module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Sets.Compacts | {
"line": 627,
"column": 2
} | {
"line": 629,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : Nonempty α\n⊢ Nonempty (PositiveCompacts α)",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Inha... | inhabit α
rcases exists_compact_mem_nhds (default : α) with ⟨K, hKc, hK⟩
exact ⟨⟨K, hKc⟩, _, mem_interior_iff_mem_nhds.2 hK⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructible | {
"line": 482,
"column": 4
} | {
"line": 483,
"column": 37
} | [
{
"pp": "case h.e'_3.a\nX : Type u_2\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\nhs : IsLocallyConstructible s\nhst : s ⊆ t\nht : IsCompact t\nU : X → Set X\nhU : ∀ (x : X), IsOpen (U x)\nhU' : ∀ (x : X), IsCompact (U x)\nhxU : ∀ (x : X), x ∈ U x\nhUs :... | rw [← Set.iUnion₂_inter, Set.subset_inter_iff]
exact ⟨hst.trans htσ, subset_rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructible | {
"line": 482,
"column": 4
} | {
"line": 483,
"column": 37
} | [
{
"pp": "case h.e'_3.a\nX : Type u_2\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\nhs : IsLocallyConstructible s\nhst : s ⊆ t\nht : IsCompact t\nU : X → Set X\nhU : ∀ (x : X), IsOpen (U x)\nhU' : ∀ (x : X), IsCompact (U x)\nhxU : ∀ (x : X), x ∈ U x\nhUs :... | rw [← Set.iUnion₂_inter, Set.subset_inter_iff]
exact ⟨hst.trans htσ, subset_rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 623,
"column": 2
} | {
"line": 623,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf : R\n⊢ IsCompact (Set.range (comap (algebraMap R (Localization (Submonoid.powers f)))))",
"usedConstants": [
"OreLocalization.instAlgebra",
"CommSemiring.toSemiring",
"algebraMap",
"Algebra.id",
"PrimeSpectrum.continuous_comap",
... | exact isCompact_range (continuous_comap _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Exponent | {
"line": 168,
"column": 2
} | {
"line": 172,
"column": 23
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ exponent G ≤ n",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponent.eq_1",
"Classical.propDecidable",
... | classical
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.Exponent | {
"line": 168,
"column": 2
} | {
"line": 172,
"column": 23
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ exponent G ≤ n",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponent.eq_1",
"Classical.propDecidable",
... | classical
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Exponent | {
"line": 168,
"column": 2
} | {
"line": 172,
"column": 23
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ exponent G ≤ n",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponent.eq_1",
"Classical.propDecidable",
... | classical
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 332,
"column": 19
} | {
"line": 332,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝³ : Group α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : IsCyclic α\nn : ℕ\nhn0 : 0 < n\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\nm : ℕ\nhm : Fintype.card α = n.gcd (Fintype.card α) * 0\nhm0 : m = 0\n⊢ False",
"usedConstants": [
"Nat.gcd",
"Nat.instMulZeroClass",... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1192,
"column": 2
} | {
"line": 1192,
"column": 87
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\n⊢ zeroLocus ∘ SetLike.coe '' minimalPrimes R = irreducibleComponents (PrimeSpectrum R)",
"usedConstants": [
"Eq.mpr",
"irreducibleComponents",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal_irreducibleC... | rw [← vanishingIdeal_irreducibleComponents, ← Set.image_comp, Set.EqOn.image_eq_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1274,
"column": 67
} | {
"line": 1276,
"column": 16
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : IsLocalRing R\n⊢ IsClosed {closedPoint R}",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"IsLocalRing.closedPoint._proof_1",
"congrArg",
"CommSemiring.toSemiring",
"IsLocalRing.closedPoint.eq_1",
"IsLocalRing... | by
rw [PrimeSpectrum.isClosed_singleton_iff_isMaximal, closedPoint]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PGroup | {
"line": 147,
"column": 91
} | {
"line": 152,
"column": 33
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\na : α\ninst✝ : Finite ↑(orbit G a)\n⊢ ∃ n, Nat.card ↑(orbit G a) = p ^ n",
"usedConstants": [
"Eq.mpr",
"Finite.of_equiv",
"congrArg",
"Nat.instMonoid",
... | by
let ϕ := orbitEquivQuotientStabilizer G a
haveI := Finite.of_equiv (orbit G a) ϕ
haveI := (stabilizer G a).finiteIndex_of_finite_quotient
rw [Nat.card_congr ϕ]
exact hG.index (stabilizer G a) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 194,
"column": 68
} | {
"line": 194,
"column": 87
} | [
{
"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... | mem_center_iff.1 ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 63
