module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.HNNExtension | {
"line": 418,
"column": 6
} | {
"line": 419,
"column": 43
} | {
"line": 419,
"column": 44
} | [
{
"pp": "case pos.cons.refl\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\ng : G\nw : NormalWord d\na✝ : ∀ (h : Cancels u w), ¬Cancels (-u) (unitsSMulWithCancel φ u w ⋯)\nh1 : w.head ∈ d.set (-u)\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubg... | [
"case pos.cons.refl\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\ng : G\nw : NormalWord d\na✝ : ∀ (h : Cancels u w), ¬Cancels (-u) (unitsSMulWithCancel φ u w ⋯)\nh1 : w.head ∈ d.set (-u)\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B (-u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 25
} | {
"line": 421,
"column": 26
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nh : ¬Cancels u w\n⊢ Cancels (-u) (cons (↑(unitsSMulGroup φ d u w.head).1) u ((↑(unitsSMulGroup φ d u w.head).2 * w.head⁻¹) • w) ⋯ ⋯) ↔\n ¬Cancels u w",
"ppTerm": "?neg✝",... | [
"case neg\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nh : ¬Cancels u w\n⊢ w.head ∈ toSubgroup A B u → ∀ (x : G), ¬w.toList.head? = some (-u, x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.MonoidLocalization.UniqueFactorization | {
"line": 53,
"column": 27
} | {
"line": 53,
"column": 38
} | {
"line": 53,
"column": 39
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝² : CommMonoidWithZero M\ninst✝¹ : CommMonoidWithZero N\nS : Submonoid M\ninst✝ : WfDvdMonoid M\nf : S.LocalizationMap N\ni : M\nhi : Irreducible i\nu m' : M\nhu : IsUnit (f u)\nhm' : Irreducible m'\nha0 : u * m' ≠ 0\nha : Irreducible (f (u * m')) → ∃ u_1 m'_1, IsUnit (... | [
"M : Type u_1\nN : Type u_2\ninst✝² : CommMonoidWithZero M\ninst✝¹ : CommMonoidWithZero N\nS : Submonoid M\ninst✝ : WfDvdMonoid M\nf : S.LocalizationMap N\ni : M\nhi : Irreducible i\nu m' : M\nhu : IsUnit (f u)\nhm' : Irreducible m'\nha0 : u * m' ≠ 0\nha : Irreducible (f (u * m')) → ∃ u_1 m'_1, IsUnit (f u_1) ∧ Irr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 428,
"column": 4
} | {
"line": 432,
"column": 28
} | {
"line": 433,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nhcan : Cancels (-u) (unitsSMul φ u w)\n⊢ unitsSMulWithCancel φ (-u) (unitsSMul φ u w) hcan = w",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"List.... | [] | have hncan : ¬ Cancels u w := (unitsSMul_cancels_iff _ _ _).1 hcan
unfold unitsSMul
simp only [dif_neg hncan]
simp [unitsSMulWithCancel, unitsSMulGroup, (d.compl u).equiv_snd_eq_inv_mul,
-SetLike.coe_sort_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.HNNExtension | {
"line": 428,
"column": 4
} | {
"line": 432,
"column": 28
} | {
"line": 433,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nhcan : Cancels (-u) (unitsSMul φ u w)\n⊢ unitsSMulWithCancel φ (-u) (unitsSMul φ u w) hcan = w",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"List.... | [] | have hncan : ¬ Cancels u w := (unitsSMul_cancels_iff _ _ _).1 hcan
unfold unitsSMul
simp only [dif_neg hncan]
simp [unitsSMulWithCancel, unitsSMulGroup, (d.compl u).equiv_snd_eq_inv_mul,
-SetLike.coe_sort_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 44
} | {
"line": 70,
"column": 45
} | [
{
"pp": "case h.right.right.hf\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt :... | [
"case h.right.right.hf\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt : ∀ l ∈ c.ran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 17
} | {
"line": 114,
"column": 18
} | [
{
"pp": "case h.h1\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhc : m.sum ≤ Fintype.card α\nh2c : ∀ a ∈ m, 2 ≤ a\nhc' : m.toList.sum ≤ Fintype.card α\np : List (List α)\nhp_length : List.map List.length p = m.toList\nhp_nodup : ∀ s ∈ p, s.Nodup\nhp_disj : List.Pairwise List.Disjoin... | [
"case h.h1\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhc : m.sum ≤ Fintype.card α\nh2c : ∀ a ∈ m, 2 ≤ a\nhc' : m.toList.sum ≤ Fintype.card α\np : List (List α)\nhp_length : List.map List.length p = m.toList\nhp_nodup : ∀ s ∈ p, s.Nodup\nhp_disj : List.Pairwise List.Disjoint p\nhp2 : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 596,
"column": 15
} | {
"line": 596,
"column": 26
} | {
"line": 596,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝¹ : Group H\nM : Type u_3\ninst✝ : Monoid M\nd : TransversalPair G A B\nm₁✝ m₂✝ : HNNExtension G A B φ\nh : ∀ (a : NormalWord d), m₁✝ • a = m₂✝ • a\n⊢ m₁✝ = m₂✝",
"ppTerm": "?m.16",
"assigned": false,
"usedCo... | [
"G : Type u_1\ninst✝² : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝¹ : Group H\nM : Type u_3\ninst✝ : Monoid M\nd : TransversalPair G A B\nm₁✝ m₂✝ : HNNExtension G A B φ\nh : ∀ (a : NormalWord d), m₁✝ • a = m₂✝ • a\n⊢ m₁✝ = m₂✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 94,
"column": 33
} | {
"line": 94,
"column": 62
} | {
"line": 94,
"column": 63
} | [
{
"pp": "case refine_4.inl\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nx y : α\nhf : x ∈ orbit (↥(closure S)) y\nh : Equiv.swap x y ∉ closure S\na : α\nha : a ∈ {x | Equiv.swap x y ∈ closure S}\nw : α\nhzw : a ≠ w\nhσ : Equiv.swap a w ∈ S\nhσa : Equiv.swap a w • a ∉ {x ... | [
"case refine_4.inl\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nx y : α\nhf : x ∈ orbit (↥(closure S)) y\nh : Equiv.swap x y ∉ closure S\na : α\nha : a ∈ {x | Equiv.swap x y ∈ closure S}\nw : α\nhzw : a ≠ w\nhσ : Equiv.swap a w ∈ S\nhσa : Equiv.swap a w • a ∉ {x | Equiv.swap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 628,
"column": 8
} | {
"line": 628,
"column": 49
} | {
"line": 628,
"column": 50
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\nthis :\n ∀ (w : ReducedWord G A B),\n w.head = 1 →\n ∃ w',\n ReducedWord.prod φ w'.toReducedWord = ReducedWord.prod φ w ∧\n List.map Prod.fst w'.toList = List.map Pr... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\nthis :\n ∀ (w : ReducedWord G A B),\n w.head = 1 →\n ∃ w',\n ReducedWord.prod φ w'.toReducedWord = ReducedWord.prod φ w ∧\n List.map Prod.fst w'.toList = List.map Prod.fst w.toL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 94,
"column": 33
} | {
"line": 94,
"column": 62
} | {
"line": 94,
"column": 63
} | [
{
"pp": "case refine_4.inr\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nx y : α\nhf : x ∈ orbit (↥(closure S)) y\nh : Equiv.swap x y ∉ closure S\na : α\nha : a ∈ {x | Equiv.swap x y ∈ closure S}\nz : α\nhzw : z ≠ a\nhσ : Equiv.swap z a ∈ S\nhσa : Equiv.swap z a • a ∉ {x ... | [
