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.Order.JordanHolder | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 73
} | {
"line": 333,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nx₁ x₂ : X\nhsat₁ : IsMaximal (last s₁) x₁\nhsat₂ : IsMaximal (last s₂) x₂\nhequiv : s₁.Equivalent s₂\nhlast : Iso (last s₁, x₁) (last s₂, x₂)\ne : Fin s₁.length.succ ≃ Fin s₂.length.succ :=\n Tra... | [] | simpa [snoc_castSucc, ← Fin.castSucc_succ] using hequiv.choose_spec i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 72
} | {
"line": 333,
"column": 72
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : Ring S\nM' : Type u_6\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nN' : Type u_7\ninst✝² : AddCommGroup N'\ninst✝¹ : Module S N'\nσ : R →+* S\nl : M' →ₛₗ[σ] N'\ninst✝ : RingHomSurjective σ\nhl : Function.Bijective ⇑l\n⊢ ComplementedLattice (Subm... | [] | (Submodule.orderIsoMapComapOfBijective l hl).complementedLattice_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 89,
"column": 32
} | {
"line": 89,
"column": 43
} | {
"line": 89,
"column": 44
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nr : R\nf : R[X]\nm : AEval R M a\n⊢ (r • (aeval a) f) • (of R M a).symm m = (of R M a).sy... | [
"R : Type u_1\nA : Type u_3\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nr : R\nf : R[X]\nm : AEval R M a\n⊢ r • (aeval a) f • (of R M a).symm m = (of R M a).symm (r • f • m)... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 546,
"column": 11
} | {
"line": 546,
"column": 35
} | {
"line": 546,
"column": 35
} | [
{
"pp": "ι : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁸ : Ring R✝\ninst✝⁷ : Ring S\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R✝ M\nm : Submodule R✝ M\nN : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR : Type u_6\ninst✝² : DivisionRing R\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nv... | [] | simpa using congr($eq 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 546,
"column": 11
} | {
"line": 546,
"column": 35
} | {
"line": 546,
"column": 35
} | [
{
"pp": "ι : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁸ : Ring R✝\ninst✝⁷ : Ring S\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R✝ M\nm : Submodule R✝ M\nN : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR : Type u_6\ninst✝² : DivisionRing R\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nv... | [] | simpa using congr($eq 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 546,
"column": 11
} | {
"line": 546,
"column": 35
} | {
"line": 546,
"column": 35
} | [
{
"pp": "ι : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁸ : Ring R✝\ninst✝⁷ : Ring S\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R✝ M\nm : Submodule R✝ M\nN : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR : Type u_6\ninst✝² : DivisionRing R\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nv... | [] | simpa using congr($eq 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | {
"line": 141,
"column": 4
} | [
{
"pp": "case single\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni : ℕ\nr : R\nn a✝ : ℕ\nb✝ : M\n⊢ ((monomial i) r • (single R a✝) b✝) n = if i ≤ n then r • ((single R a✝) b✝) (n - i) else 0",
"ppTerm": "?single",
"assigned": true,
"usedConstants": ... | [] | | single => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 100,
"column": 70
} | {
"line": 103,
"column": 25
} | {
"line": 105,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ torsionOf R M m = ⊤ ↔ m = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSM... | [] | by
refine ⟨fun h => ?_, fun h => by simp [h]⟩
rw [← one_smul R m, ← mem_torsionOf_iff m (1 : R), h]
exact Submodule.mem_top | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 177,
"column": 57
} | {
"line": 177,
"column": 65
} | {
"line": 177,
"column": 66
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nr : R[X]\nf✝ g✝ : PolynomialModule R R\nhp : (toFinsuppIso R).symm (r • f✝... | [
"case add\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nr : R[X]\nf✝ g✝ : PolynomialModule R R\nhp : (toFinsuppIso R).symm (r • f✝) = r * (toF... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 29
} | {
"line": 67,
"column": 4
} | [
{
"pp": "case h.left\nR : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns✝ : Set M\ni : M\ns : Set M\na✝ : i ∉ s\nhs✝ : s.Finite\nih :\n (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ... | [] | exact I.sub_mem hr1 hci | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 945,
"column": 2
} | {
"line": 945,
"column": 39
} | {
"line": 946,
"column": 2
} | [
{
"pp": "M : Type u_2\ninst✝ : AddCommMonoid M\n⊢ IsTorsion M ↔ Module.IsTorsion ℕ M",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nat",
"Iff.intro",
"Module.IsTorsion",
"Nat.instSemiring",
"AddCommMonoid.toNatModule",
"AddCommMonoid.toAddMonoid",
... | [
"case refine_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nh : IsTorsion M\nx : M\n⊢ ∃ a, a • x = 0",
"case refine_2\nM : Type u_2\ninst✝ : AddCommMonoid M\nh : Module.IsTorsion ℕ M\nx : M\n⊢ IsOfFinAddOrder x"
] | refine ⟨fun h x => ?_, fun h x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 954,
"column": 2
} | {
"line": 954,
"column": 39
} | {
"line": 955,
"column": 2
} | [
{
"pp": "M : Type u_2\ninst✝ : AddCommGroup M\n⊢ IsTorsion M ↔ Module.IsTorsion ℤ M",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"AddCommGroup.toAddCommMonoid",
"AddCommGroup.toAddGroup",
"Int",
"AddGroup.toSubNegMonoid",
"Iff.intro",
"AddCommGroup.toI... | [
"case refine_1\nM : Type u_2\ninst✝ : AddCommGroup M\nh : IsTorsion M\nx : M\n⊢ ∃ a, a • x = 0",
"case refine_2\nM : Type u_2\ninst✝ : AddCommGroup M\nh : Module.IsTorsion ℤ M\nx : M\n⊢ IsOfFinAddOrder x"
] | refine ⟨fun h x => ?_, fun h x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Filtration | {
"line": 355,
"column": 66
} | {
"line": 355,
"column": 79
} | {
"line": 355,
"column": 79
} | [
{
