module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Spectral.Prespectral | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 43
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nι : Type u_3\nb : ι → Set X\nbasis : IsTopologicalBasis (Set.range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\n⊢ ∀ U ∈ Set.range b, IsCompact U",
"usedConstants": [
"Membership.mem",
"Exists",
"Set.mem_range._simp_1",
"forall_e... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Spectral.Prespectral | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 43
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nι : Type u_3\nb : ι → Set X\nbasis : IsTopologicalBasis (Set.range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\n⊢ ∀ U ∈ Set.range b, IsCompact U",
"usedConstants": [
"Membership.mem",
"Exists",
"Set.mem_range._simp_1",
"forall_e... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.Compacts | {
"line": 469,
"column": 31
} | {
"line": 469,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : Continuous f\nh : Function.Injective (NonemptyCompacts.map f hf)\nx✝² x✝¹ : α\nx✝ : ((fun x ↦ {x}) ∘ f) x✝² = ((fun x ↦ {x}) ∘ f) x✝¹\n⊢ NonemptyCompacts.map f hf ((fun x ↦ {x}) x✝²) = NonemptyCompacts.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Sets.Compacts | {
"line": 469,
"column": 31
} | {
"line": 469,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : Continuous f\nh : Function.Injective (NonemptyCompacts.map f hf)\nx✝² x✝¹ : α\nx✝ : ((fun x ↦ {x}) ∘ f) x✝² = ((fun x ↦ {x}) ∘ f) x✝¹\n⊢ NonemptyCompacts.map f hf ((fun x ↦ {x}) x✝²) = NonemptyCompacts.... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.Compacts | {
"line": 469,
"column": 31
} | {
"line": 469,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : Continuous f\nh : Function.Injective (NonemptyCompacts.map f hf)\nx✝² x✝¹ : α\nx✝ : ((fun x ↦ {x}) ∘ f) x✝² = ((fun x ↦ {x}) ∘ f) x✝¹\n⊢ NonemptyCompacts.map f hf ((fun x ↦ {x}) x✝²) = NonemptyCompacts.... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.Compacts | {
"line": 634,
"column": 8
} | {
"line": 634,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nK : PositiveCompacts α\nL : PositiveCompacts β\n⊢ (interior (K.toCompacts ×ˢ L.toCompacts).carrier).Nonempty",
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
... | simp only [Compacts.carrier_eq_coe, Compacts.coe_prod, interior_prod_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.KrullDimension | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 88
} | [
{
"pp": "X : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\n⊢ topologicalKrullDim X ≤ 0",
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"WithBot",
"LE.le.antisymm'",
"CommSemiring.toSemiring",
"Order.krullDim_nonpos_iff_forall_isMax",
"Pa... | refine krullDim_nonpos_iff_forall_isMax.mpr fun Z Y h ↦ (h.antisymm' fun x hx ↦ ?_).le | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 67
} | [
{
"pp": "case refine_2\nR : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nH :\n ∀ (P : Ideal S) [inst : P.IsPrime],\n Function.Surjective (PrimeSpectrum.comap (Localization.localRingHom (Ideal.under R P) P (algebraMap R S) ⋯))\np : Ideal R\nx✝¹ : p.IsPrime\nQ : Ideal... | Ideal.under_map_of_isLocalizationAtPrime (Q.under R) hlt.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 252,
"column": 65
} | {
"line": 252,
"column": 73
} | [
{
"pp": "case negSucc\nα : Type u_1\nG : Type u_2\nG' : Type u_3\na : α\ninst✝³ : Group α\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic α\nH : Subgroup α\nthis : (a : Prop) → Decidable a\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\nhx : ∃ x ∈ H, x ≠ 1\nx : α\nhx₁ : x ∈ H\nhx₂ : x ≠ 1\nk : ℕ\nhk✝ : (fun x ↦ g... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Exponent | {
"line": 133,
"column": 25
} | {
"line": 133,
"column": 35
} | [
{
"pp": "case pos\nG : Type u\ninst✝ : Monoid G\nh : ExponentExists G\nh' : {d | 0 < d ∧ ∀ (x : G), x ^ d = 1}.Nonempty\n⊢ (if h : ExponentExists G then Nat.find h else 0) = sInf {d | 0 < d ∧ ∀ (x : G), x ^ d = 1}",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
... | dif_pos h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Exponent | {
"line": 224,
"column": 74
} | {
"line": 225,
"column": 74
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\n⊢ exponent G ∣ n ↔ ∀ (g : G), orderOf g ∣ n",
"usedConstants": [
"_private.Mathlib.GroupTheory.Exponent.0.Monoid.exponent_dvd._simp_1_1",
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"Monoid.toMulOneClass",
"congrArg",
"id",
... | by
simp_rw [exponent_dvd_iff_forall_pow_eq_one, orderOf_dvd_iff_pow_eq_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1041,
