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.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.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.SmallShiftedHom | {
"line": 320,
"column": 36
} | {
"line": 320,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D\nW : MorphismProperty C\nM : Type w'\ninst✝⁶ : AddMonoid M\ninst✝⁵ : HasShift C M\ninst✝⁴ : HasShift D M\nX Y : C\ninst✝³ : HasSmallLocalizedShiftedHom W M X Y\ninst✝² : HasSmallLocalizedShiftedHom W M X X\ninst✝¹ : Ha... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Localization.SmallShiftedHom | {
"line": 320,
"column": 36
} | {
"line": 320,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D\nW : MorphismProperty C\nM : Type w'\ninst✝⁶ : AddMonoid M\ninst✝⁵ : HasShift C M\ninst✝⁴ : HasShift D M\nX Y : C\ninst✝³ : HasSmallLocalizedShiftedHom W M X Y\ninst✝² : HasSmallLocalizedShiftedHom W M X X\ninst✝¹ : Ha... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.SmallShiftedHom | {
"line": 320,
"column": 36
} | {
"line": 320,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D\nW : MorphismProperty C\nM : Type w'\ninst✝⁶ : AddMonoid M\ninst✝⁵ : HasShift C M\ninst✝⁴ : HasShift D M\nX Y : C\ninst✝³ : HasSmallLocalizedShiftedHom W M X Y\ninst✝² : HasSmallLocalizedShiftedHom W M X X\ninst✝¹ : Ha... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologySequenceLemmas | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 31
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS : ShortComplex (HomologicalComplex C c)\nhS : S.ShortExact\nj : ι\nh₁ : Mono (HomologicalComplex.homologyMap S.g j)\nh₂ : ∀ (i : ι), c.Rel i j → Epi (HomologicalComplex.homologyMap S.g i)\n⊢ IsZero (S.X... | by_cases! hj : ∃ i, c.Rel i j | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 14
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : HasShift C A\ninst✝ : HasShift D A\na b : A\ne₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\nf₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shif... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 115,
"column": 49
} | {
"line": 115,
"column": 52
} | [
{
"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⊢ φ a₁ + b₂ = b₃",
"usedConsta... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 66
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : G.CommShift A\na : A\nX✝ : C\n⊢ (shiftFunctor C a).map (adj.unit.app X✝) =\n adj.uni... | rw [LeftAdjointCommShift.iso_hom_app adj _ _ (add_neg_cancel a)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Pullback | {
"line": 133,
"column": 47
} | {
"line": 133,
"column": 50
} | [
{
"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⊢ φ a₁ + b₂ = b₃",
"usedConsta... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives | {
"line": 84,
"column": 24
} | {
"line": 84,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasDerivedCategory C\nP : C\ninst✝¹ : Projective P\nL : CochainComplex C ℤ\ni n : ℤ\nhn : n < i\ninst✝ : L.IsStrictlyLE n\nK : CochainComplex C ℤ\nw✝ : K.IsStrictlyLE i\nπ : K ⟶ (CochainComplex.singleFunctor C i).obj P\nh : IsIso (Q.m... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives | {
"line": 84,
"column": 24
} | {
"line": 84,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : HasDerivedCategory C\nI : C\ninst✝¹ : Injective I\nK : CochainComplex C ℤ\ni n : ℤ\nhn : i < n\ninst✝ : K.IsStrictlyGE n\nL : CochainComplex C ℤ\nw✝ : L.IsStrictlyGE i\ng : K ⟶ L\nι : (CochainComplex.singleFunctor C i).obj I ⟶ L\nh : ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 78
} | [
{
"pp": "case hf.h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\nY : C\nn : ℕ\nx✝ : ↑(AddCommGrpCat.of (Ext S.X₃ Y n))\n⊢ (mk₀ S.f).comp ((mk₀ S.g).comp x✝ ⋯) ⋯ = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | simp only [mk₀_comp_mk₀_assoc, ShortComplex.zero, mk₀_zero, zero_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 269,
"column": 82
} | {
"line": 272,
"column": 43
} | [
{
"pp": "A : Type u\ninst✝² : Ring A\nrelations : Relations A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nsolution : relations.Solution M\n⊢ Function.Surjective ⇑solution.fromQuotient ↔ Function.Surjective ⇑solution.π",
"usedConstants": [
"Semiring.toModule",
"Finsupp.module",
... | by
simpa only [← fromQuotient_comp_toQuotient] using
(Function.Surjective.of_comp_iff (f := solution.fromQuotient)
relations.surjective_toQuotient).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.RingQuot | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 22
} | [
{
"pp": "R : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\na b c : R\nh : Rel r b c\n⊢ Rel r (a + b) (a + c)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Distrib.toAdd",
