module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.Focal | {
"line": 124,
"column": 67
} | {
"line": 124,
"column": 72
} | {
"line": 124,
"column": 72
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : G\nhh : h ∈ H\ng : G\nhconj : g⁻¹ * h * g ∈ H\n⊢ (g⁻¹ * h * g)⁻¹ * h = ⁅(g⁻¹ * h * g)⁻¹, g⁆",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"NonUnita... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 11
} | {
"line": 195,
"column": 4
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\nh : IsReduced L\nn : ℕ\nih :\n reduce (replicate (n + 1) L).flatten =\n conjugator L ++ (replicate (n + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)\nL₁ L₂ L₃ L₄ L₅ : List (α × Bool)\n⊢ mk L₁ * mk L₂ * (mk L₃)⁻¹ * (mk L₃ * mk L₄ *... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Nilpotent | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 9
} | {
"line": 104,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\na b : G\nha : a ∈ {x | ∀ (y : G), ⁅x, y⁆ ∈ N}\nhb : b ∈ {x | ∀ (y : G), ⁅x, y⁆ ∈ N}\ny : G\n⊢ ⁅a * b, y⁆ = ⁅a, b * y * b⁻¹⁆ * ⁅b, y⁆",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semi... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 57
} | {
"line": 115,
"column": 4
} | [
{
"pp": "G A : Type u\ninst✝³ : Group G\ninst✝² : IsFreeGroup G\ninst✝¹ : MulAction G A\nX : Type u\ninst✝ : Group X\nf : Labelling (Generators (ActionCategory G A)) X\nf' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩\n⊢ ∃! F, ∀ (a b : Generators (ActionCa... | [
"G A : Type u\ninst✝³ : Group G\ninst✝² : IsFreeGroup G\ninst✝¹ : MulAction G A\nX : Type u\ninst✝ : Group X\nf : Labelling (Generators (ActionCategory G A)) X\nf' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩\nF' : G →* (A → X) ⋊[mulAutArrow] G\nhF' : ∀ (a : ... | rcases IsFreeGroup.unique_lift f' with ⟨F', hF', uF'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.GroupTheory.Nilpotent | {
"line": 344,
"column": 8
} | {
"line": 344,
"column": 34
} | {
"line": 344,
"column": 34
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝ : Group G\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊤\nh0 : H 0 = ⊥\nhH : ∀ (x : G) (n : ℕ), x ∈ H (n + 1) → ∀ (g : G), ⁅x, g⁆ ∈ H n\nm : ℕ\ng : G\nhm : n ≤ m\n⊢ ⁅1, g⁆ ∈ (fun m ↦ H (n - m)) (m + 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [
"case pos\nG : Type u_1\ninst✝ : Group G\nH : ℕ → Subgroup G\nn : ℕ\nhn : H n = ⊤\nh0 : H 0 = ⊥\nhH : ∀ (x : G) (n : ℕ), x ∈ H (n + 1) → ∀ (g : G), ⁅x, g⁆ ∈ H n\nm : ℕ\ng : G\nhm : n ≤ m\n⊢ 1 ∈ (fun m ↦ H (n - m)) (m + 1)"
] | commutatorElement_one_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 491,
"column": 8
} | {
"line": 491,
"column": 41
} | {
"line": 491,
"column": 42
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\nH : ℕ → Subgroup ↥S\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\nh1 : ⊤.lowerCentralSeries n = ⊥\n⊢ S.lowerCentralSeries n = ⊥",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgro... | [
"case refine_1\nG : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\nH : ℕ → Subgroup ↥S\nhH : IsDescendingCentralSeries H\nhn : H n = ⊥\nh1 : ⊤.lowerCentralSeries n = ⊥\n⊢ map S.subtype (⊤.lowerCentralSeries n) = ⊥"
] | ← top_subtype_lowerCentralSeries, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 702,
"column": 6
} | {
"line": 702,
"column": 39
} | {
"line": 703,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : Group.IsNilpotent ↥S\nn : ℕ\nhn : nilpotencyClass ↥S ≤ n\n⊢ S.lowerCentralSeries n = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.map",
"congrArg",
"Subgroup.subtype",
"M... | [
"G : Type u_1\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : Group.IsNilpotent ↥S\nn : ℕ\nhn : nilpotencyClass ↥S ≤ n\n⊢ map S.subtype (⊤.lowerCentralSeries n) = ⊥"
] | ← top_subtype_lowerCentralSeries, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 724,
"column": 47
} | {
"line": 724,
"column": 66
} | {
"line": 724,
"column": 67
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\nh : S.lowerCentralSeries n ≤ center G\n⊢ closure {g | ∃ g₁ ∈ S.lowerCentralSeries n, ∃ g₂ ∈ S, ⁅g₁, g₂⁆ = g} = ⊥",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"Subgroup.closure... | [
"G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\nh : S.lowerCentralSeries n ≤ center G\n⊢ {g | ∃ g₁ ∈ S.lowerCentralSeries n, ∃ g₂ ∈ S, ⁅g₁, g₂⁆ = g} ⊆ {1}"
] | closure_eq_bot_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 984,
"column": 4
} | {
"line": 984,
"column": 46
} | {
"line": 986,
"column": 0
} | [
{
"pp": "case refine_2\nG✝ : Type u_1\ninst✝¹ : Group G✝\nG : Type u_1\ninst✝ : Group G\na b c : ℕ\nab✝ : a ≠ b\nac : a ≤ c\nhn : upperCentralSeries G a = upperCentralSeries G b\nab : a < b\n⊢ upperCentralSeries G (a + 1) ≤ upperCentralSeries G b",
"ppTerm": "?refine_2",
"assigned": true,
"usedConst... | [] | exact upperCentralSeries_mono _ (by grind) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Nilpotent | {
"line": 1068,
"column": 2
} | {
"line": 1068,
"column": 40
} | {
"line": 1069,
"column": 2
} | [
{
"pp": "G₁ : Type u_2\nG₂ : Type u_3\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : IsNilpotent G₁\ninst✝ : IsNilpotent G₂\n⊢ nilpotencyClass (G₁ × G₂) = max (nilpotencyClass G₁) (nilpotencyClass G₂)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"eq_of_forall_ge_iff",
"Prod... | [
"G₁ : Type u_2\nG₂ : Type u_3\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : IsNilpotent G₁\ninst✝ : IsNilpotent G₂\nk : ℕ\n⊢ nilpotencyClass (G₁ × G₂) ≤ k ↔ max (nilpotencyClass G₁) (nilpotencyClass G₂) ≤ k"
] | refine eq_of_forall_ge_iff fun k => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 292,
"column": 30
} | {
"line": 292,
"column": 43
} | {
"line": 292,
"column": 43
} | [
{
"pp": "G : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g ↦ FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 53
} | {
"line": 133,
"column": 0
} | [
{
"pp": "M : Type u_3\nα : Type u_4\nN : Type u_5\nβ : Type u_6\ninst✝³ : Monoid M\ninst✝² : MulAction M α\ninst✝¹ : Monoid N\ninst✝ : MulAction N β\nφ : M → N\nf : α →ₑ[φ] β\nhf : Function.Surjective ⇑f\nB : Set α\nhB : IsTrivialBlock B\n⊢ IsTrivialBlock (⇑f '' B)",
