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 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Finset.Prod | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 13
} | {
"line": 201,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Finset α\nt : Finset β\n⊢ s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"SProd.sprod",
"congrArg",
"Finset",
"id",
"Finset.... | [
"α : Type u_1\nβ : Type u_2\ns : Finset α\nt : Finset β\n⊢ (s ×ˢ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Data.Finset.Pi | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : α → Type u\ns : Finset α\nt : (a : α) → Finset (β a)\ninst✝ : DecidableEq α\n⊢ s.pi t = ∅ ↔ ∃ a ∈ s, t a = ∅",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
... | [
"α : Type u_1\nβ : α → Type u\ns : Finset α\nt : (a : α) → Finset (β a)\ninst✝ : DecidableEq α\n⊢ (s.pi t).Nonempty ↔ ∀ a ∈ s, (t a).Nonempty"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Data.Multiset.Pi | {
"line": 145,
"column": 29
} | {
"line": 145,
"column": 57
} | {
"line": 145,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u_2\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : s.Nodup → (∀ (a : α), a ∈ s → (t a).Nodup) → (s.pi t).Nodup\nhs : (a ::ₘ s).Nodup\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → (t a_1).Nodup\n⊢ ¬a ∈ s",
"ppTerm": "?m.34",
"as... | [
"α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u_2\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : s.Nodup → (∀ (a : α), a ∈ s → (t a).Nodup) → (s.pi t).Nodup\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → (t a_1).Nodup\nhs : ¬a ∈ s ∧ s.Nodup\n⊢ ¬a ∈ s"
] | simp only [nodup_cons] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Multiset.Pi | {
"line": 146,
"column": 30
} | {
"line": 146,
"column": 58
} | {
"line": 146,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u_2\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : s.Nodup → (∀ (a : α), a ∈ s → (t a).Nodup) → (s.pi t).Nodup\nhs : (a ::ₘ s).Nodup\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → (t a_1).Nodup\nhas : ¬a ∈ s\n⊢ s.Nodup",
"ppTerm": "... | [
"α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u_2\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : s.Nodup → (∀ (a : α), a ∈ s → (t a).Nodup) → (s.pi t).Nodup\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → (t a_1).Nodup\nhas : ¬a ∈ s\nhs : ¬a ∈ s ∧ s.Nodup\n⊢ s.Nodup"
] | simp only [nodup_cons] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Choose.Basic | {
"line": 318,
"column": 31
} | {
"line": 318,
"column": 53
} | {
"line": 318,
"column": 54
} | [
{
"pp": "r n : ℕ\nh : r < n / 2\n⊢ r < n - r",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.lt_sub_iff_add_lt",
"congrArg",
"HSub.hSub",
"id",
"instSubNat",
"instHAdd",
"instHSub",
"HAdd.hAdd",
"Nat",
"LT.lt"... | [
"r n : ℕ\nh : r < n / 2\n⊢ r + r < n"
] | Nat.lt_sub_iff_add_lt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Fintype.Pi | {
"line": 178,
"column": 50
} | {
"line": 178,
"column": 80
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : DecidableEq (ι → α)\ns : Finset α\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\n⊢ s.piDiag ι ⊆ Fintype.piFinset fun x ↦ s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Pr... | [] | simp [← piFinset_filter_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Fintype.Pi | {
"line": 178,
"column": 50
} | {
"line": 178,
"column": 80
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : DecidableEq (ι → α)\ns : Finset α\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\n⊢ s.piDiag ι ⊆ Fintype.piFinset fun x ↦ s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Pr... | [] | simp [← piFinset_filter_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Pi | {
"line": 178,
"column": 50
} | {
"line": 178,
"column": 80
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : DecidableEq (ι → α)\ns : Finset α\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\n⊢ s.piDiag ι ⊆ Fintype.piFinset fun x ↦ s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Pr... | [] | simp [← piFinset_filter_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Pi | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 52
} | {
"line": 152,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u_2\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : s.Nodup → (∀ (a : α), a ∈ s → (t a).Nodup) → (s.pi t).Nodup\nhs✝ : (a ::ₘ s).Nodup\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → (t a_1).Nodup\nhas : ¬a ∈ s\nhs : s.Nodup\n⊢ Pairwise (... | [
"α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u_2\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : s.Nodup → (∀ (a : α), a ∈ s → (t a).Nodup) → (s.pi t).Nodup\nhs✝ : (a ::ₘ s).Nodup\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → (t a_1).Nodup\nhas : ¬a ∈ s\nhs : s.Nodup\n⊢ ∀ (a_1 : β a),\n a_... | refine (ht a <| mem_cons_self _ _).pairwise ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Finite.Prod | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 34
} | {
"line": 229,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β → γ\ns : Set α\nt : Set β\nhfs : ∀ b ∈ t, InjOn (fun a ↦ f a b) s\nhft : ∀ a ∈ s, InjOn (f a) t\nh : (image2 f s t).Infinite\n⊢ (s ×ˢ t).Infinite",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Set.instS... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β → γ\ns : Set α\nt : Set β\nhfs : ∀ b ∈ t, InjOn (fun a ↦ f a b) s\nhft : ∀ a ∈ s, InjOn (f a) t\nh : (uncurry f '' s ×ˢ t).Infinite\n⊢ (s ×ˢ t).Infinite"
] | rw [← image_uncurry_prod] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Conj | {
"line": 63,
"column": 38
} | {
"line": 63,
"column": 52
} | {
"line": 63,
"column": 53
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nf : α →* β\na b : α\nc : αˣ\nhc : SemiconjBy (↑c) a b\n⊢ SemiconjBy (↑((Units.map f) c)) (f a) (f b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instFunLike",
