module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 22
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nR₁ R₂ : ComposableArrows C 4\nhR₁ : R₁.Exact\nhR₂ : R₂.Exact\nφ : R₁ ⟶ R₂\nh₀ : Epi (app' φ 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2)\nh₁ : IsIso (app' φ 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4)\nh₂ : IsIso (app' φ 3 is... | dsimp at h₀ h₁ h₂ h₃ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 571,
"column": 2
} | {
"line": 572,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 571,
"column": 2
} | {
"line": 572,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 579,
"column": 2
} | {
"line": 580,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 579,
"column": 2
} | {
"line": 580,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹² : Category.{v_1, u_1} C\ninst✝¹¹ : Category.{v_2, u_2} D\nE : C ≌ D\nA : Type u_3\ninst✝¹⁰ : AddMonoid A\ninst✝⁹ : HasShift C A\ninst✝⁸ : HasShift D A\ninst✝⁷ : E.functor.CommShift A\ninst✝⁶ : E.inverse.CommShift A\nF : Type u_4\ninst✝⁵ : Category.{v_3, u_4} F\ninst✝... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.ShiftedHomOpposite | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasShift C ℤ\nX Y : C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nn : ℤ\nx y : ShiftedHom (Opposite.op Y) (Opposite.op X) n\n⊢ (opEquiv n).symm (x + y) = (opEquiv n).symm x + (opEquiv n).symm y",
"usedConstants": [
... | dsimp [opEquiv_symm_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 120,
"column": 74
} | {
"line": 123,
"column": 92
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle Cᵒᵖ\n⊢ T ∈ distinguishedTriangles C ↔ T.rotate ∈ distinguishedTriangles C",
"usedConstants"... | by
simp only [mem_distinguishedTriangles_iff, Pretriangulated.rotate_distinguished_triangle
((triangleOpEquivalence C).inverse.obj (T.rotate)).unop]
exact distinguished_iff_of_iso (rotateTriangleOpEquivalenceInverseObjRotateUnopIso T).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nT : Triangle C\nhT : T ∈ distinguishedTriangles\n⊢ (triangleOpEquivalence C).functor.obj (Opposite.op T) ∈ d... | mem_distTriang_op_iff' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.RingQuot | {
"line": 284,
"column": 18
} | {
"line": 284,
"column": 72
} | [
{
"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⊢ ↑0 = 0",
"usedConstants": [
"RingQuot.instNatCast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"_private.Mathlib.Algeb... | by simp +instances [instNatCast, natCast, ← zero_quot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.RingQuot | {
"line": 445,
"column": 11
} | {
"line": 445,
"column": 35
} | [
{
"pp": "B : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\n⊢ (ringQuotToIdealQuotient r) ((mkRingHom r) x) = (Ideal.Quotient.mk (Ideal.ofRel r)) x",
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal.Quotient.mk",
"RingHom",
"RingQuot.instSemiring",
"id",
"RingQ... | ringQuotToIdealQuotient, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 28
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type u_1\ninst✝ : CommRing S\ni : n\na : ↥(rootsOfUnity (Fintype.card n) R)\n⊢ (fun A ↦ rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯) ((fun a ↦ ⟨⟨a • 1, ⋯⟩, ⋯⟩) a) = a",
"usedConstants": []
}
] | obtain ⟨⟨a, _⟩, ha⟩ := a | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 139,
"column": 2
} | {
"line": 159,
"column": 73
} | [
{
"pp": "case mpr\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 = 0\nv : ... | · letI := Classical.decEq K
obtain ⟨⟨b, hb⟩, ba_eq⟩ :=
IsLocalization.exist_integer_multiples_of_finset (nonZeroDivisors A) (Finset.univ.image v)
choose f hf using ba_eq
refine
⟨fun i => f _ (Finset.mem_image.mpr ⟨i, Finset.mem_univ i, rfl⟩),
mt (fun h => funext fun i => ?_) hv, ?_⟩
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 392,
"column": 38
} | {