"ppTerm": "?m.15",
"assigned": true,... | [] | obtain hB | hB := hB
· apply Or.intro_left; apply Set.Subsingleton.image hB
· apply Or.intro_right; rw [hB]
simp only [Set.image_univ, Set.range_eq_univ, hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 53
} | {
"line": 133,
"column": 0
} | [
{
"pp": "M : Type u_3\nα : Type u_4\nN : Type u_5\nβ : Type u_6\ninst✝³ : Monoid M\ninst✝² : MulAction M α\ninst✝¹ : Monoid N\ninst✝ : MulAction N β\nφ : M → N\nf : α →ₑ[φ] β\nhf : Function.Surjective ⇑f\nB : Set α\nhB : IsTrivialBlock B\n⊢ IsTrivialBlock (⇑f '' B)",
"ppTerm": "?m.15",
"assigned": true,... | [] | obtain hB | hB := hB
· apply Or.intro_left; apply Set.Subsingleton.image hB
· apply Or.intro_right; rw [hB]
simp only [Set.image_univ, Set.range_eq_univ, hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 136,
"column": 32
} | {
"line": 139,
"column": 65
} | {
"line": 141,
"column": 0
} | [
{
"pp": "M : Type u_3\nα : Type u_4\nN : Type u_5\nβ : Type u_6\ninst✝³ : Monoid M\ninst✝² : MulAction M α\ninst✝¹ : Monoid N\ninst✝ : MulAction N β\nφ : M → N\nf : α →ₑ[φ] β\nhf : Function.Injective ⇑f\nB : Set β\nhB : IsTrivialBlock B\n⊢ IsTrivialBlock (⇑f ⁻¹' B)",
"ppTerm": "?m.15",
"assigned": true,... | [] | by
obtain hB | hB := hB
· apply Or.intro_left; exact Set.Subsingleton.preimage hB hf
· apply Or.intro_right; simp only [hB]; apply Set.preimage_univ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 22
} | {
"line": 499,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\nS : Type u_3\nH : Type u_4\ninst✝⁴ : Group H\ninst✝³ : SetLike S H\ninst✝² : SubgroupClass S H\ns : S\ninst✝¹ : MulAction G H\ninst✝ : IsScalarTower G Hᵐᵒᵖ H\na b : G\nhab : a • ↑s ≠ b • ↑s\nc : H\nhc : c ∈ ↑s\nd : H\nhd : d ∈ ↑s\nhcd : b • d = a • c\n⊢ b • op d • ↑s = b... | [
"G : Type u_1\ninst✝⁵ : Group G\nS : Type u_3\nH : Type u_4\ninst✝⁴ : Group H\ninst✝³ : SetLike S H\ninst✝² : SubgroupClass S H\ns : S\ninst✝¹ : MulAction G H\ninst✝ : IsScalarTower G Hᵐᵒᵖ H\na b : G\nhab : a • ↑s ≠ b • ↑s\nc : H\nhc : c ∈ ↑s\nd : H\nhd : d ∈ ↑s\nhcd : b • d = a • c\n⊢ b • ↑s = b • ↑s"
] | op_smul_coe_set hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 540,
"column": 2
} | {
"line": 543,
"column": 12
} | {
"line": 544,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\nB : Set X\ninst✝ : IsPretransitive G X\nhB : IsBlock G B\na : X\nha : a ∈ B\nx : X\n⊢ x ∈ orbit (↥(stabilizer G B)) a → x ∈ B",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSM... | [
"case mpr\nG : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\nB : Set X\ninst✝ : IsPretransitive G X\nhB : IsBlock G B\na : X\nha : a ∈ B\nx : X\n⊢ x ∈ B → x ∈ orbit (↥(stabilizer G B)) a"
] | · rintro ⟨⟨k, k_mem⟩, rfl⟩
simp only [Subgroup.mk_smul]
rw [← k_mem, Set.smul_mem_smul_set_iff]
exact ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 696,
"column": 4
} | {
"line": 696,
"column": 50
} | {
"line": 702,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝³ : Group G\nX : Type u_2\ninst✝² : MulAction G X\ninst✝¹ : IsPretransitive G X\nB : Set X\ninst✝ : Finite X\nhB : IsBlock G B\nhB' : ¬2 * (orbit G B).ncard ≤ B.ncard * (orbit G B).ncard\nh : B.Nonempty\n⊢ ¬2 ≤ B.ncard",
"ppTerm": "?inr",
"assigned": true,
"used... | [] | exact fun hb ↦ hB' (Nat.mul_le_mul_right _ hb) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 177,
"column": 15
} | {
"line": 177,
"column": 48
} | {
"line": 177,
"column": 48
} | [
{
"pp": "case inr\nG : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na : X\nha : a ∉ fixedPoints G X\nH✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B\nH : orbit G a = Set.univ\nx : X\n⊢ ∃ g, g • a = x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"case inr\nG : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na : X\nha : a ∉ fixedPoints G X\nH✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B\nH : orbit G a = Set.univ\nx : X\n⊢ x ∈ Set.univ"
] | rw [← MulAction.mem_orbit_iff, H] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 87
} | {
"line": 318,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nhp : Nat.Prime (Nat.card X)\nB : Set X\nhB : IsBlock G B\nhB' : B.Nontrivial\nthis : Finite X\n⊢ B = Set.univ",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.... | [
"G : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nhp : Nat.Prime (Nat.card X)\nB : Set X\nhB : IsBlock G B\nhB' : B.Nontrivial\nthis : Finite X\n⊢ B.ncard ∣ Nat.card X"
] | rw [Set.eq_univ_iff_ncard, eq_comm, ← hp.dvd_iff_eq ((Set.one_lt_ncard).mpr hB').ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 315,
"column": 56
} | {
"line": 315,
"column": 73
} | {
"line": 315,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ Nat.card α\nthis : Nontrivial α\n⊢ 2 * 1 < (Nat.card α).factorial",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
"id",
"Nat.ca... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ Nat.card α\nthis : Nontrivial α\n⊢ (2 * 1).succ ≤ (Nat.card α).factorial"
] | ← Nat.succ_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Embedding | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 44
} | {
"line": 78,
"column": 0
} | [
{
"pp": "case h\nα : Type u_1\nm n : ℕ\nhn : ↑m + ↑n ≤ ENat.card α\nx : Fin m ↪ α\ny : Fin n ↪ α\nhxy : Disjoint (range ⇑x) (range ⇑y)\ni : Fin m\n⊢ ((fun x ↦ (castAddEmb n).trans x) (append hxy)) i = x i",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Fin.castAddEmb",
"congrArg"... | [] | simp [trans_apply, coe_castAddEmb, append] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 64
} | {
"line": 107,
"column": 2
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nH : Type u_3\nβ : Type u_4\ninst✝¹ : Group H\ninst✝ : MulAction H β\nσ : G → H\nf : α →ₑ[σ] β\nι : Type u_5\nhf : Bijective ⇑f\ny : ι ↪ β\n⊢ ∃ a, (Injective.mulActionHom_embedding ι ⋯) a = y",
"ppTerm": "?m.47",
"assigned": t... | [
