module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1125,
"column": 6
} | {
"line": 1125,
"column": 86
} | {
"line": 1126,
"column": 6
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\n⊢ Function.Bijective fun e ↦ { carrier := ↑(basicOpen (↑e).1), isClopen' := ⋯ }",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"PrimeSpectrum.isClopen_iff_mul_add",
"NonAssocSemi... | [
"case refine_1\nR : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nx✝¹ x✝ : { e // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }\nx : R × R\nhx : x.1 * x.2 = 0 ∧ x.1 + x.2 = 1\ny : R × R\nhy : y.1 * y.2 = 0 ∧ y.1 + y.2 = 1\neq :\n (fun e ↦ { carrier := ↑(basicOpen (↑e).1), isClopen' := ⋯ }) ⟨x, hx⟩ =\n ... | refine ⟨fun ⟨x, hx⟩ ⟨y, hy⟩ eq ↦ mul_eq_zero_add_eq_one_ext_left ?_, fun s ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.Rank | {
"line": 87,
"column": 95
} | {
"line": 88,
"column": 14
} | {
"line": 90,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Group.FG ↥H\ninst✝ : Group.FG ↥K\nh : H = K\n⊢ rank ↥H = rank ↥K",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Group.FG",
"Membership.mem",
"Subtype",
"Subgroup",
"Nat",
"Eq.ndrec",
... | [] | by
subst h; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 195,
"column": 61
} | {
"line": 195,
"column": 88
} | {
"line": 196,
"column": 4
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [] | by rw [mem_center_iff.1 ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Sylow | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 55
} | {
"line": 512,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : N.Normal\ninst✝ : Finite (Sylow p ↥N)\nP : Sylow p ↥N\ng : G\nx✝ : g ∈ ⊤\nn : ↥N\nhn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P\nthis : Function.Injective ⇑(Mu... | [
"G : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : N.Normal\ninst✝ : Finite (Sylow p ↥N)\nP : Sylow p ↥N\ng : G\nx✝ : g ∈ ⊤\nn : ↥N\nhn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P\nthis : Function.Injective ⇑(MulEquiv.toMon... | rw [map_map, ← congr_arg (map N.subtype) hn, map_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 547,
"column": 48
} | {
"line": 550,
"column": 37
} | {
"line": 552,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝³ : Infinite G\ninst✝² : CommGroup G\ninst✝¹ : PartialOrder G\ninst✝ : IsOrderedMonoid G\ng : G\nhg : zpowers g = ⊤\nhg1 : g < 1\n⊢ StrictAnti ⇑(intEquivOfZPowersEqTop g hg)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"zpow_right_strictAnti",
"in... | [] | by
intro x y hxy
simp only [intEquivOfZPowersEqTop, MulEquiv.ofBijective_apply, zpowersHom_apply]
exact zpow_right_strictAnti hg1 hxy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 116,
"column": 42
} | {
"line": 116,
"column": 60
} | {
"line": 116,
"column": 60
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : Submodule R M\n⊢ IsSimpleModule R ↥m ↔ IsSimpleOrder ↑(Set.Iic m)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"AddCommGroup.... | [
"R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : Submodule R M\n⊢ IsSimpleOrder (Submodule R ↥m) ↔ IsSimpleOrder ↑(Set.Iic m)"
] | isSimpleModule_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 120,
"column": 44
} | {
"line": 120,
"column": 62
} | {
"line": 120,
"column": 62
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : Submodule R M\n⊢ IsSimpleModule R (M ⧸ m) ↔ IsSimpleOrder ↑(Set.Ici m)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Set.Ici.boundedOrder",
... | [
"R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : Submodule R M\n⊢ IsSimpleOrder (Submodule R (M ⧸ m)) ↔ IsSimpleOrder ↑(Set.Ici m)"
] | isSimpleModule_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 396,
"column": 56
} | {
"line": 396,
"column": 76
} | {
"line": 396,
"column": 77
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : R[X]\nq : PolynomialModule R M\n⊢ Polynomial.eval p (Polynomial.map (algebraMap R R[X]) p') • (eval p) ((map R[X] (lsingle R 0)) q) =\n eval₂ C p p' • (comp p) q",
"ppTerm": "?m.93",
"assigne... | [
"R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : R[X]\nq : PolynomialModule R M\n⊢ eval₂ (algebraMap R R[X]) p p' • (eval p) ((map R[X] (lsingle R 0)) q) = eval₂ C p p' • (comp p) q"
] | Polynomial.eval_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nakayama | {
"line": 62,
"column": 62
} | {
"line": 70,
"column": 59
} | {
"line": 72,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN : Submodule R M\nhN : N.FG\nhIN : N ≤ I • N\nhIjac : I ≤ J.jacobson\n⊢ N = J • N",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"N... | [] | by
refine le_antisymm ?_ (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _)
intro n hn
obtain ⟨r, hr⟩ := Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hN hIN
obtain ⟨s, hs⟩ := exists_mul_sub_mem_of_sub_one_mem_jacobson r (hIjac hr.1)
have : n = -(s * r - 1) • n := by
rw [neg_su... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 873,
"column": 75
} | {
"line": 883,
"column": 29
} | {
"line": 885,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Monoid R\ninst✝¹ : AddCommMonoid M\ninst✝ : DistribMulAction R M\np : R\n⊢ IsTorsion' M ↥(Submonoid.powers p) ↔ ∀ (x : M), ∃ n, p ^ n • x = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Monoid.toMul... | [] | by
constructor
· intro h x
let ⟨⟨a, ⟨n, hn⟩⟩, hx⟩ := @h x
dsimp at hn
use n
rw [hn]
apply hx
· intro h x
let ⟨n, hn⟩ := h x
exact ⟨⟨_, ⟨n, rfl⟩⟩, hn⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Filtration | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 29
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case h.succ\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\ne : F.N 0 ≤ F'.N 0\nn₀ : ℕ\nhF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)\nn : ℕ\nhn : F.N (n + n₀) ≤ F'.N n\n⊢ F.N (n + 1 + n₀) ≤ F'.N (n + 1)",
"ppTerm": "?h.suc... | [
"case h.succ\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\ne : F.N 0 ≤ F'.N 0\nn₀ : ℕ\nhF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)\nn : ℕ\nhn : F.N (n + n₀) ≤ F'.N n\n⊢ I • F.N (n + n₀) ≤ F'.N (n + 1)",
"case h.succ.a\nR : Type u_1\nM... | rw [add_right_comm, ← hF] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Nakayama | {
"line": 224,
"column": 2
} | {
"line": 229,
"column": 46
} | {
