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MeasurableSpace.generateMeasurableRec_subset ** α : Type u s : Set (Set α) i j : (Quotient.out (ord (aleph 1))).α h : i ≤ j x : Set α hx : x ∈ generateMeasurableRec s i ⊢ x ∈ generateMeasurableRec s j ** rcases eq_or_lt_of_le h with (rfl | h) ** case inl α : Type u s : Set (Set α) i : (Quotient.out (ord (aleph 1))).α x : Set α hx : x ∈ generateMeasurableRec s i h : i ≤ i ⊢ x ∈ generateMeasurableRec s i ** exact hx ** case inr α : Type u s : Set (Set α) i j : (Quotient.out (ord (aleph 1))).α h✝ : i ≤ j x : Set α hx : x ∈ generateMeasurableRec s i h : i < j ⊢ x ∈ generateMeasurableRec s j ** convert iUnion_mem_generateMeasurableRec fun _ => ⟨i, h, hx⟩ ** case h.e'_4 α : Type u s : Set (Set α) i j : (Quotient.out (ord (aleph 1))).α h✝ : i ≤ j x : Set α hx : x ∈ generateMeasurableRec s i h : i < j ⊢ x = ⋃ n, x ** exact (iUnion_const x).symm ** Qed
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Partrec.cond ** α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ c : α → Bool f g : α →. σ hc : Computable c hf : Partrec f hg : Partrec g cf : Code ef : eval cf = fun n => Part.bind ↑(decode n) fun a => Part.map encode (f a) cg : Code eg : eval cg = fun n => Part.bind ↑(decode n) fun a => Part.map encode (g a) a : α ⊢ (Part.bind (eval (bif c a then cf else cg) (encode a)) fun b => ↑(decode (a, b).2)) = bif c a then f a else g a ** cases c a <;> simp [ef, eg, encodek] ** Qed
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Partrec.dom_re ** α : Type u_1 β : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable β f : α →. β h : Partrec f n : α x✝ : Unit ⊢ x✝ ∈ Part.map (fun b => ()) (f n) ↔ x✝ ∈ Part.assert ((fun a => (f a).Dom) n) fun x => Part.some () ** simp [Part.dom_iff_mem] ** Qed
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ComputablePred.computable_iff ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ p : α → Prop ⊢ (∃ f, Computable f ∧ p = fun a => f a = true) → ComputablePred p ** rintro ⟨f, h, rfl⟩ ** case intro.intro α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool h : Computable f ⊢ ComputablePred fun a => f a = true ** exact ⟨by infer_instance, by simpa using h⟩ ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool h : Computable f ⊢ DecidablePred fun a => f a = true ** infer_instance ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool h : Computable f ⊢ Computable fun a => decide ((fun a => f a = true) a) ** simpa using h ** Qed
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ComputablePred.not ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ p : α → Prop hp : ComputablePred p ⊢ ComputablePred fun a => ¬p a ** obtain ⟨f, hf, rfl⟩ := computable_iff.1 hp ** case intro.intro α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool hf : Computable f hp : ComputablePred fun a => f a = true ⊢ ComputablePred fun a => ¬(fun a => f a = true) a ** exact
⟨by infer_instance,
(cond hf (const false) (const true)).of_eq fun n => by
simp only [Bool.not_eq_true]
cases f n <;> rfl⟩ ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool hf : Computable f hp : ComputablePred fun a => f a = true ⊢ DecidablePred fun a => ¬(fun a => f a = true) a ** infer_instance ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool hf : Computable f hp : ComputablePred fun a => f a = true n : α ⊢ (bif f n then false else true) = decide ((fun a => ¬(fun a => f a = true) a) n) ** simp only [Bool.not_eq_true] ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → Bool hf : Computable f hp : ComputablePred fun a => f a = true n : α ⊢ (bif f n then false else true) = decide (f n = false) ** cases f n <;> rfl ** Qed
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ComputablePred.rice₂ ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ C : Set Code H : ∀ (cf cg : Code), eval cf = eval cg → (cf ∈ C ↔ cg ∈ C) ⊢ (ComputablePred fun c => c ∈ C) ↔ C = ∅ ∨ C = Set.univ ** classical exact
have hC : ∀ f, f ∈ C ↔ eval f ∈ eval '' C := fun f =>
⟨Set.mem_image_of_mem _, fun ⟨g, hg, e⟩ => (H _ _ e).1 hg⟩
⟨fun h =>
or_iff_not_imp_left.2 fun C0 =>
Set.eq_univ_of_forall fun cg =>
let ⟨cf, fC⟩ := Set.nonempty_iff_ne_empty.2 C0
(hC _).2 <|
rice (eval '' C) (h.of_eq hC)
(Partrec.nat_iff.1 <| eval_part.comp (const cf) Computable.id)
(Partrec.nat_iff.1 <| eval_part.comp (const cg) Computable.id) ((hC _).1 fC),
fun h => by {
obtain rfl | rfl := h <;> simp [ComputablePred, Set.mem_empty_iff_false] <;>
exact ⟨by infer_instance, Computable.const _⟩ }⟩ ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ C : Set Code H : ∀ (cf cg : Code), eval cf = eval cg → (cf ∈ C ↔ cg ∈ C) ⊢ (ComputablePred fun c => c ∈ C) ↔ C = ∅ ∨ C = Set.univ ** exact
have hC : ∀ f, f ∈ C ↔ eval f ∈ eval '' C := fun f =>
⟨Set.mem_image_of_mem _, fun ⟨g, hg, e⟩ => (H _ _ e).1 hg⟩
⟨fun h =>
or_iff_not_imp_left.2 fun C0 =>
Set.eq_univ_of_forall fun cg =>
let ⟨cf, fC⟩ := Set.nonempty_iff_ne_empty.2 C0
(hC _).2 <|
rice (eval '' C) (h.of_eq hC)
(Partrec.nat_iff.1 <| eval_part.comp (const cf) Computable.id)
(Partrec.nat_iff.1 <| eval_part.comp (const cg) Computable.id) ((hC _).1 fC),
fun h => by {
obtain rfl | rfl := h <;> simp [ComputablePred, Set.mem_empty_iff_false] <;>
exact ⟨by infer_instance, Computable.const _⟩ }⟩ ** α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ H : ∀ (cf cg : Code), eval cf = eval cg → (cf ∈ Set.univ ↔ cg ∈ Set.univ) hC : ∀ (f : Code), f ∈ Set.univ ↔ eval f ∈ eval '' Set.univ ⊢ DecidablePred fun c => c ∈ Set.univ ** infer_instance ** Qed
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Nat.Partrec'.to_part ** n : ℕ f : Vector ℕ n →. ℕ pf : Partrec' f ⊢ _root_.Partrec f ** induction pf ** case prim n : ℕ f : Vector ℕ n →. ℕ n✝ : ℕ f✝ : Vector ℕ n✝ → ℕ a✝ : Primrec' f✝ ⊢ _root_.Partrec ↑f✝ case comp n : ℕ f : Vector ℕ n →. ℕ m✝ n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ g✝ : Fin n✝ → Vector ℕ m✝ →. ℕ a✝¹ : Partrec' f✝ a✝ : ∀ (i : Fin n✝), Partrec' (g✝ i) a_ih✝¹ : _root_.Partrec f✝ a_ih✝ : ∀ (i : Fin n✝), _root_.Partrec (g✝ i) ⊢ _root_.Partrec fun v => (mOfFn fun i => g✝ i v) >>= f✝ case rfind n : ℕ f : Vector ℕ n →. ℕ n✝ : ℕ f✝ : Vector ℕ (n✝ + 1) → ℕ a✝ : Partrec' ↑f✝ a_ih✝ : _root_.Partrec ↑f✝ ⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0)) ** case prim n f hf => exact hf.to_prim.to_comp ** case comp n : ℕ f : Vector ℕ n →. ℕ m✝ n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ g✝ : Fin n✝ → Vector ℕ m✝ →. ℕ a✝¹ : Partrec' f✝ a✝ : ∀ (i : Fin n✝), Partrec' (g✝ i) a_ih✝¹ : _root_.Partrec f✝ a_ih✝ : ∀ (i : Fin n✝), _root_.Partrec (g✝ i) ⊢ _root_.Partrec fun v => (mOfFn fun i => g✝ i v) >>= f✝ case rfind n : ℕ f : Vector ℕ n →. ℕ n✝ : ℕ f✝ : Vector ℕ (n✝ + 1) → ℕ a✝ : Partrec' ↑f✝ a_ih✝ : _root_.Partrec ↑f✝ ⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0)) ** case comp m n f g _ _ hf hg => exact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd) ** case rfind n : ℕ f : Vector ℕ n →. ℕ n✝ : ℕ f✝ : Vector ℕ (n✝ + 1) → ℕ a✝ : Partrec' ↑f✝ a_ih✝ : _root_.Partrec ↑f✝ ⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0)) ** case rfind n f _ hf =>
have := hf.comp (vector_cons.comp snd fst)
have :=
((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp
this).to₂.partrec₂
exact _root_.Partrec.rfind this ** n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ n : ℕ f : Vector ℕ n → ℕ hf : Primrec' f ⊢ _root_.Partrec ↑f ** exact hf.to_prim.to_comp ** n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ a✝¹ : Partrec' f a✝ : ∀ (i : Fin n), Partrec' (g i) hf : _root_.Partrec f hg : ∀ (i : Fin n), _root_.Partrec (g i) ⊢ _root_.Partrec fun v => (mOfFn fun i => g i v) >>= f ** exact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd) ** n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ n : ℕ f : Vector ℕ (n + 1) → ℕ a✝ : Partrec' ↑f hf : _root_.Partrec ↑f ⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0)) ** have := hf.comp (vector_cons.comp snd fst) ** n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ n : ℕ f : Vector ℕ (n + 1) → ℕ a✝ : Partrec' ↑f hf : _root_.Partrec ↑f this : _root_.Partrec fun a => ↑f (a.2 ::ᵥ a.1) ⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0)) ** have :=
((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp
this).to₂.partrec₂ ** n✝ : ℕ f✝ : Vector ℕ n✝ →. ℕ n : ℕ f : Vector ℕ (n + 1) → ℕ a✝ : Partrec' ↑f hf : _root_.Partrec ↑f this✝ : _root_.Partrec fun a => ↑f (a.2 ::ᵥ a.1) this : Partrec₂ fun a => ↑fun b => decide ((fun a => id a = 0) (f ((a, b).2 ::ᵥ (a, b).1))) ⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0)) ** exact _root_.Partrec.rfind this ** Qed
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Nat.Partrec'.tail ** n : ℕ f : Vector ℕ n →. ℕ hf : Partrec' f v : Vector ℕ (succ n) ⊢ (mOfFn fun i => (↑fun v => Vector.get v (Fin.succ i)) v) >>= f = f (Vector.tail v) ** simp ** n : ℕ f : Vector ℕ n →. ℕ hf : Partrec' f v : Vector ℕ (succ n) ⊢ f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v) ** rw [← ofFn_get v.tail] ** n : ℕ f : Vector ℕ n →. ℕ hf : Partrec' f v : Vector ℕ (succ n) ⊢ f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v))) ** congr ** case e_a.e_a n : ℕ f : Vector ℕ n →. ℕ hf : Partrec' f v : Vector ℕ (succ n) ⊢ (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v) ** funext i ** case e_a.e_a.h n : ℕ f : Vector ℕ n →. ℕ hf : Partrec' f v : Vector ℕ (succ n) i : Fin n ⊢ Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i ** simp ** Qed