"pp": "case mpr\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF : I.Filtration M\nhF' : ∀ (i : ℕ), (F.N i).FG\nn : ℕ\nhn : F.submodule = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(F.N i))\n⊢ (⨆ i ∈ Finset.range n.succ... | [
"case mpr\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF : I.Filtration M\nhF' : ∀ (i : ℕ), (F.N i).FG\nn : ℕ\nhn : F.submodule = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n), ⇑(single R i) '' ↑(F.N i))\n⊢ (⨆ x, Submodule.span (↥(reesAlgebra I)... | iSup_subtype' | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 62,
"column": 94
} | {
"line": 64,
"column": 29
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\nx y : ↥I\n⊢ I.toCotangent x = I.toCotangent y ↔ ↑x - ↑y ∈ I ^ 2",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Semiring.toModule",
"Ideal.mem_toCotangent_ker",
"A... | [] | by
rw [← sub_eq_zero]
exact I.mem_toCotangent_ker | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 69
} | {
"line": 140,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\n⊢ I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"Submodule.Quotient.addCo... | [
"R : Type u\ninst✝ : CommRing R\nI : Ideal R\n⊢ (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range = Submodule.restrictScalars R I.cotangentIdeal"
] | · rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 49,
"column": 31
} | {
"line": 49,
"column": 50
} | {
"line": 49,
"column": 51
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms C\nF : J ⥤ HomologicalComplex C c\ns : Cone F\nhs : (i : ι) → IsLimit ((eval C c i).mapCone s)\nt : Cone F\ni i' : ι\nx✝ : c.Rel i i'\nj : J\neq : ∀ (k ... | [
"C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms C\nF : J ⥤ HomologicalComplex C c\ns : Cone F\nhs : (i : ι) → IsLimit ((eval C c i).mapCone s)\nt : Cone F\ni i' : ι\nx✝ : c.Rel i i'\nj : J\neq : ∀ (k : ι) (j : J)... | reassoc_of% (eq i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 129,
"column": 12
} | {
"line": 129,
"column": 31
} | {
"line": 129,
"column": 32
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms C\nF : J ⥤ HomologicalComplex C c\ns : Cocone F\nhs : (i : ι) → IsColimit ((eval C c i).mapCocone s)\nt : Cocone F\ni i' : ι\nx✝ : c.Rel i i'\nj : J\neq... | [
"C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms C\nF : J ⥤ HomologicalComplex C c\ns : Cocone F\nhs : (i : ι) → IsColimit ((eval C c i).mapCocone s)\nt : Cocone F\ni i' : ι\nx✝ : c.Rel i i'\nj : J\neq : ∀ (k : ι)... | reassoc_of% (eq i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 270,
"column": 42
} | {
"line": 270,
"column": 49
} | {
"line": 270,
"column": 50
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G K : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nj i : ι\nhij : c.Rel i j\nhj : ¬c.Rel j (c.next j)\n⊢ (... | [
"case neg\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G K : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nj i : ι\nhij : c.Rel i j\nhj : ¬c.Rel j (c.next j)\n⊢ (φ ≫ inr φ).f... | zero_f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 17
} | {
"line": 93,
"column": 18
} | [
{
"pp": "case refine_2.refine_1\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nD : Derivation R S M\nx y : S ⊗[R] S\nx₁ x₂ : S\n⊢ D.tensorProductTo (x₁ ⊗ₜ[R] x₂... | [
"case refine_2.refine_1\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nD : Derivation R S M\nx y : S ⊗[R] S\nx₁ x₂ : S\n⊢ D.tensorProductTo 0 =\n (TensorProduct.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 96,
"column": 8
} | {
"line": 96,
"column": 16
} | {
"line": 96,
"column": 17
} | [
{
"pp": "case refine_2.refine_3\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nD : Derivation R S M\nx y : S ⊗[R] S\nx₁✝ x₂ : S\nx₁ y₁ : S ⊗[R] S\nh₁ :\n D.ten... | [
"case refine_2.refine_3\nR : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nD : Derivation R S M\nx y : S ⊗[R] S\nx₁✝ x₂ : S\nx₁ y₁ : S ⊗[R] S\nh₁ :\n D.tensorProductTo... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 70
} | {
"line": 148,
"column": 71
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nd e : ℤ\nγ : Cochain F K d\nhe : 1 + d = e\n⊢ (inl φ).comp ((↑(fst φ)).comp γ he) ⋯ = γ",
"ppTerm": "?m.75",
"assigned": true,
"used... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nd e : ℤ\nγ : Cochain F K d\nhe : 1 + d = e\n⊢ ((inl φ).comp ↑(fst φ) ⋯).comp γ ⋯ = γ"
] | ← Cochain.comp_assoc _ _ _ (neg_add_cancel 1) (by lia) (by lia), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 64
} | {
"line": 304,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G K : CochainComplex C ℤ\nn₁ n₂ n₁₂ : ℤ\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\np q : ℤ\nhpq : p + n₁₂ = q\n⊢ (comp 0 z₂ h).v p q hpq = v 0 p q hpq",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"CochainCompl... | [] | simp only [comp_v _ _ h p _ q rfl (by lia), zero_v, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Quotient.Preadditive | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 76
} | {
"line": 54,
"column": 77
} | [
{
"pp": "C : Type ?u.2\ninst✝² : Category.{v_1, ?u.2} C\ninst✝¹ : Preadditive C\nr : HomRel C\ninst✝ : Congruence r\nhr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)\nX Y : Quotient r\nf✝ : X ⟶ Y\nf g : X.as ⟶ Y.as\nhfg : r f g\n⊢ r (-g) (-f)",
"ppTerm": "?m.73",
"assign... | [
"case e'_3\nC : Type ?u.2\ninst✝² : Category.{v_1, ?u.2} C\ninst✝¹ : Preadditive C\nr : HomRel C\ninst✝ : Congruence r\nhr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)\nX Y : Quotient r\nf✝ : X ⟶ Y\nf g : X.as ⟶ Y.as\nhfg : r f g\n⊢ -g = f + (-f - g)",
"case e'_4\nC : Type ?u.2\n... | convert! hr f g _ _ hfg (Congruence.equivalence.refl (-f - g)) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 509,
"column": 66
} | {
"line": 509,
"column": 96
} | {
"line": 510,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nr : R\nx y : S\nthis : SMulZeroClass R S := inferInstance\n⊢ ((y * (algebraMap R S) r)𝖣x) = r • y𝖣x",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finsupp.smulZeroClass... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nr : R\nx y : S\nthis : SMulZeroClass R S := inferInstance\n⊢ ((y * (algebraMap R S) r)𝖣x) = (kerTotal R S).mkQ (r • single x y)"