"column": 4
} | {
"line": 1041,
"column": 40
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : CommSemiring R\nx : R\nhx : x ∈ {e | IsIdempotentElem e}\ny : R\nhy : y ∈ {e | IsIdempotentElem e}\neq : basicOpen x = basicOpen y\nne : x ≠ y\nthis :\n ∀ x ∈ {e | IsIdempotentElem e}, ∀ y ∈ {e | IsIdempotentElem e}, basicOpen x = basicOpen y → x ≠ y → x * y ≠ x → False\n... | apply this y hy x hx eq.symm ne.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Derivation.Basic | {
"line": 501,
"column": 48
} | {
"line": 501,
"column": 60
} | [
{
"pp": "case neg.inr\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Module K M\ninst✝ : Algebra R K\nD : Derivation R K M\na : K\nn : ℤ\nhn : ¬n = 0\nha : ¬a = 0\nh : n = -↑n.natAbs\n⊢ -(a ^ n.natAbs)⁻¹ ^ 2 • D (a ^ n.nat... | leibniz_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Subgroup.Simple | {
"line": 76,
"column": 10
} | {
"line": 76,
"column": 22
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝³ : Group G\nH✝ : Type u_3\ninst✝² : Group H✝\ninst✝¹ : IsSimpleGroup G\ninst✝ : Nontrivial H✝\nf : G →* H✝\nhf : Function.Surjective ⇑f\nH : Subgroup H✝\niH : H.Normal\nh : comap f H = ⊥\n⊢ H = ⊥",
"usedConstants": [
"Eq.mpr",
"Subgroup.map",
"co... | ← map_bot f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 327,
"column": 12
} | {
"line": 327,
"column": 55
} | [
{
"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\nx : α\nhx : x ∈ {a | a ^ n = 1}\nm : ℕ\nhm : g ^ m = x\n| Fintype.card α / n.gcd (Fintype.card α) * n.gcd (Fintype.card α)",
"usedConstants": [
... | Nat.div_mul_cancel (Nat.gcd_dvd_right _ _), | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1078,
"column": 6
} | {
"line": 1079,
"column": 90
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsClopen s\nh✝ : Nontrivial R\nI : Ideal R\nhI : I.FG\nJ : Ideal R\nhJ : J.FG\nhI' : zeroLocus ↑I = sᶜ\nhJ' : zeroLocus ↑J = s\nthis : I * J ≤ nilradical R\nn : ℕ\nhn : I ^ n * J ^ n ≤ ⊥\nhnz : n ≠ 0\nx : R\nhx : x ∈ I ^... | rw [← hJ', basicOpen_eq_zeroLocus_of_mul_add _ _ mul add]
exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr <| Ideal.pow_le_self hnz hy) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1078,
"column": 6
} | {
"line": 1079,
"column": 90
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsClopen s\nh✝ : Nontrivial R\nI : Ideal R\nhI : I.FG\nJ : Ideal R\nhJ : J.FG\nhI' : zeroLocus ↑I = sᶜ\nhJ' : zeroLocus ↑J = s\nthis : I * J ≤ nilradical R\nn : ℕ\nhn : I ^ n * J ^ n ≤ ⊥\nhnz : n ≠ 0\nx : R\nhx : x ∈ I ^... | rw [← hJ', basicOpen_eq_zeroLocus_of_mul_add _ _ mul add]
exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr <| Ideal.pow_le_self hnz hy) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 331,
"column": 6
} | {
"line": 342,
"column": 48
} | [
{
"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\n⊢ #(↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset ≤ n",
"usedConstants": [
"Nat.gcd",
"Nat.gcd_dvd_left",
... | let ⟨m, hm⟩ := Nat.gcd_dvd_right n (Fintype.card α)
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by
rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm
exact hm.elim' 1
simp only [Set.toFinset_card, SetLike.coe_sort_coe]
rw [Fintype.card_zpowers, orderOf_pow g, orderOf_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 331,
"column": 6
} | {
"line": 342,
"column": 48
} | [
{
"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\n⊢ #(↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset ≤ n",
"usedConstants": [
"Nat.gcd",
"Nat.gcd_dvd_left",
... | let ⟨m, hm⟩ := Nat.gcd_dvd_right n (Fintype.card α)
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by
rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm
exact hm.elim' 1
simp only [Set.toFinset_card, SetLike.coe_sort_coe]
rw [Fintype.card_zpowers, orderOf_pow g, orderOf_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1266,
"column": 2
} | {
"line": 1268,
"column": 83
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\ne : ↑{p | p.IsPrime ∧ ⊥ ≤ p} ≃o PrimeSpectrum R :=\n { toFun := fun x ↦ { asIdeal := ↑x, isPrime := ⋯ }, invFun := fun x ↦ ⟨x.asIdeal, ⋯⟩, left_inv := ⋯, right_inv := ⋯,\n map_rel_iff' := ⋯ }\n⊢ ↑(minimalPrimes R) ≃o (↑{s | Ma... | exact OrderIso.setOfMinimalIsoSetOfMaximal
(e.trans ((PrimeSpectrum.pointsEquivIrreducibleCloseds R).trans
(TopologicalSpace.IrreducibleCloseds.orderIsoSubtype' (PrimeSpectrum R)).dual)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Sylow | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 96
} | [
{
"pp": "p✝ : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nH : Subgroup G\ncard_eq : Nat.card ↥H = p ^ (Nat.card G).factorization p\n⊢ ¬p ∣ H.index",