"line": 231,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns : Set (M ⧸ I • N)\nhN : N.FG\nhIjac : I ≤ ⊥.jacobson\nhsspan : span R s = map (I • N).mkQ N\n⊢ ∃ t, Set.InjOn (⇑(I • N).mkQ) t ∧ ⇑(I • N).mkQ '' t = s ∧ span R t = N",
"pp... | [] | use Quotient.out '' s
split_ands
· simp [Set.InjOn]
· simp [Set.image_image]
· symm; apply eq_of_map_mkQ_eq_map_mkQ_of_le_jacobson_bot hN hIjac
simp [← hsspan, map_span, Set.image_image] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nakayama | {
"line": 224,
"column": 2
} | {
"line": 229,
"column": 46
} | {
"line": 231,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns : Set (M ⧸ I • N)\nhN : N.FG\nhIjac : I ≤ ⊥.jacobson\nhsspan : span R s = map (I • N).mkQ N\n⊢ ∃ t, Set.InjOn (⇑(I • N).mkQ) t ∧ ⇑(I • N).mkQ '' t = s ∧ span R t = N",
"pp... | [] | use Quotient.out '' s
split_ands
· simp [Set.InjOn]
· simp [Set.image_image]
· symm; apply eq_of_map_mkQ_eq_map_mkQ_of_le_jacobson_bot hN hIjac
simp [← hsspan, map_span, Set.image_image] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nakayama | {
"line": 238,
"column": 56
} | {
"line": 238,
"column": 78
} | {
"line": 238,
"column": 78
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nI : Ideal R\nIle : I ≤ ⊥.jacobson\nsurj : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\n⊢ Submodule.map (I • ⊤)... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nI : Ideal R\nIle : I ≤ ⊥.jacobson\nsurj : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\n⊢ ((I • ⊤).mkQ ∘ₗ f).range = ⊤"
] | ← LinearMap.range_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 86,
"column": 43
} | {
"line": 86,
"column": 64
} | {
"line": 86,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ Submodule.map (Submodule.subtype I) (I • ⊤) ≤ I ^ 2",
"ppTerm": "?m.80",
... | [
"R : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\n⊢ I • Submodule.map (Submodule.subtype I) ⊤ ≤ I ^ 2"
] | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 188,
"column": 16
} | {
"line": 188,
"column": 65
} | {
"line": 188,
"column": 65
} | [
{
"pp": "case a\nR : Type u\ninst✝⁴ : CommRing R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A ⧸ RingHom.ker f.toRingHom ^ 2\nhx : x ∈ RingHom.ker f.kerSquareLift.toRingHom\n⊢ x ∈ (RingHom.ker f.toRingHom).cotangentIdeal",
... | [
"case a\nR : Type u\ninst✝⁴ : CommRing R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A\nhx : (Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x ∈ RingHom.ker f.kerSquareLift.toRingHom\n⊢ (Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Filtration | {
"line": 398,
"column": 2
} | {
"line": 405,
"column": 29
} | {
"line": 406,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nx : M\nN : Submodule R M := ⋯\nhN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N\n⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r,... | [
"case mpr\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nx : M\nN : Submodule R M := ⋯\nhN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N\n⊢ (∃ r, ↑r • x = x) → x ∈ ⨅ i, I ^ i •... | · obtain ⟨r, hr₁, hr₂⟩ :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 345,
"column": 20
} | {
"line": 345,
"column": 62
} | {
"line": 345,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM✝ M₁ M₂ M N : PresheafOfModules R\nf : M ⟶ N\ns : M.sections\nX Y : Cᵒᵖ\ng : X ⟶ Y\n⊢ (ConcreteCategory.hom (N.map g)) ((ConcreteCategory.hom (f.app X)) (↑s X)) = (ConcreteCategory.hom (f.app Y)) (↑s Y)",
"ppTerm": "?m.39",
"assigned... | [] | rw [← naturality_apply, sections_property] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 345,
"column": 20
} | {
"line": 345,
"column": 62
} | {
"line": 345,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM✝ M₁ M₂ M N : PresheafOfModules R\nf : M ⟶ N\ns : M.sections\nX Y : Cᵒᵖ\ng : X ⟶ Y\n⊢ (ConcreteCategory.hom (N.map g)) ((ConcreteCategory.hom (f.app X)) (↑s X)) = (ConcreteCategory.hom (f.app Y)) (↑s Y)",
"ppTerm": "?m.39",
"assigned... | [] | rw [← naturality_apply, sections_property] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 345,
"column": 20
} | {
"line": 345,
"column": 62
} | {
"line": 345,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM✝ M₁ M₂ M N : PresheafOfModules R\nf : M ⟶ N\ns : M.sections\nX Y : Cᵒᵖ\ng : X ⟶ Y\n⊢ (ConcreteCategory.hom (N.map g)) ((ConcreteCategory.hom (f.app X)) (↑s X)) = (ConcreteCategory.hom (f.app Y)) (↑s Y)",
"ppTerm": "?m.39",
"assigned... | [] | rw [← naturality_apply, sections_property] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 456,
"column": 22
} | {
"line": 458,
"column": 32
} | {
"line": 460,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM✝ M₁ M₂ : PresheafOfModules R\nX✝ : Cᵒᵖ\nhX✝ : Limits.IsInitial X✝\nX : Cᵒᵖ\nhX : Limits.IsInitial X\nM N : PresheafOfModules R\nf : M ⟶ N\nY Z : Cᵒᵖ\ng : Y ⟶ Z\n⊢ (forgetToPresheafModuleCatObj X hX M).map g ≫\n ModuleCat.ofHom { toFun ... | [] | by
ext x
exact naturality_apply f g x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.End | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 70
} | {
"line": 225,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nn : M\nX : C\ninst✝ : F.Monoidal\n⊢ (F.obj n).map ((η F).app X) = (μ F (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X",
"ppTerm": "?m.75",
"assigned": true,
"usedCo... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nn : M\nX : C\ninst✝ : F.Monoidal\n⊢ (F.obj n).map ((η F).app X ≫ (ε F).app X) =\n ((μ F (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X) ≫ (F.obj n).map ((ε F).app X)"
] | rw [← cancel_mono ((F.obj n).map ((ε F).app X)), ← Functor.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 21
} | {
"line": 105,
"column": 22
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ 𝟙 ((h.F m₃).obj ((h.F m₁ ⋙ h.F m₂).obj X)) =\n (h.add m₂ m₃).inv.app ((h.F m₁).obj X) ≫\n (h.add m₁ (m₂ + m₃)).inv.app X ≫\n 𝟙 ((h.F (m₁ + (m₂ + m₃))).obj X) ≫ (h.add ... | [
"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ 𝟙 ((h.F m₃).obj ((h.F m₁ ⋙ h.F m₂).obj X)) =\n (h.add m₂ m₃).inv.app ((h.F m₁).obj X) ≫\n (h.add m₁ (m₂ + m₃)).inv.app X ≫ (h.add m₁ (m₂ + m₃)).hom.app X ≫ (h.add m₂ m₃).hom.app ((h.F m₁)... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 21
} | {
"line": 114,
"column": 22