... | [
"α : Type u\nβ : Type v\ninst✝¹ : Monoid α\ninst✝ : Monoid β\nf : α →* β\na b : α\nc : αˣ\nhc : SemiconjBy (↑c) a b\n⊢ SemiconjBy (f ↑c) (f a) (f b)"
] | Units.coe_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Lattice.Image | {
"line": 249,
"column": 2
} | {
"line": 253,
"column": 32
} | {
"line": 255,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : ι → Set α\nhs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\n⊢ InjOn f (⋃ i, s i)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.mem_iUnion",
"Membership.mem",
"Exists",
"LE.... | [] | intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Lattice.Image | {
"line": 249,
"column": 2
} | {
"line": 253,
"column": 32
} | {
"line": 255,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : ι → Set α\nhs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\n⊢ InjOn f (⋃ i, s i)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.mem_iUnion",
"Membership.mem",
"Exists",
"LE.... | [] | intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Lattice.Image | {
"line": 500,
"column": 2
} | {
"line": 502,
"column": 77
} | {
"line": 504,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : Set α\nt : ι → Set β\nhι : Nonempty ι\n⊢ s ×ˢ ⋂ i, t i = ⋂ i, s ×ˢ t i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Set.iInter",
... | [] | ext x
simp only [mem_prod, mem_iInter]
exact ⟨fun h i => ⟨h.1, h.2 i⟩, fun h => ⟨(h hι.some).1, fun i => (h i).2⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Lattice.Image | {
"line": 500,
"column": 2
} | {
"line": 502,
"column": 77
} | {
"line": 504,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : Set α\nt : ι → Set β\nhι : Nonempty ι\n⊢ s ×ˢ ⋂ i, t i = ⋂ i, s ×ˢ t i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Set.iInter",
... | [] | ext x
simp only [mem_prod, mem_iInter]
exact ⟨fun h i => ⟨h.1, h.2 i⟩, fun h => ⟨(h hι.some).1, fun i => (h i).2⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Lattice.Image | {
"line": 610,
"column": 2
} | {
"line": 610,
"column": 58
} | {
"line": 612,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\n⊢ s ×ˢ t = (Prod.mk '' s).seq t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"id",
"Prod.mk",
"Set.image2_image_left",
"Set.im... | [] | rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Lattice.Image | {
"line": 610,
"column": 2
} | {
"line": 610,
"column": 58
} | {
"line": 612,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\n⊢ s ×ˢ t = (Prod.mk '' s).seq t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"id",
"Prod.mk",
"Set.image2_image_left",
"Set.im... | [] | rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Lattice.Image | {
"line": 610,
"column": 2
} | {
"line": 610,
"column": 58
} | {
"line": 612,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\n⊢ s ×ˢ t = (Prod.mk '' s).seq t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"id",
"Prod.mk",
"Set.image2_image_left",
"Set.im... | [] | rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 847,
"column": 4
} | {
"line": 850,
"column": 10
} | {
"line": 851,
"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",
... | [] | contrapose! +distrib
rintro (hs | rfl)
· exact hs.zpow
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 847,
"column": 4
} | {
"line": 850,
"column": 10
} | {
"line": 851,
"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",
... | [] | contrapose! +distrib
rintro (hs | rfl)
· exact hs.zpow
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 149,
"column": 4
} | {
"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\n⊢ (∃ x, x ∈ H ∧ P x⁻¹) → ∃ x, x ∈ H ∧ P x",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"Group.toDivisionMonoid... | [] | · rintro ⟨x, x_in, hx⟩
exact ⟨x⁻¹, inv_mem x_in, by simp [hx]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 149,
"column": 4
} | {
"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\n⊢ (∃ x, x ∈ H ∧ P x) → ∃ x, x ∈ H ∧ P x⁻¹",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Gr... | [] | · rintro ⟨x, x_in, hx⟩
exact ⟨x⁻¹, inv_mem x_in, by simp [hx]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 44
} | {
"line": 204,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (∃ x ∈ H, x ≠ 1) ↔ ¬∀ x ∈ H, x = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"_private.Mathlib.Algebra.Group.Subgroup.Lattice.0.Subgroup... | [] | simp only [ne_eq, not_forall, exists_prop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 26
} | {
"line": 476,
"column": 2
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝ : Group G\nx y : G\nhy : y ∈ closure {x}\n⊢ ∃ n, x ^ n = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"zpow_zero",
"DivInvMonoid.toZPow",
"Group.toDivisi... | [] | exact ⟨0, zpow_zero x⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 26
} | {
"line": 476,
"column": 2
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝ : Group G\nx y : G\nhy : y ∈ closure {x}\n⊢ ∃ n, x ^ n = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"zpow_zero",
"DivInvMonoid.toZPow",
"Group.toDivisi... | [] | exact ⟨0, zpow_zero x⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 26
} | {
"line": 476,
"column": 2
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝ : Group G\nx y : G\nhy : y ∈ closure {x}\n⊢ ∃ n, x ^ n = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"zpow_zero",
"DivInvMonoid.toZPow",
"Group.toDivisi... | [] | exact ⟨0, zpow_zero x⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 28
} | {
"line": 190,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nK : Subgroup G\nN : Type u_5\ninst✝ : Group N\nH : Subgroup N\ne : G ≃* N\n⊢ map (↑e.symm) H = K ↔ map (↑e) K = H",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MulEquiv.instEquivLike",
"Subgroup.map",
"Monoid.toMulOneClass",
