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.IndexNSmul | {
"line": 69,
"column": 80
} | {
"line": 69,
"column": 88
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : IsTorsionFree ℤ M\nn : ℕ\nhn : n ≠ 0\nx : M\nhx : x ∈ (nsmulAddMonoidHom n).ker\n⊢ x = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"False",
"instHSMul",
"eq_false",
"congrArg",
"AddCommGroup.toAddCommMonoi... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.IsSubnormal | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 22
} | [
{
"pp": "case zero\nG : Type u_1\ninst✝ : Group G\n⊢ ∀ {H : Subgroup G},\n (∃ f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f 0 = ⊤) → H.IsSubnormal",
"usedConstants": [
"Subgroup.subgroupOf",
"congrArg",
"PartialOrder.toPreorder",
"Monotone",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.IsSubnormal | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 22
} | [
{
"pp": "case zero\nG : Type u_1\ninst✝ : Group G\n⊢ ∀ {H : Subgroup G},\n (∃ f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f 0 = ⊤) → H.IsSubnormal",
"usedConstants": [
"Subgroup.subgroupOf",
"congrArg",
"PartialOrder.toPreorder",
"Monotone",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.IsSubnormal | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 22
} | [
{
"pp": "case zero\nG : Type u_1\ninst✝ : Group G\n⊢ ∀ {H : Subgroup G},\n (∃ f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f 0 = ⊤) → H.IsSubnormal",
"usedConstants": [
"Subgroup.subgroupOf",
"congrArg",
"PartialOrder.toPreorder",
"Monotone",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 56,
"column": 48
} | {
"line": 56,
"column": 60
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nS : Set G\nT : Set α\na : α\nhS : ∀ g ∈ S, g⁻¹ ∈ S\nsubset : T ⊆ orbit (↥(closure S)) a\nnotMem : a ∉ T\nnonempty : T.Nonempty\nkey0 : ∀ σ ∈ S, ∀ a ∈ T, σ • a ∈ T\nσ : G\nhσ : σ ∈ S\n⊢ σ • T = T",
"usedConstants": [
"Eq.mpr"... | Set.ext_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 10
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nmotive : HNNExtension G A B φ → Prop\nx : HNNExtension G A B φ\nof : ∀ (g : G), motive (HNNExtension.of g)\nt : motive HNNExtension.t\nmul : ∀ (x y : HNNExtension G A B φ), motive x → motive y → motive (x * y)\ninv : ∀ (x : HNNExtension G A... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 94,
"column": 33
} | {
"line": 94,
"column": 74
} | [
{
"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 : swap x y ∉ closure S\na : α\nha : a ∈ {x | swap x y ∈ closure S}\nw : α\nhzw : a ≠ w\nhσ : swap a w ∈ S\nhσa : swap a w • a ∉ {x | swap x y ∈ closure S}\... | simpa [swap_comm] using subset_closure hσ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 94,
"column": 33
} | {
"line": 94,
"column": 74
} | [
{
"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 : swap x y ∉ closure S\na : α\nha : a ∈ {x | swap x y ∈ closure S}\nz : α\nhzw : z ≠ a\nhσ : swap z a ∈ S\nhσa : swap z a • a ∉ {x | swap x y ∈ closure S}\... | simpa [swap_comm] using subset_closure hσ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_1.h.e'_2.h.h.e'_4.inl\nG : 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\nu : ℤˣ\nhu : ¬u = 1\nx✝ : G\nh✝ : u = 1\n⊢ toSubgroup A B u = B",
"usedConstants": [
"False",
"congrArg",
"False.elim",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_1.h.e'_2.h.h.e'_4.inr\nG : 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\nu : ℤˣ\nhu : ¬u = 1\nx✝ : G\nh✝ : u = -1\n⊢ toSubgroup A B u = B",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_2.h.e'_2.h.h.e'_4.inl\nG : 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\nu : ℤˣ\nhu : ¬u = 1\nx✝ : G\nh✝ : u = 1\n⊢ toSubgroup A B (-u) = A",
"usedConstants": [
"False",