"G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nH : Type u_3\nβ : Type u_4\ninst✝¹ : Group H\ninst✝ : MulAction H β\nσ : G → H\nf : α →ₑ[σ] β\nι : Type u_5\nhf : Bijective ⇑f\ny : ι ↪ β\ng : β → α\nleft✝ : LeftInverse g ⇑f\nhfg : RightInverse g ⇑f\n⊢ ∃ a, (Injective.mulActionHom_embedding ι ⋯... | obtain ⟨g, _, hfg⟩ := Function.bijective_iff_has_inverse.mp hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 237,
"column": 26
} | {
"line": 237,
"column": 31
} | {
"line": 239,
"column": 0
} | [
{
"pp": "case h.right\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\ng : M\nhg : g • t = s\nk : M\nhk : k ∈ fixingSubgroup M s\n⊢ g * (g⁻¹ * k * g) * g⁻¹ = k",
"ppTerm": "?h.right",
"assigned": true,
"usedConstants": [
"_private.Mathlib.GroupTheory.GroupActi... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 270,
"column": 18
} | {
"line": 270,
"column": 26
} | {
"line": 271,
"column": 6
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns✝ s t : Set α\ng : M\nhg : g • t = s\nx✝ : ↥(ofFixingSubgroup M t)\nx : α\nhx : x ∈ ofFixingSubgroup M t\nhgxt : g • x ∈ s\n⊢ False",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hx"... | [
"M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns✝ s t : Set α\ng : M\nhg : g • t = s\nx✝ : ↥(ofFixingSubgroup M t)\nx : α\nhx : x ∈ ofFixingSubgroup M t\nhgxt : g • x ∈ s\n⊢ x ∈ t"
] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 269,
"column": 12
} | {
"line": 272,
"column": 45
} | {
"line": 272,
"column": 45
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns✝ s t : Set α\ng : M\nhg : g • t = s\nx✝ : ↥(ofFixingSubgroup M t)\nx : α\nhx : x ∈ ofFixingSubgroup M t\n⊢ g • x ∈ ofFixingSubgroup M s",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"instHSMul",
"... | [] | by
intro hgxt; apply hx
rw [← hg] at hgxt
exact Set.smul_mem_smul_set_iff.mp hgxt | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 34
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case property\nα : Type u_2\ns : Set α\ng : Perm ↑s\n⊢ ofSubtype g ∈ stabilizer (Perm α) s",
"ppTerm": "?property",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Set.instMembership",
"Equiv.Perm.ofSubtype_mem_stabilizer",
"S... | [] | apply ofSubtype_mem_stabilizer | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 34
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case property\nα : Type u_2\ns : Set α\ng : Perm ↑s\n⊢ ofSubtype g ∈ stabilizer (Perm α) s",
"ppTerm": "?property",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Set.instMembership",
"Equiv.Perm.ofSubtype_mem_stabilizer",
"S... | [] | apply ofSubtype_mem_stabilizer | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 34
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case property\nα : Type u_2\ns : Set α\ng : Perm ↑s\n⊢ ofSubtype g ∈ stabilizer (Perm α) s",
"ppTerm": "?property",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Set.instMembership",
"Equiv.Perm.ofSubtype_mem_stabilizer",
"S... | [] | apply ofSubtype_mem_stabilizer | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 53
} | {
"line": 139,
"column": 53
} | [
{
"pp": "case h.inl\nn : ℕ\nhrec :\n ∀ m < n,\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgro... | [
"case h.inl\nn : ℕ\nhrec :\n ∀ m < n,\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Is... | ofStabilizer.isMultiplyPretransitive (a := a) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 9
} | {
"line": 167,
"column": 2
} | [
{
"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\ns : S.Splitting\nx✝¹ x✝ : N ⋊[s.conjAct] G\nn₁ : N\ng₁ : G\nn₂ : N\ng₂ : G\n⊢ S.inl n₁ * (s g₁ * S.inl n₂ * (s g₁)⁻¹) * (s g₁ * s g₂) = S.inl n₁ * s g₁ * (S.inl n₂ * s g₂)",
"ppT... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 194,
"column": 42
} | {
"line": 194,
"column": 47
} | {
"line": 194,
"column": 47
} | [
{
"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 : S.Splitting\nn : N\nhn : ⇑s₁ = fun g ↦ S.inl n * s₂ g * (S.inl n)⁻¹\n⊢ ⇑s₂ = fun g ↦ (S.inl n)⁻¹ * (S.inl n * s₂ g * (S.inl n)⁻¹) * (S.inl n)⁻¹⁻¹",
"ppTerm": "?m.42",
... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 202,
"column": 52
} | {
"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⊢ (fun g ↦ S.inl n₁ * (S.inl n₂ * s₃ g... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.IndexNSmul | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 53
} | {
"line": 92,
"column": 53
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\nA B : AddSubgroup M\ninst✝² : Module.Finite ℤ ↥B\ninst✝¹ : IsTorsionFree ℤ ↥B\nh : A ≤ B\ninst✝ : A.IsFiniteRelIndex B\nthis : (A.addSubgroupOf B).FiniteIndex\n⊢ finrank ℤ ↥A = finrank ℤ ↥B",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [
"M : Type u_1\ninst✝³ : AddCommGroup M\nA B : AddSubgroup M\ninst✝² : Module.Finite ℤ ↥B\ninst✝¹ : IsTorsionFree ℤ ↥B\nh : A ≤ B\ninst✝ : A.IsFiniteRelIndex B\nthis : (A.addSubgroupOf B).FiniteIndex\n⊢ finrank ℤ ↥A = finrank ℤ ↥(A.addSubgroupOf B)"
] | ← finrank_eq_of_finiteIndex (A.addSubgroupOf B) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝² : DecidableEq α\na c : α\nC : Type u_3\ninst✝¹ : SetLike C (Equiv.Perm α)\ninst✝ : SubmonoidClass C (Equiv.Perm α)\nM : C\nhab : swap a a ∈ M\nhbc : swap a c ∈ M\n⊢ swap a c ∈ M",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [
"case inr\nα : Type u_2\ninst✝² : DecidableEq α\na b c : α\nC : Type u_3\ninst✝¹ : SetLike C (Equiv.Perm α)\ninst✝ : SubmonoidClass C (Equiv.Perm α)\nM : C\nhab : swap a b ∈ M\nhbc : swap b c ∈ M\nhab' : a ≠ b\n⊢ swap a c ∈ M"
] | · exact hbc | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 119,
"column": 4
} | {
"line": 120,
"column": 22
} | {
"line": 121,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nsupp : Set α\nfin : supp.Finite\na : α\ns : Set α\nih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S\nf : Equiv.Perm α\nhf : ∀ (x : α), f x ∈ orbit... | [
"case refine_2\nα : Type u_2\ninst✝ : DecidableEq α\nS : Set (Equiv.Perm α)\nhS : ∀ f ∈ S, f.IsSwap\nsupp : Set α\nfin : supp.Finite\na : α\ns : Set α\nih : ∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ s → f ∈ closure S\nf : Equiv.Perm α\nhf : ∀ (x : α), f x ∈ orbit (↥(closure ... | simp_rw [notMem_compl_iff, mem_fixedBy, Perm.smul_def, Perm.mul_apply, swap_apply_def,