"RingQuot.Rel",
"add_comm",
"instDistribOfSemiring",
"NonUnitalNonAssocSemiring.toAd... | rw [add_comm a b, add_comm a c]
exact Rel.add_left h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.RingQuot | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 22
} | [
{
"pp": "R : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\na b c : R\nh : Rel r b c\n⊢ Rel r (a + b) (a + c)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Distrib.toAdd",
"RingQuot.Rel",
"add_comm",
"instDistribOfSemiring",
"NonUnitalNonAssocSemiring.toAd... | rw [add_comm a b, add_comm a c]
exact Rel.add_left h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.RingQuot | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 14
} | [
{
"pp": "case h.h.a.mp.trans\nR : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\nx₁ x₂ x✝ y✝ z✝ : R\na✝¹ : Relation.EqvGen (Rel r) x✝ y✝\na✝ : Relation.EqvGen (Rel r) y✝ z✝\na_ih✝¹ : RingConGen.Rel r x✝ y✝\na_ih✝ : RingConGen.Rel r y✝ z✝\n⊢ RingConGen.Rel r x✝ z✝",
"usedConstants": [
"RingConGen.Rel.t... | | trans => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.RingQuot | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 14
} | [
{
"pp": "case h.h.a.mpr.trans\nR : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\nx₁ x₂ x✝ y✝ z✝ : R\na✝¹ : RingConGen.Rel r x✝ y✝\na✝ : RingConGen.Rel r y✝ z✝\na_ih✝¹ : Relation.EqvGen (Rel r) x✝ y✝\na_ih✝ : Relation.EqvGen (Rel r) y✝ z✝\n⊢ Relation.EqvGen (Rel r) x✝ z✝",
"usedConstants": [
"RingQuot... | | trans => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.RingQuot | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 12
} | [
{
"pp": "case h.h.a.mpr.add\nR : Type uR\ninst✝ : Semiring R\nr : R → R → Prop\nx₁ x₂ w✝ x✝ y✝ z✝ : R\na✝¹ : RingConGen.Rel r w✝ x✝\na✝ : RingConGen.Rel r y✝ z✝\na_ih✝¹ : Relation.EqvGen (Rel r) w✝ x✝\na_ih✝ : Relation.EqvGen (Rel r) y✝ z✝\n⊢ Relation.EqvGen (Rel r) (w✝ + y✝) (x✝ + z✝)",
"usedConstants": [
... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.RingQuot | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 20
} | [
{
"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 b : RingQuot r), a + b = b + a",
"usedConstants": [
"RingQuot"
]
}
] | rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences | {
"line": 286,
"column": 60
} | {
"line": 289,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nS : ShortComplex C\nhS : S.ShortExact\nY : C\nn₁ : ℕ\nx₃ : Ext S.X₃ Y n₁\nhx₃ : (mk₀ S.g).comp x₃ ⋯ = 0\nn₀ : ℕ\nhn₀ : 1 + n₀ = n₁\n⊢ ∃ x₁, hS.extClass.comp x₁ hn₀ = x₃",
"usedConstants": [
"CategoryTheory.Abelian.t... | by
have := contravariant_sequence_exact₃' hS Y n₀ n₁ hn₀
rw [ShortComplex.ab_exact_iff] at this
exact this x₃ hx₃ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.RingQuot | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 20
} | [
{
"pp": "R✝ : 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\nR : Type uR\ninst✝ : Ring R\nr : R → R → Prop\n⊢ ∀ (a b : RingQuot r), a - b = a + -b",
"usedConstants": [
"RingQuot",
"Rin... | rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.RingQuot | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 20
} | [
{
"pp": "R✝ : 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\nR : Type uR\ninst✝ : CommSemiring R\nr : R → R → Prop\n⊢ ∀ (a b : RingQuot r), a * b = b * a",
"usedConstants": [
"CommSemiring.t... | rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.RingQuot | {
"line": 323,
"column": 14
} | {
"line": 325,
"column": 29
} | [
{
"pp": "R✝ : 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\nR : Type uR\ninst✝ : CommSemiring R\nr : R → R → Prop\n⊢ ∀ (a b : RingQuot r), a * b = b * a",
"usedConstants": [
"RingQuot.cases... | by
rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩
simp [mul_quot, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 12
} | [
{
"pp": "case refine_1.h\nm : Type u_1\nA : Type u_4\ninst✝³ : Fintype m\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : DecidableEq m\nM : Matrix m m A\nhM : M.det ≠ 0\nv : m → A\ni : m\nh : v ⬝ᵥ M *ᵥ M.cramer (Pi.single i 1) = 0\n⊢ v i = 0 i",
"usedConstants": [
"Pi.Function.module",
"False... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 186,
"column": 8
} | {
"line": 186,
"column": 23
} | [
{
"pp": "ι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_9\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² ... | toLin_toMatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 186,
"column": 43
} | {
"line": 186,
"column": 58
} | [
{
"pp": "ι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_9\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² ... | toLin_toMatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 200,