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.... | rw [← mul_dvd_mul_iff_left (Nat.card_pos (α := H)).ne', card_mul_index, card_eq, ← pow_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Sylow | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 21
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nN : Subgroup G\nP Q : Sylow p G\nhP : ↑P ≤ N\nhQ : ↑Q ≤ N\nh : P.subtype hP = Q.subtype hQ\n⊢ P = Q",
"usedConstants": [
"Sylow.instSetLike",
"Sylow",
"congrArg",
"Membership.mem",
"Eq.mp",
"Subtype",
"Subgroup",
... | SetLike.ext_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 362,
"column": 58
} | {
"line": 362,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\na : ℕ\nha : 0 < 3\nha' : p ^ 0 = orderOf g\n⊢ orderOf g = 1",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 362,
"column": 58
} | {
"line": 362,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\na : ℕ\nha : 0 < 3\nha' : p ^ 0 = orderOf g\n⊢ orderOf g = 1",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 362,
"column": 58
} | {
"line": 362,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\na : ℕ\nha : 0 < 3\nha' : p ^ 0 = orderOf g\n⊢ orderOf g = 1",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 12
} | [
{
"pp": "case «1»\nα : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\na : ℕ\nha : 1 < 3\nha' : p ^ 1 = orderOf g\n⊢ orderOf g =... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 12
} | [
{
"pp": "case «1»\nα : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\na : ℕ\nha : 1 < 3\nha' : p ^ 1 = orderOf g\n⊢ orderOf g =... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 12
} | [
{
"pp": "case «1»\nα : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\na : ℕ\nha : 1 < 3\nha' : p ^ 1 = orderOf g\n⊢ orderOf g =... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 378,
"column": 40
} | {
"line": 378,
"column": 54
} | [
{
"pp": "case h\nG : Type u_2\ninst✝ : AddGroup G\ng : G\nx✝ : ℤ\n⊢ ↑(addOrderOf g) ∣ x✝ ↔ ∃ k, k • ↑(addOrderOf g) = x✝",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instHSMul",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"addOrderOf",
"Exists",
"id",
"NonU... | zsmul_eq_mul', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Torsion | {
"line": 244,
"column": 34
} | {
"line": 246,
"column": 18
} | [
{
"pp": "G : Type u_1\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\ng : ↥(primaryComponent G p)\n⊢ ∃ n, orderOf g = p ^ n",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"orderOf_submonoid",
"CommMonoid.mem_primaryComponent_if... | by
rw [← orderOf_submonoid, ← mem_primaryComponent_iff_orderOf]
exact g.property | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Sylow | {
"line": 765,
"column": 8
} | {
"line": 765,
"column": 17
} | [
{
"pp": "case inl\nG : Type u\ninst✝³ : Group G\ninst✝² : Finite G\nhnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nheq : normalizer ↑P = ⊤\n⊢ normalizer ↑↑P = ⊤",
"usedConstants": []
}
] | exact heq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Sylow | {
"line": 765,
"column": 8
} | {
"line": 765,
"column": 17
} | [
{
"pp": "case inl\nG : Type u\ninst✝³ : Group G\ninst✝² : Finite G\nhnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nheq : normalizer ↑P = ⊤\n⊢ normalizer ↑↑P = ⊤",
"usedConstants": []
}
] | exact heq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Sylow | {
"line": 765,
"column": 8
} | {
"line": 765,
"column": 17
} | [
{
"pp": "case inl\nG : Type u\ninst✝³ : Group G\ninst✝² : Finite G\nhnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nheq : normalizer ↑P = ⊤\n⊢ normalizer ↑↑P = ⊤",
"usedConstants": []
}
] | exact heq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Sylow | {
"line": 805,
"column": 4
} | {
"line": 812,
"column": 86
} | [
{
"pp": "case refine_2.left\nG : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := ⋯\nP : (p : ℕ) → Sylow p G := ⋯\nthis : (p : ℕ) → Fintype ↥↑(P p)\nhcomm : _root_.Pairwise fun p₁ p₂ ↦ ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P... | apply Subgroup.injective_noncommPiCoprod_of_iSupIndep
apply independent_of_coprime_order hcomm
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
have hne' : p₁ ≠ p₂ := by simpa using hne
simp only [←... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Sylow | {
"line": 805,
"column": 4
} | {
"line": 812,
"column": 86
} | [
{
"pp": "case refine_2.left\nG : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := ⋯\nP : (p : ℕ) → Sylow p G := ⋯\nthis : (p : ℕ) → Fintype ↥↑(P p)\nhcomm : _root_.Pairwise fun p₁ p₂ ↦ ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P... | apply Subgroup.injective_noncommPiCoprod_of_iSupIndep
