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.Sets.OpenCover | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 49
} | [
{
"pp": "case neg\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nU : ι → Opens X\nhn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)\nh : ∀ (i : ι), IsPreirreducible ↑(U i)\ns : Set X\nhs : IsOpen[inst✝] s\nhsU : s ⊆ ⋃ i, ↑(U i)\nx : X\nhx : x ∈ s\ni : ι\nhi : x ∈ ↑(U i)\nu : Set X\nhu : u ∈ irreducib... | choose j haj using mem_iUnion.mp <| hsU ha₁ | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Topology.QuasiSeparated | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 22
} | [
{
"pp": "case insert\nα✝ : Type u_1\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : QuasiSeparatedSpace α\ns : Set (Set α)\na✝ : Set α\ns✝ : Set (Set α)\nha : a✝ ∉ s✝\nhs : s✝.Finite\nih :\n (∀ t ∈ s✝, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →\n (∀ t ∈ s✝, IsCompact t) → (∀ ... | | insert ha hs ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.KrullDimension | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 46
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : QuasiSober Y\ninst✝² : T0Space Y\ninst✝¹ : QuasiSober X\ninst✝ : T0Space X\nx : X\nf : X → Y\nhf : IsOpenEmbedding f\n⊢ coheight (irreducibleSetEquivPoints.symm (f x)) = coheight (irreducibleSetEquivPoints.sy... | ← Topology.IsOpenEmbedding.coheight_map hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Derivation.ToSquareZero | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 69
} | [
{
"pp": "case refine_3.a\nR : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nhI : I ^ 2 = ⊥\nf : A →ₐ[R] B\ne : (Ideal.Quotient.mkₐ R I).comp f = Is... | simp only [map_mul, sub_mul, mul_sub, Algebra.smul_def] at this ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 27,
"column": 2
} | {
"line": 27,
"column": 29
} | [
{
"pp": "b : ℕ\n⊢ factorizationLCMLeft 0 b = 1",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"Nat.f... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 27,
"column": 2
} | {
"line": 27,
"column": 29
} | [
{
"pp": "b : ℕ\n⊢ factorizationLCMLeft 0 b = 1",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"Nat.f... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 27,
"column": 2
} | {
"line": 27,
"column": 29
} | [
{
"pp": "b : ℕ\n⊢ factorizationLCMLeft 0 b = 1",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"Nat.f... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 29
} | [
{
"pp": "a : ℕ\n⊢ a.factorizationLCMLeft 0 = 1",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"congrArg",
"zero_le._simp_1",
"Nat.... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 29
} | [
{
"pp": "a : ℕ\n⊢ a.factorizationLCMLeft 0 = 1",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"congrArg",
"zero_le._simp_1",
"Nat.... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 29
} | [
{
"pp": "a : ℕ\n⊢ a.factorizationLCMLeft 0 = 1",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"congrArg",
"zero_le._simp_1",
"Nat.... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 31
} | [
{
"pp": "case inr.inl\na : ℕ\nha : a ≠ 0\n⊢ a.factorizationLCMLeft 0 ∣ a",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Dvd.dvd",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"cong... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 31
} | [
{
"pp": "case inr.inl\na : ℕ\nha : a ≠ 0\n⊢ a.factorizationLCMLeft 0 ∣ a",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Dvd.dvd",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"cong... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 31
} | [
{
"pp": "case inr.inl\na : ℕ\nha : a ≠ 0\n⊢ a.factorizationLCMLeft 0 ∣ a",
"usedConstants": [
"Nat.lcm",
"Finsupp.instFunLike",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Dvd.dvd",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Monoid.toMulOneClass",
"cong... | simp [factorizationLCMLeft] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.Flat | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 27
} | [
{
"pp": "R S T : CommRingCat\nf : R ⟶ S\ng : R ⟶ T\nhf : Function.Injective ⇑(ConcreteCategory.hom f)\nhg : (Hom.hom g).Flat\n⊢ Function.Injective ⇑(ConcreteCategory.hom (pushout.inr f g))",
"usedConstants": [
"CommRingCat.Hom.hom",
"CommRingCat.carrier",
"CommSemiring.toSemiring",
"... | algebraize [f.hom, g.hom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.Flat | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 27
} | [
{
"pp": "R S T : CommRingCat\nf : R ⟶ S\ng : R ⟶ T\nhf : (Hom.hom f).Flat\nhg : Function.Injective ⇑(ConcreteCategory.hom g)\n⊢ Function.Injective ⇑(ConcreteCategory.hom (pushout.inl f g))",
"usedConstants": [
"CommRingCat.Hom.hom",
"CommRingCat.carrier",
"CommSemiring.toSemiring",
"... | algebraize [f.hom, g.hom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 633,
"column": 6
} | {
"line": 633,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nι : Type u_1\nf : ι → R\n⊢ (zeroLocus (⋃ i, {f i}))ᶜ = Set.univ ↔ Ideal.span (Set.range f) = ⊤",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"congrArg",