... | [
"case mp\nG : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nH : Subgroup N\ne : G ≃* N\n⊢ map (↑e) (map (↑e.symm) H) = H",
"case mpr\nG : Type u_1\ninst✝¹ : Group G\nK : Subgroup G\nN : Type u_5\ninst✝ : Group N\ne : G ≃* N\n⊢ map (↑e.symm) (map (↑e) K) = K"
] | constructor <;> rintro rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 1041,
"column": 15
} | {
"line": 1041,
"column": 31
} | {
"line": 1043,
"column": 0
} | [
{
"pp": "ι : Type u_4\nM : ι → Type u_5\ninst✝ : (i : ι) → MulOneClass (M i)\nI : Set ι\nx : (i : ι) → M i\n⊢ (x ∈ pi I fun i ↦ ⊤) ↔ x ∈ ⊤",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Submonoid.mem_top._simp_2",
"Submonoid.instTop",... | [] | by simp [mem_pi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 1045,
"column": 15
} | {
"line": 1045,
"column": 31
} | {
"line": 1047,
"column": 0
} | [
{
"pp": "ι : Type u_4\nM : ι → Type u_5\ninst✝ : (i : ι) → MulOneClass (M i)\nH : (i : ι) → Submonoid (M i)\nx : (i : ι) → M i\n⊢ x ∈ pi ∅ H ↔ x ∈ ⊤",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.m... | [] | by simp [mem_pi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.ZPowers.Basic | {
"line": 99,
"column": 98
} | {
"line": 101,
"column": 66
} | {
"line": 103,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx : G\n⊢ ⇑Additive.ofMul '' ↑(Subgroup.zpowers x) = ↑(AddSubgroup.zmultiples (Additive.ofMul x))",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Equiv.instEquivLike",
"Additive",
"Equiv",
"Subgroup",
"A... | [] | by
ext
exact Set.mem_image_iff_of_inverse (congrFun rfl) (congrFun rfl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 192,
"column": 15
} | {
"line": 192,
"column": 31
} | {
"line": 194,
"column": 0
} | [
{
"pp": "η : Type u_7\nf : η → Type u_8\ninst✝ : (i : η) → Group (f i)\nI : Set η\nx : (i : η) → f i\n⊢ (x ∈ pi I fun i ↦ ⊤) ↔ x ∈ ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"congrArg",
"_private.Mathlib.Algebra.Group.Subgroup.Basic.0.Subgroup.pi_top._simp_1_1",
"M... | [] | by simp [mem_pi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 196,
"column": 15
} | {
"line": 196,
"column": 31
} | {
"line": 198,
"column": 0
} | [
{
"pp": "η : Type u_7\nf : η → Type u_8\ninst✝ : (i : η) → Group (f i)\nH : (i : η) → Subgroup (f i)\nx : (i : η) → f i\n⊢ x ∈ pi ∅ H ↔ x ∈ ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Membership.mem",
... | [] | by simp [mem_pi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 69,
"column": 21
} | {
"line": 69,
"column": 34
} | {
"line": 69,
"column": 35
} | [
{
"pp": "α : Type u_2\ninst✝ : Mul α\ns t : Set α\n⊢ image2 (fun x1 x2 ↦ x1 • x2) (op '' s) t = t * s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"Mul.toSMulMulOpposite",
"congrArg",
"MulOpposite",
"id",
... | [
"α : Type u_2\ninst✝ : Mul α\ns t : Set α\n⊢ image2 (fun x1 x2 ↦ x1 • x2) (op '' s) t = image2 (fun x1 x2 ↦ x1 * x2) t s"
] | ← image2_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 87
} | {
"line": 719,
"column": 87
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\ns : Set G\nH : Subgroup G\na : G\nh : a ∈ H.normalCore\nb c : G\n⊢ c * (b * a * b⁻¹) * c⁻¹ ∈ H",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | rw [mul_assoc, mul_assoc, ← mul_inv_rev, ← mul_assoc, ← mul_assoc]; exact h (c * b) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 87
} | {
"line": 719,
"column": 87
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\ns : Set G\nH : Subgroup G\na : G\nh : a ∈ H.normalCore\nb c : G\n⊢ c * (b * a * b⁻¹) * c⁻¹ ∈ H",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | rw [mul_assoc, mul_assoc, ← mul_inv_rev, ← mul_assoc, ← mul_assoc]; exact h (c * b) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Group.List | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 19
} | {
"line": 161,
"column": 4
} | [
{
"pp": "case cons\nM : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Preorder M\ninst✝ : CanonicallyOrderedMul M\nx y : M\nys : List M\nih : x ∈ ys → x ≤ ys.prod\nh₁ : x ∈ y :: ys\n⊢ x ≤ (y :: ys).prod",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
... | [] | | cons y ys ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 1187,
"column": 40
} | {
"line": 1187,
"column": 71
} | {
"line": 1187,
"column": 71
} | [
{
"pp": "M : Type u_6\ninst✝² : AddGroup M\nI : AddSubgroup M\nG : Type u_7\ninst✝¹ : Group G\ninst✝ : MulAction G M\nH : Subgroup G\n⊢ Subgroup.map H.subtype ((I.inertia G).subgroupOf H) = I.inertia G ⊓ H",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.subg... | [
"M : Type u_6\ninst✝² : AddGroup M\nI : AddSubgroup M\nG : Type u_7\ninst✝¹ : Group G\ninst✝ : MulAction G M\nH : Subgroup G\n⊢ I.inertia G ⊓ H = I.inertia G ⊓ H"
] | Subgroup.subgroupOf_map_subtype | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 383,
"column": 56
} | {
"line": 383,
"column": 77
} | {
"line": 383,
"column": 77
} | [
{
"pp": "M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : Monoid M\na x : M\nn : ℕ\nhpos : 0 < n\nhx : x ^ n = 1\nm k : ℕ\n⊢ (x ^ (m.succ % n)) ^ k = x ^ (k * (↑(m % n)).toNat) * x ^ k",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"pow_eq_pow_mod",
"HMul.hMu... | [
"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : Monoid M\na x : M\nn : ℕ\nhpos : 0 < n\nhx : x ^ n = 1\nm k : ℕ\n⊢ (x ^ m.succ) ^ k = x ^ (k * (↑(m % n)).toNat) * x ^ k"
] | ← pow_eq_pow_mod _ hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 383,
"column": 56
} | {
"line": 383,
"column": 77
} | {
"line": 383,
"column": 77
} | [
{
"pp": "M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : Monoid M\na x : M\nn : ℕ\nhpos : 0 < n\nhx : x ^ n = 1\nm k : ℕ\n⊢ (x ^ m.succ) ^ k = (x ^ (m % n)) ^ k * x ^ k",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"pow_eq_pow_mod",
"HMul.hMul",
"Mono... | [