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n⊢ 𝟙 ((h.F (0 + n)).obj X) = eqToHom ⋯ ≫ 𝟙 ((h.F n).obj ((𝟭 C).obj X)) ≫ eqToHom ⋯",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory... | [
"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n⊢ 𝟙 ((h.F (0 + n)).obj X) = eqToHom ⋯ ≫ eqToHom ⋯"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Single | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 54
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case neg\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nK : HomologicalComplex V c\nj : ι\nA : V\nφ : K.X j ⟶ A\nhφ : ∀ (i : ι), c.Rel i j → K.d i j ≫ φ = 0\ni k : ι\nhik : c.Rel i k\nhk : ¬k = j\n... | [] | · apply (isZero_single_obj_X c j A k hk).eq_of_tgt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 570,
"column": 20
} | {
"line": 570,
"column": 37
} | {
"line": 570,
"column": 38
} | [
{
"pp": "case e_a\nC : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX : C\nm n p m' n' p' : A\nhm : m' + m = 0\nhn : n' + n = 0\nhp : p' + p = 0\nh : m + n = p\n⊢ 𝟙 ((shiftFunctor C 0).obj X) =\n (shiftFunctorAdd' C n' n 0 hn).hom.app X ≫\n 𝟙 ((shiftFunc... | [
"case e_a\nC : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX : C\nm n p m' n' p' : A\nhm : m' + m = 0\nhn : n' + n = 0\nhp : p' + p = 0\nh : m + n = p\n⊢ 𝟙 ((shiftFunctor C 0).obj X) = (shiftFunctorAdd' C n' n 0 hn).hom.app X ≫ (shiftFunctorAdd' C n' n 0 ⋯).inv.app ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 56
} | {
"line": 75,
"column": 57
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝¹ : HasZeroMorphisms C\nF : J ⥤ HomologicalComplex C c\ninst✝ : ∀ (n : ι), HasLimit (F ⋙ eval C c n)\nn m : ι\nh : ¬c.Rel n m\nj : J\n⊢ limit.π (F ⋙ eval C c n) j ≫ (F.obj ... | [] | rw [(F.obj j).shape _ _ h, comp_zero, zero_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 587,
"column": 22
} | {
"line": 587,
"column": 39
} | {
"line": 587,
"column": 40
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX : C\nm n p m' n' p' : A\nhm : m' + m = 0\nhn : n' + n = 0\nhp : p' + p = 0\nh : m + n = p\n⊢ 𝟙 ((shiftFunctor C p).obj ((shiftFunctor C p').obj X)) =\n (shiftFunctor C p).map ((shiftFunctorAdd' C n' m... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX : C\nm n p m' n' p' : A\nhm : m' + m = 0\nhn : n' + n = 0\nhp : p' + p = 0\nh : m + n = p\n⊢ 𝟙 ((shiftFunctor C p).obj ((shiftFunctor C p').obj X)) =\n (shiftFunctor C p).map ((shiftFunctorAdd' C n' m' p' ⋯).hom.... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 808,
"column": 26
} | {
"line": 808,
"column": 43
} | {
"line": 808,
"column": 44
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝³ : Category.{v, u} C\nD : Type u_2\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : AddMonoid A\ninst✝ : HasShift D A\nF : C ⥤ D\nhF : F.FullyFaithful\ns : A → C ⥤ C\ni : (i : A) → s i ⋙ F ≅ F ⋙ shiftFunctor D i\nm₁ m₂ m₃ : A\nX : C\nh :\n (shiftFunctorAdd D (m₁ + m₂) m₃).hom.... | [
"C : Type u\nA : Type u_1\ninst✝³ : Category.{v, u} C\nD : Type u_2\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : AddMonoid A\ninst✝ : HasShift D A\nF : C ⥤ D\nhF : F.FullyFaithful\ns : A → C ⥤ C\ni : (i : A) → s i ⋙ F ≅ F ⋙ shiftFunctor D i\nm₁ m₂ m₃ : A\nX : C\nh :\n (shiftFunctorAdd D (m₁ + m₂) m₃).hom.app (F.obj X... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 32
} | {
"line": 92,
"column": 0
} | [
{
"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 : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\ni : ι\nhG : IsZero (G.X i)\nhF : ∀ (j : ι), c.Rel i j → IsZero (F.X j)\nh : ¬c.Rel i (c.n... | [] | · exact hG.of_iso (XIso φ i h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Homotopy | {
"line": 366,
"column": 31
} | {
"line": 366,
"column": 54
} | {
"line": 366,
"column": 55
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₁ k₀ : ι\nr₁₀ : c.Rel k₁ k₀\nhk₀ : ∀ (l : ι), ¬c.Rel k₀ l\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ (AddMonoidHom.mk' (fun f ↦ C.d k₀ (c.next k₀) ≫ f (c.next k₀) k₀) ⋯) hom + hom k₀... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₁ k₀ : ι\nr₁₀ : c.Rel k₁ k₀\nhk₀ : ∀ (l : ι), ¬c.Rel k₀ l\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀"
] | AddMonoidHom.mk'_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 383,
"column": 31
} | {
"line": 383,
"column": 54
} | {
"line": 383,
"column": 55
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₁ k₀ : ι\nr₁₀ : c.Rel k₁ k₀\nhk₁ : ∀ (l : ι), ¬c.Rel l k₁\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ C.d k₁ k₀ ≫ hom k₀ k₁ + (AddMonoidHom.mk' (fun f ↦ f k₁ (c.prev k₁) ≫ D.d (c.prev... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₁ k₀ : ι\nr₁₀ : c.Rel k₁ k₀\nhk₁ : ∀ (l : ι), ¬c.Rel l k₁\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ (c.prev k₁) ≫ D.d (c.prev k₁) k₁ = C.d k₁ k₀ ≫ hom k₀ k₁"
] | AddMonoidHom.mk'_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 41
} | {
"line": 402,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\nK : HomologicalComplex C c\nx₁ : G ⟶ K\nx₂ : Homotopy (φ ≫ x₁) 0\ny₂ : Homotopy (φ ≫ ⟨x₁, x₂⟩.fst) 0\nh : ∀ (i j : ι), c.Rel j i → ⟨x₁, x₂⟩.snd.hom i... | [] | · simp only [Homotopy.zero _ _ _ hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Homotopy | {
"line": 804,
"column": 6
} | {
"line": 805,
"column": 44
} | {
"line": 806,
"column": 4
} | [
{
"pp": "case pos\nι✝ : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : Preadditive V\nc✝ : ComplexShape ι✝\nC✝ D E : HomologicalComplex V c✝\nf✝ g✝ : C✝ ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι✝\nC : Type u_2\ninst✝² : Category.{v_1, u_2} C\ninst✝¹ : Preadditive C\nι : Type ?u.42\nc : ComplexShape ι\ninst✝ : Decid... | [] | dsimp
simp only [assoc, d_comp_d, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Homotopy | {
"line": 804,
"column": 6
} | {
"line": 805,
"column": 44
} | {
"line": 806,
"column": 4
} | [
{
"pp": "case pos\nι✝ : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : Preadditive V\nc✝ : ComplexShape ι✝\nC✝ D E : HomologicalComplex V c✝\nf✝ g✝ : C✝ ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι✝\nC : Type u_2\ninst✝² : Category.{v_1, u_2} C\ninst✝¹ : Preadditive C\nι : Type ?u.42\nc : ComplexShape ι\ninst✝ : Decid... | [] | dsimp
simp only [assoc, d_comp_d, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Quotient | {
