module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous | {
"line": 280,
"column": 4
} | {
"line": 281,
"column": 64
} | {
"line": 282,
"column": 4
} | [
{
"pp": "C✝ : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D✝\nD' : Type u₃\ninst✝⁶ : Category.{v₃, u₃} D'\nD'' : Type u₄\ninst✝⁵ : Category.{v₄, u₄} D''\nJ✝ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nF✝ : C✝ ⥤ D✝\nS✝ : Sheaf J✝ RingCat\nR✝ : Sheaf K✝ RingCat\ni... | [
"C✝ : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D✝\nD' : Type u₃\ninst✝⁶ : Category.{v₃, u₃} D'\nD'' : Type u₄\ninst✝⁵ : Category.{v₄, u₄} D''\nJ✝ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nF✝ : C✝ ⥤ D✝\nS✝ : Sheaf J✝ RingCat\nR✝ : Sheaf K✝ RingCat\ninst✝⁴ : F✝.I... | change (X.val.presheaf.map (G.map (adj.counit.app U.unop)).op ≫
X.val.presheaf.map (adj.unit.app (G.obj U.unop)).op) _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Topology.Sheaves.Presheaf | {
"line": 310,
"column": 8
} | {
"line": 313,
"column": 70
} | {
"line": 314,
"column": 8
} | [
{
"pp": "case g\nC✝ : Type u\ninst✝⁴ : Category.{v, u} C✝\nX✝ : TopCat\nC : Type u\ninst✝³ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nℱ : Presheaf C Y\nU : Ope... | [
"case w\nC✝ : Type u\ninst✝⁴ : Category.{v, u} C✝\nX✝ : TopCat\nC : Type u\ninst✝³ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nℱ : Presheaf C Y\nU : Opens ↑X\nH : I... | · change op (unop _) ⟶ op (⟨_, H⟩ : Opens _)
refine (homOfLE ?_).op
apply (Set.image_mono s.pt.hom.unop.le).trans
exact Set.image_preimage.l_u_le (SetLike.coe s.pt.left.unop) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sheaves.SheafCondition.Sites | {
"line": 150,
"column": 2
} | {
"line": 155,
"column": 16
} | {
"line": 157,
"column": 0
} | [
{
"pp": "X Y : TopCat\nf : X ⟶ Y\nhf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\n⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) hf.functor",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Topology.IsOpenEmbedding.toIsEmbedding",
"CategoryTheory.Mono",
... | [] | haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.injective
apply compatiblePreservingOfDownwardsClosed
intro U V i
refine ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ ?_⟩
obtain ⟨_, _, rfl⟩ := i.le h
exact ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.Sites | {
"line": 150,
"column": 2
} | {
"line": 155,
"column": 16
} | {
"line": 157,
"column": 0
} | [
{
"pp": "X Y : TopCat\nf : X ⟶ Y\nhf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\n⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) hf.functor",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Topology.IsOpenEmbedding.toIsEmbedding",
"CategoryTheory.Mono",
... | [] | haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.injective
apply compatiblePreservingOfDownwardsClosed
intro U V i
refine ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ ?_⟩
obtain ⟨_, _, rfl⟩ := i.le h
exact ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 52
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case e'_2.pair\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type x\nU : ι → Opens ↑X\ns : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j\nh :\n ∃! s_1,\n ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ),\n (ConcreteCategory.hom ((Functor.mapCone F (Pairwise.cocone U).op).π.app i... | [] | exact (hs <| op <| Pairwise.Hom.left i j).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 52
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case e'_2.pair\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type x\nU : ι → Opens ↑X\ns : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j\nh :\n ∃! s_1,\n ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ),\n (ConcreteCategory.hom ((Functor.mapCone F (Pairwise.cocone U).op).π.app i... | [] | exact (hs <| op <| Pairwise.Hom.left i j).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 52
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case e'_2.pair\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type x\nU : ι → Opens ↑X\ns : (j : (CategoryTheory.Pairwise ι)ᵒᵖ) → ((Pairwise.diagram U).op ⋙ F).obj j\nh :\n ∃! s_1,\n ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ),\n (ConcreteCategory.hom ((Functor.mapCone F (Pairwise.cocone U).op).π.app i... | [] | exact (hs <| op <| Pairwise.Hom.left i j).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 363,
"column": 2
} | {
"line": 366,
"column": 26
} | {
"line": 366,
"column": 27
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V\nhι : U ⊔ V = iSup ι\ni j : CategoryTheory.Pai... | [
"case refine_2.single.left.single.left.id_single\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V\nhι : U ⊔ V = iSup ι\n⊢ ... | rcases i with (⟨⟨_ | _⟩⟩ | ⟨⟨_ | _⟩, ⟨_⟩⟩) <;>
rcases j with (⟨⟨_ | _⟩⟩ | ⟨⟨_ | _⟩, ⟨_⟩⟩) <;>
rcases g with ⟨⟩ <;>
dsimp [Pairwise.diagram] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Presentable.Finite | {
"line": 42,
"column": 2
} | {
"line": 43,
"column": 50
} | {
"line": 43,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\n⊢ F.IsFinitelyAccessible ↔ PreservesFilteredColimitsOfSize.{w, w, v, v', u, u'} F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor.IsFinitelyAc... | [
"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nx✝ : F.IsFinitelyAccessible\nH : ∀ (J : Type w) [inst : SmallCategory J] [IsFiltered J], PreservesColimitsOfShape J F\n⊢ ∀ (J : Type w) [inst : Category.{w, w} J] [IsFiltered J], PreservesColimitsOfShape J F... | refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>
simp only [isCardinalFiltered_aleph0_iff] at * | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Presentable.Finite | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 50
} | {
"line": 110,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ HasCardinalFilteredColimits C ℵ₀ ↔ HasFilteredColimitsOfSize.{w, w, v, u} C",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsCardinalFiltered",
"CategoryTheory.Limits.HasColimitsOfShape",
"E... | [
"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nx✝ : HasCardinalFilteredColimits C ℵ₀\nH : ∀ (J : Type w) [inst : SmallCategory J] [IsFiltered J], HasColimitsOfShape J C\n⊢ ∀ (I : Type w) [inst : Category.{w, w} I] [IsFiltered I], HasColimitsOfShape I C",
"case refine_2\nC : Type u\ninst✝ : Category.{v, u}... | refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>
simp only [isCardinalFiltered_aleph0_iff] at * | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Presentable.Basic | {
"line": 161,
"column": 2
} | {
"line": 162,
"column": 16
} | {