apply independent_of_coprime_order hcomm
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
have hne' : p₁ ≠ p₂ := by simpa using hne
simp only [←... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coprime.Ideal | {
"line": 81,
"column": 8
} | {
"line": 81,
"column": 39
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j ↦ I i ⊔ I j = ⊤\nμ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)\nhμ : ∑ x ∈... | have := Submodule.coe_mem (μ x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 447,
"column": 8
} | {
"line": 449,
"column": 51
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\np : ι → Ideal R\nS : Finset ι\nhp : (↑S).Pairwise fun i j ↦ p i ⊔ p j = ⊤\nh : S.Nonempty\nx : M\nhx : ∀ (a : ↑↑(⨅ i ∈ S, p i)), ↑a • x = 0\nμ : (i : ι) → ↥(⨅ j ∈ S, ⨅ (_ : j ≠ i),... | have := coe_mem (μ i)
simp only [mem_iInf] at this
exact Ideal.mul_mem_left _ _ (this j hj ij) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 447,
"column": 8
} | {
"line": 449,
"column": 51
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\np : ι → Ideal R\nS : Finset ι\nhp : (↑S).Pairwise fun i j ↦ p i ⊔ p j = ⊤\nh : S.Nonempty\nx : M\nhx : ∀ (a : ↑↑(⨅ i ∈ S, p i)), ↑a • x = 0\nμ : (i : ι) → ↥(⨅ j ∈ S, ⨅ (_ : j ≠ i),... | have := coe_mem (μ i)
simp only [mem_iInf] at this
exact Ideal.mul_mem_left _ _ (this j hj ij) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Filtration | {
"line": 400,
"column": 8
} | {
"line": 400,
"column": 91
} | [
{
"pp": "R : 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 := ⨅ i, I ^ i • ⊤\nhN : ∀ (k : ℕ), (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N\n⊢ N ≤ I • N",
"used... | obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Nakayama | {
"line": 174,
"column": 2
} | {
"line": 176,
"column": 15
} | [
{
"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 N' P : Submodule R M\nhN' : N'.FG\nhN'le : N' ≤ P\nhNN' : P ≤ N ⊔ I • N'\nhNN'' : P ≤ N ⊔ N'\nh1 : map N.mkQ P = map N.mkQ N'\nh2 : map N.mkQ P = map N.mkQ (I • N')\n⊢ ∃ r, r - 1 ∈ I ∧ r • P ≤ N... | have hle : (P.map N.mkQ) ≤ I • P.map N.mkQ := by
conv_lhs => rw [h2]
simp [← h1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 910,
"column": 2
} | {
"line": 911,
"column": 68
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i ↦ p... | have : pOrder hM (s i) ≤ pOrder hM (s <| Option.get _ hoj) :=
List.le_of_mem_argmax (List.mem_finRange i) (Option.get_mem hoj) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 20
} | [
{
"pp": "case h\nC : 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\nj : J\ni : ι\n⊢ ({ f := fun i ↦ (hs ... | apply (hs i).fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 20
} | [
{
"pp": "case h\nC : 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\nj : J\ni : ι\n⊢ (s.ι.app j ≫... | apply (hs i).fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 186,
"column": 2
} | {
"line": 187,
"column": 16
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasFiniteColimits C\nK L : HomologicalComplex C c\nφ : K ⟶ L\ninst✝ : Epi φ\nn : ι\n⊢ Epi (φ.f n)",
"usedConstants": [
"CategoryT... | change Epi ((HomologicalComplex.eval C c n).map φ)
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplexLimits | {
"line": 186,
"column": 2
} | {
"line": 187,
"column": 16
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nJ : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_3} J\nc : ComplexShape ι\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasFiniteColimits C\nK L : HomologicalComplex C c\nφ : K ⟶ L\ninst✝ : Epi φ\nn : ι\n⊢ Epi (φ.f n)",
"usedConstants": [
"CategoryT... | change Epi ((HomologicalComplex.eval C c n).map φ)
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 89
} | [
{
"pp": "C : 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\n⊢ f = g",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"AddCommGrpCat.instCategory... | ext X x
exact ConcreteCategory.congr_hom (((evaluation _ _).obj X ⋙ forget Ab).congr_map h) x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 89
} | [
{
"pp": "C : 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\n⊢ f = g",
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"AddCommGrpCat.instCategory... | ext X x
exact ConcreteCategory.congr_hom (((evaluation _ _).obj X ⋙ forget Ab).congr_map h) x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 46
} | [
{