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Nat.Partrec'.map ** n : ℕ f : Vector ℕ n →. ℕ g : Vector ℕ (n + 1) → ℕ hf : Partrec' f hg : Partrec' ↑g ⊢ Partrec' fun v => Part.map (fun a => g (a ::ᵥ v)) (f v) ** simp [(Part.bind_some_eq_map _ _).symm] ** n : ℕ f : Vector ℕ n →. ℕ g : Vector ℕ (n + 1) → ℕ hf : Partrec' f hg : Partrec' ↑g ⊢ Partrec' fun v => Part.bind (f v) (Part.some ∘ fun a => g (a ::ᵥ v)) ** exact hf.bind hg ** Qed
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Nat.Partrec'.cons ** n m : ℕ f : Vector ℕ n → ℕ g : Vector ℕ n → Vector ℕ m hf : Partrec' ↑f hg : Vec g i : Fin (succ m) ⊢ Partrec' fun v => ↑(some (Vector.get ((fun v => f v ::ᵥ g v) v) 0)) ** simpa using hf ** n m : ℕ f : Vector ℕ n → ℕ g : Vector ℕ n → Vector ℕ m hf : Partrec' ↑f hg : Vec g i✝ : Fin (succ m) i : Fin m ⊢ Partrec' fun v => ↑(some (Vector.get ((fun v => f v ::ᵥ g v) v) (Fin.succ i))) ** simp only [hg i, get_cons_succ] ** Qed
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Nat.Partrec'.comp' ** n m : ℕ f : Vector ℕ m →. ℕ g : Vector ℕ n → Vector ℕ m hf : Partrec' f hg : Vec g v : Vector ℕ n ⊢ (mOfFn fun i => ↑(some (Vector.get (g v) i))) >>= f = f (g v) ** simp ** Qed
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manyOneReducible_refl ** α : Type u_1 inst✝ : Primcodable α p : α → Prop ⊢ ∀ (a : α), p a ↔ p (id a) ** simp ** Qed
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ManyOneReducible.trans ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → β c₁ : Computable f h₁ : ∀ (a : α), p a ↔ q (f a) g : β → γ c₂ : Computable g h₂ : ∀ (a : β), q a ↔ r (g a) a : α h : p a ⊢ r ((g ∘ f) a) ** erw [← h₂, ← h₁] ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → β c₁ : Computable f h₁ : ∀ (a : α), p a ↔ q (f a) g : β → γ c₂ : Computable g h₂ : ∀ (a : β), q a ↔ r (g a) a : α h : p a ⊢ p a ** assumption ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → β c₁ : Computable f h₁ : ∀ (a : α), p a ↔ q (f a) g : β → γ c₂ : Computable g h₂ : ∀ (a : β), q a ↔ r (g a) a : α h : r ((g ∘ f) a) ⊢ p a ** rwa [h₁, h₂] ** Qed
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oneOneReducible_refl ** α : Type u_1 inst✝ : Primcodable α p : α → Prop ⊢ ∀ (a : α), p a ↔ p (id a) ** simp ** Qed
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OneOneReducible.trans ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → β c₁ : Computable f i₁ : Injective f h₁ : ∀ (a : α), p a ↔ q (f a) g : β → γ c₂ : Computable g i₂ : Injective g h₂ : ∀ (a : β), q a ↔ r (g a) a : α h : p a ⊢ r ((g ∘ f) a) ** erw [← h₂, ← h₁] ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → β c₁ : Computable f i₁ : Injective f h₁ : ∀ (a : α), p a ↔ q (f a) g : β → γ c₂ : Computable g i₂ : Injective g h₂ : ∀ (a : β), q a ↔ r (g a) a : α h : p a ⊢ p a ** assumption ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → β c₁ : Computable f i₁ : Injective f h₁ : ∀ (a : α), p a ↔ q (f a) g : β → γ c₂ : Computable g i₂ : Injective g h₂ : ∀ (a : β), q a ↔ r (g a) a : α h : r ((g ∘ f) a) ⊢ p a ** rwa [h₁, h₂] ** Qed
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OneOneReducible.of_equiv_symm ** α : Type u_1 β : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable β e : α ≃ β q : β → Prop h : Computable ↑e.symm ⊢ q ≤₁ (q ∘ ↑e) ** convert OneOneReducible.of_equiv _ h ** case h.e'_5 α : Type u_1 β : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable β e : α ≃ β q : β → Prop h : Computable ↑e.symm ⊢ q = (q ∘ ↑e) ∘ ↑e.symm ** funext ** case h.e'_5.h α : Type u_1 β : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable β e : α ≃ β q : β → Prop h : Computable ↑e.symm x✝ : β ⊢ q x✝ = ((q ∘ ↑e) ∘ ↑e.symm) x✝ ** simp ** Qed
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ComputablePred.computable_of_manyOneReducible ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ p : α → Prop q : β → Prop h₁ : p ≤₀ q h₂ : ComputablePred q ⊢ ComputablePred p ** rcases h₁ with ⟨f, c, hf⟩ ** case intro.intro α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ p : α → Prop q : β → Prop h₂ : ComputablePred q f : α → β c : Computable f hf : ∀ (a : α), p a ↔ q (f a) ⊢ ComputablePred p ** rw [show p = fun a => q (f a) from Set.ext hf] ** case intro.intro α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ p : α → Prop q : β → Prop h₂ : ComputablePred q f : α → β c : Computable f hf : ∀ (a : α), p a ↔ q (f a) ⊢ ComputablePred fun a => q (f a) ** rcases computable_iff.1 h₂ with ⟨g, hg, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ p : α → Prop f : α → β c : Computable f g : β → Bool hg : Computable g h₂ : ComputablePred fun a => g a = true hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a) ⊢ ComputablePred fun a => (fun a => g a = true) (f a) ** exact ⟨by infer_instance, by simpa using hg.comp c⟩ ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ p : α → Prop f : α → β c : Computable f g : β → Bool hg : Computable g h₂ : ComputablePred fun a => g a = true hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a) ⊢ DecidablePred fun a => (fun a => g a = true) (f a) ** infer_instance ** α : Type u_1 β : Type u_2 σ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable σ p : α → Prop f : α → β c : Computable f g : β → Bool hg : Computable g h₂ : ComputablePred fun a => g a = true hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a) ⊢ Computable fun a => decide ((fun a => (fun a => g a = true) (f a)) a) ** simpa using hg.comp c ** Qed
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disjoin_manyOneReducible ** α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : Primcodable α inst✝¹ : Primcodable β inst✝ : Primcodable γ p : α → Prop q : β → Prop r : γ → Prop f : α → γ c₁ : Computable f h₁ : ∀ (a : α), p a ↔ r (f a) g : β → γ c₂ : Computable g h₂ : ∀ (a : β), q a ↔ r (g a) x : α ⊕ β ⊢ (p ⊕' q) x ↔ r ((f ⊕' g) x) ** cases x <;> [apply h₁; apply h₂] ** Qed
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manyOneReducible_toNat ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p : Set α ⊢ ∀ (a : α), p a ↔ toNat p (Encodable.encode a) ** simp [toNat, setOf] ** Qed
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toNat_manyOneEquiv ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p : Set α ⊢ ManyOneEquiv (toNat p) p ** simp [ManyOneEquiv] ** Qed
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manyOneEquiv_toNat ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p : Set α q : Set β ⊢ ManyOneEquiv (toNat p) (toNat q) ↔ ManyOneEquiv p q ** simp [ManyOneEquiv] ** Qed
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ManyOneDegree.of_eq_of ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p : α → Prop q : β → Prop ⊢ of p = of q ↔ ManyOneEquiv p q ** rw [of, of, Quotient.eq''] ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p : α → Prop q : β → Prop ⊢ Setoid.r (toNat p) (toNat q) ↔ ManyOneEquiv p q ** unfold Setoid.r ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p : α → Prop q : β → Prop ⊢ { r := ManyOneEquiv, iseqv := proof_1 }.1 (toNat p) (toNat q) ↔ ManyOneEquiv p q ** simp ** Qed
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ManyOneDegree.le_refl ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ d : ManyOneDegree ⊢ d ≤ d ** induction d using ManyOneDegree.ind_on ** case h α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p✝ : Set ℕ ⊢ of p✝ ≤ of p✝ ** simp ** case h α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p✝ : Set ℕ ⊢ p✝ ≤₀ p✝ ** rfl ** Qed
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ManyOneDegree.le_trans ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ d₁ d₂ d₃ : ManyOneDegree ⊢ d₁ ≤ d₂ → d₂ ≤ d₃ → d₁ ≤ d₃ ** induction d₁ using ManyOneDegree.ind_on ** case h α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ d₂ d₃ : ManyOneDegree p✝ : Set ℕ ⊢ of p✝ ≤ d₂ → d₂ ≤ d₃ → of p✝ ≤ d₃ ** induction d₂ using ManyOneDegree.ind_on ** case h.h α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ d₃ : ManyOneDegree p✝¹ p✝ : Set ℕ ⊢ of p✝¹ ≤ of p✝ → of p✝ ≤ d₃ → of p✝¹ ≤ d₃ ** induction d₃ using ManyOneDegree.ind_on ** case h.h.h α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ p✝² p✝¹ p✝ : Set ℕ ⊢ of p✝² ≤ of p✝¹ → of p✝¹ ≤ of p✝ → of p✝² ≤ of p✝ ** apply ManyOneReducible.trans ** Qed
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ManyOneDegree.le_add_left ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ d₁ d₂ : ManyOneDegree ⊢ d₁ + ?m.118616 d₁ d₂ ≤ d₁ + d₂ ** rfl ** Qed
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ManyOneDegree.le_add_right ** α : Type u inst✝⁵ : Primcodable α inst✝⁴ : Inhabited α β : Type v inst✝³ : Primcodable β inst✝² : Inhabited β γ : Type w inst✝¹ : Primcodable γ inst✝ : Inhabited γ d₁ d₂ : ManyOneDegree ⊢ ?m.118797 d₁ d₂ + d₂ ≤ d₁ + d₂ ** rfl ** Qed
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MeasureTheory.lintegral_mul_left_eq_self ** G : Type u_1 inst✝³ : MeasurableSpace G μ : Measure G g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulLeftInvariant μ f : G → ℝ≥0∞ g : G ⊢ ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ ** convert (lintegral_map_equiv f <| MeasurableEquiv.mulLeft g).symm ** case h.e'_3.h.e'_3 G : Type u_1 inst✝³ : MeasurableSpace G μ : Measure G g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulLeftInvariant μ f : G → ℝ≥0∞ g : G ⊢ μ = map (↑(MeasurableEquiv.mulLeft g)) μ ** simp [map_mul_left_eq_self μ g] ** Qed
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MeasureTheory.lintegral_mul_right_eq_self ** G : Type u_1 inst✝³ : MeasurableSpace G μ : Measure G g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ f : G → ℝ≥0∞ g : G ⊢ ∫⁻ (x : G), f (x * g) ∂μ = ∫⁻ (x : G), f x ∂μ ** convert (lintegral_map_equiv f <| MeasurableEquiv.mulRight g).symm using 1 ** case h.e'_3 G : Type u_1 inst✝³ : MeasurableSpace G μ : Measure G g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ f : G → ℝ≥0∞ g : G ⊢ ∫⁻ (x : G), f x ∂μ = ∫⁻ (a : G), f a ∂map (↑(MeasurableEquiv.mulRight g)) μ ** simp [map_mul_right_eq_self μ g] ** Qed