"column": 8
} | {
"line": 200,
"column": 23
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝⁹ : CommSemiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\nN : Type u_9\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁴ : Fintype ι'\ninst✝³ ... | toLin_toMatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 200,
"column": 43
} | {
"line": 200,
"column": 58
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝⁹ : CommSemiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\nN : Type u_9\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁴ : Fintype ι'\ninst✝³ ... | toLin_toMatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 138,
"column": 43
} | {
"line": 138,
"column": 56
} | [
{
"pp": "case mp.refine_2.h\nn : Type u_1\ninst✝⁶ : Fintype n\nA : Type u_4\nK : Type u_5\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommRing A\ninst✝³ : Nontrivial A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nM : Matrix n n A\nthis : (∃ v, v ≠ 0 ∧ (algebraMap A K).mapMatrix M *ᵥ v = 0) ↔ M.det... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GradedMonoid | {
"line": 472,
"column": 4
} | {
"line": 472,
"column": 83
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nα : Type u_3\ninst✝¹ : AddMonoid ι\ninst✝ : Monoid R\nl : List α\nfι : α → ι\nfA : α → R\nhead : α\ntail : List α\n⊢ (head :: tail).dProd fι fA = (map fA (head :: tail)).prod",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"AddMonoid.t... | rw [List.dProd_cons, List.map_cons, List.prod_cons, List.dProd_monoid tail _ _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 282,
"column": 52
} | {
"line": 283,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\na b : M\nh : QuadraticMap.IsOrtho Q a b\n⊢ (ι Q) a * ((ι Q) b * x) = -((ι Q) b * ((ι Q) a * x))",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg"... | by
rw [← mul_assoc, ι_mul_ι_comm_of_isOrtho h, neg_mul, mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 362,
"column": 42
} | {
"line": 362,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM₁ : Type u_4\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nf : Q₁ →qᵢ Q₂\nhf : Function.Surjective ⇑f\nx✝³ x✝² : CliffordAlgebra Q₂\nx✝¹ : ∃ a, (map f) ... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 362,
"column": 42
} | {
"line": 362,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM₁ : Type u_4\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nf : Q₁ →qᵢ Q₂\nhf : Function.Surjective ⇑f\nx✝³ x✝² : CliffordAlgebra Q₂\nx✝¹ : ∃ a, (map f) ... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 362,
"column": 42
} | {
"line": 362,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM₁ : Type u_4\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nf : Q₁ →qᵢ Q₂\nhf : Function.Surjective ⇑f\nx✝³ x✝² : CliffordAlgebra Q₂\nx✝¹ : ∃ a, (map f) ... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Internal | {
"line": 67,
"column": 89
} | {
"line": 74,
"column": 49
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddMonoidWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : GradedOne A\nn : ℕ\n⊢ ↑n ∈ A 0",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Nat.recAux",
"AddSubmonoidClass.to... | by
induction n with
| zero =>
rw [Nat.cast_zero]
exact zero_mem (A 0)
| succ _ n_ih =>
rw [Nat.cast_succ]
exact add_mem n_ih (SetLike.one_mem_graded _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 258,
"column": 84
} | {
"line": 275,
"column": 8
} | [
{
"pp": "n : Type u_11\nm : Type u_12\nn' : Type u_13\nm' : Type u_14\nR : Type u_16\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Fintype n\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq n\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nB : (n → R) →ₗ[R] (m →... | by
ext i j
simp only [LinearMap.toMatrix₂'_apply, LinearMap.compl₁₂_apply, transpose_apply, Matrix.mul_apply,
LinearMap.toMatrix', LinearEquiv.coe_mk, LinearMap.coe_mk, AddHom.coe_mk, sum_mul]
rw [sum_comm]
conv_lhs => rw [← LinearMap.sum_repr_mul_repr_mul (Pi.basisFun R n) (Pi.basisFun R m) (l _) (r _)]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 393,
"column": 29
} | {
"line": 393,
"column": 44
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_6\nM₂ : Type u_7\nn : Type u_11\nm : Type u_12\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\nσ₁ σ₂ : R →+* R\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nb₁... | ← b₂.sum_repr y | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 446,
"column": 37
} | {
"line": 446,
"column": 45
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : AddCommMonoid ι\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : IsOrderedAddMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\nr r' : ⨁ (i : ι), ↥(A i)\nm n : ι\nhr ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.DirectSum.Internal | {