"case refine_4.inr\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nx y : α\nhf : x ∈ orbit (↥(closure S)) y\nh : Equiv.swap x y ∉ closure S\na : α\nha : a ∈ {x | Equiv.swap x y ∈ closure S}\nz : α\nhzw : z ≠ a\nhσ : Equiv.swap z a ∈ S\nhσa : Equiv.swap z a • a ∉ {x | Equiv.swap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 95,
"column": 38
} | {
"line": 95,
"column": 52
} | {
"line": 96,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nx✝ y✝ : α\nhf✝ : x✝ ∈ orbit (↥(closure S)) y✝\nh : Equiv.swap x✝ y✝ ∉ closure S\nx y : α\nhf : Equiv.swap x y ∈ S\n⊢ (Equiv.swap x y)⁻¹ ∈ S",
"ppTerm": "?refine_1",
"assigned": true,
"usedCon... | [] | rwa [swap_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.HNNExtension | {
"line": 652,
"column": 10
} | {
"line": 652,
"column": 21
} | {
"line": 652,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\na : ℤˣ × G\nl : List (ℤˣ × G)\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) (a :: l)\nw' : NormalWord d\nhw'1 : ReducedWord.prod φ w'.toReducedWord = ReducedWord.pro... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\na : ℤˣ × G\nl : List (ℤˣ × G)\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) (a :: l)\nw' : NormalWord d\nhw'1 : ReducedWord.prod φ w'.toReducedWord = ReducedWord.prod φ { head :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 316,
"column": 10
} | {
"line": 316,
"column": 75
} | {
"line": 316,
"column": 76
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nc d : ↥g.cycleFactorsFinset\nhd : x ∈ (↑d).support\nm : ℤ\nhm : (g ^ m) (a d) = x\nh : ¬c = d\nH : (↑c).Disjoint ↑d\nh' : ↑(↑τ c) = ↑(↑τ d)\n⊢ c = d",
"ppTerm": "?m.206",
"assigne... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nc d : ↥g.cycleFactorsFinset\nhd : x ∈ (↑d).support\nm : ℤ\nhm : (g ^ m) (a d) = x\nh : ¬c = d\nH : (↑c).Disjoint ↑d\nh' : ↑(↑τ c) = ↑(↑τ d)\n⊢ c = d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 40
} | {
"line": 123,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nsupp : Set α\nfin : supp.Finite\na : α\ns : Set α\nih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S\nf : Equiv.Perm α\nhf : ∀ (x : α), f x ∈ orbit (↥(closure S))... | [
"α : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nsupp : Set α\nfin : supp.Finite\na : α\ns : Set α\nih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S\nf : Equiv.Perm α\nhf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x\nsupp_sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 385,
"column": 6
} | {
"line": 385,
"column": 46
} | {
"line": 385,
"column": 47
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nc : ↥g.cycleFactorsFinset\nhc : x ∈ (↑c).support\nm : ℤ\nhm : (g ^ m) (a c) = x\nH : ¬↑τ c = c\n⊢ (¬∃ a, ↑τ a ≠ a ∧ ↑a x ≠ x) ↔ ↑τ c = c",
"ppTerm": "?neg✝",
"assigned":... | [
"case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nc : ↥g.cycleFactorsFinset\nhc : x ∈ (↑c).support\nm : ℤ\nhm : (g ^ m) (a c) = x\nH : ¬↑τ c = c\n⊢ ∃ a, ↑τ a ≠ a ∧ ↑a x ≠ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.RegularWreathProduct | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 17
} | {
"line": 163,
"column": 4
} | [
{
"pp": "D : Type u_1\nQ : Type u_2\ninst✝⁵ : Group D\ninst✝⁴ : Group Q\nΛ : Type u_3\ninst✝³ : MulAction D Λ\ninst✝² : FaithfulSMul D Λ\ninst✝¹ : Nonempty Q\ninst✝ : Nonempty Λ\n⊢ ∀ {m₁ m₂ : D ≀ᵣ Q},\n (∀ (a : Λ) (b : Q), m₁.left (m₁.right * b) • a = m₂.left (m₂.right * b) • a ∧ m₁.right = m₂.right) → m₁ = ... | [
"D : Type u_1\nQ : Type u_2\ninst✝⁵ : Group D\ninst✝⁴ : Group Q\nΛ : Type u_3\ninst✝³ : MulAction D Λ\ninst✝² : FaithfulSMul D Λ\ninst✝¹ : Nonempty Q\ninst✝ : Nonempty Λ\nm₁ m₂ : D ≀ᵣ Q\nh : ∀ (a : Λ) (b : Q), m₁.left (m₁.right * b) • a = m₂.left (m₂.right * b) • a ∧ m₁.right = m₂.right\n⊢ m₁ = m₂"
] | intro m₁ m₂ h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.HNNExtension | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 18
} | {
"line": 691,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nw : ReducedWord G A B\ng : G\nhg : of g = ReducedWord.prod φ w\nw' : ReducedWord G A B :=\n let __src := ReducedWord.empty G A B;\n { head := g, toList := __src.toList, chain := ⋯ }\nthis : ReducedWord.prod φ w = ReducedWord.prod φ w'\n⊢ ... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nw : ReducedWord G A B\ng : G\nhg : of g = ReducedWord.prod φ w\nw' : ReducedWord G A B :=\n let __src := ReducedWord.empty G A B;\n { head := g, toList := __src.toList, chain := ⋯ }\nthis : ReducedWord.prod φ w = ReducedWord.prod φ w'\n⊢ w.toList = [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 446,
"column": 4
} | {
"line": 447,
"column": 85
} | {
"line": 449,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nτ : Perm ↥g.cycleFactorsFinset\n⊢ (∀ (c : ↥g.cycleFactorsFinset), #(↑(τ c)).support = #(↑c).support) → τ ∈ (toPermHom g).range",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
... | [] | obtain ⟨a⟩ := Basis.nonempty g
exact fun hτ ↦ ⟨toCentralizer a ⟨τ, hτ⟩, toPermHom_apply_toCentralizer a ⟨τ, hτ⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 446,
"column": 4
} | {
"line": 447,
"column": 85
} | {
"line": 449,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nτ : Perm ↥g.cycleFactorsFinset\n⊢ (∀ (c : ↥g.cycleFactorsFinset), #(↑(τ c)).support = #(↑c).support) → τ ∈ (toPermHom g).range",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
... | [] | obtain ⟨a⟩ := Basis.nonempty g
exact fun hτ ↦ ⟨toCentralizer a ⟨τ, hτ⟩, toPermHom_apply_toCentralizer a ⟨τ, hτ⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.ResiduallyFinite | {
"line": 97,
"column": 47
} | {
"line": 97,
"column": 58
} | {
"line": 97,
"column": 59
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nh : ∀ (g : G), g ≠ 1 → ∃ H x, ∃ (_ : Finite H), ∃ f, f g ≠ 1\ng : G\nhg : g ≠ 1\nw✝² : Type u\nw✝¹ : Group w✝²\nw✝ : Finite w✝²\nf : G →* w✝²\nhf : f g ≠ 1\n⊢ g ∉ f.ker",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toO... | [
"G : Type u_1\ninst✝ : Group G\nh : ∀ (g : G), g ≠ 1 → ∃ H x, ∃ (_ : Finite H), ∃ f, f g ≠ 1\ng : G\nhg : g ≠ 1\nw✝² : Type u\nw✝¹ : Group w✝²\nw✝ : Finite w✝²\nf : G →* w✝²\nhf : f g ≠ 1\n⊢ ¬f g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PushoutI | {
"line": 186,
"column": 15
} | {
"line": 186,
"column": 26
} | {
"line": 186,
"column": 27
} | [
{
"pp": "case H.inl.one\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nmotive : (con φ).Quotient → Prop\nof : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)\nbase : ∀ (h : H), motive (((con φ).mk'.comp inr) h)\n... | [
