module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 844,
"column": 2
} | {
"line": 847,
"column": 10
} | {
"line": 848,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝ : DivisionMonoid α\ns : Set α\nn : ℤ\n⊢ s ^ n = ∅ → s = ∅ ∧ n ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Set.one_nonempty._simp_2",
... | [
"case mpr\nα : Type u_2\ninst✝ : DivisionMonoid α\ns : Set α\nn : ℤ\n⊢ s = ∅ ∧ n ≠ 0 → s ^ n = ∅"
] | · contrapose! +distrib
rintro (hs | rfl)
· exact hs.zpow
· simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 848,
"column": 2
} | {
"line": 849,
"column": 23
} | {
"line": 851,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝ : DivisionMonoid α\ns : Set α\nn : ℤ\n⊢ s = ∅ ∧ n ≠ 0 → s ^ n = ∅",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Set.ZPow",
"Monoid.toMulOneClass",
"DivisionMonoid.to... | [] | · rintro ⟨rfl, hn⟩
exact empty_zpow hn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 905,
"column": 59
} | {
"line": 906,
"column": 56
} | {
"line": 908,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\nt : Set α\na : α\n⊢ (fun x ↦ a * x) '' t = (fun x ↦ a⁻¹ * x) ⁻¹' t",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_mul_cancel_left",
"Monoid.toMulOneClass",
... | [] | by
rw [image_eq_preimage_of_inverse] <;> intro c <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 909,
"column": 60
} | {
"line": 910,
"column": 56
} | {
"line": 912,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\nt : Set α\nb : α\n⊢ (fun x ↦ x * b) '' t = (fun x ↦ x * b⁻¹) ⁻¹' t",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"mul_inv_cancel_right",
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | [] | by
rw [image_eq_preimage_of_inverse] <;> intro c <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 953,
"column": 2
} | {
"line": 953,
"column": 75
} | {
"line": 955,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Group α\ninst✝² : DivisionMonoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nf : F\ns : Set α\n⊢ ⇑f '' s⁻¹ = (⇑f '' s)⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"map_inv",
"DivInvOn... | [] | rw [← image_inv_eq_inv, ← image_inv_eq_inv]; exact image_comm (map_inv _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 953,
"column": 2
} | {
"line": 953,
"column": 75
} | {
"line": 955,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Group α\ninst✝² : DivisionMonoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nf : F\ns : Set α\n⊢ ⇑f '' s⁻¹ = (⇑f '' s)⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"map_inv",
"DivInvOn... | [] | rw [← image_inv_eq_inv, ← image_inv_eq_inv]; exact image_comm (map_inv _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 45
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case mp\nG : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nP : G → Prop\nx : G\nx_in : x ∈ H\nhx : P x⁻¹\n⊢ ∃ x, x ∈ H ∧ P x",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"Group.toDivi... | [] | exact ⟨x⁻¹, inv_mem x_in, by simp [hx]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 45
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case mpr\nG : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nP : G → Prop\nx : G\nx_in : x ∈ H\nhx : P x\n⊢ ∃ x, x ∈ H ∧ P x⁻¹",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"congrArg",... | [] | exact ⟨x⁻¹, inv_mem x_in, by simp [hx]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 559,
"column": 4
} | {
"line": 560,
"column": 21
} | {
"line": 561,
"column": 2
} | [
{
"pp": "case mp.right\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nh : u ≤ s.prod t\n⊢ map (snd M N) u ≤ t",
"ppTerm": "?mp.right",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
... | [] | · rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Subgroup.Ker | {
"line": 356,
"column": 17
} | {
"line": 357,
"column": 94
} | {
"line": 357,
"column": 94
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\ninst✝⁴ : Group G''\nA : Type u_4\ninst✝³ : AddGroup A\nN : Type u_5\nP : Type u_6\ninst✝² : Group N\ninst✝¹ : Group P\nK : Subgroup G\nM : Type u_7\ninst✝ : MulOneClass M\nf : G →* M\nx : G\nhx : x ∈ f.ker\ny : G\n⊢ y * x... | [] | by
rw [mem_ker, map_mul, map_mul, mem_ker.1 hx, mul_one, map_mul_eq_one f (mul_inv_cancel y)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Ker | {
"line": 481,
"column": 53
} | {
"line": 481,
"column": 65
} | {
"line": 481,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\nhf : Surjective ⇑f\nH : Subgroup N\n⊢ Disjoint H ⊤ ↔ H = ⊥",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"disjoint_top",
"OrderBot.toBot",
"PartialOrder... | [
"G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\nhf : Surjective ⇑f\nH : Subgroup N\n⊢ H = ⊥ ↔ H = ⊥"
] | disjoint_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Ker | {
"line": 595,
"column": 60
} | {
"line": 595,
"column": 78
} | {
"line": 595,
"column": 78
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA A' B : Subgroup G\nhA : A ≤ B\nhA' : A' ≤ B\n⊢ A ⊔ A' = B ⊓ A ⊔ A'",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"SemilatticeSup.toMax",
... | [
"G : Type u_1\ninst✝ : Group G\nA A' B : Subgroup G\nhA : A ≤ B\nhA' : A' ≤ B\n⊢ A ⊔ A' = A ⊔ A'"
] | inf_of_le_right hA | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 922,
"column": 15
} | {
"line": 922,
"column": 65
} | {
"line": 922,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nH : Subgroup N\nhH : H.Normal\nf : G →* N\nx✝ : G\n⊢ x✝ ∈ Subgroup.comap f H → ∀ (g : G), g * x✝ * g⁻¹ ∈ Subgroup.comap f H",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"D... | [] | simp +contextual [Subgroup.mem_comap, hH.conj_mem] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 922,
"column": 15
} | {
"line": 922,
"column": 65
} | {
"line": 922,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nH : Subgroup N\nhH : H.Normal\nf : G →* N\nx✝ : G\n⊢ x✝ ∈ Subgroup.comap f H → ∀ (g : G), g * x✝ * g⁻¹ ∈ Subgroup.comap f H",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"D... | [] | simp +contextual [Subgroup.mem_comap, hH.conj_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 922,
"column": 15
} | {
"line": 922,
"column": 65
} | {
"line": 922,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nH : Subgroup N\nhH : H.Normal\nf : G →* N\nx✝ : G\n⊢ x✝ ∈ Subgroup.comap f H → ∀ (g : G), g * x✝ * g⁻¹ ∈ Subgroup.comap f H",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"D... | [] | simp +contextual [Subgroup.mem_comap, hH.conj_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 1009,