"line": 164,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nX✝ Y : C\ne : X✝ ≅ Y\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nX : (isCardinalPresentable C κ).FullSubcategory\n⊢ IsCardinalPresentable ((isCardinalPresentable C κ).ι.obj X) κ",
"ppTerm": "?m.18",
"assigned... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Basic | {
"line": 161,
"column": 2
} | {
"line": 162,
"column": 16
} | {
"line": 164,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nX✝ Y : C\ne : X✝ ≅ Y\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nX : (isCardinalPresentable C κ).FullSubcategory\n⊢ IsCardinalPresentable ((isCardinalPresentable C κ).ι.obj X) κ",
"ppTerm": "?m.18",
"assigned... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 138,
"column": 17
} | {
"line": 138,
"column": 55
} | {
"line": 138,
"column": 55
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nι : Type v'\nj : J\nk : ι → J\nf : (i : ι) → j ⟶ k i\nhι : HasCardinalLT ι κ\nφ : ι → (j ⟶ max k hι) := fun i ↦ f i ≫ toMax k hι i\n⊢ ∀ (i : ι), f i ≫ (fun i ↦ toMax k hι i ≫ coeqHom φ hι) i... | [] | by simpa [φ] using coeq_condition φ hι | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Ring.FinitePresentation | {
"line": 116,
"column": 4
} | {
"line": 118,
"column": 37
} | {
"line": 119,
"column": 4
} | [
{
"pp": "J : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nF : J ⥤ CommRingCat\nα : (Functor.const J).obj R ⟶ F\nS : CommRingCat\nc : Cocone F\nhc : IsColimit c\ninst✝ : PreservesColimit F (forget CommRingCat)\ng : S ⟶ c.pt\nhc' : IsColimit ((forget CommRingCat).mapCocone c)\nn ... | [
"J : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nF : J ⥤ CommRingCat\nα : (Functor.const J).obj R ⟶ F\nS : CommRingCat\nc : Cocone F\nhc : IsColimit c\ninst✝ : PreservesColimit F (forget CommRingCat)\ng : S ⟶ c.pt\nhc' : IsColimit ((forget CommRingCat).mapCocone c)\nn : ℕ\nP : Com... | suffices H : (g' ≫ c.ι.app i) r = 0 by
obtain ⟨k, f, g, e⟩ := this.mp (by simpa using! H)
exact ⟨k, f, by simpa using! e⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.CharP.CharAndCard | {
"line": 45,
"column": 13
} | {
"line": 45,
"column": 22
} | {
"line": 45,
"column": 23
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhR : ringChar R ≠ 0\nhch : ↑(ringChar R) = 0\nhp : Nat.Prime p\nh : ¬p ∣ ringChar R\na b : ℤ\nhab : ↑a * ↑p + ↑b * 0 = 1\n⊢ IsUnit ↑p",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Int.cast",
... | [
"case mpr\nR : Type u_1\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhR : ringChar R ≠ 0\nhch : ↑(ringChar R) = 0\nhp : Nat.Prime p\nh : ¬p ∣ ringChar R\na b : ℤ\nhab : ↑a * ↑p + 0 = 1\n⊢ IsUnit ↑p"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.CharAndCard | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 15
} | {
"line": 71,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Fintype R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nh : p ∣ Fintype.card R\nh₀ : ¬p ∣ ringChar R\nr : R\nhr : addOrderOf r = p\nu : R\nhu : u * ↑p = 1\nhr₁ : u * (↑p * r) = u * 0\n⊢ False",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Fintype R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nh : p ∣ Fintype.card R\nh₀ : ¬p ∣ ringChar R\nr : R\nhr : addOrderOf r = p\nu : R\nhu : u * ↑p = 1\nhr₁ : u * (↑p * r) = 0\n⊢ False"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Seq.Defs | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 20
} | {
"line": 119,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\na : α\ns : Seq α\n⊢ (some a :: ↑s).IsSeq",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Option.some",
"instOfNatNat",
"Nat.casesAuxOn",
"Stream'",
"Option.none",
"instHAdd",
"Stream'.IsSeq",
"HAdd... | [
"case zero\nα : Type u\nβ : Type v\nγ : Type w\na : α\ns : Seq α\nh : (some a :: ↑s) 0 = none\n⊢ (some a :: ↑s) (0 + 1) = none",
"case succ\nα : Type u\nβ : Type v\nγ : Type w\na : α\ns : Seq α\nn✝ : ℕ\nh : (some a :: ↑s) (n✝ + 1) = none\n⊢ (some a :: ↑s) (n✝ + 1 + 1) = none"
] | rintro (n | _) h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.Stream.Init | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 50
} | {
"line": 431,
"column": 50
} | [
{
"pp": "α : Type u\nn : ℕ\ns : Stream' α\n⊢ s.even.get n.succ = s.get (2 * n).succ.succ",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.get_succ",
"HMul.hMul",
"congrArg",
"Stream'.even",
"id",
"instMulNat",
"instOfNatNat"... | [
"α : Type u\nn : ℕ\ns : Stream' α\n⊢ s.tail.tail.get (2 * n) = s.tail.get (2 * n + 1)"
] | rw [get_succ, get_succ, tail_even, get_even n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.ContinuedFractions.Determinant | {
"line": 56,
"column": 86
} | {
"line": 60,
"column": 10
} | {
"line": 61,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\ng : GenContFract K\nn✝ n : ℕ\nhyp : n + 1 = 0 ∨ ¬g.TerminatedAt (n + 1 - 1)\nconts : Pair K := g.contsAux (n + 2)\npred_conts : Pair K := g.contsAux (n + 1)\npred_conts_eq : pred_conts = g.contsAux (n + 1)\nppred_conts : Pair K := g.contsAux n\nIH :\n n = 0 ∨ ¬g.Terminat... | [] | by
rw [Finset.prod_range_succ, ← this, partNum_eq_s_a s_nth_eq, Option.getD_some]
subst conts
rw [contsAux_recurrence s_nth_eq ppred_conts_eq pred_conts_eq]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Basic | {
"line": 454,
"column": 6
} | {
"line": 456,
"column": 31
} | {
"line": 457,
"column": 4
} | [
{
"pp": "α : Type u\na : α\ns✝ : Seq α\nS : Seq (Seq1 α)\ns1 s2 : Seq α\nh : s1 = s2 ∨ ∃ a s S, s1 = (cons (a, s) S).join ∧ s2 = cons a (s.append S.join)\ns : Seq α\n⊢ BisimO (fun s1 s2 ↦ s1 = s2 ∨ ∃ a s S, s1 = (cons (a, s) S).join ∧ s2 = cons a (s.append S.join)) s.destruct\n s.destruct",
"ppTerm": "?m... | [] | cases s; · trivial
· rw [destruct_cons]
exact ⟨rfl, Or.inl rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 454,
"column": 6
} | {
"line": 456,
"column": 31
} | {
"line": 457,
"column": 4
} | [
{
"pp": "α : Type u\na : α\ns✝ : Seq α\nS : Seq (Seq1 α)\ns1 s2 : Seq α\nh : s1 = s2 ∨ ∃ a s S, s1 = (cons (a, s) S).join ∧ s2 = cons a (s.append S.join)\ns : Seq α\n⊢ BisimO (fun s1 s2 ↦ s1 = s2 ∨ ∃ a s S, s1 = (cons (a, s) S).join ∧ s2 = cons a (s.append S.join)) s.destruct\n s.destruct",
"ppTerm": "?m... | [] | cases s; · trivial
· rw [destruct_cons]
exact ⟨rfl, Or.inl rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 526,
"column": 17
} | {
"line": 526,
"column": 40
} | {
"line": 528,
"column": 0
} | [
{
"pp": "case succ\nα : Type u\nm : ℕ\nih : nil.drop m = nil\n⊢ nil.drop (m + 1) = nil",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_nil._simp_1_4",
