module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 202,
"column": 21
} | {
"line": 202,
"column": 57
} | {
"line": 202,
"column": 57
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\ns₁ s₂ s₃ : S.Splitting\nn₁ : N\nhn₁ : ⇑s₁ = fun g ↦ S.inl n₁ * s₂ g * (S.inl n₁)⁻¹\nn₂ : N\nhn₂ : ⇑s₂ = fun g ↦ S.inl n₂ * s₃ g * (S.inl n₂)⁻¹\n⊢ ⇑s₁ = fun g ↦ S.inl (n₁ * n₂) * s₃ g... | [] | simp only [hn₁, hn₂, map_mul]; group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 199,
"column": 22
} | {
"line": 202,
"column": 58
} | {
"line": 204,
"column": 0
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\ns₁ s₂ s₃ : S.Splitting\nh₁ : S.IsConj s₁ s₂\nh₂ : S.IsConj s₂ s₃\n⊢ S.IsConj s₁ s₃",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid... | [] | by
obtain ⟨n₁, hn₁⟩ := h₁
obtain ⟨n₂, hn₂⟩ := h₂
exact ⟨n₁ * n₂, by simp only [hn₁, hn₂, map_mul]; group⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 438,
"column": 16
} | {
"line": 438,
"column": 40
} | {
"line": 438,
"column": 40
} | [
{
"pp": "α : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh3g : g.IsThreeCycle\nhg : g ∈ G\nhα4 : Nat.card α ≤ 3\nthis : 3 = orderOf ⟨g, hg⟩\n⊢ orderOf ⟨g, hg⟩ ∣ Nat.card ↥G",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants":... | [
"α : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh3g : g.IsThreeCycle\nhg : g ∈ G\nhα4 : Nat.card α ≤ 3\nthis : 3 = orderOf ⟨g, hg⟩\n⊢ orderOf ⟨g, hg⟩ ∣ Fintype.card ↥G"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.IsPerfect | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 23
} | {
"line": 99,
"column": 0
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsPerfect G\n⊢ IsPerfect ↥f.range",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"Subgroup.map",
"congrArg",
"Group.IsPerfect.instSubt... | [] | rw [MonoidHom.range_eq_map]
exact IsPerfect.map _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.IsPerfect | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 23
} | {
"line": 99,
"column": 0
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝² : Group G\ninst✝¹ : Group G'\nf : G →* G'\ninst✝ : IsPerfect G\n⊢ IsPerfect ↥f.range",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"Subgroup.map",
"congrArg",
"Group.IsPerfect.instSubt... | [] | rw [MonoidHom.range_eq_map]
exact IsPerfect.map _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.IsSubnormal | {
"line": 195,
"column": 4
} | {
"line": 203,
"column": 51
} | {
"line": 205,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f n = ⊤) → H.IsSubnormal",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"Subgroup.subgroupOf",
"congrArg",
"P... | [] | rintro ⟨n, hyps⟩
revert H
induction n with
| zero => simp_all
| succ n ih =>
rintro J ⟨F, hF, H_le, rfl, ih1⟩
refine step _ _ (hF <| Nat.le_add_right 0 1) ?_ (H_le _)
refine ih ⟨fun n ↦ F (n + 1), ?_⟩
grind only [Monotone, monotone_iff_forall_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.IsSubnormal | {
"line": 195,
"column": 4
} | {
"line": 203,
"column": 51
} | {
"line": 205,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f n = ⊤) → H.IsSubnormal",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"Subgroup.subgroupOf",
"congrArg",
"P... | [] | rintro ⟨n, hyps⟩
revert H
induction n with
| zero => simp_all
| succ n ih =>
rintro J ⟨F, hF, H_le, rfl, ih1⟩
refine step _ _ (hF <| Nat.le_add_right 0 1) ?_ (H_le _)
refine ih ⟨fun n ↦ F (n + 1), ?_⟩
grind only [Monotone, monotone_iff_forall_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 59
} | {
"line": 100,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhc : m.sum ≤ Fintype.card α\nh2c : ∀ a ∈ m, 2 ≤ a\nhc' : m.toList.sum ≤ Fintype.card α\np : List (List α)\nhp_length : List.map List.length p = m.toList\nhp_nodup : ∀ s ∈ p, s.Nodup\nhp_disj : List.Pairwise List.Disjoint p\nx : Li... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhc : m.sum ≤ Fintype.card α\nh2c : ∀ a ∈ m, 2 ≤ a\nhc' : m.toList.sum ≤ Fintype.card α\np : List (List α)\nhp_length : List.map List.length p = m.toList\nhp_nodup : ∀ s ∈ p, s.Nodup\nhp_disj : List.Pairwise List.Disjoint p\nx : List α\nhx : x... | rw [← Multiset.mem_toList, ← hp_length, List.mem_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 26
} | {
"line": 147,
"column": 27
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nS : Set G\nhS1 : ∀ σ ∈ S, (toPermHom G α) σ = 1 ∨ ((toPermHom G α) σ).IsSwap\nhS2 : closure S = ⊤\nh : IsPretransitive G α\n⊢ closure (⇑(toPermHom G α) '' S \\ {1}) = (toPermHom G α).range",
... | [
"G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nS : Set G\nhS1 : ∀ σ ∈ S, (toPermHom G α) σ = 1 ∨ ((toPermHom G α) σ).IsSwap\nhS2 : closure S = ⊤\nh : IsPretransitive G α\n⊢ closure (⇑(toPermHom G α) '' S) = (toPermHom G α).range"
] | closure_sdiff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 50
} | {
"line": 263,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nx : α\nhx : g.cycleOf x ∈ g.cycleFactorsFinset\n⊢ ∃ c, ∃ (_ : x ∈ (↑c).support), ∃ m, (g ^ m) (a c) = x",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instDecidableRelSameCycle",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nx : α\nhx : g.cycleOf x ∈ g.cycleFactorsFinset\n⊢ x ∈ (↑⟨g.cycleOf x, hx⟩).support"
] | refine ⟨⟨g.cycleOf x, hx⟩, ?_, (a.sameCycle hx)⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 273,
"column": 62
} | {
"line": 273,
"column": 74
} | {
"line": 273,
"column": 74
} | [
{
"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 : ↥g.cycleFactorsFinset\nhx : x ∈ (↑c).support\nm : ℤ\nhm : (g ^ m) (a c) = x\nhx' : ↑c = g.cycleOf x\nhx'' : g.cycleOf x ∈ g.cycleFactorsFinset\nn : ℕ := Nat.find ⋯\n⊢ (g ^ ↑n) (a c) =... | [
"α : 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\nhx : x ∈ (↑c).support\nm : ℤ\nhm : (g ^ m) (a c) = x\nhx' : ↑c = g.cycleOf x\nhx'' : g.cycleOf x ∈ g.cycleFactorsFinset\nn : ℕ := Nat.find ⋯\n⊢ (g ^ n) (a c) = (g ^ Nat.fin... | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 552,