"column": 6
} | {
"line": 1009,
"column": 45
} | {
"line": 1009,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B' B : Subgroup G\nhN : (B'.subgroupOf B).Normal\n⊢ ((A ⊓ B').subgroupOf (A ⊓ B)).Normal",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.subgroupOf",
"congrArg",
"PartialOrder.toPreorder",
"Preorder... | [
"G : Type u_1\ninst✝ : Group G\nA B' B : Subgroup G\nhN : B ≤ normalizer ↑(B' ⊓ B)\n⊢ A ⊓ B ≤ normalizer ↑(A ⊓ B' ⊓ (A ⊓ B))"
] | normal_subgroupOf_iff_le_normalizer_inf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 55
} | {
"line": 280,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\ns t : Set α\nx : α\n⊢ x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Mono... | [] | rw [← nonempty_iff_ne_empty, smul_inter_nonempty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 55
} | {
"line": 280,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\ns t : Set α\nx : α\n⊢ x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Mono... | [] | rw [← nonempty_iff_ne_empty, smul_inter_nonempty_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 55
} | {
"line": 280,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\ns t : Set α\nx : α\n⊢ x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Mono... | [] | rw [← nonempty_iff_ne_empty, smul_inter_nonempty_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 1016,
"column": 6
} | {
"line": 1016,
"column": 45
} | {
"line": 1016,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA' A B : Subgroup G\nhN : (A'.subgroupOf A).Normal\n⊢ ((A' ⊓ B).subgroupOf (A ⊓ B)).Normal",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.subgroupOf",
"congrArg",
"PartialOrder.toPreorder",
"Preorder... | [
"G : Type u_1\ninst✝ : Group G\nA' A B : Subgroup G\nhN : A ≤ normalizer ↑(A' ⊓ A)\n⊢ A ⊓ B ≤ normalizer ↑(A' ⊓ B ⊓ (A ⊓ B))"
] | normal_subgroupOf_iff_le_normalizer_inf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 299,
"column": 50
} | {
"line": 299,
"column": 66
} | {
"line": 299,
"column": 66
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\ns t : Set α\nx : αᵐᵒᵖ\na b : α\nH : a⁻¹ * b = unop x\nha : a ∈ s\nhb : b ∈ t\nthis : op (a⁻¹ * b) = x\n⊢ x • a = b",
"ppTerm": "?m.157",
"assigned": true,
"usedConstants": [
"instHSMul",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Mo... | [] | by simp [← this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 1058,
"column": 6
} | {
"line": 1058,
"column": 45
} | {
"line": 1058,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ normalizer ↑N\n⊢ (N.subgroupOf H).Normal",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.subgroupOf",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Members... | [
"G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ normalizer ↑N\n⊢ H ≤ normalizer ↑(N ⊓ H)"
] | normal_subgroupOf_iff_le_normalizer_inf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.FreeMonoid.Basic | {
"line": 430,
"column": 6
} | {
"line": 430,
"column": 39
} | {
"line": 431,
"column": 6
} | [
{
"pp": "case mp.mul_of\nα : Type u_1\nβ : Type u_2\nf : α → β\nfs : Function.Surjective ⇑(map f)\nd : β\nhead : α\na✝¹ : FreeMonoid α\na✝ : (map f) a✝¹ = of d → ∃ a, f a = d\nhb : (map f) (of head * a✝¹) = of d\n⊢ ∃ a, f a = d",
"ppTerm": "?mp.mul_of",
"assigned": true,
"usedConstants": [
"Mo... | [
"case mp.mul_of\nα : Type u_1\nβ : Type u_2\nf : α → β\nfs : Function.Surjective ⇑(map f)\nd : β\nhead : α\na✝¹ : FreeMonoid α\na✝ : (map f) a✝¹ = of d → ∃ a, f a = d\nhb : of (f head) * (map f) a✝¹ = of d\n⊢ ∃ a, f a = d"
] | simp only [map_mul, map_of] at hb | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sigma.Lex | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 37
} | {
"line": 124,
"column": 37
} | [
{
"pp": "case inr.inr\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b✝ : (i : ι) × α i\ninst✝¹ : Std.Trichotomous r\ninst✝ : ∀ (i : ι), Std.Total (s i)\ni : ι\na : α i\nj : ι\nb : α j\nhji : r j i\n⊢ Lex r s ⟨i, a⟩ ⟨j, b⟩ ∨ Lex r s ⟨j, b⟩ ⟨i, a⟩",
"ppTerm"... | [] | exact Or.inr (Lex.left _ _ hji) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Sigma.Lex | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 37
} | {
"line": 124,
"column": 37
} | [
{
"pp": "case inr.inr\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b✝ : (i : ι) × α i\ninst✝¹ : Std.Trichotomous r\ninst✝ : ∀ (i : ι), Std.Total (s i)\ni : ι\na : α i\nj : ι\nb : α j\nhji : r j i\n⊢ Lex r s ⟨i, a⟩ ⟨j, b⟩ ∨ Lex r s ⟨j, b⟩ ⟨i, a⟩",
"ppTerm"... | [] | exact Or.inr (Lex.left _ _ hji) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sigma.Lex | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 37
} | {
"line": 124,
"column": 37
} | [
{
"pp": "case inr.inr\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b✝ : (i : ι) × α i\ninst✝¹ : Std.Trichotomous r\ninst✝ : ∀ (i : ι), Std.Total (s i)\ni : ι\na : α i\nj : ι\nb : α j\nhji : r j i\n⊢ Lex r s ⟨i, a⟩ ⟨j, b⟩ ∨ Lex r s ⟨j, b⟩ ⟨i, a⟩",
"ppTerm"... | [] | exact Or.inr (Lex.left _ _ hji) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 291,
"column": 28
} | {
"line": 291,
"column": 52
} | {
"line": 293,
"column": 0
} | [
{
"pp": "case mul_left\nM : Type u_1\ninst✝ : Monoid M\ns : Set M\nmotive : M → Prop\nhs : closure s = ⊤\nx✝ : M\none : motive 1\nmul_left : ∀ x ∈ s, ∀ (y : M), motive y → motive (x * y)\nx : M\nhx : x ∈ s\ny : M\nhy✝ : y ∈ closure s\nih : motive y\n⊢ motive (x * y)",
"ppTerm": "?mul_left",
"assigned": ... | [] | exact mul_left x hx y ih | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 291,
"column": 28
} | {
"line": 291,
"column": 52
} | {
"line": 293,
"column": 0
} | [
{
"pp": "case mul_left\nM : Type u_1\ninst✝ : Monoid M\ns : Set M\nmotive : M → Prop\nhs : closure s = ⊤\nx✝ : M\none : motive 1\nmul_left : ∀ x ∈ s, ∀ (y : M), motive y → motive (x * y)\nx : M\nhx : x ∈ s\ny : M\nhy✝ : y ∈ closure s\nih : motive y\n⊢ motive (x * y)",
"ppTerm": "?mul_left",
"assigned": ... | [] | exact mul_left x hx y ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 291,
"column": 28
} | {
"line": 291,
"column": 52
} | {
"line": 293,
"column": 0
} | [
{