"case H.inl.one\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nmotive : (con φ).Quotient → Prop\nof : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)\nbase : ∀ (h : H), motive (((con φ).mk'.comp inr) h)\nmul : ∀ (x y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 107,
"column": 16
} | {
"line": 120,
"column": 9
} | {
"line": 122,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nm : Multiset ℕ\n⊢ #{g | (↑g).cycleType = m} =\n if (m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) ∧ Even (m.sum + m.card) then\n (Fintype.card α)! / ((Fintype.card α - m.sum)! * (m.prod * ∏ n ∈ m.toFinset, (Multiset.count n m)!))\n else 0",... | [] | by
split_ifs with hm
· -- m is an even cycle_type
rw [← Finset.card_map, map_subtype_of_cycleType, if_pos hm.2,
Equiv.Perm.card_of_cycleType α m, if_pos hm.1, mul_assoc]
· -- m does not correspond to a permutation, or to an odd one,
rw [← Finset.card_map, map_subtype_of_cycleType]
rw [apply_ite ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PushoutI | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 67
} | {
"line": 353,
"column": 68
} | [
{
"pp": "case refine_2\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 =... | [
"case refine_2\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\nhφw : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 199,
"column": 2
} | {
"line": 206,
"column": 34
} | {
"line": 207,
"column": 2
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nN : Subgroup G\ninst✝² : N.Normal\nh1 : (Nat.card ↥N).Coprime N.index\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [Finite G'],\n Nat.card G' < Nat.card G →\n ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'\nh3 : ∀ (H : S... | [
"G : Type u\ninst✝³ : Group G\nN : Subgroup G\ninst✝² : N.Normal\nh1 : (Nat.card ↥N).Coprime N.index\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [Finite G'],\n Nat.card G' < Nat.card G →\n ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'\nh3 : ∀ (H : Subgroup G), ... | have h6 :
(Nat.card (N.map (QuotientGroup.mk' K))).Coprime (N.map (QuotientGroup.mk' K)).index := by
have index_map := N.index_map_eq this (by rwa [QuotientGroup.ker_mk'])
have index_pos : 0 < N.index := Nat.pos_of_ne_zero index_ne_zero_of_finite
rw [index_map]
refine h1.coprime_dvd_left ?_
rw [... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 704,
"column": 4
} | {
"line": 704,
"column": 35
} | {
"line": 705,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhm : m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a\n⊢ (Fintype.card α)! / ((Fintype.card α - m.sum)! * m.prod * ∏ n ∈ m.toFinset, (Multiset.count n m)!) =\n #{g | g.cycleType = m}",
"ppTerm": "?pos✝",
"assigned": tr... | [
"case pos.H1\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhm : m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a\n⊢ 0 < (Fintype.card α - m.sum)! * m.prod * ∏ n ∈ m.toFinset, (Multiset.count n m)!",
"case pos.H2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhm : m.su... | apply Nat.div_eq_of_eq_mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 720,
"column": 2
} | {
"line": 721,
"column": 9
} | {
"line": 721,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nhn' : 2 ≤ n\nhα : n ≤ card α\nhn₀ : n ≠ 0\naux : n ! = (n - 1)! * n\n⊢ #{g | g.cycleType = {n}} * (n * (card α - n)!) = (card α)!",
"ppTerm": "?m.93",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nhn' : 2 ≤ n\nhα : n ≤ card α\nhn₀ : n ≠ 0\naux : n ! = (n - 1)! * n\n⊢ #{g | g.cycleType = {n}} * (n * (card α - n)!) = (card α)!"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 65
} | {
"line": 207,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1\nx : Perm α\n⊢ x ∈ (kerParam g).range ↔ x ∈ Subgroup.centralizer {g}",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MonoidHom.range",
"Equiv.ins... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1\nx : Perm α\nhx : x ∈ Subgroup.centralizer {g}\n⊢ x ∈ (kerParam g).range"
] | refine ⟨fun hx ↦ kerParam_range_le_centralizer hx, fun hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 89,
"column": 30
} | {
"line": 89,
"column": 75
} | {
"line": 89,
"column": 76
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nn : ℕ\nhg : orderOf g ∣ 2 ^ n\nk : ℕ\nhk : 4 ∈ g.cycleType\nhk4 : 4 ≤ 4\nhk1 : 1 < 4\nhg0 : 4 ≠ 2\nt : Multiset ℕ\nh1 : t = Multiset.replicate t.card 0\nht : g.cycleType = 4 ::ₘ t\nh : 0 ∉ g.cycleType\n⊢ t = 0",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nn : ℕ\nhg : orderOf g ∣ 2 ^ n\nk : ℕ\nhk : 4 ∈ g.cycleType\nhk4 : 4 ≤ 4\nhk1 : 1 < 4\nhg0 : 4 ≠ 2\nt : Multiset ℕ\nh1 : t = Multiset.replicate t.card 0\nht : g.cycleType = 4 ::ₘ t\nh : 0 ∉ g.cycleType\n⊢ t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 21
} | {
"line": 222,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\n⊢ Subgroup.centralizer {g} ≤ alternatingGroup α ↔\n (∀ c ∈ g.cycleType, Odd c) ∧ Fintype.card α ≤ g.cycleType.sum + 1 ∧ ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants":... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\n⊢ (∀ ⦃x : Perm α⦄, x ∈ Subgroup.centralizer {g} → x ∈ alternatingGroup α) ↔\n (∀ c ∈ g.cycleType, Odd c) ∧ Fintype.card α ≤ g.cycleType.sum + 1 ∧ ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1"
] | rw [SetLike.le_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 120,
"column": 54
} | {
"line": 120,
"column": 65
} | {
"line": 120,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nk : ↥(alternatingGroup α)\nhk : k ∈ ↑S\nn : ℕ\nhn : orderOf ⟨k, hk⟩ = 2 ^ n\n⊢ orderOf ↑k = 2 ^ n",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nk : ↥(alternatingGroup α)\nhk : k ∈ ↑S\nn : ℕ\nhn : orderOf ⟨k, hk⟩ = 2 ^ n\n⊢ orderOf k = 2 ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 66
} | {
"line": 177,
"column": 67
} | [
{
"pp": "case e'_2.e'_5\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ {1}\n⊢ g = 1",
"ppTerm": "?e'_2.e'_5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case e'_2.e'_5\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ {1}\n⊢ g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 18
} | {
"line": 256,
"column": 19
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh5 : 5 ≤ Nat.card α\ng : ↥(alternatingGroup α)\nhg : (↑g).IsThreeCycle\n⊢ (↑(g ^ 2)).IsThreeCycle",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.IsThreeCycle.congr_simp",
"HMul.hMul",... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh5 : 5 ≤ Nat.card α\ng : ↥(alternatingGroup α)\nhg : (↑g).IsThreeCycle\n⊢ (↑g * ↑g).IsThreeCycle"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 17
} | {
"line": 130,
"column": 18