"pp": "C : 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 j : ι\nhij : c.Rel j i\nA : C\nf g : X φ j ⟶ A\nh₁ : inlX φ i j hij ≫ f = inlX φ i j hij ≫ g\nh₂ ... | rw [← cancel_epi (XIsoBiprod φ j i hij).inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G K L : CochainComplex C ℤ\nn₁ n₂ n₃ n₁₂ n₂₃ n₁₂₃ : ℤ\nz₁ : Cochain F G n₁\nz₂ : Cochain G K n₂\nz₃ : Cochain K L n₃\nh₁₂ : n₁ + n₂ = n₁₂\nh₂₃ : n₂ + n₃ = n₂₃\nh₁₂₃ : n₁ + n₂ + n₃ = n₁₂₃\n⊢ (z₁.comp z₂ h₁₂).comp z₃ ⋯ = z₁.comp (z₂.comp z₃... | substs h₁₂ h₂₃ h₁₂₃ | Mathlib.Tactic.Substs._aux_Mathlib_Tactic_Substs___macroRules_Mathlib_Tactic_Substs_substs_1 | Mathlib.Tactic.Substs.substs |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 469,
"column": 75
} | {
"line": 483,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nn₀ n₁ n₂ : ℤ\nz : Cochain F G n₀\n⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"add_neg_cancel",
"NegZeroClass.toNeg",
"_private.Mathli... | by
by_cases h₁₂ : n₁ + 1 = n₂; swap
· rw [δ_shape _ _ h₁₂]
by_cases h₀₁ : n₀ + 1 = n₁; swap
· rw [δ_shape _ _ h₀₁, δ_zero]
ext p q hpq
dsimp
simp only [δ_v n₁ n₂ h₁₂ _ p q hpq _ _ rfl rfl,
δ_v n₀ n₁ h₀₁ z p (q - 1) (by lia) (q - 2) _ (by lia) rfl,
δ_v n₀ n₁ h₀₁ z (p + 1) q (by lia) _ (p + 2) rfl (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 852,
"column": 2
} | {
"line": 852,
"column": 22
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\np q n p' q' : ℤ\nhpq' : p' + n = q'\n⊢ (single 0 n).v p' q' hpq' = v 0 p' q' hpq'",
"usedConstants": [
"CochainComplex.HomComplex.instAddCommGroupCochain",
"CategoryTheory.CategoryStruct.toQ... | by_cases hp : p' = p | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 20
} | [
{
"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\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\n⊢ (liftCochain φ α β h).comp (↑(fst φ)) h = α",
"usedConstants": [
"Homo... | simp [liftCochain] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 20
} | [
{
"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\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\n⊢ (liftCochain φ α β h).comp (↑(fst φ)) h = α",
"usedConstants": [
"Homo... | simp [liftCochain] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 20
} | [
{
"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\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\n⊢ (liftCochain φ α β h).comp (↑(fst φ)) h = α",
"usedConstants": [
"Homo... | simp [liftCochain] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 20
} | [
{
"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\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\n⊢ (liftCochain φ α β h).comp (snd φ) ⋯ = β",
"usedConstants": [
"Homolog... | simp [liftCochain] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 20
} | [
{
"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\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\n⊢ (liftCochain φ α β h).comp (snd φ) ⋯ = β",
"usedConstants": [
"Homolog... | simp [liftCochain] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 20
} | [
{
"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\nn m : ℤ\nα : Cochain K F m\nβ : Cochain K G n\nh : n + 1 = m\n⊢ (liftCochain φ α β h).comp (snd φ) ⋯ = β",
"usedConstants": [
"Homolog... | simp [liftCochain] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 90,
"column": 68
} | {
"line": 90,
"column": 71
} | [
{
"pp": "case 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 x₁ y₁ : S ⊗[R] S\nh₁ :\n D.tensorProductTo (x₁ * y) =\n (Ten... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 96,
"column": 68
} | {
"line": 96,
"column": 71
} | [
{
"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... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 44
} | [
{
"pp": "case refine_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S ⊗[R] S\nhx✝ : x ∈ ideal R S\nhx : x ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)\n⊢ ∃ (hx : 0 ∈ ideal R S), (fromIdeal R S) ⟨0, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))",
"... | · exact ⟨zero_mem _, Submodule.zero_mem _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 431,
"column": 65
} | {
"line": 436,
"column": 62
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : Category.{v_2, u_2} D\ninst✝¹³ : Category.{v_3, u_3} E\ninst✝¹² : Category.{v_4, u_4} J\nF₁ F₂ F₃ : C ⥤ D\nτ : F₁ ⟶ F₂\nτ' : F₂ ⟶ F₃\ne : F₁ ≅ F₂\nG G' : D ⥤ E\nτ'' : G ⟶ G'\nH : E ⥤ J\nA : Type u_5\ninst... | by
ext X
simp only [Functor.whiskerRight_twice, comp_app, Functor.commShiftIso_comp_hom_app,
Functor.associator_hom_app, Functor.whiskerRight_app, Functor.comp_map,
Functor.associator_inv_app, comp_id, id_comp, assoc, ← Functor.commShiftIso_hom_naturality, ←
G.map_comp_assoc, shift_app_comm, Functor.whi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 34
} | [
{
"pp": "C : Type ?u.86310\nD : Type ?u.86313\ninst✝⁵ : Category.{v_1, ?u.86310} C\ninst✝⁴ : Category.{v_2, ?u.86313} D\nF : C ⥤ D\nA : Type ?u.86352\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✝ : (... | letI := HasShift.induced F A s i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 451,
