module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.GroupTheory.HNNExtension | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 79
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\na : ↥(toSubgroup A B (-1))\n⊢ ↑((toSubgroupEquiv φ (- -1)) ((toSubgroupEquiv φ (-1)) a)) = ↑a",
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"C... | simp only [toSubgroup_neg_one, toSubgroupEquiv_neg_one, SetLike.coe_eq_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.HNNExtension | {
"line": 311,
"column": 10
} | {
"line": 311,
"column": 21
} | [
{
"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\nmotive : NormalWord d → Sort u_4\nofGroup : (g : G) → motive (NormalWord.ofGroup g)\ncons :\n (g : G) →\n (u : ℤˣ) →\n (w : NormalWord d) →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nhw : w.head ∈ toSubgroup A B u\nx : G\nh2 : ∀ u' ∈ Option.map Prod.fst (some (-u, x)), w.head ∈ toSubgroup A B u → u = u'\nhx : w.toList.head? = some (-u, x)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 40
} | [
{
"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))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 416,
"column": 6
} | {
"line": 417,
"column": 43
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 25
} | [
{
"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",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 425,
"column": 4
} | {
"line": 429,
"column": 28
} | [
{
"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",
"usedConstants": [
"List.head?",
"mul_inv_cancel_right",
"... | 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": 425,
"column": 4
} | {
"line": 429,
"column": 28
} | [
{
"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",
"usedConstants": [
"List.head?",
"mul_inv_cancel_right",
"... | 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.Centralizer | {
"line": 316,
"column": 10
} | {
"line": 316,
"column": 75
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 590,
"column": 15
} | {
"line": 590,
"column": 26
} | [
{
"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✝",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 384,
"column": 6
} | {
"line": 384,
"column": 46
} | [
{
"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",
"usedConstants": [
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.RegularWreathProduct | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 17
} | [
{
"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 = ... | intro m₁ m₂ h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.HNNExtension | {
"line": 622,
"column": 8
} | {
"line": 622,
"column": 49
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 445,
"column": 4
} | {
"line": 446,
"column": 85
} | [
{
"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",
"usedConstants": [
"Equiv.Perm.support",
"MonoidHom.range",
"MonoidHom.in... | 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": 445,
"column": 4
} | {
"line": 446,
"column": 85
} | [
{
"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",
"usedConstants": [
"Equiv.Perm.support",
"MonoidHom.range",
"MonoidHom.in... | 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
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MonoidHom.instFunLike",
"Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 646,
"column": 10
} | {
"line": 646,
"column": 21
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Schreier | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 55
} | [
{
"pp": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬(center G).index = 0\nthis✝¹ : (center G).FiniteIndex\nh1 : (center G).relIndex (_root_.commutator G) ∣ (center G).index\nthis✝ : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex\nh2 :\n Group.rank ↥((center... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 700,
"column": 4
} | {
"line": 700,
"column": 35
} | [
{
"pp": "case pos.a\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}",
"usedConstants": [
"Multiset.... | apply Nat.div_eq_of_eq_mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.HNNExtension | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 18
} | [
{
"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⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 716,
"column": 2
} | {
"line": 717,
"column": 9
} | [
{
"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)!",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 89,
"column": 30
} | {
"line": 89,
"column": 75
} | [
{
"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",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 120,
"column": 54
} | {
"line": 120,
"column": 65
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 197,
"column": 2
} | {
"line": 204,
"column": 34
} | [
{
"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... | 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.SpecificGroups.Alternating.KleinFour | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 66
} | [
{
"pp": "case h.e'_2.h.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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 107,
"column": 16
} | {
"line": 120,
"column": 9
} | [
{
"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": 186,
"column": 15
} | {
"line": 186,
"column": 26
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 65
} | [
{
"pp": "case h\nα : 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}",
"usedConstants": [
"MonoidHom.range",
"Equiv.instEquivLike",
"Finset",
"S... | refine ⟨fun hx ↦ kerParam_range_le_centralizer hx, fun hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 21
} | [
{
"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",
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
... | rw [SetLike.le_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 22
} | [
{
"pp": "case hx\n⊢ orderOf ?x = 4",
"usedConstants": [
"HMul.hMul",
"ZMod.commRing",
"CommSemiring.toSemiring",
"instMulNat",
"instOfNatNat",
"QuaternionGroup.orderOf_xa",
"ZMod",
"CommRing.toCommSemiring",
"Nat.instNeZeroSucc",
"Nat",
"Zero... | exact orderOf_xa 0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 15
} | [
{
"pp": "case inl\nn : ℕ\nh : 0 < n\nthis : CharZero (ZMod (2 * 0))\n⊢ ¬↑n = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 234,
"column": 2
} | {
"line": 235,
"column": 86
} | [
{
"pp": "case inr\nn : ℕ\nhn : NeZero n\n⊢ orderOf (a 1) = 2 * n",
"usedConstants": [
"QuaternionGroup.a_one_pow_n",
"Nat.instMulZeroClass",
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"HMul.hMul",