"instOfNat... | [] | simp [← dropn_tail, ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Seq.Basic | {
"line": 526,
"column": 17
} | {
"line": 526,
"column": 40
} | {
"line": 528,
"column": 0
} | [
{
"pp": "case succ\nα : Type u\nm : ℕ\nih : nil.drop m = nil\n⊢ nil.drop (m + 1) = nil",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_nil._simp_1_4",
"instOfNat... | [] | simp [← dropn_tail, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 526,
"column": 17
} | {
"line": 526,
"column": 40
} | {
"line": 528,
"column": 0
} | [
{
"pp": "case succ\nα : Type u\nm : ℕ\nih : nil.drop m = nil\n⊢ nil.drop (m + 1) = nil",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_nil._simp_1_4",
"instOfNat... | [] | simp [← dropn_tail, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Colimit.Ring | {
"line": 172,
"column": 14
} | {
"line": 172,
"column": 65
} | {
"line": 174,
"column": 0
} | [
{
"pp": "case ih\nι : Type u_1\ninst✝⁴ : Preorder ι\nG : ι → Type u_2\ninst✝³ : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nP : Type u_3\ninst✝² : CommRing P\ng✝ : (i : ι) → G i →+* P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) (f i j hij x) = (g✝ i) x\ninst✝¹ : Nonempty ι\ninst✝ : IsDire... | [] | rw [lift_of] at hz; rw [injective _ g hz, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Colimit.Ring | {
"line": 172,
"column": 14
} | {
"line": 172,
"column": 65
} | {
"line": 174,
"column": 0
} | [
{
"pp": "case ih\nι : Type u_1\ninst✝⁴ : Preorder ι\nG : ι → Type u_2\ninst✝³ : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\nP : Type u_3\ninst✝² : CommRing P\ng✝ : (i : ι) → G i →+* P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) (f i j hij x) = (g✝ i) x\ninst✝¹ : Nonempty ι\ninst✝ : IsDire... | [] | rw [lift_of] at hz; rw [injective _ g hz, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 725,
"column": 2
} | {
"line": 725,
"column": 51
} | {
"line": 727,
"column": 0
} | [
{
"pp": "α : Type u\nx : α\ns : Seq α\nm n : ℕ\nh : m < n\ni : ℕ\n⊢ ((s.set m x).drop n).get? i = (s.drop n).get? i",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_set... | [] | simp [get?_set_of_ne _ _ (show n + i ≠ m by lia)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 323,
"column": 29
} | {
"line": 323,
"column": 84
} | {
"line": 323,
"column": 84
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\nv : K\nn : ℕ\ninst✝ : IsStrictOrderedRing K\nh : fract v ≠ 0\n⊢ { a := 1, b := ↑⌊(fract v)⁻¹⌋ }.b + convs'Aux (of v).s.tail n = (of (fract v)⁻¹).convs' n",
"ppTerm": "?m.189",
"assigned": true,
"usedConstan... | [] | by rw [convs', of_h_eq_floor, add_right_inj, of_s_tail] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 143,
"column": 6
} | {
"line": 146,
"column": 27
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case some.inr\nK : Type u_1\nn : ℕ\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : s.get? (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : s.get? (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\n⊢ (squashSeq s (n + 1)).tail.get? m = (s... | [] | cases s_succ_mth_eq : s.get? (m + 1)
· simp only [*, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith,
Option.map₂_none_right]
· simp [*, squashSeq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 143,
"column": 6
} | {
"line": 146,
"column": 27
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case some.inr\nK : Type u_1\nn : ℕ\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : s.get? (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : s.get? (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\n⊢ (squashSeq s (n + 1)).tail.get? m = (s... | [] | cases s_succ_mth_eq : s.get? (m + 1)
· simp only [*, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith,
Option.map₂_none_right]
· simp [*, squashSeq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.CubicDiscriminant | {
"line": 302,
"column": 32
} | {
"line": 302,
"column": 43
} | {
"line": 302,
"column": 43
} | [
{
"pp": "R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\nhc : P.c = 0\nhd : P.d = 0\n⊢ degree 0 = ⊥",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
"P... | [
"R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\nhc : P.c = 0\nhd : P.d = 0\n⊢ ⊥ = ⊥"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.Prod | {
"line": 46,
"column": 3
} | {
"line": 48,
"column": 24
} | {
"line": 48,
"column": 24
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nhM : Module.Finite R M\nhN : Module.Finite R N\n⊢ ⊤.FG",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [← Submodule.prod_top]
exact hM.1.prod hN.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Idempotents | {
"line": 231,
"column": 34
} | {
"line": 231,
"column": 43
} | {
"line": 231,
"column": 44
} | [
{
"pp": "case inr.refine_2.succ\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [
"case inr.refine_2.succ\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a * e₂ = 0\nh... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Idempotents | {
"line": 308,
"column": 4
} | {
"line": 309,
"column": 44
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn : ℕ\ne : Fin n → S\nhe : CompleteOrthogonalIdempotents e\nhe' : ∀ (i : Fin n), e i ∈ f.range\nh✝ : Subsingleton R\n⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e",
"... | [] | choose e' he' using he'
exact ⟨e', .of_subsingleton, funext he'⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Idempotents | {
"line": 308,
"column": 4
} | {
"line": 309,
"column": 44
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn : ℕ\ne : Fin n → S\nhe : CompleteOrthogonalIdempotents e\nhe' : ∀ (i : Fin n), e i ∈ f.range\nh✝ : Subsingleton R\n⊢ ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e",
"... | [] | choose e' he' using he'
exact ⟨e', .of_subsingleton, funext he'⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.GeneralLinearGroup.AlgEquiv | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 64
} | {
"line": 44,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : Semifield K\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module K V\ninst✝³ : Projective K V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module K W\ninst✝ : Projective K W\nf : End K V ≃ₐ[K] End K W\nhV : Nontrivial V\nu : V\nhu : u ≠ 0\nv : Dual K V\nhuv : v... | [
"case neg\nK : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : Semifield K\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : Module K V\ninst✝³ : Projective K V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module K W\ninst✝ : Projective K W\nf : End K V ≃ₐ[K] End K W\nhV : Nontrivial V\nu : V\nhu : u ≠ 0\nv : Dual K V\nhuv : v u ≠ 0\nz : ... | obtain ⟨z, hz⟩ : ∃ z : W, ¬ f (smulRight v u) z = (0 : End K W) z := by