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MeasureTheory.lintegral_eq_zero_of_isMulLeftInvariant ** G : Type u_1 inst✝⁶ : MeasurableSpace G μ : Measure G g : G inst✝⁵ : TopologicalSpace G inst✝⁴ : Group G inst✝³ : TopologicalGroup G inst✝² : BorelSpace G inst✝¹ : IsMulLeftInvariant μ inst✝ : Regular μ hμ : μ ≠ 0 f : G → ℝ≥0∞ hf : Continuous f ⊢ ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0 ** haveI := isOpenPosMeasure_of_mulLeftInvariant_of_regular hμ ** G : Type u_1 inst✝⁶ : MeasurableSpace G μ : Measure G g : G inst✝⁵ : TopologicalSpace G inst✝⁴ : Group G inst✝³ : TopologicalGroup G inst✝² : BorelSpace G inst✝¹ : IsMulLeftInvariant μ inst✝ : Regular μ hμ : μ ≠ 0 f : G → ℝ≥0∞ hf : Continuous f this : IsOpenPosMeasure μ ⊢ ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0 ** rw [lintegral_eq_zero_iff hf.measurable, hf.ae_eq_iff_eq μ continuous_zero] ** Qed
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MeasurableSet.const_smul ** G : Type u_1 α : Type u_2 inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSpace α inst✝ : MeasurableSMul G α s : Set α hs : MeasurableSet s a : G ⊢ MeasurableSet (a • s) ** rw [← preimage_smul_inv] ** G : Type u_1 α : Type u_2 inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSpace α inst✝ : MeasurableSMul G α s : Set α hs : MeasurableSet s a : G ⊢ MeasurableSet ((fun x => a⁻¹ • x) ⁻¹' s) ** exact measurable_const_smul _ hs ** Qed
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MeasurableSet.const_smul_of_ne_zero ** G₀ : Type u_1 α : Type u_2 inst✝⁴ : GroupWithZero G₀ inst✝³ : MulAction G₀ α inst✝² : MeasurableSpace G₀ inst✝¹ : MeasurableSpace α inst✝ : MeasurableSMul G₀ α s : Set α hs : MeasurableSet s a : G₀ ha : a ≠ 0 ⊢ MeasurableSet (a • s) ** rw [← preimage_smul_inv₀ ha] ** G₀ : Type u_1 α : Type u_2 inst✝⁴ : GroupWithZero G₀ inst✝³ : MulAction G₀ α inst✝² : MeasurableSpace G₀ inst✝¹ : MeasurableSpace α inst✝ : MeasurableSMul G₀ α s : Set α hs : MeasurableSet s a : G₀ ha : a ≠ 0 ⊢ MeasurableSet ((fun x => a⁻¹ • x) ⁻¹' s) ** exact measurable_const_smul _ hs ** Qed
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Computability.Encoding.encode_injective ** α : Type u e : Encoding α ⊢ Function.Injective e.encode ** refine' fun _ _ h => Option.some_injective _ _ ** α : Type u e : Encoding α x✝¹ x✝ : α h : encode e x✝¹ = encode e x✝ ⊢ some x✝¹ = some x✝ ** rw [← e.decode_encode, ← e.decode_encode, h] ** Qed
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Computability.decode_encodePosNum ** ⊢ ∀ (n : PosNum), decodePosNum (encodePosNum n) = n ** intro n ** n : PosNum ⊢ decodePosNum (encodePosNum n) = n ** induction' n with m hm m hm <;> unfold encodePosNum decodePosNum ** case one ⊢ (if [] = [] then PosNum.one else PosNum.bit1 (decodePosNum [])) = PosNum.one ** rfl ** case bit1 m : PosNum hm : decodePosNum (encodePosNum m) = m ⊢ (if encodePosNum m = [] then PosNum.one else PosNum.bit1 (decodePosNum (encodePosNum m))) = PosNum.bit1 m ** rw [hm] ** case bit1 m : PosNum hm : decodePosNum (encodePosNum m) = m ⊢ (if encodePosNum m = [] then PosNum.one else PosNum.bit1 m) = PosNum.bit1 m ** exact if_neg (encodePosNum_nonempty m) ** case bit0 m : PosNum hm : decodePosNum (encodePosNum m) = m ⊢ PosNum.bit0 (decodePosNum (encodePosNum m)) = PosNum.bit0 m ** exact congr_arg PosNum.bit0 hm ** Qed
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Computability.Encoding.card_le_aleph0 ** α : Type u e : Encoding α inst✝ : Encodable e.Γ ⊢ #α ≤ ℵ₀ ** refine' Cardinal.lift_le.1 (e.card_le_card_list.trans _) ** α : Type u e : Encoding α inst✝ : Encodable e.Γ ⊢ lift.{u, v} #(List e.Γ) ≤ lift.{v, u} ℵ₀ ** simp only [Cardinal.lift_aleph0, Cardinal.lift_le_aleph0] ** α : Type u e : Encoding α inst✝ : Encodable e.Γ ⊢ #(List e.Γ) ≤ ℵ₀ ** cases' isEmpty_or_nonempty e.Γ with h h ** case inl α : Type u e : Encoding α inst✝ : Encodable e.Γ h : IsEmpty e.Γ ⊢ #(List e.Γ) ≤ ℵ₀ ** simp only [Cardinal.mk_le_aleph0] ** case inr α : Type u e : Encoding α inst✝ : Encodable e.Γ h : Nonempty e.Γ ⊢ #(List e.Γ) ≤ ℵ₀ ** rw [Cardinal.mk_list_eq_aleph0] ** Qed
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ack_zero ** n : ℕ ⊢ ack 0 n = n + 1 ** rw [ack] ** Qed
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ack_succ_zero ** m : ℕ ⊢ ack (m + 1) 0 = ack m 1 ** rw [ack] ** Qed
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ack_succ_succ ** m n : ℕ ⊢ ack (m + 1) (n + 1) = ack m (ack (m + 1) n) ** rw [ack] ** Qed
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ack_one ** n : ℕ ⊢ ack 1 n = n + 2 ** induction' n with n IH ** case zero ⊢ ack 1 zero = zero + 2 ** rfl ** case succ n : ℕ IH : ack 1 n = n + 2 ⊢ ack 1 (succ n) = succ n + 2 ** simp [IH] ** Qed
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ack_two ** n : ℕ ⊢ ack 2 n = 2 * n + 3 ** induction' n with n IH ** case zero ⊢ ack 2 zero = 2 * zero + 3 ** rfl ** case succ n : ℕ IH : ack 2 n = 2 * n + 3 ⊢ ack 2 (succ n) = 2 * succ n + 3 ** simpa [mul_succ] ** Qed
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ack_pos ** n : ℕ ⊢ 0 < ack 0 n ** simp ** m : ℕ ⊢ 0 < ack (m + 1) 0 ** rw [ack_succ_zero] ** m : ℕ ⊢ 0 < ack m 1 ** apply ack_pos ** m n : ℕ ⊢ 0 < ack (m + 1) (n + 1) ** rw [ack_succ_succ] ** m n : ℕ ⊢ 0 < ack m (ack (m + 1) n) ** apply ack_pos ** Qed
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one_lt_ack_succ_left ** n : ℕ ⊢ 1 < ack (0 + 1) n ** simp ** m : ℕ ⊢ 1 < ack (m + 1 + 1) 0 ** rw [ack_succ_zero] ** m : ℕ ⊢ 1 < ack (m + 1) 1 ** apply one_lt_ack_succ_left ** m n : ℕ ⊢ 1 < ack (m + 1 + 1) (n + 1) ** rw [ack_succ_succ] ** m n : ℕ ⊢ 1 < ack (m + 1) (ack (m + 1 + 1) n) ** apply one_lt_ack_succ_left ** Qed
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one_lt_ack_succ_right ** n : ℕ ⊢ 1 < ack 0 (n + 1) ** simp ** m n : ℕ ⊢ 1 < ack (m + 1) (n + 1) ** rw [ack_succ_succ] ** m n : ℕ ⊢ 1 < ack m (ack (m + 1) n) ** cases' exists_eq_succ_of_ne_zero (ack_pos (m + 1) n).ne' with h h ** case intro m n h✝ : ℕ h : ack (m + 1) n = succ h✝ ⊢ 1 < ack m (ack (m + 1) n) ** rw [h] ** case intro m n h✝ : ℕ h : ack (m + 1) n = succ h✝ ⊢ 1 < ack m (succ h✝) ** apply one_lt_ack_succ_right ** Qed
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add_lt_ack ** n : ℕ ⊢ 0 + n < ack 0 n ** simp ** m : ℕ ⊢ m + 1 + 0 < ack (m + 1) 0 ** simpa using add_lt_ack m 1 ** m n : ℕ ⊢ m + 1 + n + 1 ≤ m + (m + n + 2) ** linarith ** m n : ℕ ⊢ m + n + 2 = succ (m + 1 + n) ** rw [succ_eq_add_one] ** m n : ℕ ⊢ m + n + 2 = m + 1 + n + 1 ** ring_nf ** Qed
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ack_succ_right_le_ack_succ_left ** m n : ℕ ⊢ ack m (n + 1) ≤ ack (m + 1) n ** cases' n with n n ** case zero m : ℕ ⊢ ack m (zero + 1) ≤ ack (m + 1) zero ** simp ** case succ m n : ℕ ⊢ ack m (succ n + 1) ≤ ack (m + 1) (succ n) ** rw [ack_succ_succ, succ_eq_add_one] ** case succ m n : ℕ ⊢ ack m (n + 1 + 1) ≤ ack m (ack (m + 1) n) ** apply ack_mono_right m (le_trans _ <| add_add_one_le_ack _ n) ** m n : ℕ ⊢ n + 1 + 1 ≤ m + 1 + n + 1 ** linarith ** Qed
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sq_le_two_pow_add_one_minus_three ** n : ℕ ⊢ n ^ 2 ≤ 2 ^ (n + 1) - 3 ** induction' n with k hk ** case zero ⊢ zero ^ 2 ≤ 2 ^ (zero + 1) - 3 ** norm_num ** case succ k : ℕ hk : k ^ 2 ≤ 2 ^ (k + 1) - 3 ⊢ succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ** cases' k with k k ** case succ.zero hk : zero ^ 2 ≤ 2 ^ (zero + 1) - 3 ⊢ succ zero ^ 2 ≤ 2 ^ (succ zero + 1) - 3 ** norm_num ** case succ.succ k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ succ (succ k) ^ 2 ≤ 2 ^ (succ (succ k) + 1) - 3 ** rw [succ_eq_add_one, add_sq, Nat.pow_succ 2, mul_comm _ 2, two_mul (2 ^ _),
add_tsub_assoc_of_le, add_comm (2 ^ _), add_assoc] ** case succ.succ k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ (k + 1) ^ 2 + (2 * (k + 1) * 1 + 1 ^ 2) ≤ 2 ^ (k + 2) - 3 + 2 ^ (k + 2) ** apply Nat.add_le_add hk ** case succ.succ k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ 2 * (k + 1) * 1 + 1 ^ 2 ≤ 2 ^ (k + 2) ** norm_num ** case succ.succ k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ 2 * (k + 1) + 1 ≤ 2 ^ (k + 2) ** apply succ_le_of_lt ** case succ.succ.h k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ 2 * (k + 1) < 2 ^ (k + 2) ** rw [Nat.pow_succ, mul_comm _ 2, mul_lt_mul_left (zero_lt_two' ℕ)] ** case succ.succ.h k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ k + 1 < 2 ^ (k + 1) ** apply lt_two_pow ** case succ.succ.h k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ 3 ≤ 2 ^ (k + 2) ** rw [Nat.pow_succ, Nat.pow_succ] ** case succ.succ.h k : ℕ hk : succ k ^ 2 ≤ 2 ^ (succ k + 1) - 3 ⊢ 3 ≤ 2 ^ k * 2 * 2 ** linarith [one_le_pow k 2 zero_lt_two] ** Qed
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MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd ** E : Type u_1 L : Type u_2 inst✝⁶ : MeasurableSpace E μ : Measure E F s : Set E inst✝⁵ : AddCommGroup L inst✝⁴ : Countable L inst✝³ : AddAction L E inst✝² : MeasurableSpace L inst✝¹ : MeasurableVAdd L E inst✝ : VAddInvariantMeasure L E μ fund : IsAddFundamentalDomain L F hS : NullMeasurableSet s h : ↑↑μ F < ↑↑μ s ⊢ ∃ x y, x ≠ y ∧ ¬Disjoint (x +ᵥ s) (y +ᵥ s) ** contrapose! h ** E : Type u_1 L : Type u_2 inst✝⁶ : MeasurableSpace E μ : Measure E F s : Set E inst✝⁵ : AddCommGroup L inst✝⁴ : Countable L inst✝³ : AddAction L E inst✝² : MeasurableSpace L inst✝¹ : MeasurableVAdd L E inst✝ : VAddInvariantMeasure L E μ fund : IsAddFundamentalDomain L F hS : NullMeasurableSet s h : ∀ (x y : L), x ≠ y → Disjoint (x +ᵥ s) (y +ᵥ s) ⊢ ↑↑μ s ≤ ↑↑μ F ** exact ((fund.measure_eq_tsum _).trans (measure_iUnion₀