"CommSemiring.toSemiring",
"Compl.compl",... | ← PrimeSpectrum.zeroLocus_empty_iff_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Nat.card α = p\nthis✝ : Finite α\nthis : Nontrivial α\n⊢ IsCyclic α",
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Group.toDivisionMonoid",
"DivisionMonoid.toDivInvOneMonoid",
... | obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := exists_ne 1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 886,
"column": 38
} | {
"line": 888,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nhf : StableUnderSpecialization (Set.range (comap f))\n⊢ IsClosed (Set.range (comap f))",
"usedConstants": [
"isClosed_univ",
"Eq.mpr",
"Set.image_univ",
"StableUnderSpecialization",
... | by
rw [← Set.image_univ] at hf ⊢
exact isClosed_image_of_stableUnderSpecialization _ _ isClosed_univ hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Exponent | {
"line": 135,
"column": 4
} | {
"line": 137,
"column": 64
} | [
{
"pp": "case neg\nG : Type u\ninst✝ : Monoid G\nh : ¬ExponentExists G\n⊢ exponent G = sInf {d | 0 < d ∧ ∀ (x : G), x ^ d = 1}",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"setOf",
"Memb... | have : {d | 0 < d ∧ ∀ (x : G), x ^ d = 1} = ∅ :=
Set.eq_empty_of_forall_notMem fun n hn ↦ h ⟨n, hn⟩
rw [Monoid.exponent_eq_zero_iff.mpr h, this, Nat.sInf_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Exponent | {
"line": 135,
"column": 4
} | {
"line": 137,
"column": 64
} | [
{
"pp": "case neg\nG : Type u\ninst✝ : Monoid G\nh : ¬ExponentExists G\n⊢ exponent G = sInf {d | 0 < d ∧ ∀ (x : G), x ^ d = 1}",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"setOf",
"Memb... | have : {d | 0 < d ∧ ∀ (x : G), x ^ d = 1} = ∅ :=
Set.eq_empty_of_forall_notMem fun n hn ↦ h ⟨n, hn⟩
rw [Monoid.exponent_eq_zero_iff.mpr h, this, Nat.sInf_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 958,
"column": 12
} | {
"line": 958,
"column": 13
} | [
{
"pp": "case mp\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nH : DenseRange (comap f)\n⊢ ∀ (I : Ideal R) (h : I ∈ minimalPrimes R), { asIdeal := I, isPrime := ⋯ } ∈ Set.range (comap f)",
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal"
]
... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 69
} | [
{
"pp": "α : Type u_1\nG : Type u_2\nG' : Type u_3\na : α\ninst✝⁵ : Group α\ninst✝⁴ : Group G\ninst✝³ : Group G'\nM : Type u_4\ninst✝² : Monoid M\ninst✝¹ : IsCyclic G\ninst✝ : MulDistribMulAction M G\nn✝ : ℕ\nhn : Nat.card G = n✝\nm n : M\ng : G\nhg : orderOf g = Nat.card G\n⊢ g ^ ⋯.choose = (toMonoidHom G m) (... | ← (MulDistribMulAction.toMonoidHom G n).map_cyclic.choose_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PGroup | {
"line": 218,
"column": 4
} | {
"line": 224,
"column": 54
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\n⊢ Nontrivial ↥(Subgroup.center G)",
"usedConstants": [
"ConjAct.fixedPoints_eq_center",
"Nontrivial",
"Dvd.dvd",
"InvOneClass.toOne",
"DivInvOneM... | have := (hG.of_equiv ConjAct.toConjAct).exists_fixed_point_of_prime_dvd_card_of_fixed_point G
rw [ConjAct.fixedPoints_eq_center] at this
have dvd : p ∣ Nat.card G := by
obtain ⟨n, hn0, hn⟩ := hG.nontrivial_iff_card.mp inferInstance
exact hn.symm ▸ dvd_pow_self _ (ne_of_gt hn0)
obtain ⟨g, hg⟩ := ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PGroup | {
"line": 218,
"column": 4
} | {
"line": 224,
"column": 54
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\n⊢ Nontrivial ↥(Subgroup.center G)",
"usedConstants": [
"ConjAct.fixedPoints_eq_center",
"Nontrivial",
"Dvd.dvd",
"InvOneClass.toOne",
"DivInvOneM... | have := (hG.of_equiv ConjAct.toConjAct).exists_fixed_point_of_prime_dvd_card_of_fixed_point G
rw [ConjAct.fixedPoints_eq_center] at this
have dvd : p ∣ Nat.card G := by
obtain ⟨n, hn0, hn⟩ := hG.nontrivial_iff_card.mp inferInstance
exact hn.symm ▸ dvd_pow_self _ (ne_of_gt hn0)
obtain ⟨g, hg⟩ := ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 92,
"column": 2
} | {
"line": 99,
"column": 32
} | [
{
"pp": "case H.inr\nα : Type u_1\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n\nd : ℕ\nIH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m\nhd : d ∣ Fintype.card α\nhpos : 0 < #{a | orderOf a = d}\nhd0 : d ≠ 0\na ... | have h1 :
(∑ m ∈ d.properDivisors, #{a : α | orderOf a = m}) =
∑ m ∈ d.properDivisors, φ m := by
refine Finset.sum_congr rfl fun m hm => ?_
simp only [mem_properDivisors] at hm
refine IH m hm.2 (hm.1.trans hd) (Finset.card_pos.2 ⟨a ^ (d / m), ?_⟩)
rw [mem_filter_univ, orderOf_pow a, ha, Nat.gc... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Sylow | {
"line": 368,
"column": 59
} | {
"line": 368,
"column": 78
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x ∈ centralizer ↑P\nhy : g⁻¹ * x * g ∈ centralizer ↑P\n⊢ ↑P ≤ centralizer ↑(zpowers x)",
"usedConstants": [
"Sylow.toSubgroup",
"Eq.mpr",
"Subgroup.le_centr... | le_centralizer_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Sylow | {
