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.CategoryTheory.Sites.Spaces | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 52
} | {
"line": 55,
"column": 2
} | [
{
"pp": "T : Type u\ninst✝ : TopologicalSpace T\nX Y : Opens T\nS : Sieve X\nf : Y ⟶ X\nhf : S ∈ {S | ∀ x ∈ X, ∃ U f, S.arrows f ∧ x ∈ U}\ny : T\nhy : y ∈ Y\n⊢ ∃ U f_1, (Sieve.pullback f S).arrows f_1 ∧ y ∈ U",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStr... | [] | rcases hf y (f.le hy) with ⟨U, g, hg, hU⟩
refine ⟨U ⊓ Y, homOfLE inf_le_right, ?_, hU, hy⟩
apply S.downward_closed hg (homOfLE inf_le_left) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Spaces | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 52
} | {
"line": 55,
"column": 2
} | [
{
"pp": "T : Type u\ninst✝ : TopologicalSpace T\nX Y : Opens T\nS : Sieve X\nf : Y ⟶ X\nhf : S ∈ {S | ∀ x ∈ X, ∃ U f, S.arrows f ∧ x ∈ U}\ny : T\nhy : y ∈ Y\n⊢ ∃ U f_1, (Sieve.pullback f S).arrows f_1 ∧ y ∈ U",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStr... | [] | rcases hf y (f.le hy) with ⟨U, g, hg, hU⟩
refine ⟨U ⊓ Y, homOfLE inf_le_right, ?_, hU, hy⟩
apply S.downward_closed hg (homOfLE inf_le_left) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Pairwise | {
"line": 113,
"column": 32
} | {
"line": 113,
"column": 44
} | {
"line": 113,
"column": 44
} | [
{
"pp": "ι : Type v\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni j k : ι\n⊢ ∀ (x : single i ⟶ pair j k), x ∈ ∅",
"ppTerm": "?m.282",
"assigned": true,
"usedConstants": [
"CategoryTheory.Pairwise.ctorIdx",
"CategoryTheory.Pairwise.Hom.right",
"CategoryTheory.Pairwise.Hom.casesOn",
... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.SheafCondition.Sites | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 35
} | {
"line": 162,
"column": 2
} | [
{
"pp": "X Y : TopCat\nf : X ⟶ Y\nhf : IsOpenMap ⇑(ConcreteCategory.hom f)\nU : Opens ↑X\nS : Sieve U\nhU : S ∈ (Opens.grothendieckTopology ↑X) U\nx : ↑X\nhx : x ∈ ↑U\n⊢ ∃ U_1 f_1, (Sieve.functorPushforward hf.functor S).arrows f_1 ∧ (ConcreteCategory.hom f) x ∈ U_1",
"ppTerm": "?m.56",
"assigned": true... | [
"X Y : TopCat\nf : X ⟶ Y\nhf : IsOpenMap ⇑(ConcreteCategory.hom f)\nU : Opens ↑X\nS : Sieve U\nhU : S ∈ (Opens.grothendieckTopology ↑X) U\nx : ↑X\nhx : x ∈ ↑U\nV : Opens ↑X\ni : V ⟶ U\nhV : S.arrows i\nhxV : x ∈ V\n⊢ ∃ U_1 f_1, (Sieve.functorPushforward hf.functor S).arrows f_1 ∧ (ConcreteCategory.hom f) x ∈ U_1"
] | obtain ⟨V, i, hV, hxV⟩ := hU x hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 59
} | {
"line": 208,
"column": 59
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Presheaf C X\nι : Type u_2\nU : ι → Opens ↑X\nh : F.IsSheaf\n⊢ Nonempty (IsLimit (Functor.mapCone F (opensLeCoverCocone U).op))",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"Eq... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Presheaf C X\nι : Type u_2\nU : ι → Opens ↑X\nh : F.IsSheaf\n⊢ Nonempty (IsLimit (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows.cocone.op))"
] | (isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent | {
"line": 440,
"column": 7
} | {
"line": 440,
"column": 71
} | {
"line": 440,
"column": 71
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR✝ : Sheaf J RingCat\ninst✝³ : ∀ (X : C), HasSheafify (J.over X) AddCommGrpCat\ninst✝² : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat\ninst✝¹ : ∀ (X : C) (Y : Over X), HasSheafify ((J.over X).over Y) AddCommGrpCat\nin... | [] | by exact e.counitIso.app ((M.over (X i.1)).over ((D i.1).X i.2)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 363,
"column": 2
} | {
"line": 365,
"column": 18
} | {
"line": 365,
"column": 19
} | [
{
"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 ⟨⟩ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Sheaves.Stalks | {
"line": 495,
"column": 61
} | {
"line": 500,
"column": 73
} | {
"line": 502,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimits C\nX : TopCat\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝ : PreservesFilteredColimits (forget C)\nB : Set (Opens ↑X)\nhB : Opens.IsBasis B\nF : Presheaf C... | [] | by
obtain ⟨W, hxW, hWU, hWV, e⟩ := F.germ_eq x mU mV _ _ h
obtain ⟨_, ⟨W', hW', rfl⟩, hxW', hW'W⟩ := hB.exists_subset_of_mem_open hxW W.2
refine ⟨W', hxW', hW', hW'W.trans hWU.le, hW'W.trans hWV.le, ?_⟩
simpa only [← ConcreteCategory.comp_apply, ← F.map_comp] using!
DFunLike.congr_arg (ConcreteCategory.hom ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 262,
"column": 2
} | {
"line": 285,
"column": 52
} | {
"line": 287,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nτR : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\n⊢ IsModuleTopology R R",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Continuous",
"instSMu... | [] | apply of_continuous_id
/-
The idea needed here is to rewrite the identity function as the composite of `r ↦ (r,1)`
from `R` to `R × R`, and multiplication `R × R → R`.
-/
rw [show (id : R → R) = (fun rs ↦ rs.1 • rs.2) ∘ (fun r ↦ (r, 1)) by ext; simp]
/-
It thus suffices to show that each of these maps are... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 262,
"column": 2
} | {
"line": 285,
"column": 52
} | {
"line": 287,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nτR : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\n⊢ IsModuleTopology R R",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Continuous",
"instSMu... | [] | apply of_continuous_id
/-
The idea needed here is to rewrite the identity function as the composite of `r ↦ (r,1)`
from `R` to `R × R`, and multiplication `R × R → R`.