"NonUnitalCommRing.toNonUnita... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_2.h.e'_2.h.h.e'_4.inr\nG : 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\nu : ℤˣ\nhu : ¬u = 1\nx✝ : G\nh✝ : u = -1\n⊢ toSubgroup A B (-u) = A",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommR... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_3.e'_3.inl\nG : 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\nu : ℤˣ\nhu : ¬u = 1\ne_1✝ : ↥(toSubgroup A B u) = ↥B\nh✝ : u = 1\n⊢ toSubgroup A B u = B",
"usedConstants": [
"False",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_3.e'_3.inr\nG : 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\nu : ℤˣ\nhu : ¬u = 1\ne_1✝ : ↥(toSubgroup A B u) = ↥B\nh✝ : u = -1\n⊢ toSubgroup A B u = B",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_4.e'_3.inl\nG : 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\nu : ℤˣ\nhu : ¬u = 1\ne_2✝ : ↥(toSubgroup A B (-u)) = ↥A\nh✝ : u = 1\n⊢ toSubgroup A B (-u) = A",
"usedConstants": [
"False",
"NonUnitalCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 164,
"column": 36
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.e'_4.e'_3.inr\nG : 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\nu : ℤˣ\nhu : ¬u = 1\ne_2✝ : ↥(toSubgroup A B (-u)) = ↥A\nh✝ : u = -1\n⊢ toSubgroup A B (-u) = A",
"usedConstants": [
"NonUnitalCommRing.toNonUn... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 236,
"column": 35
} | {
"line": 236,
"column": 43
} | [
{
"pp": "case mk\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nd : TransversalPair G A B\nhead✝ : G\ntoList✝ : List (ℤˣ × G)\nchain✝ : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) toList✝\nmem_set✝¹ : ∀ (u : ℤˣ) (g : G), (u, g) ∈ { head := head✝, toList := toList✝, chain := chain✝ }.toList ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 291,
"column": 21
} | {
"line": 291,
"column": 29
} | [
{
"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\ng : G\nu : ℤˣ\nw : NormalWord d\nh1 : w.head ∈ d.set u\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'\nu' : ℤˣ\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.HNNExtension | {
"line": 291,
"column": 21
} | {
"line": 291,
"column": 29
} | [
{
"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\ng : G\nu : ℤˣ\nw : NormalWord d\nh1 : w.head ∈ d.set u\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'\nu' : ℤˣ\... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.HNNExtension | {
"line": 291,
"column": 21
} | {
"line": 291,
"column": 29
} | [
{
"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\ng : G\nu : ℤˣ\nw : NormalWord d\nh1 : w.head ∈ d.set u\nh2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u → u = u'\nu' : ℤˣ\... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.HNNExtension | {
"line": 389,
"column": 10
} | {
"line": 389,
"column": 18
} | [
{
"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\nu : ℤˣ\nw : NormalWord d\nthis✝ : (p : Prop) → Decidable p := Classical.dec\ng' : ↥(toSubgroup A B (-u)) × ↑(d.set u) := unitsSMulGroup φ d u w.head... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_2\ninst✝ : DecidableEq α\nf : Equiv.Perm α\nx : α\nh : x = f x\n⊢ swap x (f x) ∈ closure fun x ↦ ∃ x_1 y, x_1 ≠ y ∧ x = swap x_1 y",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Subgroup.closure",
"congrArg",
"Equiv.swap",
"Membership... | · rw [← h, swap_self]
apply Subgroup.one_mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.HNNExtension | {
"line": 406,
"column": 2
} | {
"line": 419,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\n⊢ Cancels (-u) (unitsSMul φ u w) ↔ ¬Cancels u w",
"usedConstants": [
"List.head?",
"Eq.mpr",
"HNNExtension.NormalWord.ofGroup",
"HNNExtension.NormalWord.Cancel... | by_cases h : Cancels u w
· simp only [unitsSMul, h, dite_true, not_true_eq_false, iff_false]
induction w using consRecOn with
| ofGroup => simp [Cancels, unitsSMulWithCancel]
| cons g u' w h1 h2 _ =>
intro hc
apply not_cancels_of_cons_hyp _ _ h2
simp only [Cancels, cons_head, cons_toList... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.HNNExtension | {
"line": 406,
"column": 2
} | {
"line": 419,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\n⊢ Cancels (-u) (unitsSMul φ u w) ↔ ¬Cancels u w",