rw [← not_forall, ← LinearMap.ext_iff, EmbeddingLike.map_eq_zero_iff, LinearMap.ext_iff]
exact not_forall.mpr ⟨u, huv.isUnit.smul_eq_zero.not.mpr hu⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Idempotents | {
"line": 529,
"column": 73
} | {
"line": 529,
"column": 81
} | {
"line": 529,
"column": 82
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : NonUnitalSemiring R\na b : R\n⊢ e * (a + b) * e = e * a * e + e * b * e",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"AddMonoid.toA... | [
"R : Type u_1\ne : R\ninst✝ : NonUnitalSemiring R\na b : R\n⊢ (e * a + e * b) * e = e * a * e + e * b * e"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Idempotents | {
"line": 530,
"column": 31
} | {
"line": 530,
"column": 40
} | {
"line": 530,
"column": 41
} | [
{
"pp": "R : Type u_1\ne : R\ninst✝ : NonUnitalSemiring R\n⊢ e * 0 * e = 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocSemiring.toMulZeroClass",
... | [
"R : Type u_1\ne : R\ninst✝ : NonUnitalSemiring R\n⊢ 0 * e = 0"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 74,
"column": 68
} | {
"line": 74,
"column": 96
} | {
"line": 75,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : ι → Submodule R M\ninst✝³ : DecidableEq ι\ninst✝² : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝¹ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\ninst✝ : Fintype ι\nf : M →ₗ[R] M\nhf : ∀ (i : ι), ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : ι → Submodule R M\ninst✝³ : DecidableEq ι\ninst✝² : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝¹ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\ninst✝ : Fintype ι\nf : M →ₗ[R] M\nhf : ∀ (i : ι), MapsTo ⇑f ↑(... | Matrix.trace_blockDiagonal', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 23
} | {
"line": 272,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nx y : FreeGroup α\n⊢ (x * y).toWord <+ x.toWord ++ y.toWord",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"FreeGroup.toWord",
"FreeGroup.Red.sublist",
"instHAppendOfAppend",
"List",
"Bool",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\nx y : FreeGroup α\n⊢ Red (x.toWord ++ y.toWord) (x * y).toWord"
] | refine Red.sublist ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.GCDMonoid.FinsetLemmas | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 83
} | {
"line": 40,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\nh : (↑s).Pairwise (Function.onFun IsRelPrime f)\n⊢ s.lcm f = s.prod f",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"instNormalizedGCDMonoidNat",
"Nat.unique_units",
"Nat.instIsCanc... | [] | exact associated_lcm_prod h |>.eq_of_normalized (normalize_eq _) (normalize_eq _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 43
} | {
"line": 134,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP : map (al... | [] | simp_all [natDegree_pos_iff_degree_pos] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 43
} | {
"line": 134,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP : map (al... | [] | simp_all [natDegree_pos_iff_degree_pos] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 43
} | {
"line": 134,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP : map (al... | [] | simp_all [natDegree_pos_iff_degree_pos] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 155,
"column": 6
} | {
"line": 159,
"column": 51
} | {
"line": 160,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP : map (al... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_prim : f.IsPrimitive\nhfd0 : 0 < f.natDegree\nhfP : (algebraMap R K) f.leadingCoeff ≠ 0\nhfmodP : map (algebraMap R K... | have h (x : ℕ × ℕ) : (Ideal.Quotient.mk (P ^ 2)) (r.coeff x.1 * s.coeff x.2) = 0 := by
rw [eq_zero_iff_mem, pow_two]
apply mul_mem_mul
· rw [mem_ker, ← coeff_map, hr, coeff_zero]
· rw [mem_ker, ← coeff_map, hs, coeff_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Eisenstein.Basic | {
"line": 159,
"column": 36
} | {
"line": 159,
"column": 48
} | {
"line": 159,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhf : f.IsWeaklyEisensteinAt 𝓟\nx : S\nhx : eval x (Polynomial.map (algebraMap R S) f) = 0\nhmo : f.Monic\n⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟",... | [
"R : Type u\ninst✝² : CommRing R\n𝓟 : Ideal R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhf : f.IsWeaklyEisensteinAt 𝓟\nx : S\nhx : (Polynomial.map (algebraMap R S) f).IsRoot x\nhmo : f.Monic\n⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟"
] | ← IsRoot.def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Quaternion | {
"line": 1097,
"column": 46
} | {
"line": 1097,
"column": 54
} | {
"line": 1097,
"column": 55
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ (a * (star a + star b) + b * (star a + star b)).re =\n (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)",
"ppTerm": "?m.178",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
... | [
"R : Type u_3\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ (a * star a + a * star b + (b * star a + b * star b)).re =\n (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Quaternion | {
"line": 1150,
"column": 25
} | {
"line": 1150,
"column": 35
} | {
"line": 1150,
"column": 36
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ a ^ 2 = -(star a * a) ↔ star a = -a",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"Rin... | [
"case inr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ a ^ 2 = star a * -a ↔ star a = -a"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Action.Option | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 54
} | {
"line": 65,
"column": 54
} | [
{
"pp": "case none\nM : Type u_1\nN : Type u_2\nα : Type u_3\ninst✝³ : SMul M α\ninst✝² : SMul N α\na✝ : M\nb : α\nx : Option α\ninst✝¹ : SMul Mᵐᵒᵖ α\ninst✝ : IsCentralScalar M α\na : M\n⊢ MulOpposite.op a • none = a • none",
"ppTerm": "?none",
"assigned": true,
"usedConstants": [
"instHSMul",... | [] | exacts [rfl, congr_arg some (op_smul_eq_smul _ _)] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Algebra.Group.NatPowAssoc | {
"line": 79,
"column": 27
} | {
"line": 79,
"column": 36
} | {
"line": 79,
"column": 37
} | [
{
"pp": "case zero\nM : Type u_1\ninst✝² : MulOneClass M\ninst✝¹ : Pow M ℕ\ninst✝ : NatPowAssoc M\nx : M\nm : ℕ\n⊢ x ^ (m * 0) = 1",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClass",
"HMul.hMul",
"MulZeroClass.toMu... | [