(Pairwise.mono h fun i j hij => (hij.mono inf_le_left inf_le_left).aedisjoint)
fun _ => (hS.vadd _).inter fund.nullMeasurableSet).symm).trans_le
(measure_mono <| Set.iUnion_subset fun _ => Set.inter_subset_right _ _) ** Qed
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NFA.mem_stepSet ** α : Type u σ σ' : Type v M : NFA α σ s : σ S : Set σ a : α ⊢ s ∈ stepSet M S a ↔ ∃ t, t ∈ S ∧ s ∈ step M t a ** simp [stepSet] ** Qed
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NFA.stepSet_empty ** α : Type u σ σ' : Type v M : NFA α σ a : α ⊢ stepSet M ∅ a = ∅ ** simp [stepSet] ** Qed
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NFA.evalFrom_append_singleton ** α : Type u σ σ' : Type v M : NFA α σ S : Set σ x : List α a : α ⊢ evalFrom M S (x ++ [a]) = stepSet M (evalFrom M S x) a ** simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil] ** Qed
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NFA.mem_accepts ** α : Type u σ σ' : Type v M : NFA α σ x : List α ⊢ x ∈ accepts M ↔ ∃ S, S ∈ M.accept ∧ S ∈ evalFrom M M.start x ** rfl ** Qed
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NFA.toDFA_correct ** α : Type u σ σ' : Type v M : NFA α σ ⊢ DFA.accepts (toDFA M) = accepts M ** ext x ** case h α : Type u σ σ' : Type v M : NFA α σ x : List α ⊢ x ∈ DFA.accepts (toDFA M) ↔ x ∈ accepts M ** rw [mem_accepts, DFA.mem_accepts] ** case h.mpr α : Type u σ σ' : Type v M : NFA α σ x : List α ⊢ (∃ S, S ∈ M.accept ∧ S ∈ evalFrom M M.start x) → DFA.evalFrom (toDFA M) (toDFA M).start x ∈ (toDFA M).accept ** exact fun ⟨w, h2, h3⟩ => ⟨w, h3, h2⟩ ** Qed
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DFA.toNFA_evalFrom_match ** α : Type u σ σ' : Type v M✝ : NFA α σ M : DFA α σ start : σ s : List α ⊢ NFA.evalFrom (toNFA M) {start} s = {evalFrom M start s} ** change List.foldl M.toNFA.stepSet {start} s = {List.foldl M.step start s} ** α : Type u σ σ' : Type v M✝ : NFA α σ M : DFA α σ start : σ s : List α ⊢ List.foldl (NFA.stepSet (toNFA M)) {start} s = {List.foldl M.step start s} ** induction' s with a s ih generalizing start ** case nil α : Type u σ σ' : Type v M✝ : NFA α σ M : DFA α σ start✝ start : σ ⊢ List.foldl (NFA.stepSet (toNFA M)) {start} [] = {List.foldl M.step start []} ** tauto ** case cons α : Type u σ σ' : Type v M✝ : NFA α σ M : DFA α σ start✝ : σ a : α s : List α ih : ∀ (start : σ), List.foldl (NFA.stepSet (toNFA M)) {start} s = {List.foldl M.step start s} start : σ ⊢ List.foldl (NFA.stepSet (toNFA M)) {start} (a :: s) = {List.foldl M.step start (a :: s)} ** rw [List.foldl, List.foldl,
show M.toNFA.stepSet {start} a = {M.step start a} by simp [NFA.stepSet] ] ** case cons α : Type u σ σ' : Type v M✝ : NFA α σ M : DFA α σ start✝ : σ a : α s : List α ih : ∀ (start : σ), List.foldl (NFA.stepSet (toNFA M)) {start} s = {List.foldl M.step start s} start : σ ⊢ List.foldl (NFA.stepSet (toNFA M)) {step M start a} s = {List.foldl M.step (step M start a) s} ** tauto ** α : Type u σ σ' : Type v M✝ : NFA α σ M : DFA α σ start✝ : σ a : α s : List α ih : ∀ (start : σ), List.foldl (NFA.stepSet (toNFA M)) {start} s = {List.foldl M.step start s} start : σ ⊢ NFA.stepSet (toNFA M) {start} a = {step M start a} ** simp [NFA.stepSet] ** Qed
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MeasurableSet.univ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α ⊢ MeasurableSet univᶜ ** simp ** Qed
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MeasurableSet.congr ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t✝ u : Set α m : MeasurableSpace α s t : Set α hs : MeasurableSet s h : s = t ⊢ MeasurableSet t ** rwa [← h] ** Qed
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MeasurableSet.iUnion ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) ⊢ MeasurableSet (⋃ b, f b) ** cases isEmpty_or_nonempty ι ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) h✝ : IsEmpty ι ⊢ MeasurableSet (⋃ b, f b) ** simp ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) h✝ : Nonempty ι ⊢ MeasurableSet (⋃ b, f b) ** rcases exists_surjective_nat ι with ⟨e, he⟩ ** case inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) h✝ : Nonempty ι e : ℕ → ι he : Surjective e ⊢ MeasurableSet (⋃ b, f b) ** rw [← iUnion_congr_of_surjective _ he (fun _ => rfl)] ** case inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) h✝ : Nonempty ι e : ℕ → ι he : Surjective e ⊢ MeasurableSet (⋃ x, f (e x)) ** exact m.measurableSet_iUnion _ fun _ => h _ ** Qed
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MeasurableSet.biUnion ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α f : β → Set α s : Set β hs : Set.Countable s h : ∀ (b : β), b ∈ s → MeasurableSet (f b) ⊢ MeasurableSet (⋃ b ∈ s, f b) ** rw [biUnion_eq_iUnion] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α f : β → Set α s : Set β hs : Set.Countable s h : ∀ (b : β), b ∈ s → MeasurableSet (f b) ⊢ MeasurableSet (⋃ x, f ↑x) ** have := hs.to_subtype ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α f : β → Set α s : Set β hs : Set.Countable s h : ∀ (b : β), b ∈ s → MeasurableSet (f b) this : Countable ↑s ⊢ MeasurableSet (⋃ x, f ↑x) ** exact MeasurableSet.iUnion (by simpa using h) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α f : β → Set α s : Set β hs : Set.Countable s h : ∀ (b : β), b ∈ s → MeasurableSet (f b) this : Countable ↑s ⊢ ∀ (b : ↑s), MeasurableSet (f ↑b) ** simpa using h ** Qed
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MeasurableSet.sUnion ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → MeasurableSet t ⊢ MeasurableSet (⋃₀ s) ** rw [sUnion_eq_biUnion] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → MeasurableSet t ⊢ MeasurableSet (⋃ i ∈ s, i) ** exact .biUnion hs h ** Qed
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MeasurableSet.iInter ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) ⊢ MeasurableSet (⋂ b, f b)ᶜ ** rw [compl_iInter] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α inst✝ : Countable ι f : ι → Set α h : ∀ (b : ι), MeasurableSet (f b) ⊢ MeasurableSet (⋃ i, (f i)ᶜ) ** exact .iUnion fun b => (h b).compl ** Qed
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MeasurableSet.biInter ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α f : β → Set α s : Set β hs : Set.Countable s h : ∀ (b : β), b ∈ s → MeasurableSet (f b) ⊢ MeasurableSet (⋂ b ∈ s, f b)ᶜ ** rw [compl_iInter₂] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α f : β → Set α s : Set β hs : Set.Countable s h : ∀ (b : β), b ∈ s → MeasurableSet (f b) ⊢ MeasurableSet (⋃ i ∈ s, (f i)ᶜ) ** exact .biUnion hs fun b hb => (h b hb).compl ** Qed
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MeasurableSet.sInter ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → MeasurableSet t ⊢ MeasurableSet (⋂₀ s) ** rw [sInter_eq_biInter] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α m : MeasurableSpace α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → MeasurableSet t ⊢ MeasurableSet (⋂ i ∈ s, i) ** exact MeasurableSet.biInter hs h ** Qed
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MeasurableSet.union ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α s₁ s₂ : Set α h₁ : MeasurableSet s₁ h₂ : MeasurableSet s₂ ⊢ MeasurableSet (s₁ ∪ s₂) ** rw [union_eq_iUnion] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α s₁ s₂ : Set α h₁ : MeasurableSet s₁ h₂ : MeasurableSet s₂ ⊢ MeasurableSet (⋃ b, bif b then s₁ else s₂) ** exact .iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩) ** Qed
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MeasurableSet.ite' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t✝ u : Set α m : MeasurableSpace α s t : Set α p : Prop hs : p → MeasurableSet s ht : ¬p → MeasurableSet t ⊢ MeasurableSet (if p then s else t) ** split_ifs with h ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t✝ u : Set α m : MeasurableSpace α s t : Set α p : Prop hs : p → MeasurableSet s ht : ¬p → MeasurableSet t h : p ⊢ MeasurableSet s case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t✝ u : Set α m : MeasurableSpace α s t : Set α p : Prop hs : p → MeasurableSet s ht : ¬p → MeasurableSet t h : ¬p ⊢ MeasurableSet t ** exacts [hs h, ht h] ** Qed
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MeasurableSet.cond ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α s₁ s₂ : Set α h₁ : MeasurableSet s₁ h₂ : MeasurableSet s₂ i : Bool ⊢ MeasurableSet (bif i then s₁ else s₂) ** cases i ** case false α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α s₁ s₂ : Set α h₁ : MeasurableSet s₁ h₂ : MeasurableSet s₂ ⊢ MeasurableSet (bif false then s₁ else s₂) case true α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α s₁ s₂ : Set α h₁ : MeasurableSet s₁ h₂ : MeasurableSet s₂ ⊢ MeasurableSet (bif true then s₁ else s₂) ** exacts [h₂, h₁] ** Qed
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MeasurableSet.const ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α m : MeasurableSpace α p : Prop ⊢ MeasurableSet {_a | p} ** by_cases p <;> simp [*] ** Qed