"line": 392,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * 0 = 1\nhg : c = 0\n⊢ a * d = 1",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"AddGroupWithOne.toAddGroup",
"... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 858,
"column": 73
} | {
"line": 858,
"column": 84
} | [
{
"pp": "S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹³ : CommSemiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra S R\ninst✝⁸ : Module S M\ninst✝⁷ : Module R M\ninst✝⁶ : Module Rᵐᵒᵖ M\ninst✝⁵ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁴ : IsScalarTower S R M\ninst✝³ : ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 859,
"column": 20
} | {
"line": 859,
"column": 31
} | [
{
"pp": "S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹³ : CommSemiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra S R\ninst✝⁸ : Module S M\ninst✝⁷ : Module R M\ninst✝⁶ : Module Rᵐᵒᵖ M\ninst✝⁵ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁴ : IsScalarTower S R M\ninst✝³ : ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 1050,
"column": 2
} | {
"line": 1050,
"column": 33
} | [
{
"pp": "S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹⁶ : CommSemiring S\ninst✝¹⁵ : Semiring R\ninst✝¹⁴ : CommSemiring R'\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Algebra S R\ninst✝¹¹ : Module S M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower S R M\ninst✝⁶... | refine Equiv.trans ?_ liftEquiv | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.GradedMonoid | {
"line": 428,
"column": 89
} | {
"line": 429,
"column": 60
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_3\ninst✝¹ : AddMonoid ι\ninst✝ : GMonoid A\nn : ℕ\nf : Fin n → GradedMonoid A\n⊢ (List.ofFn f).prod =\n mk ((List.finRange n).dProdIndex fun i ↦ (f i).fst) ((List.finRange n).dProd (fun i ↦ (f i).fst) fun i ↦ (f i).snd)",
"usedConstants": [
"Eq.mpr",
"Add... | by
rw [List.ofFn_eq_map, GradedMonoid.list_prod_map_eq_dProd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 234,
"column": 21
} | {
"line": 234,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_4\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nhf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)\na b : M\n⊢ f a * f b + f b * f a = (f a + f b) * (f a + f b) - f a... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.Ring | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 19
} | [
{
"pp": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ((mulHom A) ((of A 0) GradedMonoid.GOne.one)) ((of A i) xi) = (AddMonoidHom.id (⨁ (i : ι), A i)) ((of A i) xi)",
... | rw [mulHom_of_of] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.DirectSum.Ring | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 21
} | [
{
"pp": "case H.h.H.h.H.h\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\na b c : ⨁ (i : ι), A i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : A ci\n⊢ (of A ai) ax * (of A bi) bx * (of A ci) cx = (of A ai) ax * ((o... | simp_rw [of_mul_of] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.DirectSum.Internal | {
"line": 160,
"column": 2
} | {
"line": 162,
"column": 7
} | [
{
"pp": "case pos\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : (i : ι) → (x : ↥(A i)) → Decidable (x ≠ 0)\nr r' : ⨁ (i : ι), ↥(A i)\nn x✝¹ x✝... | · subst h
rw [of_eq_same]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.DirectSum.Internal | {
"line": 169,
"column": 66
} | {
"line": 173,
"column": 50
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : HasAntidiagonal ι\ninst✝ : SetLike.GradedMonoid A\nr r' : ⨁ (i : ι), ↥(A i)\nn : ι\n⊢ ↑((r * r') n) = ∑ ij ∈ antidiagona... | by
classical
rw [coe_mul_apply]
apply Finset.sum_subset (fun _ ↦ by simp)
aesop (erase simp not_and) (add simp not_and_or) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.DirectSum.Internal | {
"line": 196,
"column": 38
} | {
"line": 196,
"column": 47
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (DFinsupp... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 243,