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\n⊢ Nat.card α ≠ 2 * 3",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_2\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\n⊢ ¬Fintype.card α = 6"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 131,
"column": 59
} | {
"line": 131,
"column": 70
} | {
"line": 131,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\nthis : IsPreprimitive ↥(alternatingGroup α) ↑(Set.powersetCard α 3)\n⊢ 5 ≤ Nat.card α",
"ppTerm": "?m.72",
"assigned": t... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\nthis : IsPreprimitive ↥(alternatingGroup α) ↑(Set.powersetCard α 3)\n⊢ 5 ≤ Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 22
} | {
"line": 223,
"column": 0
} | [
{
"pp": "case hx\n⊢ orderOf ?x = 4",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"ZMod.commRing",
"CommSemiring.toSemiring",
"instMulNat",
"instOfNatNat",
"QuaternionGroup.orderOf_xa",
"ZMod",
"CommRing.toCommSemiring",
"Na... | [] | exact orderOf_xa 0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 15
} | {
"line": 233,
"column": 16
} | [
{
"pp": "case inl\nn : ℕ\nh : 0 < n\nthis : CharZero (ZMod (2 * 0))\n⊢ ¬↑n = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.to... | [
"case inl\nn : ℕ\nh : 0 < n\nthis : CharZero (ZMod (2 * 0))\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 234,
"column": 2
} | {
"line": 235,
"column": 86
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case inr\nn : ℕ\nhn : NeZero n\n⊢ orderOf (a 1) = 2 * n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"QuaternionGroup.a_one_pow_n",
"Nat.instMulZeroClass",
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"HMul.hMul",
"ZMod... | [
"case inr\nn : ℕ\nhn : NeZero n\n⊢ ¬orderOf (a 1) < 2 * n"
] | apply (Nat.le_of_dvd
(NeZero.pos _) (orderOf_dvd_of_pow_eq_one (@a_one_pow_n n))).lt_or_eq.resolve_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 227,
"column": 2
} | {
"line": 241,
"column": 41
} | {
"line": 243,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ orderOf (a 1) = 2 * n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"QuaternionGroup.a.noConfusion",
"QuaternionGroup.a_one_pow_n",
"Eq.mpr",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"Preorder.toLT",
"InvOneClass.t... | [] | rcases eq_zero_or_neZero n with rfl | hn
· simp_rw [mul_zero, orderOf_eq_zero_iff']
intro n h
rw [one_def, a_one_pow]
apply mt a.inj
have : CharZero (ZMod (2 * 0)) := ZMod.charZero
simpa using h.ne'
apply (Nat.le_of_dvd
(NeZero.pos _) (orderOf_dvd_of_pow_eq_one (@a_one_pow_n n))).lt_or_eq.re... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 227,
"column": 2
} | {
"line": 241,
"column": 41
} | {
"line": 243,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ orderOf (a 1) = 2 * n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"QuaternionGroup.a.noConfusion",
"QuaternionGroup.a_one_pow_n",
"Eq.mpr",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"Preorder.toLT",
"InvOneClass.t... | [] | rcases eq_zero_or_neZero n with rfl | hn
· simp_rw [mul_zero, orderOf_eq_zero_iff']
intro n h
rw [one_def, a_one_pow]
apply mt a.inj
have : CharZero (ZMod (2 * 0)) := ZMod.charZero
simpa using h.ne'
apply (Nat.le_of_dvd
(NeZero.pos _) (orderOf_dvd_of_pow_eq_one (@a_one_pow_n n))).lt_or_eq.re... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 34
} | {
"line": 57,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : IsZGroup G\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nP : Subgroup G\nhP : IsPGroup p ↥P\nQ : Sylow p G\nhQ : P ≤ ↑Q\n⊢ IsCyclic ↥P",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Subgroup.isCyclic_of_le",
"I... | [] | exact Subgroup.isCyclic_of_le hQ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 55,
"column": 49
} | {
"line": 55,
"column": 60
} | {
"line": 55,
"column": 61
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\nn : ℕ\ng : G\nhgn : g ^ n ∈ H.toSubmonoid\n⊢ n = 0 ∨ g ∈ H.toSubmonoid",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\nn : ℕ\ng : G\nhgn : g ^ n ∈ H.toSubmonoid\n⊢ n = 0 ∨ g ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 55,
"column": 71
} | {
"line": 55,
"column": 82
} | {
"line": 55,
"column": 83
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\nn : ℕ\ng : G\nhgn : g ^ n ∈ H.toSubmonoid\n⊢ g ^ ↑n ∈ H",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZP... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\nn : ℕ\ng : G\nhgn : g ^ n ∈ H.toSubmonoid\n⊢ g ^ n ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 47
} | {
"line": 56,
"column": 58
} | {
"line": 56,
"column": 59
} | [
{
"pp": "case inl\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ ↑n ∈ H\n⊢ ↑n = 0 ∨ g ∈ H",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Subgroup",
"instOfNatNat"... | [
"case inl\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ ↑n ∈ H\n⊢ n = 0 ∨ g ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 65
} | {
"line": 56,
"column": 76
} | {
"line": 56,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ ↑n ∈ H\n⊢ g ^ n ∈ H.toSubmonoid",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"Membership.mem",
"id",
"DivInvMonoid.toMonoid... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ ↑n ∈ H\n⊢ g ^ n ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 47
} | {
"line": 56,
"column": 58
} | {
"line": 56,
"column": 59
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ (-↑n) ∈ H\n⊢ -↑n = 0 ∨ g ∈ H",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Int.neg_eq_zero._simp_1",
"Membership.mem",
"id",
... | [
"case inr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ (-↑n) ∈ H\n⊢ n = 0 ∨ g ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 65
} | {
"line": 56,
"column": 76
} | {
"line": 56,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ (-↑n) ∈ H\n⊢ g ^ n ∈ H.toSubmonoid",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"Membership.mem",
"id",
"DivInvMonoid.toMo... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : H.PowSaturated\ng : G\nn : ℕ\nhgn : g ^ (-↑n) ∈ H\n⊢ g ^ n ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 15
} | {
"line": 207,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα : 5 ≤ Nat.card α\n⊢ 3 ≤ Nat.card α",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fintype.card",
"id",
"Nat.card",
"instOfNatNat",
"LE.le",
"instLENa... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα : 5 ≤ Nat.card α\n⊢ 3 ≤ Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PushoutI | {
"line": 572,
"column": 15
} | {
"line": 572,
"column": 26
} | {
"line": 572,
"column": 27
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nm₁✝ m₂✝ : PushoutI φ\nh : ∀ (a : NormalWord d), m₁✝ • a = m₂✝ • a\n⊢ m... | [
"ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝⁴ : Monoid K\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nm₁✝ m₂✝ : PushoutI φ\nh : ∀ (a : NormalWord d), m₁✝ • a = m₂✝ • a\n⊢ m₁✝ = m₂✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.UniquelyDecodable | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 14
} | [
{
"pp": "α : Type u_1\nS : Set (List α)\nh : UniquelyDecodable S\n⊢ ¬[] ∈ S",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nS : Set (List α)\nh : UniquelyDecodable S\n⊢ ¬[] ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 89
} | {
"line": 167,
"column": 90
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\nH : Subgroup G\nh✝ : H ≠ ⊥\nhH : IsCyclic ↥⁅commutator ↥H, commutator ↥H⁆\nh : Subgroup.map (commutator ↥H).subtype (commutator ↥(commutator ↥H)) ≤ Subgroup.centralizer ↑(commutator ↥H)\n⊢ commutator ↥(commutator ↥H) ≤ Subgroup.cent... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\nH : Subgroup G\nh✝ : H ≠ ⊥\nhH : IsCyclic ↥⁅commutator ↥H, commutator ↥H⁆\nh : Subgroup.map (commutator ↥H).subtype (commutator ↥(commutator ↥H)) ≤ Subgroup.centralizer ↑(commutator ↥H)\n⊢ ∀ (a : G) (b : a ∈ H) (b_1 : ⟨a, b⟩ ∈ commutator ↥H),\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PushoutI | {
"line": 626,
"column": 42
} | {
"line": 626,
"column": 49
} | {
"line": 626,
"column": 50
} | [
{
"pp": "case cons.refine_1\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ni : ι\ng : G i\nw : Word G\nhIdx : w.fstIdx ≠ some i\nhg1 : g ≠ 1\nih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.to... | [
"case cons.refine_1\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ni : ι\ng : G i\nw : Word G\nhIdx : w.fstIdx ≠ some i\nhg1 : g ≠ 1\nih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.... | hw'map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Defs | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 57
} | {
"line": 411,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : Denumerable ι\nκs : ι → Kernel α β\nhκs : ∀ (n : ι), IsSFiniteKernel (κs n)\ne : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm\nhκ_eq : Kernel.sum κs = Kernel.sum fun n ↦ Kernel.sum (κs n).seq\na : α\ns : Se... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : Denumerable ι\nκs : ι → Kernel α β\nhκs : ∀ (n : ι), IsSFiniteKernel (κs n)\ne : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm\nhκ_eq : Kernel.sum κs = Kernel.sum fun n ↦ Kernel.sum (κs n).seq\na : α\ns : Set β\nhs : Me... | ENNReal.summable.tsum_prod' fun _ => ENNReal.summable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.Hamming | {
"line": 374,
"column": 58
} | {
"line": 374,
"column": 69
} | {
"line": 374,
"column": 70
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → DecidableEq (β i)\ns : Set (Hamming β × Hamming β)\nhs : s ∈ uniformity (Hamming β)\na✝ b✝ : Hamming β\nhab : ↑(hammingDist (ofHamming a✝) (ofHamming b✝)) < 1\n⊢ (a✝, b✝) ∈ SetRel.id",
"ppTerm": "?m.73",
"assign... | [
"α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → DecidableEq (β i)\ns : Set (Hamming β × Hamming β)\nhs : s ∈ uniformity (Hamming β)\na✝ b✝ : Hamming β\nhab : ↑(hammingDist (ofHamming a✝) (ofHamming b✝)) < 1\n⊢ a✝ = b✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 9
} | {
"line": 82,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ((Kernel.id ∘ₖ κ) a) s = (κ a) s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ∫⁻ (b : β), s.indicator 1 b ∂κ a = (κ a) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 15
} | {
"line": 89,
"column": 16
} | [
{
"pp": "case left\nα : Type u_1\nS : Finset (List α)\nr : ℕ\nw : Fin r → ↥S\nh0 : ∀ (c : ↥S), ↑c ≠ []\nthis : ∑ x, 1 ≤ ∑ i, (↑(w i)).length\n⊢ r ≤ ∑ i, (↑(w i)).length",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Finset",
"Preorder.toLE... | [
"case left\nα : Type u_1\nS : Finset (List α)\nr : ℕ\nw : Fin r → ↥S\nh0 : ∀ (c : ↥S), ↑c ≠ []\nthis : ∑ x, 1 ≤ ∑ i, (↑(w i)).length\n⊢ r ≤ ∑ i, (↑(w i)).length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 27
} | {
"line": 76,
"column": 28
} | [
{
"pp": "case iUnion\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nt : Set (α × β)\nhκs : ∀ (a : α), IsFiniteMeasure (κ a)\nf : ℕ → Set (α × β)\nh_disj : Pairwise (Disjoint on f)\nhf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nhf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (P... | [
"case iUnion\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nt : Set (α × β)\nhκs : ∀ (a : α), IsFiniteMeasure (κ a)\nf : ℕ → Set (α × β)\nh_disj : Pairwise (Disjoint on f)\nhf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nhf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 111,
"column": 43
} | {
"line": 111,
"column": 61
} | {
"line": 111,
"column": 62
} | [
{
"pp": "α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nr : ℕ\nhr : r ≥ 1\nmaxLen : ℕ := S.sup List.length\nT : Finset (List α) := Finset.image concatFn Finset.univ\nw✝ : Fin r → ↥S\nleft✝ : w✝ ∈ Finset.univ\nhx : concatFn w✝ ∈ T\nc : ↥S\nhnil : ↑c = []\n⊢ [... | [
"α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nr : ℕ\nhr : r ≥ 1\nmaxLen : ℕ := S.sup List.length\nT : Finset (List α) := Finset.image concatFn Finset.univ\nw✝ : Fin r → ↥S\nleft✝ : w✝ ∈ Finset.univ\nhx : concatFn w✝ ∈ T\nc : ↥S\nhnil : ↑c = []\n⊢ [] ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 50
} | {
"line": 118,
"column": 8
} | [
{
"pp": "α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nr : ℕ\nhr : r ≥ 1\nmaxLen : ℕ := S.sup List.length\nT : Finset (List α) := Finset.image concatFn Finset.univ\nhlen_maps : ∀ x ∈ T, x.length ∈ Finset.Icc r (r * maxLen)\nD : ℝ := ↑(Fintype.card α)\n⊢ (∑ ... | [
"α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nr : ℕ\nhr : r ≥ 1\nmaxLen : ℕ := S.sup List.length\nT : Finset (List α) := Finset.image concatFn Finset.univ\nhlen_maps : ∀ x ∈ T, x.length ∈ Finset.Icc r (r * maxLen)\nD : ℝ := ↑(Fintype.card α)\n⊢ (∑ i ∈ S.attach... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 35
} | {
"line": 123,
"column": 36
} | [
{
"pp": "α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nr : ℕ\nhr : r ≥ 1\nmaxLen : ℕ := S.sup List.length\nT : Finset (List α) := Finset.image concatFn Finset.univ\nhlen_maps : ∀ x ∈ T, x.length ∈ Finset.Icc r (r * maxLen)\nD : ℝ := ↑(Fintype.card α)\nw : F... | [
"α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nr : ℕ\nhr : r ≥ 1\nmaxLen : ℕ := S.sup List.length\nT : Finset (List α) := Finset.image concatFn Finset.univ\nhlen_maps : ∀ x ∈ T, x.length ∈ Finset.Icc r (r * maxLen)\nD : ℝ := ↑(Fintype.card α)\nw : Fin r → ↥S\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 222,
