z := add_le_add (min_le_left _ _) (min_le_left _ _) · simp only [not_exists, not_lt] at hx₂ filter_upwards [hf z₁ z₁lt] with z h₁z have A1 : min (f z) (f x) ∈ u := by by_cases H : f z ≤ f x · simp [H] exact h₁ ⟨h₁z, H⟩ · simp [le_of_not_le H] exact h₁ ⟨z₁lt, le_rfl⟩ have : (min (f z) (f x), g x) ∈ u ×ˢ v := ⟨A1, xv⟩ calc y < min (f z) (f x) + g x := h this _ ≤ f z + g z := add_le_add (min_le_left _ _) (hx₂ (g z)) · simp only [not_exists, not_lt] at hx₁ by_cases hx₂ : ∃ l, l < g x · obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v := exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂ filter_upwards [hg z₂ z₂lt] with z h₂z have A2 : min (g z) (g x) ∈ v := by by_cases H : g z ≤ g x · simp [H] exact h₂ ⟨h₂z, H⟩ · simp [le_of_not_le H] exact h₂ ⟨z₂lt, le_rfl⟩ have : (f x, min (g z) (g x)) ∈ u ×ˢ v := ⟨xu, A2⟩ calc y < f x + min (g z) (g x) := h this _ ≤ f z + g z := add_le_add (hx₁ (f z)) (min_le_left _ _) · simp only [not_exists, not_lt] at hx₁ hx₂ apply Filter.eventually_of_forall intro z have : (f x, g x) ∈ u ×ˢ v := ⟨xu, xv⟩ calc y < f x + g x := h this _ ≤ f z + g z := add_le_add (hx₁ (f z)) (hx₂ (g z)) #align lower_semicontinuous_within_at.add' LowerSemicontinuousWithinAt.add' /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem LowerSemicontinuousAt.add' {f g : α → γ} (hf : LowerSemicontinuousAt f x) (hg : LowerSemicontinuousAt g x) (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : LowerSemicontinuousAt (fun z => f z + g z) x := by simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at exact hf.add' hg hcont #align lower_semicontinuous_at.add' LowerSemicontinuousAt.add' /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem LowerSemicontinuousOn.add' {f g : α → γ} (hf : LowerSemicontinuousOn f s) (hg : LowerSemicontinuousOn g s) (hcont : ∀ x ∈ s, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : LowerSemicontinuousOn (fun z => f z + g z) s := fun x hx => (hf x hx).add' (hg x hx) (hcont x hx) #align lower_semicontinuous_on.add' LowerSemicontinuousOn.add' /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem LowerSemicontinuous.add' {f g : α → γ} (hf : LowerSemicontinuous f) (hg : LowerSemicontinuous g) (hcont : ∀ x, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : LowerSemicontinuous fun z => f z + g z := fun x => (hf x).add' (hg x) (hcont x) #align lower_semicontinuous.add' LowerSemicontinuous.add' variable [ContinuousAdd γ] /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem LowerSemicontinuousWithinAt.add {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x) (hg : LowerSemicontinuousWithinAt g s x) : LowerSemicontinuousWithinAt (fun z => f z + g z) s x := hf.add' hg continuous_add.continuousAt #align lower_semicontinuous_within_at.add LowerSemicontinuousWithinAt.add /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem LowerSemicontinuousAt.add {f g : α → γ} (hf : LowerSemicontinuousAt f x) (hg : LowerSemicontinuousAt g x) : LowerSemicontinuousAt (fun z => f z + g z) x := hf.add' hg continuous_add.continuousAt #align lower_semicontinuous_at.add LowerSemicontinuousAt.add /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem LowerSemicontinuousOn.add {f g : α → γ} (hf : LowerSemicontinuousOn f s) (hg : LowerSemicontinuousOn g s) : LowerSemicontinuousOn (fun z => f z + g z) s := hf.add' hg fun _x _hx => continuous_add.continuousAt #align lower_semicontinuous_on.add LowerSemicontinuousOn.add /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem LowerSemicontinuous.add {f g : α → γ} (hf : LowerSemicontinuous f) (hg : LowerSemicontinuous g) : LowerSemicontinuous fun z => f z + g z := hf.add' hg fun _x => continuous_add.continuousAt #align lower_semicontinuous.add LowerSemicontinuous.add theorem lowerSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, LowerSemicontinuousWithinAt (f i) s x) : LowerSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x := by classical induction' a using Finset.induction_on with i a ia IH · exact lowerSemicontinuousWithinAt_const · simp only [ia, Finset.sum_insert, not_false_iff] exact LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self i a)) (IH fun j ja => ha j (Finset.mem_insert_of_mem ja)) #align lower_semicontinuous_within_at_sum lowerSemicontinuousWithinAt_sum theorem lowerSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, LowerSemicontinuousAt (f i) x) : LowerSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x := by simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at * exact lowerSemicontinuousWithinAt_sum ha #align lower_semicontinuous_at_sum lowerSemicontinuousAt_sum theorem lowerSemicontinuousOn_sum {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, LowerSemicontinuousOn (f i) s) : LowerSemicontinuousOn (fun z => ∑ i ∈ a, f i z) s := fun x hx => lowerSemicontinuousWithinAt_sum fun i hi => ha i hi x hx #align lower_semicontinuous_on_sum lowerSemicontinuousOn_sum theorem lowerSemicontinuous_sum {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, LowerSemicontinuous (f i)) : LowerSemicontinuous fun z => ∑ i ∈ a, f i z := fun x => lowerSemicontinuousAt_sum fun i hi => ha i hi x #align lower_semicontinuous_sum lowerSemicontinuous_sum end /-! #### Supremum -/ section variable {ι : Sort} {δ δ' : Type} [CompleteLinearOrder δ] [ConditionallyCompleteLinearOrder δ'] theorem lowerSemicontinuousWithinAt_ciSup {f : ι → α → δ'} (bdd : ∀ᶠ y in 𝓝[s] x, BddAbove (range fun i => f i y)) (h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) : LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x := by cases isEmpty_or_nonempty ι · simpa only [iSup_of_empty'] using lowerSemicontinuousWithinAt_const · intro y hy rcases exists_lt_of_lt_ciSup hy with ⟨i, hi⟩ filter_upwards [h i y hi, bdd] with y hy hy' using hy.trans_le (le_ciSup hy' i) #align lower_semicontinuous_within_at_csupr lowerSemicontinuousWithinAt_ciSup theorem lowerSemicontinuousWithinAt_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) : LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x := lowerSemicontinuousWithinAt_ciSup (by simp) h #align lower_semicontinuous_within_at_supr lowerSemicontinuousWithinAt_iSup theorem lowerSemicontinuousWithinAt_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ} (h : ∀ i hi, LowerSemicontinuousWithinAt (f i hi) s x) : LowerSemicontinuousWithinAt (fun x' => ⨆ (i) (hi), f i hi x') s x := lowerSemicontinuousWithinAt_iSup fun i => lowerSemicontinuousWithinAt_iSup fun hi => h i hi #align lower_semicontinuous_within_at_bsupr lowerSemicontinuousWithinAt_biSup theorem lowerSemicontinuousAt_ciSup {f : ι → α → δ'} (bdd : ∀ᶠ y in 𝓝 x, BddAbove (range fun i => f i y)) (h : ∀ i, LowerSemicontinuousAt (f i) x) : LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x := by simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at * rw [← nhdsWithin_univ] at bdd exact lowerSemicontinuousWithinAt_ciSup bdd h #align lower_semicontinuous_at_csupr lowerSemicontinuousAt_ciSup theorem lowerSemicontinuousAt_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousAt (f i) x) : LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x := lowerSemicontinuousAt_ciSup (by simp) h #align lower_semicontinuous_at_supr lowerSemicontinuousAt_iSup theorem lowerSemicontinuousAt_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ} (h : ∀ i hi, LowerSemicontinuousAt (f i hi) x) : LowerSemicontinuousAt (fun x' => ⨆ (i) (hi), f i hi x') x := lowerSemicontinuousAt_iSup fun i => lowerSemicontinuousAt_iSup fun hi => h i hi #align lower_semicontinuous_at_bsupr lowerSemicontinuousAt_biSup theorem lowerSemicontinuousOn_ciSup {f : ι → α → δ'} (bdd : ∀ x ∈ s, BddAbove (range fun i => f i x)) (h : ∀ i, LowerSemicontinuousOn (f i) s) : LowerSemicontinuousOn (fun x' => ⨆ i, f i x') s := fun x hx => lowerSemicontinuousWithinAt_ciSup (eventually_nhdsWithin_of_forall bdd) fun i => h i x hx #align lower_semicontinuous_on_csupr lowerSemicontinuousOn_ciSup theorem lowerSemicontinuousOn_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousOn (f i) s) : LowerSemicontinuousOn (fun x' => ⨆ i, f i x') s := lowerSemicontinuousOn_ciSup (by simp) h #align lower_semicontinuous_on_supr lowerSemicontinuousOn_iSup theorem lowerSemicontinuousOn_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ} (h : ∀ i hi, LowerSemicontinuousOn (f i hi) s) : LowerSemicontinuousOn (fun x' => ⨆ (i) (hi), f i hi x') s := lowerSemicontinuousOn_iSup fun i => lowerSemicontinuousOn_iSup fun hi => h i hi #align lower_semicontinuous_on_bsupr lowerSemicontinuousOn_biSup theorem lowerSemicontinuous_ciSup {f : ι → α → δ'} (bdd : ∀ x, BddAbove (range fun i => f i x)) (h : ∀ i, LowerSemicontinuous (f i)) : LowerSemicontinuous fun x' => ⨆ i, f i x' := fun x => lowerSemicontinuousAt_ciSup (eventually_of_forall bdd) fun i => h i x #align lower_semicontinuous_csupr lowerSemicontinuous_ciSup theorem lowerSemicontinuous_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuous (f i)) : LowerSemicontinuous fun x' => ⨆ i, f i x' := lowerSemicontinuous_ciSup (by simp) h #align lower_semicontinuous_supr lowerSemicontinuous_iSup theorem lowerSemicontinuous_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ} (h : ∀ i hi, LowerSemicontinuous (f i hi)) : LowerSemicontinuous fun x' => ⨆ (i) (hi), f i hi x' := lowerSemicontinuous_iSup fun i => lowerSemicontinuous_iSup fun hi => h i hi #align lower_semicontinuous_bsupr lowerSemicontinuous_biSup end /-! #### Infinite sums -/ section variable {ι : Type} theorem lowerSemicontinuousWithinAt_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) : LowerSemicontinuousWithinAt (fun x' => ∑' i, f i x') s x := by simp_rw [ENNReal.tsum_eq_iSup_sum] refine lowerSemicontinuousWithinAt_iSup fun b => ?_ exact lowerSemicontinuousWithinAt_sum fun i _hi => h i #align lower_semicontinuous_within_at_tsum lowerSemicontinuousWithinAt_tsum theorem lowerSemicontinuousAt_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousAt (f i) x) : LowerSemicontinuousAt (fun x' => ∑' i, f i x') x := by simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at exact lowerSemicontinuousWithinAt_tsum h #align lower_semicontinuous_at_tsum lowerSemicontinuousAt_tsum theorem lowerSemicontinuousOn_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuousOn (f i) s) : LowerSemicontinuousOn (fun x' => ∑' i, f i x') s := fun x hx => lowerSemicontinuousWithinAt_tsum fun i => h i x hx #align lower_semicontinuous_on_tsum lowerSemicontinuousOn_tsum theorem lowerSemicontinuous_tsum {f : ι → α → ℝ≥0∞} (h : ∀ i, LowerSemicontinuous (f i)) : LowerSemicontinuous fun x' => ∑' i, f i x' := fun x => lowerSemicontinuousAt_tsum fun i => h i x #align lower_semicontinuous_tsum lowerSemicontinuous_tsum end /-! ### Upper semicontinuous functions -/ /-! #### Basic dot notation interface for upper semicontinuity -/ theorem UpperSemicontinuousWithinAt.mono (h : UpperSemicontinuousWithinAt f s x) (hst : t ⊆ s) : UpperSemicontinuousWithinAt f t x := fun y hy => Filter.Eventually.filter_mono (nhdsWithin_mono _ hst) (h y hy) #align upper_semicontinuous_within_at.mono UpperSemicontinuousWithinAt.mono theorem upperSemicontinuousWithinAt_univ_iff : UpperSemicontinuousWithinAt f univ x ↔ UpperSemicontinuousAt f x := by simp [UpperSemicontinuousWithinAt, UpperSemicontinuousAt, nhdsWithin_univ] #align upper_semicontinuous_within_at_univ_iff upperSemicontinuousWithinAt_univ_iff theorem UpperSemicontinuousAt.upperSemicontinuousWithinAt (s : Set α) (h : UpperSemicontinuousAt f x) : UpperSemicontinuousWithinAt f s x := fun y hy => Filter.Eventually.filter_mono nhdsWithin_le_nhds (h y hy) #align upper_semicontinuous_at.upper_semicontinuous_within_at UpperSemicontinuousAt.upperSemicontinuousWithinAt theorem UpperSemicontinuousOn.upperSemicontinuousWithinAt (h : UpperSemicontinuousOn f s) (hx : x ∈ s) : UpperSemicontinuousWithinAt f s x := h x hx #align upper_semicontinuous_on.upper_semicontinuous_within_at UpperSemicontinuousOn.upperSemicontinuousWithinAt theorem UpperSemicontinuousOn.mono (h : UpperSemicontinuousOn f s) (hst : t ⊆ s) : UpperSemicontinuousOn f t := fun x hx => (h x (hst hx)).mono hst #align upper_semicontinuous_on.mono UpperSemicontinuousOn.mono theorem upperSemicontinuousOn_univ_iff : UpperSemicontinuousOn f univ ↔ UpperSemicontinuous f := by simp [UpperSemicontinuousOn, UpperSemicontinuous, upperSemicontinuousWithinAt_univ_iff] #align upper_semicontinuous_on_univ_iff upperSemicontinuousOn_univ_iff theorem UpperSemicontinuous.upperSemicontinuousAt (h : UpperSemicontinuous f) (x : α) : UpperSemicontinuousAt f x := h x #align upper_semicontinuous.upper_semicontinuous_at UpperSemicontinuous.upperSemicontinuousAt theorem UpperSemicontinuous.upperSemicontinuousWithinAt (h : UpperSemicontinuous f) (s : Set α) (x : α) : UpperSemicontinuousWithinAt f s x := (h x).upperSemicontinuousWithinAt s #align upper_semicontinuous.upper_semicontinuous_within_at UpperSemicontinuous.upperSemicontinuousWithinAt theorem UpperSemicontinuous.upperSemicontinuousOn (h : UpperSemicontinuous f) (s : Set α) : UpperSemicontinuousOn f s := fun x _hx => h.upperSemicontinuousWithinAt s x #align upper_semicontinuous.upper_semicontinuous_on UpperSemicontinuous.upperSemicontinuousOn /-! #### Constants -/ theorem upperSemicontinuousWithinAt_const : UpperSemicontinuousWithinAt (fun _x => z) s x := fun _y hy => Filter.eventually_of_forall fun _x => hy #align upper_semicontinuous_within_at_const upperSemicontinuousWithinAt_const theorem upperSemicontinuousAt_const : UpperSemicontinuousAt (fun _x => z) x := fun _y hy => Filter.eventually_of_forall fun _x => hy #align upper_semicontinuous_at_const upperSemicontinuousAt_const theorem upperSemicontinuousOn_const : UpperSemicontinuousOn (fun _x => z) s := fun _x _hx => upperSemicontinuousWithinAt_const #align upper_semicontinuous_on_const upperSemicontinuousOn_const theorem upperSemicontinuous_const : UpperSemicontinuous fun _x : α => z := fun _x => upperSemicontinuousAt_const #align upper_semicontinuous_const upperSemicontinuous_const /-! #### Indicators -/ section variable [Zero β] theorem IsOpen.upperSemicontinuous_indicator (hs : IsOpen s) (hy : y ≤ 0) : UpperSemicontinuous (indicator s fun _x => y) := @IsOpen.lowerSemicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy #align is_open.upper_semicontinuous_indicator IsOpen.upperSemicontinuous_indicator theorem IsOpen.upperSemicontinuousOn_indicator (hs : IsOpen s) (hy : y ≤ 0) : UpperSemicontinuousOn (indicator s fun _x => y) t := (hs.upperSemicontinuous_indicator hy).upperSemicontinuousOn t #align is_open.upper_semicontinuous_on_indicator IsOpen.upperSemicontinuousOn_indicator theorem IsOpen.upperSemicontinuousAt_indicator (hs : IsOpen s) (hy : y ≤ 0) : UpperSemicontinuousAt (indicator s fun _x => y) x := (hs.upperSemicontinuous_indicator hy).upperSemicontinuousAt x #align is_open.upper_semicontinuous_at_indicator IsOpen.upperSemicontinuousAt_indicator theorem IsOpen.upperSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : y ≤ 0) : UpperSemicontinuousWithinAt (indicator s fun _x => y) t x := (hs.upperSemicontinuous_indicator hy).upperSemicontinuousWithinAt t x #align is_open.upper_semicontinuous_within_at_indicator IsOpen.upperSemicontinuousWithinAt_indicator theorem IsClosed.upperSemicontinuous_indicator (hs : IsClosed s) (hy : 0 ≤ y) : UpperSemicontinuous (indicator s fun _x => y) := @IsClosed.lowerSemicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy #align is_closed.upper_semicontinuous_indicator IsClosed.upperSemicontinuous_indicator theorem IsClosed.upperSemicontinuousOn_indicator (hs : IsClosed s) (hy : 0 ≤ y) : UpperSemicontinuousOn (indicator s fun _x => y) t := (hs.upperSemicontinuous_indicator hy).upperSemicontinuousOn t #align is_closed.upper_semicontinuous_on_indicator IsClosed.upperSemicontinuousOn_indicator theorem IsClosed.upperSemicontinuousAt_indicator (hs : IsClosed s) (hy : 0 ≤ y) : UpperSemicontinuousAt (indicator s fun _x => y) x := (hs.upperSemicontinuous_indicator hy).upperSemicontinuousAt x #align is_closed.upper_semicontinuous_at_indicator IsClosed.upperSemicontinuousAt_indicator theorem IsClosed.upperSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : 0 ≤ y) : UpperSemicontinuousWithinAt (indicator s fun _x => y) t x := (hs.upperSemicontinuous_indicator hy).upperSemicontinuousWithinAt t x #align is_closed.upper_semicontinuous_within_at_indicator IsClosed.upperSemicontinuousWithinAt_indicator end /-! #### Relationship with continuity -/ theorem upperSemicontinuous_iff_isOpen_preimage : UpperSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Iio y) := ⟨fun H y => isOpen_iff_mem_nhds.2 fun x hx => H x y hx, fun H _x y y_lt => IsOpen.mem_nhds (H y) y_lt⟩ #align upper_semicontinuous_iff_is_open_preimage upperSemicontinuous_iff_isOpen_preimage theorem UpperSemicontinuous.isOpen_preimage (hf : UpperSemicontinuous f) (y : β) : IsOpen (f ⁻¹' Iio y) := upperSemicontinuous_iff_isOpen_preimage.1 hf y #align upper_semicontinuous.is_open_preimage UpperSemicontinuous.isOpen_preimage section variable {γ : Type} [LinearOrder γ] theorem upperSemicontinuous_iff_isClosed_preimage {f : α → γ} : UpperSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Ici y) := by rw [upperSemicontinuous_iff_isOpen_preimage] simp only [← isOpen_compl_iff, ← preimage_compl, compl_Ici] #align upper_semicontinuous_iff_is_closed_preimage upperSemicontinuous_iff_isClosed_preimage theorem UpperSemicontinuous.isClosed_preimage {f : α → γ} (hf : UpperSemicontinuous f) (y : γ) : IsClosed (f ⁻¹' Ici y) := upperSemicontinuous_iff_isClosed_preimage.1 hf y #align upper_semicontinuous.is_closed_preimage UpperSemicontinuous.isClosed_preimage variable [TopologicalSpace γ] [OrderTopology γ] theorem ContinuousWithinAt.upperSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) : UpperSemicontinuousWithinAt f s x := fun _y hy => h (Iio_mem_nhds hy) #align continuous_within_at.upper_semicontinuous_within_at ContinuousWithinAt.upperSemicontinuousWithinAt theorem ContinuousAt.upperSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) : UpperSemicontinuousAt f x := fun _y hy => h (Iio_mem_nhds hy) #align continuous_at.upper_semicontinuous_at ContinuousAt.upperSemicontinuousAt theorem ContinuousOn.upperSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) : UpperSemicontinuousOn f s := fun x hx => (h x hx).upperSemicontinuousWithinAt #align continuous_on.upper_semicontinuous_on ContinuousOn.upperSemicontinuousOn theorem Continuous.upperSemicontinuous {f : α → γ} (h : Continuous f) : UpperSemicontinuous f := fun _x => h.continuousAt.upperSemicontinuousAt #align continuous.upper_semicontinuous Continuous.upperSemicontinuous end /-! #### Equivalent definitions -/ section variable {γ : Type} [CompleteLinearOrder γ] [DenselyOrdered γ] theorem upperSemicontinuousWithinAt_iff_limsup_le {f : α → γ} : UpperSemicontinuousWithinAt f s x ↔ limsup f (𝓝[s] x) ≤ f x := lowerSemicontinuousWithinAt_iff_le_liminf (γ := γᵒᵈ) alias ⟨UpperSemicontinuousWithinAt.limsup_le, _⟩ := upperSemicontinuousWithinAt_iff_limsup_le theorem upperSemicontinuousAt_iff_limsup_le {f : α → γ} : UpperSemicontinuousAt f x ↔ limsup f (𝓝 x) ≤ f x := lowerSemicontinuousAt_iff_le_liminf (γ := γᵒᵈ) alias ⟨UpperSemicontinuousAt.limsup_le, _⟩ := upperSemicontinuousAt_iff_limsup_le theorem upperSemicontinuous_iff_limsup_le {f : α → γ} : UpperSemicontinuous f ↔ ∀ x, limsup f (𝓝 x) ≤ f x := lowerSemicontinuous_iff_le_liminf (γ := γᵒᵈ) alias ⟨UpperSemicontinuous.limsup_le, _⟩ := upperSemicontinuous_iff_limsup_le theorem upperSemicontinuousOn_iff_limsup_le {f : α → γ} : UpperSemicontinuousOn f s ↔ ∀ x ∈ s, limsup f (𝓝[s] x) ≤ f x := lowerSemicontinuousOn_iff_le_liminf (γ := γᵒᵈ) alias ⟨UpperSemicontinuousOn.limsup_le, _⟩ := upperSemicontinuousOn_iff_limsup_le variable [TopologicalSpace γ] [OrderTopology γ] theorem upperSemicontinuous_iff_IsClosed_hypograph {f : α → γ} : UpperSemicontinuous f ↔ IsClosed {p : α × γ \| p.2 ≤ f p.1} := lowerSemicontinuous_iff_isClosed_epigraph (γ := γᵒᵈ) alias ⟨UpperSemicontinuous.IsClosed_hypograph, _⟩ := upperSemicontinuous_iff_IsClosed_hypograph end /-! ### Composition -/ section variable {γ : Type} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] variable {δ : Type} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] variable {ι : Type} [TopologicalSpace ι] theorem ContinuousAt.comp_upperSemicontinuousWithinAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Monotone g) : UpperSemicontinuousWithinAt (g ∘ f) s x := @ContinuousAt.comp_lowerSemicontinuousWithinAt α _ x s γᵒᵈ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon.dual #align continuous_at.comp_upper_semicontinuous_within_at ContinuousAt.comp_upperSemicontinuousWithinAt theorem ContinuousAt.comp_upperSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousAt f x) (gmon : Monotone g) : UpperSemicontinuousAt (g ∘ f) x := @ContinuousAt.comp_lowerSemicontinuousAt α _ x γᵒᵈ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon.dual #align continuous_at.comp_upper_semicontinuous_at ContinuousAt.comp_upperSemicontinuousAt theorem Continuous.comp_upperSemicontinuousOn {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuousOn f s) (gmon : Monotone g) : UpperSemicontinuousOn (g ∘ f) s := fun x hx => hg.continuousAt.comp_upperSemicontinuousWithinAt (hf x hx) gmon #align continuous.comp_upper_semicontinuous_on Continuous.comp_upperSemicontinuousOn theorem Continuous.comp_upperSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuous f) (gmon : Monotone g) : UpperSemicontinuous (g ∘ f) := fun x => hg.continuousAt.comp_upperSemicontinuousAt (hf x) gmon #align continuous.comp_upper_semicontinuous Continuous.comp_upperSemicontinuous theorem ContinuousAt.comp_upperSemicontinuousWithinAt_antitone {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousWithinAt f s x) (gmon : Antitone g) : LowerSemicontinuousWithinAt (g ∘ f) s x := @ContinuousAt.comp_upperSemicontinuousWithinAt α _ x s γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon #align continuous_at.comp_upper_semicontinuous_within_at_antitone ContinuousAt.comp_upperSemicontinuousWithinAt_antitone theorem ContinuousAt.comp_upperSemicontinuousAt_antitone {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : UpperSemicontinuousAt f x) (gmon : Antitone g) : LowerSemicontinuousAt (g ∘ f) x := @ContinuousAt.comp_upperSemicontinuousAt α _ x γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon #align continuous_at.comp_upper_semicontinuous_at_antitone ContinuousAt.comp_upperSemicontinuousAt_antitone theorem Continuous.comp_upperSemicontinuousOn_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuousOn f s) (gmon : Antitone g) : LowerSemicontinuousOn (g ∘ f) s := fun x hx => hg.continuousAt.comp_upperSemicontinuousWithinAt_antitone (hf x hx) gmon #align continuous.comp_upper_semicontinuous_on_antitone Continuous.comp_upperSemicontinuousOn_antitone theorem Continuous.comp_upperSemicontinuous_antitone {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : UpperSemicontinuous f) (gmon : Antitone g) : LowerSemicontinuous (g ∘ f) := fun x => hg.continuousAt.comp_upperSemicontinuousAt_antitone (hf x) gmon #align continuous.comp_upper_semicontinuous_antitone Continuous.comp_upperSemicontinuous_antitone theorem UpperSemicontinuousAt.comp_continuousAt {f : α → β} {g : ι → α} {x : ι} (hf : UpperSemicontinuousAt f (g x)) (hg : ContinuousAt g x) : UpperSemicontinuousAt (fun x ↦ f (g x)) x := fun _ lt ↦ hg.eventually (hf _ lt) theorem UpperSemicontinuousAt.comp_continuousAt_of_eq {f : α → β} {g : ι → α} {y : α} {x : ι} (hf : UpperSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y) : UpperSemicontinuousAt (fun x ↦ f (g x)) x := by rw [← hy] at hf exact comp_continuousAt hf hg theorem UpperSemicontinuous.comp_continuous {f : α → β} {g : ι → α} (hf : UpperSemicontinuous f) (hg : Continuous g) : UpperSemicontinuous fun x ↦ f (g x) := fun x ↦ (hf (g x)).comp_continuousAt hg.continuousAt end /-! #### Addition -/ section variable {ι : Type} {γ : Type} [LinearOrderedAddCommMonoid γ] [TopologicalSpace γ] [OrderTopology γ] /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem UpperSemicontinuousWithinAt.add' {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x) (hg : UpperSemicontinuousWithinAt g s x) (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : UpperSemicontinuousWithinAt (fun z => f z + g z) s x := @LowerSemicontinuousWithinAt.add' α _ x s γᵒᵈ _ _ _ _ _ hf hg hcont #align upper_semicontinuous_within_at.add' UpperSemicontinuousWithinAt.add' /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem UpperSemicontinuousAt.add' {f g : α → γ} (hf : UpperSemicontinuousAt f x) (hg : UpperSemicontinuousAt g x) (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : UpperSemicontinuousAt (fun z => f z + g z) x := by simp_rw [← upperSemicontinuousWithinAt_univ_iff] at exact hf.add' hg hcont #align upper_semicontinuous_at.add' UpperSemicontinuousAt.add' /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem UpperSemicontinuousOn.add' {f g : α → γ} (hf : UpperSemicontinuousOn f s) (hg : UpperSemicontinuousOn g s) (hcont : ∀ x ∈ s, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : UpperSemicontinuousOn (fun z => f z + g z) s := fun x hx => (hf x hx).add' (hg x hx) (hcont x hx) #align upper_semicontinuous_on.add' UpperSemicontinuousOn.add' /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an explicit continuity assumption on addition, for application to `EReal`. The unprimed version of the lemma uses `[ContinuousAdd]`. -/ theorem UpperSemicontinuous.add' {f g : α → γ} (hf : UpperSemicontinuous f) (hg : UpperSemicontinuous g) (hcont : ∀ x, ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) : UpperSemicontinuous fun z => f z + g z := fun x => (hf x).add' (hg x) (hcont x) #align upper_semicontinuous.add' UpperSemicontinuous.add' variable [ContinuousAdd γ] /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem UpperSemicontinuousWithinAt.add {f g : α → γ} (hf : UpperSemicontinuousWithinAt f s x) (hg : UpperSemicontinuousWithinAt g s x) : UpperSemicontinuousWithinAt (fun z => f z + g z) s x := hf.add' hg continuous_add.continuousAt #align upper_semicontinuous_within_at.add UpperSemicontinuousWithinAt.add /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem UpperSemicontinuousAt.add {f g : α → γ} (hf : UpperSemicontinuousAt f x) (hg : UpperSemicontinuousAt g x) : UpperSemicontinuousAt (fun z => f z + g z) x := hf.add' hg continuous_add.continuousAt #align upper_semicontinuous_at.add UpperSemicontinuousAt.add /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem UpperSemicontinuousOn.add {f g : α → γ} (hf : UpperSemicontinuousOn f s) (hg : UpperSemicontinuousOn g s) : UpperSemicontinuousOn (fun z => f z + g z) s := hf.add' hg fun _x _hx => continuous_add.continuousAt #align upper_semicontinuous_on.add UpperSemicontinuousOn.add /-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with `[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on addition, for application to `EReal`. -/ theorem UpperSemicontinuous.add {f g : α → γ} (hf : UpperSemicontinuous f) (hg : UpperSemicontinuous g) : UpperSemicontinuous fun z => f z + g z := hf.add' hg fun _x => continuous_add.continuousAt #align upper_semicontinuous.add UpperSemicontinuous.add theorem upperSemicontinuousWithinAt_sum {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, UpperSemicontinuousWithinAt (f i) s x) : UpperSemicontinuousWithinAt (fun z => ∑ i ∈ a, f i z) s x := @lowerSemicontinuousWithinAt_sum α _ x s ι γᵒᵈ _ _ _ _ f a ha #align upper_semicontinuous_within_at_sum upperSemicontinuousWithinAt_sum	Mathlib/Topology/Semicontinuous.lean	1,116	1,120	theorem upperSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι} (ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x) : UpperSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x := by	simp_rw [← upperSemicontinuousWithinAt_univ_iff] at * exact upperSemicontinuousWithinAt_sum ha
/- Copyright (c) 2021 Anatole Dedecker. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anatole Dedecker, Bhavik Mehta -/ import Mathlib.Analysis.Calculus.Deriv.Support import Mathlib.Analysis.SpecialFunctions.Pow.Deriv import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Order.Filter.AtTopBot import Mathlib.MeasureTheory.Function.Jacobian import Mathlib.MeasureTheory.Measure.Haar.NormedSpace import Mathlib.MeasureTheory.Measure.Haar.Unique #align_import measure_theory.integral.integral_eq_improper from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5" /-! # Links between an integral and its "improper" version In its current state, mathlib only knows how to talk about definite ("proper") integrals, in the sense that it treats integrals over `[x, +∞)` the same as it treats integrals over `[y, z]`. For example, the integral over `[1, +∞)` is not defined to be the limit of the integral over `[1, x]` as `x` tends to `+∞`, which is known as an improper integral. Indeed, the "proper" definition is stronger than the "improper" one. The usual counterexample is `x ↦ sin(x)/x`, which has an improper integral over `[1, +∞)` but no definite integral. Although definite integrals have better properties, they are hardly usable when it comes to computing integrals on unbounded sets, which is much easier using limits. Thus, in this file, we prove various ways of studying the proper integral by studying the improper one. ## Definitions The main definition of this file is `MeasureTheory.AECover`. It is a rather technical definition whose sole purpose is generalizing and factoring proofs. Given an index type `ι`, a countably generated filter `l` over `ι`, and an `ι`-indexed family `φ` of subsets of a measurable space `α` equipped with a measure `μ`, one should think of a hypothesis `hφ : MeasureTheory.AECover μ l φ` as a sufficient condition for being able to interpret `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ i, f x ∂μ` as `i` tends to `l`. When using this definition with a measure restricted to a set `s`, which happens fairly often, one should not try too hard to use a `MeasureTheory.AECover` of subsets of `s`, as it often makes proofs more complicated than necessary. See for example the proof of `MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto` where we use `(fun x ↦ oi x)` as a `MeasureTheory.AECover` w.r.t. `μ.restrict (Iic b)`, instead of using `(fun x ↦ Ioc x b)`. ## Main statements - `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated` : if `φ` is a `MeasureTheory.AECover μ l`, where `l` is a countably generated filter, and if `f` is a measurable `ENNReal`-valued function, then `∫⁻ x in φ n, f x ∂μ` tends to `∫⁻ x, f x ∂μ` as `n` tends to `l` - `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` : if `φ` is a `MeasureTheory.AECover μ l`, where `l` is a countably generated filter, if `f` is measurable and integrable on each `φ n`, and if `∫ x in φ n, ‖f x‖ ∂μ` tends to some `I : ℝ` as n tends to `l`, then `f` is integrable - `MeasureTheory.AECover.integral_tendsto_of_countably_generated` : if `φ` is a `MeasureTheory.AECover μ l`, where `l` is a countably generated filter, and if `f` is measurable and integrable (globally), then `∫ x in φ n, f x ∂μ` tends to `∫ x, f x ∂μ` as `n` tends to `+∞`. We then specialize these lemmas to various use cases involving intervals, which are frequent in analysis. In particular, - `MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto` is a version of FTC-2 on the interval `(a, +∞)`, giving the formula `∫ x in (a, +∞), g' x = l - g a` if `g'` is integrable and `g` tends to `l` at `+∞`. - `MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg` gives the same result assuming that `g'` is nonnegative instead of integrable. Its automatic integrability in this context is proved in `MeasureTheory.integrableOn_Ioi_deriv_of_nonneg`. - `MeasureTheory.integral_comp_smul_deriv_Ioi` is a version of the change of variables formula on semi-infinite intervals. - `MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi` shows that a function whose derivative is integrable on `(a, +∞)` has a limit at `+∞`. - `MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi` shows that an integrable function whose derivative is integrable on `(a, +∞)` tends to `0` at `+∞`. Versions of these results are also given on the intervals `(-∞, a]` and `(-∞, +∞)`, as well as the corresponding versions of integration by parts. -/ open MeasureTheory Filter Set TopologicalSpace open scoped ENNReal NNReal Topology namespace MeasureTheory section AECover variable {α ι : Type} [MeasurableSpace α] (μ : Measure α) (l : Filter ι) /-- A sequence `φ` of subsets of `α` is a `MeasureTheory.AECover` w.r.t. a measure `μ` and a filter `l` if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if each `φ n` is measurable. This definition is a technical way to avoid duplicating a lot of proofs. It should be thought of as a sufficient condition for being able to interpret `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ n, f x ∂μ` as `n` tends to `l`. See for example `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated`, `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` and `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/ structure AECover (φ : ι → Set α) : Prop where ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i protected measurableSet : ∀ i, MeasurableSet <\| φ i #align measure_theory.ae_cover MeasureTheory.AECover #align measure_theory.ae_cover.ae_eventually_mem MeasureTheory.AECover.ae_eventually_mem #align measure_theory.ae_cover.measurable MeasureTheory.AECover.measurableSet variable {μ} {l} namespace AECover /-! ## Operations on `AECover`s Porting note: this is a new section. -/ /-- Elementwise intersection of two `AECover`s is an `AECover`. -/ theorem inter {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hψ : AECover μ l ψ) : AECover μ l (fun i ↦ φ i ∩ ψ i) where ae_eventually_mem := hψ.1.mp <\| hφ.1.mono fun _ ↦ Eventually.and measurableSet _ := (hφ.2 _).inter (hψ.2 _) theorem superset {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hsub : ∀ i, φ i ⊆ ψ i) (hmeas : ∀ i, MeasurableSet (ψ i)) : AECover μ l ψ := ⟨hφ.1.mono fun _x hx ↦ hx.mono fun i hi ↦ hsub i hi, hmeas⟩ theorem mono_ac {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≪ μ) : AECover ν l φ := ⟨hle hφ.1, hφ.2⟩ theorem mono {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≤ μ) : AECover ν l φ := hφ.mono_ac hle.absolutelyContinuous end AECover section MetricSpace variable [PseudoMetricSpace α] [OpensMeasurableSpace α] theorem aecover_ball {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) : AECover μ l (fun i ↦ Metric.ball x (r i)) where measurableSet _ := Metric.isOpen_ball.measurableSet ae_eventually_mem := by filter_upwards with y filter_upwards [hr (Ioi_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) : AECover μ l (fun i ↦ Metric.closedBall x (r i)) where measurableSet _ := Metric.isClosed_ball.measurableSet ae_eventually_mem := by filter_upwards with y filter_upwards [hr (Ici_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha end MetricSpace section Preorderα variable [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) theorem aecover_Ici : AECover μ l fun i => Ici (a i) where ae_eventually_mem := ae_of_all μ ha.eventually_le_atBot measurableSet _ := measurableSet_Ici #align measure_theory.ae_cover_Ici MeasureTheory.aecover_Ici theorem aecover_Iic : AECover μ l fun i => Iic <\| b i := aecover_Ici (α := αᵒᵈ) hb #align measure_theory.ae_cover_Iic MeasureTheory.aecover_Iic theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) := (aecover_Ici ha).inter (aecover_Iic hb) #align measure_theory.ae_cover_Icc MeasureTheory.aecover_Icc end Preorderα section LinearOrderα variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) theorem aecover_Ioi [NoMinOrder α] : AECover μ l fun i => Ioi (a i) where ae_eventually_mem := ae_of_all μ ha.eventually_lt_atBot measurableSet _ := measurableSet_Ioi #align measure_theory.ae_cover_Ioi MeasureTheory.aecover_Ioi theorem aecover_Iio [NoMaxOrder α] : AECover μ l fun i => Iio (b i) := aecover_Ioi (α := αᵒᵈ) hb #align measure_theory.ae_cover_Iio MeasureTheory.aecover_Iio theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo (a i) (b i) := (aecover_Ioi ha).inter (aecover_Iio hb) #align measure_theory.ae_cover_Ioo MeasureTheory.aecover_Ioo theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) := (aecover_Ioi ha).inter (aecover_Iic hb) #align measure_theory.ae_cover_Ioc MeasureTheory.aecover_Ioc theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) := (aecover_Ici ha).inter (aecover_Iio hb) #align measure_theory.ae_cover_Ico MeasureTheory.aecover_Ico end LinearOrderα section FiniteIntervals variable [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) -- Porting note (#10756): new lemma theorem aecover_Ioi_of_Ioi : AECover (μ.restrict (Ioi A)) l fun i ↦ Ioi (a i) where ae_eventually_mem := (ae_restrict_mem measurableSet_Ioi).mono fun _x hx ↦ ha.eventually <\| eventually_lt_nhds hx measurableSet _ := measurableSet_Ioi -- Porting note (#10756): new lemma theorem aecover_Iio_of_Iio : AECover (μ.restrict (Iio B)) l fun i ↦ Iio (b i) := aecover_Ioi_of_Ioi (α := αᵒᵈ) hb -- Porting note (#10756): new lemma theorem aecover_Ioi_of_Ici : AECover (μ.restrict (Ioi A)) l fun i ↦ Ici (a i) := (aecover_Ioi_of_Ioi ha).superset (fun _ ↦ Ioi_subset_Ici_self) fun _ ↦ measurableSet_Ici -- Porting note (#10756): new lemma theorem aecover_Iio_of_Iic : AECover (μ.restrict (Iio B)) l fun i ↦ Iic (b i) := aecover_Ioi_of_Ici (α := αᵒᵈ) hb theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <\| Ioo A B) l fun i => Ioo (a i) (b i) := ((aecover_Ioi_of_Ioi ha).mono <\| Measure.restrict_mono Ioo_subset_Ioi_self le_rfl).inter ((aecover_Iio_of_Iio hb).mono <\| Measure.restrict_mono Ioo_subset_Iio_self le_rfl) #align measure_theory.ae_cover_Ioo_of_Ioo MeasureTheory.aecover_Ioo_of_Ioo theorem aecover_Ioo_of_Icc : AECover (μ.restrict <\| Ioo A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).superset (fun _ ↦ Ioo_subset_Icc_self) fun _ ↦ measurableSet_Icc #align measure_theory.ae_cover_Ioo_of_Icc MeasureTheory.aecover_Ioo_of_Icc theorem aecover_Ioo_of_Ico : AECover (μ.restrict <\| Ioo A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).superset (fun _ ↦ Ioo_subset_Ico_self) fun _ ↦ measurableSet_Ico #align measure_theory.ae_cover_Ioo_of_Ico MeasureTheory.aecover_Ioo_of_Ico theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <\| Ioo A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).superset (fun _ ↦ Ioo_subset_Ioc_self) fun _ ↦ measurableSet_Ioc #align measure_theory.ae_cover_Ioo_of_Ioc MeasureTheory.aecover_Ioo_of_Ioc variable [NoAtoms μ] theorem aecover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Icc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Icc MeasureTheory.aecover_Ioc_of_Icc theorem aecover_Ioc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ico ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Ico MeasureTheory.aecover_Ioc_of_Ico theorem aecover_Ioc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Ioc MeasureTheory.aecover_Ioc_of_Ioc theorem aecover_Ioc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ioc A B) l fun i => Ioo (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ioc).ge #align measure_theory.ae_cover_Ioc_of_Ioo MeasureTheory.aecover_Ioc_of_Ioo theorem aecover_Ico_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Icc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Icc MeasureTheory.aecover_Ico_of_Icc theorem aecover_Ico_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ico ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Ico MeasureTheory.aecover_Ico_of_Ico theorem aecover_Ico_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Ioc MeasureTheory.aecover_Ico_of_Ioc theorem aecover_Ico_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Ico A B) l fun i => Ioo (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Ico).ge #align measure_theory.ae_cover_Ico_of_Ioo MeasureTheory.aecover_Ico_of_Ioo theorem aecover_Icc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Icc (a i) (b i) := (aecover_Ioo_of_Icc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Icc MeasureTheory.aecover_Icc_of_Icc theorem aecover_Icc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Ico (a i) (b i) := (aecover_Ioo_of_Ico ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Ico MeasureTheory.aecover_Icc_of_Ico theorem aecover_Icc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Ioc (a i) (b i) := (aecover_Ioo_of_Ioc ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Ioc MeasureTheory.aecover_Icc_of_Ioc theorem aecover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) : AECover (μ.restrict <\| Icc A B) l fun i => Ioo (a i) (b i) := (aecover_Ioo_of_Ioo ha hb).mono (Measure.restrict_congr_set Ioo_ae_eq_Icc).ge #align measure_theory.ae_cover_Icc_of_Ioo MeasureTheory.aecover_Icc_of_Ioo end FiniteIntervals protected theorem AECover.restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} : AECover (μ.restrict s) l φ := hφ.mono Measure.restrict_le_self #align measure_theory.ae_cover.restrict MeasureTheory.AECover.restrict theorem aecover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s) (ae_eventually_mem : ∀ᵐ x ∂μ, x ∈ s → ∀ᶠ n in l, x ∈ φ n) (measurable : ∀ n, MeasurableSet <\| φ n) : AECover (μ.restrict s) l φ where ae_eventually_mem := by rwa [ae_restrict_iff' hs] measurableSet := measurable #align measure_theory.ae_cover_restrict_of_ae_imp MeasureTheory.aecover_restrict_of_ae_imp theorem AECover.inter_restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} (hs : MeasurableSet s) : AECover (μ.restrict s) l fun i => φ i ∩ s := aecover_restrict_of_ae_imp hs (hφ.ae_eventually_mem.mono fun _x hx hxs => hx.mono fun _i hi => ⟨hi, hxs⟩) fun i => (hφ.measurableSet i).inter hs #align measure_theory.ae_cover.inter_restrict MeasureTheory.AECover.inter_restrict theorem AECover.ae_tendsto_indicator {β : Type} [Zero β] [TopologicalSpace β] (f : α → β) {φ : ι → Set α} (hφ : AECover μ l φ) : ∀ᵐ x ∂μ, Tendsto (fun i => (φ i).indicator f x) l (𝓝 <\| f x) := hφ.ae_eventually_mem.mono fun _x hx => tendsto_const_nhds.congr' <\| hx.mono fun _n hn => (indicator_of_mem hn _).symm #align measure_theory.ae_cover.ae_tendsto_indicator MeasureTheory.AECover.ae_tendsto_indicator theorem AECover.aemeasurable {β : Type} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ) (hfm : ∀ i, AEMeasurable f (μ.restrict <\| φ i)) : AEMeasurable f μ := by obtain ⟨u, hu⟩ := l.exists_seq_tendsto have := aemeasurable_iUnion_iff.mpr fun n : ℕ => hfm (u n) rwa [Measure.restrict_eq_self_of_ae_mem] at this filter_upwards [hφ.ae_eventually_mem] with x hx using mem_iUnion.mpr (hu.eventually hx).exists #align measure_theory.ae_cover.ae_measurable MeasureTheory.AECover.aemeasurable theorem AECover.aestronglyMeasurable {β : Type} [TopologicalSpace β] [PseudoMetrizableSpace β] [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ) (hfm : ∀ i, AEStronglyMeasurable f (μ.restrict <\| φ i)) : AEStronglyMeasurable f μ := by obtain ⟨u, hu⟩ := l.exists_seq_tendsto have := aestronglyMeasurable_iUnion_iff.mpr fun n : ℕ => hfm (u n) rwa [Measure.restrict_eq_self_of_ae_mem] at this filter_upwards [hφ.ae_eventually_mem] with x hx using mem_iUnion.mpr (hu.eventually hx).exists #align measure_theory.ae_cover.ae_strongly_measurable MeasureTheory.AECover.aestronglyMeasurable end AECover theorem AECover.comp_tendsto {α ι ι' : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} {l' : Filter ι'} {φ : ι → Set α} (hφ : AECover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) : AECover μ l' (φ ∘ u) where ae_eventually_mem := hφ.ae_eventually_mem.mono fun _x hx => hu.eventually hx measurableSet i := hφ.measurableSet (u i) #align measure_theory.ae_cover.comp_tendsto MeasureTheory.AECover.comp_tendsto section AECoverUnionInterCountable variable {α ι : Type} [Countable ι] [MeasurableSpace α] {μ : Measure α} theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋃ (k) (_h : k ∈ Iic n), φ k := hφ.superset (fun _ ↦ subset_biUnion_of_mem right_mem_Iic) fun _ ↦ .biUnion (to_countable _) fun _ _ ↦ (hφ.2 _) #align measure_theory.ae_cover.bUnion_Iic_ae_cover MeasureTheory.AECover.biUnion_Iic_aecover -- Porting note: generalized from `[SemilatticeSup ι] [Nonempty ι]` to `[Preorder ι]` theorem AECover.biInter_Ici_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (_h : k ∈ Ici n), φ k where ae_eventually_mem := hφ.ae_eventually_mem.mono fun x h ↦ by simpa only [mem_iInter, mem_Ici, eventually_forall_ge_atTop] measurableSet i := .biInter (to_countable _) fun n _ => hφ.measurableSet n #align measure_theory.ae_cover.bInter_Ici_ae_cover MeasureTheory.AECover.biInter_Ici_aecover end AECoverUnionInterCountable section Lintegral variable {α ι : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <\| ∫⁻ x, f x ∂μ) := let F n := (φ n).indicator f have key₁ : ∀ n, AEMeasurable (F n) μ := fun n => hfm.indicator (hφ.measurableSet n) have key₂ : ∀ᵐ x : α ∂μ, Monotone fun n => F n x := ae_of_all _ fun x _i _j hij => indicator_le_indicator_of_subset (hmono hij) (fun x => zero_le <\| f x) x have key₃ : ∀ᵐ x : α ∂μ, Tendsto (fun n => F n x) atTop (𝓝 (f x)) := hφ.ae_tendsto_indicator f (lintegral_tendsto_of_tendsto_of_monotone key₁ key₂ key₃).congr fun n => lintegral_indicator f (hφ.measurableSet n) theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : Tendsto (∫⁻ x in φ ·, f x ∂μ) atTop (𝓝 <\| ∫⁻ x, f x ∂μ) := by have lim₁ := lintegral_tendsto_of_monotone_of_nat hφ.biInter_Ici_aecover (fun i j hij => biInter_subset_biInter_left (Ici_subset_Ici.mpr hij)) hfm have lim₂ := lintegral_tendsto_of_monotone_of_nat hφ.biUnion_Iic_aecover (fun i j hij => biUnion_subset_biUnion_left (Iic_subset_Iic.mpr hij)) hfm refine tendsto_of_tendsto_of_tendsto_of_le_of_le lim₁ lim₂ (fun n ↦ ?_) fun n ↦ ?_ exacts [lintegral_mono_set (biInter_subset_of_mem left_mem_Ici), lintegral_mono_set (subset_biUnion_of_mem right_mem_Iic)] #align measure_theory.ae_cover.lintegral_tendsto_of_nat MeasureTheory.AECover.lintegral_tendsto_of_nat theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 <\| ∫⁻ x, f x ∂μ) := tendsto_of_seq_tendsto fun _u hu => (hφ.comp_tendsto hu).lintegral_tendsto_of_nat hfm #align measure_theory.ae_cover.lintegral_tendsto_of_countably_generated MeasureTheory.AECover.lintegral_tendsto_of_countably_generated theorem AECover.lintegral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ) (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : ∫⁻ x, f x ∂μ = I := tendsto_nhds_unique (hφ.lintegral_tendsto_of_countably_generated hfm) htendsto #align measure_theory.ae_cover.lintegral_eq_of_tendsto MeasureTheory.AECover.lintegral_eq_of_tendsto theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) : ⨆ i : ι, ∫⁻ x in φ i, f x ∂μ = ∫⁻ x, f x ∂μ := by have := hφ.lintegral_tendsto_of_countably_generated hfm refine ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun i => lintegral_mono' Measure.restrict_le_self le_rfl) fun w hw => ?_ rcases exists_between hw with ⟨m, hm₁, hm₂⟩ rcases (eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩ exact ⟨i, lt_of_lt_of_le hm₁ hi⟩ #align measure_theory.ae_cover.supr_lintegral_eq_of_countably_generated MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated end Lintegral section Integrable variable {α ι E : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E] theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ) (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ := by refine ⟨hfm, (le_of_tendsto ?_ hbounded).trans_lt ENNReal.ofReal_lt_top⟩ exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ) (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <\| ENNReal.ofReal I)) : Integrable f μ := by refine hφ.integrable_of_lintegral_nnnorm_bounded (max 1 (I + 1)) hfm ?_ refine htendsto.eventually (ge_mem_nhds ?_) refine (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 ?_ exact lt_max_of_lt_right (lt_add_one I) #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ) (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ := hφ.integrable_of_lintegral_nnnorm_bounded I hfm (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded) #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_bounded' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded' theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ) (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ := hφ.integrable_of_lintegral_nnnorm_tendsto I hfm (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto) #align measure_theory.ae_cover.integrable_of_lintegral_nnnorm_tendsto' MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto' theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ := by have hfm : AEStronglyMeasurable f μ := hφ.aestronglyMeasurable fun i => (hfi i).aestronglyMeasurable refine hφ.integrable_of_lintegral_nnnorm_bounded I hfm ?_ conv at hbounded in integral _ _ => rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x)) hfm.norm.restrict] conv at hbounded in ENNReal.ofReal _ => rw [← coe_nnnorm] rw [ENNReal.ofReal_coe_nnreal] refine hbounded.mono fun i hi => ?_ rw [← ENNReal.ofReal_toReal (ne_top_of_lt (hfi i).2)] apply ENNReal.ofReal_le_ofReal hi #align measure_theory.ae_cover.integrable_of_integral_norm_bounded MeasureTheory.AECover.integrable_of_integral_norm_bounded theorem AECover.integrable_of_integral_norm_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ := let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le hφ.integrable_of_integral_norm_bounded I' hfi hI' #align measure_theory.ae_cover.integrable_of_integral_norm_tendsto MeasureTheory.AECover.integrable_of_integral_norm_tendsto theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ := hφ.integrable_of_integral_norm_bounded I hfi <\| hbounded.mono fun _i hi => (integral_congr_ae <\| ae_restrict_of_ae <\| hnng.mono fun _ => Real.norm_of_nonneg).le.trans hi #align measure_theory.ae_cover.integrable_of_integral_bounded_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ) (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) : Integrable f μ := let ⟨I', hI'⟩ := htendsto.isBoundedUnder_le hφ.integrable_of_integral_bounded_of_nonneg_ae I' hfi hnng hI' #align measure_theory.ae_cover.integrable_of_integral_tendsto_of_nonneg_ae MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae end Integrable section Integral variable {α ι E : Type} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (hfi : Integrable f μ) : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <\| ∫ x, f x ∂μ) := suffices h : Tendsto (fun i => ∫ x : α, (φ i).indicator f x ∂μ) l (𝓝 (∫ x : α, f x ∂μ)) from by convert h using 2; rw [integral_indicator (hφ.measurableSet _)] tendsto_integral_filter_of_dominated_convergence (fun x => ‖f x‖) (eventually_of_forall fun i => hfi.aestronglyMeasurable.indicator <\| hφ.measurableSet i) (eventually_of_forall fun i => ae_of_all _ fun x => norm_indicator_le_norm_self _ _) hfi.norm (hφ.ae_tendsto_indicator f) #align measure_theory.ae_cover.integral_tendsto_of_countably_generated MeasureTheory.AECover.integral_tendsto_of_countably_generated /-- Slight reformulation of `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/ theorem AECover.integral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ) (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I := tendsto_nhds_unique (hφ.integral_tendsto_of_countably_generated hfi) h #align measure_theory.ae_cover.integral_eq_of_tendsto MeasureTheory.AECover.integral_eq_of_tendsto theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f) (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I := have hfi' : Integrable f μ := hφ.integrable_of_integral_tendsto_of_nonneg_ae I hfi hnng htendsto hφ.integral_eq_of_tendsto I hfi' htendsto #align measure_theory.ae_cover.integral_eq_of_tendsto_of_nonneg_ae MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae end Integral section IntegrableOfIntervalIntegral variable {ι E : Type} {μ : Measure ℝ} {l : Filter ι} [Filter.NeBot l] [IsCountablyGenerated l] [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E} theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ := by have hφ : AECover μ l _ := aecover_Ioc ha hb refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_) filter_upwards [ha.eventually (eventually_le_atBot 0), hb.eventually (eventually_ge_atTop 0)] with i hai hbi ht rwa [← intervalIntegral.integral_of_le (hai.trans hbi)] #align measure_theory.integrable_of_interval_integral_norm_bounded MeasureTheory.integrable_of_intervalIntegral_norm_bounded /-- If `f` is integrable on intervals `Ioc (a i) (b i)`, where `a i` tends to -∞ and `b i` tends to ∞, and `∫ x in a i .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`, then `f` is integrable on the interval (-∞, ∞) -/ theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a i..b i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ := let ⟨I', hI'⟩ := h.isBoundedUnder_le integrable_of_intervalIntegral_norm_bounded I' hfi ha hb hI' #align measure_theory.integrable_of_interval_integral_norm_tendsto MeasureTheory.integrable_of_intervalIntegral_norm_tendsto theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot) (h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ := by have hφ : AECover (μ.restrict <\| Iic b) l _ := aecover_Ioi ha have hfi : ∀ i, IntegrableOn f (Ioi (a i)) (μ.restrict <\| Iic b) := by intro i rw [IntegrableOn, Measure.restrict_restrict (hφ.measurableSet i)] exact hfi i refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_) filter_upwards [ha.eventually (eventually_le_atBot b)] with i hai rw [intervalIntegral.integral_of_le hai, Measure.restrict_restrict (hφ.measurableSet i)] exact id #align measure_theory.integrable_on_Iic_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded /-- If `f` is integrable on intervals `Ioc (a i) b`, where `a i` tends to -∞, and `∫ x in a i .. b, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`, then `f` is integrable on the interval (-∞, b) -/ theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot) (h : Tendsto (fun i => ∫ x in a i..b, ‖f x‖ ∂μ) l (𝓝 I)) : IntegrableOn f (Iic b) μ := let ⟨I', hI'⟩ := h.isBoundedUnder_le integrableOn_Iic_of_intervalIntegral_norm_bounded I' b hfi ha hI' #align measure_theory.integrable_on_Iic_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a..b i, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Ioi a) μ := by have hφ : AECover (μ.restrict <\| Ioi a) l _ := aecover_Iic hb have hfi : ∀ i, IntegrableOn f (Iic (b i)) (μ.restrict <\| Ioi a) := by intro i rw [IntegrableOn, Measure.restrict_restrict (hφ.measurableSet i), inter_comm] exact hfi i refine hφ.integrable_of_integral_norm_bounded I hfi (h.mp ?_) filter_upwards [hb.eventually (eventually_ge_atTop a)] with i hbi rw [intervalIntegral.integral_of_le hbi, Measure.restrict_restrict (hφ.measurableSet i), inter_comm] exact id #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded /-- If `f` is integrable on intervals `Ioc a (b i)`, where `b i` tends to ∞, and `∫ x in a .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`, then `f` is integrable on the interval (a, ∞) -/ theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ) (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a..b i, ‖f x‖ ∂μ) l (𝓝 <\| I)) : IntegrableOn f (Ioi a) μ := let ⟨I', hI'⟩ := h.isBoundedUnder_le integrableOn_Ioi_of_intervalIntegral_norm_bounded I' a hfi hb hI' #align measure_theory.integrable_on_Ioi_of_interval_integral_norm_tendsto MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ} (hfi : ∀ i, IntegrableOn f <\| Ioc (a i) (b i)) (ha : Tendsto a l <\| 𝓝 a₀) (hb : Tendsto b l <\| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b₀) := by refine (aecover_Ioc_of_Ioc ha hb).integrable_of_integral_norm_bounded I (fun i => (hfi i).restrict measurableSet_Ioc) (h.mono fun i hi ↦ ?_) rw [Measure.restrict_restrict measurableSet_Ioc] refine le_trans (setIntegral_mono_set (hfi i).norm ?_ ?_) hi <;> apply ae_of_all · simp only [Pi.zero_apply, norm_nonneg, forall_const] · intro c hc; exact hc.1 #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ} (hfi : ∀ i, IntegrableOn f <\| Ioc (a i) b) (ha : Tendsto a l <\| 𝓝 a₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) := integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi ha tendsto_const_nhds h #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_left MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_right {I a b₀ : ℝ} (hfi : ∀ i, IntegrableOn f <\| Ioc a (b i)) (hb : Tendsto b l <\| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) := integrableOn_Ioc_of_intervalIntegral_norm_bounded hfi tendsto_const_nhds hb h #align measure_theory.integrable_on_Ioc_of_interval_integral_norm_bounded_right MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right @[deprecated (since := "2024-04-06")] alias integrableOn_Ioc_of_interval_integral_norm_bounded := integrableOn_Ioc_of_intervalIntegral_norm_bounded @[deprecated (since := "2024-04-06")] alias integrableOn_Ioc_of_interval_integral_norm_bounded_left := integrableOn_Ioc_of_intervalIntegral_norm_bounded_left @[deprecated (since := "2024-04-06")] alias integrableOn_Ioc_of_interval_integral_norm_bounded_right := integrableOn_Ioc_of_intervalIntegral_norm_bounded_right end IntegrableOfIntervalIntegral section IntegralOfIntervalIntegral variable {ι E : Type*} {μ : Measure ℝ} {l : Filter ι} [IsCountablyGenerated l] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {a b : ι → ℝ} {f : ℝ → E} theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto a l atBot) (hb : Tendsto b l atTop) : Tendsto (fun i => ∫ x in a i..b i, f x ∂μ) l (𝓝 <\| ∫ x, f x ∂μ) := by let φ i := Ioc (a i) (b i) have hφ : AECover μ l φ := aecover_Ioc ha hb refine (hφ.integral_tendsto_of_countably_generated hfi).congr' ?_ filter_upwards [ha.eventually (eventually_le_atBot 0), hb.eventually (eventually_ge_atTop 0)] with i hai hbi exact (intervalIntegral.integral_of_le (hai.trans hbi)).symm #align measure_theory.interval_integral_tendsto_integral MeasureTheory.intervalIntegral_tendsto_integral	Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean	669	677	theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ) (ha : Tendsto a l atBot) : Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <\| ∫ x in Iic b, f x ∂μ) := by	let φ i := Ioi (a i) have hφ : AECover (μ.restrict <\| Iic b) l φ := aecover_Ioi ha refine (hφ.integral_tendsto_of_countably_generated hfi).congr' ?_ filter_upwards [ha.eventually (eventually_le_atBot <\| b)] with i hai rw [intervalIntegral.integral_of_le hai, Measure.restrict_restrict (hφ.measurableSet i)] rfl
/- Copyright (c) 2020 Joseph Myers. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Joseph Myers -/ import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv #align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840" /-! # Affine spaces This file defines affine subspaces (over modules) and the affine span of a set of points. ## Main definitions * `AffineSubspace k P` is the type of affine subspaces. Unlike affine spaces, affine subspaces are allowed to be empty, and lemmas that do not apply to empty affine subspaces have `Nonempty` hypotheses. There is a `CompleteLattice` structure on affine subspaces. * `AffineSubspace.direction` gives the `Submodule` spanned by the pairwise differences of points in an `AffineSubspace`. There are various lemmas relating to the set of vectors in the `direction`, and relating the lattice structure on affine subspaces to that on their directions. * `AffineSubspace.parallel`, notation `∥`, gives the property of two affine subspaces being parallel (one being a translate of the other). * `affineSpan` gives the affine subspace spanned by a set of points, with `vectorSpan` giving its direction. The `affineSpan` is defined in terms of `spanPoints`, which gives an explicit description of the points contained in the affine span; `spanPoints` itself should generally only be used when that description is required, with `affineSpan` being the main definition for other purposes. Two other descriptions of the affine span are proved equivalent: it is the `sInf` of affine subspaces containing the points, and (if `[Nontrivial k]`) it contains exactly those points that are affine combinations of points in the given set. ## Implementation notes `outParam` is used in the definition of `AddTorsor V P` to make `V` an implicit argument (deduced from `P`) in most cases. As for modules, `k` is an explicit argument rather than implied by `P` or `V`. This file only provides purely algebraic definitions and results. Those depending on analysis or topology are defined elsewhere; see `Analysis.NormedSpace.AddTorsor` and `Topology.Algebra.Affine`. ## References * https://en.wikipedia.org/wiki/Affine_space * https://en.wikipedia.org/wiki/Principal_homogeneous_space -/ noncomputable section open Affine open Set section variable (k : Type) {V : Type} {P : Type} [Ring k] [AddCommGroup V] [Module k V] variable [AffineSpace V P] /-- The submodule spanning the differences of a (possibly empty) set of points. -/ def vectorSpan (s : Set P) : Submodule k V := Submodule.span k (s -ᵥ s) #align vector_span vectorSpan /-- The definition of `vectorSpan`, for rewriting. -/ theorem vectorSpan_def (s : Set P) : vectorSpan k s = Submodule.span k (s -ᵥ s) := rfl #align vector_span_def vectorSpan_def /-- `vectorSpan` is monotone. -/ theorem vectorSpan_mono {s₁ s₂ : Set P} (h : s₁ ⊆ s₂) : vectorSpan k s₁ ≤ vectorSpan k s₂ := Submodule.span_mono (vsub_self_mono h) #align vector_span_mono vectorSpan_mono variable (P) /-- The `vectorSpan` of the empty set is `⊥`. -/ @[simp] theorem vectorSpan_empty : vectorSpan k (∅ : Set P) = (⊥ : Submodule k V) := by rw [vectorSpan_def, vsub_empty, Submodule.span_empty] #align vector_span_empty vectorSpan_empty variable {P} /-- The `vectorSpan` of a single point is `⊥`. -/ @[simp] theorem vectorSpan_singleton (p : P) : vectorSpan k ({p} : Set P) = ⊥ := by simp [vectorSpan_def] #align vector_span_singleton vectorSpan_singleton /-- The `s -ᵥ s` lies within the `vectorSpan k s`. -/ theorem vsub_set_subset_vectorSpan (s : Set P) : s -ᵥ s ⊆ ↑(vectorSpan k s) := Submodule.subset_span #align vsub_set_subset_vector_span vsub_set_subset_vectorSpan /-- Each pairwise difference is in the `vectorSpan`. -/ theorem vsub_mem_vectorSpan {s : Set P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) : p1 -ᵥ p2 ∈ vectorSpan k s := vsub_set_subset_vectorSpan k s (vsub_mem_vsub hp1 hp2) #align vsub_mem_vector_span vsub_mem_vectorSpan /-- The points in the affine span of a (possibly empty) set of points. Use `affineSpan` instead to get an `AffineSubspace k P`. -/ def spanPoints (s : Set P) : Set P := { p \| ∃ p1 ∈ s, ∃ v ∈ vectorSpan k s, p = v +ᵥ p1 } #align span_points spanPoints /-- A point in a set is in its affine span. -/ theorem mem_spanPoints (p : P) (s : Set P) : p ∈ s → p ∈ spanPoints k s \| hp => ⟨p, hp, 0, Submodule.zero_mem _, (zero_vadd V p).symm⟩ #align mem_span_points mem_spanPoints /-- A set is contained in its `spanPoints`. -/ theorem subset_spanPoints (s : Set P) : s ⊆ spanPoints k s := fun p => mem_spanPoints k p s #align subset_span_points subset_spanPoints /-- The `spanPoints` of a set is nonempty if and only if that set is. -/ @[simp] theorem spanPoints_nonempty (s : Set P) : (spanPoints k s).Nonempty ↔ s.Nonempty := by constructor · contrapose rw [Set.not_nonempty_iff_eq_empty, Set.not_nonempty_iff_eq_empty] intro h simp [h, spanPoints] · exact fun h => h.mono (subset_spanPoints _ _) #align span_points_nonempty spanPoints_nonempty /-- Adding a point in the affine span and a vector in the spanning submodule produces a point in the affine span. -/ theorem vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan {s : Set P} {p : P} {v : V} (hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s) : v +ᵥ p ∈ spanPoints k s := by rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩ rw [hv2p, vadd_vadd] exact ⟨p2, hp2, v + v2, (vectorSpan k s).add_mem hv hv2, rfl⟩ #align vadd_mem_span_points_of_mem_span_points_of_mem_vector_span vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan /-- Subtracting two points in the affine span produces a vector in the spanning submodule. -/ theorem vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints {s : Set P} {p1 p2 : P} (hp1 : p1 ∈ spanPoints k s) (hp2 : p2 ∈ spanPoints k s) : p1 -ᵥ p2 ∈ vectorSpan k s := by rcases hp1 with ⟨p1a, ⟨hp1a, ⟨v1, ⟨hv1, hv1p⟩⟩⟩⟩ rcases hp2 with ⟨p2a, ⟨hp2a, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩ rw [hv1p, hv2p, vsub_vadd_eq_vsub_sub (v1 +ᵥ p1a), vadd_vsub_assoc, add_comm, add_sub_assoc] have hv1v2 : v1 - v2 ∈ vectorSpan k s := (vectorSpan k s).sub_mem hv1 hv2 refine (vectorSpan k s).add_mem ?_ hv1v2 exact vsub_mem_vectorSpan k hp1a hp2a #align vsub_mem_vector_span_of_mem_span_points_of_mem_span_points vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints end /-- An `AffineSubspace k P` is a subset of an `AffineSpace V P` that, if not empty, has an affine space structure induced by a corresponding subspace of the `Module k V`. -/ structure AffineSubspace (k : Type) {V : Type} (P : Type) [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] where /-- The affine subspace seen as a subset. -/ carrier : Set P smul_vsub_vadd_mem : ∀ (c : k) {p1 p2 p3 : P}, p1 ∈ carrier → p2 ∈ carrier → p3 ∈ carrier → c • (p1 -ᵥ p2 : V) +ᵥ p3 ∈ carrier #align affine_subspace AffineSubspace namespace Submodule variable {k V : Type} [Ring k] [AddCommGroup V] [Module k V] /-- Reinterpret `p : Submodule k V` as an `AffineSubspace k V`. -/ def toAffineSubspace (p : Submodule k V) : AffineSubspace k V where carrier := p smul_vsub_vadd_mem _ _ _ _ h₁ h₂ h₃ := p.add_mem (p.smul_mem _ (p.sub_mem h₁ h₂)) h₃ #align submodule.to_affine_subspace Submodule.toAffineSubspace end Submodule namespace AffineSubspace variable (k : Type) {V : Type} (P : Type) [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] instance : SetLike (AffineSubspace k P) P where coe := carrier coe_injective' p q _ := by cases p; cases q; congr /-- A point is in an affine subspace coerced to a set if and only if it is in that affine subspace. -/ -- Porting note: removed `simp`, proof is `simp only [SetLike.mem_coe]` theorem mem_coe (p : P) (s : AffineSubspace k P) : p ∈ (s : Set P) ↔ p ∈ s := Iff.rfl #align affine_subspace.mem_coe AffineSubspace.mem_coe variable {k P} /-- The direction of an affine subspace is the submodule spanned by the pairwise differences of points. (Except in the case of an empty affine subspace, where the direction is the zero submodule, every vector in the direction is the difference of two points in the affine subspace.) -/ def direction (s : AffineSubspace k P) : Submodule k V := vectorSpan k (s : Set P) #align affine_subspace.direction AffineSubspace.direction /-- The direction equals the `vectorSpan`. -/ theorem direction_eq_vectorSpan (s : AffineSubspace k P) : s.direction = vectorSpan k (s : Set P) := rfl #align affine_subspace.direction_eq_vector_span AffineSubspace.direction_eq_vectorSpan /-- Alternative definition of the direction when the affine subspace is nonempty. This is defined so that the order on submodules (as used in the definition of `Submodule.span`) can be used in the proof of `coe_direction_eq_vsub_set`, and is not intended to be used beyond that proof. -/ def directionOfNonempty {s : AffineSubspace k P} (h : (s : Set P).Nonempty) : Submodule k V where carrier := (s : Set P) -ᵥ s zero_mem' := by cases' h with p hp exact vsub_self p ▸ vsub_mem_vsub hp hp add_mem' := by rintro _ _ ⟨p1, hp1, p2, hp2, rfl⟩ ⟨p3, hp3, p4, hp4, rfl⟩ rw [← vadd_vsub_assoc] refine vsub_mem_vsub ?_ hp4 convert s.smul_vsub_vadd_mem 1 hp1 hp2 hp3 rw [one_smul] smul_mem' := by rintro c _ ⟨p1, hp1, p2, hp2, rfl⟩ rw [← vadd_vsub (c • (p1 -ᵥ p2)) p2] refine vsub_mem_vsub ?_ hp2 exact s.smul_vsub_vadd_mem c hp1 hp2 hp2 #align affine_subspace.direction_of_nonempty AffineSubspace.directionOfNonempty /-- `direction_of_nonempty` gives the same submodule as `direction`. -/ theorem directionOfNonempty_eq_direction {s : AffineSubspace k P} (h : (s : Set P).Nonempty) : directionOfNonempty h = s.direction := by refine le_antisymm ?_ (Submodule.span_le.2 Set.Subset.rfl) rw [← SetLike.coe_subset_coe, directionOfNonempty, direction, Submodule.coe_set_mk, AddSubmonoid.coe_set_mk] exact vsub_set_subset_vectorSpan k _ #align affine_subspace.direction_of_nonempty_eq_direction AffineSubspace.directionOfNonempty_eq_direction /-- The set of vectors in the direction of a nonempty affine subspace is given by `vsub_set`. -/ theorem coe_direction_eq_vsub_set {s : AffineSubspace k P} (h : (s : Set P).Nonempty) : (s.direction : Set V) = (s : Set P) -ᵥ s := directionOfNonempty_eq_direction h ▸ rfl #align affine_subspace.coe_direction_eq_vsub_set AffineSubspace.coe_direction_eq_vsub_set /-- A vector is in the direction of a nonempty affine subspace if and only if it is the subtraction of two vectors in the subspace. -/ theorem mem_direction_iff_eq_vsub {s : AffineSubspace k P} (h : (s : Set P).Nonempty) (v : V) : v ∈ s.direction ↔ ∃ p1 ∈ s, ∃ p2 ∈ s, v = p1 -ᵥ p2 := by rw [← SetLike.mem_coe, coe_direction_eq_vsub_set h, Set.mem_vsub] simp only [SetLike.mem_coe, eq_comm] #align affine_subspace.mem_direction_iff_eq_vsub AffineSubspace.mem_direction_iff_eq_vsub /-- Adding a vector in the direction to a point in the subspace produces a point in the subspace. -/ theorem vadd_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} (hp : p ∈ s) : v +ᵥ p ∈ s := by rw [mem_direction_iff_eq_vsub ⟨p, hp⟩] at hv rcases hv with ⟨p1, hp1, p2, hp2, hv⟩ rw [hv] convert s.smul_vsub_vadd_mem 1 hp1 hp2 hp rw [one_smul] exact s.mem_coe k P _ #align affine_subspace.vadd_mem_of_mem_direction AffineSubspace.vadd_mem_of_mem_direction /-- Subtracting two points in the subspace produces a vector in the direction. -/ theorem vsub_mem_direction {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) : p1 -ᵥ p2 ∈ s.direction := vsub_mem_vectorSpan k hp1 hp2 #align affine_subspace.vsub_mem_direction AffineSubspace.vsub_mem_direction /-- Adding a vector to a point in a subspace produces a point in the subspace if and only if the vector is in the direction. -/ theorem vadd_mem_iff_mem_direction {s : AffineSubspace k P} (v : V) {p : P} (hp : p ∈ s) : v +ᵥ p ∈ s ↔ v ∈ s.direction := ⟨fun h => by simpa using vsub_mem_direction h hp, fun h => vadd_mem_of_mem_direction h hp⟩ #align affine_subspace.vadd_mem_iff_mem_direction AffineSubspace.vadd_mem_iff_mem_direction /-- Adding a vector in the direction to a point produces a point in the subspace if and only if the original point is in the subspace. -/ theorem vadd_mem_iff_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} : v +ᵥ p ∈ s ↔ p ∈ s := by refine ⟨fun h => ?_, fun h => vadd_mem_of_mem_direction hv h⟩ convert vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) h simp #align affine_subspace.vadd_mem_iff_mem_of_mem_direction AffineSubspace.vadd_mem_iff_mem_of_mem_direction /-- Given a point in an affine subspace, the set of vectors in its direction equals the set of vectors subtracting that point on the right. -/ theorem coe_direction_eq_vsub_set_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : (s.direction : Set V) = (· -ᵥ p) '' s := by rw [coe_direction_eq_vsub_set ⟨p, hp⟩] refine le_antisymm ?_ ?_ · rintro v ⟨p1, hp1, p2, hp2, rfl⟩ exact ⟨p1 -ᵥ p2 +ᵥ p, vadd_mem_of_mem_direction (vsub_mem_direction hp1 hp2) hp, vadd_vsub _ _⟩ · rintro v ⟨p2, hp2, rfl⟩ exact ⟨p2, hp2, p, hp, rfl⟩ #align affine_subspace.coe_direction_eq_vsub_set_right AffineSubspace.coe_direction_eq_vsub_set_right /-- Given a point in an affine subspace, the set of vectors in its direction equals the set of vectors subtracting that point on the left. -/ theorem coe_direction_eq_vsub_set_left {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : (s.direction : Set V) = (p -ᵥ ·) '' s := by ext v rw [SetLike.mem_coe, ← Submodule.neg_mem_iff, ← SetLike.mem_coe, coe_direction_eq_vsub_set_right hp, Set.mem_image, Set.mem_image] conv_lhs => congr ext rw [← neg_vsub_eq_vsub_rev, neg_inj] #align affine_subspace.coe_direction_eq_vsub_set_left AffineSubspace.coe_direction_eq_vsub_set_left /-- Given a point in an affine subspace, a vector is in its direction if and only if it results from subtracting that point on the right. -/ theorem mem_direction_iff_eq_vsub_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) : v ∈ s.direction ↔ ∃ p2 ∈ s, v = p2 -ᵥ p := by rw [← SetLike.mem_coe, coe_direction_eq_vsub_set_right hp] exact ⟨fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩, fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩⟩ #align affine_subspace.mem_direction_iff_eq_vsub_right AffineSubspace.mem_direction_iff_eq_vsub_right /-- Given a point in an affine subspace, a vector is in its direction if and only if it results from subtracting that point on the left. -/ theorem mem_direction_iff_eq_vsub_left {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) : v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -ᵥ p2 := by rw [← SetLike.mem_coe, coe_direction_eq_vsub_set_left hp] exact ⟨fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩, fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩⟩ #align affine_subspace.mem_direction_iff_eq_vsub_left AffineSubspace.mem_direction_iff_eq_vsub_left /-- Given a point in an affine subspace, a result of subtracting that point on the right is in the direction if and only if the other point is in the subspace. -/ theorem vsub_right_mem_direction_iff_mem {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P) : p2 -ᵥ p ∈ s.direction ↔ p2 ∈ s := by rw [mem_direction_iff_eq_vsub_right hp] simp #align affine_subspace.vsub_right_mem_direction_iff_mem AffineSubspace.vsub_right_mem_direction_iff_mem /-- Given a point in an affine subspace, a result of subtracting that point on the left is in the direction if and only if the other point is in the subspace. -/ theorem vsub_left_mem_direction_iff_mem {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P) : p -ᵥ p2 ∈ s.direction ↔ p2 ∈ s := by rw [mem_direction_iff_eq_vsub_left hp] simp #align affine_subspace.vsub_left_mem_direction_iff_mem AffineSubspace.vsub_left_mem_direction_iff_mem /-- Two affine subspaces are equal if they have the same points. -/ theorem coe_injective : Function.Injective ((↑) : AffineSubspace k P → Set P) := SetLike.coe_injective #align affine_subspace.coe_injective AffineSubspace.coe_injective @[ext] theorem ext {p q : AffineSubspace k P} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q := SetLike.ext h #align affine_subspace.ext AffineSubspace.ext -- Porting note: removed `simp`, proof is `simp only [SetLike.ext'_iff]` theorem ext_iff (s₁ s₂ : AffineSubspace k P) : (s₁ : Set P) = s₂ ↔ s₁ = s₂ := SetLike.ext'_iff.symm #align affine_subspace.ext_iff AffineSubspace.ext_iff /-- Two affine subspaces with the same direction and nonempty intersection are equal. -/ theorem ext_of_direction_eq {s1 s2 : AffineSubspace k P} (hd : s1.direction = s2.direction) (hn : ((s1 : Set P) ∩ s2).Nonempty) : s1 = s2 := by ext p have hq1 := Set.mem_of_mem_inter_left hn.some_mem have hq2 := Set.mem_of_mem_inter_right hn.some_mem constructor · intro hp rw [← vsub_vadd p hn.some] refine vadd_mem_of_mem_direction ?_ hq2 rw [← hd] exact vsub_mem_direction hp hq1 · intro hp rw [← vsub_vadd p hn.some] refine vadd_mem_of_mem_direction ?_ hq1 rw [hd] exact vsub_mem_direction hp hq2 #align affine_subspace.ext_of_direction_eq AffineSubspace.ext_of_direction_eq -- See note [reducible non instances] /-- This is not an instance because it loops with `AddTorsor.nonempty`. -/ abbrev toAddTorsor (s : AffineSubspace k P) [Nonempty s] : AddTorsor s.direction s where vadd a b := ⟨(a : V) +ᵥ (b : P), vadd_mem_of_mem_direction a.2 b.2⟩ zero_vadd := fun a => by ext exact zero_vadd _ _ add_vadd a b c := by ext apply add_vadd vsub a b := ⟨(a : P) -ᵥ (b : P), (vsub_left_mem_direction_iff_mem a.2 _).mpr b.2⟩ vsub_vadd' a b := by ext apply AddTorsor.vsub_vadd' vadd_vsub' a b := by ext apply AddTorsor.vadd_vsub' #align affine_subspace.to_add_torsor AffineSubspace.toAddTorsor attribute [local instance] toAddTorsor @[simp, norm_cast] theorem coe_vsub (s : AffineSubspace k P) [Nonempty s] (a b : s) : ↑(a -ᵥ b) = (a : P) -ᵥ (b : P) := rfl #align affine_subspace.coe_vsub AffineSubspace.coe_vsub @[simp, norm_cast] theorem coe_vadd (s : AffineSubspace k P) [Nonempty s] (a : s.direction) (b : s) : ↑(a +ᵥ b) = (a : V) +ᵥ (b : P) := rfl #align affine_subspace.coe_vadd AffineSubspace.coe_vadd /-- Embedding of an affine subspace to the ambient space, as an affine map. -/ protected def subtype (s : AffineSubspace k P) [Nonempty s] : s →ᵃ[k] P where toFun := (↑) linear := s.direction.subtype map_vadd' _ _ := rfl #align affine_subspace.subtype AffineSubspace.subtype @[simp] theorem subtype_linear (s : AffineSubspace k P) [Nonempty s] : s.subtype.linear = s.direction.subtype := rfl #align affine_subspace.subtype_linear AffineSubspace.subtype_linear theorem subtype_apply (s : AffineSubspace k P) [Nonempty s] (p : s) : s.subtype p = p := rfl #align affine_subspace.subtype_apply AffineSubspace.subtype_apply @[simp] theorem coeSubtype (s : AffineSubspace k P) [Nonempty s] : (s.subtype : s → P) = ((↑) : s → P) := rfl #align affine_subspace.coe_subtype AffineSubspace.coeSubtype theorem injective_subtype (s : AffineSubspace k P) [Nonempty s] : Function.Injective s.subtype := Subtype.coe_injective #align affine_subspace.injective_subtype AffineSubspace.injective_subtype /-- Two affine subspaces with nonempty intersection are equal if and only if their directions are equal. -/ theorem eq_iff_direction_eq_of_mem {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) : s₁ = s₂ ↔ s₁.direction = s₂.direction := ⟨fun h => h ▸ rfl, fun h => ext_of_direction_eq h ⟨p, h₁, h₂⟩⟩ #align affine_subspace.eq_iff_direction_eq_of_mem AffineSubspace.eq_iff_direction_eq_of_mem /-- Construct an affine subspace from a point and a direction. -/ def mk' (p : P) (direction : Submodule k V) : AffineSubspace k P where carrier := { q \| ∃ v ∈ direction, q = v +ᵥ p } smul_vsub_vadd_mem c p1 p2 p3 hp1 hp2 hp3 := by rcases hp1 with ⟨v1, hv1, hp1⟩ rcases hp2 with ⟨v2, hv2, hp2⟩ rcases hp3 with ⟨v3, hv3, hp3⟩ use c • (v1 - v2) + v3, direction.add_mem (direction.smul_mem c (direction.sub_mem hv1 hv2)) hv3 simp [hp1, hp2, hp3, vadd_vadd] #align affine_subspace.mk' AffineSubspace.mk' /-- An affine subspace constructed from a point and a direction contains that point. -/ theorem self_mem_mk' (p : P) (direction : Submodule k V) : p ∈ mk' p direction := ⟨0, ⟨direction.zero_mem, (zero_vadd _ _).symm⟩⟩ #align affine_subspace.self_mem_mk' AffineSubspace.self_mem_mk' /-- An affine subspace constructed from a point and a direction contains the result of adding a vector in that direction to that point. -/ theorem vadd_mem_mk' {v : V} (p : P) {direction : Submodule k V} (hv : v ∈ direction) : v +ᵥ p ∈ mk' p direction := ⟨v, hv, rfl⟩ #align affine_subspace.vadd_mem_mk' AffineSubspace.vadd_mem_mk' /-- An affine subspace constructed from a point and a direction is nonempty. -/ theorem mk'_nonempty (p : P) (direction : Submodule k V) : (mk' p direction : Set P).Nonempty := ⟨p, self_mem_mk' p direction⟩ #align affine_subspace.mk'_nonempty AffineSubspace.mk'_nonempty /-- The direction of an affine subspace constructed from a point and a direction. -/ @[simp] theorem direction_mk' (p : P) (direction : Submodule k V) : (mk' p direction).direction = direction := by ext v rw [mem_direction_iff_eq_vsub (mk'_nonempty _ _)] constructor · rintro ⟨p1, ⟨v1, hv1, hp1⟩, p2, ⟨v2, hv2, hp2⟩, hv⟩ rw [hv, hp1, hp2, vadd_vsub_vadd_cancel_right] exact direction.sub_mem hv1 hv2 · exact fun hv => ⟨v +ᵥ p, vadd_mem_mk' _ hv, p, self_mem_mk' _ _, (vadd_vsub _ _).symm⟩ #align affine_subspace.direction_mk' AffineSubspace.direction_mk' /-- A point lies in an affine subspace constructed from another point and a direction if and only if their difference is in that direction. -/ theorem mem_mk'_iff_vsub_mem {p₁ p₂ : P} {direction : Submodule k V} : p₂ ∈ mk' p₁ direction ↔ p₂ -ᵥ p₁ ∈ direction := by refine ⟨fun h => ?_, fun h => ?_⟩ · rw [← direction_mk' p₁ direction] exact vsub_mem_direction h (self_mem_mk' _ _) · rw [← vsub_vadd p₂ p₁] exact vadd_mem_mk' p₁ h #align affine_subspace.mem_mk'_iff_vsub_mem AffineSubspace.mem_mk'_iff_vsub_mem /-- Constructing an affine subspace from a point in a subspace and that subspace's direction yields the original subspace. -/ @[simp] theorem mk'_eq {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : mk' p s.direction = s := ext_of_direction_eq (direction_mk' p s.direction) ⟨p, Set.mem_inter (self_mem_mk' _ _) hp⟩ #align affine_subspace.mk'_eq AffineSubspace.mk'_eq /-- If an affine subspace contains a set of points, it contains the `spanPoints` of that set. -/ theorem spanPoints_subset_coe_of_subset_coe {s : Set P} {s1 : AffineSubspace k P} (h : s ⊆ s1) : spanPoints k s ⊆ s1 := by rintro p ⟨p1, hp1, v, hv, hp⟩ rw [hp] have hp1s1 : p1 ∈ (s1 : Set P) := Set.mem_of_mem_of_subset hp1 h refine vadd_mem_of_mem_direction ?_ hp1s1 have hs : vectorSpan k s ≤ s1.direction := vectorSpan_mono k h rw [SetLike.le_def] at hs rw [← SetLike.mem_coe] exact Set.mem_of_mem_of_subset hv hs #align affine_subspace.span_points_subset_coe_of_subset_coe AffineSubspace.spanPoints_subset_coe_of_subset_coe end AffineSubspace namespace Submodule variable {k V : Type} [Ring k] [AddCommGroup V] [Module k V] @[simp] theorem mem_toAffineSubspace {p : Submodule k V} {x : V} : x ∈ p.toAffineSubspace ↔ x ∈ p := Iff.rfl @[simp] theorem toAffineSubspace_direction (s : Submodule k V) : s.toAffineSubspace.direction = s := by ext x; simp [← s.toAffineSubspace.vadd_mem_iff_mem_direction _ s.zero_mem] end Submodule theorem AffineMap.lineMap_mem {k V P : Type} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {Q : AffineSubspace k P} {p₀ p₁ : P} (c : k) (h₀ : p₀ ∈ Q) (h₁ : p₁ ∈ Q) : AffineMap.lineMap p₀ p₁ c ∈ Q := by rw [AffineMap.lineMap_apply] exact Q.smul_vsub_vadd_mem c h₁ h₀ h₀ #align affine_map.line_map_mem AffineMap.lineMap_mem section affineSpan variable (k : Type) {V : Type} {P : Type} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] /-- The affine span of a set of points is the smallest affine subspace containing those points. (Actually defined here in terms of spans in modules.) -/ def affineSpan (s : Set P) : AffineSubspace k P where carrier := spanPoints k s smul_vsub_vadd_mem c _ _ _ hp1 hp2 hp3 := vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan k hp3 ((vectorSpan k s).smul_mem c (vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints k hp1 hp2)) #align affine_span affineSpan /-- The affine span, converted to a set, is `spanPoints`. -/ @[simp] theorem coe_affineSpan (s : Set P) : (affineSpan k s : Set P) = spanPoints k s := rfl #align coe_affine_span coe_affineSpan /-- A set is contained in its affine span. -/ theorem subset_affineSpan (s : Set P) : s ⊆ affineSpan k s := subset_spanPoints k s #align subset_affine_span subset_affineSpan /-- The direction of the affine span is the `vectorSpan`. -/ theorem direction_affineSpan (s : Set P) : (affineSpan k s).direction = vectorSpan k s := by apply le_antisymm · refine Submodule.span_le.2 ?_ rintro v ⟨p1, ⟨p2, hp2, v1, hv1, hp1⟩, p3, ⟨p4, hp4, v2, hv2, hp3⟩, rfl⟩ simp only [SetLike.mem_coe] rw [hp1, hp3, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc] exact (vectorSpan k s).sub_mem ((vectorSpan k s).add_mem hv1 (vsub_mem_vectorSpan k hp2 hp4)) hv2 · exact vectorSpan_mono k (subset_spanPoints k s) #align direction_affine_span direction_affineSpan /-- A point in a set is in its affine span. -/ theorem mem_affineSpan {p : P} {s : Set P} (hp : p ∈ s) : p ∈ affineSpan k s := mem_spanPoints k p s hp #align mem_affine_span mem_affineSpan end affineSpan namespace AffineSubspace variable {k : Type} {V : Type} {P : Type} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] instance : CompleteLattice (AffineSubspace k P) := { PartialOrder.lift ((↑) : AffineSubspace k P → Set P) coe_injective with sup := fun s1 s2 => affineSpan k (s1 ∪ s2) le_sup_left := fun s1 s2 => Set.Subset.trans Set.subset_union_left (subset_spanPoints k _) le_sup_right := fun s1 s2 => Set.Subset.trans Set.subset_union_right (subset_spanPoints k _) sup_le := fun s1 s2 s3 hs1 hs2 => spanPoints_subset_coe_of_subset_coe (Set.union_subset hs1 hs2) inf := fun s1 s2 => mk (s1 ∩ s2) fun c p1 p2 p3 hp1 hp2 hp3 => ⟨s1.smul_vsub_vadd_mem c hp1.1 hp2.1 hp3.1, s2.smul_vsub_vadd_mem c hp1.2 hp2.2 hp3.2⟩ inf_le_left := fun _ _ => Set.inter_subset_left inf_le_right := fun _ _ => Set.inter_subset_right le_sInf := fun S s1 hs1 => by -- Porting note: surely there is an easier way? refine Set.subset_sInter (t := (s1 : Set P)) ?