"line": 446,
"column": 37
} | {
"line": 446,
"column": 45
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : AddCommMonoid ι\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : IsOrderedAddMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\nr r' : ⨁ (i : ι), ↥(A i)\nm n : ι\nhr ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Internal | {
"line": 446,
"column": 37
} | {
"line": 446,
"column": 45
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : AddCommMonoid ι\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : IsOrderedAddMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝ : SetLike.GradedMonoid A\nr r' : ⨁ (i : ι), ↥(A i)\nm n : ι\nhr ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 44
} | [
{
"pp": "S : Type u_1\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : SMul S M\ninst✝² : IsScalarTower S R M\ninst✝¹ ... | ← IsScalarTower.algebraMap_smul R a x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 332,
"column": 6
} | {
"line": 332,
"column": 44
} | [
{
"pp": "S : Type u_1\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : Module S M\ninst✝² : IsScalarTower S R M\ninst✝¹ : Mo... | ← IsScalarTower.algebraMap_smul R a x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 348,
"column": 22
} | {
"line": 348,
"column": 66
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : Module S M\... | rw [polar_comm, polar_smul_left, polar_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 348,
"column": 22
} | {
"line": 348,
"column": 66
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : Module S M\... | rw [polar_comm, polar_smul_left, polar_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 348,
"column": 22
} | {
"line": 348,
"column": 66
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : Module S M\... | rw [polar_comm, polar_smul_left, polar_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Fold | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nP : CliffordAlgebra Q → Prop\nalgebraMap : ∀ (r : R), P ((Algebra.algebraMap R (CliffordAlgebra Q)) r)\nadd : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)\nι_mul : ∀ (x : CliffordA... | refine reverse_involutive.surjective.forall.2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Alternating.Curry | {
"line": 85,
"column": 47
} | {
"line": 85,
"column": 70
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nN : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R N\nn : ℕ\ng : M₂ →ₗ[R] M\nf : M [⋀^Fin n.succ]→ₗ[R] N\nm : M₂\nv : Fin n → M₂\ni :... | cases i using Fin.cases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 63,
"column": 22
} | {
"line": 63,
"column": 45
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.cases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 64,
"column": 22
} | {
"line": 64,
"column": 45
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.cases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 76,
"column": 18
} | {
"line": 78,
"column": 26
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | by
ext m
exact cons_add f m x y | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 195,
"column": 22
} | {
"line": 195,
"column": 65
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.succAboveCases p <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 195,
"column": 22
} | {
"line": 195,
"column": 65
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.succAboveCases p <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 195,
"column": 22
} | {
"line": 195,
"column": 65
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.succAboveCases p <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 196,
"column": 22
} | {
"line": 196,
"column": 65
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.succAboveCases p <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 196,
"column": 22
} | {
"line": 196,
"column": 65
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.succAboveCases p <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 196,
"column": 22
} | {
"line": 196,
"column": 65
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nι' : Type uι'\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n.succ) → M... | cases i using Fin.succAboveCases p <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 380,
"column": 2
} | {
"line": 382,
"column": 34
} | [
{
"pp": "case h.e_5.h.h.e_6.h\nR : Type uR\nM₂ : Type v₂\nM' : Type v'\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nk l n : ℕ\ns : Finset (Fin n)\nhk : #s = k\nhl : #sᶜ = l\nf : MultilinearMap R (fun x ↦ M') (MultilinearMap R (fun x ↦... | · ext