] | ← LinearMap.map_smul_of_tower, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 153,
"column": 8
} | {
"line": 153,
"column": 71
} | {
"line": 154,
"column": 8
} | [
{
"pp": "C : Type ?u.2\nD : Type ?u.4\ninst✝⁵ : Category.{v_1, ?u.2} C\ninst✝⁴ : Category.{v_2, ?u.4} D\nF : C ⥤ D\nA : Type ?u.15\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D... | [
"C : Type ?u.2\nD : Type ?u.4\ninst✝⁵ : Category.{v_1, ?u.2} C\ninst✝⁴ : Category.{v_2, ?u.4} D\nF : C ⥤ D\nA : Type ?u.15\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D D).obj F).F... | rw [← cancel_mono ((s m₃).map ((s m₂).map ((i m₁).hom.app X)))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Shift | {
"line": 56,
"column": 8
} | {
"line": 56,
"column": 17
} | {
"line": 57,
"column": 8
} | [
{
"pp": "case a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : Preadditive D\nn : ℤ\nK : CochainComplex C ℤ\ni j : ℤ\nhij : ¬(ComplexShape.up ℤ).Rel i j\nhij' : (ComplexShape.up ℤ).Rel (i + n) (j + n)\n⊢ False",
"ppTerm": "?a✝",
"assig... | [
"case a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : Preadditive D\nn : ℤ\nK : CochainComplex C ℤ\ni j : ℤ\nhij : ¬(ComplexShape.up ℤ).Rel i j\nhij' : (ComplexShape.up ℤ).Rel (i + n) (j + n)\n⊢ (ComplexShape.up ℤ).Rel i j"
] | apply hij | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 469,
"column": 23
} | {
"line": 471,
"column": 53
} | {
"line": 472,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nF G : C ⥤ D\ne : F ≅ G\nA : Type u_4\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : F.CommShift A\n⊢ (shiftFunctor C 0).isoWhiskerLeft e.symm ≪≫ commShiftIso F 0 ≪≫ is... | [] | by
ext X
simp [F.commShiftIso_zero, ← NatTrans.naturality] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 271,
"column": 2
} | {
"line": 272,
"column": 15
} | {
"line": 274,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na : ℤ\n⊢ (-γ).shift a = -γ.shift a",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"HomologicalComplex.instCategory",
"AddMono... | [] | change shiftAddHom K L n a (-γ) = _
apply map_neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 271,
"column": 2
} | {
"line": 272,
"column": 15
} | {
"line": 274,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na : ℤ\n⊢ (-γ).shift a = -γ.shift a",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"HomologicalComplex.instCategory",
"AddMono... | [] | change shiftAddHom K L n a (-γ) = _
apply map_neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 407,
"column": 19
} | {
"line": 407,
"column": 27
} | {
"line": 407,
"column": 28
} | [
{
"pp": "case e_a.e_a\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ (a * (m + n')).negOnePow = (a * ... | [
"case e_a.e_a\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ (a * m + a * n').negOnePow = (a * m).negOneP... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 413,
"column": 53
} | {
"line": 413,
"column": 62
} | {
"line": 413,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nγ' : Cochain L M 0\n⊢ (a * 0).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯ = (γ.leftShift a n' hn').comp γ' ⋯",
"ppTerm": "?m.85",
"assigned": true,
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nγ' : Cochain L M 0\n⊢ Int.negOnePow 0 • (γ.leftShift a n' hn').comp γ' ⋯ = (γ.leftShift a n' hn').comp γ' ⋯"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Basic | {
"line": 210,
"column": 37
} | {
"line": 210,
"column": 79
} | {
"line": 211,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasShift C ℤ\nA B : Triangle C\ne : A ≅ B\n⊢ e.hom.hom₁ ≫ e.inv.hom₁ = 𝟙 A.obj₁",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [] | by rw [← comp_hom₁, e.hom_inv_id, id_hom₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 463,
"column": 31
} | {
"line": 463,
"column": 39
} | {
"line": 463,
"column": 40
} | [
{
"pp": "case pos.e_a.e_a\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' m' : ℤ\nhn' : n + a = n'\nm : ℤ\nhm' : m + a = m'\nhnm : n + 1 = m\nhnm' : n' + 1 = m'\np q : ℤ\nhpq : p + m' = q\n⊢ (a * n' + a * (a - 1) / 2).negOnePow = a.negOneP... | [
"case pos.e_a.e_a\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' m' : ℤ\nhn' : n + a = n'\nm : ℤ\nhm' : m + a = m'\nhnm : n + 1 = m\nhnm' : n' + 1 = m'\np q : ℤ\nhpq : p + m' = q\n⊢ (a * n' + a * (a - 1) / 2).negOnePow = a.negOnePow * (a * 1 ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 523,
"column": 83
} | {
"line": 523,
"column": 91
} | {
"line": 524,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' n'' : ℤ\nhn' : n' + a = n\nhn'' : n + a = n''\n⊢ (a * n + a * (a - 1) / 2).negOnePow • γ.shift a =\n (a.negOnePow * (a * (a + n) + a * (a - 1) / 2).negOnePow) • γ.shift a",
"pp... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' n'' : ℤ\nhn' : n' + a = n\nhn'' : n + a = n''\n⊢ (a * n + a * (a - 1) / 2).negOnePow • γ.shift a =\n (a.negOnePow * (a * a + a * n + a * (a - 1) / 2).negOnePow) • γ.shift a"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 258,
"column": 42
} | {
"line": 261,
"column": 31
} | {
"line": 263,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nX : C\nf : X ⟶ T.obj₂\nhf : f ≫ T.mor₂ = 0\n⊢ ∃ g, f =... | [] | by
obtain ⟨a, ⟨ha₁, _⟩⟩ := complete_distinguished_triangle_morphism₁ _ T
(contractible_distinguished X) hT f 0 (by cat_disch)
exact ⟨a, by simpa using ha₁⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 78
} | {
"line": 377,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\nh₁ : T.mor₁ = 0\nh₃ : T.mor₃ = 0\nthis : Epi ... | [] | exact ⟨f, hf.symm, by rw [← cancel_epi T.mor₂, comp_id, ← reassoc_of% hf]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 498,
"column": 4
} | {
"line": 500,
"column": 53
} | {