-/
rw [show (id : R → R) = (fun rs ↦ rs.1 • rs.2) ∘ (fun r ↦ (r, 1)) by ext; simp]
/-
It thus suffices to show that each of these maps are... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | {
"line": 44,
"column": 31
} | {
"line": 44,
"column": 43
} | {
"line": 44,
"column": 43
} | [
{
"pp": "J : MulticospanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.R\nb : J.L\n⊢ ∀ (x : right a ⟶ left b), x ∈ ∅",
"ppTerm": "?m.734",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.MulticospanShape.snd",
"Cate... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | {
"line": 72,
"column": 31
} | {
"line": 72,
"column": 43
} | {
"line": 72,
"column": 43
} | [
{
"pp": "J : MultispanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.R\nb : J.L\n⊢ ∀ (x : right a ⟶ left b), x ∈ ∅",
"ppTerm": "?m.722",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingMultispan",
"CategoryTh... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Defs | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 22
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\ns : Seq α\n⊢ s.head = none → s = nil",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Stream'.Seq.head_eq_none"
],
"usedFVars": [
"α",
"s"
],
"usedGoals": []
}
] | [] | apply head_eq_none | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Seq.Defs | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 22
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\ns : Seq α\n⊢ s.head = none → s = nil",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Stream'.Seq.head_eq_none"
],
"usedFVars": [
"α",
"s"
],
"usedGoals": []
}
] | [] | apply head_eq_none | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Defs | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 22
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\ns : Seq α\n⊢ s.head = none → s = nil",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Stream'.Seq.head_eq_none"
],
"usedFVars": [
"α",
"s"
],
"usedGoals": []
}
] | [] | apply head_eq_none | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Stream.Init | {
"line": 53,
"column": 81
} | {
"line": 55,
"column": 5
} | {
"line": 57,
"column": 0
} | [
{
"pp": "α : Type u\nn m : ℕ\ns : Stream' α\n⊢ (drop m s).get n = s.get (m + n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Stream'.drop",
"id",
"instHAdd",
"Stream'.get",
"HAdd.hAdd",
"Nat",
"instAddNat",
... | [] | by
rw [Nat.add_comm]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Stream.Init | {
"line": 356,
"column": 95
} | {
"line": 359,
"column": 77
} | {
"line": 361,
"column": 0
} | [
{
"pp": "α : Type u\ns₁ s₂ : Stream' α\n⊢ s₁ ⋈ s₂ = s₁.head :: s₂.head :: (s₁.tail ⋈ s₂.tail)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.interleave",
"congrArg",
"id",
"Prod.mk",
"Stream'.corec_eq",
"Stream'.interleave.match... | [] | by
let t := tail s₁ ⋈ tail s₂
change s₁ ⋈ s₂ = head s₁::head s₂::t
unfold interleave; unfold corecOn; rw [corec_eq]; dsimp; rw [corec_eq]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Colimit.Ring | {
"line": 89,
"column": 47
} | {
"line": 89,
"column": 85
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (... | [
"case refine_1\nι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (Ideal.Quotie... | have ⟨k, hik, hjk⟩ := exists_ge_ge i j | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Colimit.Ring | {
"line": 89,
"column": 47
} | {
"line": 89,
"column": 85
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case refine_2\nι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (... | [
"case refine_2\nι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : FreeCommRing ((i : ι) × G i)\nx' y' : FreeCommRing ((i : ι) × G i)\ni : ι\nx : G i\nhx :\n (of G f i) x =\n (Ideal.Quotie... | have ⟨k, hik, hjk⟩ := exists_ge_ge i j | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Seq.Computation | {
"line": 399,
"column": 49
} | {
"line": 399,
"column": 84
} | {
"line": 401,
"column": 0
} | [
{
"pp": "α : Type u\ns : Computation α\nh : s.Terminates\na : α\n⊢ s.get = a → a ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Computation.get_mem",
"Computation",
"Computation.instMembership",
... | [] | by intro h; rw [← h]; apply get_mem | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 70
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u\na : α\ns : Stream' α\n⊢ ↑(a :: s) = cons a ↑s",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Option.some",
"Stream'.Seq.ofStream",
"id",
"Stream'.map_cons",
"Stream'",
"Stream'.IsSeq",
"Stream... | [] | apply Subtype.ext; simp only [ofStream, cons]; rw [Stream'.map_cons] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 70
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u\na : α\ns : Stream' α\n⊢ ↑(a :: s) = cons a ↑s",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Option.some",
"Stream'.Seq.ofStream",
"id",
"Stream'.map_cons",
"Stream'",
"Stream'.IsSeq",
"Stream... | [] | apply Subtype.ext; simp only [ofStream, cons]; rw [Stream'.map_cons] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 524,
"column": 2
} | {
"line": 526,
"column": 40
} | {
"line": 528,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\n⊢ nil.drop n = nil",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Nat.recAux",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_nil._simp_1_4",
"instOfNatNat",
"Stream'.... | [] | induction n with
| zero => simp [drop]
| succ m ih => simp [← dropn_tail, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.Seq.Basic | {
"line": 524,
"column": 2
} | {
"line": 526,
"column": 40
} | {
"line": 528,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\n⊢ nil.drop n = nil",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Nat.recAux",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_nil._simp_1_4",
"instOfNatNat",
"Stream'.... | [] | induction n with
| zero => simp [drop]
| succ m ih => simp [← dropn_tail, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 524,
"column": 2
} | {
"line": 526,
"column": 40
} | {
"line": 528,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\n⊢ nil.drop n = nil",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Nat.recAux",
"Stream'.Seq.drop",
"congrArg",
"_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.drop_nil._simp_1_4",
"instOfNatNat",
"Stream'.... | [] | induction n with