"column": 37
} | {
"line": 559,
"column": 8
} | {
"line": 561,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\ng : HNNExtension G A B φ\nw : NormalWord d\n⊢ ReducedWord.prod φ (g • w).toReducedWord = g * ReducedWord.prod φ w.toReducedWord",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Units.va... | [] | by
induction g using induction_on generalizing w with
| of => simp [of_smul_eq_smul]
| t => simp [t_smul_eq_unitsSMul, prod_unitsSMul]
| mul => simp_all [mul_smul, mul_assoc]
| inv x ih =>
rw [← mul_right_inj x, ← ih]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 291,
"column": 42
} | {
"line": 291,
"column": 56
} | {
"line": 291,
"column": 56
} | [
{
"pp": "case hnc\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nhx : x ∈ Function.fixedPoints ⇑g\n⊢ x ∉ g.support",
"ppTerm": "?hnc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.P... | [
"case hnc\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nhx : x ∈ Function.fixedPoints ⇑g\n⊢ g x = x"
] | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 672,
"column": 17
} | {
"line": 672,
"column": 25
} | {
"line": 672,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nw₁ w₂ : ReducedWord G A B\nhprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂\nd : TransversalPair G A B\nw₁' : NormalWord d\nhw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁\nhw₁'2 : List.map Prod.fst w₁'.toList = Li... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nw₁ w₂ : ReducedWord G A B\nhprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂\nd : TransversalPair G A B\nw₁' : NormalWord d\nhw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁\nhw₁'2 : List.map Prod.fst w₁'.toList = List.map Prod.... | ← hw₁'2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 572,
"column": 46
} | {
"line": 572,
"column": 60
} | {
"line": 572,
"column": 60
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\np : ↥(centralizer {g})\nu : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯\nhp :\n ∀ c ∈ g.cycleFactorsFinset,\n ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c\nv... | [
"case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\np : ↥(centralizer {g})\nu : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯\nhp :\n ∀ c ∈ g.cycleFactorsFinset,\n ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c\nv : (c : ↥g.c... | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 648,
"column": 57
} | {
"line": 648,
"column": 81
} | {
"line": 648,
"column": 81
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ Fintype.card ↑{h | IsConj g h} * Nat.card ↥(stabilizer (ConjAct (Perm α)) g) = (Fintype.card α)!",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"setOf... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ Fintype.card ↑{h | IsConj g h} * Fintype.card ↥(stabilizer (ConjAct (Perm α)) g) = (Fintype.card α)!"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 264,
"column": 43
} | {
"line": 270,
"column": 80
} | {
"line": 272,
"column": 0
} | [
{
"pp": "n : ℕ\nG : Type u\ninst✝² : Group G\ninst✝¹ : Finite G\nhG : Nat.card G = n\nN : Subgroup G\ninst✝ : N.Normal\nhN : (Nat.card ↥N).Coprime N.index\n⊢ ∃ H, N.IsComplement' H",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"not_forall_not",
"_private.... | [] | by
revert G
induction n using Nat.strongRecOn with | ind n ih => ?_
rintro G _ _ rfl N _ hN
refine not_forall_not.mp fun h3 => ?_
haveI := SchurZassenhausInduction.step7 hN (fun G' _ _ hG' => by apply ih _ hG'; rfl) h3
exact not_exists_of_forall_not h3 (exists_right_complement'_of_coprime_aux hN) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 74,
"column": 30
} | {
"line": 74,
"column": 54
} | {
"line": 74,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nhg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1\nn : ℕ\nhg : orderOf g ∣ 2 ^ n\n⊢ g.support.card ≤ Nat.card α",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equ... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\ng : Perm α\nhg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1\nn : ℕ\nhg : orderOf g ∣ 2 ^ n\n⊢ g.support.card ≤ Fintype.card α"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PushoutI | {
"line": 510,
"column": 8
} | {
"line": 510,
"column": 27
} | {
"line": 511,
"column": 8
} | [
{
"pp": "case e'_9\nι : 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)\nmotive : NormalWord d → Sort ?u.126\nempty : motive NormalW... | [
"case e'_9\nι : 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)\nmotive : NormalWord d → Sort ?u.126\nempty : motive NormalWord.empty\nc... | apply d.injective i | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 220,
"column": 37
} | {
"line": 220,
"column": 52
} | {
"line": 220,
"column": 52
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nx✝ : (kleinFour α).Normal\nthis : Nat.card (↥(alternatingGroup α) ⧸ kleinFour α) = 3\ncomm_le : commutator ↥(alternatingGroup α) ≤ kleinFour α\ncomm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥\nk : ↥(alternatingG... | [
"case inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nx✝ : (kleinFour α).Normal\nthis : Nat.card (↥(alternatingGroup α) ⧸ kleinFour α) = 3\ncomm_le : commutator ↥(alternatingGroup α) ≤ kleinFour α\ncomm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥\nk : ↥(alternatingGroup α)\nhk ... | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case a.a.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (i + j + k) = a (i + (j + k))",
"ppTerm": "?a.a.a",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.RingNF.add_assoc_rev",
"HMul.hMul",
"Nat.rawCast",
"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case a.a.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (k - (i + j)) = xa (k - j - i)",
"ppTerm": "?a.a.xa",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case a.xa.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (j - i + k) = xa (j + k - i)",