rw [finSumEquivOfFinset_inl, Finset.piecewise_eq_of_mem]
apply Finset.orderEmbOfFin_mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating | {
"line": 99,
"column": 4
} | {
"line": 100,
"column": 43
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nn : ℕ\nih :\n ∀ (f : (i : ℕ) → M [⋀^Fin i]→ₗ[R] N) (v : Fin n → M),\n (liftAlternating f) (List.ofFn fun i ↦ (ι R) (v i)).prod = (f n)... | rw [List.ofFn_succ, List.prod_cons, liftAlternating_ι_mul, ih,
AlternatingMap.curryLeft_apply_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Free | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 47
} | [
{
"pp": "case h\nR : Type u_3\ninst✝⁵ : Ring R\nS : ShortComplex (ModuleCat R)\nhS' : S.ShortExact\nn p : ℕ\ninst✝⁴ : Free R ↑S.X₁\ninst✝³ : Free R ↑S.X₃\ninst✝² : Module.Finite R ↑S.X₁\ninst✝¹ : Module.Finite R ↑S.X₃\nhN : finrank R ↑S.X₁ = n\nhP : finrank R ↑S.X₃ = p\ninst✝ : StrongRankCondition R\n⊢ Module.r... | rw [free_shortExact_rank_add hS', ← hN, ← hP] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 16
} | [
{
"pp": "case a\nR : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\n⊢ span R (⇑(ExteriorAlgebra.ιMulti R n) '' {a | range a ⊆ s}) ≤ ⋀[R]^n M",
"usedConstants": [
"AlternatingMap",
"Eq.mpr",
"Submodule",
"Se... | rw [span_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits | {
"line": 44,
"column": 8
} | {
"line": 50,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections\nc : Cone F\nhc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone... | apply (isLimitOfPreserves (ModuleCat.restrictScalars (R.map f).hom) (hc Y)).hom_ext
intro j
have h₁ := (c.π.app j).naturality f
have h₂ := (hc X).fac ((evaluation R X).mapCone s) j
rw [Functor.mapCone_π_app, assoc, assoc, ← Functor.map_comp, IsLimit.fac]
dsimp at h₁ h₂ ⊢
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits | {
"line": 44,
"column": 8
} | {
"line": 50,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections\nc : Cone F\nhc : (X : Cᵒᵖ) → IsLimit ((evaluation R X).mapCone... | apply (isLimitOfPreserves (ModuleCat.restrictScalars (R.map f).hom) (hc Y)).hom_ext
intro j
have h₁ := (c.π.app j).naturality f
have h₂ := (hc X).fac ((evaluation R X).mapCone s) j
rw [Functor.mapCone_π_app, assoc, assoc, ← Functor.map_comp, IsLimit.fac]
dsimp at h₁ h₂ ⊢
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 292,
"column": 55
} | {
"line": 295,
"column": 77
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\n⊢ F.Final ↔ ∀ (d : D), IsFiltered (StructuredArrow d F)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congr... | by
refine ⟨?_, fun h => final_of_isFiltered_structuredArrow F⟩
rw [final_iff_of_isFiltered]
exact fun h => isFiltered_structuredArrow_of_isFiltered_of_exists F h.1 h.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 269,
"column": 14
} | {
"line": 269,
"column": 63
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nP : IsColimit c\ns : Cocone D.coconePoints\nj : J... | simp only [Functor.const_obj_map, Category.assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 297,
"column": 13
} | {
"line": 297,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM'✝ : PresheafOfModules R\ncM'✝ : Cocone M'✝.presheaf\... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Grothendieck | {
"line": 411,
"column": 22
} | {
"line": 415,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\nhro : RightOreCondition C\n⊢ ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ {S | ∃ Y f, S.arrows f} → Sieve.pullback f S ∈ {S | ∃ Y_1 f, S.arrows f}",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQui... | by
rintro X Y S h ⟨Z, f, hf⟩
rcases hro h f with ⟨W, g, k, comm⟩
refine ⟨_, g, ?_⟩
simp [comm, hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.SheafOfTypes | {
"line": 237,
"column": 8
} | {
"line": 237,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\nH✝ : ∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows\ns : Cocone S.arrows.diagram\nf : S.arrows.category\nH : ∃! t, (S.arrows.yonedaFamilyOfElements_fromCocone s).IsAmalgamation t\n⊢ S.arrows.cocone.ι.app f ≫ Exists.choose ⋯ = s.ι.app f",
"us... | have ht := H.choose_spec.1 f.obj.hom f.property | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Types.Equalizers | {
"line": 53,
"column": 2
} | {
"line": 54,
"column": 15
} | [
{
"pp": "X Y Z : Type u\nf : X ⟶ Y\ng h : Y ⟶ Z\nw : f ≫ g = f ≫ h\nt : IsLimit (Fork.ofι f w)\ny : Y\nhy : (hom g) y = (hom h) y\ny' : PUnit.{u + 1} ⟶ Y := ↾fun x ↦ y\nhy' : y' ≫ g = y' ≫ h\nx' : X\nhx' : (hom f) x' = y\n⊢ x' = (hom ↑(Fork.IsLimit.lift' t y' hy')) PUnit.unit",
"usedConstants": [