"column": 38
} | {
"line": 243,
"column": 47
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Semiring R\ninst✝⁵ : SetLike σ R\ninst✝⁴ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : PartialOrder ι\ninst✝¹ : CanonicallyOrderedAdd ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' :... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 44
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"NonUnitalC... | simpa using separatingLeft_iff_det_ne_zero | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 44
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"NonUnitalC... | simpa using separatingLeft_iff_det_ne_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 44
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"NonUnitalC... | simpa using separatingLeft_iff_det_ne_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 290,
"column": 54
} | {
"line": 290,
"column": 68
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nx x' y : M\n⊢ polar (⇑Q) x y + polar (⇑Q) (-x') y = polar (⇑Q) x y + -polar (⇑Q) x' y",
"usedConstants": [
"Eq.mpr"... | polar_neg_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1106,
"column": 45
} | {
"line": 1107,
"column": 60
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\n⊢ ¬Q.Anisotropic ↔ ∃ x, x ≠ 0 ∧ Q x = 0",
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
... | by
simp only [Anisotropic, not_forall, exists_prop, and_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 63
} | [
{
"pp": "case h.e_6.h.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 ↦ M') M... | rw [finSumEquivOfFinset_inr, Finset.piecewise_eq_of_notMem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating | {
"line": 97,
"column": 16
} | {
"line": 97,
"column": 31
} | [
{
"pp": "case zero\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\nf : (i : ℕ) → M [⋀^Fin i]→ₗ[R] N\nv : Fin 0 → M\n⊢ (liftAlternating f) (List.ofFn fun i ↦ (ι R) (v i)).prod = (f 0) v",
"usedConstants... | List.ofFn_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.FilteredColimits | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nJ : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ ModuleCat R\nr s : R\nj : J\nx : ↑(F.obj j)\n⊢ (r * s) • M.mk F ⟨j, x⟩ = r • s • M.mk F ⟨j, x⟩",
"usedConstants": [
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"ModuleCat",
... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.LeftResolution.Basic | {
"line": 157,
"column": 2
} | {
"line": 159,
"column": 48
} | [
{
"pp": "case h\nA : Type u_1\nC : Type u_2\ninst✝⁵ : Category.{v_1, u_2} C\ninst✝⁴ : Category.{v_2, u_1} A\nι : C ⥤ A\nΛ : LeftResolution ι\nX : A\ninst✝³ : ι.Full\ninst✝² : ι.Faithful\ninst✝¹ : HasZeroMorphisms C\ninst✝ : Abelian A\nn : ℕ\n⊢ (Λ.chainComplexMap (𝟙 X)).f n = (𝟙 (Λ.chainComplex X)).f n",
"... | induction n with
| zero => simp
| succ n hn => obtain _ | n := n <;> simp [hn] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Homology.LeftResolution.Basic | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 48
} | [
{
"pp": "case h\nA : Type u_1\nC : Type u_2\ninst✝⁶ : Category.{v_1, u_2} C\ninst✝⁵ : Category.{v_2, u_1} A\nι : C ⥤ A\nΛ : LeftResolution ι\nX Y : A\ninst✝⁴ : ι.Full\ninst✝³ : ι.Faithful\ninst✝² : HasZeroMorphisms C\ninst✝¹ : Abelian A\ninst✝ : Λ.F.PreservesZeroMorphisms\nn : ℕ\n⊢ (Λ.chainComplexMap 0).f n = H... | induction n with
| zero => simp
| succ n hn => obtain _ | n := n <;> simp [hn] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 74
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nn : ℕ\nM : Type u_1\nN✝ : Type u_2\nN' : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N✝\ninst✝² : Module R N✝\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nN : Type ?u.111840\nx✝¹ : AddCommGroup N\nx✝ : Module R N\nf f' : ↥(⋀[R]^n M) ... | rw [Submodule.linearMap_eq_iff_of_span_eq_top _ _ (ιMulti_span R n M)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 451,