"ppTerm": "?a.xa.a",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Ta... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case a.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (↑n + k - (j - i)) = a (i + (↑n + k - j))",
"ppTerm": "?a.xa.xa",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case xa.a.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (i + j + k) = xa (i + (j + k))",
"ppTerm": "?xa.a.a",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.RingNF.add_assoc_rev",
"HMul.hMul",
"Nat.rawCast",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case xa.a.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (↑n + k - (i + j)) = a (↑n + (k - j) - i)",
"ppTerm": "?xa.a.xa",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case xa.xa.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (↑n + j - i + k) = a (↑n + (j + k) - i)",
"ppTerm": "?xa.xa.a",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 66
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case xa.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (k - (↑n + j - i)) = xa (i + (↑n + k - j))",
"ppTerm": "?xa.xa.xa",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",... | [
"case xa.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (k - ↑n - j + i) = xa (k + ↑n - j + i)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 257,
"column": 2
} | {
"line": 263,
"column": 35
} | {
"line": 264,
"column": 2
} | [
{
"pp": "case inr.a\nn : ℕ\nhn : NeZero n\n⊢ Monoid.exponent (QuaternionGroup n) ∣ lcm (2 * n) 4",
"ppTerm": "?inr.a✝",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Nat.div_mul_cancel",
"MulOne.toOne",
"Dvd.dvd",
"instHDiv",... | [
"case inr.a\nn : ℕ\nhn : NeZero n\n⊢ lcm (2 * n) 4 ∣ Monoid.exponent (QuaternionGroup n)"
] | · apply Monoid.exponent_dvd_of_forall_pow_eq_one
rintro (m | m)
· rw [← orderOf_dvd_iff_pow_eq_one, orderOf_a]
refine Nat.dvd_trans ⟨gcd (2 * n) m.val, ?_⟩ (dvd_lcm_left (2 * n) 4)
exact (Nat.div_mul_cancel (Nat.gcd_dvd_left (2 * n) m.val)).symm
· rw [← orderOf_dvd_iff_pow_eq_one, orderOf_xa]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Basic | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 50
} | {
"line": 269,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ns : Set β\nμ : Measure β\nhs : MeasurableSet s\na : α\ns✝ : Set β\na✝ : MeasurableSet s✝\n⊢ (((const α μ).restrict hs) a) s✝ = ((const α (μ.restrict s)) a) s✝",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": ... | [] | simp [Kernel.restrict_apply, Kernel.const_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 87
} | {
"line": 59,
"column": 88
} | [
{
"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\nκ✝ : Kernel α β\nη✝ : Kernel γ δ\nx : α × γ\nκ : Kernel α β\nη : Kernel γ δ\nh : IsSFiniteKernel κ ∧ IsSFiniteKernel η\nhκ : IsSFiniteKernel κ\nhη : Is... | [] | exact measurable_kernel_prodMk_left (measurable_fst.snd.prodMk measurable_snd hs) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 67
} | {
"line": 80,
"column": 0
} | [
{
"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\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel γ δ\ninst✝ : IsSFiniteKernel η\nx : α × γ\n⊢ (κ ∥ₖ η) x = (κ x.1).prod (η x.2)",
"ppTerm": "... | [] | rw [parallelComp, dif_pos ⟨inferInstance, inferInstance⟩, coe_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 67
} | {
"line": 80,
"column": 0
} | [
{
"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\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel γ δ\ninst✝ : IsSFiniteKernel η\nx : α × γ\n⊢ (κ ∥ₖ η) x = (κ x.1).prod (η x.2)",
"ppTerm": "... | [] | rw [parallelComp, dif_pos ⟨inferInstance, inferInstance⟩, coe_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 67
} | {
"line": 80,
"column": 0
} | [
{
"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\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel γ δ\ninst✝ : IsSFiniteKernel η\nx : α × γ\n⊢ (κ ∥ₖ η) x = (κ x.1).prod (η x.2)",
"ppTerm": "... | [] | rw [parallelComp, dif_pos ⟨inferInstance, inferInstance⟩, coe_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 37
} | {
"line": 202,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\na : α\ns t : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ((copy β ∘ₖ κ) a) (s ×ˢ t) = (κ a) (s ∩ t)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\na : α\ns t : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ∫⁻ (b : β), ((copy β) b) (s ×ˢ t) ∂κ a = (κ a) (s ∩ t)"
] | comp_apply' _ _ _ <| hs.prod ht | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 432,
"column": 6
} | {
"line": 432,
"column": 19
} | {
"line": 432,
"column": 19
} | [
{
"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 × γ)\ninst✝ : IsMarkovKernel κ\n⊢ IsMarkovKernel κ.fst",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.... | [
"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 × γ)\ninst✝ : IsMarkovKernel κ\n⊢ IsMarkovKernel (κ.map Prod.fst)"
] | Kernel.fst_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 436,
"column": 6
} | {
"line": 436,
"column": 19
} | {
"line": 436,
"column": 19
} | [
{
"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 × γ)\ninst✝ : IsZeroOrMarkovKernel κ\n⊢ IsZeroOrMarkovKernel κ.fst",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": ... | [
"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 × γ)\ninst✝ : IsZeroOrMarkovKernel κ\n⊢ IsZeroOrMarkovKernel (κ.map Prod.fst)"
] | Kernel.fst_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 439,
"column": 6
} | {
"line": 439,
"column": 19
} | {
"line": 439,
"column": 19
} | [
{
"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 × γ)\ninst✝ : IsFiniteKernel κ\n⊢ IsFiniteKernel κ.fst",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.... | [
"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 × γ)\ninst✝ : IsFiniteKernel κ\n⊢ IsFiniteKernel (κ.map Prod.fst)"