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MeasurableSpace.measurableSet_injective ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α MeasurableSet'✝¹ : Set α → Prop measurableSet_empty✝¹ : MeasurableSet'✝¹ ∅ measurableSet_compl✝¹ : ∀ (s : Set α), MeasurableSet'✝¹ s → MeasurableSet'✝¹ sᶜ measurableSet_iUnion✝¹ : ∀ (f : ℕ → Set α), (∀ (i : ℕ), MeasurableSet'✝¹ (f i)) → MeasurableSet'✝¹ (⋃ i, f i) MeasurableSet'✝ : Set α → Prop measurableSet_empty✝ : MeasurableSet'✝ ∅ measurableSet_compl✝ : ∀ (s : Set α), MeasurableSet'✝ s → MeasurableSet'✝ sᶜ measurableSet_iUnion✝ : ∀ (f : ℕ → Set α), (∀ (i : ℕ), MeasurableSet'✝ (f i)) → MeasurableSet'✝ (⋃ i, f i) x✝ : MeasurableSet = MeasurableSet ⊢ { MeasurableSet' := MeasurableSet'✝¹, measurableSet_empty := measurableSet_empty✝¹, measurableSet_compl := measurableSet_compl✝¹, measurableSet_iUnion := measurableSet_iUnion✝¹ } = { MeasurableSet' := MeasurableSet'✝, measurableSet_empty := measurableSet_empty✝, measurableSet_compl := measurableSet_compl✝, measurableSet_iUnion := measurableSet_iUnion✝ } ** congr ** Qed
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measurableSet_insert ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α a : α s : Set α h : MeasurableSet (insert a s) ha : a ∈ s ⊢ MeasurableSet s ** rwa [← insert_eq_of_mem ha] ** Qed
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Set.Countable.measurableSet ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α s : Set α hs : Set.Countable s ⊢ MeasurableSet s ** rw [← biUnion_of_singleton s] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α s : Set α hs : Set.Countable s ⊢ MeasurableSet (⋃ x ∈ s, {x}) ** exact .biUnion hs fun b _ => .singleton b ** Qed
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MeasurableSpace.generateFrom_induction ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α p : Set α → Prop C : Set (Set α) hC : ∀ (t : Set α), t ∈ C → p t h_empty : p ∅ h_compl : ∀ (t : Set α), p t → p tᶜ h_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ i, f i) s : Set α hs : MeasurableSet s ⊢ p s ** induction hs ** case basic α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α p : Set α → Prop C : Set (Set α) hC : ∀ (t : Set α), t ∈ C → p t h_empty : p ∅ h_compl : ∀ (t : Set α), p t → p tᶜ h_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ i, f i) s u✝ : Set α a✝ : u✝ ∈ C ⊢ p u✝ case empty α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α p : Set α → Prop C : Set (Set α) hC : ∀ (t : Set α), t ∈ C → p t h_empty : p ∅ h_compl : ∀ (t : Set α), p t → p tᶜ h_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ i, f i) s : Set α ⊢ p ∅ case compl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α p : Set α → Prop C : Set (Set α) hC : ∀ (t : Set α), t ∈ C → p t h_empty : p ∅ h_compl : ∀ (t : Set α), p t → p tᶜ h_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ i, f i) s t✝ : Set α a✝ : GenerateMeasurable C t✝ a_ih✝ : p t✝ ⊢ p t✝ᶜ case iUnion α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α p : Set α → Prop C : Set (Set α) hC : ∀ (t : Set α), t ∈ C → p t h_empty : p ∅ h_compl : ∀ (t : Set α), p t → p tᶜ h_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ i, f i) s : Set α f✝ : ℕ → Set α a✝ : ∀ (n : ℕ), GenerateMeasurable C (f✝ n) a_ih✝ : ∀ (n : ℕ), p (f✝ n) ⊢ p (⋃ i, f✝ i) ** exacts [hC _ ‹_›, h_empty, h_compl _ ‹_›, h_Union ‹_› ‹_›] ** Qed
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MeasurableSpace.forall_generateFrom_mem_iff_mem_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α S : Set (Set α) x y : α ⊢ (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) ↔ ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) ** refine ⟨fun H s hs ↦ H s (.basic s hs), fun H s ↦ ?_⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α S : Set (Set α) x y : α H : ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) s : Set α ⊢ MeasurableSet s → (x ∈ s ↔ y ∈ s) ** apply generateFrom_induction ** case hC α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α S : Set (Set α) x y : α H : ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) s : Set α ⊢ ∀ (t : Set α), t ∈ S → (x ∈ t ↔ y ∈ t) ** exact H ** case h_empty α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α S : Set (Set α) x y : α H : ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) s : Set α ⊢ x ∈ ∅ ↔ y ∈ ∅ ** rfl ** case h_compl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α S : Set (Set α) x y : α H : ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) s : Set α ⊢ ∀ (t : Set α), (x ∈ t ↔ y ∈ t) → (x ∈ tᶜ ↔ y ∈ tᶜ) ** exact fun _ ↦ Iff.not ** case h_Union α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α S : Set (Set α) x y : α H : ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) s : Set α ⊢ ∀ (f : ℕ → Set α), (∀ (n : ℕ), x ∈ f n ↔ y ∈ f n) → (x ∈ ⋃ i, f i ↔ y ∈ ⋃ i, f i) ** intro f hf ** case h_Union α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α S : Set (Set α) x y : α H : ∀ (s : Set α), s ∈ S → (x ∈ s ↔ y ∈ s) s : Set α f : ℕ → Set α hf : ∀ (n : ℕ), x ∈ f n ↔ y ∈ f n ⊢ x ∈ ⋃ i, f i ↔ y ∈ ⋃ i, f i ** simp only [mem_iUnion, hf] ** Qed
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MeasurableSpace.generateFrom_singleton_empty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α ⊢ ∀ (t : Set α), t ∈ {∅} → MeasurableSet t ** simp [@MeasurableSet.empty α ⊥] ** Qed
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MeasurableSpace.generateFrom_singleton_univ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α ⊢ ∀ (t : Set α), t ∈ {univ} → MeasurableSet t ** simp ** Qed
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MeasurableSpace.generateFrom_insert_empty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s t u : Set α S : Set (Set α) ⊢ generateFrom (insert ∅ S) = generateFrom S ** rw [insert_eq, ← generateFrom_sup_generateFrom, generateFrom_singleton_empty, bot_sup_eq] ** Qed
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MeasurableSpace.measurableSet_bot_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u s : Set α ⊢ ∀ (s : Set α), (fun s => s = ∅ ∨ s = univ) s → (fun s => s = ∅ ∨ s = univ) sᶜ ** simp (config := { contextual := true }) [or_imp] ** Qed
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MeasurableSpace.measurableSet_sInf ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α ms : Set (MeasurableSpace α) s : Set α ⊢ s ∈ ⋂₀ ((fun m => {t | MeasurableSet t}) '' ms) ↔ ∀ (m : MeasurableSpace α), m ∈ ms → MeasurableSet s ** simp ** Qed
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MeasurableSpace.measurableSet_sSup ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α ms : Set (MeasurableSpace α) s : Set α ⊢ MeasurableSet s ↔ GenerateMeasurable {s | ∃ m, m ∈ ms ∧ MeasurableSet s} s ** change GenerateMeasurable (⋃₀ _) _ ↔ _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort u_6 s✝ t u : Set α ms : Set (MeasurableSpace α) s : Set α ⊢ GenerateMeasurable (⋃₀ ((fun m => {t | MeasurableSet t}) '' ms)) s ↔ GenerateMeasurable {s | ∃ m, m ∈ ms ∧ MeasurableSet s} s ** simp [← setOf_exists] ** Qed
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MeasureTheory.integral_inv_eq_self ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ✝ : Measure G f✝ : G → E g : G inst✝² : Group G inst✝¹ : MeasurableInv G f : G → E μ : Measure G inst✝ : IsInvInvariant μ ⊢ ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ ** have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ✝ : Measure G f✝ : G → E g : G inst✝² : Group G inst✝¹ : MeasurableInv G f : G → E μ : Measure G inst✝ : IsInvInvariant μ h : MeasurableEmbedding fun x => x⁻¹ ⊢ ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ ** rw [← h.integral_map, map_inv_eq_self] ** Qed
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MeasureTheory.integral_mul_left_eq_self ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulLeftInvariant μ f : G → E g : G ⊢ ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ ** have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulLeftInvariant μ f : G → E g : G h_mul : MeasurableEmbedding fun x => g * x ⊢ ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ ** rw [← h_mul.integral_map, map_mul_left_eq_self] ** Qed
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MeasureTheory.integral_mul_right_eq_self ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ f : G → E g : G ⊢ ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ ** have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ f : G → E g : G h_mul : MeasurableEmbedding fun x => x * g ⊢ ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ ** rw [← h_mul.integral_map, map_mul_right_eq_self] ** Qed
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MeasureTheory.integral_div_right_eq_self ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ f : G → E g : G ⊢ ∫ (x : G), f (x / g) ∂μ = ∫ (x : G), f x ∂μ ** simp_rw [div_eq_mul_inv] ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ f : G → E g : G ⊢ ∫ (x : G), f (x * g⁻¹) ∂μ = ∫ (x : G), f x ∂μ ** rw [integral_mul_right_eq_self f g⁻¹] ** Qed
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MeasureTheory.integral_eq_zero_of_mul_left_eq_neg ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f : G → E g : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulLeftInvariant μ hf' : ∀ (x : G), f (g * x) = -f x ⊢ ∫ (x : G), f x ∂μ = 0 ** simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_left_eq_self] ** Qed
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MeasureTheory.integral_eq_zero_of_mul_right_eq_neg ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f : G → E g : G inst✝² : Group G inst✝¹ : MeasurableMul G inst✝ : IsMulRightInvariant μ hf' : ∀ (x : G), f (x * g) = -f x ⊢ ∫ (x : G), f x ∂μ = 0 ** simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_right_eq_self] ** Qed