"usedConstants": [
"List.head?",
"Eq.mpr",
"HNNExtension.NormalWord.ofGroup",
"HNNExtension.NormalWord.Cancel... | by_cases h : Cancels u w
· simp only [unitsSMul, h, dite_true, not_true_eq_false, iff_false]
induction w using consRecOn with
| ofGroup => simp [Cancels, unitsSMulWithCancel]
| cons g u' w h1 h2 _ =>
intro hc
apply not_cancels_of_cons_hyp _ _ h2
simp only [Cancels, cons_head, cons_toList... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.HNNExtension | {
"line": 649,
"column": 45
} | {
"line": 649,
"column": 53
} | [
{
"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\nw' : NormalWord d\nhS : a.2 ∈ toSubgroup A B a.1\nx : G\nhx : w'.toList.head? = some (-a.1, x)\nthis : w'.head ∈ toSubgroup A B a.1\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 54,
"column": 6
} | {
"line": 55,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\nthis : (-1) ^ #c.support ≠ 1\n⊢ Odd ((card ∘ support) c)",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"NonUnit... | rw [← Nat.not_even_iff_odd, comp_apply]
exact fun h ↦ this h.neg_one_pow | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 54,
"column": 6
} | {
"line": 55,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\nthis : (-1) ^ #c.support ≠ 1\n⊢ Odd ((card ∘ support) c)",
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"NonUnit... | rw [← Nat.not_even_iff_odd, comp_apply]
exact fun h ↦ this h.neg_one_pow | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 10
} | [
{
"pp": "ι : 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\nhead✝¹ : H\ntoList✝¹ : List ((i : ι) × G i)\nne_one✝¹ : ∀ l ∈ toList✝¹, l.snd ≠ 1\nchain_ne✝¹ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) toList✝¹\nnormalized✝¹ :\n ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 22
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh_odd : ∀ c ∈ g.cycleType, Odd c\nh_fixed : Fintype.card α ≤ g.cycleType.sum + 1\nh_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1\ny : Perm ↑(Function.fixedPoints ⇑g)\nuv : (c : ↥g.cycleFactorsFinset) → ↥(Subgroup.zpow... | convert! mul_one _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.GroupTheory.PushoutI | {
"line": 362,
"column": 10
} | {
"line": 362,
"column": 13
} | [
{
"pp": "case refine_1.left\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... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PushoutI | {
"line": 370,
"column": 8
} | {
"line": 370,
"column": 16
} | [
{
"pp": "case pos\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\nh : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\nhh1 : ¬h... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.InformationTheory.Hamming | {
"line": 45,
"column": 65
} | {
"line": 47,
"column": 26
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_3\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → DecidableEq (β i)\nx : (i : ι) → β i\n⊢ hammingDist x x = 0",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Finset.univ",
"hammingDist.eq_1",
"congrArg",
"Finset",
"Membership.mem",
... | by
rw [hammingDist, card_eq_zero, filter_eq_empty_iff]
exact fun _ _ H => H rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PushoutI | {
"line": 659,
"column": 43
} | {
"line": 659,
"column": 51
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\n⊢ (of i) g₁... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.PushoutI | {
"line": 659,
"column": 43
} | {
"line": 659,
"column": 51
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\n⊢ (of i) g₁... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PushoutI | {
"line": 659,
"column": 43
} | {
"line": 659,
"column": 51
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\n⊢ (of i) g₁... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 661,
"column": 43
} | {
"line": 661,
"column": 51
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.PushoutI | {
"line": 661,
"column": 43
} | {
"line": 661,