] | Kernel.fst_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 442,
"column": 38
} | {
"line": 442,
"column": 51
} | {
"line": 442,
"column": 51
} | [
{
"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 × γ)\ninst✝ : IsSFiniteKernel κ\n⊢ IsSFiniteKernel κ.fst",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"E... | [
"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 × γ)\ninst✝ : IsSFiniteKernel κ\n⊢ IsSFiniteKernel (κ.map Prod.fst)"
] | Kernel.fst_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.MapComap | {
"line": 459,
"column": 4
} | {
"line": 460,
"column": 33
} | {
"line": 461,
"column": 4
} | [
{
"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\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",
"ppTerm": "?neg... | [
"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\nf : β → γ\ng : β → δ\nhg : Measurable g\nhf : ¬Measurable f\nthis : ¬Measurable fun x ↦ (f x, g x)\n⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.... | have : ¬ Measurable (fun x ↦ (f x, g x)) := by
contrapose hf; exact hf.fst | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 74
} | {
"line": 96,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\n⊢ ⇑(Kernel.discard α) ∘ₘ μ = μ Set.univ • dirac ()",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lintegral_const",
"Unit.unit",
"instHSMul",
"Mea... | [] | ext s hs; simp [Measure.bind_apply hs (Kernel.aemeasurable _), mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 74
} | {
"line": 96,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\n⊢ ⇑(Kernel.discard α) ∘ₘ μ = μ Set.univ • dirac ()",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.bind_apply",
"MeasureTheory.lintegral_const",
"Unit.unit",
"instHSMul",
"Mea... | [] | ext s hs; simp [Measure.bind_apply hs (Kernel.aemeasurable _), mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 47
} | {
"line": 303,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\nt : Set γ\nht : MeasurableSet t\n⊢ κ ⊗ₖ η.restrict ht = (κ ⊗ₖ η).restrict ⋯",
"ppTerm": "?m.... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\nt : Set γ\nht : MeasurableSet t\n⊢ κ ⊗ₖ η.restrict ht = κ.restrict ⋯ ⊗ₖ η.restrict ht"
] | ← compProd_restrict MeasurableSet.univ ht | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 58
} | {
"line": 313,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\nf : γ → E\nhf : StronglyMeasurable f\nthis : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))\n⊢ ∫⁻... | [] | exact fun h ↦ ae_lt_top hf.enorm.lintegral_kernel h.ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 58
} | {
"line": 313,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\nf : γ → E\nhf : StronglyMeasurable f\nthis : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))\n⊢ ∫⁻... | [] | exact fun h ↦ ae_lt_top hf.enorm.lintegral_kernel h.ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 58
} | {
"line": 313,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\nf : γ → E\nhf : StronglyMeasurable f\nthis : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))\n⊢ ∫⁻... | [] | exact fun h ↦ ae_lt_top hf.enorm.lintegral_kernel h.ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 438,
"column": 2
} | {
"line": 439,
"column": 16
} | {
"line": 441,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsZeroOrMarkovKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsZeroOrMarkovKernel η\n⊢ IsZeroOrMarkovKernel (κ ⊗ₖ η)",
"ppTerm": "?m.23",
"assigne... | [] | rw [compProd_def]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 438,
"column": 2
} | {
"line": 439,
"column": 16
} | {
"line": 441,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsZeroOrMarkovKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsZeroOrMarkovKernel η\n⊢ IsZeroOrMarkovKernel (κ ⊗ₖ η)",
"ppTerm": "?m.23",
"assigne... | [] | rw [compProd_def]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 457,
"column": 2
} | {
"line": 458,
"column": 16
} | {
"line": 460,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsFiniteKernel η\n⊢ IsFiniteKernel (κ ⊗ₖ η)",
"ppTerm": "?m.23",
"assigned": true,
"use... | [] | rw [compProd_def]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 457,
"column": 2
} | {
"line": 458,
"column": 16
} | {
"line": 460,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsFiniteKernel η\n⊢ IsFiniteKernel (κ ⊗ₖ η)",
"ppTerm": "?m.23",
"assigned": true,
"use... | [] | rw [compProd_def]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 462,
"column": 2
} | {
"line": 463,
"column": 16
} | {
"line": 465,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\nη : Kernel (α × β) γ\n⊢ IsSFiniteKernel (κ ⊗ₖ η)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.Kernel.instI... | [] | rw [compProd_def]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 462,
"column": 2
} | {
"line": 463,
"column": 16
} | {
"line": 465,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\nη : Kernel (α × β) γ\n⊢ IsSFiniteKernel (κ ⊗ₖ η)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.Kernel.instI... | [] | rw [compProd_def]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 490,
"column": 10
} | {
"line": 490,
"column": 51
} | {
"line": 491,
"column": 2
} | [
{
"pp": "case pos.hf\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\nhη : IsSFiniteKernel η\nhξ : IsSFiniteKernel ξ\na ... | [] | exact measurable_kernel_prodMk_left' hs a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 490,
"column": 10
} | {
"line": 490,
"column": 51
} | {
"line": 491,
"column": 2
} | [
{
"pp": "case pos.hf\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\nhη : IsSFiniteKernel η\nhξ : IsSFiniteKernel ξ\na ... | [] | exact measurable_kernel_prodMk_left' hs a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 490,
"column": 10
} | {
"line": 490,
"column": 51
} | {
"line": 491,
"column": 2
} | [
{