"pp": "case mul_left\nM : Type u_1\ninst✝ : Monoid M\ns : Set M\nmotive : M → Prop\nhs : closure s = ⊤\nx✝ : M\none : motive 1\nmul_left : ∀ x ∈ s, ∀ (y : M), motive y → motive (x * y)\nx : M\nhx : x ∈ s\ny : M\nhy✝ : y ∈ closure s\nih : motive y\n⊢ motive (x * y)",
"ppTerm": "?mul_left",
"assigned": ... | [] | exact mul_left x hx y ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 314,
"column": 43
} | {
"line": 314,
"column": 55
} | {
"line": 316,
"column": 0
} | [
{
"pp": "M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : Monoid M\na n✝ n : M\ni : ℕ\n⊢ (fun x ↦ n✝ ^ x) i = n ↔ ((powersHom M) n✝) i = n",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"Equiv.instEquivLike",
"MonoidHom",
"Monoid.toMulOneC... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.MinMax | {
"line": 67,
"column": 17
} | {
"line": 67,
"column": 19
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case append_singleton\nα : Type u_1\nr : α → α → Prop\ninst✝ : DecidableRel r\nl : List α\nhr₀ : Std.Irrefl r\nhr₁ : IsTrans α r\ntl : List α\na : α\nih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m\nb m : α\no : Option α\nhb : b ∈ tl ++ [a]\n⊢ m ∈ foldl (argAux r) o (tl ++ ... | [
"case append_singleton\nα : Type u_1\nr : α → α → Prop\ninst✝ : DecidableRel r\nl : List α\nhr₀ : Std.Irrefl r\nhr₁ : IsTrans α r\ntl : List α\na : α\nih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m\nb m : α\no : Option α\nhb : b ∈ tl ++ [a]\nho : m ∈ foldl (argAux r) o (tl ++ [a])\n⊢ ¬... | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.List.MinMax | {
"line": 473,
"column": 29
} | {
"line": 473,
"column": 44
} | {
"line": 473,
"column": 44
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl : List α\nhd : α\ntl : List α\nIH : tl ≠ [] → ↑(foldr max ⊥ tl) = tl.maximum\nh : hd :: tl ≠ []\n⊢ ↑(max hd (foldr max ⊥ tl)) = max (↑hd) tl.maximum",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"List.m... | [
"case cons\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl : List α\nhd : α\ntl : List α\nIH : tl ≠ [] → ↑(foldr max ⊥ tl) = tl.maximum\nh : hd :: tl ≠ []\n⊢ max ↑hd ↑(foldr max ⊥ tl) = max (↑hd) tl.maximum"
] | WithBot.coe_max | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Lattice.Union | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 27
} | {
"line": 84,
"column": 0
} | [
{
"pp": "α : Type u_7\nβ : Type u_8\ninst✝ : DecidableEq β\ns : Finset α\nt : α → Finset β\na✝ : β\n⊢ a✝ ∈ s.sup t ↔ a✝ ∈ s.biUnion t",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Finset.mem_sup",
"congrArg",
"Finset",
... | [] | rw [mem_sup, mem_biUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.Sublists | {
"line": 134,
"column": 8
} | {
"line": 134,
"column": 19
} | {
"line": 134,
"column": 20
} | [
{
"pp": "case cons\nα : Type u\nl₂ : List α\na : α\nl₁ : List α\nih :\n foldr (fun a acc ↦ flatMap (fun x ↦ [x, a :: x]) acc) (foldr (fun a acc ↦ flatMap (fun x ↦ [x, a :: x]) acc) [[]] l₂)\n l₁ =\n do\n let x ← foldr (fun a acc ↦ flatMap (fun x ↦ [x, a :: x]) acc) [[]] l₂\n map (fun x_1 ↦ x_1 ++... | [
"case cons\nα : Type u\nl₂ : List α\na : α\nl₁ : List α\nih :\n foldr (fun a acc ↦ flatMap (fun x ↦ [x, a :: x]) acc) (foldr (fun a acc ↦ flatMap (fun x ↦ [x, a :: x]) acc) [[]] l₂)\n l₁ =\n do\n let x ← foldr (fun a acc ↦ flatMap (fun x ↦ [x, a :: x]) acc) [[]] l₂\n map (fun x_1 ↦ x_1 ++ x) (foldr (... | foldr_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Zip | {
"line": 135,
"column": 24
} | {
"line": 135,
"column": 33
} | {
"line": 135,
"column": 34
} | [
{
"pp": "case mp\nα : Type u\nhd : α\ntl : List α\nih : ∀ {init tail : List α}, (init, tail) ∈ tl.inits.zip tl.tails ↔ init ++ tail = tl\ninit tail : List α\n⊢ (init, tail) ∈ ([], hd :: tl) :: (map (fun t ↦ hd :: t) tl.inits).zip tl.tails → init ++ tail = hd :: tl",
"ppTerm": "?mp",
"assigned": true,
... | [
"case mp\nα : Type u\nhd : α\ntl : List α\nih : ∀ {init tail : List α}, (init, tail) ∈ tl.inits.zip tl.tails ↔ init ++ tail = tl\ninit tail : List α\n⊢ (init, tail) = ([], hd :: tl) ∨ (init, tail) ∈ (map (fun t ↦ hd :: t) tl.inits).zip tl.tails → init ++ tail = hd :: tl"
] | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Zip | {
"line": 135,
"column": 24
} | {
"line": 135,
"column": 33
} | {
"line": 135,
"column": 34
} | [
{
"pp": "case mpr\nα : Type u\nhd : α\ntl : List α\nih : ∀ {init tail : List α}, (init, tail) ∈ tl.inits.zip tl.tails ↔ init ++ tail = tl\ninit tail : List α\n⊢ init ++ tail = hd :: tl → (init, tail) ∈ ([], hd :: tl) :: (map (fun t ↦ hd :: t) tl.inits).zip tl.tails",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\nα : Type u\nhd : α\ntl : List α\nih : ∀ {init tail : List α}, (init, tail) ∈ tl.inits.zip tl.tails ↔ init ++ tail = tl\ninit tail : List α\n⊢ init ++ tail = hd :: tl → (init, tail) = ([], hd :: tl) ∨ (init, tail) ∈ (map (fun t ↦ hd :: t) tl.inits).zip tl.tails"
] | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Sublists | {
"line": 277,
"column": 6
} | {
"line": 278,
"column": 32
} | {
"line": 279,
"column": 2
} | [
{
"pp": "case cons.cons\nα : Type u\nl l' l₂ : List α\na b : α\nl₁ : List α\ns : b :: l₁ <+ l₂\nIH : b :: l₁ ∈ sublistsLen (b :: l₁).length l₂\n⊢ b :: l₁ ∈ sublistsLen (b :: l₁).length (a :: l₂)",
"ppTerm": "?cons.cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"L... | [] | rw [length, sublistsLen_succ_cons]
exact mem_append_left _ IH | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Sublists | {
"line": 277,
"column": 6
} | {
"line": 278,
"column": 32
} | {
"line": 279,
"column": 2
} | [
{
"pp": "case cons.cons\nα : Type u\nl l' l₂ : List α\na b : α\nl₁ : List α\ns : b :: l₁ <+ l₂\nIH : b :: l₁ ∈ sublistsLen (b :: l₁).length l₂\n⊢ b :: l₁ ∈ sublistsLen (b :: l₁).length (a :: l₂)",
"ppTerm": "?cons.cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"L... | [] | rw [length, sublistsLen_succ_cons]
exact mem_append_left _ IH | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Powerset | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 71
} | {
"line": 280,
"column": 4
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : Multiset α\nl : List α\n⊢ powersetCard n ⟦l⟧ ≤ powerset ⟦l⟧",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"List.sublists'",
"Eq.mpr",
"Multiset.powerset_coe'",
"congrArg",
"List.map",
"PartialOrder.toPreorder",
"... | [
"α : Type u_1\nn : ℕ\ns : Multiset α\nl : List α\n⊢ List.map ofList (sublistsLen n l) <+~ List.map ofList l.sublists'"