apply_eq_iff_eq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 12
} | {
"line": 304,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nc : ↥g.cycleFactorsFinset\nhx : x ∉ g.support\nd : ↥g.cycleFactorsFinset\nhx' : x ∈ (↑d).support\n⊢ False",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [],
... | [
"case inl\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nτ : ↥(range_toPermHom' g)\nx : α\nc : ↥g.cycleFactorsFinset\nhx : x ∉ g.support\nd : ↥g.cycleFactorsFinset\nhx' : x ∈ (↑d).support\n⊢ x ∈ g.support"
] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 356,
"column": 25
} | {
"line": 356,
"column": 85
} | {
"line": 356,
"column": 86
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nc : ↥g.cycleFactorsFinset\nτ✝ τ : ↥(range_toPermHom' g)\nx : α\n⊢ a.ofPermHomFun τ (a.ofPermHomFun τ⁻¹ x) = x",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | by rw [← ofPermHomFun_mul, mul_inv_cancel, ofPermHomFun_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.HNNExtension | {
"line": 567,
"column": 4
} | {
"line": 568,
"column": 33
} | {
"line": 569,
"column": 4
} | [
{
"pp": "case cons\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\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'\nih : ReducedWord.prod φ w.toReducedWord • empty = w\n⊢ Reduced... | [
"case cons\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\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'\nih : ReducedWord.prod φ w.toReducedWord • empty = w\n⊢ (g •\n if h : ... | rw [prod_cons, ← mul_assoc, mul_smul, ih, mul_smul, t_pow_smul_eq_unitsSMul,
of_smul_eq_smul, unitsSMul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Schreier | {
"line": 77,
"column": 72
} | {
"line": 77,
"column": 77
} | {
"line": 77,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ u * r * s = u * ... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Schreier | {
"line": 77,
"column": 72
} | {
"line": 77,
"column": 77
} | {
"line": 77,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ u * r * s = u * ... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Schreier | {
"line": 77,
"column": 72
} | {
"line": 77,
"column": 77
} | {
"line": 77,
"column": 77
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ u * r * s = u * ... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Schreier | {
"line": 81,
"column": 74
} | {
"line": 81,
"column": 79
} | {
"line": 81,
"column": 79
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ u * r * s⁻¹ = u ... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Schreier | {
"line": 81,
"column": 74
} | {
"line": 81,
"column": 79
} | {
"line": 81,
"column": 79
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ u * r * s⁻¹ = u ... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Schreier | {
"line": 81,
"column": 74
} | {
"line": 81,
"column": 79
} | {
"line": 81,
"column": 79
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ u * r * s⁻¹ = u ... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Schreier | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 64
} | {
"line": 224,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite ↑(commutatorSet G)\nh1 :\n (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))\nh2 :\n Nat.card ↥(_root_.commutator G) ∣\n (center ↥(closureCommutatorRepresentatives G)).index ^\n ... | [] | exact pow_pos (Nat.pos_of_ne_zero FiniteIndex.index_ne_zero) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.PushoutI | {
"line": 160,
"column": 36
} | {
"line": 160,
"column": 74
} | {
"line": 160,
"column": 75
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝² : Monoid K\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nx✝ : { f // ∀ (i : ι), (f.1 i).comp (φ i) = f.2 }\nfst✝ : (i : ι) → G i →* K\nsnd✝ : H →* K\nproperty✝ : ∀ (i : ι), ((fst✝, snd✝).1 i).comp (φ i) = (... | [] | by simp [DFunLike.ext_iff, funext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PushoutI | {
"line": 184,
"column": 8
} | {
"line": 184,
"column": 20
} | {
"line": 185,
"column": 8
} | [
{
"pp": "case H.inl.mul\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nmotive : (con φ).Quotient → Prop\nof : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)\nbase : ∀ (h : H), motive (((con φ).mk'.comp inr) h)\n... | [
"case H.inl.mul\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Monoid (G i)\ninst✝ : Monoid H\nφ : (i : ι) → H →* G i\nmotive : (con φ).Quotient → Prop\nof : ∀ (i : ι) (g : G i), motive (((con φ).mk'.comp (inl.comp CoprodI.of)) g)\nbase : ∀ (h : H), motive (((con φ).mk'.comp inr) h)\nmul : ∀ (x y... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 323,
"column": 32
} | {
"line": 323,
"column": 45
} | {
"line": 323,
"column": 45
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\nK : Type u_4\ninst✝² : Monoid K\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ni✝ : ι\ng✝ : G i✝\nw : NormalWord d\nhmw : w.fstIdx ≠ some i✝\nhgr : g✝ ∉ (φ i✝).range\nn : ↑↑(φ i✝).range × ↑(d.set i✝) := ⋯.equiv ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 42
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nm : Multiset ℕ\nhm : ¬Even (m.sum + m.card)\n⊢ map (Embedding.subtype fun x ↦ x ∈ alternatingGroup α) {g | (↑g).cycleType = m} = ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.c... | [
"case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nm : Multiset ℕ\nhm : ¬Even (m.sum + m.card)\n⊢ ∀ (x : Perm α), x ∉ map (Embedding.subtype fun x ↦ x ∈ alternatingGroup α) {g | (↑g).cycleType = m}"
] | rw [Finset.eq_empty_iff_forall_notMem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 26
} | {
"line": 638,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ Nat.card ↥(OnCycleFactors.toPermHom g).ker * ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)! =\n (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod * ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ Nat.card ↥(OnCycleFactors.toPermHom g).ker * ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)! =\n Fintype.card ↥(kerParam g).range * ∏ n ∈ g.cycleType.toFinset, (Multiset.count n g.cycleType)!"