"pp": "case pos.hf\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\nhη : IsSFiniteKernel η\nhξ : IsSFiniteKernel ξ\na ... | [] | exact measurable_kernel_prodMk_left' hs a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 43
} | {
"line": 514,
"column": 0
} | [
{
"pp": "case pos.hf\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nhμ : IsSFiniteKernel μ\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ Measurabl... | [] | exact measurable_kernel_prodMk_left' hs a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 168,
"column": 4
} | {
"line": 173,
"column": 61
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case refine_2\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_nonpos_g), Measure.restrict_restrict h_meas_nonpos_f,
hgf _ (@MeasurableSet.inter α m _ _ h_meas_nonpos_g hs)
((measure_mono Set.inter_subset_right).trans_lt (lt_top_iff_ne_top.mpr hμs)),
← Measure.restrict_restrict (hm _ h_meas_nonpos_g), ←
Measure.r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 168,
"column": 4
} | {
"line": 173,
"column": 61
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case refine_2\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_nonpos_g), Measure.restrict_restrict h_meas_nonpos_f,
hgf _ (@MeasurableSet.inter α m _ _ h_meas_nonpos_g hs)
((measure_mono Set.inter_subset_right).trans_lt (lt_top_iff_ne_top.mpr hμs)),
← Measure.restrict_restrict (hm _ h_meas_nonpos_g), ←
Measure.r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 50
} | {
"line": 187,
"column": 0
} | [
{
"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 =ᵐ[μ.restrict s] 0\n⊢ Measurable fun x ↦ ↑‖↑↑f x‖₊",
"ppTerm": "?m.180",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommR... | [] | · exact (Lp.stronglyMeasurable _).enorm (ε := ℝ) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 348,
"column": 2
} | {
"line": 351,
"column": 51
} | {
"line": 352,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nF' : Type u_3\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ns : Set α\ninst✝ : CompleteSpace F'\nf : ↥(Lp F' 1 μ)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\n⊢ ∀ (c : F') {s_1 : ... | [
"case refine_2\nα : Type u_1\nF' : Type u_3\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ns : Set α\ninst✝ : CompleteSpace F'\nf : ↥(Lp F' 1 μ)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\n⊢ ∀ ⦃f g : α → F'⦄ (hf : MemLp f... | · intro x t ht hμt
simp_rw [condExpL1CLM_indicatorConst ht hμt.ne x]
rw [Lp.simpleFunc.coe_indicatorConst, setIntegral_indicatorConstLp (hm _ hs)]
exact setIntegral_condExpInd hs ht hμs hμt.ne x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 320,
"column": 10
} | {
"line": 320,
"column": 29
} | {
"line": 320,
"column": 29
} | [
{
"pp": "α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\nt : Set α\... | [
"α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\nt : Set α\nht : Measur... | lintegral_mul_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 16
} | {
"line": 149,
"column": 0
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ η : Kernel 𝓧 𝓨\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nh_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η\nh_ac_μν : μ ≪ ν\nh_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η\nh_iff_... | [] | · exact h_int1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 388,
"column": 10
} | {
"line": 388,
"column": 29
} | {
"line": 388,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\n⊢ ∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp ... | [
"α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\n⊢ (∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)... | lintegral_mul_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 403,
"column": 58
} | {
"line": 416,
"column": 42
} | {
"line": 418,
"column": 0
} | [
{
"pp": "α : Type u_1\nF' : Type u_3\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : CompleteSpace F'\nf : ↥(Lp F' 1 μ)\n⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' hm μ) f)) μ",
"ppTerm": "?m.37",
... | [] | by
refine @Lp.induction _ _ _ _ _ _ _ ENNReal.one_ne_top
(fun f : α →₁[μ] F' => AEStronglyMeasurable[m] (condExpL1CLM F' hm μ f) μ) ?_ ?_ ?_ f
· intro c s hs hμs
rw [condExpL1CLM_indicatorConst hs hμs.ne c]
exact aestronglyMeasurable_condExpInd hs hμs.ne c
· intro f g hf hg _ hfm hgm
rw [(condExpL... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 233,
"column": 79
} | {
"line": 233,
"column": 86
} | {
"line": 235,
"column": 2
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ η : Kernel 𝓧 𝓨\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nh_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η\nh_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 39
} | {
"line": 118,
"column": 39
} | [
{
"pp": "case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nh✝ : Nontrivial k\nj : ι\nhj : j ≠ i\n⊢ (univ.weightedVSub p... | [
"case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nh✝ : Nontrivial k\nj : ι\nhj : j ≠ i\n⊢ (affineCombination k univ p) (-(... | weightedVSub_vadd_affineCombination | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeFree | {
"line": 91,
"column": 2
} | {
"line": 103,
"column": 44
} | {
"line": 105,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nV : Type u_2\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module R V\nA : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra A R\ninst✝¹ : Module A V\ninst✝ : IsScalarTower A R V\nι : Type u_4\nb : Module.Basis ι R V\n⊢ IsBaseChange R (Finsupp.linearCombination A ⇑b)",
... | [] | let j := TensorProduct.finsuppScalarRight A R R ι
refine of_equiv ?_ ?_
· apply LinearEquiv.ofBijective (Finsupp.linearCombination R b ∘ₗ j)
rw [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, j.bijective.of_comp_iff]
simp [Function.Bijective,
← span_range_eq_top_iff_surjective_finsuppLinearCombina... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.IsBaseChangeFree | {
"line": 91,
"column": 2