] | simp only [quot_mk_to_coe, powersetCard_coe, powerset_coe', coe_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Pairwise.Lattice | {
"line": 108,
"column": 2
} | {
"line": 110,
"column": 45
} | {
"line": 112,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nι : Type u_2\nι' : Type u_3\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i ↦ ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' ↦ ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t\nj : ι\nj' : ι'\nhj : (j, j... | [] | · refine (hs hi.1 hj.1 hij).mono ?_ ?_
· convert! le_iSup₂ (α := α) i' hi.2; rfl
· convert! le_iSup₂ (α := α) j' hj.2; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Interval.Set.WithBotTop | {
"line": 54,
"column": 78
} | {
"line": 54,
"column": 100
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ some ⁻¹' Icc ↑a ↑b = Icc a b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.Ici",
"WithTop.instPreorder",
"congrArg",
"WithTop.preimage_coe_Ici",
"Set.instInter",
"WithTop.some",
"Inter.int... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.WithBotTop | {
"line": 54,
"column": 78
} | {
"line": 54,
"column": 100
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ some ⁻¹' Icc ↑a ↑b = Icc a b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.Ici",
"WithTop.instPreorder",
"congrArg",
"WithTop.preimage_coe_Ici",
"Set.instInter",
"WithTop.some",
"Inter.int... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.WithBotTop | {
"line": 54,
"column": 78
} | {
"line": 54,
"column": 100
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ some ⁻¹' Icc ↑a ↑b = Icc a b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.Ici",
"WithTop.instPreorder",
"congrArg",
"WithTop.preimage_coe_Ici",
"Set.instInter",
"WithTop.some",
"Inter.int... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 235,
"column": 2
} | {
"line": 252,
"column": 20
} | {
"line": 254,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : ConditionallyCompleteLattice α\nf : β × γ → α\nhf : BddAbove (range f)\n⊢ ⨆ p, f p = ⨆ b, ⨆ c, f (b, c)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Set.mem_range",
"_private.Mathlib... | [] | rcases isEmpty_or_nonempty β
· simp [iSup_of_empty']
rcases isEmpty_or_nonempty γ
· simp [iSup_of_empty']
have h₁ : BddAbove (Set.range fun b ↦ ⨆ c, f (b, c)) := by
rw [bddAbove_def] at hf ⊢
obtain ⟨B, hB⟩ := hf
refine ⟨B, fun y hy ↦ ?_⟩
obtain ⟨z, rfl⟩ := Set.mem_range.mp hy
exact ciSup_le ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 235,
"column": 2
} | {
"line": 252,
"column": 20
} | {
"line": 254,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : ConditionallyCompleteLattice α\nf : β × γ → α\nhf : BddAbove (range f)\n⊢ ⨆ p, f p = ⨆ b, ⨆ c, f (b, c)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Set.mem_range",
"_private.Mathlib... | [] | rcases isEmpty_or_nonempty β
· simp [iSup_of_empty']
rcases isEmpty_or_nonempty γ
· simp [iSup_of_empty']
have h₁ : BddAbove (Set.range fun b ↦ ⨆ c, f (b, c)) := by
rw [bddAbove_def] at hf ⊢
obtain ⟨B, hB⟩ := hf
refine ⟨B, fun y hy ↦ ?_⟩
obtain ⟨z, rfl⟩ := Set.mem_range.mp hy
exact ciSup_le ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 33
} | {
"line": 365,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLattice α\nι : Type u_5\nι' : Type u_6\ns : Set ι\nf : ι → ι'\ng : ι' → α\nhf : BddAbove (range fun i ↦ g (f ↑i))\nhg' : sSup ∅ ≤ ⨆ i, g (f ↑i)\nhs : s.Nonempty\n⊢ BddAbove (range fun i ↦ g ↑i)",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants"... | [] | simpa [bddAbove_def] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 375,
"column": 4
} | {
"line": 375,
"column": 33
} | {
"line": 376,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLattice α\nι : Type u_5\nι' : Type u_6\ns : Set ι\nf : ι → ι'\ng : ι' → α\nhf : BddAbove (range fun i ↦ g (f ↑i))\nhg' : sSup ∅ ≤ ⨆ i, g (f ↑i)\nhs : s.Nonempty\nhg : BddAbove (range fun i ↦ g ↑i)\nthis : Nonempty ↑s\ni : ι\nh : i ∈ s\nt : ↑(f '' s)\nht : g ↑t... | [] | simpa [bddAbove_def] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Finset.Preimage | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 33
} | {
"line": 127,
"column": 33
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : DecidableEq β\nf : α → β\ns : Finset β\nt : Finset α\nhs : s ⊆ image f t\nhf : InjOn f (f ⁻¹' ↑s)\n⊢ s.preimage f hf ⊆ t",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrder.toPreorder... | [
"α : Type u\nβ : Type v\ninst✝ : DecidableEq β\nf : α → β\ns : Finset β\nt : Finset α\nhs : s ⊆ image f t\nhf : InjOn f (f ⁻¹' ↑s)\n⊢ f ⁻¹' ↑s ⊆ ↑t"
] | rw [← coe_subset, coe_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Cover | {
"line": 147,
"column": 81
} | {
"line": 150,
"column": 24
} | {
"line": 152,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2 : a ≤ c\nh3 : c ≤ b\n⊢ c = a ∨ c = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"LE.le.eq_or_lt",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"Preorder.toLE",
"LE.l... | [] | by
rcases h2.eq_or_lt with (h2 | h2); · exact Or.inl h2.symm
rcases h3.eq_or_lt with (h3 | h3); · exact Or.inr h3
exact (h.2 h2 h3).elim | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Cover | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 36
} | {
"line": 392,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b c : α\nh : ∀ (x : α), a ≤ x ∧ x < b ↔ x = c\n⊢ a = c ∧ a ⋖ b",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"CovBy",
"PartialOrder.toPreorder",
"Preorder.toLE",
"_private.Mat... | [
"α : Type u_1\ninst✝ : PartialOrder α\na b c : α\nh : ∀ (x : α), a ≤ x ∧ x < b ↔ x = c\nhac : a ≤ c\nhcb : c < b\n⊢ a = c ∧ a ⋖ b"
] | have ⟨hac, hcb⟩ := (h c).mpr rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Cover | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 36
} | {
"line": 430,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b c : α\nh : ∀ (x : α), a < x ∧ x < b ↔ x = c\n⊢ a ⋖ c ∧ c ⋖ b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"CovBy",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b c : α\nh : ∀ (x : α), a < x ∧ x < b ↔ x = c\nhac : a < c\nhcb : c < b\n⊢ a ⋖ c ∧ c ⋖ b"
] | have ⟨hac, hcb⟩ := (h c).mpr rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Cover | {
"line": 582,
"column": 2
} | {
"line": 585,
"column": 15
} | {
"line": 587,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na₁ a₂ : α\nb : β\n⊢ (a₁, b) ⩿ (a₂, b) ↔ a₁ ⩿ a₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"And.imp",