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nh : IsCyclic G\nthis : NeZero (Nat.card G)\n⊢ Nat.card (MulAut G) = (Nat.card G).totient",
"usedConstants": [
"Eq.mpr",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"ZMod.fintype... | rw [← ZMod.card_units_eq_totient, ← Nat.card_eq_fintype_card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.CompactlyGenerated.Intervals | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 9
} | [
{
"pp": "α : Type u_2\ninst✝ : CompleteLattice α\na : α\nb : ↑(Iic a)\nh : ∀ (ι : Type u_2) (s : ι → α), ↑b ≤ iSup s → ∃ t, ↑b ≤ ⨆ a ∈ t, s a\n⊢ ∀ (ι : Type u_2) (s : ι → ↑(Iic a)), b ≤ iSup s → ∃ t, b ≤ ⨆ a_2 ∈ t, s a_2",
"usedConstants": []
}
] | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.GroupTheory.Sylow | {
"line": 737,
"column": 41
} | {
"line": 739,
"column": 40
} | [
{
"pp": "G : Type u\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : (↑P).Normal\n⊢ (↑P).Characteristic",
"usedConstants": [
"Sylow",
"Unique",
"Sylow.characteristic_of_subsingleton",
"Unique.instSubsingleton",
"Sylow.unique_... | by
have _ := unique_of_normal P h
exact characteristic_of_subsingleton _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 327,
"column": 4
} | {
"line": 327,
"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.RingTheory.SimpleModule.Basic | {
"line": 540,
"column": 11
} | {
"line": 540,
"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($eq 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 540,
"column": 11
} | {
"line": 540,
"column": 35
} | [
{
"pp": "ι : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁸ : Ring R✝\ninst✝⁷ : Ring S\nM : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R✝ M\nm : Submodule R✝ M\nN : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R✝ N\nR : Type u_6\ninst✝² : DivisionRing R\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\nv... | simpa using congr($eq 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 540,
"column": 11
} | {
"line": 540,
"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": 142,
"column": 2
} | {
"line": 142,
"column": 13
} | [
{
"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",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
... | | single => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 152,
"column": 18
} | {
"line": 152,
"column": 36
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : (p • (single R i) m) n = if i ≤ n then p.coeff (n - i) • m else 0\nhq : (q • (single R i) m) n = if i ≤ n then q.coeff (n - i) • m else 0\n⊢ (p • (single R i) m ... | Finsupp.add_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 162,
"column": 18
} | {
"line": 162,
"column": 36
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ng : PolynomialModule R M\nn : ℕ\np q : R[X]\nhp : (p • g) n = ∑ x ∈ Finset.antidiagonal n, p.coeff x.1 • g x.2\nhq : (q • g) n = ∑ x ∈ Finset.antidiagonal n, q.coeff x.1 • g x.2\n⊢ (p • g + q • g) n ... | Finsupp.add_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 181,
"column": 57
} | {
"line": 181,
"column": 65
} | [
{
"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✝... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 29
} | [
{
"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": 101,
"column": 70
} | {
"line": 104,
"column": 25
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"Semiring.toModule",
"Monoid... | 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.AEval | {
"line": 92,
"column": 32
} | {
"line": 92,
"column": 43
} | [
{
"pp": "case a\nR : 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... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Filtration | {
"line": 355,
"column": 66
} | {
"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... | iSup_subtype' | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 930,
"column": 2
} | {
"line": 930,
"column": 39
} | [
{
"pp": "M : Type u_2\ninst✝ : AddCommMonoid M\n⊢ IsTorsion M ↔ Module.IsTorsion ℕ M",
"usedConstants": [
"Nat",
"Iff.intro",
"Module.IsTorsion",
"Nat.instSemiring",
"AddCommMonoid.toNatModule",
"AddCommMonoid.toAddMonoid",
"AddMonoid.IsTorsion"
]
}
] | refine ⟨fun h x => ?_, fun h x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 939,
"column": 2
} | {
"line": 939,
"column": 39
} | [
{
"pp": "M : Type u_2\ninst✝ : AddCommGroup M\n⊢ IsTorsion M ↔ Module.IsTorsion ℤ M",
"usedConstants": [
"AddCommGroup.toAddCommMonoid",
"AddCommGroup.toAddGroup",
"Int",
"AddGroup.toSubNegMonoid",
"Iff.intro",
"AddCommGroup.toIntModule",
"Module.IsTorsion",
"... | refine ⟨fun h x => ?_, fun h x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 62,
"column": 94
} | {
"line": 64,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\nx y : ↥I\n⊢ I.toCotangent x = I.toCotangent y ↔ ↑x - ↑y ∈ I ^ 2",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Semiring.toModule",
"Ideal.mem_toCotangent_ker",