"column": 51
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PushoutI | {
"line": 661,
"column": 43
} | {
"line": 661,
"column": 51
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 670,
"column": 50
} | {
"line": 670,
"column": 58
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.PushoutI | {
"line": 670,
"column": 50
} | {
"line": 670,
"column": 58
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PushoutI | {
"line": 670,
"column": 50
} | {
"line": 670,
"column": 58
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 670,
"column": 63
} | {
"line": 670,
"column": 71
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.PushoutI | {
"line": 670,
"column": 63
} | {
"line": 670,
"column": 71
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PushoutI | {
"line": 670,
"column": 63
} | {
"line": 670,
"column": 71
} | [
{
"pp": "ι : 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\nhφ : ∀ (i : ι), Injective ⇑(φ i)\ni j : ι\nhij : i ≠ j\nx : PushoutI φ\ng₁ : G i\nhg₁ : (of i) g₁ = x\ng₂ : G j\nhg₂ : (of j) g₂ = x\nhx : ¬x ∈ (base φ).range\nhx1 : x ≠ 1\nhg₁1 : g₁ ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 221,
"column": 8
} | {
"line": 221,
"column": 24
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP K : Subgroup G\ninst✝ : IsCyclic ↥P\nhP : IsPGroup p ↥P\nhKP : K ≤ Subgroup.normalizer ↑P\nhPK : (Nat.card ↥P).Coprime (Nat.card ↥K)\nx✝ : MulDistribMulAction ↥K ↥P := MulDistribMulAction.compHom (↥P) (P.normalizerMono... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Basic | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nab : α × β\n⊢ (swap α β) ab = Measure.dirac ab.swap",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.deterministic_apply",
"ProbabilityTheory.Kernel... | rw [swap, deterministic_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Basic | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nab : α × β\n⊢ (swap α β) ab = Measure.dirac ab.swap",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.deterministic_apply",
"ProbabilityTheory.Kernel... | rw [swap, deterministic_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Basic | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nab : α × β\n⊢ (swap α β) ab = Measure.dirac ab.swap",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"ProbabilityTheory.Kernel.deterministic_apply",
"ProbabilityTheory.Kernel... | rw [swap, deterministic_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Defs | {
"line": 395,
"column": 8
} | {
"line": 395,
"column": 37
} | [
{
"pp": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκs : ι → Kernel α β\ni : ι\nI : Finset ι\nhi_notMem_I : i ∉ I\nh_ind : (∀ i ∈ I, IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i ∈ I, κs i)\nh : ∀ i_1 ∈ insert i I, IsSFiniteKernel (κs i_1)\n⊢ IsSFinit... | Finset.sum_insert hi_notMem_I | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 108,
"column": 2
} | {
"line": 109,
"column": 32
} | [
{
"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\nη : Kernel β γ\na : α\ns : Set γ\nh : ((η ∘ₖ κ) a) s = 0\nt : Set γ\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun y ↦ (η y) t) =ᶠ[ae (κ a)] 0\n⊢ (fun x ↦ (η x) s) ≤ᶠ[ae (κ a... | exact ⟨Filter.EventuallyLE.trans_eq (ae_of_all _ fun _ ↦ measure_mono hst) ht,
ae_of_all _ fun _ ↦ zero_le⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 116,
"column": 56
} | {
"line": 118,
"column": 87
} | [
{
"pp": "α : 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\nf : α → γ\ng : β → δ\nhf : Measurable f\nhg : Measurable g\n⊢ deterministic f hf ∥ₖ deterministic g hg = deterministic (Prod.map f g) ⋯",
"usedCons... | by
ext x : 1
simp_rw [parallelComp_apply, deterministic_apply, Prod.map, Measure.dirac_prod_dirac] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 12
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nμ : Measure α\nf : α → β\nhf : Measurable f\n⊢ (const γ μ).map f = const γ (Measure.map f μ)",
"usedConstants": [