_ rintro t ⟨s, _hs, rfl⟩ exact Set.subset_iInter (hs1 s) top := { carrier := Set.univ smul_vsub_vadd_mem := fun _ _ _ _ _ _ _ => Set.mem_univ _ } le_top := fun _ _ _ => Set.mem_univ _ bot := { carrier := ∅ smul_vsub_vadd_mem := fun _ _ _ _ => False.elim } bot_le := fun _ _ => False.elim sSup := fun s => affineSpan k (⋃ s' ∈ s, (s' : Set P)) sInf := fun s => mk (⋂ s' ∈ s, (s' : Set P)) fun c p1 p2 p3 hp1 hp2 hp3 => Set.mem_iInter₂.2 fun s2 hs2 => by rw [Set.mem_iInter₂] at * exact s2.smul_vsub_vadd_mem c (hp1 s2 hs2) (hp2 s2 hs2) (hp3 s2 hs2) le_sSup := fun _ _ h => Set.Subset.trans (Set.subset_biUnion_of_mem h) (subset_spanPoints k _) sSup_le := fun _ _ h => spanPoints_subset_coe_of_subset_coe (Set.iUnion₂_subset h) sInf_le := fun _ _ => Set.biInter_subset_of_mem le_inf := fun _ _ _ => Set.subset_inter } instance : Inhabited (AffineSubspace k P) := ⟨⊤⟩ /-- The `≤` order on subspaces is the same as that on the corresponding sets. -/ theorem le_def (s1 s2 : AffineSubspace k P) : s1 ≤ s2 ↔ (s1 : Set P) ⊆ s2 := Iff.rfl #align affine_subspace.le_def AffineSubspace.le_def /-- One subspace is less than or equal to another if and only if all its points are in the second subspace. -/ theorem le_def' (s1 s2 : AffineSubspace k P) : s1 ≤ s2 ↔ ∀ p ∈ s1, p ∈ s2 := Iff.rfl #align affine_subspace.le_def' AffineSubspace.le_def' /-- The `<` order on subspaces is the same as that on the corresponding sets. -/ theorem lt_def (s1 s2 : AffineSubspace k P) : s1 < s2 ↔ (s1 : Set P) ⊂ s2 := Iff.rfl #align affine_subspace.lt_def AffineSubspace.lt_def /-- One subspace is not less than or equal to another if and only if it has a point not in the second subspace. -/ theorem not_le_iff_exists (s1 s2 : AffineSubspace k P) : ¬s1 ≤ s2 ↔ ∃ p ∈ s1, p ∉ s2 := Set.not_subset #align affine_subspace.not_le_iff_exists AffineSubspace.not_le_iff_exists /-- If a subspace is less than another, there is a point only in the second. -/ theorem exists_of_lt {s1 s2 : AffineSubspace k P} (h : s1 < s2) : ∃ p ∈ s2, p ∉ s1 := Set.exists_of_ssubset h #align affine_subspace.exists_of_lt AffineSubspace.exists_of_lt /-- A subspace is less than another if and only if it is less than or equal to the second subspace and there is a point only in the second. -/ theorem lt_iff_le_and_exists (s1 s2 : AffineSubspace k P) : s1 < s2 ↔ s1 ≤ s2 ∧ ∃ p ∈ s2, p ∉ s1 := by rw [lt_iff_le_not_le, not_le_iff_exists] #align affine_subspace.lt_iff_le_and_exists AffineSubspace.lt_iff_le_and_exists /-- If an affine subspace is nonempty and contained in another with the same direction, they are equal. -/ theorem eq_of_direction_eq_of_nonempty_of_le {s₁ s₂ : AffineSubspace k P} (hd : s₁.direction = s₂.direction) (hn : (s₁ : Set P).Nonempty) (hle : s₁ ≤ s₂) : s₁ = s₂ := let ⟨p, hp⟩ := hn ext_of_direction_eq hd ⟨p, hp, hle hp⟩ #align affine_subspace.eq_of_direction_eq_of_nonempty_of_le AffineSubspace.eq_of_direction_eq_of_nonempty_of_le variable (k V) /-- The affine span is the `sInf` of subspaces containing the given points. -/ theorem affineSpan_eq_sInf (s : Set P) : affineSpan k s = sInf { s' : AffineSubspace k P \| s ⊆ s' } := le_antisymm (spanPoints_subset_coe_of_subset_coe <\| Set.subset_iInter₂ fun _ => id) (sInf_le (subset_spanPoints k _)) #align affine_subspace.affine_span_eq_Inf AffineSubspace.affineSpan_eq_sInf variable (P) /-- The Galois insertion formed by `affineSpan` and coercion back to a set. -/ protected def gi : GaloisInsertion (affineSpan k) ((↑) : AffineSubspace k P → Set P) where choice s _ := affineSpan k s gc s1 _s2 := ⟨fun h => Set.Subset.trans (subset_spanPoints k s1) h, spanPoints_subset_coe_of_subset_coe⟩ le_l_u _ := subset_spanPoints k _ choice_eq _ _ := rfl #align affine_subspace.gi AffineSubspace.gi /-- The span of the empty set is `⊥`. -/ @[simp] theorem span_empty : affineSpan k (∅ : Set P) = ⊥ := (AffineSubspace.gi k V P).gc.l_bot #align affine_subspace.span_empty AffineSubspace.span_empty /-- The span of `univ` is `⊤`. -/ @[simp] theorem span_univ : affineSpan k (Set.univ : Set P) = ⊤ := eq_top_iff.2 <\| subset_spanPoints k _ #align affine_subspace.span_univ AffineSubspace.span_univ variable {k V P} theorem _root_.affineSpan_le {s : Set P} {Q : AffineSubspace k P} : affineSpan k s ≤ Q ↔ s ⊆ (Q : Set P) := (AffineSubspace.gi k V P).gc _ _ #align affine_span_le affineSpan_le variable (k V) {p₁ p₂ : P} /-- The affine span of a single point, coerced to a set, contains just that point. -/ @[simp 1001] -- Porting note: this needs to take priority over `coe_affineSpan` theorem coe_affineSpan_singleton (p : P) : (affineSpan k ({p} : Set P) : Set P) = {p} := by ext x rw [mem_coe, ← vsub_right_mem_direction_iff_mem (mem_affineSpan k (Set.mem_singleton p)) _, direction_affineSpan] simp #align affine_subspace.coe_affine_span_singleton AffineSubspace.coe_affineSpan_singleton /-- A point is in the affine span of a single point if and only if they are equal. -/ @[simp] theorem mem_affineSpan_singleton : p₁ ∈ affineSpan k ({p₂} : Set P) ↔ p₁ = p₂ := by simp [← mem_coe] #align affine_subspace.mem_affine_span_singleton AffineSubspace.mem_affineSpan_singleton @[simp] theorem preimage_coe_affineSpan_singleton (x : P) : ((↑) : affineSpan k ({x} : Set P) → P) ⁻¹' {x} = univ := eq_univ_of_forall fun y => (AffineSubspace.mem_affineSpan_singleton _ _).1 y.2 #align affine_subspace.preimage_coe_affine_span_singleton AffineSubspace.preimage_coe_affineSpan_singleton /-- The span of a union of sets is the sup of their spans. -/ theorem span_union (s t : Set P) : affineSpan k (s ∪ t) = affineSpan k s ⊔ affineSpan k t := (AffineSubspace.gi k V P).gc.l_sup #align affine_subspace.span_union AffineSubspace.span_union /-- The span of a union of an indexed family of sets is the sup of their spans. -/ theorem span_iUnion {ι : Type} (s : ι → Set P) : affineSpan k (⋃ i, s i) = ⨆ i, affineSpan k (s i) := (AffineSubspace.gi k V P).gc.l_iSup #align affine_subspace.span_Union AffineSubspace.span_iUnion variable (P) /-- `⊤`, coerced to a set, is the whole set of points. -/ @[simp] theorem top_coe : ((⊤ : AffineSubspace k P) : Set P) = Set.univ := rfl #align affine_subspace.top_coe AffineSubspace.top_coe variable {P} /-- All points are in `⊤`. -/ @[simp] theorem mem_top (p : P) : p ∈ (⊤ : AffineSubspace k P) := Set.mem_univ p #align affine_subspace.mem_top AffineSubspace.mem_top variable (P) /-- The direction of `⊤` is the whole module as a submodule. -/ @[simp] theorem direction_top : (⊤ : AffineSubspace k P).direction = ⊤ := by cases' S.nonempty with p ext v refine ⟨imp_intro Submodule.mem_top, fun _hv => ?_⟩ have hpv : (v +ᵥ p -ᵥ p : V) ∈ (⊤ : AffineSubspace k P).direction := vsub_mem_direction (mem_top k V _) (mem_top k V _) rwa [vadd_vsub] at hpv #align affine_subspace.direction_top AffineSubspace.direction_top /-- `⊥`, coerced to a set, is the empty set. -/ @[simp] theorem bot_coe : ((⊥ : AffineSubspace k P) : Set P) = ∅ := rfl #align affine_subspace.bot_coe AffineSubspace.bot_coe theorem bot_ne_top : (⊥ : AffineSubspace k P) ≠ ⊤ := by intro contra rw [← ext_iff, bot_coe, top_coe] at contra exact Set.empty_ne_univ contra #align affine_subspace.bot_ne_top AffineSubspace.bot_ne_top instance : Nontrivial (AffineSubspace k P) := ⟨⟨⊥, ⊤, bot_ne_top k V P⟩⟩ theorem nonempty_of_affineSpan_eq_top {s : Set P} (h : affineSpan k s = ⊤) : s.Nonempty := by rw [Set.nonempty_iff_ne_empty] rintro rfl rw [AffineSubspace.span_empty] at h exact bot_ne_top k V P h #align affine_subspace.nonempty_of_affine_span_eq_top AffineSubspace.nonempty_of_affineSpan_eq_top /-- If the affine span of a set is `⊤`, then the vector span of the same set is the `⊤`. -/ theorem vectorSpan_eq_top_of_affineSpan_eq_top {s : Set P} (h : affineSpan k s = ⊤) : vectorSpan k s = ⊤ := by rw [← direction_affineSpan, h, direction_top] #align affine_subspace.vector_span_eq_top_of_affine_span_eq_top AffineSubspace.vectorSpan_eq_top_of_affineSpan_eq_top /-- For a nonempty set, the affine span is `⊤` iff its vector span is `⊤`. -/ theorem affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty {s : Set P} (hs : s.Nonempty) : affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤ := by refine ⟨vectorSpan_eq_top_of_affineSpan_eq_top k V P, ?_⟩ intro h suffices Nonempty (affineSpan k s) by obtain ⟨p, hp : p ∈ affineSpan k s⟩ := this rw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affineSpan, h, direction_top] obtain ⟨x, hx⟩ := hs exact ⟨⟨x, mem_affineSpan k hx⟩⟩ #align affine_subspace.affine_span_eq_top_iff_vector_span_eq_top_of_nonempty AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty /-- For a non-trivial space, the affine span of a set is `⊤` iff its vector span is `⊤`. -/ theorem affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial {s : Set P} [Nontrivial P] : affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤ := by rcases s.eq_empty_or_nonempty with hs \| hs · simp [hs, subsingleton_iff_bot_eq_top, AddTorsor.subsingleton_iff V P, not_subsingleton] · rw [affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty k V P hs] #align affine_subspace.affine_span_eq_top_iff_vector_span_eq_top_of_nontrivial AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial theorem card_pos_of_affineSpan_eq_top {ι : Type} [Fintype ι] {p : ι → P} (h : affineSpan k (range p) = ⊤) : 0 < Fintype.card ι := by obtain ⟨-, ⟨i, -⟩⟩ := nonempty_of_affineSpan_eq_top k V P h exact Fintype.card_pos_iff.mpr ⟨i⟩ #align affine_subspace.card_pos_of_affine_span_eq_top AffineSubspace.card_pos_of_affineSpan_eq_top attribute [local instance] toAddTorsor /-- The top affine subspace is linearly equivalent to the affine space. This is the affine version of `Submodule.topEquiv`. -/ @[simps! linear apply symm_apply_coe] def topEquiv : (⊤ : AffineSubspace k P) ≃ᵃ[k] P where toEquiv := Equiv.Set.univ P linear := .ofEq _ _ (direction_top _ _ _) ≪≫ₗ Submodule.topEquiv map_vadd' _p _v := rfl variable {P} /-- No points are in `⊥`. -/ theorem not_mem_bot (p : P) : p ∉ (⊥ : AffineSubspace k P) := Set.not_mem_empty p #align affine_subspace.not_mem_bot AffineSubspace.not_mem_bot variable (P) /-- The direction of `⊥` is the submodule `⊥`. -/ @[simp] theorem direction_bot : (⊥ : AffineSubspace k P).direction = ⊥ := by rw [direction_eq_vectorSpan, bot_coe, vectorSpan_def, vsub_empty, Submodule.span_empty] #align affine_subspace.direction_bot AffineSubspace.direction_bot variable {k V P} @[simp] theorem coe_eq_bot_iff (Q : AffineSubspace k P) : (Q : Set P) = ∅ ↔ Q = ⊥ := coe_injective.eq_iff' (bot_coe _ _ _) #align affine_subspace.coe_eq_bot_iff AffineSubspace.coe_eq_bot_iff @[simp] theorem coe_eq_univ_iff (Q : AffineSubspace k P) : (Q : Set P) = univ ↔ Q = ⊤ := coe_injective.eq_iff' (top_coe _ _ _) #align affine_subspace.coe_eq_univ_iff AffineSubspace.coe_eq_univ_iff theorem nonempty_iff_ne_bot (Q : AffineSubspace k P) : (Q : Set P).Nonempty ↔ Q ≠ ⊥ := by rw [nonempty_iff_ne_empty] exact not_congr Q.coe_eq_bot_iff #align affine_subspace.nonempty_iff_ne_bot AffineSubspace.nonempty_iff_ne_bot theorem eq_bot_or_nonempty (Q : AffineSubspace k P) : Q = ⊥ ∨ (Q : Set P).Nonempty := by rw [nonempty_iff_ne_bot] apply eq_or_ne #align affine_subspace.eq_bot_or_nonempty AffineSubspace.eq_bot_or_nonempty theorem subsingleton_of_subsingleton_span_eq_top {s : Set P} (h₁ : s.Subsingleton) (h₂ : affineSpan k s = ⊤) : Subsingleton P := by obtain ⟨p, hp⟩ := AffineSubspace.nonempty_of_affineSpan_eq_top k V P h₂ have : s = {p} := Subset.antisymm (fun q hq => h₁ hq hp) (by simp [hp]) rw [this, ← AffineSubspace.ext_iff, AffineSubspace.coe_affineSpan_singleton, AffineSubspace.top_coe, eq_comm, ← subsingleton_iff_singleton (mem_univ _)] at h₂ exact subsingleton_of_univ_subsingleton h₂ #align affine_subspace.subsingleton_of_subsingleton_span_eq_top AffineSubspace.subsingleton_of_subsingleton_span_eq_top theorem eq_univ_of_subsingleton_span_eq_top {s : Set P} (h₁ : s.Subsingleton) (h₂ : affineSpan k s = ⊤) : s = (univ : Set P) := by obtain ⟨p, hp⟩ := AffineSubspace.nonempty_of_affineSpan_eq_top k V P h₂ have : s = {p} := Subset.antisymm (fun q hq => h₁ hq hp) (by simp [hp]) rw [this, eq_comm, ← subsingleton_iff_singleton (mem_univ p), subsingleton_univ_iff] exact subsingleton_of_subsingleton_span_eq_top h₁ h₂ #align affine_subspace.eq_univ_of_subsingleton_span_eq_top AffineSubspace.eq_univ_of_subsingleton_span_eq_top /-- A nonempty affine subspace is `⊤` if and only if its direction is `⊤`. -/ @[simp] theorem direction_eq_top_iff_of_nonempty {s : AffineSubspace k P} (h : (s : Set P).Nonempty) : s.direction = ⊤ ↔ s = ⊤ := by constructor · intro hd rw [← direction_top k V P] at hd refine ext_of_direction_eq hd ?_ simp [h] · rintro rfl simp #align affine_subspace.direction_eq_top_iff_of_nonempty AffineSubspace.direction_eq_top_iff_of_nonempty /-- The inf of two affine subspaces, coerced to a set, is the intersection of the two sets of points. -/ @[simp] theorem inf_coe (s1 s2 : AffineSubspace k P) : (s1 ⊓ s2 : Set P) = (s1 : Set P) ∩ s2 := rfl #align affine_subspace.inf_coe AffineSubspace.inf_coe /-- A point is in the inf of two affine subspaces if and only if it is in both of them. -/ theorem mem_inf_iff (p : P) (s1 s2 : AffineSubspace k P) : p ∈ s1 ⊓ s2 ↔ p ∈ s1 ∧ p ∈ s2 := Iff.rfl #align affine_subspace.mem_inf_iff AffineSubspace.mem_inf_iff /-- The direction of the inf of two affine subspaces is less than or equal to the inf of their directions. -/ theorem direction_inf (s1 s2 : AffineSubspace k P) : (s1 ⊓ s2).direction ≤ s1.direction ⊓ s2.direction := by simp only [direction_eq_vectorSpan, vectorSpan_def] exact le_inf (sInf_le_sInf fun p hp => trans (vsub_self_mono inter_subset_left) hp) (sInf_le_sInf fun p hp => trans (vsub_self_mono inter_subset_right) hp) #align affine_subspace.direction_inf AffineSubspace.direction_inf /-- If two affine subspaces have a point in common, the direction of their inf equals the inf of their directions. -/ theorem direction_inf_of_mem {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) : (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction := by ext v rw [Submodule.mem_inf, ← vadd_mem_iff_mem_direction v h₁, ← vadd_mem_iff_mem_direction v h₂, ← vadd_mem_iff_mem_direction v ((mem_inf_iff p s₁ s₂).2 ⟨h₁, h₂⟩), mem_inf_iff] #align affine_subspace.direction_inf_of_mem AffineSubspace.direction_inf_of_mem /-- If two affine subspaces have a point in their inf, the direction of their inf equals the inf of their directions. -/ theorem direction_inf_of_mem_inf {s₁ s₂ : AffineSubspace k P} {p : P} (h : p ∈ s₁ ⊓ s₂) : (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction := direction_inf_of_mem ((mem_inf_iff p s₁ s₂).1 h).1 ((mem_inf_iff p s₁ s₂).1 h).2 #align affine_subspace.direction_inf_of_mem_inf AffineSubspace.direction_inf_of_mem_inf /-- If one affine subspace is less than or equal to another, the same applies to their directions. -/ theorem direction_le {s1 s2 : AffineSubspace k P} (h : s1 ≤ s2) : s1.direction ≤ s2.direction := by simp only [direction_eq_vectorSpan, vectorSpan_def] exact vectorSpan_mono k h #align affine_subspace.direction_le AffineSubspace.direction_le /-- If one nonempty affine subspace is less than another, the same applies to their directions -/ theorem direction_lt_of_nonempty {s1 s2 : AffineSubspace k P} (h : s1 < s2) (hn : (s1 : Set P).Nonempty) : s1.direction < s2.direction := by cases' hn with p hp rw [lt_iff_le_and_exists] at h rcases h with ⟨hle, p2, hp2, hp2s1⟩ rw [SetLike.lt_iff_le_and_exists] use direction_le hle, p2 -ᵥ p, vsub_mem_direction hp2 (hle hp) intro hm rw [vsub_right_mem_direction_iff_mem hp p2] at hm exact hp2s1 hm #align affine_subspace.direction_lt_of_nonempty AffineSubspace.direction_lt_of_nonempty /-- The sup of the directions of two affine subspaces is less than or equal to the direction of their sup. -/ theorem sup_direction_le (s1 s2 : AffineSubspace k P) : s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction := by simp only [direction_eq_vectorSpan, vectorSpan_def] exact sup_le (sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_left : s1 ≤ s1 ⊔ s2)) hp) (sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_right : s2 ≤ s1 ⊔ s2)) hp) #align affine_subspace.sup_direction_le AffineSubspace.sup_direction_le /-- The sup of the directions of two nonempty affine subspaces with empty intersection is less than the direction of their sup. -/ theorem sup_direction_lt_of_nonempty_of_inter_empty {s1 s2 : AffineSubspace k P} (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty) (he : (s1 ∩ s2 : Set P) = ∅) : s1.direction ⊔ s2.direction < (s1 ⊔ s2).direction := by cases' h1 with p1 hp1 cases' h2 with p2 hp2 rw [SetLike.lt_iff_le_and_exists] use sup_direction_le s1 s2, p2 -ᵥ p1, vsub_mem_direction ((le_sup_right : s2 ≤ s1 ⊔ s2) hp2) ((le_sup_left : s1 ≤ s1 ⊔ s2) hp1) intro h rw [Submodule.mem_sup] at h rcases h with ⟨v1, hv1, v2, hv2, hv1v2⟩ rw [← sub_eq_zero, sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm v1, add_assoc, ← vadd_vsub_assoc, ← neg_neg v2, add_comm, ← sub_eq_add_neg, ← vsub_vadd_eq_vsub_sub, vsub_eq_zero_iff_eq] at hv1v2 refine Set.Nonempty.ne_empty ?_ he use v1 +ᵥ p1, vadd_mem_of_mem_direction hv1 hp1 rw [hv1v2] exact vadd_mem_of_mem_direction (Submodule.neg_mem _ hv2) hp2 #align affine_subspace.sup_direction_lt_of_nonempty_of_inter_empty AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty /-- If the directions of two nonempty affine subspaces span the whole module, they have nonempty intersection. -/ theorem inter_nonempty_of_nonempty_of_sup_direction_eq_top {s1 s2 : AffineSubspace k P} (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty) (hd : s1.direction ⊔ s2.direction = ⊤) : ((s1 : Set P) ∩ s2).Nonempty := by by_contra h rw [Set.not_nonempty_iff_eq_empty] at h have hlt := sup_direction_lt_of_nonempty_of_inter_empty h1 h2 h rw [hd] at hlt exact not_top_lt hlt #align affine_subspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top /-- If the directions of two nonempty affine subspaces are complements of each other, they intersect in exactly one point. -/ theorem inter_eq_singleton_of_nonempty_of_isCompl {s1 s2 : AffineSubspace k P} (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty) (hd : IsCompl s1.direction s2.direction) : ∃ p, (s1 : Set P) ∩ s2 = {p} := by cases' inter_nonempty_of_nonempty_of_sup_direction_eq_top h1 h2 hd.sup_eq_top with p hp use p ext q rw [Set.mem_singleton_iff] constructor · rintro ⟨hq1, hq2⟩ have hqp : q -ᵥ p ∈ s1.direction ⊓ s2.direction := ⟨vsub_mem_direction hq1 hp.1, vsub_mem_direction hq2 hp.2⟩ rwa [hd.inf_eq_bot, Submodule.mem_bot, vsub_eq_zero_iff_eq] at hqp · exact fun h => h.symm ▸ hp #align affine_subspace.inter_eq_singleton_of_nonempty_of_is_compl AffineSubspace.inter_eq_singleton_of_nonempty_of_isCompl /-- Coercing a subspace to a set then taking the affine span produces the original subspace. -/ @[simp] theorem affineSpan_coe (s : AffineSubspace k P) : affineSpan k (s : Set P) = s := by refine le_antisymm ?_ (subset_spanPoints _ _) rintro p ⟨p1, hp1, v, hv, rfl⟩ exact vadd_mem_of_mem_direction hv hp1 #align affine_subspace.affine_span_coe AffineSubspace.affineSpan_coe end AffineSubspace section AffineSpace' variable (k : Type) {V : Type} {P : Type} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] variable {ι : Type} open AffineSubspace Set /-- The `vectorSpan` is the span of the pairwise subtractions with a given point on the left. -/ theorem vectorSpan_eq_span_vsub_set_left {s : Set P} {p : P} (hp : p ∈ s) : vectorSpan k s = Submodule.span k ((p -ᵥ ·) '' s) := by rw [vectorSpan_def] refine le_antisymm ?_ (Submodule.span_mono ?_) · rw [Submodule.span_le] rintro v ⟨p1, hp1, p2, hp2, hv⟩ simp_rw [← vsub_sub_vsub_cancel_left p1 p2 p] at hv rw [← hv, SetLike.mem_coe, Submodule.mem_span] exact fun m hm => Submodule.sub_mem _ (hm ⟨p2, hp2, rfl⟩) (hm ⟨p1, hp1, rfl⟩) · rintro v ⟨p2, hp2, hv⟩ exact ⟨p, hp, p2, hp2, hv⟩ #align vector_span_eq_span_vsub_set_left vectorSpan_eq_span_vsub_set_left /-- The `vectorSpan` is the span of the pairwise subtractions with a given point on the right. -/ theorem vectorSpan_eq_span_vsub_set_right {s : Set P} {p : P} (hp : p ∈ s) : vectorSpan k s = Submodule.span k ((· -ᵥ p) '' s) := by rw [vectorSpan_def] refine le_antisymm ?_ (Submodule.span_mono ?_) · rw [Submodule.span_le] rintro v ⟨p1, hp1, p2, hp2, hv⟩ simp_rw [← vsub_sub_vsub_cancel_right p1 p2 p] at hv rw [← hv, SetLike.mem_coe, Submodule.mem_span] exact fun m hm => Submodule.sub_mem _ (hm ⟨p1, hp1, rfl⟩) (hm ⟨p2, hp2, rfl⟩) · rintro v ⟨p2, hp2, hv⟩ exact ⟨p2, hp2, p, hp, hv⟩ #align vector_span_eq_span_vsub_set_right vectorSpan_eq_span_vsub_set_right /-- The `vectorSpan` is the span of the pairwise subtractions with a given point on the left, excluding the subtraction of that point from itself. -/ theorem vectorSpan_eq_span_vsub_set_left_ne {s : Set P} {p : P} (hp : p ∈ s) : vectorSpan k s = Submodule.span k ((p -ᵥ ·) '' (s \ {p})) := by conv_lhs => rw [vectorSpan_eq_span_vsub_set_left k hp, ← Set.insert_eq_of_mem hp, ← Set.insert_diff_singleton, Set.image_insert_eq] simp [Submodule.span_insert_eq_span] #align vector_span_eq_span_vsub_set_left_ne vectorSpan_eq_span_vsub_set_left_ne /-- The `vectorSpan` is the span of the pairwise subtractions with a given point on the right, excluding the subtraction of that point from itself. -/ theorem vectorSpan_eq_span_vsub_set_right_ne {s : Set P} {p : P} (hp : p ∈ s) : vectorSpan k s = Submodule.span k ((· -ᵥ p) '' (s \ {p})) := by conv_lhs => rw [vectorSpan_eq_span_vsub_set_right k hp, ← Set.insert_eq_of_mem hp, ← Set.insert_diff_singleton, Set.image_insert_eq] simp [Submodule.span_insert_eq_span] #align vector_span_eq_span_vsub_set_right_ne vectorSpan_eq_span_vsub_set_right_ne /-- The `vectorSpan` is the span of the pairwise subtractions with a given point on the right, excluding the subtraction of that point from itself. -/ theorem vectorSpan_eq_span_vsub_finset_right_ne [DecidableEq P] [DecidableEq V] {s : Finset P} {p : P} (hp : p ∈ s) : vectorSpan k (s : Set P) = Submodule.span k ((s.erase p).image (· -ᵥ p)) := by simp [vectorSpan_eq_span_vsub_set_right_ne _ (Finset.mem_coe.mpr hp)] #align vector_span_eq_span_vsub_finset_right_ne vectorSpan_eq_span_vsub_finset_right_ne /-- The `vectorSpan` of the image of a function is the span of the pairwise subtractions with a given point on the left, excluding the subtraction of that point from itself. -/ theorem vectorSpan_image_eq_span_vsub_set_left_ne (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s) : vectorSpan k (p '' s) = Submodule.span k ((p i -ᵥ ·) '' (p '' (s \ {i}))) := by conv_lhs => rw [vectorSpan_eq_span_vsub_set_left k (Set.mem_image_of_mem p hi), ← Set.insert_eq_of_mem hi, ← Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq] simp [Submodule.span_insert_eq_span] #align vector_span_image_eq_span_vsub_set_left_ne vectorSpan_image_eq_span_vsub_set_left_ne /-- The `vectorSpan` of the image of a function is the span of the pairwise subtractions with a given point on the right, excluding the subtraction of that point from itself. -/ theorem vectorSpan_image_eq_span_vsub_set_right_ne (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s) : vectorSpan k (p '' s) = Submodule.span k ((· -ᵥ p i) '' (p '' (s \ {i}))) := by conv_lhs => rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), ← Set.insert_eq_of_mem hi, ← Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq] simp [Submodule.span_insert_eq_span] #align vector_span_image_eq_span_vsub_set_right_ne vectorSpan_image_eq_span_vsub_set_right_ne /-- The `vectorSpan` of an indexed family is the span of the pairwise subtractions with a given point on the left. -/ theorem vectorSpan_range_eq_span_range_vsub_left (p : ι → P) (i0 : ι) : vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : ι => p i0 -ᵥ p i) := by rw [vectorSpan_eq_span_vsub_set_left k (Set.mem_range_self i0), ← Set.range_comp] congr #align vector_span_range_eq_span_range_vsub_left vectorSpan_range_eq_span_range_vsub_left /-- The `vectorSpan` of an indexed family is the span of the pairwise subtractions with a given point on the right. -/ theorem vectorSpan_range_eq_span_range_vsub_right (p : ι → P) (i0 : ι) : vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : ι => p i -ᵥ p i0) := by rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_range_self i0), ← Set.range_comp] congr #align vector_span_range_eq_span_range_vsub_right vectorSpan_range_eq_span_range_vsub_right /-- The `vectorSpan` of an indexed family is the span of the pairwise subtractions with a given point on the left, excluding the subtraction of that point from itself. -/ theorem vectorSpan_range_eq_span_range_vsub_left_ne (p : ι → P) (i₀ : ι) : vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : { x // x ≠ i₀ } => p i₀ -ᵥ p i) := by rw [← Set.image_univ, vectorSpan_image_eq_span_vsub_set_left_ne k _ (Set.mem_univ i₀)] congr with v simp only [Set.mem_range, Set.mem_image, Set.mem_diff, Set.mem_singleton_iff, Subtype.exists, Subtype.coe_mk] constructor · rintro ⟨x, ⟨i₁, ⟨⟨_, hi₁⟩, rfl⟩⟩, hv⟩ exact ⟨i₁, hi₁, hv⟩ · exact fun ⟨i₁, hi₁, hv⟩ => ⟨p i₁, ⟨i₁, ⟨Set.mem_univ _, hi₁⟩, rfl⟩, hv⟩ #align vector_span_range_eq_span_range_vsub_left_ne vectorSpan_range_eq_span_range_vsub_left_ne /-- The `vectorSpan` of an indexed family is the span of the pairwise subtractions with a given point on the right, excluding the subtraction of that point from itself. -/ theorem vectorSpan_range_eq_span_range_vsub_right_ne (p : ι → P) (i₀ : ι) : vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : { x // x ≠ i₀ } => p i -ᵥ p i₀) := by rw [← Set.image_univ, vectorSpan_image_eq_span_vsub_set_right_ne k _ (Set.mem_univ i₀)] congr with v simp only [Set.mem_range, Set.mem_image, Set.mem_diff, Set.mem_singleton_iff, Subtype.exists, Subtype.coe_mk] constructor · rintro ⟨x, ⟨i₁, ⟨⟨_, hi₁⟩, rfl⟩⟩, hv⟩ exact ⟨i₁, hi₁, hv⟩ · exact fun ⟨i₁, hi₁, hv⟩ => ⟨p i₁, ⟨i₁, ⟨Set.mem_univ _, hi₁⟩, rfl⟩, hv⟩ #align vector_span_range_eq_span_range_vsub_right_ne vectorSpan_range_eq_span_range_vsub_right_ne section variable {s : Set P} /-- The affine span of a set is nonempty if and only if that set is. -/ theorem affineSpan_nonempty : (affineSpan k s : Set P).Nonempty ↔ s.Nonempty := spanPoints_nonempty k s #align affine_span_nonempty affineSpan_nonempty alias ⟨_, _root_.Set.Nonempty.affineSpan⟩ := affineSpan_nonempty #align set.nonempty.affine_span Set.Nonempty.affineSpan /-- The affine span of a nonempty set is nonempty. -/ instance [Nonempty s] : Nonempty (affineSpan k s) := ((nonempty_coe_sort.1 ‹_›).affineSpan _).to_subtype /-- The affine span of a set is `⊥` if and only if that set is empty. -/ @[simp] theorem affineSpan_eq_bot : affineSpan k s = ⊥ ↔ s = ∅ := by rw [← not_iff_not, ← Ne, ← Ne, ← nonempty_iff_ne_bot, affineSpan_nonempty, nonempty_iff_ne_empty] #align affine_span_eq_bot affineSpan_eq_bot @[simp] theorem bot_lt_affineSpan : ⊥ < affineSpan k s ↔ s.Nonempty := by rw [bot_lt_iff_ne_bot, nonempty_iff_ne_empty] exact (affineSpan_eq_bot _).not #align bot_lt_affine_span bot_lt_affineSpan end variable {k} /-- An induction principle for span membership. If `p` holds for all elements of `s` and is preserved under certain affine combinations, then `p` holds for all elements of the span of `s`. -/ theorem affineSpan_induction {x : P} {s : Set P} {p : P → Prop} (h : x ∈ affineSpan k s) (mem : ∀ x : P, x ∈ s → p x) (smul_vsub_vadd : ∀ (c : k) (u v w : P), p u → p v → p w → p (c • (u -ᵥ v) +ᵥ w)) : p x := (affineSpan_le (Q := ⟨p, smul_vsub_vadd⟩)).mpr mem h #align affine_span_induction affineSpan_induction /-- A dependent version of `affineSpan_induction`. -/ @[elab_as_elim] theorem affineSpan_induction' {s : Set P} {p : ∀ x, x ∈ affineSpan k s → Prop} (mem : ∀ (y) (hys : y ∈ s), p y (subset_affineSpan k _ hys)) (smul_vsub_vadd : ∀ (c : k) (u hu v hv w hw), p u hu → p v hv → p w hw → p (c • (u -ᵥ v) +ᵥ w) (AffineSubspace.smul_vsub_vadd_mem _ _ hu hv hw)) {x : P} (h : x ∈ affineSpan k s) : p x h := by refine Exists.elim ?_ fun (hx : x ∈ affineSpan k s) (hc : p x hx) => hc -- Porting note: Lean couldn't infer the motive refine affineSpan_induction (p := fun y => ∃ z, p y z) h ?_ ?_ · exact fun y hy => ⟨subset_affineSpan _ _ hy, mem y hy⟩ · exact fun c u v w hu hv hw => Exists.elim hu fun hu' hu => Exists.elim hv fun hv' hv => Exists.elim hw fun hw' hw => ⟨AffineSubspace.smul_vsub_vadd_mem _ _ hu' hv' hw', smul_vsub_vadd _ _ _ _ _ _ _ hu hv hw⟩ #align affine_span_induction' affineSpan_induction' section WithLocalInstance attribute [local instance] AffineSubspace.toAddTorsor /-- A set, considered as a subset of its spanned affine subspace, spans the whole subspace. -/ @[simp] theorem affineSpan_coe_preimage_eq_top (A : Set P) [Nonempty A] : affineSpan k (((↑) : affineSpan k A → P) ⁻¹' A) = ⊤ := by rw [eq_top_iff] rintro ⟨x, hx⟩ - refine affineSpan_induction' (fun y hy ↦ ?_) (fun c u hu v hv w hw ↦ ?_) hx · exact subset_affineSpan _ _ hy · exact AffineSubspace.smul_vsub_vadd_mem _ _ #align affine_span_coe_preimage_eq_top affineSpan_coe_preimage_eq_top end WithLocalInstance /-- Suppose a set of vectors spans `V`. Then a point `p`, together with those vectors added to `p`, spans `P`. -/ theorem affineSpan_singleton_union_vadd_eq_top_of_span_eq_top {s : Set V} (p : P) (h : Submodule.span k (Set.range ((↑) : s → V)) = ⊤) : affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s) = ⊤ := by convert ext_of_direction_eq _ ⟨p, mem_affineSpan k (Set.mem_union_left _ (Set.mem_singleton _)), mem_top k V p⟩ rw [direction_affineSpan, direction_top, vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ (Set.mem_singleton _) : p ∈ _), eq_top_iff, ← h] apply Submodule.span_mono rintro v ⟨v', rfl⟩ use (v' : V) +ᵥ p simp #align affine_span_singleton_union_vadd_eq_top_of_span_eq_top affineSpan_singleton_union_vadd_eq_top_of_span_eq_top variable (k) /-- The `vectorSpan` of two points is the span of their difference. -/ theorem vectorSpan_pair (p₁ p₂ : P) : vectorSpan k ({p₁, p₂} : Set P) = k ∙ p₁ -ᵥ p₂ := by simp_rw [vectorSpan_eq_span_vsub_set_left k (mem_insert p₁ _), image_pair, vsub_self, Submodule.span_insert_zero] #align vector_span_pair vectorSpan_pair /-- The `vectorSpan` of two points is the span of their difference (reversed). -/ theorem vectorSpan_pair_rev (p₁ p₂ : P) : vectorSpan k ({p₁, p₂} : Set P) = k ∙ p₂ -ᵥ p₁ := by rw [pair_comm, vectorSpan_pair] #align vector_span_pair_rev vectorSpan_pair_rev /-- The difference between two points lies in their `vectorSpan`. -/ theorem vsub_mem_vectorSpan_pair (p₁ p₂ : P) : p₁ -ᵥ p₂ ∈ vectorSpan k ({p₁, p₂} : Set P) := vsub_mem_vectorSpan _ (Set.mem_insert _ _) (Set.mem_insert_of_mem _ (Set.mem_singleton _)) #align vsub_mem_vector_span_pair vsub_mem_vectorSpan_pair /-- The difference between two points (reversed) lies in their `vectorSpan`. -/ theorem vsub_rev_mem_vectorSpan_pair (p₁ p₂ : P) : p₂ -ᵥ p₁ ∈ vectorSpan k ({p₁, p₂} : Set P) := vsub_mem_vectorSpan _ (Set.mem_insert_of_mem _ (Set.mem_singleton _)) (Set.mem_insert _ _) #align vsub_rev_mem_vector_span_pair vsub_rev_mem_vectorSpan_pair variable {k} /-- A multiple of the difference between two points lies in their `vectorSpan`. -/ theorem smul_vsub_mem_vectorSpan_pair (r : k) (p₁ p₂ : P) : r • (p₁ -ᵥ p₂) ∈ vectorSpan k ({p₁, p₂} : Set P) := Submodule.smul_mem _ _ (vsub_mem_vectorSpan_pair k p₁ p₂) #align smul_vsub_mem_vector_span_pair smul_vsub_mem_vectorSpan_pair /-- A multiple of the difference between two points (reversed) lies in their `vectorSpan`. -/ theorem smul_vsub_rev_mem_vectorSpan_pair (r : k) (p₁ p₂ : P) : r • (p₂ -ᵥ p₁) ∈ vectorSpan k ({p₁, p₂} : Set P) := Submodule.smul_mem _ _ (vsub_rev_mem_vectorSpan_pair k p₁ p₂) #align smul_vsub_rev_mem_vector_span_pair smul_vsub_rev_mem_vectorSpan_pair /-- A vector lies in the `vectorSpan` of two points if and only if it is a multiple of their difference. -/ theorem mem_vectorSpan_pair {p₁ p₂ : P} {v : V} : v ∈ vectorSpan k ({p₁, p₂} : Set P) ↔ ∃ r : k, r • (p₁ -ᵥ p₂) = v := by rw [vectorSpan_pair, Submodule.mem_span_singleton] #align mem_vector_span_pair mem_vectorSpan_pair /-- A vector lies in the `vectorSpan` of two points if and only if it is a multiple of their difference (reversed). -/ theorem mem_vectorSpan_pair_rev {p₁ p₂ : P} {v : V} : v ∈ vectorSpan k ({p₁, p₂} : Set P) ↔ ∃ r : k, r • (p₂ -ᵥ p₁) = v := by rw [vectorSpan_pair_rev, Submodule.mem_span_singleton] #align mem_vector_span_pair_rev mem_vectorSpan_pair_rev variable (k) /-- The line between two points, as an affine subspace. -/ notation "line[" k ", " p₁ ", " p₂ "]" => affineSpan k (insert p₁ (@singleton _ _ Set.instSingletonSet p₂)) /-- The first of two points lies in their affine span. -/ theorem left_mem_affineSpan_pair (p₁ p₂ : P) : p₁ ∈ line[k, p₁, p₂] := mem_affineSpan _ (Set.mem_insert _ _) #align left_mem_affine_span_pair left_mem_affineSpan_pair /-- The second of two points lies in their affine span. -/ theorem right_mem_affineSpan_pair (p₁ p₂ : P) : p₂ ∈ line[k, p₁, p₂] := mem_affineSpan _ (Set.mem_insert_of_mem _ (Set.mem_singleton _)) #align right_mem_affine_span_pair right_mem_affineSpan_pair variable {k} /-- A combination of two points expressed with `lineMap` lies in their affine span. -/ theorem AffineMap.lineMap_mem_affineSpan_pair (r : k) (p₁ p₂ : P) : AffineMap.lineMap p₁ p₂ r ∈ line[k, p₁, p₂] := AffineMap.lineMap_mem _ (left_mem_affineSpan_pair _ _ _) (right_mem_affineSpan_pair _ _ _) #align affine_map.line_map_mem_affine_span_pair AffineMap.lineMap_mem_affineSpan_pair /-- A combination of two points expressed with `lineMap` (with the two points reversed) lies in their affine span. -/ theorem AffineMap.lineMap_rev_mem_affineSpan_pair (r : k) (p₁ p₂ : P) : AffineMap.lineMap p₂ p₁ r ∈ line[k, p₁, p₂] := AffineMap.lineMap_mem _ (right_mem_affineSpan_pair _ _ _) (left_mem_affineSpan_pair _ _ _) #align affine_map.line_map_rev_mem_affine_span_pair AffineMap.lineMap_rev_mem_affineSpan_pair /-- A multiple of the difference of two points added to the first point lies in their affine span. -/ theorem smul_vsub_vadd_mem_affineSpan_pair (r : k) (p₁ p₂ : P) : r • (p₂ -ᵥ p₁) +ᵥ p₁ ∈ line[k, p₁, p₂] := AffineMap.lineMap_mem_affineSpan_pair _ _ _ #align smul_vsub_vadd_mem_affine_span_pair smul_vsub_vadd_mem_affineSpan_pair /-- A multiple of the difference of two points added to the second point lies in their affine span. -/ theorem smul_vsub_rev_vadd_mem_affineSpan_pair (r : k) (p₁ p₂ : P) : r • (p₁ -ᵥ p₂) +ᵥ p₂ ∈ line[k, p₁, p₂] := AffineMap.lineMap_rev_mem_affineSpan_pair _ _ _ #align smul_vsub_rev_vadd_mem_affine_span_pair smul_vsub_rev_vadd_mem_affineSpan_pair /-- A vector added to the first point lies in the affine span of two points if and only if it is a multiple of their difference. -/ theorem vadd_left_mem_affineSpan_pair {p₁ p₂ : P} {v : V} : v +ᵥ p₁ ∈ line[k, p₁, p₂] ↔ ∃ r : k, r • (p₂ -ᵥ p₁) = v := by rw [vadd_mem_iff_mem_direction _ (left_mem_affineSpan_pair _ _ _), direction_affineSpan, mem_vectorSpan_pair_rev] #align vadd_left_mem_affine_span_pair vadd_left_mem_affineSpan_pair /-- A vector added to the second point lies in the affine span of two points if and only if it is a multiple of their difference. -/ theorem vadd_right_mem_affineSpan_pair {p₁ p₂ : P} {v : V} : v +ᵥ p₂ ∈ line[k, p₁, p₂] ↔ ∃ r : k, r • (p₁ -ᵥ p₂) = v := by rw [vadd_mem_iff_mem_direction _ (right_mem_affineSpan_pair _ _ _), direction_affineSpan, mem_vectorSpan_pair] #align vadd_right_mem_affine_span_pair vadd_right_mem_affineSpan_pair /-- The span of two points that lie in an affine subspace is contained in that subspace. -/ theorem affineSpan_pair_le_of_mem_of_mem {p₁ p₂ : P} {s : AffineSubspace k P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : line[k, p₁, p₂] ≤ s := by rw [affineSpan_le, Set.insert_subset_iff, Set.singleton_subset_iff] exact ⟨hp₁, hp₂⟩ #align affine_span_pair_le_of_mem_of_mem affineSpan_pair_le_of_mem_of_mem /-- One line is contained in another differing in the first point if the first point of the first line is contained in the second line. -/ theorem affineSpan_pair_le_of_left_mem {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) : line[k, p₁, p₃] ≤ line[k, p₂, p₃] := affineSpan_pair_le_of_mem_of_mem h (right_mem_affineSpan_pair _ _ _) #align affine_span_pair_le_of_left_mem affineSpan_pair_le_of_left_mem /-- One line is contained in another differing in the second point if the second point of the first line is contained in the second line. -/ theorem affineSpan_pair_le_of_right_mem {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) : line[k, p₂, p₁] ≤ line[k, p₂, p₃] := affineSpan_pair_le_of_mem_of_mem (left_mem_affineSpan_pair _ _ _) h #align affine_span_pair_le_of_right_mem affineSpan_pair_le_of_right_mem variable (k) /-- `affineSpan` is monotone. -/ @[mono] theorem affineSpan_mono {s₁ s₂ : Set P} (h : s₁ ⊆ s₂) : affineSpan k s₁ ≤ affineSpan k s₂ := spanPoints_subset_coe_of_subset_coe (Set.Subset.trans h (subset_affineSpan k _)) #align affine_span_mono affineSpan_mono /-- Taking the affine span of a set, adding a point and taking the span again produces the same results as adding the point to the set and taking the span. -/ theorem affineSpan_insert_affineSpan (p : P) (ps : Set P) : affineSpan k (insert p (affineSpan k ps : Set P)) = affineSpan k (insert p ps) := by rw [Set.insert_eq, Set.insert_eq, span_union, span_union, affineSpan_coe] #align affine_span_insert_affine_span affineSpan_insert_affineSpan /-- If a point is in the affine span of a set, adding it to that set does not change the affine span. -/ theorem affineSpan_insert_eq_affineSpan {p : P} {ps : Set P} (h : p ∈ affineSpan k ps) : affineSpan k (insert p ps) = affineSpan k ps := by rw [← mem_coe] at h rw [← affineSpan_insert_affineSpan, Set.insert_eq_of_mem h, affineSpan_coe] #align affine_span_insert_eq_affine_span affineSpan_insert_eq_affineSpan variable {k} /-- If a point is in the affine span of a set, adding it to that set does not change the vector span. -/ theorem vectorSpan_insert_eq_vectorSpan {p : P} {ps : Set P} (h : p ∈ affineSpan k ps) : vectorSpan k (insert p ps) = vectorSpan k ps := by simp_rw [← direction_affineSpan, affineSpan_insert_eq_affineSpan _ h] #align vector_span_insert_eq_vector_span vectorSpan_insert_eq_vectorSpan /-- When the affine space is also a vector space, the affine span is contained within the linear span. -/ lemma affineSpan_le_toAffineSubspace_span {s : Set V} : affineSpan k s ≤ (Submodule.span k s).toAffineSubspace := by intro x hx show x ∈ Submodule.span k s induction hx using affineSpan_induction' with \| mem x hx => exact Submodule.subset_span hx \| smul_vsub_vadd c u _ v _ w _ hu hv hw => simp only [vsub_eq_sub, vadd_eq_add] apply Submodule.add_mem _ _ hw exact Submodule.smul_mem _ _ (Submodule.sub_mem _ hu hv) lemma affineSpan_subset_span {s : Set V} : (affineSpan k s : Set V) ⊆ Submodule.span k s := affineSpan_le_toAffineSubspace_span end AffineSpace' namespace AffineSubspace variable {k : Type} {V : Type} {P : Type} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] /-- The direction of the sup of two nonempty affine subspaces is the sup of the two directions and of any one difference between points in the two subspaces. -/ theorem direction_sup {s1 s2 : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s1) (hp2 : p2 ∈ s2) : (s1 ⊔ s2).direction = s1.direction ⊔ s2.direction ⊔ k ∙ p2 -ᵥ p1 := by refine le_antisymm ?_ ?_ · change (affineSpan k ((s1 : Set P) ∪ s2)).direction ≤ _ rw [← mem_coe] at hp1 rw [direction_affineSpan, vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ hp1), Submodule.span_le] rintro v ⟨p3, hp3, rfl⟩ cases' hp3 with hp3 hp3 · rw [sup_assoc, sup_comm, SetLike.mem_coe, Submodule.mem_sup] use 0, Submodule.zero_mem _, p3 -ᵥ p1, vsub_mem_direction hp3 hp1 rw [zero_add] · rw [sup_assoc, SetLike.mem_coe, Submodule.mem_sup] use 0, Submodule.zero_mem _, p3 -ᵥ p1 rw [and_comm, zero_add] use rfl rw [← vsub_add_vsub_cancel p3 p2 p1, Submodule.mem_sup] use p3 -ᵥ p2, vsub_mem_direction hp3 hp2, p2 -ᵥ p1, Submodule.mem_span_singleton_self _ · refine sup_le (sup_direction_le _ _) ?_ rw [direction_eq_vectorSpan, vectorSpan_def] exact sInf_le_sInf fun p hp => Set.Subset.trans (Set.singleton_subset_iff.2 (vsub_mem_vsub (mem_spanPoints k p2 _ (Set.mem_union_right _ hp2)) (mem_spanPoints k p1 _ (Set.mem_union_left _ hp1)))) hp #align affine_subspace.direction_sup AffineSubspace.direction_sup /-- The direction of the span of the result of adding a point to a nonempty affine subspace is the sup of the direction of that subspace and of any one difference between that point and a point in the subspace. -/ theorem direction_affineSpan_insert {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) : (affineSpan k (insert p2 (s : Set P))).direction = Submodule.span k {p2 -ᵥ p1} ⊔ s.direction := by rw [sup_comm, ← Set.union_singleton, ← coe_affineSpan_singleton k V p2] change (s ⊔ affineSpan k {p2}).direction = _ rw [direction_sup hp1 (mem_affineSpan k (Set.mem_singleton _)), direction_affineSpan] simp #align affine_subspace.direction_affine_span_insert AffineSubspace.direction_affineSpan_insert /-- Given a point `p1` in an affine subspace `s`, and a point `p2`, a point `p` is in the span of `s` with `p2` added if and only if it is a multiple of `p2 -ᵥ p1` added to a point in `s`. -/ theorem mem_affineSpan_insert_iff {s : AffineSubspace k P} {p1 : P} (hp1 : p1 ∈ s) (p2 p : P) : p ∈ affineSpan k (insert p2 (s : Set P)) ↔ ∃ r : k, ∃ p0 ∈ s, p = r • (p2 -ᵥ p1 : V) +ᵥ p0 := by rw [← mem_coe] at hp1 rw [← vsub_right_mem_direction_iff_mem (mem_affineSpan k (Set.mem_insert_of_mem _ hp1)), direction_affineSpan_insert hp1, Submodule.mem_sup] constructor · rintro ⟨v1, hv1, v2, hv2, hp⟩ rw [Submodule.mem_span_singleton] at hv1 rcases hv1 with ⟨r, rfl⟩ use r, v2 +ᵥ p1, vadd_mem_of_mem_direction hv2 hp1 symm at hp rw [← sub_eq_zero, ← vsub_vadd_eq_vsub_sub, vsub_eq_zero_iff_eq] at hp rw [hp, vadd_vadd] · rintro ⟨r, p3, hp3, rfl⟩ use r • (p2 -ᵥ p1), Submodule.mem_span_singleton.2 ⟨r, rfl⟩, p3 -ᵥ p1, vsub_mem_direction hp3 hp1 rw [vadd_vsub_assoc] #align affine_subspace.mem_affine_span_insert_iff AffineSubspace.mem_affineSpan_insert_iff end AffineSubspace section MapComap variable {k V₁ P₁ V₂ P₂ V₃ P₃ : Type} [Ring k] variable [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] variable [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] variable [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] section variable (f : P₁ →ᵃ[k] P₂) @[simp] theorem AffineMap.vectorSpan_image_eq_submodule_map {s : Set P₁} : Submodule.map f.linear (vectorSpan k s) = vectorSpan k (f '' s) := by rw [vectorSpan_def, vectorSpan_def, f.image_vsub_image, Submodule.span_image] -- Porting note: Lean unfolds things too far with `simp` here. #align affine_map.vector_span_image_eq_submodule_map AffineMap.vectorSpan_image_eq_submodule_map namespace AffineSubspace /-- The image of an affine subspace under an affine map as an affine subspace. -/ def map (s : AffineSubspace k P₁) : AffineSubspace k P₂ where carrier := f '' s smul_vsub_vadd_mem := by rintro t - - - ⟨p₁, h₁, rfl⟩ ⟨p₂, h₂, rfl⟩ ⟨p₃, h₃, rfl⟩ use t • (p₁ -ᵥ p₂) +ᵥ p₃ suffices t • (p₁ -ᵥ p₂) +ᵥ p₃ ∈ s by { simp only [SetLike.mem_coe, true_and, this] rw [AffineMap.map_vadd, map_smul, AffineMap.linearMap_vsub] } exact s.smul_vsub_vadd_mem t h₁ h₂ h₃ #align affine_subspace.map AffineSubspace.map @[simp] theorem coe_map (s : AffineSubspace k P₁) : (s.map f : Set P₂) = f '' s := rfl #align affine_subspace.coe_map AffineSubspace.coe_map @[simp] theorem mem_map {f : P₁ →ᵃ[k] P₂} {x : P₂} {s : AffineSubspace k P₁} : x ∈ s.map f ↔ ∃ y ∈ s, f y = x := Iff.rfl #align affine_subspace.mem_map AffineSubspace.mem_map theorem mem_map_of_mem {x : P₁} {s : AffineSubspace k P₁} (h : x ∈ s) : f x ∈ s.map f := Set.mem_image_of_mem _ h #align affine_subspace.mem_map_of_mem AffineSubspace.mem_map_of_mem -- The simpNF linter says that the LHS can be simplified via `AffineSubspace.mem_map`. -- However this is a higher priority lemma. -- https://github.com/leanprover/std4/issues/207 @[simp 1100, nolint simpNF] theorem mem_map_iff_mem_of_injective {f : P₁ →ᵃ[k] P₂} {x : P₁} {s : AffineSubspace k P₁} (hf : Function.Injective f) : f x ∈ s.map f ↔ x ∈ s := hf.mem_set_image #align affine_subspace.mem_map_iff_mem_of_injective AffineSubspace.mem_map_iff_mem_of_injective @[simp] theorem map_bot : (⊥ : AffineSubspace k P₁).map f = ⊥ := coe_injective <\| image_empty f #align affine_subspace.map_bot AffineSubspace.map_bot @[simp] theorem map_eq_bot_iff {s : AffineSubspace k P₁} : s.map f = ⊥ ↔ s = ⊥ := by refine ⟨fun h => ?_, fun h => ?_⟩ · rwa [← coe_eq_bot_iff, coe_map, image_eq_empty, coe_eq_bot_iff] at h · rw [h, map_bot] #align affine_subspace.map_eq_bot_iff AffineSubspace.map_eq_bot_iff @[simp] theorem map_id (s : AffineSubspace k P₁) : s.map (AffineMap.id k P₁) = s := coe_injective <\| image_id _ #align affine_subspace.map_id AffineSubspace.map_id theorem map_map (s : AffineSubspace k P₁) (f : P₁ →ᵃ[k] P₂) (g : P₂ →ᵃ[k] P₃) : (s.map f).map g = s.map (g.comp f) := coe_injective <\| image_image _ _ _ #align affine_subspace.map_map AffineSubspace.map_map @[simp] theorem map_direction (s : AffineSubspace k P₁) : (s.map f).direction = s.direction.map f.linear := by rw [direction_eq_vectorSpan, direction_eq_vectorSpan, coe_map, AffineMap.vectorSpan_image_eq_submodule_map] -- Porting note: again, Lean unfolds too aggressively with `simp` #align affine_subspace.map_direction AffineSubspace.map_direction theorem map_span (s : Set P₁) : (affineSpan k s).map f = affineSpan k (f '' s) := by rcases s.eq_empty_or_nonempty with (rfl \| ⟨p, hp⟩); · rw [image_empty, span_empty, span_empty, map_bot] -- Porting note: I don't know exactly why this `simp` was broken. apply ext_of_direction_eq · simp [direction_affineSpan] · exact ⟨f p, mem_image_of_mem f (subset_affineSpan k _ hp), subset_affineSpan k _ (mem_image_of_mem f hp)⟩ #align affine_subspace.map_span AffineSubspace.map_span section inclusion variable {S₁ S₂ : AffineSubspace k P₁} [Nonempty S₁] [Nonempty S₂] attribute [local instance] AffineSubspace.toAddTorsor /-- Affine map from a smaller to a larger subspace of the same space. This is the affine version of `Submodule.inclusion`. -/ @[simps linear] def inclusion (h : S₁ ≤ S₂) : S₁ →ᵃ[k] S₂ where toFun := Set.inclusion h linear := Submodule.inclusion <\| AffineSubspace.direction_le h map_vadd' _ _ := rfl @[simp] theorem coe_inclusion_apply (h : S₁ ≤ S₂) (x : S₁) : (inclusion h x : P₁) = x := rfl @[simp] theorem inclusion_rfl : inclusion (le_refl S₁) = AffineMap.id k S₁ := rfl end inclusion end AffineSubspace namespace AffineMap @[simp] theorem map_top_of_surjective (hf : Function.Surjective f) : AffineSubspace.map f ⊤ = ⊤ := by rw [← AffineSubspace.ext_iff] exact image_univ_of_surjective hf #align affine_map.map_top_of_surjective AffineMap.map_top_of_surjective	Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean	1,641	1,643	theorem span_eq_top_of_surjective {s : Set P₁} (hf : Function.Surjective f) (h : affineSpan k s = ⊤) : affineSpan k (f '' s) = ⊤ := by	rw [← AffineSubspace.map_span, h, map_top_of_surjective f hf]
/- Copyright (c) 2022 Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies -/ import Mathlib.Data.Fintype.Card import Mathlib.Order.UpperLower.Basic #align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46" /-! # Intersecting families This file defines intersecting families and proves their basic properties. ## Main declarations * `Set.Intersecting`: Predicate for a set of elements in a generalized boolean algebra to be an intersecting family. * `Set.Intersecting.card_le`: An intersecting family can only take up to half the elements, because `a` and `aᶜ` cannot simultaneously be in it. * `Set.Intersecting.is_max_iff_card_eq`: Any maximal intersecting family takes up half the elements. ## References * [D. J. Kleitman, Families of non-disjoint subsets][kleitman1966] -/ open Finset variable {α : Type*} namespace Set section SemilatticeInf variable [SemilatticeInf α] [OrderBot α] {s t : Set α} {a b c : α} /-- A set family is intersecting if every pair of elements is non-disjoint. -/ def Intersecting (s : Set α) : Prop := ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ¬Disjoint a b #align set.intersecting Set.Intersecting @[mono] theorem Intersecting.mono (h : t ⊆ s) (hs : s.Intersecting) : t.Intersecting := fun _a ha _b hb => hs (h ha) (h hb) #align set.intersecting.mono Set.Intersecting.mono theorem Intersecting.not_bot_mem (hs : s.Intersecting) : ⊥ ∉ s := fun h => hs h h disjoint_bot_left #align set.intersecting.not_bot_mem Set.Intersecting.not_bot_mem theorem Intersecting.ne_bot (hs : s.Intersecting) (ha : a ∈ s) : a ≠ ⊥ := ne_of_mem_of_not_mem ha hs.not_bot_mem #align set.intersecting.ne_bot Set.Intersecting.ne_bot theorem intersecting_empty : (∅ : Set α).Intersecting := fun _ => False.elim #align set.intersecting_empty Set.intersecting_empty @[simp]	Mathlib/Combinatorics/SetFamily/Intersecting.lean	61	61	theorem intersecting_singleton : ({a} : Set α).Intersecting ↔ a ≠ ⊥ := by	simp [Intersecting]
/- Copyright (c) 2021 Kexing Ying. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kexing Ying, Rémy Degenne -/ import Mathlib.Probability.Process.Adapted import Mathlib.MeasureTheory.Constructions.BorelSpace.Order #align_import probability.process.stopping from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca" /-! # Stopping times, stopped processes and stopped values Definition and properties of stopping times. ## Main definitions * `MeasureTheory.IsStoppingTime`: a stopping time with respect to some filtration `f` is a function `τ` such that for all `i`, the preimage of `{j \| j ≤ i}` along `τ` is `f i`-measurable * `MeasureTheory.IsStoppingTime.measurableSpace`: the σ-algebra associated with a stopping time ## Main results * `ProgMeasurable.stoppedProcess`: the stopped process of a progressively measurable process is progressively measurable. * `memℒp_stoppedProcess`: if a process belongs to `ℒp` at every time in `ℕ`, then its stopped process belongs to `ℒp` as well. ## Tags stopping time, stochastic process -/ open Filter Order TopologicalSpace open scoped Classical MeasureTheory NNReal ENNReal Topology namespace MeasureTheory variable {Ω β ι : Type*} {m : MeasurableSpace Ω} /-! ### Stopping times -/ /-- A stopping time with respect to some filtration `f` is a function `τ` such that for all `i`, the preimage of `{j \| j ≤ i}` along `τ` is measurable with respect to `f i`. Intuitively, the stopping time `τ` describes some stopping rule such that at time `i`, we may determine it with the information we have at time `i`. -/ def IsStoppingTime [Preorder ι] (f : Filtration ι m) (τ : Ω → ι) := ∀ i : ι, MeasurableSet[f i] <\| {ω \| τ ω ≤ i} #align measure_theory.is_stopping_time MeasureTheory.IsStoppingTime theorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) : IsStoppingTime f fun _ => i := fun j => by simp only [MeasurableSet.const] #align measure_theory.is_stopping_time_const MeasureTheory.isStoppingTime_const section MeasurableSet section Preorder variable [Preorder ι] {f : Filtration ι m} {τ : Ω → ι} protected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω ≤ i} := hτ i #align measure_theory.is_stopping_time.measurable_set_le MeasureTheory.IsStoppingTime.measurableSet_le theorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := by by_cases hi_min : IsMin i · suffices {ω : Ω \| τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i) ext1 ω simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] rw [isMin_iff_forall_not_lt] at hi_min exact hi_min (τ ω) have : {ω : Ω \| τ ω < i} = τ ⁻¹' Set.Iic (pred i) := by ext; simp [Iic_pred_of_not_isMin hi_min] rw [this] exact f.mono (pred_le i) _ (hτ.measurableSet_le <\| pred i) #align measure_theory.is_stopping_time.measurable_set_lt_of_pred MeasureTheory.IsStoppingTime.measurableSet_lt_of_pred end Preorder section CountableStoppingTime namespace IsStoppingTime variable [PartialOrder ι] {τ : Ω → ι} {f : Filtration ι m} protected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω \| τ ω = i} := by have : {ω \| τ ω = i} = {ω \| τ ω ≤ i} \ ⋃ (j ∈ Set.range τ) (_ : j < i), {ω \| τ ω ≤ j} := by ext1 a simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq', Set.mem_diff, Set.mem_iUnion, exists_prop, not_exists, not_and, not_le] constructor <;> intro h · simp only [h, lt_iff_le_not_le, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff] · exact h.1.eq_or_lt.resolve_right fun h_lt => h.2 a h_lt le_rfl rw [this] refine (hτ.measurableSet_le i).diff ?_ refine MeasurableSet.biUnion h_countable fun j _ => ?_ rw [Set.iUnion_eq_if] split_ifs with hji · exact f.mono hji.le _ (hτ.measurableSet_le j) · exact @MeasurableSet.empty _ (f i) #align measure_theory.is_stopping_time.measurable_set_eq_of_countable_range MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range protected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω = i} := hτ.measurableSet_eq_of_countable_range (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_eq_of_countable MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable protected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := by have : {ω \| τ ω < i} = {ω \| τ ω ≤ i} \ {ω \| τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne] rw [this] exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i) #align measure_theory.is_stopping_time.measurable_set_lt_of_countable_range MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range protected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := hτ.measurableSet_lt_of_countable_range (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_lt_of_countable MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable protected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → ι} {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω < i}ᶜ := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] rw [this] exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl #align measure_theory.is_stopping_time.measurable_set_ge_of_countable_range MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range protected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → ι} {f : Filtration ι m} [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| i ≤ τ ω} := hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_ge_of_countable MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable end IsStoppingTime end CountableStoppingTime section LinearOrder variable [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι} theorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| i < τ ω} := by have : {ω \| i < τ ω} = {ω \| τ ω ≤ i}ᶜ := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le] rw [this] exact (hτ.measurableSet_le i).compl #align measure_theory.is_stopping_time.measurable_set_gt MeasureTheory.IsStoppingTime.measurableSet_gt section TopologicalSpace variable [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] /-- Auxiliary lemma for `MeasureTheory.IsStoppingTime.measurableSet_lt`. -/ theorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι) (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω \| τ ω < i} := by by_cases hi_min : IsMin i · suffices {ω \| τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i) ext1 ω simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] exact isMin_iff_forall_not_lt.mp hi_min (τ ω) obtain ⟨seq, -, -, h_tendsto, h_bound⟩ : ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i := h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min) have h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k \| k ≤ seq j} := by ext1 k simp only [Set.mem_Iio, Set.mem_iUnion, Set.mem_setOf_eq] refine ⟨fun hk_lt_i => ?_, fun h_exists_k_le_seq => ?_⟩ · rw [tendsto_atTop'] at h_tendsto have h_nhds : Set.Ici k ∈ 𝓝 i := mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩ obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds exact ⟨a, ha a le_rfl⟩ · obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq exact hk_seq_j.trans_lt (h_bound j) have h_lt_eq_preimage : {ω \| τ ω < i} = τ ⁻¹' Set.Iio i := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iio] rw [h_lt_eq_preimage, h_Ioi_eq_Union] simp only [Set.preimage_iUnion, Set.preimage_setOf_eq] exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n)) #align measure_theory.is_stopping_time.measurable_set_lt_of_is_lub MeasureTheory.IsStoppingTime.measurableSet_lt_of_isLUB theorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω < i} := by obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i cases' lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i h_Iio_eq_Iic · rw [← hi'_eq_i] at hi'_lub ⊢ exact hτ.measurableSet_lt_of_isLUB i' hi'_lub · have h_lt_eq_preimage : {ω : Ω \| τ ω < i} = τ ⁻¹' Set.Iio i := rfl rw [h_lt_eq_preimage, h_Iio_eq_Iic] exact f.mono (lub_Iio_le i hi'_lub) _ (hτ.measurableSet_le i') #align measure_theory.is_stopping_time.measurable_set_lt MeasureTheory.IsStoppingTime.measurableSet_lt theorem IsStoppingTime.measurableSet_ge (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω < i}ᶜ := by ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] rw [this] exact (hτ.measurableSet_lt i).compl #align measure_theory.is_stopping_time.measurable_set_ge MeasureTheory.IsStoppingTime.measurableSet_ge theorem IsStoppingTime.measurableSet_eq (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω \| τ ω = i} := by have : {ω \| τ ω = i} = {ω \| τ ω ≤ i} ∩ {ω \| τ ω ≥ i} := by ext1 ω; simp only [Set.mem_setOf_eq, ge_iff_le, Set.mem_inter_iff, le_antisymm_iff] rw [this] exact (hτ.measurableSet_le i).inter (hτ.measurableSet_ge i) #align measure_theory.is_stopping_time.measurable_set_eq MeasureTheory.IsStoppingTime.measurableSet_eq theorem IsStoppingTime.measurableSet_eq_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) : MeasurableSet[f j] {ω \| τ ω = i} := f.mono hle _ <\| hτ.measurableSet_eq i #align measure_theory.is_stopping_time.measurable_set_eq_le MeasureTheory.IsStoppingTime.measurableSet_eq_le theorem IsStoppingTime.measurableSet_lt_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) : MeasurableSet[f j] {ω \| τ ω < i} := f.mono hle _ <\| hτ.measurableSet_lt i #align measure_theory.is_stopping_time.measurable_set_lt_le MeasureTheory.IsStoppingTime.measurableSet_lt_le end TopologicalSpace end LinearOrder section Countable theorem isStoppingTime_of_measurableSet_eq [Preorder ι] [Countable ι] {f : Filtration ι m} {τ : Ω → ι} (hτ : ∀ i, MeasurableSet[f i] {ω \| τ ω = i}) : IsStoppingTime f τ := by intro i rw [show {ω \| τ ω ≤ i} = ⋃ k ≤ i, {ω \| τ ω = k} by ext; simp] refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_ exact f.mono hk _ (hτ k) #align measure_theory.is_stopping_time_of_measurable_set_eq MeasureTheory.isStoppingTime_of_measurableSet_eq end Countable end MeasurableSet namespace IsStoppingTime protected theorem max [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : IsStoppingTime f fun ω => max (τ ω) (π ω) := by intro i simp_rw [max_le_iff, Set.setOf_and] exact (hτ i).inter (hπ i) #align measure_theory.is_stopping_time.max MeasureTheory.IsStoppingTime.max protected theorem max_const [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ) (i : ι) : IsStoppingTime f fun ω => max (τ ω) i := hτ.max (isStoppingTime_const f i) #align measure_theory.is_stopping_time.max_const MeasureTheory.IsStoppingTime.max_const protected theorem min [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : IsStoppingTime f fun ω => min (τ ω) (π ω) := by intro i simp_rw [min_le_iff, Set.setOf_or] exact (hτ i).union (hπ i) #align measure_theory.is_stopping_time.min MeasureTheory.IsStoppingTime.min protected theorem min_const [LinearOrder ι] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ) (i : ι) : IsStoppingTime f fun ω => min (τ ω) i := hτ.min (isStoppingTime_const f i) #align measure_theory.is_stopping_time.min_const MeasureTheory.IsStoppingTime.min_const theorem add_const [AddGroup ι] [Preorder ι] [CovariantClass ι ι (Function.swap (· + ·)) (· ≤ ·)] [CovariantClass ι ι (· + ·) (· ≤ ·)] {f : Filtration ι m} {τ : Ω → ι} (hτ : IsStoppingTime f τ) {i : ι} (hi : 0 ≤ i) : IsStoppingTime f fun ω => τ ω + i := by intro j simp_rw [← le_sub_iff_add_le] exact f.mono (sub_le_self j hi) _ (hτ (j - i)) #align measure_theory.is_stopping_time.add_const MeasureTheory.IsStoppingTime.add_const theorem add_const_nat {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) {i : ℕ} : IsStoppingTime f fun ω => τ ω + i := by refine isStoppingTime_of_measurableSet_eq fun j => ?_ by_cases hij : i ≤ j · simp_rw [eq_comm, ← Nat.sub_eq_iff_eq_add hij, eq_comm] exact f.mono (j.sub_le i) _ (hτ.measurableSet_eq (j - i)) · rw [not_le] at hij convert @MeasurableSet.empty _ (f.1 j) ext ω simp only [Set.mem_empty_iff_false, iff_false_iff, Set.mem_setOf] omega #align measure_theory.is_stopping_time.add_const_nat MeasureTheory.IsStoppingTime.add_const_nat -- generalize to certain countable type? theorem add {f : Filtration ℕ m} {τ π : Ω → ℕ} (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : IsStoppingTime f (τ + π) := by intro i rw [(_ : {ω \| (τ + π) ω ≤ i} = ⋃ k ≤ i, {ω \| π ω = k} ∩ {ω \| τ ω + k ≤ i})] · exact MeasurableSet.iUnion fun k => MeasurableSet.iUnion fun hk => (hπ.measurableSet_eq_le hk).inter (hτ.add_const_nat i) ext ω simp only [Pi.add_apply, Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_prop] refine ⟨fun h => ⟨π ω, by omega, rfl, h⟩, ?_⟩ rintro ⟨j, hj, rfl, h⟩ assumption #align measure_theory.is_stopping_time.add MeasureTheory.IsStoppingTime.add section Preorder variable [Preorder ι] {f : Filtration ι m} {τ π : Ω → ι} /-- The associated σ-algebra with a stopping time. -/ protected def measurableSpace (hτ : IsStoppingTime f τ) : MeasurableSpace Ω where MeasurableSet' s := ∀ i : ι, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) measurableSet_empty i := (Set.empty_inter {ω \| τ ω ≤ i}).symm ▸ @MeasurableSet.empty _ (f i) measurableSet_compl s hs i := by rw [(_ : sᶜ ∩ {ω \| τ ω ≤ i} = (sᶜ ∪ {ω \| τ ω ≤ i}ᶜ) ∩ {ω \| τ ω ≤ i})] · refine MeasurableSet.inter ?_ ?_ · rw [← Set.compl_inter] exact (hs i).compl · exact hτ i · rw [Set.union_inter_distrib_right] simp only [Set.compl_inter_self, Set.union_empty] measurableSet_iUnion s hs i := by rw [forall_swap] at hs rw [Set.iUnion_inter] exact MeasurableSet.iUnion (hs i) #align measure_theory.is_stopping_time.measurable_space MeasureTheory.IsStoppingTime.measurableSpace protected theorem measurableSet (hτ : IsStoppingTime f τ) (s : Set Ω) : MeasurableSet[hτ.measurableSpace] s ↔ ∀ i : ι, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) := Iff.rfl #align measure_theory.is_stopping_time.measurable_set MeasureTheory.IsStoppingTime.measurableSet theorem measurableSpace_mono (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) (hle : τ ≤ π) : hτ.measurableSpace ≤ hπ.measurableSpace := by intro s hs i rw [(_ : s ∩ {ω \| π ω ≤ i} = s ∩ {ω \| τ ω ≤ i} ∩ {ω \| π ω ≤ i})] · exact (hs i).inter (hπ i) · ext simp only [Set.mem_inter_iff, iff_self_and, and_congr_left_iff, Set.mem_setOf_eq] intro hle' _ exact le_trans (hle _) hle' #align measure_theory.is_stopping_time.measurable_space_mono MeasureTheory.IsStoppingTime.measurableSpace_mono theorem measurableSpace_le_of_countable [Countable ι] (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := by intro s hs change ∀ i, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) at hs rw [(_ : s = ⋃ i, s ∩ {ω \| τ ω ≤ i})] · exact MeasurableSet.iUnion fun i => f.le i _ (hs i) · ext ω; constructor <;> rw [Set.mem_iUnion] · exact fun hx => ⟨τ ω, hx, le_rfl⟩ · rintro ⟨_, hx, _⟩ exact hx #align measure_theory.is_stopping_time.measurable_space_le_of_countable MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable theorem measurableSpace_le' [IsCountablyGenerated (atTop : Filter ι)] [(atTop : Filter ι).NeBot] (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := by intro s hs change ∀ i, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) at hs obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := (atTop : Filter ι).exists_seq_tendsto rw [(_ : s = ⋃ n, s ∩ {ω \| τ ω ≤ seq n})] · exact MeasurableSet.iUnion fun i => f.le (seq i) _ (hs (seq i)) · ext ω; constructor <;> rw [Set.mem_iUnion] · intro hx suffices ∃ i, τ ω ≤ seq i from ⟨this.choose, hx, this.choose_spec⟩ rw [tendsto_atTop] at h_seq_tendsto exact (h_seq_tendsto (τ ω)).exists · rintro ⟨_, hx, _⟩ exact hx #align measure_theory.is_stopping_time.measurable_space_le' MeasureTheory.IsStoppingTime.measurableSpace_le' theorem measurableSpace_le {ι} [SemilatticeSup ι] {f : Filtration ι m} {τ : Ω → ι} [IsCountablyGenerated (atTop : Filter ι)] (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := by cases isEmpty_or_nonempty ι · haveI : IsEmpty Ω := ⟨fun ω => IsEmpty.false (τ ω)⟩ intro s _ suffices hs : s = ∅ by rw [hs]; exact MeasurableSet.empty haveI : Unique (Set Ω) := Set.uniqueEmpty rw [Unique.eq_default s, Unique.eq_default ∅] exact measurableSpace_le' hτ #align measure_theory.is_stopping_time.measurable_space_le MeasureTheory.IsStoppingTime.measurableSpace_le example {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := hτ.measurableSpace_le example {f : Filtration ℝ m} {τ : Ω → ℝ} (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m := hτ.measurableSpace_le @[simp] theorem measurableSpace_const (f : Filtration ι m) (i : ι) : (isStoppingTime_const f i).measurableSpace = f i := by ext1 s change MeasurableSet[(isStoppingTime_const f i).measurableSpace] s ↔ MeasurableSet[f i] s rw [IsStoppingTime.measurableSet] constructor <;> intro h · specialize h i simpa only [le_refl, Set.setOf_true, Set.inter_univ] using h · intro j by_cases hij : i ≤ j · simp only [hij, Set.setOf_true, Set.inter_univ] exact f.mono hij _ h · simp only [hij, Set.setOf_false, Set.inter_empty, @MeasurableSet.empty _ (f.1 j)] #align measure_theory.is_stopping_time.measurable_space_const MeasureTheory.IsStoppingTime.measurableSpace_const theorem measurableSet_inter_eq_iff (hτ : IsStoppingTime f τ) (s : Set Ω) (i : ι) : MeasurableSet[hτ.measurableSpace] (s ∩ {ω \| τ ω = i}) ↔ MeasurableSet[f i] (s ∩ {ω \| τ ω = i}) := by have : ∀ j, {ω : Ω \| τ ω = i} ∩ {ω : Ω \| τ ω ≤ j} = {ω : Ω \| τ ω = i} ∩ {_ω \| i ≤ j} := by intro j ext1 ω simp only [Set.mem_inter_iff, Set.mem_setOf_eq, and_congr_right_iff] intro hxi rw [hxi] constructor <;> intro h · specialize h i simpa only [Set.inter_assoc, this, le_refl, Set.setOf_true, Set.inter_univ] using h · intro j rw [Set.inter_assoc, this] by_cases hij : i ≤ j · simp only [hij, Set.setOf_true, Set.inter_univ] exact f.mono hij _ h · set_option tactic.skipAssignedInstances false in simp [hij] convert @MeasurableSet.empty _ (Filtration.seq f j) #align measure_theory.is_stopping_time.measurable_set_inter_eq_iff MeasureTheory.IsStoppingTime.measurableSet_inter_eq_iff theorem measurableSpace_le_of_le_const (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) : hτ.measurableSpace ≤ f i := (measurableSpace_mono hτ _ hτ_le).trans (measurableSpace_const _ _).le #align measure_theory.is_stopping_time.measurable_space_le_of_le_const MeasureTheory.IsStoppingTime.measurableSpace_le_of_le_const theorem measurableSpace_le_of_le (hτ : IsStoppingTime f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) : hτ.measurableSpace ≤ m := (hτ.measurableSpace_le_of_le_const hτ_le).trans (f.le n) #align measure_theory.is_stopping_time.measurable_space_le_of_le MeasureTheory.IsStoppingTime.measurableSpace_le_of_le theorem le_measurableSpace_of_const_le (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, i ≤ τ ω) : f i ≤ hτ.measurableSpace := (measurableSpace_const _ _).symm.le.trans (measurableSpace_mono _ hτ hτ_le) #align measure_theory.is_stopping_time.le_measurable_space_of_const_le MeasureTheory.IsStoppingTime.le_measurableSpace_of_const_le end Preorder instance sigmaFinite_stopping_time {ι} [SemilatticeSup ι] [OrderBot ι] [(Filter.atTop : Filter ι).IsCountablyGenerated] {μ : Measure Ω} {f : Filtration ι m} {τ : Ω → ι} [SigmaFiniteFiltration μ f] (hτ : IsStoppingTime f τ) : SigmaFinite (μ.trim hτ.measurableSpace_le) := by refine @sigmaFiniteTrim_mono _ _ ?_ _ _ _ ?_ ?_ · exact f ⊥ · exact hτ.le_measurableSpace_of_const_le fun _ => bot_le · infer_instance #align measure_theory.is_stopping_time.sigma_finite_stopping_time MeasureTheory.IsStoppingTime.sigmaFinite_stopping_time instance sigmaFinite_stopping_time_of_le {ι} [SemilatticeSup ι] [OrderBot ι] {μ : Measure Ω} {f : Filtration ι m} {τ : Ω → ι} [SigmaFiniteFiltration μ f] (hτ : IsStoppingTime f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) : SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le)) := by refine @sigmaFiniteTrim_mono _ _ ?_ _ _ _ ?_ ?_ · exact f ⊥ · exact hτ.le_measurableSpace_of_const_le fun _ => bot_le · infer_instance #align measure_theory.is_stopping_time.sigma_finite_stopping_time_of_le MeasureTheory.IsStoppingTime.sigmaFinite_stopping_time_of_le section LinearOrder variable [LinearOrder ι] {f : Filtration ι m} {τ π : Ω → ι} protected theorem measurableSet_le' (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω ≤ i} := by intro j have : {ω : Ω \| τ ω ≤ i} ∩ {ω : Ω \| τ ω ≤ j} = {ω : Ω \| τ ω ≤ min i j} := by ext1 ω; simp only [Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff] rw [this] exact f.mono (min_le_right i j) _ (hτ _) #align measure_theory.is_stopping_time.measurable_set_le' MeasureTheory.IsStoppingTime.measurableSet_le' protected theorem measurableSet_gt' (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i < τ ω} := by have : {ω : Ω \| i < τ ω} = {ω : Ω \| τ ω ≤ i}ᶜ := by ext1 ω; simp rw [this] exact (hτ.measurableSet_le' i).compl #align measure_theory.is_stopping_time.measurable_set_gt' MeasureTheory.IsStoppingTime.measurableSet_gt' protected theorem measurableSet_eq' [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω = i} := by rw [← Set.univ_inter {ω \| τ ω = i}, measurableSet_inter_eq_iff, Set.univ_inter] exact hτ.measurableSet_eq i #align measure_theory.is_stopping_time.measurable_set_eq' MeasureTheory.IsStoppingTime.measurableSet_eq' protected theorem measurableSet_ge' [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω = i} ∪ {ω \| i < τ ω} := by ext1 ω simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union] rw [@eq_comm _ i, or_comm] rw [this] exact (hτ.measurableSet_eq' i).union (hτ.measurableSet_gt' i) #align measure_theory.is_stopping_time.measurable_set_ge' MeasureTheory.IsStoppingTime.measurableSet_ge' protected theorem measurableSet_lt' [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω < i} := by have : {ω \| τ ω < i} = {ω \| τ ω ≤ i} \ {ω \| τ ω = i} := by ext1 ω simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] rw [this] exact (hτ.measurableSet_le' i).diff (hτ.measurableSet_eq' i) #align measure_theory.is_stopping_time.measurable_set_lt' MeasureTheory.IsStoppingTime.measurableSet_lt' section Countable protected theorem measurableSet_eq_of_countable_range' (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω = i} := by rw [← Set.univ_inter {ω \| τ ω = i}, measurableSet_inter_eq_iff, Set.univ_inter] exact hτ.measurableSet_eq_of_countable_range h_countable i #align measure_theory.is_stopping_time.measurable_set_eq_of_countable_range' MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range' protected theorem measurableSet_eq_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω = i} := hτ.measurableSet_eq_of_countable_range' (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_eq_of_countable' MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable' protected theorem measurableSet_ge_of_countable_range' (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i ≤ τ ω} := by have : {ω \| i ≤ τ ω} = {ω \| τ ω = i} ∪ {ω \| i < τ ω} := by ext1 ω simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union] rw [@eq_comm _ i, or_comm] rw [this] exact (hτ.measurableSet_eq_of_countable_range' h_countable i).union (hτ.measurableSet_gt' i) #align measure_theory.is_stopping_time.measurable_set_ge_of_countable_range' MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range' protected theorem measurableSet_ge_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| i ≤ τ ω} := hτ.measurableSet_ge_of_countable_range' (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_ge_of_countable' MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable' protected theorem measurableSet_lt_of_countable_range' (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω < i} := by have : {ω \| τ ω < i} = {ω \| τ ω ≤ i} \ {ω \| τ ω = i} := by ext1 ω simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] rw [this] exact (hτ.measurableSet_le' i).diff (hτ.measurableSet_eq_of_countable_range' h_countable i) #align measure_theory.is_stopping_time.measurable_set_lt_of_countable_range' MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range' protected theorem measurableSet_lt_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[hτ.measurableSpace] {ω \| τ ω < i} := hτ.measurableSet_lt_of_countable_range' (Set.to_countable _) i #align measure_theory.is_stopping_time.measurable_set_lt_of_countable' MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable' protected theorem measurableSpace_le_of_countable_range (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) : hτ.measurableSpace ≤ m := by intro s hs change ∀ i, MeasurableSet[f i] (s ∩ {ω \| τ ω ≤ i}) at hs rw [(_ : s = ⋃ i ∈ Set.range τ, s ∩ {ω \| τ ω ≤ i})] · exact MeasurableSet.biUnion h_countable fun i _ => f.le i _ (hs i) · ext ω constructor <;> rw [Set.mem_iUnion] · exact fun hx => ⟨τ ω, by simpa using hx⟩ · rintro ⟨i, hx⟩ simp only [Set.mem_range, Set.iUnion_exists, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_setOf_eq, exists_prop, exists_and_right] at hx exact hx.2.1 #align measure_theory.is_stopping_time.measurable_space_le_of_countable_range MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable_range end Countable protected theorem measurable [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) : Measurable[hτ.measurableSpace] τ := @measurable_of_Iic ι Ω _ _ _ hτ.measurableSpace _ _ _ _ fun i => hτ.measurableSet_le' i #align measure_theory.is_stopping_time.measurable MeasureTheory.IsStoppingTime.measurable protected theorem measurable_of_le [TopologicalSpace ι] [MeasurableSpace ι] [BorelSpace ι] [OrderTopology ι] [SecondCountableTopology ι] (hτ : IsStoppingTime f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) : Measurable[f i] τ := hτ.measurable.mono (measurableSpace_le_of_le_const _ hτ_le) le_rfl #align measure_theory.is_stopping_time.measurable_of_le MeasureTheory.IsStoppingTime.measurable_of_le theorem measurableSpace_min (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) : (hτ.min hπ).measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace := by refine le_antisymm ?_ ?_ · exact le_inf (measurableSpace_mono _ hτ fun _ => min_le_left _ _) (measurableSpace_mono _ hπ fun _ => min_le_right _ _) · intro s change MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s → MeasurableSet[(hτ.min hπ).measurableSpace] s simp_rw [IsStoppingTime.measurableSet] have : ∀ i, {ω \| min (τ ω) (π ω) ≤ i} = {ω \| τ ω ≤ i} ∪ {ω \| π ω ≤ i} := by intro i; ext1 ω; simp simp_rw [this, Set.inter_union_distrib_left] exact fun h i => (h.left i).union (h.right i) #align measure_theory.is_stopping_time.measurable_space_min MeasureTheory.IsStoppingTime.measurableSpace_min theorem measurableSet_min_iff (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) (s : Set Ω) : MeasurableSet[(hτ.min hπ).measurableSpace] s ↔ MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s := by rw [measurableSpace_min hτ hπ]; rfl #align measure_theory.is_stopping_time.measurable_set_min_iff MeasureTheory.IsStoppingTime.measurableSet_min_iff	Mathlib/Probability/Process/Stopping.lean	607	609	theorem measurableSpace_min_const (hτ : IsStoppingTime f τ) {i : ι} : (hτ.min_const i).measurableSpace = hτ.measurableSpace ⊓ f i := by	rw [hτ.measurableSpace_min (isStoppingTime_const _ i), measurableSpace_const]
/- Copyright (c) 2020 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes -/ import Mathlib.Algebra.Group.Aut import Mathlib.Algebra.Group.Subgroup.Basic import Mathlib.Logic.Function.Basic #align_import group_theory.semidirect_product from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c" /-! # Semidirect product This file defines semidirect products of groups, and the canonical maps in and out of the semidirect product. The semidirect product of `N` and `G` given a hom `φ` from `G` to the automorphism group of `N` is the product of sets with the group `⟨n₁, g₁⟩ * ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩` ## Key definitions There are two homs into the semidirect product `inl : N →* N ⋊[φ] G` and `inr : G →* N ⋊[φ] G`, and `lift` can be used to define maps `N ⋊[φ] G →* H` out of the semidirect product given maps `f₁ : N →* H` and `f₂ : G →* H` that satisfy the condition `∀ n g, f₁ (φ g n) = f₂ g * f₁ n * f₂ g⁻¹` ## Notation This file introduces the global notation `N ⋊[φ] G` for `SemidirectProduct N G φ` ## Tags group, semidirect product -/ variable (N : Type) (G : Type) {H : Type} [Group N] [Group G] [Group H] /-- The semidirect product of groups `N` and `G`, given a map `φ` from `G` to the automorphism group of `N`. It the product of sets with the group operation `⟨n₁, g₁⟩ ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩` -/ @[ext] structure SemidirectProduct (φ : G →* MulAut N) where /-- The element of N -/ left : N /-- The element of G -/ right : G deriving DecidableEq #align semidirect_product SemidirectProduct -- Porting note: these lemmas are autogenerated by the inductive definition and are not -- in simple form due to the existence of mk_eq_inl_mul_inr attribute [nolint simpNF] SemidirectProduct.mk.injEq attribute [nolint simpNF] SemidirectProduct.mk.sizeOf_spec -- Porting note: unknown attribute -- attribute [pp_using_anonymous_constructor] SemidirectProduct @[inherit_doc] notation:35 N " ⋊[" φ:35 "] " G:35 => SemidirectProduct N G φ namespace SemidirectProduct variable {N G} variable {φ : G →* MulAut N} instance : Mul (SemidirectProduct N G φ) where mul a b := ⟨a.1 * φ a.2 b.1, a.2 * b.2⟩ lemma mul_def (a b : SemidirectProduct N G φ) : a * b = ⟨a.1 * φ a.2 b.1, a.2 * b.2⟩ := rfl @[simp] theorem mul_left (a b : N ⋊[φ] G) : (a * b).left = a.left * φ a.right b.left := rfl #align semidirect_product.mul_left SemidirectProduct.mul_left @[simp] theorem mul_right (a b : N ⋊[φ] G) : (a * b).right = a.right * b.right := rfl #align semidirect_product.mul_right SemidirectProduct.mul_right instance : One (SemidirectProduct N G φ) where one := ⟨1, 1⟩ @[simp] theorem one_left : (1 : N ⋊[φ] G).left = 1 := rfl #align semidirect_product.one_left SemidirectProduct.one_left @[simp] theorem one_right : (1 : N ⋊[φ] G).right = 1 := rfl #align semidirect_product.one_right SemidirectProduct.one_right instance : Inv (SemidirectProduct N G φ) where inv x := ⟨φ x.2⁻¹ x.1⁻¹, x.2⁻¹⟩ @[simp] theorem inv_left (a : N ⋊[φ] G) : a⁻¹.left = φ a.right⁻¹ a.left⁻¹ := rfl #align semidirect_product.inv_left SemidirectProduct.inv_left @[simp] theorem inv_right (a : N ⋊[φ] G) : a⁻¹.right = a.right⁻¹ := rfl #align semidirect_product.inv_right SemidirectProduct.inv_right instance : Group (N ⋊[φ] G) where mul_assoc a b c := SemidirectProduct.ext _ _ (by simp [mul_assoc]) (by simp [mul_assoc]) one_mul a := SemidirectProduct.ext _ _ (by simp) (one_mul a.2) mul_one a := SemidirectProduct.ext _ _ (by simp) (mul_one _) mul_left_inv a := SemidirectProduct.ext _ _ (by simp) (by simp) instance : Inhabited (N ⋊[φ] G) := ⟨1⟩ /-- The canonical map `N →* N ⋊[φ] G` sending `n` to `⟨n, 1⟩` -/ def inl : N →* N ⋊[φ] G where toFun n := ⟨n, 1⟩ map_one' := rfl map_mul' := by intros; ext <;> simp only [mul_left, map_one, MulAut.one_apply, mul_right, mul_one] #align semidirect_product.inl SemidirectProduct.inl @[simp] theorem left_inl (n : N) : (inl n : N ⋊[φ] G).left = n := rfl #align semidirect_product.left_inl SemidirectProduct.left_inl @[simp] theorem right_inl (n : N) : (inl n : N ⋊[φ] G).right = 1 := rfl #align semidirect_product.right_inl SemidirectProduct.right_inl theorem inl_injective : Function.Injective (inl : N → N ⋊[φ] G) := Function.injective_iff_hasLeftInverse.2 ⟨left, left_inl⟩ #align semidirect_product.inl_injective SemidirectProduct.inl_injective @[simp] theorem inl_inj {n₁ n₂ : N} : (inl n₁ : N ⋊[φ] G) = inl n₂ ↔ n₁ = n₂ := inl_injective.eq_iff #align semidirect_product.inl_inj SemidirectProduct.inl_inj /-- The canonical map `G →* N ⋊[φ] G` sending `g` to `⟨1, g⟩` -/ def inr : G →* N ⋊[φ] G where toFun g := ⟨1, g⟩ map_one' := rfl map_mul' := by intros; ext <;> simp #align semidirect_product.inr SemidirectProduct.inr @[simp] theorem left_inr (g : G) : (inr g : N ⋊[φ] G).left = 1 := rfl #align semidirect_product.left_inr SemidirectProduct.left_inr @[simp] theorem right_inr (g : G) : (inr g : N ⋊[φ] G).right = g := rfl #align semidirect_product.right_inr SemidirectProduct.right_inr theorem inr_injective : Function.Injective (inr : G → N ⋊[φ] G) := Function.injective_iff_hasLeftInverse.2 ⟨right, right_inr⟩ #align semidirect_product.inr_injective SemidirectProduct.inr_injective @[simp] theorem inr_inj {g₁ g₂ : G} : (inr g₁ : N ⋊[φ] G) = inr g₂ ↔ g₁ = g₂ := inr_injective.eq_iff #align semidirect_product.inr_inj SemidirectProduct.inr_inj theorem inl_aut (g : G) (n : N) : (inl (φ g n) : N ⋊[φ] G) = inr g * inl n * inr g⁻¹ := by ext <;> simp #align semidirect_product.inl_aut SemidirectProduct.inl_aut theorem inl_aut_inv (g : G) (n : N) : (inl ((φ g)⁻¹ n) : N ⋊[φ] G) = inr g⁻¹ * inl n * inr g := by rw [← MonoidHom.map_inv, inl_aut, inv_inv] #align semidirect_product.inl_aut_inv SemidirectProduct.inl_aut_inv @[simp] theorem mk_eq_inl_mul_inr (g : G) (n : N) : (⟨n, g⟩ : N ⋊[φ] G) = inl n * inr g := by ext <;> simp #align semidirect_product.mk_eq_inl_mul_inr SemidirectProduct.mk_eq_inl_mul_inr @[simp] theorem inl_left_mul_inr_right (x : N ⋊[φ] G) : inl x.left * inr x.right = x := by ext <;> simp #align semidirect_product.inl_left_mul_inr_right SemidirectProduct.inl_left_mul_inr_right /-- The canonical projection map `N ⋊[φ] G →* G`, as a group hom. -/ def rightHom : N ⋊[φ] G →* G where toFun := SemidirectProduct.right map_one' := rfl map_mul' _ _ := rfl #align semidirect_product.right_hom SemidirectProduct.rightHom @[simp] theorem rightHom_eq_right : (rightHom : N ⋊[φ] G → G) = right := rfl #align semidirect_product.right_hom_eq_right SemidirectProduct.rightHom_eq_right @[simp] theorem rightHom_comp_inl : (rightHom : N ⋊[φ] G →* G).comp inl = 1 := by ext; simp [rightHom] #align semidirect_product.right_hom_comp_inl SemidirectProduct.rightHom_comp_inl @[simp] theorem rightHom_comp_inr : (rightHom : N ⋊[φ] G →* G).comp inr = MonoidHom.id _ := by ext; simp [rightHom] #align semidirect_product.right_hom_comp_inr SemidirectProduct.rightHom_comp_inr @[simp] theorem rightHom_inl (n : N) : rightHom (inl n : N ⋊[φ] G) = 1 := by simp [rightHom] #align semidirect_product.right_hom_inl SemidirectProduct.rightHom_inl @[simp]	Mathlib/GroupTheory/SemidirectProduct.lean	198	198	theorem rightHom_inr (g : G) : rightHom (inr g : N ⋊[φ] G) = g := by	simp [rightHom]