| zero => simp [drop]
| succ m ih => simp [← dropn_tail, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 286,
"column": 13
} | {
"line": 286,
"column": 39
} | {
"line": 287,
"column": 6
} | [
{
"pp": "case inr.inl\nK : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\nv : K\ninst✝ : IsStrictOrderedRing K\nn : ℕ\nh : fract v ≠ 0\nh₁ : (of (fract v)⁻¹).s.get? n = none\n⊢ (of v).s.get? (n + 1) = none",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [... | [
"case inr.inl\nK : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\nv : K\ninst✝ : IsStrictOrderedRing K\nn : ℕ\nh : fract v ≠ 0\nh₁ : (of (fract v)⁻¹).s.get? n = none\n⊢ (of v).TerminatedAt (n + 1)"
] | ← terminatedAt_iff_s_none, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.AddChar | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 91
} | {
"line": 35,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nG : ι → Type u_3\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → AddCommGroup (G i)\ninst✝ : CommMonoid R\n⊢ Injective directSum",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddChar.toAddMonoidHomEquiv",
"Equiv.instEquivLike",
"Monoid.... | [
"ι : Type u_1\nR : Type u_2\nG : ι → Type u_3\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → AddCommGroup (G i)\ninst✝ : CommMonoid R\n⊢ Injective fun ψ i ↦ toAddMonoidHomEquiv (ψ i)"
] | refine toAddMonoidHomEquiv.symm.injective.comp <| DirectSum.toAddMonoid_injective.comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.CubicDiscriminant | {
"line": 119,
"column": 81
} | {
"line": 119,
"column": 91
} | {
"line": 119,
"column": 91
} | [
{
"pp": "R : Type u_1\nP Q : Cubic R\ninst✝ : Semiring R\nh : P.toPoly = Q.toPoly\n⊢ Q.toPoly.coeff 1 = Q.c",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cubic.coeff_eq_c",
"congrArg",
"id",
"Cubic.c",
"instOfNatNat",
"Cubic.toPoly",
... | [
"R : Type u_1\nP Q : Cubic R\ninst✝ : Semiring R\nh : P.toPoly = Q.toPoly\n⊢ Q.c = Q.c"
] | coeff_eq_c | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CubicDiscriminant | {
"line": 206,
"column": 59
} | {
"line": 206,
"column": 61
} | {
"line": 206,
"column": 61
} | [
{
"pp": "R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 1\na✝ : Nontrivial R\n⊢ P.a = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"id",
"AddCommMonoidWithOne.toAddMonoidWithOne",... | [
"R : Type u_1\nP : Cubic R\ninst✝ : Semiring R\nha : P.a = 1\na✝ : Nontrivial R\n⊢ 1 = 1"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 22
} | {
"line": 132,
"column": 23
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nx : M\n⊢ x ∈ (f.genEigenspace μ) 0 ↔ x = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instAddMonoidWithOneENat",
"cong... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nx : M\n⊢ x ∈ (f.genEigenspace μ) ↑0 ↔ x = 0"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 45
} | {
"line": 83,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : ι → Submodule R M\ninst✝² : DecidableEq ι\ninst✝¹ : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\nhN : {i | N i ≠ ⊥}.Finite\nf : M →ₗ[R] M\nhf : ∀ (i... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : ι → Submodule R M\ninst✝² : DecidableEq ι\ninst✝¹ : ∀ (i : ι), Module.Finite R ↥(N i)\ninst✝ : ∀ (i : ι), Free R ↥(N i)\nh : IsInternal N\nhN : {i | N i ≠ ⊥}.Finite\nf : M →ₗ[R] M\nhf : ∀ (i : ι), MapsT... | let _ : Fintype {i | N i ≠ ⊥} := hN.fintype | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Idempotents | {
"line": 599,
"column": 16
} | {
"line": 600,
"column": 99
} | {
"line": 601,
"column": 2
} | [
{
"pp": "R : Type u_1\ne✝ : R\nI : Type u_2\ninst✝¹ : Fintype I\ne : I → R\ninst✝ : Semiring R\nhe : CompleteOrthogonalIdempotents e\nhc : ∀ (i : I), IsMulCentral (e i)\nr : R\n⊢ (fun r ↦ ∑ i, ↑(r i)) ((fun r i ↦ ⟨(fun x ↦ e i * x * e i) r, ⋯⟩) r) = r",
"ppTerm": "?m.153",
"assigned": true,
"usedCon... | [] | by
simp_rw [((hc _).comm _).eq, mul_assoc, (he.idem _).eq, ← Finset.mul_sum, he.complete, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Trace | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 60
} | {
"line": 221,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] M\ng : N →ₗ[R] N\n⊢ (trace R (M × N)) (f.pr... | [
"R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] M\ng : N →ₗ[R] N\nh : (trace R (M × N) ∘ₗ prodMapLinear... | have h := LinearMap.ext_iff.1 (trace_prodMap R M N) (f, g) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Trace | {
"line": 259,
"column": 61
} | {
"line": 263,
"column": 9
} | {
"line": 265,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] M\ng : N →ₗ[R] N\n⊢ (trace R (M ⊗[R] N)) (m... | [] | by
have h := LinearMap.ext_iff.1 (LinearMap.ext_iff.1 (trace_tensorProduct R M N) f) g
simp only [compr₂_apply, mapBilinear_apply, compl₁₂_apply, lsmul_apply,
smul_eq_mul] at h
exact h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 112,
"column": 6
} | {
"line": 115,
"column": 17
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case neg.cons\nα : Type u_1\ninst✝ : DecidableEq α\np : Prop\nx : α\nb : Bool\nL1 L3 : List (α × Bool)\nx' : α\nb' : Bool\ntail : List (α × Bool)\ny : α\nc : Bool\nr : reduce L1 = (y, c) :: tail\nh : ¬(x = y ∧ b = !c)\na : α × Bool\nL2 : List (α × Bool)\nH : (x, b) :: (y, c) :: tail = a :: L2 ++ (x', b... | [] | · refine @reduce.not _ L1 L2 L3 x' b' ?_
rw [List.cons_append] at H
injection H with _ H
rw [r, H] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.FiveLemma | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 46
} | {
"line": 161,
"column": 2
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝⁵ : Group M₁\ninst✝⁴ : Group M₂\ninst✝³ : Group M₃\ninst✝² : Group N₁\ninst✝¹ : Group N₂\ninst✝ : Group N₃\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\ng₁ : N₁ →* N₂\ng₂ : N₂ →* N₃\ni₁ : M₁ →* N₁\ni₂ : M₂ →* N₂\ni₃ : M₃ →*... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝⁵ : Group M₁\ninst✝⁴ : Group M₂\ninst✝³ : Group M₃\ninst✝² : Group N₁\ninst✝¹ : Group N₂\ninst✝ : Group N₃\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\ng₁ : N₁ →* N₂\ng₂ : N₂ →* N₃\ni₁ : M₁ →* N₁\ni₂ : M₂ →* N₂\ni₃ : M₃ →* N₃\nhc₁ : g... | refine ⟨injective_of_surjective_of_injective_of_right_exact f₁ f₂ g₁ g₂ i₁ i₂ i₃