"column": 6
} | {
"line": 451,
"column": 61
} | [
{
"pp": "case pos.h\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⊢ (γ.leftShift a n' hn').v p (p + n') ⋯ ≫ L.d (p + n') q... | γ.leftShift_v a n' hn' p (p + n') rfl (p + a) (by lia), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 740,
"column": 4
} | {
"line": 740,
"column": 12
} | [
{
"pp": "case a.add\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nx✝ y✝ : B ⊗[A] Ω[A⁄R]\na✝¹ : (mapBaseChange R A B) x✝ ∈ (map R A B B).ker\na✝ : (mapBaseChang... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 181,
"column": 15
} | {
"line": 181,
"column": 71
} | [
{
"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), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nZ X : C\nh : Z ⟶ (shiftFunctor C 1).obj X\nY' : C\nf' : (shiftFunctor C 1).obj X ⟶ Y'\ng' : Y' ⟶ (shiftFunctor C 1).ob... | simp only [shift_shiftFunctorCompIsoId_inv_app, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 232,
"column": 15
} | {
"line": 232,
"column": 71
} | [
{
"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), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nX : C\n⊢ 𝟙 ((shiftFunctor C 1).obj X) ≫ (shiftFunctor C 1).map ((shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app X) =\n ... | simp only [shift_shiftFunctorCompIsoId_inv_app, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 16
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 327,
"column": 57
} | {
"line": 327,
"column": 60
} | [
{
"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\nh₁ : T.mor₁ = 0\nh₂ : T.mor₂ = 0\n⊢ 𝟙 T.obj₂ ≫ T.mor₂... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 16
} | [
{
"pp": "case refine_3\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).obj... | change T' ∈ distTriang C at hT' | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 637,
"column": 26
} | {
"line": 637,
"column": 29
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).obj... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 647,
"column": 12
} | {
"line": 647,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).obj... | ← cancel_mono (φ'.hom₂⟦1⟧'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 664,
"column": 37
} | {
"line": 664,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).obj... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Functor | {
"line": 194,
"column": 46
} | {
"line": 194,
"column": 49
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝²⁰ : Category.{v_1, u_1} C\ninst✝¹⁹ : Category.{v_2, u_2} D\ninst✝¹⁸ : Category.{v_3, u_3} E\ninst✝¹⁷ : HasShift C ℤ\ninst✝¹⁶ : HasShift D ℤ\ninst✝¹⁵ : HasShift E ℤ\nF : C ⥤ D\ninst✝¹⁴ : F.CommShift ℤ\nG : D ⥤ E\ninst✝¹³ : G.CommShift ℤ\ninst✝¹² : HasZeroO... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Functor | {
"line": 214,
"column": 10
} | {
"line": 216,
"column": 37
} | [
{
"pp": "case h₂\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝²⁰ : Category.{v_1, u_1} C\ninst✝¹⁹ : Category.{v_2, u_2} D\ninst✝¹⁸ : Category.{v_3, u_3} E\ninst✝¹⁷ : HasShift C ℤ\ninst✝¹⁶ : HasShift D ℤ\ninst✝¹⁵ : HasShift E ℤ\nF : C ⥤ D\ninst✝¹⁴ : F.CommShift ℤ\nG : D ⥤ E\ninst✝¹³ : G.CommShift ℤ\ninst✝¹² :... | simp only [assoc, prodComparison_snd, prod.comp_lift, comp_id, comp_zero,
limit.lift_π, BinaryFan.mk_pt, BinaryFan.π_app_right, BinaryFan.mk_snd,
← F.map_comp, F.map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.Functor | {
"line": 214,
"column": 10
} | {
"line": 216,
"column": 37
} | [
{
"pp": "case h₂\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝²⁰ : Category.{v_1, u_1} C\ninst✝¹⁹ : Category.{v_2, u_2} D\ninst✝¹⁸ : Category.{v_3, u_3} E\ninst✝¹⁷ : HasShift C ℤ\ninst✝¹⁶ : HasShift D ℤ\ninst✝¹⁵ : HasShift E ℤ\nF : C ⥤ D\ninst✝¹⁴ : F.CommShift ℤ\nG : D ⥤ E\ninst✝¹³ : G.CommShift ℤ\ninst✝¹² :... | simp only [assoc, prodComparison_snd, prod.comp_lift, comp_id, comp_zero,
limit.lift_π, BinaryFan.mk_pt, BinaryFan.π_app_right, BinaryFan.mk_snd,
← F.map_comp, F.map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Functor | {
"line": 214,
"column": 10
} | {
"line": 216,
"column": 37
} | [
{
"pp": "case h₂\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝²⁰ : Category.{v_1, u_1} C\ninst✝¹⁹ : Category.{v_2, u_2} D\ninst✝¹⁸ : Category.{v_3, u_3} E\ninst✝¹⁷ : HasShift C ℤ\ninst✝¹⁶ : HasShift D ℤ\ninst✝¹⁵ : HasShift E ℤ\nF : C ⥤ D\ninst✝¹⁴ : F.CommShift ℤ\nG : D ⥤ E\ninst✝¹³ : G.CommShift ℤ\ninst✝¹² :... | simp only [assoc, prodComparison_snd, prod.comp_lift, comp_id, comp_zero,
limit.lift_π, BinaryFan.mk_pt, BinaryFan.π_app_right, BinaryFan.mk_snd,