/- Copyright (c) 2014 Parikshit Khanna. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro -/ import Mathlib.Data.Nat.Defs import Mathlib.Data.Option.Basic import Mathlib.Data.List.Defs import Mathlib.Init.Data.List.Basic import Mathlib.Init.Data.List.Instances import Mathlib.Init.Data.List.Lemmas import Mathlib.Logic.Unique import Mathlib.Order.Basic import Mathlib.Tactic.Common #align_import data.list.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83" /-! # Basic properties of lists -/ assert_not_exists Set.range assert_not_exists GroupWithZero assert_not_exists Ring open Function open Nat hiding one_pos namespace List universe u v w variable {ι : Type} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α} -- Porting note: Delete this attribute -- attribute [inline] List.head! /-- There is only one list of an empty type -/ instance uniqueOfIsEmpty [IsEmpty α] : Unique (List α) := { instInhabitedList with uniq := fun l => match l with \| [] => rfl \| a :: _ => isEmptyElim a } #align list.unique_of_is_empty List.uniqueOfIsEmpty instance : Std.LawfulIdentity (α := List α) Append.append [] where left_id := nil_append right_id := append_nil instance : Std.Associative (α := List α) Append.append where assoc := append_assoc #align list.cons_ne_nil List.cons_ne_nil #align list.cons_ne_self List.cons_ne_self #align list.head_eq_of_cons_eq List.head_eq_of_cons_eqₓ -- implicits order #align list.tail_eq_of_cons_eq List.tail_eq_of_cons_eqₓ -- implicits order @[simp] theorem cons_injective {a : α} : Injective (cons a) := fun _ _ => tail_eq_of_cons_eq #align list.cons_injective List.cons_injective #align list.cons_inj List.cons_inj #align list.cons_eq_cons List.cons_eq_cons theorem singleton_injective : Injective fun a : α => [a] := fun _ _ h => (cons_eq_cons.1 h).1 #align list.singleton_injective List.singleton_injective theorem singleton_inj {a b : α} : [a] = [b] ↔ a = b := singleton_injective.eq_iff #align list.singleton_inj List.singleton_inj #align list.exists_cons_of_ne_nil List.exists_cons_of_ne_nil theorem set_of_mem_cons (l : List α) (a : α) : { x \| x ∈ a :: l } = insert a { x \| x ∈ l } := Set.ext fun _ => mem_cons #align list.set_of_mem_cons List.set_of_mem_cons /-! ### mem -/ #align list.mem_singleton_self List.mem_singleton_self #align list.eq_of_mem_singleton List.eq_of_mem_singleton #align list.mem_singleton List.mem_singleton #align list.mem_of_mem_cons_of_mem List.mem_of_mem_cons_of_mem theorem _root_.Decidable.List.eq_or_ne_mem_of_mem [DecidableEq α] {a b : α} {l : List α} (h : a ∈ b :: l) : a = b ∨ a ≠ b ∧ a ∈ l := by by_cases hab : a = b · exact Or.inl hab · exact ((List.mem_cons.1 h).elim Or.inl (fun h => Or.inr ⟨hab, h⟩)) #align decidable.list.eq_or_ne_mem_of_mem Decidable.List.eq_or_ne_mem_of_mem #align list.eq_or_ne_mem_of_mem List.eq_or_ne_mem_of_mem #align list.not_mem_append List.not_mem_append #align list.ne_nil_of_mem List.ne_nil_of_mem lemma mem_pair {a b c : α} : a ∈ [b, c] ↔ a = b ∨ a = c := by rw [mem_cons, mem_singleton] @[deprecated (since := "2024-03-23")] alias mem_split := append_of_mem #align list.mem_split List.append_of_mem #align list.mem_of_ne_of_mem List.mem_of_ne_of_mem #align list.ne_of_not_mem_cons List.ne_of_not_mem_cons #align list.not_mem_of_not_mem_cons List.not_mem_of_not_mem_cons #align list.not_mem_cons_of_ne_of_not_mem List.not_mem_cons_of_ne_of_not_mem #align list.ne_and_not_mem_of_not_mem_cons List.ne_and_not_mem_of_not_mem_cons #align list.mem_map List.mem_map #align list.exists_of_mem_map List.exists_of_mem_map #align list.mem_map_of_mem List.mem_map_of_memₓ -- implicits order -- The simpNF linter says that the LHS can be simplified via `List.mem_map`. -- However this is a higher priority lemma. -- https://github.com/leanprover/std4/issues/207 @[simp 1100, nolint simpNF] theorem mem_map_of_injective {f : α → β} (H : Injective f) {a : α} {l : List α} : f a ∈ map f l ↔ a ∈ l := ⟨fun m => let ⟨_, m', e⟩ := exists_of_mem_map m; H e ▸ m', mem_map_of_mem _⟩ #align list.mem_map_of_injective List.mem_map_of_injective @[simp] theorem _root_.Function.Involutive.exists_mem_and_apply_eq_iff {f : α → α} (hf : Function.Involutive f) (x : α) (l : List α) : (∃ y : α, y ∈ l ∧ f y = x) ↔ f x ∈ l := ⟨by rintro ⟨y, h, rfl⟩; rwa [hf y], fun h => ⟨f x, h, hf _⟩⟩ #align function.involutive.exists_mem_and_apply_eq_iff Function.Involutive.exists_mem_and_apply_eq_iff theorem mem_map_of_involutive {f : α → α} (hf : Involutive f) {a : α} {l : List α} : a ∈ map f l ↔ f a ∈ l := by rw [mem_map, hf.exists_mem_and_apply_eq_iff] #align list.mem_map_of_involutive List.mem_map_of_involutive #align list.forall_mem_map_iff List.forall_mem_map_iffₓ -- universe order #align list.map_eq_nil List.map_eq_nilₓ -- universe order attribute [simp] List.mem_join #align list.mem_join List.mem_join #align list.exists_of_mem_join List.exists_of_mem_join #align list.mem_join_of_mem List.mem_join_of_memₓ -- implicits order attribute [simp] List.mem_bind #align list.mem_bind List.mem_bindₓ -- implicits order -- Porting note: bExists in Lean3, And in Lean4 #align list.exists_of_mem_bind List.exists_of_mem_bindₓ -- implicits order #align list.mem_bind_of_mem List.mem_bind_of_memₓ -- implicits order #align list.bind_map List.bind_mapₓ -- implicits order theorem map_bind (g : β → List γ) (f : α → β) : ∀ l : List α, (List.map f l).bind g = l.bind fun a => g (f a) \| [] => rfl \| a :: l => by simp only [cons_bind, map_cons, map_bind _ _ l] #align list.map_bind List.map_bind /-! ### length -/ #align list.length_eq_zero List.length_eq_zero #align list.length_singleton List.length_singleton #align list.length_pos_of_mem List.length_pos_of_mem #align list.exists_mem_of_length_pos List.exists_mem_of_length_pos #align list.length_pos_iff_exists_mem List.length_pos_iff_exists_mem alias ⟨ne_nil_of_length_pos, length_pos_of_ne_nil⟩ := length_pos #align list.ne_nil_of_length_pos List.ne_nil_of_length_pos #align list.length_pos_of_ne_nil List.length_pos_of_ne_nil theorem length_pos_iff_ne_nil {l : List α} : 0 < length l ↔ l ≠ [] := ⟨ne_nil_of_length_pos, length_pos_of_ne_nil⟩ #align list.length_pos_iff_ne_nil List.length_pos_iff_ne_nil #align list.exists_mem_of_ne_nil List.exists_mem_of_ne_nil #align list.length_eq_one List.length_eq_one theorem exists_of_length_succ {n} : ∀ l : List α, l.length = n + 1 → ∃ h t, l = h :: t \| [], H => absurd H.symm <\| succ_ne_zero n \| h :: t, _ => ⟨h, t, rfl⟩ #align list.exists_of_length_succ List.exists_of_length_succ @[simp] lemma length_injective_iff : Injective (List.length : List α → ℕ) ↔ Subsingleton α := by constructor · intro h; refine ⟨fun x y => ?_⟩; (suffices [x] = [y] by simpa using this); apply h; rfl · intros hα l1 l2 hl induction l1 generalizing l2 <;> cases l2 · rfl · cases hl · cases hl · next ih _ _ => congr · exact Subsingleton.elim _ _ · apply ih; simpa using hl #align list.length_injective_iff List.length_injective_iff @[simp default+1] -- Porting note: this used to be just @[simp] lemma length_injective [Subsingleton α] : Injective (length : List α → ℕ) := length_injective_iff.mpr inferInstance #align list.length_injective List.length_injective theorem length_eq_two {l : List α} : l.length = 2 ↔ ∃ a b, l = [a, b] := ⟨fun _ => let [a, b] := l; ⟨a, b, rfl⟩, fun ⟨_, _, e⟩ => e ▸ rfl⟩ #align list.length_eq_two List.length_eq_two theorem length_eq_three {l : List α} : l.length = 3 ↔ ∃ a b c, l = [a, b, c] := ⟨fun _ => let [a, b, c] := l; ⟨a, b, c, rfl⟩, fun ⟨_, _, _, e⟩ => e ▸ rfl⟩ #align list.length_eq_three List.length_eq_three #align list.sublist.length_le List.Sublist.length_le /-! ### set-theoretic notation of lists -/ -- ADHOC Porting note: instance from Lean3 core instance instSingletonList : Singleton α (List α) := ⟨fun x => [x]⟩ #align list.has_singleton List.instSingletonList -- ADHOC Porting note: instance from Lean3 core instance [DecidableEq α] : Insert α (List α) := ⟨List.insert⟩ -- ADHOC Porting note: instance from Lean3 core instance [DecidableEq α] : LawfulSingleton α (List α) := { insert_emptyc_eq := fun x => show (if x ∈ ([] : List α) then [] else [x]) = [x] from if_neg (not_mem_nil _) } #align list.empty_eq List.empty_eq theorem singleton_eq (x : α) : ({x} : List α) = [x] := rfl #align list.singleton_eq List.singleton_eq theorem insert_neg [DecidableEq α] {x : α} {l : List α} (h : x ∉ l) : Insert.insert x l = x :: l := insert_of_not_mem h #align list.insert_neg List.insert_neg theorem insert_pos [DecidableEq α] {x : α} {l : List α} (h : x ∈ l) : Insert.insert x l = l := insert_of_mem h #align list.insert_pos List.insert_pos theorem doubleton_eq [DecidableEq α] {x y : α} (h : x ≠ y) : ({x, y} : List α) = [x, y] := by rw [insert_neg, singleton_eq] rwa [singleton_eq, mem_singleton] #align list.doubleton_eq List.doubleton_eq /-! ### bounded quantifiers over lists -/ #align list.forall_mem_nil List.forall_mem_nil #align list.forall_mem_cons List.forall_mem_cons theorem forall_mem_of_forall_mem_cons {p : α → Prop} {a : α} {l : List α} (h : ∀ x ∈ a :: l, p x) : ∀ x ∈ l, p x := (forall_mem_cons.1 h).2 #align list.forall_mem_of_forall_mem_cons List.forall_mem_of_forall_mem_cons #align list.forall_mem_singleton List.forall_mem_singleton #align list.forall_mem_append List.forall_mem_append #align list.not_exists_mem_nil List.not_exists_mem_nilₓ -- bExists change -- Porting note: bExists in Lean3 and And in Lean4 theorem exists_mem_cons_of {p : α → Prop} {a : α} (l : List α) (h : p a) : ∃ x ∈ a :: l, p x := ⟨a, mem_cons_self _ _, h⟩ #align list.exists_mem_cons_of List.exists_mem_cons_ofₓ -- bExists change -- Porting note: bExists in Lean3 and And in Lean4 theorem exists_mem_cons_of_exists {p : α → Prop} {a : α} {l : List α} : (∃ x ∈ l, p x) → ∃ x ∈ a :: l, p x := fun ⟨x, xl, px⟩ => ⟨x, mem_cons_of_mem _ xl, px⟩ #align list.exists_mem_cons_of_exists List.exists_mem_cons_of_existsₓ -- bExists change -- Porting note: bExists in Lean3 and And in Lean4 theorem or_exists_of_exists_mem_cons {p : α → Prop} {a : α} {l : List α} : (∃ x ∈ a :: l, p x) → p a ∨ ∃ x ∈ l, p x := fun ⟨x, xal, px⟩ => Or.elim (eq_or_mem_of_mem_cons xal) (fun h : x = a => by rw [← h]; left; exact px) fun h : x ∈ l => Or.inr ⟨x, h, px⟩ #align list.or_exists_of_exists_mem_cons List.or_exists_of_exists_mem_consₓ -- bExists change theorem exists_mem_cons_iff (p : α → Prop) (a : α) (l : List α) : (∃ x ∈ a :: l, p x) ↔ p a ∨ ∃ x ∈ l, p x := Iff.intro or_exists_of_exists_mem_cons fun h => Or.elim h (exists_mem_cons_of l) exists_mem_cons_of_exists #align list.exists_mem_cons_iff List.exists_mem_cons_iff /-! ### list subset -/ instance : IsTrans (List α) Subset where trans := fun _ _ _ => List.Subset.trans #align list.subset_def List.subset_def #align list.subset_append_of_subset_left List.subset_append_of_subset_left #align list.subset_append_of_subset_right List.subset_append_of_subset_right #align list.cons_subset List.cons_subset theorem cons_subset_of_subset_of_mem {a : α} {l m : List α} (ainm : a ∈ m) (lsubm : l ⊆ m) : a::l ⊆ m := cons_subset.2 ⟨ainm, lsubm⟩ #align list.cons_subset_of_subset_of_mem List.cons_subset_of_subset_of_mem theorem append_subset_of_subset_of_subset {l₁ l₂ l : List α} (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) : l₁ ++ l₂ ⊆ l := fun _ h ↦ (mem_append.1 h).elim (@l₁subl _) (@l₂subl _) #align list.append_subset_of_subset_of_subset List.append_subset_of_subset_of_subset -- Porting note: in Batteries #align list.append_subset_iff List.append_subset alias ⟨eq_nil_of_subset_nil, _⟩ := subset_nil #align list.eq_nil_of_subset_nil List.eq_nil_of_subset_nil #align list.eq_nil_iff_forall_not_mem List.eq_nil_iff_forall_not_mem #align list.map_subset List.map_subset theorem map_subset_iff {l₁ l₂ : List α} (f : α → β) (h : Injective f) : map f l₁ ⊆ map f l₂ ↔ l₁ ⊆ l₂ := by refine ⟨?_, map_subset f⟩; intro h2 x hx rcases mem_map.1 (h2 (mem_map_of_mem f hx)) with ⟨x', hx', hxx'⟩ cases h hxx'; exact hx' #align list.map_subset_iff List.map_subset_iff /-! ### append -/ theorem append_eq_has_append {L₁ L₂ : List α} : List.append L₁ L₂ = L₁ ++ L₂ := rfl #align list.append_eq_has_append List.append_eq_has_append #align list.singleton_append List.singleton_append #align list.append_ne_nil_of_ne_nil_left List.append_ne_nil_of_ne_nil_left #align list.append_ne_nil_of_ne_nil_right List.append_ne_nil_of_ne_nil_right #align list.append_eq_nil List.append_eq_nil -- Porting note: in Batteries #align list.nil_eq_append_iff List.nil_eq_append @[deprecated (since := "2024-03-24")] alias append_eq_cons_iff := append_eq_cons #align list.append_eq_cons_iff List.append_eq_cons @[deprecated (since := "2024-03-24")] alias cons_eq_append_iff := cons_eq_append #align list.cons_eq_append_iff List.cons_eq_append #align list.append_eq_append_iff List.append_eq_append_iff #align list.take_append_drop List.take_append_drop #align list.append_inj List.append_inj #align list.append_inj_right List.append_inj_rightₓ -- implicits order #align list.append_inj_left List.append_inj_leftₓ -- implicits order #align list.append_inj' List.append_inj'ₓ -- implicits order #align list.append_inj_right' List.append_inj_right'ₓ -- implicits order #align list.append_inj_left' List.append_inj_left'ₓ -- implicits order @[deprecated (since := "2024-01-18")] alias append_left_cancel := append_cancel_left #align list.append_left_cancel List.append_cancel_left @[deprecated (since := "2024-01-18")] alias append_right_cancel := append_cancel_right #align list.append_right_cancel List.append_cancel_right @[simp] theorem append_left_eq_self {x y : List α} : x ++ y = y ↔ x = [] := by rw [← append_left_inj (s₁ := x), nil_append] @[simp] theorem self_eq_append_left {x y : List α} : y = x ++ y ↔ x = [] := by rw [eq_comm, append_left_eq_self] @[simp] theorem append_right_eq_self {x y : List α} : x ++ y = x ↔ y = [] := by rw [← append_right_inj (t₁ := y), append_nil] @[simp] theorem self_eq_append_right {x y : List α} : x = x ++ y ↔ y = [] := by rw [eq_comm, append_right_eq_self] theorem append_right_injective (s : List α) : Injective fun t ↦ s ++ t := fun _ _ ↦ append_cancel_left #align list.append_right_injective List.append_right_injective #align list.append_right_inj List.append_right_inj theorem append_left_injective (t : List α) : Injective fun s ↦ s ++ t := fun _ _ ↦ append_cancel_right #align list.append_left_injective List.append_left_injective #align list.append_left_inj List.append_left_inj #align list.map_eq_append_split List.map_eq_append_split /-! ### replicate -/ @[simp] lemma replicate_zero (a : α) : replicate 0 a = [] := rfl #align list.replicate_zero List.replicate_zero attribute [simp] replicate_succ #align list.replicate_succ List.replicate_succ lemma replicate_one (a : α) : replicate 1 a = [a] := rfl #align list.replicate_one List.replicate_one #align list.length_replicate List.length_replicate #align list.mem_replicate List.mem_replicate #align list.eq_of_mem_replicate List.eq_of_mem_replicate theorem eq_replicate_length {a : α} : ∀ {l : List α}, l = replicate l.length a ↔ ∀ b ∈ l, b = a \| [] => by simp \| (b :: l) => by simp [eq_replicate_length] #align list.eq_replicate_length List.eq_replicate_length #align list.eq_replicate_of_mem List.eq_replicate_of_mem #align list.eq_replicate List.eq_replicate theorem replicate_add (m n) (a : α) : replicate (m + n) a = replicate m a ++ replicate n a := by induction m <;> simp [, succ_add, replicate] #align list.replicate_add List.replicate_add theorem replicate_succ' (n) (a : α) : replicate (n + 1) a = replicate n a ++ [a] := replicate_add n 1 a #align list.replicate_succ' List.replicate_succ' theorem replicate_subset_singleton (n) (a : α) : replicate n a ⊆ [a] := fun _ h => mem_singleton.2 (eq_of_mem_replicate h) #align list.replicate_subset_singleton List.replicate_subset_singleton theorem subset_singleton_iff {a : α} {L : List α} : L ⊆ [a] ↔ ∃ n, L = replicate n a := by simp only [eq_replicate, subset_def, mem_singleton, exists_eq_left'] #align list.subset_singleton_iff List.subset_singleton_iff @[simp] theorem map_replicate (f : α → β) (n) (a : α) : map f (replicate n a) = replicate n (f a) := by induction n <;> [rfl; simp only [, replicate, map]] #align list.map_replicate List.map_replicate @[simp] theorem tail_replicate (a : α) (n) : tail (replicate n a) = replicate (n - 1) a := by cases n <;> rfl #align list.tail_replicate List.tail_replicate @[simp] theorem join_replicate_nil (n : ℕ) : join (replicate n []) = @nil α := by induction n <;> [rfl; simp only [, replicate, join, append_nil]] #align list.join_replicate_nil List.join_replicate_nil theorem replicate_right_injective {n : ℕ} (hn : n ≠ 0) : Injective (@replicate α n) := fun _ _ h => (eq_replicate.1 h).2 _ <\| mem_replicate.2 ⟨hn, rfl⟩ #align list.replicate_right_injective List.replicate_right_injective theorem replicate_right_inj {a b : α} {n : ℕ} (hn : n ≠ 0) : replicate n a = replicate n b ↔ a = b := (replicate_right_injective hn).eq_iff #align list.replicate_right_inj List.replicate_right_inj @[simp] theorem replicate_right_inj' {a b : α} : ∀ {n}, replicate n a = replicate n b ↔ n = 0 ∨ a = b \| 0 => by simp \| n + 1 => (replicate_right_inj n.succ_ne_zero).trans <\| by simp only [n.succ_ne_zero, false_or] #align list.replicate_right_inj' List.replicate_right_inj' theorem replicate_left_injective (a : α) : Injective (replicate · a) := LeftInverse.injective (length_replicate · a) #align list.replicate_left_injective List.replicate_left_injective @[simp] theorem replicate_left_inj {a : α} {n m : ℕ} : replicate n a = replicate m a ↔ n = m := (replicate_left_injective a).eq_iff #align list.replicate_left_inj List.replicate_left_inj @[simp] theorem head_replicate (n : ℕ) (a : α) (h) : head (replicate n a) h = a := by cases n <;> simp at h ⊢ /-! ### pure -/ theorem mem_pure (x y : α) : x ∈ (pure y : List α) ↔ x = y := by simp #align list.mem_pure List.mem_pure /-! ### bind -/ @[simp] theorem bind_eq_bind {α β} (f : α → List β) (l : List α) : l >>= f = l.bind f := rfl #align list.bind_eq_bind List.bind_eq_bind #align list.bind_append List.append_bind /-! ### concat -/ #align list.concat_nil List.concat_nil #align list.concat_cons List.concat_cons #align list.concat_eq_append List.concat_eq_append #align list.init_eq_of_concat_eq List.init_eq_of_concat_eq #align list.last_eq_of_concat_eq List.last_eq_of_concat_eq #align list.concat_ne_nil List.concat_ne_nil #align list.concat_append List.concat_append #align list.length_concat List.length_concat #align list.append_concat List.append_concat /-! ### reverse -/ #align list.reverse_nil List.reverse_nil #align list.reverse_core List.reverseAux -- Porting note: Do we need this? attribute [local simp] reverseAux #align list.reverse_cons List.reverse_cons #align list.reverse_core_eq List.reverseAux_eq theorem reverse_cons' (a : α) (l : List α) : reverse (a :: l) = concat (reverse l) a := by simp only [reverse_cons, concat_eq_append] #align list.reverse_cons' List.reverse_cons' theorem reverse_concat' (l : List α) (a : α) : (l ++ [a]).reverse = a :: l.reverse := by rw [reverse_append]; rfl -- Porting note (#10618): simp can prove this -- @[simp] theorem reverse_singleton (a : α) : reverse [a] = [a] := rfl #align list.reverse_singleton List.reverse_singleton #align list.reverse_append List.reverse_append #align list.reverse_concat List.reverse_concat #align list.reverse_reverse List.reverse_reverse @[simp] theorem reverse_involutive : Involutive (@reverse α) := reverse_reverse #align list.reverse_involutive List.reverse_involutive @[simp] theorem reverse_injective : Injective (@reverse α) := reverse_involutive.injective #align list.reverse_injective List.reverse_injective theorem reverse_surjective : Surjective (@reverse α) := reverse_involutive.surjective #align list.reverse_surjective List.reverse_surjective theorem reverse_bijective : Bijective (@reverse α) := reverse_involutive.bijective #align list.reverse_bijective List.reverse_bijective @[simp] theorem reverse_inj {l₁ l₂ : List α} : reverse l₁ = reverse l₂ ↔ l₁ = l₂ := reverse_injective.eq_iff #align list.reverse_inj List.reverse_inj theorem reverse_eq_iff {l l' : List α} : l.reverse = l' ↔ l = l'.reverse := reverse_involutive.eq_iff #align list.reverse_eq_iff List.reverse_eq_iff #align list.reverse_eq_nil List.reverse_eq_nil_iff theorem concat_eq_reverse_cons (a : α) (l : List α) : concat l a = reverse (a :: reverse l) := by simp only [concat_eq_append, reverse_cons, reverse_reverse] #align list.concat_eq_reverse_cons List.concat_eq_reverse_cons #align list.length_reverse List.length_reverse -- Porting note: This one was @[simp] in mathlib 3, -- but Lean contains a competing simp lemma reverse_map. -- For now we remove @[simp] to avoid simplification loops. -- TODO: Change Lean lemma to match mathlib 3? theorem map_reverse (f : α → β) (l : List α) : map f (reverse l) = reverse (map f l) := (reverse_map f l).symm #align list.map_reverse List.map_reverse theorem map_reverseAux (f : α → β) (l₁ l₂ : List α) : map f (reverseAux l₁ l₂) = reverseAux (map f l₁) (map f l₂) := by simp only [reverseAux_eq, map_append, map_reverse] #align list.map_reverse_core List.map_reverseAux #align list.mem_reverse List.mem_reverse @[simp] theorem reverse_replicate (n) (a : α) : reverse (replicate n a) = replicate n a := eq_replicate.2 ⟨by rw [length_reverse, length_replicate], fun b h => eq_of_mem_replicate (mem_reverse.1 h)⟩ #align list.reverse_replicate List.reverse_replicate /-! ### empty -/ -- Porting note: this does not work as desired -- attribute [simp] List.isEmpty theorem isEmpty_iff_eq_nil {l : List α} : l.isEmpty ↔ l = [] := by cases l <;> simp [isEmpty] #align list.empty_iff_eq_nil List.isEmpty_iff_eq_nil /-! ### dropLast -/ #align list.length_init List.length_dropLast /-! ### getLast -/ @[simp] theorem getLast_cons {a : α} {l : List α} : ∀ h : l ≠ nil, getLast (a :: l) (cons_ne_nil a l) = getLast l h := by induction l <;> intros · contradiction · rfl #align list.last_cons List.getLast_cons theorem getLast_append_singleton {a : α} (l : List α) : getLast (l ++ [a]) (append_ne_nil_of_ne_nil_right l _ (cons_ne_nil a _)) = a := by simp only [getLast_append] #align list.last_append_singleton List.getLast_append_singleton -- Porting note: name should be fixed upstream theorem getLast_append' (l₁ l₂ : List α) (h : l₂ ≠ []) : getLast (l₁ ++ l₂) (append_ne_nil_of_ne_nil_right l₁ l₂ h) = getLast l₂ h := by induction' l₁ with _ _ ih · simp · simp only [cons_append] rw [List.getLast_cons] exact ih #align list.last_append List.getLast_append' theorem getLast_concat' {a : α} (l : List α) : getLast (concat l a) (concat_ne_nil a l) = a := getLast_concat .. #align list.last_concat List.getLast_concat' @[simp] theorem getLast_singleton' (a : α) : getLast [a] (cons_ne_nil a []) = a := rfl #align list.last_singleton List.getLast_singleton' -- Porting note (#10618): simp can prove this -- @[simp] theorem getLast_cons_cons (a₁ a₂ : α) (l : List α) : getLast (a₁ :: a₂ :: l) (cons_ne_nil _ _) = getLast (a₂ :: l) (cons_ne_nil a₂ l) := rfl #align list.last_cons_cons List.getLast_cons_cons theorem dropLast_append_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l ++ [getLast l h] = l \| [], h => absurd rfl h \| [a], h => rfl \| a :: b :: l, h => by rw [dropLast_cons₂, cons_append, getLast_cons (cons_ne_nil _ _)] congr exact dropLast_append_getLast (cons_ne_nil b l) #align list.init_append_last List.dropLast_append_getLast	Mathlib/Data/List/Basic.lean	662	663	theorem getLast_congr {l₁ l₂ : List α} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : getLast l₁ h₁ = getLast l₂ h₂ := by	subst l₁; rfl
/- Copyright (c) 2019 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.FDeriv.Linear import Mathlib.Analysis.Calculus.FDeriv.Comp #align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee" /-! # Additive operations on derivatives For detailed documentation of the Fréchet derivative, see the module docstring of `Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of * sum of finitely many functions * multiplication of a function by a scalar constant * negative of a function * subtraction of two functions -/ open Filter Asymptotics ContinuousLinearMap Set Metric open scoped Classical open Topology NNReal Filter Asymptotics ENNReal noncomputable section section variable {𝕜 : Type} [NontriviallyNormedField 𝕜] variable {E : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E] variable {F : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F] variable {G : Type} [NormedAddCommGroup G] [NormedSpace 𝕜 G] variable {G' : Type} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] variable {f f₀ f₁ g : E → F} variable {f' f₀' f₁' g' : E →L[𝕜] F} variable (e : E →L[𝕜] F) variable {x : E} variable {s t : Set E} variable {L L₁ L₂ : Filter E} section ConstSMul variable {R : Type} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] /-! ### Derivative of a function multiplied by a constant -/ @[fun_prop] theorem HasStrictFDerivAt.const_smul (h : HasStrictFDerivAt f f' x) (c : R) : HasStrictFDerivAt (fun x => c • f x) (c • f') x := (c • (1 : F →L[𝕜] F)).hasStrictFDerivAt.comp x h #align has_strict_fderiv_at.const_smul HasStrictFDerivAt.const_smul theorem HasFDerivAtFilter.const_smul (h : HasFDerivAtFilter f f' x L) (c : R) : HasFDerivAtFilter (fun x => c • f x) (c • f') x L := (c • (1 : F →L[𝕜] F)).hasFDerivAtFilter.comp x h tendsto_map #align has_fderiv_at_filter.const_smul HasFDerivAtFilter.const_smul @[fun_prop] nonrec theorem HasFDerivWithinAt.const_smul (h : HasFDerivWithinAt f f' s x) (c : R) : HasFDerivWithinAt (fun x => c • f x) (c • f') s x := h.const_smul c #align has_fderiv_within_at.const_smul HasFDerivWithinAt.const_smul @[fun_prop] nonrec theorem HasFDerivAt.const_smul (h : HasFDerivAt f f' x) (c : R) : HasFDerivAt (fun x => c • f x) (c • f') x := h.const_smul c #align has_fderiv_at.const_smul HasFDerivAt.const_smul @[fun_prop] theorem DifferentiableWithinAt.const_smul (h : DifferentiableWithinAt 𝕜 f s x) (c : R) : DifferentiableWithinAt 𝕜 (fun y => c • f y) s x := (h.hasFDerivWithinAt.const_smul c).differentiableWithinAt #align differentiable_within_at.const_smul DifferentiableWithinAt.const_smul @[fun_prop] theorem DifferentiableAt.const_smul (h : DifferentiableAt 𝕜 f x) (c : R) : DifferentiableAt 𝕜 (fun y => c • f y) x := (h.hasFDerivAt.const_smul c).differentiableAt #align differentiable_at.const_smul DifferentiableAt.const_smul @[fun_prop] theorem DifferentiableOn.const_smul (h : DifferentiableOn 𝕜 f s) (c : R) : DifferentiableOn 𝕜 (fun y => c • f y) s := fun x hx => (h x hx).const_smul c #align differentiable_on.const_smul DifferentiableOn.const_smul @[fun_prop] theorem Differentiable.const_smul (h : Differentiable 𝕜 f) (c : R) : Differentiable 𝕜 fun y => c • f y := fun x => (h x).const_smul c #align differentiable.const_smul Differentiable.const_smul theorem fderivWithin_const_smul (hxs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f s x) (c : R) : fderivWithin 𝕜 (fun y => c • f y) s x = c • fderivWithin 𝕜 f s x := (h.hasFDerivWithinAt.const_smul c).fderivWithin hxs #align fderiv_within_const_smul fderivWithin_const_smul theorem fderiv_const_smul (h : DifferentiableAt 𝕜 f x) (c : R) : fderiv 𝕜 (fun y => c • f y) x = c • fderiv 𝕜 f x := (h.hasFDerivAt.const_smul c).fderiv #align fderiv_const_smul fderiv_const_smul end ConstSMul section Add /-! ### Derivative of the sum of two functions -/ @[fun_prop] nonrec theorem HasStrictFDerivAt.add (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) : HasStrictFDerivAt (fun y => f y + g y) (f' + g') x := (hf.add hg).congr_left fun y => by simp only [LinearMap.sub_apply, LinearMap.add_apply, map_sub, map_add, add_apply] abel #align has_strict_fderiv_at.add HasStrictFDerivAt.add theorem HasFDerivAtFilter.add (hf : HasFDerivAtFilter f f' x L) (hg : HasFDerivAtFilter g g' x L) : HasFDerivAtFilter (fun y => f y + g y) (f' + g') x L := .of_isLittleO <\| (hf.isLittleO.add hg.isLittleO).congr_left fun _ => by simp only [LinearMap.sub_apply, LinearMap.add_apply, map_sub, map_add, add_apply] abel #align has_fderiv_at_filter.add HasFDerivAtFilter.add @[fun_prop] nonrec theorem HasFDerivWithinAt.add (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x) : HasFDerivWithinAt (fun y => f y + g y) (f' + g') s x := hf.add hg #align has_fderiv_within_at.add HasFDerivWithinAt.add @[fun_prop] nonrec theorem HasFDerivAt.add (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) : HasFDerivAt (fun x => f x + g x) (f' + g') x := hf.add hg #align has_fderiv_at.add HasFDerivAt.add @[fun_prop] theorem DifferentiableWithinAt.add (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) : DifferentiableWithinAt 𝕜 (fun y => f y + g y) s x := (hf.hasFDerivWithinAt.add hg.hasFDerivWithinAt).differentiableWithinAt #align differentiable_within_at.add DifferentiableWithinAt.add @[simp, fun_prop] theorem DifferentiableAt.add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : DifferentiableAt 𝕜 (fun y => f y + g y) x := (hf.hasFDerivAt.add hg.hasFDerivAt).differentiableAt #align differentiable_at.add DifferentiableAt.add @[fun_prop] theorem DifferentiableOn.add (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) : DifferentiableOn 𝕜 (fun y => f y + g y) s := fun x hx => (hf x hx).add (hg x hx) #align differentiable_on.add DifferentiableOn.add @[simp, fun_prop] theorem Differentiable.add (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) : Differentiable 𝕜 fun y => f y + g y := fun x => (hf x).add (hg x) #align differentiable.add Differentiable.add theorem fderivWithin_add (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) : fderivWithin 𝕜 (fun y => f y + g y) s x = fderivWithin 𝕜 f s x + fderivWithin 𝕜 g s x := (hf.hasFDerivWithinAt.add hg.hasFDerivWithinAt).fderivWithin hxs #align fderiv_within_add fderivWithin_add theorem fderiv_add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : fderiv 𝕜 (fun y => f y + g y) x = fderiv 𝕜 f x + fderiv 𝕜 g x := (hf.hasFDerivAt.add hg.hasFDerivAt).fderiv #align fderiv_add fderiv_add @[fun_prop] theorem HasStrictFDerivAt.add_const (hf : HasStrictFDerivAt f f' x) (c : F) : HasStrictFDerivAt (fun y => f y + c) f' x := add_zero f' ▸ hf.add (hasStrictFDerivAt_const _ _) #align has_strict_fderiv_at.add_const HasStrictFDerivAt.add_const theorem HasFDerivAtFilter.add_const (hf : HasFDerivAtFilter f f' x L) (c : F) : HasFDerivAtFilter (fun y => f y + c) f' x L := add_zero f' ▸ hf.add (hasFDerivAtFilter_const _ _ _) #align has_fderiv_at_filter.add_const HasFDerivAtFilter.add_const @[fun_prop] nonrec theorem HasFDerivWithinAt.add_const (hf : HasFDerivWithinAt f f' s x) (c : F) : HasFDerivWithinAt (fun y => f y + c) f' s x := hf.add_const c #align has_fderiv_within_at.add_const HasFDerivWithinAt.add_const @[fun_prop] nonrec theorem HasFDerivAt.add_const (hf : HasFDerivAt f f' x) (c : F) : HasFDerivAt (fun x => f x + c) f' x := hf.add_const c #align has_fderiv_at.add_const HasFDerivAt.add_const @[fun_prop] theorem DifferentiableWithinAt.add_const (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) : DifferentiableWithinAt 𝕜 (fun y => f y + c) s x := (hf.hasFDerivWithinAt.add_const c).differentiableWithinAt #align differentiable_within_at.add_const DifferentiableWithinAt.add_const @[simp] theorem differentiableWithinAt_add_const_iff (c : F) : DifferentiableWithinAt 𝕜 (fun y => f y + c) s x ↔ DifferentiableWithinAt 𝕜 f s x := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_within_at_add_const_iff differentiableWithinAt_add_const_iff @[fun_prop] theorem DifferentiableAt.add_const (hf : DifferentiableAt 𝕜 f x) (c : F) : DifferentiableAt 𝕜 (fun y => f y + c) x := (hf.hasFDerivAt.add_const c).differentiableAt #align differentiable_at.add_const DifferentiableAt.add_const @[simp] theorem differentiableAt_add_const_iff (c : F) : DifferentiableAt 𝕜 (fun y => f y + c) x ↔ DifferentiableAt 𝕜 f x := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_at_add_const_iff differentiableAt_add_const_iff @[fun_prop] theorem DifferentiableOn.add_const (hf : DifferentiableOn 𝕜 f s) (c : F) : DifferentiableOn 𝕜 (fun y => f y + c) s := fun x hx => (hf x hx).add_const c #align differentiable_on.add_const DifferentiableOn.add_const @[simp] theorem differentiableOn_add_const_iff (c : F) : DifferentiableOn 𝕜 (fun y => f y + c) s ↔ DifferentiableOn 𝕜 f s := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_on_add_const_iff differentiableOn_add_const_iff @[fun_prop] theorem Differentiable.add_const (hf : Differentiable 𝕜 f) (c : F) : Differentiable 𝕜 fun y => f y + c := fun x => (hf x).add_const c #align differentiable.add_const Differentiable.add_const @[simp] theorem differentiable_add_const_iff (c : F) : (Differentiable 𝕜 fun y => f y + c) ↔ Differentiable 𝕜 f := ⟨fun h => by simpa using h.add_const (-c), fun h => h.add_const c⟩ #align differentiable_add_const_iff differentiable_add_const_iff theorem fderivWithin_add_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) : fderivWithin 𝕜 (fun y => f y + c) s x = fderivWithin 𝕜 f s x := if hf : DifferentiableWithinAt 𝕜 f s x then (hf.hasFDerivWithinAt.add_const c).fderivWithin hxs else by rw [fderivWithin_zero_of_not_differentiableWithinAt hf, fderivWithin_zero_of_not_differentiableWithinAt] simpa #align fderiv_within_add_const fderivWithin_add_const theorem fderiv_add_const (c : F) : fderiv 𝕜 (fun y => f y + c) x = fderiv 𝕜 f x := by simp only [← fderivWithin_univ, fderivWithin_add_const uniqueDiffWithinAt_univ] #align fderiv_add_const fderiv_add_const @[fun_prop] theorem HasStrictFDerivAt.const_add (hf : HasStrictFDerivAt f f' x) (c : F) : HasStrictFDerivAt (fun y => c + f y) f' x := zero_add f' ▸ (hasStrictFDerivAt_const _ _).add hf #align has_strict_fderiv_at.const_add HasStrictFDerivAt.const_add theorem HasFDerivAtFilter.const_add (hf : HasFDerivAtFilter f f' x L) (c : F) : HasFDerivAtFilter (fun y => c + f y) f' x L := zero_add f' ▸ (hasFDerivAtFilter_const _ _ _).add hf #align has_fderiv_at_filter.const_add HasFDerivAtFilter.const_add @[fun_prop] nonrec theorem HasFDerivWithinAt.const_add (hf : HasFDerivWithinAt f f' s x) (c : F) : HasFDerivWithinAt (fun y => c + f y) f' s x := hf.const_add c #align has_fderiv_within_at.const_add HasFDerivWithinAt.const_add @[fun_prop] nonrec theorem HasFDerivAt.const_add (hf : HasFDerivAt f f' x) (c : F) : HasFDerivAt (fun x => c + f x) f' x := hf.const_add c #align has_fderiv_at.const_add HasFDerivAt.const_add @[fun_prop] theorem DifferentiableWithinAt.const_add (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) : DifferentiableWithinAt 𝕜 (fun y => c + f y) s x := (hf.hasFDerivWithinAt.const_add c).differentiableWithinAt #align differentiable_within_at.const_add DifferentiableWithinAt.const_add @[simp] theorem differentiableWithinAt_const_add_iff (c : F) : DifferentiableWithinAt 𝕜 (fun y => c + f y) s x ↔ DifferentiableWithinAt 𝕜 f s x := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_within_at_const_add_iff differentiableWithinAt_const_add_iff @[fun_prop] theorem DifferentiableAt.const_add (hf : DifferentiableAt 𝕜 f x) (c : F) : DifferentiableAt 𝕜 (fun y => c + f y) x := (hf.hasFDerivAt.const_add c).differentiableAt #align differentiable_at.const_add DifferentiableAt.const_add @[simp] theorem differentiableAt_const_add_iff (c : F) : DifferentiableAt 𝕜 (fun y => c + f y) x ↔ DifferentiableAt 𝕜 f x := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_at_const_add_iff differentiableAt_const_add_iff @[fun_prop] theorem DifferentiableOn.const_add (hf : DifferentiableOn 𝕜 f s) (c : F) : DifferentiableOn 𝕜 (fun y => c + f y) s := fun x hx => (hf x hx).const_add c #align differentiable_on.const_add DifferentiableOn.const_add @[simp] theorem differentiableOn_const_add_iff (c : F) : DifferentiableOn 𝕜 (fun y => c + f y) s ↔ DifferentiableOn 𝕜 f s := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_on_const_add_iff differentiableOn_const_add_iff @[fun_prop] theorem Differentiable.const_add (hf : Differentiable 𝕜 f) (c : F) : Differentiable 𝕜 fun y => c + f y := fun x => (hf x).const_add c #align differentiable.const_add Differentiable.const_add @[simp] theorem differentiable_const_add_iff (c : F) : (Differentiable 𝕜 fun y => c + f y) ↔ Differentiable 𝕜 f := ⟨fun h => by simpa using h.const_add (-c), fun h => h.const_add c⟩ #align differentiable_const_add_iff differentiable_const_add_iff theorem fderivWithin_const_add (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) : fderivWithin 𝕜 (fun y => c + f y) s x = fderivWithin 𝕜 f s x := by simpa only [add_comm] using fderivWithin_add_const hxs c #align fderiv_within_const_add fderivWithin_const_add theorem fderiv_const_add (c : F) : fderiv 𝕜 (fun y => c + f y) x = fderiv 𝕜 f x := by simp only [add_comm c, fderiv_add_const] #align fderiv_const_add fderiv_const_add end Add section Sum /-! ### Derivative of a finite sum of functions -/ variable {ι : Type} {u : Finset ι} {A : ι → E → F} {A' : ι → E →L[𝕜] F} @[fun_prop] theorem HasStrictFDerivAt.sum (h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x) : HasStrictFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x := by dsimp [HasStrictFDerivAt] at convert IsLittleO.sum h simp [Finset.sum_sub_distrib, ContinuousLinearMap.sum_apply] #align has_strict_fderiv_at.sum HasStrictFDerivAt.sum theorem HasFDerivAtFilter.sum (h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) x L) : HasFDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L := by simp only [hasFDerivAtFilter_iff_isLittleO] at * convert IsLittleO.sum h simp [ContinuousLinearMap.sum_apply] #align has_fderiv_at_filter.sum HasFDerivAtFilter.sum @[fun_prop] theorem HasFDerivWithinAt.sum (h : ∀ i ∈ u, HasFDerivWithinAt (A i) (A' i) s x) : HasFDerivWithinAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) s x := HasFDerivAtFilter.sum h #align has_fderiv_within_at.sum HasFDerivWithinAt.sum @[fun_prop] theorem HasFDerivAt.sum (h : ∀ i ∈ u, HasFDerivAt (A i) (A' i) x) : HasFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x := HasFDerivAtFilter.sum h #align has_fderiv_at.sum HasFDerivAt.sum @[fun_prop] theorem DifferentiableWithinAt.sum (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x) : DifferentiableWithinAt 𝕜 (fun y => ∑ i ∈ u, A i y) s x := HasFDerivWithinAt.differentiableWithinAt <\| HasFDerivWithinAt.sum fun i hi => (h i hi).hasFDerivWithinAt #align differentiable_within_at.sum DifferentiableWithinAt.sum @[simp, fun_prop] theorem DifferentiableAt.sum (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x) : DifferentiableAt 𝕜 (fun y => ∑ i ∈ u, A i y) x := HasFDerivAt.differentiableAt <\| HasFDerivAt.sum fun i hi => (h i hi).hasFDerivAt #align differentiable_at.sum DifferentiableAt.sum @[fun_prop] theorem DifferentiableOn.sum (h : ∀ i ∈ u, DifferentiableOn 𝕜 (A i) s) : DifferentiableOn 𝕜 (fun y => ∑ i ∈ u, A i y) s := fun x hx => DifferentiableWithinAt.sum fun i hi => h i hi x hx #align differentiable_on.sum DifferentiableOn.sum @[simp, fun_prop] theorem Differentiable.sum (h : ∀ i ∈ u, Differentiable 𝕜 (A i)) : Differentiable 𝕜 fun y => ∑ i ∈ u, A i y := fun x => DifferentiableAt.sum fun i hi => h i hi x #align differentiable.sum Differentiable.sum theorem fderivWithin_sum (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x) : fderivWithin 𝕜 (fun y => ∑ i ∈ u, A i y) s x = ∑ i ∈ u, fderivWithin 𝕜 (A i) s x := (HasFDerivWithinAt.sum fun i hi => (h i hi).hasFDerivWithinAt).fderivWithin hxs #align fderiv_within_sum fderivWithin_sum theorem fderiv_sum (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x) : fderiv 𝕜 (fun y => ∑ i ∈ u, A i y) x = ∑ i ∈ u, fderiv 𝕜 (A i) x := (HasFDerivAt.sum fun i hi => (h i hi).hasFDerivAt).fderiv #align fderiv_sum fderiv_sum end Sum section Neg /-! ### Derivative of the negative of a function -/ @[fun_prop] theorem HasStrictFDerivAt.neg (h : HasStrictFDerivAt f f' x) : HasStrictFDerivAt (fun x => -f x) (-f') x := (-1 : F →L[𝕜] F).hasStrictFDerivAt.comp x h #align has_strict_fderiv_at.neg HasStrictFDerivAt.neg theorem HasFDerivAtFilter.neg (h : HasFDerivAtFilter f f' x L) : HasFDerivAtFilter (fun x => -f x) (-f') x L := (-1 : F →L[𝕜] F).hasFDerivAtFilter.comp x h tendsto_map #align has_fderiv_at_filter.neg HasFDerivAtFilter.neg @[fun_prop] nonrec theorem HasFDerivWithinAt.neg (h : HasFDerivWithinAt f f' s x) : HasFDerivWithinAt (fun x => -f x) (-f') s x := h.neg #align has_fderiv_within_at.neg HasFDerivWithinAt.neg @[fun_prop] nonrec theorem HasFDerivAt.neg (h : HasFDerivAt f f' x) : HasFDerivAt (fun x => -f x) (-f') x := h.neg #align has_fderiv_at.neg HasFDerivAt.neg @[fun_prop] theorem DifferentiableWithinAt.neg (h : DifferentiableWithinAt 𝕜 f s x) : DifferentiableWithinAt 𝕜 (fun y => -f y) s x := h.hasFDerivWithinAt.neg.differentiableWithinAt #align differentiable_within_at.neg DifferentiableWithinAt.neg @[simp] theorem differentiableWithinAt_neg_iff : DifferentiableWithinAt 𝕜 (fun y => -f y) s x ↔ DifferentiableWithinAt 𝕜 f s x := ⟨fun h => by simpa only [neg_neg] using h.neg, fun h => h.neg⟩ #align differentiable_within_at_neg_iff differentiableWithinAt_neg_iff @[fun_prop] theorem DifferentiableAt.neg (h : DifferentiableAt 𝕜 f x) : DifferentiableAt 𝕜 (fun y => -f y) x := h.hasFDerivAt.neg.differentiableAt #align differentiable_at.neg DifferentiableAt.neg @[simp] theorem differentiableAt_neg_iff : DifferentiableAt 𝕜 (fun y => -f y) x ↔ DifferentiableAt 𝕜 f x := ⟨fun h => by simpa only [neg_neg] using h.neg, fun h => h.neg⟩ #align differentiable_at_neg_iff differentiableAt_neg_iff @[fun_prop] theorem DifferentiableOn.neg (h : DifferentiableOn 𝕜 f s) : DifferentiableOn 𝕜 (fun y => -f y) s := fun x hx => (h x hx).neg #align differentiable_on.neg DifferentiableOn.neg @[simp] theorem differentiableOn_neg_iff : DifferentiableOn 𝕜 (fun y => -f y) s ↔ DifferentiableOn 𝕜 f s := ⟨fun h => by simpa only [neg_neg] using h.neg, fun h => h.neg⟩ #align differentiable_on_neg_iff differentiableOn_neg_iff @[fun_prop] theorem Differentiable.neg (h : Differentiable 𝕜 f) : Differentiable 𝕜 fun y => -f y := fun x => (h x).neg #align differentiable.neg Differentiable.neg @[simp] theorem differentiable_neg_iff : (Differentiable 𝕜 fun y => -f y) ↔ Differentiable 𝕜 f := ⟨fun h => by simpa only [neg_neg] using h.neg, fun h => h.neg⟩ #align differentiable_neg_iff differentiable_neg_iff theorem fderivWithin_neg (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 (fun y => -f y) s x = -fderivWithin 𝕜 f s x := if h : DifferentiableWithinAt 𝕜 f s x then h.hasFDerivWithinAt.neg.fderivWithin hxs else by rw [fderivWithin_zero_of_not_differentiableWithinAt h, fderivWithin_zero_of_not_differentiableWithinAt, neg_zero] simpa #align fderiv_within_neg fderivWithin_neg @[simp] theorem fderiv_neg : fderiv 𝕜 (fun y => -f y) x = -fderiv 𝕜 f x := by simp only [← fderivWithin_univ, fderivWithin_neg uniqueDiffWithinAt_univ] #align fderiv_neg fderiv_neg end Neg section Sub /-! ### Derivative of the difference of two functions -/ @[fun_prop] theorem HasStrictFDerivAt.sub (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) : HasStrictFDerivAt (fun x => f x - g x) (f' - g') x := by simpa only [sub_eq_add_neg] using hf.add hg.neg #align has_strict_fderiv_at.sub HasStrictFDerivAt.sub theorem HasFDerivAtFilter.sub (hf : HasFDerivAtFilter f f' x L) (hg : HasFDerivAtFilter g g' x L) : HasFDerivAtFilter (fun x => f x - g x) (f' - g') x L := by simpa only [sub_eq_add_neg] using hf.add hg.neg #align has_fderiv_at_filter.sub HasFDerivAtFilter.sub @[fun_prop] nonrec theorem HasFDerivWithinAt.sub (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x) : HasFDerivWithinAt (fun x => f x - g x) (f' - g') s x := hf.sub hg #align has_fderiv_within_at.sub HasFDerivWithinAt.sub @[fun_prop] nonrec theorem HasFDerivAt.sub (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) : HasFDerivAt (fun x => f x - g x) (f' - g') x := hf.sub hg #align has_fderiv_at.sub HasFDerivAt.sub @[fun_prop] theorem DifferentiableWithinAt.sub (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) : DifferentiableWithinAt 𝕜 (fun y => f y - g y) s x := (hf.hasFDerivWithinAt.sub hg.hasFDerivWithinAt).differentiableWithinAt #align differentiable_within_at.sub DifferentiableWithinAt.sub @[simp, fun_prop] theorem DifferentiableAt.sub (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : DifferentiableAt 𝕜 (fun y => f y - g y) x := (hf.hasFDerivAt.sub hg.hasFDerivAt).differentiableAt #align differentiable_at.sub DifferentiableAt.sub @[fun_prop] theorem DifferentiableOn.sub (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) : DifferentiableOn 𝕜 (fun y => f y - g y) s := fun x hx => (hf x hx).sub (hg x hx) #align differentiable_on.sub DifferentiableOn.sub @[simp, fun_prop] theorem Differentiable.sub (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) : Differentiable 𝕜 fun y => f y - g y := fun x => (hf x).sub (hg x) #align differentiable.sub Differentiable.sub theorem fderivWithin_sub (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) : fderivWithin 𝕜 (fun y => f y - g y) s x = fderivWithin 𝕜 f s x - fderivWithin 𝕜 g s x := (hf.hasFDerivWithinAt.sub hg.hasFDerivWithinAt).fderivWithin hxs #align fderiv_within_sub fderivWithin_sub theorem fderiv_sub (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : fderiv 𝕜 (fun y => f y - g y) x = fderiv 𝕜 f x - fderiv 𝕜 g x := (hf.hasFDerivAt.sub hg.hasFDerivAt).fderiv #align fderiv_sub fderiv_sub @[fun_prop] theorem HasStrictFDerivAt.sub_const (hf : HasStrictFDerivAt f f' x) (c : F) : HasStrictFDerivAt (fun x => f x - c) f' x := by simpa only [sub_eq_add_neg] using hf.add_const (-c) #align has_strict_fderiv_at.sub_const HasStrictFDerivAt.sub_const theorem HasFDerivAtFilter.sub_const (hf : HasFDerivAtFilter f f' x L) (c : F) : HasFDerivAtFilter (fun x => f x - c) f' x L := by simpa only [sub_eq_add_neg] using hf.add_const (-c) #align has_fderiv_at_filter.sub_const HasFDerivAtFilter.sub_const @[fun_prop] nonrec theorem HasFDerivWithinAt.sub_const (hf : HasFDerivWithinAt f f' s x) (c : F) : HasFDerivWithinAt (fun x => f x - c) f' s x := hf.sub_const c #align has_fderiv_within_at.sub_const HasFDerivWithinAt.sub_const @[fun_prop] nonrec theorem HasFDerivAt.sub_const (hf : HasFDerivAt f f' x) (c : F) : HasFDerivAt (fun x => f x - c) f' x := hf.sub_const c #align has_fderiv_at.sub_const HasFDerivAt.sub_const @[fun_prop] theorem hasStrictFDerivAt_sub_const {x : F} (c : F) : HasStrictFDerivAt (· - c) (id 𝕜 F) x := (hasStrictFDerivAt_id x).sub_const c @[fun_prop] theorem hasFDerivAt_sub_const {x : F} (c : F) : HasFDerivAt (· - c) (id 𝕜 F) x := (hasFDerivAt_id x).sub_const c @[fun_prop] theorem DifferentiableWithinAt.sub_const (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) : DifferentiableWithinAt 𝕜 (fun y => f y - c) s x := (hf.hasFDerivWithinAt.sub_const c).differentiableWithinAt #align differentiable_within_at.sub_const DifferentiableWithinAt.sub_const @[simp] theorem differentiableWithinAt_sub_const_iff (c : F) : DifferentiableWithinAt 𝕜 (fun y => f y - c) s x ↔ DifferentiableWithinAt 𝕜 f s x := by simp only [sub_eq_add_neg, differentiableWithinAt_add_const_iff] #align differentiable_within_at_sub_const_iff differentiableWithinAt_sub_const_iff @[fun_prop] theorem DifferentiableAt.sub_const (hf : DifferentiableAt 𝕜 f x) (c : F) : DifferentiableAt 𝕜 (fun y => f y - c) x := (hf.hasFDerivAt.sub_const c).differentiableAt #align differentiable_at.sub_const DifferentiableAt.sub_const @[simp] theorem differentiableAt_sub_const_iff (c : F) : DifferentiableAt 𝕜 (fun y => f y - c) x ↔ DifferentiableAt 𝕜 f x := by simp only [sub_eq_add_neg, differentiableAt_add_const_iff] #align differentiable_at_sub_const_iff differentiableAt_sub_const_iff @[fun_prop] theorem DifferentiableOn.sub_const (hf : DifferentiableOn 𝕜 f s) (c : F) : DifferentiableOn 𝕜 (fun y => f y - c) s := fun x hx => (hf x hx).sub_const c #align differentiable_on.sub_const DifferentiableOn.sub_const @[simp] theorem differentiableOn_sub_const_iff (c : F) : DifferentiableOn 𝕜 (fun y => f y - c) s ↔ DifferentiableOn 𝕜 f s := by simp only [sub_eq_add_neg, differentiableOn_add_const_iff] #align differentiable_on_sub_const_iff differentiableOn_sub_const_iff @[fun_prop] theorem Differentiable.sub_const (hf : Differentiable 𝕜 f) (c : F) : Differentiable 𝕜 fun y => f y - c := fun x => (hf x).sub_const c #align differentiable.sub_const Differentiable.sub_const @[simp] theorem differentiable_sub_const_iff (c : F) : (Differentiable 𝕜 fun y => f y - c) ↔ Differentiable 𝕜 f := by simp only [sub_eq_add_neg, differentiable_add_const_iff] #align differentiable_sub_const_iff differentiable_sub_const_iff theorem fderivWithin_sub_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) : fderivWithin 𝕜 (fun y => f y - c) s x = fderivWithin 𝕜 f s x := by simp only [sub_eq_add_neg, fderivWithin_add_const hxs] #align fderiv_within_sub_const fderivWithin_sub_const theorem fderiv_sub_const (c : F) : fderiv 𝕜 (fun y => f y - c) x = fderiv 𝕜 f x := by simp only [sub_eq_add_neg, fderiv_add_const] #align fderiv_sub_const fderiv_sub_const @[fun_prop] theorem HasStrictFDerivAt.const_sub (hf : HasStrictFDerivAt f f' x) (c : F) : HasStrictFDerivAt (fun x => c - f x) (-f') x := by simpa only [sub_eq_add_neg] using hf.neg.const_add c #align has_strict_fderiv_at.const_sub HasStrictFDerivAt.const_sub theorem HasFDerivAtFilter.const_sub (hf : HasFDerivAtFilter f f' x L) (c : F) : HasFDerivAtFilter (fun x => c - f x) (-f') x L := by simpa only [sub_eq_add_neg] using hf.neg.const_add c #align has_fderiv_at_filter.const_sub HasFDerivAtFilter.const_sub @[fun_prop] nonrec theorem HasFDerivWithinAt.const_sub (hf : HasFDerivWithinAt f f' s x) (c : F) : HasFDerivWithinAt (fun x => c - f x) (-f') s x := hf.const_sub c #align has_fderiv_within_at.const_sub HasFDerivWithinAt.const_sub @[fun_prop] nonrec theorem HasFDerivAt.const_sub (hf : HasFDerivAt f f' x) (c : F) : HasFDerivAt (fun x => c - f x) (-f') x := hf.const_sub c #align has_fderiv_at.const_sub HasFDerivAt.const_sub @[fun_prop] theorem DifferentiableWithinAt.const_sub (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) : DifferentiableWithinAt 𝕜 (fun y => c - f y) s x := (hf.hasFDerivWithinAt.const_sub c).differentiableWithinAt #align differentiable_within_at.const_sub DifferentiableWithinAt.const_sub @[simp] theorem differentiableWithinAt_const_sub_iff (c : F) : DifferentiableWithinAt 𝕜 (fun y => c - f y) s x ↔ DifferentiableWithinAt 𝕜 f s x := by simp [sub_eq_add_neg] #align differentiable_within_at_const_sub_iff differentiableWithinAt_const_sub_iff @[fun_prop] theorem DifferentiableAt.const_sub (hf : DifferentiableAt 𝕜 f x) (c : F) : DifferentiableAt 𝕜 (fun y => c - f y) x := (hf.hasFDerivAt.const_sub c).differentiableAt #align differentiable_at.const_sub DifferentiableAt.const_sub @[simp] theorem differentiableAt_const_sub_iff (c : F) : DifferentiableAt 𝕜 (fun y => c - f y) x ↔ DifferentiableAt 𝕜 f x := by simp [sub_eq_add_neg] #align differentiable_at_const_sub_iff differentiableAt_const_sub_iff @[fun_prop] theorem DifferentiableOn.const_sub (hf : DifferentiableOn 𝕜 f s) (c : F) : DifferentiableOn 𝕜 (fun y => c - f y) s := fun x hx => (hf x hx).const_sub c #align differentiable_on.const_sub DifferentiableOn.const_sub @[simp]	Mathlib/Analysis/Calculus/FDeriv/Add.lean	693	694	theorem differentiableOn_const_sub_iff (c : F) : DifferentiableOn 𝕜 (fun y => c - f y) s ↔ DifferentiableOn 𝕜 f s := by	simp [sub_eq_add_neg]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Scott Morrison -/ import Mathlib.Algebra.BigOperators.Finsupp import Mathlib.Algebra.Module.Basic import Mathlib.Algebra.Regular.SMul import Mathlib.Data.Finset.Preimage import Mathlib.Data.Rat.BigOperators import Mathlib.GroupTheory.GroupAction.Hom import Mathlib.Data.Set.Subsingleton #align_import data.finsupp.basic from "leanprover-community/mathlib"@"f69db8cecc668e2d5894d7e9bfc491da60db3b9f" /-! # Miscellaneous definitions, lemmas, and constructions using finsupp ## Main declarations * `Finsupp.graph`: the finset of input and output pairs with non-zero outputs. * `Finsupp.mapRange.equiv`: `Finsupp.mapRange` as an equiv. * `Finsupp.mapDomain`: maps the domain of a `Finsupp` by a function and by summing. * `Finsupp.comapDomain`: postcomposition of a `Finsupp` with a function injective on the preimage of its support. * `Finsupp.some`: restrict a finitely supported function on `Option α` to a finitely supported function on `α`. * `Finsupp.filter`: `filter p f` is the finitely supported function that is `f a` if `p a` is true and 0 otherwise. * `Finsupp.frange`: the image of a finitely supported function on its support. * `Finsupp.subtype_domain`: the restriction of a finitely supported function `f` to a subtype. ## Implementation notes This file is a `noncomputable theory` and uses classical logic throughout. ## TODO * This file is currently ~1600 lines long and is quite a miscellany of definitions and lemmas, so it should be divided into smaller pieces. * Expand the list of definitions and important lemmas to the module docstring. -/ noncomputable section open Finset Function variable {α β γ ι M M' N P G H R S : Type} namespace Finsupp /-! ### Declarations about `graph` -/ section Graph variable [Zero M] /-- The graph of a finitely supported function over its support, i.e. the finset of input and output pairs with non-zero outputs. -/ def graph (f : α →₀ M) : Finset (α × M) := f.support.map ⟨fun a => Prod.mk a (f a), fun _ _ h => (Prod.mk.inj h).1⟩ #align finsupp.graph Finsupp.graph theorem mk_mem_graph_iff {a : α} {m : M} {f : α →₀ M} : (a, m) ∈ f.graph ↔ f a = m ∧ m ≠ 0 := by simp_rw [graph, mem_map, mem_support_iff] constructor · rintro ⟨b, ha, rfl, -⟩ exact ⟨rfl, ha⟩ · rintro ⟨rfl, ha⟩ exact ⟨a, ha, rfl⟩ #align finsupp.mk_mem_graph_iff Finsupp.mk_mem_graph_iff @[simp] theorem mem_graph_iff {c : α × M} {f : α →₀ M} : c ∈ f.graph ↔ f c.1 = c.2 ∧ c.2 ≠ 0 := by cases c exact mk_mem_graph_iff #align finsupp.mem_graph_iff Finsupp.mem_graph_iff theorem mk_mem_graph (f : α →₀ M) {a : α} (ha : a ∈ f.support) : (a, f a) ∈ f.graph := mk_mem_graph_iff.2 ⟨rfl, mem_support_iff.1 ha⟩ #align finsupp.mk_mem_graph Finsupp.mk_mem_graph theorem apply_eq_of_mem_graph {a : α} {m : M} {f : α →₀ M} (h : (a, m) ∈ f.graph) : f a = m := (mem_graph_iff.1 h).1 #align finsupp.apply_eq_of_mem_graph Finsupp.apply_eq_of_mem_graph @[simp 1100] -- Porting note: change priority to appease `simpNF` theorem not_mem_graph_snd_zero (a : α) (f : α →₀ M) : (a, (0 : M)) ∉ f.graph := fun h => (mem_graph_iff.1 h).2.irrefl #align finsupp.not_mem_graph_snd_zero Finsupp.not_mem_graph_snd_zero @[simp] theorem image_fst_graph [DecidableEq α] (f : α →₀ M) : f.graph.image Prod.fst = f.support := by classical simp only [graph, map_eq_image, image_image, Embedding.coeFn_mk, (· ∘ ·), image_id'] #align finsupp.image_fst_graph Finsupp.image_fst_graph theorem graph_injective (α M) [Zero M] : Injective (@graph α M _) := by intro f g h classical have hsup : f.support = g.support := by rw [← image_fst_graph, h, image_fst_graph] refine ext_iff'.2 ⟨hsup, fun x hx => apply_eq_of_mem_graph <\| h.symm ▸ ?_⟩ exact mk_mem_graph _ (hsup ▸ hx) #align finsupp.graph_injective Finsupp.graph_injective @[simp] theorem graph_inj {f g : α →₀ M} : f.graph = g.graph ↔ f = g := (graph_injective α M).eq_iff #align finsupp.graph_inj Finsupp.graph_inj @[simp] theorem graph_zero : graph (0 : α →₀ M) = ∅ := by simp [graph] #align finsupp.graph_zero Finsupp.graph_zero @[simp] theorem graph_eq_empty {f : α →₀ M} : f.graph = ∅ ↔ f = 0 := (graph_injective α M).eq_iff' graph_zero #align finsupp.graph_eq_empty Finsupp.graph_eq_empty end Graph end Finsupp /-! ### Declarations about `mapRange` -/ section MapRange namespace Finsupp section Equiv variable [Zero M] [Zero N] [Zero P] /-- `Finsupp.mapRange` as an equiv. -/ @[simps apply] def mapRange.equiv (f : M ≃ N) (hf : f 0 = 0) (hf' : f.symm 0 = 0) : (α →₀ M) ≃ (α →₀ N) where toFun := (mapRange f hf : (α →₀ M) → α →₀ N) invFun := (mapRange f.symm hf' : (α →₀ N) → α →₀ M) left_inv x := by rw [← mapRange_comp _ _ _ _] <;> simp_rw [Equiv.symm_comp_self] · exact mapRange_id _ · rfl right_inv x := by rw [← mapRange_comp _ _ _ _] <;> simp_rw [Equiv.self_comp_symm] · exact mapRange_id _ · rfl #align finsupp.map_range.equiv Finsupp.mapRange.equiv @[simp] theorem mapRange.equiv_refl : mapRange.equiv (Equiv.refl M) rfl rfl = Equiv.refl (α →₀ M) := Equiv.ext mapRange_id #align finsupp.map_range.equiv_refl Finsupp.mapRange.equiv_refl theorem mapRange.equiv_trans (f : M ≃ N) (hf : f 0 = 0) (hf') (f₂ : N ≃ P) (hf₂ : f₂ 0 = 0) (hf₂') : (mapRange.equiv (f.trans f₂) (by rw [Equiv.trans_apply, hf, hf₂]) (by rw [Equiv.symm_trans_apply, hf₂', hf']) : (α →₀ _) ≃ _) = (mapRange.equiv f hf hf').trans (mapRange.equiv f₂ hf₂ hf₂') := Equiv.ext <\| mapRange_comp f₂ hf₂ f hf ((congrArg f₂ hf).trans hf₂) #align finsupp.map_range.equiv_trans Finsupp.mapRange.equiv_trans @[simp] theorem mapRange.equiv_symm (f : M ≃ N) (hf hf') : ((mapRange.equiv f hf hf').symm : (α →₀ _) ≃ _) = mapRange.equiv f.symm hf' hf := Equiv.ext fun _ => rfl #align finsupp.map_range.equiv_symm Finsupp.mapRange.equiv_symm end Equiv section ZeroHom variable [Zero M] [Zero N] [Zero P] /-- Composition with a fixed zero-preserving homomorphism is itself a zero-preserving homomorphism on functions. -/ @[simps] def mapRange.zeroHom (f : ZeroHom M N) : ZeroHom (α →₀ M) (α →₀ N) where toFun := (mapRange f f.map_zero : (α →₀ M) → α →₀ N) map_zero' := mapRange_zero #align finsupp.map_range.zero_hom Finsupp.mapRange.zeroHom @[simp] theorem mapRange.zeroHom_id : mapRange.zeroHom (ZeroHom.id M) = ZeroHom.id (α →₀ M) := ZeroHom.ext mapRange_id #align finsupp.map_range.zero_hom_id Finsupp.mapRange.zeroHom_id theorem mapRange.zeroHom_comp (f : ZeroHom N P) (f₂ : ZeroHom M N) : (mapRange.zeroHom (f.comp f₂) : ZeroHom (α →₀ _) _) = (mapRange.zeroHom f).comp (mapRange.zeroHom f₂) := ZeroHom.ext <\| mapRange_comp f (map_zero f) f₂ (map_zero f₂) (by simp only [comp_apply, map_zero]) #align finsupp.map_range.zero_hom_comp Finsupp.mapRange.zeroHom_comp end ZeroHom section AddMonoidHom variable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] variable {F : Type} [FunLike F M N] [AddMonoidHomClass F M N] /-- Composition with a fixed additive homomorphism is itself an additive homomorphism on functions. -/ @[simps] def mapRange.addMonoidHom (f : M →+ N) : (α →₀ M) →+ α →₀ N where toFun := (mapRange f f.map_zero : (α →₀ M) → α →₀ N) map_zero' := mapRange_zero map_add' a b := by dsimp only; exact mapRange_add f.map_add _ _; -- Porting note: `dsimp` needed #align finsupp.map_range.add_monoid_hom Finsupp.mapRange.addMonoidHom @[simp] theorem mapRange.addMonoidHom_id : mapRange.addMonoidHom (AddMonoidHom.id M) = AddMonoidHom.id (α →₀ M) := AddMonoidHom.ext mapRange_id #align finsupp.map_range.add_monoid_hom_id Finsupp.mapRange.addMonoidHom_id theorem mapRange.addMonoidHom_comp (f : N →+ P) (f₂ : M →+ N) : (mapRange.addMonoidHom (f.comp f₂) : (α →₀ _) →+ _) = (mapRange.addMonoidHom f).comp (mapRange.addMonoidHom f₂) := AddMonoidHom.ext <\| mapRange_comp f (map_zero f) f₂ (map_zero f₂) (by simp only [comp_apply, map_zero]) #align finsupp.map_range.add_monoid_hom_comp Finsupp.mapRange.addMonoidHom_comp @[simp] theorem mapRange.addMonoidHom_toZeroHom (f : M →+ N) : (mapRange.addMonoidHom f).toZeroHom = (mapRange.zeroHom f.toZeroHom : ZeroHom (α →₀ _) _) := ZeroHom.ext fun _ => rfl #align finsupp.map_range.add_monoid_hom_to_zero_hom Finsupp.mapRange.addMonoidHom_toZeroHom theorem mapRange_multiset_sum (f : F) (m : Multiset (α →₀ M)) : mapRange f (map_zero f) m.sum = (m.map fun x => mapRange f (map_zero f) x).sum := (mapRange.addMonoidHom (f : M →+ N) : (α →₀ _) →+ _).map_multiset_sum _ #align finsupp.map_range_multiset_sum Finsupp.mapRange_multiset_sum theorem mapRange_finset_sum (f : F) (s : Finset ι) (g : ι → α →₀ M) : mapRange f (map_zero f) (∑ x ∈ s, g x) = ∑ x ∈ s, mapRange f (map_zero f) (g x) := map_sum (mapRange.addMonoidHom (f : M →+ N)) _ _ #align finsupp.map_range_finset_sum Finsupp.mapRange_finset_sum /-- `Finsupp.mapRange.AddMonoidHom` as an equiv. -/ @[simps apply] def mapRange.addEquiv (f : M ≃+ N) : (α →₀ M) ≃+ (α →₀ N) := { mapRange.addMonoidHom f.toAddMonoidHom with toFun := (mapRange f f.map_zero : (α →₀ M) → α →₀ N) invFun := (mapRange f.symm f.symm.map_zero : (α →₀ N) → α →₀ M) left_inv := fun x => by rw [← mapRange_comp _ _ _ _] <;> simp_rw [AddEquiv.symm_comp_self] · exact mapRange_id _ · rfl right_inv := fun x => by rw [← mapRange_comp _ _ _ _] <;> simp_rw [AddEquiv.self_comp_symm] · exact mapRange_id _ · rfl } #align finsupp.map_range.add_equiv Finsupp.mapRange.addEquiv @[simp] theorem mapRange.addEquiv_refl : mapRange.addEquiv (AddEquiv.refl M) = AddEquiv.refl (α →₀ M) := AddEquiv.ext mapRange_id #align finsupp.map_range.add_equiv_refl Finsupp.mapRange.addEquiv_refl theorem mapRange.addEquiv_trans (f : M ≃+ N) (f₂ : N ≃+ P) : (mapRange.addEquiv (f.trans f₂) : (α →₀ M) ≃+ (α →₀ P)) = (mapRange.addEquiv f).trans (mapRange.addEquiv f₂) := AddEquiv.ext (mapRange_comp _ f₂.map_zero _ f.map_zero (by simp)) #align finsupp.map_range.add_equiv_trans Finsupp.mapRange.addEquiv_trans @[simp] theorem mapRange.addEquiv_symm (f : M ≃+ N) : ((mapRange.addEquiv f).symm : (α →₀ _) ≃+ _) = mapRange.addEquiv f.symm := AddEquiv.ext fun _ => rfl #align finsupp.map_range.add_equiv_symm Finsupp.mapRange.addEquiv_symm @[simp] theorem mapRange.addEquiv_toAddMonoidHom (f : M ≃+ N) : ((mapRange.addEquiv f : (α →₀ _) ≃+ _) : _ →+ _) = (mapRange.addMonoidHom f.toAddMonoidHom : (α →₀ _) →+ _) := AddMonoidHom.ext fun _ => rfl #align finsupp.map_range.add_equiv_to_add_monoid_hom Finsupp.mapRange.addEquiv_toAddMonoidHom @[simp] theorem mapRange.addEquiv_toEquiv (f : M ≃+ N) : ↑(mapRange.addEquiv f : (α →₀ _) ≃+ _) = (mapRange.equiv (f : M ≃ N) f.map_zero f.symm.map_zero : (α →₀ _) ≃ _) := Equiv.ext fun _ => rfl #align finsupp.map_range.add_equiv_to_equiv Finsupp.mapRange.addEquiv_toEquiv end AddMonoidHom end Finsupp end MapRange /-! ### Declarations about `equivCongrLeft` -/ section EquivCongrLeft variable [Zero M] namespace Finsupp /-- Given `f : α ≃ β`, we can map `l : α →₀ M` to `equivMapDomain f l : β →₀ M` (computably) by mapping the support forwards and the function backwards. -/ def equivMapDomain (f : α ≃ β) (l : α →₀ M) : β →₀ M where support := l.support.map f.toEmbedding toFun a := l (f.symm a) mem_support_toFun a := by simp only [Finset.mem_map_equiv, mem_support_toFun]; rfl #align finsupp.equiv_map_domain Finsupp.equivMapDomain @[simp] theorem equivMapDomain_apply (f : α ≃ β) (l : α →₀ M) (b : β) : equivMapDomain f l b = l (f.symm b) := rfl #align finsupp.equiv_map_domain_apply Finsupp.equivMapDomain_apply theorem equivMapDomain_symm_apply (f : α ≃ β) (l : β →₀ M) (a : α) : equivMapDomain f.symm l a = l (f a) := rfl #align finsupp.equiv_map_domain_symm_apply Finsupp.equivMapDomain_symm_apply @[simp] theorem equivMapDomain_refl (l : α →₀ M) : equivMapDomain (Equiv.refl _) l = l := by ext x; rfl #align finsupp.equiv_map_domain_refl Finsupp.equivMapDomain_refl theorem equivMapDomain_refl' : equivMapDomain (Equiv.refl _) = @id (α →₀ M) := by ext x; rfl #align finsupp.equiv_map_domain_refl' Finsupp.equivMapDomain_refl' theorem equivMapDomain_trans (f : α ≃ β) (g : β ≃ γ) (l : α →₀ M) : equivMapDomain (f.trans g) l = equivMapDomain g (equivMapDomain f l) := by ext x; rfl #align finsupp.equiv_map_domain_trans Finsupp.equivMapDomain_trans theorem equivMapDomain_trans' (f : α ≃ β) (g : β ≃ γ) : @equivMapDomain _ _ M _ (f.trans g) = equivMapDomain g ∘ equivMapDomain f := by ext x; rfl #align finsupp.equiv_map_domain_trans' Finsupp.equivMapDomain_trans' @[simp] theorem equivMapDomain_single (f : α ≃ β) (a : α) (b : M) : equivMapDomain f (single a b) = single (f a) b := by classical ext x simp only [single_apply, Equiv.apply_eq_iff_eq_symm_apply, equivMapDomain_apply] #align finsupp.equiv_map_domain_single Finsupp.equivMapDomain_single @[simp] theorem equivMapDomain_zero {f : α ≃ β} : equivMapDomain f (0 : α →₀ M) = (0 : β →₀ M) := by ext; simp only [equivMapDomain_apply, coe_zero, Pi.zero_apply] #align finsupp.equiv_map_domain_zero Finsupp.equivMapDomain_zero @[to_additive (attr := simp)] theorem prod_equivMapDomain [CommMonoid N] (f : α ≃ β) (l : α →₀ M) (g : β → M → N): prod (equivMapDomain f l) g = prod l (fun a m => g (f a) m) := by simp [prod, equivMapDomain] /-- Given `f : α ≃ β`, the finitely supported function spaces are also in bijection: `(α →₀ M) ≃ (β →₀ M)`. This is the finitely-supported version of `Equiv.piCongrLeft`. -/ def equivCongrLeft (f : α ≃ β) : (α →₀ M) ≃ (β →₀ M) := by refine ⟨equivMapDomain f, equivMapDomain f.symm, fun f => ?_, fun f => ?_⟩ <;> ext x <;> simp only [equivMapDomain_apply, Equiv.symm_symm, Equiv.symm_apply_apply, Equiv.apply_symm_apply] #align finsupp.equiv_congr_left Finsupp.equivCongrLeft @[simp] theorem equivCongrLeft_apply (f : α ≃ β) (l : α →₀ M) : equivCongrLeft f l = equivMapDomain f l := rfl #align finsupp.equiv_congr_left_apply Finsupp.equivCongrLeft_apply @[simp] theorem equivCongrLeft_symm (f : α ≃ β) : (@equivCongrLeft _ _ M _ f).symm = equivCongrLeft f.symm := rfl #align finsupp.equiv_congr_left_symm Finsupp.equivCongrLeft_symm end Finsupp end EquivCongrLeft section CastFinsupp variable [Zero M] (f : α →₀ M) namespace Nat @[simp, norm_cast] theorem cast_finsupp_prod [CommSemiring R] (g : α → M → ℕ) : (↑(f.prod g) : R) = f.prod fun a b => ↑(g a b) := Nat.cast_prod _ _ #align nat.cast_finsupp_prod Nat.cast_finsupp_prod @[simp, norm_cast] theorem cast_finsupp_sum [CommSemiring R] (g : α → M → ℕ) : (↑(f.sum g) : R) = f.sum fun a b => ↑(g a b) := Nat.cast_sum _ _ #align nat.cast_finsupp_sum Nat.cast_finsupp_sum end Nat namespace Int @[simp, norm_cast] theorem cast_finsupp_prod [CommRing R] (g : α → M → ℤ) : (↑(f.prod g) : R) = f.prod fun a b => ↑(g a b) := Int.cast_prod _ _ #align int.cast_finsupp_prod Int.cast_finsupp_prod @[simp, norm_cast] theorem cast_finsupp_sum [CommRing R] (g : α → M → ℤ) : (↑(f.sum g) : R) = f.sum fun a b => ↑(g a b) := Int.cast_sum _ _ #align int.cast_finsupp_sum Int.cast_finsupp_sum end Int namespace Rat @[simp, norm_cast] theorem cast_finsupp_sum [DivisionRing R] [CharZero R] (g : α → M → ℚ) : (↑(f.sum g) : R) = f.sum fun a b => ↑(g a b) := cast_sum _ _ #align rat.cast_finsupp_sum Rat.cast_finsupp_sum @[simp, norm_cast] theorem cast_finsupp_prod [Field R] [CharZero R] (g : α → M → ℚ) : (↑(f.prod g) : R) = f.prod fun a b => ↑(g a b) := cast_prod _ _ #align rat.cast_finsupp_prod Rat.cast_finsupp_prod end Rat end CastFinsupp /-! ### Declarations about `mapDomain` -/ namespace Finsupp section MapDomain variable [AddCommMonoid M] {v v₁ v₂ : α →₀ M} /-- Given `f : α → β` and `v : α →₀ M`, `mapDomain f v : β →₀ M` is the finitely supported function whose value at `a : β` is the sum of `v x` over all `x` such that `f x = a`. -/ def mapDomain (f : α → β) (v : α →₀ M) : β →₀ M := v.sum fun a => single (f a) #align finsupp.map_domain Finsupp.mapDomain theorem mapDomain_apply {f : α → β} (hf : Function.Injective f) (x : α →₀ M) (a : α) : mapDomain f x (f a) = x a := by rw [mapDomain, sum_apply, sum_eq_single a, single_eq_same] · intro b _ hba exact single_eq_of_ne (hf.ne hba) · intro _ rw [single_zero, coe_zero, Pi.zero_apply] #align finsupp.map_domain_apply Finsupp.mapDomain_apply	Mathlib/Data/Finsupp/Basic.lean	460	463	theorem mapDomain_notin_range {f : α → β} (x : α →₀ M) (a : β) (h : a ∉ Set.range f) : mapDomain f x a = 0 := by	rw [mapDomain, sum_apply, sum] exact Finset.sum_eq_zero fun a' _ => single_eq_of_ne fun eq => h <\| eq ▸ Set.mem_range_self _
/- Copyright (c) 2022 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov, Yaël Dillies -/ import Mathlib.MeasureTheory.Integral.SetIntegral #align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" /-! # Integral average of a function In this file we define `MeasureTheory.average μ f` (notation: `⨍ x, f x ∂μ`) to be the average value of `f` with respect to measure `μ`. It is defined as `∫ x, f x ∂((μ univ)⁻¹ • μ)`, so it is equal to zero if `f` is not integrable or if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is equal to its integral. For the average on a set, we use `⨍ x in s, f x ∂μ` (notation for `⨍ x, f x ∂(μ.restrict s)`). For average w.r.t. the volume, one can omit `∂volume`. Both have a version for the Lebesgue integral rather than Bochner. We prove several version of the first moment method: An integrable function is below/above its average on a set of positive measure. ## Implementation notes The average is defined as an integral over `(μ univ)⁻¹ • μ` so that all theorems about Bochner integrals work for the average without modifications. For theorems that require integrability of a function, we provide a convenience lemma `MeasureTheory.Integrable.to_average`. ## TODO Provide the first moment method for the Lebesgue integral as well. A draft is available on branch `first_moment_lintegral` in mathlib3 repository. ## Tags integral, center mass, average value -/ open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function open scoped Topology ENNReal Convex variable {α E F : Type} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {μ ν : Measure α} {s t : Set α} /-! ### Average value of a function w.r.t. a measure The (Bochner, Lebesgue) average value of a function `f` w.r.t. a measure `μ` (notation: `⨍ x, f x ∂μ`, `⨍⁻ x, f x ∂μ`) is defined as the (Bochner, Lebesgue) integral divided by the total measure, so it is equal to zero if `μ` is an infinite measure, and (typically) equal to infinity if `f` is not integrable. If `μ` is a probability measure, then the average of any function is equal to its integral. -/ namespace MeasureTheory section ENNReal variable (μ) {f g : α → ℝ≥0∞} /-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`, denoted `⨍⁻ x, f x ∂μ`. It is equal to `(μ univ)⁻¹ ∫⁻ x, f x ∂μ`, so it takes value zero if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is equal to its integral. For the average on a set, use `⨍⁻ x in s, f x ∂μ`, defined as `⨍⁻ x, f x ∂(μ.restrict s)`. For the average w.r.t. the volume, one can omit `∂volume`. -/ noncomputable def laverage (f : α → ℝ≥0∞) := ∫⁻ x, f x ∂(μ univ)⁻¹ • μ #align measure_theory.laverage MeasureTheory.laverage /-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ`. It is equal to `(μ univ)⁻¹ * ∫⁻ x, f x ∂μ`, so it takes value zero if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is equal to its integral. For the average on a set, use `⨍⁻ x in s, f x ∂μ`, defined as `⨍⁻ x, f x ∂(μ.restrict s)`. For the average w.r.t. the volume, one can omit `∂volume`. -/ notation3 "⨍⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => laverage μ r /-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. to the standard measure. It is equal to `(volume univ)⁻¹ * ∫⁻ x, f x`, so it takes value zero if the space has infinite measure. In a probability space, the average of any function is equal to its integral. For the average on a set, use `⨍⁻ x in s, f x`, defined as `⨍⁻ x, f x ∂(volume.restrict s)`. -/ notation3 "⨍⁻ "(...)", "r:60:(scoped f => laverage volume f) => r /-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. a measure `μ` on a set `s`. It is equal to `(μ s)⁻¹ * ∫⁻ x, f x ∂μ`, so it takes value zero if `s` has infinite measure. If `s` has measure `1`, then the average of any function is equal to its integral. For the average w.r.t. the volume, one can omit `∂volume`. -/ notation3 "⨍⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => laverage (Measure.restrict μ s) r /-- Average value of an `ℝ≥0∞`-valued function `f` w.r.t. to the standard measure on a set `s`. It is equal to `(volume s)⁻¹ * ∫⁻ x, f x`, so it takes value zero if `s` has infinite measure. If `s` has measure `1`, then the average of any function is equal to its integral. -/ notation3 (prettyPrint := false) "⨍⁻ "(...)" in "s", "r:60:(scoped f => laverage Measure.restrict volume s f) => r @[simp] theorem laverage_zero : ⨍⁻ _x, (0 : ℝ≥0∞) ∂μ = 0 := by rw [laverage, lintegral_zero] #align measure_theory.laverage_zero MeasureTheory.laverage_zero @[simp] theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by simp [laverage] #align measure_theory.laverage_zero_measure MeasureTheory.laverage_zero_measure theorem laverage_eq' (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂(μ univ)⁻¹ • μ := rfl #align measure_theory.laverage_eq' MeasureTheory.laverage_eq' theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul] #align measure_theory.laverage_eq MeasureTheory.laverage_eq theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by rw [laverage, measure_univ, inv_one, one_smul] #align measure_theory.laverage_eq_lintegral MeasureTheory.laverage_eq_lintegral @[simp] theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) : μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ := by rcases eq_or_ne μ 0 with hμ \| hμ · rw [hμ, lintegral_zero_measure, laverage_zero_measure, mul_zero] · rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)] #align measure_theory.measure_mul_laverage MeasureTheory.measure_mul_laverage theorem setLaverage_eq (f : α → ℝ≥0∞) (s : Set α) : ⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s := by rw [laverage_eq, restrict_apply_univ] #align measure_theory.set_laverage_eq MeasureTheory.setLaverage_eq theorem setLaverage_eq' (f : α → ℝ≥0∞) (s : Set α) : ⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s := by simp only [laverage_eq', restrict_apply_univ] #align measure_theory.set_laverage_eq' MeasureTheory.setLaverage_eq' variable {μ} theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by simp only [laverage_eq, lintegral_congr_ae h] #align measure_theory.laverage_congr MeasureTheory.laverage_congr theorem setLaverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h] #align measure_theory.set_laverage_congr MeasureTheory.setLaverage_congr theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ := by simp only [laverage_eq, set_lintegral_congr_fun hs h] #align measure_theory.set_laverage_congr_fun MeasureTheory.setLaverage_congr_fun theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by obtain rfl \| hμ := eq_or_ne μ 0 · simp · rw [laverage_eq] exact div_lt_top hf (measure_univ_ne_zero.2 hμ) #align measure_theory.laverage_lt_top MeasureTheory.laverage_lt_top theorem setLaverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ := laverage_lt_top #align measure_theory.set_laverage_lt_top MeasureTheory.setLaverage_lt_top theorem laverage_add_measure : ⨍⁻ x, f x ∂(μ + ν) = μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν := by by_cases hμ : IsFiniteMeasure μ; swap · rw [not_isFiniteMeasure_iff] at hμ simp [laverage_eq, hμ] by_cases hν : IsFiniteMeasure ν; swap · rw [not_isFiniteMeasure_iff] at hν simp [laverage_eq, hν] haveI := hμ; haveI := hν simp only [← ENNReal.mul_div_right_comm, measure_mul_laverage, ← ENNReal.add_div, ← lintegral_add_measure, ← Measure.add_apply, ← laverage_eq] #align measure_theory.laverage_add_measure MeasureTheory.laverage_add_measure theorem measure_mul_setLaverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) : μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ := by have := Fact.mk h.lt_top rw [← measure_mul_laverage, restrict_apply_univ] #align measure_theory.measure_mul_set_laverage MeasureTheory.measure_mul_setLaverage theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) : ⨍⁻ x in s ∪ t, f x ∂μ = μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ := by rw [restrict_union₀ hd ht, laverage_add_measure, restrict_apply_univ, restrict_apply_univ] #align measure_theory.laverage_union MeasureTheory.laverage_union theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) : ⨍⁻ x in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in t, f x ∂μ) := by refine ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), ENNReal.div_pos hs₀ <\| add_ne_top.2 ⟨hsμ, htμ⟩, ENNReal.div_pos ht₀ <\| add_ne_top.2 ⟨hsμ, htμ⟩, ?_, (laverage_union hd ht).symm⟩ rw [← ENNReal.add_div, ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)] #align measure_theory.laverage_union_mem_open_segment MeasureTheory.laverage_union_mem_openSegment theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) : ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] := by by_cases hs₀ : μ s = 0 · rw [← ae_eq_empty] at hs₀ rw [restrict_congr_set (hs₀.union EventuallyEq.rfl), empty_union] exact right_mem_segment _ _ _ · refine ⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, ?_, (laverage_union hd ht).symm⟩ rw [← ENNReal.add_div, ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)] #align measure_theory.laverage_union_mem_segment MeasureTheory.laverage_union_mem_segment theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) : ⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) := by simpa only [union_compl_self, restrict_univ] using laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _) (measure_ne_top _ _) #align measure_theory.laverage_mem_open_segment_compl_self MeasureTheory.laverage_mem_openSegment_compl_self @[simp] theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : NeZero μ] (c : ℝ≥0∞) : ⨍⁻ _x, c ∂μ = c := by simp only [laverage, lintegral_const, measure_univ, mul_one] #align measure_theory.laverage_const MeasureTheory.laverage_const theorem setLaverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ _x in s, c ∂μ = c := by simp only [setLaverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one] #align measure_theory.set_laverage_const MeasureTheory.setLaverage_const theorem laverage_one [IsFiniteMeasure μ] [NeZero μ] : ⨍⁻ _x, (1 : ℝ≥0∞) ∂μ = 1 := laverage_const _ _ #align measure_theory.laverage_one MeasureTheory.laverage_one theorem setLaverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ _x in s, (1 : ℝ≥0∞) ∂μ = 1 := setLaverage_const hs₀ hs _ #align measure_theory.set_laverage_one MeasureTheory.setLaverage_one -- Porting note: Dropped `simp` because of `simp` seeing through `1 : α → ℝ≥0∞` and applying -- `lintegral_const`. This is suboptimal. theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) : ∫⁻ _x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ := by obtain rfl \| hμ := eq_or_ne μ 0 · simp · rw [lintegral_const, laverage_eq, ENNReal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)] #align measure_theory.lintegral_laverage MeasureTheory.lintegral_laverage theorem setLintegral_setLaverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α) : ∫⁻ _x in s, ⨍⁻ a in s, f a ∂μ ∂μ = ∫⁻ x in s, f x ∂μ := lintegral_laverage _ _ #align measure_theory.set_lintegral_set_laverage MeasureTheory.setLintegral_setLaverage end ENNReal section NormedAddCommGroup variable (μ) variable {f g : α → E} /-- Average value of a function `f` w.r.t. a measure `μ`, denoted `⨍ x, f x ∂μ`. It is equal to `(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it takes value zero if `f` is not integrable or if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is equal to its integral. For the average on a set, use `⨍ x in s, f x ∂μ`, defined as `⨍ x, f x ∂(μ.restrict s)`. For the average w.r.t. the volume, one can omit `∂volume`. -/ noncomputable def average (f : α → E) := ∫ x, f x ∂(μ univ)⁻¹ • μ #align measure_theory.average MeasureTheory.average /-- Average value of a function `f` w.r.t. a measure `μ`. It is equal to `(μ univ).toReal⁻¹ • ∫ x, f x ∂μ`, so it takes value zero if `f` is not integrable or if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is equal to its integral. For the average on a set, use `⨍ x in s, f x ∂μ`, defined as `⨍ x, f x ∂(μ.restrict s)`. For the average w.r.t. the volume, one can omit `∂volume`. -/ notation3 "⨍ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => average μ r /-- Average value of a function `f` w.r.t. to the standard measure. It is equal to `(volume univ).toReal⁻¹ * ∫ x, f x`, so it takes value zero if `f` is not integrable or if the space has infinite measure. In a probability space, the average of any function is equal to its integral. For the average on a set, use `⨍ x in s, f x`, defined as `⨍ x, f x ∂(volume.restrict s)`. -/ notation3 "⨍ "(...)", "r:60:(scoped f => average volume f) => r /-- Average value of a function `f` w.r.t. a measure `μ` on a set `s`. It is equal to `(μ s).toReal⁻¹ * ∫ x, f x ∂μ`, so it takes value zero if `f` is not integrable on `s` or if `s` has infinite measure. If `s` has measure `1`, then the average of any function is equal to its integral. For the average w.r.t. the volume, one can omit `∂volume`. -/ notation3 "⨍ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => average (Measure.restrict μ s) r /-- Average value of a function `f` w.r.t. to the standard measure on a set `s`. It is equal to `(volume s).toReal⁻¹ * ∫ x, f x`, so it takes value zero `f` is not integrable on `s` or if `s` has infinite measure. If `s` has measure `1`, then the average of any function is equal to its integral. -/ notation3 "⨍ "(...)" in "s", "r:60:(scoped f => average (Measure.restrict volume s) f) => r @[simp] theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 := by rw [average, integral_zero] #align measure_theory.average_zero MeasureTheory.average_zero @[simp] theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by rw [average, smul_zero, integral_zero_measure] #align measure_theory.average_zero_measure MeasureTheory.average_zero_measure @[simp] theorem average_neg (f : α → E) : ⨍ x, -f x ∂μ = -⨍ x, f x ∂μ := integral_neg f #align measure_theory.average_neg MeasureTheory.average_neg theorem average_eq' (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂(μ univ)⁻¹ • μ := rfl #align measure_theory.average_eq' MeasureTheory.average_eq' theorem average_eq (f : α → E) : ⨍ x, f x ∂μ = (μ univ).toReal⁻¹ • ∫ x, f x ∂μ := by rw [average_eq', integral_smul_measure, ENNReal.toReal_inv] #align measure_theory.average_eq MeasureTheory.average_eq theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by rw [average, measure_univ, inv_one, one_smul] #align measure_theory.average_eq_integral MeasureTheory.average_eq_integral @[simp] theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) : (μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by rcases eq_or_ne μ 0 with hμ \| hμ · rw [hμ, integral_zero_measure, average_zero_measure, smul_zero] · rw [average_eq, smul_inv_smul₀] refine (ENNReal.toReal_pos ?_ <\| measure_ne_top _ _).ne' rwa [Ne, measure_univ_eq_zero] #align measure_theory.measure_smul_average MeasureTheory.measure_smul_average theorem setAverage_eq (f : α → E) (s : Set α) : ⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by rw [average_eq, restrict_apply_univ] #align measure_theory.set_average_eq MeasureTheory.setAverage_eq theorem setAverage_eq' (f : α → E) (s : Set α) : ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by simp only [average_eq', restrict_apply_univ] #align measure_theory.set_average_eq' MeasureTheory.setAverage_eq' variable {μ} theorem average_congr {f g : α → E} (h : f =ᵐ[μ] g) : ⨍ x, f x ∂μ = ⨍ x, g x ∂μ := by simp only [average_eq, integral_congr_ae h] #align measure_theory.average_congr MeasureTheory.average_congr theorem setAverage_congr (h : s =ᵐ[μ] t) : ⨍ x in s, f x ∂μ = ⨍ x in t, f x ∂μ := by simp only [setAverage_eq, setIntegral_congr_set_ae h, measure_congr h] #align measure_theory.set_average_congr MeasureTheory.setAverage_congr theorem setAverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by simp only [average_eq, setIntegral_congr_ae hs h] #align measure_theory.set_average_congr_fun MeasureTheory.setAverage_congr_fun theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E} (hμ : Integrable f μ) (hν : Integrable f ν) : ⨍ x, f x ∂(μ + ν) = ((μ univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂μ + ((ν univ).toReal / ((μ univ).toReal + (ν univ).toReal)) • ⨍ x, f x ∂ν := by simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ← smul_add, ← integral_add_measure hμ hν, ← ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)] rw [average_eq, Measure.add_apply] #align measure_theory.average_add_measure MeasureTheory.average_add_measure theorem average_pair {f : α → E} {g : α → F} (hfi : Integrable f μ) (hgi : Integrable g μ) : ⨍ x, (f x, g x) ∂μ = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) := integral_pair hfi.to_average hgi.to_average #align measure_theory.average_pair MeasureTheory.average_pair theorem measure_smul_setAverage (f : α → E) {s : Set α} (h : μ s ≠ ∞) : (μ s).toReal • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by haveI := Fact.mk h.lt_top rw [← measure_smul_average, restrict_apply_univ] #align measure_theory.measure_smul_set_average MeasureTheory.measure_smul_setAverage theorem average_union {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) : ⨍ x in s ∪ t, f x ∂μ = ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in s, f x ∂μ + ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ x in t, f x ∂μ := by haveI := Fact.mk hsμ.lt_top; haveI := Fact.mk htμ.lt_top rw [restrict_union₀ hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ] #align measure_theory.average_union MeasureTheory.average_union theorem average_union_mem_openSegment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) : ⨍ x in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in t, f x ∂μ) := by replace hs₀ : 0 < (μ s).toReal := ENNReal.toReal_pos hs₀ hsμ replace ht₀ : 0 < (μ t).toReal := ENNReal.toReal_pos ht₀ htμ exact mem_openSegment_iff_div.mpr ⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩ #align measure_theory.average_union_mem_open_segment MeasureTheory.average_union_mem_openSegment theorem average_union_mem_segment {f : α → E} {s t : Set α} (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) : ⨍ x in s ∪ t, f x ∂μ ∈ [⨍ x in s, f x ∂μ -[ℝ] ⨍ x in t, f x ∂μ] := by by_cases hse : μ s = 0 · rw [← ae_eq_empty] at hse rw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union] exact right_mem_segment _ _ _ · refine mem_segment_iff_div.mpr ⟨(μ s).toReal, (μ t).toReal, ENNReal.toReal_nonneg, ENNReal.toReal_nonneg, ?_, (average_union hd ht hsμ htμ hfs hft).symm⟩ calc 0 < (μ s).toReal := ENNReal.toReal_pos hse hsμ _ ≤ _ := le_add_of_nonneg_right ENNReal.toReal_nonneg #align measure_theory.average_union_mem_segment MeasureTheory.average_union_mem_segment theorem average_mem_openSegment_compl_self [IsFiniteMeasure μ] {f : α → E} {s : Set α} (hs : NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) (hfi : Integrable f μ) : ⨍ x, f x ∂μ ∈ openSegment ℝ (⨍ x in s, f x ∂μ) (⨍ x in sᶜ, f x ∂μ) := by simpa only [union_compl_self, restrict_univ] using average_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _) (measure_ne_top _ _) hfi.integrableOn hfi.integrableOn #align measure_theory.average_mem_open_segment_compl_self MeasureTheory.average_mem_openSegment_compl_self @[simp] theorem average_const (μ : Measure α) [IsFiniteMeasure μ] [h : NeZero μ] (c : E) : ⨍ _x, c ∂μ = c := by rw [average, integral_const, measure_univ, ENNReal.one_toReal, one_smul] #align measure_theory.average_const MeasureTheory.average_const theorem setAverage_const {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : E) : ⨍ _ in s, c ∂μ = c := have := NeZero.mk hs₀; have := Fact.mk hs.lt_top; average_const _ _ #align measure_theory.set_average_const MeasureTheory.setAverage_const -- Porting note (#10618): was `@[simp]` but `simp` can prove it theorem integral_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) : ∫ _, ⨍ a, f a ∂μ ∂μ = ∫ x, f x ∂μ := by simp #align measure_theory.integral_average MeasureTheory.integral_average theorem setIntegral_setAverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) (s : Set α) : ∫ _ in s, ⨍ a in s, f a ∂μ ∂μ = ∫ x in s, f x ∂μ := integral_average _ _ #align measure_theory.set_integral_set_average MeasureTheory.setIntegral_setAverage theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) : ∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 := by by_cases hf : Integrable f μ · rw [integral_sub hf (integrable_const _), integral_average, sub_self] refine integral_undef fun h => hf ?_ convert h.add (integrable_const (⨍ a, f a ∂μ)) exact (sub_add_cancel _ _).symm #align measure_theory.integral_sub_average MeasureTheory.integral_sub_average theorem setAverage_sub_setAverage (hs : μ s ≠ ∞) (f : α → E) : ∫ x in s, f x - ⨍ a in s, f a ∂μ ∂μ = 0 := haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩ integral_sub_average _ _ #align measure_theory.set_integral_sub_set_average MeasureTheory.setAverage_sub_setAverage theorem integral_average_sub [IsFiniteMeasure μ] (hf : Integrable f μ) : ∫ x, ⨍ a, f a ∂μ - f x ∂μ = 0 := by rw [integral_sub (integrable_const _) hf, integral_average, sub_self] #align measure_theory.integral_average_sub MeasureTheory.integral_average_sub theorem setIntegral_setAverage_sub (hs : μ s ≠ ∞) (hf : IntegrableOn f s μ) : ∫ x in s, ⨍ a in s, f a ∂μ - f x ∂μ = 0 := haveI : Fact (μ s < ∞) := ⟨lt_top_iff_ne_top.2 hs⟩ integral_average_sub hf #align measure_theory.set_integral_set_average_sub MeasureTheory.setIntegral_setAverage_sub end NormedAddCommGroup theorem ofReal_average {f : α → ℝ} (hf : Integrable f μ) (hf₀ : 0 ≤ᵐ[μ] f) : ENNReal.ofReal (⨍ x, f x ∂μ) = (∫⁻ x, ENNReal.ofReal (f x) ∂μ) / μ univ := by obtain rfl \| hμ := eq_or_ne μ 0 · simp · rw [average_eq, smul_eq_mul, ← toReal_inv, ofReal_mul toReal_nonneg, ofReal_toReal (inv_ne_top.2 <\| measure_univ_ne_zero.2 hμ), ofReal_integral_eq_lintegral_ofReal hf hf₀, ENNReal.div_eq_inv_mul] #align measure_theory.of_real_average MeasureTheory.ofReal_average theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) : ENNReal.ofReal (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) / μ s := by simpa using ofReal_average hf hf₀ #align measure_theory.of_real_set_average MeasureTheory.ofReal_setAverage theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) : (⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ := by rw [average_eq, laverage_eq, smul_eq_mul, toReal_div, div_eq_inv_mul, ← integral_toReal hf (hf'.mono fun _ => lt_top_iff_ne_top.2)] #align measure_theory.to_real_laverage MeasureTheory.toReal_laverage theorem toReal_setLaverage {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s)) (hf' : ∀ᵐ x ∂μ.restrict s, f x ≠ ∞) : (⨍⁻ x in s, f x ∂μ).toReal = ⨍ x in s, (f x).toReal ∂μ := by simpa [laverage_eq] using toReal_laverage hf hf' #align measure_theory.to_real_set_laverage MeasureTheory.toReal_setLaverage /-! ### First moment method -/ section FirstMomentReal variable {N : Set α} {f : α → ℝ} /-- First moment method. An integrable function is smaller than its mean on a set of positive measure. -/ theorem measure_le_setAverage_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) : 0 < μ ({x ∈ s \| f x ≤ ⨍ a in s, f a ∂μ}) := by refine pos_iff_ne_zero.2 fun H => ?_ replace H : (μ.restrict s) {x \| f x ≤ ⨍ a in s, f a ∂μ} = 0 := by rwa [restrict_apply₀, inter_comm] exact AEStronglyMeasurable.nullMeasurableSet_le hf.1 aestronglyMeasurable_const haveI := Fact.mk hμ₁.lt_top refine (integral_sub_average (μ.restrict s) f).not_gt ?_ refine (setIntegral_pos_iff_support_of_nonneg_ae ?_ ?_).2 ?_ · refine measure_mono_null (fun x hx ↦ ?_) H simp only [Pi.zero_apply, sub_nonneg, mem_compl_iff, mem_setOf_eq, not_le] at hx exact hx.le · exact hf.sub (integrableOn_const.2 <\| Or.inr <\| lt_top_iff_ne_top.2 hμ₁) · rwa [pos_iff_ne_zero, inter_comm, ← diff_compl, ← diff_inter_self_eq_diff, measure_diff_null] refine measure_mono_null ?_ (measure_inter_eq_zero_of_restrict H) exact inter_subset_inter_left _ fun a ha => (sub_eq_zero.1 <\| of_not_not ha).le #align measure_theory.measure_le_set_average_pos MeasureTheory.measure_le_setAverage_pos /-- First moment method. An integrable function is greater than its mean on a set of positive measure. -/ theorem measure_setAverage_le_pos (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) : 0 < μ ({x ∈ s \| ⨍ a in s, f a ∂μ ≤ f x}) := by simpa [integral_neg, neg_div] using measure_le_setAverage_pos hμ hμ₁ hf.neg #align measure_theory.measure_set_average_le_pos MeasureTheory.measure_setAverage_le_pos /-- First moment method. The minimum of an integrable function is smaller than its mean. -/ theorem exists_le_setAverage (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) : ∃ x ∈ s, f x ≤ ⨍ a in s, f a ∂μ := let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_le_setAverage_pos hμ hμ₁ hf).ne' ⟨x, hx, h⟩ #align measure_theory.exists_le_set_average MeasureTheory.exists_le_setAverage /-- First moment method. The maximum of an integrable function is greater than its mean. -/ theorem exists_setAverage_le (hμ : μ s ≠ 0) (hμ₁ : μ s ≠ ∞) (hf : IntegrableOn f s μ) : ∃ x ∈ s, ⨍ a in s, f a ∂μ ≤ f x := let ⟨x, hx, h⟩ := nonempty_of_measure_ne_zero (measure_setAverage_le_pos hμ hμ₁ hf).ne' ⟨x, hx, h⟩ #align measure_theory.exists_set_average_le MeasureTheory.exists_setAverage_le section FiniteMeasure variable [IsFiniteMeasure μ] /-- First moment method. An integrable function is smaller than its mean on a set of positive measure. -/ theorem measure_le_average_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ {x \| f x ≤ ⨍ a, f a ∂μ} := by simpa using measure_le_setAverage_pos (Measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.integrableOn #align measure_theory.measure_le_average_pos MeasureTheory.measure_le_average_pos /-- First moment method. An integrable function is greater than its mean on a set of positive measure. -/ theorem measure_average_le_pos (hμ : μ ≠ 0) (hf : Integrable f μ) : 0 < μ {x \| ⨍ a, f a ∂μ ≤ f x} := by simpa using measure_setAverage_le_pos (Measure.measure_univ_ne_zero.2 hμ) (measure_ne_top _ _) hf.integrableOn #align measure_theory.measure_average_le_pos MeasureTheory.measure_average_le_pos /-- First moment method. The minimum of an integrable function is smaller than its mean. -/ theorem exists_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ := let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_average_pos hμ hf).ne' ⟨x, hx⟩ #align measure_theory.exists_le_average MeasureTheory.exists_le_average /-- First moment method. The maximum of an integrable function is greater than its mean. -/ theorem exists_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, ⨍ a, f a ∂μ ≤ f x := let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_average_le_pos hμ hf).ne' ⟨x, hx⟩ #align measure_theory.exists_average_le MeasureTheory.exists_average_le /-- First moment method. The minimum of an integrable function is smaller than its mean, while avoiding a null set. -/ theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) : ∃ x, x ∉ N ∧ f x ≤ ⨍ a, f a ∂μ := by have := measure_le_average_pos hμ hf rw [← measure_diff_null hN] at this obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne' exact ⟨x, hxN, hx⟩ #align measure_theory.exists_not_mem_null_le_average MeasureTheory.exists_not_mem_null_le_average /-- First moment method. The maximum of an integrable function is greater than its mean, while avoiding a null set. -/ theorem exists_not_mem_null_average_le (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) : ∃ x, x ∉ N ∧ ⨍ a, f a ∂μ ≤ f x := by simpa [integral_neg, neg_div] using exists_not_mem_null_le_average hμ hf.neg hN #align measure_theory.exists_not_mem_null_average_le MeasureTheory.exists_not_mem_null_average_le end FiniteMeasure section ProbabilityMeasure variable [IsProbabilityMeasure μ] /-- First moment method. An integrable function is smaller than its integral on a set of positive measure. -/ theorem measure_le_integral_pos (hf : Integrable f μ) : 0 < μ {x \| f x ≤ ∫ a, f a ∂μ} := by simpa only [average_eq_integral] using measure_le_average_pos (IsProbabilityMeasure.ne_zero μ) hf #align measure_theory.measure_le_integral_pos MeasureTheory.measure_le_integral_pos /-- First moment method. An integrable function is greater than its integral on a set of positive measure. -/ theorem measure_integral_le_pos (hf : Integrable f μ) : 0 < μ {x \| ∫ a, f a ∂μ ≤ f x} := by simpa only [average_eq_integral] using measure_average_le_pos (IsProbabilityMeasure.ne_zero μ) hf #align measure_theory.measure_integral_le_pos MeasureTheory.measure_integral_le_pos /-- First moment method. The minimum of an integrable function is smaller than its integral. -/ theorem exists_le_integral (hf : Integrable f μ) : ∃ x, f x ≤ ∫ a, f a ∂μ := by simpa only [average_eq_integral] using exists_le_average (IsProbabilityMeasure.ne_zero μ) hf #align measure_theory.exists_le_integral MeasureTheory.exists_le_integral /-- First moment method. The maximum of an integrable function is greater than its integral. -/	Mathlib/MeasureTheory/Integral/Average.lean	634	635	theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x := by	simpa only [average_eq_integral] using exists_average_le (IsProbabilityMeasure.ne_zero μ) hf