"case zero\nM : Type u_1\ninst✝² : MulOneClass M\ninst✝¹ : Pow M ℕ\ninst✝ : NatPowAssoc M\nx : M\nm : ℕ\n⊢ x ^ 0 = 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.NatPowAssoc | {
"line": 80,
"column": 21
} | {
"line": 80,
"column": 29
} | {
"line": 80,
"column": 30
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝² : MulOneClass M\ninst✝¹ : Pow M ℕ\ninst✝ : NatPowAssoc M\nx : M\nm n : ℕ\nih : x ^ (m * n) = (x ^ m) ^ n\n⊢ x ^ (m * (n + 1)) = (x ^ m) ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
... | [
"case succ\nM : Type u_1\ninst✝² : MulOneClass M\ninst✝¹ : Pow M ℕ\ninst✝ : NatPowAssoc M\nx : M\nm n : ℕ\nih : x ^ (m * n) = (x ^ m) ^ n\n⊢ x ^ (m * n + m * 1) = (x ^ m) ^ (n + 1)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 74,
"column": 41
} | {
"line": 74,
"column": 49
} | {
"line": 74,
"column": 50
} | [
{
"pp": "case succ\nM : Type u_1\ninst✝² : Mul M\ninst✝¹ : Pow M ℕ+\ninst✝ : PNatPowAssoc M\nx : M\nm k : ℕ+\nhk : x ^ (m * k) = (x ^ m) ^ k\n⊢ x ^ (m * (k + 1)) = (x ^ m) ^ k * x ^ m",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"H... | [
"case succ\nM : Type u_1\ninst✝² : Mul M\ninst✝¹ : Pow M ℕ+\ninst✝ : PNatPowAssoc M\nx : M\nm k : ℕ+\nhk : x ^ (m * k) = (x ^ m) ^ k\n⊢ x ^ (m * k + m * 1) = (x ^ m) ^ k * x ^ m"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Action.Center | {
"line": 27,
"column": 14
} | {
"line": 27,
"column": 23
} | {
"line": 27,
"column": 24
} | [
{
"pp": "case inl\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\nu : ↥(center G₀ˣ)\n⊢ 0 * ↑↑u = ↑↑u * 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOne... | [
"case inl\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\nu : ↥(center G₀ˣ)\n⊢ 0 * ↑↑u = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 59
} | {
"line": 111,
"column": 2
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nh... | [
"ι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝⁴ : Monoid M\ninst✝³ : LinearOrder S\ninst✝² : Finite ι\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\nt : Set ι := {i | IsMulIndecomposable v {j | 1 < f (v j)} i}\ns : Set ι := {j | 1 < f (v j) ∧ v j ∉ closure (v '' t)}\nhne : s.Nonem... | have hk' : v k ∈ closure (v '' t) := hi₂ k hk <| by aesop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.MooreComplex | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 41
} | {
"line": 103,
"column": 42
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX : SimplicialObject C\n⊢ (Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).factorThru\n ((Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).arrow ≫ X.δ 0) ⋯ ≫\n (Finset.univ.inf fun k ↦ kernelSubobject (X.... | [
"case zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX : SimplicialObject C\n⊢ ((Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).arrow ≫ X.δ 0) ≫ X.δ 0 ≫ inv ⊤.arrow = 0"
] | Subobject.factorThru_arrow_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 163,
"column": 17
} | {
"line": 163,
"column": 49
} | {
"line": 163,
"column": 49
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\n⊢ P.p ≫ 0 ≫ Q.p = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.Limits.comp_zero",
... | [] | simp only [comp_zero, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 163,
"column": 17
} | {
"line": 163,
"column": 49
} | {
"line": 163,
"column": 49
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\n⊢ P.p ≫ 0 ≫ Q.p = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.Limits.comp_zero",
... | [] | simp only [comp_zero, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 163,
"column": 17
} | {
"line": 163,
"column": 49
} | {
"line": 163,
"column": 49
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\n⊢ P.p ≫ 0 ≫ Q.p = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.Limits.comp_zero",
... | [] | simp only [comp_zero, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.FunctorCategories | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 18
} | {
"line": 86,
"column": 0
} | [
{
"pp": "case h.right\nJ : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nP Q : Karoubi (J ⥤ C)\nf : P ⟶ Q\nX : J\ninst✝ : IsIdempotentComplete C\nF : J ⥤ C\np : F ⟶ F\nhp : p ≫ p = p\nhC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nthis : ∀ (j : J), HasEq... | [] | simp [Y, i, e] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 340,
"column": 6
} | {
"line": 342,
"column": 96
} | {
"line": 343,
"column": 6
} | [
{
"pp": "case inl.inl\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : j.castSucc < i\nk : Fin (⦋n + 1⦌.len + 1)\nhik : k ≤ i\nhjk : k ≤ j.castSucc\n⊢ j.castSucc.predAbove k.castSucc = i.succAbove (j.predAbove k)",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Fin.s... | [
"case inl.inl.h\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : j.castSucc < i\nk : Fin (⦋n + 1⦌.len + 1)\nhik : k ≤ i\nhjk : k ≤ j.castSucc\n⊢ (k.castPred ⋯).castSucc < i"
] | rw [Fin.predAbove_of_le_castSucc _ _
(Fin.castSucc_le_castSucc_iff.mpr hjk), Fin.castPred_castSucc,
Fin.predAbove_of_le_castSucc _ _ hjk, Fin.succAbove_of_castSucc_lt, Fin.castSucc_castPred] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 510,
"column": 8
} | {
"line": 510,
"column": 94
} | {
"line": 510,
"column": 95
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ X : SimplicialObject C\nX₀ : C\nf : X _⦋0⦌ ⟶ X₀\nw : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), X.map g₁.op ≫ f = X.map g₂.op ≫ f\ni j : SimplexCategoryᵒᵖ\ng : i ⟶ j\n⊢ X.map g.unop.op ≫ X.map (⦋0⦌.const (unop j) 0).op ≫ f = (X.map (⦋0⦌.const (unop i) 0).op ≫ f... | [] | simpa only [← X.map_comp, ← Category.assoc, Category.comp_id, ← op_comp] using w _ _ _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 5
} | {
"line": 59,
"column": 0
} | [
{
"pp": "i n m : ℕ\nh : n + (i + 1) = m\n⊢ δ₀Iter (i + 1) h = δ 0 ≫ δ₀Iter i ⋯",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"SimplexCategory.coe_δ",
"_private.Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter.0.SimplexCategory.δ₀I... | [] | refine ConcreteCategory.hom_ext _ _ (fun k ↦ ?_)
ext
rw [dsimp% ConcreteCategory.comp_apply (δ 0) (δ₀Iter i), coe_δ,
δ₀Iter_apply .., δ₀Iter_apply ..]
dsimp
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 5
} | {
"line": 59,
"column": 0
} | [
{
"pp": "i n m : ℕ\nh : n + (i + 1) = m\n⊢ δ₀Iter (i + 1) h = δ 0 ≫ δ₀Iter i ⋯",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"SimplexCategory.coe_δ",
"_private.Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter.0.SimplexCategory.δ₀I... | [] | refine ConcreteCategory.hom_ext _ _ (fun k ↦ ?_)
ext
rw [dsimp% ConcreteCategory.comp_apply (δ 0) (δ₀Iter i), coe_δ,
δ₀Iter_apply .., δ₀Iter_apply ..]