"M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝ : Monoid M\na x : M\nn : ℕ\nhpos : 0 < n\nhx : x ^ n = 1\nm k : ℕ\n⊢ (x ^ m.succ) ^ k = (x ^ m) ^ k * x ^ k"
] | ← pow_eq_pow_mod _ hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 555,
"column": 98
} | {
"line": 557,
"column": 66
} | {
"line": 559,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nx : M\n⊢ ⇑Additive.ofMul '' ↑(Submonoid.powers x) = ↑(AddSubmonoid.multiples (Additive.ofMul x))",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Equiv.instEquivLike",
"Monoid.toMulOneClass",
"Additive",
"Add... | [] | by
ext
exact Set.mem_image_iff_of_inverse (congrFun rfl) (congrFun rfl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Denumerable | {
"line": 197,
"column": 56
} | {
"line": 197,
"column": 83
} | {
"line": 197,
"column": 83
} | [
{
"pp": "s : Set ℕ\ninst✝ : Infinite ↑s\nx : ↑s\nh : ¬∃ n, ↑x + n + 1 ∈ s\nthis : ∀ a ∈ s, a < ↑x + 1\n⊢ ∀ a ∈ Multiset.filter (fun x ↦ x ∈ s) (Multiset.range (↑x).succ), a ∈ s",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Multiset.mem_filter._simp_1",
"congrArg",
"Cla... | [] | simp [-Multiset.range_succ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Denumerable | {
"line": 197,
"column": 56
} | {
"line": 197,
"column": 83
} | {
"line": 197,
"column": 83
} | [
{
"pp": "s : Set ℕ\ninst✝ : Infinite ↑s\nx : ↑s\nh : ¬∃ n, ↑x + n + 1 ∈ s\nthis : ∀ a ∈ s, a < ↑x + 1\n⊢ ∀ a ∈ Multiset.filter (fun x ↦ x ∈ s) (Multiset.range (↑x).succ), a ∈ s",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Multiset.mem_filter._simp_1",
"congrArg",
"Cla... | [] | simp [-Multiset.range_succ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Denumerable | {
"line": 197,
"column": 56
} | {
"line": 197,
"column": 83
} | {
"line": 197,
"column": 83
} | [
{
"pp": "s : Set ℕ\ninst✝ : Infinite ↑s\nx : ↑s\nh : ¬∃ n, ↑x + n + 1 ∈ s\nthis : ∀ a ∈ s, a < ↑x + 1\n⊢ ∀ a ∈ Multiset.filter (fun x ↦ x ∈ s) (Multiset.range (↑x).succ), a ∈ s",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Multiset.mem_filter._simp_1",
"congrArg",
"Cla... | [] | simp [-Multiset.range_succ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Hom.Lattice | {
"line": 143,
"column": 9
} | {
"line": 143,
"column": 23
} | {
"line": 143,
"column": 24
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : FunLike F α β\ninst✝² : SemilatticeInf α\ninst✝¹ : SemilatticeInf β\nf : F\ninst✝ : InfHomClass F α β\nhf : Injective ⇑f\nx y : α\nh : f x ≤ f y\n⊢ x ≤ y",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : FunLike F α β\ninst✝² : SemilatticeInf α\ninst✝¹ : SemilatticeInf β\nf : F\ninst✝ : InfHomClass F α β\nhf : Injective ⇑f\nx y : α\nh : f x ≤ f y\n⊢ x ⊓ y = x"
] | ← inf_eq_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.Powerset | {
"line": 172,
"column": 6
} | {
"line": 172,
"column": 34
} | {
"line": 174,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Multiset α\nle : s ≤ t\nl₁✝ l₂✝ : List α\nhsub : l₁✝ <+ l₂✝\n⊢ l₁✝.sublists' <+ l₂✝.sublists'",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"List.Sublist.sublists'"
],
"usedFVars": [
"α",
"l₁✝",
"l₂✝",
"hsub"
],
"... | [] | exact Sublist.sublists' hsub | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Multiset.Powerset | {
"line": 328,
"column": 6
} | {
"line": 329,
"column": 97
} | {
"line": 329,
"column": 97
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nl : List α\nh : Nodup ⟦l⟧\n⊢ ∀ (x : List α), x ∈ l.sublists' → ∀ (y : List α), y ∈ l.sublists' → ↑x = ↑y → x = y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"List.sublists'",
"Membership.mem",
"Multiset",
"List.Perm",
... | [] | exact fun x sx y sy e =>
(h.perm_iff_eq_of_sublist (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1 (Quotient.exact e) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 19
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\n⊢ (∅.sup fun x ↦ ⊥) = ⊥",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Finset.sup_empty",
"Bot.bot",
... | [] | exact sup_empty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 19
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\n⊢ (∅.sup fun x ↦ ⊥) = ⊥",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Finset.sup_empty",
"Bot.bot",
... | [] | exact sup_empty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 19
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\n⊢ (∅.sup fun x ↦ ⊥) = ⊥",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Finset.sup_empty",
"Bot.bot",
... | [] | exact sup_empty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Powerset | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 25
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Finset α\na : α\nht : t ∈ s.powerset\nh : a ∉ s\n⊢ a ∈ t → a ∈ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"LE.le",
"Finset.mem_powerset",
"F... | [] | apply mem_powerset.1 ht | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Finset.Powerset | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 13
} | {
"line": 267,
"column": 2
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : Finset α\n⊢ powersetCard n s = ∅ → #s < n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Ne",
"LE.le",
"Finset.in... | [
"α : Type u_1\nn : ℕ\ns : Finset α\n⊢ n ≤ #s → (powersetCard n s).Nonempty"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Data.Finset.Powerset | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 69
} | {
"line": 322,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq (Finset α)\ns : Finset α\n⊢ s.powerset = (range (#s + 1)).biUnion fun i ↦ powersetCard i s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.disjiUnion_eq_biUnion",
"congrArg",
"Finset",
"Disjoint",
"Eq.mp",
... | [] | simpa only [disjiUnion_eq_biUnion] using powerset_card_disjiUnion s | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Finset.Powerset | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 69
} | {