"Eq.... | suffices (fun _ : PUnit => x') = (Fork.IsLimit.lift' t y' hy').1 by
rw [← this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 39
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nS : Sieve X\nx : FamilyOfElements P S.arrows\n⊢ x.SieveCompatible → x.Compatible",
"usedConstants": [
"CategoryTheory.Presieve.FamilyOfElements.SieveCompatible"
]
}
] | intro h Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ k | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 927,
"column": 7
} | {
"line": 927,
"column": 11
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z✝ : C\nf✝ : Y ⟶ X\nS✝ R : Sieve X\nf : Y ⟶ X\ninst✝ : Mono f\nS : Sieve Y\nZ : C\ng g₁ : Z ⟶ Y\nhf : g₁ = g\nhg₁ : S.arrows g₁\n⊢ S.arrows g",
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 749,
"column": 2
} | {
"line": 756,
"column": 8
} | [
{
"pp": "case right\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type w\nS : Sieve X\nR : Presieve X\nh : S.arrows ≤ R\nhS : IsSheafFor P S.arrows\ntrans : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R f → IsSeparatedFor P (Sieve.pullback f S).arrows\n⊢ ∀ (x : FamilyOfElements P R), x.Compatible → ∃ t, x.IsAmalgamati... | · intro x hx
use hS.amalgamate _ (hx.restrict h)
intro W j hj
apply (trans hj).ext
intro Y f hf
rw [← comp_apply, ← Functor.map_comp, ← op_comp, hS.valid_glue (hx.restrict h) _ hf,
FamilyOfElements.restrict, ← hx (𝟙 _) f (h _ _ hf) _ (id_comp _)]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Subfunctor.Image | {
"line": 188,
"column": 2
} | {
"line": 191,
"column": 22
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nF F' : C ⥤ Type w\nG : Subfunctor F\np : F' ⟶ F\nhp : Epi p\n⊢ G ≤ (G.preimage p).image p",
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfSize",
"CategoryTheory.Subfunctor.image",
"CategoryTheory.Functor",
"Catego... | · intro i x hx
simp only [NatTrans.epi_iff_epi_app, epi_iff_surjective] at hp
obtain ⟨y, rfl⟩ := hp _ x
exact ⟨y, hx, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Limits | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 74
} | [
{
"pp": "case h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{w', w} D\nK : Type z\ninst✝¹ : Category.{z', z} K\ninst✝ : HasLimitsOfShape K D\nF : K ⥤ Sheaf J D\nE : Cone (F ⋙ sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : J.Cover X\nS : Multifork (W.index... | erw [← hm, Category.assoc, ← (E.π.app k).naturality, Category.assoc] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 272,
"column": 8
} | {
"line": 272,
"column": 45
} | [
{
"pp": "X Y : TopCat\nc : BinaryCofan X Y\nh₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl)\nh₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))\n⊢ ∀ (x : ↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := W... | rw [eq_compl_iff_isCompl.mpr h₃.symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 85,
"column": 26
} | {
"line": 88,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' : H.IsVanKampen\n⊢ ⋯.IsVanKampen",
"usedConstants": [
"_private.Mathlib.CategoryTheory.Adhesive.Basic.0.CategoryTheory.IsPushout.IsVanKampen.flip._simp_1_2",
"Eq.mpr... | by
introv W' hf hg hh hi w
simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using
H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 79
} | [
{
"pp": "case intro\nJ : Type u₁\nK : Type u₂\ninst✝⁴ : Category.{v₁, u₁} J\ninst✝³ : Category.{v₂, u₂} K\ninst✝² : Small.{v, u₂} K\nF : J × K ⥤ Type v\ninst✝¹ : IsFiltered K\ninst✝ : Finite J\nval✝ : Fintype J\nkx : K\nx : limit ((curry.obj (swap K J ⋙ F)).obj kx)\nky : K\ny : limit ((curry.obj (swap K J ⋙ F))... | have kjO : ∀ j, k j ∈ O := fun j => Finset.mem_union.mpr (Or.inl (by simp)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 25
} | [
{
"pp": "J : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nK : Type u_3\ninst✝ : Category.{v_3, u_3} K\ne : J ≌ K\nF : K ⥤ C\nc : Cocone F\nhc : IsUniversalColimit c\n⊢ IsUniversalColimit (Cocone.whisker e.functor c)",
"usedConstants": [
"CategoryTheory.Functor"
]
... | intro F' c' α f e' hα H | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 195,
"column": 6
} | {
"line": 195,
"column": 53
} | [
{
"pp": "case mp.refine_2.h₂\nC : Type u\ninst✝ : Category.{v, u} C\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f c.inl iY fE\nh : IsColimit (BinaryCofan.mk (c.inr ≫ fE) iY)\ns : PushoutCocone f c.inl\nm : YE ⟶ s.pt\ne₁ : iY ≫ m = s.inl\ne₂ : fE ≫ m =... | · refine e₁.trans (Eq.symm ?_); exact h.fac _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Extensive | {
"line": 376,
"column": 6
} | {
"line": 376,
"column": 48
} | [
{