hc₁ hc₂ hf₁ hg₁ hi₁ hi₂.1 hf₂, fun y ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.FiveLemma | {
"line": 257,
"column": 2
} | {
"line": 258,
"column": 46
} | {
"line": 259,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type u_7\nN₂ : Type u_8\nN₃ : Type u_9\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : AddCommGroup M₂\ninst✝⁹ : AddCommGroup M₃\ninst✝⁸ : Module R M₁\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\ninst✝⁵ : AddCommGroup N₁\ninst✝... | [
"R : Type u_1\ninst✝¹² : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type u_7\nN₂ : Type u_8\nN₃ : Type u_9\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : AddCommGroup M₂\ninst✝⁹ : AddCommGroup M₃\ninst✝⁸ : Module R M₁\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : AddCommG... | refine ⟨injective_of_surjective_of_injective_of_right_exact f₁ f₂ g₁ g₂ i₁ i₂ i₃
hc₁ hc₂ hf₁ hg₁ hi₁ hi₂.1 hf₂, fun y ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.FreeMonoid.FreeSemigroup | {
"line": 76,
"column": 9
} | {
"line": 76,
"column": 41
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u_1\nx : FreeMonoid α\n⊢ x ∈ Set.range ⇑toFreeMonoid ↔ x ∈ {1}ᶜ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.FreeMonoid.FreeSemigroup.0.FreeSemigroup.range_toFreeMonoid._proof_1_1"
],
"usedFVars": [
"α",
"x"
],
... | [] | grind [eq_one_or_toFreeMonoid x] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Algebra.Quaternion | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 70
} | {
"line": 347,
"column": 0
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝ : AddCommGroup R\n⊢ AddCommGroup ℍ[R,c₁,c₂,c₃]",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"QuaternionAlgebra.instSub",
"instHSMul",
"Equiv.ins... | [] | apply (equivProd c₁ c₂ c₃).injective.addCommGroup <;> intros <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Quaternion | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 70
} | {
"line": 347,
"column": 0
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝ : AddCommGroup R\n⊢ AddCommGroup ℍ[R,c₁,c₂,c₃]",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"QuaternionAlgebra.instSub",
"instHSMul",
"Equiv.ins... | [] | apply (equivProd c₁ c₂ c₃).injective.addCommGroup <;> intros <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Quaternion | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 70
} | {
"line": 347,
"column": 0
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝ : AddCommGroup R\n⊢ AddCommGroup ℍ[R,c₁,c₂,c₃]",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"QuaternionAlgebra.instSub",
"instHSMul",
"Equiv.ins... | [] | apply (equivProd c₁ c₂ c₃).injective.addCommGroup <;> intros <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Action.Equidecomp | {
"line": 109,
"column": 10
} | {
"line": 109,
"column": 12
} | {
"line": 110,
"column": 2
} | [
{
"pp": "X : Type u_1\nG : Type u_2\ninst✝ : SMul G X\nf f' : X → X\nA A' : Set X\nS : Finset G\nh : IsDecompOn f A S\nhA' : A' ⊆ A\nhf' : EqOn f f' A'\na : X\n⊢ a ∈ A' → ∃ g ∈ S, f' a = g • a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership... | [
"X : Type u_1\nG : Type u_2\ninst✝ : SMul G X\nf f' : X → X\nA A' : Set X\nS : Finset G\nh : IsDecompOn f A S\nhA' : A' ⊆ A\nhf' : EqOn f f' A'\na : X\nha : a ∈ A'\n⊢ ∃ g ∈ S, f' a = g • a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Group.Action.Equidecomp | {
"line": 148,
"column": 26
} | {
"line": 148,
"column": 43
} | {
"line": 148,
"column": 43
} | [
{
"pp": "X : Type u_1\nG : Type u_2\nA B C : Set X\ninst✝¹ : Monoid G\ninst✝ : MulAction G X\n⊢ IsDecompOn (↑(PartialEquiv.refl X)) (PartialEquiv.refl X).source {1}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
... | [] | simp [IsDecompOn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Action.Equidecomp | {
"line": 148,
"column": 26
} | {
"line": 148,
"column": 43
} | {
"line": 148,
"column": 43
} | [
{
"pp": "X : Type u_1\nG : Type u_2\nA B C : Set X\ninst✝¹ : Monoid G\ninst✝ : MulAction G X\n⊢ IsDecompOn (↑(PartialEquiv.refl X)) (PartialEquiv.refl X).source {1}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
... | [] | simp [IsDecompOn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Equidecomp | {
"line": 148,
"column": 26
} | {
"line": 148,
"column": 43
} | {
"line": 148,
"column": 43
} | [
{
"pp": "X : Type u_1\nG : Type u_2\nA B C : Set X\ninst✝¹ : Monoid G\ninst✝ : MulAction G X\n⊢ IsDecompOn (↑(PartialEquiv.refl X)) (PartialEquiv.refl X).source {1}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
... | [] | simp [IsDecompOn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Saturation | {
"line": 255,
"column": 19
} | {
"line": 255,
"column": 61
} | {
"line": 255,
"column": 61
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\ns : Submonoid M\nx x✝ y✝ : M\nhx✝ : x✝ ∈ s.saturation\nhy✝ : y✝ ∈ s.saturation\nih₁ : ∃ y, x✝ * y ∈ s\nih₂ : ∃ y, y✝ * y ∈ s\ny₁ : M\nh₁ : x✝ * y₁ ∈ s\ny₂ : M\nh₂ : y✝ * y₂ ∈ s\n⊢ x✝ * y✝ * (y₁ * y₂) ∈ s",
"ppTerm": "?m.91",
"assigned": true,
"usedConstan... | [] | rw [mul_mul_mul_comm]; exact mul_mem h₁ h₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Saturation | {
"line": 255,
"column": 19
} | {
"line": 255,
"column": 61
} | {
"line": 255,
"column": 61
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\ns : Submonoid M\nx x✝ y✝ : M\nhx✝ : x✝ ∈ s.saturation\nhy✝ : y✝ ∈ s.saturation\nih₁ : ∃ y, x✝ * y ∈ s\nih₂ : ∃ y, y✝ * y ∈ s\ny₁ : M\nh₁ : x✝ * y₁ ∈ s\ny₂ : M\nh₂ : y✝ * y₂ ∈ s\n⊢ x✝ * y✝ * (y₁ * y₂) ∈ s",
"ppTerm": "?m.91",
"assigned": true,
"usedConstan... | [] | rw [mul_mul_mul_comm]; exact mul_mem h₁ h₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Saturation | {
"line": 309,
"column": 75
} | {
"line": 310,
"column": 91
} | {
"line": 312,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nx : M\n⊢ x ∈ ⊥ ↔ IsUnit x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"SaturatedSubmonoid",
"CompleteLattice.toLattice",
"Monoid.toMu... | [] | by
simp_rw [bot_def, Submonoid.mem_saturation_iff, Submonoid.mem_bot, isUnit_iff_exists_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Pointwise.Finset | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 22
} | {
"line": 41,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : Zero α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : IsLeftCancelMulZero α\nhs : (s.erase 0).Nonempty\n⊢ #s ≤ #(s * s)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Finset",