/- Copyright (c) 2019 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes -/ import Mathlib.Algebra.Group.Basic import Mathlib.Algebra.Group.Pi.Basic import Mathlib.Order.Fin import Mathlib.Order.PiLex import Mathlib.Order.Interval.Set.Basic #align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b" /-! # Operation on tuples We interpret maps `∀ i : Fin n, α i` as `n`-tuples of elements of possibly varying type `α i`, `(α 0, …, α (n-1))`. A particular case is `Fin n → α` of elements with all the same type. In this case when `α i` is a constant map, then tuples are isomorphic (but not definitionally equal) to `Vector`s. We define the following operations: * `Fin.tail` : the tail of an `n+1` tuple, i.e., its last `n` entries; * `Fin.cons` : adding an element at the beginning of an `n`-tuple, to get an `n+1`-tuple; * `Fin.init` : the beginning of an `n+1` tuple, i.e., its first `n` entries; * `Fin.snoc` : adding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc` comes from `cons` (i.e., adding an element to the left of a tuple) read in reverse order. * `Fin.insertNth` : insert an element to a tuple at a given position. * `Fin.find p` : returns the first index `n` where `p n` is satisfied, and `none` if it is never satisfied. * `Fin.append a b` : append two tuples. * `Fin.repeat n a` : repeat a tuple `n` times. -/ assert_not_exists MonoidWithZero universe u v namespace Fin variable {m n : ℕ} open Function section Tuple /-- There is exactly one tuple of size zero. -/ example (α : Fin 0 → Sort u) : Unique (∀ i : Fin 0, α i) := by infer_instance theorem tuple0_le {α : Fin 0 → Type} [∀ i, Preorder (α i)] (f g : ∀ i, α i) : f ≤ g := finZeroElim #align fin.tuple0_le Fin.tuple0_le variable {α : Fin (n + 1) → Type u} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ) (i : Fin n) (y : α i.succ) (z : α 0) /-- The tail of an `n+1` tuple, i.e., its last `n` entries. -/ def tail (q : ∀ i, α i) : ∀ i : Fin n, α i.succ := fun i ↦ q i.succ #align fin.tail Fin.tail theorem tail_def {n : ℕ} {α : Fin (n + 1) → Type} {q : ∀ i, α i} : (tail fun k : Fin (n + 1) ↦ q k) = fun k : Fin n ↦ q k.succ := rfl #align fin.tail_def Fin.tail_def /-- Adding an element at the beginning of an `n`-tuple, to get an `n+1`-tuple. -/ def cons (x : α 0) (p : ∀ i : Fin n, α i.succ) : ∀ i, α i := fun j ↦ Fin.cases x p j #align fin.cons Fin.cons @[simp] theorem tail_cons : tail (cons x p) = p := by simp (config := { unfoldPartialApp := true }) [tail, cons] #align fin.tail_cons Fin.tail_cons @[simp] theorem cons_succ : cons x p i.succ = p i := by simp [cons] #align fin.cons_succ Fin.cons_succ @[simp] theorem cons_zero : cons x p 0 = x := by simp [cons] #align fin.cons_zero Fin.cons_zero @[simp] theorem cons_one {α : Fin (n + 2) → Type} (x : α 0) (p : ∀ i : Fin n.succ, α i.succ) : cons x p 1 = p 0 := by rw [← cons_succ x p]; rfl /-- Updating a tuple and adding an element at the beginning commute. -/ @[simp] theorem cons_update : cons x (update p i y) = update (cons x p) i.succ y := by ext j by_cases h : j = 0 · rw [h] simp [Ne.symm (succ_ne_zero i)] · let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this, cons_succ] by_cases h' : j' = i · rw [h'] simp · have : j'.succ ≠ i.succ := by rwa [Ne, succ_inj] rw [update_noteq h', update_noteq this, cons_succ] #align fin.cons_update Fin.cons_update /-- As a binary function, `Fin.cons` is injective. -/ theorem cons_injective2 : Function.Injective2 (@cons n α) := fun x₀ y₀ x y h ↦ ⟨congr_fun h 0, funext fun i ↦ by simpa using congr_fun h (Fin.succ i)⟩ #align fin.cons_injective2 Fin.cons_injective2 @[simp] theorem cons_eq_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} : cons x₀ x = cons y₀ y ↔ x₀ = y₀ ∧ x = y := cons_injective2.eq_iff #align fin.cons_eq_cons Fin.cons_eq_cons theorem cons_left_injective (x : ∀ i : Fin n, α i.succ) : Function.Injective fun x₀ ↦ cons x₀ x := cons_injective2.left _ #align fin.cons_left_injective Fin.cons_left_injective theorem cons_right_injective (x₀ : α 0) : Function.Injective (cons x₀) := cons_injective2.right _ #align fin.cons_right_injective Fin.cons_right_injective /-- Adding an element at the beginning of a tuple and then updating it amounts to adding it directly. -/ theorem update_cons_zero : update (cons x p) 0 z = cons z p := by ext j by_cases h : j = 0 · rw [h] simp · simp only [h, update_noteq, Ne, not_false_iff] let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this, cons_succ, cons_succ] #align fin.update_cons_zero Fin.update_cons_zero /-- Concatenating the first element of a tuple with its tail gives back the original tuple -/ @[simp, nolint simpNF] -- Porting note: linter claims LHS doesn't simplify theorem cons_self_tail : cons (q 0) (tail q) = q := by ext j by_cases h : j = 0 · rw [h] simp · let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this] unfold tail rw [cons_succ] #align fin.cons_self_tail Fin.cons_self_tail -- Porting note: Mathport removes `_root_`? /-- Recurse on an `n+1`-tuple by splitting it into a single element and an `n`-tuple. -/ @[elab_as_elim] def consCases {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x)) (x : ∀ i : Fin n.succ, α i) : P x := _root_.cast (by rw [cons_self_tail]) <\| h (x 0) (tail x) #align fin.cons_cases Fin.consCases @[simp] theorem consCases_cons {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x)) (x₀ : α 0) (x : ∀ i : Fin n, α i.succ) : @consCases _ _ _ h (cons x₀ x) = h x₀ x := by rw [consCases, cast_eq] congr #align fin.cons_cases_cons Fin.consCases_cons /-- Recurse on a tuple by splitting into `Fin.elim0` and `Fin.cons`. -/ @[elab_as_elim] def consInduction {α : Type} {P : ∀ {n : ℕ}, (Fin n → α) → Sort v} (h0 : P Fin.elim0) (h : ∀ {n} (x₀) (x : Fin n → α), P x → P (Fin.cons x₀ x)) : ∀ {n : ℕ} (x : Fin n → α), P x \| 0, x => by convert h0 \| n + 1, x => consCases (fun x₀ x ↦ h _ _ <\| consInduction h0 h _) x #align fin.cons_induction Fin.consInductionₓ -- Porting note: universes theorem cons_injective_of_injective {α} {x₀ : α} {x : Fin n → α} (hx₀ : x₀ ∉ Set.range x) (hx : Function.Injective x) : Function.Injective (cons x₀ x : Fin n.succ → α) := by refine Fin.cases ?_ ?_ · refine Fin.cases ?_ ?_ · intro rfl · intro j h rw [cons_zero, cons_succ] at h exact hx₀.elim ⟨_, h.symm⟩ · intro i refine Fin.cases ?_ ?_ · intro h rw [cons_zero, cons_succ] at h exact hx₀.elim ⟨_, h⟩ · intro j h rw [cons_succ, cons_succ] at h exact congr_arg _ (hx h) #align fin.cons_injective_of_injective Fin.cons_injective_of_injective theorem cons_injective_iff {α} {x₀ : α} {x : Fin n → α} : Function.Injective (cons x₀ x : Fin n.succ → α) ↔ x₀ ∉ Set.range x ∧ Function.Injective x := by refine ⟨fun h ↦ ⟨?_, ?_⟩, fun h ↦ cons_injective_of_injective h.1 h.2⟩ · rintro ⟨i, hi⟩ replace h := @h i.succ 0 simp [hi, succ_ne_zero] at h · simpa [Function.comp] using h.comp (Fin.succ_injective _) #align fin.cons_injective_iff Fin.cons_injective_iff @[simp] theorem forall_fin_zero_pi {α : Fin 0 → Sort} {P : (∀ i, α i) → Prop} : (∀ x, P x) ↔ P finZeroElim := ⟨fun h ↦ h _, fun h x ↦ Subsingleton.elim finZeroElim x ▸ h⟩ #align fin.forall_fin_zero_pi Fin.forall_fin_zero_pi @[simp] theorem exists_fin_zero_pi {α : Fin 0 → Sort} {P : (∀ i, α i) → Prop} : (∃ x, P x) ↔ P finZeroElim := ⟨fun ⟨x, h⟩ ↦ Subsingleton.elim x finZeroElim ▸ h, fun h ↦ ⟨_, h⟩⟩ #align fin.exists_fin_zero_pi Fin.exists_fin_zero_pi theorem forall_fin_succ_pi {P : (∀ i, α i) → Prop} : (∀ x, P x) ↔ ∀ a v, P (Fin.cons a v) := ⟨fun h a v ↦ h (Fin.cons a v), consCases⟩ #align fin.forall_fin_succ_pi Fin.forall_fin_succ_pi theorem exists_fin_succ_pi {P : (∀ i, α i) → Prop} : (∃ x, P x) ↔ ∃ a v, P (Fin.cons a v) := ⟨fun ⟨x, h⟩ ↦ ⟨x 0, tail x, (cons_self_tail x).symm ▸ h⟩, fun ⟨_, _, h⟩ ↦ ⟨_, h⟩⟩ #align fin.exists_fin_succ_pi Fin.exists_fin_succ_pi /-- Updating the first element of a tuple does not change the tail. -/ @[simp] theorem tail_update_zero : tail (update q 0 z) = tail q := by ext j simp [tail, Fin.succ_ne_zero] #align fin.tail_update_zero Fin.tail_update_zero /-- Updating a nonzero element and taking the tail commute. -/ @[simp] theorem tail_update_succ : tail (update q i.succ y) = update (tail q) i y := by ext j by_cases h : j = i · rw [h] simp [tail] · simp [tail, (Fin.succ_injective n).ne h, h] #align fin.tail_update_succ Fin.tail_update_succ theorem comp_cons {α : Type} {β : Type} (g : α → β) (y : α) (q : Fin n → α) : g ∘ cons y q = cons (g y) (g ∘ q) := by ext j by_cases h : j = 0 · rw [h] rfl · let j' := pred j h have : j'.succ = j := succ_pred j h rw [← this, cons_succ, comp_apply, comp_apply, cons_succ] #align fin.comp_cons Fin.comp_cons theorem comp_tail {α : Type} {β : Type} (g : α → β) (q : Fin n.succ → α) : g ∘ tail q = tail (g ∘ q) := by ext j simp [tail] #align fin.comp_tail Fin.comp_tail theorem le_cons [∀ i, Preorder (α i)] {x : α 0} {q : ∀ i, α i} {p : ∀ i : Fin n, α i.succ} : q ≤ cons x p ↔ q 0 ≤ x ∧ tail q ≤ p := forall_fin_succ.trans <\| and_congr Iff.rfl <\| forall_congr' fun j ↦ by simp [tail] #align fin.le_cons Fin.le_cons theorem cons_le [∀ i, Preorder (α i)] {x : α 0} {q : ∀ i, α i} {p : ∀ i : Fin n, α i.succ} : cons x p ≤ q ↔ x ≤ q 0 ∧ p ≤ tail q := @le_cons _ (fun i ↦ (α i)ᵒᵈ) _ x q p #align fin.cons_le Fin.cons_le theorem cons_le_cons [∀ i, Preorder (α i)] {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} : cons x₀ x ≤ cons y₀ y ↔ x₀ ≤ y₀ ∧ x ≤ y := forall_fin_succ.trans <\| and_congr_right' <\| by simp only [cons_succ, Pi.le_def] #align fin.cons_le_cons Fin.cons_le_cons theorem pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} (s : ∀ {i : Fin n.succ}, α i → α i → Prop) : Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔ s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_fin_succ] simp [and_assoc, exists_and_left] #align fin.pi_lex_lt_cons_cons Fin.pi_lex_lt_cons_cons theorem range_fin_succ {α} (f : Fin (n + 1) → α) : Set.range f = insert (f 0) (Set.range (Fin.tail f)) := Set.ext fun _ ↦ exists_fin_succ.trans <\| eq_comm.or Iff.rfl #align fin.range_fin_succ Fin.range_fin_succ @[simp] theorem range_cons {α : Type} {n : ℕ} (x : α) (b : Fin n → α) : Set.range (Fin.cons x b : Fin n.succ → α) = insert x (Set.range b) := by rw [range_fin_succ, cons_zero, tail_cons] #align fin.range_cons Fin.range_cons section Append /-- Append a tuple of length `m` to a tuple of length `n` to get a tuple of length `m + n`. This is a non-dependent version of `Fin.add_cases`. -/ def append {α : Type} (a : Fin m → α) (b : Fin n → α) : Fin (m + n) → α := @Fin.addCases _ _ (fun _ => α) a b #align fin.append Fin.append @[simp] theorem append_left {α : Type} (u : Fin m → α) (v : Fin n → α) (i : Fin m) : append u v (Fin.castAdd n i) = u i := addCases_left _ #align fin.append_left Fin.append_left @[simp] theorem append_right {α : Type} (u : Fin m → α) (v : Fin n → α) (i : Fin n) : append u v (natAdd m i) = v i := addCases_right _ #align fin.append_right Fin.append_right theorem append_right_nil {α : Type} (u : Fin m → α) (v : Fin n → α) (hv : n = 0) : append u v = u ∘ Fin.cast (by rw [hv, Nat.add_zero]) := by refine funext (Fin.addCases (fun l => ?_) fun r => ?_) · rw [append_left, Function.comp_apply] refine congr_arg u (Fin.ext ?_) simp · exact (Fin.cast hv r).elim0 #align fin.append_right_nil Fin.append_right_nil @[simp] theorem append_elim0 {α : Type} (u : Fin m → α) : append u Fin.elim0 = u ∘ Fin.cast (Nat.add_zero _) := append_right_nil _ _ rfl #align fin.append_elim0 Fin.append_elim0 theorem append_left_nil {α : Type} (u : Fin m → α) (v : Fin n → α) (hu : m = 0) : append u v = v ∘ Fin.cast (by rw [hu, Nat.zero_add]) := by refine funext (Fin.addCases (fun l => ?_) fun r => ?_) · exact (Fin.cast hu l).elim0 · rw [append_right, Function.comp_apply] refine congr_arg v (Fin.ext ?_) simp [hu] #align fin.append_left_nil Fin.append_left_nil @[simp] theorem elim0_append {α : Type} (v : Fin n → α) : append Fin.elim0 v = v ∘ Fin.cast (Nat.zero_add _) := append_left_nil _ _ rfl #align fin.elim0_append Fin.elim0_append theorem append_assoc {p : ℕ} {α : Type} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) : append (append a b) c = append a (append b c) ∘ Fin.cast (Nat.add_assoc ..) := by ext i rw [Function.comp_apply] refine Fin.addCases (fun l => ?_) (fun r => ?_) i · rw [append_left] refine Fin.addCases (fun ll => ?_) (fun lr => ?_) l · rw [append_left] simp [castAdd_castAdd] · rw [append_right] simp [castAdd_natAdd] · rw [append_right] simp [← natAdd_natAdd] #align fin.append_assoc Fin.append_assoc /-- Appending a one-tuple to the left is the same as `Fin.cons`. -/ theorem append_left_eq_cons {α : Type} {n : ℕ} (x₀ : Fin 1 → α) (x : Fin n → α) : Fin.append x₀ x = Fin.cons (x₀ 0) x ∘ Fin.cast (Nat.add_comm ..) := by ext i refine Fin.addCases ?_ ?_ i <;> clear i · intro i rw [Subsingleton.elim i 0, Fin.append_left, Function.comp_apply, eq_comm] exact Fin.cons_zero _ _ · intro i rw [Fin.append_right, Function.comp_apply, Fin.cast_natAdd, eq_comm, Fin.addNat_one] exact Fin.cons_succ _ _ _ #align fin.append_left_eq_cons Fin.append_left_eq_cons /-- `Fin.cons` is the same as appending a one-tuple to the left. -/ theorem cons_eq_append {α : Type} (x : α) (xs : Fin n → α) : cons x xs = append (cons x Fin.elim0) xs ∘ Fin.cast (Nat.add_comm ..) := by funext i; simp [append_left_eq_cons] @[simp] lemma append_cast_left {n m} {α : Type} (xs : Fin n → α) (ys : Fin m → α) (n' : ℕ) (h : n' = n) : Fin.append (xs ∘ Fin.cast h) ys = Fin.append xs ys ∘ (Fin.cast <\| by rw [h]) := by subst h; simp @[simp] lemma append_cast_right {n m} {α : Type} (xs : Fin n → α) (ys : Fin m → α) (m' : ℕ) (h : m' = m) : Fin.append xs (ys ∘ Fin.cast h) = Fin.append xs ys ∘ (Fin.cast <\| by rw [h]) := by subst h; simp lemma append_rev {m n} {α : Type} (xs : Fin m → α) (ys : Fin n → α) (i : Fin (m + n)) : append xs ys (rev i) = append (ys ∘ rev) (xs ∘ rev) (cast (Nat.add_comm ..) i) := by rcases rev_surjective i with ⟨i, rfl⟩ rw [rev_rev] induction i using Fin.addCases · simp [rev_castAdd] · simp [cast_rev, rev_addNat] lemma append_comp_rev {m n} {α : Type} (xs : Fin m → α) (ys : Fin n → α) : append xs ys ∘ rev = append (ys ∘ rev) (xs ∘ rev) ∘ cast (Nat.add_comm ..) := funext <\| append_rev xs ys end Append section Repeat /-- Repeat `a` `m` times. For example `Fin.repeat 2 ![0, 3, 7] = ![0, 3, 7, 0, 3, 7]`. -/ -- Porting note: removed @[simp] def «repeat» {α : Type} (m : ℕ) (a : Fin n → α) : Fin (m * n) → α \| i => a i.modNat #align fin.repeat Fin.repeat -- Porting note: added (leanprover/lean4#2042) @[simp] theorem repeat_apply {α : Type} (a : Fin n → α) (i : Fin (m n)) : Fin.repeat m a i = a i.modNat := rfl @[simp] theorem repeat_zero {α : Type} (a : Fin n → α) : Fin.repeat 0 a = Fin.elim0 ∘ cast (Nat.zero_mul _) := funext fun x => (cast (Nat.zero_mul _) x).elim0 #align fin.repeat_zero Fin.repeat_zero @[simp] theorem repeat_one {α : Type} (a : Fin n → α) : Fin.repeat 1 a = a ∘ cast (Nat.one_mul _) := by generalize_proofs h apply funext rw [(Fin.rightInverse_cast h.symm).surjective.forall] intro i simp [modNat, Nat.mod_eq_of_lt i.is_lt] #align fin.repeat_one Fin.repeat_one theorem repeat_succ {α : Type} (a : Fin n → α) (m : ℕ) : Fin.repeat m.succ a = append a (Fin.repeat m a) ∘ cast ((Nat.succ_mul _ _).trans (Nat.add_comm ..)) := by generalize_proofs h apply funext rw [(Fin.rightInverse_cast h.symm).surjective.forall] refine Fin.addCases (fun l => ?_) fun r => ?_ · simp [modNat, Nat.mod_eq_of_lt l.is_lt] · simp [modNat] #align fin.repeat_succ Fin.repeat_succ @[simp] theorem repeat_add {α : Type} (a : Fin n → α) (m₁ m₂ : ℕ) : Fin.repeat (m₁ + m₂) a = append (Fin.repeat m₁ a) (Fin.repeat m₂ a) ∘ cast (Nat.add_mul ..) := by generalize_proofs h apply funext rw [(Fin.rightInverse_cast h.symm).surjective.forall] refine Fin.addCases (fun l => ?_) fun r => ?_ · simp [modNat, Nat.mod_eq_of_lt l.is_lt] · simp [modNat, Nat.add_mod] #align fin.repeat_add Fin.repeat_add theorem repeat_rev {α : Type} (a : Fin n → α) (k : Fin (m n)) : Fin.repeat m a k.rev = Fin.repeat m (a ∘ Fin.rev) k := congr_arg a k.modNat_rev theorem repeat_comp_rev {α} (a : Fin n → α) : Fin.repeat m a ∘ Fin.rev = Fin.repeat m (a ∘ Fin.rev) := funext <\| repeat_rev a end Repeat end Tuple section TupleRight /-! In the previous section, we have discussed inserting or removing elements on the left of a tuple. In this section, we do the same on the right. A difference is that `Fin (n+1)` is constructed inductively from `Fin n` starting from the left, not from the right. This implies that Lean needs more help to realize that elements belong to the right types, i.e., we need to insert casts at several places. -/ -- Porting note: `i.castSucc` does not work like it did in Lean 3; -- `(castSucc i)` must be used. variable {α : Fin (n + 1) → Type u} (x : α (last n)) (q : ∀ i, α i) (p : ∀ i : Fin n, α (castSucc i)) (i : Fin n) (y : α (castSucc i)) (z : α (last n)) /-- The beginning of an `n+1` tuple, i.e., its first `n` entries -/ def init (q : ∀ i, α i) (i : Fin n) : α (castSucc i) := q (castSucc i) #align fin.init Fin.init theorem init_def {n : ℕ} {α : Fin (n + 1) → Type} {q : ∀ i, α i} : (init fun k : Fin (n + 1) ↦ q k) = fun k : Fin n ↦ q (castSucc k) := rfl #align fin.init_def Fin.init_def /-- Adding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc` comes from `cons` (i.e., adding an element to the left of a tuple) read in reverse order. -/ def snoc (p : ∀ i : Fin n, α (castSucc i)) (x : α (last n)) (i : Fin (n + 1)) : α i := if h : i.val < n then _root_.cast (by rw [Fin.castSucc_castLT i h]) (p (castLT i h)) else _root_.cast (by rw [eq_last_of_not_lt h]) x #align fin.snoc Fin.snoc @[simp] theorem init_snoc : init (snoc p x) = p := by ext i simp only [init, snoc, coe_castSucc, is_lt, cast_eq, dite_true] convert cast_eq rfl (p i) #align fin.init_snoc Fin.init_snoc @[simp] theorem snoc_castSucc : snoc p x (castSucc i) = p i := by simp only [snoc, coe_castSucc, is_lt, cast_eq, dite_true] convert cast_eq rfl (p i) #align fin.snoc_cast_succ Fin.snoc_castSucc @[simp] theorem snoc_comp_castSucc {n : ℕ} {α : Sort _} {a : α} {f : Fin n → α} : (snoc f a : Fin (n + 1) → α) ∘ castSucc = f := funext fun i ↦ by rw [Function.comp_apply, snoc_castSucc] #align fin.snoc_comp_cast_succ Fin.snoc_comp_castSucc @[simp] theorem snoc_last : snoc p x (last n) = x := by simp [snoc] #align fin.snoc_last Fin.snoc_last lemma snoc_zero {α : Type} (p : Fin 0 → α) (x : α) : Fin.snoc p x = fun _ ↦ x := by ext y have : Subsingleton (Fin (0 + 1)) := Fin.subsingleton_one simp only [Subsingleton.elim y (Fin.last 0), snoc_last] @[simp] theorem snoc_comp_nat_add {n m : ℕ} {α : Sort _} (f : Fin (m + n) → α) (a : α) : (snoc f a : Fin _ → α) ∘ (natAdd m : Fin (n + 1) → Fin (m + n + 1)) = snoc (f ∘ natAdd m) a := by ext i refine Fin.lastCases ?_ (fun i ↦ ?_) i · simp only [Function.comp_apply] rw [snoc_last, natAdd_last, snoc_last] · simp only [comp_apply, snoc_castSucc] rw [natAdd_castSucc, snoc_castSucc] #align fin.snoc_comp_nat_add Fin.snoc_comp_nat_add @[simp] theorem snoc_cast_add {α : Fin (n + m + 1) → Type} (f : ∀ i : Fin (n + m), α (castSucc i)) (a : α (last (n + m))) (i : Fin n) : (snoc f a) (castAdd (m + 1) i) = f (castAdd m i) := dif_pos _ #align fin.snoc_cast_add Fin.snoc_cast_add -- Porting note: Had to `unfold comp` @[simp] theorem snoc_comp_cast_add {n m : ℕ} {α : Sort _} (f : Fin (n + m) → α) (a : α) : (snoc f a : Fin _ → α) ∘ castAdd (m + 1) = f ∘ castAdd m := funext (by unfold comp; exact snoc_cast_add _ _) #align fin.snoc_comp_cast_add Fin.snoc_comp_cast_add /-- Updating a tuple and adding an element at the end commute. -/ @[simp] theorem snoc_update : snoc (update p i y) x = update (snoc p x) (castSucc i) y := by ext j by_cases h : j.val < n · rw [snoc] simp only [h] simp only [dif_pos] by_cases h' : j = castSucc i · have C1 : α (castSucc i) = α j := by rw [h'] have E1 : update (snoc p x) (castSucc i) y j = _root_.cast C1 y := by have : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y := by simp convert this · exact h'.symm · exact heq_of_cast_eq (congr_arg α (Eq.symm h')) rfl have C2 : α (castSucc i) = α (castSucc (castLT j h)) := by rw [castSucc_castLT, h'] have E2 : update p i y (castLT j h) = _root_.cast C2 y := by have : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y := by simp convert this · simp [h, h'] · exact heq_of_cast_eq C2 rfl rw [E1, E2] exact eq_rec_compose (Eq.trans C2.symm C1) C2 y · have : ¬castLT j h = i := by intro E apply h' rw [← E, castSucc_castLT] simp [h', this, snoc, h] · rw [eq_last_of_not_lt h] simp [Ne.symm (ne_of_lt (castSucc_lt_last i))] #align fin.snoc_update Fin.snoc_update /-- Adding an element at the beginning of a tuple and then updating it amounts to adding it directly. -/ theorem update_snoc_last : update (snoc p x) (last n) z = snoc p z := by ext j by_cases h : j.val < n · have : j ≠ last n := ne_of_lt h simp [h, update_noteq, this, snoc] · rw [eq_last_of_not_lt h] simp #align fin.update_snoc_last Fin.update_snoc_last /-- Concatenating the first element of a tuple with its tail gives back the original tuple -/ @[simp] theorem snoc_init_self : snoc (init q) (q (last n)) = q := by ext j by_cases h : j.val < n · simp only [init, snoc, h, cast_eq, dite_true, castSucc_castLT] · rw [eq_last_of_not_lt h] simp #align fin.snoc_init_self Fin.snoc_init_self /-- Updating the last element of a tuple does not change the beginning. -/ @[simp] theorem init_update_last : init (update q (last n) z) = init q := by ext j simp [init, ne_of_lt, castSucc_lt_last] #align fin.init_update_last Fin.init_update_last /-- Updating an element and taking the beginning commute. -/ @[simp] theorem init_update_castSucc : init (update q (castSucc i) y) = update (init q) i y := by ext j by_cases h : j = i · rw [h] simp [init] · simp [init, h, castSucc_inj] #align fin.init_update_cast_succ Fin.init_update_castSucc /-- `tail` and `init` commute. We state this lemma in a non-dependent setting, as otherwise it would involve a cast to convince Lean that the two types are equal, making it harder to use. -/ theorem tail_init_eq_init_tail {β : Type} (q : Fin (n + 2) → β) : tail (init q) = init (tail q) := by ext i simp [tail, init, castSucc_fin_succ] #align fin.tail_init_eq_init_tail Fin.tail_init_eq_init_tail /-- `cons` and `snoc` commute. We state this lemma in a non-dependent setting, as otherwise it would involve a cast to convince Lean that the two types are equal, making it harder to use. -/ theorem cons_snoc_eq_snoc_cons {β : Type} (a : β) (q : Fin n → β) (b : β) : @cons n.succ (fun _ ↦ β) a (snoc q b) = snoc (cons a q) b := by ext i by_cases h : i = 0 · rw [h] -- Porting note: `refl` finished it here in Lean 3, but I had to add more. simp [snoc, castLT] set j := pred i h with ji have : i = j.succ := by rw [ji, succ_pred] rw [this, cons_succ] by_cases h' : j.val < n · set k := castLT j h' with jk have : j = castSucc k := by rw [jk, castSucc_castLT] rw [this, ← castSucc_fin_succ, snoc] simp [pred, snoc, cons] rw [eq_last_of_not_lt h', succ_last] simp #align fin.cons_snoc_eq_snoc_cons Fin.cons_snoc_eq_snoc_cons theorem comp_snoc {α : Type} {β : Type} (g : α → β) (q : Fin n → α) (y : α) : g ∘ snoc q y = snoc (g ∘ q) (g y) := by ext j by_cases h : j.val < n · simp [h, snoc, castSucc_castLT] · rw [eq_last_of_not_lt h] simp #align fin.comp_snoc Fin.comp_snoc /-- Appending a one-tuple to the right is the same as `Fin.snoc`. -/ theorem append_right_eq_snoc {α : Type} {n : ℕ} (x : Fin n → α) (x₀ : Fin 1 → α) : Fin.append x x₀ = Fin.snoc x (x₀ 0) := by ext i refine Fin.addCases ?_ ?_ i <;> clear i · intro i rw [Fin.append_left] exact (@snoc_castSucc _ (fun _ => α) _ _ i).symm · intro i rw [Subsingleton.elim i 0, Fin.append_right] exact (@snoc_last _ (fun _ => α) _ _).symm #align fin.append_right_eq_snoc Fin.append_right_eq_snoc /-- `Fin.snoc` is the same as appending a one-tuple -/ theorem snoc_eq_append {α : Type} (xs : Fin n → α) (x : α) : snoc xs x = append xs (cons x Fin.elim0) := (append_right_eq_snoc xs (cons x Fin.elim0)).symm theorem append_left_snoc {n m} {α : Type} (xs : Fin n → α) (x : α) (ys : Fin m → α) : Fin.append (Fin.snoc xs x) ys = Fin.append xs (Fin.cons x ys) ∘ Fin.cast (Nat.succ_add_eq_add_succ ..) := by rw [snoc_eq_append, append_assoc, append_left_eq_cons, append_cast_right]; rfl theorem append_right_cons {n m} {α : Type} (xs : Fin n → α) (y : α) (ys : Fin m → α) : Fin.append xs (Fin.cons y ys) = Fin.append (Fin.snoc xs y) ys ∘ Fin.cast (Nat.succ_add_eq_add_succ ..).symm := by rw [append_left_snoc]; rfl theorem append_cons {α} (a : α) (as : Fin n → α) (bs : Fin m → α) : Fin.append (cons a as) bs = cons a (Fin.append as bs) ∘ (Fin.cast <\| Nat.add_right_comm n 1 m) := by funext i rcases i with ⟨i, -⟩ simp only [append, addCases, cons, castLT, cast, comp_apply] cases' i with i · simp · split_ifs with h · have : i < n := Nat.lt_of_succ_lt_succ h simp [addCases, this] · have : ¬i < n := Nat.not_le.mpr <\| Nat.lt_succ.mp <\| Nat.not_le.mp h simp [addCases, this] theorem append_snoc {α} (as : Fin n → α) (bs : Fin m → α) (b : α) : Fin.append as (snoc bs b) = snoc (Fin.append as bs) b := by funext i rcases i with ⟨i, isLt⟩ simp only [append, addCases, castLT, cast_mk, subNat_mk, natAdd_mk, cast, ge_iff_le, snoc.eq_1, cast_eq, eq_rec_constant, Nat.add_eq, Nat.add_zero, castLT_mk] split_ifs with lt_n lt_add sub_lt nlt_add lt_add <;> (try rfl) · have := Nat.lt_add_right m lt_n contradiction · obtain rfl := Nat.eq_of_le_of_lt_succ (Nat.not_lt.mp nlt_add) isLt simp [Nat.add_comm n m] at sub_lt · have := Nat.sub_lt_left_of_lt_add (Nat.not_lt.mp lt_n) lt_add contradiction theorem comp_init {α : Type} {β : Type} (g : α → β) (q : Fin n.succ → α) : g ∘ init q = init (g ∘ q) := by ext j simp [init] #align fin.comp_init Fin.comp_init /-- Recurse on an `n+1`-tuple by splitting it its initial `n`-tuple and its last element. -/ @[elab_as_elim, inline] def snocCases {P : (∀ i : Fin n.succ, α i) → Sort} (h : ∀ xs x, P (Fin.snoc xs x)) (x : ∀ i : Fin n.succ, α i) : P x := _root_.cast (by rw [Fin.snoc_init_self]) <\| h (Fin.init x) (x <\| Fin.last _) @[simp] lemma snocCases_snoc {P : (∀ i : Fin (n+1), α i) → Sort} (h : ∀ x x₀, P (Fin.snoc x x₀)) (x : ∀ i : Fin n, (Fin.init α) i) (x₀ : α (Fin.last _)) : snocCases h (Fin.snoc x x₀) = h x x₀ := by rw [snocCases, cast_eq_iff_heq, Fin.init_snoc, Fin.snoc_last] /-- Recurse on a tuple by splitting into `Fin.elim0` and `Fin.snoc`. -/ @[elab_as_elim] def snocInduction {α : Type} {P : ∀ {n : ℕ}, (Fin n → α) → Sort*} (h0 : P Fin.elim0) (h : ∀ {n} (x : Fin n → α) (x₀), P x → P (Fin.snoc x x₀)) : ∀ {n : ℕ} (x : Fin n → α), P x \| 0, x => by convert h0 \| n + 1, x => snocCases (fun x₀ x ↦ h _ _ <\| snocInduction h0 h _) x end TupleRight section InsertNth variable {α : Fin (n + 1) → Type u} {β : Type v} /- Porting note: Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling automatic insertion and specifying that motive seems to work. -/ /-- Define a function on `Fin (n + 1)` from a value on `i : Fin (n + 1)` and values on each `Fin.succAbove i j`, `j : Fin n`. This version is elaborated as eliminator and works for propositions, see also `Fin.insertNth` for a version without an `@[elab_as_elim]` attribute. -/ @[elab_as_elim] def succAboveCases {α : Fin (n + 1) → Sort u} (i : Fin (n + 1)) (x : α i) (p : ∀ j : Fin n, α (i.succAbove j)) (j : Fin (n + 1)) : α j := if hj : j = i then Eq.rec x hj.symm else if hlt : j < i then @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_castPred_of_lt _ _ hlt) (p _) else @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_pred_of_lt _ _ <\| (Ne.lt_or_lt hj).resolve_left hlt) (p _) #align fin.succ_above_cases Fin.succAboveCases theorem forall_iff_succAbove {p : Fin (n + 1) → Prop} (i : Fin (n + 1)) : (∀ j, p j) ↔ p i ∧ ∀ j, p (i.succAbove j) := ⟨fun h ↦ ⟨h _, fun _ ↦ h _⟩, fun h ↦ succAboveCases i h.1 h.2⟩ #align fin.forall_iff_succ_above Fin.forall_iff_succAbove /-- Insert an element into a tuple at a given position. For `i = 0` see `Fin.cons`, for `i = Fin.last n` see `Fin.snoc`. See also `Fin.succAboveCases` for a version elaborated as an eliminator. -/ def insertNth (i : Fin (n + 1)) (x : α i) (p : ∀ j : Fin n, α (i.succAbove j)) (j : Fin (n + 1)) : α j := succAboveCases i x p j #align fin.insert_nth Fin.insertNth @[simp] theorem insertNth_apply_same (i : Fin (n + 1)) (x : α i) (p : ∀ j, α (i.succAbove j)) : insertNth i x p i = x := by simp [insertNth, succAboveCases] #align fin.insert_nth_apply_same Fin.insertNth_apply_same @[simp] theorem insertNth_apply_succAbove (i : Fin (n + 1)) (x : α i) (p : ∀ j, α (i.succAbove j)) (j : Fin n) : insertNth i x p (i.succAbove j) = p j := by simp only [insertNth, succAboveCases, dif_neg (succAbove_ne _ _), succAbove_lt_iff_castSucc_lt] split_ifs with hlt · generalize_proofs H₁ H₂; revert H₂ generalize hk : castPred ((succAbove i) j) H₁ = k rw [castPred_succAbove _ _ hlt] at hk; cases hk intro; rfl · generalize_proofs H₁ H₂; revert H₂ generalize hk : pred (succAbove i j) H₁ = k erw [pred_succAbove _ _ (le_of_not_lt hlt)] at hk; cases hk intro; rfl #align fin.insert_nth_apply_succ_above Fin.insertNth_apply_succAbove @[simp] theorem succAbove_cases_eq_insertNth : @succAboveCases.{u + 1} = @insertNth.{u} := rfl #align fin.succ_above_cases_eq_insert_nth Fin.succAbove_cases_eq_insertNth /- Porting note: Had to `unfold comp`. Sometimes, when I use a placeholder, if I try to insert what Lean says it synthesized, it gives me a type error anyway. In this case, it's `x` and `p`. -/ @[simp] theorem insertNth_comp_succAbove (i : Fin (n + 1)) (x : β) (p : Fin n → β) : insertNth i x p ∘ i.succAbove = p := funext (by unfold comp; exact insertNth_apply_succAbove i _ _) #align fin.insert_nth_comp_succ_above Fin.insertNth_comp_succAbove theorem insertNth_eq_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} : i.insertNth x p = q ↔ q i = x ∧ p = fun j ↦ q (i.succAbove j) := by simp [funext_iff, forall_iff_succAbove i, eq_comm] #align fin.insert_nth_eq_iff Fin.insertNth_eq_iff theorem eq_insertNth_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} : q = i.insertNth x p ↔ q i = x ∧ p = fun j ↦ q (i.succAbove j) := eq_comm.trans insertNth_eq_iff #align fin.eq_insert_nth_iff Fin.eq_insertNth_iff /- Porting note: Once again, Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling automatic insertion and specifying that motive seems to work. -/ theorem insertNth_apply_below {i j : Fin (n + 1)} (h : j < i) (x : α i) (p : ∀ k, α (i.succAbove k)) : i.insertNth x p j = @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_castPred_of_lt _ _ h) (p <\| j.castPred _) := by rw [insertNth, succAboveCases, dif_neg h.ne, dif_pos h] #align fin.insert_nth_apply_below Fin.insertNth_apply_below /- Porting note: Once again, Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling automatic insertion and specifying that motive seems to work. -/	Mathlib/Data/Fin/Tuple/Basic.lean	827	831	theorem insertNth_apply_above {i j : Fin (n + 1)} (h : i < j) (x : α i) (p : ∀ k, α (i.succAbove k)) : i.insertNth x p j = @Eq.recOn _ _ (fun x _ ↦ α x) _ (succAbove_pred_of_lt _ _ h) (p <\| j.pred _) := by	rw [insertNth, succAboveCases, dif_neg h.ne', dif_neg h.not_lt]
/- Copyright (c) 2021 Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies [`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/data/finset/sym?range=98e83c3d541c77cdb7da20d79611a780ff8e7d90..02ba8949f486ebecf93fe7460f1ed0564b5e442c) -/ import Mathlib.Data.Finset.Lattice import Mathlib.Data.Fintype.Vector import Mathlib.Data.Multiset.Sym #align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c" /-! # Symmetric powers of a finset This file defines the symmetric powers of a finset as `Finset (Sym α n)` and `Finset (Sym2 α)`. ## Main declarations * `Finset.sym`: The symmetric power of a finset. `s.sym n` is all the multisets of cardinality `n` whose elements are in `s`. * `Finset.sym2`: The symmetric square of a finset. `s.sym2` is all the pairs whose elements are in `s`. * A `Fintype (Sym2 α)` instance that does not require `DecidableEq α`. ## TODO `Finset.sym` forms a Galois connection between `Finset α` and `Finset (Sym α n)`. Similar for `Finset.sym2`. -/ namespace Finset variable {α : Type} /-- `s.sym2` is the finset of all unordered pairs of elements from `s`. It is the image of `s ×ˢ s` under the quotient `α × α → Sym2 α`. -/ @[simps] protected def sym2 (s : Finset α) : Finset (Sym2 α) := ⟨s.1.sym2, s.2.sym2⟩ #align finset.sym2 Finset.sym2 section variable {s t : Finset α} {a b : α} theorem mk_mem_sym2_iff : s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s := by rw [mem_mk, sym2_val, Multiset.mk_mem_sym2_iff, mem_mk, mem_mk] #align finset.mk_mem_sym2_iff Finset.mk_mem_sym2_iff @[simp] theorem mem_sym2_iff {m : Sym2 α} : m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s := by rw [mem_mk, sym2_val, Multiset.mem_sym2_iff] simp only [mem_val] #align finset.mem_sym2_iff Finset.mem_sym2_iff instance _root_.Sym2.instFintype [Fintype α] : Fintype (Sym2 α) where elems := Finset.univ.sym2 complete := fun x ↦ by rw [mem_sym2_iff]; exact (fun a _ ↦ mem_univ a) -- Note(kmill): Using a default argument to make this simp lemma more general. @[simp] theorem sym2_univ [Fintype α] (inst : Fintype (Sym2 α) := Sym2.instFintype) : (univ : Finset α).sym2 = univ := by ext simp only [mem_sym2_iff, mem_univ, implies_true] #align finset.sym2_univ Finset.sym2_univ @[simp, mono] theorem sym2_mono (h : s ⊆ t) : s.sym2 ⊆ t.sym2 := by rw [← val_le_iff, sym2_val, sym2_val] apply Multiset.sym2_mono rwa [val_le_iff] #align finset.sym2_mono Finset.sym2_mono theorem monotone_sym2 : Monotone (Finset.sym2 : Finset α → _) := fun _ _ => sym2_mono theorem injective_sym2 : Function.Injective (Finset.sym2 : Finset α → _) := by intro s t h ext x simpa using congr(s(x, x) ∈ $h) theorem strictMono_sym2 : StrictMono (Finset.sym2 : Finset α → _) := monotone_sym2.strictMono_of_injective injective_sym2 theorem sym2_toFinset [DecidableEq α] (m : Multiset α) : m.toFinset.sym2 = m.sym2.toFinset := by ext z refine z.ind fun x y ↦ ?_ simp only [mk_mem_sym2_iff, Multiset.mem_toFinset, Multiset.mk_mem_sym2_iff] @[simp] theorem sym2_empty : (∅ : Finset α).sym2 = ∅ := rfl #align finset.sym2_empty Finset.sym2_empty @[simp] theorem sym2_eq_empty : s.sym2 = ∅ ↔ s = ∅ := by rw [← val_eq_zero, sym2_val, Multiset.sym2_eq_zero_iff, val_eq_zero] #align finset.sym2_eq_empty Finset.sym2_eq_empty @[simp, aesop safe apply (rule_sets := [finsetNonempty])] theorem sym2_nonempty : s.sym2.Nonempty ↔ s.Nonempty := by rw [← not_iff_not] simp_rw [not_nonempty_iff_eq_empty, sym2_eq_empty] #align finset.sym2_nonempty Finset.sym2_nonempty protected alias ⟨_, Nonempty.sym2⟩ := sym2_nonempty #align finset.nonempty.sym2 Finset.Nonempty.sym2 @[simp] theorem sym2_singleton (a : α) : ({a} : Finset α).sym2 = {Sym2.diag a} := rfl #align finset.sym2_singleton Finset.sym2_singleton /-- Finset stars and bars* for the case `n = 2`. -/ theorem card_sym2 (s : Finset α) : s.sym2.card = Nat.choose (s.card + 1) 2 := by rw [card_def, sym2_val, Multiset.card_sym2, ← card_def] #align finset.card_sym2 Finset.card_sym2 end variable [DecidableEq α] {s t : Finset α} {a b : α} theorem sym2_eq_image : s.sym2 = (s ×ˢ s).image Sym2.mk := by ext z refine z.ind fun x y ↦ ?_ rw [mk_mem_sym2_iff, mem_image] constructor · intro h use (x, y) simp only [mem_product, h, and_self, true_and] · rintro ⟨⟨a, b⟩, h⟩ simp only [mem_product, Sym2.eq_iff] at h obtain ⟨h, (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩)⟩ := h <;> simp [h] theorem isDiag_mk_of_mem_diag {a : α × α} (h : a ∈ s.diag) : (Sym2.mk a).IsDiag := (Sym2.isDiag_iff_proj_eq _).2 (mem_diag.1 h).2 #align finset.is_diag_mk_of_mem_diag Finset.isDiag_mk_of_mem_diag theorem not_isDiag_mk_of_mem_offDiag {a : α × α} (h : a ∈ s.offDiag) : ¬ (Sym2.mk a).IsDiag := by rw [Sym2.isDiag_iff_proj_eq] exact (mem_offDiag.1 h).2.2 #align finset.not_is_diag_mk_of_mem_off_diag Finset.not_isDiag_mk_of_mem_offDiag section Sym2 variable {m : Sym2 α} -- Porting note: add this lemma and remove simp in the next lemma since simpNF lint -- warns that its LHS is not in normal form @[simp]	Mathlib/Data/Finset/Sym.lean	152	154	theorem diag_mem_sym2_mem_iff : (∀ b, b ∈ Sym2.diag a → b ∈ s) ↔ a ∈ s := by	rw [← mem_sym2_iff] exact mk_mem_sym2_iff.trans <\| and_self_iff
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import Mathlib.Topology.Defs.Induced import Mathlib.Topology.Basic #align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4" /-! # Ordering on topologies and (co)induced topologies Topologies on a fixed type `α` are ordered, by reverse inclusion. That is, for topologies `t₁` and `t₂` on `α`, we write `t₁ ≤ t₂` if every set open in `t₂` is also open in `t₁`. (One also calls `t₁` finer than `t₂`, and `t₂` coarser than `t₁`.) Any function `f : α → β` induces * `TopologicalSpace.induced f : TopologicalSpace β → TopologicalSpace α`; * `TopologicalSpace.coinduced f : TopologicalSpace α → TopologicalSpace β`. Continuity, the ordering on topologies and (co)induced topologies are related as follows: * The identity map `(α, t₁) → (α, t₂)` is continuous iff `t₁ ≤ t₂`. * A map `f : (α, t) → (β, u)` is continuous * iff `t ≤ TopologicalSpace.induced f u` (`continuous_iff_le_induced`) * iff `TopologicalSpace.coinduced f t ≤ u` (`continuous_iff_coinduced_le`). Topologies on `α` form a complete lattice, with `⊥` the discrete topology and `⊤` the indiscrete topology. For a function `f : α → β`, `(TopologicalSpace.coinduced f, TopologicalSpace.induced f)` is a Galois connection between topologies on `α` and topologies on `β`. ## Implementation notes There is a Galois insertion between topologies on `α` (with the inclusion ordering) and all collections of sets in `α`. The complete lattice structure on topologies on `α` is defined as the reverse of the one obtained via this Galois insertion. More precisely, we use the corresponding Galois coinsertion between topologies on `α` (with the reversed inclusion ordering) and collections of sets in `α` (with the reversed inclusion ordering). ## Tags finer, coarser, induced topology, coinduced topology -/ open Function Set Filter Topology universe u v w namespace TopologicalSpace variable {α : Type u} /-- The open sets of the least topology containing a collection of basic sets. -/ inductive GenerateOpen (g : Set (Set α)) : Set α → Prop \| basic : ∀ s ∈ g, GenerateOpen g s \| univ : GenerateOpen g univ \| inter : ∀ s t, GenerateOpen g s → GenerateOpen g t → GenerateOpen g (s ∩ t) \| sUnion : ∀ S : Set (Set α), (∀ s ∈ S, GenerateOpen g s) → GenerateOpen g (⋃₀ S) #align topological_space.generate_open TopologicalSpace.GenerateOpen /-- The smallest topological space containing the collection `g` of basic sets -/ def generateFrom (g : Set (Set α)) : TopologicalSpace α where IsOpen := GenerateOpen g isOpen_univ := GenerateOpen.univ isOpen_inter := GenerateOpen.inter isOpen_sUnion := GenerateOpen.sUnion #align topological_space.generate_from TopologicalSpace.generateFrom theorem isOpen_generateFrom_of_mem {g : Set (Set α)} {s : Set α} (hs : s ∈ g) : IsOpen[generateFrom g] s := GenerateOpen.basic s hs #align topological_space.is_open_generate_from_of_mem TopologicalSpace.isOpen_generateFrom_of_mem theorem nhds_generateFrom {g : Set (Set α)} {a : α} : @nhds α (generateFrom g) a = ⨅ s ∈ { s \| a ∈ s ∧ s ∈ g }, 𝓟 s := by letI := generateFrom g rw [nhds_def] refine le_antisymm (biInf_mono fun s ⟨as, sg⟩ => ⟨as, .basic _ sg⟩) <\| le_iInf₂ ?_ rintro s ⟨ha, hs⟩ induction hs with \| basic _ hs => exact iInf₂_le _ ⟨ha, hs⟩ \| univ => exact le_top.trans_eq principal_univ.symm \| inter _ _ _ _ hs ht => exact (le_inf (hs ha.1) (ht ha.2)).trans_eq inf_principal \| sUnion _ _ hS => let ⟨t, htS, hat⟩ := ha exact (hS t htS hat).trans (principal_mono.2 <\| subset_sUnion_of_mem htS) #align topological_space.nhds_generate_from TopologicalSpace.nhds_generateFrom lemma tendsto_nhds_generateFrom_iff {β : Type} {m : α → β} {f : Filter α} {g : Set (Set β)} {b : β} : Tendsto m f (@nhds β (generateFrom g) b) ↔ ∀ s ∈ g, b ∈ s → m ⁻¹' s ∈ f := by simp only [nhds_generateFrom, @forall_swap (b ∈ _), tendsto_iInf, mem_setOf_eq, and_imp, tendsto_principal]; rfl @[deprecated] alias ⟨_, tendsto_nhds_generateFrom⟩ := tendsto_nhds_generateFrom_iff #align topological_space.tendsto_nhds_generate_from TopologicalSpace.tendsto_nhds_generateFrom /-- Construct a topology on α given the filter of neighborhoods of each point of α. -/ protected def mkOfNhds (n : α → Filter α) : TopologicalSpace α where IsOpen s := ∀ a ∈ s, s ∈ n a isOpen_univ _ _ := univ_mem isOpen_inter := fun _s _t hs ht x ⟨hxs, hxt⟩ => inter_mem (hs x hxs) (ht x hxt) isOpen_sUnion := fun _s hs _a ⟨x, hx, hxa⟩ => mem_of_superset (hs x hx _ hxa) (subset_sUnion_of_mem hx) #align topological_space.mk_of_nhds TopologicalSpace.mkOfNhds theorem nhds_mkOfNhds_of_hasBasis {n : α → Filter α} {ι : α → Sort} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α} (hb : ∀ a, (n a).HasBasis (p a) (s a)) (hpure : ∀ a i, p a i → a ∈ s a i) (hopen : ∀ a i, p a i → ∀ᶠ x in n a, s a i ∈ n x) (a : α) : @nhds α (.mkOfNhds n) a = n a := by let t : TopologicalSpace α := .mkOfNhds n apply le_antisymm · intro U hU replace hpure : pure ≤ n := fun x ↦ (hb x).ge_iff.2 (hpure x) refine mem_nhds_iff.2 ⟨{x \| U ∈ n x}, fun x hx ↦ hpure x hx, fun x hx ↦ ?_, hU⟩ rcases (hb x).mem_iff.1 hx with ⟨i, hpi, hi⟩ exact (hopen x i hpi).mono fun y hy ↦ mem_of_superset hy hi · exact (nhds_basis_opens a).ge_iff.2 fun U ⟨haU, hUo⟩ ↦ hUo a haU theorem nhds_mkOfNhds (n : α → Filter α) (a : α) (h₀ : pure ≤ n) (h₁ : ∀ a, ∀ s ∈ n a, ∀ᶠ y in n a, s ∈ n y) : @nhds α (TopologicalSpace.mkOfNhds n) a = n a := nhds_mkOfNhds_of_hasBasis (fun a ↦ (n a).basis_sets) h₀ h₁ _ #align topological_space.nhds_mk_of_nhds TopologicalSpace.nhds_mkOfNhds theorem nhds_mkOfNhds_single [DecidableEq α] {a₀ : α} {l : Filter α} (h : pure a₀ ≤ l) (b : α) : @nhds α (TopologicalSpace.mkOfNhds (update pure a₀ l)) b = (update pure a₀ l : α → Filter α) b := by refine nhds_mkOfNhds _ _ (le_update_iff.mpr ⟨h, fun _ _ => le_rfl⟩) fun a s hs => ?_ rcases eq_or_ne a a₀ with (rfl \| ha) · filter_upwards [hs] with b hb rcases eq_or_ne b a with (rfl \| hb) · exact hs · rwa [update_noteq hb] · simpa only [update_noteq ha, mem_pure, eventually_pure] using hs #align topological_space.nhds_mk_of_nhds_single TopologicalSpace.nhds_mkOfNhds_single theorem nhds_mkOfNhds_filterBasis (B : α → FilterBasis α) (a : α) (h₀ : ∀ x, ∀ n ∈ B x, x ∈ n) (h₁ : ∀ x, ∀ n ∈ B x, ∃ n₁ ∈ B x, ∀ x' ∈ n₁, ∃ n₂ ∈ B x', n₂ ⊆ n) : @nhds α (TopologicalSpace.mkOfNhds fun x => (B x).filter) a = (B a).filter := nhds_mkOfNhds_of_hasBasis (fun a ↦ (B a).hasBasis) h₀ h₁ a #align topological_space.nhds_mk_of_nhds_filter_basis TopologicalSpace.nhds_mkOfNhds_filterBasis section Lattice variable {α : Type u} {β : Type v} /-- The ordering on topologies on the type `α`. `t ≤ s` if every set open in `s` is also open in `t` (`t` is finer than `s`). -/ instance : PartialOrder (TopologicalSpace α) := { PartialOrder.lift (fun t => OrderDual.toDual IsOpen[t]) (fun _ _ => TopologicalSpace.ext) with le := fun s t => ∀ U, IsOpen[t] U → IsOpen[s] U } protected theorem le_def {α} {t s : TopologicalSpace α} : t ≤ s ↔ IsOpen[s] ≤ IsOpen[t] := Iff.rfl #align topological_space.le_def TopologicalSpace.le_def theorem le_generateFrom_iff_subset_isOpen {g : Set (Set α)} {t : TopologicalSpace α} : t ≤ generateFrom g ↔ g ⊆ { s \| IsOpen[t] s } := ⟨fun ht s hs => ht _ <\| .basic s hs, fun hg _s hs => hs.recOn (fun _ h => hg h) isOpen_univ (fun _ _ _ _ => IsOpen.inter) fun _ _ => isOpen_sUnion⟩ #align topological_space.le_generate_from_iff_subset_is_open TopologicalSpace.le_generateFrom_iff_subset_isOpen /-- If `s` equals the collection of open sets in the topology it generates, then `s` defines a topology. -/ protected def mkOfClosure (s : Set (Set α)) (hs : { u \| GenerateOpen s u } = s) : TopologicalSpace α where IsOpen u := u ∈ s isOpen_univ := hs ▸ TopologicalSpace.GenerateOpen.univ isOpen_inter := hs ▸ TopologicalSpace.GenerateOpen.inter isOpen_sUnion := hs ▸ TopologicalSpace.GenerateOpen.sUnion #align topological_space.mk_of_closure TopologicalSpace.mkOfClosure theorem mkOfClosure_sets {s : Set (Set α)} {hs : { u \| GenerateOpen s u } = s} : TopologicalSpace.mkOfClosure s hs = generateFrom s := TopologicalSpace.ext hs.symm #align topological_space.mk_of_closure_sets TopologicalSpace.mkOfClosure_sets theorem gc_generateFrom (α) : GaloisConnection (fun t : TopologicalSpace α => OrderDual.toDual { s \| IsOpen[t] s }) (generateFrom ∘ OrderDual.ofDual) := fun _ _ => le_generateFrom_iff_subset_isOpen.symm /-- The Galois coinsertion between `TopologicalSpace α` and `(Set (Set α))ᵒᵈ` whose lower part sends a topology to its collection of open subsets, and whose upper part sends a collection of subsets of `α` to the topology they generate. -/ def gciGenerateFrom (α : Type) : GaloisCoinsertion (fun t : TopologicalSpace α => OrderDual.toDual { s \| IsOpen[t] s }) (generateFrom ∘ OrderDual.ofDual) where gc := gc_generateFrom α u_l_le _ s hs := TopologicalSpace.GenerateOpen.basic s hs choice g hg := TopologicalSpace.mkOfClosure g (Subset.antisymm hg <\| le_generateFrom_iff_subset_isOpen.1 <\| le_rfl) choice_eq _ _ := mkOfClosure_sets #align gi_generate_from TopologicalSpace.gciGenerateFrom /-- Topologies on `α` form a complete lattice, with `⊥` the discrete topology and `⊤` the indiscrete topology. The infimum of a collection of topologies is the topology generated by all their open sets, while the supremum is the topology whose open sets are those sets open in every member of the collection. -/ instance : CompleteLattice (TopologicalSpace α) := (gciGenerateFrom α).liftCompleteLattice @[mono] theorem generateFrom_anti {α} {g₁ g₂ : Set (Set α)} (h : g₁ ⊆ g₂) : generateFrom g₂ ≤ generateFrom g₁ := (gc_generateFrom _).monotone_u h #align topological_space.generate_from_anti TopologicalSpace.generateFrom_anti theorem generateFrom_setOf_isOpen (t : TopologicalSpace α) : generateFrom { s \| IsOpen[t] s } = t := (gciGenerateFrom α).u_l_eq t #align topological_space.generate_from_set_of_is_open TopologicalSpace.generateFrom_setOf_isOpen theorem leftInverse_generateFrom : LeftInverse generateFrom fun t : TopologicalSpace α => { s \| IsOpen[t] s } := (gciGenerateFrom α).u_l_leftInverse #align topological_space.left_inverse_generate_from TopologicalSpace.leftInverse_generateFrom theorem generateFrom_surjective : Surjective (generateFrom : Set (Set α) → TopologicalSpace α) := (gciGenerateFrom α).u_surjective #align topological_space.generate_from_surjective TopologicalSpace.generateFrom_surjective theorem setOf_isOpen_injective : Injective fun t : TopologicalSpace α => { s \| IsOpen[t] s } := (gciGenerateFrom α).l_injective #align topological_space.set_of_is_open_injective TopologicalSpace.setOf_isOpen_injective end Lattice end TopologicalSpace section Lattice variable {α : Type} {t t₁ t₂ : TopologicalSpace α} {s : Set α} theorem IsOpen.mono (hs : IsOpen[t₂] s) (h : t₁ ≤ t₂) : IsOpen[t₁] s := h s hs #align is_open.mono IsOpen.mono theorem IsClosed.mono (hs : IsClosed[t₂] s) (h : t₁ ≤ t₂) : IsClosed[t₁] s := (@isOpen_compl_iff α s t₁).mp <\| hs.isOpen_compl.mono h #align is_closed.mono IsClosed.mono theorem closure.mono (h : t₁ ≤ t₂) : closure[t₁] s ⊆ closure[t₂] s := @closure_minimal _ s (@closure _ t₂ s) t₁ subset_closure (IsClosed.mono isClosed_closure h) theorem isOpen_implies_isOpen_iff : (∀ s, IsOpen[t₁] s → IsOpen[t₂] s) ↔ t₂ ≤ t₁ := Iff.rfl #align is_open_implies_is_open_iff isOpen_implies_isOpen_iff /-- The only open sets in the indiscrete topology are the empty set and the whole space. -/ theorem TopologicalSpace.isOpen_top_iff {α} (U : Set α) : IsOpen[⊤] U ↔ U = ∅ ∨ U = univ := ⟨fun h => by induction h with \| basic _ h => exact False.elim h \| univ => exact .inr rfl \| inter _ _ _ _ h₁ h₂ => rcases h₁ with (rfl \| rfl) <;> rcases h₂ with (rfl \| rfl) <;> simp \| sUnion _ _ ih => exact sUnion_mem_empty_univ ih, by rintro (rfl \| rfl) exacts [@isOpen_empty _ ⊤, @isOpen_univ _ ⊤]⟩ #align topological_space.is_open_top_iff TopologicalSpace.isOpen_top_iff /-- A topological space is discrete if every set is open, that is, its topology equals the discrete topology `⊥`. -/ class DiscreteTopology (α : Type) [t : TopologicalSpace α] : Prop where /-- The `TopologicalSpace` structure on a type with discrete topology is equal to `⊥`. -/ eq_bot : t = ⊥ #align discrete_topology DiscreteTopology theorem discreteTopology_bot (α : Type) : @DiscreteTopology α ⊥ := @DiscreteTopology.mk α ⊥ rfl #align discrete_topology_bot discreteTopology_bot section DiscreteTopology variable [TopologicalSpace α] [DiscreteTopology α] {β : Type} @[simp] theorem isOpen_discrete (s : Set α) : IsOpen s := (@DiscreteTopology.eq_bot α _).symm ▸ trivial #align is_open_discrete isOpen_discrete @[simp] theorem isClosed_discrete (s : Set α) : IsClosed s := ⟨isOpen_discrete _⟩ #align is_closed_discrete isClosed_discrete @[simp] theorem closure_discrete (s : Set α) : closure s = s := (isClosed_discrete _).closure_eq @[simp] theorem dense_discrete {s : Set α} : Dense s ↔ s = univ := by simp [dense_iff_closure_eq] @[simp] theorem denseRange_discrete {ι : Type} {f : ι → α} : DenseRange f ↔ Surjective f := by rw [DenseRange, dense_discrete, range_iff_surjective] @[nontriviality, continuity] theorem continuous_of_discreteTopology [TopologicalSpace β] {f : α → β} : Continuous f := continuous_def.2 fun _ _ => isOpen_discrete _ #align continuous_of_discrete_topology continuous_of_discreteTopology /-- A function to a discrete topological space is continuous if and only if the preimage of every singleton is open. -/ theorem continuous_discrete_rng [TopologicalSpace β] [DiscreteTopology β] {f : α → β} : Continuous f ↔ ∀ b : β, IsOpen (f ⁻¹' {b}) := ⟨fun h b => (isOpen_discrete _).preimage h, fun h => ⟨fun s _ => by rw [← biUnion_of_singleton s, preimage_iUnion₂] exact isOpen_biUnion fun _ _ => h _⟩⟩ @[simp] theorem nhds_discrete (α : Type) [TopologicalSpace α] [DiscreteTopology α] : @nhds α _ = pure := le_antisymm (fun _ s hs => (isOpen_discrete s).mem_nhds hs) pure_le_nhds #align nhds_discrete nhds_discrete theorem mem_nhds_discrete {x : α} {s : Set α} : s ∈ 𝓝 x ↔ x ∈ s := by rw [nhds_discrete, mem_pure] #align mem_nhds_discrete mem_nhds_discrete end DiscreteTopology theorem le_of_nhds_le_nhds (h : ∀ x, @nhds α t₁ x ≤ @nhds α t₂ x) : t₁ ≤ t₂ := fun s => by rw [@isOpen_iff_mem_nhds _ _ t₁, @isOpen_iff_mem_nhds α _ t₂] exact fun hs a ha => h _ (hs _ ha) #align le_of_nhds_le_nhds le_of_nhds_le_nhds @[deprecated (since := "2024-03-01")] alias eq_of_nhds_eq_nhds := TopologicalSpace.ext_nhds #align eq_of_nhds_eq_nhds TopologicalSpace.ext_nhds theorem eq_bot_of_singletons_open {t : TopologicalSpace α} (h : ∀ x, IsOpen[t] {x}) : t = ⊥ := bot_unique fun s _ => biUnion_of_singleton s ▸ isOpen_biUnion fun x _ => h x #align eq_bot_of_singletons_open eq_bot_of_singletons_open theorem forall_open_iff_discrete {X : Type} [TopologicalSpace X] : (∀ s : Set X, IsOpen s) ↔ DiscreteTopology X := ⟨fun h => ⟨eq_bot_of_singletons_open fun _ => h _⟩, @isOpen_discrete _ _⟩ #align forall_open_iff_discrete forall_open_iff_discrete theorem discreteTopology_iff_forall_isClosed [TopologicalSpace α] : DiscreteTopology α ↔ ∀ s : Set α, IsClosed s := forall_open_iff_discrete.symm.trans <\| compl_surjective.forall.trans <\| forall_congr' fun _ ↦ isOpen_compl_iff theorem singletons_open_iff_discrete {X : Type} [TopologicalSpace X] : (∀ a : X, IsOpen ({a} : Set X)) ↔ DiscreteTopology X := ⟨fun h => ⟨eq_bot_of_singletons_open h⟩, fun a _ => @isOpen_discrete _ _ a _⟩ #align singletons_open_iff_discrete singletons_open_iff_discrete theorem discreteTopology_iff_singleton_mem_nhds [TopologicalSpace α] : DiscreteTopology α ↔ ∀ x : α, {x} ∈ 𝓝 x := by simp only [← singletons_open_iff_discrete, isOpen_iff_mem_nhds, mem_singleton_iff, forall_eq] #align discrete_topology_iff_singleton_mem_nhds discreteTopology_iff_singleton_mem_nhds /-- This lemma characterizes discrete topological spaces as those whose singletons are neighbourhoods. -/ theorem discreteTopology_iff_nhds [TopologicalSpace α] : DiscreteTopology α ↔ ∀ x : α, 𝓝 x = pure x := by simp only [discreteTopology_iff_singleton_mem_nhds, ← nhds_neBot.le_pure_iff, le_pure_iff] #align discrete_topology_iff_nhds discreteTopology_iff_nhds theorem discreteTopology_iff_nhds_ne [TopologicalSpace α] : DiscreteTopology α ↔ ∀ x : α, 𝓝[≠] x = ⊥ := by simp only [discreteTopology_iff_singleton_mem_nhds, nhdsWithin, inf_principal_eq_bot, compl_compl] #align discrete_topology_iff_nhds_ne discreteTopology_iff_nhds_ne /-- If the codomain of a continuous injective function has discrete topology, then so does the domain. See also `Embedding.discreteTopology` for an important special case. -/ theorem DiscreteTopology.of_continuous_injective {β : Type} [TopologicalSpace α] [TopologicalSpace β] [DiscreteTopology β] {f : α → β} (hc : Continuous f) (hinj : Injective f) : DiscreteTopology α := forall_open_iff_discrete.1 fun s ↦ hinj.preimage_image s ▸ (isOpen_discrete _).preimage hc end Lattice section GaloisConnection variable {α β γ : Type*} theorem isOpen_induced_iff [t : TopologicalSpace β] {s : Set α} {f : α → β} : IsOpen[t.induced f] s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := Iff.rfl #align is_open_induced_iff isOpen_induced_iff theorem isClosed_induced_iff [t : TopologicalSpace β] {s : Set α} {f : α → β} : IsClosed[t.induced f] s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by letI := t.induced f simp only [← isOpen_compl_iff, isOpen_induced_iff] exact compl_surjective.exists.trans (by simp only [preimage_compl, compl_inj_iff]) #align is_closed_induced_iff isClosed_induced_iff theorem isOpen_coinduced {t : TopologicalSpace α} {s : Set β} {f : α → β} : IsOpen[t.coinduced f] s ↔ IsOpen (f ⁻¹' s) := Iff.rfl #align is_open_coinduced isOpen_coinduced theorem preimage_nhds_coinduced [TopologicalSpace α] {π : α → β} {s : Set β} {a : α} (hs : s ∈ @nhds β (TopologicalSpace.coinduced π ‹_›) (π a)) : π ⁻¹' s ∈ 𝓝 a := by letI := TopologicalSpace.coinduced π ‹_› rcases mem_nhds_iff.mp hs with ⟨V, hVs, V_op, mem_V⟩ exact mem_nhds_iff.mpr ⟨π ⁻¹' V, Set.preimage_mono hVs, V_op, mem_V⟩ #align preimage_nhds_coinduced preimage_nhds_coinduced variable {t t₁ t₂ : TopologicalSpace α} {t' : TopologicalSpace β} {f : α → β} {g : β → α} theorem Continuous.coinduced_le (h : Continuous[t, t'] f) : t.coinduced f ≤ t' := (@continuous_def α β t t').1 h #align continuous.coinduced_le Continuous.coinduced_le theorem coinduced_le_iff_le_induced {f : α → β} {tα : TopologicalSpace α} {tβ : TopologicalSpace β} : tα.coinduced f ≤ tβ ↔ tα ≤ tβ.induced f := ⟨fun h _s ⟨_t, ht, hst⟩ => hst ▸ h _ ht, fun h s hs => h _ ⟨s, hs, rfl⟩⟩ #align coinduced_le_iff_le_induced coinduced_le_iff_le_induced theorem Continuous.le_induced (h : Continuous[t, t'] f) : t ≤ t'.induced f := coinduced_le_iff_le_induced.1 h.coinduced_le #align continuous.le_induced Continuous.le_induced theorem gc_coinduced_induced (f : α → β) : GaloisConnection (TopologicalSpace.coinduced f) (TopologicalSpace.induced f) := fun _ _ => coinduced_le_iff_le_induced #align gc_coinduced_induced gc_coinduced_induced theorem induced_mono (h : t₁ ≤ t₂) : t₁.induced g ≤ t₂.induced g := (gc_coinduced_induced g).monotone_u h #align induced_mono induced_mono theorem coinduced_mono (h : t₁ ≤ t₂) : t₁.coinduced f ≤ t₂.coinduced f := (gc_coinduced_induced f).monotone_l h #align coinduced_mono coinduced_mono @[simp] theorem induced_top : (⊤ : TopologicalSpace α).induced g = ⊤ := (gc_coinduced_induced g).u_top #align induced_top induced_top @[simp] theorem induced_inf : (t₁ ⊓ t₂).induced g = t₁.induced g ⊓ t₂.induced g := (gc_coinduced_induced g).u_inf #align induced_inf induced_inf @[simp] theorem induced_iInf {ι : Sort w} {t : ι → TopologicalSpace α} : (⨅ i, t i).induced g = ⨅ i, (t i).induced g := (gc_coinduced_induced g).u_iInf #align induced_infi induced_iInf @[simp] theorem coinduced_bot : (⊥ : TopologicalSpace α).coinduced f = ⊥ := (gc_coinduced_induced f).l_bot #align coinduced_bot coinduced_bot @[simp] theorem coinduced_sup : (t₁ ⊔ t₂).coinduced f = t₁.coinduced f ⊔ t₂.coinduced f := (gc_coinduced_induced f).l_sup #align coinduced_sup coinduced_sup @[simp] theorem coinduced_iSup {ι : Sort w} {t : ι → TopologicalSpace α} : (⨆ i, t i).coinduced f = ⨆ i, (t i).coinduced f := (gc_coinduced_induced f).l_iSup #align coinduced_supr coinduced_iSup theorem induced_id [t : TopologicalSpace α] : t.induced id = t := TopologicalSpace.ext <\| funext fun s => propext <\| ⟨fun ⟨_, hs, h⟩ => h ▸ hs, fun hs => ⟨s, hs, rfl⟩⟩ #align induced_id induced_id theorem induced_compose {tγ : TopologicalSpace γ} {f : α → β} {g : β → γ} : (tγ.induced g).induced f = tγ.induced (g ∘ f) := TopologicalSpace.ext <\| funext fun _ => propext ⟨fun ⟨_, ⟨s, hs, h₂⟩, h₁⟩ => h₁ ▸ h₂ ▸ ⟨s, hs, rfl⟩, fun ⟨s, hs, h⟩ => ⟨preimage g s, ⟨s, hs, rfl⟩, h ▸ rfl⟩⟩ #align induced_compose induced_compose theorem induced_const [t : TopologicalSpace α] {x : α} : (t.induced fun _ : β => x) = ⊤ := le_antisymm le_top (@continuous_const β α ⊤ t x).le_induced #align induced_const induced_const theorem coinduced_id [t : TopologicalSpace α] : t.coinduced id = t := TopologicalSpace.ext rfl #align coinduced_id coinduced_id theorem coinduced_compose [tα : TopologicalSpace α] {f : α → β} {g : β → γ} : (tα.coinduced f).coinduced g = tα.coinduced (g ∘ f) := TopologicalSpace.ext rfl #align coinduced_compose coinduced_compose	Mathlib/Topology/Order.lean	489	493	theorem Equiv.induced_symm {α β : Type*} (e : α ≃ β) : TopologicalSpace.induced e.symm = TopologicalSpace.coinduced e := by	ext t U rw [isOpen_induced_iff, isOpen_coinduced] simp only [e.symm.preimage_eq_iff_eq_image, exists_eq_right, ← preimage_equiv_eq_image_symm]
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne -/ import Mathlib.Order.MinMax import Mathlib.Data.Set.Subsingleton import Mathlib.Tactic.Says #align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c" /-! # Intervals In any preorder `α`, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. This file contains these definitions, and basic facts on inclusion, intersection, difference of intervals (where the precise statements may depend on the properties of the order, in particular for some statements it should be `LinearOrder` or `DenselyOrdered`). TODO: This is just the beginning; a lot of rules are missing -/ open Function open OrderDual (toDual ofDual) variable {α β : Type} namespace Set section Preorder variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α} /-- Left-open right-open interval -/ def Ioo (a b : α) := { x \| a < x ∧ x < b } #align set.Ioo Set.Ioo /-- Left-closed right-open interval -/ def Ico (a b : α) := { x \| a ≤ x ∧ x < b } #align set.Ico Set.Ico /-- Left-infinite right-open interval -/ def Iio (a : α) := { x \| x < a } #align set.Iio Set.Iio /-- Left-closed right-closed interval -/ def Icc (a b : α) := { x \| a ≤ x ∧ x ≤ b } #align set.Icc Set.Icc /-- Left-infinite right-closed interval -/ def Iic (b : α) := { x \| x ≤ b } #align set.Iic Set.Iic /-- Left-open right-closed interval -/ def Ioc (a b : α) := { x \| a < x ∧ x ≤ b } #align set.Ioc Set.Ioc /-- Left-closed right-infinite interval -/ def Ici (a : α) := { x \| a ≤ x } #align set.Ici Set.Ici /-- Left-open right-infinite interval -/ def Ioi (a : α) := { x \| a < x } #align set.Ioi Set.Ioi theorem Ioo_def (a b : α) : { x \| a < x ∧ x < b } = Ioo a b := rfl #align set.Ioo_def Set.Ioo_def theorem Ico_def (a b : α) : { x \| a ≤ x ∧ x < b } = Ico a b := rfl #align set.Ico_def Set.Ico_def theorem Iio_def (a : α) : { x \| x < a } = Iio a := rfl #align set.Iio_def Set.Iio_def theorem Icc_def (a b : α) : { x \| a ≤ x ∧ x ≤ b } = Icc a b := rfl #align set.Icc_def Set.Icc_def theorem Iic_def (b : α) : { x \| x ≤ b } = Iic b := rfl #align set.Iic_def Set.Iic_def theorem Ioc_def (a b : α) : { x \| a < x ∧ x ≤ b } = Ioc a b := rfl #align set.Ioc_def Set.Ioc_def theorem Ici_def (a : α) : { x \| a ≤ x } = Ici a := rfl #align set.Ici_def Set.Ici_def theorem Ioi_def (a : α) : { x \| a < x } = Ioi a := rfl #align set.Ioi_def Set.Ioi_def @[simp] theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b := Iff.rfl #align set.mem_Ioo Set.mem_Ioo @[simp] theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b := Iff.rfl #align set.mem_Ico Set.mem_Ico @[simp] theorem mem_Iio : x ∈ Iio b ↔ x < b := Iff.rfl #align set.mem_Iio Set.mem_Iio @[simp] theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b := Iff.rfl #align set.mem_Icc Set.mem_Icc @[simp] theorem mem_Iic : x ∈ Iic b ↔ x ≤ b := Iff.rfl #align set.mem_Iic Set.mem_Iic @[simp] theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b := Iff.rfl #align set.mem_Ioc Set.mem_Ioc @[simp] theorem mem_Ici : x ∈ Ici a ↔ a ≤ x := Iff.rfl #align set.mem_Ici Set.mem_Ici @[simp] theorem mem_Ioi : x ∈ Ioi a ↔ a < x := Iff.rfl #align set.mem_Ioi Set.mem_Ioi instance decidableMemIoo [Decidable (a < x ∧ x < b)] : Decidable (x ∈ Ioo a b) := by assumption #align set.decidable_mem_Ioo Set.decidableMemIoo instance decidableMemIco [Decidable (a ≤ x ∧ x < b)] : Decidable (x ∈ Ico a b) := by assumption #align set.decidable_mem_Ico Set.decidableMemIco instance decidableMemIio [Decidable (x < b)] : Decidable (x ∈ Iio b) := by assumption #align set.decidable_mem_Iio Set.decidableMemIio instance decidableMemIcc [Decidable (a ≤ x ∧ x ≤ b)] : Decidable (x ∈ Icc a b) := by assumption #align set.decidable_mem_Icc Set.decidableMemIcc instance decidableMemIic [Decidable (x ≤ b)] : Decidable (x ∈ Iic b) := by assumption #align set.decidable_mem_Iic Set.decidableMemIic instance decidableMemIoc [Decidable (a < x ∧ x ≤ b)] : Decidable (x ∈ Ioc a b) := by assumption #align set.decidable_mem_Ioc Set.decidableMemIoc instance decidableMemIci [Decidable (a ≤ x)] : Decidable (x ∈ Ici a) := by assumption #align set.decidable_mem_Ici Set.decidableMemIci instance decidableMemIoi [Decidable (a < x)] : Decidable (x ∈ Ioi a) := by assumption #align set.decidable_mem_Ioi Set.decidableMemIoi -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioo Set.left_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl] #align set.left_mem_Ico Set.left_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.left_mem_Icc Set.left_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by simp [lt_irrefl] #align set.left_mem_Ioc Set.left_mem_Ioc theorem left_mem_Ici : a ∈ Ici a := by simp #align set.left_mem_Ici Set.left_mem_Ici -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioo : b ∈ Ioo a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ioo Set.right_mem_Ioo -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ico : b ∈ Ico a b ↔ False := by simp [lt_irrefl] #align set.right_mem_Ico Set.right_mem_Ico -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl] #align set.right_mem_Icc Set.right_mem_Icc -- Porting note (#10618): `simp` can prove this -- @[simp] theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl] #align set.right_mem_Ioc Set.right_mem_Ioc theorem right_mem_Iic : a ∈ Iic a := by simp #align set.right_mem_Iic Set.right_mem_Iic @[simp] theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a := rfl #align set.dual_Ici Set.dual_Ici @[simp] theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a := rfl #align set.dual_Iic Set.dual_Iic @[simp] theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a := rfl #align set.dual_Ioi Set.dual_Ioi @[simp] theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a := rfl #align set.dual_Iio Set.dual_Iio @[simp] theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a := Set.ext fun _ => and_comm #align set.dual_Icc Set.dual_Icc @[simp] theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a := Set.ext fun _ => and_comm #align set.dual_Ioc Set.dual_Ioc @[simp] theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a := Set.ext fun _ => and_comm #align set.dual_Ico Set.dual_Ico @[simp] theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a := Set.ext fun _ => and_comm #align set.dual_Ioo Set.dual_Ioo @[simp] theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := ⟨fun ⟨_, hx⟩ => hx.1.trans hx.2, fun h => ⟨a, left_mem_Icc.2 h⟩⟩ #align set.nonempty_Icc Set.nonempty_Icc @[simp] theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_lt hx.2, fun h => ⟨a, left_mem_Ico.2 h⟩⟩ #align set.nonempty_Ico Set.nonempty_Ico @[simp] theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := ⟨fun ⟨_, hx⟩ => hx.1.trans_le hx.2, fun h => ⟨b, right_mem_Ioc.2 h⟩⟩ #align set.nonempty_Ioc Set.nonempty_Ioc @[simp] theorem nonempty_Ici : (Ici a).Nonempty := ⟨a, left_mem_Ici⟩ #align set.nonempty_Ici Set.nonempty_Ici @[simp] theorem nonempty_Iic : (Iic a).Nonempty := ⟨a, right_mem_Iic⟩ #align set.nonempty_Iic Set.nonempty_Iic @[simp] theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := ⟨fun ⟨_, ha, hb⟩ => ha.trans hb, exists_between⟩ #align set.nonempty_Ioo Set.nonempty_Ioo @[simp] theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty := exists_gt a #align set.nonempty_Ioi Set.nonempty_Ioi @[simp] theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty := exists_lt a #align set.nonempty_Iio Set.nonempty_Iio theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) := Nonempty.to_subtype (nonempty_Icc.mpr h) #align set.nonempty_Icc_subtype Set.nonempty_Icc_subtype theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) := Nonempty.to_subtype (nonempty_Ico.mpr h) #align set.nonempty_Ico_subtype Set.nonempty_Ico_subtype theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) := Nonempty.to_subtype (nonempty_Ioc.mpr h) #align set.nonempty_Ioc_subtype Set.nonempty_Ioc_subtype /-- An interval `Ici a` is nonempty. -/ instance nonempty_Ici_subtype : Nonempty (Ici a) := Nonempty.to_subtype nonempty_Ici #align set.nonempty_Ici_subtype Set.nonempty_Ici_subtype /-- An interval `Iic a` is nonempty. -/ instance nonempty_Iic_subtype : Nonempty (Iic a) := Nonempty.to_subtype nonempty_Iic #align set.nonempty_Iic_subtype Set.nonempty_Iic_subtype theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) := Nonempty.to_subtype (nonempty_Ioo.mpr h) #align set.nonempty_Ioo_subtype Set.nonempty_Ioo_subtype /-- In an order without maximal elements, the intervals `Ioi` are nonempty. -/ instance nonempty_Ioi_subtype [NoMaxOrder α] : Nonempty (Ioi a) := Nonempty.to_subtype nonempty_Ioi #align set.nonempty_Ioi_subtype Set.nonempty_Ioi_subtype /-- In an order without minimal elements, the intervals `Iio` are nonempty. -/ instance nonempty_Iio_subtype [NoMinOrder α] : Nonempty (Iio a) := Nonempty.to_subtype nonempty_Iio #align set.nonempty_Iio_subtype Set.nonempty_Iio_subtype instance [NoMinOrder α] : NoMinOrder (Iio a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, lt_trans hb a.2⟩, hb⟩⟩ instance [NoMinOrder α] : NoMinOrder (Iic a) := ⟨fun a => let ⟨b, hb⟩ := exists_lt (a : α) ⟨⟨b, hb.le.trans a.2⟩, hb⟩⟩ instance [NoMaxOrder α] : NoMaxOrder (Ioi a) := OrderDual.noMaxOrder (α := Iio (toDual a)) instance [NoMaxOrder α] : NoMaxOrder (Ici a) := OrderDual.noMaxOrder (α := Iic (toDual a)) @[simp] theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Icc_eq_empty Set.Icc_eq_empty @[simp] theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_lt hb) #align set.Ico_eq_empty Set.Ico_eq_empty @[simp] theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_le hb) #align set.Ioc_eq_empty Set.Ioc_eq_empty @[simp] theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ := eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb) #align set.Ioo_eq_empty Set.Ioo_eq_empty @[simp] theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ := Icc_eq_empty h.not_le #align set.Icc_eq_empty_of_lt Set.Icc_eq_empty_of_lt @[simp] theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ := Ico_eq_empty h.not_lt #align set.Ico_eq_empty_of_le Set.Ico_eq_empty_of_le @[simp] theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ := Ioc_eq_empty h.not_lt #align set.Ioc_eq_empty_of_le Set.Ioc_eq_empty_of_le @[simp] theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ := Ioo_eq_empty h.not_lt #align set.Ioo_eq_empty_of_le Set.Ioo_eq_empty_of_le -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ico_self (a : α) : Ico a a = ∅ := Ico_eq_empty <\| lt_irrefl _ #align set.Ico_self Set.Ico_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioc_self (a : α) : Ioc a a = ∅ := Ioc_eq_empty <\| lt_irrefl _ #align set.Ioc_self Set.Ioc_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioo_self (a : α) : Ioo a a = ∅ := Ioo_eq_empty <\| lt_irrefl _ #align set.Ioo_self Set.Ioo_self theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a := ⟨fun h => h <\| left_mem_Ici, fun h _ hx => h.trans hx⟩ #align set.Ici_subset_Ici Set.Ici_subset_Ici @[gcongr] alias ⟨_, _root_.GCongr.Ici_subset_Ici_of_le⟩ := Ici_subset_Ici theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b := @Ici_subset_Ici αᵒᵈ _ _ _ #align set.Iic_subset_Iic Set.Iic_subset_Iic @[gcongr] alias ⟨_, _root_.GCongr.Iic_subset_Iic_of_le⟩ := Iic_subset_Iic theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a := ⟨fun h => h left_mem_Ici, fun h _ hx => h.trans_le hx⟩ #align set.Ici_subset_Ioi Set.Ici_subset_Ioi theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b := ⟨fun h => h right_mem_Iic, fun h _ hx => lt_of_le_of_lt hx h⟩ #align set.Iic_subset_Iio Set.Iic_subset_Iio @[gcongr] theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans_le h₂⟩ #align set.Ioo_subset_Ioo Set.Ioo_subset_Ioo @[gcongr] theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b := Ioo_subset_Ioo h le_rfl #align set.Ioo_subset_Ioo_left Set.Ioo_subset_Ioo_left @[gcongr] theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ := Ioo_subset_Ioo le_rfl h #align set.Ioo_subset_Ioo_right Set.Ioo_subset_Ioo_right @[gcongr] theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, hx₂.trans_le h₂⟩ #align set.Ico_subset_Ico Set.Ico_subset_Ico @[gcongr] theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b := Ico_subset_Ico h le_rfl #align set.Ico_subset_Ico_left Set.Ico_subset_Ico_left @[gcongr] theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ := Ico_subset_Ico le_rfl h #align set.Ico_subset_Ico_right Set.Ico_subset_Ico_right @[gcongr] theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans hx₁, le_trans hx₂ h₂⟩ #align set.Icc_subset_Icc Set.Icc_subset_Icc @[gcongr] theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b := Icc_subset_Icc h le_rfl #align set.Icc_subset_Icc_left Set.Icc_subset_Icc_left @[gcongr] theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ := Icc_subset_Icc le_rfl h #align set.Icc_subset_Icc_right Set.Icc_subset_Icc_right theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ hx => ⟨ha.trans_le hx.1, hx.2.trans_lt hb⟩ #align set.Icc_subset_Ioo Set.Icc_subset_Ioo theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := fun _ => And.left #align set.Icc_subset_Ici_self Set.Icc_subset_Ici_self theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := fun _ => And.right #align set.Icc_subset_Iic_self Set.Icc_subset_Iic_self theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := fun _ => And.right #align set.Ioc_subset_Iic_self Set.Ioc_subset_Iic_self @[gcongr] theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ => ⟨h₁.trans_lt hx₁, hx₂.trans h₂⟩ #align set.Ioc_subset_Ioc Set.Ioc_subset_Ioc @[gcongr] theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b := Ioc_subset_Ioc h le_rfl #align set.Ioc_subset_Ioc_left Set.Ioc_subset_Ioc_left @[gcongr] theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ := Ioc_subset_Ioc le_rfl h #align set.Ioc_subset_Ioc_right Set.Ioc_subset_Ioc_right theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := fun _ => And.imp_left h₁.trans_le #align set.Ico_subset_Ioo_left Set.Ico_subset_Ioo_left theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := fun _ => And.imp_right fun h' => h'.trans_lt h #align set.Ioc_subset_Ioo_right Set.Ioc_subset_Ioo_right theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := fun _ => And.imp_right fun h₂ => h₂.trans_lt h₁ #align set.Icc_subset_Ico_right Set.Icc_subset_Ico_right theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := fun _ => And.imp_left le_of_lt #align set.Ioo_subset_Ico_self Set.Ioo_subset_Ico_self theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := fun _ => And.imp_right le_of_lt #align set.Ioo_subset_Ioc_self Set.Ioo_subset_Ioc_self theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := fun _ => And.imp_right le_of_lt #align set.Ico_subset_Icc_self Set.Ico_subset_Icc_self theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := fun _ => And.imp_left le_of_lt #align set.Ioc_subset_Icc_self Set.Ioc_subset_Icc_self theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b := Subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self #align set.Ioo_subset_Icc_self Set.Ioo_subset_Icc_self theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := fun _ => And.right #align set.Ico_subset_Iio_self Set.Ico_subset_Iio_self theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := fun _ => And.right #align set.Ioo_subset_Iio_self Set.Ioo_subset_Iio_self theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := fun _ => And.left #align set.Ioc_subset_Ioi_self Set.Ioc_subset_Ioi_self theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := fun _ => And.left #align set.Ioo_subset_Ioi_self Set.Ioo_subset_Ioi_self theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a := fun _ hx => le_of_lt hx #align set.Ioi_subset_Ici_self Set.Ioi_subset_Ici_self theorem Iio_subset_Iic_self : Iio a ⊆ Iic a := fun _ hx => le_of_lt hx #align set.Iio_subset_Iic_self Set.Iio_subset_Iic_self theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := fun _ => And.left #align set.Ico_subset_Ici_self Set.Ico_subset_Ici_self theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a := ⟨Ioi_subset_Ici_self, fun h => lt_irrefl a (h le_rfl)⟩ #align set.Ioi_ssubset_Ici_self Set.Ioi_ssubset_Ici_self theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a := @Ioi_ssubset_Ici_self αᵒᵈ _ _ #align set.Iio_ssubset_Iic_self Set.Iio_ssubset_Iic_self theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Icc_iff Set.Icc_subset_Icc_iff theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ioo_iff Set.Icc_subset_Ioo_iff theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans hx, hx'.trans_lt h'⟩⟩ #align set.Icc_subset_Ico_iff Set.Icc_subset_Ico_iff theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ := ⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ => ⟨h.trans_le hx, hx'.trans h'⟩⟩ #align set.Icc_subset_Ioc_iff Set.Icc_subset_Ioc_iff theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans_lt h⟩ #align set.Icc_subset_Iio_iff Set.Icc_subset_Iio_iff theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans_le hx⟩ #align set.Icc_subset_Ioi_iff Set.Icc_subset_Ioi_iff theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ := ⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans h⟩ #align set.Icc_subset_Iic_iff Set.Icc_subset_Iic_iff theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ := ⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans hx⟩ #align set.Icc_subset_Ici_iff Set.Icc_subset_Ici_iff theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc (le_of_lt ha) hb)).mpr ⟨a₂, left_mem_Icc.mpr hI, not_and.mpr fun f _ => lt_irrefl a₂ (ha.trans_le f)⟩ #align set.Icc_ssubset_Icc_left Set.Icc_ssubset_Icc_left theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ := (ssubset_iff_of_subset (Icc_subset_Icc ha (le_of_lt hb))).mpr ⟨b₂, right_mem_Icc.mpr hI, fun f => lt_irrefl b₁ (hb.trans_le f.2)⟩ #align set.Icc_ssubset_Icc_right Set.Icc_ssubset_Icc_right /-- If `a ≤ b`, then `(b, +∞) ⊆ (a, +∞)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Ioi_subset_Ioi_iff`. -/ @[gcongr] theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := fun _ hx => h.trans_lt hx #align set.Ioi_subset_Ioi Set.Ioi_subset_Ioi /-- If `a ≤ b`, then `(b, +∞) ⊆ [a, +∞)`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Ioi_subset_Ici_iff`. -/ theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a := Subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self #align set.Ioi_subset_Ici Set.Ioi_subset_Ici /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b)`. In preorders, this is just an implication. If you need the equivalence in linear orders, use `Iio_subset_Iio_iff`. -/ @[gcongr] theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := fun _ hx => lt_of_lt_of_le hx h #align set.Iio_subset_Iio Set.Iio_subset_Iio /-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b]`. In preorders, this is just an implication. If you need the equivalence in dense linear orders, use `Iio_subset_Iic_iff`. -/ theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b := Subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self #align set.Iio_subset_Iic Set.Iio_subset_Iic theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b := rfl #align set.Ici_inter_Iic Set.Ici_inter_Iic theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b := rfl #align set.Ici_inter_Iio Set.Ici_inter_Iio theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b := rfl #align set.Ioi_inter_Iic Set.Ioi_inter_Iic theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b := rfl #align set.Ioi_inter_Iio Set.Ioi_inter_Iio theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a := inter_comm _ _ #align set.Iic_inter_Ici Set.Iic_inter_Ici theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a := inter_comm _ _ #align set.Iio_inter_Ici Set.Iio_inter_Ici theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a := inter_comm _ _ #align set.Iic_inter_Ioi Set.Iic_inter_Ioi theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a := inter_comm _ _ #align set.Iio_inter_Ioi Set.Iio_inter_Ioi theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b := Ioo_subset_Icc_self h #align set.mem_Icc_of_Ioo Set.mem_Icc_of_Ioo theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b := Ioo_subset_Ico_self h #align set.mem_Ico_of_Ioo Set.mem_Ico_of_Ioo theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b := Ioo_subset_Ioc_self h #align set.mem_Ioc_of_Ioo Set.mem_Ioc_of_Ioo theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b := Ico_subset_Icc_self h #align set.mem_Icc_of_Ico Set.mem_Icc_of_Ico theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b := Ioc_subset_Icc_self h #align set.mem_Icc_of_Ioc Set.mem_Icc_of_Ioc theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a := Ioi_subset_Ici_self h #align set.mem_Ici_of_Ioi Set.mem_Ici_of_Ioi theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a := Iio_subset_Iic_self h #align set.mem_Iic_of_Iio Set.mem_Iic_of_Iio theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc] #align set.Icc_eq_empty_iff Set.Icc_eq_empty_iff theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico] #align set.Ico_eq_empty_iff Set.Ico_eq_empty_iff theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioc] #align set.Ioc_eq_empty_iff Set.Ioc_eq_empty_iff theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo] #align set.Ioo_eq_empty_iff Set.Ioo_eq_empty_iff theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ := eq_univ_of_forall h #align is_top.Iic_eq IsTop.Iic_eq theorem _root_.IsBot.Ici_eq (h : IsBot a) : Ici a = univ := eq_univ_of_forall h #align is_bot.Ici_eq IsBot.Ici_eq theorem _root_.IsMax.Ioi_eq (h : IsMax a) : Ioi a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_max.Ioi_eq IsMax.Ioi_eq theorem _root_.IsMin.Iio_eq (h : IsMin a) : Iio a = ∅ := eq_empty_of_subset_empty fun _ => h.not_lt #align is_min.Iio_eq IsMin.Iio_eq theorem Iic_inter_Ioc_of_le (h : a ≤ c) : Iic a ∩ Ioc b c = Ioc b a := ext fun _ => ⟨fun H => ⟨H.2.1, H.1⟩, fun H => ⟨H.2, H.1, H.2.trans h⟩⟩ #align set.Iic_inter_Ioc_of_le Set.Iic_inter_Ioc_of_le theorem not_mem_Icc_of_lt (ha : c < a) : c ∉ Icc a b := fun h => ha.not_le h.1 #align set.not_mem_Icc_of_lt Set.not_mem_Icc_of_lt theorem not_mem_Icc_of_gt (hb : b < c) : c ∉ Icc a b := fun h => hb.not_le h.2 #align set.not_mem_Icc_of_gt Set.not_mem_Icc_of_gt theorem not_mem_Ico_of_lt (ha : c < a) : c ∉ Ico a b := fun h => ha.not_le h.1 #align set.not_mem_Ico_of_lt Set.not_mem_Ico_of_lt theorem not_mem_Ioc_of_gt (hb : b < c) : c ∉ Ioc a b := fun h => hb.not_le h.2 #align set.not_mem_Ioc_of_gt Set.not_mem_Ioc_of_gt -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Ioi_self : a ∉ Ioi a := lt_irrefl _ #align set.not_mem_Ioi_self Set.not_mem_Ioi_self -- Porting note (#10618): `simp` can prove this -- @[simp] theorem not_mem_Iio_self : b ∉ Iio b := lt_irrefl _ #align set.not_mem_Iio_self Set.not_mem_Iio_self theorem not_mem_Ioc_of_le (ha : c ≤ a) : c ∉ Ioc a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioc_of_le Set.not_mem_Ioc_of_le theorem not_mem_Ico_of_ge (hb : b ≤ c) : c ∉ Ico a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ico_of_ge Set.not_mem_Ico_of_ge theorem not_mem_Ioo_of_le (ha : c ≤ a) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.1.trans_le ha #align set.not_mem_Ioo_of_le Set.not_mem_Ioo_of_le theorem not_mem_Ioo_of_ge (hb : b ≤ c) : c ∉ Ioo a b := fun h => lt_irrefl _ <\| h.2.trans_le hb #align set.not_mem_Ioo_of_ge Set.not_mem_Ioo_of_ge end Preorder section PartialOrder variable [PartialOrder α] {a b c : α} @[simp] theorem Icc_self (a : α) : Icc a a = {a} := Set.ext <\| by simp [Icc, le_antisymm_iff, and_comm] #align set.Icc_self Set.Icc_self instance instIccUnique : Unique (Set.Icc a a) where default := ⟨a, by simp⟩ uniq y := Subtype.ext <\| by simpa using y.2 @[simp] theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by refine ⟨fun h => ?