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MeasureTheory.Integrable.comp_div_right ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G f : G → F inst✝ : IsMulRightInvariant μ hf : Integrable f g : G ⊢ Integrable fun t => f (t / g) ** simp_rw [div_eq_mul_inv] ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁷ : MeasurableSpace G inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝² : Group G inst✝¹ : MeasurableMul G f : G → F inst✝ : IsMulRightInvariant μ hf : Integrable f g : G ⊢ Integrable fun t => f (t * g⁻¹) ** exact hf.comp_mul_right g⁻¹ ** Qed
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MeasureTheory.integrable_comp_div_left ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁹ : MeasurableSpace G inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : CompleteSpace E inst✝⁵ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝⁴ : Group G inst✝³ : MeasurableMul G inst✝² : MeasurableInv G f : G → F inst✝¹ : IsInvInvariant μ inst✝ : IsMulLeftInvariant μ g : G ⊢ (Integrable fun t => f (g / t)) ↔ Integrable f ** refine' ⟨fun h => _, fun h => h.comp_div_left g⟩ ** 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁹ : MeasurableSpace G inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : CompleteSpace E inst✝⁵ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝⁴ : Group G inst✝³ : MeasurableMul G inst✝² : MeasurableInv G f : G → F inst✝¹ : IsInvInvariant μ inst✝ : IsMulLeftInvariant μ g : G h : Integrable fun t => f (g / t) ⊢ Integrable f ** convert h.comp_inv.comp_mul_left g⁻¹ ** case h.e'_5.h.h.e'_1 𝕜 : Type u_1 M : Type u_2 α : Type u_3 G : Type u_4 E : Type u_5 F : Type u_6 inst✝⁹ : MeasurableSpace G inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : CompleteSpace E inst✝⁵ : NormedAddCommGroup F μ : Measure G f✝ : G → E g✝ : G inst✝⁴ : Group G inst✝³ : MeasurableMul G inst✝² : MeasurableInv G f : G → F inst✝¹ : IsInvInvariant μ inst✝ : IsMulLeftInvariant μ g : G h : Integrable fun t => f (g / t) x✝ : G ⊢ x✝ = g / (g⁻¹ * x✝)⁻¹ ** simp_rw [div_inv_eq_mul, mul_inv_cancel_left] ** Qed
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AddCircle.closedBall_ae_eq_ball ** T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ ⊢ closedBall x ε =ᶠ[ae volume] ball x ε ** cases' le_or_lt ε 0 with hε hε ** case inl T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : ε ≤ 0 ⊢ closedBall x ε =ᶠ[ae volume] ball x ε ** rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closedBall,
min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ENNReal.ofReal_eq_zero] ** case inl T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : ε ≤ 0 ⊢ 2 * ε ≤ 0 ** exact mul_nonpos_of_nonneg_of_nonpos zero_le_two hε ** T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : ε ≤ 0 ⊢ 2 * ε ≤ T ** linarith [hT.out] ** case inr T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε ⊢ closedBall x ε =ᶠ[ae volume] ball x ε ** suffices volume (closedBall x ε) ≤ volume (ball x ε) by
exact (ae_eq_of_subset_of_measure_ge ball_subset_closedBall this measurableSet_ball
(measure_ne_top _ _)).symm ** case inr T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε ⊢ ↑↑volume (closedBall x ε) ≤ ↑↑volume (ball x ε) ** have : Tendsto (fun δ => volume (closedBall x δ)) (𝓝[<] ε) (𝓝 <| volume (closedBall x ε)) := by
simp_rw [volume_closedBall]
refine' ENNReal.tendsto_ofReal (Tendsto.min tendsto_const_nhds <| Tendsto.const_mul _ _)
convert (@monotone_id ℝ _).tendsto_nhdsWithin_Iio ε
simp ** case inr T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε this : Tendsto (fun δ => ↑↑volume (closedBall x δ)) (𝓝[Iio ε] ε) (𝓝 (↑↑volume (closedBall x ε))) ⊢ ↑↑volume (closedBall x ε) ≤ ↑↑volume (ball x ε) ** refine' le_of_tendsto this (mem_nhdsWithin_Iio_iff_exists_Ioo_subset.mpr ⟨0, hε, fun r hr => _⟩) ** case inr T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε this : Tendsto (fun δ => ↑↑volume (closedBall x δ)) (𝓝[Iio ε] ε) (𝓝 (↑↑volume (closedBall x ε))) r : ℝ hr : r ∈ Ioo 0 ε ⊢ r ∈ {x_1 | (fun c => ↑↑volume (closedBall x c) ≤ ↑↑volume (ball x ε)) x_1} ** exact measure_mono (closedBall_subset_ball hr.2) ** T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε this : ↑↑volume (closedBall x ε) ≤ ↑↑volume (ball x ε) ⊢ closedBall x ε =ᶠ[ae volume] ball x ε ** exact (ae_eq_of_subset_of_measure_ge ball_subset_closedBall this measurableSet_ball
(measure_ne_top _ _)).symm ** T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε ⊢ Tendsto (fun δ => ↑↑volume (closedBall x δ)) (𝓝[Iio ε] ε) (𝓝 (↑↑volume (closedBall x ε))) ** simp_rw [volume_closedBall] ** T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε ⊢ Tendsto (fun δ => ENNReal.ofReal (min T (2 * δ))) (𝓝[Iio ε] ε) (𝓝 (ENNReal.ofReal (min T (2 * ε)))) ** refine' ENNReal.tendsto_ofReal (Tendsto.min tendsto_const_nhds <| Tendsto.const_mul _ _) ** T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε ⊢ Tendsto (fun k => k) (𝓝[Iio ε] ε) (𝓝 ε) ** convert (@monotone_id ℝ _).tendsto_nhdsWithin_Iio ε ** case h.e'_5.h.e'_3 T : ℝ hT : Fact (0 < T) x : AddCircle T ε : ℝ hε : 0 < ε ⊢ ε = sSup (id '' Iio ε) ** simp ** Qed
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εNFA.εClosure_empty ** α : Type u σ σ' : Type v M : εNFA α σ S : Set σ x : List α s✝ : σ a : α s : σ hs : s ∈ εClosure M ∅ ⊢ False ** induction hs <;> assumption ** Qed
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εNFA.mem_stepSet_iff ** α : Type u σ σ' : Type v M : εNFA α σ S : Set σ x : List α s : σ a : α ⊢ s ∈ stepSet M S a ↔ ∃ t, t ∈ S ∧ s ∈ εClosure M (step M t (some a)) ** simp_rw [stepSet, mem_iUnion₂, exists_prop] ** Qed
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εNFA.stepSet_empty ** α : Type u σ σ' : Type v M : εNFA α σ S : Set σ x : List α s : σ a✝ a : α ⊢ stepSet M ∅ a = ∅ ** simp_rw [stepSet, mem_empty_iff_false, iUnion_false, iUnion_empty] ** Qed
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εNFA.evalFrom_append_singleton ** α : Type u σ σ' : Type v M : εNFA α σ S✝ : Set σ x✝ : List α s : σ a✝ : α S : Set σ x : List α a : α ⊢ evalFrom M S (x ++ [a]) = stepSet M (evalFrom M S x) a ** rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil] ** Qed
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NFA.toεNFA_εClosure ** α : Type u σ σ' : Type v M✝ : εNFA α σ S✝ : Set σ x : List α s : σ a : α M : NFA α σ S : Set σ ⊢ εNFA.εClosure (toεNFA M) S = S ** ext a ** case h α : Type u σ σ' : Type v M✝ : εNFA α σ S✝ : Set σ x : List α s : σ a✝ : α M : NFA α σ S : Set σ a : σ ⊢ a ∈ εNFA.εClosure (toεNFA M) S ↔ a ∈ S ** refine' ⟨_, εNFA.εClosure.base _⟩ ** case h α : Type u σ σ' : Type v M✝ : εNFA α σ S✝ : Set σ x : List α s : σ a✝ : α M : NFA α σ S : Set σ a : σ ⊢ a ∈ εNFA.εClosure (toεNFA M) S → a ∈ S ** rintro (⟨_, h⟩ | ⟨_, _, h, _⟩) ** case h.base α : Type u σ σ' : Type v M✝ : εNFA α σ S✝ : Set σ x : List α s : σ a✝ : α M : NFA α σ S : Set σ a : σ h : a ∈ S ⊢ a ∈ S ** exact h ** case h.step α : Type u σ σ' : Type v M✝ : εNFA α σ S✝ : Set σ x : List α s : σ a✝¹ : α M : NFA α σ S : Set σ a s✝ : σ a✝ : εNFA.εClosure (toεNFA M) S s✝ h : a ∈ εNFA.step (toεNFA M) s✝ none ⊢ a ∈ S ** cases h ** Qed
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DFA.mem_accepts ** α : Type u σ : Type v M : DFA α σ x : List α ⊢ x ∈ accepts M ↔ evalFrom M M.start x ∈ M.accept ** rfl ** Qed
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DFA.evalFrom_of_pow ** α : Type u σ : Type v M : DFA α σ x y : List α s : σ hx : evalFrom M s x = s hy : y ∈ {x}∗ ⊢ evalFrom M s y = s ** rw [Language.mem_kstar] at hy ** α : Type u σ : Type v M : DFA α σ x y : List α s : σ hx : evalFrom M s x = s hy : ∃ L, y = List.join L ∧ ∀ (y : List α), y ∈ L → y ∈ {x} ⊢ evalFrom M s y = s ** rcases hy with ⟨S, rfl, hS⟩ ** case intro.intro α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S : List (List α) hS : ∀ (y : List α), y ∈ S → y ∈ {x} ⊢ evalFrom M s (List.join S) = s ** induction' S with a S ih ** case intro.intro.nil α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S : List (List α) hS✝ : ∀ (y : List α), y ∈ S → y ∈ {x} hS : ∀ (y : List α), y ∈ [] → y ∈ {x} ⊢ evalFrom M s (List.join []) = s ** rfl ** case intro.intro.cons α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S✝ : List (List α) hS✝ : ∀ (y : List α), y ∈ S✝ → y ∈ {x} a : List α S : List (List α) ih : (∀ (y : List α), y ∈ S → y ∈ {x}) → evalFrom M s (List.join S) = s hS : ∀ (y : List α), y ∈ a :: S → y ∈ {x} ⊢ evalFrom M s (List.join (a :: S)) = s ** have ha := hS a (List.mem_cons_self _ _) ** case intro.intro.cons α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S✝ : List (List α) hS✝ : ∀ (y : List α), y ∈ S✝ → y ∈ {x} a : List α S : List (List α) ih : (∀ (y : List α), y ∈ S → y ∈ {x}) → evalFrom M s (List.join S) = s hS : ∀ (y : List α), y ∈ a :: S → y ∈ {x} ha : a ∈ {x} ⊢ evalFrom M s (List.join (a :: S)) = s ** rw [Set.mem_singleton_iff] at ha ** case intro.intro.cons α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S✝ : List (List α) hS✝ : ∀ (y : List α), y ∈ S✝ → y ∈ {x} a : List α S : List (List α) ih : (∀ (y : List α), y ∈ S → y ∈ {x}) → evalFrom M s (List.join S) = s hS : ∀ (y : List α), y ∈ a :: S → y ∈ {x} ha : a = x ⊢ evalFrom M s (List.join (a :: S)) = s ** rw [List.join, evalFrom_of_append, ha, hx] ** case intro.intro.cons α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S✝ : List (List α) hS✝ : ∀ (y : List α), y ∈ S✝ → y ∈ {x} a : List α S : List (List α) ih : (∀ (y : List α), y ∈ S → y ∈ {x}) → evalFrom M s (List.join S) = s hS : ∀ (y : List α), y ∈ a :: S → y ∈ {x} ha : a = x ⊢ evalFrom M s (List.join S) = s ** apply ih ** case intro.intro.cons α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S✝ : List (List α) hS✝ : ∀ (y : List α), y ∈ S✝ → y ∈ {x} a : List α S : List (List α) ih : (∀ (y : List α), y ∈ S → y ∈ {x}) → evalFrom M s (List.join S) = s hS : ∀ (y : List α), y ∈ a :: S → y ∈ {x} ha : a = x ⊢ ∀ (y : List α), y ∈ S → y ∈ {x} ** intro z hz ** case intro.intro.cons α : Type u σ : Type v M : DFA α σ x : List α s : σ hx : evalFrom M s x = s S✝ : List (List α) hS✝ : ∀ (y : List α), y ∈ S✝ → y ∈ {x} a : List α S : List (List α) ih : (∀ (y : List α), y ∈ S → y ∈ {x}) → evalFrom M s (List.join S) = s hS : ∀ (y : List α), y ∈ a :: S → y ∈ {x} ha : a = x z : List α hz : z ∈ S ⊢ z ∈ {x} ** exact hS z (List.mem_cons_of_mem a hz) ** Qed