"pp": "case h_map\nC : Type u\ninst✝⁹ : Category.{v, u} C\nD : Type u''\ninst✝⁸ : Category.{v'', u''} D\ninst✝⁷ : HasFiniteCoproducts D\ninst✝⁶ : HasPullbacksOfInclusions D\ninst✝⁵ : FinitaryExtensive C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\ninst✝² : ∀ (X : D) (Y : C) ... | · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 386,
"column": 6
} | {
"line": 387,
"column": 20
} | [
{
"pp": "J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z✝ : C\nf✝ : W ⟶ X\ng✝ : W ⟶ Y\nh : X ⟶ Z✝\ni : Y ⟶ Z✝\ninst✝² : Adhesive C\nZ A B : C\na : A ⟶ Z\nb : B ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nK : C\nf g : K ⟶ pushout (fst a b) (snd a b)\nw : f ≫ pushout.desc a b ... | rw [← p₁.isoPullback_hom_fst]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 386,
"column": 6
} | {
"line": 387,
"column": 20
} | [
{
"pp": "J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z✝ : C\nf✝ : W ⟶ X\ng✝ : W ⟶ Y\nh : X ⟶ Z✝\ni : Y ⟶ Z✝\ninst✝² : Adhesive C\nZ A B : C\na : A ⟶ Z\nb : B ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nK : C\nf g : K ⟶ pushout (fst a b) (snd a b)\nw : f ≫ pushout.desc a b ... | rw [← p₁.isoPullback_hom_fst]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 15
} | [
{
"pp": "case refine_1.w.h\nJ : Type v'\ninst✝⁶ : Category.{u', v'} J\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝³ : Gr.Full\ninst✝² : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsUniversalColimit c\ninst✝¹ : ∀ (X : D)... | dsimp [c''] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 350,
"column": 6
} | {
"line": 350,
"column": 17
} | [
{
"pp": "J : Type v'\ninst✝⁶ : Category.{u', v'} J\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝³ : Gr.Full\ninst✝² : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsUniversalColimit c\ninst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.... | dsimp [c''] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 72,
"column": 2
} | {
"line": 80,
"column": 76
} | [
{
"pp": "case h.mpr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nX Y : C\nf : X ⟶ Y\n⊢ P.isLocal f → P.isoClosure.isLocal f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",... | · rintro hf Z ⟨Z', hZ', ⟨e⟩⟩
constructor
· intro g₁ g₂ eq
rw [← cancel_mono e.hom]
apply (hf _ hZ').1
simp only [reassoc_of% eq]
· intro g
obtain ⟨a, h⟩ := (hf _ hZ').2 (g ≫ e.hom)
exact ⟨a ≫ e.inv, by simp only [reassoc_of% h, e.hom_inv_id, comp_id]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 426,
"column": 6
} | {
"line": 427,
"column": 94
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ ∀ (j : Discrete WalkingPair),\n IsPullback ((BinaryCofan.mk (𝟙 A) (𝟙 A)).ι.app j) ((mapPair f f).app j) f\n ((BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥... | rintro ⟨⟨⟩⟩ <;> dsimp <;>
exact IsPullback.of_horiz_isIso ⟨(Category.id_comp _).trans (Category.comp_id _).symm⟩ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 493,
"column": 2
} | {
"line": 508,
"column": 59
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nc : BinaryCofan X Y\ncofans : (X Y : C) → BinaryCofan X Y\ncolimits : (X Y : C) → IsColimit (cofans X Y)\ncones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g\nlimits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)\n... | · rintro ⟨h⟩
let e := h.coconePointUniqueUpToIso (colimits _ _)
obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [e, hX]) (by simp [e, hY])
constructor
· rw [← Category.id_comp αX, ← Iso.hom_inv_id_assoc e f]
haveI : IsIso (𝟙 X') := inferInstance
have : c'.inl ≫ e.hom = 𝟙 X' ≫ (cofans X' Y... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 480,
"column": 6
} | {
"line": 482,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nS : C\nι : Type u_2\ninst✝ : Small.{max w v, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\nc : Cofan fun i ↦ uliftYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\ne : (Discrete.functor fun i ↦ uliftYoneda.{w, v, u}.obj (X i)) ≅\n Discrete.fu... | rw [Cofan.IsColimit.fac_assoc, NatTrans.naturality,
← Cofan.IsColimit.fac hc' (fun i ↦ shrinkYoneda.map (f i)) i]
simp [Cofan.inj, e] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 480,
"column": 6
} | {
"line": 482,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nS : C\nι : Type u_2\ninst✝ : Small.{max w v, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\nc : Cofan fun i ↦ uliftYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\ne : (Discrete.functor fun i ↦ uliftYoneda.{w, v, u}.obj (X i)) ≅\n Discrete.fu... | rw [Cofan.IsColimit.fac_assoc, NatTrans.naturality,
← Cofan.IsColimit.fac hc' (fun i ↦ shrinkYoneda.map (f i)) i]
simp [Cofan.inj, e] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 747,
"column": 30
} | {
"line": 758,
"column": 53