"column": 12
} | {
"line": 222,
"column": 23
} | {
"line": 222,
"column": 24
} | [
{
"pp": "case inl.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : IsZeroOrMarkovKernel 0\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel (0 ∘ₖ 0)",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
... | [
"case inl.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : IsZeroOrMarkovKernel 0\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 222,
"column": 12
} | {
"line": 222,
"column": 23
} | {
"line": 222,
"column": 24
} | [
{
"pp": "case inl.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nη : Kernel β γ\ninst✝¹ : IsZeroOrMarkovKernel η\ninst✝ : IsZeroOrMarkovKernel 0\nh✝ : IsMarkovKernel η\n⊢ IsZeroOrMarkovKernel (η ∘ₖ 0)",
"ppTerm": "?inl.inr",
"assign... | [
"case inl.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nη : Kernel β γ\ninst✝¹ : IsZeroOrMarkovKernel η\ninst✝ : IsZeroOrMarkovKernel 0\nh✝ : IsMarkovKernel η\n⊢ IsZeroOrMarkovKernel 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 222,
"column": 12
} | {
"line": 222,
"column": 23
} | {
"line": 222,
"column": 24
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsZeroOrMarkovKernel κ\nh✝ : IsMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel (0 ∘ₖ κ)",
"ppTerm": "?inr.inl",
"assign... | [
"case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsZeroOrMarkovKernel κ\nh✝ : IsMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 222,
"column": 12
} | {
"line": 222,
"column": 23
} | {
"line": 222,
"column": 24
} | [
{
"pp": "case inr.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsZeroOrMarkovKernel κ\nη : Kernel β γ\ninst✝ : IsZeroOrMarkovKernel η\nh✝¹ : IsMarkovKernel κ\nh✝ : IsMarkovKernel η\n⊢ IsZeroOrMarkovKernel (η ∘ₖ κ)... | [
"case inr.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsZeroOrMarkovKernel κ\nη : Kernel β γ\ninst✝ : IsZeroOrMarkovKernel η\nh✝¹ : IsMarkovKernel κ\nh✝ : IsMarkovKernel η\n⊢ IsZeroOrMarkovKernel (η ∘ₖ κ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 263,
"column": 14
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nκ : Kernel α α\nn : ℕ\na : α\ns : Set α\nhs : MeasurableSet s\n⊢ ((κ ^ (n + 1)) a) s = ∫⁻ (b : α), (κ b) s ∂(κ ^ n) a",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nmα : MeasurableSpace α\nκ : Kernel α α\nn : ℕ\na : α\ns : Set α\nhs : MeasurableSet s\n⊢ ((κ ^ (n + 1)) a) s = ∫⁻ (b : α), (κ b) s ∂(κ ^ n) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 78,
"column": 35
} | {
"line": 78,
"column": 82
} | {
"line": 80,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nf : β → γ\nκ : Kernel α β\nhf : Measurable f\na : α\ns : Set γ\nhs : MeasurableSet s\n⊢ ((κ.map f) a) s = (κ a) (f ⁻¹' s)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants":... | [] | by rw [map_apply _ hf, Measure.map_apply hf hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 24
} | [
{
"pp": "case inl.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel 0\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel (0 ∥ₖ 0)",
"ppTerm": "?inl.inl",
... | [
"case inl.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel 0\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 24
} | [
{
"pp": "case inl.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel η\ninst✝ : IsZeroOrMarkovKernel 0\nh✝ : IsMarkovKernel η\n⊢ IsZeroOrMarkovKernel... | [
"case inl.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel η\ninst✝ : IsZeroOrMarkovKernel 0\nh✝ : IsMarkovKernel η\n⊢ IsZeroOrMarkovKernel 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 24
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel κ\nh✝ : IsMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel... | [
"case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel κ\nh✝ : IsMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel 0\n⊢ IsZeroOrMarkovKernel 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 24
} | [
{
"pp": "case inr.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel η\nh✝¹ : IsMarkovKernel κ\nh✝ : ... | [
"case inr.inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsZeroOrMarkovKernel κ\ninst✝ : IsZeroOrMarkovKernel η\nh✝¹ : IsMarkovKernel κ\nh✝ : IsMarkovKern... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 97
} | {
"line": 172,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\nh✝ : IsSFiniteKernel κ\nh : IsSFiniteKernel η\n⊢ IsSFiniteKernel (κ ∥ₖ η)",
"ppTerm": "?pos✝",... | [
"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\nh✝ : IsSFiniteKernel κ\nh : IsSFiniteKernel η\n⊢ IsSFiniteKernel (Kernel.sum fun i ↦ Kernel.sum fun i_1 ↦ κ.se... | simp_rw [← kernel_sum_seq κ, ← kernel_sum_seq η, parallelComp_sum_left, parallelComp_sum_right] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Composition.CompNotation | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 40
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\ninst✝ : IsMarkovKernel κ\n⊢ (⇑κ ∘ₘ μ) Set.univ = μ Set.univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lin... | [] | simp [bind_apply .univ κ.aemeasurable] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Kernel.Composition.CompNotation | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 40
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\ninst✝ : IsMarkovKernel κ\n⊢ (⇑κ ∘ₘ μ) Set.univ = μ Set.univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lin... | [] | simp [bind_apply .univ κ.aemeasurable] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompNotation | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 40
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\ninst✝ : IsMarkovKernel κ\n⊢ (⇑κ ∘ₘ μ) Set.univ = μ Set.univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lin... | [] | simp [bind_apply .univ κ.aemeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 295,
"column": 8
} | {
"line": 295,
"column": 58
} | {
"line": 296,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝¹ : SFinite μ\ninst✝ : IsSFiniteKernel κ\nh_zero : ∀ (a : α), NeZero (κ a)\nh : μ ⊗ₘ κ ≪ ν ⊗ₘ η\ns : Set α\nhs : MeasurableSet s\nhs0 : ν s = 0\nh1 : (ν ⊗ₘ η) (s ×ˢ univ) = 0\nh2 : ∫⁻ (a ... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ η : Kernel α β\ninst✝¹ : SFinite μ\ninst✝ : IsSFiniteKernel κ\nh_zero : ∀ (a : α), NeZero (κ a)\nh : μ ⊗ₘ κ ≪ ν ⊗ₘ η\ns : Set α\nhs : MeasurableSet s\nhs0 : ν s = 0\nh1 : (ν ⊗ₘ η) (s ×ˢ univ) = 0\nh2 : (fun a ↦ (κ a) uni... | · exact Kernel.measurable_coe _ MeasurableSet.univ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 66