"line": 370,
"column": 8
} | {
"line": 370,
"column": 27
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x ∈ centralizer ↑P\nhy : g⁻¹ * x * g ∈ centralizer ↑P\nh1 : ↑P ≤ centralizer ↑(zpowers x)\n⊢ ↑(g • P) ≤ centralizer ↑(zpowers x)",
"usedConstants": [
"Sylow.toSubgroup"... | le_centralizer_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 232,
"column": 52
} | {
"line": 233,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : IsSimpleGroup α\ng : α\nhg : ∀ (x : α), x ∈ Subgroup.zpowers g\nhα : orderOf g ≠ 1\nn : ℕ\nh : Subgroup.zpowers (g ^ n) = ⊤\n⊢ g ∈ Subgroup.zpowers (g ^ n)",
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | simp_all only [ne_eq, orderOf_eq_one_iff,
Subgroup.mem_top] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.JordanHolder | {
"line": 337,
"column": 70
} | {
"line": 340,
"column": 21
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : head s₁ = head s₂\nht : last s₁ = last s₂\nhs₁ : s₁.length = 0\n⊢ s₂.length = 0",
"usedConstants": [
"RelSeries.last",
"congrArg",
"AddMonoid.toAddZeroClass",
"setOf",
"Nat... | by
have : Fin.last s₂.length = (0 : Fin s₂.length.succ) :=
s₂.injective (hb.symm.trans ((congr_arg s₁ (Fin.ext (by simp [hs₁]))).trans ht)).symm
simpa [Fin.ext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 423,
"column": 8
} | {
"line": 423,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np q : Submodule R M\nx✝¹ : IsSemisimpleModule R ↥p\nx✝ : IsSemisimpleModule R ↥q\nf : Bool → Submodule R M := fun t ↦ Bool.rec q p t\n⊢ ∀ i ∈ Set.univ, IsSemisimpleModule R ↥(f i)",
"usedConstants": [
"S... | rintro (_ | _) _ <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 423,
"column": 8
} | {
"line": 423,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np q : Submodule R M\nx✝¹ : IsSemisimpleModule R ↥p\nx✝ : IsSemisimpleModule R ↥q\nf : Bool → Submodule R M := fun t ↦ Bool.rec q p t\n⊢ ∀ i ∈ Set.univ, IsSemisimpleModule R ↥(f i)",
"usedConstants": [
"S... | rintro (_ | _) _ <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 423,
"column": 8
} | {
"line": 423,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np q : Submodule R M\nx✝¹ : IsSemisimpleModule R ↥p\nx✝ : IsSemisimpleModule R ↥q\nf : Bool → Submodule R M := fun t ↦ Bool.rec q p t\n⊢ ∀ i ∈ Set.univ, IsSemisimpleModule R ↥(f i)",
"usedConstants": [
"S... | rintro (_ | _) _ <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 105,
"column": 30
} | {
"line": 105,
"column": 57
} | [
{
"pp": "R : Type ?u.50900\nA : Type ?u.50903\nM : Type ?u.50925\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A\na : A\ninst✝⁸ : Algebra R A\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module A M\ninst✝⁵ : Module R M\ninst✝⁴ : IsScalarTower R A M\nN : Type ?u.51854\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹... | LinearEquiv.coe_toLinearMap | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 56,
"column": 2
} | {
"line": 68,
"column": 14
} | [
{
"pp": "case insert\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 ∧ ... | have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
specialize hrn hs.1
rw [mem_comap, mem_sup] at hrn
rcases hrn with ⟨y, hy, z, hz, hyz⟩
dsimp at hyz
rw [mem_smul_span_singleton] at hy
rcases hy with ⟨c, hci, rfl⟩
use r - c
constructor
· rw [sub_right_comm]
exact I.sub_mem... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\n⊢ W.IsInvertedBy F ↔ W.map F ≤ isomorphisms D",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.instCompleteBoolean... | rw [iff_le_inverseImage_isomorphisms, map_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\n⊢ W.IsInvertedBy F ↔ W.map F ≤ isomorphisms D",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.instCompleteBoolean... | rw [iff_le_inverseImage_isomorphisms, map_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\n⊢ W.IsInvertedBy F ↔ W.map F ≤ isomorphisms D",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.instCompleteBoolean... | rw [iff_le_inverseImage_isomorphisms, map_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologicalComplexBiprod | {
"line": 152,
"column": 16
} | {
"line": 152,
"column": 31
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK L : HomologicalComplex C c\ninst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)\nM : HomologicalComplex C c\nα : M ⟶ K\nβ : M ⟶ L\ni : ι\n⊢ (biprod.lift α β ≫ biprod.snd).f i = β.f i",
"usedC... | biprod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 667,
"column": 4
} | {
"line": 667,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m i : ℤ\nz : Cochain F G 0\nhz : z ∈ cocycle F G 0\n⊢ z.v i i ⋯ ≫ G.d i (i + 1) = F.d i (i + 1) ≫ z.v (i + 1) (i + 1) ⋯",
"usedConstants": [
"Coch... | rw [mem_iff 0 1 (zero_add 1)] at hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 90
} | [
{
"pp": "case refine_2\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 : ↥(KaehlerDifferential.ideal R S)\nhx : x ∈ Submodule.restrictSca... | refine Submodule.smul_induction_on ((Submodule.restrictScalars_mem _ _ _).mp hx) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 568,