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nι : Type u_3\nS B : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nf : (i : ι) → X i ⟶ S\nu : a.pt ⟶ S\nv : B ⟶ S\ns : (i : ι) → PullbackCone v (f i)\nhs : (i : ι) → IsLimit (s i)\nt : PullbackCone v u\nht : IsLimit t\nd : Cofan fun i ↦ (s i).pt\ne : d.pt ... | by
let iso : d ≅ (Cofan.mk _ fun i : ι ↦ PullbackCone.IsLimit.lift ht
(s i).fst ((s i).snd ≫ a.inj i) (by simp [hu, (s i).condition])) :=
Cofan.ext e <| fun p ↦ PullbackCone.IsLimit.hom_ext ht (by simp [he₁]) (by simp [he₂])
rw [(IsColimit.equivIsoColimit iso).nonempty_congr]
refine hau _ (Discrete.natT... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 890,
"column": 6
} | {
"line": 890,
"column": 23
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nι : Type u_3\nι' : Type u_4\nS : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nY : ι' → C\nb : Cofan Y\nhbu : IsUniversalColimit b\nf : (i : ι) → X i ⟶ S\ng : (i : ι') → Y i ⟶ S\nu : a.pt ⟶ S\nv : b.pt ⟶ S\ninst✝ : ∀ (i : ι), HasPullback (... | simp [c', c, he₁] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 890,
"column": 6
} | {
"line": 890,
"column": 23
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nι : Type u_3\nι' : Type u_4\nS : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nY : ι' → C\nb : Cofan Y\nhbu : IsUniversalColimit b\nf : (i : ι) → X i ⟶ S\ng : (i : ι') → Y i ⟶ S\nu : a.pt ⟶ S\nv : b.pt ⟶ S\ninst✝ : ∀ (i : ι), HasPullback (... | simp [c', c, he₁] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 890,
"column": 6
} | {
"line": 890,
"column": 23
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nι : Type u_3\nι' : Type u_4\nS : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nY : ι' → C\nb : Cofan Y\nhbu : IsUniversalColimit b\nf : (i : ι) → X i ⟶ S\ng : (i : ι') → Y i ⟶ S\nu : a.pt ⟶ S\nv : b.pt ⟶ S\ninst✝ : ∀ (i : ι), HasPullback (... | simp [c', c, he₁] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 67
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝ : HasWeakSheafify J AddCommGrpCat\... | rw [← isIso_iff_of_reflects_iso _ (SheafOfModules.toSheaf.{v} R)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 180,
"column": 2
} | {
"line": 198,
"column": 90
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrpCat\nφ : M₀.presheaf ⟶ A.obj\ninst✝¹ : Pr... | let S := (Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m)
have hS : S ∈ J _ := by
apply J.intersection_covering
all_goals apply Presheaf.imageSieve_mem
have h₁ : S ≤ Presheaf.imageSieve α r := fun _ _ h => h.1
have h₂ : S ≤ Presheaf.imageSieve φ m := fun _ _ h => h.2
let r₀ := (Presieve.FamilyOfEleme... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 180,
"column": 2
} | {
"line": 198,
"column": 90
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrpCat\nφ : M₀.presheaf ⟶ A.obj\ninst✝¹ : Pr... | let S := (Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m)
have hS : S ∈ J _ := by
apply J.intersection_covering
all_goals apply Presheaf.imageSieve_mem
have h₁ : S ≤ Presheaf.imageSieve α r := fun _ _ h => h.1
have h₂ : S ≤ Presheaf.imageSieve φ m := fun _ _ h => h.2
let r₀ := (Presieve.FamilyOfEleme... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 67
} | [
{
"pp": "case a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrpCat\nφ : M₀.presheaf ⟶ A.obj\nins... | map_smul_eq α φ r m (π ≫ f.op) r₀ (by rw [hr₀, Functor.map_comp, RingCat.comp_apply]) m₀
(by rw [hm₀, Functor.map_comp, ConcreteCategory.comp_apply]), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 89
} | [
{
"pp": "case mpr.transitive\nC : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nH : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ... | simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 85
} | [
{
"pp": "C : Type u_2\ninst✝ : Category.{v_1, u_2} C\nX : C\nS T : Presieve X\nP : Cᵒᵖ ⥤ Type u_1\nH : S.FactorsThru T\nhS : IsSheafFor P S\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong S f\n⊢ IsSheafFor P T",
"usedConstants": [
"Eq.mpr",
"Opposite",
"Categ... | simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 872,
"column": 24
} | {
"line": 878,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Precoverage C\ninst✝ : J.IsStableUnderBaseChange\nS : C\nE F : PreZeroHypercover S\ne : E ≅ F\nh : E.presieve₀ ∈ J.coverings S\n⊢ F.presieve₀ ∈ J.coverings S",
"usedConstants": [
"CategoryTheory.Presieve.ofArrows_comp_eq_of_surjective",
"Eq.mp... | by
refine J.mem_coverings_of_isPullback (fun i ↦ E.f (e.inv.s₀ i)) ?_ (𝟙 S) _ (fun i ↦ ?_) ?_
· convert! h
exact Presieve.ofArrows_comp_eq_of_surjective _ (fun i ↦ ⟨e.hom.s₀ i, by simp⟩)
· exact e.inv.h₀ i
· intro i
exact CategoryTheory.IsPullback.of_vert_isIso (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.