"Membership.mem",
... | [
"case inr\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : Zero α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : IsLeftCancelMulZero α\na : α\nha : a ∈ s.erase 0\n⊢ #s ≤ #(s * s)"
] | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 105,
"column": 25
} | {
"line": 105,
"column": 39
} | {
"line": 105,
"column": 39
} | [
{
"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... | contrapose hi₁ | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Algebra.GroupWithZero.ProdHom | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 69
} | {
"line": 163,
"column": 0
} | [
{
"pp": "G₀ : Type u_1\nH₀ : Type u_2\ninst✝¹ : GroupWithZero G₀\ninst✝ : GroupWithZero H₀\n⊢ Function.Surjective ⇑(snd G₀ H₀)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"MonoidWithZeroHom.snd",
"Function.HasRightInv... | [] | exact Function.HasRightInverse.surjective ⟨inr .., fun _ ↦ by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.GroupWithZero.ProdHom | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 69
} | {
"line": 163,
"column": 0
} | [
{
"pp": "G₀ : Type u_1\nH₀ : Type u_2\ninst✝¹ : GroupWithZero G₀\ninst✝ : GroupWithZero H₀\n⊢ Function.Surjective ⇑(snd G₀ H₀)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"MonoidWithZeroHom.snd",
"Function.HasRightInv... | [] | exact Function.HasRightInverse.surjective ⟨inr .., fun _ ↦ by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.ProdHom | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 69
} | {
"line": 163,
"column": 0
} | [
{
"pp": "G₀ : Type u_1\nH₀ : Type u_2\ninst✝¹ : GroupWithZero G₀\ninst✝ : GroupWithZero H₀\n⊢ Function.Surjective ⇑(snd G₀ H₀)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"MonoidWithZeroHom.snd",
"Function.HasRightInv... | [] | exact Function.HasRightInverse.surjective ⟨inr .., fun _ ↦ by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 61,
"column": 9
} | {
"line": 61,
"column": 34
} | {
"line": 61,
"column": 34
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝² : MulZeroOneClass G\ninst✝¹ : MulZeroOneClass H\ninst✝ : Nontrivial H\nf : G →*₀ H\n⊢ 1 ≠ 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
"NeZero.one",
"MonoidWithZeroHom.instMulZeroOneClassSubty... | [] | by simp [Subtype.ext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 16
} | {
"line": 237,
"column": 6
} | [
{
"pp": "case refine_1.mem\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\ny x✝ : Bˣ\nh : x✝ ∈ ↑{ carrier := Units.val ⁻¹' range ⇑f, mul_mem' := ⋯, one_mem' := ⋯ }\n⊢ ∃ a, f a ≠ 0 ∧ ∃ x, f a * ↑x✝ = f x",
"ppTerm": "?refine_1.mem",
"assigned": true,
... | [] | | mem _ h => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 18
} | {
"line": 178,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\nf g : P ⟶ Q\n⊢ f + g = g + f",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"CategoryTheory.Idempotents.instAdd",
"CategoryTheory.I... | [] | ext
apply add_comm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 18
} | {
"line": 178,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\nf g : P ⟶ Q\n⊢ f + g = g + f",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"CategoryTheory.Idempotents.instAdd",
"CategoryTheory.I... | [] | ext
apply add_comm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 218,
"column": 19
} | {
"line": 218,
"column": 28
} | {
"line": 218,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi C\np : P ⟶ P\nhp : p.f ≫ p.f = p.f\n⊢ { X := P.X, p := p.f, idem := hp }.p ≫ p.f ≫ P.p = p.f",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"CategoryTh... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : Karoubi C\np : P ⟶ P\nhp : p.f ≫ p.f = p.f\n⊢ { X := P.X, p := p.f, idem := hp }.p ≫ p.f = p.f"
] | comp_p p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.AlternatingFaceMapComplex | {
"line": 91,
"column": 16
} | {
"line": 91,
"column": 18
} | {
"line": 91,
"column": 19
} | [
{
"pp": "case hi\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\nP : Type := Fin (n + 2) × Fin (n + 3)\nS : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}\nφ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)\nij : P\nhij : ij ∈ {ij | ↑ij.2 ≤ ↑ij.1}\n⊢ φ ij... | [
"case hi\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\nP : Type := Fin (n + 2) × Fin (n + 3)\nS : Finset P := {ij | ↑ij.2 ≤ ↑ij.1}\nφ : (ij : P) → ij ∈ S → P := fun ij hij ↦ (ij.2.castLT ⋯, ij.1.succ)\nij : P\nhij : ij ∈ {ij | ↑ij.2 ≤ ↑ij.1}\n⊢ (ij.2.castLT ⋯, ... | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 71
} | {
"line": 288,
"column": 4
} | [
{
"pp": "case inl.h.inl\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i ≤ j.castSucc\nk : Fin (⦋n + 1⦌.len + 1)\nhik : i ≤ k\nhjk : k ≤ j.castSucc\n⊢ i ≤ (j.predAbove k).castSucc",
"ppTerm": "?inl.h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fin.ne_of_lt",
... | [
"case inl.h.inr\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i ≤ j.castSucc\nk : Fin (⦋n + 1⦌.len + 1)\nhik : i ≤ k\nhjk : j.castSucc < k\n⊢ i ≤ (j.predAbove k).castSucc"
] | · rwa [Fin.predAbove_of_le_castSucc _ _ hjk, Fin.castSucc_castPred] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 419,
"column": 30
} | {
"line": 419,
"column": 55
} | {
"line": 421,
"column": 0
} | [
{
"pp": "case zero\nm n : ℕ\nf : ⦋m⦌ ⟶ ⦋n + 1⦌\nk : Fin (⦋m⦌.len + 1)\nhj : ∀ (k : Fin (m + 1)), (Hom.toOrderHom f) k ≠ 0\n⊢ (Hom.toOrderHom (factor_δ f 0 ≫ δ 0)) k = (Hom.toOrderHom f) k",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"CategoryTheory.Category.... | [] | simp_all [factor_δ, δ, σ] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 419,
"column": 30
} | {
"line": 419,
"column": 55
} | {
"line": 421,
"column": 0
} | [
{
"pp": "case succ\nm n : ℕ\nf : ⦋m⦌ ⟶ ⦋n + 1⦌\nk : Fin (⦋m⦌.len + 1)\ni✝ : Fin (n + 1)\nhj : ∀ (k : Fin (m + 1)), (Hom.toOrderHom f) k ≠ i✝.succ\n⊢ (Hom.toOrderHom (factor_δ f i✝.succ ≫ δ i✝.succ)) k = (Hom.toOrderHom f) k",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Fin.succAbo... | [] | simp_all [factor_δ, δ, σ] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 819,
"column": 90
} | {
"line": 828,
"column": 44
} | {
"line": 830,
"column": 0
} | [
{