"column": 6
} | {
"line": 568,
"column": 39
} | [
{
"pp": "R : 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\nx : S →₀ S\nthis : ((derivationQuotKerTotal R S).liftKaehlerDifferential ∘ₗ linearCombination S ⇑(D R S))... | rw [LinearMap.comp_apply] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 393,
"column": 7
} | {
"line": 393,
"column": 66
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\n⊢ n' + m = t'",
"usedConstants"... | by rw [← ht', ← h, ← hn', add_assoc, add_comm a, add_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃\nv₁₂ : X₂ ⟶ Z₁... | have rel₁₂ := H.triangleMorphism₁.comm₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 678,
"column": 18
} | {
"line": 678,
"column": 29
} | [
{
"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\nT₁ T₂ : Triangle C\nhT₁ : T₁ ∈ distinguishedTriangles\nhT₂ : T₂ ∈ distinguishedTriangles\ne : Arrow.mk T₁.mor₁ ≅ Arrow... | by simp [φ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 678,
"column": 31
} | {
"line": 678,
"column": 42
} | [
{
"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\nT₁ T₂ : Triangle C\nhT₁ : T₁ ∈ distinguishedTriangles\nhT₂ : T₂ ∈ distinguishedTriangles\ne : Arrow.mk T₁.mor₁ ≅ Arrow... | by simp [φ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | {
"line": 163,
"column": 88
} | {
"line": 164,
"column": 27
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nK₁ L₁ K₂ L₂ : CochainComplex C ℤ\nφ₁ : K₁ ⟶ L₁\nφ₂ : K₂ ⟶ L₂\na : K₁ ⟶ K₂\nb : L₁ ⟶ L₂\ncomm : φ₁ ≫ b = a ≫ φ₂\n⊢ map φ₁ φ₂ a b comm = mapOfHomotopy (Homotopy.ofEq comm)",
"usedConstants": [
"... | by
simp [map, mapOfHomotopy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 147,
"column": 89
} | {
"line": 152,
"column": 92
} | [
{
"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\n⊢ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.f ≫ (mappingConeHomOfDegreewiseSplitIso ... | by
ext n
have h := (σ (n + 1)).f_r
dsimp at h
dsimp [mappingConeHomOfDegreewiseSplitXIso]
rw [id_comp, comp_sub, ← comp_f_assoc, S.zero, zero_f, zero_comp, zero_sub, reassoc_of% h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 33
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\n⊢ (∃ S σ, Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ)) → T ∈ distinguishedTriangles",
"usedC... | · rintro ⟨S, σ, ⟨e⟩⟩
rw [rotate_distinguished_triangle, rotate_distinguished_triangle]
refine isomorphic_distinguished _ ?_ _
((rotate _ ⋙ rotate _).mapIso e ≪≫
CochainComplex.trianglehOfDegreewiseSplitRotateRotateIso S σ)
exact ⟨_, _, _, ⟨Iso.refl _⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 77
} | [
{
"pp": "C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C :=\n { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.cycles j, f := K.homologyι i, g := ... | let T := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) (by simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Category.Quiv | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 44
} | [
{
"pp": "V W : Type u\ninst✝¹ : Quiver V\ninst✝ : Quiver W\ne : V ≃ W\nhe : (X Y : V) → (X ⟶ Y) ≃ (e X ⟶ e Y)\nX Y : ↑(of V)\nf : X ⟶ Y\n⊢ ({ obj := ⇑e, map := fun {X Y} ↦ ⇑(he X Y) } ≫\n { obj := ⇑e.symm, map := fun {X Y} f ↦ (he (e.symm X) (e.symm Y)).symm (Quiver.homOfEq f ⋯ ⋯) }).map\n f =\n ... | dsimp [Quiv.id_eq_id, Quiv.comp_eq_comp] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Category.Quiv | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 35
} | [
{
"pp": "V : Type u\nC : Type u₁\ninst✝¹ : Quiver V\ninst✝ : Category.{v₁, u₁} C\nG : V ⥤q C\n⊢ Paths.of V ⋙q (Cat.freeMap G).toPrefunctor ⋙q (pathComposition C).toPrefunctor = G",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"congrArg",
"CategoryTheory.Path... | pathsOf_freeMap_toPrefunctor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 408,
"column": 2
} | {
"line": 408,
"column": 77
} | [
{
"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_4\ninst✝¹ : Category.{v_4, u_4} E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : L.IsLocalization W\ne : L ⋙ eq.functor ≅ L'\nh : W.IsInvertedBy L'\nF₁ : W.Localization ⥤ D := Con... | let e' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso W.Q W (L ⋙ eq.functor) L' _ _ e | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 12
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nW : MorphismProperty C\nX Y : C\nφ : W.LeftFraction X Y\nL : C ⥤ D\nhL : W.IsInvertedBy L\n⊢ φ.map L hL ≫ L.map φ.s = L.map φ.f",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheor... | letI := hL _ φ.hs
simp [map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 12
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nW : MorphismProperty C\nX Y : C\nφ : W.LeftFraction X Y\nL : C ⥤ D\nhL : W.IsInvertedBy L\n⊢ φ.map L hL ≫ L.map φ.s = L.map φ.f",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheor... | letI := hL _ φ.hs