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DFA.pumping_lemma ** α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx : x ∈ accepts M hlen : Fintype.card σ ≤ List.length x ⊢ ∃ a b c, x = a ++ b ++ c ∧ List.length a + List.length b ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ accepts M ** obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split hlen rfl ** case intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x ⊢ ∃ a b c, x = a ++ b ++ c ∧ List.length a + List.length b ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ accepts M ** use a, b, c, hx, hlen, hnil ** case right α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x ⊢ {a} * {b}∗ * {c} ≤ accepts M ** intro y hy ** case right α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x y : List α hy : y ∈ {a} * {b}∗ * {c} ⊢ y ∈ accepts M ** rw [Language.mem_mul] at hy ** case right α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x y : List α hy : ∃ a_1 b_1, a_1 ∈ {a} * {b}∗ ∧ b_1 ∈ {c} ∧ a_1 ++ b_1 = y ⊢ y ∈ accepts M ** rcases hy with ⟨ab, c', hab, hc', rfl⟩ ** case right.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x ab c' : List α hab : ab ∈ {a} * {b}∗ hc' : c' ∈ {c} ⊢ ab ++ c' ∈ accepts M ** rw [Language.mem_mul] at hab ** case right.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x ab c' : List α hab : ∃ a_1 b_1, a_1 ∈ {a} ∧ b_1 ∈ {b}∗ ∧ a_1 ++ b_1 = ab hc' : c' ∈ {c} ⊢ ab ++ c' ∈ accepts M ** rcases hab with ⟨a', b', ha', hb', rfl⟩ ** case right.intro.intro.intro.intro.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x c' : List α hc' : c' ∈ {c} a' b' : List α ha' : a' ∈ {a} hb' : b' ∈ {b}∗ ⊢ a' ++ b' ++ c' ∈ accepts M ** rw [Set.mem_singleton_iff] at ha' hc' ** case right.intro.intro.intro.intro.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x a b c : List α hx : x = a ++ b ++ c hlen : List.length a + List.length b ≤ Fintype.card σ hnil : b ≠ [] hb : evalFrom M (evalFrom M ?m.8079 a) b = evalFrom M ?m.8079 a hc : evalFrom M (evalFrom M ?m.8079 a) c = evalFrom M ?m.8079 x c' : List α hc' : c' = c a' b' : List α ha' : a' = a hb' : b' ∈ {b}∗ ⊢ a' ++ b' ++ c' ∈ accepts M ** substs ha' hc' ** case right.intro.intro.intro.intro.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x b : List α hnil : b ≠ [] c' a' b' : List α hb' : b' ∈ {b}∗ hlen : List.length a' + List.length b ≤ Fintype.card σ hb : evalFrom M (evalFrom M ?m.8079 a') b = evalFrom M ?m.8079 a' hx : x = a' ++ b ++ c' hc : evalFrom M (evalFrom M ?m.8079 a') c' = evalFrom M ?m.8079 x ⊢ a' ++ b' ++ c' ∈ accepts M ** have h := M.evalFrom_of_pow hb hb' ** case right.intro.intro.intro.intro.intro.intro.intro.intro α : Type u σ : Type v M : DFA α σ inst✝ : Fintype σ x : List α hx✝ : x ∈ accepts M hlen✝ : Fintype.card σ ≤ List.length x b : List α hnil : b ≠ [] c' a' b' : List α hb' : b' ∈ {b}∗ hlen : List.length a' + List.length b ≤ Fintype.card σ hb : evalFrom M (evalFrom M ?m.8079 a') b = evalFrom M ?m.8079 a' hx : x = a' ++ b ++ c' hc : evalFrom M (evalFrom M ?m.8079 a') c' = evalFrom M ?m.8079 x h : evalFrom M (evalFrom M ?m.8079 a') b' = evalFrom M ?m.8079 a' ⊢ a' ++ b' ++ c' ∈ accepts M ** rwa [mem_accepts, evalFrom_of_append, evalFrom_of_append, h, hc] ** Qed
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measurable_of_tendsto_ennreal' ** α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : Tendsto f u (𝓝 g) ⊢ Measurable g ** rcases u.exists_seq_tendsto with ⟨x, hx⟩ ** case intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : Tendsto f u (𝓝 g) x : ℕ → ι hx : Tendsto x atTop u ⊢ Measurable g ** rw [tendsto_pi_nhds] at lim ** case intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x)) x : ℕ → ι hx : Tendsto x atTop u ⊢ Measurable g ** have : (fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop) = g := by
ext1 y
exact ((lim y).comp hx).liminf_eq ** case intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x)) x : ℕ → ι hx : Tendsto x atTop u this : (fun y => liminf (fun n => f (x n) y) atTop) = g ⊢ Measurable g ** rw [← this] ** case intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x)) x : ℕ → ι hx : Tendsto x atTop u this : (fun y => liminf (fun n => f (x n) y) atTop) = g ⊢ Measurable fun y => liminf (fun n => f (x n) y) atTop ** exact measurable_liminf fun n => hf (x n) ** α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x)) x : ℕ → ι hx : Tendsto x atTop u ⊢ (fun y => liminf (fun n => f (x n) y) atTop) = g ** ext1 y ** case h α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : MeasurableSpace β inst✝² : BorelSpace β ι : Type u_3 f : ι → α → ℝ≥0∞ g : α → ℝ≥0∞ u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (i : ι), Measurable (f i) lim : ∀ (x : α), Tendsto (fun i => f i x) u (𝓝 (g x)) x : ℕ → ι hx : Tendsto x atTop u y : α ⊢ liminf (fun n => f (x n) y) atTop = g y ** exact ((lim y).comp hx).liminf_eq ** Qed
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aemeasurable_of_tendsto_metrizable_ae ** α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) ⊢ AEMeasurable g ** rcases u.exists_seq_tendsto with ⟨v, hv⟩ ** case intro α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u ⊢ AEMeasurable g ** have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n) ** case intro α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) ⊢ AEMeasurable g ** set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) ** case intro α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) ⊢ AEMeasurable g ** have hp : ∀ᵐ x ∂μ, p x fun n => f (v n) x := by
filter_upwards [h_tendsto] with x hx using hx.comp hv ** case intro α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x ⊢ AEMeasurable g ** set aeSeqLim := fun x => ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty β).some ** case intro α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) ⊢ AEMeasurable g ** refine'
⟨aeSeqLim,
measurable_of_tendsto_metrizable' atTop (aeSeq.measurable h'f p)
(tendsto_pi_nhds.mpr fun x => _),
_⟩ ** α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) ⊢ ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x ** filter_upwards [h_tendsto] with x hx using hx.comp hv ** case intro.refine'_1 α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) x : α ⊢ Tendsto (fun i => aeSeq h'f p i x) atTop (𝓝 (aeSeqLim x)) ** simp_rw [aeSeq] ** case intro.refine'_1 α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) x : α ⊢ Tendsto (fun i => if x ∈ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) then AEMeasurable.mk (f (v i)) (_ : AEMeasurable (f (v i))) x else Nonempty.some (_ : Nonempty β)) atTop (𝓝 (if x ∈ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) then g x else Nonempty.some (_ : Nonempty β))) ** split_ifs with hx ** case pos α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) x : α hx : x ∈ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) ⊢ Tendsto (fun i => AEMeasurable.mk (f (v i)) (_ : AEMeasurable (f (v i))) x) atTop (𝓝 (g x)) ** simp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx] ** case pos α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) x : α hx : x ∈ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) ⊢ Tendsto (fun i => f (v i) x) atTop (𝓝 (g x)) ** exact @aeSeq.fun_prop_of_mem_aeSeqSet _ α β _ _ _ _ _ h'f x hx ** case neg α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) x : α hx : ¬x ∈ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) ⊢ Tendsto (fun i => Nonempty.some (_ : Nonempty β)) atTop (𝓝 (Nonempty.some (_ : Nonempty β))) ** exact tendsto_const_nhds ** case intro.refine'_2 α : Type u_1 β : Type u_2 inst✝⁵ : MeasurableSpace α inst✝⁴ : TopologicalSpace β inst✝³ : PseudoMetrizableSpace β inst✝² : MeasurableSpace β inst✝¹ : BorelSpace β ι : Type u_3 μ : Measure α f : ι → α → β g : α → β u : Filter ι hu : NeBot u inst✝ : IsCountablyGenerated u hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x)) v : ℕ → ι hv : Tendsto v atTop u h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x)) hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else Nonempty.some (_ : Nonempty β) ⊢ g =ᵐ[μ] aeSeqLim ** exact
(ite_ae_eq_of_measure_compl_zero g (fun x => (⟨f (v 0) x⟩ : Nonempty β).some) (aeSeqSet h'f p)
(aeSeq.measure_compl_aeSeqSet_eq_zero h'f hp)).symm ** Qed
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aemeasurable_of_unif_approx ** α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε ⊢ AEMeasurable g ** obtain ⟨u, -, u_pos, u_lim⟩ :
∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
exists_seq_strictAnti_tendsto (0 : ℝ) ** case intro.intro.intro α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) ⊢ AEMeasurable g ** choose f Hf using fun n : ℕ => hf (u n) (u_pos n) ** case intro.intro.intro α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n ⊢ AEMeasurable g ** have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
filter_upwards [this]
intro x hx
rw [tendsto_iff_dist_tendsto_zero]