"pp": "n : ℕ\nθ : ⦋n⦌ ⟶ ⦋n + 1⦌\ninst✝ : Mono θ\n⊢ ∃ i, θ = δ i",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Epi",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"PartialOrd... | [] | by
obtain ⟨i, θ', h⟩ := eq_comp_δ_of_not_surjective θ (by
rw [← epi_iff_surjective]
grind [→ le_of_epi])
use i
have : Mono (θ' ≫ δ i) := by
rw [← h]
infer_instance
have := CategoryTheory.mono_of_mono θ' (δ i)
rw [h, eq_id_of_mono θ', Category.id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Simplices | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 18
} | {
"line": 85,
"column": 2
} | [
{
"pp": "X : SSet\nd✝ d : ℕ\nw✝ : X _⦋d⦌\nhd : { dim := d, simplex := w✝ }.dim = d✝\n⊢ { dim := d, simplex := w✝ }.cast hd = { dim := d, simplex := w✝ }",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"SSet.S",
"Eq.rec",
"Nat",
"SSet.S.dim",
"SSet.S.cast",
... | [
"X : SSet\nd : ℕ\nw✝ : X _⦋d⦌\n⊢ { dim := d, simplex := w✝ }.cast ⋯ = { dim := d, simplex := w✝ }"
] | obtain rfl := hd | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.Finite | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 67
} | {
"line": 79,
"column": 2
} | [
{
"pp": "case mk\nX : SSet\ninst✝ : X.Finite\nn : ℕ\n⊢ Finite (X _⦋n⦌)",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.ConcreteCategory.hom",
"Quiver.Hom.op",
"SSet.nonDege... | [
"case mk\nX : SSet\ninst✝ : X.Finite\nn : ℕ\nφ : (m : Fin (n + 1)) × (_ : ⦋n⦌ ⟶ ⦋↑m⦌) × ↑(X.nonDegenerate ↑m) → X _⦋n⦌ :=\n fun x ↦\n match x with\n | ⟨m, ⟨f, x⟩⟩ => (ConcreteCategory.hom (X.map f.op)) ↑x\n⊢ Finite (X _⦋n⦌)"
] | let φ : (Σ (m : Fin (n + 1)) (f : ⦋n⦌ ⟶ ⦋m.1⦌),
X.nonDegenerate m.1) → X _⦋n⦌ := fun ⟨m, f, x⟩ ↦ X.map f.op x.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 472,
"column": 4
} | {
"line": 472,
"column": 38
} | {
"line": 472,
"column": 38
} | [
{
"pp": "n d : ℕ\ns : Δ[n] _⦋d⦌\n⊢ StrictMono ⇑s ↔ Mono (objEquiv s)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"Equiv.instEquivLike",
"StrictMono",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver... | [
"n d : ℕ\ns : Δ[n] _⦋d⦌\n⊢ StrictMono ⇑s ↔ Function.Injective ⇑(Hom.toOrderHom (objEquiv s))"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 470,
"column": 52
} | {
"line": 478,
"column": 34
} | {
"line": 480,
"column": 0
} | [
{
"pp": "n d : ℕ\ns : Δ[n] _⦋d⦌\n⊢ s ∈ Δ[n].nonDegenerate d ↔ Mono (objEquiv s)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SSet.stdSimplex.mem_nonDegenerate_iff_strictMono",
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Fin.castSucc_le_succ",
"Preord... | [] | by
rw [mem_nonDegenerate_iff_strictMono,
SimplexCategory.mono_iff_injective]
refine ⟨fun h ↦ h.injective, fun h ↦ ?_⟩
rw [Fin.strictMono_iff_lt_succ]
intro i
obtain h' | h' := (stdSimplex.monotone_apply s i.castSucc_le_succ).lt_or_eq
· exact h'
· simpa [Fin.ext_iff] using h h' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 626,
"column": 40
} | {
"line": 626,
"column": 74
} | {
"line": 626,
"column": 74
} | [
{
"pp": "n m : ℕ\nx✝ : { x // x ∈ Δ[n].nonDegenerate m }\nx : Δ[n] _⦋m⦌\nhx : Mono (objEquiv x)\n⊢ #(Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ) = m + 1",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Opposite",
"Equiv.instEquivLike",
"CategoryTheory.Mono",
... | [
"n m : ℕ\nx✝ : { x // x ∈ Δ[n].nonDegenerate m }\nx : Δ[n] _⦋m⦌\nhx : Function.Injective ⇑(Hom.toOrderHom (objEquiv x))\n⊢ #(Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ) = m + 1"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 645,
"column": 38
} | {
"line": 645,
"column": 72
} | {
"line": 645,
"column": 72
} | [
{
"pp": "case right.refine_1\nn m : ℕ\nS : Finset (Fin (n + 1))\nhS : #S = m + 1\ne : Fin (m + 1) ≃o ↥S := monoEquivOfFin ↥S ⋯\n⊢ Mono (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom)))",
"ppTerm": "?right.refine_1",
"assigned": true,
"usedConstants": [
... | [
"case right.refine_1\nn m : ℕ\nS : Finset (Fin (n + 1))\nhS : #S = m + 1\ne : Fin (m + 1) ≃o ↥S := monoEquivOfFin ↥S ⋯\n⊢ Function.Injective\n ⇑(Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom))))"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 671,
"column": 10
} | {
"line": 671,
"column": 44
} | {
"line": 671,
"column": 44
} | [
{
"pp": "case left\nn d : ℕ\nx : ↑(Δ[n].nonDegenerate d)\nthis : Mono (objEquiv ↑x)\n⊢ Function.Injective fun i ↦ ⟨↑x i, ⋯⟩",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Opposite",
"Equiv.instEquivLike",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQui... | [
"case left\nn d : ℕ\nx : ↑(Δ[n].nonDegenerate d)\nthis : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))\n⊢ Function.Injective fun i ↦ ⟨↑x i, ⋯⟩"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {
"line": 678,
"column": 8
} | {
"line": 678,
"column": 42
} | {
"line": 678,
"column": 42
} | [
{
"pp": "n d : ℕ\nx : ↑(Δ[n].nonDegenerate d)\nthis : Mono (objEquiv ↑x)\n⊢ ∀ {a b : Fin (d + 1)},\n (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) a ≤ (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) b ↔ a ≤ b",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Opposite",
"Equiv.instEqui... | [
"n d : ℕ\nx : ↑(Δ[n].nonDegenerate d)\nthis : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))\n⊢ ∀ {a b : Fin (d + 1)},\n (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) a ≤ (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) b ↔ a ≤ b"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ExtraDegeneracy | {
"line": 297,
"column": 4
} | {
"line": 303,
"column": 36
} | {
"line": 304,
"column": 2
} | [
{
"pp": "Δ : SimplexCategory\nn : ℕ\ni : Fin (n + 2)\n⊢ (↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.δ i.succ =\n (stdSimplex.obj Δ).left.δ i ≫ ↾fun f ↦ objEquiv.symm (shift (objEquiv f))",
"ppTerm": "?m.293",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
... | [] | ext φ
apply objEquiv.injective
apply SimplexCategory.Hom.ext
ext j : 2
dsimp [SimplicialObject.δ, SimplexCategory.δ, SSet.stdSimplex,
objEquiv, Equiv.ulift, uliftFunctor]
cases j using Fin.cases <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ExtraDegeneracy | {
"line": 297,
"column": 4
} | {
"line": 303,
"column": 36
} | {
"line": 304,
"column": 2
} | [
{
"pp": "Δ : SimplexCategory\nn : ℕ\ni : Fin (n + 2)\n⊢ (↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.δ i.succ =\n (stdSimplex.obj Δ).left.δ i ≫ ↾fun f ↦ objEquiv.symm (shift (objEquiv f))",