simp [map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TriangleShift | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\na b : ℤ\n⊢ CategoryTheory.shiftFunctorAdd' (Triangle C) a b (a + b) ⋯ = shiftFunctorAdd' C a b (a + b) ⋯",
"usedConstants": [
"Eq.mpr",
"... | rw [shiftFunctorAdd'_eq_shiftFunctorAdd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 12
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nW : MorphismProperty C\nX Y : C\nφ : W.RightFraction X Y\nL : C ⥤ D\nhL : W.IsInvertedBy L\n⊢ L.map φ.s ≫ φ.map L hL = L.map φ.f",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Qui... | letI := hL _ φ.hs
simp [map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 12
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nW : MorphismProperty C\nX Y : C\nφ : W.RightFraction X Y\nL : C ⥤ D\nhL : W.IsInvertedBy L\n⊢ L.map φ.s ≫ φ.map L hL = L.map φ.f",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Qui... | letI := hL _ φ.hs
simp [map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 404,
"column": 4
} | {
"line": 417,
"column": 58
} | [
{
"pp": "case refine_2.a.refine_2\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z\nU : C\nt₁ : a₁.Y' ⟶ U\nt₂ ... | · obtain ⟨q, fac₃⟩ := exists_leftFraction (RightFraction.mk (z₁.s ≫ w₁.s)
(W.comp_mem _ _ z₁.hs w₁.hs) (z₂.s ≫ w₂.s))
dsimp at fac₃
simp only [assoc] at fac₃
have eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s := by
rw [← reassoc_of% fac₁', ← fac₃, reassoc_of% hst, reassoc_of% fa... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.ObjectProperty.FiniteProducts | {
"line": 95,
"column": 8
} | {
"line": 95,
"column": 98
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nH : ∀ (J : Type w) [Finite J], P.IsClosedUnderLimitsOfShape (Discrete J)\nJ : Type\nx✝ : Finite J\n⊢ P.IsClosedUnderLimitsOfShape (Discrete J)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"UnivLE.small",
... | P.isClosedUnderLimitsOfShape_iff_of_equivalence (Discrete.equivalence (equivShrink.{w} _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 952,
"column": 75
} | {
"line": 956,
"column": 5
} | [
{
"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\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasRightCalculusOfFractions\nX Y : C\nf : L.obj X ⟶ L.obj Y\n⊢ ∃ φ, f = φ.map L ⋯",
"usedConstants": [
"CategoryTheory.Functor.... | by
obtain ⟨φ, eq⟩ := Localization.exists_leftFraction L.op W.op f.op
refine ⟨φ.unop, Quiver.Hom.op_inj ?_⟩
rw [eq, MorphismProperty.RightFraction.op_map]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 362,
"column": 4
} | {
"line": 363,
"column": 29
} | [
{
"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 : ObjectProperty C\ninst✝ : IsTriangulated C\nn n' : ℕ\nh : n = n' + 1\nm : ℕ\nhm : P.extensio... | rw [← add_assoc, extensionProductIter_succ', extensionProductIter_succ', hm,
extensionProduct_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 362,
"column": 4
} | {
"line": 363,
"column": 29
} | [
{
"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 : ObjectProperty C\ninst✝ : IsTriangulated C\nn n' : ℕ\nh : n = n' + 1\nm : ℕ\nhm : P.extensio... | rw [← add_assoc, extensionProductIter_succ', extensionProductIter_succ', hm,
extensionProduct_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 362,
"column": 4
} | {
"line": 363,
"column": 29
} | [
{
"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 : ObjectProperty C\ninst✝ : IsTriangulated C\nn n' : ℕ\nh : n = n' + 1\nm : ℕ\nhm : P.extensio... | rw [← add_assoc, extensionProductIter_succ', extensionProductIter_succ', hm,
extensionProduct_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Opposite | {
"line": 56,
"column": 40
} | {
"line": 56,
"column": 52
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\ninst✝ : Abelian V\nX Y Z : Vᵒᵖ\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ g.unop ≫ f.unop = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Opposite | {
"line": 99,
"column": 32
} | {
"line": 99,
"column": 44
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\nx✝⁴ x✝³ x✝² : ι\nx✝¹ : c.symm.Rel x✝⁴ x✝³\nx✝ : c.symm.Rel x✝³ x✝²\n⊢ (X.d x✝³ x✝⁴).unop ≫ (X.d x✝² x✝³).unop = 0",
"usedConstants": [
"Eq.mpr",
"Opp... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Opposite | {
"line": 107,
"column": 32
} | {
"line": 107,
"column": 44
} | [
{
"pp": "ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{v_1, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c.symm\nx✝⁴ x✝³ x✝² : ι\nx✝¹ : c.Rel x✝⁴ x✝³\nx✝ : c.Rel x✝³ x✝²\n⊢ (X.d x✝³ x✝⁴).unop ≫ (X.d x✝² x✝³).unop = 0",
"usedConstants": [
"Eq.mpr",
"Opposite... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone | {
"line": 134,
"column": 57
} | {
"line": 136,
"column": 64
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ninst✝ : HasHomotopyCofiber φ\nM : CochainComplex C ℤ\nn m : ℤ\nα : Cochain K M m\nβ : Cochain L M n\nh : m + 1 = n\n⊢ (↑(inr φ)).comp (descCochain φ α β h) ⋯ = β",
"usedConstants": [
"Co... | by
ext p q hpq
simp [Cochain.comp_v (n₂ := m) _ _ _ _ (p + 1) q rfl (by lia)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 220,