exact squeeze_zero (fun n => dist_nonneg) hx u_lim ** case intro.intro.intro α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n this : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ AEMeasurable g ** exact aemeasurable_of_tendsto_metrizable_ae' (fun n => (Hf n).1) this ** α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ** have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2 ** α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ** filter_upwards [this] ** case h α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n ⊢ ∀ (a : α), (∀ (n : ℕ), dist (f n a) (g a) ≤ u n) → Tendsto (fun n => f n a) atTop (𝓝 (g a)) ** intro x hx ** case h α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n x : α hx : ∀ (n : ℕ), dist (f n x) (g x) ≤ u n ⊢ Tendsto (fun n => f n x) atTop (𝓝 (g x)) ** rw [tendsto_iff_dist_tendsto_zero] ** case h α : Type u_1 β✝ : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β✝ inst✝⁵ : PseudoMetrizableSpace β✝ inst✝⁴ : MeasurableSpace β✝ inst✝³ : BorelSpace β✝ β : Type u_3 inst✝² : MeasurableSpace β inst✝¹ : PseudoMetricSpace β inst✝ : BorelSpace β μ : Measure α g : α → β hf : ∀ (ε : ℝ), ε > 0 → ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε u : ℕ → ℝ u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) f : ℕ → α → β Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n x : α hx : ∀ (n : ℕ), dist (f n x) (g x) ≤ u n ⊢ Tendsto (fun b => dist (f b x) (g x)) atTop (𝓝 0) ** exact squeeze_zero (fun n => dist_nonneg) hx u_lim ** Qed
| |
measurable_limit_of_tendsto_metrizable_ae ** α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** inhabit ι ** α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** rcases eq_or_neBot L with (rfl | hL) ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** let p : α → (ι → β) → Prop := fun x f' => ∃ l : β, Tendsto (fun n => f' n) L (𝓝 l) ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** have hp_mem : ∀ x ∈ aeSeqSet hf p, p x fun n => f n x := fun x hx =>
aeSeq.fun_prop_of_mem_aeSeqSet hf hx ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** have h_ae_eq : ∀ᵐ x ∂μ, ∀ n, aeSeq hf p n x = f n x := aeSeq.aeSeq_eq_fun_ae hf h_ae_tendsto ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** set f_lim : α → β := fun x => dite (x ∈ aeSeqSet hf p) (fun h => (hp_mem x h).choose)
fun _ => (⟨f default x⟩ : Nonempty β).some ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) hf_lim : ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x)) ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** have h_ae_tendsto_f_lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) :=
h_ae_eq.mono fun x hx => (hf_lim x).congr hx ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) hf_lim : ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x)) h_ae_tendsto_f_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** have h_f_lim_meas : Measurable f_lim :=
measurable_of_tendsto_metrizable' L (aeSeq.measurable hf p)
(tendsto_pi_nhds.mpr fun x => hf_lim x) ** case inr α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) hf_lim : ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x)) h_ae_tendsto_f_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) h_f_lim_meas : Measurable f_lim ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) ** exact ⟨f_lim, h_f_lim_meas, h_ae_tendsto_f_lim⟩ ** case inl α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β hf : ∀ (n : ι), AEMeasurable (f n) inhabited_h : Inhabited ι inst✝ : IsCountablyGenerated ⊥ h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) ⊥ (𝓝 l) ⊢ ∃ f_lim hf_lim_meas, ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) ⊥ (𝓝 (f_lim x)) ** exact ⟨(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot⟩ ** α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) ⊢ ∀ (x : α), Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x)) ** intro x ** α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) x : α ⊢ Tendsto (fun n => aeSeq hf p n x) L (𝓝 (f_lim x)) ** simp only [aeSeq] ** α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) x : α ⊢ Tendsto (fun n => if x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) then AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x else Nonempty.some (_ : Nonempty β)) L (𝓝 (if h : x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β))) ** split_ifs with h ** case pos α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) x : α h : x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) ⊢ Tendsto (fun n => AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x) L (𝓝 (Exists.choose (_ : p x fun n => f n x))) ** refine' (hp_mem x h).choose_spec.congr fun n => _ ** case pos α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) x : α h : x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) n : ι ⊢ (fun n => f n x) n = AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x ** exact (aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h n).symm ** case neg α : Type u_1 β : Type u_2 inst✝⁷ : MeasurableSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β ι : Type u_3 inst✝² : Countable ι inst✝¹ : Nonempty ι μ : Measure α f : ι → α → β L : Filter ι inst✝ : IsCountablyGenerated L hf : ∀ (n : ι), AEMeasurable (f n) h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l) inhabited_h : Inhabited ι hL : NeBot L p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) hp_mem : ∀ (x : α), x ∈ aeSeqSet hf p → p x fun n => f n x h_ae_eq : ∀ᵐ (x : α) ∂μ, ∀ (n : ι), aeSeq hf p n x = f n x f_lim : α → β := fun x => if h : x ∈ aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty β) x : α h : ¬x ∈ aeSeqSet hf fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l) ⊢ Tendsto (fun n => Nonempty.some (_ : Nonempty β)) L (𝓝 (Nonempty.some (_ : Nonempty β))) ** exact tendsto_const_nhds ** Qed
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MeasureTheory.measurePreserving_smul ** G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝³ : MeasurableSpace M inst✝² : SMul M α inst✝¹ : MeasurableSMul M α c : M μ : Measure α inst✝ : SMulInvariantMeasure M α μ ⊢ map (fun x => c • x) μ = μ ** ext1 s hs ** case h G : Type u M : Type v α : Type w s✝ : Set α m : MeasurableSpace α inst✝³ : MeasurableSpace M inst✝² : SMul M α inst✝¹ : MeasurableSMul M α c : M μ : Measure α inst✝ : SMulInvariantMeasure M α μ s : Set α hs : MeasurableSet s ⊢ ↑↑(map (fun x => c • x) μ) s = ↑↑μ s ** rw [map_apply (measurable_const_smul c) hs] ** case h G : Type u M : Type v α : Type w s✝ : Set α m : MeasurableSpace α inst✝³ : MeasurableSpace M inst✝² : SMul M α inst✝¹ : MeasurableSMul M α c : M μ : Measure α inst✝ : SMulInvariantMeasure M α μ s : Set α hs : MeasurableSet s ⊢ ↑↑μ ((fun x x_1 => x • x_1) c ⁻¹' s) = ↑↑μ s ** exact SMulInvariantMeasure.measure_preimage_smul c hs ** Qed
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MeasureTheory.smulInvariantMeasure_map ** G : Type u M✝ : Type v α✝ : Type w s : Set α✝ M : Type uM N : Type uN α : Type uα β : Type uβ inst✝⁷ : MeasurableSpace M inst✝⁶ : MeasurableSpace N inst✝⁵ : MeasurableSpace α inst✝⁴ : MeasurableSpace β inst✝³ : SMul M α inst✝² : SMul M β inst✝¹ : MeasurableSMul M β μ : Measure α inst✝ : SMulInvariantMeasure M α μ f : α → β hsmul : ∀ (m : M) (a : α), f (m • a) = m • f a hf : Measurable f m : M S : Set β hS : MeasurableSet S ⊢ ↑↑μ (f ⁻¹' ((fun x => m • x) ⁻¹' S)) = ↑↑μ ((fun x => m • f x) ⁻¹' S) ** rw [preimage_preimage] ** G : Type u M✝ : Type v α✝ : Type w s : Set α✝ M : Type uM N : Type uN α : Type uα β : Type uβ inst✝⁷ : MeasurableSpace M inst✝⁶ : MeasurableSpace N inst✝⁵ : MeasurableSpace α inst✝⁴ : MeasurableSpace β inst✝³ : SMul M α inst✝² : SMul M β inst✝¹ : MeasurableSMul M β μ : Measure α inst✝ : SMulInvariantMeasure M α μ f : α → β hsmul : ∀ (m : M) (a : α), f (m • a) = m • f a hf : Measurable f m : M S : Set β hS : MeasurableSet S ⊢ ↑↑μ ((fun x => m • f x) ⁻¹' S) = ↑↑μ ((fun x => f (m • x)) ⁻¹' S) ** simp_rw [hsmul] ** G : Type u M✝ : Type v α✝ : Type w s : Set α✝ M : Type uM N : Type uN α : Type uα β : Type uβ inst✝⁷ : MeasurableSpace M inst✝⁶ : MeasurableSpace N inst✝⁵ : MeasurableSpace α inst✝⁴ : MeasurableSpace β inst✝³ : SMul M α inst✝² : SMul M β inst✝¹ : MeasurableSMul M β μ : Measure α inst✝ : SMulInvariantMeasure M α μ f : α → β hsmul : ∀ (m : M) (a : α), f (m • a) = m • f a hf : Measurable f m : M S : Set β hS : MeasurableSet S ⊢ ↑↑μ ((fun x => f (m • x)) ⁻¹' S) = ↑↑μ ((fun x => m • x) ⁻¹' (f ⁻¹' S)) ** rw [←preimage_preimage] ** G : Type u M✝ : Type v α✝ : Type w s : Set α✝ M : Type uM N : Type uN α : Type uα β : Type uβ inst✝⁷ : MeasurableSpace M inst✝⁶ : MeasurableSpace N inst✝⁵ : MeasurableSpace α inst✝⁴ : MeasurableSpace β inst✝³ : SMul M α inst✝² : SMul M β inst✝¹ : MeasurableSMul M β μ : Measure α inst✝ : SMulInvariantMeasure M α μ f : α → β hsmul : ∀ (m : M) (a : α), f (m • a) = m • f a hf : Measurable f m : M S : Set β hS : MeasurableSet S ⊢ ↑↑μ ((fun x => m • x) ⁻¹' (f ⁻¹' S)) = ↑↑μ (f ⁻¹' S) ** rw [SMulInvariantMeasure.measure_preimage_smul m (hS.preimage hf)] ** Qed
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MeasureTheory.NullMeasurableSet.smul ** G : Type u M : Type v α : Type w s✝ : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c✝ : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ s : Set α hs : NullMeasurableSet s c : G ⊢ NullMeasurableSet (c • s) ** simpa only [← preimage_smul_inv] using
hs.preimage (measurePreserving_smul _ _).quasiMeasurePreserving ** Qed
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MeasureTheory.measure_smul_null ** G : Type u M : Type v α : Type w s✝ : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c✝ : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ s : Set α h : ↑↑μ s = 0 c : G ⊢ ↑↑μ (c • s) = 0 ** rwa [measure_smul] ** Qed
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