"ProbabilityTheory.Kernel.map",
"MeasureTheory.Measure.map"... | ext x s hs | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 14
} | [
{
"pp": "case pos\nα : 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\nhf : Measurable f\n⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f",
"usedConstants":... | ext x s hs | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 14
} | [
{
"pp": "case pos\nα : 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\nhf : Measurable f\nhg : Measurable g\n⊢ (κ.map fun x ↦ (f x, g x)).snd = κ.map g",
"usedConstants":... | ext x s hs | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\ninst✝¹ : SFinite μ\ninst✝ : SFinite ν\nκ : Kernel α β\nhκ : ¬IsSFiniteKernel κ\n⊢ (μ + ν) ⊗ₘ κ = μ ⊗ₘ κ + ν ⊗ₘ κ",
"usedConstants": [
"False",
"MeasureTheory.Measure",
"eq_false"... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 58
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nγ : Type u_4\nmγ : MeasurableSpace γ\nh : ¬IsSFiniteKernel κ\n⊢ κ ⊗ₖ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ProbabilityTheory.Kernel.compProd_of_not_isSFiniteKernel_lef... | rw [Kernel.compProd_of_not_isSFiniteKernel_left _ _ h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 58
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nγ : Type u_4\nmγ : MeasurableSpace γ\nh : ¬IsSFiniteKernel κ\n⊢ κ ⊗ₖ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ProbabilityTheory.Kernel.compProd_of_not_isSFiniteKernel_lef... | rw [Kernel.compProd_of_not_isSFiniteKernel_left _ _ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 58
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nγ : Type u_4\nmγ : MeasurableSpace γ\nh : ¬IsSFiniteKernel κ\n⊢ κ ⊗ₖ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ProbabilityTheory.Kernel.compProd_of_not_isSFiniteKernel_lef... | rw [Kernel.compProd_of_not_isSFiniteKernel_left _ _ h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 175,
"column": 51
} | {
"line": 183,
"column": 33
} | [
{
"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\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\n⊢ κ ⊗ₖ η = 0 ↔ ∀ (a : α), ∀ᵐ (b : β) ∂κ a, η (a, b) = 0",
"usedConstants": [
"MeasureT... | by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· simp_rw [← Measure.measure_univ_eq_zero]
refine fun a ↦ (lintegral_eq_zero_iff ?_).mp ?_
· exact (η.measurable_coe .univ).comp measurable_prodMk_left
· rw [← setLIntegral_univ, ← Kernel.compProd_apply_prod .univ .univ, h]
simp
· rw [← Kernel.compProd_zero_rig... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.KernelLemmas | {
"line": 61,
"column": 8
} | {
"line": 61,
"column": 19
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nη : Kernel Z T\nhκ : ¬IsSFiniteKernel κ\n⊢ swap Y T ∘ₖ (κ ∥ₖ η) = η ∥ₖ κ ∘ₖ swap X Z",
"usedConstants": [
"False",
... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 19
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\nγ : Type u_3\nmγ : MeasurableSpace γ\nη : Kernel (α × β) γ\nhμ : SFinite μ\nhκ : ¬IsSFiniteKernel κ\n⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η",
"usedCons... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.KernelLemmas | {
"line": 84,
"column": 8
} | {
"line": 84,
"column": 19
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nX' : Type u_5\nmX' : MeasurableSpace X'\nη : Kernel X' Z\ninst✝¹ : IsSFiniteKernel η\nξ : Kernel Z T\ninst✝ : IsSFiniteKernel... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.KernelLemmas | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 19
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nη : Kernel Z T\nhκ : ¬IsSFiniteKernel κ\n⊢ Kernel.id ∥ₖ κ ∘ₖ (η ∥ₖ Kernel.id) = η ∥ₖ Kernel.id ∘ₖ (Kernel.id ∥ₖ κ)",
"use... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.RadonNikodym | {
"line": 48,
"column": 2
} | {
"line": 74,
"column": 17
} | [
{
"pp": "α : 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\n⊢ (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ κ) =ᶠ[ae (ν ⊗ₘ κ)] fun p ↦ μ.rnDeriv ν p.1",