"column": 41
} | {
"line": 452,
"column": 17
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : Fin 1 → M\n⊢ (oneEquiv R M) ((ιMulti R 1) f) = f 0",
"usedConstants": [
"LinearMap.id",
"AlternatingMap",
"Submodule",
"Semiring.toModule",
"Equiv.instEquivLike",
"Qua... | by
simp [oneEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal | {
"line": 138,
"column": 39
} | {
"line": 138,
"column": 67
} | [
{
"pp": "case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nR : Cᵒᵖ ⥤ CommRingCat\nX✝¹ Y✝ : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nx✝ : X✝¹ ⟶ Y✝\nX✝ : Cᵒᵖ\n⊢ (x✝ ▷ 𝟙_ (PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)) ≫ (ρ_ Y✝).hom).app X✝ = ((ρ_ X✝¹).hom ≫ x✝).app X✝",
"usedConstants": ... | apply rightUnitor_naturality | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 540,
"column": 24
} | {
"line": 540,
"column": 59
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ S R : Sieve X\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S.arrows f ∨ R.arrows f → ∀ (g : Z ⟶ Y), S.arrows (g ≫ f) ∨ R.arrows (g ≫ f)",
"usedConstants": [
"CategoryTheory.CategoryStruc... | rintro _ _ _ (h | h) g <;> simp [h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 540,
"column": 24
} | {
"line": 540,
"column": 59
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ S R : Sieve X\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S.arrows f ∨ R.arrows f → ∀ (g : Z ⟶ Y), S.arrows (g ≫ f) ∨ R.arrows (g ≫ f)",
"usedConstants": [
"CategoryTheory.CategoryStruc... | rintro _ _ _ (h | h) g <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 540,
"column": 24
} | {
"line": 540,
"column": 59
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ S R : Sieve X\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S.arrows f ∨ R.arrows f → ∀ (g : Z ⟶ Y), S.arrows (g ≫ f) ∨ R.arrows (g ≫ f)",
"usedConstants": [
"CategoryTheory.CategoryStruc... | rintro _ _ _ (h | h) g <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 932,
"column": 2
} | {
"line": 934,
"column": 34
} | [
{
"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✝ : IsSplitEpi f\n⊢ GaloisInsertion (pushforward f) (pullback f)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.H... | apply (galoisConnection f).toGaloisInsertion
intro S Z g hg
exact ⟨g ≫ section_ f, by simpa⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 932,
"column": 2
} | {
"line": 934,
"column": 34
} | [
{
"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✝ : IsSplitEpi f\n⊢ GaloisInsertion (pushforward f) (pullback f)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.H... | apply (galoisConnection f).toGaloisInsertion
intro S Z g hg
exact ⟨g ≫ section_ f, by simpa⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1110,
"column": 2
} | {
"line": 1110,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : Y✝ ⟶ X✝\nS✝ R : Sieve X✝\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\nG : D ⥤ E\ninst✝¹ : F.EssSurj\ninst✝ : F.Full\nX : C\nS : Sieve (F.obj X)\nY : D\nf : Y ⟶ F.obj X\nhf : S.arrows f\n⊢ (... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 212,
"column": 49
} | {
"line": 219,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{w', w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\n⊢ η ≫ J.toPlus Q = J.toPlus P ≫ ... | by
ext
dsimp [toPlus, plusMap]
delta Cover.toMultiequalizer
simp only [ι_colimMap, Category.assoc]
simp_rw [← Category.assoc]
congr 1
exact Multiequalizer.hom_ext _ _ _ (fun I => by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 978,
"column": 2
} | {
"line": 979,
"column": 99
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : X ⟶ Y\n⊢ (∀ (b : P.obj (op X)),\n ((FamilyOfElements.singletonEquiv P f).symm b).Compatible →\n ∃! t, ((FamilyOfElements.singletonEquiv P f).symm b).IsAmalgamation t) ↔\n ∀ (x : P.obj (op X)),\n (∀ {Z : C} (p₁ ... | simp_rw [FamilyOfElements.compatible_singleton_iff,