"line": 322,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq (Finset α)\ns : Finset α\n⊢ s.powerset = (range (#s + 1)).biUnion fun i ↦ powersetCard i s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.disjiUnion_eq_biUnion",
"congrArg",
"Finset",
"Disjoint",
"Eq.mp",
... | [] | simpa only [disjiUnion_eq_biUnion] using powerset_card_disjiUnion s | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Powerset | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 69
} | {
"line": 322,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq (Finset α)\ns : Finset α\n⊢ s.powerset = (range (#s + 1)).biUnion fun i ↦ powersetCard i s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.disjiUnion_eq_biUnion",
"congrArg",
"Finset",
"Disjoint",
"Eq.mp",
... | [] | simpa only [disjiUnion_eq_biUnion] using powerset_card_disjiUnion s | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Hom.WithTopBot | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 5
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : OrderTop α\ninst✝ : DecidablePred fun x ↦ x = ⊤\na : α\nh : a ≠ ⊤\n⊢ subtypeOrderIso.symm a = ↑⟨a, h⟩",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithTop.instPreorder",
"congrArg",
"WithTop.subt... | [] | rw [OrderIso.symm_apply_eq]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Hom.WithTopBot | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 5
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : OrderTop α\ninst✝ : DecidablePred fun x ↦ x = ⊤\na : α\nh : a ≠ ⊤\n⊢ subtypeOrderIso.symm a = ↑⟨a, h⟩",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithTop.instPreorder",
"congrArg",
"WithTop.subt... | [] | rw [OrderIso.symm_apply_eq]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteLattice.Finset | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 64
} | {
"line": 95,
"column": 0
} | [
{
"pp": "ι : Type u_5\ns : Set ι\nx : ι\n⊢ (∃ i, ∃ (x_1 : x ∈ s), ⟨x, ⋯⟩ ∈ i) ↔ x ∈ s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Iff.of_eq",
"Finset",
"Membership.mem",
"Exists",
"Set.Elem",
"Subtype",
"Finset.mem_singleton._... | [] | exact ⟨fun ⟨_, hx, _⟩ ↦ hx, fun hx ↦ ⟨{⟨x, hx⟩}, hx, by simp⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Cover | {
"line": 199,
"column": 11
} | {
"line": 199,
"column": 17
} | {
"line": 199,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝ : LT α\na b : α\nh : a < b\n⊢ ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CovBy",
"Exists",
"id",
"And",
"Iff",
"LT.lt",
"Not"
],
"usedFVars": [
"α",
"inst✝",
"... | [
"α : Type u_1\ninst✝ : LT α\na b : α\nh : a < b\n⊢ ¬(a < b ∧ ∀ ⦃c : α⦄, a < c → ¬c < b) ↔ ∃ c, a < c ∧ c < b"
] | CovBy, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.Sigma | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 13
} | {
"line": 67,
"column": 13
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ns : Finset ι\nt : (i : ι) → Finset (α i)\n⊢ s.sigma t = ∅ ↔ ∀ i ∈ s, t i = ∅",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"congrAr... | [
"ι : Type u_1\nα : ι → Type u_2\ns : Finset ι\nt : (i : ι) → Finset (α i)\n⊢ (s.sigma t).Nonempty ↔ ∃ i ∈ s, (t i).Nonempty"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.Minimal | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 79
} | {
"line": 380,
"column": 0
} | [
{
"pp": "α : Type u_2\nP : Set α → Prop\ns : Set α\nhP : ∀ ⦃s t : Set α⦄, P t → s ⊆ t → P s\n⊢ Maximal P s ↔ P s ∧ ∀ (x : α), P (insert x s) → x ∈ s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Insert.insert",
"LE.le",
"Maximal.mem_of_prop_inse... | [] | exact ⟨fun h ↦ ⟨h.1, fun x ↦ h.mem_of_prop_insert⟩,
fun h ↦ ⟨h.1, fun t ht hst x hxt ↦ h.2 x (hP ht <| insert_subset hxt hst)⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Lattice.Nat | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 34
} | {
"line": 75,
"column": 0
} | [
{
"pp": "ι : Sort u_1\ninst✝ : IsEmpty ι\nf : ι → ℕ\n⊢ iInf f = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
"id",
"instOfNatNat",
"Nat.instInfSet",
"Nat",
"Nat.sInf_empty",
"Set.instEmptyCollection",
... | [] | rw [iInf_of_isEmpty, sInf_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Lattice.Nat | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 34
} | {
"line": 75,
"column": 0
} | [
{
"pp": "ι : Sort u_1\ninst✝ : IsEmpty ι\nf : ι → ℕ\n⊢ iInf f = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
"id",
"instOfNatNat",
"Nat.instInfSet",
"Nat",
"Nat.sInf_empty",
"Set.instEmptyCollection",
... | [] | rw [iInf_of_isEmpty, sInf_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Lattice.Nat | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 34
} | {
"line": 75,
"column": 0
} | [
{
"pp": "ι : Sort u_1\ninst✝ : IsEmpty ι\nf : ι → ℕ\n⊢ iInf f = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
"id",
"instOfNatNat",
"Nat.instInfSet",
"Nat",
"Nat.sInf_empty",
"Set.instEmptyCollection",
... | [] | rw [iInf_of_isEmpty, sInf_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Lattice.Nat | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 44
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nu : ℕ → α\nn : ℕ\n⊢ ⨆ k, ⨆ (_ : k ≤ n + 1), u k = u 0 ⊔ ⨆ k, ⨆ (_ : k ≤ n), u (k + 1)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Nat.iSup_lt_succ'",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Iff.of_eq",
"con... | [] | simp_rw [← Nat.lt_succ_iff, iSup_lt_succ'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.Lattice.Nat | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 44
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nu : ℕ → α\nn : ℕ\n⊢ ⨆ k, ⨆ (_ : k ≤ n + 1), u k = u 0 ⊔ ⨆ k, ⨆ (_ : k ≤ n), u (k + 1)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Nat.iSup_lt_succ'",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Iff.of_eq",