_, ?_⟩ · have hab : a ≤ b := nonempty_Icc.1 (h.symm.subst <\| singleton_nonempty c) exact ⟨eq_of_mem_singleton <\| h.subst <\| left_mem_Icc.2 hab, eq_of_mem_singleton <\| h.subst <\| right_mem_Icc.2 hab⟩ · rintro ⟨rfl, rfl⟩ exact Icc_self _ #align set.Icc_eq_singleton_iff Set.Icc_eq_singleton_iff lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) := fun _x ⟨hax, hxb⟩ _y ⟨hay, hyb⟩ ↦ le_antisymm (le_implies_le_of_le_of_le hxb hay hba) (le_implies_le_of_le_of_le hyb hax hba) #align set.subsingleton_Icc_of_ge Set.subsingleton_Icc_of_ge @[simp] lemma subsingleton_Icc_iff {α : Type} [LinearOrder α] {a b : α} : Set.Subsingleton (Icc a b) ↔ b ≤ a := by refine ⟨fun h ↦ ?_, subsingleton_Icc_of_ge⟩ contrapose! h simp only [ge_iff_le, gt_iff_lt, not_subsingleton_iff] exact ⟨a, ⟨le_refl _, h.le⟩, b, ⟨h.le, le_refl _⟩, h.ne⟩ @[simp] theorem Icc_diff_left : Icc a b \ {a} = Ioc a b := ext fun x => by simp [lt_iff_le_and_ne, eq_comm, and_right_comm] #align set.Icc_diff_left Set.Icc_diff_left @[simp] theorem Icc_diff_right : Icc a b \ {b} = Ico a b := ext fun x => by simp [lt_iff_le_and_ne, and_assoc] #align set.Icc_diff_right Set.Icc_diff_right @[simp] theorem Ico_diff_left : Ico a b \ {a} = Ioo a b := ext fun x => by simp [and_right_comm, ← lt_iff_le_and_ne, eq_comm] #align set.Ico_diff_left Set.Ico_diff_left @[simp] theorem Ioc_diff_right : Ioc a b \ {b} = Ioo a b := ext fun x => by simp [and_assoc, ← lt_iff_le_and_ne] #align set.Ioc_diff_right Set.Ioc_diff_right @[simp] theorem Icc_diff_both : Icc a b \ {a, b} = Ioo a b := by rw [insert_eq, ← diff_diff, Icc_diff_left, Ioc_diff_right] #align set.Icc_diff_both Set.Icc_diff_both @[simp] theorem Ici_diff_left : Ici a \ {a} = Ioi a := ext fun x => by simp [lt_iff_le_and_ne, eq_comm] #align set.Ici_diff_left Set.Ici_diff_left @[simp] theorem Iic_diff_right : Iic a \ {a} = Iio a := ext fun x => by simp [lt_iff_le_and_ne] #align set.Iic_diff_right Set.Iic_diff_right @[simp] theorem Ico_diff_Ioo_same (h : a < b) : Ico a b \ Ioo a b = {a} := by rw [← Ico_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Ico.2 h)] #align set.Ico_diff_Ioo_same Set.Ico_diff_Ioo_same @[simp] theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} := by rw [← Ioc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Ioc.2 h)] #align set.Ioc_diff_Ioo_same Set.Ioc_diff_Ioo_same @[simp] theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <\| right_mem_Icc.2 h)] #align set.Icc_diff_Ico_same Set.Icc_diff_Ico_same @[simp] theorem Icc_diff_Ioc_same (h : a ≤ b) : Icc a b \ Ioc a b = {a} := by rw [← Icc_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 <\| left_mem_Icc.2 h)] #align set.Icc_diff_Ioc_same Set.Icc_diff_Ioc_same @[simp] theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} := by rw [← Icc_diff_both, diff_diff_cancel_left] simp [insert_subset_iff, h] #align set.Icc_diff_Ioo_same Set.Icc_diff_Ioo_same @[simp] theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} := by rw [← Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)] #align set.Ici_diff_Ioi_same Set.Ici_diff_Ioi_same @[simp] theorem Iic_diff_Iio_same : Iic a \ Iio a = {a} := by rw [← Iic_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 right_mem_Iic)] #align set.Iic_diff_Iio_same Set.Iic_diff_Iio_same -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Ioi_union_left : Ioi a ∪ {a} = Ici a := ext fun x => by simp [eq_comm, le_iff_eq_or_lt] #align set.Ioi_union_left Set.Ioi_union_left -- Porting note (#10618): `simp` can prove this -- @[simp] theorem Iio_union_right : Iio a ∪ {a} = Iic a := ext fun _ => le_iff_lt_or_eq.symm #align set.Iio_union_right Set.Iio_union_right theorem Ioo_union_left (hab : a < b) : Ioo a b ∪ {a} = Ico a b := by rw [← Ico_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Ico.2 hab)] #align set.Ioo_union_left Set.Ioo_union_left theorem Ioo_union_right (hab : a < b) : Ioo a b ∪ {b} = Ioc a b := by simpa only [dual_Ioo, dual_Ico] using Ioo_union_left hab.dual #align set.Ioo_union_right Set.Ioo_union_right theorem Ioo_union_both (h : a ≤ b) : Ioo a b ∪ {a, b} = Icc a b := by have : (Icc a b \ {a, b}) ∪ {a, b} = Icc a b := diff_union_of_subset fun \| x, .inl rfl => left_mem_Icc.mpr h \| x, .inr rfl => right_mem_Icc.mpr h rw [← this, Icc_diff_both] theorem Ioc_union_left (hab : a ≤ b) : Ioc a b ∪ {a} = Icc a b := by rw [← Icc_diff_left, diff_union_self, union_eq_self_of_subset_right (singleton_subset_iff.2 <\| left_mem_Icc.2 hab)] #align set.Ioc_union_left Set.Ioc_union_left theorem Ico_union_right (hab : a ≤ b) : Ico a b ∪ {b} = Icc a b := by simpa only [dual_Ioc, dual_Icc] using Ioc_union_left hab.dual #align set.Ico_union_right Set.Ico_union_right @[simp] theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b := by rw [insert_eq, union_comm, Ico_union_right h] #align set.Ico_insert_right Set.Ico_insert_right @[simp] theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b := by rw [insert_eq, union_comm, Ioc_union_left h] #align set.Ioc_insert_left Set.Ioc_insert_left @[simp] theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b := by rw [insert_eq, union_comm, Ioo_union_left h] #align set.Ioo_insert_left Set.Ioo_insert_left @[simp] theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b := by rw [insert_eq, union_comm, Ioo_union_right h] #align set.Ioo_insert_right Set.Ioo_insert_right @[simp] theorem Iio_insert : insert a (Iio a) = Iic a := ext fun _ => le_iff_eq_or_lt.symm #align set.Iio_insert Set.Iio_insert @[simp] theorem Ioi_insert : insert a (Ioi a) = Ici a := ext fun _ => (or_congr_left eq_comm).trans le_iff_eq_or_lt.symm #align set.Ioi_insert Set.Ioi_insert theorem mem_Ici_Ioi_of_subset_of_subset {s : Set α} (ho : Ioi a ⊆ s) (hc : s ⊆ Ici a) : s ∈ ({Ici a, Ioi a} : Set (Set α)) := by_cases (fun h : a ∈ s => Or.inl <\| Subset.antisymm hc <\| by rw [← Ioi_union_left, union_subset_iff]; simp []) fun h => Or.inr <\| Subset.antisymm (fun x hx => lt_of_le_of_ne (hc hx) fun heq => h <\| heq.symm ▸ hx) ho #align set.mem_Ici_Ioi_of_subset_of_subset Set.mem_Ici_Ioi_of_subset_of_subset theorem mem_Iic_Iio_of_subset_of_subset {s : Set α} (ho : Iio a ⊆ s) (hc : s ⊆ Iic a) : s ∈ ({Iic a, Iio a} : Set (Set α)) := @mem_Ici_Ioi_of_subset_of_subset αᵒᵈ _ a s ho hc #align set.mem_Iic_Iio_of_subset_of_subset Set.mem_Iic_Iio_of_subset_of_subset theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) : s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α)) := by classical by_cases ha : a ∈ s <;> by_cases hb : b ∈ s · refine Or.inl (Subset.antisymm hc ?_) rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, ← Icc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_right] exact subset_diff_singleton hc hb · rwa [← Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho · refine Or.inr <\| Or.inr <\| Or.inl <\| Subset.antisymm ?_ ?_ · rw [← Icc_diff_left] exact subset_diff_singleton hc ha · rwa [← Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho · refine Or.inr <\| Or.inr <\| Or.inr <\| Subset.antisymm ?_ ho rw [← Ico_diff_left, ← Icc_diff_right] apply_rules [subset_diff_singleton] #align set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset theorem eq_left_or_mem_Ioo_of_mem_Ico {x : α} (hmem : x ∈ Ico a b) : x = a ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => ⟨h, hmem.2⟩ #align set.eq_left_or_mem_Ioo_of_mem_Ico Set.eq_left_or_mem_Ioo_of_mem_Ico theorem eq_right_or_mem_Ioo_of_mem_Ioc {x : α} (hmem : x ∈ Ioc a b) : x = b ∨ x ∈ Ioo a b := hmem.2.eq_or_lt.imp_right <\| And.intro hmem.1 #align set.eq_right_or_mem_Ioo_of_mem_Ioc Set.eq_right_or_mem_Ioo_of_mem_Ioc theorem eq_endpoints_or_mem_Ioo_of_mem_Icc {x : α} (hmem : x ∈ Icc a b) : x = a ∨ x = b ∨ x ∈ Ioo a b := hmem.1.eq_or_gt.imp_right fun h => eq_right_or_mem_Ioo_of_mem_Ioc ⟨h, hmem.2⟩ #align set.eq_endpoints_or_mem_Ioo_of_mem_Icc Set.eq_endpoints_or_mem_Ioo_of_mem_Icc theorem _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} := eq_singleton_iff_unique_mem.2 ⟨left_mem_Ici, fun _ => h.eq_of_ge⟩ #align is_max.Ici_eq IsMax.Ici_eq theorem _root_.IsMin.Iic_eq (h : IsMin a) : Iic a = {a} := h.toDual.Ici_eq #align is_min.Iic_eq IsMin.Iic_eq theorem Ici_injective : Injective (Ici : α → Set α) := fun _ _ => eq_of_forall_ge_iff ∘ Set.ext_iff.1 #align set.Ici_injective Set.Ici_injective theorem Iic_injective : Injective (Iic : α → Set α) := fun _ _ => eq_of_forall_le_iff ∘ Set.ext_iff.1 #align set.Iic_injective Set.Iic_injective theorem Ici_inj : Ici a = Ici b ↔ a = b := Ici_injective.eq_iff #align set.Ici_inj Set.Ici_inj theorem Iic_inj : Iic a = Iic b ↔ a = b := Iic_injective.eq_iff #align set.Iic_inj Set.Iic_inj end PartialOrder section OrderTop @[simp] theorem Ici_top [PartialOrder α] [OrderTop α] : Ici (⊤ : α) = {⊤} := isMax_top.Ici_eq #align set.Ici_top Set.Ici_top variable [Preorder α] [OrderTop α] {a : α} @[simp] theorem Ioi_top : Ioi (⊤ : α) = ∅ := isMax_top.Ioi_eq #align set.Ioi_top Set.Ioi_top @[simp] theorem Iic_top : Iic (⊤ : α) = univ := isTop_top.Iic_eq #align set.Iic_top Set.Iic_top @[simp] theorem Icc_top : Icc a ⊤ = Ici a := by simp [← Ici_inter_Iic] #align set.Icc_top Set.Icc_top @[simp] theorem Ioc_top : Ioc a ⊤ = Ioi a := by simp [← Ioi_inter_Iic] #align set.Ioc_top Set.Ioc_top end OrderTop section OrderBot @[simp] theorem Iic_bot [PartialOrder α] [OrderBot α] : Iic (⊥ : α) = {⊥} := isMin_bot.Iic_eq #align set.Iic_bot Set.Iic_bot variable [Preorder α] [OrderBot α] {a : α} @[simp] theorem Iio_bot : Iio (⊥ : α) = ∅ := isMin_bot.Iio_eq #align set.Iio_bot Set.Iio_bot @[simp] theorem Ici_bot : Ici (⊥ : α) = univ := isBot_bot.Ici_eq #align set.Ici_bot Set.Ici_bot @[simp] theorem Icc_bot : Icc ⊥ a = Iic a := by simp [← Ici_inter_Iic] #align set.Icc_bot Set.Icc_bot @[simp] theorem Ico_bot : Ico ⊥ a = Iio a := by simp [← Ici_inter_Iio] #align set.Ico_bot Set.Ico_bot end OrderBot theorem Icc_bot_top [PartialOrder α] [BoundedOrder α] : Icc (⊥ : α) ⊤ = univ := by simp #align set.Icc_bot_top Set.Icc_bot_top section LinearOrder variable [LinearOrder α] {a a₁ a₂ b b₁ b₂ c d : α} theorem not_mem_Ici : c ∉ Ici a ↔ c < a := not_le #align set.not_mem_Ici Set.not_mem_Ici theorem not_mem_Iic : c ∉ Iic b ↔ b < c := not_le #align set.not_mem_Iic Set.not_mem_Iic theorem not_mem_Ioi : c ∉ Ioi a ↔ c ≤ a := not_lt #align set.not_mem_Ioi Set.not_mem_Ioi theorem not_mem_Iio : c ∉ Iio b ↔ b ≤ c := not_lt #align set.not_mem_Iio Set.not_mem_Iio @[simp] theorem compl_Iic : (Iic a)ᶜ = Ioi a := ext fun _ => not_le #align set.compl_Iic Set.compl_Iic @[simp] theorem compl_Ici : (Ici a)ᶜ = Iio a := ext fun _ => not_le #align set.compl_Ici Set.compl_Ici @[simp] theorem compl_Iio : (Iio a)ᶜ = Ici a := ext fun _ => not_lt #align set.compl_Iio Set.compl_Iio @[simp] theorem compl_Ioi : (Ioi a)ᶜ = Iic a := ext fun _ => not_lt #align set.compl_Ioi Set.compl_Ioi @[simp] theorem Ici_diff_Ici : Ici a \ Ici b = Ico a b := by rw [diff_eq, compl_Ici, Ici_inter_Iio] #align set.Ici_diff_Ici Set.Ici_diff_Ici @[simp] theorem Ici_diff_Ioi : Ici a \ Ioi b = Icc a b := by rw [diff_eq, compl_Ioi, Ici_inter_Iic] #align set.Ici_diff_Ioi Set.Ici_diff_Ioi @[simp] theorem Ioi_diff_Ioi : Ioi a \ Ioi b = Ioc a b := by rw [diff_eq, compl_Ioi, Ioi_inter_Iic] #align set.Ioi_diff_Ioi Set.Ioi_diff_Ioi @[simp] theorem Ioi_diff_Ici : Ioi a \ Ici b = Ioo a b := by rw [diff_eq, compl_Ici, Ioi_inter_Iio] #align set.Ioi_diff_Ici Set.Ioi_diff_Ici @[simp] theorem Iic_diff_Iic : Iic b \ Iic a = Ioc a b := by rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iic] #align set.Iic_diff_Iic Set.Iic_diff_Iic @[simp] theorem Iio_diff_Iic : Iio b \ Iic a = Ioo a b := by rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iio] #align set.Iio_diff_Iic Set.Iio_diff_Iic @[simp] theorem Iic_diff_Iio : Iic b \ Iio a = Icc a b := by rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iic] #align set.Iic_diff_Iio Set.Iic_diff_Iio @[simp] theorem Iio_diff_Iio : Iio b \ Iio a = Ico a b := by rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iio] #align set.Iio_diff_Iio Set.Iio_diff_Iio theorem Ioi_injective : Injective (Ioi : α → Set α) := fun _ _ => eq_of_forall_gt_iff ∘ Set.ext_iff.1 #align set.Ioi_injective Set.Ioi_injective theorem Iio_injective : Injective (Iio : α → Set α) := fun _ _ => eq_of_forall_lt_iff ∘ Set.ext_iff.1 #align set.Iio_injective Set.Iio_injective theorem Ioi_inj : Ioi a = Ioi b ↔ a = b := Ioi_injective.eq_iff #align set.Ioi_inj Set.Ioi_inj theorem Iio_inj : Iio a = Iio b ↔ a = b := Iio_injective.eq_iff #align set.Iio_inj Set.Iio_inj theorem Ico_subset_Ico_iff (h₁ : a₁ < b₁) : Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => have : a₂ ≤ a₁ ∧ a₁ < b₂ := h ⟨le_rfl, h₁⟩ ⟨this.1, le_of_not_lt fun h' => lt_irrefl b₂ (h ⟨this.2.le, h'⟩).2⟩, fun ⟨h₁, h₂⟩ => Ico_subset_Ico h₁ h₂⟩ #align set.Ico_subset_Ico_iff Set.Ico_subset_Ico_iff theorem Ioc_subset_Ioc_iff (h₁ : a₁ < b₁) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ b₁ ≤ b₂ ∧ a₂ ≤ a₁ := by convert @Ico_subset_Ico_iff αᵒᵈ _ b₁ b₂ a₁ a₂ h₁ using 2 <;> exact (@dual_Ico α _ _ _).symm #align set.Ioc_subset_Ioc_iff Set.Ioc_subset_Ioc_iff theorem Ioo_subset_Ioo_iff [DenselyOrdered α] (h₁ : a₁ < b₁) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := ⟨fun h => by rcases exists_between h₁ with ⟨x, xa, xb⟩ constructor <;> refine le_of_not_lt fun h' => ?_ · have ab := (h ⟨xa, xb⟩).1.trans xb exact lt_irrefl _ (h ⟨h', ab⟩).1 · have ab := xa.trans (h ⟨xa, xb⟩).2 exact lt_irrefl _ (h ⟨ab, h'⟩).2, fun ⟨h₁, h₂⟩ => Ioo_subset_Ioo h₁ h₂⟩ #align set.Ioo_subset_Ioo_iff Set.Ioo_subset_Ioo_iff theorem Ico_eq_Ico_iff (h : a₁ < b₁ ∨ a₂ < b₂) : Ico a₁ b₁ = Ico a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ := ⟨fun e => by simp only [Subset.antisymm_iff] at e simp only [le_antisymm_iff] cases' h with h h <;> simp only [gt_iff_lt, not_lt, ge_iff_le, Ico_subset_Ico_iff h] at e <;> [ rcases e with ⟨⟨h₁, h₂⟩, e'⟩; rcases e with ⟨e', ⟨h₁, h₂⟩⟩ ] <;> -- Porting note: restore `tauto` have hab := (Ico_subset_Ico_iff <\| h₁.trans_lt <\| h.trans_le h₂).1 e' <;> [ exact ⟨⟨hab.left, h₁⟩, ⟨h₂, hab.right⟩⟩; exact ⟨⟨h₁, hab.left⟩, ⟨hab.right, h₂⟩⟩ ], fun ⟨h₁, h₂⟩ => by rw [h₁, h₂]⟩ #align set.Ico_eq_Ico_iff Set.Ico_eq_Ico_iff lemma Ici_eq_singleton_iff_isTop {x : α} : (Ici x = {x}) ↔ IsTop x := by refine ⟨fun h y ↦ ?_, fun h ↦ by ext y; simp [(h y).ge_iff_eq]⟩ by_contra! H have : y ∈ Ici x := H.le rw [h, mem_singleton_iff] at this exact lt_irrefl y (this.le.trans_lt H) open scoped Classical @[simp] theorem Ioi_subset_Ioi_iff : Ioi b ⊆ Ioi a ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Ioi_subset_Ioi h⟩ by_contra ba exact lt_irrefl _ (h (not_le.mp ba)) #align set.Ioi_subset_Ioi_iff Set.Ioi_subset_Ioi_iff @[simp] theorem Ioi_subset_Ici_iff [DenselyOrdered α] : Ioi b ⊆ Ici a ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Ioi_subset_Ici h⟩ by_contra ba obtain ⟨c, bc, ca⟩ : ∃ c, b < c ∧ c < a := exists_between (not_le.mp ba) exact lt_irrefl _ (ca.trans_le (h bc)) #align set.Ioi_subset_Ici_iff Set.Ioi_subset_Ici_iff @[simp] theorem Iio_subset_Iio_iff : Iio a ⊆ Iio b ↔ a ≤ b := by refine ⟨fun h => ?_, fun h => Iio_subset_Iio h⟩ by_contra ab exact lt_irrefl _ (h (not_le.mp ab)) #align set.Iio_subset_Iio_iff Set.Iio_subset_Iio_iff @[simp] theorem Iio_subset_Iic_iff [DenselyOrdered α] : Iio a ⊆ Iic b ↔ a ≤ b := by rw [← diff_eq_empty, Iio_diff_Iic, Ioo_eq_empty_iff, not_lt] #align set.Iio_subset_Iic_iff Set.Iio_subset_Iic_iff /-! ### Unions of adjacent intervals -/ /-! #### Two infinite intervals -/ theorem Iic_union_Ioi_of_le (h : a ≤ b) : Iic b ∪ Ioi a = univ := eq_univ_of_forall fun x => (h.lt_or_le x).symm #align set.Iic_union_Ioi_of_le Set.Iic_union_Ioi_of_le theorem Iio_union_Ici_of_le (h : a ≤ b) : Iio b ∪ Ici a = univ := eq_univ_of_forall fun x => (h.le_or_lt x).symm #align set.Iio_union_Ici_of_le Set.Iio_union_Ici_of_le theorem Iic_union_Ici_of_le (h : a ≤ b) : Iic b ∪ Ici a = univ := eq_univ_of_forall fun x => (h.le_or_le x).symm #align set.Iic_union_Ici_of_le Set.Iic_union_Ici_of_le theorem Iio_union_Ioi_of_lt (h : a < b) : Iio b ∪ Ioi a = univ := eq_univ_of_forall fun x => (h.lt_or_lt x).symm #align set.Iio_union_Ioi_of_lt Set.Iio_union_Ioi_of_lt @[simp] theorem Iic_union_Ici : Iic a ∪ Ici a = univ := Iic_union_Ici_of_le le_rfl #align set.Iic_union_Ici Set.Iic_union_Ici @[simp] theorem Iio_union_Ici : Iio a ∪ Ici a = univ := Iio_union_Ici_of_le le_rfl #align set.Iio_union_Ici Set.Iio_union_Ici @[simp] theorem Iic_union_Ioi : Iic a ∪ Ioi a = univ := Iic_union_Ioi_of_le le_rfl #align set.Iic_union_Ioi Set.Iic_union_Ioi @[simp] theorem Iio_union_Ioi : Iio a ∪ Ioi a = {a}ᶜ := ext fun _ => lt_or_lt_iff_ne #align set.Iio_union_Ioi Set.Iio_union_Ioi /-! #### A finite and an infinite interval -/ theorem Ioo_union_Ioi' (h₁ : c < b) : Ioo a b ∪ Ioi c = Ioi (min a c) := by ext1 x simp_rw [mem_union, mem_Ioo, mem_Ioi, min_lt_iff] by_cases hc : c < x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x < b := (le_of_not_gt hc).trans_lt h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ioo_union_Ioi' Set.Ioo_union_Ioi' theorem Ioo_union_Ioi (h : c < max a b) : Ioo a b ∪ Ioi c = Ioi (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ioo_union_Ioi' h · rw [min_comm] simp [, min_eq_left_of_lt] #align set.Ioo_union_Ioi Set.Ioo_union_Ioi theorem Ioi_subset_Ioo_union_Ici : Ioi a ⊆ Ioo a b ∪ Ici b := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ioi_subset_Ioo_union_Ici Set.Ioi_subset_Ioo_union_Ici @[simp] theorem Ioo_union_Ici_eq_Ioi (h : a < b) : Ioo a b ∪ Ici b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_le) Ioi_subset_Ioo_union_Ici #align set.Ioo_union_Ici_eq_Ioi Set.Ioo_union_Ici_eq_Ioi theorem Ici_subset_Ico_union_Ici : Ici a ⊆ Ico a b ∪ Ici b := fun x hx => (lt_or_le x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ici_subset_Ico_union_Ici Set.Ici_subset_Ico_union_Ici @[simp] theorem Ico_union_Ici_eq_Ici (h : a ≤ b) : Ico a b ∪ Ici b = Ici a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans) Ici_subset_Ico_union_Ici #align set.Ico_union_Ici_eq_Ici Set.Ico_union_Ici_eq_Ici theorem Ico_union_Ici' (h₁ : c ≤ b) : Ico a b ∪ Ici c = Ici (min a c) := by ext1 x simp_rw [mem_union, mem_Ico, mem_Ici, min_le_iff] by_cases hc : c ≤ x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ico_union_Ici' Set.Ico_union_Ici' theorem Ico_union_Ici (h : c ≤ max a b) : Ico a b ∪ Ici c = Ici (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ico_union_Ici' h · simp [] #align set.Ico_union_Ici Set.Ico_union_Ici theorem Ioi_subset_Ioc_union_Ioi : Ioi a ⊆ Ioc a b ∪ Ioi b := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ioi_subset_Ioc_union_Ioi Set.Ioi_subset_Ioc_union_Ioi @[simp] theorem Ioc_union_Ioi_eq_Ioi (h : a ≤ b) : Ioc a b ∪ Ioi b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_lt) Ioi_subset_Ioc_union_Ioi #align set.Ioc_union_Ioi_eq_Ioi Set.Ioc_union_Ioi_eq_Ioi theorem Ioc_union_Ioi' (h₁ : c ≤ b) : Ioc a b ∪ Ioi c = Ioi (min a c) := by ext1 x simp_rw [mem_union, mem_Ioc, mem_Ioi, min_lt_iff] by_cases hc : c < x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Ioc_union_Ioi' Set.Ioc_union_Ioi' theorem Ioc_union_Ioi (h : c ≤ max a b) : Ioc a b ∪ Ioi c = Ioi (min a c) := by rcases le_total a b with hab \| hab <;> simp [hab] at h · exact Ioc_union_Ioi' h · simp [] #align set.Ioc_union_Ioi Set.Ioc_union_Ioi theorem Ici_subset_Icc_union_Ioi : Ici a ⊆ Icc a b ∪ Ioi b := fun x hx => (le_or_lt x b).elim (fun hxb => Or.inl ⟨hx, hxb⟩) fun hxb => Or.inr hxb #align set.Ici_subset_Icc_union_Ioi Set.Ici_subset_Icc_union_Ioi @[simp] theorem Icc_union_Ioi_eq_Ici (h : a ≤ b) : Icc a b ∪ Ioi b = Ici a := Subset.antisymm (fun _ hx => (hx.elim And.left) fun hx' => h.trans <\| le_of_lt hx') Ici_subset_Icc_union_Ioi #align set.Icc_union_Ioi_eq_Ici Set.Icc_union_Ioi_eq_Ici theorem Ioi_subset_Ioc_union_Ici : Ioi a ⊆ Ioc a b ∪ Ici b := Subset.trans Ioi_subset_Ioo_union_Ici (union_subset_union_left _ Ioo_subset_Ioc_self) #align set.Ioi_subset_Ioc_union_Ici Set.Ioi_subset_Ioc_union_Ici @[simp] theorem Ioc_union_Ici_eq_Ioi (h : a < b) : Ioc a b ∪ Ici b = Ioi a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans_le) Ioi_subset_Ioc_union_Ici #align set.Ioc_union_Ici_eq_Ioi Set.Ioc_union_Ici_eq_Ioi theorem Ici_subset_Icc_union_Ici : Ici a ⊆ Icc a b ∪ Ici b := Subset.trans Ici_subset_Ico_union_Ici (union_subset_union_left _ Ico_subset_Icc_self) #align set.Ici_subset_Icc_union_Ici Set.Ici_subset_Icc_union_Ici @[simp] theorem Icc_union_Ici_eq_Ici (h : a ≤ b) : Icc a b ∪ Ici b = Ici a := Subset.antisymm (fun _ hx => hx.elim And.left h.trans) Ici_subset_Icc_union_Ici #align set.Icc_union_Ici_eq_Ici Set.Icc_union_Ici_eq_Ici theorem Icc_union_Ici' (h₁ : c ≤ b) : Icc a b ∪ Ici c = Ici (min a c) := by ext1 x simp_rw [mem_union, mem_Icc, mem_Ici, min_le_iff] by_cases hc : c ≤ x · simp only [hc, or_true] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_ge hc).trans h₁ simp only [hxb, and_true] -- Porting note: restore `tauto` #align set.Icc_union_Ici' Set.Icc_union_Ici' theorem Icc_union_Ici (h : c ≤ max a b) : Icc a b ∪ Ici c = Ici (min a c) := by rcases le_or_lt a b with hab \| hab <;> simp [hab] at h · exact Icc_union_Ici' h · cases' h with h h · simp [] · have hca : c ≤ a := h.trans hab.le simp [] #align set.Icc_union_Ici Set.Icc_union_Ici /-! #### An infinite and a finite interval -/ theorem Iic_subset_Iio_union_Icc : Iic b ⊆ Iio a ∪ Icc a b := fun x hx => (lt_or_le x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iic_subset_Iio_union_Icc Set.Iic_subset_Iio_union_Icc @[simp] theorem Iio_union_Icc_eq_Iic (h : a ≤ b) : Iio a ∪ Icc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx => (le_of_lt hx).trans h) And.right) Iic_subset_Iio_union_Icc #align set.Iio_union_Icc_eq_Iic Set.Iio_union_Icc_eq_Iic theorem Iio_subset_Iio_union_Ico : Iio b ⊆ Iio a ∪ Ico a b := fun x hx => (lt_or_le x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iio_subset_Iio_union_Ico Set.Iio_subset_Iio_union_Ico @[simp] theorem Iio_union_Ico_eq_Iio (h : a ≤ b) : Iio a ∪ Ico a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_lt_of_le hx' h) And.right) Iio_subset_Iio_union_Ico #align set.Iio_union_Ico_eq_Iio Set.Iio_union_Ico_eq_Iio theorem Iio_union_Ico' (h₁ : c ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by ext1 x simp_rw [mem_union, mem_Iio, mem_Ico, lt_max_iff] by_cases hc : c ≤ x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x < b := (lt_of_not_ge hc).trans_le h₁ simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iio_union_Ico' Set.Iio_union_Ico' theorem Iio_union_Ico (h : min c d ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iio_union_Ico' h · simp [] #align set.Iio_union_Ico Set.Iio_union_Ico theorem Iic_subset_Iic_union_Ioc : Iic b ⊆ Iic a ∪ Ioc a b := fun x hx => (le_or_lt x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iic_subset_Iic_union_Ioc Set.Iic_subset_Iic_union_Ioc @[simp] theorem Iic_union_Ioc_eq_Iic (h : a ≤ b) : Iic a ∪ Ioc a b = Iic b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => le_trans hx' h) And.right) Iic_subset_Iic_union_Ioc #align set.Iic_union_Ioc_eq_Iic Set.Iic_union_Ioc_eq_Iic theorem Iic_union_Ioc' (h₁ : c < b) : Iic b ∪ Ioc c d = Iic (max b d) := by ext1 x simp_rw [mem_union, mem_Iic, mem_Ioc, le_max_iff] by_cases hc : c < x · simp only [hc, true_and] -- Porting note: restore `tauto` · have hxb : x ≤ b := (le_of_not_gt hc).trans h₁.le simp only [hxb, true_or] -- Porting note: restore `tauto` #align set.Iic_union_Ioc' Set.Iic_union_Ioc' theorem Iic_union_Ioc (h : min c d < b) : Iic b ∪ Ioc c d = Iic (max b d) := by rcases le_total c d with hcd \| hcd <;> simp [hcd] at h · exact Iic_union_Ioc' h · rw [max_comm] simp [, max_eq_right_of_lt h] #align set.Iic_union_Ioc Set.Iic_union_Ioc theorem Iio_subset_Iic_union_Ioo : Iio b ⊆ Iic a ∪ Ioo a b := fun x hx => (le_or_lt x a).elim (fun hxa => Or.inl hxa) fun hxa => Or.inr ⟨hxa, hx⟩ #align set.Iio_subset_Iic_union_Ioo Set.Iio_subset_Iic_union_Ioo @[simp] theorem Iic_union_Ioo_eq_Iio (h : a < b) : Iic a ∪ Ioo a b = Iio b := Subset.antisymm (fun _ hx => hx.elim (fun hx' => lt_of_le_of_lt hx' h) And.right) Iio_subset_Iic_union_Ioo #align set.Iic_union_Ioo_eq_Iio Set.Iic_union_Ioo_eq_Iio	Mathlib/Order/Interval/Set/Basic.lean	1,477	1,483	theorem Iio_union_Ioo' (h₁ : c < b) : Iio b ∪ Ioo c d = Iio (max b d) := by	ext x cases' lt_or_le x b with hba hba · simp [hba, h₁] · simp only [mem_Iio, mem_union, mem_Ioo, lt_max_iff] refine or_congr Iff.rfl ⟨And.right, ?_⟩ exact fun h₂ => ⟨h₁.trans_le hba, h₂⟩
/- Copyright (c) 2021 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov, Alistair Tucker, Wen Yang -/ import Mathlib.Order.Interval.Set.Image import Mathlib.Order.CompleteLatticeIntervals import Mathlib.Topology.Order.DenselyOrdered import Mathlib.Topology.Order.Monotone #align_import topology.algebra.order.intermediate_value from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514" /-! # Intermediate Value Theorem In this file we prove the Intermediate Value Theorem: if `f : α → β` is a function defined on a connected set `s` that takes both values `≤ a` and values `≥ a` on `s`, then it is equal to `a` at some point of `s`. We also prove that intervals in a dense conditionally complete order are preconnected and any preconnected set is an interval. Then we specialize IVT to functions continuous on intervals. ## Main results * `IsPreconnected_I??` : all intervals `I??` are preconnected, * `IsPreconnected.intermediate_value`, `intermediate_value_univ` : Intermediate Value Theorem for connected sets and connected spaces, respectively; * `intermediate_value_Icc`, `intermediate_value_Icc'`: Intermediate Value Theorem for functions on closed intervals. ### Miscellaneous facts * `IsClosed.Icc_subset_of_forall_mem_nhdsWithin` : “Continuous induction” principle; if `s ∩ [a, b]` is closed, `a ∈ s`, and for each `x ∈ [a, b) ∩ s` some of its right neighborhoods is included `s`, then `[a, b] ⊆ s`. * `IsClosed.Icc_subset_of_forall_exists_gt`, `IsClosed.mem_of_ge_of_forall_exists_gt` : two other versions of the “continuous induction” principle. * `ContinuousOn.StrictMonoOn_of_InjOn_Ioo` : Every continuous injective `f : (a, b) → δ` is strictly monotone or antitone (increasing or decreasing). ## Tags intermediate value theorem, connected space, connected set -/ open Filter OrderDual TopologicalSpace Function Set open Topology Filter universe u v w /-! ### Intermediate value theorem on a (pre)connected space In this section we prove the following theorem (see `IsPreconnected.intermediate_value₂`): if `f` and `g` are two functions continuous on a preconnected set `s`, `f a ≤ g a` at some `a ∈ s` and `g b ≤ f b` at some `b ∈ s`, then `f c = g c` at some `c ∈ s`. We prove several versions of this statement, including the classical IVT that corresponds to a constant function `g`. -/ section variable {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] /-- Intermediate value theorem for two functions: if `f` and `g` are two continuous functions on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. -/ theorem intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x := by obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x \| f x ≤ g x ∧ g x ≤ f x }).Nonempty := isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg) (isClosed_le hg hf) (fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩ exact ⟨x, le_antisymm hfg hgf⟩ #align intermediate_value_univ₂ intermediate_value_univ₂ theorem intermediate_value_univ₂_eventually₁ [PreconnectedSpace X] {a : X} {l : Filter X} [NeBot l] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x, f x = g x := let ⟨_, h⟩ := he.exists; intermediate_value_univ₂ hf hg ha h #align intermediate_value_univ₂_eventually₁ intermediate_value_univ₂_eventually₁ theorem intermediate_value_univ₂_eventually₂ [PreconnectedSpace X] {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) : ∃ x, f x = g x := let ⟨_, h₁⟩ := he₁.exists let ⟨_, h₂⟩ := he₂.exists intermediate_value_univ₂ hf hg h₁ h₂ #align intermediate_value_univ₂_eventually₂ intermediate_value_univ₂_eventually₂ /-- Intermediate value theorem for two functions: if `f` and `g` are two functions continuous on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`, then for some `x ∈ s` we have `f x = g x`. -/ theorem IsPreconnected.intermediate_value₂ {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a) (hb' : g b ≤ f b) : ∃ x ∈ s, f x = g x := let ⟨x, hx⟩ := @intermediate_value_univ₂ s α _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ ⟨b, hb⟩ _ _ (continuousOn_iff_continuous_restrict.1 hf) (continuousOn_iff_continuous_restrict.1 hg) ha' hb' ⟨x, x.2, hx⟩ #align is_preconnected.intermediate_value₂ IsPreconnected.intermediate_value₂ theorem IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x := by rw [continuousOn_iff_continuous_restrict] at hf hg obtain ⟨b, h⟩ := @intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _ (comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _) exact ⟨b, b.prop, h⟩ #align is_preconnected.intermediate_value₂_eventually₁ IsPreconnected.intermediate_value₂_eventually₁ theorem IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) : ∃ x ∈ s, f x = g x := by rw [continuousOn_iff_continuous_restrict] at hf hg obtain ⟨b, h⟩ := @intermediate_value_univ₂_eventually₂ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) _ _ (comap_coe_neBot_of_le_principal hl₁) (comap_coe_neBot_of_le_principal hl₂) _ _ hf hg (he₁.comap _) (he₂.comap _) exact ⟨b, b.prop, h⟩ #align is_preconnected.intermediate_value₂_eventually₂ IsPreconnected.intermediate_value₂_eventually₂ /-- Intermediate Value Theorem for continuous functions on connected sets. -/ theorem IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s) (hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s) : Icc (f a) (f b) ⊆ f '' s := fun _x hx => hs.intermediate_value₂ ha hb hf continuousOn_const hx.1 hx.2 #align is_preconnected.intermediate_value IsPreconnected.intermediate_value theorem IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s := fun _ h => hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1 (eventually_ge_of_tendsto_gt h.2 ht) #align is_preconnected.intermediate_value_Ico IsPreconnected.intermediate_value_Ico theorem IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s := fun _ h => (hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h.2 (eventually_le_of_tendsto_lt h.1 ht)).imp fun _ h => h.imp_right Eq.symm #align is_preconnected.intermediate_value_Ioc IsPreconnected.intermediate_value_Ioc theorem IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) : Ioo v₁ v₂ ⊆ f '' s := fun _ h => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (eventually_le_of_tendsto_lt h.1 ht₁) (eventually_ge_of_tendsto_gt h.2 ht₂) #align is_preconnected.intermediate_value_Ioo IsPreconnected.intermediate_value_Ioo theorem IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht : Tendsto f l atTop) : Ici (f a) ⊆ f '' s := fun y h => hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h (tendsto_atTop.1 ht y) #align is_preconnected.intermediate_value_Ici IsPreconnected.intermediate_value_Ici theorem IsPreconnected.intermediate_value_Iic {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht : Tendsto f l atBot) : Iic (f a) ⊆ f '' s := fun y h => (hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h (tendsto_atBot.1 ht y)).imp fun _ h => h.imp_right Eq.symm #align is_preconnected.intermediate_value_Iic IsPreconnected.intermediate_value_Iic theorem IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s := fun y h => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (eventually_le_of_tendsto_lt h ht₁) (tendsto_atTop.1 ht₂ y) #align is_preconnected.intermediate_value_Ioi IsPreconnected.intermediate_value_Ioi theorem IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s := fun y h => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (tendsto_atBot.1 ht₁ y) (eventually_ge_of_tendsto_gt h ht₂) #align is_preconnected.intermediate_value_Iio IsPreconnected.intermediate_value_Iio theorem IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s := fun y _ => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (tendsto_atBot.1 ht₁ y) (tendsto_atTop.1 ht₂ y) set_option linter.uppercaseLean3 false in #align is_preconnected.intermediate_value_Iii IsPreconnected.intermediate_value_Iii /-- Intermediate Value Theorem for continuous functions on connected spaces. -/ theorem intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) : Icc (f a) (f b) ⊆ range f := fun _ hx => intermediate_value_univ₂ hf continuous_const hx.1 hx.2 #align intermediate_value_univ intermediate_value_univ /-- Intermediate Value Theorem for continuous functions on connected spaces. -/ theorem mem_range_of_exists_le_of_exists_ge [PreconnectedSpace X] {c : α} {f : X → α} (hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) : c ∈ range f := let ⟨a, ha⟩ := h₁; let ⟨b, hb⟩ := h₂; intermediate_value_univ a b hf ⟨ha, hb⟩ #align mem_range_of_exists_le_of_exists_ge mem_range_of_exists_le_of_exists_ge /-! ### (Pre)connected sets in a linear order In this section we prove the following results: * `IsPreconnected.ordConnected`: any preconnected set `s` in a linear order is `OrdConnected`, i.e. `a ∈ s` and `b ∈ s` imply `Icc a b ⊆ s`; * `IsPreconnected.mem_intervals`: any preconnected set `s` in a conditionally complete linear order is one of the intervals `Set.Icc`, `set.`Ico`, `set.Ioc`, `set.Ioo`, ``Set.Ici`, `Set.Iic`, `Set.Ioi`, `Set.Iio`; note that this is false for non-complete orders: e.g., in `ℝ \ {0}`, the set of positive numbers cannot be represented as `Set.Ioi _`. -/ /-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/ theorem IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : Icc a b ⊆ s := by simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id #align is_preconnected.Icc_subset IsPreconnected.Icc_subset theorem IsPreconnected.ordConnected {s : Set α} (h : IsPreconnected s) : OrdConnected s := ⟨fun _ hx _ hy => h.Icc_subset hx hy⟩ #align is_preconnected.ord_connected IsPreconnected.ordConnected /-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/ theorem IsConnected.Icc_subset {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : Icc a b ⊆ s := hs.2.Icc_subset ha hb #align is_connected.Icc_subset IsConnected.Icc_subset /-- If preconnected set in a linear order space is unbounded below and above, then it is the whole space. -/ theorem IsPreconnected.eq_univ_of_unbounded {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s) (ha : ¬BddAbove s) : s = univ := by refine eq_univ_of_forall fun x => ?_ obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bddBelow_iff.1 hb x obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩ #align is_preconnected.eq_univ_of_unbounded IsPreconnected.eq_univ_of_unbounded end variable {α : Type u} {β : Type v} {γ : Type w} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] [Nonempty γ] /-- A bounded connected subset of a conditionally complete linear order includes the open interval `(Inf s, Sup s)`. -/ theorem IsConnected.Ioo_csInf_csSup_subset {s : Set α} (hs : IsConnected s) (hb : BddBelow s) (ha : BddAbove s) : Ioo (sInf s) (sSup s) ⊆ s := fun _x hx => let ⟨_y, ys, hy⟩ := (isGLB_lt_iff (isGLB_csInf hs.nonempty hb)).1 hx.1 let ⟨_z, zs, hz⟩ := (lt_isLUB_iff (isLUB_csSup hs.nonempty ha)).1 hx.2 hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ #align is_connected.Ioo_cInf_cSup_subset IsConnected.Ioo_csInf_csSup_subset theorem eq_Icc_csInf_csSup_of_connected_bdd_closed {s : Set α} (hc : IsConnected s) (hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s) : s = Icc (sInf s) (sSup s) := (subset_Icc_csInf_csSup hb ha).antisymm <\| hc.Icc_subset (hcl.csInf_mem hc.nonempty hb) (hcl.csSup_mem hc.nonempty ha) #align eq_Icc_cInf_cSup_of_connected_bdd_closed eq_Icc_csInf_csSup_of_connected_bdd_closed theorem IsPreconnected.Ioi_csInf_subset {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s) (ha : ¬BddAbove s) : Ioi (sInf s) ⊆ s := fun x hx => have sne : s.Nonempty := nonempty_of_not_bddAbove ha let ⟨_y, ys, hy⟩ : ∃ y ∈ s, y < x := (isGLB_lt_iff (isGLB_csInf sne hb)).1 hx let ⟨_z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ #align is_preconnected.Ioi_cInf_subset IsPreconnected.Ioi_csInf_subset theorem IsPreconnected.Iio_csSup_subset {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s) (ha : BddAbove s) : Iio (sSup s) ⊆ s := IsPreconnected.Ioi_csInf_subset (α := αᵒᵈ) hs ha hb #align is_preconnected.Iio_cSup_subset IsPreconnected.Iio_csSup_subset /-- A preconnected set in a conditionally complete linear order is either one of the intervals `[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`, `(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires `α` to be densely ordered. -/ theorem IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) : s ∈ ({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) := by rcases s.eq_empty_or_nonempty with (rfl \| hne) · apply_rules [Or.inr, mem_singleton] have hs' : IsConnected s := ⟨hne, hs⟩ by_cases hb : BddBelow s <;> by_cases ha : BddAbove s · refine mem_of_subset_of_mem ?_ <\| mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset (hs'.Ioo_csInf_csSup_subset hb ha) (subset_Icc_csInf_csSup hb ha) simp only [insert_subset_iff, mem_insert_iff, mem_singleton_iff, true_or, or_true, singleton_subset_iff, and_self] · refine Or.inr <\| Or.inr <\| Or.inr <\| Or.inr ?_ cases' mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_csInf_subset hb ha) fun x hx => csInf_le hb hx with hs hs · exact Or.inl hs · exact Or.inr (Or.inl hs) · iterate 6 apply Or.inr cases' mem_Iic_Iio_of_subset_of_subset (hs.Iio_csSup_subset hb ha) fun x hx => le_csSup ha hx with hs hs · exact Or.inl hs · exact Or.inr (Or.inl hs) · iterate 8 apply Or.inr exact Or.inl (hs.eq_univ_of_unbounded hb ha) #align is_preconnected.mem_intervals IsPreconnected.mem_intervals /-- A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordered. Though one can represent `∅` as `(Inf ∅, Inf ∅)`, we include it into the list of possible cases to improve readability. -/ theorem setOf_isPreconnected_subset_of_ordered : { s : Set α \| IsPreconnected s } ⊆ -- bounded intervals (range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪ -- unbounded intervals and `univ` (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by intro s hs rcases hs.mem_intervals with (hs \| hs \| hs \| hs \| hs \| hs \| hs \| hs \| hs \| hs) <;> rw [hs] <;> simp only [union_insert, union_singleton, mem_insert_iff, mem_union, mem_range, Prod.exists, uncurry_apply_pair, exists_apply_eq_apply, true_or, or_true, exists_apply_eq_apply2] #align set_of_is_preconnected_subset_of_ordered setOf_isPreconnected_subset_of_ordered /-! ### Intervals are connected In this section we prove that a closed interval (hence, any `OrdConnected` set) in a dense conditionally complete linear order is preconnected. -/ /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. -/ theorem IsClosed.mem_of_ge_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty) : b ∈ s := by let S := s ∩ Icc a b replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩ have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩ let c := sSup (s ∩ Icc a b) have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2 cases' eq_or_lt_of_le c_le with hc hc · exact hc ▸ c_mem.1 exfalso rcases hgt c ⟨c_mem.1, c_mem.2.1, hc⟩ with ⟨x, xs, cx, xb⟩ exact not_lt_of_le (le_csSup Sbd ⟨xs, le_trans (le_csSup Sbd ha) (le_of_lt cx), xb⟩) cx #align is_closed.mem_of_ge_of_forall_exists_gt IsClosed.mem_of_ge_of_forall_exists_gt /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]` is not empty, then `[a, b] ⊆ s`. -/ theorem IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s := by intro y hy have : IsClosed (s ∩ Icc a y) := by suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y by rw [this] exact IsClosed.inter hs isClosed_Icc rw [inter_assoc] congr exact (inter_eq_self_of_subset_right <\| Icc_subset_Icc_right hy.2).symm exact IsClosed.mem_of_ge_of_forall_exists_gt this ha hy.1 fun x hx => hgt x ⟨hx.1, Ico_subset_Ico_right hy.2 hx.2⟩ y hx.2.2 #align is_closed.Icc_subset_of_forall_exists_gt IsClosed.Icc_subset_of_forall_exists_gt variable [DenselyOrdered α] {a b : α} /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. -/ theorem IsClosed.Icc_subset_of_forall_mem_nhdsWithin {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) : Icc a b ⊆ s := by apply hs.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxs, hxab⟩ y hyxb have : s ∩ Ioc x y ∈ 𝓝[>] x := inter_mem (hgt x ⟨hxs, hxab⟩) (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hyxb⟩) exact (nhdsWithin_Ioi_self_neBot' ⟨b, hxab.2⟩).nonempty_of_mem this #align is_closed.Icc_subset_of_forall_mem_nhds_within IsClosed.Icc_subset_of_forall_mem_nhdsWithin theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s) (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) : (Icc a b ∩ (s ∩ t)).Nonempty := by have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2 by_contra hst suffices Icc x y ⊆ s from hst ⟨y, xyab <\| right_mem_Icc.2 hxy, this <\| right_mem_Icc.2 hxy, hy.2⟩ apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2 rintro z ⟨zs, hz⟩ have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <\| Ico_subset_Icc_self hz, zs, zt⟩ have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2] exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩ apply mem_of_superset this have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩) exact fun w ⟨wt, wzy⟩ => (this wzy).elim id fun h => (wt h).elim #align is_preconnected_Icc_aux isPreconnected_Icc_aux /-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/ theorem isPreconnected_Icc : IsPreconnected (Icc a b) := isPreconnected_closed_iff.2 (by rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩ -- This used to use `wlog`, but it was causing timeouts. rcases le_total x y with h \| h · exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy · rw [inter_comm s t] rw [union_comm s t] at hab exact isPreconnected_Icc_aux y x t s h ht hs hab hy hx) #align is_preconnected_Icc isPreconnected_Icc theorem isPreconnected_uIcc : IsPreconnected (uIcc a b) := isPreconnected_Icc #align is_preconnected_uIcc isPreconnected_uIcc theorem Set.OrdConnected.isPreconnected {s : Set α} (h : s.OrdConnected) : IsPreconnected s := isPreconnected_of_forall_pair fun x hx y hy => ⟨uIcc x y, h.uIcc_subset hx hy, left_mem_uIcc, right_mem_uIcc, isPreconnected_uIcc⟩ #align set.ord_connected.is_preconnected Set.OrdConnected.isPreconnected theorem isPreconnected_iff_ordConnected {s : Set α} : IsPreconnected s ↔ OrdConnected s := ⟨IsPreconnected.ordConnected, Set.OrdConnected.isPreconnected⟩ #align is_preconnected_iff_ord_connected isPreconnected_iff_ordConnected theorem isPreconnected_Ici : IsPreconnected (Ici a) := ordConnected_Ici.isPreconnected #align is_preconnected_Ici isPreconnected_Ici theorem isPreconnected_Iic : IsPreconnected (Iic a) := ordConnected_Iic.isPreconnected #align is_preconnected_Iic isPreconnected_Iic theorem isPreconnected_Iio : IsPreconnected (Iio a) := ordConnected_Iio.isPreconnected #align is_preconnected_Iio isPreconnected_Iio theorem isPreconnected_Ioi : IsPreconnected (Ioi a) := ordConnected_Ioi.isPreconnected #align is_preconnected_Ioi isPreconnected_Ioi theorem isPreconnected_Ioo : IsPreconnected (Ioo a b) := ordConnected_Ioo.isPreconnected #align is_preconnected_Ioo isPreconnected_Ioo theorem isPreconnected_Ioc : IsPreconnected (Ioc a b) := ordConnected_Ioc.isPreconnected #align is_preconnected_Ioc isPreconnected_Ioc theorem isPreconnected_Ico : IsPreconnected (Ico a b) := ordConnected_Ico.isPreconnected #align is_preconnected_Ico isPreconnected_Ico theorem isConnected_Ici : IsConnected (Ici a) := ⟨nonempty_Ici, isPreconnected_Ici⟩ #align is_connected_Ici isConnected_Ici theorem isConnected_Iic : IsConnected (Iic a) := ⟨nonempty_Iic, isPreconnected_Iic⟩ #align is_connected_Iic isConnected_Iic theorem isConnected_Ioi [NoMaxOrder α] : IsConnected (Ioi a) := ⟨nonempty_Ioi, isPreconnected_Ioi⟩ #align is_connected_Ioi isConnected_Ioi theorem isConnected_Iio [NoMinOrder α] : IsConnected (Iio a) := ⟨nonempty_Iio, isPreconnected_Iio⟩ #align is_connected_Iio isConnected_Iio theorem isConnected_Icc (h : a ≤ b) : IsConnected (Icc a b) := ⟨nonempty_Icc.2 h, isPreconnected_Icc⟩ #align is_connected_Icc isConnected_Icc theorem isConnected_Ioo (h : a < b) : IsConnected (Ioo a b) := ⟨nonempty_Ioo.2 h, isPreconnected_Ioo⟩ #align is_connected_Ioo isConnected_Ioo theorem isConnected_Ioc (h : a < b) : IsConnected (Ioc a b) := ⟨nonempty_Ioc.2 h, isPreconnected_Ioc⟩ #align is_connected_Ioc isConnected_Ioc theorem isConnected_Ico (h : a < b) : IsConnected (Ico a b) := ⟨nonempty_Ico.2 h, isPreconnected_Ico⟩ #align is_connected_Ico isConnected_Ico instance (priority := 100) ordered_connected_space : PreconnectedSpace α := ⟨ordConnected_univ.isPreconnected⟩ #align ordered_connected_space ordered_connected_space /-- In a dense conditionally complete linear order, the set of preconnected sets is exactly the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`, or `∅`. Though one can represent `∅` as `(sInf s, sInf s)`, we include it into the list of possible cases to improve readability. -/ theorem setOf_isPreconnected_eq_of_ordered : { s : Set α \| IsPreconnected s } = -- bounded intervals range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪ -- unbounded intervals and `univ` (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by refine Subset.antisymm setOf_isPreconnected_subset_of_ordered ?_ simp only [subset_def, forall_mem_range, uncurry, or_imp, forall_and, mem_union, mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true_iff, isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Ioo, isPreconnected_Ioi, isPreconnected_Iio, isPreconnected_Ici, isPreconnected_Iic, isPreconnected_univ, isPreconnected_empty] #align set_of_is_preconnected_eq_of_ordered setOf_isPreconnected_eq_of_ordered /-- This lemmas characterizes when a subset `s` of a densely ordered conditionally complete linear order is totally disconnected with respect to the order topology: between any two distinct points of `s` must lie a point not in `s`. -/ lemma isTotallyDisconnected_iff_lt {s : Set α} : IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y := by simp only [IsTotallyDisconnected, isPreconnected_iff_ordConnected, ← not_nontrivial_iff, nontrivial_iff_exists_lt, not_exists, not_and] refine ⟨fun h x hx y hy hxy ↦ ?_, fun h t hts ht x hx y hy hxy ↦ ?_⟩ · simp_rw [← not_ordConnected_inter_Icc_iff hx hy] exact fun hs ↦ h _ inter_subset_left hs _ ⟨hx, le_rfl, hxy.le⟩ _ ⟨hy, hxy.le, le_rfl⟩ hxy · obtain ⟨z, h1z, h2z⟩ := h x (hts hx) y (hts hy) hxy exact h1z <\| hts <\| ht.1 hx hy ⟨h2z.1.le, h2z.2.le⟩ /-! ### Intermediate Value Theorem on an interval In this section we prove several versions of the Intermediate Value Theorem for a function continuous on an interval. -/ variable {δ : Type} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] /-- Intermediate Value Theorem* for continuous functions on closed intervals, case `f a ≤ t ≤ f b`. -/ theorem intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Icc (f a) (f b) ⊆ f '' Icc a b := isPreconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf #align intermediate_value_Icc intermediate_value_Icc /-- Intermediate Value Theorem for continuous functions on closed intervals, case `f a ≥ t ≥ f b`. -/ theorem intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Icc (f b) (f a) ⊆ f '' Icc a b := isPreconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf #align intermediate_value_Icc' intermediate_value_Icc' /-- Intermediate Value Theorem for continuous functions on closed intervals, unordered case. -/ theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f (uIcc a b)) : uIcc (f a) (f b) ⊆ f '' uIcc a b := by cases le_total (f a) (f b) <;> simp [, isPreconnected_uIcc.intermediate_value] #align intermediate_value_uIcc intermediate_value_uIcc theorem intermediate_value_Ico {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ico (f a) (f b) ⊆ f '' Ico a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_le (he ▸ h.1))) fun hlt => @IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩ (right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _ ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ico_subset_Icc_self) #align intermediate_value_Ico intermediate_value_Ico theorem intermediate_value_Ico' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioc (f b) (f a) ⊆ f '' Ico a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_le (he ▸ h.2))) fun hlt => @IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩ (right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _ ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ico_subset_Icc_self) #align intermediate_value_Ico' intermediate_value_Ico' theorem intermediate_value_Ioc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioc (f a) (f b) ⊆ f '' Ioc a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_le_of_lt (he ▸ h.1))) fun hlt => @IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩ (left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _ ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioc_subset_Icc_self) #align intermediate_value_Ioc intermediate_value_Ioc theorem intermediate_value_Ioc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ico (f b) (f a) ⊆ f '' Ioc a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_le_of_lt (he ▸ h.2))) fun hlt => @IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩ (left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _ ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioc_subset_Icc_self) #align intermediate_value_Ioc' intermediate_value_Ioc' theorem intermediate_value_Ioo {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioo (f a) (f b) ⊆ f '' Ioo a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_lt (he ▸ h.1))) fun hlt => @IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _ (left_nhdsWithin_Ioo_neBot hlt) (right_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _ (hf.mono Ioo_subset_Icc_self) _ _ ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioo_subset_Icc_self) ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ioo_subset_Icc_self) #align intermediate_value_Ioo intermediate_value_Ioo theorem intermediate_value_Ioo' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioo (f b) (f a) ⊆ f '' Ioo a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_lt (he ▸ h.2))) fun hlt => @IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _ (right_nhdsWithin_Ioo_neBot hlt) (left_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _ (hf.mono Ioo_subset_Icc_self) _ _ ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ioo_subset_Icc_self) ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioo_subset_Icc_self) #align intermediate_value_Ioo' intermediate_value_Ioo' /-- Intermediate value theorem: if `f` is continuous on an order-connected set `s` and `a`, `b` are two points of this set, then `f` sends `s` to a superset of `Icc (f x) (f y)`. -/ theorem ContinuousOn.surjOn_Icc {s : Set α} [hs : OrdConnected s] {f : α → δ} (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (Icc (f a) (f b)) := hs.isPreconnected.intermediate_value ha hb hf #align continuous_on.surj_on_Icc ContinuousOn.surjOn_Icc /-- Intermediate value theorem: if `f` is continuous on an order-connected set `s` and `a`, `b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. -/ theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ} (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (uIcc (f a) (f b)) := by rcases le_total (f a) (f b) with hab \| hab <;> simp [hf.surjOn_Icc, ] #align continuous_on.surj_on_uIcc ContinuousOn.surjOn_uIcc /-- A continuous function which tendsto `Filter.atTop` along `Filter.atTop` and to `atBot` along `at_bot` is surjective. -/ theorem Continuous.surjective {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atTop atTop) (h_bot : Tendsto f atBot atBot) : Function.Surjective f := fun p => mem_range_of_exists_le_of_exists_ge hf (h_bot.eventually (eventually_le_atBot p)).exists (h_top.eventually (eventually_ge_atTop p)).exists #align continuous.surjective Continuous.surjective /-- A continuous function which tendsto `Filter.atBot` along `Filter.atTop` and to `Filter.atTop` along `atBot` is surjective. -/ theorem Continuous.surjective' {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atBot atTop) (h_bot : Tendsto f atTop atBot) : Function.Surjective f := Continuous.surjective (α := αᵒᵈ) hf h_top h_bot #align continuous.surjective' Continuous.surjective' /-- If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s` tends to `at_bot : Filter β` along `at_bot : Filter ↥s` and tends to `Filter.atTop : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the conclusion as `Function.surjOn f s Set.univ`. -/ theorem ContinuousOn.surjOn_of_tendsto {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty) (hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atBot) (htop : Tendsto (fun x : s => f x) atTop atTop) : SurjOn f s univ := haveI := Classical.inhabited_of_nonempty hs.to_subtype surjOn_iff_surjective.2 <\| hf.restrict.surjective htop hbot #align continuous_on.surj_on_of_tendsto ContinuousOn.surjOn_of_tendsto /-- If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s` tends to `Filter.atTop : Filter β` along `Filter.atBot : Filter ↥s` and tends to `Filter.atBot : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the conclusion as `Function.surjOn f s Set.univ`. -/ theorem ContinuousOn.surjOn_of_tendsto' {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty) (hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atTop) (htop : Tendsto (fun x : s => f x) atTop atBot) : SurjOn f s univ := ContinuousOn.surjOn_of_tendsto (δ := δᵒᵈ) hs hf hbot htop #align continuous_on.surj_on_of_tendsto' ContinuousOn.surjOn_of_tendsto' theorem Continuous.strictMono_of_inj_boundedOrder [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf : f ⊥ ≤ f ⊤) (hf_i : Injective f) : StrictMono f := by intro a b hab by_contra! h have H : f b < f a := lt_of_le_of_ne h <\| hf_i.ne hab.ne' by_cases ha : f a ≤ f ⊥ · obtain ⟨u, hu⟩ := intermediate_value_Ioc le_top hf_c.continuousOn ⟨H.trans_le ha, hf⟩ have : u = ⊥ := hf_i hu.2 aesop · by_cases hb : f ⊥ < f b · obtain ⟨u, hu⟩ := intermediate_value_Ioo bot_le hf_c.continuousOn ⟨hb, H⟩ rw [hf_i hu.2] at hu exact (hab.trans hu.1.2).false · push_neg at ha hb replace hb : f b < f ⊥ := lt_of_le_of_ne hb <\| hf_i.ne (lt_of_lt_of_le' hab bot_le).ne' obtain ⟨u, hu⟩ := intermediate_value_Ioo' hab.le hf_c.continuousOn ⟨hb, ha⟩ have : u = ⊥ := hf_i hu.2 aesop theorem Continuous.strictAnti_of_inj_boundedOrder [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf : f ⊤ ≤ f ⊥) (hf_i : Injective f) : StrictAnti f := hf_c.strictMono_of_inj_boundedOrder (δ := δᵒᵈ) hf hf_i theorem Continuous.strictMono_of_inj_boundedOrder' [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f := (le_total (f ⊥) (f ⊤)).imp (hf_c.strictMono_of_inj_boundedOrder · hf_i) (hf_c.strictAnti_of_inj_boundedOrder · hf_i) /-- Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is continuous and injective. Then `f` is strictly monotone (increasing) if it is strictly monotone (increasing) on some closed interval `[a, b]`. -/ theorem Continuous.strictMonoOn_of_inj_rigidity {f : α → δ} (hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b) (hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f := by intro x y hxy let s := min a x let t := max b y have hsa : s ≤ a := min_le_left a x have hbt : b ≤ t := le_max_left b y have hst : s ≤ t := hsa.trans $ hbt.trans' hab.le have hf_mono_st : StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t) := by letI := Icc.completeLinearOrder hst have := Continuous.strictMono_of_inj_boundedOrder' (f := Set.restrict (Icc s t) f) hf_c.continuousOn.restrict hf_i.injOn.injective exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr have (h : StrictAntiOn f (Icc s t)) : False := by have : Icc a b ⊆ Icc s t := Icc_subset_Icc hsa hbt replace : StrictAntiOn f (Icc a b) := StrictAntiOn.mono h this replace : IsAntichain (· ≤ ·) (Icc a b) := IsAntichain.of_strictMonoOn_antitoneOn hf_mono this.antitoneOn exact this.not_lt (left_mem_Icc.mpr (le_of_lt hab)) (right_mem_Icc.mpr (le_of_lt hab)) hab replace hf_mono_st : StrictMonoOn f (Icc s t) := hf_mono_st.resolve_right this have hsx : s ≤ x := min_le_right a x have hyt : y ≤ t := le_max_right b y replace : Icc x y ⊆ Icc s t := Icc_subset_Icc hsx hyt replace : StrictMonoOn f (Icc x y) := StrictMonoOn.mono hf_mono_st this exact this (left_mem_Icc.mpr (le_of_lt hxy)) (right_mem_Icc.mpr (le_of_lt hxy)) hxy /-- Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone (increasing) if `f(a) ≤ f(b)`. -/ theorem ContinuousOn.strictMonoOn_of_injOn_Icc {a b : α} {f : α → δ} (hab : a ≤ b) (hfab : f a ≤ f b) (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) : StrictMonoOn f (Icc a b) := by letI := Icc.completeLinearOrder hab refine StrictMono.of_restrict ?_ set g : Icc a b → δ := Set.restrict (Icc a b) f have hgab : g ⊥ ≤ g ⊤ := by aesop exact Continuous.strictMono_of_inj_boundedOrder (f := g) hf_c.restrict hgab hf_i.injective /-- Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly antitone (decreasing) if `f(b) ≤ f(a)`. -/ theorem ContinuousOn.strictAntiOn_of_injOn_Icc {a b : α} {f : α → δ} (hab : a ≤ b) (hfab : f b ≤ f a) (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) : StrictAntiOn f (Icc a b) := ContinuousOn.strictMonoOn_of_injOn_Icc (δ := δᵒᵈ) hab hfab hf_c hf_i /-- Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone or antitone (increasing or decreasing). -/ theorem ContinuousOn.strictMonoOn_of_injOn_Icc' {a b : α} {f : α → δ} (hab : a ≤ b) (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) : StrictMonoOn f (Icc a b) ∨ StrictAntiOn f (Icc a b) := (le_total (f a) (f b)).imp (ContinuousOn.strictMonoOn_of_injOn_Icc hab · hf_c hf_i) (ContinuousOn.strictAntiOn_of_injOn_Icc hab · hf_c hf_i) /-- Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is continuous and injective. Then `f` is strictly monotone or antitone (increasing or decreasing). -/ theorem Continuous.strictMono_of_inj {f : α → δ} (hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f := by have H {c d : α} (hcd : c < d) : StrictMono f ∨ StrictAnti f := (hf_c.continuousOn.strictMonoOn_of_injOn_Icc' hcd.le hf_i.injOn).imp (hf_c.strictMonoOn_of_inj_rigidity hf_i hcd) (hf_c.strictMonoOn_of_inj_rigidity (δ := δᵒᵈ) hf_i hcd) by_cases hn : Nonempty α · let a : α := Classical.choice ‹_› by_cases h : ∃ b : α, a ≠ b · choose b hb using h by_cases hab : a < b · exact H hab · push_neg at hab have : b < a := by exact Ne.lt_of_le (id (Ne.symm hb)) hab exact H this · push_neg at h haveI : Subsingleton α := ⟨fun c d => Trans.trans (h c).symm (h d)⟩ exact Or.inl <\| Subsingleton.strictMono f · aesop /-- Every continuous injective `f : (a, b) → δ` is strictly monotone or antitone (increasing or decreasing). -/	Mathlib/Topology/Order/IntermediateValue.lean	765	772	theorem ContinuousOn.strictMonoOn_of_injOn_Ioo {a b : α} {f : α → δ} (hab : a < b) (hf_c : ContinuousOn f (Ioo a b)) (hf_i : InjOn f (Ioo a b)) : StrictMonoOn f (Ioo a b) ∨ StrictAntiOn f (Ioo a b) := by	haveI : Inhabited (Ioo a b) := Classical.inhabited_of_nonempty (nonempty_Ioo_subtype hab) let g : Ioo a b → δ := Set.restrict (Ioo a b) f have : StrictMono g ∨ StrictAnti g := Continuous.strictMono_of_inj hf_c.restrict hf_i.injective exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr
/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import Mathlib.Logic.Equiv.PartialEquiv import Mathlib.Topology.Sets.Opens #align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db" /-! # Partial homeomorphisms This file defines homeomorphisms between open subsets of topological spaces. An element `e` of `PartialHomeomorph X Y` is an extension of `PartialEquiv X Y`, i.e., it is a pair of functions `e.toFun` and `e.invFun`, inverse of each other on the sets `e.source` and `e.target`. Additionally, we require that these sets are open, and that the functions are continuous on them. Equivalently, they are homeomorphisms there. As in equivs, we register a coercion to functions, and we use `e x` and `e.symm x` throughout instead of `e.toFun x` and `e.invFun x`. ## Main definitions * `Homeomorph.toPartialHomeomorph`: associating a partial homeomorphism to a homeomorphism, with `source = target = Set.univ`; * `PartialHomeomorph.symm`: the inverse of a partial homeomorphism * `PartialHomeomorph.trans`: the composition of two partial homeomorphisms * `PartialHomeomorph.refl`: the identity partial homeomorphism * `PartialHomeomorph.ofSet`: the identity on a set `s` * `PartialHomeomorph.EqOnSource`: equivalence relation describing the "right" notion of equality for partial homeomorphisms ## Implementation notes Most statements are copied from their `PartialEquiv` versions, although some care is required especially when restricting to subsets, as these should be open subsets. For design notes, see `PartialEquiv.lean`. ### Local coding conventions If a lemma deals with the intersection of a set with either source or target of a `PartialEquiv`, then it should use `e.source ∩ s` or `e.target ∩ t`, not `s ∩ e.source` or `t ∩ e.target`. -/ open Function Set Filter Topology variable {X X' : Type} {Y Y' : Type} {Z Z' : Type} [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y'] [TopologicalSpace Z] [TopologicalSpace Z'] /-- Partial homeomorphisms, defined on open subsets of the space -/ -- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance] structure PartialHomeomorph (X : Type) (Y : Type*) [TopologicalSpace X] [TopologicalSpace Y] extends PartialEquiv X Y where open_source : IsOpen source open_target : IsOpen target continuousOn_toFun : ContinuousOn toFun source continuousOn_invFun : ContinuousOn invFun target #align local_homeomorph PartialHomeomorph namespace PartialHomeomorph variable (e : PartialHomeomorph X Y) /-! Basic properties; inverse (symm instance) -/ section Basic /-- Coercion of a partial homeomorphisms to a function. We don't use `e.toFun` because it is actually `e.toPartialEquiv.toFun`, so `simp` will apply lemmas about `toPartialEquiv`. While we may want to switch to this behavior later, doing it mid-port will break a lot of proofs. -/ @[coe] def toFun' : X → Y := e.toFun /-- Coercion of a `PartialHomeomorph` to function. Note that a `PartialHomeomorph` is not `DFunLike`. -/ instance : CoeFun (PartialHomeomorph X Y) fun _ => X → Y := ⟨fun e => e.toFun'⟩ /-- The inverse of a partial homeomorphism -/ @[symm] protected def symm : PartialHomeomorph Y X where toPartialEquiv := e.toPartialEquiv.symm open_source := e.open_target open_target := e.open_source continuousOn_toFun := e.continuousOn_invFun continuousOn_invFun := e.continuousOn_toFun #align local_homeomorph.symm PartialHomeomorph.symm /-- See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections. -/ def Simps.apply (e : PartialHomeomorph X Y) : X → Y := e #align local_homeomorph.simps.apply PartialHomeomorph.Simps.apply /-- See Note [custom simps projection] -/ def Simps.symm_apply (e : PartialHomeomorph X Y) : Y → X := e.symm #align local_homeomorph.simps.symm_apply PartialHomeomorph.Simps.symm_apply initialize_simps_projections PartialHomeomorph (toFun → apply, invFun → symm_apply) protected theorem continuousOn : ContinuousOn e e.source := e.continuousOn_toFun #align local_homeomorph.continuous_on PartialHomeomorph.continuousOn theorem continuousOn_symm : ContinuousOn e.symm e.target := e.continuousOn_invFun #align local_homeomorph.continuous_on_symm PartialHomeomorph.continuousOn_symm @[simp, mfld_simps] theorem mk_coe (e : PartialEquiv X Y) (a b c d) : (PartialHomeomorph.mk e a b c d : X → Y) = e := rfl #align local_homeomorph.mk_coe PartialHomeomorph.mk_coe @[simp, mfld_simps] theorem mk_coe_symm (e : PartialEquiv X Y) (a b c d) : ((PartialHomeomorph.mk e a b c d).symm : Y → X) = e.symm := rfl #align local_homeomorph.mk_coe_symm PartialHomeomorph.mk_coe_symm theorem toPartialEquiv_injective : Injective (toPartialEquiv : PartialHomeomorph X Y → PartialEquiv X Y) \| ⟨_, _, _, _, _⟩, ⟨_, _, _, _, _⟩, rfl => rfl #align local_homeomorph.to_local_equiv_injective PartialHomeomorph.toPartialEquiv_injective /- Register a few simp lemmas to make sure that `simp` puts the application of a local homeomorphism in its normal form, i.e., in terms of its coercion to a function. -/ @[simp, mfld_simps] theorem toFun_eq_coe (e : PartialHomeomorph X Y) : e.toFun = e := rfl #align local_homeomorph.to_fun_eq_coe PartialHomeomorph.toFun_eq_coe @[simp, mfld_simps] theorem invFun_eq_coe (e : PartialHomeomorph X Y) : e.invFun = e.symm := rfl #align local_homeomorph.inv_fun_eq_coe PartialHomeomorph.invFun_eq_coe @[simp, mfld_simps] theorem coe_coe : (e.toPartialEquiv : X → Y) = e := rfl #align local_homeomorph.coe_coe PartialHomeomorph.coe_coe @[simp, mfld_simps] theorem coe_coe_symm : (e.toPartialEquiv.symm : Y → X) = e.symm := rfl #align local_homeomorph.coe_coe_symm PartialHomeomorph.coe_coe_symm @[simp, mfld_simps] theorem map_source {x : X} (h : x ∈ e.source) : e x ∈ e.target := e.map_source' h #align local_homeomorph.map_source PartialHomeomorph.map_source /-- Variant of `map_source`, stated for images of subsets of `source`. -/ lemma map_source'' : e '' e.source ⊆ e.target := fun _ ⟨_, hx, hex⟩ ↦ mem_of_eq_of_mem (id hex.symm) (e.map_source' hx) @[simp, mfld_simps] theorem map_target {x : Y} (h : x ∈ e.target) : e.symm x ∈ e.source := e.map_target' h #align local_homeomorph.map_target PartialHomeomorph.map_target @[simp, mfld_simps] theorem left_inv {x : X} (h : x ∈ e.source) : e.symm (e x) = x := e.left_inv' h #align local_homeomorph.left_inv PartialHomeomorph.left_inv @[simp, mfld_simps] theorem right_inv {x : Y} (h : x ∈ e.target) : e (e.symm x) = x := e.right_inv' h #align local_homeomorph.right_inv PartialHomeomorph.right_inv theorem eq_symm_apply {x : X} {y : Y} (hx : x ∈ e.source) (hy : y ∈ e.target) : x = e.symm y ↔ e x = y := e.toPartialEquiv.eq_symm_apply hx hy #align local_homeomorph.eq_symm_apply PartialHomeomorph.eq_symm_apply protected theorem mapsTo : MapsTo e e.source e.target := fun _ => e.map_source #align local_homeomorph.maps_to PartialHomeomorph.mapsTo protected theorem symm_mapsTo : MapsTo e.symm e.target e.source := e.symm.mapsTo #align local_homeomorph.symm_maps_to PartialHomeomorph.symm_mapsTo protected theorem leftInvOn : LeftInvOn e.symm e e.source := fun _ => e.left_inv #align local_homeomorph.left_inv_on PartialHomeomorph.leftInvOn protected theorem rightInvOn : RightInvOn e.symm e e.target := fun _ => e.right_inv #align local_homeomorph.right_inv_on PartialHomeomorph.rightInvOn protected theorem invOn : InvOn e.symm e e.source e.target := ⟨e.leftInvOn, e.rightInvOn⟩ #align local_homeomorph.inv_on PartialHomeomorph.invOn protected theorem injOn : InjOn e e.source := e.leftInvOn.injOn #align local_homeomorph.inj_on PartialHomeomorph.injOn protected theorem bijOn : BijOn e e.source e.target := e.invOn.bijOn e.mapsTo e.symm_mapsTo #align local_homeomorph.bij_on PartialHomeomorph.bijOn protected theorem surjOn : SurjOn e e.source e.target := e.bijOn.surjOn #align local_homeomorph.surj_on PartialHomeomorph.surjOn end Basic /-- Interpret a `Homeomorph` as a `PartialHomeomorph` by restricting it to an open set `s` in the domain and to `t` in the codomain. -/ @[simps! (config := .asFn) apply symm_apply toPartialEquiv, simps! (config := .lemmasOnly) source target] def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y where toPartialEquiv := e.toPartialEquivOfImageEq s t h open_source := hs open_target := by simpa [← h] continuousOn_toFun := e.continuous.continuousOn continuousOn_invFun := e.symm.continuous.continuousOn /-- A homeomorphism induces a partial homeomorphism on the whole space -/ @[simps! (config := mfld_cfg)] def _root_.Homeomorph.toPartialHomeomorph (e : X ≃ₜ Y) : PartialHomeomorph X Y := e.toPartialHomeomorphOfImageEq univ isOpen_univ univ <\| by rw [image_univ, e.surjective.range_eq] #align homeomorph.to_local_homeomorph Homeomorph.toPartialHomeomorph /-- Replace `toPartialEquiv` field to provide better definitional equalities. -/ def replaceEquiv (e : PartialHomeomorph X Y) (e' : PartialEquiv X Y) (h : e.toPartialEquiv = e') : PartialHomeomorph X Y where toPartialEquiv := e' open_source := h ▸ e.open_source open_target := h ▸ e.open_target continuousOn_toFun := h ▸ e.continuousOn_toFun continuousOn_invFun := h ▸ e.continuousOn_invFun #align local_homeomorph.replace_equiv PartialHomeomorph.replaceEquiv theorem replaceEquiv_eq_self (e' : PartialEquiv X Y) (h : e.toPartialEquiv = e') : e.replaceEquiv e' h = e := by cases e subst e' rfl #align local_homeomorph.replace_equiv_eq_self PartialHomeomorph.replaceEquiv_eq_self theorem source_preimage_target : e.source ⊆ e ⁻¹' e.target := e.mapsTo #align local_homeomorph.source_preimage_target PartialHomeomorph.source_preimage_target @[deprecated toPartialEquiv_injective (since := "2023-02-18")] theorem eq_of_partialEquiv_eq {e e' : PartialHomeomorph X Y} (h : e.toPartialEquiv = e'.toPartialEquiv) : e = e' := toPartialEquiv_injective h #align local_homeomorph.eq_of_local_equiv_eq PartialHomeomorph.eq_of_partialEquiv_eq theorem eventually_left_inverse {x} (hx : x ∈ e.source) : ∀ᶠ y in 𝓝 x, e.symm (e y) = y := (e.open_source.eventually_mem hx).mono e.left_inv' #align local_homeomorph.eventually_left_inverse PartialHomeomorph.eventually_left_inverse theorem eventually_left_inverse' {x} (hx : x ∈ e.target) : ∀ᶠ y in 𝓝 (e.symm x), e.symm (e y) = y := e.eventually_left_inverse (e.map_target hx) #align local_homeomorph.eventually_left_inverse' PartialHomeomorph.eventually_left_inverse' theorem eventually_right_inverse {x} (hx : x ∈ e.target) : ∀ᶠ y in 𝓝 x, e (e.symm y) = y := (e.open_target.eventually_mem hx).mono e.right_inv' #align local_homeomorph.eventually_right_inverse PartialHomeomorph.eventually_right_inverse theorem eventually_right_inverse' {x} (hx : x ∈ e.source) : ∀ᶠ y in 𝓝 (e x), e (e.symm y) = y := e.eventually_right_inverse (e.map_source hx) #align local_homeomorph.eventually_right_inverse' PartialHomeomorph.eventually_right_inverse' theorem eventually_ne_nhdsWithin {x} (hx : x ∈ e.source) : ∀ᶠ x' in 𝓝[≠] x, e x' ≠ e x := eventually_nhdsWithin_iff.2 <\| (e.eventually_left_inverse hx).mono fun x' hx' => mt fun h => by rw [mem_singleton_iff, ← e.left_inv hx, ← h, hx'] #align local_homeomorph.eventually_ne_nhds_within PartialHomeomorph.eventually_ne_nhdsWithin theorem nhdsWithin_source_inter {x} (hx : x ∈ e.source) (s : Set X) : 𝓝[e.source ∩ s] x = 𝓝[s] x := nhdsWithin_inter_of_mem (mem_nhdsWithin_of_mem_nhds <\| IsOpen.mem_nhds e.open_source hx) #align local_homeomorph.nhds_within_source_inter PartialHomeomorph.nhdsWithin_source_inter theorem nhdsWithin_target_inter {x} (hx : x ∈ e.target) (s : Set Y) : 𝓝[e.target ∩ s] x = 𝓝[s] x := e.symm.nhdsWithin_source_inter hx s #align local_homeomorph.nhds_within_target_inter PartialHomeomorph.nhdsWithin_target_inter theorem image_eq_target_inter_inv_preimage {s : Set X} (h : s ⊆ e.source) : e '' s = e.target ∩ e.symm ⁻¹' s := e.toPartialEquiv.image_eq_target_inter_inv_preimage h #align local_homeomorph.image_eq_target_inter_inv_preimage PartialHomeomorph.image_eq_target_inter_inv_preimage theorem image_source_inter_eq' (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' s := e.toPartialEquiv.image_source_inter_eq' s #align local_homeomorph.image_source_inter_eq' PartialHomeomorph.image_source_inter_eq' theorem image_source_inter_eq (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' (e.source ∩ s) := e.toPartialEquiv.image_source_inter_eq s #align local_homeomorph.image_source_inter_eq PartialHomeomorph.image_source_inter_eq theorem symm_image_eq_source_inter_preimage {s : Set Y} (h : s ⊆ e.target) : e.symm '' s = e.source ∩ e ⁻¹' s := e.symm.image_eq_target_inter_inv_preimage h #align local_homeomorph.symm_image_eq_source_inter_preimage PartialHomeomorph.symm_image_eq_source_inter_preimage theorem symm_image_target_inter_eq (s : Set Y) : e.symm '' (e.target ∩ s) = e.source ∩ e ⁻¹' (e.target ∩ s) := e.symm.image_source_inter_eq _ #align local_homeomorph.symm_image_target_inter_eq PartialHomeomorph.symm_image_target_inter_eq theorem source_inter_preimage_inv_preimage (s : Set X) : e.source ∩ e ⁻¹' (e.symm ⁻¹' s) = e.source ∩ s := e.toPartialEquiv.source_inter_preimage_inv_preimage s #align local_homeomorph.source_inter_preimage_inv_preimage PartialHomeomorph.source_inter_preimage_inv_preimage theorem target_inter_inv_preimage_preimage (s : Set Y) : e.target ∩ e.symm ⁻¹' (e ⁻¹' s) = e.target ∩ s := e.symm.source_inter_preimage_inv_preimage _ #align local_homeomorph.target_inter_inv_preimage_preimage PartialHomeomorph.target_inter_inv_preimage_preimage theorem source_inter_preimage_target_inter (s : Set Y) : e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s := e.toPartialEquiv.source_inter_preimage_target_inter s #align local_homeomorph.source_inter_preimage_target_inter PartialHomeomorph.source_inter_preimage_target_inter theorem image_source_eq_target : e '' e.source = e.target := e.toPartialEquiv.image_source_eq_target #align local_homeomorph.image_source_eq_target PartialHomeomorph.image_source_eq_target theorem symm_image_target_eq_source : e.symm '' e.target = e.source := e.symm.image_source_eq_target #align local_homeomorph.symm_image_target_eq_source PartialHomeomorph.symm_image_target_eq_source /-- Two partial homeomorphisms are equal when they have equal `toFun`, `invFun` and `source`. It is not sufficient to have equal `toFun` and `source`, as this only determines `invFun` on the target. This would only be true for a weaker notion of equality, arguably the right one, called `EqOnSource`. -/ @[ext] protected theorem ext (e' : PartialHomeomorph X Y) (h : ∀ x, e x = e' x) (hinv : ∀ x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' := toPartialEquiv_injective (PartialEquiv.ext h hinv hs) #align local_homeomorph.ext PartialHomeomorph.ext protected theorem ext_iff {e e' : PartialHomeomorph X Y} : e = e' ↔ (∀ x, e x = e' x) ∧ (∀ x, e.symm x = e'.symm x) ∧ e.source = e'.source := ⟨by rintro rfl exact ⟨fun x => rfl, fun x => rfl, rfl⟩, fun h => e.ext e' h.1 h.2.1 h.2.2⟩ #align local_homeomorph.ext_iff PartialHomeomorph.ext_iff @[simp, mfld_simps] theorem symm_toPartialEquiv : e.symm.toPartialEquiv = e.toPartialEquiv.symm := rfl #align local_homeomorph.symm_to_local_equiv PartialHomeomorph.symm_toPartialEquiv -- The following lemmas are already simp via `PartialEquiv` theorem symm_source : e.symm.source = e.target := rfl #align local_homeomorph.symm_source PartialHomeomorph.symm_source theorem symm_target : e.symm.target = e.source := rfl #align local_homeomorph.symm_target PartialHomeomorph.symm_target @[simp, mfld_simps] theorem symm_symm : e.symm.symm = e := rfl #align local_homeomorph.symm_symm PartialHomeomorph.symm_symm theorem symm_bijective : Function.Bijective (PartialHomeomorph.symm : PartialHomeomorph X Y → PartialHomeomorph Y X) := Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ /-- A partial homeomorphism is continuous at any point of its source -/ protected theorem continuousAt {x : X} (h : x ∈ e.source) : ContinuousAt e x := (e.continuousOn x h).continuousAt (e.open_source.mem_nhds h) #align local_homeomorph.continuous_at PartialHomeomorph.continuousAt /-- A partial homeomorphism inverse is continuous at any point of its target -/ theorem continuousAt_symm {x : Y} (h : x ∈ e.target) : ContinuousAt e.symm x := e.symm.continuousAt h #align local_homeomorph.continuous_at_symm PartialHomeomorph.continuousAt_symm theorem tendsto_symm {x} (hx : x ∈ e.source) : Tendsto e.symm (𝓝 (e x)) (𝓝 x) := by simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx) #align local_homeomorph.tendsto_symm PartialHomeomorph.tendsto_symm theorem map_nhds_eq {x} (hx : x ∈ e.source) : map e (𝓝 x) = 𝓝 (e x) := le_antisymm (e.continuousAt hx) <\| le_map_of_right_inverse (e.eventually_right_inverse' hx) (e.tendsto_symm hx) #align local_homeomorph.map_nhds_eq PartialHomeomorph.map_nhds_eq theorem symm_map_nhds_eq {x} (hx : x ∈ e.source) : map e.symm (𝓝 (e x)) = 𝓝 x := (e.symm.map_nhds_eq <\| e.map_source hx).trans <\| by rw [e.left_inv hx] #align local_homeomorph.symm_map_nhds_eq PartialHomeomorph.symm_map_nhds_eq theorem image_mem_nhds {x} (hx : x ∈ e.source) {s : Set X} (hs : s ∈ 𝓝 x) : e '' s ∈ 𝓝 (e x) := e.map_nhds_eq hx ▸ Filter.image_mem_map hs #align local_homeomorph.image_mem_nhds PartialHomeomorph.image_mem_nhds theorem map_nhdsWithin_eq {x} (hx : x ∈ e.source) (s : Set X) : map e (𝓝[s] x) = 𝓝[e '' (e.source ∩ s)] e x := calc map e (𝓝[s] x) = map e (𝓝[e.source ∩ s] x) := congr_arg (map e) (e.nhdsWithin_source_inter hx _).symm _ = 𝓝[e '' (e.source ∩ s)] e x := (e.leftInvOn.mono inter_subset_left).map_nhdsWithin_eq (e.left_inv hx) (e.continuousAt_symm (e.map_source hx)).continuousWithinAt (e.continuousAt hx).continuousWithinAt #align local_homeomorph.map_nhds_within_eq PartialHomeomorph.map_nhdsWithin_eq theorem map_nhdsWithin_preimage_eq {x} (hx : x ∈ e.source) (s : Set Y) : map e (𝓝[e ⁻¹' s] x) = 𝓝[s] e x := by rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage, e.nhdsWithin_target_inter (e.map_source hx)] #align local_homeomorph.map_nhds_within_preimage_eq PartialHomeomorph.map_nhdsWithin_preimage_eq theorem eventually_nhds {x : X} (p : Y → Prop) (hx : x ∈ e.source) : (∀ᶠ y in 𝓝 (e x), p y) ↔ ∀ᶠ x in 𝓝 x, p (e x) := Iff.trans (by rw [e.map_nhds_eq hx]) eventually_map #align local_homeomorph.eventually_nhds PartialHomeomorph.eventually_nhds theorem eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) : (∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x := by rw [e.eventually_nhds _ hx] refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_) rw [hy] #align local_homeomorph.eventually_nhds' PartialHomeomorph.eventually_nhds'	Mathlib/Topology/PartialHomeomorph.lean	427	430	theorem eventually_nhdsWithin {x : X} (p : Y → Prop) {s : Set X} (hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) := by	refine Iff.trans ?_ eventually_map rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)]