"line": 239,
"column": 77
} | {
"line": 242,
"column": 43
} | {
"line": 244,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\n⊢ Congruence fun X Y ↦ Relation.EqvGen (HomRel.CompClosure r)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.Quotient",
"Quiver.Ho... | [] | by
convert! (inferInstance : Congruence (functor r).homRel)
ext
rw [functor_homRel_eq_compClosure_eqvGen] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 652,
"column": 6
} | {
"line": 655,
"column": 49
} | {
"line": 655,
"column": 49
} | [
{
"pp": "case right\nC : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v, u_1} C\ninst✝⁵ : Category.{v', u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (... | [] | simp only [comp_add, add_comp, assoc, inl_v_fst_v_assoc, inr_f_fst_v_assoc,
Functor.mapHomologicalComplex_obj_X, zero_comp, comp_zero, add_zero, inl_v_snd_v_assoc,
inr_f_snd_v_assoc, zero_add, inl_v_snd_v, inr_f_snd_v, comp_id, ← H.map_comp,
d_snd_v φ n (n + 1) rfl, Functor.map_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 77
} | {
"line": 155,
"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... | simp only [Induced.add_hom_app_obj, Category.assoc, Functor.map_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 60,
"column": 37
} | {
"line": 60,
"column": 75
} | {
"line": 60,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n' + a = n\np q : ℤ\nhpq : p + n' = q\np' : ℤ\nhp' : p + n = p'\n⊢ p' = q + a",
"ppTerm": "?m.73",
"assi... | [] | by rw [← hp', ← hpq, ← hn', add_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 65
} | {
"line": 363,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L : CochainComplex C ℤ\nn' a : ℤ\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'\nn : ℤ\nhn : n' + a = n\nx : R\n⊢ (x • γ).rightUnshift n hn = x • γ.rightU... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L : CochainComplex C ℤ\nn' a : ℤ\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'\nn : ℤ\nhn : n' + a = n\nx : R\n⊢ (rightShiftLinearEquiv R K L n a n' hn).symm (x • γ)... | change (rightShiftLinearEquiv R K L n a n' hn).symm (x • γ) = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 239,
"column": 28
} | {
"line": 239,
"column": 45
} | {
"line": 239,
"column": 46
} | [
{
"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... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 320,
"column": 59
} | {
"line": 324,
"column": 75
} | {
"line": 326,
"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\n⊢ T.mor₃ = 0 ↔ Mono T.mor₁",
"ppTerm": "?m.44",
... | [] | by
have h := mor₁_eq_zero_iff_mono₂ _ (inv_rot_of_distTriang _ hT)
dsimp at h
rw [← h, neg_eq_zero, IsIso.comp_right_eq_zero]
exact (Functor.map_eq_zero_iff (CategoryTheory.shiftFunctor C (-1))).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 415,
"column": 14
} | {
"line": 415,
"column": 26
} | {
"line": 416,
"column": 4
} | [
{
"pp": "case ofNat.zero\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\nH : ℤ → Prop := fun n ↦ ∀ T ∈ distinguishedTriangles, (shiftFunctor C n).obj T ∈ disti... | [] | exact H_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 415,
"column": 14
} | {
"line": 415,
"column": 26
} | {
"line": 416,
"column": 4
} | [
{
"pp": "case ofNat.zero\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\nH : ℤ → Prop := fun n ↦ ∀ T ∈ distinguishedTriangles, (shiftFunctor C n).obj T ∈ disti... | [] | exact H_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 415,
"column": 14
} | {
"line": 415,
"column": 26
} | {
"line": 416,
"column": 4
} | [
{
"pp": "case ofNat.zero\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\nH : ℤ → Prop := fun n ↦ ∀ T ∈ distinguishedTriangles, (shiftFunctor C n).obj T ∈ disti... | [] | exact H_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 449,
"column": 2
} | {
"line": 450,
"column": 6
} | {
"line": 452,
"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\nh : IsZero T.obj₂\n⊢ T ∈ distinguishedTriangles ↔ IsIso T.mor₃",
"ppTerm": "?m.55",... | [] | rw [rotate_distinguished_triangle, distinguished_iff_of_isZero₁ _ h]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 449,
"column": 2
} | {
"line": 450,
"column": 6
} | {
"line": 452,
"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\nh : IsZero T.obj₂\n⊢ T ∈ distinguishedTriangles ↔ IsIso T.mor₃",
"ppTerm": "?m.55",... | [] | rw [rotate_distinguished_triangle, distinguished_iff_of_isZero₁ _ h]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 493,
"column": 13
} | {
"line": 493,
"column": 15
} | {
"line": 493,
"column": 16
} | [
{
"pp": "case refine_1\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_1\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... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 36
} | {
"line": 590,
"column": 0
} | [
{
"pp": "case h₁\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 : Triangle C\nhT : T ∈ distinguishedTriangles\nzero : T.mor₃ = 0\nthis✝ : Epi T.mor₂\nthis : IsSplitEpi T.m... | [] | · simpa [d] using d.bicone.inl_snd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 37
} | {
"line": 150,
"column": 2
} | [
{
"pp": "case h\nC : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ⋯\nS' : ShortComplex C := ⋯\nι : S ⟶ S' := ⋯\nhS : S.Exact\nT : ShortComplex C :... | [] | exact hS.exact_toComposableArrows | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 37
} | {
"line": 150,
"column": 2
} | [
{
"pp": "case h\nC : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ⋯\nS' : ShortComplex C := ⋯\nι : S ⟶ S' := ⋯\nhS : S.Exact\nT : ShortComplex C :... | [] | exact hS.exact_toComposableArrows | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 37
} | {
"line": 150,
"column": 2
} | [
{
"pp": "case h\nC : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ⋯\nS' : ShortComplex C := ⋯\nι : S ⟶ S' := ⋯\nhS : S.Exact\nT : ShortComplex C :... | [] | exact hS.exact_toComposableArrows | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 184,
"column": 38
} | {
"line": 184,
"column": 55
} | {
"line": 184,
"column": 56
} | [
{
"pp": "case h_map.nil\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯\nX Y : Paths V\n⊢ 𝟙 (F.obj X) = eqToHom ⋯ ≫ 𝟙 (G.obj X) ≫ eqToHom ⋯",
"ppTerm... | [