"column": 4
} | {
"line": 224,
"column": 51
} | [
{
"pp": "case refine_5\nC : 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 ⟶ mapping... | · refine isomorphic_distinguished _ (mappingConeCompTriangleh_distinguished u₁₂ u₂₃) _ ?_
exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _)
(by dsimp [α, mappingConeCompTriangleh]; simp)
(by dsimp [β, mappingConeCompTriangleh]; simp)
(by dsimp [mappingConeCompTriangleh]; simp... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 46
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nj k : ι\nhj : ¬e.BoundaryGE j\nhk : c.next j = k\nhjk : ¬c.Rel j k\n⊢ ¬e.BoundaryGE k",
"usedConstants": [
"Eq.mpr",
"ComplexShape.Embedding.BoundaryGE",
"congr... | subst hk
simpa only [c.next_eq_self j hjk] using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 46
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nj k : ι\nhj : ¬e.BoundaryGE j\nhk : c.next j = k\nhjk : ¬c.Rel j k\n⊢ ¬e.BoundaryGE k",
"usedConstants": [
"Eq.mpr",
"ComplexShape.Embedding.BoundaryGE",
"congr... | subst hk
simpa only [c.next_eq_self j hjk] using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 81,
"column": 2
} | {
"line": 91,
"column": 45
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij : c.prev j = i\nhj : ¬e.BoundaryGE j\n⊢ c'.prev (e.f j) = e.f i",
"usedConstants": [
"Eq.mpr",
"ComplexShape.Embedding.BoundaryGE",
"False",
"congrArg"... | by_cases hij' : c.Rel i j
· exact c'.prev_eq' (by simpa only [e.rel_iff] using hij')
· obtain rfl : j = i := by
simpa only [c.prev_eq_self j (by simpa only [hij] using hij')] using hij
apply c'.prev_eq_self
intro hj'
simp only [BoundaryGE, not_and, not_forall, not_not] at hj
obtain ⟨i, hi⟩ := ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 81,
"column": 2
} | {
"line": 91,
"column": 45
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij : c.prev j = i\nhj : ¬e.BoundaryGE j\n⊢ c'.prev (e.f j) = e.f i",
"usedConstants": [
"Eq.mpr",
"ComplexShape.Embedding.BoundaryGE",
"False",
"congrArg"... | by_cases hij' : c.Rel i j
· exact c'.prev_eq' (by simpa only [e.rel_iff] using hij')
· obtain rfl : j = i := by
simpa only [c.prev_eq_self j (by simpa only [hij] using hij')] using hij
apply c'.prev_eq_self
intro hj'
simp only [BoundaryGE, not_and, not_forall, not_not] at hj
obtain ⟨i, hi⟩ := ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.TruncGE | {
"line": 174,
"column": 4
} | {
"line": 180,
"column": 45
} | [
{
"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)]
by_cases hi : e.BoundaryGE i
· rw [dif_pos hi]
simp [truncGE'_d_eq_fromOpcycles _ e hij rfl rfl hi,
← cancel_epi (K.pOpcycles (e.f i))]
· rw [dif_neg hi]
simp [truncGE'_d_eq _ e hij rfl rfl hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.TruncGE | {
"line": 174,
"column": 4
} | {
"line": 180,
"column": 45
} | [
{
"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)]
by_cases hi : e.BoundaryGE i
· rw [dif_pos hi]
simp [truncGE'_d_eq_fromOpcycles _ e hij rfl rfl hi,
← cancel_epi (K.pOpcycles (e.f i))]
· rw [dif_neg hi]
simp [truncGE'_d_eq _ e hij rfl rfl hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Fractions | {
"line": 130,
"column": 4
} | {
"line": 133,
"column": 57
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\ninst✝³ : HasDerivedCategory C\nX Y : CochainComplex C ℤ\na b : ℤ\ninst✝² : X.IsStrictlyGE a\ninst✝¹ : X.IsStrictlyLE b\ninst✝ : Y.IsStrictlyGE a\nf : Q.obj X ⟶ Q.obj Y\nX' : CochainComplex C ℤ\nhX' : X'.IsStrictlyLE b\ns : X' ⟶ ... | rw [← Functor.map_comp_assoc, ← CochainComplex.πTruncGE_naturality s a,
Functor.map_comp, assoc, IsIso.hom_inv_id_assoc,
← Functor.map_comp_assoc, CochainComplex.πTruncGE_naturality g a,
Functor.map_comp, assoc, IsIso.hom_inv_id, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.SmallHom | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 52
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nW : MorphismProperty C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nL : C ⥤ D\ninst✝² : L.IsLocalization W\nX Y : C\ninst✝¹ : HasSmallLocalizedHom W X Y\ninst✝ : HasSmallLocalizedHom W X Y\ne : SmallHom W X Y\n⊢ (equiv W L) (chgUniv e) = (equiv W L) e",
"us... | obtain ⟨f, rfl⟩ := (equiv W W.Q).symm.surjective e | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 93,
"column": 46
} | {
"line": 95,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\n⊢ HasExt C",
"usedConstants": [
"CategoryTheory.hasExt_of_hasDerivedCategory",
"HasDerivedCategory.standard"
]
}
] | by
letI := HasDerivedCategory.standard
exact hasExt_of_hasDerivedCategory _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 3\nφ : R₁ ⟶ R₂\nhR₁ :\n (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)\n (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._pro... | rw [comp_sub] at h₅ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 374,
"column": 4
} | {
"line": 378,
"column": 62
} | [
{