"con... | [] | simp_rw [← Nat.lt_succ_iff, iSup_lt_succ'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Lattice.Nat | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 44
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nu : ℕ → α\nn : ℕ\n⊢ ⨆ k, ⨆ (_ : k ≤ n + 1), u k = u 0 ⊔ ⨆ k, ⨆ (_ : k ≤ n), u (k + 1)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Nat.iSup_lt_succ'",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Iff.of_eq",
"con... | [] | simp_rw [← Nat.lt_succ_iff, iSup_lt_succ'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Finset.Basic | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | {
"line": 339,
"column": 13
} | [
{
"pp": "α : Type u_2\na : α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderTop α\n⊢ (Ioi a).Nonempty ↔ ¬IsMax a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.Ioi",
"congrArg",
"Finset",
"Preorder.toLE",
"id",
"Finset.instEmptyC... | [
"α : Type u_2\na : α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderTop α\n⊢ Ioi a = ∅ ↔ IsMax a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.Interval.Finset.Basic | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 13
} | {
"line": 402,
"column": 13
} | [
{
"pp": "α : Type u_2\na : α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderBot α\n⊢ (Iio a).Nonempty ↔ ¬IsMin a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.Iio",
"Preorder.toLE",
"id",
"Finset.instEmptyC... | [
"α : Type u_2\na : α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderBot α\n⊢ Iio a = ∅ ↔ IsMin a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 16
} | {
"line": 98,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : Monoid M\ns : Set M\nm n : ℕ\nhmn : m ∣ n\n⊢ closure (s ^ n) ≤ closure (s ^ m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Submonoid.pow_subset._simp_2",
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [] | obtain ⟨k, rfl⟩ := hmn
simp [pow_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 16
} | {
"line": 98,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : Monoid M\ns : Set M\nm n : ℕ\nhmn : m ∣ n\n⊢ closure (s ^ n) ≤ closure (s ^ m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Submonoid.pow_subset._simp_2",
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [] | obtain ⟨k, rfl⟩ := hmn
simp [pow_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellFoundedSet | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 17
} | {
"line": 359,
"column": 2
} | [
{
"pp": "case refine_2.inl\nα : Type u_2\nr : α → α → Prop\ns : Set α\ninst✝¹ : Std.Refl r\ninst✝ : Std.Symm r\nhs : ∀ t ⊆ s, IsAntichain r t → t.Finite\nf : ℕ → α\nhf : ∀ (n : ℕ), f n ∈ s\nH : ∀ (m n : ℕ), m < n → ¬r (f m) (f n)\nm n : ℕ\nhmn : f m ≠ f n\nh : m < n\n⊢ rᶜ (f m) (f n)",
"ppTerm": "?refine_2.... | [
"case refine_2.inr\nα : Type u_2\nr : α → α → Prop\ns : Set α\ninst✝¹ : Std.Refl r\ninst✝ : Std.Symm r\nhs : ∀ t ⊆ s, IsAntichain r t → t.Finite\nf : ℕ → α\nhf : ∀ (n : ℕ), f n ∈ s\nH : ∀ (m n : ℕ), m < n → ¬r (f m) (f n)\nm n : ℕ\nhmn : f m ≠ f n\nh : n < m\n⊢ rᶜ (f m) (f n)"
] | · exact H _ _ h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Interval.Finset.Basic | {
"line": 743,
"column": 2
} | {
"line": 744,
"column": 82
} | {
"line": 746,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrderTop α\ninst✝ : DecidableEq α\na : α\n⊢ insert a (Ioi a) = Ici a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.Ioi_insert._simp_1_3",
"Eq.mpr",
... | [] | ext
simp_rw [Finset.mem_insert, mem_Ici, mem_Ioi, le_iff_lt_or_eq, or_comm, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Finset.Basic | {
"line": 743,
"column": 2
} | {
"line": 744,
"column": 82
} | {
"line": 746,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrderTop α\ninst✝ : DecidableEq α\na : α\n⊢ insert a (Ioi a) = Ici a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.Ioi_insert._simp_1_3",
"Eq.mpr",
... | [] | ext
simp_rw [Finset.mem_insert, mem_Ici, mem_Ioi, le_iff_lt_or_eq, or_comm, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Finset.Basic | {
"line": 750,
"column": 76
} | {
"line": 751,
"column": 43
} | {
"line": 753,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrderTop α\na : α\n⊢ Ici a = cons a (Ioi a) ⋯",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.notMem_Ioi_self",
"Eq.mpr",
"Finset.Ioi_insert",
"Finset.Ioi",
"Finset.cons",
"cong... | [] | by
classical rw [cons_eq_insert, Ioi_insert] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Basic | {
"line": 1091,
"column": 4
} | {
"line": 1091,
"column": 21
} | {
"line": 1092,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\nx y : α\nhxy : x ≤ y\nthis : #(Ico x y) < #(Icc x y)\nhxy' : ¬y ≤ x\nz : α\nhxz : x ≤ z\nhz : Maximal (fun x_1 ↦ x_1 ∈ Set.Ico x y) z\nz_card : #(Icc x z) < #(Icc x y)\nh₁ : TransGen (fun x1 x2 ↦ x1 ⩿ x2) x z\nh₂ : z ⩿ y\n⊢ Tran... | [] | exact .tail h₁ h₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Submonoid.Center | {
"line": 146,
"column": 27
} | {
"line": 146,
"column": 41
} | {
"line": 146,
"column": 42
} | [
{
"pp": "M : Type ?u.3\ninst✝ : Monoid M\nu : (↥(Submonoid.center M))ˣ\nr : Mˣ\n⊢ ↑r * ↑((Units.map (Submonoid.center M).subtype) u) = ↑((Units.map (Submonoid.center M).subtype) u * r)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instF... | [
"M : Type ?u.3\ninst✝ : Monoid M\nu : (↥(Submonoid.center M))ˣ\nr : Mˣ\n⊢ ↑r * (Submonoid.center M).subtype ↑u = ↑((Units.map (Submonoid.center M).subtype) u * r)"
] | Units.coe_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Submonoid.Center | {
"line": 146,
"column": 80
} | {
"line": 146,
"column": 94
} | {
"line": 147,
"column": 10
} | [
{
"pp": "M : Type ?u.3\ninst✝ : Monoid M\nu : (↥(Submonoid.center M))ˣ\nr : Mˣ\n⊢ ↑r * ↑↑u = ↑((Units.map (Submonoid.center M).subtype) u) * ↑r",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instFunLike",
"HMul.hMul",
"Monoid... | [