"case h_map.nil\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nF G : Paths V ⥤ C\nh_obj : F.obj = G.obj\nh : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯\nX Y : Paths V\n⊢ 𝟙 (F.obj X) = eqToHom ⋯ ≫ eqToHom ⋯"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Equivalence | {
"line": 78,
"column": 2
} | {
"line": 81,
"column": 70
} | {
"line": 82,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\ninst✝¹ : Category.{v_3, u_3} D\nL₁ : C₁ ⥤ D\nW₁ : MorphismProperty C₁\nL₂ : C₂ ⥤ D\nW₂ : MorphismProperty C₂\nE : C₁ ≌ C₂\nhW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor\nhW₂ : W₂.IsInvertedBy... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\ninst✝¹ : Category.{v_3, u_3} D\nL₁ : C₁ ⥤ D\nW₁ : MorphismProperty C₁\nL₂ : C₂ ⥤ D\nW₂ : MorphismProperty C₂\nE : C₁ ≌ C₂\nhW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor\nhW₂ : W₂.IsInvertedBy L₂\ninst✝ :... | have h : W₁.IsInvertedBy (E.functor ⋙ W₂.Q) := fun _ _ f hf => by
obtain ⟨_, _, f', hf', ⟨e⟩⟩ := hW₁ f hf
exact ((MorphismProperty.isomorphisms _).arrow_mk_iso_iff
(W₂.Q.mapArrow.mapIso e)).1 (Localization.inverts W₂.Q W₂ _ hf') | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 89
} | {
"line": 226,
"column": 8
} | [
{
"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\nX✝ Y✝ : D ⥤ E\nτ : X✝ ⟶ Y✝\nx✝ : C\n⊢ (((whiskeringLeft W.Localization D E).obj (equivalenceFromModel ... | [
"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\nX✝ Y✝ : D ⥤ E\nτ : X✝ ⟶ Y✝\nx✝ : C\n⊢ τ.app (L.obj (W.Q.obj x✝).as.obj) ≫ (eqToHom ⋯).hom.app x✝ =\n (eqToHom ⋯... | dsimp [Construction.whiskeringLeftEquivalence, equivalenceFromModel, whiskerLeft] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions | {
"line": 254,
"column": 2
} | {
"line": 262,
"column": 24
} | {
"line": 264,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"CategoryTheory.MorphismPrope... | [] | obtain ⟨ψ₁, hψ₁⟩ := φ.fst.exists_leftFraction
obtain ⟨ψ₂, hψ₂⟩ := φ.snd.exists_leftFraction
obtain ⟨α, hα⟩ := (RightFraction.mk _ ψ₁.hs ψ₂.s).exists_leftFraction
dsimp at hψ₁ hψ₂ hα
refine ⟨LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s)
(W.comp_mem _ _ ψ₂.hs α.hs), ?_, ?_⟩
· dsimp
rw [hα, r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions | {
"line": 254,
"column": 2
} | {
"line": 262,
"column": 24
} | {
"line": 264,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"CategoryTheory.MorphismPrope... | [] | obtain ⟨ψ₁, hψ₁⟩ := φ.fst.exists_leftFraction
obtain ⟨ψ₂, hψ₂⟩ := φ.snd.exists_leftFraction
obtain ⟨α, hα⟩ := (RightFraction.mk _ ψ₁.hs ψ₂.s).exists_leftFraction
dsimp at hψ₁ hψ₂ hα
refine ⟨LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s)
(W.comp_mem _ _ ψ₂.hs α.hs), ?_, ?_⟩
· dsimp
rw [hα, r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Localization | {
"line": 215,
"column": 26
} | {
"line": 215,
"column": 43
} | {
"line": 215,
"column": 44
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹⁰ : Category.{v₁, u₁} C\ninst✝⁹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝⁸ : Category.{v₃, u₃} E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁷ : L.IsLocalization W\nA : Type w\ninst✝⁶ : AddMonoid A\ninst✝⁵ : HasShift C A\nF : C ⥤ E\nF' : D ⥤ E\ninst✝⁴ : Lifting L W F F'\nin... | [
"C : Type u₁\nD : Type u₂\ninst✝¹⁰ : Category.{v₁, u₁} C\ninst✝⁹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝⁸ : Category.{v₃, u₃} E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁷ : L.IsLocalization W\nA : Type w\ninst✝⁶ : AddMonoid A\ninst✝⁵ : HasShift C A\nF : C ⥤ E\nF' : D ⥤ E\ninst✝⁴ : Lifting L W F F'\ninst✝³ : HasSh... | Category.id_comp, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 421,
"column": 6
} | {
"line": 423,
"column": 58
} | {
"line": 424,
"column": 4
} | [
{
"pp": "case refine_2.refine_2.refine_3\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z✝ : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z✝\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z✝\nU : C\nt₁ : a₁.Y... | [] | · simp only [p₁, assoc, ← reassoc_of% fac₃]
exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z₂.hs
(W.comp_mem _ _ w₂.hs (W.comp_mem _ _ q.hs hu))) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 455,
"column": 4
} | {
"line": 459,
"column": 32
} | {
"line": 460,
"column": 2
} | [
{
"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\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\n⊢ ∀ {X Y : Localization W} (f : Hom W X Y), f.comp (Hom.mk (ofHom W (𝟙 Y))) = f",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants":... | [] | rintro (X Y : C) f
obtain ⟨z, rfl⟩ := Hom.mk_surjective f
rw [Hom.comp_eq, comp_eq z (ofHom W (𝟙 Y)) (ofInv z.s z.hs) (by simp)]
dsimp [comp₀]
simp only [comp_id, id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 455,
"column": 4
} | {
"line": 459,
"column": 32
} | {
"line": 460,
"column": 2
} | [
{
"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\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\n⊢ ∀ {X Y : Localization W} (f : Hom W X Y), f.comp (Hom.mk (ofHom W (𝟙 Y))) = f",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants":... | [] | rintro (X Y : C) f
obtain ⟨z, rfl⟩ := Hom.mk_surjective f
rw [Hom.comp_eq, comp_eq z (ofHom W (𝟙 Y)) (ofInv z.s z.hs) (by simp)]
dsimp [comp₀]
simp only [comp_id, id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 57
} | {
"line": 107,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : P.IsTriangulatedClosed₁\nY Z : C\nb : Y ⟶ Z\nhY : P Y\nhZ : P Z\n⊢ ∃ X, ∃ (_ ... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : P.IsTriangulatedClosed₁\nY Z : C\nb : Y ⟶ Z\nhY : P Y\nhZ : P Z\nX : C\na : X ⟶ Y\nc : Z ... | obtain ⟨X, a, c, h⟩ := distinguished_cocone_triangle₁ b | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 70
} | {
"line": 234,
"column": 0
} | [
{
"pp": "case neg\nι : 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 : HomologicalComplex C c\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap (𝟙 K) e).f i' = 𝟙 ((K.exte... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 70
} | {
"line": 234,
"column": 0
} | [
{
"pp": "case neg\nι : 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 : HomologicalComplex C c\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap (𝟙 K) e).f i' = 𝟙 ((K.exte... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 70
} | {
"line": 234,
"column": 0
} | [
{