"usedConstants": [
... | refine ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite (by fun_prop) (by fun_prop) fun s hs _ ↦ ?_
have h_key t₁ t₂ : MeasurableSet t₁ → MeasurableSet t₂ →
∫⁻ x in t₁ ×ˢ t₂, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ κ) x ∂ν ⊗ₘ κ =
∫⁻ x in t₁ ×ˢ t₂, μ.rnDeriv ν x.1 ∂ν ⊗ₘ κ := by
intro ht₁ ht₂
rw [Measure.setLIntegra... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.RadonNikodym | {
"line": 48,
"column": 2
} | {
"line": 74,
"column": 17
} | [
{
"pp": "α : 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\n⊢ (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ κ) =ᶠ[ae (ν ⊗ₘ κ)] fun p ↦ μ.rnDeriv ν p.1",
"usedConstants": [
... | refine ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite (by fun_prop) (by fun_prop) fun s hs _ ↦ ?_
have h_key t₁ t₂ : MeasurableSet t₁ → MeasurableSet t₂ →
∫⁻ x in t₁ ×ˢ t₂, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ κ) x ∂ν ⊗ₘ κ =
∫⁻ x in t₁ ×ˢ t₂, μ.rnDeriv ν x.1 ∂ν ⊗ₘ κ := by
intro ht₁ ht₂
rw [Measure.setLIntegra... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 473,
"column": 8
} | {
"line": 473,
"column": 19
} | [
{
"pp": "case neg\nα : 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\nη : Kernel (α × β) γ\nξ : Kernel (α × β × γ) δ\nhκ : ¬IsSFiniteKernel κ\n⊢ (κ ⊗ₖ (η ⊗ₖ ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯))... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 527,
"column": 8
} | {
"line": 527,
"column": 19
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nι : Type u_4\ninst✝¹ : Countable ι\nκ : Kernel α β\nη : ι → Kernel (α × β) γ\ninst✝ : ∀ (i : ι), IsSFiniteKernel (η i)\nhκ : ¬IsSFiniteKernel κ\n⊢ κ ⊗ₖ Kernel.sum η = Kernel.sum f... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 12
} | [
{
"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\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsMarkovKernel η\n⊢ (κ ⊗ₖ η).fst = κ",
"usedConstants": [
"ProbabilityTheory.Kernel.ext",
"Probabil... | ext x s hs | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 61
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\ng : α → E\ns : Set α\nhs : MeasurableSet s\nhf : AEMeasurable f μ\n⊢ AEMeasurable (fun x ↦ rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) μ",
"usedConstants": [
"T6Spa... | exact (measurable_exp.comp_aemeasurable hf).div_const _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 61
} | [
{
"pp": "case pos.hf\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nf : α → ℝ\ng : α → E\ns : Set α\nhf : AEMeasurable f μ\n⊢ AEMeasurable (fun x ↦ rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) μ",
"usedConstants": [
... | exact (measurable_exp.comp_aemeasurable hf).div_const _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 284,
"column": 2
} | {
"line": 291,
"column": 61
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ μ ≪ μ.tilted f",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"T6Space.toT5Space",
"Real.partialOrder",
"Real.inst... | cases eq_zero_or_neZero μ with
| inl h => simp only [h, tilted_zero_measure]; exact fun _ _ ↦ by simp
| inr h0 =>
refine withDensity_absolutelyContinuous' ?_ ?_
· exact (hf.1.aemeasurable.div_const _).ennreal_ofReal
· filter_upwards
simp only [ne_eq, ENNReal.ofReal_eq_zero, not_le]
exact fun... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 284,
"column": 2
} | {
"line": 291,
"column": 61
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ μ ≪ μ.tilted f",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"T6Space.toT5Space",
"Real.partialOrder",
"Real.inst... | cases eq_zero_or_neZero μ with
| inl h => simp only [h, tilted_zero_measure]; exact fun _ _ ↦ by simp
| inr h0 =>
refine withDensity_absolutelyContinuous' ?_ ?_
· exact (hf.1.aemeasurable.div_const _).ennreal_ofReal
· filter_upwards
simp only [ne_eq, ENNReal.ofReal_eq_zero, not_le]
exact fun... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 284,