FamilyOfElements.isAmalgamation_singleton_iff, FamilyOfElements.singletonEquiv_symm_apply_self] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 367,
"column": 8
} | {
"line": 368,
"column": 18
} | [
{
"pp": "case a\nC : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (W... | · cases I
exact hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Sheafification | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP : Cᵒᵖ ⥤ D\nhP : Presheaf.IsSheaf J P\n⊢ (isoSheafify J hP).inv = sheafifyLift J (𝟙 P) hP",
"usedConstants": [
"CategoryTheory.Functor",
"O... | apply sheafifyLift_unique | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Sheafification | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nhR : Presheaf.IsSheaf J R\n⊢ sheafifyMap J η ≫ sheafifyLift J γ hR = sheafifyLift J (η ≫ γ) hR",
"usedConstants": ... | apply sheafifyLift_unique | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Limits | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 44
} | [
{
"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... | rw [(F.obj k).property.amalgamate_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 185,
"column": 2
} | {
"line": 204,
"column": 14
} | [
{
"pp": "W X Y Z S T : TopCat\nf₁ : W ⟶ S\nf₂ : X ⟶ S\ng₁ : Y ⟶ T\ng₂ : Z ⟶ T\ni₁ : W ⟶ Y\ni₂ : X ⟶ Z\ni₃ : S ⟶ T\nH₃ : Mono i₃\neq₁ : f₁ ≫ i₃ = i₁ ≫ g₁\neq₂ : f₂ ≫ i₃ = i₂ ≫ g₂\n⊢ Set.range ⇑(ConcreteCategory.hom (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)) =\n ⇑(ConcreteCategory.hom (pullback.fst g₁ g₂)) ⁻... | ext
constructor
· rintro ⟨y, rfl⟩
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range]
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply]
simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app]
exact ⟨exists_apply_eq_apply _ _, exists_apply_eq_apply _ _⟩
rintro... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 185,
"column": 2
} | {
"line": 204,
"column": 14
} | [
{
"pp": "W X Y Z S T : TopCat\nf₁ : W ⟶ S\nf₂ : X ⟶ S\ng₁ : Y ⟶ T\ng₂ : Z ⟶ T\ni₁ : W ⟶ Y\ni₂ : X ⟶ Z\ni₃ : S ⟶ T\nH₃ : Mono i₃\neq₁ : f₁ ≫ i₃ = i₁ ≫ g₁\neq₂ : f₂ ≫ i₃ = i₂ ≫ g₂\n⊢ Set.range ⇑(ConcreteCategory.hom (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)) =\n ⇑(ConcreteCategory.hom (pullback.fst g₁ g₂)) ⁻... | ext
constructor
· rintro ⟨y, rfl⟩
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range]
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply]
simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app]
exact ⟨exists_apply_eq_apply _ _, exists_apply_eq_apply _ _⟩
rintro... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 269,
"column": 2
} | {
"line": 277,
"column": 27
} | [
{
"pp": "W X Y Z S T : TopCat\nf₁ : W ⟶ S\nf₂ : X ⟶ S\ng₁ : Y ⟶ T\ng₂ : Z ⟶ T\ni₁ : W ⟶ Y\ni₂ : X ⟶ Z\nH₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₁)\nH₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₂)\ni₃ : S ⟶ T\nH₃ : Mono i₃\neq₁ : f₁ ≫ i₃ = i₁ ≫ g₁\neq₂ : f₂ ≫ i₃ = i₂ ≫ g₂\n⊢ IsOpenEmbedding ⇑(ConcreteCategor... | constructor
· apply
pullback_map_isEmbedding f₁ f₂ g₁ g₂ H₁.isEmbedding H₂.isEmbedding i₃ eq₁ eq₂
· rw [range_pullback_map]
apply IsOpen.inter <;> apply Continuous.isOpen_preimage
· apply ContinuousMap.continuous_toFun
· exact H₁.isOpen_range
· apply ContinuousMap.continuous_toFun
· exact ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 269,
"column": 2
} | {
"line": 277,
"column": 27
} | [
{
"pp": "W X Y Z S T : TopCat\nf₁ : W ⟶ S\nf₂ : X ⟶ S\ng₁ : Y ⟶ T\ng₂ : Z ⟶ T\ni₁ : W ⟶ Y\ni₂ : X ⟶ Z\nH₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₁)\nH₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₂)\ni₃ : S ⟶ T\nH₃ : Mono i₃\neq₁ : f₁ ≫ i₃ = i₁ ≫ g₁\neq₂ : f₂ ≫ i₃ = i₂ ≫ g₂\n⊢ IsOpenEmbedding ⇑(ConcreteCategor... | constructor
· apply
pullback_map_isEmbedding f₁ f₂ g₁ g₂ H₁.isEmbedding H₂.isEmbedding i₃ eq₁ eq₂
· rw [range_pullback_map]