"ppTerm": "?m.293",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
... | [] | ext φ
apply objEquiv.injective
apply SimplexCategory.Hom.ext
ext j : 2
dsimp [SimplicialObject.δ, SimplexCategory.δ, SSet.stdSimplex,
objEquiv, Equiv.ulift, uliftFunctor]
cases j using Fin.cases <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologicalBicomplex | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 30
} | {
"line": 73,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nI₁ : Type u_2\nI₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nx✝¹ x✝ : HomologicalComplex₂ C c₁ c₂\nφ₁ φ₂ : x✝¹ ⟶ x✝\nh : (toGradedObjectFunctor C c₁ c₂).map φ₁ = (toGradedObjectFunctor C c₁ c₂).map φ₂\ni₁ : I₁\ni₂ : ... | [] | exact congr_fun h ⟨i₁, i₂⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 24
} | {
"line": 207,
"column": 8
} | [
{
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | [
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄\n... | ext j i₁ i₂ i₃ h | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 210,
"column": 8
} | {
"line": 210,
"column": 24
} | {
"line": 211,
"column": 8
} | [
{
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | [
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄\n... | ext j i₁ i₂ i₃ h | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 24
} | {
"line": 218,
"column": 8
} | [
{
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | [
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄\n... | ext j i₁ i₂ i₃ h | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 248,
"column": 12
} | {
"line": 248,
"column": 28
} | {
"line": 249,
"column": 12
} | [
{
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | [
"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✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄\n... | ext j i₁ i₂ i₃ h | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 11
} | {
"line": 498,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₂₃ : Type u_6\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂₃ ⥤ C₄\nG₂₃ : C₂ ⥤ C₃ ⥤ C₂₃\nI₁ : Type u_7\nI₂ : T... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₂₃ : Type u_6\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂₃ ⥤ C₄\nG₂₃ : C₂ ⥤ C₃ ⥤ C₂₃\nI₁ : Type u_7\nI₂ : Type u_8\nI₃ ... | subst h₂₃ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 289,
"column": 30
} | {
"line": 289,
"column": 49
} | {
"line": 289,
"column": 49
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : 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✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : De... | [] | by rw [← h, h₁, h₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 11
} | {
"line": 438,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²² : Category.{v_1, u_1} C₁\ninst✝²¹ : Category.{v_2, u_2} C₂\ninst✝²⁰ : Category.{v_3, u_5} C₃\ninst✝¹⁹ : Category.{v_4, u_6} C₄\ninst✝¹⁸ : Category.{v_6, u_4} C₂₃\ninst✝¹⁷ : HasZeroMorphisms C₁\ninst✝¹⁶ : HasZeroMorphism... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²² : Category.{v_1, u_1} C₁\ninst✝²¹ : Category.{v_2, u_2} C₂\ninst✝²⁰ : Category.{v_3, u_5} C₃\ninst✝¹⁹ : Category.{v_4, u_6} C₄\ninst✝¹⁸ : Category.{v_6, u_4} C₂₃\ninst✝¹⁷ : HasZeroMorphisms C₁\ninst✝¹⁶ : HasZeroMorphisms C₂\ninst✝¹... | subst h₂₃ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.HomotopyCategory.KInjective | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 72
} | {
"line": 83,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nL : CochainComplex C ℤ\nhL : (HomotopyCategory.subcategoryAcyclic C).rightOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)\nK : CochainComplex C ℤ\nf : K ⟶ L\nhK : HomologicalComplex.Acyclic K\n⊢ (Homoto... | [] | rw [← HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_acyclic] at hK
rw [hL ((HomotopyCategory.quotient _ _).map f) hK, Functor.map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.KInjective | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 72
} | {
"line": 83,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nL : CochainComplex C ℤ\nhL : (HomotopyCategory.subcategoryAcyclic C).rightOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj L)\nK : CochainComplex C ℤ\nf : K ⟶ L\nhK : HomologicalComplex.Acyclic K\n⊢ (Homoto... | [] | rw [← HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_acyclic] at hK
rw [hL ((HomotopyCategory.quotient _ _).map f) hK, Functor.map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 95,
"column": 12
} | {
"line": 95,
"column": 71
} | {
"line": 96,
"column": 10
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni : ι\nhi : ¬c.Rel (c.pre... | [] | simp [prevD_eq _ hi', L.extend_d_to_eq_zero _ _ _ _ rfl hi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 95,
"column": 12
} | {
"line": 95,
"column": 71
} | {
"line": 96,
"column": 10
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni : ι\nhi : ¬c.Rel (c.pre... | [] | simp [prevD_eq _ hi', L.extend_d_to_eq_zero _ _ _ _ rfl hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 95,
"column": 12
} | {
"line": 95,
"column": 71
} | {
"line": 96,
"column": 10
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni : ι\nhi : ¬c.Rel (c.pre... | [] | simp [prevD_eq _ hi', L.extend_d_to_eq_zero _ _ _ _ rfl hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 127,
"column": 18
} | {
"line": 127,
"column": 20
} | {
"line": 127,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)\nH_zero : H 0\nH_one : H 1\na b ... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)\nH_zero : H 0\nH_one : H 1\na b c : ℕ\nh : a... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 150,
"column": 18
} | {
"line": 150,
"column": 20
} | {
"line": 150,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n\nH_zero : H 0\nH_one : H 1\na b ... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nH : ℕ → Prop := fun a ↦ ∀ (n : ℤ), t.ge (n + ↑a) ≤ t.ge n\nH_zero : H 0\nH_one : H 1\na b c : ℕ\nh : a... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.HomotopyCategory.KInjective | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 27
} | {