] | ← kerParam_range_card, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 70
} | {
"line": 243,
"column": 2
} | [
{
"pp": "G : Type u\ninst✝² : Group G\nN : Subgroup G\ninst✝¹ : N.Normal\nh1 : (Nat.card ↥N).Coprime N.index\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [Finite G'],\n Nat.card G' < Nat.card G →\n ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'\nh3 : ∀ (H : S... | [
"G : Type u\ninst✝² : Group G\nN : Subgroup G\ninst✝¹ : N.Normal\nh1 : (Nat.card ↥N).Coprime N.index\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [Finite G'],\n Nat.card G' < Nat.card G →\n ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'\nh3 : ∀ (H : Subgroup G), ... | refine Sylow.nonempty.elim fun P => P.2.of_surjective P.1.subtype ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 28
} | {
"line": 183,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nthis : Monoid.exponent ↥(kleinFour α) = 1 ∨ Monoid.exponent ↥(kleinFour α) = 2\n⊢ ¬Monoid.exponent ↥(kleinFour α) = 1",
"ppTerm": "?m.396",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nthis : Monoid.exponent ↥(kleinFour α) = 1 ∨ Monoid.exponent ↥(kleinFour α) = 2\n⊢ ¬Subsingleton ↥(kleinFour α)"
] | Monoid.exp_eq_one_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 276,
"column": 85
} | {
"line": 290,
"column": 55
} | {
"line": 292,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nhN : (Nat.card ↥N).Coprime N.index\n⊢ ∃ H, N.IsComplement' H",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"Nat.instMulZeroClass",
"Nat.coprime_zero_right",
"Is... | [] | by
by_cases hN1 : Nat.card N = 0
· rw [hN1, Nat.coprime_zero_left, index_eq_one] at hN
rw [hN]
exact ⟨⊥, isComplement'_top_bot⟩
by_cases hN2 : N.index = 0
· rw [hN2, Nat.coprime_zero_right, Nat.card_eq_one_iff_unique] at hN
have := hN.1
rw [N.eq_bot_of_subsingleton]
exact ⟨⊤, isComplement'_b... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 69
} | {
"line": 125,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\n⊢ N = ⊥ ∨ N = ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"congrArg",
"M... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\n⊢ Nontrivial ↥N → N = ⊤"
] | rw [or_iff_not_imp_left, ← ne_eq, ← Subgroup.nontrivial_iff_ne_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 11
} | {
"line": 199,
"column": 2
} | [
{
"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\nd : Perm α\nhd : d ∈ g.cycleFactorsFinset\nhm : #c.support = #d.support\nhm' : c ≠ d\nτ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩... | [
"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\nd : Perm α\nhd : d ∈ g.cycleFactorsFinset\nhm : #c.support = #d.support\nhm' : c ≠ d\nτ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩\na : g.Basi... | rw [that] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 184,
"column": 22
} | {
"line": 184,
"column": 29
} | {
"line": 184,
"column": 29
} | [
{
"pp": "case succ\nn k : ℕ\nIH : a 1 ^ k = a ↑k\n⊢ a ↑k * a 1 = a ↑(k + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAd... | [
"case succ\nn k : ℕ\nIH : a 1 ^ k = a ↑k\n⊢ a (↑k + 1) = a ↑(k + 1)"
] | a_mul_a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 69
} | {
"line": 182,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 8\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\n⊢ N = ⊥ ∨ N = ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"congrArg",
"M... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 8\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\n⊢ Nontrivial ↥N → N = ⊤"
] | rw [or_iff_not_imp_left, ← ne_eq, ← Subgroup.nontrivial_iff_ne_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 655,
"column": 6
} | {
"line": 655,
"column": 36
} | {
"line": 656,
"column": 6
} | [
{
"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\n⊢ (of i).range ⊓ (of j).range ≤ (base φ).range",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Mono... | [
"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\n⊢ x ∈ (base φ).range"
] | intro x ⟨⟨g₁, hg₁⟩, ⟨g₂, hg₂⟩⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.PushoutI | {
"line": 687,
"column": 10
} | {
"line": 687,
"column": 50
} | {
"line": 687,
"column": 50
} | [
{
"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\nh : H\n⊢ ((of j).comp (φ j)) h ∈ (of j).range",
"ppTerm": "?m.351",
"assigned": true,
"usedConstants": [
"Iff.... | [] | exact MonoidHom.mem_range.2 ⟨φ j h, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 219,
"column": 2
} | {
"line": 220,
"column": 64
} | {
"line": 221,
"column": 2
} | [
{
"pp": "case refine_1\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... | [
"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.normalizerMonoidHom.comp (... | · rw [eq_bot_iff, Subgroup.commutator_le]
exact fun k hk g hg ↦ Subtype.ext_iff.mp (h ⟨g, hg⟩ ⟨k, hk⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 55,
"column": 8
} | {
"line": 55,
"column": 16
} | {
"line": 55,
"column": 16
} | [
{
"pp": "case basic\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nt : Set (α × β)\nhκs : ∀ (a : α), IsFiniteMeasure (κ a)\nt₁ : Set α\nht₁ : MeasurableSet t₁\nt₂ : Set β\nht₂ : MeasurableSet t₂\nh_eq_ite : (fun a ↦ (κ a) (if a ∈ t₁ then t₂ else ∅)) = fun a ↦ if a ∈... | [
"case basic\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nt : Set (α × β)\nhκs : ∀ (a : α), IsFiniteMeasure (κ a)\nt₁ : Set α\nht₁ : MeasurableSet t₁\nt₂ : Set β\nht₂ : MeasurableSet t₂\nh_eq_ite : (fun a ↦ (κ a) (if a ∈ t₁ then t₂ else ∅)) = fun a ↦ if a ∈ t₁ then (κ ... | h_eq_ite | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Basic | {
"line": 412,
"column": 6
} | {
"line": 412,
"column": 46
} | {
"line": 412,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nμ : Measure α\nh : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)\nh' : μ ≠ 0\n⊢ μ (toMeasurable μ {a | ¬IsProbabilityMeasure (κ a)}) = 0",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"E... | [] | by simpa [measure_toMeasurable] using! h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.CompMap | {
"line": 72,
"column": 2
} | {
"line": 74,
"column": 24
} | {