"column": 2
} | {
"line": 291,
"column": 61
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ μ ≪ μ.tilted f",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"T6Space.toT5Space",
"Real.partialOrder",
"Real.inst... | cases eq_zero_or_neZero μ with
| inl h => simp only [h, tilted_zero_measure]; exact fun _ _ ↦ by simp
| inr h0 =>
refine withDensity_absolutelyContinuous' ?_ ?_
· exact (hf.1.aemeasurable.div_const _).ennreal_ofReal
· filter_upwards
simp only [ne_eq, ENNReal.ofReal_eq_zero, not_le]
exact fun... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 95,
"column": 2
} | {
"line": 98,
"column": 52
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\n⊢ klDiv μ ν = ∞ ↔ μ ≪ ν → ¬Integrable (llr μ ν) μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"or_not_of_imp",
"eq_false",
... | constructor <;> intro h
· contrapose! h
simp [klDiv_of_ac_of_integrable h.1 h.2]
· rcases or_not_of_imp h with (h | h) <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 95,
"column": 2
} | {
"line": 98,
"column": 52
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\n⊢ klDiv μ ν = ∞ ↔ μ ≪ ν → ¬Integrable (llr μ ν) μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"or_not_of_imp",
"eq_false",
... | constructor <;> intro h
· contrapose! h
simp [klDiv_of_ac_of_integrable h.1 h.2]
· rcases or_not_of_imp h with (h | h) <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 168,
"column": 39
} | {
"line": 168,
"column": 60
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : μ ≪ ν\nh_eq : μ univ = ν univ\nh_int : ¬Integrable (llr μ ν) μ\n⊢ ∞.toReal = ∫ (a : α), llr μ ν a ∂μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpac... | integral_undef h_int, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 162,
"column": 4
} | {
"line": 167,
"column": 61
} | [
{
"pp": "case refine_1\nα : 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, ... | rw [Measure.restrict_restrict (hm _ h_meas_nonneg_g), Measure.restrict_restrict h_meas_nonneg_f,
hgf _ (@MeasurableSet.inter α m _ _ h_meas_nonneg_g hs)
((measure_mono Set.inter_subset_right).trans_lt (lt_top_iff_ne_top.mpr hμs)),
← Measure.restrict_restrict (hm _ h_meas_nonneg_g), ←
Measure.r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 162,
"column": 4
} | {
"line": 167,
"column": 61
} | [
{
"pp": "case refine_1\nα : 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, ... | rw [Measure.restrict_restrict (hm _ h_meas_nonneg_g), Measure.restrict_restrict h_meas_nonneg_f,
hgf _ (@MeasurableSet.inter α m _ _ h_meas_nonneg_g hs)
((measure_mono Set.inter_subset_right).trans_lt (lt_top_iff_ne_top.mpr hμs)),
← Measure.restrict_restrict (hm _ h_meas_nonneg_g), ←
Measure.r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 62
} | [
{
"pp": "α : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nhp_ne_top : p ≠ ∞\nP : ↥(Lp F p μ) → Prop\nh_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indi... | refine ((indicator_ae_eq_of_ae_eq_set this).trans ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable | {
"line": 447,
"column": 4
} | {
"line": 447,
"column": 62
} | [
{
"pp": "α : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nhp_ne_top : p ≠ ∞\nP : ↥(Lp F p μ) → Prop\nh_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indi... | refine ((indicator_ae_eq_of_ae_eq_set this).trans ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 175,
"column": 10
} | {
"line": 175,
"column": 23
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nf : ↥(Lp ℝ 2 μ)\nhf : ↑↑f =ᶠ[ae (μ.restrict s)] 0\nh_nnnorm_eq_zero : (fun x ↦ ↑‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊) =ᶠ[ae (μ.restrict s)] 0\nx : α\nhx : ↑‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊ = 0 x\... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 327,
"column": 4
} | {
"line": 329,
"column": 46
} | [] | μ[-f | m] = μ[(-1 : ℝ) • f | m] := by rw [neg_one_smul ℝ f]
_ =ᵐ[μ] (-1 : ℝ) • μ[f | m] := condExp_smul ..