dsimp
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 754,
"column": 10
} | {
"line": 754,
"column": 13
} | {
"line": 754,
"column": 13
} | [
{
"pp": "case pos\nn : ℕ\nΔ' : SimplexCategory\nθ : ⦋n + 1⦌ ⟶ Δ'\ni : Fin (n + 1)\nhi : (Hom.toOrderHom θ) i.castSucc = (Hom.toOrderHom θ) i.succ\nx : Fin (⦋n + 1⦌.len + 1)\nh'✝ : i.castSucc < x\ny : Fin ⦋n + 1⦌.len := x.pred ⋯\nh' : i.castSucc < y.succ\nhy : x = y.succ\nh'' : y = i\n⊢ (Hom.toOrderHom θ) y.succ... | [
"case pos\nn : ℕ\nΔ' : SimplexCategory\nθ : ⦋n + 1⦌ ⟶ Δ'\ni : Fin (n + 1)\nhi : (Hom.toOrderHom θ) i.castSucc = (Hom.toOrderHom θ) i.succ\nx : Fin (⦋n + 1⦌.len + 1)\nh'✝ : i.castSucc < x\ny : Fin ⦋n + 1⦌.len := x.pred ⋯\nh' : i.castSucc < y.succ\nhy : x = y.succ\nh'' : y = i\n⊢ (Hom.toOrderHom θ) i.succ = (Hom.toOr... | h'' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 214,
"column": 53
} | {
"line": 224,
"column": 41
} | {
"line": 226,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 2)\nj m : ℕ\ni' : Fin (m + 2)\nh : m + j = n\nhi' : j < ↑i\nhi'' : ↑i = ↑i' + j\n⊢ δ i ≫ σ₀Iter j ⋯ = σ₀Iter j ⋯ ≫ δ i'",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"SimplexCategory.σ₀Iter_succ._proof_2",
"Eq.mpr",
"CategoryTheory.Category.... | [] | by
induction j generalizing n m with
| zero =>
obtain rfl : n = m := by lia
obtain rfl : i = i' := by lia
simp
| succ j hj =>
rw [σ₀Iter_succ _, σ₀Iter_succ_assoc _,
reassoc_of% hj _ i'.succ (by lia) (by lia) (by grind),
dsimp% δ_comp_σ_of_gt (i := i') (j := 0)
(by rw [Fin.lt_d... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex | {
"line": 32,
"column": 32
} | {
"line": 40,
"column": 20
} | {
"line": 42,
"column": 0
} | [
{
"pp": "X✝ Y✝ : SSet\nf : X✝ ⟶ Y✝\nx✝¹ : Mono f\nx✝ : Epi f\n⊢ IsIso f",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfSize",
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Eq.mpr",
"CategoryTheory.Functor",
"C... | [] | by
rw [NatTrans.isIso_iff_isIso_app]
intro
rw [isIso_iff_bijective]
constructor
· rw [← mono_iff_injective]
infer_instance
· rw [← epi_iff_surjective]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 103,
"column": 29
} | {
"line": 103,
"column": 43
} | {
"line": 104,
"column": 4
} | [
{
"pp": "X : SSet\nd : ℕ\nx : X _⦋d⦌\nhx : x ∈ X.nonDegenerate d\ny : X _⦋d⦌\nhy : y ∈ X.nonDegenerate d\nh : mk ↑⟨x, hx⟩ ⋯ < mk ↑⟨y, hy⟩ ⋯\nw✝ : Mono (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌)\nhf : (ConcreteCategory.hom (X.map (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌).op)) (mk ↑⟨y, hy⟩ ⋯).simplex = (mk ↑⟨x, hx⟩ ⋯).simplex\n⊢ y = x",
"p... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 103,
"column": 29
} | {
"line": 103,
"column": 43
} | {
"line": 104,
"column": 4
} | [
{
"pp": "X : SSet\nd : ℕ\nx : X _⦋d⦌\nhx : x ∈ X.nonDegenerate d\ny : X _⦋d⦌\nhy : y ∈ X.nonDegenerate d\nh : mk ↑⟨x, hx⟩ ⋯ < mk ↑⟨y, hy⟩ ⋯\nw✝ : Mono (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌)\nhf : (ConcreteCategory.hom (X.map (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌).op)) (mk ↑⟨y, hy⟩ ⋯).simplex = (mk ↑⟨x, hx⟩ ⋯).simplex\n⊢ y = x",
"p... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 103,
"column": 29
} | {
"line": 103,
"column": 43
} | {
"line": 104,
"column": 4
} | [
{
"pp": "X : SSet\nd : ℕ\nx : X _⦋d⦌\nhx : x ∈ X.nonDegenerate d\ny : X _⦋d⦌\nhy : y ∈ X.nonDegenerate d\nh : mk ↑⟨x, hx⟩ ⋯ < mk ↑⟨y, hy⟩ ⋯\nw✝ : Mono (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌)\nhf : (ConcreteCategory.hom (X.map (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌).op)) (mk ↑⟨y, hy⟩ ⋯).simplex = (mk ↑⟨x, hx⟩ ⋯).simplex\n⊢ y = x",
"p... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Finite | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 39
} | {
"line": 107,
"column": 4
} | [
{
"pp": "X Y : SSet\ninst✝ : X.Finite\nf : X ⟶ Y\nhf : Epi f\nd : ℕ\nh✝ : X.HasDimensionLT d\nthis : Y.HasDimensionLT d\ni : ℕ\nhi : i < d\n⊢ Finite (Y _⦋i⦌)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfSize",
"CategoryTheory.Functor... | [
"X Y : SSet\ninst✝ : X.Finite\nf : X ⟶ Y\nhf : ∀ (k : SimplexCategoryᵒᵖ), Epi (f.app k)\nd : ℕ\nh✝ : X.HasDimensionLT d\nthis : Y.HasDimensionLT d\ni : ℕ\nhi : i < d\n⊢ Finite (Y _⦋i⦌)"
] | rw [NatTrans.epi_iff_epi_app] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 26
} | {
"line": 115,
"column": 4
} | [
{
"pp": "X : SSet\nn₁ : ℕ\nx₁ : X _⦋n₁⦌\nhx₁ : x₁ ∈ X.nonDegenerate n₁\nx₂ : X _⦋n₁⦌\nhx₂ : x₂ ∈ X.nonDegenerate n₁\nh : ∃ f, ∃ (_ : Mono f), (ConcreteCategory.hom (X.map f.op)) (mk ↑⟨x₂, hx₂⟩ ⋯).simplex = (mk ↑⟨x₁, hx₁⟩ ⋯).simplex\nh' : mk ↑⟨x₂, hx₂⟩ ⋯ ≤ mk ↑⟨x₁, hx₁⟩ ⋯\n⊢ mk ↑⟨x₁, hx₁⟩ ⋯ = mk ↑⟨x₂, hx₂⟩ ⋯",
... | [
"X : SSet\nn₁ : ℕ\nx₁ : X _⦋n₁⦌\nhx₁ : x₁ ∈ X.nonDegenerate n₁\nx₂ : X _⦋n₁⦌\nhx₂ : x₂ ∈ X.nonDegenerate n₁\nh' : mk ↑⟨x₂, hx₂⟩ ⋯ ≤ mk ↑⟨x₁, hx₁⟩ ⋯\nf : ⦋(mk ↑⟨x₁, hx₁⟩ ⋯).dim⦌ ⟶ ⦋(mk ↑⟨x₂, hx₂⟩ ⋯).dim⦌\nhf : Mono f\nh : (ConcreteCategory.hom (X.map f.op)) (mk ↑⟨x₂, hx₂⟩ ⋯).simplex = (mk ↑⟨x₁, hx₁⟩ ⋯).simplex\n⊢ mk... | obtain ⟨f, hf, h⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.ExtraDegeneracy | {
"line": 207,
"column": 40
} | {
"line": 207,
"column": 70
} | {
"line": 207,
"column": 70
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : Augmented C\ned : X.ExtraDegeneracy\nn : ℕ\ni : Fin (n + 1)\nk : Fin (n + 1 + 1)\nhk : i.succ.rev = k\n⊢ X.left.δ₀Iter ↑i ⋯ ≫ X.left.δ 0 ≫ ed.s ↑k ≫ X.left.σ₀Iter (↑i + 1) ⋯ ≫ X.left.δ i.castSucc.succ =\n X.left.δ₀Iter ↑i ⋯ ≫ X.left.δ 0 ≫ ed.s ↑k ≫ X.... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : Augmented C\ned : X.ExtraDegeneracy\nn : ℕ\ni : Fin (n + 1)\nk : Fin (n + 1 + 1)\nhk : i.succ.rev = k\n⊢ X.left.δ₀Iter ↑i ⋯ ≫ X.left.δ 0 ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑i ⋯ =\n X.left.δ₀Iter ↑i ⋯ ≫ X.left.δ 0 ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑i ⋯"
] | X.left.σ₀Iter_δ _ _ (by grind) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ComplexShapeSigns | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 27
} | {
"line": 190,
"column": 0
} | [
{
"pp": "I₁ : Type u_1\nI₂ : Type u_2\nI₃ : Type u_3\nI₁₂ : Type u_4\nI₂₃ : Type u_5\nJ : Type u_6\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₃ : ComplexShape I₃\nc₁₂ : ComplexShape I₁₂\nc₂₃ : ComplexShape I₂₃\nc✝ : ComplexShape J\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\nI : Type u_7\ninst✝¹ : AddMonoid I\nc :... | [] | rw [Int.negOnePow_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ComplexShapeSigns | {