apply IsOpen.inter <;> apply Continuous.isOpen_preimage
· apply ContinuousMap.continuous_toFun
· exact H₁.isOpen_range
· apply ContinuousMap.continuous_toFun
· exact ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 151,
"column": 4
} | {
"line": 152,
"column": 83
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nI₁ : Type u_2\nI₂ : Type u_3\nX : I₁ ⊕ I₂ → C\nc : Cofan X\nc₁ : Cofan (X ∘ Sum.inl)\nc₂ : Cofan (X ∘ Sum.inr)\nhc : IsColimit c\nhc₁ : IsColimit c₁\nhc₂ : IsColimit c₂\ninst✝ : MonoCoprod C\ninl : c₁.pt ⟶ c.pt\nhinl : ∀ (i₁ : I₁), c₁.inj i₁ ≫ inl = c.inj (... | rw [this]
exact MonoCoprod.binaryCofan_inl _ (isColimitBinaryCofanSum c c₁ c₂ hc hc₁ hc₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 151,
"column": 4
} | {
"line": 152,
"column": 83
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nI₁ : Type u_2\nI₂ : Type u_3\nX : I₁ ⊕ I₂ → C\nc : Cofan X\nc₁ : Cofan (X ∘ Sum.inl)\nc₂ : Cofan (X ∘ Sum.inr)\nhc : IsColimit c\nhc₁ : IsColimit c₁\nhc₂ : IsColimit c₂\ninst✝ : MonoCoprod C\ninl : c₁.pt ⟶ c.pt\nhinl : ∀ (i₁ : I₁), c₁.inj i₁ ≫ inl = c.inj (... | rw [this]
exact MonoCoprod.binaryCofan_inl _ (isColimitBinaryCofanSum c c₁ c₂ hc hc₁ hc₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Extensive | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 23
} | [
{
"pp": "case h₂.fac_left.h\nJ : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u''\ninst✝ : Category.{v'', u''} D\nX Y : C\nZ : Type u\nf : Z ⟶ (Types.binaryCoproductCocone PUnit.{u + 1} PUnit.{u + 1}).pt\nthis : ∀ (x : Z), f x = Sum.inl PUnit.unit ∨ f x = Sum.inr PUnit... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 11
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\n⊢ P.isoClosure.isColocal = P.isColocal",
"usedConstants": [
"CategoryTheory.ObjectProperty.isoClosure",
"CategoryTheory.Category.toCategoryStruct",
"CategoryTheory.MorphismProperty.ext",
"CategoryTheory.Objec... | ext Y Z g | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 229,
"column": 10
} | {
"line": 229,
"column": 87
} | [
{
"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\nF : C ⥤ D\nG : D ⥤ C\nadj : G ⊣ F\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nX Y : D\nf : X ⟶ Y\nthis : f ≫ adj.unit.app Y = adj.unit.app X ≫ F.map (G.map f)\n⊢ isLocal (fun x ↦ x ∈ Set.range F.obj) (adj.unit.app X ≫... | (isLocal (· ∈ Set.range F.obj)).precomp_iff _ _ (isLocal_adj_unit_app adj X), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 100,
"column": 89
} | {
"line": 105,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nJ : GrothendieckTopology C\nF₁ F₂ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\n⊢ IsLocallyInjective J (Functor.whiske... | by
constructor
· intro
exact ⟨fun x y h => equalizerSieve_mem J (Functor.whiskerRight φ (forget D)) x y h⟩
· intro
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 145,
"column": 58
} | {
"line": 147,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝² : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝¹ : ConcreteCategory D FD\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nφψ : F₁ ⟶ F₃\nfac : φ... | by
subst fac
exact isLocallyInjective_of_isLocallyInjective J φ ψ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝ : ConcreteCategory D FD\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nP : Cᵒᵖ ⥤ Type (max u ... | rw [GrothendieckTopology.plusMap_toPlus] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 483,
"column": 2
} | {
"line": 484,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u''\ninst✝⁶ : Category.{v'', u''} D\nF : C ⥤ D\ninst✝⁵ : Adhesive D\ninst✝⁴ : HasPullbacks C\ninst✝³ : HasPushouts C\ninst✝² : PreservesLimitsOfShape WalkingCospan F\ninst✝¹ : PreservesColimitsOfShape WalkingSpan F\ninst✝ : F.ReflectsIsomorphisms\nthis :... | haveI : ReflectsColimitsOfShape WalkingSpan F :=