"pp": "case neg\nι : 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 : HomologicalComplex C c\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap (𝟙 K) e).f i' = 𝟙 ((K.exte... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 70
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case neg\nι : 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 : HomologicalComplex C c\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap 0 e).f i' = Hom.f 0 i'",
... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 70
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case neg\nι : 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 : HomologicalComplex C c\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap 0 e).f i' = Hom.f 0 i'",
... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 70
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case neg\nι : 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 : HomologicalComplex C c\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap 0 e).f i' = Hom.f 0 i'",
... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 40
} | {
"line": 108,
"column": 2
} | [
{
"pp": "case left\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\nk' : ι'\nj : ι\nhj : c'.Rel (e.f j) k'\nhk' : ∀ (i : ι), e.f i ≠ k'\n⊢ c'.Rel (e.f j) (c'.next (e.f j))",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"cong... | [
"case right\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\nk' : ι'\nj : ι\nhj : c'.Rel (e.f j) k'\nhk' : ∀ (i : ι), e.f i ≠ k'\n⊢ ∀ (k : ι), ¬c'.Rel (e.f j) (e.f k)"
] | · simpa only [c'.next_eq' hj] using hj | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 70
} | {
"line": 274,
"column": 0
} | [
{
"pp": "case neg\nι : 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✝ : Preadditive C\nK L : HomologicalComplex C c\nφ φ' : K ⟶ L\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap (φ + φ') e).f i' ... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 70
} | {
"line": 274,
"column": 0
} | [
{
"pp": "case neg\nι : 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✝ : Preadditive C\nK L : HomologicalComplex C c\nφ φ' : K ⟶ L\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap (φ + φ') e).f i' ... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 70
} | {
"line": 274,
"column": 0
} | [
{
"pp": "case neg\nι : 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✝ : Preadditive C\nK L : HomologicalComplex C c\nφ φ' : K ⟶ L\ne : c.Embedding c'\ni' : ι'\nhi' : ¬∃ i, e.f i = i'\n⊢ (extendMap (φ + φ') e).f i' ... | [] | apply (K.isZero_extend_X e i' (fun i hi => hi' ⟨i, hi⟩)).eq_of_src | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 151,
"column": 2
} | {
"line": 155,
"column": 42
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case neg\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\nhj : ¬c'.Rel (e.f j) (c'.next (e.f j))\n⊢ c'.next (e.f j) = e.f k",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | · rw [c'.next_eq_self _ hj, ← hjk, c.next_eq_self j]
intro hj'
apply hj
rw [← e.rel_iff] at hj'
simpa only [c'.next_eq' hj'] using hj' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 306,
"column": 32
} | {
"line": 306,
"column": 52
} | {
"line": 307,
"column": 8
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁴ : Category.{v_1, u_3} C\ninst✝³ : HasZeroObject C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : DecidableEq ι\ne : c.Embedding c'\nX : C\ninst✝ : DecidableEq ι'\ni : ι\ni' : ι'\nh : e.f i = i'\nj' : ι'\nj : ι\nhj : e.f... | [] | by rw [← hj, hij, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.TruncGEHomology | {
"line": 180,
"column": 4
} | {
"line": 186,
"column": 68
} | {
"line": 187,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁴ : Category.{v_1, u_3} C\ninst✝³ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝² : e.IsTruncGE\ninst✝¹ : ∀ (i' : ι'), K.HasHomology i'\ninst✝ : HasZeroObject C\nj : ι\nj' :... | [
"case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁴ : Category.{v_1, u_3} C\ninst✝³ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝² : e.IsTruncGE\ninst✝¹ : ∀ (i' : ι'), K.HasHomology i'\ninst✝ : HasZeroObject C\nj : ι\nj' : ι'\nhj' : e... | have : Epi φ.τ₁ := by
by_cases hi : ∃ i, e.f i = c'.prev j'
· obtain ⟨i, hi⟩ := hi
dsimp [φ, πTruncGE]
rw [e.epi_liftExtend_f_iff _ _ hi]
infer_instance
· apply IsZero.epi (isZero_extend_X _ _ _ (by simpa using hi)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 41
} | {
"line": 82,
"column": 4
} | [
{
"pp": "case left.inr.inl\nι : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nc : ComplexShape ι\nc₁ : ComplexShape ι₁\nc₂ : ComplexShape ι₂\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\nac : e₁.AreComplementary e₂\ni₂ : ι₂\nj₁ : ι₁\nh : fromSum e₁ e₂ (Sum.inr i₂) = fromSum e₁ e₂ (Sum.inl j₁)\n⊢ Sum.inr i₂ = Sum.inl j₁"... | [
"case left.inr.inr\nι : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nc : ComplexShape ι\nc₁ : ComplexShape ι₁\nc₂ : ComplexShape ι₂\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\nac : e₁.AreComplementary e₂\ni₂ j₂ : ι₂\nh : fromSum e₁ e₂ (Sum.inr i₂) = fromSum e₁ e₂ (Sum.inr j₂)\n⊢ Sum.inr i₂ = Sum.inr j₂"
] | · exact (ac.disjoint _ _ h.symm).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 76,
"column": 2
} | {
"line": 87,
"column": 29
} | {
"line": 89,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nc : ComplexShape ι\nc₁ : ComplexShape ι₁\nc₂ : ComplexShape ι₂\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\nac : e₁.AreComplementary e₂\n⊢ Function.Bijective (fromSum e₁ e₂)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False.elim... | [] | constructor
· rintro (i₁ | i₂) (j₁ | j₂) h
· obtain rfl := e₁.injective_f h
rfl
· exact (ac.disjoint _ _ h).elim
· exact (ac.disjoint _ _ h.symm).elim
· obtain rfl := e₂.injective_f h
rfl
· intro n
obtain ⟨i₁, rfl⟩ | ⟨i₂, rfl⟩ := ac.union n
· exact ⟨Sum.inl i₁, rfl⟩
· exact ⟨... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 76,
"column": 2
} | {
"line": 87,
"column": 29
} | {
"line": 89,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nc : ComplexShape ι\nc₁ : ComplexShape ι₁\nc₂ : ComplexShape ι₂\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\nac : e₁.AreComplementary e₂\n⊢ Function.Bijective (fromSum e₁ e₂)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False.elim... | [] | constructor