"line": 357,
"column": 38
} | {
"line": 357,
"column": 46
} | {
"line": 357,
"column": 47
} | [
{
"pp": "I₁ : Type u_1\nI₂ : Type u_2\nI₃ : Type u_3\nI₁₂ : Type u_4\nI₂₃ : Type u_5\nJ : Type u_6\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₃ : ComplexShape I₃\nc₁₂ : ComplexShape I₁₂\nc₂₃ : ComplexShape I₂₃\nc : ComplexShape J\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : TotalComplexShape c₂ c₁ c₁₂\nin... | [
"I₁ : Type u_1\nI₂ : Type u_2\nI₃ : Type u_3\nI₁₂ : Type u_4\nI₂₃ : Type u_5\nJ : Type u_6\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₃ : ComplexShape I₃\nc₁₂ : ComplexShape I₁₂\nc₂₃ : ComplexShape I₂₃\nc : ComplexShape J\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : TotalComplexShape c₂ c₁ c₁₂\ninst✝ : TotalC... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ExtraDegeneracy | {
"line": 419,
"column": 51
} | {
"line": 419,
"column": 65
} | {
"line": 419,
"column": 65
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : Augmented C\ned : X.ExtraDegeneracy\n⊢ point.obj X ⟶ (AlternatingFaceMapComplex.obj (drop.obj X)).X 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryTheory.Simplicial... | [] | by exact ed.s' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 39
} | {
"line": 705,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ objEquiv.symm (𝟙 (unop (op ⦋n⦌))) ∈ Δ[n].nonDegenerate n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SSet.stdSimplex.mem_nonDegenerate_iff_strictMono",
"Eq.mpr",
"Opposite",
"Equiv.instEquivLike",
"StrictMono",
"CategoryTheory.Cat... | [
"n : ℕ\n⊢ StrictMono ⇑(objEquiv.symm (𝟙 (unop (op ⦋n⦌))))"
] | rw [mem_nonDegenerate_iff_strictMono] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 712,
"column": 4
} | {
"line": 714,
"column": 84
} | {
"line": 715,
"column": 2
} | [
{
"pp": "case refine_1\nn : ℕ\nx : Δ[n] _⦋n⦌\nh : x ∈ Δ[n].nonDegenerate n\n⊢ x = objEquiv.symm (𝟙 ⦋n⦌)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.apply_symm_apply",
"Opposite",
"Equiv.instEquivLike",
"CategoryTheory.Mono",
"Ca... | [] | obtain ⟨f, rfl⟩ := objEquiv.symm.surjective x
have : Mono f := by simpa using (mem_nonDegenerate_iff_mono _).mp h
simpa only [EmbeddingLike.apply_eq_iff_eq] using SimplexCategory.eq_id_of_mono f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 712,
"column": 4
} | {
"line": 714,
"column": 84
} | {
"line": 715,
"column": 2
} | [
{
"pp": "case refine_1\nn : ℕ\nx : Δ[n] _⦋n⦌\nh : x ∈ Δ[n].nonDegenerate n\n⊢ x = objEquiv.symm (𝟙 ⦋n⦌)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.apply_symm_apply",
"Opposite",
"Equiv.instEquivLike",
"CategoryTheory.Mono",
"Ca... | [] | obtain ⟨f, rfl⟩ := objEquiv.symm.surjective x
have : Mono f := by simpa using (mem_nonDegenerate_iff_mono _).mp h
simpa only [EmbeddingLike.apply_eq_iff_eq] using SimplexCategory.eq_id_of_mono f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 297,
"column": 19
} | {
"line": 297,
"column": 28
} | {
"line": 297,
"column": 28
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} C₃\ninst✝⁴ : Category.{v_4, u_4} C₄\ninst✝³ : Category.{v_5, u_5} C₁₂\ninst✝² : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ ... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} C₃\ninst✝⁴ : Category.{v_4, u_4} C₄\ninst✝³ : Category.{v_5, u_5} C₁₂\ninst✝² : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ C₂ ⥤ C₁₂\nG ... | ← ρ₁₂.hpq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 305,
"column": 28
} | {
"line": 305,
"column": 37
} | {
"line": 305,
"column": 37
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} C₃\ninst✝⁴ : Category.{v_4, u_4} C₄\ninst✝³ : Category.{v_5, u_5} C₁₂\ninst✝² : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ ... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} C₃\ninst✝⁴ : Category.{v_4, u_4} C₄\ninst✝³ : Category.{v_5, u_5} C₁₂\ninst✝² : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ C₂ ⥤ C₁₂\nG ... | ← ρ₁₂.hpq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.TotalComplexSymmetry | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 46
} | {
"line": 104,
"column": 6
} | [
{
"pp": "case pos\nC : Type u_1\nI₁ : Type u_2\nI₂ : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc : ComplexShape J\ninst✝⁴ : TotalComplexShape c₁ c₂ c\ninst✝³ : TotalComplexShape c₂ c₁ c\ninst✝² : T... | [
"case pos\nC : Type u_1\nI₁ : Type u_2\nI₂ : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc : ComplexShape J\ninst✝⁴ : TotalComplexShape c₁ c₂ c\ninst✝³ : TotalComplexShape c₂ c₁ c\ninst✝² : TotalComplexS... | have h₄ : ComplexShape.π c₁ c₂ c (i₁, ComplexShape.next c₂ i₂) = j' := by
rw [← h₃, ComplexShape.π_symm c₁ c₂ c] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.TotalComplexSymmetry | {
"line": 158,
"column": 33
} | {
"line": 158,
"column": 52
} | {
"line": 158,
"column": 52
} | [
{
"pp": "C : Type u_1\nI₁ : Type u_2\nI₂ : Type u_3\nJ : Type u_4\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc : ComplexShape J\ninst✝⁶ : TotalComplexShape c₁ c₂ c\ninst✝⁵ : TotalComplexShape c₂ c₁ c\ninst✝⁴ : TotalComple... | [
"C : Type u_1\nI₁ : Type u_2\nI₂ : Type u_3\nJ : Type u_4\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc : ComplexShape J\ninst✝⁶ : TotalComplexShape c₁ c₂ c\ninst✝⁵ : TotalComplexShape c₂ c₁ c\ninst✝⁴ : TotalComplexShapeSymmet... | ComplexShape.σ_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 34
} | {
"line": 378,
"column": 2
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\nc₁₂ : ComplexShape I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : DecidableEq I₁₂\... | [
"case neg\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\nc₁₂ : ComplexShape I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : DecidableEq I₁₂\ninst✝¹ : K.... | · simp [totalAux.d₁_eq' _ c₁₂ h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 63
} | {
"line": 413,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : De... | [] | rw [comp_add, add_comp, mapAux.mapMap_D₁, mapAux.mapMap_D₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 16
} | {
"line": 121,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\n⊢ ((shiftFunctor₂ C y ⋙ shiftFunctor₁ C x).obj K).HasTotal (up ℤ)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 16
} | {
"line": 121,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\n⊢ ((shiftFunctor₂ C y ⋙ shiftFunctor₁ C x).obj K).HasTotal (up ℤ)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 16
} | {