"Preorder.toLT",
"Prod.instLE_mathlib",
"congrArg",
... | [] | refine ⟨WCovBy.fst, (And.imp mk_le_mk_iff_left.2) fun h c h₁ h₂ => ?_⟩
have : c.2 = b := h₂.le.2.antisymm h₁.le.2
rw [← @Prod.mk.eta _ _ c, this, mk_lt_mk_iff_left] at h₁ h₂
exact h h₁ h₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Cover | {
"line": 582,
"column": 2
} | {
"line": 585,
"column": 15
} | {
"line": 587,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\na₁ a₂ : α\nb : β\n⊢ (a₁, b) ⩿ (a₂, b) ↔ a₁ ⩿ a₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"And.imp",
"Preorder.toLT",
"Prod.instLE_mathlib",
"congrArg",
... | [] | refine ⟨WCovBy.fst, (And.imp mk_le_mk_iff_left.2) fun h c h₁ h₂ => ?_⟩
have : c.2 = b := h₂.le.2.antisymm h₁.le.2
rw [← @Prod.mk.eta _ _ c, this, mk_lt_mk_iff_left] at h₁ h₂
exact h h₁ h₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Multiset | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 68
} | {
"line": 302,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ Ico a b - Ico c b = Ico a (min b c)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrder.toDecidableEq",
"Finset.sdiff_val",
"congrArg",
"Finset",
... | [] | rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_sdiff_Ico_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Interval.Multiset | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 68
} | {
"line": 302,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ Ico a b - Ico c b = Ico a (min b c)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrder.toDecidableEq",
"Finset.sdiff_val",
"congrArg",
"Finset",
... | [] | rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_sdiff_Ico_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Multiset | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 68
} | {
"line": 302,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ Ico a b - Ico c b = Ico a (min b c)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrder.toDecidableEq",
"Finset.sdiff_val",
"congrArg",
"Finset",
... | [] | rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_sdiff_Ico_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Finset.Nat | {
"line": 95,
"column": 42
} | {
"line": 95,
"column": 53
} | {
"line": 95,
"column": 54
} | [
{
"pp": "b : ℕ\n⊢ #(Iio b) = b",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset.Iio_eq_Ico",
"Finset",
"OrderBot.toBot",
"Finset.Iio",
"Preorder.toLE",
"Nat.instLocallyFiniteOrder",
"LocallyFiniteOrder.toLoca... | [
"b : ℕ\n⊢ #(Ico ⊥ b) = b"
] | Iio_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Lattice.Nat | {
"line": 142,
"column": 6
} | {
"line": 143,
"column": 87
} | {
"line": 144,
"column": 6
} | [
{
"pp": "s : Set ℕ\nhs : ¬BddAbove s\n⊢ sSup s = sSup ∅",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"dite_congr",
"instDecidableTrue",
"congrArg",
"_private.Mathlib.Order.Lattice.Nat.0.Nat.instConditionallyCompleteLinearOrderBot._s... | [
"s : Set ℕ\nhs : ¬BddAbove s\n⊢ (if h : ∃ n, ∀ a ∈ s, a ≤ n then Nat.find h else 0) = Nat.find ⋯"
] | simp only [sSup,
mem_empty_iff_false, IsEmpty.forall_iff, forall_const, exists_const, dite_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.OrderIsoNat | {
"line": 135,
"column": 2
} | {
"line": 141,
"column": 93
} | {
"line": 143,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\ne : ℕ → α\nhe : ∀ (n : ℕ), e n ∈ s ∪ t\n⊢ ∃ g, (∀ (n : ℕ), e (g n) ∈ s) ∨ ∀ (n : ℕ), e (g n) ∈ t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Classical.propDecidable",
"_private.Mathlib... | [] | classical
have : Infinite (e ⁻¹' s) ∨ Infinite (e ⁻¹' t) := by
simp only [Set.infinite_coe_iff, ← Set.infinite_union, ← Set.preimage_union,
Set.eq_univ_of_forall fun n => Set.mem_preimage.2 (he n), Set.infinite_univ]
cases this
exacts [⟨Nat.orderEmbeddingOfSet (e ⁻¹' s), Or.inl fun n => (Nat.S... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Order.OrderIsoNat | {
"line": 135,
"column": 2
} | {
"line": 141,
"column": 93
} | {
"line": 143,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\ne : ℕ → α\nhe : ∀ (n : ℕ), e n ∈ s ∪ t\n⊢ ∃ g, (∀ (n : ℕ), e (g n) ∈ s) ∨ ∀ (n : ℕ), e (g n) ∈ t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Classical.propDecidable",
"_private.Mathlib... | [] | classical
have : Infinite (e ⁻¹' s) ∨ Infinite (e ⁻¹' t) := by
simp only [Set.infinite_coe_iff, ← Set.infinite_union, ← Set.preimage_union,
Set.eq_univ_of_forall fun n => Set.mem_preimage.2 (he n), Set.infinite_univ]
cases this
exacts [⟨Nat.orderEmbeddingOfSet (e ⁻¹' s), Or.inl fun n => (Nat.S... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OrderIsoNat | {
"line": 135,
"column": 2
} | {
"line": 141,
"column": 93
} | {
"line": 143,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\ne : ℕ → α\nhe : ∀ (n : ℕ), e n ∈ s ∪ t\n⊢ ∃ g, (∀ (n : ℕ), e (g n) ∈ s) ∨ ∀ (n : ℕ), e (g n) ∈ t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Classical.propDecidable",
"_private.Mathlib... | [] | classical
have : Infinite (e ⁻¹' s) ∨ Infinite (e ⁻¹' t) := by
simp only [Set.infinite_coe_iff, ← Set.infinite_union, ← Set.preimage_union,
Set.eq_univ_of_forall fun n => Set.mem_preimage.2 (he n), Set.infinite_univ]
cases this
exacts [⟨Nat.orderEmbeddingOfSet (e ⁻¹' s), Or.inl fun n => (Nat.S... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellQuasiOrder | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 72
} | {
"line": 78,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsPreorder α r\nh : ∀ (f : ℕ → α), ∃ g, ∀ (m n : ℕ), m ≤ n → r (f (g m)) (f (g n))\nf : ℕ → α\n⊢ ∃ m n, m < n ∧ r (f m) (f n)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"Preorder.toLE",
"Exists"... | [] | obtain ⟨g, gmon⟩ := h f
exact ⟨_, _, g.strictMono Nat.zero_lt_one, gmon _ _ (Nat.zero_le 1)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.WellQuasiOrder | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 72
} | {
"line": 78,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nr : α → α → Prop\ninst✝ : IsPreorder α r\nh : ∀ (f : ℕ → α), ∃ g, ∀ (m n : ℕ), m ≤ n → r (f (g m)) (f (g n))\nf : ℕ → α\n⊢ ∃ m n, m < n ∧ r (f m) (f n)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"Preorder.toLE",
"Exists"... | [] | obtain ⟨g, gmon⟩ := h f
exact ⟨_, _, g.strictMono Nat.zero_lt_one, gmon _ _ (Nat.zero_le 1)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellQuasiOrder | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 69