} | {
"line": 77,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nh2s : ((κ ⊗ₖ η) a) s ≠ ∞\nt : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s... | [] | exact measure_mono (preimage_mono (subset_toMeasurable _ _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.MeasurableIntegral | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 15
} | {
"line": 139,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\na : α\nE : Type u_4\ninst✝² : NormedAddCommGroup E\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : β × γ → E\nhf : StronglyMeasurable f\nthis : StronglyMeasu... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\na : α\nE : Type u_4\ninst✝² : NormedAddCommGroup E\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : β × γ → E\nhf : StronglyMeasurable f\nthis : StronglyMeasurable fun x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.MeasurableIntegral | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 15
} | {
"line": 157,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\na : α\nE : Type u_4\ninst✝² : NormedAddCommGroup E\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : γ × β → E\nhf : StronglyMeasurable f\nthis : StronglyMeasu... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\na : α\nE : Type u_4\ninst✝² : NormedAddCommGroup E\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\nf : γ × β → E\nhf : StronglyMeasurable f\nthis : StronglyMeasurable fun y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.RadonNikodym | {
"line": 67,
"column": 6
} | {
"line": 67,
"column": 62
} | {
"line": 67,
"column": 63
} | [
{
"pp": "case refine_3.e_a\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nhμν : μ ≪ ν\nκ : Kernel α β\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsFiniteKernel κ\ns : Set (α × β)\nhs : MeasurableSet s\nx✝ : (ν ⊗ₘ κ) s < ∞\nh_key :\n ∫⁻ (x : α... | [
"case refine_3.e_a\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nhμν : μ ≪ ν\nκ : Kernel α β\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsFiniteKernel κ\ns : Set (α × β)\nhs : MeasurableSet s\nx✝ : (ν ⊗ₘ κ) s < ∞\nh_key :\n ∫⁻ (x : α × β) in uni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 320,
"column": 4
} | {
"line": 321,
"column": 52
} | {
"line": 323,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\nf : γ → E\nhf : AEStronglyMeasurable f ((η ∘ₖ κ) a)\n⊢ (fun x ↦ ∫ (y : γ), ‖AEStronglyMeasu... | [] | filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 320,
"column": 4
} | {
"line": 321,
"column": 52
} | {
"line": 323,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\nf : γ → E\nhf : AEStronglyMeasurable f ((η ∘ₖ κ) a)\n⊢ (fun x ↦ ∫ (y : γ), ‖AEStronglyMeasu... | [] | filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 320,
"column": 4
} | {
"line": 321,
"column": 52
} | {
"line": 323,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\nf : γ → E\nhf : AEStronglyMeasurable f ((η ∘ₖ κ) a)\n⊢ (fun x ↦ ∫ (y : γ), ‖AEStronglyMeasu... | [] | filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 259,
"column": 4
} | {
"line": 260,
"column": 51
} | {
"line": 261,
"column": 4
} | [
{
"pp": "case inr.e_f\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x ↦ rexp (f x)) μ\ng : α → ℝ\nh0 : NeZero μ\ns : Set α\nhs : MeasurableSet s\nx : α\n⊢ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) *\n ENNReal.ofReal (rexp (g x) / ∫ (x : α), rexp (g x) ∂μ... | [
"case inr.e_f\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x ↦ rexp (f x)) μ\ng : α → ℝ\nh0 : NeZero μ\ns : Set α\nhs : MeasurableSet s\nx : α\n⊢ ENNReal.ofReal\n ((rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) *\n (rexp (g x) / ((∫ (x : α), rexp ((f + g) x) ∂μ) / ∫ (x ... | rw [← ENNReal.ofReal_mul (by positivity),
integral_exp_tilted f, Pi.add_apply, exp_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 413,
"column": 54
} | {
"line": 431,
"column": 97
} | {
"line": 433,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝ : NormedSpace ℝ E\n⊢ ∀ {f : γ → E}, Integrable f ((η ∘ₖ κ) a) → ∫ (z : γ), f z ∂(η ∘ₖ κ) a = ∫ (x :... | [] | by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE]
apply Integrable.induction
· intro c s hs ms
simp_rw [integral_indicator hs, MeasureTheory.setIntegral_const, integral_smul_const,
measureReal_def]
congr
rw [integral_toReal, Kernel.comp_apply' _ _ _ hs]
· exact (Kernel.measurabl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 13
} | {
"line": 459,
"column": 14
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\ninst✝² : SFinite μ\ninst✝¹ : IsSFiniteKernel κ\nE : Type u_8\ninst✝ : NormedAddCommGroup E\nf : α → β → E\nhf : AEStronglyMeasurable (uncurry f) (μ ⊗ₘ κ)\n⊢ ∀ᵐ (x : α) ∂μ, AEStronglyMeasurable (f ... | [
"α : Type u_5\nβ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\ninst✝² : SFinite μ\ninst✝¹ : IsSFiniteKernel κ\nE : Type u_8\ninst✝ : NormedAddCommGroup E\nf : α → β → E\nhf : AEStronglyMeasurable (uncurry f) (μ ⊗ₘ κ)\n⊢ ∀ᵐ (x : α) ∂μ, AEStronglyMeasurable (f x) (κ x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 94,
"column": 16
} | {
"line": 94,
"column": 27
} | {
"line": 94,
"column": 28
} | [
{
"pp": "⊢ ¬DifferentiableAt ℝ (fun x ↦ x * log x + 1 - x) 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"differentiableAt_fun_id._simp_1",
"differentiableAt_add_const_iff._simp_1",
"Real",
"Semiring.toMod... | [
"⊢ ¬DifferentiableAt ℝ (fun y ↦ y * log y) 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 31
} | {
"line": 111,
"column": 0
} | [
{
"pp": "⊢ Tendsto log (nhdsWithin 0 (Ioi 0)) atBot",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real.tendsto_log_nhdsGT_zero"
],
"usedFVars": [],
"usedGoals": []
}
] | [] | exact tendsto_log_nhdsGT_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 67
} | {
"line": 113,
"column": 2
} | [
{
"pp": "⊢ ¬DifferentiableWithinAt ℝ klFun (Iio 0) 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Zero.toOfNat0",
"not_differentiableWithinAt_of_deriv_tendsto_atBot_Iio",
"InformationTheory.klFun",
"OfNat.ofNat"
],
"u... | [
"⊢ Tendsto (deriv klFun) (nhdsWithin 0 (Iio 0)) atBot"
] | refine not_differentiableWithinAt_of_deriv_tendsto_atBot_Iio _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 173,
"column": 4
} | {
"line": 174,
"column": 29
} | {
"line": 175,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nthis :\n Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal + (1 - (μ.rnDeriv ν x).toReal)) ν ↔\n Integrable (llr μ ν) μ\n⊢ Integrable (fun x ↦ klFun (μ.rn... | [] | convert! this using 3 with x
rw [klFun, add_sub_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 173,