} | {
"line": 103,
"column": 44
} | {
"line": 105,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nV : Type u_2\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module R V\nA : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Algebra A R\ninst✝¹ : Module A V\ninst✝ : IsScalarTower A R V\nι : Type u_4\nb : Module.Basis ι R V\n⊢ IsBaseChange R (Finsupp.linearCombination A ⇑b)",
... | [] | let j := TensorProduct.finsuppScalarRight A R R ι
refine of_equiv ?_ ?_
· apply LinearEquiv.ofBijective (Finsupp.linearCombination R b ∘ₗ j)
rw [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, j.bijective.of_comp_iff]
simp [Function.Bijective,
← span_range_eq_top_iff_surjective_finsuppLinearCombina... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 46
} | {
"line": 216,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommSemiring R\nS : Type u_2\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra R S\nM : Type u_3\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S P\ninst✝² : IsScalarTower R S P\nα : M →ₗ[R] P\nibcM : IsBaseC... | [
"R : Type u_1\ninst✝¹⁰ : CommSemiring R\nS : Type u_2\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra R S\nM : Type u_3\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nP : Type u_5\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : Module R P\ninst✝³ : Module S P\ninst✝² : IsScalarTower R S P\nα : M →ₗ[R] P\nibcM : IsBaseChange S α\nι... | simp only [toMatrix_apply, Matrix.map_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 195,
"column": 35
} | {
"line": 195,
"column": 58
} | {
"line": 195,
"column": 59
} | [
{
"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' ... | [
"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' Mᵢ\ninst✝¹ :... | TensorProduct.tmul_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Center | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 21
} | {
"line": 121,
"column": 4
} | [
{
"pp": "R : 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 (b i) = 0... | [
"case pos\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 (b i) = 0\n... | split_ifs with hj | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 399,
"column": 40
} | {
"line": 399,
"column": 55
} | {
"line": 399,
"column": 55
} | [
{
"pp": "V : Type u_2\ninst✝² : AddCommGroup V\nK : Type u_3\ninst✝¹ : DivisionRing K\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nu : V →ₗ[K] V := ↑e - LinearMap.id\nhu : u + LinearMap.id = ↑e\nhr : 1 ≤ Module.rank K ↥u.range\nb : Basis Unit K ↥u.range\nx : V\nthis : ↑(u.rangeRestrict x) = u x\n⊢ ↑(u.rangeRestrict x) =... | [
"V : Type u_2\ninst✝² : AddCommGroup V\nK : Type u_3\ninst✝¹ : DivisionRing K\ninst✝ : Module K V\ne : V ≃ₗ[K] V\nu : V →ₗ[K] V := ↑e - LinearMap.id\nhu : u + LinearMap.id = ↑e\nhr : 1 ≤ Module.rank K ↥u.range\nb : Basis Unit K ↥u.range\nx : V\nthis : ↑(u.rangeRestrict x) = u x\n⊢ u.rangeRestrict x = (b.repr (u.ran... | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 50
} | {
"line": 280,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\nc : CliffordAlgebra (Q c₁ c₂)\n⊢ toQuaternion (involute (reverse c)) = star (toQuaternion c)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"QuaternionAlgebra.instModule",
"CliffordAlgebra.ι",
... | [] | induction c using CliffordAlgebra.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 326,
"column": 34
} | {
"line": 326,
"column": 52
} | {
"line": 326,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\nq : ℍ[R,c₁,c₂]\n⊢ toQuaternion (ofQuaternion (star q)) = toQuaternion (star (ofQuaternion q))",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CliffordAlgebraQuaternion.toQuaternion_star",
"Semiring.toModule"... | [
"R : Type u_1\ninst✝ : CommRing R\nc₁ c₂ : R\nq : ℍ[R,c₁,c₂]\n⊢ toQuaternion (ofQuaternion (star q)) = star (toQuaternion (ofQuaternion q))"
] | toQuaternion_star, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 174,
"column": 87
} | {
"line": 175,
"column": 76
} | {
"line": 177,
"column": 0
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nr : R\n⊢ (contractRight ((algebraMap R (CliffordAlgebra Q)) r)) d = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CliffordAlgebra.cont... | [] | by
rw [contractRight_eq, reverse.commutes, contractLeft_algebraMap, map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.Even | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 56
} | {
"line": 89,
"column": 56
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nQ : QuadraticForm R M\nA : Type u_3\nB : Type u_4\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx✝² x✝¹ x✝ : M\n⊢ ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) (x✝¹ + x✝),... | [] | simp only [map_add, mul_add]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Even | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 56
} | {
"line": 89,
"column": 56
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nQ : QuadraticForm R M\nA : Type u_3\nB : Type u_4\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx✝² x✝¹ x✝ : M\n⊢ ⟨(CliffordAlgebra.ι Q) x✝² * (CliffordAlgebra.ι Q) (x✝¹ + x✝),... | [] | simp only [map_add, mul_add]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Even | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 41
} | {
"line": 182,
"column": 4
} | [
{
"pp": "case snd\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : EvenHom Q A\nm : M\na : A\ng : M →ₗ[R] A\nhg : g ∈ S f\nm₁ : M\n⊢ ↑(((fFold f) m) (((fFold f) m) (a, ⟨g, hg⟩))).2 m₁ = ... | [
"case snd\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : EvenHom Q A\nm : M\na : A\ng : M →ₗ[R] A\nhg : g ∈ S f\nm₁ : M\n⊢ (f.bilin m₁) m * g m = Q m • g m₁"
] | change f.bilin _ _ * g m = Q m • g m₁ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 126,
"column": 68
} | {
"line": 152,
"column": 40
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\n⊢ ↥(even (Q' Q)) →ₐ[R] CliffordAlgebra Q",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"CliffordAlgebra.EquivEven.Q'",