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_3\nα : ι → Type u_4\ninst✝¹ : Finite ι\nr : (i : ι) → α i → α i → Prop\ninst✝ : ∀ (i : ι), IsPreorder (α i) (r i)\nhr : ∀ (i : ι), WellQuasiOrdered (r i)\nthis✝ : Fintype ι\nthis : IsPreorder ((i : ι) → α i) fun a b ↦ ∀ (i : ι), r i (a i) (b i)\n⊢ ∀ (f : ℕ → (i : ι) → α i), ∃ ... | [
"case refine_2\nι : Type u_3\nα : ι → Type u_4\ninst✝¹ : Finite ι\nr : (i : ι) → α i → α i → Prop\ninst✝ : ∀ (i : ι), IsPreorder (α i) (r i)\nhr : ∀ (i : ι), WellQuasiOrdered (r i)\nthis✝ : Fintype ι\nthis : IsPreorder ((i : ι) → α i) fun a b ↦ ∀ (i : ι), r i (a i) (b i)\n⊢ ∀ (a : ι) (s : Finset ι) (h : a ∉ s),\n ... | · intro f
exists RelEmbedding.refl (· ≤ ·)
simp only [IsEmpty.forall_iff, imp_true_iff, Finset.notMem_empty] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.WellFoundedSet | {
"line": 81,
"column": 2
} | {
"line": 91,
"column": 58
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_2\nr : α → α → Prop\ns : Set α\n⊢ s.WellFoundedOn r ↔ WellFounded fun a b ↦ r a b ∧ a ∈ s ∧ b ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RelEmbedding.mk",
"False",
"and_true",
"Subtype.preimage_coe_nonempty",
"congr... | [] | have f : RelEmbedding (Subrel r (· ∈ s)) fun a b : α => r a b ∧ a ∈ s ∧ b ∈ s :=
⟨⟨(↑), Subtype.coe_injective⟩, by simp⟩
refine ⟨fun h => ?_, f.wellFounded⟩
rw [WellFounded.wellFounded_iff_has_min]
intro t ht
by_cases hst : (s ∩ t).Nonempty
· rw [← Subtype.preimage_coe_nonempty] at hst
rcases h.has_mi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.WellFoundedSet | {
"line": 81,
"column": 2
} | {
"line": 91,
"column": 58
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_2\nr : α → α → Prop\ns : Set α\n⊢ s.WellFoundedOn r ↔ WellFounded fun a b ↦ r a b ∧ a ∈ s ∧ b ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RelEmbedding.mk",
"False",
"and_true",
"Subtype.preimage_coe_nonempty",
"congr... | [] | have f : RelEmbedding (Subrel r (· ∈ s)) fun a b : α => r a b ∧ a ∈ s ∧ b ∈ s :=
⟨⟨(↑), Subtype.coe_injective⟩, by simp⟩
refine ⟨fun h => ?_, f.wellFounded⟩
rw [WellFounded.wellFounded_iff_has_min]
intro t ht
by_cases hst : (s ∩ t).Nonempty
· rw [← Subtype.preimage_coe_nonempty] at hst
rcases h.has_mi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Finset.Basic | {
"line": 594,
"column": 71
} | {
"line": 595,
"column": 100
} | {
"line": 597,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a < b\n⊢ insert b (Ioo a b) = Ioc a b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Set.insert_eq",
"congrArg",
"Finset",... | [] | by
rw [← coe_inj, coe_insert, coe_Ioo, coe_Ioc, Set.insert_eq, Set.union_comm, Set.Ioo_union_right h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 37
} | {
"line": 297,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns : Set α\nhpos : ∀ x ∈ s, 1 ≤ x\nh : s.IsPWO\n⊢ (↑(Submonoid.closure s)).IsPWO",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.IsPWO",
"Eq.mpr",
"MulOne.toOne",
... | [
"α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns : Set α\nhpos : ∀ x ∈ s, 1 ≤ x\nh : s.IsPWO\n⊢ (List.prod '' {l | ∀ x ∈ l, x ∈ s}).IsPWO"
] | Submonoid.closure_eq_image_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Finset.Basic | {
"line": 1014,
"column": 2
} | {
"line": 1014,
"column": 39
} | {
"line": 1016,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DistribLattice α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ a ⊓ c ≤ b ∧ b ≤ a ⊔ c → a ⊓ b ≤ c ∧ c ≤ a ⊔ b → b = c",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.eq_of_mem_uIcc_of_mem_uIcc'"
],
"usedFVars": [
"α",
"inst✝¹",
... | [] | exact Set.eq_of_mem_uIcc_of_mem_uIcc' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.WellFoundedSet | {
"line": 731,
"column": 2
} | {
"line": 731,
"column": 42
} | {
"line": 732,
"column": 2
} | [
{
"pp": "α : Type u_2\ns : Set α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\n⊢ BddBelow s → s.WellFoundedOn fun x1 x2 ↦ x1 < x2",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Preorder.toLE",
"Membership.mem",
... | [
"α : Type u_2\ns : Set α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\n⊢ BddBelow s → ∀ (f : (fun x1 x2 ↦ x1 > x2) ↪r fun x1 x2 ↦ x1 < x2), ¬∀ (n : ℕ), f n ∈ s"
] | rw [wellFoundedOn_iff_no_descending_seq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.WellFoundedSet | {
"line": 861,
"column": 9
} | {
"line": 861,
"column": 19
} | {
"line": 861,
"column": 20
} | [
{
"pp": "case pos\nα : Type u_2\nr : α → α → Prop\ninst✝ : IsPreorder α r\ns : Set α\nh : s.PartiallyWellOrderedOn r\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nf : ℕ → List α\nhf1 : IsBadSeq (List.SublistForall₂ r) {l | ∀ x ∈ l, x ∈ s} f\nhf2 : ∀ (n : ℕ), IsMinBadSeq (List.SublistForall₂ r) List.length {l | ∀... | [
"case pos\nα : Type u_2\nr : α → α → Prop\ninst✝ : IsPreorder α r\ns : Set α\nh : s.PartiallyWellOrderedOn r\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nf : ℕ → List α\nhf1 : IsBadSeq (List.SublistForall₂ r) {l | ∀ x ∈ l, x ∈ s} f\nhf2 : ∀ (n : ℕ), IsMinBadSeq (List.SublistForall₂ r) List.length {l | ∀ x ∈ l, x ∈ ... | if_pos hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 88,
"column": 41
} | {
"line": 88,
"column": 64
} | {
"line": 88,
"column": 65
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\ns : Set G\nhs : s.Nonempty\nn : ℕ\n⊢ s ^ n * (s * ↑(closure s)) = ↑(closure s)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"Subgroup.closure",
"Monoid.toMulOneClass",
"... | [
"G : Type u_2\ninst✝ : Group G\ns : Set G\nhs : s.Nonempty\nn : ℕ\n⊢ s ^ n * ↑(closure s) = ↑(closure s)"
] | mul_subgroupClosure hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 95
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case refine_2\nG : Type u_2\ninst✝ : Group G\nS : Set G\n⊢ S ∪ S⁻¹ ⊆ ↑(closure S).toSubmonoid",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Subgroup.closure",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Group.toDivisionMonoid",
"DivisionMonoi... | [] | · simp only [true_and, coe_toSubmonoid, union_subset_iff, subset_closure, inv_subset_closure] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 325,
"column": 4
} | {
"line": 327,
"column": 30
} | {
"line": 328,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_2\ninst✝ : Group G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝) → x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"Members... | [] | rintro ⟨y, ⟨hyA, hyB⟩, z, hz, rfl⟩