"AddGroupWithOne.toAddGroup",
"congrArg",... | 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
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\n⊢ I.cotangentToQuotientSquare.range = (I.cotangentToQuotientSquare ∘ₗ I.toCotangent).range",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toMod... | · rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 48,
"column": 31
} | {
"line": 48,
"column": 50
} | [
{
"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 ... | reassoc_of% (eq i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 126,
"column": 12
} | {
"line": 126,
"column": 31
} | [
{
"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... | reassoc_of% (eq i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 72
} | [
{
"pp": "case h.hf.h\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\nx✝¹ x✝ : PresheafOfModules R\nf g : x✝¹ ⟶ x✝\nh : (toPresheaf R).map f = (toPresheaf R).map g\nX : Cᵒᵖ\nx : ↑(x✝¹.obj X)\n⊢ (ModuleCat.Hom.hom (f.app X)) x = (ModuleCat.Hom.hom (g.app X)) x",
"u... | exact congr_fun (((evaluation _ _).obj X ⋙ forget Ab).congr_map h) x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 260,
"column": 42
} | {
"line": 260,
"column": 49
} | [
{
"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⊢ (... | zero_f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 70
} | [
{
"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",
"usedConstants": [
"CochainComplex.HomComple... | ← 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
} | [
{
"pp": "case h\nC : 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",
"usedConstants": [
"CochainComplex.HomComplex.instAddCommGroupCochain... | simp only [comp_v _ _ h p _ q rfl (by lia), zero_v, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 679,
"column": 53
} | {
"line": 680,
"column": 58
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nz : Cocycle F G 0\n⊢ Cochain.ofHom z.homOf = ↑z",
"usedConstants": [
"CochainComplex.HomComplex.instAddCommGroupCochain",
"CochainComplex.HomComplex.Cocycle.ofHom_homOf_eq_self",
"AddCommGroup... | by
simpa only [Cocycle.ext_iff] using ofHom_homOf_eq_self z | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 148,
"column": 8
} | {
"line": 148,
"column": 71
} | [
{
"pp": "case a.w.h\nC : Type ?u.17436\nD : Type ?u.17439\ninst✝⁵ : Category.{v_1, ?u.17436} C\ninst✝⁴ : Category.{v_2, ?u.17439} D\nF : C ⥤ D\nA : Type ?u.17478\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).Ful... | 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": 55,
"column": 8
} | {
"line": 55,
"column": 17
} | [
{
"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",
"usedConstants": []
}
] | apply hij | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 17
} | [
{
"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... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 471,
"column": 23
} | {
"line": 473,
"column": 53
} | [
{
"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.RingTheory.Kaehler.Basic | {
"line": 96,
"column": 8
} | {
"line": 96,
"column": 16
} | [
{
"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... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 270,
"column": 2
} | {
"line": 271,
"column": 15
} | [
{
"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",
"usedConstants": [
"NegZeroClass.toNeg",
"HomologicalComplex.instCategory",
"AddMonoidHom.instAddMonoidHomClass",
"CochainC... | change shiftAddHom K L n a (-γ) = _
apply map_neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 270,
"column": 2
} | {
"line": 271,
"column": 15
} | [
{
"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",
"usedConstants": [
"NegZeroClass.toNeg",
"HomologicalComplex.instCategory",
"AddMonoidHom.instAddMonoidHomClass",
"CochainC... | change shiftAddHom K L n a (-γ) = _
apply map_neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 403,
"column": 19
} | {
"line": 403,
"column": 27
} | [
{
"pp": "case h.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 ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 409,
"column": 53
} | {
"line": 409,
"column": 62
} | [
{
"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' ⋯",
"usedConstants": [
"CochainComplex.H... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Basic | {
"line": 213,
"column": 37
} | {
"line": 213,
"column": 79
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.Pretriangulated.Triangle.... | by rw [← comp_hom₁, e.hom_inv_id, id_hom₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 457,
"column": 31
} | {
"line": 457,
"column": 39
} | [
{
"pp": "case pos.h.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.negOn... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 507,
"column": 66
} | {
"line": 507,
"column": 96
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Finsupp.smulZeroClass",