"line": 76,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nf : β → γ\nhf : Measurable f\n⊢ η ∘ₖ κ.map f = η.comap f hf ∘ₖ κ",
"ppTerm": "?m.34",
"assigned": true,
"us... | [] | ext x s ms
rw [comp_apply' _ _ _ ms, lintegral_map _ hf _ (η.measurable_coe ms), comp_apply' _ _ _ ms]
simp_rw [comap_apply'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompMap | {
"line": 72,
"column": 2
} | {
"line": 74,
"column": 24
} | {
"line": 76,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nδ : Type u_5\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nf : β → γ\nhf : Measurable f\n⊢ η ∘ₖ κ.map f = η.comap f hf ∘ₖ κ",
"ppTerm": "?m.34",
"assigned": true,
"us... | [] | ext x s ms
rw [comp_apply' _ _ _ ms, lintegral_map _ hf _ (η.measurable_coe ms), comp_apply' _ _ _ ms]
simp_rw [comap_apply'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 89
} | {
"line": 176,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\ninst✝¹ : SFinite μ\nκ : Kernel α β\ninst✝ : IsMarkovKernel κ\n⊢ (μ ⊗ₘ κ).fst = μ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"MeasureTheory.Measu... | [] | ext s
rw [compProd, Measure.fst, ← Kernel.fst_apply, Kernel.fst_compProd, Kernel.const_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 89
} | {
"line": 176,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\ninst✝¹ : SFinite μ\nκ : Kernel α β\ninst✝ : IsMarkovKernel κ\n⊢ (μ ⊗ₘ κ).fst = μ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"MeasureTheory.Measu... | [] | ext s
rw [compProd, Measure.fst, ← Kernel.fst_apply, Kernel.fst_compProd, Kernel.const_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 65
} | {
"line": 49,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\ninst✝¹ : SFinite μ\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\ns : Set β\nhs : MeasurableSet s\n⊢ (μ ⊗ₘ κ).snd s = (⇑κ ∘ₘ μ) s",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"MeasureT... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\ninst✝¹ : SFinite μ\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\ns : Set β\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α), (κ a) (Prod.mk a ⁻¹' Prod.snd ⁻¹' s) ∂μ = ∫⁻ (a : α), (κ a) s ∂μ",
"α : Type u_1\nβ : Type u_2\nmα : MeasurableS... | rw [bind_apply hs κ.aemeasurable, snd_apply hs, compProd_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 95
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ : Kernel α β\n⊢ ⇑κ ∘ₘ (μ + ν) = ⇑κ ∘ₘ μ + ⇑κ ∘ₘ ν",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"MeasureTheory.Measure",
"congrArg",
... | [] | simp_rw [comp_eq_comp_const_apply, Kernel.const_add, Kernel.comp_add_right, Kernel.add_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 95
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ : Kernel α β\n⊢ ⇑κ ∘ₘ (μ + ν) = ⇑κ ∘ₘ μ + ⇑κ ∘ₘ ν",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"MeasureTheory.Measure",
"congrArg",
... | [] | simp_rw [comp_eq_comp_const_apply, Kernel.const_add, Kernel.comp_add_right, Kernel.add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 95
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nκ : Kernel α β\n⊢ ⇑κ ∘ₘ (μ + ν) = ⇑κ ∘ₘ μ + ⇑κ ∘ₘ ν",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"MeasureTheory.Measure",
"congrArg",
... | [] | simp_rw [comp_eq_comp_const_apply, Kernel.const_add, Kernel.comp_add_right, Kernel.add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 58
} | {
"line": 245,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nδ : Type u_6\nmδ : MeasurableSpace δ\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel β γ\ninst✝¹ : IsSFiniteKernel η\nμ : Measure δ\ninst✝ : SFinite μ\nx : α\ns : Set (γ × δ)\nms : ... | [] | lintegral_comp _ _ _ (measurable_measure_prodMk_left ms) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.MeasureCompProd | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 72
} | {
"line": 333,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nhμν : μ ⟂ₘ ν\nκ η : Kernel α β\nhμ : SFinite μ\nhν : SFinite ν\nhκ : IsSFiniteKernel κ\nhη : IsSFiniteKernel η\n⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [... | [
"case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ ν : Measure α\nhμν : μ ⟂ₘ ν\nκ η : Kernel α β\nhμ : SFinite μ\nhν : SFinite ν\nhκ : IsSFiniteKernel κ\nhη : IsSFiniteKernel η\n⊢ (μ ⊗ₘ κ) (hμν.nullSet ×ˢ univ) = 0 ∧ (ν ⊗ₘ η) (hμν.nullSet ×ˢ univ)ᶜ = 0"
] | refine ⟨hμν.nullSet ×ˢ univ, hμν.measurableSet_nullSet.prod .univ, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv | {
"line": 204,
"column": 2
} | {
"line": 205,
"column": 75
} | {
"line": 206,
"column": 2
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\n𝓨 : Type u_2\nm𝓨 : MeasurableSpace 𝓨\nκ η : Kernel 𝓧 𝓨\nf : ℝ → ℝ\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_co... | [
"𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\n𝓨 : Type u_2\nm𝓨 : MeasurableSpace 𝓨\nκ η : Kernel 𝓧 𝓨\nf : ℝ → ℝ\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\nhf : StronglyMeasurable f\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont_at : Cont... | obtain ⟨c, c', h⟩ : ∃ c c', ∀ x, 0 ≤ x → c * x + c' ≤ f x :=
hf_cvx.exists_affine_le_real isClosed_Ici hf_cont.lowerSemicontinuousOn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 21
} | {
"line": 93,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nmα : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedSpace ℝ E\nhp : Fact (1 ≤ p)\ninst✝¹ : CompleteSpace ↥(Lp E p μ)\ninst✝ : Nontrivial ↥(Lp E p μ)\n⊢ ∃ f, f ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [... | [] | have : Nontrivial (Lp ℝ p μ) := nontrivial_Lp_real_of_nontrivial_Lp E p μ
exact exists_ne 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 21
} | {
"line": 93,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nmα : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedSpace ℝ E\nhp : Fact (1 ≤ p)\ninst✝¹ : CompleteSpace ↥(Lp E p μ)\ninst✝ : Nontrivial ↥(Lp E p μ)\n⊢ ∃ f, f ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [... | [] | have : Nontrivial (Lp ℝ p μ) := nontrivial_Lp_real_of_nontrivial_Lp E p μ
exact exists_ne 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum | {
"line": 80,
"column": 2
} | {
"line": 89,
"column": 24
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case inr\nX : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\nι : Type u_3\ninst✝ : Countable ι\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ι → X → E\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nh'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞\nh'p : p < ∞\nA : ∀ (s : Set X), Meas... | [
"case inr\nX : Type u_1\nE : Type u_2\nx✝ : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\nι : Type u_3\ninst✝ : Countable ι\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ι → X → E\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\nh'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞\nh'p : p < ∞\nA : ∀ (s : Set X), MeasurableSet s ... | have C : ∀ᵐ x ∂μ, ∀ n, x ∈ s n → ∑' n, ‖f n x‖ₑ < ∞ := by
apply ae_all_iff.2 (fun n ↦ ?_)
have : ∀ᵐ x ∂μ, ∀ i, x ∈ s n ∩ spanningSets (μ.restrict (s n)) i → ∑' n, ‖f n x‖ₑ < ∞ := by
apply ae_all_iff.2 (fun i ↦ ?_)
apply A _ ((s_meas n).inter (measurableSet_spanningSets _ _))
rw [Set.inter_comm... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 183,
"column": 10
} | {
"line": 183,
"column": 72
} | {
"line": 184,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nc : ℝ\nx : G\nhs : ¬MeasurableSet s\n⊢ condExpIndL1 hm μ s (c • x) = c • condExpIndL1 hm μ s x",
"ppTerm":... | [] | simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [smul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 183,
"column": 10
} | {
"line": 183,
"column": 72
} | {
"line": 184,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nc : ℝ\nx : G\nhs : ¬MeasurableSet s\n⊢ condExpIndL1 hm μ s (c • x) = c • condExpIndL1 hm μ s x",
"ppTerm":... | [] | simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [smul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 192,
"column": 10
} | {
"line": 192,
"column": 72
} | {
"line": 193,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nF : Type u_2\n𝕜 : Type u_6\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nc : 𝕜\nx : F\nhs... | [] | simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [smul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 192,
"column": 10
} | {
"line": 192,
"column": 72
} | {
"line": 193,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nF : Type u_2\n𝕜 : Type u_6\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\ninst✝² : SigmaFinite (μ.trim hm)\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SMulCommClass ℝ 𝕜 F\nc : 𝕜\nx : F\nhs... | [] | simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [smul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 48
} | {
"line": 179,
"column": 2
} | [
{
"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\nh_nnnorm_eq_zero : ∫⁻ (x : α) in s, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊ ∂μ = 0\n⊢ Measurable ↑↑↑((condExpL2 ℝ ℝ hm) f)",
"ppTerm": "?m.138",
... | [] | exact (Lp.stronglyMeasurable _).measurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.TensorProduct.IsBaseChangeFree | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 33
} | {
"line": 132,
"column": 33
} | [
{
"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\na : ι → A\nv : V\n⊢ (Fintype.linea... | [
"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\na : ι → A\nv : V\n⊢ (Fintype.linearCombination... | ← LinearEquiv.symm_apply_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 23
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_4\nV : Type u_5\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\na : R\n⊢ (∀ (x : V ⧸ (↑e).fixedSubmodule), e.fixedReduce x = a • x) →\n ∀ (v : V), e.fixedReduce ((↑e).fixedSubmodule.mkQ v) = (↑e).fixedSubmodule.mkQ (a • v)",
"ppTerm": "?mp",
... | [
"case mpr\nR : Type u_4\nV : Type u_5\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\na : R\n⊢ (∀ (v : V), e.fixedReduce ((↑e).fixedSubmodule.mkQ v) = (↑e).fixedSubmodule.mkQ (a • v)) →\n ∀ (x : V ⧸ (↑e).fixedSubmodule), e.fixedReduce x = a • x"
] | · intro H x; simp [H] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Center | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 27
} | {
"line": 74,
"column": 4
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι) (r : R), i ≠ j → f * (b.coord i).transvection (r • b j) = (b.coord i).transvection (r • b j) * f\ni j : ι... | [
"R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι) (r : R), i ≠ j → f * (b.coord i).transvection (r • b j) = (b.coord i).transvection (r • b j) * f\ni j : ι\nhij : i ≠ ... | have := hcomm i j r hij | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 224,
"column": 71
} | {
"line": 226,
"column": 53
} | {
"line": 227,
"column": 4
} | [
{
"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' ... | [] | by
simp_rw [f, Subgroup.inv_mem_iff, MonoidHom.mem_range, Finset.univ_filter_exists,
Finset.sum_image sumCongrHom_injective.injOn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 471,
"column": 23
} | {
"line": 471,
"column": 46
} | {
"line": 471,
"column": 47
} | [
{
"pp": "case h\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_top... | [
"case h\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_top : (↑e).fixe... | ← LinearMap.ker_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 142,
"column": 57
} | {
"line": 143,
"column": 72
} | {
"line": 145,
"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\na : CliffordAlgebra Q\nr : R\n⊢ (contractLeft d) (a * (algebraMap R (CliffordAlgebra Q)) r) =\n (contractLeft d) a * (algebraMap R (CliffordAlgebra Q)) r",
"ppT... | [] | by
rw [← Algebra.commutes, contractLeft_algebraMap_mul, Algebra.commutes] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 44
} | {
"line": 55,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : Fin i → M\nb : Fin j → M\n⊢ (List.ofFn fun i_1 ↦ (TensorAlgebra.ι R) (Fin.append a b i_1)) =\n (List.ofFn fun i ↦ (TensorAlgebra.ι R) (a i)) ++ List.ofFn fun i ↦ (TensorAlgebra.ι R) (b i)"... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : Fin i → M\nb : Fin j → M\n⊢ List.map (⇑(TensorAlgebra.ι R)) (List.ofFn (Fin.append a b)) =\n (List.ofFn fun i ↦ (TensorAlgebra.ι R) (a i)) ++ List.ofFn fun i ↦ (TensorAlgebra.ι R) (b i)"