... | [] | by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
-/
let f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap.comp <|
(ι Q).comp (LinearMap.fst _ _ _... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 72,
"column": 30
} | {
"line": 72,
"column": 34
} | {
"line": 72,
"column": 35
} | [
{
"pp": "case mem.mem_mul\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticF... | [
"case mem.mem_mul\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup N\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQₙ : QuadraticForm R N\nf₁ ... | ih₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 29
} | {
"line": 145,
"column": 2
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\... | [
"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\ninst✝ : Mod... | letI fAB1 := auxEquiv R 𝒜 ℬ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.LinearAlgebra.CliffordAlgebra.Prod | {
"line": 157,
"column": 2
} | {
"line": 158,
"column": 25
} | {
"line": 159,
"column": 2
} | [
{
"pp": "case hl\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm : M₁\n⊢ (toProd Q₁ Q₂) ((ofProd Q₁ Q₂) ((ι (QuadraticMap.prod Q₁ Q₂)) (m, 0))) = (... | [
"case hr\nR : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nm : M₂\n⊢ (toProd Q₁ Q₂) ((ofProd Q₁ Q₂) ((ι (QuadraticMap.prod Q₁ Q₂)) (0, m))) = (ι (Quadratic... | · rw [ofProd_ι_mk, map_add, toProd_one_tmul_ι, toProd_ι_tmul_one, Prod.mk_zero_zero,
map_zero, add_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 38
} | {
"line": 349,
"column": 4
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : Ring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝⁴ : GradedAlgebra 𝒜\ninst✝³ : GradedAlgebra ... | [
"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : Ring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝⁴ : GradedAlgebra 𝒜\ninst✝³ : GradedAlgebra ℬ\ninst✝² : ... | apply AlgHom.toLinearMap_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Goursat | {
"line": 52,
"column": 65
} | {
"line": 54,
"column": 47
} | {
"line": 56,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nL : Submodule R (M × N)\n⊢ L.goursatFst.toAddSubgroup = L.toAddSubgroup.goursatFst",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
... | [] | by
ext x
simp [goursatFst, AddSubgroup.mem_goursatFst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 29
} | {
"line": 137,
"column": 6
} | [
{
"pp": "m : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\nB : Matrix m' n R\nhA : A.IsTotallyUnimodular\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → Function... | [
"m : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : DecidableEq n\nA : Matrix m n R\nB : Matrix m' n R\nhA : A.IsTotallyUnimodular\nhB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s\nk : ℕ\nih :\n ∀ (f : Fin k → m ⊕ m') (g : Fin k → n),\n Function.Injective f → Function.Injective g... | choose f' hf' using hfr | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.MvPolynomial | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 29
} | {
"line": 39,
"column": 4
} | [
{
"pp": "m : Type u_1\nk : Type u_2\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\ninst✝¹ : Field k\ninst✝ : Infinite k\np q : MvPolynomial (m × m) k\nh : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q\n⊢ (p - q) * (Matrix.mvPolynomialX m m k).det = 0",
"ppTerm": "?m.42",
... | [
"m : Type u_1\nk : Type u_2\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\ninst✝¹ : Field k\ninst✝ : Infinite k\np q : MvPolynomial (m × m) k\nh : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q\n⊢ ∀ (x : m × m → k), (eval x) ((p - q) * (Matrix.mvPolynomialX m m k).det) = (eval x) 0"
] | apply MvPolynomial.funext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.MvPolynomial | {
"line": 43,
"column": 6
} | {
"line": 46,
"column": 33
} | {
"line": 47,
"column": 2
} | [
{
"pp": "case neg\nm : Type u_1\nk : Type u_2\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\ninst✝¹ : Field k\ninst✝ : Infinite k\np q : MvPolynomial (m × m) k\nh : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q\ns : m × m → k\nhs_det : ¬(Matrix.of fun i j ↦ s (i, j)).det = 0\n⊢ (... | [] | have hh : (eval s) p = (eval s) q :=
h (Matrix.GeneralLinearGroup.mkOfDetNeZero
(Matrix.of fun i j : m => s (i, j)) hs_det)
rw [hh, sub_self, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.MvPolynomial | {
"line": 43,
"column": 6
} | {
"line": 46,
"column": 33
} | {
"line": 47,
"column": 2
} | [
{
"pp": "case neg\nm : Type u_1\nk : Type u_2\ninst✝³ : Fintype m\ninst✝² : DecidableEq m\ninst✝¹ : Field k\ninst✝ : Infinite k\np q : MvPolynomial (m × m) k\nh : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q\ns : m × m → k\nhs_det : ¬(Matrix.of fun i j ↦ s (i, j)).det = 0\n⊢ (... | [] | have hh : (eval s) p = (eval s) q :=
h (Matrix.GeneralLinearGroup.mkOfDetNeZero
(Matrix.of fun i j : m => s (i, j)) hs_det)
rw [hh, sub_self, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 30
} | {
"line": 42,
"column": 30
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nk : ℕ\nhk : k ≤ n\n⊢ Nat.card { s // LinearIndependent K s } = ∏ i, (q ^ n - q ^ ↑i)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"K : Type u_1\nV : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Fintype K\ninst✝ : Finite V\nk : ℕ\nhk : k ≤ n\n⊢ card { s // LinearIndependent K s } = ∏ i, (q ^ n - q ^ ↑i)"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Gershgorin | {
"line": 32,
"column": 2
} | {
"line": 60,
"column": 94
} | {
"line": 62,
"column": 0
} | [
{
"pp": "K : Type u_1\nn : Type u_2\ninst✝² : NormedField K\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n K\nμ : K\nhμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ\n⊢ ∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖)",
"ppTerm": "?m.32",
"assigned": true,
"usedCo... | [] | cases isEmpty_or_nonempty n