/- Copyright (c) 2018 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Simon Hudon -/ import Mathlib.Data.PFunctor.Multivariate.Basic #align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" /-! # Multivariate quotients of polynomial functors. Basic definition of multivariate QPF. QPFs form a compositional framework for defining inductive and coinductive types, their quotients and nesting. The idea is based on building ever larger functors. For instance, we can define a list using a shape functor: ```lean inductive ListShape (a b : Type) \| nil : ListShape \| cons : a -> b -> ListShape ``` This shape can itself be decomposed as a sum of product which are themselves QPFs. It follows that the shape is a QPF and we can take its fixed point and create the list itself: ```lean def List (a : Type) := fix ListShape a -- not the actual notation ``` We can continue and define the quotient on permutation of lists and create the multiset type: ```lean def Multiset (a : Type) := QPF.quot List.perm List a -- not the actual notion ``` And `Multiset` is also a QPF. We can then create a novel data type (for Lean): ```lean inductive Tree (a : Type) \| node : a -> Multiset Tree -> Tree ``` An unordered tree. This is currently not supported by Lean because it nests an inductive type inside of a quotient. We can go further and define unordered, possibly infinite trees: ```lean coinductive Tree' (a : Type) \| node : a -> Multiset Tree' -> Tree' ``` by using the `cofix` construct. Those options can all be mixed and matched because they preserve the properties of QPF. The latter example, `Tree'`, combines fixed point, co-fixed point and quotients. ## Related modules * constructions * Fix * Cofix * Quot * Comp * Sigma / Pi * Prj * Const each proves that some operations on functors preserves the QPF structure ## Reference ! * [Jeremy Avigad, Mario M. Carneiro and Simon Hudon, Data Types as Quotients of Polynomial Functors][avigad-carneiro-hudon2019] -/ universe u open MvFunctor /-- Multivariate quotients of polynomial functors. -/ class MvQPF {n : ℕ} (F : TypeVec.{u} n → Type) [MvFunctor F] where P : MvPFunctor.{u} n abs : ∀ {α}, P α → F α repr : ∀ {α}, F α → P α abs_repr : ∀ {α} (x : F α), abs (repr x) = x abs_map : ∀ {α β} (f : α ⟹ β) (p : P α), abs (f <$$> p) = f <$$> abs p #align mvqpf MvQPF namespace MvQPF variable {n : ℕ} {F : TypeVec.{u} n → Type} [MvFunctor F] [q : MvQPF F] open MvFunctor (LiftP LiftR) /-! ### Show that every MvQPF is a lawful MvFunctor. -/ protected theorem id_map {α : TypeVec n} (x : F α) : TypeVec.id <$$> x = x := by rw [← abs_repr x] cases' repr x with a f rw [← abs_map] rfl #align mvqpf.id_map MvQPF.id_map @[simp] theorem comp_map {α β γ : TypeVec n} (f : α ⟹ β) (g : β ⟹ γ) (x : F α) : (g ⊚ f) <$$> x = g <$$> f <$$> x := by rw [← abs_repr x] cases' repr x with a f rw [← abs_map, ← abs_map, ← abs_map] rfl #align mvqpf.comp_map MvQPF.comp_map instance (priority := 100) lawfulMvFunctor : LawfulMvFunctor F where id_map := @MvQPF.id_map n F _ _ comp_map := @comp_map n F _ _ #align mvqpf.is_lawful_mvfunctor MvQPF.lawfulMvFunctor -- Lifting predicates and relations theorem liftP_iff {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (x : F α) : LiftP p x ↔ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ i j, p (f i j) := by constructor · rintro ⟨y, hy⟩ cases' h : repr y with a f use a, fun i j => (f i j).val constructor · rw [← hy, ← abs_repr y, h, ← abs_map]; rfl intro i j apply (f i j).property rintro ⟨a, f, h₀, h₁⟩ use abs ⟨a, fun i j => ⟨f i j, h₁ i j⟩⟩ rw [← abs_map, h₀]; rfl #align mvqpf.liftp_iff MvQPF.liftP_iff	Mathlib/Data/QPF/Multivariate/Basic.lean	141	157	theorem liftR_iff {α : TypeVec n} (r : ∀ /- ⦃i⦄ -/ {i}, α i → α i → Prop) (x y : F α) : LiftR r x y ↔ ∃ a f₀ f₁, x = abs ⟨a, f₀⟩ ∧ y = abs ⟨a, f₁⟩ ∧ ∀ i j, r (f₀ i j) (f₁ i j) := by	constructor · rintro ⟨u, xeq, yeq⟩ cases' h : repr u with a f use a, fun i j => (f i j).val.fst, fun i j => (f i j).val.snd constructor · rw [← xeq, ← abs_repr u, h, ← abs_map]; rfl constructor · rw [← yeq, ← abs_repr u, h, ← abs_map]; rfl intro i j exact (f i j).property rintro ⟨a, f₀, f₁, xeq, yeq, h⟩ use abs ⟨a, fun i j => ⟨(f₀ i j, f₁ i j), h i j⟩⟩ dsimp; constructor · rw [xeq, ← abs_map]; rfl rw [yeq, ← abs_map]; rfl
/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import Mathlib.Algebra.Group.Indicator import Mathlib.Data.Finset.Piecewise import Mathlib.Data.Finset.Preimage #align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83" /-! # Big operators In this file we define products and sums indexed by finite sets (specifically, `Finset`). ## Notation We introduce the following notation. Let `s` be a `Finset α`, and `f : α → β` a function. * `∏ x ∈ s, f x` is notation for `Finset.prod s f` (assuming `β` is a `CommMonoid`) * `∑ x ∈ s, f x` is notation for `Finset.sum s f` (assuming `β` is an `AddCommMonoid`) * `∏ x, f x` is notation for `Finset.prod Finset.univ f` (assuming `α` is a `Fintype` and `β` is a `CommMonoid`) * `∑ x, f x` is notation for `Finset.sum Finset.univ f` (assuming `α` is a `Fintype` and `β` is an `AddCommMonoid`) ## Implementation Notes The first arguments in all definitions and lemmas is the codomain of the function of the big operator. This is necessary for the heuristic in `@[to_additive]`. See the documentation of `to_additive.attr` for more information. -/ -- TODO -- assert_not_exists AddCommMonoidWithOne assert_not_exists MonoidWithZero assert_not_exists MulAction variable {ι κ α β γ : Type} open Fin Function namespace Finset /-- `∏ x ∈ s, f x` is the product of `f x` as `x` ranges over the elements of the finite set `s`. -/ @[to_additive "`∑ x ∈ s, f x` is the sum of `f x` as `x` ranges over the elements of the finite set `s`."] protected def prod [CommMonoid β] (s : Finset α) (f : α → β) : β := (s.1.map f).prod #align finset.prod Finset.prod #align finset.sum Finset.sum @[to_additive (attr := simp)] theorem prod_mk [CommMonoid β] (s : Multiset α) (hs : s.Nodup) (f : α → β) : (⟨s, hs⟩ : Finset α).prod f = (s.map f).prod := rfl #align finset.prod_mk Finset.prod_mk #align finset.sum_mk Finset.sum_mk @[to_additive (attr := simp)] theorem prod_val [CommMonoid α] (s : Finset α) : s.1.prod = s.prod id := by rw [Finset.prod, Multiset.map_id] #align finset.prod_val Finset.prod_val #align finset.sum_val Finset.sum_val end Finset library_note "operator precedence of big operators"/-- There is no established mathematical convention for the operator precedence of big operators like `∏` and `∑`. We will have to make a choice. Online discussions, such as https://math.stackexchange.com/q/185538/30839 seem to suggest that `∏` and `∑` should have the same precedence, and that this should be somewhere between `` and `+`. The latter have precedence levels `70` and `65` respectively, and we therefore choose the level `67`. In practice, this means that parentheses should be placed as follows: ```lean ∑ k ∈ K, (a k + b k) = ∑ k ∈ K, a k + ∑ k ∈ K, b k → ∏ k ∈ K, a k * b k = (∏ k ∈ K, a k) * (∏ k ∈ K, b k) ``` (Example taken from page 490 of Knuth's Concrete Mathematics.) -/ namespace BigOperators open Batteries.ExtendedBinder Lean Meta -- TODO: contribute this modification back to `extBinder` /-- A `bigOpBinder` is like an `extBinder` and has the form `x`, `x : ty`, or `x pred` where `pred` is a `binderPred` like `< 2`. Unlike `extBinder`, `x` is a term. -/ syntax bigOpBinder := term:max ((" : " term) <\|> binderPred)? /-- A BigOperator binder in parentheses -/ syntax bigOpBinderParenthesized := " (" bigOpBinder ")" /-- A list of parenthesized binders -/ syntax bigOpBinderCollection := bigOpBinderParenthesized+ /-- A single (unparenthesized) binder, or a list of parenthesized binders -/ syntax bigOpBinders := bigOpBinderCollection <\|> (ppSpace bigOpBinder) /-- Collects additional binder/Finset pairs for the given `bigOpBinder`. Note: this is not extensible at the moment, unlike the usual `bigOpBinder` expansions. -/ def processBigOpBinder (processed : (Array (Term × Term))) (binder : TSyntax ``bigOpBinder) : MacroM (Array (Term × Term)) := set_option hygiene false in withRef binder do match binder with \| `(bigOpBinder\| $x:term) => match x with \| `(($a + $b = $n)) => -- Maybe this is too cute. return processed \|>.push (← `(⟨$a, $b⟩), ← `(Finset.Nat.antidiagonal $n)) \| _ => return processed \|>.push (x, ← ``(Finset.univ)) \| `(bigOpBinder\| $x : $t) => return processed \|>.push (x, ← ``((Finset.univ : Finset $t))) \| `(bigOpBinder\| $x ∈ $s) => return processed \|>.push (x, ← `(finset% $s)) \| `(bigOpBinder\| $x < $n) => return processed \|>.push (x, ← `(Finset.Iio $n)) \| `(bigOpBinder\| $x ≤ $n) => return processed \|>.push (x, ← `(Finset.Iic $n)) \| `(bigOpBinder\| $x > $n) => return processed \|>.push (x, ← `(Finset.Ioi $n)) \| `(bigOpBinder\| $x ≥ $n) => return processed \|>.push (x, ← `(Finset.Ici $n)) \| _ => Macro.throwUnsupported /-- Collects the binder/Finset pairs for the given `bigOpBinders`. -/ def processBigOpBinders (binders : TSyntax ``bigOpBinders) : MacroM (Array (Term × Term)) := match binders with \| `(bigOpBinders\| $b:bigOpBinder) => processBigOpBinder #[] b \| `(bigOpBinders\| $[($bs:bigOpBinder)]) => bs.foldlM processBigOpBinder #[] \| _ => Macro.throwUnsupported /-- Collect the binderIdents into a `⟨...⟩` expression. -/ def bigOpBindersPattern (processed : (Array (Term × Term))) : MacroM Term := do let ts := processed.map Prod.fst if ts.size == 1 then return ts[0]! else `(⟨$ts,⟩) /-- Collect the terms into a product of sets. -/ def bigOpBindersProd (processed : (Array (Term × Term))) : MacroM Term := do if processed.isEmpty then `((Finset.univ : Finset Unit)) else if processed.size == 1 then return processed[0]!.2 else processed.foldrM (fun s p => `(SProd.sprod $(s.2) $p)) processed.back.2 (start := processed.size - 1) /-- - `∑ x, f x` is notation for `Finset.sum Finset.univ f`. It is the sum of `f x`, where `x` ranges over the finite domain of `f`. - `∑ x ∈ s, f x` is notation for `Finset.sum s f`. It is the sum of `f x`, where `x` ranges over the finite set `s` (either a `Finset` or a `Set` with a `Fintype` instance). - `∑ x ∈ s with p x, f x` is notation for `Finset.sum (Finset.filter p s) f`. - `∑ (x ∈ s) (y ∈ t), f x y` is notation for `Finset.sum (s ×ˢ t) (fun ⟨x, y⟩ ↦ f x y)`. These support destructuring, for example `∑ ⟨x, y⟩ ∈ s ×ˢ t, f x y`. Notation: `"∑" bigOpBinders* ("with" term)? "," term` -/ syntax (name := bigsum) "∑ " bigOpBinders ("with " term)? ", " term:67 : term /-- - `∏ x, f x` is notation for `Finset.prod Finset.univ f`. It is the product of `f x`, where `x` ranges over the finite domain of `f`. - `∏ x ∈ s, f x` is notation for `Finset.prod s f`. It is the product of `f x`, where `x` ranges over the finite set `s` (either a `Finset` or a `Set` with a `Fintype` instance). - `∏ x ∈ s with p x, f x` is notation for `Finset.prod (Finset.filter p s) f`. - `∏ (x ∈ s) (y ∈ t), f x y` is notation for `Finset.prod (s ×ˢ t) (fun ⟨x, y⟩ ↦ f x y)`. These support destructuring, for example `∏ ⟨x, y⟩ ∈ s ×ˢ t, f x y`. Notation: `"∏" bigOpBinders* ("with" term)? "," term` -/ syntax (name := bigprod) "∏ " bigOpBinders ("with " term)? ", " term:67 : term macro_rules (kind := bigsum) \| `(∑ $bs:bigOpBinders $[with $p?]?, $v) => do let processed ← processBigOpBinders bs let x ← bigOpBindersPattern processed let s ← bigOpBindersProd processed match p? with \| some p => `(Finset.sum (Finset.filter (fun $x ↦ $p) $s) (fun $x ↦ $v)) \| none => `(Finset.sum $s (fun $x ↦ $v)) macro_rules (kind := bigprod) \| `(∏ $bs:bigOpBinders $[with $p?]?, $v) => do let processed ← processBigOpBinders bs let x ← bigOpBindersPattern processed let s ← bigOpBindersProd processed match p? with \| some p => `(Finset.prod (Finset.filter (fun $x ↦ $p) $s) (fun $x ↦ $v)) \| none => `(Finset.prod $s (fun $x ↦ $v)) /-- (Deprecated, use `∑ x ∈ s, f x`) `∑ x in s, f x` is notation for `Finset.sum s f`. It is the sum of `f x`, where `x` ranges over the finite set `s`. -/ syntax (name := bigsumin) "∑ " extBinder " in " term ", " term:67 : term macro_rules (kind := bigsumin) \| `(∑ $x:ident in $s, $r) => `(∑ $x:ident ∈ $s, $r) \| `(∑ $x:ident : $t in $s, $r) => `(∑ $x:ident ∈ ($s : Finset $t), $r) /-- (Deprecated, use `∏ x ∈ s, f x`) `∏ x in s, f x` is notation for `Finset.prod s f`. It is the product of `f x`, where `x` ranges over the finite set `s`. -/ syntax (name := bigprodin) "∏ " extBinder " in " term ", " term:67 : term macro_rules (kind := bigprodin) \| `(∏ $x:ident in $s, $r) => `(∏ $x:ident ∈ $s, $r) \| `(∏ $x:ident : $t in $s, $r) => `(∏ $x:ident ∈ ($s : Finset $t), $r) open Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr open Batteries.ExtendedBinder /-- Delaborator for `Finset.prod`. The `pp.piBinderTypes` option controls whether to show the domain type when the product is over `Finset.univ`. -/ @[delab app.Finset.prod] def delabFinsetProd : Delab := whenPPOption getPPNotation <\| withOverApp 5 <\| do let #[_, _, _, s, f] := (← getExpr).getAppArgs \| failure guard <\| f.isLambda let ppDomain ← getPPOption getPPPiBinderTypes let (i, body) ← withAppArg <\| withBindingBodyUnusedName fun i => do return (i, ← delab) if s.isAppOfArity ``Finset.univ 2 then let binder ← if ppDomain then let ty ← withNaryArg 0 delab `(bigOpBinder\| $(.mk i):ident : $ty) else `(bigOpBinder\| $(.mk i):ident) `(∏ $binder:bigOpBinder, $body) else let ss ← withNaryArg 3 <\| delab `(∏ $(.mk i):ident ∈ $ss, $body) /-- Delaborator for `Finset.sum`. The `pp.piBinderTypes` option controls whether to show the domain type when the sum is over `Finset.univ`. -/ @[delab app.Finset.sum] def delabFinsetSum : Delab := whenPPOption getPPNotation <\| withOverApp 5 <\| do let #[_, _, _, s, f] := (← getExpr).getAppArgs \| failure guard <\| f.isLambda let ppDomain ← getPPOption getPPPiBinderTypes let (i, body) ← withAppArg <\| withBindingBodyUnusedName fun i => do return (i, ← delab) if s.isAppOfArity ``Finset.univ 2 then let binder ← if ppDomain then let ty ← withNaryArg 0 delab `(bigOpBinder\| $(.mk i):ident : $ty) else `(bigOpBinder\| $(.mk i):ident) `(∑ $binder:bigOpBinder, $body) else let ss ← withNaryArg 3 <\| delab `(∑ $(.mk i):ident ∈ $ss, $body) end BigOperators namespace Finset variable {s s₁ s₂ : Finset α} {a : α} {f g : α → β} @[to_additive] theorem prod_eq_multiset_prod [CommMonoid β] (s : Finset α) (f : α → β) : ∏ x ∈ s, f x = (s.1.map f).prod := rfl #align finset.prod_eq_multiset_prod Finset.prod_eq_multiset_prod #align finset.sum_eq_multiset_sum Finset.sum_eq_multiset_sum @[to_additive (attr := simp)] lemma prod_map_val [CommMonoid β] (s : Finset α) (f : α → β) : (s.1.map f).prod = ∏ a ∈ s, f a := rfl #align finset.prod_map_val Finset.prod_map_val #align finset.sum_map_val Finset.sum_map_val @[to_additive] theorem prod_eq_fold [CommMonoid β] (s : Finset α) (f : α → β) : ∏ x ∈ s, f x = s.fold ((· * ·) : β → β → β) 1 f := rfl #align finset.prod_eq_fold Finset.prod_eq_fold #align finset.sum_eq_fold Finset.sum_eq_fold @[simp] theorem sum_multiset_singleton (s : Finset α) : (s.sum fun x => {x}) = s.val := by simp only [sum_eq_multiset_sum, Multiset.sum_map_singleton] #align finset.sum_multiset_singleton Finset.sum_multiset_singleton end Finset @[to_additive (attr := simp)] theorem map_prod [CommMonoid β] [CommMonoid γ] {G : Type} [FunLike G β γ] [MonoidHomClass G β γ] (g : G) (f : α → β) (s : Finset α) : g (∏ x ∈ s, f x) = ∏ x ∈ s, g (f x) := by simp only [Finset.prod_eq_multiset_prod, map_multiset_prod, Multiset.map_map]; rfl #align map_prod map_prod #align map_sum map_sum @[to_additive] theorem MonoidHom.coe_finset_prod [MulOneClass β] [CommMonoid γ] (f : α → β → γ) (s : Finset α) : ⇑(∏ x ∈ s, f x) = ∏ x ∈ s, ⇑(f x) := map_prod (MonoidHom.coeFn β γ) _ _ #align monoid_hom.coe_finset_prod MonoidHom.coe_finset_prod #align add_monoid_hom.coe_finset_sum AddMonoidHom.coe_finset_sum /-- See also `Finset.prod_apply`, with the same conclusion but with the weaker hypothesis `f : α → β → γ` -/ @[to_additive (attr := simp) "See also `Finset.sum_apply`, with the same conclusion but with the weaker hypothesis `f : α → β → γ`"] theorem MonoidHom.finset_prod_apply [MulOneClass β] [CommMonoid γ] (f : α → β →* γ) (s : Finset α) (b : β) : (∏ x ∈ s, f x) b = ∏ x ∈ s, f x b := map_prod (MonoidHom.eval b) _ _ #align monoid_hom.finset_prod_apply MonoidHom.finset_prod_apply #align add_monoid_hom.finset_sum_apply AddMonoidHom.finset_sum_apply variable {s s₁ s₂ : Finset α} {a : α} {f g : α → β} namespace Finset section CommMonoid variable [CommMonoid β] @[to_additive (attr := simp)] theorem prod_empty : ∏ x ∈ ∅, f x = 1 := rfl #align finset.prod_empty Finset.prod_empty #align finset.sum_empty Finset.sum_empty @[to_additive] theorem prod_of_empty [IsEmpty α] (s : Finset α) : ∏ i ∈ s, f i = 1 := by rw [eq_empty_of_isEmpty s, prod_empty] #align finset.prod_of_empty Finset.prod_of_empty #align finset.sum_of_empty Finset.sum_of_empty @[to_additive (attr := simp)] theorem prod_cons (h : a ∉ s) : ∏ x ∈ cons a s h, f x = f a * ∏ x ∈ s, f x := fold_cons h #align finset.prod_cons Finset.prod_cons #align finset.sum_cons Finset.sum_cons @[to_additive (attr := simp)] theorem prod_insert [DecidableEq α] : a ∉ s → ∏ x ∈ insert a s, f x = f a * ∏ x ∈ s, f x := fold_insert #align finset.prod_insert Finset.prod_insert #align finset.sum_insert Finset.sum_insert /-- The product of `f` over `insert a s` is the same as the product over `s`, as long as `a` is in `s` or `f a = 1`. -/ @[to_additive (attr := simp) "The sum of `f` over `insert a s` is the same as the sum over `s`, as long as `a` is in `s` or `f a = 0`."] theorem prod_insert_of_eq_one_if_not_mem [DecidableEq α] (h : a ∉ s → f a = 1) : ∏ x ∈ insert a s, f x = ∏ x ∈ s, f x := by by_cases hm : a ∈ s · simp_rw [insert_eq_of_mem hm] · rw [prod_insert hm, h hm, one_mul] #align finset.prod_insert_of_eq_one_if_not_mem Finset.prod_insert_of_eq_one_if_not_mem #align finset.sum_insert_of_eq_zero_if_not_mem Finset.sum_insert_of_eq_zero_if_not_mem /-- The product of `f` over `insert a s` is the same as the product over `s`, as long as `f a = 1`. -/ @[to_additive (attr := simp) "The sum of `f` over `insert a s` is the same as the sum over `s`, as long as `f a = 0`."] theorem prod_insert_one [DecidableEq α] (h : f a = 1) : ∏ x ∈ insert a s, f x = ∏ x ∈ s, f x := prod_insert_of_eq_one_if_not_mem fun _ => h #align finset.prod_insert_one Finset.prod_insert_one #align finset.sum_insert_zero Finset.sum_insert_zero @[to_additive] theorem prod_insert_div {M : Type} [CommGroup M] [DecidableEq α] (ha : a ∉ s) {f : α → M} : (∏ x ∈ insert a s, f x) / f a = ∏ x ∈ s, f x := by simp [ha] @[to_additive (attr := simp)] theorem prod_singleton (f : α → β) (a : α) : ∏ x ∈ singleton a, f x = f a := Eq.trans fold_singleton <\| mul_one _ #align finset.prod_singleton Finset.prod_singleton #align finset.sum_singleton Finset.sum_singleton @[to_additive] theorem prod_pair [DecidableEq α] {a b : α} (h : a ≠ b) : (∏ x ∈ ({a, b} : Finset α), f x) = f a f b := by rw [prod_insert (not_mem_singleton.2 h), prod_singleton] #align finset.prod_pair Finset.prod_pair #align finset.sum_pair Finset.sum_pair @[to_additive (attr := simp)] theorem prod_const_one : (∏ _x ∈ s, (1 : β)) = 1 := by simp only [Finset.prod, Multiset.map_const', Multiset.prod_replicate, one_pow] #align finset.prod_const_one Finset.prod_const_one #align finset.sum_const_zero Finset.sum_const_zero @[to_additive (attr := simp)] theorem prod_image [DecidableEq α] {s : Finset γ} {g : γ → α} : (∀ x ∈ s, ∀ y ∈ s, g x = g y → x = y) → ∏ x ∈ s.image g, f x = ∏ x ∈ s, f (g x) := fold_image #align finset.prod_image Finset.prod_image #align finset.sum_image Finset.sum_image @[to_additive (attr := simp)] theorem prod_map (s : Finset α) (e : α ↪ γ) (f : γ → β) : ∏ x ∈ s.map e, f x = ∏ x ∈ s, f (e x) := by rw [Finset.prod, Finset.map_val, Multiset.map_map]; rfl #align finset.prod_map Finset.prod_map #align finset.sum_map Finset.sum_map @[to_additive] lemma prod_attach (s : Finset α) (f : α → β) : ∏ x ∈ s.attach, f x = ∏ x ∈ s, f x := by classical rw [← prod_image Subtype.coe_injective.injOn, attach_image_val] #align finset.prod_attach Finset.prod_attach #align finset.sum_attach Finset.sum_attach @[to_additive (attr := congr)] theorem prod_congr (h : s₁ = s₂) : (∀ x ∈ s₂, f x = g x) → s₁.prod f = s₂.prod g := by rw [h]; exact fold_congr #align finset.prod_congr Finset.prod_congr #align finset.sum_congr Finset.sum_congr @[to_additive] theorem prod_eq_one {f : α → β} {s : Finset α} (h : ∀ x ∈ s, f x = 1) : ∏ x ∈ s, f x = 1 := calc ∏ x ∈ s, f x = ∏ _x ∈ s, 1 := Finset.prod_congr rfl h _ = 1 := Finset.prod_const_one #align finset.prod_eq_one Finset.prod_eq_one #align finset.sum_eq_zero Finset.sum_eq_zero @[to_additive] theorem prod_disjUnion (h) : ∏ x ∈ s₁.disjUnion s₂ h, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := by refine Eq.trans ?_ (fold_disjUnion h) rw [one_mul] rfl #align finset.prod_disj_union Finset.prod_disjUnion #align finset.sum_disj_union Finset.sum_disjUnion @[to_additive] theorem prod_disjiUnion (s : Finset ι) (t : ι → Finset α) (h) : ∏ x ∈ s.disjiUnion t h, f x = ∏ i ∈ s, ∏ x ∈ t i, f x := by refine Eq.trans ?_ (fold_disjiUnion h) dsimp [Finset.prod, Multiset.prod, Multiset.fold, Finset.disjUnion, Finset.fold] congr exact prod_const_one.symm #align finset.prod_disj_Union Finset.prod_disjiUnion #align finset.sum_disj_Union Finset.sum_disjiUnion @[to_additive] theorem prod_union_inter [DecidableEq α] : (∏ x ∈ s₁ ∪ s₂, f x) * ∏ x ∈ s₁ ∩ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := fold_union_inter #align finset.prod_union_inter Finset.prod_union_inter #align finset.sum_union_inter Finset.sum_union_inter @[to_additive] theorem prod_union [DecidableEq α] (h : Disjoint s₁ s₂) : ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := by rw [← prod_union_inter, disjoint_iff_inter_eq_empty.mp h]; exact (mul_one _).symm #align finset.prod_union Finset.prod_union #align finset.sum_union Finset.sum_union @[to_additive] theorem prod_filter_mul_prod_filter_not (s : Finset α) (p : α → Prop) [DecidablePred p] [∀ x, Decidable (¬p x)] (f : α → β) : (∏ x ∈ s.filter p, f x) * ∏ x ∈ s.filter fun x => ¬p x, f x = ∏ x ∈ s, f x := by have := Classical.decEq α rw [← prod_union (disjoint_filter_filter_neg s s p), filter_union_filter_neg_eq] #align finset.prod_filter_mul_prod_filter_not Finset.prod_filter_mul_prod_filter_not #align finset.sum_filter_add_sum_filter_not Finset.sum_filter_add_sum_filter_not section ToList @[to_additive (attr := simp)] theorem prod_to_list (s : Finset α) (f : α → β) : (s.toList.map f).prod = s.prod f := by rw [Finset.prod, ← Multiset.prod_coe, ← Multiset.map_coe, Finset.coe_toList] #align finset.prod_to_list Finset.prod_to_list #align finset.sum_to_list Finset.sum_to_list end ToList @[to_additive] theorem _root_.Equiv.Perm.prod_comp (σ : Equiv.Perm α) (s : Finset α) (f : α → β) (hs : { a \| σ a ≠ a } ⊆ s) : (∏ x ∈ s, f (σ x)) = ∏ x ∈ s, f x := by convert (prod_map s σ.toEmbedding f).symm exact (map_perm hs).symm #align equiv.perm.prod_comp Equiv.Perm.prod_comp #align equiv.perm.sum_comp Equiv.Perm.sum_comp @[to_additive] theorem _root_.Equiv.Perm.prod_comp' (σ : Equiv.Perm α) (s : Finset α) (f : α → α → β) (hs : { a \| σ a ≠ a } ⊆ s) : (∏ x ∈ s, f (σ x) x) = ∏ x ∈ s, f x (σ.symm x) := by convert σ.prod_comp s (fun x => f x (σ.symm x)) hs rw [Equiv.symm_apply_apply] #align equiv.perm.prod_comp' Equiv.Perm.prod_comp' #align equiv.perm.sum_comp' Equiv.Perm.sum_comp' /-- A product over all subsets of `s ∪ {x}` is obtained by multiplying the product over all subsets of `s`, and over all subsets of `s` to which one adds `x`. -/ @[to_additive "A sum over all subsets of `s ∪ {x}` is obtained by summing the sum over all subsets of `s`, and over all subsets of `s` to which one adds `x`."] lemma prod_powerset_insert [DecidableEq α] (ha : a ∉ s) (f : Finset α → β) : ∏ t ∈ (insert a s).powerset, f t = (∏ t ∈ s.powerset, f t) * ∏ t ∈ s.powerset, f (insert a t) := by rw [powerset_insert, prod_union, prod_image] · exact insert_erase_invOn.2.injOn.mono fun t ht ↦ not_mem_mono (mem_powerset.1 ht) ha · aesop (add simp [disjoint_left, insert_subset_iff]) #align finset.prod_powerset_insert Finset.prod_powerset_insert #align finset.sum_powerset_insert Finset.sum_powerset_insert /-- A product over all subsets of `s ∪ {x}` is obtained by multiplying the product over all subsets of `s`, and over all subsets of `s` to which one adds `x`. -/ @[to_additive "A sum over all subsets of `s ∪ {x}` is obtained by summing the sum over all subsets of `s`, and over all subsets of `s` to which one adds `x`."] lemma prod_powerset_cons (ha : a ∉ s) (f : Finset α → β) : ∏ t ∈ (s.cons a ha).powerset, f t = (∏ t ∈ s.powerset, f t) * ∏ t ∈ s.powerset.attach, f (cons a t $ not_mem_mono (mem_powerset.1 t.2) ha) := by classical simp_rw [cons_eq_insert] rw [prod_powerset_insert ha, prod_attach _ fun t ↦ f (insert a t)] /-- A product over `powerset s` is equal to the double product over sets of subsets of `s` with `card s = k`, for `k = 1, ..., card s`. -/ @[to_additive "A sum over `powerset s` is equal to the double sum over sets of subsets of `s` with `card s = k`, for `k = 1, ..., card s`"] lemma prod_powerset (s : Finset α) (f : Finset α → β) : ∏ t ∈ powerset s, f t = ∏ j ∈ range (card s + 1), ∏ t ∈ powersetCard j s, f t := by rw [powerset_card_disjiUnion, prod_disjiUnion] #align finset.prod_powerset Finset.prod_powerset #align finset.sum_powerset Finset.sum_powerset end CommMonoid end Finset section open Finset variable [Fintype α] [CommMonoid β] @[to_additive] theorem IsCompl.prod_mul_prod {s t : Finset α} (h : IsCompl s t) (f : α → β) : (∏ i ∈ s, f i) * ∏ i ∈ t, f i = ∏ i, f i := (Finset.prod_disjUnion h.disjoint).symm.trans <\| by classical rw [Finset.disjUnion_eq_union, ← Finset.sup_eq_union, h.sup_eq_top]; rfl #align is_compl.prod_mul_prod IsCompl.prod_mul_prod #align is_compl.sum_add_sum IsCompl.sum_add_sum end namespace Finset section CommMonoid variable [CommMonoid β] /-- Multiplying the products of a function over `s` and over `sᶜ` gives the whole product. For a version expressed with subtypes, see `Fintype.prod_subtype_mul_prod_subtype`. -/ @[to_additive "Adding the sums of a function over `s` and over `sᶜ` gives the whole sum. For a version expressed with subtypes, see `Fintype.sum_subtype_add_sum_subtype`. "] theorem prod_mul_prod_compl [Fintype α] [DecidableEq α] (s : Finset α) (f : α → β) : (∏ i ∈ s, f i) * ∏ i ∈ sᶜ, f i = ∏ i, f i := IsCompl.prod_mul_prod isCompl_compl f #align finset.prod_mul_prod_compl Finset.prod_mul_prod_compl #align finset.sum_add_sum_compl Finset.sum_add_sum_compl @[to_additive] theorem prod_compl_mul_prod [Fintype α] [DecidableEq α] (s : Finset α) (f : α → β) : (∏ i ∈ sᶜ, f i) * ∏ i ∈ s, f i = ∏ i, f i := (@isCompl_compl _ s _).symm.prod_mul_prod f #align finset.prod_compl_mul_prod Finset.prod_compl_mul_prod #align finset.sum_compl_add_sum Finset.sum_compl_add_sum @[to_additive] theorem prod_sdiff [DecidableEq α] (h : s₁ ⊆ s₂) : (∏ x ∈ s₂ \ s₁, f x) * ∏ x ∈ s₁, f x = ∏ x ∈ s₂, f x := by rw [← prod_union sdiff_disjoint, sdiff_union_of_subset h] #align finset.prod_sdiff Finset.prod_sdiff #align finset.sum_sdiff Finset.sum_sdiff @[to_additive] theorem prod_subset_one_on_sdiff [DecidableEq α] (h : s₁ ⊆ s₂) (hg : ∀ x ∈ s₂ \ s₁, g x = 1) (hfg : ∀ x ∈ s₁, f x = g x) : ∏ i ∈ s₁, f i = ∏ i ∈ s₂, g i := by rw [← prod_sdiff h, prod_eq_one hg, one_mul] exact prod_congr rfl hfg #align finset.prod_subset_one_on_sdiff Finset.prod_subset_one_on_sdiff #align finset.sum_subset_zero_on_sdiff Finset.sum_subset_zero_on_sdiff @[to_additive] theorem prod_subset (h : s₁ ⊆ s₂) (hf : ∀ x ∈ s₂, x ∉ s₁ → f x = 1) : ∏ x ∈ s₁, f x = ∏ x ∈ s₂, f x := haveI := Classical.decEq α prod_subset_one_on_sdiff h (by simpa) fun _ _ => rfl #align finset.prod_subset Finset.prod_subset #align finset.sum_subset Finset.sum_subset @[to_additive (attr := simp)] theorem prod_disj_sum (s : Finset α) (t : Finset γ) (f : Sum α γ → β) : ∏ x ∈ s.disjSum t, f x = (∏ x ∈ s, f (Sum.inl x)) * ∏ x ∈ t, f (Sum.inr x) := by rw [← map_inl_disjUnion_map_inr, prod_disjUnion, prod_map, prod_map] rfl #align finset.prod_disj_sum Finset.prod_disj_sum #align finset.sum_disj_sum Finset.sum_disj_sum @[to_additive] theorem prod_sum_elim (s : Finset α) (t : Finset γ) (f : α → β) (g : γ → β) : ∏ x ∈ s.disjSum t, Sum.elim f g x = (∏ x ∈ s, f x) * ∏ x ∈ t, g x := by simp #align finset.prod_sum_elim Finset.prod_sum_elim #align finset.sum_sum_elim Finset.sum_sum_elim @[to_additive] theorem prod_biUnion [DecidableEq α] {s : Finset γ} {t : γ → Finset α} (hs : Set.PairwiseDisjoint (↑s) t) : ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i := by rw [← disjiUnion_eq_biUnion _ _ hs, prod_disjiUnion] #align finset.prod_bUnion Finset.prod_biUnion #align finset.sum_bUnion Finset.sum_biUnion /-- Product over a sigma type equals the product of fiberwise products. For rewriting in the reverse direction, use `Finset.prod_sigma'`. -/ @[to_additive "Sum over a sigma type equals the sum of fiberwise sums. For rewriting in the reverse direction, use `Finset.sum_sigma'`"] theorem prod_sigma {σ : α → Type} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : Sigma σ → β) : ∏ x ∈ s.sigma t, f x = ∏ a ∈ s, ∏ s ∈ t a, f ⟨a, s⟩ := by simp_rw [← disjiUnion_map_sigma_mk, prod_disjiUnion, prod_map, Function.Embedding.sigmaMk_apply] #align finset.prod_sigma Finset.prod_sigma #align finset.sum_sigma Finset.sum_sigma @[to_additive] theorem prod_sigma' {σ : α → Type} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : ∀ a, σ a → β) : (∏ a ∈ s, ∏ s ∈ t a, f a s) = ∏ x ∈ s.sigma t, f x.1 x.2 := Eq.symm <\| prod_sigma s t fun x => f x.1 x.2 #align finset.prod_sigma' Finset.prod_sigma' #align finset.sum_sigma' Finset.sum_sigma' section bij variable {ι κ α : Type} [CommMonoid α] {s : Finset ι} {t : Finset κ} {f : ι → α} {g : κ → α} /-- Reorder a product. The difference with `Finset.prod_bij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.prod_nbij` is that the bijection is allowed to use membership of the domain of the product, rather than being a non-dependent function. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_bij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.sum_nbij` is that the bijection is allowed to use membership of the domain of the sum, rather than being a non-dependent function."] theorem prod_bij (i : ∀ a ∈ s, κ) (hi : ∀ a ha, i a ha ∈ t) (i_inj : ∀ a₁ ha₁ a₂ ha₂, i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂) (i_surj : ∀ b ∈ t, ∃ a ha, i a ha = b) (h : ∀ a ha, f a = g (i a ha)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := congr_arg Multiset.prod (Multiset.map_eq_map_of_bij_of_nodup f g s.2 t.2 i hi i_inj i_surj h) #align finset.prod_bij Finset.prod_bij #align finset.sum_bij Finset.sum_bij /-- Reorder a product. The difference with `Finset.prod_bij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.prod_nbij'` is that the bijection and its inverse are allowed to use membership of the domains of the products, rather than being non-dependent functions. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_bij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.sum_nbij'` is that the bijection and its inverse are allowed to use membership of the domains of the sums, rather than being non-dependent functions."] theorem prod_bij' (i : ∀ a ∈ s, κ) (j : ∀ a ∈ t, ι) (hi : ∀ a ha, i a ha ∈ t) (hj : ∀ a ha, j a ha ∈ s) (left_inv : ∀ a ha, j (i a ha) (hi a ha) = a) (right_inv : ∀ a ha, i (j a ha) (hj a ha) = a) (h : ∀ a ha, f a = g (i a ha)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := by refine prod_bij i hi (fun a1 h1 a2 h2 eq ↦ ?_) (fun b hb ↦ ⟨_, hj b hb, right_inv b hb⟩) h rw [← left_inv a1 h1, ← left_inv a2 h2] simp only [eq] #align finset.prod_bij' Finset.prod_bij' #align finset.sum_bij' Finset.sum_bij' /-- Reorder a product. The difference with `Finset.prod_nbij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.prod_bij` is that the bijection is a non-dependent function, rather than being allowed to use membership of the domain of the product. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_nbij'` is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with `Finset.sum_bij` is that the bijection is a non-dependent function, rather than being allowed to use membership of the domain of the sum."] lemma prod_nbij (i : ι → κ) (hi : ∀ a ∈ s, i a ∈ t) (i_inj : (s : Set ι).InjOn i) (i_surj : (s : Set ι).SurjOn i t) (h : ∀ a ∈ s, f a = g (i a)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := prod_bij (fun a _ ↦ i a) hi i_inj (by simpa using i_surj) h /-- Reorder a product. The difference with `Finset.prod_nbij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.prod_bij'` is that the bijection and its inverse are non-dependent functions, rather than being allowed to use membership of the domains of the products. The difference with `Finset.prod_equiv` is that bijectivity is only required to hold on the domains of the products, rather than on the entire types. -/ @[to_additive "Reorder a sum. The difference with `Finset.sum_nbij` is that the bijection is specified with an inverse, rather than as a surjective injection. The difference with `Finset.sum_bij'` is that the bijection and its inverse are non-dependent functions, rather than being allowed to use membership of the domains of the sums. The difference with `Finset.sum_equiv` is that bijectivity is only required to hold on the domains of the sums, rather than on the entire types."] lemma prod_nbij' (i : ι → κ) (j : κ → ι) (hi : ∀ a ∈ s, i a ∈ t) (hj : ∀ a ∈ t, j a ∈ s) (left_inv : ∀ a ∈ s, j (i a) = a) (right_inv : ∀ a ∈ t, i (j a) = a) (h : ∀ a ∈ s, f a = g (i a)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := prod_bij' (fun a _ ↦ i a) (fun b _ ↦ j b) hi hj left_inv right_inv h /-- Specialization of `Finset.prod_nbij'` that automatically fills in most arguments. See `Fintype.prod_equiv` for the version where `s` and `t` are `univ`. -/ @[to_additive "`Specialization of `Finset.sum_nbij'` that automatically fills in most arguments. See `Fintype.sum_equiv` for the version where `s` and `t` are `univ`."] lemma prod_equiv (e : ι ≃ κ) (hst : ∀ i, i ∈ s ↔ e i ∈ t) (hfg : ∀ i ∈ s, f i = g (e i)) : ∏ i ∈ s, f i = ∏ i ∈ t, g i := by refine prod_nbij' e e.symm ?_ ?_ ?_ ?_ hfg <;> simp [hst] #align finset.equiv.prod_comp_finset Finset.prod_equiv #align finset.equiv.sum_comp_finset Finset.sum_equiv /-- Specialization of `Finset.prod_bij` that automatically fills in most arguments. See `Fintype.prod_bijective` for the version where `s` and `t` are `univ`. -/ @[to_additive "`Specialization of `Finset.sum_bij` that automatically fills in most arguments. See `Fintype.sum_bijective` for the version where `s` and `t` are `univ`."] lemma prod_bijective (e : ι → κ) (he : e.Bijective) (hst : ∀ i, i ∈ s ↔ e i ∈ t) (hfg : ∀ i ∈ s, f i = g (e i)) : ∏ i ∈ s, f i = ∏ i ∈ t, g i := prod_equiv (.ofBijective e he) hst hfg @[to_additive] lemma prod_of_injOn (e : ι → κ) (he : Set.InjOn e s) (hest : Set.MapsTo e s t) (h' : ∀ i ∈ t, i ∉ e '' s → g i = 1) (h : ∀ i ∈ s, f i = g (e i)) : ∏ i ∈ s, f i = ∏ j ∈ t, g j := by classical exact (prod_nbij e (fun a ↦ mem_image_of_mem e) he (by simp [Set.surjOn_image]) h).trans <\| prod_subset (image_subset_iff.2 hest) <\| by simpa using h' variable [DecidableEq κ] @[to_additive] lemma prod_fiberwise_eq_prod_filter (s : Finset ι) (t : Finset κ) (g : ι → κ) (f : ι → α) : ∏ j ∈ t, ∏ i ∈ s.filter fun i ↦ g i = j, f i = ∏ i ∈ s.filter fun i ↦ g i ∈ t, f i := by rw [← prod_disjiUnion, disjiUnion_filter_eq] @[to_additive] lemma prod_fiberwise_eq_prod_filter' (s : Finset ι) (t : Finset κ) (g : ι → κ) (f : κ → α) : ∏ j ∈ t, ∏ _i ∈ s.filter fun i ↦ g i = j, f j = ∏ i ∈ s.filter fun i ↦ g i ∈ t, f (g i) := by calc _ = ∏ j ∈ t, ∏ i ∈ s.filter fun i ↦ g i = j, f (g i) := prod_congr rfl fun j _ ↦ prod_congr rfl fun i hi ↦ by rw [(mem_filter.1 hi).2] _ = _ := prod_fiberwise_eq_prod_filter _ _ _ _ @[to_additive] lemma prod_fiberwise_of_maps_to {g : ι → κ} (h : ∀ i ∈ s, g i ∈ t) (f : ι → α) : ∏ j ∈ t, ∏ i ∈ s.filter fun i ↦ g i = j, f i = ∏ i ∈ s, f i := by rw [← prod_disjiUnion, disjiUnion_filter_eq_of_maps_to h] #align finset.prod_fiberwise_of_maps_to Finset.prod_fiberwise_of_maps_to #align finset.sum_fiberwise_of_maps_to Finset.sum_fiberwise_of_maps_to @[to_additive] lemma prod_fiberwise_of_maps_to' {g : ι → κ} (h : ∀ i ∈ s, g i ∈ t) (f : κ → α) : ∏ j ∈ t, ∏ _i ∈ s.filter fun i ↦ g i = j, f j = ∏ i ∈ s, f (g i) := by calc _ = ∏ y ∈ t, ∏ x ∈ s.filter fun x ↦ g x = y, f (g x) := prod_congr rfl fun y _ ↦ prod_congr rfl fun x hx ↦ by rw [(mem_filter.1 hx).2] _ = _ := prod_fiberwise_of_maps_to h _ variable [Fintype κ] @[to_additive] lemma prod_fiberwise (s : Finset ι) (g : ι → κ) (f : ι → α) : ∏ j, ∏ i ∈ s.filter fun i ↦ g i = j, f i = ∏ i ∈ s, f i := prod_fiberwise_of_maps_to (fun _ _ ↦ mem_univ _) _ #align finset.prod_fiberwise Finset.prod_fiberwise #align finset.sum_fiberwise Finset.sum_fiberwise @[to_additive] lemma prod_fiberwise' (s : Finset ι) (g : ι → κ) (f : κ → α) : ∏ j, ∏ _i ∈ s.filter fun i ↦ g i = j, f j = ∏ i ∈ s, f (g i) := prod_fiberwise_of_maps_to' (fun _ _ ↦ mem_univ _) _ end bij /-- Taking a product over `univ.pi t` is the same as taking the product over `Fintype.piFinset t`. `univ.pi t` and `Fintype.piFinset t` are essentially the same `Finset`, but differ in the type of their element, `univ.pi t` is a `Finset (Π a ∈ univ, t a)` and `Fintype.piFinset t` is a `Finset (Π a, t a)`. -/ @[to_additive "Taking a sum over `univ.pi t` is the same as taking the sum over `Fintype.piFinset t`. `univ.pi t` and `Fintype.piFinset t` are essentially the same `Finset`, but differ in the type of their element, `univ.pi t` is a `Finset (Π a ∈ univ, t a)` and `Fintype.piFinset t` is a `Finset (Π a, t a)`."] lemma prod_univ_pi [DecidableEq ι] [Fintype ι] {κ : ι → Type} (t : ∀ i, Finset (κ i)) (f : (∀ i ∈ (univ : Finset ι), κ i) → β) : ∏ x ∈ univ.pi t, f x = ∏ x ∈ Fintype.piFinset t, f fun a _ ↦ x a := by apply prod_nbij' (fun x i ↦ x i $ mem_univ _) (fun x i _ ↦ x i) <;> simp #align finset.prod_univ_pi Finset.prod_univ_pi #align finset.sum_univ_pi Finset.sum_univ_pi @[to_additive (attr := simp)] lemma prod_diag [DecidableEq α] (s : Finset α) (f : α × α → β) : ∏ i ∈ s.diag, f i = ∏ i ∈ s, f (i, i) := by apply prod_nbij' Prod.fst (fun i ↦ (i, i)) <;> simp @[to_additive] theorem prod_finset_product (r : Finset (γ × α)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : γ × α, p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1) {f : γ × α → β} : ∏ p ∈ r, f p = ∏ c ∈ s, ∏ a ∈ t c, f (c, a) := by refine Eq.trans ?_ (prod_sigma s t fun p => f (p.1, p.2)) apply prod_equiv (Equiv.sigmaEquivProd _ _).symm <;> simp [h] #align finset.prod_finset_product Finset.prod_finset_product #align finset.sum_finset_product Finset.sum_finset_product @[to_additive] theorem prod_finset_product' (r : Finset (γ × α)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : γ × α, p ∈ r ↔ p.1 ∈ s ∧ p.2 ∈ t p.1) {f : γ → α → β} : ∏ p ∈ r, f p.1 p.2 = ∏ c ∈ s, ∏ a ∈ t c, f c a := prod_finset_product r s t h #align finset.prod_finset_product' Finset.prod_finset_product' #align finset.sum_finset_product' Finset.sum_finset_product' @[to_additive] theorem prod_finset_product_right (r : Finset (α × γ)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : α × γ, p ∈ r ↔ p.2 ∈ s ∧ p.1 ∈ t p.2) {f : α × γ → β} : ∏ p ∈ r, f p = ∏ c ∈ s, ∏ a ∈ t c, f (a, c) := by refine Eq.trans ?_ (prod_sigma s t fun p => f (p.2, p.1)) apply prod_equiv ((Equiv.prodComm _ _).trans (Equiv.sigmaEquivProd _ _).symm) <;> simp [h] #align finset.prod_finset_product_right Finset.prod_finset_product_right #align finset.sum_finset_product_right Finset.sum_finset_product_right @[to_additive] theorem prod_finset_product_right' (r : Finset (α × γ)) (s : Finset γ) (t : γ → Finset α) (h : ∀ p : α × γ, p ∈ r ↔ p.2 ∈ s ∧ p.1 ∈ t p.2) {f : α → γ → β} : ∏ p ∈ r, f p.1 p.2 = ∏ c ∈ s, ∏ a ∈ t c, f a c := prod_finset_product_right r s t h #align finset.prod_finset_product_right' Finset.prod_finset_product_right' #align finset.sum_finset_product_right' Finset.sum_finset_product_right' @[to_additive] theorem prod_image' [DecidableEq α] {s : Finset γ} {g : γ → α} (h : γ → β) (eq : ∀ c ∈ s, f (g c) = ∏ x ∈ s.filter fun c' => g c' = g c, h x) : ∏ x ∈ s.image g, f x = ∏ x ∈ s, h x := calc ∏ x ∈ s.image g, f x = ∏ x ∈ s.image g, ∏ x ∈ s.filter fun c' => g c' = x, h x := (prod_congr rfl) fun _x hx => let ⟨c, hcs, hc⟩ := mem_image.1 hx hc ▸ eq c hcs _ = ∏ x ∈ s, h x := prod_fiberwise_of_maps_to (fun _x => mem_image_of_mem g) _ #align finset.prod_image' Finset.prod_image' #align finset.sum_image' Finset.sum_image' @[to_additive] theorem prod_mul_distrib : ∏ x ∈ s, f x * g x = (∏ x ∈ s, f x) * ∏ x ∈ s, g x := Eq.trans (by rw [one_mul]; rfl) fold_op_distrib #align finset.prod_mul_distrib Finset.prod_mul_distrib #align finset.sum_add_distrib Finset.sum_add_distrib @[to_additive] lemma prod_mul_prod_comm (f g h i : α → β) : (∏ a ∈ s, f a * g a) * ∏ a ∈ s, h a * i a = (∏ a ∈ s, f a * h a) * ∏ a ∈ s, g a * i a := by simp_rw [prod_mul_distrib, mul_mul_mul_comm] @[to_additive] theorem prod_product {s : Finset γ} {t : Finset α} {f : γ × α → β} : ∏ x ∈ s ×ˢ t, f x = ∏ x ∈ s, ∏ y ∈ t, f (x, y) := prod_finset_product (s ×ˢ t) s (fun _a => t) fun _p => mem_product #align finset.prod_product Finset.prod_product #align finset.sum_product Finset.sum_product /-- An uncurried version of `Finset.prod_product`. -/ @[to_additive "An uncurried version of `Finset.sum_product`"] theorem prod_product' {s : Finset γ} {t : Finset α} {f : γ → α → β} : ∏ x ∈ s ×ˢ t, f x.1 x.2 = ∏ x ∈ s, ∏ y ∈ t, f x y := prod_product #align finset.prod_product' Finset.prod_product' #align finset.sum_product' Finset.sum_product' @[to_additive] theorem prod_product_right {s : Finset γ} {t : Finset α} {f : γ × α → β} : ∏ x ∈ s ×ˢ t, f x = ∏ y ∈ t, ∏ x ∈ s, f (x, y) := prod_finset_product_right (s ×ˢ t) t (fun _a => s) fun _p => mem_product.trans and_comm #align finset.prod_product_right Finset.prod_product_right #align finset.sum_product_right Finset.sum_product_right /-- An uncurried version of `Finset.prod_product_right`. -/ @[to_additive "An uncurried version of `Finset.sum_product_right`"] theorem prod_product_right' {s : Finset γ} {t : Finset α} {f : γ → α → β} : ∏ x ∈ s ×ˢ t, f x.1 x.2 = ∏ y ∈ t, ∏ x ∈ s, f x y := prod_product_right #align finset.prod_product_right' Finset.prod_product_right' #align finset.sum_product_right' Finset.sum_product_right' /-- Generalization of `Finset.prod_comm` to the case when the inner `Finset`s depend on the outer variable. -/ @[to_additive "Generalization of `Finset.sum_comm` to the case when the inner `Finset`s depend on the outer variable."] theorem prod_comm' {s : Finset γ} {t : γ → Finset α} {t' : Finset α} {s' : α → Finset γ} (h : ∀ x y, x ∈ s ∧ y ∈ t x ↔ x ∈ s' y ∧ y ∈ t') {f : γ → α → β} : (∏ x ∈ s, ∏ y ∈ t x, f x y) = ∏ y ∈ t', ∏ x ∈ s' y, f x y := by classical have : ∀ z : γ × α, (z ∈ s.biUnion fun x => (t x).map <\| Function.Embedding.sectr x _) ↔ z.1 ∈ s ∧ z.2 ∈ t z.1 := by rintro ⟨x, y⟩ simp only [mem_biUnion, mem_map, Function.Embedding.sectr_apply, Prod.mk.injEq, exists_eq_right, ← and_assoc] exact (prod_finset_product' _ _ _ this).symm.trans ((prod_finset_product_right' _ _ _) fun ⟨x, y⟩ => (this _).trans ((h x y).trans and_comm)) #align finset.prod_comm' Finset.prod_comm' #align finset.sum_comm' Finset.sum_comm' @[to_additive] theorem prod_comm {s : Finset γ} {t : Finset α} {f : γ → α → β} : (∏ x ∈ s, ∏ y ∈ t, f x y) = ∏ y ∈ t, ∏ x ∈ s, f x y := prod_comm' fun _ _ => Iff.rfl #align finset.prod_comm Finset.prod_comm #align finset.sum_comm Finset.sum_comm @[to_additive] theorem prod_hom_rel [CommMonoid γ] {r : β → γ → Prop} {f : α → β} {g : α → γ} {s : Finset α} (h₁ : r 1 1) (h₂ : ∀ a b c, r b c → r (f a * b) (g a * c)) : r (∏ x ∈ s, f x) (∏ x ∈ s, g x) := by delta Finset.prod apply Multiset.prod_hom_rel <;> assumption #align finset.prod_hom_rel Finset.prod_hom_rel #align finset.sum_hom_rel Finset.sum_hom_rel @[to_additive] theorem prod_filter_of_ne {p : α → Prop} [DecidablePred p] (hp : ∀ x ∈ s, f x ≠ 1 → p x) : ∏ x ∈ s.filter p, f x = ∏ x ∈ s, f x := (prod_subset (filter_subset _ _)) fun x => by classical rw [not_imp_comm, mem_filter] exact fun h₁ h₂ => ⟨h₁, by simpa using hp _ h₁ h₂⟩ #align finset.prod_filter_of_ne Finset.prod_filter_of_ne #align finset.sum_filter_of_ne Finset.sum_filter_of_ne -- If we use `[DecidableEq β]` here, some rewrites fail because they find a wrong `Decidable` -- instance first; `{∀ x, Decidable (f x ≠ 1)}` doesn't work with `rw ← prod_filter_ne_one` @[to_additive] theorem prod_filter_ne_one (s : Finset α) [∀ x, Decidable (f x ≠ 1)] : ∏ x ∈ s.filter fun x => f x ≠ 1, f x = ∏ x ∈ s, f x := prod_filter_of_ne fun _ _ => id #align finset.prod_filter_ne_one Finset.prod_filter_ne_one #align finset.sum_filter_ne_zero Finset.sum_filter_ne_zero @[to_additive] theorem prod_filter (p : α → Prop) [DecidablePred p] (f : α → β) : ∏ a ∈ s.filter p, f a = ∏ a ∈ s, if p a then f a else 1 := calc ∏ a ∈ s.filter p, f a = ∏ a ∈ s.filter p, if p a then f a else 1 := prod_congr rfl fun a h => by rw [if_pos]; simpa using (mem_filter.1 h).2 _ = ∏ a ∈ s, if p a then f a else 1 := by { refine prod_subset (filter_subset _ s) fun x hs h => ?_ rw [mem_filter, not_and] at h exact if_neg (by simpa using h hs) } #align finset.prod_filter Finset.prod_filter #align finset.sum_filter Finset.sum_filter @[to_additive] theorem prod_eq_single_of_mem {s : Finset α} {f : α → β} (a : α) (h : a ∈ s) (h₀ : ∀ b ∈ s, b ≠ a → f b = 1) : ∏ x ∈ s, f x = f a := by haveI := Classical.decEq α calc ∏ x ∈ s, f x = ∏ x ∈ {a}, f x := by { refine (prod_subset ?_ ?_).symm · intro _ H rwa [mem_singleton.1 H] · simpa only [mem_singleton] } _ = f a := prod_singleton _ _ #align finset.prod_eq_single_of_mem Finset.prod_eq_single_of_mem #align finset.sum_eq_single_of_mem Finset.sum_eq_single_of_mem @[to_additive] theorem prod_eq_single {s : Finset α} {f : α → β} (a : α) (h₀ : ∀ b ∈ s, b ≠ a → f b = 1) (h₁ : a ∉ s → f a = 1) : ∏ x ∈ s, f x = f a := haveI := Classical.decEq α by_cases (prod_eq_single_of_mem a · h₀) fun this => (prod_congr rfl fun b hb => h₀ b hb <\| by rintro rfl; exact this hb).trans <\| prod_const_one.trans (h₁ this).symm #align finset.prod_eq_single Finset.prod_eq_single #align finset.sum_eq_single Finset.sum_eq_single @[to_additive] lemma prod_union_eq_left [DecidableEq α] (hs : ∀ a ∈ s₂, a ∉ s₁ → f a = 1) : ∏ a ∈ s₁ ∪ s₂, f a = ∏ a ∈ s₁, f a := Eq.symm <\| prod_subset subset_union_left fun _a ha ha' ↦ hs _ ((mem_union.1 ha).resolve_left ha') ha' @[to_additive] lemma prod_union_eq_right [DecidableEq α] (hs : ∀ a ∈ s₁, a ∉ s₂ → f a = 1) : ∏ a ∈ s₁ ∪ s₂, f a = ∏ a ∈ s₂, f a := by rw [union_comm, prod_union_eq_left hs] @[to_additive] theorem prod_eq_mul_of_mem {s : Finset α} {f : α → β} (a b : α) (ha : a ∈ s) (hb : b ∈ s) (hn : a ≠ b) (h₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1) : ∏ x ∈ s, f x = f a * f b := by haveI := Classical.decEq α; let s' := ({a, b} : Finset α) have hu : s' ⊆ s := by refine insert_subset_iff.mpr ?_ apply And.intro ha apply singleton_subset_iff.mpr hb have hf : ∀ c ∈ s, c ∉ s' → f c = 1 := by intro c hc hcs apply h₀ c hc apply not_or.mp intro hab apply hcs rw [mem_insert, mem_singleton] exact hab rw [← prod_subset hu hf] exact Finset.prod_pair hn #align finset.prod_eq_mul_of_mem Finset.prod_eq_mul_of_mem #align finset.sum_eq_add_of_mem Finset.sum_eq_add_of_mem @[to_additive] theorem prod_eq_mul {s : Finset α} {f : α → β} (a b : α) (hn : a ≠ b) (h₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1) (ha : a ∉ s → f a = 1) (hb : b ∉ s → f b = 1) : ∏ x ∈ s, f x = f a * f b := by haveI := Classical.decEq α; by_cases h₁ : a ∈ s <;> by_cases h₂ : b ∈ s · exact prod_eq_mul_of_mem a b h₁ h₂ hn h₀ · rw [hb h₂, mul_one] apply prod_eq_single_of_mem a h₁ exact fun c hc hca => h₀ c hc ⟨hca, ne_of_mem_of_not_mem hc h₂⟩ · rw [ha h₁, one_mul] apply prod_eq_single_of_mem b h₂ exact fun c hc hcb => h₀ c hc ⟨ne_of_mem_of_not_mem hc h₁, hcb⟩ · rw [ha h₁, hb h₂, mul_one] exact _root_.trans (prod_congr rfl fun c hc => h₀ c hc ⟨ne_of_mem_of_not_mem hc h₁, ne_of_mem_of_not_mem hc h₂⟩) prod_const_one #align finset.prod_eq_mul Finset.prod_eq_mul #align finset.sum_eq_add Finset.sum_eq_add -- Porting note: simpNF linter complains that LHS doesn't simplify, but it does /-- A product over `s.subtype p` equals one over `s.filter p`. -/ @[to_additive (attr := simp, nolint simpNF) "A sum over `s.subtype p` equals one over `s.filter p`."] theorem prod_subtype_eq_prod_filter (f : α → β) {p : α → Prop} [DecidablePred p] : ∏ x ∈ s.subtype p, f x = ∏ x ∈ s.filter p, f x := by conv_lhs => erw [← prod_map (s.subtype p) (Function.Embedding.subtype _) f] exact prod_congr (subtype_map _) fun x _hx => rfl #align finset.prod_subtype_eq_prod_filter Finset.prod_subtype_eq_prod_filter #align finset.sum_subtype_eq_sum_filter Finset.sum_subtype_eq_sum_filter /-- If all elements of a `Finset` satisfy the predicate `p`, a product over `s.subtype p` equals that product over `s`. -/ @[to_additive "If all elements of a `Finset` satisfy the predicate `p`, a sum over `s.subtype p` equals that sum over `s`."] theorem prod_subtype_of_mem (f : α → β) {p : α → Prop} [DecidablePred p] (h : ∀ x ∈ s, p x) : ∏ x ∈ s.subtype p, f x = ∏ x ∈ s, f x := by rw [prod_subtype_eq_prod_filter, filter_true_of_mem] simpa using h #align finset.prod_subtype_of_mem Finset.prod_subtype_of_mem #align finset.sum_subtype_of_mem Finset.sum_subtype_of_mem /-- A product of a function over a `Finset` in a subtype equals a product in the main type of a function that agrees with the first function on that `Finset`. -/ @[to_additive "A sum of a function over a `Finset` in a subtype equals a sum in the main type of a function that agrees with the first function on that `Finset`."] theorem prod_subtype_map_embedding {p : α → Prop} {s : Finset { x // p x }} {f : { x // p x } → β} {g : α → β} (h : ∀ x : { x // p x }, x ∈ s → g x = f x) : (∏ x ∈ s.map (Function.Embedding.subtype _), g x) = ∏ x ∈ s, f x := by rw [Finset.prod_map] exact Finset.prod_congr rfl h #align finset.prod_subtype_map_embedding Finset.prod_subtype_map_embedding #align finset.sum_subtype_map_embedding Finset.sum_subtype_map_embedding variable (f s) @[to_additive] theorem prod_coe_sort_eq_attach (f : s → β) : ∏ i : s, f i = ∏ i ∈ s.attach, f i := rfl #align finset.prod_coe_sort_eq_attach Finset.prod_coe_sort_eq_attach #align finset.sum_coe_sort_eq_attach Finset.sum_coe_sort_eq_attach @[to_additive] theorem prod_coe_sort : ∏ i : s, f i = ∏ i ∈ s, f i := prod_attach _ _ #align finset.prod_coe_sort Finset.prod_coe_sort #align finset.sum_coe_sort Finset.sum_coe_sort @[to_additive] theorem prod_finset_coe (f : α → β) (s : Finset α) : (∏ i : (s : Set α), f i) = ∏ i ∈ s, f i := prod_coe_sort s f #align finset.prod_finset_coe Finset.prod_finset_coe #align finset.sum_finset_coe Finset.sum_finset_coe variable {f s} @[to_additive] theorem prod_subtype {p : α → Prop} {F : Fintype (Subtype p)} (s : Finset α) (h : ∀ x, x ∈ s ↔ p x) (f : α → β) : ∏ a ∈ s, f a = ∏ a : Subtype p, f a := by have : (· ∈ s) = p := Set.ext h subst p rw [← prod_coe_sort] congr! #align finset.prod_subtype Finset.prod_subtype #align finset.sum_subtype Finset.sum_subtype @[to_additive] lemma prod_preimage' (f : ι → κ) [DecidablePred (· ∈ Set.range f)] (s : Finset κ) (hf) (g : κ → β) : ∏ x ∈ s.preimage f hf, g (f x) = ∏ x ∈ s.filter (· ∈ Set.range f), g x := by classical calc ∏ x ∈ preimage s f hf, g (f x) = ∏ x ∈ image f (preimage s f hf), g x := Eq.symm <\| prod_image <\| by simpa only [mem_preimage, Set.InjOn] using hf _ = ∏ x ∈ s.filter fun x => x ∈ Set.range f, g x := by rw [image_preimage] #align finset.prod_preimage' Finset.prod_preimage' #align finset.sum_preimage' Finset.sum_preimage' @[to_additive] lemma prod_preimage (f : ι → κ) (s : Finset κ) (hf) (g : κ → β) (hg : ∀ x ∈ s, x ∉ Set.range f → g x = 1) : ∏ x ∈ s.preimage f hf, g (f x) = ∏ x ∈ s, g x := by classical rw [prod_preimage', prod_filter_of_ne]; exact fun x hx ↦ Not.imp_symm (hg x hx) #align finset.prod_preimage Finset.prod_preimage #align finset.sum_preimage Finset.sum_preimage @[to_additive] lemma prod_preimage_of_bij (f : ι → κ) (s : Finset κ) (hf : Set.BijOn f (f ⁻¹' ↑s) ↑s) (g : κ → β) : ∏ x ∈ s.preimage f hf.injOn, g (f x) = ∏ x ∈ s, g x := prod_preimage _ _ hf.injOn g fun _ hs h_f ↦ (h_f <\| hf.subset_range hs).elim #align finset.prod_preimage_of_bij Finset.prod_preimage_of_bij #align finset.sum_preimage_of_bij Finset.sum_preimage_of_bij @[to_additive] theorem prod_set_coe (s : Set α) [Fintype s] : (∏ i : s, f i) = ∏ i ∈ s.toFinset, f i := (Finset.prod_subtype s.toFinset (fun _ ↦ Set.mem_toFinset) f).symm /-- The product of a function `g` defined only on a set `s` is equal to the product of a function `f` defined everywhere, as long as `f` and `g` agree on `s`, and `f = 1` off `s`. -/ @[to_additive "The sum of a function `g` defined only on a set `s` is equal to the sum of a function `f` defined everywhere, as long as `f` and `g` agree on `s`, and `f = 0` off `s`."] theorem prod_congr_set {α : Type} [CommMonoid α] {β : Type} [Fintype β] (s : Set β) [DecidablePred (· ∈ s)] (f : β → α) (g : s → α) (w : ∀ (x : β) (h : x ∈ s), f x = g ⟨x, h⟩) (w' : ∀ x : β, x ∉ s → f x = 1) : Finset.univ.prod f = Finset.univ.prod g := by rw [← @Finset.prod_subset _ _ s.toFinset Finset.univ f _ (by simp)] · rw [Finset.prod_subtype] · apply Finset.prod_congr rfl exact fun ⟨x, h⟩ _ => w x h · simp · rintro x _ h exact w' x (by simpa using h) #align finset.prod_congr_set Finset.prod_congr_set #align finset.sum_congr_set Finset.sum_congr_set @[to_additive] theorem prod_apply_dite {s : Finset α} {p : α → Prop} {hp : DecidablePred p} [DecidablePred fun x => ¬p x] (f : ∀ x : α, p x → γ) (g : ∀ x : α, ¬p x → γ) (h : γ → β) : (∏ x ∈ s, h (if hx : p x then f x hx else g x hx)) = (∏ x ∈ (s.filter p).attach, h (f x.1 <\| by simpa using (mem_filter.mp x.2).2)) * ∏ x ∈ (s.filter fun x => ¬p x).attach, h (g x.1 <\| by simpa using (mem_filter.mp x.2).2) := calc (∏ x ∈ s, h (if hx : p x then f x hx else g x hx)) = (∏ x ∈ s.filter p, h (if hx : p x then f x hx else g x hx)) * ∏ x ∈ s.filter (¬p ·), h (if hx : p x then f x hx else g x hx) := (prod_filter_mul_prod_filter_not s p _).symm _ = (∏ x ∈ (s.filter p).attach, h (if hx : p x.1 then f x.1 hx else g x.1 hx)) * ∏ x ∈ (s.filter (¬p ·)).attach, h (if hx : p x.1 then f x.1 hx else g x.1 hx) := congr_arg₂ _ (prod_attach _ _).symm (prod_attach _ _).symm _ = (∏ x ∈ (s.filter p).attach, h (f x.1 <\| by simpa using (mem_filter.mp x.2).2)) * ∏ x ∈ (s.filter (¬p ·)).attach, h (g x.1 <\| by simpa using (mem_filter.mp x.2).2) := congr_arg₂ _ (prod_congr rfl fun x _hx ↦ congr_arg h (dif_pos <\| by simpa using (mem_filter.mp x.2).2)) (prod_congr rfl fun x _hx => congr_arg h (dif_neg <\| by simpa using (mem_filter.mp x.2).2)) #align finset.prod_apply_dite Finset.prod_apply_dite #align finset.sum_apply_dite Finset.sum_apply_dite @[to_additive] theorem prod_apply_ite {s : Finset α} {p : α → Prop} {_hp : DecidablePred p} (f g : α → γ) (h : γ → β) : (∏ x ∈ s, h (if p x then f x else g x)) = (∏ x ∈ s.filter p, h (f x)) * ∏ x ∈ s.filter fun x => ¬p x, h (g x) := (prod_apply_dite _ _ _).trans <\| congr_arg₂ _ (prod_attach _ (h ∘ f)) (prod_attach _ (h ∘ g)) #align finset.prod_apply_ite Finset.prod_apply_ite #align finset.sum_apply_ite Finset.sum_apply_ite @[to_additive] theorem prod_dite {s : Finset α} {p : α → Prop} {hp : DecidablePred p} (f : ∀ x : α, p x → β) (g : ∀ x : α, ¬p x → β) : ∏ x ∈ s, (if hx : p x then f x hx else g x hx) = (∏ x ∈ (s.filter p).attach, f x.1 (by simpa using (mem_filter.mp x.2).2)) * ∏ x ∈ (s.filter fun x => ¬p x).attach, g x.1 (by simpa using (mem_filter.mp x.2).2) := by simp [prod_apply_dite _ _ fun x => x] #align finset.prod_dite Finset.prod_dite #align finset.sum_dite Finset.sum_dite @[to_additive] theorem prod_ite {s : Finset α} {p : α → Prop} {hp : DecidablePred p} (f g : α → β) : ∏ x ∈ s, (if p x then f x else g x) = (∏ x ∈ s.filter p, f x) * ∏ x ∈ s.filter fun x => ¬p x, g x := by simp [prod_apply_ite _ _ fun x => x] #align finset.prod_ite Finset.prod_ite #align finset.sum_ite Finset.sum_ite @[to_additive] theorem prod_ite_of_false {p : α → Prop} {hp : DecidablePred p} (f g : α → β) (h : ∀ x ∈ s, ¬p x) : ∏ x ∈ s, (if p x then f x else g x) = ∏ x ∈ s, g x := by rw [prod_ite, filter_false_of_mem, filter_true_of_mem] · simp only [prod_empty, one_mul] all_goals intros; apply h; assumption #align finset.prod_ite_of_false Finset.prod_ite_of_false #align finset.sum_ite_of_false Finset.sum_ite_of_false @[to_additive] theorem prod_ite_of_true {p : α → Prop} {hp : DecidablePred p} (f g : α → β) (h : ∀ x ∈ s, p x) : ∏ x ∈ s, (if p x then f x else g x) = ∏ x ∈ s, f x := by simp_rw [← ite_not (p _)] apply prod_ite_of_false simpa #align finset.prod_ite_of_true Finset.prod_ite_of_true #align finset.sum_ite_of_true Finset.sum_ite_of_true @[to_additive] theorem prod_apply_ite_of_false {p : α → Prop} {hp : DecidablePred p} (f g : α → γ) (k : γ → β) (h : ∀ x ∈ s, ¬p x) : (∏ x ∈ s, k (if p x then f x else g x)) = ∏ x ∈ s, k (g x) := by simp_rw [apply_ite k] exact prod_ite_of_false _ _ h #align finset.prod_apply_ite_of_false Finset.prod_apply_ite_of_false #align finset.sum_apply_ite_of_false Finset.sum_apply_ite_of_false @[to_additive] theorem prod_apply_ite_of_true {p : α → Prop} {hp : DecidablePred p} (f g : α → γ) (k : γ → β) (h : ∀ x ∈ s, p x) : (∏ x ∈ s, k (if p x then f x else g x)) = ∏ x ∈ s, k (f x) := by simp_rw [apply_ite k] exact prod_ite_of_true _ _ h #align finset.prod_apply_ite_of_true Finset.prod_apply_ite_of_true #align finset.sum_apply_ite_of_true Finset.sum_apply_ite_of_true @[to_additive] theorem prod_extend_by_one [DecidableEq α] (s : Finset α) (f : α → β) : ∏ i ∈ s, (if i ∈ s then f i else 1) = ∏ i ∈ s, f i := (prod_congr rfl) fun _i hi => if_pos hi #align finset.prod_extend_by_one Finset.prod_extend_by_one #align finset.sum_extend_by_zero Finset.sum_extend_by_zero @[to_additive (attr := simp)] theorem prod_ite_mem [DecidableEq α] (s t : Finset α) (f : α → β) : ∏ i ∈ s, (if i ∈ t then f i else 1) = ∏ i ∈ s ∩ t, f i := by rw [← Finset.prod_filter, Finset.filter_mem_eq_inter] #align finset.prod_ite_mem Finset.prod_ite_mem #align finset.sum_ite_mem Finset.sum_ite_mem @[to_additive (attr := simp)] theorem prod_dite_eq [DecidableEq α] (s : Finset α) (a : α) (b : ∀ x : α, a = x → β) : ∏ x ∈ s, (if h : a = x then b x h else 1) = ite (a ∈ s) (b a rfl) 1 := by split_ifs with h · rw [Finset.prod_eq_single a, dif_pos rfl] · intros _ _ h rw [dif_neg] exact h.symm · simp [h] · rw [Finset.prod_eq_one] intros rw [dif_neg] rintro rfl contradiction #align finset.prod_dite_eq Finset.prod_dite_eq #align finset.sum_dite_eq Finset.sum_dite_eq @[to_additive (attr := simp)] theorem prod_dite_eq' [DecidableEq α] (s : Finset α) (a : α) (b : ∀ x : α, x = a → β) : ∏ x ∈ s, (if h : x = a then b x h else 1) = ite (a ∈ s) (b a rfl) 1 := by split_ifs with h · rw [Finset.prod_eq_single a, dif_pos rfl] · intros _ _ h rw [dif_neg] exact h · simp [h] · rw [Finset.prod_eq_one] intros rw [dif_neg] rintro rfl contradiction #align finset.prod_dite_eq' Finset.prod_dite_eq' #align finset.sum_dite_eq' Finset.sum_dite_eq' @[to_additive (attr := simp)] theorem prod_ite_eq [DecidableEq α] (s : Finset α) (a : α) (b : α → β) : (∏ x ∈ s, ite (a = x) (b x) 1) = ite (a ∈ s) (b a) 1 := prod_dite_eq s a fun x _ => b x #align finset.prod_ite_eq Finset.prod_ite_eq #align finset.sum_ite_eq Finset.sum_ite_eq /-- A product taken over a conditional whose condition is an equality test on the index and whose alternative is `1` has value either the term at that index or `1`. The difference with `Finset.prod_ite_eq` is that the arguments to `Eq` are swapped. -/ @[to_additive (attr := simp) "A sum taken over a conditional whose condition is an equality test on the index and whose alternative is `0` has value either the term at that index or `0`. The difference with `Finset.sum_ite_eq` is that the arguments to `Eq` are swapped."] theorem prod_ite_eq' [DecidableEq α] (s : Finset α) (a : α) (b : α → β) : (∏ x ∈ s, ite (x = a) (b x) 1) = ite (a ∈ s) (b a) 1 := prod_dite_eq' s a fun x _ => b x #align finset.prod_ite_eq' Finset.prod_ite_eq' #align finset.sum_ite_eq' Finset.sum_ite_eq' @[to_additive] theorem prod_ite_index (p : Prop) [Decidable p] (s t : Finset α) (f : α → β) : ∏ x ∈ if p then s else t, f x = if p then ∏ x ∈ s, f x else ∏ x ∈ t, f x := apply_ite (fun s => ∏ x ∈ s, f x) _ _ _ #align finset.prod_ite_index Finset.prod_ite_index #align finset.sum_ite_index Finset.sum_ite_index @[to_additive (attr := simp)] theorem prod_ite_irrel (p : Prop) [Decidable p] (s : Finset α) (f g : α → β) : ∏ x ∈ s, (if p then f x else g x) = if p then ∏ x ∈ s, f x else ∏ x ∈ s, g x := by split_ifs with h <;> rfl #align finset.prod_ite_irrel Finset.prod_ite_irrel #align finset.sum_ite_irrel Finset.sum_ite_irrel @[to_additive (attr := simp)] theorem prod_dite_irrel (p : Prop) [Decidable p] (s : Finset α) (f : p → α → β) (g : ¬p → α → β) : ∏ x ∈ s, (if h : p then f h x else g h x) = if h : p then ∏ x ∈ s, f h x else ∏ x ∈ s, g h x := by split_ifs with h <;> rfl #align finset.prod_dite_irrel Finset.prod_dite_irrel #align finset.sum_dite_irrel Finset.sum_dite_irrel @[to_additive (attr := simp)] theorem prod_pi_mulSingle' [DecidableEq α] (a : α) (x : β) (s : Finset α) : ∏ a' ∈ s, Pi.mulSingle a x a' = if a ∈ s then x else 1 := prod_dite_eq' _ _ _ #align finset.prod_pi_mul_single' Finset.prod_pi_mulSingle' #align finset.sum_pi_single' Finset.sum_pi_single' @[to_additive (attr := simp)] theorem prod_pi_mulSingle {β : α → Type} [DecidableEq α] [∀ a, CommMonoid (β a)] (a : α) (f : ∀ a, β a) (s : Finset α) : (∏ a' ∈ s, Pi.mulSingle a' (f a') a) = if a ∈ s then f a else 1 := prod_dite_eq _ _ _ #align finset.prod_pi_mul_single Finset.prod_pi_mulSingle @[to_additive] lemma mulSupport_prod (s : Finset ι) (f : ι → α → β) : mulSupport (fun x ↦ ∏ i ∈ s, f i x) ⊆ ⋃ i ∈ s, mulSupport (f i) := by simp only [mulSupport_subset_iff', Set.mem_iUnion, not_exists, nmem_mulSupport] exact fun x ↦ prod_eq_one #align function.mul_support_prod Finset.mulSupport_prod #align function.support_sum Finset.support_sum section indicator open Set variable {κ : Type} /-- Consider a product of `g i (f i)` over a finset. Suppose `g` is a function such as `n ↦ (· ^ n)`, which maps a second argument of `1` to `1`. Then if `f` is replaced by the corresponding multiplicative indicator function, the finset may be replaced by a possibly larger finset without changing the value of the product. -/ @[to_additive "Consider a sum of `g i (f i)` over a finset. Suppose `g` is a function such as `n ↦ (n • ·)`, which maps a second argument of `0` to `0` (or a weighted sum of `f i * h i` or `f i • h i`, where `f` gives the weights that are multiplied by some other function `h`). Then if `f` is replaced by the corresponding indicator function, the finset may be replaced by a possibly larger finset without changing the value of the sum."] lemma prod_mulIndicator_subset_of_eq_one [One α] (f : ι → α) (g : ι → α → β) {s t : Finset ι} (h : s ⊆ t) (hg : ∀ a, g a 1 = 1) : ∏ i ∈ t, g i (mulIndicator ↑s f i) = ∏ i ∈ s, g i (f i) := by calc _ = ∏ i ∈ s, g i (mulIndicator ↑s f i) := by rw [prod_subset h fun i _ hn ↦ by simp [hn, hg]] -- Porting note: This did not use to need the implicit argument _ = _ := prod_congr rfl fun i hi ↦ congr_arg _ <\| mulIndicator_of_mem (α := ι) hi f #align set.prod_mul_indicator_subset_of_eq_one Finset.prod_mulIndicator_subset_of_eq_one #align set.sum_indicator_subset_of_eq_zero Finset.sum_indicator_subset_of_eq_zero /-- Taking the product of an indicator function over a possibly larger finset is the same as taking the original function over the original finset. -/ @[to_additive "Summing an indicator function over a possibly larger `Finset` is the same as summing the original function over the original finset."] lemma prod_mulIndicator_subset (f : ι → β) {s t : Finset ι} (h : s ⊆ t) : ∏ i ∈ t, mulIndicator (↑s) f i = ∏ i ∈ s, f i := prod_mulIndicator_subset_of_eq_one _ (fun _ ↦ id) h fun _ ↦ rfl #align set.prod_mul_indicator_subset Finset.prod_mulIndicator_subset #align set.sum_indicator_subset Finset.sum_indicator_subset @[to_additive] lemma prod_mulIndicator_eq_prod_filter (s : Finset ι) (f : ι → κ → β) (t : ι → Set κ) (g : ι → κ) [DecidablePred fun i ↦ g i ∈ t i] : ∏ i ∈ s, mulIndicator (t i) (f i) (g i) = ∏ i ∈ s.filter fun i ↦ g i ∈ t i, f i (g i) := by refine (prod_filter_mul_prod_filter_not s (fun i ↦ g i ∈ t i) _).symm.trans <\| Eq.trans (congr_arg₂ (· * ·) ?_ ?_) (mul_one _) · exact prod_congr rfl fun x hx ↦ mulIndicator_of_mem (mem_filter.1 hx).2 _ · exact prod_eq_one fun x hx ↦ mulIndicator_of_not_mem (mem_filter.1 hx).2 _ #align finset.prod_mul_indicator_eq_prod_filter Finset.prod_mulIndicator_eq_prod_filter #align finset.sum_indicator_eq_sum_filter Finset.sum_indicator_eq_sum_filter @[to_additive] lemma prod_mulIndicator_eq_prod_inter [DecidableEq ι] (s t : Finset ι) (f : ι → β) : ∏ i ∈ s, (t : Set ι).mulIndicator f i = ∏ i ∈ s ∩ t, f i := by rw [← filter_mem_eq_inter, prod_mulIndicator_eq_prod_filter]; rfl @[to_additive] lemma mulIndicator_prod (s : Finset ι) (t : Set κ) (f : ι → κ → β) : mulIndicator t (∏ i ∈ s, f i) = ∏ i ∈ s, mulIndicator t (f i) := map_prod (mulIndicatorHom _ _) _ _ #align set.mul_indicator_finset_prod Finset.mulIndicator_prod #align set.indicator_finset_sum Finset.indicator_sum variable {κ : Type} @[to_additive] lemma mulIndicator_biUnion (s : Finset ι) (t : ι → Set κ) {f : κ → β} : ((s : Set ι).PairwiseDisjoint t) → mulIndicator (⋃ i ∈ s, t i) f = fun a ↦ ∏ i ∈ s, mulIndicator (t i) f a := by classical refine Finset.induction_on s (by simp) fun i s hi ih hs ↦ funext fun j ↦ ?_ rw [prod_insert hi, set_biUnion_insert, mulIndicator_union_of_not_mem_inter, ih (hs.subset <\| subset_insert _ _)] simp only [not_exists, exists_prop, mem_iUnion, mem_inter_iff, not_and] exact fun hji i' hi' hji' ↦ (ne_of_mem_of_not_mem hi' hi).symm <\| hs.elim_set (mem_insert_self _ _) (mem_insert_of_mem hi') _ hji hji' #align set.mul_indicator_finset_bUnion Finset.mulIndicator_biUnion #align set.indicator_finset_bUnion Finset.indicator_biUnion @[to_additive] lemma mulIndicator_biUnion_apply (s : Finset ι) (t : ι → Set κ) {f : κ → β} (h : (s : Set ι).PairwiseDisjoint t) (x : κ) : mulIndicator (⋃ i ∈ s, t i) f x = ∏ i ∈ s, mulIndicator (t i) f x := by rw [mulIndicator_biUnion s t h] #align set.mul_indicator_finset_bUnion_apply Finset.mulIndicator_biUnion_apply #align set.indicator_finset_bUnion_apply Finset.indicator_biUnion_apply end indicator @[to_additive] theorem prod_bij_ne_one {s : Finset α} {t : Finset γ} {f : α → β} {g : γ → β} (i : ∀ a ∈ s, f a ≠ 1 → γ) (hi : ∀ a h₁ h₂, i a h₁ h₂ ∈ t) (i_inj : ∀ a₁ h₁₁ h₁₂ a₂ h₂₁ h₂₂, i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂) (i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ a h₁ h₂, i a h₁ h₂ = b) (h : ∀ a h₁ h₂, f a = g (i a h₁ h₂)) : ∏ x ∈ s, f x = ∏ x ∈ t, g x := by classical calc ∏ x ∈ s, f x = ∏ x ∈ s.filter fun x => f x ≠ 1, f x := by rw [prod_filter_ne_one] _ = ∏ x ∈ t.filter fun x => g x ≠ 1, g x := prod_bij (fun a ha => i a (mem_filter.mp ha).1 <\| by simpa using (mem_filter.mp ha).2) ?_ ?_ ?_ ?_ _ = ∏ x ∈ t, g x := prod_filter_ne_one _ · intros a ha refine (mem_filter.mp ha).elim ?_ intros h₁ h₂ refine (mem_filter.mpr ⟨hi a h₁ _, ?_⟩) specialize h a h₁ fun H ↦ by rw [H] at h₂; simp at h₂ rwa [← h] · intros a₁ ha₁ a₂ ha₂ refine (mem_filter.mp ha₁).elim fun _ha₁₁ _ha₁₂ ↦ ?_ refine (mem_filter.mp ha₂).elim fun _ha₂₁ _ha₂₂ ↦ ?_ apply i_inj · intros b hb refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_ obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂ exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩ · refine (fun a ha => (mem_filter.mp ha).elim fun h₁ h₂ ↦ ?_) exact h a h₁ fun H ↦ by rw [H] at h₂; simp at h₂ #align finset.prod_bij_ne_one Finset.prod_bij_ne_one #align finset.sum_bij_ne_zero Finset.sum_bij_ne_zero @[to_additive] theorem prod_dite_of_false {p : α → Prop} {hp : DecidablePred p} (h : ∀ x ∈ s, ¬p x) (f : ∀ x : α, p x → β) (g : ∀ x : α, ¬p x → β) : ∏ x ∈ s, (if hx : p x then f x hx else g x hx) = ∏ x : s, g x.val (h x.val x.property) := by refine prod_bij' (fun x hx => ⟨x, hx⟩) (fun x _ ↦ x) ?_ ?_ ?_ ?_ ?_ <;> aesop #align finset.prod_dite_of_false Finset.prod_dite_of_false #align finset.sum_dite_of_false Finset.sum_dite_of_false @[to_additive] theorem prod_dite_of_true {p : α → Prop} {hp : DecidablePred p} (h : ∀ x ∈ s, p x) (f : ∀ x : α, p x → β) (g : ∀ x : α, ¬p x → β) : ∏ x ∈ s, (if hx : p x then f x hx else g x hx) = ∏ x : s, f x.val (h x.val x.property) := by refine prod_bij' (fun x hx => ⟨x, hx⟩) (fun x _ ↦ x) ?_ ?_ ?_ ?_ ?_ <;> aesop #align finset.prod_dite_of_true Finset.prod_dite_of_true #align finset.sum_dite_of_true Finset.sum_dite_of_true @[to_additive] theorem nonempty_of_prod_ne_one (h : ∏ x ∈ s, f x ≠ 1) : s.Nonempty := s.eq_empty_or_nonempty.elim (fun H => False.elim <\| h <\| H.symm ▸ prod_empty) id #align finset.nonempty_of_prod_ne_one Finset.nonempty_of_prod_ne_one #align finset.nonempty_of_sum_ne_zero Finset.nonempty_of_sum_ne_zero @[to_additive] theorem exists_ne_one_of_prod_ne_one (h : ∏ x ∈ s, f x ≠ 1) : ∃ a ∈ s, f a ≠ 1 := by classical rw [← prod_filter_ne_one] at h rcases nonempty_of_prod_ne_one h with ⟨x, hx⟩ exact ⟨x, (mem_filter.1 hx).1, by simpa using (mem_filter.1 hx).2⟩ #align finset.exists_ne_one_of_prod_ne_one Finset.exists_ne_one_of_prod_ne_one #align finset.exists_ne_zero_of_sum_ne_zero Finset.exists_ne_zero_of_sum_ne_zero @[to_additive] theorem prod_range_succ_comm (f : ℕ → β) (n : ℕ) : (∏ x ∈ range (n + 1), f x) = f n ∏ x ∈ range n, f x := by rw [range_succ, prod_insert not_mem_range_self] #align finset.prod_range_succ_comm Finset.prod_range_succ_comm #align finset.sum_range_succ_comm Finset.sum_range_succ_comm @[to_additive] theorem prod_range_succ (f : ℕ → β) (n : ℕ) : (∏ x ∈ range (n + 1), f x) = (∏ x ∈ range n, f x) * f n := by simp only [mul_comm, prod_range_succ_comm] #align finset.prod_range_succ Finset.prod_range_succ #align finset.sum_range_succ Finset.sum_range_succ @[to_additive] theorem prod_range_succ' (f : ℕ → β) : ∀ n : ℕ, (∏ k ∈ range (n + 1), f k) = (∏ k ∈ range n, f (k + 1)) * f 0 \| 0 => prod_range_succ _ _ \| n + 1 => by rw [prod_range_succ _ n, mul_right_comm, ← prod_range_succ' _ n, prod_range_succ] #align finset.prod_range_succ' Finset.prod_range_succ' #align finset.sum_range_succ' Finset.sum_range_succ' @[to_additive] theorem eventually_constant_prod {u : ℕ → β} {N : ℕ} (hu : ∀ n ≥ N, u n = 1) {n : ℕ} (hn : N ≤ n) : (∏ k ∈ range n, u k) = ∏ k ∈ range N, u k := by obtain ⟨m, rfl : n = N + m⟩ := Nat.exists_eq_add_of_le hn clear hn induction' m with m hm · simp · simp [← add_assoc, prod_range_succ, hm, hu] #align finset.eventually_constant_prod Finset.eventually_constant_prod #align finset.eventually_constant_sum Finset.eventually_constant_sum @[to_additive] theorem prod_range_add (f : ℕ → β) (n m : ℕ) : (∏ x ∈ range (n + m), f x) = (∏ x ∈ range n, f x) * ∏ x ∈ range m, f (n + x) := by induction' m with m hm · simp · erw [Nat.add_succ, prod_range_succ, prod_range_succ, hm, mul_assoc] #align finset.prod_range_add Finset.prod_range_add #align finset.sum_range_add Finset.sum_range_add @[to_additive] theorem prod_range_add_div_prod_range {α : Type} [CommGroup α] (f : ℕ → α) (n m : ℕ) : (∏ k ∈ range (n + m), f k) / ∏ k ∈ range n, f k = ∏ k ∈ Finset.range m, f (n + k) := div_eq_of_eq_mul' (prod_range_add f n m) #align finset.prod_range_add_div_prod_range Finset.prod_range_add_div_prod_range #align finset.sum_range_add_sub_sum_range Finset.sum_range_add_sub_sum_range @[to_additive] theorem prod_range_zero (f : ℕ → β) : ∏ k ∈ range 0, f k = 1 := by rw [range_zero, prod_empty] #align finset.prod_range_zero Finset.prod_range_zero #align finset.sum_range_zero Finset.sum_range_zero @[to_additive sum_range_one] theorem prod_range_one (f : ℕ → β) : ∏ k ∈ range 1, f k = f 0 := by rw [range_one, prod_singleton] #align finset.prod_range_one Finset.prod_range_one #align finset.sum_range_one Finset.sum_range_one open List @[to_additive] theorem prod_list_map_count [DecidableEq α] (l : List α) {M : Type} [CommMonoid M] (f : α → M) : (l.map f).prod = ∏ m ∈ l.toFinset, f m ^ l.count m := by induction' l with a s IH; · simp only [map_nil, prod_nil, count_nil, pow_zero, prod_const_one] simp only [List.map, List.prod_cons, toFinset_cons, IH] by_cases has : a ∈ s.toFinset · rw [insert_eq_of_mem has, ← insert_erase has, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _), ← mul_assoc, count_cons_self, pow_succ'] congr 1 refine prod_congr rfl fun x hx => ?_ rw [count_cons_of_ne (ne_of_mem_erase hx)] rw [prod_insert has, count_cons_self, count_eq_zero_of_not_mem (mt mem_toFinset.2 has), pow_one] congr 1 refine prod_congr rfl fun x hx => ?_ rw [count_cons_of_ne] rintro rfl exact has hx #align finset.prod_list_map_count Finset.prod_list_map_count #align finset.sum_list_map_count Finset.sum_list_map_count @[to_additive] theorem prod_list_count [DecidableEq α] [CommMonoid α] (s : List α) : s.prod = ∏ m ∈ s.toFinset, m ^ s.count m := by simpa using prod_list_map_count s id #align finset.prod_list_count Finset.prod_list_count #align finset.sum_list_count Finset.sum_list_count @[to_additive] theorem prod_list_count_of_subset [DecidableEq α] [CommMonoid α] (m : List α) (s : Finset α) (hs : m.toFinset ⊆ s) : m.prod = ∏ i ∈ s, i ^ m.count i := by rw [prod_list_count] refine prod_subset hs fun x _ hx => ?_ rw [mem_toFinset] at hx rw [count_eq_zero_of_not_mem hx, pow_zero] #align finset.prod_list_count_of_subset Finset.prod_list_count_of_subset #align finset.sum_list_count_of_subset Finset.sum_list_count_of_subset theorem sum_filter_count_eq_countP [DecidableEq α] (p : α → Prop) [DecidablePred p] (l : List α) : ∑ x ∈ l.toFinset.filter p, l.count x = l.countP p := by simp [Finset.sum, sum_map_count_dedup_filter_eq_countP p l] #align finset.sum_filter_count_eq_countp Finset.sum_filter_count_eq_countP open Multiset @[to_additive] theorem prod_multiset_map_count [DecidableEq α] (s : Multiset α) {M : Type} [CommMonoid M] (f : α → M) : (s.map f).prod = ∏ m ∈ s.toFinset, f m ^ s.count m := by refine Quot.induction_on s fun l => ?_ simp [prod_list_map_count l f] #align finset.prod_multiset_map_count Finset.prod_multiset_map_count #align finset.sum_multiset_map_count Finset.sum_multiset_map_count @[to_additive] theorem prod_multiset_count [DecidableEq α] [CommMonoid α] (s : Multiset α) : s.prod = ∏ m ∈ s.toFinset, m ^ s.count m := by convert prod_multiset_map_count s id rw [Multiset.map_id] #align finset.prod_multiset_count Finset.prod_multiset_count #align finset.sum_multiset_count Finset.sum_multiset_count @[to_additive] theorem prod_multiset_count_of_subset [DecidableEq α] [CommMonoid α] (m : Multiset α) (s : Finset α) (hs : m.toFinset ⊆ s) : m.prod = ∏ i ∈ s, i ^ m.count i := by revert hs refine Quot.induction_on m fun l => ?_ simp only [quot_mk_to_coe'', prod_coe, coe_count] apply prod_list_count_of_subset l s #align finset.prod_multiset_count_of_subset Finset.prod_multiset_count_of_subset #align finset.sum_multiset_count_of_subset Finset.sum_multiset_count_of_subset @[to_additive] theorem prod_mem_multiset [DecidableEq α] (m : Multiset α) (f : { x // x ∈ m } → β) (g : α → β) (hfg : ∀ x, f x = g x) : ∏ x : { x // x ∈ m }, f x = ∏ x ∈ m.toFinset, g x := by refine prod_bij' (fun x _ ↦ x) (fun x hx ↦ ⟨x, Multiset.mem_toFinset.1 hx⟩) ?_ ?_ ?_ ?_ ?_ <;> simp [hfg] #align finset.prod_mem_multiset Finset.prod_mem_multiset #align finset.sum_mem_multiset Finset.sum_mem_multiset /-- To prove a property of a product, it suffices to prove that the property is multiplicative and holds on factors. -/ @[to_additive "To prove a property of a sum, it suffices to prove that the property is additive and holds on summands."] theorem prod_induction {M : Type} [CommMonoid M] (f : α → M) (p : M → Prop) (hom : ∀ a b, p a → p b → p (a * b)) (unit : p 1) (base : ∀ x ∈ s, p <\| f x) : p <\| ∏ x ∈ s, f x := Multiset.prod_induction _ _ hom unit (Multiset.forall_mem_map_iff.mpr base) #align finset.prod_induction Finset.prod_induction #align finset.sum_induction Finset.sum_induction /-- To prove a property of a product, it suffices to prove that the property is multiplicative and holds on factors. -/ @[to_additive "To prove a property of a sum, it suffices to prove that the property is additive and holds on summands."] theorem prod_induction_nonempty {M : Type} [CommMonoid M] (f : α → M) (p : M → Prop) (hom : ∀ a b, p a → p b → p (a b)) (nonempty : s.Nonempty) (base : ∀ x ∈ s, p <\| f x) : p <\| ∏ x ∈ s, f x := Multiset.prod_induction_nonempty p hom (by simp [nonempty_iff_ne_empty.mp nonempty]) (Multiset.forall_mem_map_iff.mpr base) #align finset.prod_induction_nonempty Finset.prod_induction_nonempty #align finset.sum_induction_nonempty Finset.sum_induction_nonempty /-- For any product along `{0, ..., n - 1}` of a commutative-monoid-valued function, we can verify that it's equal to a different function just by checking ratios of adjacent terms. This is a multiplicative discrete analogue of the fundamental theorem of calculus. -/ @[to_additive "For any sum along `{0, ..., n - 1}` of a commutative-monoid-valued function, we can verify that it's equal to a different function just by checking differences of adjacent terms. This is a discrete analogue of the fundamental theorem of calculus."] theorem prod_range_induction (f s : ℕ → β) (base : s 0 = 1) (step : ∀ n, s (n + 1) = s n * f n) (n : ℕ) : ∏ k ∈ Finset.range n, f k = s n := by induction' n with k hk · rw [Finset.prod_range_zero, base] · simp only [hk, Finset.prod_range_succ, step, mul_comm] #align finset.prod_range_induction Finset.prod_range_induction #align finset.sum_range_induction Finset.sum_range_induction /-- A telescoping product along `{0, ..., n - 1}` of a commutative group valued function reduces to the ratio of the last and first factors. -/ @[to_additive "A telescoping sum along `{0, ..., n - 1}` of an additive commutative group valued function reduces to the difference of the last and first terms."] theorem prod_range_div {M : Type} [CommGroup M] (f : ℕ → M) (n : ℕ) : (∏ i ∈ range n, f (i + 1) / f i) = f n / f 0 := by apply prod_range_induction <;> simp #align finset.prod_range_div Finset.prod_range_div #align finset.sum_range_sub Finset.sum_range_sub @[to_additive] theorem prod_range_div' {M : Type} [CommGroup M] (f : ℕ → M) (n : ℕ) : (∏ i ∈ range n, f i / f (i + 1)) = f 0 / f n := by apply prod_range_induction <;> simp #align finset.prod_range_div' Finset.prod_range_div' #align finset.sum_range_sub' Finset.sum_range_sub' @[to_additive] theorem eq_prod_range_div {M : Type} [CommGroup M] (f : ℕ → M) (n : ℕ) : f n = f 0 ∏ i ∈ range n, f (i + 1) / f i := by rw [prod_range_div, mul_div_cancel] #align finset.eq_prod_range_div Finset.eq_prod_range_div #align finset.eq_sum_range_sub Finset.eq_sum_range_sub @[to_additive]	Mathlib/Algebra/BigOperators/Group/Finset.lean	1,713	1,716	theorem eq_prod_range_div' {M : Type*} [CommGroup M] (f : ℕ → M) (n : ℕ) : f n = ∏ i ∈ range (n + 1), if i = 0 then f 0 else f i / f (i - 1) := by	conv_lhs => rw [Finset.eq_prod_range_div f] simp [Finset.prod_range_succ', mul_comm]
/- Copyright (c) 2020 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury G. Kudryashov -/ import Mathlib.Analysis.NormedSpace.Basic #align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2" /-! # Extended norm In this file we define a structure `ENorm 𝕜 V` representing an extended norm (i.e., a norm that can take the value `∞`) on a vector space `V` over a normed field `𝕜`. We do not use `class` for an `ENorm` because the same space can have more than one extended norm. For example, the space of measurable functions `f : α → ℝ` has a family of `L_p` extended norms. We prove some basic inequalities, then define * `EMetricSpace` structure on `V` corresponding to `e : ENorm 𝕜 V`; * the subspace of vectors with finite norm, called `e.finiteSubspace`; * a `NormedSpace` structure on this space. The last definition is an instance because the type involves `e`. ## Implementation notes We do not define extended normed groups. They can be added to the chain once someone will need them. ## Tags normed space, extended norm -/ noncomputable section attribute [local instance] Classical.propDecidable open ENNReal /-- Extended norm on a vector space. As in the case of normed spaces, we require only `‖c • x‖ ≤ ‖c‖ * ‖x‖` in the definition, then prove an equality in `map_smul`. -/ structure ENorm (𝕜 : Type) (V : Type) [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] where toFun : V → ℝ≥0∞ eq_zero' : ∀ x, toFun x = 0 → x = 0 map_add_le' : ∀ x y : V, toFun (x + y) ≤ toFun x + toFun y map_smul_le' : ∀ (c : 𝕜) (x : V), toFun (c • x) ≤ ‖c‖₊ * toFun x #align enorm ENorm namespace ENorm variable {𝕜 : Type} {V : Type} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) -- Porting note: added to appease norm_cast complaints attribute [coe] ENorm.toFun instance : CoeFun (ENorm 𝕜 V) fun _ => V → ℝ≥0∞ := ⟨ENorm.toFun⟩ theorem coeFn_injective : Function.Injective ((↑) : ENorm 𝕜 V → V → ℝ≥0∞) := fun e₁ e₂ h => by cases e₁ cases e₂ congr #align enorm.coe_fn_injective ENorm.coeFn_injective @[ext] theorem ext {e₁ e₂ : ENorm 𝕜 V} (h : ∀ x, e₁ x = e₂ x) : e₁ = e₂ := coeFn_injective <\| funext h #align enorm.ext ENorm.ext theorem ext_iff {e₁ e₂ : ENorm 𝕜 V} : e₁ = e₂ ↔ ∀ x, e₁ x = e₂ x := ⟨fun h _ => h ▸ rfl, ext⟩ #align enorm.ext_iff ENorm.ext_iff @[simp, norm_cast] theorem coe_inj {e₁ e₂ : ENorm 𝕜 V} : (e₁ : V → ℝ≥0∞) = e₂ ↔ e₁ = e₂ := coeFn_injective.eq_iff #align enorm.coe_inj ENorm.coe_inj @[simp]	Mathlib/Analysis/NormedSpace/ENorm.lean	82	92	theorem map_smul (c : 𝕜) (x : V) : e (c • x) = ‖c‖₊ * e x := by	apply le_antisymm (e.map_smul_le' c x) by_cases hc : c = 0 · simp [hc] calc (‖c‖₊ : ℝ≥0∞) * e x = ‖c‖₊ * e (c⁻¹ • c • x) := by rw [inv_smul_smul₀ hc] _ ≤ ‖c‖₊ * (‖c⁻¹‖₊ * e (c • x)) := mul_le_mul_left' (e.map_smul_le' _ _) _ _ = e (c • x) := by rw [← mul_assoc, nnnorm_inv, ENNReal.coe_inv, ENNReal.mul_inv_cancel _ ENNReal.coe_ne_top, one_mul] <;> simp [hc]
/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Kevin Kappelmann -/ import Mathlib.Algebra.CharZero.Lemmas import Mathlib.Algebra.Order.Interval.Set.Group import Mathlib.Algebra.Group.Int import Mathlib.Data.Int.Lemmas import Mathlib.Data.Set.Subsingleton import Mathlib.Init.Data.Nat.Lemmas import Mathlib.Order.GaloisConnection import Mathlib.Tactic.Abel import Mathlib.Tactic.Linarith import Mathlib.Tactic.Positivity #align_import algebra.order.floor from "leanprover-community/mathlib"@"afdb43429311b885a7988ea15d0bac2aac80f69c" /-! # Floor and ceil ## Summary We define the natural- and integer-valued floor and ceil functions on linearly ordered rings. ## Main Definitions * `FloorSemiring`: An ordered semiring with natural-valued floor and ceil. * `Nat.floor a`: Greatest natural `n` such that `n ≤ a`. Equal to `0` if `a < 0`. * `Nat.ceil a`: Least natural `n` such that `a ≤ n`. * `FloorRing`: A linearly ordered ring with integer-valued floor and ceil. * `Int.floor a`: Greatest integer `z` such that `z ≤ a`. * `Int.ceil a`: Least integer `z` such that `a ≤ z`. * `Int.fract a`: Fractional part of `a`, defined as `a - floor a`. * `round a`: Nearest integer to `a`. It rounds halves towards infinity. ## Notations * `⌊a⌋₊` is `Nat.floor a`. * `⌈a⌉₊` is `Nat.ceil a`. * `⌊a⌋` is `Int.floor a`. * `⌈a⌉` is `Int.ceil a`. The index `₊` in the notations for `Nat.floor` and `Nat.ceil` is used in analogy to the notation for `nnnorm`. ## TODO `LinearOrderedRing`/`LinearOrderedSemiring` can be relaxed to `OrderedRing`/`OrderedSemiring` in many lemmas. ## Tags rounding, floor, ceil -/ open Set variable {F α β : Type*} /-! ### Floor semiring -/ /-- A `FloorSemiring` is an ordered semiring over `α` with a function `floor : α → ℕ` satisfying `∀ (n : ℕ) (x : α), n ≤ ⌊x⌋ ↔ (n : α) ≤ x)`. Note that many lemmas require a `LinearOrder`. Please see the above `TODO`. -/ class FloorSemiring (α) [OrderedSemiring α] where /-- `FloorSemiring.floor a` computes the greatest natural `n` such that `(n : α) ≤ a`. -/ floor : α → ℕ /-- `FloorSemiring.ceil a` computes the least natural `n` such that `a ≤ (n : α)`. -/ ceil : α → ℕ /-- `FloorSemiring.floor` of a negative element is zero. -/ floor_of_neg {a : α} (ha : a < 0) : floor a = 0 /-- A natural number `n` is smaller than `FloorSemiring.floor a` iff its coercion to `α` is smaller than `a`. -/ gc_floor {a : α} {n : ℕ} (ha : 0 ≤ a) : n ≤ floor a ↔ (n : α) ≤ a /-- `FloorSemiring.ceil` is the lower adjoint of the coercion `↑ : ℕ → α`. -/ gc_ceil : GaloisConnection ceil (↑) #align floor_semiring FloorSemiring instance : FloorSemiring ℕ where floor := id ceil := id floor_of_neg ha := (Nat.not_lt_zero _ ha).elim gc_floor _ := by rw [Nat.cast_id] rfl gc_ceil n a := by rw [Nat.cast_id] rfl namespace Nat section OrderedSemiring variable [OrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} /-- `⌊a⌋₊` is the greatest natural `n` such that `n ≤ a`. If `a` is negative, then `⌊a⌋₊ = 0`. -/ def floor : α → ℕ := FloorSemiring.floor #align nat.floor Nat.floor /-- `⌈a⌉₊` is the least natural `n` such that `a ≤ n` -/ def ceil : α → ℕ := FloorSemiring.ceil #align nat.ceil Nat.ceil @[simp] theorem floor_nat : (Nat.floor : ℕ → ℕ) = id := rfl #align nat.floor_nat Nat.floor_nat @[simp] theorem ceil_nat : (Nat.ceil : ℕ → ℕ) = id := rfl #align nat.ceil_nat Nat.ceil_nat @[inherit_doc] notation "⌊" a "⌋₊" => Nat.floor a @[inherit_doc] notation "⌈" a "⌉₊" => Nat.ceil a end OrderedSemiring section LinearOrderedSemiring variable [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} theorem le_floor_iff (ha : 0 ≤ a) : n ≤ ⌊a⌋₊ ↔ (n : α) ≤ a := FloorSemiring.gc_floor ha #align nat.le_floor_iff Nat.le_floor_iff theorem le_floor (h : (n : α) ≤ a) : n ≤ ⌊a⌋₊ := (le_floor_iff <\| n.cast_nonneg.trans h).2 h #align nat.le_floor Nat.le_floor theorem floor_lt (ha : 0 ≤ a) : ⌊a⌋₊ < n ↔ a < n := lt_iff_lt_of_le_iff_le <\| le_floor_iff ha #align nat.floor_lt Nat.floor_lt theorem floor_lt_one (ha : 0 ≤ a) : ⌊a⌋₊ < 1 ↔ a < 1 := (floor_lt ha).trans <\| by rw [Nat.cast_one] #align nat.floor_lt_one Nat.floor_lt_one theorem lt_of_floor_lt (h : ⌊a⌋₊ < n) : a < n := lt_of_not_le fun h' => (le_floor h').not_lt h #align nat.lt_of_floor_lt Nat.lt_of_floor_lt theorem lt_one_of_floor_lt_one (h : ⌊a⌋₊ < 1) : a < 1 := mod_cast lt_of_floor_lt h #align nat.lt_one_of_floor_lt_one Nat.lt_one_of_floor_lt_one theorem floor_le (ha : 0 ≤ a) : (⌊a⌋₊ : α) ≤ a := (le_floor_iff ha).1 le_rfl #align nat.floor_le Nat.floor_le theorem lt_succ_floor (a : α) : a < ⌊a⌋₊.succ := lt_of_floor_lt <\| Nat.lt_succ_self _ #align nat.lt_succ_floor Nat.lt_succ_floor	Mathlib/Algebra/Order/Floor.lean	162	162	theorem lt_floor_add_one (a : α) : a < ⌊a⌋₊ + 1 := by	simpa using lt_succ_floor a