_ = -μ[f | m] := neg_one_smul ℝ (μ[f | m]) | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 78
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_7\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : ↥(Lp E 2 μ)\nc : E\nh_mem_Lp : MemLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a... | exact (integrableOn_condExpL2_of_measure_ne_top hm hμs.ne _).const_inner _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 67
} | [
{
"pp": "case refine_3\nα : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\ninst✝⁶ : CompleteSpace E'\ninst✝⁵ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\nE'' : Type u_8\n𝕜' : Type u_9\ninst✝⁴ : RCLike 𝕜'\ninst✝³ : N... | have h_coe := T.coeFn_compLp (condExpL2 E' 𝕜 hm f : α →₂[μ] E') | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 45
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nx : G\n⊢ condExpIndSMul hm ⋯ ⋯ x = 0",
"usedConstants": [
"MeasureTheory.condExpIndSMul.eq_1",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"... | rw [condExpIndSMul, indicatorConstLp_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 186,
"column": 8
} | {
"line": 186,
"column": 29
} | [
{
"pp": "case neg\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ : Kernel 𝓧 𝓨\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsMarkovKernel κ\nh_ac : ¬μ ⊗ₘ κ ≪ ν ⊗ₘ κ\n⊢ klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ) = klDiv μ ν",
"usedConstants": [
... | klDiv_of_not_ac h_ac, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 187,
"column": 9
} | {
"line": 187,
"column": 55
} | [
{
"pp": "case neg\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ : Kernel 𝓧 𝓨\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsMarkovKernel κ\nh_ac : ¬μ ⊗ₘ κ ≪ ν ⊗ₘ κ\n⊢ ¬μ ≪ ν",
"usedConstants": [
"ProbabilityTheory.IsZeroOrM... | Measure.absolutelyContinuous_compProd_left_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 52
} | [
{
"pp": "case pos\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ : Kernel 𝓧 𝓨\ninst✝² : IsFiniteMeasure μ\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsMarkovKernel κ\nh_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ κ\n⊢ ∫⁻ (x : 𝓧 × 𝓨), ENNReal.ofReal (klFun ((∂μ ⊗ₘ κ/∂ν ⊗ₘ κ) x).toRea... | Measure.absolutelyContinuous_compProd_left_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeFree | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nV : Type u_2\ninst✝⁶ : AddCommMonoid V\ninst✝⁵ : Module R V\nA : Type u_3\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra A R\ninst✝² : Module A V\ninst✝¹ : IsScalarTower A R V\nι : Type u_4\nb : Module.Basis ι R V\ninst✝ : Fintype ι\n⊢ IsBaseChange R (Fintype.linearCo... | have : DecidableEq ι := Classical.typeDecidableEq ι | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 54,
"column": 9
} | {
"line": 54,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\nf g : V →ₗ[R] V\nx✝ : V\n⊢ x✝ ∈ f.fixedSubmodule ⊓ g.fixedSubmodule → x✝ ∈ (f ∘ₗ g).fixedSubmodule",
"usedConstants": [
"Submodule",
"congrArg",
"LinearMap.instFunLike",
"Membershi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 109,
"column": 50
} | {
"line": 109,
"column": 58
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 109,
"column": 50
} | {
"line": 109,
"column": 58
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 109,
"column": 50
} | {
"line": 109,
"column": 58
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 112,
"column": 50
} | {
"line": 112,
"column": 58
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 112,
"column": 50
} | {
"line": 112,
"column": 58
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 112,
"column": 50
} | {
"line": 112,
"column": 58
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Center | {
"line": 122,
"column": 4
} | {
"line": 131,
"column": 19
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\ni j : ι\nh✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)\ns t : R\nh : s • b i + t • f... | · have : t = 0 ∨ b.repr (f (b i)) j = 0 := by
rw [b.ext_elem_iff] at h
simpa [single_eq_of_ne' hj] using h j
apply Or.resolve_left this
contrapose h'
refine ⟨?_, h'⟩
simp only [h', zero_smul, add_zero] at h
contrapose hj
apply b.linearIndependent.eq_of_smul_apply_eq_s... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 452,
"column": 46
} | {
"line": 452,
"column": 50
} | [
{
"pp": "case hle\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_t... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 481,
"column": 8
} | {
"line": 481,
"column": 50
} | [
{
"pp": "case h.right\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_... | simp_all [v, LinearMap.transvection.apply] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 481,
"column": 8
} | {
"line": 481,
"column": 50
} | [
{
"pp": "case h.right\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_... | simp_all [v, LinearMap.transvection.apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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