"column": 4
} | {
"line": 174,
"column": 29
} | {
"line": 175,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nthis :\n Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal + (1 - (μ.rnDeriv ν x).toReal)) ν ↔\n Integrable (llr μ ν) μ\n⊢ Integrable (fun x ↦ klFun (μ.rn... | [] | convert! this using 3 with x
rw [klFun, add_sub_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 21
} | {
"line": 36,
"column": 2
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ limsup f (ae (μ.trim hm)) = limsup f (ae μ)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"MeasureTheory.Measure.... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ sInf {a | ∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a} = sInf {a | ∀ᵐ (n : α) ∂μ, f n ≤ a}"
] | simp_rw [limsup_eq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp | {
"line": 58,
"column": 51
} | {
"line": 58,
"column": 62
} | {
"line": 58,
"column": 63
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : Nontrivial ↥(Lp E 0 μ)\nf : ↥(Lp E 0 μ)\nhf : f ≠ 0\nhfne : ¬↑↑f =ᵐ[μ] 0\nh'p : 0 < ∞\n⊢ MemLp (fun x ↦ 1) 0 μ",
"ppTerm": "?m.302",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : Nontrivial ↥(Lp E 0 μ)\nf : ↥(Lp E 0 μ)\nhf : f ≠ 0\nhfne : ¬↑↑f =ᵐ[μ] 0\nh'p : 0 < ∞\n⊢ AEStronglyMeasurable (fun x ↦ 1) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 50
} | {
"line": 58,
"column": 51
} | [
{
"pp": "case pos\nα : Type u_1\nε : Type u_3\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhm : m ≤ m0\nf : α → ε\nhf : StronglyMeasurable f\nh0 : ¬p = 0\nh_top : p = ∞\n⊢ eLpNorm f p (μ.trim hm) = eLpNorm f p μ",
"ppTerm": "?pos✝",
"assigne... | [
"case pos\nα : Type u_1\nε : Type u_3\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhm : m ≤ m0\nf : α → ε\nhf : StronglyMeasurable f\nh0 : ¬p = 0\nh_top : p = ∞\n⊢ eLpNormEssSup f (μ.trim hm) = eLpNormEssSup f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 49
} | {
"line": 59,
"column": 50
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_3\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhm : m ≤ m0\nf : α → ε\nhf : StronglyMeasurable f\nh0 : ¬p = 0\nh_top : ¬p = ∞\n⊢ eLpNorm f p (μ.trim hm) = eLpNorm f p μ",
"ppTerm": "?neg✝",
"assign... | [
"case neg\nα : Type u_1\nε : Type u_3\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhm : m ≤ m0\nf : α → ε\nhf : StronglyMeasurable f\nh0 : ¬p = 0\nh_top : ¬p = ∞\n⊢ eLpNorm' f p.toReal (μ.trim hm) = eLpNorm' f p.toReal μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 48
} | {
"line": 161,
"column": 2
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nf g : α → ℝ\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s μ\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s μ\nhgf : ∀ (t : Set α), MeasurableSet t → μ t < ∞ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ\nhs : Me... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nf g : α → ℝ\nhf : StronglyMeasurable f\nhfi : IntegrableOn f s μ\nhg : StronglyMeasurable g\nhgi : IntegrableOn g s μ\nhgf : ∀ (t : Set α), MeasurableSet t → μ t < ∞ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ\nhs : MeasurableSet ... | have h_meas_nonpos_f : MeasurableSet {x | f x ≤ 0} :=
hf.measurableSet_le stronglyMeasurable_const | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 378,
"column": 29
} | {
"line": 387,
"column": 81
} | {
"line": 389,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ klDiv μ ν = 0 ↔ μ = ν",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toSeminormedCommRing",
"MeasureTheory.lintegr... | [] | by
refine ⟨fun h ↦ ?_, fun h ↦ h ▸ klDiv_self _⟩
have h_ne : klDiv μ ν ≠ ⊤ := by simp [h]
rw [klDiv_ne_top_iff] at h_ne
rw [klDiv_eq_lintegral_klFun, if_pos h_ne.1, lintegral_eq_zero_iff (by fun_prop)] at h
refine (Measure.rnDeriv_eq_one_iff_eq h_ne.1).mp ?_
filter_upwards [h] with x hx
simp only [Pi.zero... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 29
} | {
"line": 75,
"column": 30
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\nι : Type u_3\ninst✝ : Countable ι\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ι → X → E\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nh'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞\nh'p : p < ∞\nA : ∀ (s : Set X), MeasurableSet ... | [
"X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\nι : Type u_3\ninst✝ : Countable ι\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ι → X → E\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nh'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞\nh'p : p < ∞\nA : ∀ (s : Set X), MeasurableSet s → μ s ≠ ∞ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum | {
"line": 85,
"column": 10
} | {
"line": 85,
"column": 25
} | {
"line": 85,
"column": 26
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\nι : Type u_3\ninst✝ : Countable ι\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ι → X → E\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nh'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞\nh'p : p < ∞\nA : ∀ (s : Set X), MeasurableSet ... | [
"X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\nι : Type u_3\ninst✝ : Countable ι\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ι → X → E\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nh'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞\nh'p : p < ∞\nA : ∀ (s : Set X), MeasurableSet s → μ s ≠ ∞ ... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 27
} | {
"line": 108,
"column": 28
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\ninst✝ : CompleteSpace E\nf : ℕ → ↥(Lp E p μ)\nhf : ∑' (n : ℕ), ‖f n‖ₑ ≠ ∞\n⊢ ∑' (n : ℕ), eLpNorm (↑↑(f n)) p μ ≠ ∞",
"ppTerm": "?m.76",
"assigned": true,
"usedConst... | [
"X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\ninst✝ : CompleteSpace E\nf : ℕ → ↥(Lp E p μ)\nhf : ∑' (n : ℕ), ‖f n‖ₑ ≠ ∞\n⊢ ¬∑' (n : ℕ), eLpNorm (↑↑(f n)) p μ = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 39
} | {
"line": 119,
"column": 40
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\ninst✝ : CompleteSpace E\nf : ℕ → ↥(Lp E p μ)\nhf : ∑' (n : ℕ), ‖f n‖ₑ ≠ ∞\nA : ∀ᵐ (a : X) ∂μ, Summable fun n ↦ ‖↑↑(f n) a‖\nB : ∀ᵐ (x : X) ∂μ, ∀ (n : ℕ), ↑↑(∑ i ∈ range n, f i)... | [
"X : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\ninst✝ : CompleteSpace E\nf : ℕ → ↥(Lp E p μ)\nhf : ∑' (n : ℕ), ‖f n‖ₑ ≠ ∞\nA : ∀ᵐ (a : X) ∂μ, Summable fun n ↦ ‖↑↑(f n) a‖\nB : ∀ᵐ (x : X) ∂μ, ∀ (n : ℕ), ↑↑(∑ i ∈ range n, f i) x = ∑ i ∈ r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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