refine ⟨A.mul_mem hyA (h hz), ?_⟩
exact ⟨y, hyB, z, hz, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 325,
"column": 4
} | {
"line": 327,
"column": 30
} | {
"line": 328,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_2\ninst✝ : Group G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ (∃ x, (x ∈ ↑A ∧ x ∈ ↑B) ∧ ∃ y ∈ ↑C, x * y = x✝) → x✝ ∈ ↑A ∧ ∃ x ∈ ↑B, ∃ y ∈ ↑C, x * y = x✝",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"Members... | [] | rintro ⟨y, ⟨hyA, hyB⟩, z, hz, rfl⟩
refine ⟨A.mul_mem hyA (h hz), ?_⟩
exact ⟨y, hyB, z, hz, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 334,
"column": 31
} | {
"line": 334,
"column": 40
} | {
"line": 334,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝ : Mul M\nS : Set (Con M)\nx y : M\nh : (sInf S).toSetoid x y\nr : Setoid M\nx✝ : r ∈ toSetoid '' S\nc : Con M\nhS : c ∈ S\nhr : c.toSetoid = r\n⊢ r x y",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Setoid",... | [
"M : Type u_1\ninst✝ : Mul M\nS : Set (Con M)\nx y : M\nh : (sInf S).toSetoid x y\nr : Setoid M\nx✝ : r ∈ toSetoid '' S\nc : Con M\nhS : c ∈ S\nhr : c.toSetoid = r\n⊢ c.toSetoid x y"
] | rw [← hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 179,
"column": 57
} | {
"line": 179,
"column": 86
} | {
"line": 179,
"column": 86
} | [
{
"pp": "α : Type u\nx3 : α\nb3 : Bool\ntl x✝⁵ : List (α × Bool)\nx4 : α\nb4 : Bool\ntl2 x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\nH : (x3, b3) :: tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁵ = (x4, b4) :: tl2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: x✝⁴\nH1 : (x3, b3) = (x4, b4)\nH2 : tl.append ((x✝³, x✝²... | [] | simpa [H1] using Step.cons H4 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 179,
"column": 57
} | {
"line": 179,
"column": 86
} | {
"line": 179,
"column": 86
} | [
{
"pp": "α : Type u\nx3 : α\nb3 : Bool\ntl x✝⁵ : List (α × Bool)\nx4 : α\nb4 : Bool\ntl2 x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\nH : (x3, b3) :: tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁵ = (x4, b4) :: tl2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: x✝⁴\nH1 : (x3, b3) = (x4, b4)\nH2 : tl.append ((x✝³, x✝²... | [] | simpa [H1] using Step.cons H4 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 179,
"column": 57
} | {
"line": 179,
"column": 86
} | {
"line": 179,
"column": 86
} | [
{
"pp": "α : Type u\nx3 : α\nb3 : Bool\ntl x✝⁵ : List (α × Bool)\nx4 : α\nb4 : Bool\ntl2 x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\nH : (x3, b3) :: tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁵ = (x4, b4) :: tl2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: x✝⁴\nH1 : (x3, b3) = (x4, b4)\nH2 : tl.append ((x✝³, x✝²... | [] | simpa [H1] using Step.cons H4 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Finiteness | {
"line": 89,
"column": 19
} | {
"line": 89,
"column": 39
} | {
"line": 89,
"column": 39
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\ns t : Finset M\n⊢ closure ↑(s ∪ t) = closure ↑s ⊔ closure ↑t",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset.instUnion",
"CompleteLattice.toLattice",
"Monoid.toMulOneClass",
"congrArg... | [] | simp [closure_union] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Finiteness | {
"line": 89,
"column": 19
} | {
"line": 89,
"column": 39
} | {
"line": 89,
"column": 39
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\ns t : Finset M\n⊢ closure ↑(s ∪ t) = closure ↑s ⊔ closure ↑t",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset.instUnion",
"CompleteLattice.toLattice",
"Monoid.toMulOneClass",
"congrArg... | [] | simp [closure_union] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Finiteness | {
"line": 89,
"column": 19
} | {
"line": 89,
"column": 39
} | {
"line": 89,
"column": 39
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\ns t : Finset M\n⊢ closure ↑(s ∪ t) = closure ↑s ⊔ closure ↑t",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset.instUnion",
"CompleteLattice.toLattice",
"Monoid.toMulOneClass",
"congrArg... | [] | simp [closure_union] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Finiteness | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 39
} | {
"line": 351,
"column": 39
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\ns t : Finset G\n⊢ closure ↑(s ∪ t) = closure ↑s ⊔ closure ↑t",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Subgroup.closure",
"Finset.instUnion",
"CompleteLattice.toLattice",
"congrArg",
... | [] | simp [closure_union] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Finiteness | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 39
} | {
"line": 351,
"column": 39
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\ns t : Finset G\n⊢ closure ↑(s ∪ t) = closure ↑s ⊔ closure ↑t",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Subgroup.closure",
"Finset.instUnion",
"CompleteLattice.toLattice",
"congrArg",
... | [] | simp [closure_union] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Finiteness | {
"line": 351,
"column": 19
} | {
"line": 351,
"column": 39
} | {
"line": 351,
"column": 39
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\ns t : Finset G\n⊢ closure ↑(s ∪ t) = closure ↑s ⊔ closure ↑t",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Subgroup.closure",
"Finset.instUnion",
"CompleteLattice.toLattice",
"congrArg",
... | [] | simp [closure_union] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Finiteness | {
"line": 528,
"column": 8
} | {
"line": 528,
"column": 32
} | {
"line": 528,
"column": 32
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁶ : Monoid M\nG : Type u_3\nH : Type u_4\ninst✝⁵ : Group G\ninst✝⁴ : AddGroup H\ninst✝³ : Monoid N\nG' : Type u_5\ninst✝² : Group G'\ninst✝¹ : FG M\ninst✝ : FG N\n⊢ ⊤.FG",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.t... | [
"M : Type u_1\nN : Type u_2\ninst✝⁶ : Monoid M\nG : Type u_3\nH : Type u_4\ninst✝⁵ : Group G\ninst✝⁴ : AddGroup H\ninst✝³ : Monoid N\nG' : Type u_5\ninst✝² : Group G'\ninst✝¹ : FG M\ninst✝ : FG N\n⊢ (⊤.prod ⊤).FG"
] | ← Submonoid.top_prod_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 68,
"column": 33
} | {
"line": 69,
"column": 63
} | {
"line": 71,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\na b c d : G\nhab : a⁻¹ * b ∈ N\nhcd : c⁻¹ * d ∈ N\n⊢ c⁻¹ * (a⁻¹ * b) * c⁻¹⁻¹ * (c⁻¹ * d) = (a * c)⁻¹ * (b * d)",