/- Copyright (c) 2014 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Leonardo de Moura -/ import Mathlib.Data.Set.Subsingleton import Mathlib.Order.WithBot #align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29" /-! # Images and preimages of sets ## Main definitions * `preimage f t : Set α` : the preimage f⁻¹(t) (written `f ⁻¹' t` in Lean) of a subset of β. * `range f : Set β` : the image of `univ` under `f`. Also works for `{p : Prop} (f : p → α)` (unlike `image`) ## Notation * `f ⁻¹' t` for `Set.preimage f t` * `f '' s` for `Set.image f s` ## Tags set, sets, image, preimage, pre-image, range -/ universe u v open Function Set namespace Set variable {α β γ : Type} {ι ι' : Sort} /-! ### Inverse image -/ section Preimage variable {f : α → β} {g : β → γ} @[simp] theorem preimage_empty : f ⁻¹' ∅ = ∅ := rfl #align set.preimage_empty Set.preimage_empty theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by congr with x simp [h] #align set.preimage_congr Set.preimage_congr @[gcongr] theorem preimage_mono {s t : Set β} (h : s ⊆ t) : f ⁻¹' s ⊆ f ⁻¹' t := fun _ hx => h hx #align set.preimage_mono Set.preimage_mono @[simp, mfld_simps] theorem preimage_univ : f ⁻¹' univ = univ := rfl #align set.preimage_univ Set.preimage_univ theorem subset_preimage_univ {s : Set α} : s ⊆ f ⁻¹' univ := subset_univ _ #align set.subset_preimage_univ Set.subset_preimage_univ @[simp, mfld_simps] theorem preimage_inter {s t : Set β} : f ⁻¹' (s ∩ t) = f ⁻¹' s ∩ f ⁻¹' t := rfl #align set.preimage_inter Set.preimage_inter @[simp] theorem preimage_union {s t : Set β} : f ⁻¹' (s ∪ t) = f ⁻¹' s ∪ f ⁻¹' t := rfl #align set.preimage_union Set.preimage_union @[simp] theorem preimage_compl {s : Set β} : f ⁻¹' sᶜ = (f ⁻¹' s)ᶜ := rfl #align set.preimage_compl Set.preimage_compl @[simp] theorem preimage_diff (f : α → β) (s t : Set β) : f ⁻¹' (s \ t) = f ⁻¹' s \ f ⁻¹' t := rfl #align set.preimage_diff Set.preimage_diff open scoped symmDiff in @[simp] lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) := rfl #align set.preimage_symm_diff Set.preimage_symmDiff @[simp] theorem preimage_ite (f : α → β) (s t₁ t₂ : Set β) : f ⁻¹' s.ite t₁ t₂ = (f ⁻¹' s).ite (f ⁻¹' t₁) (f ⁻¹' t₂) := rfl #align set.preimage_ite Set.preimage_ite @[simp] theorem preimage_setOf_eq {p : α → Prop} {f : β → α} : f ⁻¹' { a \| p a } = { a \| p (f a) } := rfl #align set.preimage_set_of_eq Set.preimage_setOf_eq @[simp] theorem preimage_id_eq : preimage (id : α → α) = id := rfl #align set.preimage_id_eq Set.preimage_id_eq @[mfld_simps] theorem preimage_id {s : Set α} : id ⁻¹' s = s := rfl #align set.preimage_id Set.preimage_id @[simp, mfld_simps] theorem preimage_id' {s : Set α} : (fun x => x) ⁻¹' s = s := rfl #align set.preimage_id' Set.preimage_id' @[simp] theorem preimage_const_of_mem {b : β} {s : Set β} (h : b ∈ s) : (fun _ : α => b) ⁻¹' s = univ := eq_univ_of_forall fun _ => h #align set.preimage_const_of_mem Set.preimage_const_of_mem @[simp] theorem preimage_const_of_not_mem {b : β} {s : Set β} (h : b ∉ s) : (fun _ : α => b) ⁻¹' s = ∅ := eq_empty_of_subset_empty fun _ hx => h hx #align set.preimage_const_of_not_mem Set.preimage_const_of_not_mem theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] : (fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by split_ifs with hb exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb] #align set.preimage_const Set.preimage_const /-- If preimage of each singleton under `f : α → β` is either empty or the whole type, then `f` is a constant. -/ lemma exists_eq_const_of_preimage_singleton [Nonempty β] {f : α → β} (hf : ∀ b : β, f ⁻¹' {b} = ∅ ∨ f ⁻¹' {b} = univ) : ∃ b, f = const α b := by rcases em (∃ b, f ⁻¹' {b} = univ) with ⟨b, hb⟩ \| hf' · exact ⟨b, funext fun x ↦ eq_univ_iff_forall.1 hb x⟩ · have : ∀ x b, f x ≠ b := fun x b ↦ eq_empty_iff_forall_not_mem.1 ((hf b).resolve_right fun h ↦ hf' ⟨b, h⟩) x exact ⟨Classical.arbitrary β, funext fun x ↦ absurd rfl (this x _)⟩ theorem preimage_comp {s : Set γ} : g ∘ f ⁻¹' s = f ⁻¹' (g ⁻¹' s) := rfl #align set.preimage_comp Set.preimage_comp theorem preimage_comp_eq : preimage (g ∘ f) = preimage f ∘ preimage g := rfl #align set.preimage_comp_eq Set.preimage_comp_eq theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by induction' n with n ih; · simp rw [iterate_succ, iterate_succ', preimage_comp_eq, ih] #align set.preimage_iterate_eq Set.preimage_iterate_eq theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} : f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻¹' s := preimage_comp.symm #align set.preimage_preimage Set.preimage_preimage theorem eq_preimage_subtype_val_iff {p : α → Prop} {s : Set (Subtype p)} {t : Set α} : s = Subtype.val ⁻¹' t ↔ ∀ (x) (h : p x), (⟨x, h⟩ : Subtype p) ∈ s ↔ x ∈ t := ⟨fun s_eq x h => by rw [s_eq] simp, fun h => ext fun ⟨x, hx⟩ => by simp [h]⟩ #align set.eq_preimage_subtype_val_iff Set.eq_preimage_subtype_val_iff theorem nonempty_of_nonempty_preimage {s : Set β} {f : α → β} (hf : (f ⁻¹' s).Nonempty) : s.Nonempty := let ⟨x, hx⟩ := hf ⟨f x, hx⟩ #align set.nonempty_of_nonempty_preimage Set.nonempty_of_nonempty_preimage @[simp] theorem preimage_singleton_true (p : α → Prop) : p ⁻¹' {True} = {a \| p a} := by ext; simp #align set.preimage_singleton_true Set.preimage_singleton_true @[simp] theorem preimage_singleton_false (p : α → Prop) : p ⁻¹' {False} = {a \| ¬p a} := by ext; simp #align set.preimage_singleton_false Set.preimage_singleton_false theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v) (H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by ext ⟨x, x_in_s⟩ constructor · intro x_in_u x_in_v exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩ · intro hx exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx' #align set.preimage_subtype_coe_eq_compl Set.preimage_subtype_coe_eq_compl end Preimage /-! ### Image of a set under a function -/ section Image variable {f : α → β} {s t : Set α} -- Porting note: `Set.image` is already defined in `Init.Set` #align set.image Set.image @[deprecated mem_image (since := "2024-03-23")] theorem mem_image_iff_bex {f : α → β} {s : Set α} {y : β} : y ∈ f '' s ↔ ∃ (x : _) (_ : x ∈ s), f x = y := bex_def.symm #align set.mem_image_iff_bex Set.mem_image_iff_bex theorem image_eta (f : α → β) : f '' s = (fun x => f x) '' s := rfl #align set.image_eta Set.image_eta theorem _root_.Function.Injective.mem_set_image {f : α → β} (hf : Injective f) {s : Set α} {a : α} : f a ∈ f '' s ↔ a ∈ s := ⟨fun ⟨_, hb, Eq⟩ => hf Eq ▸ hb, mem_image_of_mem f⟩ #align function.injective.mem_set_image Function.Injective.mem_set_image theorem forall_mem_image {f : α → β} {s : Set α} {p : β → Prop} : (∀ y ∈ f '' s, p y) ↔ ∀ ⦃x⦄, x ∈ s → p (f x) := by simp #align set.ball_image_iff Set.forall_mem_image theorem exists_mem_image {f : α → β} {s : Set α} {p : β → Prop} : (∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by simp #align set.bex_image_iff Set.exists_mem_image @[deprecated (since := "2024-02-21")] alias ball_image_iff := forall_mem_image @[deprecated (since := "2024-02-21")] alias bex_image_iff := exists_mem_image @[deprecated (since := "2024-02-21")] alias ⟨_, ball_image_of_ball⟩ := forall_mem_image #align set.ball_image_of_ball Set.ball_image_of_ball @[deprecated forall_mem_image (since := "2024-02-21")] theorem mem_image_elim {f : α → β} {s : Set α} {C : β → Prop} (h : ∀ x : α, x ∈ s → C (f x)) : ∀ {y : β}, y ∈ f '' s → C y := forall_mem_image.2 h _ #align set.mem_image_elim Set.mem_image_elim @[deprecated forall_mem_image (since := "2024-02-21")] theorem mem_image_elim_on {f : α → β} {s : Set α} {C : β → Prop} {y : β} (h_y : y ∈ f '' s) (h : ∀ x : α, x ∈ s → C (f x)) : C y := forall_mem_image.2 h _ h_y #align set.mem_image_elim_on Set.mem_image_elim_on -- Porting note: used to be `safe` @[congr] theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by ext x exact exists_congr fun a ↦ and_congr_right fun ha ↦ by rw [h a ha] #align set.image_congr Set.image_congr /-- A common special case of `image_congr` -/ theorem image_congr' {f g : α → β} {s : Set α} (h : ∀ x : α, f x = g x) : f '' s = g '' s := image_congr fun x _ => h x #align set.image_congr' Set.image_congr' @[gcongr] lemma image_mono (h : s ⊆ t) : f '' s ⊆ f '' t := by rintro - ⟨a, ha, rfl⟩; exact mem_image_of_mem f (h ha) theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) := by aesop #align set.image_comp Set.image_comp theorem image_comp_eq {g : β → γ} : image (g ∘ f) = image g ∘ image f := by ext; simp /-- A variant of `image_comp`, useful for rewriting -/ theorem image_image (g : β → γ) (f : α → β) (s : Set α) : g '' (f '' s) = (fun x => g (f x)) '' s := (image_comp g f s).symm #align set.image_image Set.image_image theorem image_comm {β'} {f : β → γ} {g : α → β} {f' : α → β'} {g' : β' → γ} (h_comm : ∀ a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by simp_rw [image_image, h_comm] #align set.image_comm Set.image_comm theorem _root_.Function.Semiconj.set_image {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ => image_comm h #align function.semiconj.set_image Function.Semiconj.set_image theorem _root_.Function.Commute.set_image {f g : α → α} (h : Function.Commute f g) : Function.Commute (image f) (image g) := Function.Semiconj.set_image h #align function.commute.set_image Function.Commute.set_image /-- Image is monotone with respect to `⊆`. See `Set.monotone_image` for the statement in terms of `≤`. -/ @[gcongr] theorem image_subset {a b : Set α} (f : α → β) (h : a ⊆ b) : f '' a ⊆ f '' b := by simp only [subset_def, mem_image] exact fun x => fun ⟨w, h1, h2⟩ => ⟨w, h h1, h2⟩ #align set.image_subset Set.image_subset /-- `Set.image` is monotone. See `Set.image_subset` for the statement in terms of `⊆`. -/ lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _ #align set.monotone_image Set.monotone_image theorem image_union (f : α → β) (s t : Set α) : f '' (s ∪ t) = f '' s ∪ f '' t := ext fun x => ⟨by rintro ⟨a, h \| h, rfl⟩ <;> [left; right] <;> exact ⟨_, h, rfl⟩, by rintro (⟨a, h, rfl⟩ \| ⟨a, h, rfl⟩) <;> refine ⟨_, ?_, rfl⟩ · exact mem_union_left t h · exact mem_union_right s h⟩ #align set.image_union Set.image_union @[simp] theorem image_empty (f : α → β) : f '' ∅ = ∅ := by ext simp #align set.image_empty Set.image_empty theorem image_inter_subset (f : α → β) (s t : Set α) : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := subset_inter (image_subset _ inter_subset_left) (image_subset _ inter_subset_right) #align set.image_inter_subset Set.image_inter_subset theorem image_inter_on {f : α → β} {s t : Set α} (h : ∀ x ∈ t, ∀ y ∈ s, f x = f y → x = y) : f '' (s ∩ t) = f '' s ∩ f '' t := (image_inter_subset _ _ _).antisymm fun b ⟨⟨a₁, ha₁, h₁⟩, ⟨a₂, ha₂, h₂⟩⟩ ↦ have : a₂ = a₁ := h _ ha₂ _ ha₁ (by simp []) ⟨a₁, ⟨ha₁, this ▸ ha₂⟩, h₁⟩ #align set.image_inter_on Set.image_inter_on theorem image_inter {f : α → β} {s t : Set α} (H : Injective f) : f '' (s ∩ t) = f '' s ∩ f '' t := image_inter_on fun _ _ _ _ h => H h #align set.image_inter Set.image_inter theorem image_univ_of_surjective {ι : Type} {f : ι → β} (H : Surjective f) : f '' univ = univ := eq_univ_of_forall <\| by simpa [image] #align set.image_univ_of_surjective Set.image_univ_of_surjective @[simp] theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by ext simp [image, eq_comm] #align set.image_singleton Set.image_singleton @[simp] theorem Nonempty.image_const {s : Set α} (hs : s.Nonempty) (a : β) : (fun _ => a) '' s = {a} := ext fun _ => ⟨fun ⟨_, _, h⟩ => h ▸ mem_singleton _, fun h => (eq_of_mem_singleton h).symm ▸ hs.imp fun _ hy => ⟨hy, rfl⟩⟩ #align set.nonempty.image_const Set.Nonempty.image_const @[simp, mfld_simps] theorem image_eq_empty {α β} {f : α → β} {s : Set α} : f '' s = ∅ ↔ s = ∅ := by simp only [eq_empty_iff_forall_not_mem] exact ⟨fun H a ha => H _ ⟨_, ha, rfl⟩, fun H b ⟨_, ha, _⟩ => H _ ha⟩ #align set.image_eq_empty Set.image_eq_empty -- Porting note: `compl` is already defined in `Init.Set` theorem preimage_compl_eq_image_compl [BooleanAlgebra α] (S : Set α) : HasCompl.compl ⁻¹' S = HasCompl.compl '' S := Set.ext fun x => ⟨fun h => ⟨xᶜ, h, compl_compl x⟩, fun h => Exists.elim h fun _ hy => (compl_eq_comm.mp hy.2).symm.subst hy.1⟩ #align set.preimage_compl_eq_image_compl Set.preimage_compl_eq_image_compl theorem mem_compl_image [BooleanAlgebra α] (t : α) (S : Set α) : t ∈ HasCompl.compl '' S ↔ tᶜ ∈ S := by simp [← preimage_compl_eq_image_compl] #align set.mem_compl_image Set.mem_compl_image @[simp] theorem image_id_eq : image (id : α → α) = id := by ext; simp /-- A variant of `image_id` -/ @[simp]	Mathlib/Data/Set/Image.lean	371	373	theorem image_id' (s : Set α) : (fun x => x) '' s = s := by	ext simp
/- Copyright (c) 2022 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn -/ import Mathlib.Analysis.Calculus.ContDiff.Basic import Mathlib.Analysis.Calculus.ParametricIntegral import Mathlib.MeasureTheory.Constructions.Prod.Integral import Mathlib.MeasureTheory.Function.LocallyIntegrable import Mathlib.MeasureTheory.Group.Integral import Mathlib.MeasureTheory.Group.Prod import Mathlib.MeasureTheory.Integral.IntervalIntegral #align_import analysis.convolution from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95" /-! # Convolution of functions This file defines the convolution on two functions, i.e. `x ↦ ∫ f(t)g(x - t) ∂t`. In the general case, these functions can be vector-valued, and have an arbitrary (additive) group as domain. We use a continuous bilinear operation `L` on these function values as "multiplication". The domain must be equipped with a Haar measure `μ` (though many individual results have weaker conditions on `μ`). For many applications we can take `L = ContinuousLinearMap.lsmul ℝ ℝ` or `L = ContinuousLinearMap.mul ℝ ℝ`. We also define `ConvolutionExists` and `ConvolutionExistsAt` to state that the convolution is well-defined (everywhere or at a single point). These conditions are needed for pointwise computations (e.g. `ConvolutionExistsAt.distrib_add`), but are generally not strong enough for any local (or global) properties of the convolution. For this we need stronger assumptions on `f` and/or `g`, and generally if we impose stronger conditions on one of the functions, we can impose weaker conditions on the other. We have proven many of the properties of the convolution assuming one of these functions has compact support (in which case the other function only needs to be locally integrable). We still need to prove the properties for other pairs of conditions (e.g. both functions are rapidly decreasing) # Design Decisions We use a bilinear map `L` to "multiply" the two functions in the integrand. This generality has several advantages * This allows us to compute the total derivative of the convolution, in case the functions are multivariate. The total derivative is again a convolution, but where the codomains of the functions can be higher-dimensional. See `HasCompactSupport.hasFDerivAt_convolution_right`. * This allows us to use `@[to_additive]` everywhere (which would not be possible if we would use `mul`/`smul` in the integral, since `@[to_additive]` will incorrectly also try to additivize those definitions). * We need to support the case where at least one of the functions is vector-valued, but if we use `smul` to multiply the functions, that would be an asymmetric definition. # Main Definitions * `convolution f g L μ x = (f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ` is the convolution of `f` and `g` w.r.t. the continuous bilinear map `L` and measure `μ`. * `ConvolutionExistsAt f g x L μ` states that the convolution `(f ⋆[L, μ] g) x` is well-defined (i.e. the integral exists). * `ConvolutionExists f g L μ` states that the convolution `f ⋆[L, μ] g` is well-defined at each point. # Main Results * `HasCompactSupport.hasFDerivAt_convolution_right` and `HasCompactSupport.hasFDerivAt_convolution_left`: we can compute the total derivative of the convolution as a convolution with the total derivative of the right (left) function. * `HasCompactSupport.contDiff_convolution_right` and `HasCompactSupport.contDiff_convolution_left`: the convolution is `𝒞ⁿ` if one of the functions is `𝒞ⁿ` with compact support and the other function in locally integrable. Versions of these statements for functions depending on a parameter are also given. * `convolution_tendsto_right`: Given a sequence of nonnegative normalized functions whose support tends to a small neighborhood around `0`, the convolution tends to the right argument. This is specialized to bump functions in `ContDiffBump.convolution_tendsto_right`. # Notation The following notations are localized in the locale `convolution`: * `f ⋆[L, μ] g` for the convolution. Note: you have to use parentheses to apply the convolution to an argument: `(f ⋆[L, μ] g) x`. * `f ⋆[L] g := f ⋆[L, volume] g` * `f ⋆ g := f ⋆[lsmul ℝ ℝ] g` # To do * Existence and (uniform) continuity of the convolution if one of the maps is in `ℒ^p` and the other in `ℒ^q` with `1 / p + 1 / q = 1`. This might require a generalization of `MeasureTheory.Memℒp.smul` where `smul` is generalized to a continuous bilinear map. (see e.g. [Fremlin, Measure Theory (volume 2)][fremlin_vol2], 255K) * The convolution is an `AEStronglyMeasurable` function (see e.g. [Fremlin, Measure Theory (volume 2)][fremlin_vol2], 255I). * Prove properties about the convolution if both functions are rapidly decreasing. * Use `@[to_additive]` everywhere (this likely requires changes in `to_additive`) -/ open Set Function Filter MeasureTheory MeasureTheory.Measure TopologicalSpace open ContinuousLinearMap Metric Bornology open scoped Pointwise Topology NNReal Filter universe u𝕜 uG uE uE' uE'' uF uF' uF'' uP variable {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''} {F : Type uF} {F' : Type uF'} {F'' : Type uF''} {P : Type uP} variable [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup E''] [NormedAddCommGroup F] {f f' : G → E} {g g' : G → E'} {x x' : G} {y y' : E} namespace MeasureTheory section NontriviallyNormedField variable [NontriviallyNormedField 𝕜] variable [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 E''] [NormedSpace 𝕜 F] variable (L : E →L[𝕜] E' →L[𝕜] F) section NoMeasurability variable [AddGroup G] [TopologicalSpace G] theorem convolution_integrand_bound_right_of_le_of_subset {C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) : ‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * C) t := by -- Porting note: had to add `f := _` refine le_indicator (f := fun t ↦ ‖L (f t) (g (x - t))‖) (fun t _ => ?_) (fun t ht => ?_) t · apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl] · have : x - t ∉ support g := by refine mt (fun hxt => hu ?_) ht refine ⟨_, Set.neg_mem_neg.mpr (subset_closure hxt), _, hx, ?_⟩ simp only [neg_sub, sub_add_cancel] simp only [nmem_support.mp this, (L _).map_zero, norm_zero, le_rfl] #align convolution_integrand_bound_right_of_le_of_subset MeasureTheory.convolution_integrand_bound_right_of_le_of_subset theorem _root_.HasCompactSupport.convolution_integrand_bound_right_of_subset (hcg : HasCompactSupport g) (hg : Continuous g) {x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) : ‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t := by refine convolution_integrand_bound_right_of_le_of_subset _ (fun i => ?_) hx hu exact le_ciSup (hg.norm.bddAbove_range_of_hasCompactSupport hcg.norm) _ #align has_compact_support.convolution_integrand_bound_right_of_subset HasCompactSupport.convolution_integrand_bound_right_of_subset theorem _root_.HasCompactSupport.convolution_integrand_bound_right (hcg : HasCompactSupport g) (hg : Continuous g) {x t : G} {s : Set G} (hx : x ∈ s) : ‖L (f t) (g (x - t))‖ ≤ (-tsupport g + s).indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t := hcg.convolution_integrand_bound_right_of_subset L hg hx Subset.rfl #align has_compact_support.convolution_integrand_bound_right HasCompactSupport.convolution_integrand_bound_right theorem _root_.Continuous.convolution_integrand_fst [ContinuousSub G] (hg : Continuous g) (t : G) : Continuous fun x => L (f t) (g (x - t)) := L.continuous₂.comp₂ continuous_const <\| hg.comp <\| continuous_id.sub continuous_const #align continuous.convolution_integrand_fst Continuous.convolution_integrand_fst	Mathlib/Analysis/Convolution.lean	150	155	theorem _root_.HasCompactSupport.convolution_integrand_bound_left (hcf : HasCompactSupport f) (hf : Continuous f) {x t : G} {s : Set G} (hx : x ∈ s) : ‖L (f (x - t)) (g t)‖ ≤ (-tsupport f + s).indicator (fun t => (‖L‖ * ⨆ i, ‖f i‖) * ‖g t‖) t := by	convert hcf.convolution_integrand_bound_right L.flip hf hx using 1 simp_rw [L.opNorm_flip, mul_right_comm]
/- Copyright (c) 2020 Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bhavik Mehta, Jakob von Raumer -/ import Mathlib.Data.List.Chain import Mathlib.CategoryTheory.PUnit import Mathlib.CategoryTheory.Groupoid import Mathlib.CategoryTheory.Category.ULift #align_import category_theory.is_connected from "leanprover-community/mathlib"@"024a4231815538ac739f52d08dd20a55da0d6b23" /-! # Connected category Define a connected category as a _nonempty_ category for which every functor to a discrete category is isomorphic to the constant functor. NB. Some authors include the empty category as connected, we do not. We instead are interested in categories with exactly one 'connected component'. We give some equivalent definitions: - A nonempty category for which every functor to a discrete category is constant on objects. See `any_functor_const_on_obj` and `Connected.of_any_functor_const_on_obj`. - A nonempty category for which every function `F` for which the presence of a morphism `f : j₁ ⟶ j₂` implies `F j₁ = F j₂` must be constant everywhere. See `constant_of_preserves_morphisms` and `Connected.of_constant_of_preserves_morphisms`. - A nonempty category for which any subset of its elements containing the default and closed under morphisms is everything. See `induct_on_objects` and `Connected.of_induct`. - A nonempty category for which every object is related under the reflexive transitive closure of the relation "there is a morphism in some direction from `j₁` to `j₂`". See `connected_zigzag` and `zigzag_connected`. - A nonempty category for which for any two objects there is a sequence of morphisms (some reversed) from one to the other. See `exists_zigzag'` and `connected_of_zigzag`. We also prove the result that the functor given by `(X × -)` preserves any connected limit. That is, any limit of shape `J` where `J` is a connected category is preserved by the functor `(X × -)`. This appears in `CategoryTheory.Limits.Connected`. -/ universe v₁ v₂ u₁ u₂ noncomputable section open CategoryTheory.Category open Opposite namespace CategoryTheory /-- A possibly empty category for which every functor to a discrete category is constant. -/ class IsPreconnected (J : Type u₁) [Category.{v₁} J] : Prop where iso_constant : ∀ {α : Type u₁} (F : J ⥤ Discrete α) (j : J), Nonempty (F ≅ (Functor.const J).obj (F.obj j)) #align category_theory.is_preconnected CategoryTheory.IsPreconnected attribute [inherit_doc IsPreconnected] IsPreconnected.iso_constant /-- We define a connected category as a _nonempty_ category for which every functor to a discrete category is constant. NB. Some authors include the empty category as connected, we do not. We instead are interested in categories with exactly one 'connected component'. This allows us to show that the functor X ⨯ - preserves connected limits. See <https://stacks.math.columbia.edu/tag/002S> -/ class IsConnected (J : Type u₁) [Category.{v₁} J] extends IsPreconnected J : Prop where [is_nonempty : Nonempty J] #align category_theory.is_connected CategoryTheory.IsConnected attribute [instance 100] IsConnected.is_nonempty variable {J : Type u₁} [Category.{v₁} J] variable {K : Type u₂} [Category.{v₂} K] namespace IsPreconnected.IsoConstantAux /-- Implementation detail of `isoConstant`. -/ private def liftToDiscrete {α : Type u₂} (F : J ⥤ Discrete α) : J ⥤ Discrete J where obj j := have := Nonempty.intro j Discrete.mk (Function.invFun F.obj (F.obj j)) map {j _} f := have := Nonempty.intro j ⟨⟨congr_arg (Function.invFun F.obj) (Discrete.ext _ _ (Discrete.eq_of_hom (F.map f)))⟩⟩ /-- Implementation detail of `isoConstant`. -/ private def factorThroughDiscrete {α : Type u₂} (F : J ⥤ Discrete α) : liftToDiscrete F ⋙ Discrete.functor F.obj ≅ F := NatIso.ofComponents (fun j => eqToIso Function.apply_invFun_apply) (by aesop_cat) end IsPreconnected.IsoConstantAux /-- If `J` is connected, any functor `F : J ⥤ Discrete α` is isomorphic to the constant functor with value `F.obj j` (for any choice of `j`). -/ def isoConstant [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j : J) : F ≅ (Functor.const J).obj (F.obj j) := (IsPreconnected.IsoConstantAux.factorThroughDiscrete F).symm ≪≫ isoWhiskerRight (IsPreconnected.iso_constant _ j).some _ ≪≫ NatIso.ofComponents (fun j' => eqToIso Function.apply_invFun_apply) (by aesop_cat) #align category_theory.iso_constant CategoryTheory.isoConstant /-- If `J` is connected, any functor to a discrete category is constant on objects. The converse is given in `IsConnected.of_any_functor_const_on_obj`. -/ theorem any_functor_const_on_obj [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j j' : J) : F.obj j = F.obj j' := by ext; exact ((isoConstant F j').hom.app j).down.1 #align category_theory.any_functor_const_on_obj CategoryTheory.any_functor_const_on_obj /-- If any functor to a discrete category is constant on objects, J is connected. The converse of `any_functor_const_on_obj`. -/ theorem IsPreconnected.of_any_functor_const_on_obj (h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') : IsPreconnected J where iso_constant := fun F j' => ⟨NatIso.ofComponents fun j => eqToIso (h F j j')⟩ /-- If any functor to a discrete category is constant on objects, J is connected. The converse of `any_functor_const_on_obj`. -/ theorem IsConnected.of_any_functor_const_on_obj [Nonempty J] (h : ∀ {α : Type u₁} (F : J ⥤ Discrete α), ∀ j j' : J, F.obj j = F.obj j') : IsConnected J := { IsPreconnected.of_any_functor_const_on_obj h with } #align category_theory.is_connected.of_any_functor_const_on_obj CategoryTheory.IsConnected.of_any_functor_const_on_obj /-- If `J` is connected, then given any function `F` such that the presence of a morphism `j₁ ⟶ j₂` implies `F j₁ = F j₂`, we have that `F` is constant. This can be thought of as a local-to-global property. The converse is shown in `IsConnected.of_constant_of_preserves_morphisms` -/ theorem constant_of_preserves_morphisms [IsPreconnected J] {α : Type u₂} (F : J → α) (h : ∀ (j₁ j₂ : J) (_ : j₁ ⟶ j₂), F j₁ = F j₂) (j j' : J) : F j = F j' := by simpa using any_functor_const_on_obj { obj := Discrete.mk ∘ F map := fun f => eqToHom (by ext; exact h _ _ f) } j j' #align category_theory.constant_of_preserves_morphisms CategoryTheory.constant_of_preserves_morphisms /-- `J` is connected if: given any function `F : J → α` which is constant for any `j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant. This can be thought of as a local-to-global property. The converse of `constant_of_preserves_morphisms`. -/ theorem IsPreconnected.of_constant_of_preserves_morphisms (h : ∀ {α : Type u₁} (F : J → α), (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ j j' : J, F j = F j') : IsPreconnected J := IsPreconnected.of_any_functor_const_on_obj fun F => h F.obj fun f => by ext; exact Discrete.eq_of_hom (F.map f) /-- `J` is connected if: given any function `F : J → α` which is constant for any `j₁, j₂` for which there is a morphism `j₁ ⟶ j₂`, then `F` is constant. This can be thought of as a local-to-global property. The converse of `constant_of_preserves_morphisms`. -/ theorem IsConnected.of_constant_of_preserves_morphisms [Nonempty J] (h : ∀ {α : Type u₁} (F : J → α), (∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ j j' : J, F j = F j') : IsConnected J := { IsPreconnected.of_constant_of_preserves_morphisms h with } #align category_theory.is_connected.of_constant_of_preserves_morphisms CategoryTheory.IsConnected.of_constant_of_preserves_morphisms /-- An inductive-like property for the objects of a connected category. If the set `p` is nonempty, and `p` is closed under morphisms of `J`, then `p` contains all of `J`. The converse is given in `IsConnected.of_induct`. -/	Mathlib/CategoryTheory/IsConnected.lean	183	187	theorem induct_on_objects [IsPreconnected J] (p : Set J) {j₀ : J} (h0 : j₀ ∈ p) (h1 : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) (j : J) : j ∈ p := by	let aux (j₁ j₂ : J) (f : j₁ ⟶ j₂) := congrArg ULift.up <\| (h1 f).eq injection constant_of_preserves_morphisms (fun k => ULift.up.{u₁} (k ∈ p)) aux j j₀ with i rwa [i]
/- Copyright (c) 2020 Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bhavik Mehta -/ import Mathlib.CategoryTheory.Sites.IsSheafFor import Mathlib.CategoryTheory.Limits.Shapes.Types import Mathlib.Tactic.ApplyFun #align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" /-! # The equalizer diagram sheaf condition for a presieve In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a sheaf for a particular presieve. In this file we provide equivalent conditions in terms of equalizer diagrams. * In `Equalizer.Presieve.sheaf_condition`, the sheaf condition at a presieve is shown to be equivalent to that of https://stacks.math.columbia.edu/tag/00VM (and combined with `isSheaf_pretopology`, this shows the notions of `IsSheaf` are exactly equivalent.) * In `Equalizer.Sieve.equalizer_sheaf_condition`, the sheaf condition at a sieve is shown to be equivalent to that of Equation (3) p. 122 in Maclane-Moerdijk [MM92]. ## References * [MM92]: Sheaves in geometry and logic, Saunders MacLane, and Ieke Moerdijk: Chapter III, Section 4. * https://stacks.math.columbia.edu/tag/00VL (sheaves on a pretopology or site) -/ universe w v u namespace CategoryTheory open Opposite CategoryTheory Category Limits Sieve namespace Equalizer variable {C : Type u} [Category.{v} C] (P : Cᵒᵖ ⥤ Type max v u) {X : C} (R : Presieve X) (S : Sieve X) noncomputable section /-- The middle object of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram of <https://stacks.math.columbia.edu/tag/00VM>. -/ def FirstObj : Type max v u := ∏ᶜ fun f : ΣY, { f : Y ⟶ X // R f } => P.obj (op f.1) #align category_theory.equalizer.first_obj CategoryTheory.Equalizer.FirstObj variable {P R} -- Porting note (#10688): added to ease automation @[ext] lemma FirstObj.ext (z₁ z₂ : FirstObj P R) (h : ∀ (Y : C) (f : Y ⟶ X) (hf : R f), (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₁ = (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₂) : z₁ = z₂ := by apply Limits.Types.limit_ext rintro ⟨⟨Y, f, hf⟩⟩ exact h Y f hf variable (P R) /-- Show that `FirstObj` is isomorphic to `FamilyOfElements`. -/ @[simps] def firstObjEqFamily : FirstObj P R ≅ R.FamilyOfElements P where hom t Y f hf := Pi.π (fun f : ΣY, { f : Y ⟶ X // R f } => P.obj (op f.1)) ⟨_, _, hf⟩ t inv := Pi.lift fun f x => x _ f.2.2 #align category_theory.equalizer.first_obj_eq_family CategoryTheory.Equalizer.firstObjEqFamily instance : Inhabited (FirstObj P (⊥ : Presieve X)) := (firstObjEqFamily P _).toEquiv.inhabited -- Porting note: was not needed in mathlib instance : Inhabited (FirstObj P ((⊥ : Sieve X) : Presieve X)) := (inferInstance : Inhabited (FirstObj P (⊥ : Presieve X))) /-- The left morphism of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram of <https://stacks.math.columbia.edu/tag/00VM>. -/ def forkMap : P.obj (op X) ⟶ FirstObj P R := Pi.lift fun f => P.map f.2.1.op #align category_theory.equalizer.fork_map CategoryTheory.Equalizer.forkMap /-! This section establishes the equivalence between the sheaf condition of Equation (3) [MM92] and the definition of `IsSheafFor`. -/ namespace Sieve /-- The rightmost object of the fork diagram of Equation (3) [MM92], which contains the data used to check a family is compatible. -/ def SecondObj : Type max v u := ∏ᶜ fun f : Σ(Y Z : _) (_ : Z ⟶ Y), { f' : Y ⟶ X // S f' } => P.obj (op f.2.1) #align category_theory.equalizer.sieve.second_obj CategoryTheory.Equalizer.Sieve.SecondObj variable {P S} -- Porting note (#10688): added to ease automation @[ext] lemma SecondObj.ext (z₁ z₂ : SecondObj P S) (h : ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X) (hf : S.arrows f), (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₁ = (Pi.π _ ⟨Y, Z, g, f, hf⟩ : SecondObj P S ⟶ _) z₂) : z₁ = z₂ := by apply Limits.Types.limit_ext rintro ⟨⟨Y, Z, g, f, hf⟩⟩ apply h variable (P S) /-- The map `p` of Equations (3,4) [MM92]. -/ def firstMap : FirstObj P (S : Presieve X) ⟶ SecondObj P S := Pi.lift fun fg => Pi.π _ (⟨_, _, S.downward_closed fg.2.2.2.2 fg.2.2.1⟩ : ΣY, { f : Y ⟶ X // S f }) #align category_theory.equalizer.sieve.first_map CategoryTheory.Equalizer.Sieve.firstMap instance : Inhabited (SecondObj P (⊥ : Sieve X)) := ⟨firstMap _ _ default⟩ /-- The map `a` of Equations (3,4) [MM92]. -/ def secondMap : FirstObj P (S : Presieve X) ⟶ SecondObj P S := Pi.lift fun fg => Pi.π _ ⟨_, fg.2.2.2⟩ ≫ P.map fg.2.2.1.op #align category_theory.equalizer.sieve.second_map CategoryTheory.Equalizer.Sieve.secondMap theorem w : forkMap P (S : Presieve X) ≫ firstMap P S = forkMap P S ≫ secondMap P S := by ext simp [firstMap, secondMap, forkMap] #align category_theory.equalizer.sieve.w CategoryTheory.Equalizer.Sieve.w /-- The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap` map it to the same point. -/ theorem compatible_iff (x : FirstObj P S) : ((firstObjEqFamily P S).hom x).Compatible ↔ firstMap P S x = secondMap P S x := by rw [Presieve.compatible_iff_sieveCompatible] constructor · intro t apply SecondObj.ext intros Y Z g f hf simpa [firstMap, secondMap] using t _ g hf · intro t Y Z f g hf rw [Types.limit_ext_iff'] at t simpa [firstMap, secondMap] using t ⟨⟨Y, Z, g, f, hf⟩⟩ #align category_theory.equalizer.sieve.compatible_iff CategoryTheory.Equalizer.Sieve.compatible_iff /-- `P` is a sheaf for `S`, iff the fork given by `w` is an equalizer. -/ theorem equalizer_sheaf_condition : Presieve.IsSheafFor P (S : Presieve X) ↔ Nonempty (IsLimit (Fork.ofι _ (w P S))) := by rw [Types.type_equalizer_iff_unique, ← Equiv.forall_congr_left (firstObjEqFamily P (S : Presieve X)).toEquiv.symm] simp_rw [← compatible_iff] simp only [inv_hom_id_apply, Iso.toEquiv_symm_fun] apply forall₂_congr intro x _ apply exists_unique_congr intro t rw [← Iso.toEquiv_symm_fun] rw [Equiv.eq_symm_apply] constructor · intro q funext Y f hf simpa [firstObjEqFamily, forkMap] using q _ _ · intro q Y f hf rw [← q] simp [firstObjEqFamily, forkMap] #align category_theory.equalizer.sieve.equalizer_sheaf_condition CategoryTheory.Equalizer.Sieve.equalizer_sheaf_condition end Sieve /-! This section establishes the equivalence between the sheaf condition of https://stacks.math.columbia.edu/tag/00VM and the definition of `isSheafFor`. -/ namespace Presieve variable [R.hasPullbacks] /-- The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag/00VM, which contains the data used to check a family of elements for a presieve is compatible. -/ @[simp] def SecondObj : Type max v u := ∏ᶜ fun fg : (ΣY, { f : Y ⟶ X // R f }) × ΣZ, { g : Z ⟶ X // R g } => haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2 P.obj (op (pullback fg.1.2.1 fg.2.2.1)) #align category_theory.equalizer.presieve.second_obj CategoryTheory.Equalizer.Presieve.SecondObj /-- The map `pr₀` of <https://stacks.math.columbia.edu/tag/00VL>. -/ def firstMap : FirstObj P R ⟶ SecondObj P R := Pi.lift fun fg => haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2 Pi.π _ _ ≫ P.map pullback.fst.op #align category_theory.equalizer.presieve.first_map CategoryTheory.Equalizer.Presieve.firstMap instance [HasPullbacks C] : Inhabited (SecondObj P (⊥ : Presieve X)) := ⟨firstMap _ _ default⟩ /-- The map `pr₁` of <https://stacks.math.columbia.edu/tag/00VL>. -/ def secondMap : FirstObj P R ⟶ SecondObj P R := Pi.lift fun fg => haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2 Pi.π _ _ ≫ P.map pullback.snd.op #align category_theory.equalizer.presieve.second_map CategoryTheory.Equalizer.Presieve.secondMap theorem w : forkMap P R ≫ firstMap P R = forkMap P R ≫ secondMap P R := by dsimp ext fg simp only [firstMap, secondMap, forkMap] simp only [limit.lift_π, limit.lift_π_assoc, assoc, Fan.mk_π_app] haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2 rw [← P.map_comp, ← op_comp, pullback.condition] simp #align category_theory.equalizer.presieve.w CategoryTheory.Equalizer.Presieve.w /-- The family of elements given by `x : FirstObj P S` is compatible iff `firstMap` and `secondMap` map it to the same point. -/ theorem compatible_iff (x : FirstObj P R) : ((firstObjEqFamily P R).hom x).Compatible ↔ firstMap P R x = secondMap P R x := by rw [Presieve.pullbackCompatible_iff] constructor · intro t apply Limits.Types.limit_ext rintro ⟨⟨Y, f, hf⟩, Z, g, hg⟩ simpa [firstMap, secondMap] using t hf hg · intro t Y Z f g hf hg rw [Types.limit_ext_iff'] at t simpa [firstMap, secondMap] using t ⟨⟨⟨Y, f, hf⟩, Z, g, hg⟩⟩ #align category_theory.equalizer.presieve.compatible_iff CategoryTheory.Equalizer.Presieve.compatible_iff /-- `P` is a sheaf for `R`, iff the fork given by `w` is an equalizer. See <https://stacks.math.columbia.edu/tag/00VM>. -/ theorem sheaf_condition : R.IsSheafFor P ↔ Nonempty (IsLimit (Fork.ofι _ (w P R))) := by rw [Types.type_equalizer_iff_unique] erw [← Equiv.forall_congr_left (firstObjEqFamily P R).toEquiv.symm] simp_rw [← compatible_iff, ← Iso.toEquiv_fun, Equiv.apply_symm_apply] apply forall₂_congr intro x _ apply exists_unique_congr intro t rw [Equiv.eq_symm_apply] constructor · intro q funext Y f hf simpa [forkMap] using q _ _ · intro q Y f hf rw [← q] simp [forkMap] #align category_theory.equalizer.presieve.sheaf_condition CategoryTheory.Equalizer.Presieve.sheaf_condition namespace Arrows variable (P : Cᵒᵖ ⥤ Type w) {X : C} (R : Presieve X) (S : Sieve X) open Presieve variable {B : C} {I : Type} (X : I → C) (π : (i : I) → X i ⟶ B) [(Presieve.ofArrows X π).hasPullbacks] -- TODO: allow `I : Type w` /-- The middle object of the fork diagram of <https://stacks.math.columbia.edu/tag/00VM>. The difference between this and `Equalizer.FirstObj P (ofArrows X π)` arrises if the family of arrows `π` contains duplicates. The `Presieve.ofArrows` doesn't see those. -/ def FirstObj : Type w := ∏ᶜ (fun i ↦ P.obj (op (X i))) @[ext] lemma FirstObj.ext (z₁ z₂ : FirstObj P X) (h : ∀ i, (Pi.π _ i : FirstObj P X ⟶ _) z₁ = (Pi.π _ i : FirstObj P X ⟶ _) z₂) : z₁ = z₂ := by apply Limits.Types.limit_ext rintro ⟨i⟩ exact h i /-- The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag/00VM. The difference between this and `Equalizer.Presieve.SecondObj P (ofArrows X π)` arrises if the family of arrows `π` contains duplicates. The `Presieve.ofArrows` doesn't see those. -/ def SecondObj : Type w := ∏ᶜ (fun (ij : I × I) ↦ P.obj (op (pullback (π ij.1) (π ij.2)))) @[ext] lemma SecondObj.ext (z₁ z₂ : SecondObj P X π) (h : ∀ ij, (Pi.π _ ij : SecondObj P X π ⟶ _) z₁ = (Pi.π _ ij : SecondObj P X π ⟶ _) z₂) : z₁ = z₂ := by apply Limits.Types.limit_ext rintro ⟨i⟩ exact h i /-- The left morphism of the fork diagram. -/ def forkMap : P.obj (op B) ⟶ FirstObj P X := Pi.lift (fun i ↦ P.map (π i).op) /-- The first of the two parallel morphisms of the fork diagram, induced by the first projection in each pullback. -/ def firstMap : FirstObj P X ⟶ SecondObj P X π := Pi.lift fun _ => Pi.π _ _ ≫ P.map pullback.fst.op /-- The second of the two parallel morphisms of the fork diagram, induced by the second projection in each pullback. -/ def secondMap : FirstObj P X ⟶ SecondObj P X π := Pi.lift fun _ => Pi.π _ _ ≫ P.map pullback.snd.op	Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean	320	323	theorem w : forkMap P X π ≫ firstMap P X π = forkMap P X π ≫ secondMap P X π := by	ext x ij simp only [firstMap, secondMap, forkMap, types_comp_apply, Types.pi_lift_π_apply, ← FunctorToTypes.map_comp_apply, ← op_comp, pullback.condition]
/- Copyright (c) 2018 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon -/ import Mathlib.Data.Fin.Fin2 import Mathlib.Logic.Function.Basic import Mathlib.Tactic.Common #align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4" /-! # Tuples of types, and their categorical structure. ## Features * `TypeVec n` - n-tuples of types * `α ⟹ β` - n-tuples of maps * `f ⊚ g` - composition Also, support functions for operating with n-tuples of types, such as: * `append1 α β` - append type `β` to n-tuple `α` to obtain an (n+1)-tuple * `drop α` - drops the last element of an (n+1)-tuple * `last α` - returns the last element of an (n+1)-tuple * `appendFun f g` - appends a function g to an n-tuple of functions * `dropFun f` - drops the last function from an n+1-tuple * `lastFun f` - returns the last function of a tuple. Since e.g. `append1 α.drop α.last` is propositionally equal to `α` but not definitionally equal to it, we need support functions and lemmas to mediate between constructions. -/ universe u v w /-- n-tuples of types, as a category -/ @[pp_with_univ] def TypeVec (n : ℕ) := Fin2 n → Type* #align typevec TypeVec instance {n} : Inhabited (TypeVec.{u} n) := ⟨fun _ => PUnit⟩ namespace TypeVec variable {n : ℕ} /-- arrow in the category of `TypeVec` -/ def Arrow (α β : TypeVec n) := ∀ i : Fin2 n, α i → β i #align typevec.arrow TypeVec.Arrow @[inherit_doc] scoped[MvFunctor] infixl:40 " ⟹ " => TypeVec.Arrow open MvFunctor /-- Extensionality for arrows -/ @[ext] theorem Arrow.ext {α β : TypeVec n} (f g : α ⟹ β) : (∀ i, f i = g i) → f = g := by intro h; funext i; apply h instance Arrow.inhabited (α β : TypeVec n) [∀ i, Inhabited (β i)] : Inhabited (α ⟹ β) := ⟨fun _ _ => default⟩ #align typevec.arrow.inhabited TypeVec.Arrow.inhabited /-- identity of arrow composition -/ def id {α : TypeVec n} : α ⟹ α := fun _ x => x #align typevec.id TypeVec.id /-- arrow composition in the category of `TypeVec` -/ def comp {α β γ : TypeVec n} (g : β ⟹ γ) (f : α ⟹ β) : α ⟹ γ := fun i x => g i (f i x) #align typevec.comp TypeVec.comp @[inherit_doc] scoped[MvFunctor] infixr:80 " ⊚ " => TypeVec.comp -- type as \oo @[simp] theorem id_comp {α β : TypeVec n} (f : α ⟹ β) : id ⊚ f = f := rfl #align typevec.id_comp TypeVec.id_comp @[simp] theorem comp_id {α β : TypeVec n} (f : α ⟹ β) : f ⊚ id = f := rfl #align typevec.comp_id TypeVec.comp_id theorem comp_assoc {α β γ δ : TypeVec n} (h : γ ⟹ δ) (g : β ⟹ γ) (f : α ⟹ β) : (h ⊚ g) ⊚ f = h ⊚ g ⊚ f := rfl #align typevec.comp_assoc TypeVec.comp_assoc /-- Support for extending a `TypeVec` by one element. -/ def append1 (α : TypeVec n) (β : Type) : TypeVec (n + 1) \| Fin2.fs i => α i \| Fin2.fz => β #align typevec.append1 TypeVec.append1 @[inherit_doc] infixl:67 " ::: " => append1 /-- retain only a `n-length` prefix of the argument -/ def drop (α : TypeVec.{u} (n + 1)) : TypeVec n := fun i => α i.fs #align typevec.drop TypeVec.drop /-- take the last value of a `(n+1)-length` vector -/ def last (α : TypeVec.{u} (n + 1)) : Type _ := α Fin2.fz #align typevec.last TypeVec.last instance last.inhabited (α : TypeVec (n + 1)) [Inhabited (α Fin2.fz)] : Inhabited (last α) := ⟨show α Fin2.fz from default⟩ #align typevec.last.inhabited TypeVec.last.inhabited theorem drop_append1 {α : TypeVec n} {β : Type} {i : Fin2 n} : drop (append1 α β) i = α i := rfl #align typevec.drop_append1 TypeVec.drop_append1 theorem drop_append1' {α : TypeVec n} {β : Type} : drop (append1 α β) = α := funext fun _ => drop_append1 #align typevec.drop_append1' TypeVec.drop_append1' theorem last_append1 {α : TypeVec n} {β : Type} : last (append1 α β) = β := rfl #align typevec.last_append1 TypeVec.last_append1 @[simp] theorem append1_drop_last (α : TypeVec (n + 1)) : append1 (drop α) (last α) = α := funext fun i => by cases i <;> rfl #align typevec.append1_drop_last TypeVec.append1_drop_last /-- cases on `(n+1)-length` vectors -/ @[elab_as_elim] def append1Cases {C : TypeVec (n + 1) → Sort u} (H : ∀ α β, C (append1 α β)) (γ) : C γ := by rw [← @append1_drop_last _ γ]; apply H #align typevec.append1_cases TypeVec.append1Cases @[simp] theorem append1_cases_append1 {C : TypeVec (n + 1) → Sort u} (H : ∀ α β, C (append1 α β)) (α β) : @append1Cases _ C H (append1 α β) = H α β := rfl #align typevec.append1_cases_append1 TypeVec.append1_cases_append1 /-- append an arrow and a function for arbitrary source and target type vectors -/ def splitFun {α α' : TypeVec (n + 1)} (f : drop α ⟹ drop α') (g : last α → last α') : α ⟹ α' \| Fin2.fs i => f i \| Fin2.fz => g #align typevec.split_fun TypeVec.splitFun /-- append an arrow and a function as well as their respective source and target types / typevecs -/ def appendFun {α α' : TypeVec n} {β β' : Type} (f : α ⟹ α') (g : β → β') : append1 α β ⟹ append1 α' β' := splitFun f g #align typevec.append_fun TypeVec.appendFun @[inherit_doc] infixl:0 " ::: " => appendFun /-- split off the prefix of an arrow -/ def dropFun {α β : TypeVec (n + 1)} (f : α ⟹ β) : drop α ⟹ drop β := fun i => f i.fs #align typevec.drop_fun TypeVec.dropFun /-- split off the last function of an arrow -/ def lastFun {α β : TypeVec (n + 1)} (f : α ⟹ β) : last α → last β := f Fin2.fz #align typevec.last_fun TypeVec.lastFun -- Porting note: Lean wasn't able to infer the motive in term mode /-- arrow in the category of `0-length` vectors -/ def nilFun {α : TypeVec 0} {β : TypeVec 0} : α ⟹ β := fun i => by apply Fin2.elim0 i #align typevec.nil_fun TypeVec.nilFun theorem eq_of_drop_last_eq {α β : TypeVec (n + 1)} {f g : α ⟹ β} (h₀ : dropFun f = dropFun g) (h₁ : lastFun f = lastFun g) : f = g := by -- Porting note: FIXME: congr_fun h₀ <;> ext1 ⟨⟩ <;> apply_assumption refine funext (fun x => ?_) cases x · apply h₁ · apply congr_fun h₀ #align typevec.eq_of_drop_last_eq TypeVec.eq_of_drop_last_eq @[simp] theorem dropFun_splitFun {α α' : TypeVec (n + 1)} (f : drop α ⟹ drop α') (g : last α → last α') : dropFun (splitFun f g) = f := rfl #align typevec.drop_fun_split_fun TypeVec.dropFun_splitFun /-- turn an equality into an arrow -/ def Arrow.mp {α β : TypeVec n} (h : α = β) : α ⟹ β \| _ => Eq.mp (congr_fun h _) #align typevec.arrow.mp TypeVec.Arrow.mp /-- turn an equality into an arrow, with reverse direction -/ def Arrow.mpr {α β : TypeVec n} (h : α = β) : β ⟹ α \| _ => Eq.mpr (congr_fun h _) #align typevec.arrow.mpr TypeVec.Arrow.mpr /-- decompose a vector into its prefix appended with its last element -/ def toAppend1DropLast {α : TypeVec (n + 1)} : α ⟹ (drop α ::: last α) := Arrow.mpr (append1_drop_last _) #align typevec.to_append1_drop_last TypeVec.toAppend1DropLast /-- stitch two bits of a vector back together -/ def fromAppend1DropLast {α : TypeVec (n + 1)} : (drop α ::: last α) ⟹ α := Arrow.mp (append1_drop_last _) #align typevec.from_append1_drop_last TypeVec.fromAppend1DropLast @[simp] theorem lastFun_splitFun {α α' : TypeVec (n + 1)} (f : drop α ⟹ drop α') (g : last α → last α') : lastFun (splitFun f g) = g := rfl #align typevec.last_fun_split_fun TypeVec.lastFun_splitFun @[simp] theorem dropFun_appendFun {α α' : TypeVec n} {β β' : Type} (f : α ⟹ α') (g : β → β') : dropFun (f ::: g) = f := rfl #align typevec.drop_fun_append_fun TypeVec.dropFun_appendFun @[simp] theorem lastFun_appendFun {α α' : TypeVec n} {β β' : Type} (f : α ⟹ α') (g : β → β') : lastFun (f ::: g) = g := rfl #align typevec.last_fun_append_fun TypeVec.lastFun_appendFun theorem split_dropFun_lastFun {α α' : TypeVec (n + 1)} (f : α ⟹ α') : splitFun (dropFun f) (lastFun f) = f := eq_of_drop_last_eq rfl rfl #align typevec.split_drop_fun_last_fun TypeVec.split_dropFun_lastFun theorem splitFun_inj {α α' : TypeVec (n + 1)} {f f' : drop α ⟹ drop α'} {g g' : last α → last α'} (H : splitFun f g = splitFun f' g') : f = f' ∧ g = g' := by rw [← dropFun_splitFun f g, H, ← lastFun_splitFun f g, H]; simp #align typevec.split_fun_inj TypeVec.splitFun_inj theorem appendFun_inj {α α' : TypeVec n} {β β' : Type} {f f' : α ⟹ α'} {g g' : β → β'} : (f ::: g : (α ::: β) ⟹ _) = (f' ::: g' : (α ::: β) ⟹ _) → f = f' ∧ g = g' := splitFun_inj #align typevec.append_fun_inj TypeVec.appendFun_inj theorem splitFun_comp {α₀ α₁ α₂ : TypeVec (n + 1)} (f₀ : drop α₀ ⟹ drop α₁) (f₁ : drop α₁ ⟹ drop α₂) (g₀ : last α₀ → last α₁) (g₁ : last α₁ → last α₂) : splitFun (f₁ ⊚ f₀) (g₁ ∘ g₀) = splitFun f₁ g₁ ⊚ splitFun f₀ g₀ := eq_of_drop_last_eq rfl rfl #align typevec.split_fun_comp TypeVec.splitFun_comp theorem appendFun_comp_splitFun {α γ : TypeVec n} {β δ : Type} {ε : TypeVec (n + 1)} (f₀ : drop ε ⟹ α) (f₁ : α ⟹ γ) (g₀ : last ε → β) (g₁ : β → δ) : appendFun f₁ g₁ ⊚ splitFun f₀ g₀ = splitFun (α' := γ.append1 δ) (f₁ ⊚ f₀) (g₁ ∘ g₀) := (splitFun_comp _ _ _ _).symm #align typevec.append_fun_comp_split_fun TypeVec.appendFun_comp_splitFun theorem appendFun_comp {α₀ α₁ α₂ : TypeVec n} {β₀ β₁ β₂ : Type} (f₀ : α₀ ⟹ α₁) (f₁ : α₁ ⟹ α₂) (g₀ : β₀ → β₁) (g₁ : β₁ → β₂) : (f₁ ⊚ f₀ ::: g₁ ∘ g₀) = (f₁ ::: g₁) ⊚ (f₀ ::: g₀) := eq_of_drop_last_eq rfl rfl #align typevec.append_fun_comp TypeVec.appendFun_comp theorem appendFun_comp' {α₀ α₁ α₂ : TypeVec n} {β₀ β₁ β₂ : Type} (f₀ : α₀ ⟹ α₁) (f₁ : α₁ ⟹ α₂) (g₀ : β₀ → β₁) (g₁ : β₁ → β₂) : (f₁ ::: g₁) ⊚ (f₀ ::: g₀) = (f₁ ⊚ f₀ ::: g₁ ∘ g₀) := eq_of_drop_last_eq rfl rfl #align typevec.append_fun_comp' TypeVec.appendFun_comp' theorem nilFun_comp {α₀ : TypeVec 0} (f₀ : α₀ ⟹ Fin2.elim0) : nilFun ⊚ f₀ = f₀ := funext fun x => by apply Fin2.elim0 x -- Porting note: `by apply` is necessary? #align typevec.nil_fun_comp TypeVec.nilFun_comp theorem appendFun_comp_id {α : TypeVec n} {β₀ β₁ β₂ : Type u} (g₀ : β₀ → β₁) (g₁ : β₁ → β₂) : (@id _ α ::: g₁ ∘ g₀) = (id ::: g₁) ⊚ (id ::: g₀) := eq_of_drop_last_eq rfl rfl #align typevec.append_fun_comp_id TypeVec.appendFun_comp_id @[simp] theorem dropFun_comp {α₀ α₁ α₂ : TypeVec (n + 1)} (f₀ : α₀ ⟹ α₁) (f₁ : α₁ ⟹ α₂) : dropFun (f₁ ⊚ f₀) = dropFun f₁ ⊚ dropFun f₀ := rfl #align typevec.drop_fun_comp TypeVec.dropFun_comp @[simp] theorem lastFun_comp {α₀ α₁ α₂ : TypeVec (n + 1)} (f₀ : α₀ ⟹ α₁) (f₁ : α₁ ⟹ α₂) : lastFun (f₁ ⊚ f₀) = lastFun f₁ ∘ lastFun f₀ := rfl #align typevec.last_fun_comp TypeVec.lastFun_comp theorem appendFun_aux {α α' : TypeVec n} {β β' : Type} (f : (α ::: β) ⟹ (α' ::: β')) : (dropFun f ::: lastFun f) = f := eq_of_drop_last_eq rfl rfl #align typevec.append_fun_aux TypeVec.appendFun_aux theorem appendFun_id_id {α : TypeVec n} {β : Type} : (@TypeVec.id n α ::: @_root_.id β) = TypeVec.id := eq_of_drop_last_eq rfl rfl #align typevec.append_fun_id_id TypeVec.appendFun_id_id instance subsingleton0 : Subsingleton (TypeVec 0) := ⟨fun a b => funext fun a => by apply Fin2.elim0 a⟩ -- Porting note: `by apply` necessary? #align typevec.subsingleton0 TypeVec.subsingleton0 -- Porting note: `simp` attribute `TypeVec` moved to file `Tactic/Attr/Register.lean` /-- cases distinction for 0-length type vector -/ protected def casesNil {β : TypeVec 0 → Sort} (f : β Fin2.elim0) : ∀ v, β v := fun v => cast (by congr; funext i; cases i) f #align typevec.cases_nil TypeVec.casesNil /-- cases distinction for (n+1)-length type vector -/ protected def casesCons (n : ℕ) {β : TypeVec (n + 1) → Sort} (f : ∀ (t) (v : TypeVec n), β (v ::: t)) : ∀ v, β v := fun v : TypeVec (n + 1) => cast (by simp) (f v.last v.drop) #align typevec.cases_cons TypeVec.casesCons protected theorem casesNil_append1 {β : TypeVec 0 → Sort} (f : β Fin2.elim0) : TypeVec.casesNil f Fin2.elim0 = f := rfl #align typevec.cases_nil_append1 TypeVec.casesNil_append1 protected theorem casesCons_append1 (n : ℕ) {β : TypeVec (n + 1) → Sort} (f : ∀ (t) (v : TypeVec n), β (v ::: t)) (v : TypeVec n) (α) : TypeVec.casesCons n f (v ::: α) = f α v := rfl #align typevec.cases_cons_append1 TypeVec.casesCons_append1 /-- cases distinction for an arrow in the category of 0-length type vectors -/ def typevecCasesNil₃ {β : ∀ v v' : TypeVec 0, v ⟹ v' → Sort} (f : β Fin2.elim0 Fin2.elim0 nilFun) : ∀ v v' fs, β v v' fs := fun v v' fs => by refine cast ?_ f have eq₁ : v = Fin2.elim0 := by funext i; contradiction have eq₂ : v' = Fin2.elim0 := by funext i; contradiction have eq₃ : fs = nilFun := by funext i; contradiction cases eq₁; cases eq₂; cases eq₃; rfl #align typevec.typevec_cases_nil₃ TypeVec.typevecCasesNil₃ /-- cases distinction for an arrow in the category of (n+1)-length type vectors -/ def typevecCasesCons₃ (n : ℕ) {β : ∀ v v' : TypeVec (n + 1), v ⟹ v' → Sort} (F : ∀ (t t') (f : t → t') (v v' : TypeVec n) (fs : v ⟹ v'), β (v ::: t) (v' ::: t') (fs ::: f)) : ∀ v v' fs, β v v' fs := by intro v v' rw [← append1_drop_last v, ← append1_drop_last v'] intro fs rw [← split_dropFun_lastFun fs] apply F #align typevec.typevec_cases_cons₃ TypeVec.typevecCasesCons₃ /-- specialized cases distinction for an arrow in the category of 0-length type vectors -/ def typevecCasesNil₂ {β : Fin2.elim0 ⟹ Fin2.elim0 → Sort} (f : β nilFun) : ∀ f, β f := by intro g suffices g = nilFun by rwa [this] ext ⟨⟩ #align typevec.typevec_cases_nil₂ TypeVec.typevecCasesNil₂ /-- specialized cases distinction for an arrow in the category of (n+1)-length type vectors -/ def typevecCasesCons₂ (n : ℕ) (t t' : Type) (v v' : TypeVec n) {β : (v ::: t) ⟹ (v' ::: t') → Sort} (F : ∀ (f : t → t') (fs : v ⟹ v'), β (fs ::: f)) : ∀ fs, β fs := by intro fs rw [← split_dropFun_lastFun fs] apply F #align typevec.typevec_cases_cons₂ TypeVec.typevecCasesCons₂ theorem typevecCasesNil₂_appendFun {β : Fin2.elim0 ⟹ Fin2.elim0 → Sort} (f : β nilFun) : typevecCasesNil₂ f nilFun = f := rfl #align typevec.typevec_cases_nil₂_append_fun TypeVec.typevecCasesNil₂_appendFun theorem typevecCasesCons₂_appendFun (n : ℕ) (t t' : Type) (v v' : TypeVec n) {β : (v ::: t) ⟹ (v' ::: t') → Sort} (F : ∀ (f : t → t') (fs : v ⟹ v'), β (fs ::: f)) (f fs) : typevecCasesCons₂ n t t' v v' F (fs ::: f) = F f fs := rfl #align typevec.typevec_cases_cons₂_append_fun TypeVec.typevecCasesCons₂_appendFun -- for lifting predicates and relations /-- `PredLast α p x` predicates `p` of the last element of `x : α.append1 β`. -/ def PredLast (α : TypeVec n) {β : Type} (p : β → Prop) : ∀ ⦃i⦄, (α.append1 β) i → Prop \| Fin2.fs _ => fun _ => True \| Fin2.fz => p #align typevec.pred_last TypeVec.PredLast /-- `RelLast α r x y` says that `p` the last elements of `x y : α.append1 β` are related by `r` and all the other elements are equal. -/ def RelLast (α : TypeVec n) {β γ : Type u} (r : β → γ → Prop) : ∀ ⦃i⦄, (α.append1 β) i → (α.append1 γ) i → Prop \| Fin2.fs _ => Eq \| Fin2.fz => r #align typevec.rel_last TypeVec.RelLast section Liftp' open Nat /-- `repeat n t` is a `n-length` type vector that contains `n` occurrences of `t` -/ def «repeat» : ∀ (n : ℕ), Sort _ → TypeVec n \| 0, _ => Fin2.elim0 \| Nat.succ i, t => append1 («repeat» i t) t #align typevec.repeat TypeVec.repeat /-- `prod α β` is the pointwise product of the components of `α` and `β` -/ def prod : ∀ {n}, TypeVec.{u} n → TypeVec.{u} n → TypeVec n \| 0, _, _ => Fin2.elim0 \| n + 1, α, β => (@prod n (drop α) (drop β)) ::: (last α × last β) #align typevec.prod TypeVec.prod @[inherit_doc] scoped[MvFunctor] infixl:45 " ⊗ " => TypeVec.prod /- porting note: the order of universes in `const` is reversed w.r.t. mathlib3 -/ /-- `const x α` is an arrow that ignores its source and constructs a `TypeVec` that contains nothing but `x` -/ protected def const {β} (x : β) : ∀ {n} (α : TypeVec n), α ⟹ «repeat» _ β \| succ _, α, Fin2.fs _ => TypeVec.const x (drop α) _ \| succ _, _, Fin2.fz => fun _ => x #align typevec.const TypeVec.const open Function (uncurry) /-- vector of equality on a product of vectors -/ def repeatEq : ∀ {n} (α : TypeVec n), (α ⊗ α) ⟹ «repeat» _ Prop \| 0, _ => nilFun \| succ _, α => repeatEq (drop α) ::: uncurry Eq #align typevec.repeat_eq TypeVec.repeatEq theorem const_append1 {β γ} (x : γ) {n} (α : TypeVec n) : TypeVec.const x (α ::: β) = appendFun (TypeVec.const x α) fun _ => x := by ext i : 1; cases i <;> rfl #align typevec.const_append1 TypeVec.const_append1 theorem eq_nilFun {α β : TypeVec 0} (f : α ⟹ β) : f = nilFun := by ext x; cases x #align typevec.eq_nil_fun TypeVec.eq_nilFun theorem id_eq_nilFun {α : TypeVec 0} : @id _ α = nilFun := by ext x; cases x #align typevec.id_eq_nil_fun TypeVec.id_eq_nilFun theorem const_nil {β} (x : β) (α : TypeVec 0) : TypeVec.const x α = nilFun := by ext i : 1; cases i #align typevec.const_nil TypeVec.const_nil @[typevec] theorem repeat_eq_append1 {β} {n} (α : TypeVec n) : repeatEq (α ::: β) = splitFun (α := (α ⊗ α) ::: _ ) (α' := («repeat» n Prop) ::: _) (repeatEq α) (uncurry Eq) := by induction n <;> rfl #align typevec.repeat_eq_append1 TypeVec.repeat_eq_append1 @[typevec] theorem repeat_eq_nil (α : TypeVec 0) : repeatEq α = nilFun := by ext i; cases i #align typevec.repeat_eq_nil TypeVec.repeat_eq_nil /-- predicate on a type vector to constrain only the last object -/ def PredLast' (α : TypeVec n) {β : Type} (p : β → Prop) : (α ::: β) ⟹ «repeat» (n + 1) Prop := splitFun (TypeVec.const True α) p #align typevec.pred_last' TypeVec.PredLast' /-- predicate on the product of two type vectors to constrain only their last object -/ def RelLast' (α : TypeVec n) {β : Type} (p : β → β → Prop) : (α ::: β) ⊗ (α ::: β) ⟹ «repeat» (n + 1) Prop := splitFun (repeatEq α) (uncurry p) #align typevec.rel_last' TypeVec.RelLast' /-- given `F : TypeVec.{u} (n+1) → Type u`, `curry F : Type u → TypeVec.{u} → Type u`, i.e. its first argument can be fed in separately from the rest of the vector of arguments -/ def Curry (F : TypeVec.{u} (n + 1) → Type) (α : Type u) (β : TypeVec.{u} n) : Type _ := F (β ::: α) #align typevec.curry TypeVec.Curry instance Curry.inhabited (F : TypeVec.{u} (n + 1) → Type) (α : Type u) (β : TypeVec.{u} n) [I : Inhabited (F <\| (β ::: α))] : Inhabited (Curry F α β) := I #align typevec.curry.inhabited TypeVec.Curry.inhabited /-- arrow to remove one element of a `repeat` vector -/ def dropRepeat (α : Type*) : ∀ {n}, drop («repeat» (succ n) α) ⟹ «repeat» n α \| succ _, Fin2.fs i => dropRepeat α i \| succ _, Fin2.fz => fun (a : α) => a #align typevec.drop_repeat TypeVec.dropRepeat /-- projection for a repeat vector -/ def ofRepeat {α : Sort _} : ∀ {n i}, «repeat» n α i → α \| _, Fin2.fz => fun (a : α) => a \| _, Fin2.fs i => @ofRepeat _ _ i #align typevec.of_repeat TypeVec.ofRepeat theorem const_iff_true {α : TypeVec n} {i x p} : ofRepeat (TypeVec.const p α i x) ↔ p := by induction i with \| fz => rfl \| fs _ ih => erw [TypeVec.const, @ih (drop α) x] #align typevec.const_iff_true TypeVec.const_iff_true section variable {α β γ : TypeVec.{u} n} variable (p : α ⟹ «repeat» n Prop) (r : α ⊗ α ⟹ «repeat» n Prop) /-- left projection of a `prod` vector -/ def prod.fst : ∀ {n} {α β : TypeVec.{u} n}, α ⊗ β ⟹ α \| succ _, α, β, Fin2.fs i => @prod.fst _ (drop α) (drop β) i \| succ _, _, _, Fin2.fz => Prod.fst #align typevec.prod.fst TypeVec.prod.fst /-- right projection of a `prod` vector -/ def prod.snd : ∀ {n} {α β : TypeVec.{u} n}, α ⊗ β ⟹ β \| succ _, α, β, Fin2.fs i => @prod.snd _ (drop α) (drop β) i \| succ _, _, _, Fin2.fz => Prod.snd #align typevec.prod.snd TypeVec.prod.snd /-- introduce a product where both components are the same -/ def prod.diag : ∀ {n} {α : TypeVec.{u} n}, α ⟹ α ⊗ α \| succ _, α, Fin2.fs _, x => @prod.diag _ (drop α) _ x \| succ _, _, Fin2.fz, x => (x, x) #align typevec.prod.diag TypeVec.prod.diag /-- constructor for `prod` -/ def prod.mk : ∀ {n} {α β : TypeVec.{u} n} (i : Fin2 n), α i → β i → (α ⊗ β) i \| succ _, α, β, Fin2.fs i => mk (α := fun i => α i.fs) (β := fun i => β i.fs) i \| succ _, _, _, Fin2.fz => Prod.mk #align typevec.prod.mk TypeVec.prod.mk end @[simp] theorem prod_fst_mk {α β : TypeVec n} (i : Fin2 n) (a : α i) (b : β i) : TypeVec.prod.fst i (prod.mk i a b) = a := by induction' i with _ _ _ i_ih · simp_all only [prod.fst, prod.mk] apply i_ih #align typevec.prod_fst_mk TypeVec.prod_fst_mk @[simp] theorem prod_snd_mk {α β : TypeVec n} (i : Fin2 n) (a : α i) (b : β i) : TypeVec.prod.snd i (prod.mk i a b) = b := by induction' i with _ _ _ i_ih · simp_all [prod.snd, prod.mk] apply i_ih #align typevec.prod_snd_mk TypeVec.prod_snd_mk /-- `prod` is functorial -/ protected def prod.map : ∀ {n} {α α' β β' : TypeVec.{u} n}, α ⟹ β → α' ⟹ β' → α ⊗ α' ⟹ β ⊗ β' \| succ _, α, α', β, β', x, y, Fin2.fs _, a => @prod.map _ (drop α) (drop α') (drop β) (drop β') (dropFun x) (dropFun y) _ a \| succ _, _, _, _, _, x, y, Fin2.fz, a => (x _ a.1, y _ a.2) #align typevec.prod.map TypeVec.prod.map @[inherit_doc] scoped[MvFunctor] infixl:45 " ⊗' " => TypeVec.prod.map theorem fst_prod_mk {α α' β β' : TypeVec n} (f : α ⟹ β) (g : α' ⟹ β') : TypeVec.prod.fst ⊚ (f ⊗' g) = f ⊚ TypeVec.prod.fst := by funext i; induction i with \| fz => rfl \| fs _ i_ih => apply i_ih #align typevec.fst_prod_mk TypeVec.fst_prod_mk theorem snd_prod_mk {α α' β β' : TypeVec n} (f : α ⟹ β) (g : α' ⟹ β') : TypeVec.prod.snd ⊚ (f ⊗' g) = g ⊚ TypeVec.prod.snd := by funext i; induction i with \| fz => rfl \| fs _ i_ih => apply i_ih #align typevec.snd_prod_mk TypeVec.snd_prod_mk theorem fst_diag {α : TypeVec n} : TypeVec.prod.fst ⊚ (prod.diag : α ⟹ _) = id := by funext i; induction i with \| fz => rfl \| fs _ i_ih => apply i_ih #align typevec.fst_diag TypeVec.fst_diag theorem snd_diag {α : TypeVec n} : TypeVec.prod.snd ⊚ (prod.diag : α ⟹ _) = id := by funext i; induction i with \| fz => rfl \| fs _ i_ih => apply i_ih #align typevec.snd_diag TypeVec.snd_diag	Mathlib/Data/TypeVec.lean	582	586	theorem repeatEq_iff_eq {α : TypeVec n} {i x y} : ofRepeat (repeatEq α i (prod.mk _ x y)) ↔ x = y := by	induction' i with _ _ _ i_ih · rfl erw [repeatEq, i_ih]
/- Copyright (c) 2019 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne -/ import Mathlib.Analysis.Convex.Jensen import Mathlib.Analysis.Convex.Mul import Mathlib.Analysis.Convex.SpecificFunctions.Basic import Mathlib.Analysis.SpecialFunctions.Pow.NNReal #align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f" /-! # Mean value inequalities In this file we prove several mean inequalities for finite sums. Versions for integrals of some of these inequalities are available in `MeasureTheory.MeanInequalities`. ## Main theorems: generalized mean inequality The inequality says that for two non-negative vectors $w$ and $z$ with $\sum_{i\in s} w_i=1$ and $p ≤ q$ we have $$ \sqrt[p]{\sum_{i\in s} w_i z_i^p} ≤ \sqrt[q]{\sum_{i\in s} w_i z_i^q}. $$ Currently we only prove this inequality for $p=1$. As in the rest of `Mathlib`, we provide different theorems for natural exponents (`pow_arith_mean_le_arith_mean_pow`), integer exponents (`zpow_arith_mean_le_arith_mean_zpow`), and real exponents (`rpow_arith_mean_le_arith_mean_rpow` and `arith_mean_le_rpow_mean`). In the first two cases we prove $$ \left(\sum_{i\in s} w_i z_i\right)^n ≤ \sum_{i\in s} w_i z_i^n $$ in order to avoid using real exponents. For real exponents we prove both this and standard versions. ## TODO - each inequality `A ≤ B` should come with a theorem `A = B ↔ _`; one of the ways to prove them is to define `StrictConvexOn` functions. - generalized mean inequality with any `p ≤ q`, including negative numbers; - prove that the power mean tends to the geometric mean as the exponent tends to zero. -/ universe u v open Finset open scoped Classical open NNReal ENNReal noncomputable section variable {ι : Type u} (s : Finset ι) namespace Real theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (n : ℕ) : (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n := (convexOn_pow n).map_sum_le hw hw' hz #align real.pow_arith_mean_le_arith_mean_pow Real.pow_arith_mean_le_arith_mean_pow theorem pow_arith_mean_le_arith_mean_pow_of_even (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) {n : ℕ} (hn : Even n) : (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n := hn.convexOn_pow.map_sum_le hw hw' fun _ _ => Set.mem_univ _ #align real.pow_arith_mean_le_arith_mean_pow_of_even Real.pow_arith_mean_le_arith_mean_pow_of_even /-- Specific case of Jensen's inequality for sums of powers -/ theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) : (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by rcases s.eq_empty_or_nonempty with (rfl \| hs) · simp_rw [Finset.sum_empty, zero_pow n.succ_ne_zero, zero_div]; rfl · have hs0 : 0 < (s.card : ℝ) := Nat.cast_pos.2 hs.card_pos suffices (∑ x ∈ s, f x / s.card) ^ (n + 1) ≤ ∑ x ∈ s, f x ^ (n + 1) / s.card by rwa [← Finset.sum_div, ← Finset.sum_div, div_pow, pow_succ (s.card : ℝ), ← div_div, div_le_iff hs0, div_mul, div_self hs0.ne', div_one] at this have := @ConvexOn.map_sum_le ℝ ℝ ℝ ι _ _ _ _ _ _ (Set.Ici 0) (fun x => x ^ (n + 1)) s (fun _ => 1 / s.card) ((↑) ∘ f) (convexOn_pow (n + 1)) ?_ ?_ fun i hi => Set.mem_Ici.2 (hf i hi) · simpa only [inv_mul_eq_div, one_div, Algebra.id.smul_eq_mul] using this · simp only [one_div, inv_nonneg, Nat.cast_nonneg, imp_true_iff] · simpa only [one_div, Finset.sum_const, nsmul_eq_mul] using mul_inv_cancel hs0.ne' #align real.pow_sum_div_card_le_sum_pow Real.pow_sum_div_card_le_sum_pow theorem zpow_arith_mean_le_arith_mean_zpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ) : (∑ i ∈ s, w i * z i) ^ m ≤ ∑ i ∈ s, w i * z i ^ m := (convexOn_zpow m).map_sum_le hw hw' hz #align real.zpow_arith_mean_le_arith_mean_zpow Real.zpow_arith_mean_le_arith_mean_zpow theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p := (convexOn_rpow hp).map_sum_le hw hw' hz #align real.rpow_arith_mean_le_arith_mean_rpow Real.rpow_arith_mean_le_arith_mean_rpow	Mathlib/Analysis/MeanInequalitiesPow.lean	101	110	theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) : ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) := by	have : 0 < p := by positivity rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one] · exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp all_goals apply_rules [sum_nonneg, rpow_nonneg] intro i hi apply_rules [mul_nonneg, rpow_nonneg, hw i hi, hz i hi]
/- Copyright (c) 2018 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard, Amelia Livingston, Yury Kudryashov -/ import Mathlib.Algebra.Group.Hom.Defs import Mathlib.Algebra.Group.Subsemigroup.Basic import Mathlib.Algebra.Group.Units #align_import group_theory.submonoid.basic from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3" /-! # Submonoids: definition and `CompleteLattice` structure This file defines bundled multiplicative and additive submonoids. We also define a `CompleteLattice` structure on `Submonoid`s, define the closure of a set as the minimal submonoid that includes this set, and prove a few results about extending properties from a dense set (i.e. a set with `closure s = ⊤`) to the whole monoid, see `Submonoid.dense_induction` and `MonoidHom.ofClosureEqTopLeft`/`MonoidHom.ofClosureEqTopRight`. ## Main definitions * `Submonoid M`: the type of bundled submonoids of a monoid `M`; the underlying set is given in the `carrier` field of the structure, and should be accessed through coercion as in `(S : Set M)`. * `AddSubmonoid M` : the type of bundled submonoids of an additive monoid `M`. For each of the following definitions in the `Submonoid` namespace, there is a corresponding definition in the `AddSubmonoid` namespace. * `Submonoid.copy` : copy of a submonoid with `carrier` replaced by a set that is equal but possibly not definitionally equal to the carrier of the original `Submonoid`. * `Submonoid.closure` : monoid closure of a set, i.e., the least submonoid that includes the set. * `Submonoid.gi` : `closure : Set M → Submonoid M` and coercion `coe : Submonoid M → Set M` form a `GaloisInsertion`; * `MonoidHom.eqLocus`: the submonoid of elements `x : M` such that `f x = g x`; * `MonoidHom.ofClosureEqTopRight`: if a map `f : M → N` between two monoids satisfies `f 1 = 1` and `f (x * y) = f x * f y` for `y` from some dense set `s`, then `f` is a monoid homomorphism. E.g., if `f : ℕ → M` satisfies `f 0 = 0` and `f (x + 1) = f x + f 1`, then `f` is an additive monoid homomorphism. ## Implementation notes Submonoid inclusion is denoted `≤` rather than `⊆`, although `∈` is defined as membership of a submonoid's underlying set. Note that `Submonoid M` does not actually require `Monoid M`, instead requiring only the weaker `MulOneClass M`. This file is designed to have very few dependencies. In particular, it should not use natural numbers. `Submonoid` is implemented by extending `Subsemigroup` requiring `one_mem'`. ## Tags submonoid, submonoids -/ assert_not_exists MonoidWithZero -- Only needed for notation -- Only needed for notation variable {M : Type} {N : Type} variable {A : Type} section NonAssoc variable [MulOneClass M] {s : Set M} variable [AddZeroClass A] {t : Set A} /-- `OneMemClass S M` says `S` is a type of subsets `s ≤ M`, such that `1 ∈ s` for all `s`. -/ class OneMemClass (S : Type) (M : Type) [One M] [SetLike S M] : Prop where /-- By definition, if we have `OneMemClass S M`, we have `1 ∈ s` for all `s : S`. -/ one_mem : ∀ s : S, (1 : M) ∈ s #align one_mem_class OneMemClass export OneMemClass (one_mem) /-- `ZeroMemClass S M` says `S` is a type of subsets `s ≤ M`, such that `0 ∈ s` for all `s`. -/ class ZeroMemClass (S : Type) (M : Type) [Zero M] [SetLike S M] : Prop where /-- By definition, if we have `ZeroMemClass S M`, we have `0 ∈ s` for all `s : S`. -/ zero_mem : ∀ s : S, (0 : M) ∈ s #align zero_mem_class ZeroMemClass export ZeroMemClass (zero_mem) attribute [to_additive] OneMemClass attribute [aesop safe apply (rule_sets := [SetLike])] one_mem zero_mem section /-- A submonoid of a monoid `M` is a subset containing 1 and closed under multiplication. -/ structure Submonoid (M : Type) [MulOneClass M] extends Subsemigroup M where /-- A submonoid contains `1`. -/ one_mem' : (1 : M) ∈ carrier #align submonoid Submonoid end /-- A submonoid of a monoid `M` can be considered as a subsemigroup of that monoid. -/ add_decl_doc Submonoid.toSubsemigroup #align submonoid.to_subsemigroup Submonoid.toSubsemigroup /-- `SubmonoidClass S M` says `S` is a type of subsets `s ≤ M` that contain `1` and are closed under `()` -/ class SubmonoidClass (S : Type) (M : Type) [MulOneClass M] [SetLike S M] extends MulMemClass S M, OneMemClass S M : Prop #align submonoid_class SubmonoidClass section /-- An additive submonoid of an additive monoid `M` is a subset containing 0 and closed under addition. -/ structure AddSubmonoid (M : Type) [AddZeroClass M] extends AddSubsemigroup M where /-- An additive submonoid contains `0`. -/ zero_mem' : (0 : M) ∈ carrier #align add_submonoid AddSubmonoid end /-- An additive submonoid of an additive monoid `M` can be considered as an additive subsemigroup of that additive monoid. -/ add_decl_doc AddSubmonoid.toAddSubsemigroup #align add_submonoid.to_add_subsemigroup AddSubmonoid.toAddSubsemigroup /-- `AddSubmonoidClass S M` says `S` is a type of subsets `s ≤ M` that contain `0` and are closed under `(+)` -/ class AddSubmonoidClass (S : Type) (M : Type) [AddZeroClass M] [SetLike S M] extends AddMemClass S M, ZeroMemClass S M : Prop #align add_submonoid_class AddSubmonoidClass attribute [to_additive] Submonoid SubmonoidClass @[to_additive (attr := aesop safe apply (rule_sets := [SetLike]))] theorem pow_mem {M A} [Monoid M] [SetLike A M] [SubmonoidClass A M] {S : A} {x : M} (hx : x ∈ S) : ∀ n : ℕ, x ^ n ∈ S \| 0 => by rw [pow_zero] exact OneMemClass.one_mem S \| n + 1 => by rw [pow_succ] exact mul_mem (pow_mem hx n) hx #align pow_mem pow_mem #align nsmul_mem nsmul_mem namespace Submonoid @[to_additive] instance : SetLike (Submonoid M) M where coe s := s.carrier coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective' h @[to_additive] instance : SubmonoidClass (Submonoid M) M where one_mem := Submonoid.one_mem' mul_mem {s} := s.mul_mem' initialize_simps_projections Submonoid (carrier → coe) initialize_simps_projections AddSubmonoid (carrier → coe) @[to_additive (attr := simp)] theorem mem_toSubsemigroup {s : Submonoid M} {x : M} : x ∈ s.toSubsemigroup ↔ x ∈ s := Iff.rfl -- Porting note: `x ∈ s.carrier` is now syntactically `x ∈ s.toSubsemigroup.carrier`, -- which `simp` already simplifies to `x ∈ s.toSubsemigroup`. So we remove the `@[simp]` attribute -- here, and instead add the simp lemma `mem_toSubsemigroup` to allow `simp` to do this exact -- simplification transitively. @[to_additive] theorem mem_carrier {s : Submonoid M} {x : M} : x ∈ s.carrier ↔ x ∈ s := Iff.rfl #align submonoid.mem_carrier Submonoid.mem_carrier #align add_submonoid.mem_carrier AddSubmonoid.mem_carrier @[to_additive (attr := simp)] theorem mem_mk {s : Subsemigroup M} {x : M} (h_one) : x ∈ mk s h_one ↔ x ∈ s := Iff.rfl #align submonoid.mem_mk Submonoid.mem_mk #align add_submonoid.mem_mk AddSubmonoid.mem_mk @[to_additive (attr := simp)] theorem coe_set_mk {s : Subsemigroup M} (h_one) : (mk s h_one : Set M) = s := rfl #align submonoid.coe_set_mk Submonoid.coe_set_mk #align add_submonoid.coe_set_mk AddSubmonoid.coe_set_mk @[to_additive (attr := simp)] theorem mk_le_mk {s t : Subsemigroup M} (h_one) (h_one') : mk s h_one ≤ mk t h_one' ↔ s ≤ t := Iff.rfl #align submonoid.mk_le_mk Submonoid.mk_le_mk #align add_submonoid.mk_le_mk AddSubmonoid.mk_le_mk /-- Two submonoids are equal if they have the same elements. -/ @[to_additive (attr := ext) "Two `AddSubmonoid`s are equal if they have the same elements."] theorem ext {S T : Submonoid M} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := SetLike.ext h #align submonoid.ext Submonoid.ext #align add_submonoid.ext AddSubmonoid.ext /-- Copy a submonoid replacing `carrier` with a set that is equal to it. -/ @[to_additive "Copy an additive submonoid replacing `carrier` with a set that is equal to it."] protected def copy (S : Submonoid M) (s : Set M) (hs : s = S) : Submonoid M where carrier := s one_mem' := show 1 ∈ s from hs.symm ▸ S.one_mem' mul_mem' := hs.symm ▸ S.mul_mem' #align submonoid.copy Submonoid.copy #align add_submonoid.copy AddSubmonoid.copy variable {S : Submonoid M} @[to_additive (attr := simp)] theorem coe_copy {s : Set M} (hs : s = S) : (S.copy s hs : Set M) = s := rfl #align submonoid.coe_copy Submonoid.coe_copy #align add_submonoid.coe_copy AddSubmonoid.coe_copy @[to_additive] theorem copy_eq {s : Set M} (hs : s = S) : S.copy s hs = S := SetLike.coe_injective hs #align submonoid.copy_eq Submonoid.copy_eq #align add_submonoid.copy_eq AddSubmonoid.copy_eq variable (S) /-- A submonoid contains the monoid's 1. -/ @[to_additive "An `AddSubmonoid` contains the monoid's 0."] protected theorem one_mem : (1 : M) ∈ S := one_mem S #align submonoid.one_mem Submonoid.one_mem #align add_submonoid.zero_mem AddSubmonoid.zero_mem /-- A submonoid is closed under multiplication. -/ @[to_additive "An `AddSubmonoid` is closed under addition."] protected theorem mul_mem {x y : M} : x ∈ S → y ∈ S → x * y ∈ S := mul_mem #align submonoid.mul_mem Submonoid.mul_mem #align add_submonoid.add_mem AddSubmonoid.add_mem /-- The submonoid `M` of the monoid `M`. -/ @[to_additive "The additive submonoid `M` of the `AddMonoid M`."] instance : Top (Submonoid M) := ⟨{ carrier := Set.univ one_mem' := Set.mem_univ 1 mul_mem' := fun _ _ => Set.mem_univ _ }⟩ /-- The trivial submonoid `{1}` of a monoid `M`. -/ @[to_additive "The trivial `AddSubmonoid` `{0}` of an `AddMonoid` `M`."] instance : Bot (Submonoid M) := ⟨{ carrier := {1} one_mem' := Set.mem_singleton 1 mul_mem' := fun ha hb => by simp only [Set.mem_singleton_iff] at * rw [ha, hb, mul_one] }⟩ @[to_additive] instance : Inhabited (Submonoid M) := ⟨⊥⟩ @[to_additive (attr := simp)] theorem mem_bot {x : M} : x ∈ (⊥ : Submonoid M) ↔ x = 1 := Set.mem_singleton_iff #align submonoid.mem_bot Submonoid.mem_bot #align add_submonoid.mem_bot AddSubmonoid.mem_bot @[to_additive (attr := simp)] theorem mem_top (x : M) : x ∈ (⊤ : Submonoid M) := Set.mem_univ x #align submonoid.mem_top Submonoid.mem_top #align add_submonoid.mem_top AddSubmonoid.mem_top @[to_additive (attr := simp)] theorem coe_top : ((⊤ : Submonoid M) : Set M) = Set.univ := rfl #align submonoid.coe_top Submonoid.coe_top #align add_submonoid.coe_top AddSubmonoid.coe_top @[to_additive (attr := simp)] theorem coe_bot : ((⊥ : Submonoid M) : Set M) = {1} := rfl #align submonoid.coe_bot Submonoid.coe_bot #align add_submonoid.coe_bot AddSubmonoid.coe_bot /-- The inf of two submonoids is their intersection. -/ @[to_additive "The inf of two `AddSubmonoid`s is their intersection."] instance : Inf (Submonoid M) := ⟨fun S₁ S₂ => { carrier := S₁ ∩ S₂ one_mem' := ⟨S₁.one_mem, S₂.one_mem⟩ mul_mem' := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ => ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩ @[to_additive (attr := simp)] theorem coe_inf (p p' : Submonoid M) : ((p ⊓ p' : Submonoid M) : Set M) = (p : Set M) ∩ p' := rfl #align submonoid.coe_inf Submonoid.coe_inf #align add_submonoid.coe_inf AddSubmonoid.coe_inf @[to_additive (attr := simp)] theorem mem_inf {p p' : Submonoid M} {x : M} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' := Iff.rfl #align submonoid.mem_inf Submonoid.mem_inf #align add_submonoid.mem_inf AddSubmonoid.mem_inf @[to_additive] instance : InfSet (Submonoid M) := ⟨fun s => { carrier := ⋂ t ∈ s, ↑t one_mem' := Set.mem_biInter fun i _ => i.one_mem mul_mem' := fun hx hy => Set.mem_biInter fun i h => i.mul_mem (by apply Set.mem_iInter₂.1 hx i h) (by apply Set.mem_iInter₂.1 hy i h) }⟩ @[to_additive (attr := simp, norm_cast)] theorem coe_sInf (S : Set (Submonoid M)) : ((sInf S : Submonoid M) : Set M) = ⋂ s ∈ S, ↑s := rfl #align submonoid.coe_Inf Submonoid.coe_sInf #align add_submonoid.coe_Inf AddSubmonoid.coe_sInf @[to_additive] theorem mem_sInf {S : Set (Submonoid M)} {x : M} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := Set.mem_iInter₂ #align submonoid.mem_Inf Submonoid.mem_sInf #align add_submonoid.mem_Inf AddSubmonoid.mem_sInf @[to_additive] theorem mem_iInf {ι : Sort} {S : ι → Submonoid M} {x : M} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_mem_range] #align submonoid.mem_infi Submonoid.mem_iInf #align add_submonoid.mem_infi AddSubmonoid.mem_iInf @[to_additive (attr := simp, norm_cast)] theorem coe_iInf {ι : Sort} {S : ι → Submonoid M} : (↑(⨅ i, S i) : Set M) = ⋂ i, S i := by simp only [iInf, coe_sInf, Set.biInter_range] #align submonoid.coe_infi Submonoid.coe_iInf #align add_submonoid.coe_infi AddSubmonoid.coe_iInf /-- Submonoids of a monoid form a complete lattice. -/ @[to_additive "The `AddSubmonoid`s of an `AddMonoid` form a complete lattice."] instance : CompleteLattice (Submonoid M) := { (completeLatticeOfInf (Submonoid M)) fun _ => IsGLB.of_image (f := (SetLike.coe : Submonoid M → Set M)) (@fun S T => show (S : Set M) ≤ T ↔ S ≤ T from SetLike.coe_subset_coe) isGLB_biInf with le := (· ≤ ·) lt := (· < ·) bot := ⊥ bot_le := fun S _ hx => (mem_bot.1 hx).symm ▸ S.one_mem top := ⊤ le_top := fun _ x _ => mem_top x inf := (· ⊓ ·) sInf := InfSet.sInf le_inf := fun _ _ _ ha hb _ hx => ⟨ha hx, hb hx⟩ inf_le_left := fun _ _ _ => And.left inf_le_right := fun _ _ _ => And.right } @[to_additive (attr := simp)] theorem subsingleton_iff : Subsingleton (Submonoid M) ↔ Subsingleton M := ⟨fun h => ⟨fun x y => have : ∀ i : M, i = 1 := fun i => mem_bot.mp <\| Subsingleton.elim (⊤ : Submonoid M) ⊥ ▸ mem_top i (this x).trans (this y).symm⟩, fun h => ⟨fun x y => Submonoid.ext fun i => Subsingleton.elim 1 i ▸ by simp [Submonoid.one_mem]⟩⟩ #align submonoid.subsingleton_iff Submonoid.subsingleton_iff #align add_submonoid.subsingleton_iff AddSubmonoid.subsingleton_iff @[to_additive (attr := simp)] theorem nontrivial_iff : Nontrivial (Submonoid M) ↔ Nontrivial M := not_iff_not.mp ((not_nontrivial_iff_subsingleton.trans subsingleton_iff).trans not_nontrivial_iff_subsingleton.symm) #align submonoid.nontrivial_iff Submonoid.nontrivial_iff #align add_submonoid.nontrivial_iff AddSubmonoid.nontrivial_iff @[to_additive] instance [Subsingleton M] : Unique (Submonoid M) := ⟨⟨⊥⟩, fun a => @Subsingleton.elim _ (subsingleton_iff.mpr ‹_›) a _⟩ @[to_additive] instance [Nontrivial M] : Nontrivial (Submonoid M) := nontrivial_iff.mpr ‹_› /-- The `Submonoid` generated by a set. -/ @[to_additive "The `AddSubmonoid` generated by a set"] def closure (s : Set M) : Submonoid M := sInf { S \| s ⊆ S } #align submonoid.closure Submonoid.closure #align add_submonoid.closure AddSubmonoid.closure @[to_additive] theorem mem_closure {x : M} : x ∈ closure s ↔ ∀ S : Submonoid M, s ⊆ S → x ∈ S := mem_sInf #align submonoid.mem_closure Submonoid.mem_closure #align add_submonoid.mem_closure AddSubmonoid.mem_closure /-- The submonoid generated by a set includes the set. -/ @[to_additive (attr := simp, aesop safe 20 apply (rule_sets := [SetLike])) "The `AddSubmonoid` generated by a set includes the set."] theorem subset_closure : s ⊆ closure s := fun _ hx => mem_closure.2 fun _ hS => hS hx #align submonoid.subset_closure Submonoid.subset_closure #align add_submonoid.subset_closure AddSubmonoid.subset_closure @[to_additive] theorem not_mem_of_not_mem_closure {P : M} (hP : P ∉ closure s) : P ∉ s := fun h => hP (subset_closure h) #align submonoid.not_mem_of_not_mem_closure Submonoid.not_mem_of_not_mem_closure #align add_submonoid.not_mem_of_not_mem_closure AddSubmonoid.not_mem_of_not_mem_closure variable {S} open Set /-- A submonoid `S` includes `closure s` if and only if it includes `s`. -/ @[to_additive (attr := simp) "An additive submonoid `S` includes `closure s` if and only if it includes `s`"] theorem closure_le : closure s ≤ S ↔ s ⊆ S := ⟨Subset.trans subset_closure, fun h => sInf_le h⟩ #align submonoid.closure_le Submonoid.closure_le #align add_submonoid.closure_le AddSubmonoid.closure_le /-- Submonoid closure of a set is monotone in its argument: if `s ⊆ t`, then `closure s ≤ closure t`. -/ @[to_additive "Additive submonoid closure of a set is monotone in its argument: if `s ⊆ t`, then `closure s ≤ closure t`"] theorem closure_mono ⦃s t : Set M⦄ (h : s ⊆ t) : closure s ≤ closure t := closure_le.2 <\| Subset.trans h subset_closure #align submonoid.closure_mono Submonoid.closure_mono #align add_submonoid.closure_mono AddSubmonoid.closure_mono @[to_additive] theorem closure_eq_of_le (h₁ : s ⊆ S) (h₂ : S ≤ closure s) : closure s = S := le_antisymm (closure_le.2 h₁) h₂ #align submonoid.closure_eq_of_le Submonoid.closure_eq_of_le #align add_submonoid.closure_eq_of_le AddSubmonoid.closure_eq_of_le variable (S) /-- An induction principle for closure membership. If `p` holds for `1` and all elements of `s`, and is preserved under multiplication, then `p` holds for all elements of the closure of `s`. -/ @[to_additive (attr := elab_as_elim) "An induction principle for additive closure membership. If `p` holds for `0` and all elements of `s`, and is preserved under addition, then `p` holds for all elements of the additive closure of `s`."] theorem closure_induction {p : M → Prop} {x} (h : x ∈ closure s) (mem : ∀ x ∈ s, p x) (one : p 1) (mul : ∀ x y, p x → p y → p (x * y)) : p x := (@closure_le _ _ _ ⟨⟨p, mul _ _⟩, one⟩).2 mem h #align submonoid.closure_induction Submonoid.closure_induction #align add_submonoid.closure_induction AddSubmonoid.closure_induction /-- A dependent version of `Submonoid.closure_induction`. -/ @[to_additive (attr := elab_as_elim) "A dependent version of `AddSubmonoid.closure_induction`. "] theorem closure_induction' (s : Set M) {p : ∀ x, x ∈ closure s → Prop} (mem : ∀ (x) (h : x ∈ s), p x (subset_closure h)) (one : p 1 (one_mem _)) (mul : ∀ x hx y hy, p x hx → p y hy → p (x * y) (mul_mem hx hy)) {x} (hx : x ∈ closure s) : p x hx := by refine Exists.elim ?_ fun (hx : x ∈ closure s) (hc : p x hx) => hc exact closure_induction hx (fun x hx => ⟨_, mem x hx⟩) ⟨_, one⟩ fun x y ⟨hx', hx⟩ ⟨hy', hy⟩ => ⟨_, mul _ _ _ _ hx hy⟩ #align submonoid.closure_induction' Submonoid.closure_induction' #align add_submonoid.closure_induction' AddSubmonoid.closure_induction' /-- An induction principle for closure membership for predicates with two arguments. -/ @[to_additive (attr := elab_as_elim) "An induction principle for additive closure membership for predicates with two arguments."] theorem closure_induction₂ {p : M → M → Prop} {x} {y : M} (hx : x ∈ closure s) (hy : y ∈ closure s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H1_left : ∀ x, p 1 x) (H1_right : ∀ x, p x 1) (Hmul_left : ∀ x y z, p x z → p y z → p (x * y) z) (Hmul_right : ∀ x y z, p z x → p z y → p z (x * y)) : p x y := closure_induction hx (fun x xs => closure_induction hy (Hs x xs) (H1_right x) fun z _ h₁ h₂ => Hmul_right z _ _ h₁ h₂) (H1_left y) fun _ _ h₁ h₂ => Hmul_left _ _ _ h₁ h₂ #align submonoid.closure_induction₂ Submonoid.closure_induction₂ #align add_submonoid.closure_induction₂ AddSubmonoid.closure_induction₂ /-- If `s` is a dense set in a monoid `M`, `Submonoid.closure s = ⊤`, then in order to prove that some predicate `p` holds for all `x : M` it suffices to verify `p x` for `x ∈ s`, verify `p 1`, and verify that `p x` and `p y` imply `p (x * y)`. -/ @[to_additive (attr := elab_as_elim) "If `s` is a dense set in an additive monoid `M`, `AddSubmonoid.closure s = ⊤`, then in order to prove that some predicate `p` holds for all `x : M` it suffices to verify `p x` for `x ∈ s`, verify `p 0`, and verify that `p x` and `p y` imply `p (x + y)`."] theorem dense_induction {p : M → Prop} (x : M) {s : Set M} (hs : closure s = ⊤) (mem : ∀ x ∈ s, p x) (one : p 1) (mul : ∀ x y, p x → p y → p (x * y)) : p x := by have : ∀ x ∈ closure s, p x := fun x hx => closure_induction hx mem one mul simpa [hs] using this x #align submonoid.dense_induction Submonoid.dense_induction #align add_submonoid.dense_induction AddSubmonoid.dense_induction /-- The `Submonoid.closure` of a set is the union of `{1}` and its `Subsemigroup.closure`. -/ lemma closure_eq_one_union (s : Set M) : closure s = {(1 : M)} ∪ (Subsemigroup.closure s : Set M) := by apply le_antisymm · intro x hx induction hx using closure_induction' with \| mem x hx => exact Or.inr <\| Subsemigroup.subset_closure hx \| one => exact Or.inl <\| by simp \| mul x hx y hy hx hy => simp only [singleton_union, mem_insert_iff, SetLike.mem_coe] at hx hy obtain ⟨(rfl \| hx), (rfl \| hy)⟩ := And.intro hx hy all_goals simp_all exact Or.inr <\| mul_mem hx hy · rintro x (hx \| hx) · exact (show x = 1 by simpa using hx) ▸ one_mem (closure s) · exact Subsemigroup.closure_le.mpr subset_closure hx variable (M) /-- `closure` forms a Galois insertion with the coercion to set. -/ @[to_additive "`closure` forms a Galois insertion with the coercion to set."] protected def gi : GaloisInsertion (@closure M _) SetLike.coe where choice s _ := closure s gc _ _ := closure_le le_l_u _ := subset_closure choice_eq _ _ := rfl #align submonoid.gi Submonoid.gi #align add_submonoid.gi AddSubmonoid.gi variable {M} /-- Closure of a submonoid `S` equals `S`. -/ @[to_additive (attr := simp) "Additive closure of an additive submonoid `S` equals `S`"] theorem closure_eq : closure (S : Set M) = S := (Submonoid.gi M).l_u_eq S #align submonoid.closure_eq Submonoid.closure_eq #align add_submonoid.closure_eq AddSubmonoid.closure_eq @[to_additive (attr := simp)] theorem closure_empty : closure (∅ : Set M) = ⊥ := (Submonoid.gi M).gc.l_bot #align submonoid.closure_empty Submonoid.closure_empty #align add_submonoid.closure_empty AddSubmonoid.closure_empty @[to_additive (attr := simp)] theorem closure_univ : closure (univ : Set M) = ⊤ := @coe_top M _ ▸ closure_eq ⊤ #align submonoid.closure_univ Submonoid.closure_univ #align add_submonoid.closure_univ AddSubmonoid.closure_univ @[to_additive] theorem closure_union (s t : Set M) : closure (s ∪ t) = closure s ⊔ closure t := (Submonoid.gi M).gc.l_sup #align submonoid.closure_union Submonoid.closure_union #align add_submonoid.closure_union AddSubmonoid.closure_union @[to_additive] theorem sup_eq_closure (N N' : Submonoid M) : N ⊔ N' = closure ((N : Set M) ∪ (N' : Set M)) := by simp_rw [closure_union, closure_eq] @[to_additive] theorem closure_iUnion {ι} (s : ι → Set M) : closure (⋃ i, s i) = ⨆ i, closure (s i) := (Submonoid.gi M).gc.l_iSup #align submonoid.closure_Union Submonoid.closure_iUnion #align add_submonoid.closure_Union AddSubmonoid.closure_iUnion -- Porting note (#10618): `simp` can now prove this, so we remove the `@[simp]` attribute @[to_additive]	Mathlib/Algebra/Group/Submonoid/Basic.lean	561	562	theorem closure_singleton_le_iff_mem (m : M) (p : Submonoid M) : closure {m} ≤ p ↔ m ∈ p := by	rw [closure_le, singleton_subset_iff, SetLike.mem_coe]