· exfalso
exact hμ Submodule.eq_bot_of_subsingleton
· obtain ⟨v, h_eg, h_nz⟩ := hμ.exists_hasEigenvector
obtain ⟨i, -, h_i⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun i => ‖v i‖)
have h_nz : v i ≠ 0 := by
contrapose h_nz
ext j
rw [Pi.zero_apply, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Gershgorin | {
"line": 32,
"column": 2
} | {
"line": 60,
"column": 94
} | {
"line": 62,
"column": 0
} | [
{
"pp": "K : Type u_1\nn : Type u_2\ninst✝² : NormedField K\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n K\nμ : K\nhμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ\n⊢ ∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖)",
"ppTerm": "?m.32",
"assigned": true,
"usedCo... | [] | cases isEmpty_or_nonempty n
· exfalso
exact hμ Submodule.eq_bot_of_subsingleton
· obtain ⟨v, h_eg, h_nz⟩ := hμ.exists_hasEigenvector
obtain ⟨i, -, h_i⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun i => ‖v i‖)
have h_nz : v i ≠ 0 := by
contrapose h_nz
ext j
rw [Pi.zero_apply, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 24
} | {
"line": 129,
"column": 0
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nhc : ↑A 1 0 ≠ 0\n⊢ reduce (reduceStep A) = reduce A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instDiv",
"instHSMul",
"Matrix.SpecialLinearGroup",
"instHDiv",
"congrArg",
"Matrix",... | [] | symm
rw [reduce, if_neg hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 24
} | {
"line": 129,
"column": 0
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nhc : ↑A 1 0 ≠ 0\n⊢ reduce (reduceStep A) = reduce A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instDiv",
"instHSMul",
"Matrix.SpecialLinearGroup",
"instHDiv",
"congrArg",
"Matrix",... | [] | symm
rw [reduce, if_neg hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 86
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case «0».«0»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)| ≤ |m|",
"ppTerm": "?«0».«0»",
"assigned": true,
"usedConstants": [
... | [
"case «0».«1»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)| ≤ |m|",
"case «1».«0»\nm : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A... | · simpa only [← abs_mul, A_c_eq_zero h10] using! (le_mul_iff_one_le_right h2).mpr h1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 119,
"column": 65
} | {
"line": 154,
"column": 42
} | {
"line": 156,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\nA : Matrix n n R\ninst✝⁴ : Fintype n\ninst✝³ : IsOrderedRing R\ninst✝² : PosMulStrictMono R\ninst✝¹ : Nontrivial R\ninst✝ : DecidableEq n\nhA : ∀ (i j : n), 0 ≤ A i j\nk : ℕ\ni j : n\n⊢ 0 < (A ^ k) i j ↔ Nonempty { p // p.length = k }... | [] | by
letI := toQuiver A
induction k generalizing i j with
| zero =>
refine ⟨fun h_pos ↦ ?_, fun ⟨p, hp⟩ ↦ ?_⟩
· rcases eq_or_ne i j with rfl | h_eq
· exact ⟨⟨Quiver.Path.nil, rfl⟩⟩
· simp_all
· simp [Quiver.Path.eq_of_length_zero p hp]
| succ m ih =>
rw [pow_succ, mul_apply]
constr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Module | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 46
} | {
"line": 65,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nr : R\nv : ι → M\ni' : ι\n⊢ ∑ j_1, single i j r i' j_1 • v j_1 = Pi.single i (r • v j) i'",
"ppTerm": "?m.35",
"assigned": true,
"used... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nr : R\nv : ι → M\ni' : ι\n⊢ single i j r i' j • v j = Pi.single i (r • v j) i'",
"ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : Fintyp... | rw [Fintype.sum_eq_single j fun j' hj => ?_] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 57,
"column": 11
} | {
"line": 57,
"column": 33
} | {
"line": 57,
"column": 33
} | [
{
"pp": "case a\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : Type uN\ninst✝⁷ : Semiring R\ninst✝⁶ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝³ : Module R N\ninst✝² : Finite ι\ninst✝¹ : ∀ (i : ι), ... | [
"case a\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : Type uN\ninst✝⁷ : Semiring R\ninst✝⁶ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝³ : Module R N\ninst✝² : Finite ι\ninst✝¹ : ∀ (i : ι), Finite (κ i)... | MultilinearMap.ext_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp | {
"line": 59,
"column": 2
} | {
"line": 61,
"column": 65
} | {
"line": 63,
"column": 0
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nκ : ι → Type u_3\nM : (i : ι) → κ i → Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝³ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (κ i)\nx : (i : ι) → Π₀ (... | [] | classical
simpa [ofDFinsuppEquiv, MultilinearMap.fromDFinsuppEquiv_apply] using fun i hi ↦
((tprod R).map_coord_zero (m := fun i ↦ x i (p i)) i hi).symm | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp | {
"line": 59,
"column": 2
} | {
"line": 61,
"column": 65
} | {
"line": 63,
"column": 0
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nκ : ι → Type u_3\nM : (i : ι) → κ i → Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝³ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (κ i)\nx : (i : ι) → Π₀ (... | [] | classical
simpa [ofDFinsuppEquiv, MultilinearMap.fromDFinsuppEquiv_apply] using fun i hi ↦
((tprod R).map_coord_zero (m := fun i ↦ x i (p i)) i hi).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp | {
"line": 59,
"column": 2
} | {
"line": 61,
"column": 65
} | {
"line": 63,
"column": 0
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nκ : ι → Type u_3\nM : (i : ι) → κ i → Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → (j : κ i) → AddCommMonoid (M i j)\ninst✝³ : (i : ι) → (j : κ i) → Module R (M i j)\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (κ i)\nx : (i : ι) → Π₀ (... | [] | classical
simpa [ofDFinsuppEquiv, MultilinearMap.fromDFinsuppEquiv_apply] using fun i hi ↦
((tprod R).map_coord_zero (m := fun i ↦ x i (p i)) i hi).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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