"M : Type ?u.3\ninst✝ : Monoid M\nu : (↥(Submonoid.center M))ˣ\nr : Mˣ\n⊢ ↑r * ↑↑u = (Submonoid.center M).subtype ↑u * ↑r"
] | Units.coe_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.WellFoundedSet | {
"line": 882,
"column": 4
} | {
"line": 887,
"column": 50
} | {
"line": 888,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\ns : Set (Lex (α × β))\nhα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x ↦ (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)\nhβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO\nf : ℕ → Lex (α × β)\nhf : ∀ (n : ℕ), f n ∈ s\ng : ℕ ↪o ℕ\nh... | [] | obtain ⟨n, hn⟩ := hc
use (g 0), (g n)
constructor
· by_contra hx
simp_all
· exact Prod.Lex.toLex_le_toLex.mpr <| .inl hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.WellFoundedSet | {
"line": 882,
"column": 4
} | {
"line": 887,
"column": 50
} | {
"line": 888,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\ns : Set (Lex (α × β))\nhα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x ↦ (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)\nhβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO\nf : ℕ → Lex (α × β)\nhf : ∀ (n : ℕ), f n ∈ s\ng : ℕ ↪o ℕ\nh... | [] | obtain ⟨n, hn⟩ := hc
use (g 0), (g n)
constructor
· by_contra hx
simp_all
· exact Prod.Lex.toLex_le_toLex.mpr <| .inl hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 237,
"column": 2
} | {
"line": 238,
"column": 16
} | {
"line": 240,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\ns : Set G\nm n : ℕ\nhmn : m ∣ n\n⊢ closure (s ^ n) ≤ closure (s ^ m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"InvOneClass.toOne",
"HMul.hMul",
"Subgroup.closure",
"DivInvOneMonoid.toInvOneClass",
... | [] | obtain ⟨k, rfl⟩ := hmn
simp [pow_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 237,
"column": 2
} | {
"line": 238,
"column": 16
} | {
"line": 240,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\ns : Set G\nm n : ℕ\nhmn : m ∣ n\n⊢ closure (s ^ n) ≤ closure (s ^ m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"InvOneClass.toOne",
"HMul.hMul",
"Subgroup.closure",
"DivInvOneMonoid.toInvOneClass",
... | [] | obtain ⟨k, rfl⟩ := hmn
simp [pow_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.NoncommProd | {
"line": 264,
"column": 8
} | {
"line": 267,
"column": 26
} | {
"line": 267,
"column": 27
} | [
{
"pp": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ x ∈ s₂, f x = g x\ncomm : (↑s₁).Pairwise (Commute on f)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ ... | [] | dsimp only [Function.onFun]
rw [← h₂ _ hx, ← h₂ _ hy]
subst h₁
exact comm hx hy h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NoncommProd | {
"line": 264,
"column": 8
} | {
"line": 267,
"column": 26
} | {
"line": 267,
"column": 27
} | [
{
"pp": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf✝ : α → β → β\nop : α → α → α\ninst✝¹ : Monoid β\ninst✝ : Monoid γ\ns₁ s₂ : Finset α\nf g : α → β\nh₁ : s₁ = s₂\nh₂ : ∀ x ∈ s₂, f x = g x\ncomm : (↑s₁).Pairwise (Commute on f)\nx : α\nhx : x ∈ ↑s₂\ny : α\nhy : y ∈ ↑s₂\nh : x ≠ y\n⊢ ... | [] | dsimp only [Function.onFun]
rw [← h₂ _ hx, ← h₂ _ hy]
subst h₁
exact comm hx hy h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 643,
"column": 42
} | {
"line": 643,
"column": 61
} | {
"line": 644,
"column": 6
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nc : Con M\nf : M → M\nx✝ y : M\nhf : ∀ (x : M), ↑(f x) * ↑x = 1\nh : ↑x✝ = ↑y\nx : M\n⊢ ↑(f (f x)) * ↑(f x) * (↑x * ↑(f x)) = ↑(f (f x)) * (↑(f x) * ↑x) * ↑(f x)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hM... | [] | simp_rw [mul_assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 643,
"column": 42
} | {
"line": 643,
"column": 61
} | {
"line": 644,
"column": 6
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nc : Con M\nf : M → M\nx✝ y : M\nhf : ∀ (x : M), ↑(f x) * ↑x = 1\nh : ↑x✝ = ↑y\nx : M\n⊢ ↑(f (f x)) * ↑(f x) * (↑x * ↑(f x)) = ↑(f (f x)) * (↑(f x) * ↑x) * ↑(f x)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hM... | [] | simp_rw [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 643,
"column": 42
} | {
"line": 643,
"column": 61
} | {
"line": 644,
"column": 6
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nc : Con M\nf : M → M\nx✝ y : M\nhf : ∀ (x : M), ↑(f x) * ↑x = 1\nh : ↑x✝ = ↑y\nx : M\n⊢ ↑(f (f x)) * ↑(f x) * (↑x * ↑(f x)) = ↑(f (f x)) * (↑(f x) * ↑x) * ↑(f x)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hM... | [] | simp_rw [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 140,
"column": 60
} | {
"line": 141,
"column": 83
} | {
"line": 143,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\nh : Disjoint s₁ s₂\n⊢ ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul"... | [] | by
rw [← prod_union_inter, disjoint_iff_inter_eq_empty.mp h]; exact (mul_one _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coset.Defs | {
"line": 241,
"column": 57
} | {
"line": 241,
"column": 70
} | {
"line": 241,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\n⊢ ⋃ x, ⋃ (hx : x ∈ N), (fun x_1 ↦ x_1 * ↑⟨x, hx⟩) '' s = s * ↑N",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Membership.mem",
... | [
"α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\n⊢ ⋃ x, ⋃ (hx : x ∈ N), (fun x_1 ↦ x_1 * ↑⟨x, hx⟩) '' s = image2 (fun x1 x2 ↦ x1 * x2) s ↑N"
] | ← image2_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 963,
"column": 4
} | {
"line": 967,
"column": 46
} | {
"line": 968,
"column": 2
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : FreeGroup α\n⊢ (fun x ↦ of default ^ x) ((fun x ↦ ((map 1) x).sum) x) = x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"FreeGroup.of",
"NegZeroClass.toNeg",
... | [] | induction x with
| C1 => simp
| of x => simp [Unique.default_eq x]
| inv_of x hx => simp [Unique.default_eq x]
| mul x y hx hy => simp [zpow_add, hx, hy] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 963,
"column": 4
} | {