/- Copyright (c) 2018 Michael Jendrusch. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer -/ import Mathlib.Tactic.CategoryTheory.Coherence import Mathlib.CategoryTheory.Monoidal.Free.Coherence #align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe" /-! # Lemmas which are consequences of monoidal coherence These lemmas are all proved `by coherence`. ## Future work Investigate whether these lemmas are really needed, or if they can be replaced by use of the `coherence` tactic. -/ open CategoryTheory Category Iso namespace CategoryTheory.MonoidalCategory variable {C : Type*} [Category C] [MonoidalCategory C] -- See Proposition 2.2.4 of <http://www-math.mit.edu/~etingof/egnobookfinal.pdf> @[reassoc] theorem leftUnitor_tensor'' (X Y : C) : (α_ (𝟙_ C) X Y).hom ≫ (λ_ (X ⊗ Y)).hom = (λ_ X).hom ⊗ 𝟙 Y := by coherence #align category_theory.monoidal_category.left_unitor_tensor' CategoryTheory.MonoidalCategory.leftUnitor_tensor'' @[reassoc] theorem leftUnitor_tensor' (X Y : C) : (λ_ (X ⊗ Y)).hom = (α_ (𝟙_ C) X Y).inv ≫ ((λ_ X).hom ⊗ 𝟙 Y) := by coherence #align category_theory.monoidal_category.left_unitor_tensor CategoryTheory.MonoidalCategory.leftUnitor_tensor' @[reassoc] theorem leftUnitor_tensor_inv' (X Y : C) : (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom := by coherence #align category_theory.monoidal_category.left_unitor_tensor_inv CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv' @[reassoc] theorem id_tensor_rightUnitor_inv (X Y : C) : 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ _).inv ≫ (α_ _ _ _).hom := by coherence #align category_theory.monoidal_category.id_tensor_right_unitor_inv CategoryTheory.MonoidalCategory.id_tensor_rightUnitor_inv @[reassoc] theorem leftUnitor_inv_tensor_id (X Y : C) : (λ_ X).inv ⊗ 𝟙 Y = (λ_ _).inv ≫ (α_ _ _ _).inv := by coherence #align category_theory.monoidal_category.left_unitor_inv_tensor_id CategoryTheory.MonoidalCategory.leftUnitor_inv_tensor_id @[reassoc]	Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean	57	60	theorem pentagon_inv_inv_hom (W X Y Z : C) : (α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ X) Y Z).hom = (𝟙 W ⊗ (α_ X Y Z).hom) ≫ (α_ W X (Y ⊗ Z)).inv := by	coherence
/- Copyright (c) 2022 Jireh Loreaux. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jireh Loreaux -/ import Mathlib.Algebra.Star.Subalgebra import Mathlib.Topology.Algebra.Algebra import Mathlib.Topology.Algebra.Star #align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd" /-! # Topological star (sub)algebras A topological star algebra over a topological semiring `R` is a topological semiring with a compatible continuous scalar multiplication by elements of `R` and a continuous star operation. We reuse typeclass `ContinuousSMul` for topological algebras. ## Results This is just a minimal stub for now! The topological closure of a star subalgebra is still a star subalgebra, which as a star algebra is a topological star algebra. -/ open scoped Classical open Set TopologicalSpace open scoped Classical namespace StarSubalgebra section TopologicalStarAlgebra variable {R A B : Type} [CommSemiring R] [StarRing R] variable [TopologicalSpace A] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] instance [TopologicalSemiring A] (s : StarSubalgebra R A) : TopologicalSemiring s := s.toSubalgebra.topologicalSemiring /-- The `StarSubalgebra.inclusion` of a star subalgebra is an `Embedding`. -/ theorem embedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) : Embedding (inclusion h) := { induced := Eq.symm induced_compose inj := Subtype.map_injective h Function.injective_id } #align star_subalgebra.embedding_inclusion StarSubalgebra.embedding_inclusion /-- The `StarSubalgebra.inclusion` of a closed star subalgebra is a `ClosedEmbedding`. -/ theorem closedEmbedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) (hS₁ : IsClosed (S₁ : Set A)) : ClosedEmbedding (inclusion h) := { embedding_inclusion h with isClosed_range := isClosed_induced_iff.2 ⟨S₁, hS₁, by convert (Set.range_subtype_map id _).symm · rw [Set.image_id]; rfl · intro _ h' apply h h' ⟩ } #align star_subalgebra.closed_embedding_inclusion StarSubalgebra.closedEmbedding_inclusion variable [TopologicalSemiring A] [ContinuousStar A] variable [TopologicalSpace B] [Semiring B] [Algebra R B] [StarRing B] /-- The closure of a star subalgebra in a topological star algebra as a star subalgebra. -/ def topologicalClosure (s : StarSubalgebra R A) : StarSubalgebra R A := { s.toSubalgebra.topologicalClosure with carrier := closure (s : Set A) star_mem' := fun ha => map_mem_closure continuous_star ha fun x => (star_mem : x ∈ s → star x ∈ s) } #align star_subalgebra.topological_closure StarSubalgebra.topologicalClosure theorem topologicalClosure_toSubalgebra_comm (s : StarSubalgebra R A) : s.topologicalClosure.toSubalgebra = s.toSubalgebra.topologicalClosure := SetLike.coe_injective rfl @[simp] theorem topologicalClosure_coe (s : StarSubalgebra R A) : (s.topologicalClosure : Set A) = closure (s : Set A) := rfl #align star_subalgebra.topological_closure_coe StarSubalgebra.topologicalClosure_coe theorem le_topologicalClosure (s : StarSubalgebra R A) : s ≤ s.topologicalClosure := subset_closure #align star_subalgebra.le_topological_closure StarSubalgebra.le_topologicalClosure theorem isClosed_topologicalClosure (s : StarSubalgebra R A) : IsClosed (s.topologicalClosure : Set A) := isClosed_closure #align star_subalgebra.is_closed_topological_closure StarSubalgebra.isClosed_topologicalClosure instance {A : Type} [UniformSpace A] [CompleteSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] {S : StarSubalgebra R A} : CompleteSpace S.topologicalClosure := isClosed_closure.completeSpace_coe theorem topologicalClosure_minimal {s t : StarSubalgebra R A} (h : s ≤ t) (ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t := closure_minimal h ht #align star_subalgebra.topological_closure_minimal StarSubalgebra.topologicalClosure_minimal theorem topologicalClosure_mono : Monotone (topologicalClosure : _ → StarSubalgebra R A) := fun _ S₂ h => topologicalClosure_minimal (h.trans <\| le_topologicalClosure S₂) (isClosed_topologicalClosure S₂) #align star_subalgebra.topological_closure_mono StarSubalgebra.topologicalClosure_mono theorem topologicalClosure_map_le [StarModule R B] [TopologicalSemiring B] [ContinuousStar B] (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : IsClosedMap φ) : (map φ s).topologicalClosure ≤ map φ s.topologicalClosure := hφ.closure_image_subset _ theorem map_topologicalClosure_le [StarModule R B] [TopologicalSemiring B] [ContinuousStar B] (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : Continuous φ) : map φ s.topologicalClosure ≤ (map φ s).topologicalClosure := image_closure_subset_closure_image hφ theorem topologicalClosure_map [StarModule R B] [TopologicalSemiring B] [ContinuousStar B] (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : ClosedEmbedding φ) : (map φ s).topologicalClosure = map φ s.topologicalClosure := SetLike.coe_injective <\| hφ.closure_image_eq _ theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) : (star s).topologicalClosure = star s.topologicalClosure := by suffices ∀ t : Subalgebra R A, (star t).topologicalClosure ≤ star t.topologicalClosure from le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s))) exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure) (isClosed_closure.preimage continuous_star) /-- If a star subalgebra of a topological star algebra is commutative, then so is its topological closure. See note [reducible non-instances]. -/ abbrev commSemiringTopologicalClosure [T2Space A] (s : StarSubalgebra R A) (hs : ∀ x y : s, x * y = y * x) : CommSemiring s.topologicalClosure := s.toSubalgebra.commSemiringTopologicalClosure hs #align star_subalgebra.comm_semiring_topological_closure StarSubalgebra.commSemiringTopologicalClosure /-- If a star subalgebra of a topological star algebra is commutative, then so is its topological closure. See note [reducible non-instances]. -/ abbrev commRingTopologicalClosure {R A} [CommRing R] [StarRing R] [TopologicalSpace A] [Ring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalRing A] [ContinuousStar A] [T2Space A] (s : StarSubalgebra R A) (hs : ∀ x y : s, x * y = y * x) : CommRing s.topologicalClosure := s.toSubalgebra.commRingTopologicalClosure hs #align star_subalgebra.comm_ring_topological_closure StarSubalgebra.commRingTopologicalClosure /-- Continuous `StarAlgHom`s from the topological closure of a `StarSubalgebra` whose compositions with the `StarSubalgebra.inclusion` map agree are, in fact, equal. -/ theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A} {φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ) (h : φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) : φ = ψ := by rw [DFunLike.ext'_iff] have : Dense (Set.range <\| inclusion (le_topologicalClosure S)) := by refine embedding_subtype_val.toInducing.dense_iff.2 fun x => ?_ convert show ↑x ∈ closure (S : Set A) from x.prop rw [← Set.range_comp] exact Set.ext fun y => ⟨by rintro ⟨y, rfl⟩ exact y.prop, fun hy => ⟨⟨y, hy⟩, rfl⟩⟩ refine Continuous.ext_on this hφ hψ ?_ rintro _ ⟨x, rfl⟩ simpa only using DFunLike.congr_fun h x #align star_alg_hom.ext_topological_closure StarAlgHom.ext_topologicalClosure theorem _root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type} {S : StarSubalgebra R A} [FunLike F S.topologicalClosure B] [AlgHomClass F R S.topologicalClosure B] [StarAlgHomClass F R S.topologicalClosure B] {φ ψ : F} (hφ : Continuous φ) (hψ : Continuous ψ) (h : ∀ x : S, φ (inclusion (le_topologicalClosure S) x) = ψ ((inclusion (le_topologicalClosure S)) x)) : φ = ψ := by -- Porting note: an intervening coercion seems to have appeared since ML3 have : (φ : S.topologicalClosure →⋆ₐ[R] B) = (ψ : S.topologicalClosure →⋆ₐ[R] B) := by refine StarAlgHom.ext_topologicalClosure (R := R) (A := A) (B := B) hφ hψ (StarAlgHom.ext ?_) simpa only [StarAlgHom.coe_comp, StarAlgHom.coe_coe] using h rw [DFunLike.ext'_iff, ← StarAlgHom.coe_coe] apply congrArg _ this #align star_alg_hom_class.ext_topological_closure StarAlgHomClass.ext_topologicalClosure end TopologicalStarAlgebra end StarSubalgebra section Elemental open StarSubalgebra StarAlgebra variable (R : Type) {A B : Type} [CommSemiring R] [StarRing R] variable [TopologicalSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] variable [ContinuousStar A] [Algebra R A] [StarModule R A] variable [TopologicalSpace B] [Semiring B] [StarRing B] [Algebra R B] /-- The topological closure of the subalgebra generated by a single element. -/ def elementalStarAlgebra (x : A) : StarSubalgebra R A := (adjoin R ({x} : Set A)).topologicalClosure #align elemental_star_algebra elementalStarAlgebra namespace elementalStarAlgebra @[aesop safe apply (rule_sets := [SetLike])] theorem self_mem (x : A) : x ∈ elementalStarAlgebra R x := SetLike.le_def.mp (le_topologicalClosure _) (self_mem_adjoin_singleton R x) #align elemental_star_algebra.self_mem elementalStarAlgebra.self_mem theorem star_self_mem (x : A) : star x ∈ elementalStarAlgebra R x := star_mem <\| self_mem R x #align elemental_star_algebra.star_self_mem elementalStarAlgebra.star_self_mem /-- The `elementalStarAlgebra` generated by a normal element is commutative. -/ instance [T2Space A] {x : A} [IsStarNormal x] : CommSemiring (elementalStarAlgebra R x) := StarSubalgebra.commSemiringTopologicalClosure _ mul_comm /-- The `elementalStarAlgebra` generated by a normal element is commutative. -/ instance {R A} [CommRing R] [StarRing R] [TopologicalSpace A] [Ring A] [Algebra R A] [StarRing A] [StarModule R A] [TopologicalRing A] [ContinuousStar A] [T2Space A] {x : A} [IsStarNormal x] : CommRing (elementalStarAlgebra R x) := StarSubalgebra.commRingTopologicalClosure _ mul_comm theorem isClosed (x : A) : IsClosed (elementalStarAlgebra R x : Set A) := isClosed_closure #align elemental_star_algebra.is_closed elementalStarAlgebra.isClosed instance {A : Type} [UniformSpace A] [CompleteSpace A] [Semiring A] [StarRing A] [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] (x : A) : CompleteSpace (elementalStarAlgebra R x) := isClosed_closure.completeSpace_coe theorem le_of_isClosed_of_mem {S : StarSubalgebra R A} (hS : IsClosed (S : Set A)) {x : A} (hx : x ∈ S) : elementalStarAlgebra R x ≤ S := topologicalClosure_minimal (adjoin_le <\| Set.singleton_subset_iff.2 hx) hS #align elemental_star_algebra.le_of_is_closed_of_mem elementalStarAlgebra.le_of_isClosed_of_mem /-- The coercion from an elemental algebra to the full algebra as a `ClosedEmbedding`. -/ theorem closedEmbedding_coe (x : A) : ClosedEmbedding ((↑) : elementalStarAlgebra R x → A) := { induced := rfl inj := Subtype.coe_injective isClosed_range := by convert isClosed R x exact Set.ext fun y => ⟨by rintro ⟨y, rfl⟩ exact y.prop, fun hy => ⟨⟨y, hy⟩, rfl⟩⟩ } #align elemental_star_algebra.closed_embedding_coe elementalStarAlgebra.closedEmbedding_coe @[elab_as_elim]	Mathlib/Topology/Algebra/StarSubalgebra.lean	247	268	theorem induction_on {x y : A} (hy : y ∈ elementalStarAlgebra R x) {P : (u : A) → u ∈ elementalStarAlgebra R x → Prop} (self : P x (self_mem R x)) (star_self : P (star x) (star_self_mem R x)) (algebraMap : ∀ r, P (algebraMap R A r) (_root_.algebraMap_mem _ r)) (add : ∀ u hu v hv, P u hu → P v hv → P (u + v) (add_mem hu hv)) (mul : ∀ u hu v hv, P u hu → P v hv → P (u * v) (mul_mem hu hv)) (closure : ∀ s : Set A, (hs : s ⊆ elementalStarAlgebra R x) → (∀ u, (hu : u ∈ s) → P u (hs hu)) → ∀ v, (hv : v ∈ closure s) → P v (closure_minimal hs (isClosed R x) hv)) : P y hy := by	apply closure (adjoin R {x} : Set A) subset_closure (fun y hy ↦ ?_) y hy rw [SetLike.mem_coe, ← mem_toSubalgebra, adjoin_toSubalgebra] at hy induction hy using Algebra.adjoin_induction'' with \| mem u hu => obtain ((rfl : u = x) \| (hu : star u = x)) := by simpa using hu · exact self · simp_rw [← hu, star_star] at star_self exact star_self \| algebraMap r => exact algebraMap r \| add u hu_mem v hv_mem hu hv => exact add u (subset_closure hu_mem) v (subset_closure hv_mem) (hu hu_mem) (hv hv_mem) \| mul u hu_mem v hv_mem hu hv => exact mul u (subset_closure hu_mem) v (subset_closure hv_mem) (hu hu_mem) (hv hv_mem)
/- Copyright (c) 2023 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import Mathlib.Algebra.Group.Submonoid.Operations import Mathlib.Algebra.Star.SelfAdjoint #align_import algebra.star.order from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004" /-! # Star ordered rings We define the class `StarOrderedRing R`, which says that the order on `R` respects the star operation, i.e. an element `r` is nonnegative iff it is in the `AddSubmonoid` generated by elements of the form `star s * s`. In many cases, including all C⋆-algebras, this can be reduced to `0 ≤ r ↔ ∃ s, r = star s * s`. However, this generality is slightly more convenient (e.g., it allows us to register a `StarOrderedRing` instance for `ℚ`), and more closely resembles the literature (see the seminal paper [The positive cone in Banach algebras][kelleyVaught1953]) In order to accommodate `NonUnitalSemiring R`, we actually don't characterize nonnegativity, but rather the entire `≤` relation with `StarOrderedRing.le_iff`. However, notice that when `R` is a `NonUnitalRing`, these are equivalent (see `StarOrderedRing.nonneg_iff` and `StarOrderedRing.of_nonneg_iff`). It is important to note that while a `StarOrderedRing` is an `OrderedAddCommMonoid` it is often not an `OrderedSemiring`. ## TODO * In a Banach star algebra without a well-defined square root, the natural ordering is given by the positive cone which is the _closure_ of the sums of elements `star r * r`. A weaker version of `StarOrderedRing` could be defined for this case (again, see [The positive cone in Banach algebras][kelleyVaught1953]). Note that the current definition has the advantage of not requiring a topology. -/ open Set open scoped NNRat universe u variable {R : Type u} /-- An ordered ``-ring is a ``ring with a partial order such that the nonnegative elements constitute precisely the `AddSubmonoid` generated by elements of the form `star s * s`. If you are working with a `NonUnitalRing` and not a `NonUnitalSemiring`, it may be more convenient to declare instances using `StarOrderedRing.of_nonneg_iff`. Porting note: dropped an unneeded assumption `add_le_add_left : ∀ {x y}, x ≤ y → ∀ z, z + x ≤ z + y` -/ class StarOrderedRing (R : Type u) [NonUnitalSemiring R] [PartialOrder R] [StarRing R] : Prop where /-- characterization of the order in terms of the `StarRing` structure. -/ le_iff : ∀ x y : R, x ≤ y ↔ ∃ p, p ∈ AddSubmonoid.closure (Set.range fun s => star s * s) ∧ y = x + p #align star_ordered_ring StarOrderedRing namespace StarOrderedRing -- see note [lower instance priority] instance (priority := 100) toOrderedAddCommMonoid [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] : OrderedAddCommMonoid R where add_le_add_left := fun x y hle z ↦ by rw [StarOrderedRing.le_iff] at hle ⊢ refine hle.imp fun s hs ↦ ?_ rw [hs.2, add_assoc] exact ⟨hs.1, rfl⟩ #align star_ordered_ring.to_ordered_add_comm_monoid StarOrderedRing.toOrderedAddCommMonoid -- see note [lower instance priority] instance (priority := 100) toExistsAddOfLE [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] : ExistsAddOfLE R where exists_add_of_le h := match (le_iff _ _).mp h with \| ⟨p, _, hp⟩ => ⟨p, hp⟩ #align star_ordered_ring.to_has_exists_add_of_le StarOrderedRing.toExistsAddOfLE -- see note [lower instance priority] instance (priority := 100) toOrderedAddCommGroup [NonUnitalRing R] [PartialOrder R] [StarRing R] [StarOrderedRing R] : OrderedAddCommGroup R where add_le_add_left := @add_le_add_left _ _ _ _ #align star_ordered_ring.to_ordered_add_comm_group StarOrderedRing.toOrderedAddCommGroup /-- To construct a `StarOrderedRing` instance it suffices to show that `x ≤ y` if and only if `y = x + star s * s` for some `s : R`. This is provided for convenience because it holds in some common scenarios (e.g.,`ℝ≥0`, `C(X, ℝ≥0)`) and obviates the hassle of `AddSubmonoid.closure_induction` when creating those instances. If you are working with a `NonUnitalRing` and not a `NonUnitalSemiring`, see `StarOrderedRing.of_nonneg_iff` for a more convenient version. -/ lemma of_le_iff [NonUnitalSemiring R] [PartialOrder R] [StarRing R] (h_le_iff : ∀ x y : R, x ≤ y ↔ ∃ s, y = x + star s * s) : StarOrderedRing R where le_iff x y := by refine ⟨fun h => ?_, ?_⟩ · obtain ⟨p, hp⟩ := (h_le_iff x y).mp h exact ⟨star p * p, AddSubmonoid.subset_closure ⟨p, rfl⟩, hp⟩ · rintro ⟨p, hp, hpxy⟩ revert x y hpxy refine AddSubmonoid.closure_induction hp ?_ (fun x y h => add_zero x ▸ h.ge) ?_ · rintro _ ⟨s, rfl⟩ x y rfl exact (h_le_iff _ _).mpr ⟨s, rfl⟩ · rintro a b ha hb x y rfl rw [← add_assoc] exact (ha _ _ rfl).trans (hb _ _ rfl) #align star_ordered_ring.of_le_iff StarOrderedRing.of_le_iffₓ /-- When `R` is a non-unital ring, to construct a `StarOrderedRing` instance it suffices to show that the nonnegative elements are precisely those elements in the `AddSubmonoid` generated by `star s * s` for `s : R`. -/ lemma of_nonneg_iff [NonUnitalRing R] [PartialOrder R] [StarRing R] (h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y) (h_nonneg_iff : ∀ x : R, 0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s : R => star s * s)) : StarOrderedRing R where le_iff x y := by haveI : CovariantClass R R (· + ·) (· ≤ ·) := ⟨fun _ _ _ h => h_add h _⟩ simpa only [← sub_eq_iff_eq_add', sub_nonneg, exists_eq_right'] using h_nonneg_iff (y - x) #align star_ordered_ring.of_nonneg_iff StarOrderedRing.of_nonneg_iff /-- When `R` is a non-unital ring, to construct a `StarOrderedRing` instance it suffices to show that the nonnegative elements are precisely those elements of the form `star s * s` for `s : R`. This is provided for convenience because it holds in many common scenarios (e.g.,`ℝ`, `ℂ`, or any C⋆-algebra), and obviates the hassle of `AddSubmonoid.closure_induction` when creating those instances. -/ lemma of_nonneg_iff' [NonUnitalRing R] [PartialOrder R] [StarRing R] (h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y) (h_nonneg_iff : ∀ x : R, 0 ≤ x ↔ ∃ s, x = star s * s) : StarOrderedRing R := of_le_iff <\| by haveI : CovariantClass R R (· + ·) (· ≤ ·) := ⟨fun _ _ _ h => h_add h _⟩ simpa [sub_eq_iff_eq_add', sub_nonneg] using fun x y => h_nonneg_iff (y - x) #align star_ordered_ring.of_nonneg_iff' StarOrderedRing.of_nonneg_iff' theorem nonneg_iff [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {x : R} : 0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s : R => star s * s) := by simp only [le_iff, zero_add, exists_eq_right'] #align star_ordered_ring.nonneg_iff StarOrderedRing.nonneg_iff end StarOrderedRing section NonUnitalSemiring variable [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] theorem star_mul_self_nonneg (r : R) : 0 ≤ star r * r := StarOrderedRing.nonneg_iff.mpr <\| AddSubmonoid.subset_closure ⟨r, rfl⟩ #align star_mul_self_nonneg star_mul_self_nonneg theorem mul_star_self_nonneg (r : R) : 0 ≤ r * star r := by simpa only [star_star] using star_mul_self_nonneg (star r) #align star_mul_self_nonneg' mul_star_self_nonneg	Mathlib/Algebra/Star/Order.lean	156	166	theorem conjugate_nonneg {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ star c * a * c := by	rw [StarOrderedRing.nonneg_iff] at ha refine AddSubmonoid.closure_induction ha (fun x hx => ?_) (by rw [mul_zero, zero_mul]) fun x y hx hy => ?_ · obtain ⟨x, rfl⟩ := hx convert star_mul_self_nonneg (x * c) using 1 rw [star_mul, ← mul_assoc, mul_assoc _ _ c] · calc 0 ≤ star c * x * c + 0 := by rw [add_zero]; exact hx _ ≤ star c * x * c + star c * y * c := add_le_add_left hy _ _ ≤ _ := by rw [mul_add, add_mul]
/- Copyright (c) 2023 Peter Nelson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Peter Nelson -/ import Mathlib.Data.Matroid.IndepAxioms /-! # Matroid Duality For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the collection of bases of another matroid on `E` called the 'dual' of `M`. The map from `M` to its dual is an involution, interacts nicely with minors, and preserves many important matroid properties such as representability and connectivity. This file defines the dual matroid `M✶` of `M`, and gives associated API. The definition is in terms of its independent sets, using `IndepMatroid.matroid`. We also define 'Co-independence' (independence in the dual) of a set as a predicate `M.Coindep X`. This is an abbreviation for `M✶.Indep X`, but has its own name for the sake of dot notation. ## Main Definitions * `M.Dual`, written `M✶`, is the matroid in which a set `B` is a base if and only if `B ⊆ M.E` and `M.E \ B` is a base for `M`. * `M.Coindep X` means `M✶.Indep X`, or equivalently that `X` is contained in `M.E \ B` for some base `B` of `M`. -/ open Set namespace Matroid variable {α : Type} {M : Matroid α} {I B X : Set α} section dual /-- Given `M : Matroid α`, the `IndepMatroid α` whose independent sets are the subsets of `M.E` that are disjoint from some base of `M` -/ @[simps] def dualIndepMatroid (M : Matroid α) : IndepMatroid α where E := M.E Indep I := I ⊆ M.E ∧ ∃ B, M.Base B ∧ Disjoint I B indep_empty := ⟨empty_subset M.E, M.exists_base.imp (fun B hB ↦ ⟨hB, empty_disjoint _⟩)⟩ indep_subset := by rintro I J ⟨hJE, B, hB, hJB⟩ hIJ exact ⟨hIJ.trans hJE, ⟨B, hB, disjoint_of_subset_left hIJ hJB⟩⟩ indep_aug := by rintro I X ⟨hIE, B, hB, hIB⟩ hI_not_max hX_max have hXE := hX_max.1.1 have hB' := (base_compl_iff_mem_maximals_disjoint_base hXE).mpr hX_max set B' := M.E \ X with hX have hI := (not_iff_not.mpr (base_compl_iff_mem_maximals_disjoint_base)).mpr hI_not_max obtain ⟨B'', hB'', hB''₁, hB''₂⟩ := (hB'.indep.diff I).exists_base_subset_union_base hB rw [← compl_subset_compl, ← hIB.sdiff_eq_right, ← union_diff_distrib, diff_eq, compl_inter, compl_compl, union_subset_iff, compl_subset_compl] at hB''₂ have hssu := (subset_inter (hB''₂.2) hIE).ssubset_of_ne (by { rintro rfl; apply hI; convert hB''; simp [hB''.subset_ground] }) obtain ⟨e, ⟨(heB'' : e ∉ _), heE⟩, heI⟩ := exists_of_ssubset hssu use e simp_rw [mem_diff, insert_subset_iff, and_iff_left heI, and_iff_right heE, and_iff_right hIE] refine ⟨by_contra (fun heX ↦ heB'' (hB''₁ ⟨?_, heI⟩)), ⟨B'', hB'', ?_⟩⟩ · rw [hX]; exact ⟨heE, heX⟩ rw [← union_singleton, disjoint_union_left, disjoint_singleton_left, and_iff_left heB''] exact disjoint_of_subset_left hB''₂.2 disjoint_compl_left indep_maximal := by rintro X - I'⟨hI'E, B, hB, hI'B⟩ hI'X obtain ⟨I, hI⟩ := M.exists_basis (M.E \ X) obtain ⟨B', hB', hIB', hB'IB⟩ := hI.indep.exists_base_subset_union_base hB refine ⟨(X \ B') ∩ M.E, ⟨?_, subset_inter (subset_diff.mpr ?_) hI'E, inter_subset_left.trans diff_subset⟩, ?_⟩ · simp only [inter_subset_right, true_and] exact ⟨B', hB', disjoint_of_subset_left inter_subset_left disjoint_sdiff_left⟩ · rw [and_iff_right hI'X] refine disjoint_of_subset_right hB'IB ?_ rw [disjoint_union_right, and_iff_left hI'B] exact disjoint_of_subset hI'X hI.subset disjoint_sdiff_right simp only [mem_setOf_eq, subset_inter_iff, and_imp, forall_exists_index] intros J hJE B'' hB'' hdj _ hJX hssJ rw [and_iff_left hJE] rw [diff_eq, inter_right_comm, ← diff_eq, diff_subset_iff] at hssJ have hI' : (B'' ∩ X) ∪ (B' \ X) ⊆ B' := by rw [union_subset_iff, and_iff_left diff_subset, ← inter_eq_self_of_subset_left hB''.subset_ground, inter_right_comm, inter_assoc] calc _ ⊆ _ := inter_subset_inter_right _ hssJ _ ⊆ _ := by rw [inter_union_distrib_left, hdj.symm.inter_eq, union_empty] _ ⊆ _ := inter_subset_right obtain ⟨B₁,hB₁,hI'B₁,hB₁I⟩ := (hB'.indep.subset hI').exists_base_subset_union_base hB'' rw [union_comm, ← union_assoc, union_eq_self_of_subset_right inter_subset_left] at hB₁I have : B₁ = B' := by refine hB₁.eq_of_subset_indep hB'.indep (fun e he ↦ ?_) refine (hB₁I he).elim (fun heB'' ↦ ?_) (fun h ↦ h.1) refine (em (e ∈ X)).elim (fun heX ↦ hI' (Or.inl ⟨heB'', heX⟩)) (fun heX ↦ hIB' ?_) refine hI.mem_of_insert_indep ⟨hB₁.subset_ground he, heX⟩ (hB₁.indep.subset (insert_subset he ?_)) refine (subset_union_of_subset_right (subset_diff.mpr ⟨hIB',?_⟩) _).trans hI'B₁ exact disjoint_of_subset_left hI.subset disjoint_sdiff_left subst this refine subset_diff.mpr ⟨hJX, by_contra (fun hne ↦ ?_)⟩ obtain ⟨e, heJ, heB'⟩ := not_disjoint_iff.mp hne obtain (heB'' \| ⟨-,heX⟩ ) := hB₁I heB' · exact hdj.ne_of_mem heJ heB'' rfl exact heX (hJX heJ) subset_ground := by tauto /-- The dual of a matroid; the bases are the complements (w.r.t `M.E`) of the bases of `M`. -/ def dual (M : Matroid α) : Matroid α := M.dualIndepMatroid.matroid /-- The `✶` symbol, which denotes matroid duality. (This is distinct from the usual `` symbol for multiplication, due to precedence issues. )-/ postfix:max "✶" => Matroid.dual theorem dual_indep_iff_exists' : (M✶.Indep I) ↔ I ⊆ M.E ∧ (∃ B, M.Base B ∧ Disjoint I B) := Iff.rfl @[simp] theorem dual_ground : M✶.E = M.E := rfl @[simp] theorem dual_indep_iff_exists (hI : I ⊆ M.E := by aesop_mat) : M✶.Indep I ↔ (∃ B, M.Base B ∧ Disjoint I B) := by rw [dual_indep_iff_exists', and_iff_right hI] theorem dual_dep_iff_forall : (M✶.Dep I) ↔ (∀ B, M.Base B → (I ∩ B).Nonempty) ∧ I ⊆ M.E := by simp_rw [dep_iff, dual_indep_iff_exists', dual_ground, and_congr_left_iff, not_and, not_exists, not_and, not_disjoint_iff_nonempty_inter, Classical.imp_iff_right_iff, iff_true_intro Or.inl] instance dual_finite [M.Finite] : M✶.Finite := ⟨M.ground_finite⟩ instance dual_nonempty [M.Nonempty] : M✶.Nonempty := ⟨M.ground_nonempty⟩ @[simp] theorem dual_base_iff (hB : B ⊆ M.E := by aesop_mat) : M✶.Base B ↔ M.Base (M.E \ B) := by rw [base_compl_iff_mem_maximals_disjoint_base, base_iff_maximal_indep, dual_indep_iff_exists', mem_maximals_setOf_iff] simp [dual_indep_iff_exists'] theorem dual_base_iff' : M✶.Base B ↔ M.Base (M.E \ B) ∧ B ⊆ M.E := (em (B ⊆ M.E)).elim (fun h ↦ by rw [dual_base_iff, and_iff_left h]) (fun h ↦ iff_of_false (h ∘ (fun h' ↦ h'.subset_ground)) (h ∘ And.right)) theorem setOf_dual_base_eq : {B \| M✶.Base B} = (fun X ↦ M.E \ X) '' {B \| M.Base B} := by ext B simp only [mem_setOf_eq, mem_image, dual_base_iff'] refine ⟨fun h ↦ ⟨_, h.1, diff_diff_cancel_left h.2⟩, fun ⟨B', hB', h⟩ ↦ ⟨?_,h.symm.trans_subset diff_subset⟩⟩ rwa [← h, diff_diff_cancel_left hB'.subset_ground] @[simp] theorem dual_dual (M : Matroid α) : M✶✶ = M := eq_of_base_iff_base_forall rfl (fun B (h : B ⊆ M.E) ↦ by rw [dual_base_iff, dual_base_iff, dual_ground, diff_diff_cancel_left h]) theorem dual_involutive : Function.Involutive (dual : Matroid α → Matroid α) := dual_dual theorem dual_injective : Function.Injective (dual : Matroid α → Matroid α) := dual_involutive.injective @[simp] theorem dual_inj {M₁ M₂ : Matroid α} : M₁✶ = M₂✶ ↔ M₁ = M₂ := dual_injective.eq_iff theorem eq_dual_comm {M₁ M₂ : Matroid α} : M₁ = M₂✶ ↔ M₂ = M₁✶ := by rw [← dual_inj, dual_dual, eq_comm] theorem eq_dual_iff_dual_eq {M₁ M₂ : Matroid α} : M₁ = M₂✶ ↔ M₁✶ = M₂ := dual_involutive.eq_iff.symm theorem Base.compl_base_of_dual (h : M✶.Base B) : M.Base (M.E \ B) := (dual_base_iff'.1 h).1 theorem Base.compl_base_dual (h : M.Base B) : M✶.Base (M.E \ B) := by rwa [dual_base_iff, diff_diff_cancel_left h.subset_ground] theorem Base.compl_inter_basis_of_inter_basis (hB : M.Base B) (hBX : M.Basis (B ∩ X) X) : M✶.Basis ((M.E \ B) ∩ (M.E \ X)) (M.E \ X) := by refine Indep.basis_of_forall_insert ?_ inter_subset_right (fun e he ↦ ?_) · rw [dual_indep_iff_exists] exact ⟨B, hB, disjoint_of_subset_left inter_subset_left disjoint_sdiff_left⟩ simp only [diff_inter_self_eq_diff, mem_diff, not_and, not_not, imp_iff_right he.1.1] at he simp_rw [dual_dep_iff_forall, insert_subset_iff, and_iff_right he.1.1, and_iff_left (inter_subset_left.trans diff_subset)] refine fun B' hB' ↦ by_contra (fun hem ↦ ?_) rw [nonempty_iff_ne_empty, not_ne_iff, ← union_singleton, diff_inter_diff, union_inter_distrib_right, union_empty_iff, singleton_inter_eq_empty, diff_eq, inter_right_comm, inter_eq_self_of_subset_right hB'.subset_ground, ← diff_eq, diff_eq_empty] at hem obtain ⟨f, hfb, hBf⟩ := hB.exchange hB' ⟨he.2, hem.2⟩ have hi : M.Indep (insert f (B ∩ X)) := by refine hBf.indep.subset (insert_subset_insert ?_) simp_rw [subset_diff, and_iff_right inter_subset_left, disjoint_singleton_right, mem_inter_iff, iff_false_intro he.1.2, and_false, not_false_iff] exact hfb.2 (hBX.mem_of_insert_indep (Or.elim (hem.1 hfb.1) (False.elim ∘ hfb.2) id) hi).1 theorem Base.inter_basis_iff_compl_inter_basis_dual (hB : M.Base B) (hX : X ⊆ M.E := by aesop_mat): M.Basis (B ∩ X) X ↔ M✶.Basis ((M.E \ B) ∩ (M.E \ X)) (M.E \ X) := by refine ⟨hB.compl_inter_basis_of_inter_basis, fun h ↦ ?_⟩ simpa [inter_eq_self_of_subset_right hX, inter_eq_self_of_subset_right hB.subset_ground] using hB.compl_base_dual.compl_inter_basis_of_inter_basis h theorem base_iff_dual_base_compl (hB : B ⊆ M.E := by aesop_mat) : M.Base B ↔ M✶.Base (M.E \ B) := by rw [dual_base_iff, diff_diff_cancel_left hB] theorem ground_not_base (M : Matroid α) [h : RkPos M✶] : ¬M.Base M.E := by rwa [rkPos_iff_empty_not_base, dual_base_iff, diff_empty] at h theorem Base.ssubset_ground [h : RkPos M✶] (hB : M.Base B) : B ⊂ M.E := hB.subset_ground.ssubset_of_ne (by rintro rfl; exact M.ground_not_base hB) theorem Indep.ssubset_ground [h : RkPos M✶] (hI : M.Indep I) : I ⊂ M.E := by obtain ⟨B, hB⟩ := hI.exists_base_superset; exact hB.2.trans_ssubset hB.1.ssubset_ground /-- A coindependent set of `M` is an independent set of the dual of `M✶`. we give it a separate definition to enable dot notation. Which spelling is better depends on context. -/ abbrev Coindep (M : Matroid α) (I : Set α) : Prop := M✶.Indep I theorem coindep_def : M.Coindep X ↔ M✶.Indep X := Iff.rfl theorem Coindep.indep (hX : M.Coindep X) : M✶.Indep X := hX @[simp] theorem dual_coindep_iff : M✶.Coindep X ↔ M.Indep X := by rw [Coindep, dual_dual] theorem Indep.coindep (hI : M.Indep I) : M✶.Coindep I := dual_coindep_iff.2 hI	Mathlib/Data/Matroid/Dual.lean	237	240	theorem coindep_iff_exists' : M.Coindep X ↔ (∃ B, M.Base B ∧ B ⊆ M.E \ X) ∧ X ⊆ M.E := by	simp_rw [Coindep, dual_indep_iff_exists', and_comm (a := _ ⊆ _), and_congr_left_iff, subset_diff] exact fun _ ↦ ⟨fun ⟨B, hB, hXB⟩ ↦ ⟨B, hB, hB.subset_ground, hXB.symm⟩, fun ⟨B, hB, _, hBX⟩ ↦ ⟨B, hB, hBX.symm⟩⟩
/- Copyright (c) 2020 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import Mathlib.Algebra.Algebra.Hom import Mathlib.RingTheory.Ideal.Quotient #align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72" /-! # Quotients of non-commutative rings Unfortunately, ideals have only been developed in the commutative case as `Ideal`, and it's not immediately clear how one should formalise ideals in the non-commutative case. In this file, we directly define the quotient of a semiring by any relation, by building a bigger relation that represents the ideal generated by that relation. We prove the universal properties of the quotient, and recommend avoiding relying on the actual definition, which is made irreducible for this purpose. Since everything runs in parallel for quotients of `R`-algebras, we do that case at the same time. -/ universe uR uS uT uA u₄ variable {R : Type uR} [Semiring R] variable {S : Type uS} [CommSemiring S] variable {T : Type uT} variable {A : Type uA} [Semiring A] [Algebra S A] namespace RingCon instance (c : RingCon A) : Algebra S c.Quotient where smul := (· • ·) toRingHom := c.mk'.comp (algebraMap S A) commutes' _ := Quotient.ind' fun _ ↦ congr_arg Quotient.mk'' <\| Algebra.commutes _ _ smul_def' _ := Quotient.ind' fun _ ↦ congr_arg Quotient.mk'' <\| Algebra.smul_def _ _ @[simp, norm_cast] theorem coe_algebraMap (c : RingCon A) (s : S) : (algebraMap S A s : c.Quotient) = algebraMap S _ s := rfl #align ring_con.coe_algebra_map RingCon.coe_algebraMap end RingCon namespace RingQuot /-- Given an arbitrary relation `r` on a ring, we strengthen it to a relation `Rel r`, such that the equivalence relation generated by `Rel r` has `x ~ y` if and only if `x - y` is in the ideal generated by elements `a - b` such that `r a b`. -/ inductive Rel (r : R → R → Prop) : R → R → Prop \| of ⦃x y : R⦄ (h : r x y) : Rel r x y \| add_left ⦃a b c⦄ : Rel r a b → Rel r (a + c) (b + c) \| mul_left ⦃a b c⦄ : Rel r a b → Rel r (a * c) (b * c) \| mul_right ⦃a b c⦄ : Rel r b c → Rel r (a * b) (a * c) #align ring_quot.rel RingQuot.Rel theorem Rel.add_right {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) : Rel r (a + b) (a + c) := by rw [add_comm a b, add_comm a c] exact Rel.add_left h #align ring_quot.rel.add_right RingQuot.Rel.add_right theorem Rel.neg {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b : R⦄ (h : Rel r a b) : Rel r (-a) (-b) := by simp only [neg_eq_neg_one_mul a, neg_eq_neg_one_mul b, Rel.mul_right h] #align ring_quot.rel.neg RingQuot.Rel.neg theorem Rel.sub_left {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r a b) : Rel r (a - c) (b - c) := by simp only [sub_eq_add_neg, h.add_left] #align ring_quot.rel.sub_left RingQuot.Rel.sub_left theorem Rel.sub_right {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) : Rel r (a - b) (a - c) := by simp only [sub_eq_add_neg, h.neg.add_right] #align ring_quot.rel.sub_right RingQuot.Rel.sub_right theorem Rel.smul {r : A → A → Prop} (k : S) ⦃a b : A⦄ (h : Rel r a b) : Rel r (k • a) (k • b) := by simp only [Algebra.smul_def, Rel.mul_right h] #align ring_quot.rel.smul RingQuot.Rel.smul /-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/ def ringCon (r : R → R → Prop) : RingCon R where r := EqvGen (Rel r) iseqv := EqvGen.is_equivalence _ add' {a b c d} hab hcd := by induction hab generalizing c d with \| rel _ _ hab => refine (EqvGen.rel _ _ hab.add_left).trans _ _ _ ?_ induction hcd with \| rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right \| refl => exact EqvGen.refl _ \| symm _ _ _ h => exact h.symm _ _ \| trans _ _ _ _ _ h h' => exact h.trans _ _ _ h' \| refl => induction hcd with \| rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right \| refl => exact EqvGen.refl _ \| symm _ _ _ h => exact h.symm _ _ \| trans _ _ _ _ _ h h' => exact h.trans _ _ _ h' \| symm x y _ hxy => exact (hxy hcd.symm).symm \| trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <\| EqvGen.refl _) mul' {a b c d} hab hcd := by induction hab generalizing c d with \| rel _ _ hab => refine (EqvGen.rel _ _ hab.mul_left).trans _ _ _ ?_ induction hcd with \| rel _ _ hcd => exact EqvGen.rel _ _ hcd.mul_right \| refl => exact EqvGen.refl _ \| symm _ _ _ h => exact h.symm _ _ \| trans _ _ _ _ _ h h' => exact h.trans _ _ _ h' \| refl => induction hcd with \| rel _ _ hcd => exact EqvGen.rel _ _ hcd.mul_right \| refl => exact EqvGen.refl _ \| symm _ _ _ h => exact h.symm _ _ \| trans _ _ _ _ _ h h' => exact h.trans _ _ _ h' \| symm x y _ hxy => exact (hxy hcd.symm).symm \| trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <\| EqvGen.refl _) #align ring_quot.ring_con RingQuot.ringCon theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r := by ext x₁ x₂ constructor · intro h induction h with \| rel _ _ h => induction h with \| of => exact RingConGen.Rel.of _ _ ‹_› \| add_left _ h => exact h.add (RingConGen.Rel.refl _) \| mul_left _ h => exact h.mul (RingConGen.Rel.refl _) \| mul_right _ h => exact (RingConGen.Rel.refl _).mul h \| refl => exact RingConGen.Rel.refl _ \| symm => exact RingConGen.Rel.symm ‹_› \| trans => exact RingConGen.Rel.trans ‹_› ‹_› · intro h induction h with \| of => exact EqvGen.rel _ _ (Rel.of ‹_›) \| refl => exact (RingQuot.ringCon r).refl _ \| symm => exact (RingQuot.ringCon r).symm ‹_› \| trans => exact (RingQuot.ringCon r).trans ‹_› ‹_› \| add => exact (RingQuot.ringCon r).add ‹_› ‹_› \| mul => exact (RingQuot.ringCon r).mul ‹_› ‹_› #align ring_quot.eqv_gen_rel_eq RingQuot.eqvGen_rel_eq end RingQuot /-- The quotient of a ring by an arbitrary relation. -/ structure RingQuot (r : R → R → Prop) where toQuot : Quot (RingQuot.Rel r) #align ring_quot RingQuot namespace RingQuot variable (r : R → R → Prop) -- can't be irreducible, causes diamonds in ℕ-algebras private def natCast (n : ℕ) : RingQuot r := ⟨Quot.mk _ n⟩ private irreducible_def zero : RingQuot r := ⟨Quot.mk _ 0⟩ private irreducible_def one : RingQuot r := ⟨Quot.mk _ 1⟩ private irreducible_def add : RingQuot r → RingQuot r → RingQuot r \| ⟨a⟩, ⟨b⟩ => ⟨Quot.map₂ (· + ·) Rel.add_right Rel.add_left a b⟩ private irreducible_def mul : RingQuot r → RingQuot r → RingQuot r \| ⟨a⟩, ⟨b⟩ => ⟨Quot.map₂ (· * ·) Rel.mul_right Rel.mul_left a b⟩ private irreducible_def neg {R : Type uR} [Ring R] (r : R → R → Prop) : RingQuot r → RingQuot r \| ⟨a⟩ => ⟨Quot.map (fun a ↦ -a) Rel.neg a⟩ private irreducible_def sub {R : Type uR} [Ring R] (r : R → R → Prop) : RingQuot r → RingQuot r → RingQuot r \| ⟨a⟩, ⟨b⟩ => ⟨Quot.map₂ Sub.sub Rel.sub_right Rel.sub_left a b⟩ private irreducible_def npow (n : ℕ) : RingQuot r → RingQuot r \| ⟨a⟩ => ⟨Quot.lift (fun a ↦ Quot.mk (RingQuot.Rel r) (a ^ n)) (fun a b (h : Rel r a b) ↦ by -- note we can't define a `Rel.pow` as `Rel` isn't reflexive so `Rel r 1 1` isn't true dsimp only induction n with \| zero => rw [pow_zero, pow_zero] \| succ n ih => rw [pow_succ, pow_succ] -- Porting note: -- `simpa [mul_def] using congr_arg₂ (fun x y ↦ mul r ⟨x⟩ ⟨y⟩) (Quot.sound h) ih` -- mysteriously doesn't work have := congr_arg₂ (fun x y ↦ mul r ⟨x⟩ ⟨y⟩) ih (Quot.sound h) dsimp only at this simp? [mul_def] at this says simp only [mul_def, Quot.map₂_mk, mk.injEq] at this exact this) a⟩ -- note: this cannot be irreducible, as otherwise diamonds don't commute. private def smul [Algebra S R] (n : S) : RingQuot r → RingQuot r \| ⟨a⟩ => ⟨Quot.map (fun a ↦ n • a) (Rel.smul n) a⟩ instance : NatCast (RingQuot r) := ⟨natCast r⟩ instance : Zero (RingQuot r) := ⟨zero r⟩ instance : One (RingQuot r) := ⟨one r⟩ instance : Add (RingQuot r) := ⟨add r⟩ instance : Mul (RingQuot r) := ⟨mul r⟩ instance : NatPow (RingQuot r) := ⟨fun x n ↦ npow r n x⟩ instance {R : Type uR} [Ring R] (r : R → R → Prop) : Neg (RingQuot r) := ⟨neg r⟩ instance {R : Type uR} [Ring R] (r : R → R → Prop) : Sub (RingQuot r) := ⟨sub r⟩ instance [Algebra S R] : SMul S (RingQuot r) := ⟨smul r⟩ theorem zero_quot : (⟨Quot.mk _ 0⟩ : RingQuot r) = 0 := show _ = zero r by rw [zero_def] #align ring_quot.zero_quot RingQuot.zero_quot theorem one_quot : (⟨Quot.mk _ 1⟩ : RingQuot r) = 1 := show _ = one r by rw [one_def] #align ring_quot.one_quot RingQuot.one_quot theorem add_quot {a b} : (⟨Quot.mk _ a⟩ + ⟨Quot.mk _ b⟩ : RingQuot r) = ⟨Quot.mk _ (a + b)⟩ := by show add r _ _ = _ rw [add_def] rfl #align ring_quot.add_quot RingQuot.add_quot theorem mul_quot {a b} : (⟨Quot.mk _ a⟩ * ⟨Quot.mk _ b⟩ : RingQuot r) = ⟨Quot.mk _ (a * b)⟩ := by show mul r _ _ = _ rw [mul_def] rfl #align ring_quot.mul_quot RingQuot.mul_quot theorem pow_quot {a} {n : ℕ} : (⟨Quot.mk _ a⟩ ^ n : RingQuot r) = ⟨Quot.mk _ (a ^ n)⟩ := by show npow r _ _ = _ rw [npow_def] #align ring_quot.pow_quot RingQuot.pow_quot theorem neg_quot {R : Type uR} [Ring R] (r : R → R → Prop) {a} : (-⟨Quot.mk _ a⟩ : RingQuot r) = ⟨Quot.mk _ (-a)⟩ := by show neg r _ = _ rw [neg_def] rfl #align ring_quot.neg_quot RingQuot.neg_quot theorem sub_quot {R : Type uR} [Ring R] (r : R → R → Prop) {a b} : (⟨Quot.mk _ a⟩ - ⟨Quot.mk _ b⟩ : RingQuot r) = ⟨Quot.mk _ (a - b)⟩ := by show sub r _ _ = _ rw [sub_def] rfl #align ring_quot.sub_quot RingQuot.sub_quot theorem smul_quot [Algebra S R] {n : S} {a : R} : (n • ⟨Quot.mk _ a⟩ : RingQuot r) = ⟨Quot.mk _ (n • a)⟩ := by show smul r _ _ = _ rw [smul] rfl #align ring_quot.smul_quot RingQuot.smul_quot instance instIsScalarTower [CommSemiring T] [SMul S T] [Algebra S R] [Algebra T R] [IsScalarTower S T R] : IsScalarTower S T (RingQuot r) := ⟨fun s t ⟨a⟩ => Quot.inductionOn a fun a' => by simp only [RingQuot.smul_quot, smul_assoc]⟩ instance instSMulCommClass [CommSemiring T] [Algebra S R] [Algebra T R] [SMulCommClass S T R] : SMulCommClass S T (RingQuot r) := ⟨fun s t ⟨a⟩ => Quot.inductionOn a fun a' => by simp only [RingQuot.smul_quot, smul_comm]⟩ instance instAddCommMonoid (r : R → R → Prop) : AddCommMonoid (RingQuot r) where add := (· + ·) zero := 0 add_assoc := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp only [add_quot, add_assoc] zero_add := by rintro ⟨⟨⟩⟩ simp [add_quot, ← zero_quot, zero_add] add_zero := by rintro ⟨⟨⟩⟩ simp only [add_quot, ← zero_quot, add_zero] add_comm := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp only [add_quot, add_comm] nsmul := (· • ·) nsmul_zero := by rintro ⟨⟨⟩⟩ simp only [smul_quot, zero_smul, zero_quot] nsmul_succ := by rintro n ⟨⟨⟩⟩ simp only [smul_quot, nsmul_eq_mul, Nat.cast_add, Nat.cast_one, add_mul, one_mul, add_comm, add_quot] instance instMonoidWithZero (r : R → R → Prop) : MonoidWithZero (RingQuot r) where mul_assoc := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp only [mul_quot, mul_assoc] one_mul := by rintro ⟨⟨⟩⟩ simp only [mul_quot, ← one_quot, one_mul] mul_one := by rintro ⟨⟨⟩⟩ simp only [mul_quot, ← one_quot, mul_one] zero_mul := by rintro ⟨⟨⟩⟩ simp only [mul_quot, ← zero_quot, zero_mul] mul_zero := by rintro ⟨⟨⟩⟩ simp only [mul_quot, ← zero_quot, mul_zero] npow n x := x ^ n npow_zero := by rintro ⟨⟨⟩⟩ simp only [pow_quot, ← one_quot, pow_zero] npow_succ := by rintro n ⟨⟨⟩⟩ simp only [pow_quot, mul_quot, pow_succ] instance instSemiring (r : R → R → Prop) : Semiring (RingQuot r) where natCast := natCast r natCast_zero := by simp [Nat.cast, natCast, ← zero_quot] natCast_succ := by simp [Nat.cast, natCast, ← one_quot, add_quot] left_distrib := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp only [mul_quot, add_quot, left_distrib] right_distrib := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp only [mul_quot, add_quot, right_distrib] nsmul := (· • ·) nsmul_zero := by rintro ⟨⟨⟩⟩ simp only [smul_quot, zero_smul, zero_quot] nsmul_succ := by rintro n ⟨⟨⟩⟩ simp only [smul_quot, nsmul_eq_mul, Nat.cast_add, Nat.cast_one, add_mul, one_mul, add_comm, add_quot] __ := instAddCommMonoid r __ := instMonoidWithZero r -- can't be irreducible, causes diamonds in ℤ-algebras private def intCast {R : Type uR} [Ring R] (r : R → R → Prop) (z : ℤ) : RingQuot r := ⟨Quot.mk _ z⟩ instance instRing {R : Type uR} [Ring R] (r : R → R → Prop) : Ring (RingQuot r) := { RingQuot.instSemiring r with neg := Neg.neg add_left_neg := by rintro ⟨⟨⟩⟩ simp [neg_quot, add_quot, ← zero_quot] sub := Sub.sub sub_eq_add_neg := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp [neg_quot, sub_quot, add_quot, sub_eq_add_neg] zsmul := (· • ·) zsmul_zero' := by rintro ⟨⟨⟩⟩ simp [smul_quot, ← zero_quot] zsmul_succ' := by rintro n ⟨⟨⟩⟩ simp [smul_quot, add_quot, add_mul, add_comm] zsmul_neg' := by rintro n ⟨⟨⟩⟩ simp [smul_quot, neg_quot, add_mul] intCast := intCast r intCast_ofNat := fun n => congrArg RingQuot.mk <\| by exact congrArg (Quot.mk _) (Int.cast_natCast _) intCast_negSucc := fun n => congrArg RingQuot.mk <\| by simp_rw [neg_def] exact congrArg (Quot.mk _) (Int.cast_negSucc n) } instance instCommSemiring {R : Type uR} [CommSemiring R] (r : R → R → Prop) : CommSemiring (RingQuot r) := { RingQuot.instSemiring r with mul_comm := by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ simp [mul_quot, mul_comm] } instance {R : Type uR} [CommRing R] (r : R → R → Prop) : CommRing (RingQuot r) := { RingQuot.instCommSemiring r, RingQuot.instRing r with } instance instInhabited (r : R → R → Prop) : Inhabited (RingQuot r) := ⟨0⟩ instance instAlgebra [Algebra S R] (r : R → R → Prop) : Algebra S (RingQuot r) where smul := (· • ·) toFun r := ⟨Quot.mk _ (algebraMap S R r)⟩ map_one' := by simp [← one_quot] map_mul' := by simp [mul_quot] map_zero' := by simp [← zero_quot] map_add' := by simp [add_quot] commutes' r := by rintro ⟨⟨a⟩⟩ simp [Algebra.commutes, mul_quot] smul_def' r := by rintro ⟨⟨a⟩⟩ simp [smul_quot, Algebra.smul_def, mul_quot] /-- The quotient map from a ring to its quotient, as a homomorphism of rings. -/ irreducible_def mkRingHom (r : R → R → Prop) : R →+* RingQuot r := { toFun := fun x ↦ ⟨Quot.mk _ x⟩ map_one' := by simp [← one_quot] map_mul' := by simp [mul_quot] map_zero' := by simp [← zero_quot] map_add' := by simp [add_quot] } #align ring_quot.mk_ring_hom RingQuot.mkRingHom	Mathlib/Algebra/RingQuot.lean	419	420	theorem mkRingHom_rel {r : R → R → Prop} {x y : R} (w : r x y) : mkRingHom r x = mkRingHom r y := by	simp [mkRingHom_def, Quot.sound (Rel.of w)]
/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth -/ import Mathlib.Analysis.Convex.Basic import Mathlib.Analysis.InnerProductSpace.Orthogonal import Mathlib.Analysis.InnerProductSpace.Symmetric import Mathlib.Analysis.NormedSpace.RCLike import Mathlib.Analysis.RCLike.Lemmas import Mathlib.Algebra.DirectSum.Decomposition #align_import analysis.inner_product_space.projection from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b" /-! # The orthogonal projection Given a nonempty complete subspace `K` of an inner product space `E`, this file constructs `orthogonalProjection K : E →L[𝕜] K`, the orthogonal projection of `E` onto `K`. This map satisfies: for any point `u` in `E`, the point `v = orthogonalProjection K u` in `K` minimizes the distance `‖u - v‖` to `u`. Also a linear isometry equivalence `reflection K : E ≃ₗᵢ[𝕜] E` is constructed, by choosing, for each `u : E`, the point `reflection K u` to satisfy `u + (reflection K u) = 2 • orthogonalProjection K u`. Basic API for `orthogonalProjection` and `reflection` is developed. Next, the orthogonal projection is used to prove a series of more subtle lemmas about the orthogonal complement of complete subspaces of `E` (the orthogonal complement itself was defined in `Analysis.InnerProductSpace.Orthogonal`); the lemma `Submodule.sup_orthogonal_of_completeSpace`, stating that for a complete subspace `K` of `E` we have `K ⊔ Kᗮ = ⊤`, is a typical example. ## References The orthogonal projection construction is adapted from * [Clément & Martin, The Lax-Milgram Theorem. A detailed proof to be formalized in Coq] * [Clément & Martin, A Coq formal proof of the Lax–Milgram theorem] The Coq code is available at the following address: <http://www.lri.fr/~sboldo/elfic/index.html> -/ noncomputable section open RCLike Real Filter open LinearMap (ker range) open Topology variable {𝕜 E F : Type} [RCLike 𝕜] variable [NormedAddCommGroup E] [NormedAddCommGroup F] variable [InnerProductSpace 𝕜 E] [InnerProductSpace ℝ F] local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y local notation "absR" => abs /-! ### Orthogonal projection in inner product spaces -/ -- FIXME this monolithic proof causes a deterministic timeout with `-T50000` -- It should be broken in a sequence of more manageable pieces, -- perhaps with individual statements for the three steps below. /-- Existence of minimizers Let `u` be a point in a real inner product space, and let `K` be a nonempty complete convex subset. Then there exists a (unique) `v` in `K` that minimizes the distance `‖u - v‖` to `u`. -/ theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K) (h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by let δ := ⨅ w : K, ‖u - w‖ letI : Nonempty K := ne.to_subtype have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _ have δ_le : ∀ w : K, δ ≤ ‖u - w‖ := ciInf_le ⟨0, Set.forall_mem_range.2 fun _ => norm_nonneg _⟩ have δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖ := fun w hw => δ_le ⟨w, hw⟩ -- Step 1: since `δ` is the infimum, can find a sequence `w : ℕ → K` in `K` -- such that `‖u - w n‖ < δ + 1 / (n + 1)` (which implies `‖u - w n‖ --> δ`); -- maybe this should be a separate lemma have exists_seq : ∃ w : ℕ → K, ∀ n, ‖u - w n‖ < δ + 1 / (n + 1) := by have hδ : ∀ n : ℕ, δ < δ + 1 / (n + 1) := fun n => lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat have h := fun n => exists_lt_of_ciInf_lt (hδ n) let w : ℕ → K := fun n => Classical.choose (h n) exact ⟨w, fun n => Classical.choose_spec (h n)⟩ rcases exists_seq with ⟨w, hw⟩ have norm_tendsto : Tendsto (fun n => ‖u - w n‖) atTop (𝓝 δ) := by have h : Tendsto (fun _ : ℕ => δ) atTop (𝓝 δ) := tendsto_const_nhds have h' : Tendsto (fun n : ℕ => δ + 1 / (n + 1)) atTop (𝓝 δ) := by convert h.add tendsto_one_div_add_atTop_nhds_zero_nat simp only [add_zero] exact tendsto_of_tendsto_of_tendsto_of_le_of_le h h' (fun x => δ_le _) fun x => le_of_lt (hw _) -- Step 2: Prove that the sequence `w : ℕ → K` is a Cauchy sequence have seq_is_cauchy : CauchySeq fun n => (w n : F) := by rw [cauchySeq_iff_le_tendsto_0] -- splits into three goals let b := fun n : ℕ => 8 δ * (1 / (n + 1)) + 4 * (1 / (n + 1)) * (1 / (n + 1)) use fun n => √(b n) constructor -- first goal : `∀ (n : ℕ), 0 ≤ √(b n)` · intro n exact sqrt_nonneg _ constructor -- second goal : `∀ (n m N : ℕ), N ≤ n → N ≤ m → dist ↑(w n) ↑(w m) ≤ √(b N)` · intro p q N hp hq let wp := (w p : F) let wq := (w q : F) let a := u - wq let b := u - wp let half := 1 / (2 : ℝ) let div := 1 / ((N : ℝ) + 1) have : 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) := calc 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = 2 * ‖u - half • (wq + wp)‖ * (2 * ‖u - half • (wq + wp)‖) + ‖wp - wq‖ * ‖wp - wq‖ := by ring _ = absR (2 : ℝ) * ‖u - half • (wq + wp)‖ * (absR (2 : ℝ) * ‖u - half • (wq + wp)‖) + ‖wp - wq‖ * ‖wp - wq‖ := by rw [_root_.abs_of_nonneg] exact zero_le_two _ = ‖(2 : ℝ) • (u - half • (wq + wp))‖ * ‖(2 : ℝ) • (u - half • (wq + wp))‖ + ‖wp - wq‖ * ‖wp - wq‖ := by simp [norm_smul] _ = ‖a + b‖ * ‖a + b‖ + ‖a - b‖ * ‖a - b‖ := by rw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : ℝ) ≠ 0), ← one_add_one_eq_two, add_smul] simp only [one_smul] have eq₁ : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm have eq₂ : u + u - (wq + wp) = a + b := by show u + u - (wq + wp) = u - wq + (u - wp) abel rw [eq₁, eq₂] _ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) := parallelogram_law_with_norm ℝ _ _ have eq : δ ≤ ‖u - half • (wq + wp)‖ := by rw [smul_add] apply δ_le' apply h₂ repeat' exact Subtype.mem _ repeat' exact le_of_lt one_half_pos exact add_halves 1 have eq₁ : 4 * δ * δ ≤ 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ := by simp_rw [mul_assoc] gcongr have eq₂ : ‖a‖ ≤ δ + div := le_trans (le_of_lt <\| hw q) (add_le_add_left (Nat.one_div_le_one_div hq) _) have eq₂' : ‖b‖ ≤ δ + div := le_trans (le_of_lt <\| hw p) (add_le_add_left (Nat.one_div_le_one_div hp) _) rw [dist_eq_norm] apply nonneg_le_nonneg_of_sq_le_sq · exact sqrt_nonneg _ rw [mul_self_sqrt] · calc ‖wp - wq‖ * ‖wp - wq‖ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) - 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ := by simp [← this] _ ≤ 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) - 4 * δ * δ := by gcongr _ ≤ 2 * ((δ + div) * (δ + div) + (δ + div) * (δ + div)) - 4 * δ * δ := by gcongr _ = 8 * δ * div + 4 * div * div := by ring positivity -- third goal : `Tendsto (fun (n : ℕ) => √(b n)) atTop (𝓝 0)` suffices Tendsto (fun x ↦ √(8 * δ * x + 4 * x * x) : ℝ → ℝ) (𝓝 0) (𝓝 0) from this.comp tendsto_one_div_add_atTop_nhds_zero_nat exact Continuous.tendsto' (by continuity) _ _ (by simp) -- Step 3: By completeness of `K`, let `w : ℕ → K` converge to some `v : K`. -- Prove that it satisfies all requirements. rcases cauchySeq_tendsto_of_isComplete h₁ (fun n => Subtype.mem _) seq_is_cauchy with ⟨v, hv, w_tendsto⟩ use v use hv have h_cont : Continuous fun v => ‖u - v‖ := Continuous.comp continuous_norm (Continuous.sub continuous_const continuous_id) have : Tendsto (fun n => ‖u - w n‖) atTop (𝓝 ‖u - v‖) := by convert Tendsto.comp h_cont.continuousAt w_tendsto exact tendsto_nhds_unique this norm_tendsto #align exists_norm_eq_infi_of_complete_convex exists_norm_eq_iInf_of_complete_convex /-- Characterization of minimizers for the projection on a convex set in a real inner product space. -/ theorem norm_eq_iInf_iff_real_inner_le_zero {K : Set F} (h : Convex ℝ K) {u : F} {v : F} (hv : v ∈ K) : (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by letI : Nonempty K := ⟨⟨v, hv⟩⟩ constructor · intro eq w hw let δ := ⨅ w : K, ‖u - w‖ let p := ⟪u - v, w - v⟫_ℝ let q := ‖w - v‖ ^ 2 have δ_le (w : K) : δ ≤ ‖u - w‖ := ciInf_le ⟨0, fun _ ⟨_, h⟩ => h ▸ norm_nonneg _⟩ _ have δ_le' (w) (hw : w ∈ K) : δ ≤ ‖u - w‖ := δ_le ⟨w, hw⟩ have (θ : ℝ) (hθ₁ : 0 < θ) (hθ₂ : θ ≤ 1) : 2 * p ≤ θ * q := by have : ‖u - v‖ ^ 2 ≤ ‖u - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ + θ * θ * ‖w - v‖ ^ 2 := calc ‖u - v‖ ^ 2 _ ≤ ‖u - (θ • w + (1 - θ) • v)‖ ^ 2 := by simp only [sq]; apply mul_self_le_mul_self (norm_nonneg _) rw [eq]; apply δ_le' apply h hw hv exacts [le_of_lt hθ₁, sub_nonneg.2 hθ₂, add_sub_cancel _ _] _ = ‖u - v - θ • (w - v)‖ ^ 2 := by have : u - (θ • w + (1 - θ) • v) = u - v - θ • (w - v) := by rw [smul_sub, sub_smul, one_smul] simp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev] rw [this] _ = ‖u - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) + θ * θ * ‖w - v‖ ^ 2 := by rw [@norm_sub_sq ℝ, inner_smul_right, norm_smul] simp only [sq] show ‖u - v‖ * ‖u - v‖ - 2 * (θ * inner (u - v) (w - v)) + absR θ * ‖w - v‖ * (absR θ * ‖w - v‖) = ‖u - v‖ * ‖u - v‖ - 2 * θ * inner (u - v) (w - v) + θ * θ * (‖w - v‖ * ‖w - v‖) rw [abs_of_pos hθ₁]; ring have eq₁ : ‖u - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) + θ * θ * ‖w - v‖ ^ 2 = ‖u - v‖ ^ 2 + (θ * θ * ‖w - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v)) := by abel rw [eq₁, le_add_iff_nonneg_right] at this have eq₂ : θ * θ * ‖w - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) = θ * (θ * ‖w - v‖ ^ 2 - 2 * inner (u - v) (w - v)) := by ring rw [eq₂] at this have := le_of_sub_nonneg (nonneg_of_mul_nonneg_right this hθ₁) exact this by_cases hq : q = 0 · rw [hq] at this have : p ≤ 0 := by have := this (1 : ℝ) (by norm_num) (by norm_num) linarith exact this · have q_pos : 0 < q := lt_of_le_of_ne (sq_nonneg _) fun h ↦ hq h.symm by_contra hp rw [not_le] at hp let θ := min (1 : ℝ) (p / q) have eq₁ : θ * q ≤ p := calc θ * q ≤ p / q * q := mul_le_mul_of_nonneg_right (min_le_right _ _) (sq_nonneg _) _ = p := div_mul_cancel₀ _ hq have : 2 * p ≤ p := calc 2 * p ≤ θ * q := by set_option tactic.skipAssignedInstances false in exact this θ (lt_min (by norm_num) (div_pos hp q_pos)) (by norm_num [θ]) _ ≤ p := eq₁ linarith · intro h apply le_antisymm · apply le_ciInf intro w apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _) have := h w w.2 calc ‖u - v‖ * ‖u - v‖ ≤ ‖u - v‖ * ‖u - v‖ - 2 * inner (u - v) ((w : F) - v) := by linarith _ ≤ ‖u - v‖ ^ 2 - 2 * inner (u - v) ((w : F) - v) + ‖(w : F) - v‖ ^ 2 := by rw [sq] refine le_add_of_nonneg_right ?_ exact sq_nonneg _ _ = ‖u - v - (w - v)‖ ^ 2 := (@norm_sub_sq ℝ _ _ _ _ _ _).symm _ = ‖u - w‖ * ‖u - w‖ := by have : u - v - (w - v) = u - w := by abel rw [this, sq] · show ⨅ w : K, ‖u - w‖ ≤ (fun w : K => ‖u - w‖) ⟨v, hv⟩ apply ciInf_le use 0 rintro y ⟨z, rfl⟩ exact norm_nonneg _ #align norm_eq_infi_iff_real_inner_le_zero norm_eq_iInf_iff_real_inner_le_zero variable (K : Submodule 𝕜 E) /-- Existence of projections on complete subspaces. Let `u` be a point in an inner product space, and let `K` be a nonempty complete subspace. Then there exists a (unique) `v` in `K` that minimizes the distance `‖u - v‖` to `u`. This point `v` is usually called the orthogonal projection of `u` onto `K`. -/ theorem exists_norm_eq_iInf_of_complete_subspace (h : IsComplete (↑K : Set E)) : ∀ u : E, ∃ v ∈ K, ‖u - v‖ = ⨅ w : (K : Set E), ‖u - w‖ := by letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E letI : Module ℝ E := RestrictScalars.module ℝ 𝕜 E let K' : Submodule ℝ E := Submodule.restrictScalars ℝ K exact exists_norm_eq_iInf_of_complete_convex ⟨0, K'.zero_mem⟩ h K'.convex #align exists_norm_eq_infi_of_complete_subspace exists_norm_eq_iInf_of_complete_subspace /-- Characterization of minimizers in the projection on a subspace, in the real case. Let `u` be a point in a real inner product space, and let `K` be a nonempty subspace. Then point `v` minimizes the distance `‖u - v‖` over points in `K` if and only if for all `w ∈ K`, `⟪u - v, w⟫ = 0` (i.e., `u - v` is orthogonal to the subspace `K`). This is superceded by `norm_eq_iInf_iff_inner_eq_zero` that gives the same conclusion over any `RCLike` field. -/ theorem norm_eq_iInf_iff_real_inner_eq_zero (K : Submodule ℝ F) {u : F} {v : F} (hv : v ∈ K) : (‖u - v‖ = ⨅ w : (↑K : Set F), ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0 := Iff.intro (by intro h have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by rwa [norm_eq_iInf_iff_real_inner_le_zero] at h exacts [K.convex, hv] intro w hw have le : ⟪u - v, w⟫_ℝ ≤ 0 := by let w' := w + v have : w' ∈ K := Submodule.add_mem _ hw hv have h₁ := h w' this have h₂ : w' - v = w := by simp only [w', add_neg_cancel_right, sub_eq_add_neg] rw [h₂] at h₁ exact h₁ have ge : ⟪u - v, w⟫_ℝ ≥ 0 := by let w'' := -w + v have : w'' ∈ K := Submodule.add_mem _ (Submodule.neg_mem _ hw) hv have h₁ := h w'' this have h₂ : w'' - v = -w := by simp only [w'', neg_inj, add_neg_cancel_right, sub_eq_add_neg] rw [h₂, inner_neg_right] at h₁ linarith exact le_antisymm le ge) (by intro h have : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by intro w hw let w' := w - v have : w' ∈ K := Submodule.sub_mem _ hw hv have h₁ := h w' this exact le_of_eq h₁ rwa [norm_eq_iInf_iff_real_inner_le_zero] exacts [Submodule.convex _, hv]) #align norm_eq_infi_iff_real_inner_eq_zero norm_eq_iInf_iff_real_inner_eq_zero /-- Characterization of minimizers in the projection on a subspace. Let `u` be a point in an inner product space, and let `K` be a nonempty subspace. Then point `v` minimizes the distance `‖u - v‖` over points in `K` if and only if for all `w ∈ K`, `⟪u - v, w⟫ = 0` (i.e., `u - v` is orthogonal to the subspace `K`) -/ theorem norm_eq_iInf_iff_inner_eq_zero {u : E} {v : E} (hv : v ∈ K) : (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫ = 0 := by letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E letI : Module ℝ E := RestrictScalars.module ℝ 𝕜 E let K' : Submodule ℝ E := K.restrictScalars ℝ constructor · intro H have A : ∀ w ∈ K, re ⟪u - v, w⟫ = 0 := (norm_eq_iInf_iff_real_inner_eq_zero K' hv).1 H intro w hw apply ext · simp [A w hw] · symm calc im (0 : 𝕜) = 0 := im.map_zero _ = re ⟪u - v, (-I : 𝕜) • w⟫ := (A _ (K.smul_mem (-I) hw)).symm _ = re (-I * ⟪u - v, w⟫) := by rw [inner_smul_right] _ = im ⟪u - v, w⟫ := by simp · intro H have : ∀ w ∈ K', ⟪u - v, w⟫_ℝ = 0 := by intro w hw rw [real_inner_eq_re_inner, H w hw] exact zero_re' exact (norm_eq_iInf_iff_real_inner_eq_zero K' hv).2 this #align norm_eq_infi_iff_inner_eq_zero norm_eq_iInf_iff_inner_eq_zero /-- A subspace `K : Submodule 𝕜 E` has an orthogonal projection if evey vector `v : E` admits an orthogonal projection to `K`. -/ class HasOrthogonalProjection (K : Submodule 𝕜 E) : Prop where exists_orthogonal (v : E) : ∃ w ∈ K, v - w ∈ Kᗮ instance (priority := 100) HasOrthogonalProjection.ofCompleteSpace [CompleteSpace K] : HasOrthogonalProjection K where exists_orthogonal v := by rcases exists_norm_eq_iInf_of_complete_subspace K (completeSpace_coe_iff_isComplete.mp ‹_›) v with ⟨w, hwK, hw⟩ refine ⟨w, hwK, (K.mem_orthogonal' _).2 ?_⟩ rwa [← norm_eq_iInf_iff_inner_eq_zero K hwK] instance [HasOrthogonalProjection K] : HasOrthogonalProjection Kᗮ where exists_orthogonal v := by rcases HasOrthogonalProjection.exists_orthogonal (K := K) v with ⟨w, hwK, hw⟩ refine ⟨_, hw, ?_⟩ rw [sub_sub_cancel] exact K.le_orthogonal_orthogonal hwK instance HasOrthogonalProjection.map_linearIsometryEquiv [HasOrthogonalProjection K] {E' : Type} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') : HasOrthogonalProjection (K.map (f.toLinearEquiv : E →ₗ[𝕜] E')) where exists_orthogonal v := by rcases HasOrthogonalProjection.exists_orthogonal (K := K) (f.symm v) with ⟨w, hwK, hw⟩ refine ⟨f w, Submodule.mem_map_of_mem hwK, Set.forall_mem_image.2 fun u hu ↦ ?_⟩ erw [← f.symm.inner_map_map, f.symm_apply_apply, map_sub, f.symm_apply_apply, hw u hu] instance HasOrthogonalProjection.map_linearIsometryEquiv' [HasOrthogonalProjection K] {E' : Type} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') : HasOrthogonalProjection (K.map f.toLinearIsometry) := HasOrthogonalProjection.map_linearIsometryEquiv K f instance : HasOrthogonalProjection (⊤ : Submodule 𝕜 E) := ⟨fun v ↦ ⟨v, trivial, by simp⟩⟩ section orthogonalProjection variable [HasOrthogonalProjection K] /-- The orthogonal projection onto a complete subspace, as an unbundled function. This definition is only intended for use in setting up the bundled version `orthogonalProjection` and should not be used once that is defined. -/ def orthogonalProjectionFn (v : E) := (HasOrthogonalProjection.exists_orthogonal (K := K) v).choose #align orthogonal_projection_fn orthogonalProjectionFn variable {K} /-- The unbundled orthogonal projection is in the given subspace. This lemma is only intended for use in setting up the bundled version and should not be used once that is defined. -/ theorem orthogonalProjectionFn_mem (v : E) : orthogonalProjectionFn K v ∈ K := (HasOrthogonalProjection.exists_orthogonal (K := K) v).choose_spec.left #align orthogonal_projection_fn_mem orthogonalProjectionFn_mem /-- The characterization of the unbundled orthogonal projection. This lemma is only intended for use in setting up the bundled version and should not be used once that is defined. -/ theorem orthogonalProjectionFn_inner_eq_zero (v : E) : ∀ w ∈ K, ⟪v - orthogonalProjectionFn K v, w⟫ = 0 := (K.mem_orthogonal' _).1 (HasOrthogonalProjection.exists_orthogonal (K := K) v).choose_spec.right #align orthogonal_projection_fn_inner_eq_zero orthogonalProjectionFn_inner_eq_zero /-- The unbundled orthogonal projection is the unique point in `K` with the orthogonality property. This lemma is only intended for use in setting up the bundled version and should not be used once that is defined. -/ theorem eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero {u v : E} (hvm : v ∈ K) (hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0) : orthogonalProjectionFn K u = v := by rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜] have hvs : orthogonalProjectionFn K u - v ∈ K := Submodule.sub_mem K (orthogonalProjectionFn_mem u) hvm have huo : ⟪u - orthogonalProjectionFn K u, orthogonalProjectionFn K u - v⟫ = 0 := orthogonalProjectionFn_inner_eq_zero u _ hvs have huv : ⟪u - v, orthogonalProjectionFn K u - v⟫ = 0 := hvo _ hvs have houv : ⟪u - v - (u - orthogonalProjectionFn K u), orthogonalProjectionFn K u - v⟫ = 0 := by rw [inner_sub_left, huo, huv, sub_zero] rwa [sub_sub_sub_cancel_left] at houv #align eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero variable (K) theorem orthogonalProjectionFn_norm_sq (v : E) : ‖v‖ * ‖v‖ = ‖v - orthogonalProjectionFn K v‖ * ‖v - orthogonalProjectionFn K v‖ + ‖orthogonalProjectionFn K v‖ * ‖orthogonalProjectionFn K v‖ := by set p := orthogonalProjectionFn K v have h' : ⟪v - p, p⟫ = 0 := orthogonalProjectionFn_inner_eq_zero _ _ (orthogonalProjectionFn_mem v) convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2 <;> simp #align orthogonal_projection_fn_norm_sq orthogonalProjectionFn_norm_sq /-- The orthogonal projection onto a complete subspace. -/ def orthogonalProjection : E →L[𝕜] K := LinearMap.mkContinuous { toFun := fun v => ⟨orthogonalProjectionFn K v, orthogonalProjectionFn_mem v⟩ map_add' := fun x y => by have hm : orthogonalProjectionFn K x + orthogonalProjectionFn K y ∈ K := Submodule.add_mem K (orthogonalProjectionFn_mem x) (orthogonalProjectionFn_mem y) have ho : ∀ w ∈ K, ⟪x + y - (orthogonalProjectionFn K x + orthogonalProjectionFn K y), w⟫ = 0 := by intro w hw rw [add_sub_add_comm, inner_add_left, orthogonalProjectionFn_inner_eq_zero _ w hw, orthogonalProjectionFn_inner_eq_zero _ w hw, add_zero] ext simp [eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero hm ho] map_smul' := fun c x => by have hm : c • orthogonalProjectionFn K x ∈ K := Submodule.smul_mem K _ (orthogonalProjectionFn_mem x) have ho : ∀ w ∈ K, ⟪c • x - c • orthogonalProjectionFn K x, w⟫ = 0 := by intro w hw rw [← smul_sub, inner_smul_left, orthogonalProjectionFn_inner_eq_zero _ w hw, mul_zero] ext simp [eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero hm ho] } 1 fun x => by simp only [one_mul, LinearMap.coe_mk] refine le_of_pow_le_pow_left two_ne_zero (norm_nonneg _) ?_ change ‖orthogonalProjectionFn K x‖ ^ 2 ≤ ‖x‖ ^ 2 nlinarith [orthogonalProjectionFn_norm_sq K x] #align orthogonal_projection orthogonalProjection variable {K} @[simp] theorem orthogonalProjectionFn_eq (v : E) : orthogonalProjectionFn K v = (orthogonalProjection K v : E) := rfl #align orthogonal_projection_fn_eq orthogonalProjectionFn_eq /-- The characterization of the orthogonal projection. -/ @[simp] theorem orthogonalProjection_inner_eq_zero (v : E) : ∀ w ∈ K, ⟪v - orthogonalProjection K v, w⟫ = 0 := orthogonalProjectionFn_inner_eq_zero v #align orthogonal_projection_inner_eq_zero orthogonalProjection_inner_eq_zero /-- The difference of `v` from its orthogonal projection onto `K` is in `Kᗮ`. -/ @[simp] theorem sub_orthogonalProjection_mem_orthogonal (v : E) : v - orthogonalProjection K v ∈ Kᗮ := by intro w hw rw [inner_eq_zero_symm] exact orthogonalProjection_inner_eq_zero _ _ hw #align sub_orthogonal_projection_mem_orthogonal sub_orthogonalProjection_mem_orthogonal /-- The orthogonal projection is the unique point in `K` with the orthogonality property. -/ theorem eq_orthogonalProjection_of_mem_of_inner_eq_zero {u v : E} (hvm : v ∈ K) (hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0) : (orthogonalProjection K u : E) = v := eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero hvm hvo #align eq_orthogonal_projection_of_mem_of_inner_eq_zero eq_orthogonalProjection_of_mem_of_inner_eq_zero /-- A point in `K` with the orthogonality property (here characterized in terms of `Kᗮ`) must be the orthogonal projection. -/ theorem eq_orthogonalProjection_of_mem_orthogonal {u v : E} (hv : v ∈ K) (hvo : u - v ∈ Kᗮ) : (orthogonalProjection K u : E) = v := eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero hv <\| (Submodule.mem_orthogonal' _ _).1 hvo #align eq_orthogonal_projection_of_mem_orthogonal eq_orthogonalProjection_of_mem_orthogonal /-- A point in `K` with the orthogonality property (here characterized in terms of `Kᗮ`) must be the orthogonal projection. -/ theorem eq_orthogonalProjection_of_mem_orthogonal' {u v z : E} (hv : v ∈ K) (hz : z ∈ Kᗮ) (hu : u = v + z) : (orthogonalProjection K u : E) = v := eq_orthogonalProjection_of_mem_orthogonal hv (by simpa [hu] ) #align eq_orthogonal_projection_of_mem_orthogonal' eq_orthogonalProjection_of_mem_orthogonal' @[simp] theorem orthogonalProjection_orthogonal_val (u : E) : (orthogonalProjection Kᗮ u : E) = u - orthogonalProjection K u := eq_orthogonalProjection_of_mem_orthogonal' (sub_orthogonalProjection_mem_orthogonal _) (K.le_orthogonal_orthogonal (orthogonalProjection K u).2) <\| by simp theorem orthogonalProjection_orthogonal (u : E) : orthogonalProjection Kᗮ u = ⟨u - orthogonalProjection K u, sub_orthogonalProjection_mem_orthogonal _⟩ := Subtype.eq <\| orthogonalProjection_orthogonal_val _ /-- The orthogonal projection of `y` on `U` minimizes the distance `‖y - x‖` for `x ∈ U`. -/ theorem orthogonalProjection_minimal {U : Submodule 𝕜 E} [HasOrthogonalProjection U] (y : E) : ‖y - orthogonalProjection U y‖ = ⨅ x : U, ‖y - x‖ := by rw [norm_eq_iInf_iff_inner_eq_zero _ (Submodule.coe_mem _)] exact orthogonalProjection_inner_eq_zero _ #align orthogonal_projection_minimal orthogonalProjection_minimal /-- The orthogonal projections onto equal subspaces are coerced back to the same point in `E`. -/	Mathlib/Analysis/InnerProductSpace/Projection.lean	544	546	theorem eq_orthogonalProjection_of_eq_submodule {K' : Submodule 𝕜 E} [HasOrthogonalProjection K'] (h : K = K') (u : E) : (orthogonalProjection K u : E) = (orthogonalProjection K' u : E) := by	subst h; rfl
/- Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn, Heather Macbeth -/ import Mathlib.Geometry.Manifold.ContMDiff.Atlas import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear import Mathlib.Topology.VectorBundle.Constructions #align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833" /-! # Smooth vector bundles This file defines smooth vector bundles over a smooth manifold. Let `E` be a topological vector bundle, with model fiber `F` and base space `B`. We consider `E` as carrying a charted space structure given by its trivializations -- these are charts to `B × F`. Then, by "composition", if `B` is itself a charted space over `H` (e.g. a smooth manifold), then `E` is also a charted space over `H × F`. Now, we define `SmoothVectorBundle` as the `Prop` of having smooth transition functions. Recall the structure groupoid `smoothFiberwiseLinear` on `B × F` consisting of smooth, fiberwise linear partial homeomorphisms. We show that our definition of "smooth vector bundle" implies `HasGroupoid` for this groupoid, and show (by a "composition" of `HasGroupoid` instances) that this means that a smooth vector bundle is a smooth manifold. Since `SmoothVectorBundle` is a mixin, it should be easy to make variants and for many such variants to coexist -- vector bundles can be smooth vector bundles over several different base fields, they can also be C^k vector bundles, etc. ## Main definitions and constructions * `FiberBundle.chartedSpace`: A fiber bundle `E` over a base `B` with model fiber `F` is naturally a charted space modelled on `B × F`. * `FiberBundle.chartedSpace'`: Let `B` be a charted space modelled on `HB`. Then a fiber bundle `E` over a base `B` with model fiber `F` is naturally a charted space modelled on `HB.prod F`. * `SmoothVectorBundle`: Mixin class stating that a (topological) `VectorBundle` is smooth, in the sense of having smooth transition functions. * `SmoothFiberwiseLinear.hasGroupoid`: For a smooth vector bundle `E` over `B` with fiber modelled on `F`, the change-of-co-ordinates between two trivializations `e`, `e'` for `E`, considered as charts to `B × F`, is smooth and fiberwise linear, in the sense of belonging to the structure groupoid `smoothFiberwiseLinear`. * `Bundle.TotalSpace.smoothManifoldWithCorners`: A smooth vector bundle is naturally a smooth manifold. * `VectorBundleCore.smoothVectorBundle`: If a (topological) `VectorBundleCore` is smooth, in the sense of having smooth transition functions (cf. `VectorBundleCore.IsSmooth`), then the vector bundle constructed from it is a smooth vector bundle. * `VectorPrebundle.smoothVectorBundle`: If a `VectorPrebundle` is smooth, in the sense of having smooth transition functions (cf. `VectorPrebundle.IsSmooth`), then the vector bundle constructed from it is a smooth vector bundle. * `Bundle.Prod.smoothVectorBundle`: The direct sum of two smooth vector bundles is a smooth vector bundle. -/ assert_not_exists mfderiv open Bundle Set PartialHomeomorph open Function (id_def) open Filter open scoped Manifold Bundle Topology variable {𝕜 B B' F M : Type} {E : B → Type} /-! ### Charted space structure on a fiber bundle -/ section variable [TopologicalSpace F] [TopologicalSpace (TotalSpace F E)] [∀ x, TopologicalSpace (E x)] {HB : Type} [TopologicalSpace HB] [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] /-- A fiber bundle `E` over a base `B` with model fiber `F` is naturally a charted space modelled on `B × F`. -/ instance FiberBundle.chartedSpace' : ChartedSpace (B × F) (TotalSpace F E) where atlas := (fun e : Trivialization F (π F E) => e.toPartialHomeomorph) '' trivializationAtlas F E chartAt x := (trivializationAt F E x.proj).toPartialHomeomorph mem_chart_source x := (trivializationAt F E x.proj).mem_source.mpr (mem_baseSet_trivializationAt F E x.proj) chart_mem_atlas _ := mem_image_of_mem _ (trivialization_mem_atlas F E _) #align fiber_bundle.charted_space FiberBundle.chartedSpace' theorem FiberBundle.chartedSpace'_chartAt (x : TotalSpace F E) : chartAt (B × F) x = (trivializationAt F E x.proj).toPartialHomeomorph := rfl /- Porting note: In Lean 3, the next instance was inside a section with locally reducible `ModelProd` and it used `ModelProd B F` as the intermediate space. Using `B × F` in the middle gives the same instance. -/ --attribute [local reducible] ModelProd /-- Let `B` be a charted space modelled on `HB`. Then a fiber bundle `E` over a base `B` with model fiber `F` is naturally a charted space modelled on `HB.prod F`. -/ instance FiberBundle.chartedSpace : ChartedSpace (ModelProd HB F) (TotalSpace F E) := ChartedSpace.comp _ (B × F) _ #align fiber_bundle.charted_space' FiberBundle.chartedSpace theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) : chartAt (ModelProd HB F) x = (trivializationAt F E x.proj).toPartialHomeomorph ≫ₕ (chartAt HB x.proj).prod (PartialHomeomorph.refl F) := by dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt, chartAt_self_eq] rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)] #align fiber_bundle.charted_space_chart_at FiberBundle.chartedSpace_chartAt theorem FiberBundle.chartedSpace_chartAt_symm_fst (x : TotalSpace F E) (y : ModelProd HB F) (hy : y ∈ (chartAt (ModelProd HB F) x).target) : ((chartAt (ModelProd HB F) x).symm y).proj = (chartAt HB x.proj).symm y.1 := by simp only [FiberBundle.chartedSpace_chartAt, mfld_simps] at hy ⊢ exact (trivializationAt F E x.proj).proj_symm_apply hy.2 #align fiber_bundle.charted_space_chart_at_symm_fst FiberBundle.chartedSpace_chartAt_symm_fst end section variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (TotalSpace F E)] [∀ x, TopologicalSpace (E x)] {EB : Type} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) (E' : B → Type) [∀ x, Zero (E' x)] {EM : Type} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [Is : SmoothManifoldWithCorners IM M] {n : ℕ∞} variable [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] protected theorem FiberBundle.extChartAt (x : TotalSpace F E) : extChartAt (IB.prod 𝓘(𝕜, F)) x = (trivializationAt F E x.proj).toPartialEquiv ≫ (extChartAt IB x.proj).prod (PartialEquiv.refl F) := by simp_rw [extChartAt, FiberBundle.chartedSpace_chartAt, extend] simp only [PartialEquiv.trans_assoc, mfld_simps] -- Porting note: should not be needed rw [PartialEquiv.prod_trans, PartialEquiv.refl_trans] #align fiber_bundle.ext_chart_at FiberBundle.extChartAt protected theorem FiberBundle.extChartAt_target (x : TotalSpace F E) : (extChartAt (IB.prod 𝓘(𝕜, F)) x).target = ((extChartAt IB x.proj).target ∩ (extChartAt IB x.proj).symm ⁻¹' (trivializationAt F E x.proj).baseSet) ×ˢ univ := by rw [FiberBundle.extChartAt, PartialEquiv.trans_target, Trivialization.target_eq, inter_prod] rfl theorem FiberBundle.writtenInExtChartAt_trivializationAt {x : TotalSpace F E} {y} (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target) : writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) x (trivializationAt F E x.proj) y = y := writtenInExtChartAt_chartAt_comp _ _ hy theorem FiberBundle.writtenInExtChartAt_trivializationAt_symm {x : TotalSpace F E} {y} (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target) : writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (trivializationAt F E x.proj x) (trivializationAt F E x.proj).toPartialHomeomorph.symm y = y := writtenInExtChartAt_chartAt_symm_comp _ _ hy /-! ### Smoothness of maps in/out fiber bundles Note: For these results we don't need that the bundle is a smooth vector bundle, or even a vector bundle at all, just that it is a fiber bundle over a charted base space. -/ namespace Bundle variable {IB} /-- Characterization of C^n functions into a smooth vector bundle. -/ theorem contMDiffWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x₀ : M} : ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f s x₀ ↔ ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧ ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x ↦ (trivializationAt F E (f x₀).proj (f x)).2) s x₀ := by simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target] rw [and_and_and_comm, ← FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff] intro hf simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp, PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEquiv.refl_coe, extChartAt_self_apply, modelWithCornersSelf_coe, Function.id_def, ← chartedSpaceSelf_prod] refine (contMDiffWithinAt_prod_iff _).trans (and_congr ?_ Iff.rfl) have h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀ := ((FiberBundle.continuous_proj F E).continuousWithinAt.comp hf (mapsTo_image f s)) ((Trivialization.open_baseSet _).mem_nhds (mem_baseSet_trivializationAt F E _)) refine EventuallyEq.contMDiffWithinAt_iff (eventually_of_mem h1 fun x hx => ?_) ?_ · simp_rw [Function.comp, PartialHomeomorph.coe_coe, Trivialization.coe_coe] rw [Trivialization.coe_fst'] exact hx · simp only [mfld_simps] #align bundle.cont_mdiff_within_at_total_space Bundle.contMDiffWithinAt_totalSpace /-- Characterization of C^n functions into a smooth vector bundle. -/	Mathlib/Geometry/Manifold/VectorBundle/Basic.lean	200	204	theorem contMDiffAt_totalSpace (f : M → TotalSpace F E) (x₀ : M) : ContMDiffAt IM (IB.prod 𝓘(𝕜, F)) n f x₀ ↔ ContMDiffAt IM IB n (fun x => (f x).proj) x₀ ∧ ContMDiffAt IM 𝓘(𝕜, F) n (fun x => (trivializationAt F E (f x₀).proj (f x)).2) x₀ := by	simp_rw [← contMDiffWithinAt_univ]; exact contMDiffWithinAt_totalSpace f

Subsets and Splits

No community queries yet

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