"line": 156,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\nL : CochainComplex C ℤ\nd : ℤ\ninst✝¹ : L.IsStrictlyGE d\ninst✝ : ∀ (n : ℤ), Injective (L.X n)\nK : CochainComplex C ℤ\nf : K ⟶ L\nhK : HomologicalComplex.Acyclic K\nX : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Coc... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\nL : CochainComplex C ℤ\nd : ℤ\ninst✝¹ : L.IsStrictlyGE d\ninst✝ : ∀ (n : ℤ), Injective (L.X n)\nK : CochainComplex C ℤ\nf : K ⟶ L\nhK : HomologicalComplex.Acyclic K\nX : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f... | Cochain.ofHom_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 343,
"column": 14
} | {
"line": 345,
"column": 18
} | {
"line": 347,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : CategoryWithCofibrations C\ninst✝³ : CategoryWithFibrations C\ninst✝² : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)\ninst✝¹ : (weakEquivalences C).IsStableUnderRetracts\ninst✝ : (weakEquivalen... | [] | by
rw [← weakEquivalence_iff]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 351,
"column": 86
} | {
"line": 353,
"column": 18
} | {
"line": 353,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : CategoryWithCofibrations C\ninst✝³ : CategoryWithFibrations C\ninst✝² : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)\ninst✝¹ : (weakEquivalences C).IsStableUnderRetracts\ninst✝ : (weakEquivalen... | [] | by
rw [← weakEquivalence_iff]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Quotient | {
"line": 65,
"column": 21
} | {
"line": 65,
"column": 35
} | {
"line": 65,
"column": 36
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nhomRel : HomRel C\nW : MorphismProperty C\nh : homRel.FactorsThroughLocalization W\nW' : MorphismProperty (CategoryTheory.Quotient homRel)\nhW : W = W'.inverseImage (Quotient.functor homRel)\nE : Type u_3\ninst✝... | [
"C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nhomRel : HomRel C\nW : MorphismProperty C\nh : homRel.FactorsThroughLocalization W\nW' : MorphismProperty (CategoryTheory.Quotient homRel)\nhW : W = W'.inverseImage (Quotient.functor homRel)\nE : Type u_3\ninst✝ : Category.... | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy | {
"line": 214,
"column": 59
} | {
"line": 216,
"column": 18
} | {
"line": 216,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : CategoryWithWeakEquivalences C\nX Y : C\nf g : X ⟶ Y\nP : Cylinder X\nh : P.LeftHomotopy f g\nL : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q\n⊢ weakEquivalences C P.π",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants":... | [] | by
rw [← weakEquivalence_iff]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy | {
"line": 217,
"column": 59
} | {
"line": 219,
"column": 18
} | {
"line": 219,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : CategoryWithWeakEquivalences C\nX Y : C\nf g : X ⟶ Y\nP : PathObject Y\nh : P.RightHomotopy f g\nL : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q\n⊢ weakEquivalences C P.ι",
"ppTerm": "?m.87",
"assigned": true,
"usedConstant... | [] | by
rw [← weakEquivalence_iff]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.Plus | {
"line": 161,
"column": 4
} | {
"line": 162,
"column": 46
} | {
"line": 163,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\nA : Type u_3\ninst✝¹ : Category.{v_3, u_3} A\ninst✝ : Abelian A\na : ℤ\nX Y : Plus A\nf : X ⟶ Y\n⊢ (quasiIso A).inverseImage (shiftFunctor (Plus A) a) f ↔ quasiIso... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\nA : Type u_3\ninst✝¹ : Category.{v_3, u_3} A\ninst✝ : Abelian A\na : ℤ\nX Y : Plus A\nf : X ⟶ Y\n⊢ (quasiIso A).inverseImage (shiftFunctor (Plus A) a) f ↔\n HomotopyCategor... | simp only [quasiIso_iff, ← MorphismProperty.IsCompatibleWithShift.iff
(HomotopyCategory.quasiIso _ _) f.hom a] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.DerivedCategory.Plus | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 62
} | {
"line": 125,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\n⊢ Qh.IsLocalization (HomotopyCategory.Plus.quasiIso C)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"HomotopyCategory.Plus",
"CategoryTheory... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\n⊢ Qh.IsLocalization (HomotopyCategory.Plus.subcategoryAcyclic C).trW"
] | HomotopyCategory.Plus.quasiIso_eq_subcategoryAcyclic_trW | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.OfSequence | {
"line": 60,
"column": 6
} | {
"line": 60,
"column": 15
} | {
"line": 61,
"column": 6
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn✝ : ℕ\nhi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n✝ n✝ ⋯ = 𝟙 (X n✝)\n⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f (n✝ + 1) (n✝ + 1) ⋯ = 𝟙 (X (n✝ + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants":... | [
"case succ\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn✝ : ℕ\nhi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n✝ n✝ ⋯ = 𝟙 (X n✝)\nX : ℕ → C\nf : (n : ℕ) → X n ⟶ X (n + 1)\n⊢ map f (n✝ + 1) (n✝ + 1) ⋯ = 𝟙 (X (n✝ + 1))"
] | intro X f | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Functor.OfSequence | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 15
} | {
"line": 69,
"column": 6
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn✝ : ℕ\nhi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n✝ (n✝ + 1) ⋯ = f n✝\n⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f (n✝ + 1) (n✝ + 1 + 1) ⋯ = f (n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants":... | [
"case succ\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn✝ : ℕ\nhi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n✝ (n✝ + 1) ⋯ = f n✝\nX : ℕ → C\nf : (n : ℕ) → X n ⟶ X (n + 1)\n⊢ map f (n✝ + 1) (n✝ + 1 + 1) ⋯ = f (n✝ + 1)"
] | intro X f | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 45
} | {
"line": 634,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nn : ℤ\nh₁ : t.IsGE T.obj₁ n\nh₃ : t.IsGE T.obj₃... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nt : TStructure C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nn : ℤ\nh₁ : t.IsGE T.obj₁ n\nh₃ : t.IsGE T.obj₃ n\n⊢ ∀ (Y :... | rw [t.isGE_iff_orthogonal (n-1) n (by lia)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 36
} | {
"line": 89,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C ℤ\nι : K ⟶ ... | dsimp [CokernelCofork.map] at l hl | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
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