"Submodule",
"instHSMul",
... | ← LinearMap.map_smul_of_tower, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 514,
"column": 83
} | {
"line": 514,
"column": 91
} | [
{
"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",
"us... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 250,
"column": 42
} | {
"line": 253,
"column": 31
} | [
{
"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": 363,
"column": 4
} | {
"line": 363,
"column": 78
} | [
{
"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": 480,
"column": 4
} | {
"line": 482,
"column": 53
} | [
{
"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 :... | 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": 660,
"column": 4
} | {
"line": 660,
"column": 63
} | [
{
"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... | 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.HomotopyCategory.DegreewiseSplit | {
"line": 111,
"column": 32
} | {
"line": 111,
"column": 39
} | [
{
"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... | zero_f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 56
} | [
{
"pp": "case h\nC : 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) ≫ (mappingC... | zero_f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 16
} | [
{
"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_2... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 16
} | [
{
"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_2... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 28
} | [
{
"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",
"usedConst... | 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": 124,
"column": 2
} | {
"line": 125,
"column": 54
} | [
{
"pp": "case iso.w.w.h\nC₁ : 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 : C... | 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": 252,
"column": 2
} | {
"line": 252,
"column": 32
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.F... | rw [whiskeringLeftFunctor'_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 32
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | rw [whiskeringLeftFunctor'_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 311,
"column": 7
} | {
"line": 316,
"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 ⋯)",
"usedConstants": [
"Ca... | 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": 191,
"column": 9
} | {
"line": 191,
"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",... | W₁.isoClosure.inverseImage_equivalence_functor_eq_map_inverse | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.HasCardinalLT | {
"line": 52,
"column": 2
} | {
"line": 57,
"column": 59
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",
"Preorder.toLT",
"HasCardinalLT",
"Cardinal",
"Cardinal.lift_lift",
"congrArg",
... | 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
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",
"Preorder.toLT",
"HasCardinalLT",
"Cardinal",
"Cardinal.lift_lift",
"congrArg",
... | 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": 133,
"column": 4
} | {
"line": 134,
"column": 8
} | [
{
"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": 133,
"column": 4
} | {
"line": 134,
"column": 8
} | [
{
"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.Triangulated | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 30
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹¹ : Category.{v_1, u_1} C\ninst✝¹⁰ : Category.{v_2, u_2} D\nL : C ⥤ D\ninst✝⁹ : HasShift C ℤ\ninst✝⁸ : Preadditive C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁵ : Pretriangulated C\ninst✝⁴ : HasShift D ℤ\ninst✝³ : L.CommShift ℤ\nW... | have := inverts L W γ.s γ.hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Triangulated.HomologicalFunctor | {
"line": 142,
"column": 35
} | {
"line": 142,
"column": 66
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : HasShift C ℤ\ninst✝¹² : Category.{v_2, u_2} D\ninst✝¹¹ : HasZeroObject D\ninst✝¹⁰ : HasShift D ℤ\ninst✝⁹ : Preadditive D\ninst✝⁸ : ∀ (n : ℤ), (shiftFunctor D n).Additive\ninst✝⁷ : Pretriangulated D\ninst✝⁶ : Category.{... | by simpa using hf =≫ biprod.snd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.ShiftedHom | {
"line": 138,
"column": 54
} | {
"line": 139,
"column": 45
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
... | by
rw [comp, comp, comp, Preadditive.add_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 131,
"column": 11
} | {
"line": 140,
"column": 46
} | [
{
"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.Algebra.Homology.Localization | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 18
} | [
{
"pp": "case hP\nC : 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",
"usedConstants": [
"instCategoryHomotopyCategory",
... | intro f g e hf i | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Homology.Localization | {