] | List.ofFn_comp' _ (TensorAlgebra.ι R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower | {
"line": 55,
"column": 45
} | {
"line": 55,
"column": 83
} | {
"line": 56,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : Fin i → M\nb : Fin j → M\n⊢ List.map (⇑(TensorAlgebra.ι R)) (List.ofFn (Fin.append a b)) =\n (List.ofFn fun i ↦ (TensorAlgebra.ι R) (a i)) ++ List.ofFn fun i ↦ (TensorAlgebra.ι R) (b i)",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : Fin i → M\nb : Fin j → M\n⊢ List.map (⇑(TensorAlgebra.ι R)) (List.ofFn (Fin.append a b)) =\n List.map (⇑(TensorAlgebra.ι R)) (List.ofFn a) ++ List.ofFn fun i ↦ (TensorAlgebra.ι R) (b i)"
] | List.ofFn_comp' _ (TensorAlgebra.ι R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 42
} | {
"line": 56,
"column": 43
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : Fin i → M\nb : Fin j → M\n⊢ List.map (⇑(TensorAlgebra.ι R)) (List.ofFn (Fin.append a b)) =\n List.map (⇑(TensorAlgebra.ι R)) (List.ofFn a) ++ List.ofFn fun i ↦ (TensorAlgebra.ι R) (b i)",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : Fin i → M\nb : Fin j → M\n⊢ List.map (⇑(TensorAlgebra.ι R)) (List.ofFn (Fin.append a b)) =\n List.map (⇑(TensorAlgebra.ι R)) (List.ofFn a) ++ List.map (⇑(TensorAlgebra.ι R)) (List.ofFn b)"
] | List.ofFn_comp' _ (TensorAlgebra.ι R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 48
} | {
"line": 140,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nx : Fin n → M\n⊢ toDirectSum ((tprod R M n) x) = (DirectSum.of (fun i ↦ ⨂[R]^i M) n) ((PiTensorProduct.tprod R) x)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"PiTen... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nx : Fin n → M\n⊢ (List.ofFn (⇑toDirectSum ∘ fun i ↦ (ι R) (x i))).prod =\n (DirectSum.of (fun i ↦ ⨂[R]^i M) n) ((PiTensorProduct.tprod R) x)"
] | rw [tprod_apply, map_list_prod, List.map_ofFn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 22
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝³ : Semiring R\nι : Type u_2\ninst✝² : DecidableEq ι\nM : ι → Type u_3\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nmotive : (⨁ (i : ι), M i) → Prop\nzero : motive 0\nlof : ∀ (i : ι) (x : M i), motive ((DirectSum.lof R ι M i) x)\nadd : ∀ (x y ... | [] | exact zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 22
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝³ : Semiring R\nι : Type u_2\ninst✝² : DecidableEq ι\nM : ι → Type u_3\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nmotive : (⨁ (i : ι), M i) → Prop\nzero : motive 0\nlof : ∀ (i : ι) (x : M i), motive ((DirectSum.lof R ι M i) x)\nadd : ∀ (x y ... | [] | exact zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 22
} | {
"line": 63,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝³ : Semiring R\nι : Type u_2\ninst✝² : DecidableEq ι\nM : ι → Type u_3\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nmotive : (⨁ (i : ι), M i) → Prop\nzero : motive 0\nlof : ∀ (i : ι) (x : M i), motive ((DirectSum.lof R ι M i) x)\nadd : ∀ (x y ... | [] | exact zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Goursat | {
"line": 95,
"column": 25
} | {
"line": 95,
"column": 39
} | {
"line": 95,
"column": 40
} | [
{
"pp": "case e'_3\nR : 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)\nhL₁ : Surjective (Prod.fst ∘ ⇑L.subtype)\nhL₂ : Surjective (Prod.snd ∘ ⇑L.subtype)\ne : M ⧸ L.goursatFst ≃+ N ⧸ L.gou... | [
"case e'_3\nR : 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)\nhL₁ : Surjective (Prod.fst ∘ ⇑L.subtype)\nhL₂ : Surjective (Prod.snd ∘ ⇑L.subtype)\ne : M ⧸ L.goursatFst ≃+ N ⧸ L.goursatSnd\nhe ... | mem_graph_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.AbsoluteValue | {
"line": 59,
"column": 36
} | {
"line": 59,
"column": 44
} | {
"line": 60,
"column": 6
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : Nontrivial R\ninst✝⁴ : CommRing S\ninst✝³ : LinearOrder S\ninst✝² : IsStrictOrderedRing S\nn : Type u_3\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nι : Type u_4\ns : Finset ι\nA : ι → Matrix n n R\nabv : AbsoluteValue R S\nx : S\nhx : ∀ ... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 205,
"column": 8
} | {
"line": 205,
"column": 40
} | {
"line": 206,
"column": 8
} | [
{
"pp": "case neg.refine_1\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh : ↑A 1 0 = 0\nhd : ↑A 1 1 ≠ 0\nh1 : ¬0 < ↑A 0 0\n⊢ ↑A 0 0 < 0",
"ppTerm": "?neg.refine_1✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Matrix",
"instDecidableEqFin",
... | [
"case neg.refine_1\nm : ℤ\nhm : m ≠ 0\nA : FixedDetMatrix (Fin 2) ℤ m\nh : ↑A 1 0 = 0\nhd : ↑A 1 1 ≠ 0\nh1 : ¬0 < ↑A 0 0\n⊢ ↑A 0 0 ≤ 0 ∧ ↑A 0 0 ≠ 0"
] | simp only [Int.lt_iff_le_and_ne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Irreducible.Defs | {
"line": 133,
"column": 69
} | {
"line": 136,
"column": 21
} | {
"line": 137,
"column": 6
} | [
{
"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\nthis : Quiver n := A.toQuiver\nm : ℕ\nih : ∀ (i j : n), 0 < (A ^ m... | [] | by
simpa [Finset.sum_pos_iff_of_nonneg
(fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))]
using h_pos | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 27
} | {
"line": 54,
"column": 2
} | [
{
"pp": "ι : 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 (... | [
"ι : 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)\ninst✝ ... | have := Classical.decEq ι | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 96
} | {
"line": 87,
"column": 0
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nS : Type uS\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁴ : Semiring R\ninst✝³ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝² : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝¹ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝ : (p ... | [] | simp_rw [Function.apply_update (fun i m => m (p i)) m, Pi.smul_apply, (f p).map_update_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
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