"line": 501,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT T' : Triangle C\nφ : T ⟶ T'\nhT : T ∈ distinguishedTriangles\nhT' : T' ∈ distinguishedTriangles\nh₁ :... | [
"case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT T' : Triangle C\nφ : T ⟶ T'\nhT : T ∈ distinguishedTriangles\nhT' : T' ∈ distinguishedTriangles\nh₁ : IsIso φ.hom... | obtain ⟨x₂, hx₂⟩ := Triangle.coyoneda_exact₃ _ hT x₃
(by rw [← cancel_mono (φ.hom₁⟦(1 : ℤ)⟧'), assoc, zero_comp, φ.comm₃, reassoc_of% hx₃,
comp_distTriang_mor_zero₂₃ _ hT', comp_zero]) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 63
} | {
"line": 685,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fu... | [
"case refine_2\nC : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).... | have hb : ∀ j, _ = b j ≫ _ := fun j => (ha'' j).choose_spec | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 115,
"column": 2
} | {
"line": 116,
"column": 16
} | {
"line": 118,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝¹ : K.HasHomology i\ninst✝ : K.HasHomology j\n⊢ Mono\n ((composableArrows₃ K i j).map' 0 1 instMonoMap'ComposableArrows₃OfNatNat._proof_1... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 115,
"column": 2
} | {
"line": 116,
"column": 16
} | {
"line": 118,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhij : c.Rel i j\ninst✝¹ : K.HasHomology i\ninst✝ : K.HasHomology j\n⊢ Mono\n ((composableArrows₃ K i j).map' 0 1 instMonoMap'ComposableArrows₃OfNatNat._proof_1... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 114,
"column": 32
} | {
"line": 114,
"column": 39
} | {
"line": 114,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\ns_g : (σ (p + 1)).s ≫ S.g.f (p + 1) = 𝟙 (S.X₃.X (p + 1))\nf_r : S.f.f (p + 1) ≫ (σ (p + 1)).r... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\ns_g : (σ (p + 1)).s ≫ S.g.f (p + 1) = 𝟙 (S.X₃.X (p + 1))\nf_r : S.f.f (p + 1) ≫ (σ (p + 1)).r = 𝟙 (S.X₁.... | zero_f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 157,
"column": 49
} | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 57
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\nn : ℤ\nh : S.f.f (n + 1) ≫ (σ (n + 1)).r = 𝟙 (S.X₁.X (n + 1))\n⊢ Hom.f 0 (n + 1) ≫ (mappingCone.inl ... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\nn : ℤ\nh : S.f.f (n + 1) ≫ (σ (n + 1)).r = 𝟙 (S.X₁.X (n + 1))\n⊢ 0 ≫ (mappingCone.inl (homOfDegreewiseSplit S σ)... | zero_f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 28
} | {
"line": 165,
"column": 6
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\n⊢ ∀ (x y : Paths (LocQuiver W)) (f₁ f₂ : x ⟶ y),\n relations W f₁ f₂ → (liftToPathCategory G hG).map f₁ = (liftToPathCategory G hG).map f₂",
"ppTerm": ... | [
"C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\nX Y : C\nf₁ f₂ : { obj := X } ⟶ { obj := Y }\nr : relations W f₁ f₂\n⊢ (liftToPathCategory G hG).map f₁ = (liftToPathCategory G hG).map f₂"
] | rintro ⟨X⟩ ⟨Y⟩ f₁ f₂ r | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.CatCommSq | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 54
} | {
"line": 129,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\ninst✝¹ : Category.{v_3, u_3} C₃\ninst✝ : Category.{v_4, u_4} C₄\nT : C₁ ≌ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ≌ C₄\nh : CatCommSq T.functor L R B.functor\nX : C₁\n⊢ (B.counitIs... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\ninst✝¹ : Category.{v_3, u_3} C₃\ninst✝ : Category.{v_4, u_4} C₄\nT : C₁ ≌ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ≌ C₄\nh : CatCommSq T.functor L R B.functor\nX : C₁\n⊢ B.counitIso.inv.app (R.... | simp only [Functor.comp_obj, assoc, ← Functor.map_comp, Iso.inv_hom_id_app,
Equivalence.counitInv_app_functor, Functor.map_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 32
} | {
"line": 255,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\nE : Type u_3\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : L.IsLocalization W\n⊢ (whiskeringLeftFunctor' L W E).Full",
"ppTerm": "?m.18",
"assigned": true,
"usedConstant... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\nE : Type u_3\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : L.IsLocalization W\n⊢ (whiskeringLeftFunctor L W E ⋙ inducedFunctor ObjectProperty.FullSubcategory.obj).Full"
] | rw [whiskeringLeftFunctor'_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 32
} | {
"line": 261,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\nE : Type u_3\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : L.IsLocalization W\n⊢ (whiskeringLeftFunctor' L W E).Faithful",
"ppTerm": "?m.18",
"assigned": true,
"usedCons... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\nE : Type u_3\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : L.IsLocalization W\n⊢ (whiskeringLeftFunctor L W E ⋙ inducedFunctor ObjectProperty.FullSubcategory.obj).Faithful"
] | rw [whiskeringLeftFunctor'_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 321,
"column": 7
} | {
"line": 326,
"column": 11
} | {
"line": 326,
"column": 11
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG✝ : C ⥤ D\nhG : W.IsInvertedBy G✝\nG : W.FunctorsInverting D\n⊢ 𝟙 W.Q ◫ natTransExtension (eqToHom ⋯ ≫ (𝟙 G).hom ≫ eqToHom ⋯) = 𝟙 W.Q ◫ 𝟙 (lift G.obj ⋯)",
"ppTerm": "?m.115",
"ass... | [] | by
rw [natTransExtension_hcomp]
ext X
simp only [NatTrans.comp_app, eqToHom_app, eqToHom_refl, comp_id, id_comp,
NatTrans.hcomp_id_app, NatTrans.id_app, Functor.map_id]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.LocalizerMorphism | {
"line": 213,
"column": 9
} | {
"line": 213,
"column": 70
} | {
"line": 213,
"column": 70
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\ninst✝ : Φ.functor.IsEquivalence\nh : W₂ ≤ W₁.map Φ.functor\n⊢ W₂ ≤ W₁.isoClosure.inverseImage Φ.functor.asEquivalence.symm.functor",... | [
"C₁ : Type u₁\nC₂ : Type u₂\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\ninst✝ : Φ.functor.IsEquivalence\nh : W₂ ≤ W₁.map Φ.functor\n⊢ W₂ ≤ W₁.isoClosure.map Φ.functor.asEquivalence.functor"