"ZMod.commRing",
"PartialOrder.toPreorder",... | 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.PushoutI | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 67
} | [
{
"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 =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 18
} | [
{
"pp": "case hg\nα : 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",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.IsThreeCycle.congr_simp",
"HMul.hMul",
"Monoid.toMulOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 55,
"column": 49
} | {
"line": 55,
"column": 60
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"Membership.mem",
"id",
"Div... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 55,
"column": 71
} | {
"line": 55,
"column": 82
} | [
{
"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",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"Membership.mem",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 47
} | {
"line": 56,
"column": 58
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Subgroup",
"instOfNatNat",
"Int",
"Nat.cast",
"inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 65
} | {
"line": 56,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"Membership.mem",
"id",
"DivInvMonoid.toMonoid",
"Subgroup",
"Monoid.toPow",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 47
} | {
"line": 56,
"column": 58
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Int.neg_eq_zero._simp_1",
"Membership.mem",
"id",
"Int.instNegInt",
"Subgroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Subgroup.Saturated | {
"line": 56,
"column": 65
} | {
"line": 56,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"Membership.mem",
"id",
"DivInvMonoid.toMonoid",
"Subgroup",
"Monoid.toPow",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.UniquelyDecodable | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | [
{
"pp": "α : Type u_1\nS : Set (List α)\nh : UniquelyDecodable S\n⊢ ¬[] ∈ S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PushoutI | {
"line": 569,
"column": 15
} | {
"line": 569,
"column": 26
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 17
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Fintype... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 130,
"column": 59
} | {
"line": 130,
"column": 70
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PushoutI | {
"line": 624,
"column": 42
} | {
"line": 624,
"column": 49
} | [
{
"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... | hw'map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 34
} | [
{
"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",
"usedConstants": [
"Sylow.toSubgroup",
"Subgroup.isCyclic_of_le",
"IsZGroup.instIsCyclicSubtypeMemSubgroupOfFactP... | exact Subgroup.isCyclic_of_le hQ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 89
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα : 5 ≤ Nat.card α\n⊢ 3 ≤ Nat.card α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fintype.card",
"id",
"Nat.card",
"instOfNatNat",
"LE.le",
"instLENat",
"Nat",
"OfNat.ofNat",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 15
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Finset",
"Preorder.toLE",
"Membership.mem",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 113,
"column": 43
} | {
"line": 113,
"column": 61
} | [
{
"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⊢ [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 50
} | [
{
"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⊢ (∑ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 35
} | [
{
"pp": "case h\nα : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Hamming | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 49
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_3\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝² : (i : ι) → DecidableEq (γ i)\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → Zero (γ i)\nf : (i : ι) → γ i → β i\nx : (i : ι) → γ i\nhf : ∀ (i : ι), f i 0 = 0\n⊢ (hammingNorm fun i ↦ f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Hamming | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 50
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_3\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝² : (i : ι) → DecidableEq (γ i)\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → Zero (γ i)\nf : (i : ι) → γ i → β i\nx : (i : ι) → γ i\nhf₁ : ∀ (i : ι), Injective (f i)\nhf₂ : ∀ (i : ι), f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.Hamming | {
"line": 373,
"column": 58
} | {
"line": 373,
"column": 69
} | [
{
"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",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Defs | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 57
} | [
{
"pp": "case h.h\nα : 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 :... | ENNReal.summable.tsum_prod' fun _ => ENNReal.summable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 46
} | [
{
"pp": "α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nh_kraft : 1 < ∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length\nK : ℝ := ∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length\nmaxLen : ℕ := S.sup List.length\nhAbs : |1 / K| < 1\n⊢ Tendsto (fun r ↦ ↑r * ↑maxLen / K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 27
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Basic | {
"line": 412,
"column": 9
} | {
"line": 412,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\n⊢ μ (toMeasurable μ {a | ¬IsProbabilityMeasure (κ a)}) = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Basic | {
"line": 417,
"column": 20
} | {
"line": 417,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\ns : Set α\ns_meas : MeasurableSet s\nμs : μ s = 0\nhs : ∀ a ∉ s, IsProbabilityMeasure (κ a)\nh' : sᶜ = ∅\n⊢ μ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Basic | {
"line": 423,
"column": 6
} | {
"line": 423,
"column": 33
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\ns : Set α\ns_meas : MeasurableSet s\nμs : μ s = 0\nhs : ∀ a ∉ s, IsProbabilityMeasure (κ a)\na : α\nha : a ∈ sᶜ\nb : α\nhb : b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Basic | {
"line": 424,
"column": 6
} | {
"line": 424,
"column": 33
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\ns : Set α\ns_meas : MeasurableSet s\nμs : μ s = 0\nhs : ∀ a ∉ s, IsProbabilityMeasure (κ a)\na : α\nha : a ∈ sᶜ\nb : α\nhb : b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 80,
"column": 2