· rintro (i₁ | i₂) (j₁ | j₂) h
· obtain rfl := e₁.injective_f h
rfl
· exact (ac.disjoint _ _ h).elim
· exact (ac.disjoint _ _ h.symm).elim
· obtain rfl := e₂.injective_f h
rfl
· intro n
obtain ⟨i₁, rfl⟩ | ⟨i₂, rfl⟩ := ac.union n
· exact ⟨Sum.inl i₁, rfl⟩
· exact ⟨... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 266,
"column": 29
} | {
"line": 270,
"column": 24
} | {
"line": 271,
"column": 2
} | [
{
"pp": "ι : 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✝¹ : Category.{v_1, u_4} C\ninst✝ : HasZeroMorphisms C\ne₁ : c₁.Embedding c\ne₂ : c₂.Embedding c\nac : e₁.AreComplementary e₂\nK L : HomologicalComplex C c\nx✝ : Subtype e₁.Boun... | [] | by
ext
have h' := of_boundaryLE ac h
have h'' := of_boundaryGE ac h'.snd
exact fst_inj h'' h' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SingleHomology | {
"line": 83,
"column": 44
} | {
"line": 85,
"column": 68
} | {
"line": 87,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA : C\n⊢ (singleObjCyclesSelfIso c j A).hom ≫ (singleObjHomologySelfIso c j A).inv = ((single C c j).obj A).homologyπ j",
"ppTerm": "?m.44",... | [] | by
simp only [← cancel_mono (singleObjHomologySelfIso _ _ _).hom, assoc,
Iso.inv_hom_id, comp_id, homologyπ_singleObjHomologySelfIso_hom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 31
} | {
"line": 311,
"column": 2
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ninst✝² : HasZeroObject C\ninst✝¹ : ∀ (i : ℤ), HasHomology K i\ninst✝ : ∀ (i : ℤ), HasHomology L i\ni : ℤ\nk : ℕ\nh : ∀ (i_1 : ℕ) (i' : ℤ), (embeddingUpIntLE (i + ↑k)).f i_1 = i' → Qu... | [] | exact h k _ (by dsimp; lia) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four | {
"line": 72,
"column": 10
} | {
"line": 72,
"column": 12
} | {
"line": 72,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 3\nφ : R₁ ⟶ R₂\nhR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0\nhR₁' :\n (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 3\nφ : R₁ ⟶ R₂\nhR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0\nhR₁' :\n (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)\n ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.Refinements | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 60
} | {
"line": 43,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nK : HomologicalComplex C c\nA : C\ni : ι\nγ : A ⟶ K.homology i\n⊢ ∃ A' π, ∃ (_ : Epi π), ∃ z, ∃ (hz : z ≫ K.d i (c.next i) = 0), π ≫ γ = K.liftCycles z (c.next i) ⋯ hz ≫ K.homologyπ i",
"ppTerm": "?m.... | [] | exact (K.sc i).eq_liftCycles_homologyπ_up_to_refinements γ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomologySequenceLemmas | {
"line": 223,
"column": 4
} | {
"line": 228,
"column": 48
} | {
"line": 230,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS : ShortComplex (HomologicalComplex C c)\nhS : S.ShortExact\ni : ι\nh₁ : Epi (HomologicalComplex.homologyMap S.f i)\nh₂ : ∀ (j : ι), c.Rel i j → Mono (HomologicalComplex.homologyMap S.f j)\nhi ... | [] | have := hS.epi_g
have := HomologicalComplex.epi_homologyMap_of_epi_of_not_rel S.g i hi
rw [IsZero.iff_id_eq_zero,
← cancel_epi (HomologicalComplex.homologyMap S.g i),
← cancel_epi (HomologicalComplex.homologyMap S.f i)]
simp [← HomologicalComplex.homologyMap_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologySequenceLemmas | {
"line": 223,
"column": 4
} | {
"line": 228,
"column": 48
} | {
"line": 230,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS : ShortComplex (HomologicalComplex C c)\nhS : S.ShortExact\ni : ι\nh₁ : Epi (HomologicalComplex.homologyMap S.f i)\nh₂ : ∀ (j : ι), c.Rel i j → Mono (HomologicalComplex.homologyMap S.f j)\nhi ... | [] | have := hS.epi_g
have := HomologicalComplex.epi_homologyMap_of_epi_of_not_rel S.g i hi
rw [IsZero.iff_id_eq_zero,
← cancel_epi (HomologicalComplex.homologyMap S.g i),
← cancel_epi (HomologicalComplex.homologyMap S.f i)]
simp [← HomologicalComplex.homologyMap_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | {
"line": 95,
"column": 21
} | {
"line": 95,
"column": 38
} | {
"line": 95,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : HasDerivedCategory C\n⊢ (isoOfHom Q W (CochainComplex.mappingCone.descShortComplex (S.map (CochainComplex.singleFunctor C 0))) ⋯).inv ≫\n Q.map (CochainComplex.mappingCone.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : HasDerivedCategory C\n⊢ (isoOfHom Q W (CochainComplex.mappingCone.descShortComplex (S.map (CochainComplex.singleFunctor C 0))) ⋯).inv ≫\n Q.map (CochainComplex.mappingCone.triangle ((C... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Opposite | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 76
} | {
"line": 182,
"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\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\nF : C ⥤ D\ninst✝ : F.CommShift A\nx✝ : Cᵒᵖ\n⊢ (F.map ((shiftFunctorZero C A).inv.app (Opposite.unop x✝))).op ≫\n ((shiftFunc... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\nF : C ⥤ D\ninst✝ : F.CommShift A\nx✝ : Cᵒᵖ\n⊢ (F.map ((shiftFunctorZero C A).inv.app (Opposite.unop x✝))).op ≫\n ((shiftFunctorZero D A)... | erw [oppositeShiftFunctorZero_inv_app, oppositeShiftFunctorZero_hom_app] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.CategoryTheory.Shift.Opposite | {
"line": 208,
"column": 9
} | {
"line": 208,
"column": 42
} | {
"line": 208,
"column": 43
} | [
{
"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\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\nF : C ⥤ D\ninst✝ : (OppositeShift.functor A F).CommShift A\nx✝ : C\n⊢ ((OppositeShift.functor A F).map ((shiftFunctorZero Cᵒᵖ A).... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nA : Type u_3\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\nF : C ⥤ D\ninst✝ : (OppositeShift.functor A F).CommShift A\nx✝ : C\n⊢ ((OppositeShift.functor A F).map ((shiftFunctorZero Cᵒᵖ A).inv.app (Opp... | oppositeShiftFunctorZero_hom_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 36
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case e_a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\na₁ a₂ : A\nh₃ : φ a₁ + φ a₂ = φ (a₁ + a₂)\n⊢ ((shiftMonoidalFunctor C B).map (Functor.LaxMonoidal.μ (Discrete.addMonoi... | [