"line": 165,
"column": 12
} | {
"line": 165,
"column": 13
} | [
{
"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 :... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.Localization | {
"line": 248,
"column": 12
} | {
"line": 248,
"column": 13
} | [
{
"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... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 389,
"column": 4
} | {
"line": 390,
"column": 56
} | [
{
"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._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 389,
"column": 4
} | {
"line": 390,
"column": 56
} | [
{
"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": 389,
"column": 4
} | {
"line": 390,
"column": 56
} | [
{
"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.DerivedCategory.Basic | {
"line": 228,
"column": 51
} | {
"line": 230,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nn : ℤ\n⊢ (singleFunctor C n).Additive",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"instCategoryDerivedCategory",
"DerivedCategory",
"HasDerivedCategory._proof_1",
... | by
dsimp [singleFunctor, singleFunctors]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.IsSupported | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 69
} | [
{
"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
} | [
{
"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
} | [
{
"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
} | [
{
"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": 224,
"column": 2
} | {
"line": 225,
"column": 16
} | [
{
"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",
"usedConstants": [
"HomologicalComplex.instCategory",
"Opposite",
... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Opposite | {
"line": 224,
"column": 2
} | {
"line": 225,
"column": 16
} | [
{
"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",
"usedConstants": [
"HomologicalComplex.instCategory",
"Opposite",
... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Opposite | {
"line": 229,
"column": 2
} | {
"line": 230,
"column": 16
} | [
{
"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",
"usedConstants": [
"HomologicalComplex.unop",
"HomologicalComplex.i... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Opposite | {
"line": 229,
"column": 2
} | {
"line": 230,
"column": 16
} | [
{
"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",
"usedConstants": [
"HomologicalComplex.unop",
"HomologicalComplex.i... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Opposite | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 33
} | [
{
"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)",
"usedConstants"... | 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
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id"... | 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
} | [
{
"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... | 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.AreComplementary | {
"line": 322,
"column": 6
} | {
"line": 326,
"column": 20
} | [
{
"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": 322,
"column": 6
} | {
"line": 326,
"column": 20
} | [
{
"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
} | [
{
"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 ⋯",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Abelian.Ext.comp_assoc._proof_2",
"Cat... | by
rw [← mk₀_comp_mk₀, comp_assoc]
lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 37
} | [
{
"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",
"usedConstants": [
"DerivedCategory.instIsLocalizationCochainComplexIntQQuasiIsoUp",
"Cate... | apply SmallShiftedHom.equiv_chgUniv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 37
} | [
{
"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",
"usedConstants": [
"DerivedCategory.instIsLocalizationCochainComplexIntQQuasiIsoUp",
"Cate... | apply SmallShiftedHom.equiv_chgUniv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 37
} | [
{
"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",
"usedConstants": [
"DerivedCategory.instIsLocalizationCochainComplexIntQQuasiIsoUp",
"Cate... | apply SmallShiftedHom.equiv_chgUniv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 16
} | [
{
"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",
"usedConstants": [
"CategoryTheory.Abelian.... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 16
} | [
{
"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",
"usedConstants": [
"CategoryTheory.Abelian.... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 193,
"column": 26
} | {
"line": 193,
"column": 76
} | [
{
"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... | ← cancel_mono ((shiftFunctorAdd C a b).hom.app _), | Lean.Elab.Tactic.evalRewriteSeq | null |
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