"pp": "case e_a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasExt C\nX Y : C\nn : ℕ\ninst✝ : HasDerivedCategory C\nα β : Ext X Y n\nα' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯\nβ' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯\neq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).c... | dsimp [α']
rw [comp_hom, mk₀_hom, mk₀_hom]
dsimp
rw [ShiftedHom.mk₀_comp_mk₀_assoc, ← Functor.map_comp,
biprod.lift_fst, Functor.map_id, ShiftedHom.mk₀_id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 374,
"column": 4
} | {
"line": 378,
"column": 62
} | [
{
"pp": "case e_a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasExt C\nX Y : C\nn : ℕ\ninst✝ : HasDerivedCategory C\nα β : Ext X Y n\nα' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯\nβ' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯\neq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).c... | dsimp [α']
rw [comp_hom, mk₀_hom, mk₀_hom]
dsimp
rw [ShiftedHom.mk₀_comp_mk₀_assoc, ← Functor.map_comp,
biprod.lift_fst, Functor.map_id, ShiftedHom.mk₀_id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.ShortExact | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\n⊢ Q.map (CochainComplex.mappingCone.descShortComplex S) ≫ triangleOfSESδ hS =\n Q.map (CochainComplex.mappingCone.triangle S.f).mor₃ ≫ (Functor.commShif... | simp [triangleOfSESδ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.DerivedCategory.ShortExact | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\n⊢ Q.map (CochainComplex.mappingCone.descShortComplex S) ≫ triangleOfSESδ hS =\n Q.map (CochainComplex.mappingCone.triangle S.f).mor₃ ≫ (Functor.commShif... | simp [triangleOfSESδ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.ShortExact | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\n⊢ Q.map (CochainComplex.mappingCone.descShortComplex S) ≫ triangleOfSESδ hS =\n Q.map (CochainComplex.mappingCone.triangle S.f).mor₃ ≫ (Functor.commShif... | simp [triangleOfSESδ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.ShortExact | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 71
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\nthis : QuasiIso (CochainComplex.mappingCone.descShortComplex S)\n⊢ (Q.mapTriangle.obj (CochainComplex.mappingCone.triangle S.f)).mor₃ ≫\n ... | dsimp [triangleOfSESδ]
rw [CategoryTheory.Functor.map_id, comp_id, IsIso.hom_inv_id_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.ShortExact | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 71
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nS : ShortComplex (CochainComplex C ℤ)\nhS : S.ShortExact\nthis : QuasiIso (CochainComplex.mappingCone.descShortComplex S)\n⊢ (Q.mapTriangle.obj (CochainComplex.mappingCone.triangle S.f)).mor₃ ≫\n ... | dsimp [triangleOfSESδ]
rw [CategoryTheory.Functor.map_id, comp_id, IsIso.hom_inv_id_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\ninst✝ : HasDerivedCategory C\n⊢ hS.extClass.hom = hS.singleδ",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"instCategoryDerivedCategory",
"DerivedCa... | change SmallShiftedHom.equiv W Q hS.extClass = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 136,
"column": 2
} | {
"line": 139,
"column": 59
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nA : Type u_2\nB : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : HasShift C B\nφ : A →+ B\nX : PullbackShift C φ\na₁ a₂ a₃ : A\nh : a₁ + a₂ = a₃\nb₁ b₂ b₃ : B\nh₁ : b₁ = φ a₁\nh₂ : b₂ = φ a₂\nh₃ : b₃ = φ a₃\n⊢ (shiftFunctorAdd' (PullbackShift... | rw [← cancel_epi ((shiftFunctorAdd' _ a₁ a₂ a₃ h).inv.app X), Iso.inv_hom_id_app,
pullbackShiftFunctorAdd'_inv_app φ X a₁ a₂ a₃ h b₁ b₂ b₃ h₁ h₂ h₃, assoc, assoc, assoc,
Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app_assoc, Iso.hom_inv_id_app_assoc,
← Functor.map_comp, Iso.hom_inv_id_app, Functor.map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Opposite.Triangle | {
"line": 74,
"column": 47
} | {
"line": 74,
"column": 59
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nT₁ T₂ : Triangle Cᵒᵖ\nφ : T₁ ⟶ T₂\n⊢ ((shiftFunctor C 1).map (T₂.mor₃.unop ≫ φ.hom₃.unop)).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app T₂.obj₁ =\n ((shiftFunctor C 1).map T₁.mor₃.unop).op ≫\n ((shiftFunctor C 1).map ((... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting | {
"line": 40,
"column": 2
} | {
"line": 42,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM : ModuleCat R\n⊢ M.projectiveShortComplex.ShortExact",
"usedConstants": [
"Pi.Function.module",
"Module.Basis.ofRepr",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"P... | apply LinearMap.shortExact_shortComplexKer
refine fun m ↦ ⟨Finsupp.single m 1, ?_⟩
simp [Module.Basis.constr_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting | {
"line": 40,
"column": 2
} | {
"line": 42,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM : ModuleCat R\n⊢ M.projectiveShortComplex.ShortExact",
"usedConstants": [
"Pi.Function.module",
"Module.Basis.ofRepr",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"P... | apply LinearMap.shortExact_shortComplexKer
refine fun m ↦ ⟨Finsupp.single m 1, ?_⟩