] | W₁.isoClosure.inverseImage_equivalence_functor_eq_map_inverse | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.LocalizerMorphism | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 16
} | {
"line": 405,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁹ : Category.{v₁, u₁} C₁\ninst✝⁸ : Category.{v₂, u₂} C₂\ninst✝⁷ : Category.{v₃, u₃} C₃\ninst✝⁶ : Category.{v₄, u₄} D₁\ninst✝⁵ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\n... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.LocalizerMorphism | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 16
} | {
"line": 405,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁹ : Category.{v₁, u₁} C₁\ninst✝⁸ : Category.{v₂, u₂} C₂\ninst✝⁷ : Category.{v₃, u₃} C₃\ninst✝⁶ : Category.{v₄, u₄} D₁\ninst✝⁵ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\n... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.HasCardinalLT | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 59
} | {
"line": 59,
"column": 0
} | [
{
"pp": "X : Type u\nκ : Cardinal.{v}\nh : HasCardinalLT X κ\nY : Type u'\nf : Y → X\nhf : Function.Injective f\n⊢ HasCardinalLT Y κ",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",
"Preorder.toLT",
"HasCardinalLT",
"Cardinal",
... | [] | dsimp [HasCardinalLT] at h ⊢
rw [← Cardinal.lift_lt.{_, u}, Cardinal.lift_lift, Cardinal.lift_lift]
rw [← Cardinal.lift_lt.{_, u'}, Cardinal.lift_lift, Cardinal.lift_lift] at h
exact lt_of_le_of_lt (Cardinal.mk_le_of_injective
(Function.Injective.comp ULift.up_injective
(Function.Injective.comp hf ULift... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.HasCardinalLT | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 59
} | {
"line": 59,
"column": 0
} | [
{
"pp": "X : Type u\nκ : Cardinal.{v}\nh : HasCardinalLT X κ\nY : Type u'\nf : Y → X\nhf : Function.Injective f\n⊢ HasCardinalLT Y κ",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",
"Preorder.toLT",
"HasCardinalLT",
"Cardinal",
... | [] | dsimp [HasCardinalLT] at h ⊢
rw [← Cardinal.lift_lt.{_, u}, Cardinal.lift_lift, Cardinal.lift_lift]
rw [← Cardinal.lift_lt.{_, u'}, Cardinal.lift_lift, Cardinal.lift_lift] at h
exact lt_of_le_of_lt (Cardinal.mk_le_of_injective
(Function.Injective.comp ULift.up_injective
(Function.Injective.comp hf ULift... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.LimitsClosure | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 8
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case isMin\nC : Type u\ninst✝⁵ : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝⁴ : (a : α) → Category.{v', u'} (J a)\nβ : Type w'\ninst✝³ : LinearOrder β\ninst✝² : OrderBot β\ninst✝¹ : SuccOrder β\ninst✝ : WellFoundedLT β\nb : β\nhb : IsMin b\n⊢ P.strictLimitsClosureIter J ... | [] | obtain rfl := hb.eq_bot
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.LimitsClosure | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 8
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case isMin\nC : Type u\ninst✝⁵ : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝⁴ : (a : α) → Category.{v', u'} (J a)\nβ : Type w'\ninst✝³ : LinearOrder β\ninst✝² : OrderBot β\ninst✝¹ : SuccOrder β\ninst✝ : WellFoundedLT β\nb : β\nhb : IsMin b\n⊢ P.strictLimitsClosureIter J ... | [] | obtain rfl := hb.eq_bot
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 719,
"column": 59
} | {
"line": 726,
"column": 49
} | {
"line": 728,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝¹ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁ : W.LeftFraction X Y\nz₂ : W.LeftFraction Y Z\nz₃ : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\nL : C ⥤ D\ninst✝ : L.I... | [] | by
have : IsIso (L.map (z₂.s ≫ z₃.s)) := by
rw [L.map_comp]
infer_instance
dsimp [LeftFraction.comp₀]
rw [← cancel_mono (L.map (z₂.s ≫ z₃.s)), map_comp_map_s,
L.map_comp, assoc, map_comp_map_s_assoc, ← L.map_comp, h₃,
L.map_comp, map_comp_map_s_assoc, L.map_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 166,
"column": 11
} | {
"line": 175,
"column": 46
} | {
"line": 177,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasShift C ℤ\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁹ : Pretriangulated C\nD : Type u_2\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Preadditive D\ninst✝⁶ : HasZeroObject D\ninst✝⁵... | [] | by
rintro T hT ⟨X₁, h₁, ⟨e₁⟩⟩ ⟨X₃, h₃, ⟨e₃⟩⟩
exact ObjectProperty.le_isoClosure _ _
(P.ext_of_isTriangulatedClosed₂'
(Triangle.mk (e₁.inv ≫ T.mor₁) (T.mor₂ ≫ e₃.hom) (e₃.inv ≫ T.mor₃ ≫ e₁.hom⟦1⟧'))
(isomorphic_distinguished _ hT _
(Triangle.isoMk _ _ e₁.symm (Iso.refl _) e₃.symm (by ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.ShiftedHom | {
"line": 138,
"column": 54
} | {
"line": 139,
"column": 45
} | {
"line": 141,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nM : Type u_4\ninst✝² : AddMonoid M\ninst✝¹ : HasShift C M\nX Y Z : C\ninst✝ : Preadditive C\na b c : M\nα₁ α₂ : ShiftedHom X Y a\nβ : ShiftedHom Y Z b\nh : b + a = c\n⊢ (α₁ + α₂).comp β h = α₁.comp β h + α₂.comp β h",
"ppTerm": "?m.70",
"assigned": ... | [] | by
rw [comp, comp, comp, Preadditive.add_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 443,
"column": 4
} | {
"line": 444,
"column": 56
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\nH : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIte... | [] | grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
extensionProduct_retractClosure_retractClosure_le] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 443,
"column": 4
} | {
"line": 444,
"column": 56
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\nH : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIte... | [] | grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
extensionProduct_retractClosure_retractClosure_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 443,
"column": 4
} | {
"line": 444,
"column": 56
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\nH : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIte... | [] | grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