"ppTerm": "?m.100",
"assigned": true,
... | [] | by
simp only [mul_inv_rev, mul_assoc, inv_mul_cancel_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 34
} | {
"line": 122,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nx : G\n⊢ ↑x = 1 ↔ x ∈ N",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"QuotientGroup.mk",
"... | [
"G : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nx : G\n⊢ x⁻¹ * 1 ∈ N ↔ x ∈ N"
] | refine QuotientGroup.eq.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 330,
"column": 14
} | {
"line": 330,
"column": 64
} | {
"line": 331,
"column": 4
} | [
{
"pp": "case zero\nα : Type u\nG : Type u_1\nβ α✝ β✝ : Type u\ny : FreeAbelianGroup β✝\n⊢ Prod.mk <$> 0 <*> y = (fun b a ↦ (a, b)) <$> y <*> 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FreeAbelianGroup.seq_zero",
"congrArg",
"Monad.toApplicative",
... | [] | rw [FreeAbelianGroup.map_zero, zero_seq, seq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 330,
"column": 14
} | {
"line": 330,
"column": 64
} | {
"line": 331,
"column": 4
} | [
{
"pp": "case zero\nα : Type u\nG : Type u_1\nβ α✝ β✝ : Type u\ny : FreeAbelianGroup β✝\n⊢ Prod.mk <$> 0 <*> y = (fun b a ↦ (a, b)) <$> y <*> 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FreeAbelianGroup.seq_zero",
"congrArg",
"Monad.toApplicative",
... | [] | rw [FreeAbelianGroup.map_zero, zero_seq, seq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 330,
"column": 14
} | {
"line": 330,
"column": 64
} | {
"line": 331,
"column": 4
} | [
{
"pp": "case zero\nα : Type u\nG : Type u_1\nβ α✝ β✝ : Type u\ny : FreeAbelianGroup β✝\n⊢ Prod.mk <$> 0 <*> y = (fun b a ↦ (a, b)) <$> y <*> 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FreeAbelianGroup.seq_zero",
"congrArg",
"Monad.toApplicative",
... | [] | rw [FreeAbelianGroup.map_zero, zero_seq, seq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 531,
"column": 6
} | {
"line": 531,
"column": 15
} | {
"line": 532,
"column": 8
} | [
{
"pp": "case of.of\nα : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\ninst✝ : CommMonoid α\ns t : α\n⊢ of s * of t = of t * of s",
"ppTerm": "?of.of",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Equiv.instEquivLike",
"HMul.hMul",
"CommMonoid.toCo... | [] | | of t => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 31
} | {
"line": 74,
"column": 31
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nm₁ m₂ : M\ns₁ s₂ : ↥⊤\n⊢ mk m₁ s₁ / mk m₂ s₂ = mk (m₁ * ↑s₂) ⟨↑s₁ * m₂, ⋯⟩",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Localization.mk",
"instHDiv",
"Submonoid.mem_top",
"S... | [
"M : Type u_1\ninst✝ : CommMonoid M\nm₁ m₂ : M\ns₁ s₂ : ↥⊤\n⊢ mk (m₁ * ↑s₂) (s₁ * ⟨m₂, ⋯⟩) = mk (m₁ * ↑s₂) ⟨↑s₁ * m₂, ⋯⟩"
] | simp [div_eq_mul_inv, mk_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 59
} | {
"line": 253,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.10\ninst✝ : MulAction R X\nr : R\ns : ↥S\nr₁ : X\nr₂ : R\ns' : ↥S\nhs : r₂ * ↑s' ∈ S\nr₁' : R\ns₁' : ↥S\nh₁ : ↑s₁' * r = r₁' * ↑s'\nr₂' : R\ns₂' : ↥S\nh₂ : ↑s₂' * r = r₂' * ↑⟨r₂ * ↑s', hs⟩\n⊢ r₁' • r₁ /ₒ (s₁' * s) = r₂' • ... | [
"R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.10\ninst✝ : MulAction R X\nr : R\ns : ↥S\nr₁ : X\nr₂ : R\ns' : ↥S\nhs : r₂ * ↑s' ∈ S\nr₁' : R\ns₁' : ↥S\nh₁ : ↑s₁' * r = r₁' * ↑s'\nr₂' : R\ns₂' : ↥S\nh₂ : ↑s₂' * r = r₂' * ↑⟨r₂ * ↑s', hs⟩\nr₃' : R\ns₃' : ↥S\nh₃ : ↑s₃' * ↑s₁' = r₃' * ... | rcases oreCondition (s₁' : R) (s₂') with ⟨r₃', s₃', h₃⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 399,
"column": 29
} | {
"line": 399,
"column": 38
} | {
"line": 399,
"column": 38
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝³ : CommMonoid N\nP : Type u_3\ninst✝² : CommMonoid P\nf : M → N\nhf : S.IsLocalizationMap f\nE : Type u_4\ninst✝¹ : EquivLike E N P\ninst✝ : MulEquivClass E N P\ne✝ : E\ny : P\ne : N ≃* P := ↑e✝\nx : M × ↥S\neq : e.symm y * f ↑x.... | [] | simpa [e] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 399,
"column": 29
} | {
"line": 399,
"column": 38
} | {
"line": 399,
"column": 38
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝³ : CommMonoid N\nP : Type u_3\ninst✝² : CommMonoid P\nf : M → N\nhf : S.IsLocalizationMap f\nE : Type u_4\ninst✝¹ : EquivLike E N P\ninst✝ : MulEquivClass E N P\ne✝ : E\ny : P\ne : N ≃* P := ↑e✝\nx : M × ↥S\neq : e.symm y * f ↑x.... | [] | simpa [e] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 399,
"column": 29
} | {
"line": 399,
"column": 38
} | {
"line": 399,
"column": 38
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝³ : CommMonoid N\nP : Type u_3\ninst✝² : CommMonoid P\nf : M → N\nhf : S.IsLocalizationMap f\nE : Type u_4\ninst✝¹ : EquivLike E N P\ninst✝ : MulEquivClass E N P\ne✝ : E\ny : P\ne : N ≃* P := ↑e✝\nx : M × ↥S\neq : e.symm y * f ↑x.... | [] | simpa [e] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 78,
"column": 41
} | {
"line": 78,
"column": 51
} | {
"line": 78,
"column": 51
} | [
{
"pp": "case neg.coe.coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝¹ : α\nha : ¬↑a✝¹ = 0\na✝ : α\nhb : ¬↑a✝ = 0\n⊢ a✝¹ * a✝ = a✝¹ * untopD 0 ↑a✝",
"ppTerm": "?neg.coe.coe✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"case neg.coe.coe\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝¹ : α\nha : ¬↑a✝¹ = 0\na✝ : α\nhb : ¬↑a✝ = 0\n⊢ a✝¹ * a✝ = a✝¹ * a✝"
] | untopD_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 272,
"column": 17
} | {
"line": 272,
"column": 59
} | {
"line": 272,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\na₂ : WithTop α\nthis : MulPosStrictMono α\na₁ : α\nha : ↑a₁ < a₂\nb₁ : α\nha₂ : a₂ ≠ ⊤\nhb : ↑b₁ < ⊤\n⊢ a₂ ≠ 0",
"ppTerm": "?m.1... | [] | by simpa [bot_eq_zero] using ha.bot_lt.ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 336,
"column": 29
} | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 41
} | [
{
"pp": "case neg.bot\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\nb : WithBot α\nhb : ¬b = 0\nha : ¬⊥ = 0\n⊢ unbotD 0 ⊥ = unbotD 0 ⊥ * unbotD 0 b",
"ppTerm": "?neg.bot✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"HMul.hMul",
"MulZeroClass.t... | [
"case neg.bot\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\nb : WithBot α\nhb : ¬b = 0\nha : ¬⊥ = 0\n⊢ 0 = 0 * unbotD 0 b"
] | unbotD_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
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