"line": 126,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\n⊢ ((shiftFunctor₁ C x ⋙ shiftFunctor₂ C y).obj K).HasTotal (up ℤ)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 16
} | {
"line": 126,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\n⊢ ((shiftFunctor₁ C x ⋙ shiftFunctor₂ C y).obj K).HasTotal (up ℤ)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 19
} | {
"line": 156,
"column": 20
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\nx : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + x = n₀'\nh₁ : n₁ + x = n₁'\nh : n₀ + 1 = n₁\np q : ℤ\nhpq : p + q = n₀\n⊢ (((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) p ... | [
"case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\nx : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + x = n₀'\nh₁ : n₁ + x = n₁'\nh : n₀ + 1 = n₁\np q : ℤ\nhpq : p + q = n₀\n⊢ (((shiftFunctor₁ C x).obj K).d₁ (up ℤ) p q n₁ ≫\n ((... | ι_D₁_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 70
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\nx : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + x = n₀'\nh₁ : n₁ + x = n₁'\nh : ¬(up ℤ).Rel n₀ n₁\n⊢ ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x... | [
"case neg.h₁₂\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\nx : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + x = n₀'\nh₁ : n₁ + x = n₁'\nh : ¬(up ℤ).Rel n₀ n₁\n⊢ ¬(up ℤ).Rel n₀' n₁'"
] | rw [D₂_shape _ _ _ _ h, zero_comp, D₂_shape, comp_zero, smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 271,
"column": 8
} | {
"line": 271,
"column": 19
} | {
"line": 271,
"column": 20
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\ny : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + y = n₀'\nh₁ : n₁ + y = n₁'\nh : n₀ + 1 = n₁\np q : ℤ\nhpq : p + q = n₀\n⊢ (((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p ... | [
"case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\ny : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + y = n₀'\nh₁ : n₁ + y = n₁'\nh : n₀ + 1 = n₁\np q : ℤ\nhpq : p + q = n₀\n⊢ (((shiftFunctor₂ C y).obj K).d₁ (up ℤ) p q n₁ ≫\n ((... | ι_D₁_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 70
} | {
"line": 304,
"column": 4
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\ny : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + y = n₀'\nh₁ : n₁ + y = n₁'\nh : ¬(up ℤ).Rel n₀ n₁\n⊢ ((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y... | [
"case neg.h₁₂\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\ny : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + y = n₀'\nh₁ : n₁ + y = n₁'\nh : ¬(up ℤ).Rel n₀ n₁\n⊢ ¬(up ℤ).Rel n₀' n₁'"
] | rw [D₂_shape _ _ _ _ h, zero_comp, D₂_shape, comp_zero, smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 202,
"column": 4
} | {
"line": 203,
"column": 55
} | {
"line": 204,
"column": 4
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝³ : e.IsRelIff\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : Preadditive C\nK L : HomologicalComplex C c\nφ :\n (e.extendHomotopyFunctor C).obj ((HomotopyCategory.quotient... | [
"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝³ : e.IsRelIff\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : Preadditive C\nK L : HomologicalComplex C c\nφ : K.extend e ⟶ L.extend e\n⊢ ∃ a, (e.extendHomotopyFunctor C).map a = (Homot... | obtain ⟨φ : K.extend e ⟶ L.extend e, rfl⟩ :=
(HomotopyCategory.quotient C c').map_surjective φ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 673,
"column": 62
} | {
"line": 673,
"column": 95
} | {
"line": 673,
"column": 95
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : HasZeroMorphism... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : HasZeroMorphisms C₂\ninst✝¹... | ComplexShape.next_π₂ c₂ c₂₃ i₂ h₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.EulerCharacteristic | {
"line": 75,
"column": 31
} | {
"line": 75,
"column": 54
} | {
"line": 77,
"column": 0
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\ninst✝ : c.EulerCharSigns\ni✝ : ℤ\n⊢ (i✝ + 1).negOnePow = -i✝.negOnePow",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
... | [] | rw [Int.negOnePow_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 223,
"column": 91
} | {
"line": 231,
"column": 6
} | {
"line": 233,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : HasZeroMorphisms C\nK K' : ChainComplex C ℕ\nL L' : CochainComplex C ℕ\nh : ConnectData K L\nh' : ConnectData K' L'\nfK : K ⟶ K'\nfL : L ⟶ L'\nf_comm : fK.f 0 ≫ h'.d₀ = h.d₀ ≫ fL.f 0\nn : ℕ\ninst✝⁴ : NeZero n\nm : ℤ\nhmn : m = ↑n\ninst✝³ : HasHomology h.... | [] | by
rw [← cancel_mono (HomologicalComplex.homologyι ..)]
dsimp [homologyIsoPos]
simp only [homologyι_naturality, Category.assoc, restrictionHomologyIso_hom_homologyι,
homologyι_naturality_assoc, restrictionHomologyIso_inv_homologyι_assoc]
congr 1
rw [← cancel_epi (HomologicalComplex.pOpcycles ..)]
subst ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.LiftingProperty | {
"line": 199,
"column": 2
} | {
"line": 202,
"column": 54
} | {
"line": 204,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\nG : C ⥤ D\ninst✝ : G.IsEquivalence\nA B X Y : C\ni : A ⟶ B\np : X ⟶ Y\n⊢ (MorphismProperty.ofHoms fun x ↦ i).rlp (G.asEquivalence.inverse.map (G.asEquivalence.functor.map p)) ↔\n (MorphismProperty.ofHoms fun x ↦ i)... | [] | exact MorphismProperty.arrow_mk_iso_iff _
(Arrow.isoMk (G.asEquivalence.unitIso.symm.app _)
(G.asEquivalence.unitIso.symm.app _)
(G.asEquivalence.unitIso.inv.naturality p).symm) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 696,
"column": 56
} | {
"line": 696,
"column": 89
} | {
"line": 696,
"column": 89
} | [
{
"pp": "case pos.h\nC₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : Has... | [
"case pos.h\nC₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : HasZeroMorphism... | ComplexShape.next_π₂ c₂ c₂₃ i₂ h₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 46,
"column": 2
} | {
"line": 47,
"column": 16
} | {
"line": 49,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ Injective ((I K).X n)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Inj... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5b | {
"line": 46,
"column": 2
} | {
"line": 47,
"column": 16
} | {
"line": 49,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\n⊢ Injective ((I K).X n)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Inj... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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