← F.map_comp, F.map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Equivalence | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 14
} | [
{
"pp": "case w\nC₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_4\nD₂ : Type u_5\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_4, u_4} D₁\ninst✝⁴ : Category.{v_5, u_5} D₂\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\ninst✝³ : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismPrope... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.CatCommSq | {
"line": 166,
"column": 42
} | {
"line": 166,
"column": 63
} | [
{
"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 L.functor R.functor B\nX : C... | ← L.functor.map_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | {
"line": 190,
"column": 8
} | {
"line": 190,
"column": 11
} | [
{
"pp": "case hφ₂\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf₁ f₂ : L.obj X ⟶ L.obj Y\ng : L.obj Y ⟶ L.obj Z\nα : W.LeftFrac... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.ObjectProperty.Shift | {
"line": 116,
"column": 2
} | {
"line": 118,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : ObjectProperty C\nG : Type u_4\ninst✝³ : AddGroup G\ninst✝² : HasShift C G\ninst✝¹ : P.IsStableUnderShift G\ninst✝ : P.IsClosedUnderIsomorphisms\nX : C\ng : G\n⊢ P ((shiftFunctor C g).obj X) ↔ P X",
"usedConstants": [
"AddGroup.toSubtractionMo... | refine ⟨fun hX ↦ ?_, P.le_shift g _⟩
rw [← P.shift_zero G, ← P.shift_shift g (-g) 0 (by simp)]
exact P.le_shift (-g) _ hX | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.Shift | {
"line": 116,
"column": 2
} | {
"line": 118,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : ObjectProperty C\nG : Type u_4\ninst✝³ : AddGroup G\ninst✝² : HasShift C G\ninst✝¹ : P.IsStableUnderShift G\ninst✝ : P.IsClosedUnderIsomorphisms\nX : C\ng : G\n⊢ P ((shiftFunctor C g).obj X) ↔ P X",
"usedConstants": [
"AddGroup.toSubtractionMo... | refine ⟨fun hX ↦ ?_, P.le_shift g _⟩
rw [← P.shift_zero G, ← P.shift_shift g (-g) 0 (by simp)]
exact P.le_shift (-g) _ hX | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Composition | {
"line": 121,
"column": 9
} | {
"line": 121,
"column": 21
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝⁴ : Category.{v₁, u₁} C₁\ninst✝³ : Category.{v₂, u₂} C₂\ninst✝² : Category.{v₃, u₃} C₃\nL₁ : C₁ ⥤ C₂\nL₂ : C₂ ⥤ C₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₁\ninst✝¹ : L₁.IsLocalization W₁\ninst✝ : (L₁ ⋙ L₂).IsLocalizatio... | by rw [hW₂₃] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 82
} | [
{
"pp": "case succ\nC : 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 Q : ObjectProperty C\ninst✝ : Q.IsTriangulatedClosed₂\nh : P ≤ Q\nn : ℕ\nH : P.extensionProduc... | exact extensionProduct_le_of_isTriangulatedClosed₂ (h.trans Q.le_isoClosure) H | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 42
} | [
{
"pp": "C : Type u_1\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : HasShift C ℤ\ninst✝⁴ : Preadditive C\ninst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝² : Pretriangulated C\nP : ObjectProperty C\ninst✝¹ : P.ContainsZero\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Arrow.mk (𝟙 X) ≅ Arr... | exact Arrow.isoMk (Iso.refl _) (asIso f) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 539,
"column": 14
} | {
"line": 544,
"column": 57
} | [
{
"pp": "C : Type u_1\ninst✝¹⁵ : Category.{v_1, u_1} C\ninst✝¹⁴ : HasZeroObject C\ninst✝¹³ : HasShift C ℤ\ninst✝¹² : Preadditive C\ninst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹⁰ : Pretriangulated C\nD : Type u_2\ninst✝⁹ : Category.{v_2, u_2} D\ninst✝⁸ : Preadditive D\ninst✝⁷ : HasZeroObject D\ninst✝... | by
rw [← trW_isoClosure]
rintro X₁ X₂ X₃ u₁₂ u₂₃ ⟨Z₁₂, v₁₂, w₁₂, H₁₂, mem₁₂⟩ ⟨Z₂₃, v₂₃, w₂₃, H₂₃, mem₂₃⟩
obtain ⟨Z₁₃, v₁₃, w₁₂, H₁₃⟩ := distinguished_cocone_triangle (u₁₂ ≫ u₂₃)
exact ⟨_, _, _, H₁₃, P.isoClosure.ext_of_isTriangulatedClosed₂
_ (someOctahedron rfl H₁₂ H₂₃ H₁₃).mem mem₁₂ mem₂₃⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 42,
"column": 32
} | {
"line": 42,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ (f ≫ g) ≫ 𝟙 X₃ = f ≫ g",
"usedConstants": [
"Eq.mpr",
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryStruct.to... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 42,
"column": 32
} | {
"line": 42,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ (f ≫ g) ≫ 𝟙 X₃ = f ≫ g",
"usedConstants": [
"Eq.mpr",
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryStruct.to... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 42,
"column": 32
} | {
"line": 42,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ (f ≫ g) ≫ 𝟙 X₃ = f ≫ g",