extensionProduct_retractClosure_retractClosure_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Localization | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 18
} | {
"line": 126,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nι : Type u_2\nc : ComplexShape ι\ninst✝¹ : Preadditive C\ninst✝ : CategoryWithHomology C\n⊢ ∀ (f g : Arrow (HomotopyCategory C c)) (x : f ≅ g), quasiIso C c f.hom → quasiIso C c g.hom",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nι : Type u_2\nc : ComplexShape ι\ninst✝¹ : Preadditive C\ninst✝ : CategoryWithHomology C\nf g : Arrow (HomotopyCategory C c)\ne : f ≅ g\nhf : quasiIso C c f.hom\ni : ι\n⊢ IsIso ((homologyFunctor C c i).map g.hom)"
] | intro f g e hf i | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Homology.Localization | {
"line": 177,
"column": 12
} | {
"line": 177,
"column": 13
} | {
"line": 177,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nι : Type u_2\nc : ComplexShape ι\ninst✝³ : Preadditive C\ninst✝² : CategoryWithHomology C\ninst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization\ninst✝ : c.QFactorsThroughHomotopy C\nK : HomologicalComplex C c\n⊢ ∀ (y : HomologicalComplex C c) (f₁ f₂ :... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nι : Type u_2\nc : ComplexShape ι\ninst✝³ : Preadditive C\ninst✝² : CategoryWithHomology C\ninst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization\ninst✝ : c.QFactorsThroughHomotopy C\nK L : HomologicalComplex C c\n⊢ ∀ (f₁ f₂ : K ⟶ L), homotopic C c f₁ f₂ → Q.map f₁... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.Localization | {
"line": 260,
"column": 12
} | {
"line": 260,
"column": 13
} | {
"line": 260,
"column": 14
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nC : Type u_2\ninst✝³ : Category.{v_1, u_2} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nE : Type u_3\ninst✝ : Category.{v_2, u_3} E\nF : HomologicalComplex C c ⥤ E\nhF : (HomologicalComplex.homotopyEquivalences C c).IsInver... | [
"ι : Type u_1\nc : ComplexShape ι\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nC : Type u_2\ninst✝³ : Category.{v_1, u_2} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nE : Type u_3\ninst✝ : Category.{v_2, u_3} E\nF : HomologicalComplex C c ⥤ E\nhF : (HomologicalComplex.homotopyEquivalences C c).IsInvertedBy F\nK L... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.DerivedCategory.Basic | {
"line": 245,
"column": 51
} | {
"line": 247,
"column": 16
} | {
"line": 250,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nn : ℤ\n⊢ (singleFunctor C n).Additive",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"instCategoryDerivedCategory",
"DerivedCategory... | [] | by
dsimp [singleFunctor, singleFunctors]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.IsSupported | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝¹ : Category.{v_1, u_3} C\ninst✝ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\nh₁ : K.IsStrictlySupported e\nh₂ : K.IsStrictlySupportedOutside e\nn : ι'\nhn : ¬∃ i, e.f i = n\n⊢ ... | [] | exact K.isZero_X_of_isStrictlySupported e _ (by simpa using hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.Embedding.IsSupported | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝¹ : Category.{v_1, u_3} C\ninst✝ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\nh₁ : K.IsStrictlySupported e\nh₂ : K.IsStrictlySupportedOutside e\nn : ι'\nhn : ¬∃ i, e.f i = n\n⊢ ... | [] | exact K.isZero_X_of_isStrictlySupported e _ (by simpa using hn) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.IsSupported | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝¹ : Category.{v_1, u_3} C\ninst✝ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\nh₁ : K.IsStrictlySupported e\nh₂ : K.IsStrictlySupportedOutside e\nn : ι'\nhn : ¬∃ i, e.f i = n\n⊢ ... | [] | exact K.isZero_X_of_isStrictlySupported e _ (by simpa using hn) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 47,
"column": 30
} | {
"line": 47,
"column": 46
} | {
"line": 47,
"column": 46
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni : Option ι\nj : ι\nhj : i = some j\n⊢ X K i = K.X j",
... | [] | by subst hj; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Opposite | {
"line": 235,
"column": 2
} | {
"line": 236,
"column": 16
} | {
"line": 238,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝² : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms V\nK : HomologicalComplex V c\ni : ι\ninst✝ : K.HasHomology i\n⊢ ((opFunctor V c).obj (op K)).HasHomology i",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"HomologicalC... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Opposite | {
"line": 235,
"column": 2
} | {
"line": 236,
"column": 16
} | {
"line": 238,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝² : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms V\nK : HomologicalComplex V c\ni : ι\ninst✝ : K.HasHomology i\n⊢ ((opFunctor V c).obj (op K)).HasHomology i",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"HomologicalC... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Opposite | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 16
} | {
"line": 244,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝² : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms V\nK : HomologicalComplex Vᵒᵖ c\ni : ι\ninst✝ : K.HasHomology i\n⊢ ((unopFunctor V c).obj (op K)).HasHomology i",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Homologi... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Opposite | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 16
} | {
"line": 244,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝² : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms V\nK : HomologicalComplex Vᵒᵖ c\ni : ι\ninst✝ : K.HasHomology i\n⊢ ((unopFunctor V c).obj (op K)).HasHomology i",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Homologi... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Opposite | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 33