simp [Module.Basis.constr_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.RingQuot | {
"line": 244,
"column": 14
} | {
"line": 246,
"column": 47
} | [
{
"pp": "R : Type uR\ninst✝³ : Semiring R\nS : Type uS\ninst✝² : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\n⊢ ∀ (a : RingQuot r), a + 0 = a",
"usedConstants": [
"RingQuot.add_quot",
"NonAssocSemiring.toAddCommMonoidWithOne",
"_p... | by
rintro ⟨⟨⟩⟩
simp only [add_quot, ← zero_quot, add_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.RingQuot | {
"line": 380,
"column": 29
} | {
"line": 380,
"column": 54
} | [
{
"pp": "case add_left\nR : Type uR\ninst✝⁴ : Semiring R\nS : Type uS\ninst✝³ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\ninst✝ : Semiring T\nr : R → R → Prop\nf : R →+* T\nh : ∀ ⦃x y : R⦄, r x y → f x = f y\nx : RingQuot r\na✝² b✝¹ a✝¹ b✝ c✝ : R\na✝... | rw [map_add, map_add, r'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.RingQuot | {
"line": 380,
"column": 29
} | {
"line": 380,
"column": 54
} | [
{
"pp": "case add_left\nR : Type uR\ninst✝⁴ : Semiring R\nS : Type uS\ninst✝³ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\ninst✝ : Semiring T\nr : R → R → Prop\nf : R →+* T\nh : ∀ ⦃x y : R⦄, r x y → f x = f y\nx : RingQuot r\na✝² b✝¹ a✝¹ b✝ c✝ : R\na✝... | rw [map_add, map_add, r'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.RingQuot | {
"line": 380,
"column": 29
} | {
"line": 380,
"column": 54
} | [
{
"pp": "case add_left\nR : Type uR\ninst✝⁴ : Semiring R\nS : Type uS\ninst✝³ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr✝ : R → R → Prop\ninst✝ : Semiring T\nr : R → R → Prop\nf : R →+* T\nh : ∀ ⦃x y : R⦄, r x y → f x = f y\nx : RingQuot r\na✝² b✝¹ a✝¹ b✝ c✝ : R\na✝... | rw [map_add, map_add, r'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralDomain | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 29
} | [
{
"pp": "case calc_2\nR : Type u_1\nG : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Group G\ninst✝ : Fintype G\nf : G →* R\nhf : f ≠ 1\nx : ↥f.toHomUnits.range\nhx : ∀ (y : ↥f.toHomUnits.range), y ∈ Submonoid.powers x\nhx1 : ↑↑x - 1 ≠ 0\nc : ℕ := #{g | f.toHomUnits g = 1}\n⊢ ↑↑(x ^ orderOf x) -... | simp [pow_orderOf_eq_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 71
} | [
{
"pp": "case succ\nn : ℕ\nh : ↑(T ^ ↑n) = !![1, ↑n; 0, 1]\n⊢ ↑(T ^ (↑n + 1)) = !![1, ↑n + 1; 0, 1]",
"usedConstants": [
"Eq.mpr",
"Matrix.SpecialLinearGroup",
"zpow_add",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Equiv.instEquivLike",
"HMul.hMul",
"CommRing... | simp_rw [zpow_add, zpow_one, coe_mul, h, coe_T, Matrix.mul_fin_two] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 62
} | [
{
"pp": "K : Type u_3\nV : Type u_8\ninst✝⁴ : Field K\ninst✝³ : Invertible 2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nQ : QuadraticForm K V\nhQ : LinearMap.SeparatingLeft (associated Q)\nv : Basis (Fin (finrank K V)) K V\nhv₁ : LinearMap.IsOrthoᵢ ((associatedHom K) Q) ⇑v\nhv... | simp_rw [LinearMap.IsOrtho, associated_eq_self_apply] at hv₂ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 753,
"column": 74
} | {
"line": 754,
"column": 95
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : AddCommGroup M\ninst✝³ : Semiring R\ninst✝² : Module Rᵐᵒᵖ M\ninst✝¹ : Module R M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\n⊢ IsUnit x ↔ IsUnit x.fst",
"usedConstants": [
"TrivSqZeroExt.one",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.t... | by
simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivInvertibleFst x).nonempty_congr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 807,
"column": 2
} | {
"line": 816,
"column": 58
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : DivisionSemiring R\ninst✝³ : AddCommGroup M\ninst✝² : Module Rᵐᵒᵖ M\ninst✝¹ : Module R M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\na b : tsze R M\n⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoid... | ext
· rw [fst_inv, fst_mul, fst_mul, mul_inv_rev, fst_inv, fst_inv]
· simp only [snd_inv, snd_mul, fst_mul, fst_inv]
simp only [smul_neg, smul_add]
simp_rw [mul_inv_rev, smul_comm (_ : R), op_smul_op_smul, smul_smul, add_comm, neg_add]
obtain ha0 | ha := eq_or_ne (fst a) 0
· simp [ha0]
obtain hb... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 807,
"column": 2
} | {
"line": 816,
"column": 58
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : DivisionSemiring R\ninst✝³ : AddCommGroup M\ninst✝² : Module Rᵐᵒᵖ M\ninst✝¹ : Module R M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\na b : tsze R M\n⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoid... | ext
· rw [fst_inv, fst_mul, fst_mul, mul_inv_rev, fst_inv, fst_inv]
· simp only [snd_inv, snd_mul, fst_mul, fst_inv]
simp only [smul_neg, smul_add]
simp_rw [mul_inv_rev, smul_comm (_ : R), op_smul_op_smul, smul_smul, add_comm, neg_add]
obtain ha0 | ha := eq_or_ne (fst a) 0
· simp [ha0]
obtain hb... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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