"usedConstants": [
"Eq.mpr",
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryStruct.to... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 111,
"column": 10
} | {
"line": 111,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ Cochain.ofHom (inv f g ≫ hom f g) =\n δ (-1) 0\n (-(snd (mappingConeCompTriangle f g).mor₁).comp\n ((↑(fst (f ≫ g))).comp ((... | δ_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 213,
"column": 68
} | {
"line": 213,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nX₁✝ X₂✝ X₃✝ : CochainComplex C ℤ\nf : X₁✝ ⟶ X₂✝\ng : X₂✝ ⟶ X₃✝\ninst✝ : HasZeroObject C\nX₁ X₂ X₃ : CochainComplex C ℤ\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nα : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 213,
"column": 68
} | {
"line": 213,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nX₁✝ X₂✝ X₃✝ : CochainComplex C ℤ\nf : X₁✝ ⟶ X₂✝\ng : X₂✝ ⟶ X₃✝\ninst✝ : HasZeroObject C\nX₁ X₂ X₃ : CochainComplex C ℤ\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nα : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 213,
"column": 68
} | {
"line": 213,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\nX₁✝ X₂✝ X₃✝ : CochainComplex C ℤ\nf : X₁✝ ⟶ X₂✝\ng : X₂✝ ⟶ X₃✝\ninst✝ : HasZeroObject C\nX₁ X₂ X₃ : CochainComplex C ℤ\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nα : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 48,
"column": 2
} | {
"line": 51,
"column": 13
} | [
{
"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 : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nK : HomologicalComplex C c\ne : c.Embedding c'\nj k : ι\nj' k' : ι'\nhj' : e.f j = j'\nhk : c.next j = k\nhk' : c'.next j'... | · simp only [K.shape _ _ hjk, comp_zero, true_iff]
rw [K.extend_d_from_eq_zero e j' k' j hj', comp_zero, comp_zero]
rw [hk]
exact hjk | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.TruncGE | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 44
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁵ : Category.{v_1, u_3} C\ninst✝⁴ : HasZeroMorphisms C\nK L M : HomologicalComplex C c'\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ninst✝³ : e.IsTruncGE\ninst✝² : ∀ (i' : ι'), K.HasHomology i'\ninst✝¹ : ∀ (i' : ι'... | rw [dif_neg (e.not_boundaryGE_next hij)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.RestrictionHomology | {
"line": 95,
"column": 90
} | {
"line": 97,
"column": 31
} | [
{
"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 : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝² : e.IsRelIff\ni j k : ι\nhi : c.prev j = i\nhk : c.next j = k\ni' j' k' : ι'\nhi' : e.f i = i'\nhj' :... | by
rw [restriction_d_eq _ _ hi' hj', assoc, assoc, Iso.inv_hom_id_assoc,
d_pOpcycles, comp_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.AreComplementary | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 29
} | [
{
"pp": "case right.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\ni₁ : ι₁\n⊢ ∃ a, fromSum e₁ e₂ a = e₁.f i₁",
"usedConstants": [
"Sum",
"Sum.inl",
"Comp... | · exact ⟨Sum.inl i₁, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.CochainComplex | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroMorphisms C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ne : K ≅ L\ninst✝³ : HasZeroObject C\ninst✝² : ∀ (i : ℤ), HasHomology K i\ninst✝¹ : ∀ (i : ℤ), HasHomology L i\nn : ℤ\ninst✝ : K.IsGE n\n⊢ QuasiIso (K.πTruncGE n)",
"usedConstants": [
... | rw [quasiIso_πTruncGE_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.SmallShiftedHom | {
"line": 244,
"column": 41
} | {
"line": 250,
"column": 90
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nW : MorphismProperty C\nM : Type w'\ninst✝⁵ : AddMonoid M\ninst✝⁴ : HasShift C M\ninst✝³ : HasShift D M\nL : C ⥤ D\ninst✝² : L.IsLocalization W\ninst✝¹ : L.CommShift M\nX Y : C\ninst✝ : HasSmallLocalizedShiftedHom W M... | by
subst hm₀
dsimp [equiv, mk₀]
erw [SmallHom.equiv_mk, Functor.map_comp]
dsimp [equiv, mk₀, ShiftedHom.mk₀, shiftFunctorZero']
simp only [comp_id, L.commShiftIso_zero, Functor.CommShift.isoZero_hom_app, assoc,
← Functor.map_comp_assoc, Iso.inv_hom_id_app, Functor.id_obj, Functor.map_id, id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.SmallShiftedHom | {
"line": 310,
"column": 36
} | {
"line": 310,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\nW : MorphismProperty C\nM : Type w'\ninst✝⁵ : AddMonoid M\ninst✝⁴ : HasShift C M\ninst✝³ : HasShift D M\nX Y : C\ninst✝² : HasSmallLocalizedShiftedHom W M X Y\ninst✝¹ : HasSmallLocalizedShiftedHom W M Y Y\ninst✝ : W.I... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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