"line": 967,
"column": 46
} | {
"line": 968,
"column": 2
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : FreeGroup α\n⊢ (fun x ↦ of default ^ x) ((fun x ↦ ((map 1) x).sum) x) = x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"FreeGroup.of",
"NegZeroClass.toNeg",
... | [] | induction x with
| C1 => simp
| of x => simp [Unique.default_eq x]
| inv_of x hx => simp [Unique.default_eq x]
| mul x y hx hy => simp [zpow_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 963,
"column": 4
} | {
"line": 967,
"column": 46
} | {
"line": 968,
"column": 2
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : FreeGroup α\n⊢ (fun x ↦ of default ^ x) ((fun x ↦ ((map 1) x).sum) x) = x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"FreeGroup.of",
"NegZeroClass.toNeg",
... | [] | induction x with
| C1 => simp
| of x => simp [Unique.default_eq x]
| inv_of x hx => simp [Unique.default_eq x]
| mul x y hx hy => simp [zpow_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 504,
"column": 48
} | {
"line": 504,
"column": 74
} | {
"line": 505,
"column": 4
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nt : Finset κ\nf : ι → M\ng : κ → M\ni : (a : ι) → a ∈ s → f a ≠ 1 → κ\nhi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),... | [] | by rw [prod_filter_ne_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Finiteness | {
"line": 236,
"column": 4
} | {
"line": 237,
"column": 30
} | {
"line": 239,
"column": 0
} | [
{
"pp": "case h\nM : Type u_1\ninst✝² : Monoid M\nM' : Type u_3\ninst✝¹ : Monoid M'\ninst✝ : FG M\nf : M →* M'\nhf : Function.Surjective ⇑f\ns : Finset M\nhs : Submonoid.closure ↑s = ⊤\n⊢ Submonoid.closure ↑(Finset.image (⇑f) s) = ⊤",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | rwa [Finset.coe_image, ← MonoidHom.map_mclosure, hs, ← MonoidHom.mrange_eq_map,
MonoidHom.mrange_eq_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 229,
"column": 4
} | {
"line": 232,
"column": 12
} | {
"line": 234,
"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\n⊢ ∀ {a b : { N // N.Normal }},\n { toFun := fun N ↦ QuotientGroup.con ↑N, invFun := fun c ↦ ⟨c.subgroup, ⋯⟩, left_inv := ⋯, right_inv := ⋯ } a ≤\n { toF... | [] | simp only [QuotientGroup.con, Equiv.coe_fn_mk, Con.le_def, Con.rel_mk, leftRel_apply]
refine ⟨fun h x _ ↦ ?_, fun hle _ _ h ↦ hle h⟩
specialize @h 1 x
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 229,
"column": 4
} | {
"line": 232,
"column": 12
} | {
"line": 234,
"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\n⊢ ∀ {a b : { N // N.Normal }},\n { toFun := fun N ↦ QuotientGroup.con ↑N, invFun := fun c ↦ ⟨c.subgroup, ⋯⟩, left_inv := ⋯, right_inv := ⋯ } a ≤\n { toF... | [] | simp only [QuotientGroup.con, Equiv.coe_fn_mk, Con.le_def, Con.rel_mk, leftRel_apply]
refine ⟨fun h x _ ↦ ?_, fun hle _ _ h ↦ hle h⟩
specialize @h 1 x
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 43
} | {
"line": 140,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ (lift (⇑g ∘ f)) a = g ((lift f) a)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrA... | [
"α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ (lift (⇑g ∘ f)) a = (g.comp (lift f)) a"
] | rw [← AddMonoidHom.comp_apply g (lift f)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Commutator.Basic | {
"line": 172,
"column": 2
} | {
"line": 173,
"column": 20
} | {
"line": 175,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH₁ H₂ H₃ : Subgroup G\nh1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1\nh2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1\nx : G\nhx : x ∈ H₁\ny : G\nhy : y ∈ H₂\nz : G\nhz : z ∈ ↑H₃\n⊢ x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹ = 1"... | [] | · rw [h1 _ (H₂.inv_mem hy) _ hz _ (H₁.inv_mem hx), h2 _ (H₃.inv_mem hz) _ (H₁.inv_mem hx) _ hy]
simp [mul_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Commutator.Basic | {
"line": 501,
"column": 4
} | {
"line": 501,
"column": 58
} | {
"line": 502,
"column": 4
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nhcomm : IsMulCommutative (G ⧸ N)\np q : G\n⊢ ↑(p * q * p⁻¹ * q⁻¹) = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toInv",
"InvOneClass.toOne",
"HMul.hMul",
... | [
"case refine_1\nG : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nhcomm : IsMulCommutative (G ⧸ N)\np q : G\n⊢ ↑p * ↑q * (↑p)⁻¹ * (↑q)⁻¹ = 1"
] | simp only [QuotientGroup.mk_mul, QuotientGroup.mk_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Commutator.Basic | {
"line": 518,
"column": 2
} | {
"line": 518,
"column": 49
} | {
"line": 520,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nH : Subgroup G\nhHN : N ⊔ H = ⊤\nhH : IsMulCommutative ↥H\n⊢ _root_.commutator G ≤ N",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Monoid.toMulOneClass",
"Subgroup.Normal.quotient_commutative_iff_comm... | [
"G : Type u_1\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nH : Subgroup G\nhHN : N ⊔ H = ⊤\nhH : IsMulCommutative ↥H\n⊢ IsMulCommutative (G ⧸ N)"
] | apply quotient_commutative_iff_commutator_le.mp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 203,
"column": 24
} | {
"line": 203,
"column": 33
} | {
"line": 203,
"column": 33
} | [
{
"pp": "case c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.10\ninst✝ : MulAction R X\nC : Sort u_2\nP : X → ↥S → X → ↥S → C\nhP :\n ∀ (r₁ : X) (t₁ : R) (s₁ : ↥S) (ht₁ : t₁ * ↑s₁ ∈ S) (r₂ : X) (t₂ : R) (s₂ : ↥S) (ht₂ : t₂ * ↑s₂ ∈ S),\n P r₁ s₁ r₂ s₂ = P (t₁ • r₁) ⟨t₁ * ... | [
"case c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.10\ninst✝ : MulAction R X\nC : Sort u_2\nP : X → ↥S → X → ↥S → C\nhP :\n ∀ (r₁ : X) (t₁ : R) (s₁ : ↥S) (ht₁ : t₁ * ↑s₁ ∈ S) (r₂ : X) (t₂ : R) (s₂ : ↥S) (ht₂ : t₂ * ↑s₂ ∈ S),\n P r₁ s₁ r₂ s₂ = P (t₁ • r₁) ⟨t₁ * ↑s₁, ht₁⟩ (t... | | _ r₂ s₂
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
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