} | {
"line": 292,
"column": 2
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝⁴ : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝³ : HasZeroMorphisms V\nK L : HomologicalComplex V c\nφ : K ⟶ L\ninst✝² : ∀ (i : ι), K.HasHomology i\ninst✝¹ : ∀ (i : ι), L.HasHomology i\ninst✝ : QuasiIso φ\n⊢ QuasiIso ((opFunctor V c).map φ.op)",
"ppTerm": "?m.5... | [
"ι : Type u_1\nV : Type u_2\ninst✝⁴ : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝³ : HasZeroMorphisms V\nK L : HomologicalComplex V c\nφ : K ⟶ L\ninst✝² : ∀ (i : ι), K.HasHomology i\ninst✝¹ : ∀ (i : ι), L.HasHomology i\ninst✝ : QuasiIso φ\n⊢ QuasiIso φ"
] | rw [quasiIso_opFunctor_map_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 36
} | {
"line": 151,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsTruncGE\nj k : ι\nhjk : c.next j = k\nk' : ι\nhj : c.Rel j k'\nhk' : e.f k' = c'.next (e.f j)\n⊢ c'.next (e.f j) = e.f k",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [... | [] | rw [← hk', ← c.next_eq' hj, hjk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 63
} | {
"line": 155,
"column": 4
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni j k : ι\ni' j' k' : ι'\nhj' : e.f j = j'\nhi : c.prev j... | [
"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni j k : ι\ni' j' k' : ι'\nhj' : e.f j = j'\nhi : c.prev j = i\nhi' : ... | rw [← lift_d_comp_eq_zero_iff K e hj' hi hi' hk hk' _ h.hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 210,
"column": 4
} | {
"line": 211,
"column": 29
} | {
"line": 211,
"column": 29
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nK : HomologicalComplex C c\ne : c.Embedding c'\ni j k : ι\ni' j' k' : ι'\nhj' : e.f j = j'\nhi : c.prev j = i\nhi' : c'.prev j' = i'... | [] | simpa using! (isColimitCokernelCofork K e hj' hi hi' cocone hcocone).fac _
WalkingParallelPair.one | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 210,
"column": 4
} | {
"line": 211,
"column": 29
} | {
"line": 211,
"column": 29
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nK : HomologicalComplex C c\ne : c.Embedding c'\ni j k : ι\ni' j' k' : ι'\nhj' : e.f j = j'\nhi : c.prev j = i\nhi' : c'.prev j' = i'... | [] | simpa using! (isColimitCokernelCofork K e hj' hi hi' cocone hcocone).fac _
WalkingParallelPair.one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 210,
"column": 4
} | {
"line": 211,
"column": 29
} | {
"line": 211,
"column": 29
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nK : HomologicalComplex C c\ne : c.Embedding c'\ni j k : ι\ni' j' k' : ι'\nhj' : e.f j = j'\nhi : c.prev j = i\nhi' : c'.prev j' = i'... | [] | simpa using! (isColimitCokernelCofork K e hj' hi hi' cocone hcocone).fac _
WalkingParallelPair.one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Fractions | {
"line": 34,
"column": 91
} | {
"line": 36,
"column": 16
} | {
"line": 38,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\n⊢ (HomotopyCategory.quasiIso C (ComplexShape.up ℤ)).HasLeftCalculusOfFractions",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"CategoryTheory.Abelian.toPr... | [] | by
rw [HomotopyCategory.quasiIso_eq_trW_subcategoryAcyclic]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Fractions | {
"line": 38,
"column": 92
} | {
"line": 40,
"column": 16
} | {
"line": 42,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\n⊢ (HomotopyCategory.quasiIso C (ComplexShape.up ℤ)).HasRightCalculusOfFractions",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"CategoryTheory.Abelian.toP... | [] | by
rw [HomotopyCategory.quasiIso_eq_trW_subcategoryAcyclic]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 324,
"column": 6
} | {
"line": 328,
"column": 20
} | {
"line": 330,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nc : ComplexShape ι\nc₁ : ComplexShape ι₁\nc₂ : ComplexShape ι₂\nC : Type u_4\ninst✝³ : Category.{v_1, u_4} C\ninst✝² : Abelian C\nK : HomologicalComplex C c\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\ninst✝¹ : e₁.IsTruncLE\ninst✝ : e₂.IsTruncGE\nac :... | [] | have := quasiIsoAt_shortComplexTruncLE_g K e₁ (e₂.f i₂) (fun _ => ac.disjoint _ _)
rw [← quasiIsoAt_iff_comp_left (K.shortComplexTruncLE e₁).g
(K.shortComplexTruncLEX₃ToTruncGE ac), g_shortComplexTruncLEX₃ToTruncGE]
dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 324,
"column": 6
} | {
"line": 328,
"column": 20
} | {
"line": 330,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nc : ComplexShape ι\nc₁ : ComplexShape ι₁\nc₂ : ComplexShape ι₂\nC : Type u_4\ninst✝³ : Category.{v_1, u_4} C\ninst✝² : Abelian C\nK : HomologicalComplex C c\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\ninst✝¹ : e₁.IsTruncLE\ninst✝ : e₂.IsTruncGE\nac :... | [] | have := quasiIsoAt_shortComplexTruncLE_g K e₁ (e₂.f i₂) (fun _ => ac.disjoint _ _)
rw [← quasiIsoAt_iff_comp_left (K.shortComplexTruncLE e₁).g
(K.shortComplexTruncLEX₃ToTruncGE ac), g_shortComplexTruncLEX₃ToTruncGE]
dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 200,
"column": 43
} | {
"line": 202,
"column": 5
} | {
"line": 205,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y Z T : C\nf : X ⟶ Y\ng : Y ⟶ Z\nn : ℕ\nα : Ext Z T n\n⊢ (mk₀ f).comp ((mk₀ g).comp α ⋯) ⋯ = (mk₀ (f ≫ g)).comp α ⋯",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory... | [] | by
rw [← mk₀_comp_mk₀, comp_assoc]
lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 16
} | {
"line": 58,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\n⊢ HasSmallLocalizedShiftedHom W ℤ (S.map (CochainComplex.singleFunctor C 0)).X₃\n (S.map (CochainComplex.singleFunctor C 0)).X₁",
"ppTerm": "?m.50",
"assigned": true,
"use... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 16
} | {
"line": 58,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\n⊢ HasSmallLocalizedShiftedHom W ℤ (S.map (CochainComplex.singleFunctor C 0)).X₃\n (S.map (CochainComplex.singleFunctor C 0)).X₁",
"ppTerm": "?m.50",
"assigned": true,
"use... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 37