"case e_a\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\na₁ a₂ : A\nh₃ : φ a₁ + φ a₂ = φ (a₁ + a₂)\n⊢ ((shiftMonoidalFunctor C B).map (Discrete.eqToHom ⋯)).app X =\n (pullbackShiftIso ... | rw [Discrete.addMonoidalFunctor_μ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 445,
"column": 47
} | {
"line": 445,
"column": 85
} | {
"line": 445,
"column": 85
} | [
{
"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\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\na b : A\nh : a + b = 0\ninst✝ : G.CommShift A\nX : C\n⊢ b + a = 0",
"ppTerm": "?m.251",
... | [] | by simp [eq_neg_of_add_eq_zero_left h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 456,
"column": 40
} | {
"line": 456,
"column": 78
} | {
"line": 456,
"column": 78
} | [
{
"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\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\na b : A\nh : a + b = 0\ninst✝ : G.CommShift A\nY : C\n⊢ b + a = 0",
"ppTerm": "?m.271",
... | [] | by simp [eq_neg_of_add_eq_zero_left h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 468,
"column": 70
} | {
"line": 477,
"column": 6
} | {
"line": 479,
"column": 0
} | [
{
"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\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : G.CommShift A\na : A\n⊢ CommShift.CompatibilityUnit adj (iso adj a) (Functor.commShiftI... | [] | by
intro
rw [LeftAdjointCommShift.iso_hom_app adj _ _ (add_neg_cancel a)]
simp only [Functor.id_obj, Functor.comp_obj, Functor.map_shiftFunctorCompIsoId_inv_app,
Functor.map_comp, assoc, unit_naturality_assoc, right_triangle_components_assoc]
slice_rhs 4 5 => rw [← Functor.map_comp, Iso.inv_hom_id_app]
si... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 76,
"column": 2
} | {
"line": 79,
"column": 54
} | {
"line": 81,
"column": 0
} | [
{
"pp": "case mpr\nC : 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',\n ∃ (_ : T' ∈ Pretriangulated.distinguishedTriangles),\n No... | [] | · rintro ⟨T', hT', ⟨e⟩⟩
refine isomorphic_distinguished _ hT' _ ?_
exact Iso.unop ((triangleOpEquivalence C).unitIso.app (Opposite.op T') ≪≫
(triangleOpEquivalence C).inverse.mapIso e.symm) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Localization.Linear | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 25
} | {
"line": 92,
"column": 26
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} D\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : Preadditive D\nE : Type u_1\ninst✝⁸ : Category.{v_1, u_1} E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁷ : L.IsLocalization W\ninst✝⁶ : Preadditive E\nR : Type u_2\ninst✝⁵ : ... | [
"case mp\nC : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} D\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : Preadditive D\nE : Type u_1\ninst✝⁸ : Category.{v_1, u_1} E\nL : C ⥤ D\nW : MorphismProperty C\ninst✝⁷ : L.IsLocalization W\ninst✝⁶ : Preadditive E\nR : Type u_2\ninst✝⁵ : Ring R\ninst... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 123,
"column": 62
} | {
"line": 123,
"column": 71
} | {
"line": 123,
"column": 72
} | [
{
"pp": "case intro\nι : Type u_1\nι' : Type u_2\nR : Type u_5\nM : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ne : Basis ι R M\ninst✝² : Finite ι\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\nv : Basis ι' R M\ni : ι'\nval✝ : Fintype ι\n⊢ ∑ j, e.toMatrix (⇑v) j i • e j = Li... | [
"case intro\nι : Type u_1\nι' : Type u_2\nR : Type u_5\nM : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ne : Basis ι R M\ninst✝² : Finite ι\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\nv : Basis ι' R M\ni : ι'\nval✝ : Fintype ι\n⊢ ∑ j, e.toMatrix (⇑v) j i • e j = v i"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralDomain | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 62
} | {
"line": 195,
"column": 4
} | [
{
"pp": "R : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhf : ↑↑x = 1\ng : G\nn : ℕ\nhn : (fun x_1 ↦ x ^ x_1) n = ⟨f.toHomUnits g, ⋯⟩\n⊢ f g = 1 g",
"ppTe... | [] | simpa [hf, Subtype.ext_iff, Units.ext_iff] using hn.symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.IntegralDomain | {
"line": 208,
"column": 4
} | {
"line": 211,
"column": 39
} | {
"line": 213,
"column": 4
} | [
{
"pp": "case calc_1\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\nu : Rˣ\nhu : u ∈ imag... | [
"case calc_2\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\n⊢ ∑ b, ↑↑b = 0"
] | · -- remaining goal 1
apply MonoidHom.card_fiber_eq_of_mem_range f.toHomUnits
· simpa only [mem_image, mem_univ, true_and, Set.mem_range] using hu
· exact ⟨1, f.toHomUnits.map_one⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.IntegralDomain | {
"line": 213,
"column": 4
} | {
"line": 222,
"column": 17
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case calc_2\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\n⊢ ∑ b, ↑↑b = 0",
... | [
"case calc_2\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\n⊢ ∑ n ∈ range (orderOf x), ↑↑x ^ ... | calc
(∑ b : MonoidHom.range f.toHomUnits, (b.1 : R))
= ∑ n ∈ range (orderOf x), (x.1 : R) ^ n :=
Eq.symm <|
sum_nbij (x ^ ·) (by simp)
(by simpa using pow_injOn_Iio_orderOf)
(fun b _ => let ⟨n, hn⟩ := hx b
⟨n % orderOf x, mem_range.2 (Nat.mod_lt _ (o... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 91,
"column": 17
} | {
"line": 93,
"column": 36
} | {
"line": 95,
"column": 0
} | [
{
"pp": "n : Type u_1\ninst✝⁴ : Fintype n\nR : Type u_2\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb : Basis n R M\ninst✝ : DecidableEq n\nA : Matrix n n R\nhA : IsUnit A.det\nx : M\n⊢ ((toLin b b) A) (((toLin b b) A⁻¹) x) = x",
"ppTerm": "?m.98",
"assigned": true,... | [] | by
simp_rw [← LinearMap.comp_apply, ← Matrix.toLin_mul b b b, Matrix.mul_nonsing_inv _ hA,
toLin_one, LinearMap.id_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 203,
"column": 47
} | {
"line": 206,
"column": 31
} | {
"line": 208,
"column": 0
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nM N : Matrix n n A\nh : N.Nondegenerate\n⊢ (M * N).Nondegenerate ↔ M.Nondegenerate",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"No... | [] | by
classical
simp only [nondegenerate_iff_det_ne_zero, det_mul] at h ⊢
exact mul_ne_zero_iff_right h | [anonymous] | Lean.Parser.Term.byTactic |
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