} | {
"line": 81,
"column": 9
} | [
{
"pp": "case h.h\nα : 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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 221,
"column": 12
} | {
"line": 221,
"column": 23
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ProbabilityT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 221,
"column": 12
} | {
"line": 221,
"column": 23
} | [
{
"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)",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 221,
"column": 12
} | {
"line": 221,
"column": 23
} | [
{
"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)",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 221,
"column": 12
} | {
"line": 221,
"column": 23
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 78,
"column": 35
} | {
"line": 78,
"column": 82
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"ProbabilityTheory.K... | 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
} | [
{
"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)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 151,
"column": 12
} | {
"line": 151,
"column": 23
} | [
{
"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✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 97
} | [
{
"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)",
"usedConstants": [... | 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
} | [
{
"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",
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",... | 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
} | [
{
"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",
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",... | 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
} | [
{
"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",
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",... | simp [bind_apply .univ κ.aemeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 453,
"column": 50
} | {
"line": 461,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nδ : Type u_4\nmδ : MeasurableSpace δ\nκ : Kernel α β\nf : β → γ\ng : β → δ\nhg : Measurable g\n⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f",
"usedConstants": [
"Eq.mpr",
"Mea... | by
by_cases hf : Measurable f
· ext x s hs
rw [fst_apply' _ _ hs, map_apply' _ (hf.prod hg) _, map_apply' _ hf _ hs]
· simp only [Set.preimage, Set.mem_setOf]
· exact measurable_fst hs
· have : ¬ Measurable (fun x ↦ (f x, g x)) := by
contrapose hf; exact hf.fst
simp [map_of_not_measurable _ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 55
} | [
{
"pp": "case hf\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nhs : MeasurableSet s\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nb : β\nx : γ\n⊢ Measurable fun y ↦ ((swap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 295,
"column": 8
} | {
"line": 295,
"column": 58
} | [
{
"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 ... | · 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
} | [
{
"pp": "case h\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 : 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... | exact measure_mono (preimage_mono (subset_toMeasurable _ _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.RadonNikodym | {
"line": 67,
"column": 6
} | {
"line": 67,
"column": 62
} | [
{
"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 < ∞\nt : Set (α × β)\nht ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.MeasurableIntegral | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.MeasurableIntegral | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 259,
"column": 4
} | {
"line": 260,
"column": 51
} | [
{
"pp": "case inr.h.e_f.h\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... | 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": 324,
"column": 4
} | {
"line": 325,
"column": 52
} | [
{
"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": 324,
"column": 4
} | {
"line": 325,
"column": 52
} | [
{
"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": 324,
"column": 4
} | {
"line": 325,
"column": 52
} | [
{
"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.InformationTheory.KullbackLeibler.KLFun | {
"line": 94,
"column": 16
} | {
"line": 94,
"column": 27
} | [
{
"pp": "⊢ ¬DifferentiableAt ℝ (fun x ↦ x * log x + 1 - x) 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"differentiableAt_fun_id._simp_1",
"differentiableAt_add_const_iff._simp_1",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.KullbackLeibler.KLFun | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 31
} | [
{
"pp": "⊢ Tendsto log (nhdsWithin 0 (Ioi 0)) atBot",
"usedConstants": [
"Real.tendsto_log_nhdsGT_zero"
]
}
] | 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
} | [
{
"pp": "⊢ ¬DifferentiableWithinAt ℝ klFun (Iio 0) 0",
"usedConstants": [
"Real",
"Real.instZero",
"Zero.toOfNat0",
"not_differentiableWithinAt_of_deriv_tendsto_atBot_Iio",
"InformationTheory.klFun",
"OfNat.ofNat"
]
}
] | 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
} | [
{
"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
} | [
{
"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.Probability.Kernel.Composition.IntegralCompProd | {
"line": 417,
"column": 54
} | {
"line": 435,
"column": 97
} | [
{
"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": 463,
"column": 2
} | {
"line": 463,
"column": 13
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 21
} | [
{
"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)",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"MeasureTheory.Measure.trim",
"id",
"ConditionallyComple... | simp_rw [limsup_eq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 50
} | [
{
"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",
"usedConstants": [
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 49
} | [
{
"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",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp | {
"line": 61,
"column": 51
} | {
"line": 61,
"column": 62
} | [
{
"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 =ᶠ[ae μ] 0\nh'p : 0 < ∞\n⊢ MemLp (fun x ↦ 1) 0 μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 13
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\nhμ : ¬μ = 0\nhν : ¬ν = 0\nthis : NeZero ν\nν' : Measure α := (ν univ)⁻¹ • ν\nhμν' : μ ≪ ν'\nh : 0 ≤ ∫ (a : α), llr μ ν a ∂μ - μ.r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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