} | {
"line": 549,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\ninst✝¹ : HasExt C\nX Y : C\nn : ℕ\ninst✝ : HasDerivedCategory C\ne : Ext X Y n\n⊢ homEquiv (chgUniv e) = homEquiv e",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"DerivedCategory.instIsLocalizat... | [] | apply SmallShiftedHom.equiv_chgUniv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 37
} | {
"line": 549,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\ninst✝¹ : HasExt C\nX Y : C\nn : ℕ\ninst✝ : HasDerivedCategory C\ne : Ext X Y n\n⊢ homEquiv (chgUniv e) = homEquiv e",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"DerivedCategory.instIsLocalizat... | [] | apply SmallShiftedHom.equiv_chgUniv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 37
} | {
"line": 549,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasExt C\ninst✝¹ : HasExt C\nX Y : C\nn : ℕ\ninst✝ : HasDerivedCategory C\ne : Ext X Y n\n⊢ homEquiv (chgUniv e) = homEquiv e",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"DerivedCategory.instIsLocalizat... | [] | apply SmallShiftedHom.equiv_chgUniv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 197,
"column": 26
} | {
"line": 197,
"column": 76
} | {
"line": 198,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : HasShift C A\ninst✝ : HasShift D A\na b : A\ne₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\nf₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shif... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : HasShift C A\ninst✝ : HasShift D A\na b : A\ne₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\nf₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b... | ← cancel_mono ((shiftFunctorAdd C a b).hom.app _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 588,
"column": 2
} | {
"line": 589,
"column": 16
} | {
"line": 591,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 588,
"column": 2
} | {
"line": 589,
"column": 16
} | {
"line": 591,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 597,
"column": 2
} | {
"line": 598,
"column": 16
} | {
"line": 600,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 597,
"column": 2
} | {
"line": 598,
"column": 16
} | {
"line": 600,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.ShiftedHomOpposite | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 28
} | {
"line": 150,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasShift C ℤ\nX Y : C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nn : ℤ\nx y : ShiftedHom (Opposite.op Y) (Opposite.op X) n\n⊢ (opEquiv n).symm (x + y) = (opEquiv n).symm x + (opEquiv n).symm y",
"ppTerm": "?m.61",
... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasShift C ℤ\nX Y : C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nn : ℤ\nx y : ShiftedHom (Opposite.op Y) (Opposite.op X) n\n⊢ ((opShiftFunctorEquivalence C n).unitIso.inv.app (Opposite.op X)).unop ≫\n (shiftFunctor C n).map (... | dsimp [opEquiv_symm_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 122,
"column": 74
} | {
"line": 125,
"column": 92
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle Cᵒᵖ\n⊢ T ∈ distinguishedTriangles C ↔ T.rotate ∈ distinguishedTriangles C",
"ppTerm": "?m.4... | [] | by
simp only [mem_distinguishedTriangles_iff, Pretriangulated.rotate_distinguished_triangle
((triangleOpEquivalence C).inverse.obj (T.rotate)).unop]
exact distinguished_iff_of_iso (rotateTriangleOpEquivalenceInverseObjRotateUnopIso T).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 28
} | {
"line": 191,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ (triangleOpEquivalence C).functor.obj (Opposite.op T) ∈ d... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ ∃ T',\n ∃ (_ : T' ∈ distinguishedTriangles),\n Nonempty\n ... | mem_distTriang_op_iff' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.RingQuot | {
"line": 260,
"column": 18
} | {
"line": 260,
"column": 72
} | {
"line": 261,
"column": 2
} | [
{
"pp": "R : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\n⊢ ↑0 = 0",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"RingQuot.instNatCast",
"NonAssocSemiring.toAddCom... | [] | by simp +instances [instNatCast, natCast, ← zero_quot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.RingQuot | {
"line": 421,
"column": 11
} | {
"line": 421,
"column": 35
} | {
"line": 421,
"column": 36
} | [
{
"pp": "B : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\n⊢ (ringQuotToIdealQuotient r) ((mkRingHom r) x) = (Ideal.Quotient.mk (Ideal.ofRel r)) x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal.Quotient.mk",
"RingHom",
"Rin... | [
"B : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\n⊢ (lift ⟨Ideal.Quotient.mk (Ideal.ofRel r), ⋯⟩) ((mkRingHom r) x) = (Ideal.Quotient.mk (Ideal.ofRel r)) x"
] | ringQuotToIdealQuotient, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 317,
"column": 4
} | {
"line": 318,
"column": 65
} | {
"line": 319,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG✝ : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G✝\nG : Type u_7\ninst✝¹ : CommGroup G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nG' : Type u_8\ninst✝ : CommGroup G'\nζ : ↥(rootsOfUnity (... | [] | simpa only [orderOf_eq_card_of_forall_mem_zpowers hg, orderOf_dvd_iff_pow_eq_one,
← Units.val_pow_eq_pow_val, Units.val_eq_one] using! ζ.prop | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 317,
"column": 4
} | {
"line": 318,
"column": 65
} | {
"line": 319,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG✝ : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G✝\nG : Type u_7\ninst✝¹ : CommGroup G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nG' : Type u_8\ninst✝ : CommGroup G'\nζ : ↥(rootsOfUnity (... | [] | simpa only [orderOf_eq_card_of_forall_mem_zpowers hg, orderOf_dvd_iff_pow_eq_one,
← Units.val_pow_eq_pow_val, Units.val_eq_one] using! ζ.prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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