reflectsColimitsOfShape_of_reflectsIsomorphisms | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝ : ConcreteCategory A FA\nP : Cᵒᵖ ⥤ Type (max u v)\n⊢ IsLocallySurjective J (J.toPlus P ≫ J.p... | rw [GrothendieckTopology.plusMap_toPlus] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 75
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nR : Cᵒᵖ ⥤ RingCat\nF : D ⥤ PresheafOfModules R\ninst✝¹ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections\nc : Cone F\ninst✝ : HasLimitsOfSh... | exact Sheaf.isSheaf_of_isLimit G _ (isLimitOfPreserves (toPresheaf R) hc) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 694,
"column": 8
} | {
"line": 694,
"column": 21
} | [
{
"pp": "case refine_2.refine_1.isTrue\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasInitial C\nι : Type u_3\nX : ι → C\nc : Cofan X\nhc : IsVanKampenColimit c\ni j✝ : ι\ninst✝ : DecidableEq ι\nt : Cofan fun k ↦ if k = i then X i else ⊥_ C\nj : ι\nh✝ : j = i\n⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj i = t.inj j",... | subst ‹j = i› | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 66
} | [
{
"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\... | ← isIso_iff_of_reflects_iso _ (SheafOfModules.toSheaf.{v} R) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 36
} | [
{
"pp": "case diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝¹ val✝ : J\nf : val✝¹ ⟶ val✝\n⊢ BiconeHom.diagram f ∈ Finset.image BiconeHom.diagram Fintype.elems",
"usedConstants": [
"Eq.mpr",
"Fintype.elems",
"CategoryTheory.Bicone.diagram",
"CategoryTheory... | simp only [Finset.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 389,
"column": 2
} | {
"line": 389,
"column": 27
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝⁵ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝⁴ : G.IsCocontinuous J K\ninst✝³ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\ninst✝² :... | apply sheafifyLift_unique | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 19
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nX : C\nS : Sieve X\n⊢ (∃ T, Sieve.functorPullback G ↑T = S) → Sieve.functorPushforward G S ∈ ... | rintro ⟨T, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 65
} | [
{
"pp": "case w.h.a.w.h.h\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.... | apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 65
} | [
{
"pp": "case w.h.a.w.h.h\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.obj\nX : Dᵒᵖ\n... | apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.IsConnected | {
"line": 80,
"column": 15
} | {
"line": 84,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : IsPreconnected C\ninst✝ : HasColimit (constPUnitFunctor C)\na b : colimit (constPUnitFunctor C)\n⊢ a = b",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Exists",
"CategoryTheory.Limits.Types.colim... | by
obtain ⟨c, ⟨⟩, rfl⟩ := jointly_surjective' a
obtain ⟨d, ⟨⟩, rfl⟩ := jointly_surjective' b
apply constant_of_preserves_morphisms (colimit.ι (constPUnitFunctor C) · PUnit.unit)
exact fun c d f => colimit_sound f rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 53
} | [
{
"pp": "case w\nC : Type u\ninst✝ : SmallCategory C\nX Y : C ⥤ Type u\nj : C\n⊢ colimit.ι ((tensor (C ⥤ Type u)).obj (X, Y)) j ≫\n (HasColimit.isoOfNatIso ((externalProductCompDiagIso C (Type u)).app (X, Y)).symm).hom ≫\n colimit.pre (X ⊠ Y) (Functor.diag C) ≫\n (PreservesColimit₂.isoColim... | dsimp [externalProductBifunctor, CartesianMonoidalCategory.prodComparison,
externalProductBifunctorCurried, externalProduct] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 260,
"column": 2
} | {
"line": 261,
"column": 65
} | [
{
"pp": "C : Type u\ninst✝³ : SmallCategory C\nD : Type u\ninst✝² : SmallCategory D\ninst✝¹ : IsSifted C\nF : C ⥤ D\ninst✝ : F.Final\n⊢ IsSifted D",
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfShape",
"CategoryTheory.Functor",
"CategoryTheory.whiskeringLeft_preservesLimits... | have : PreservesFiniteProducts (colim : (D ⥤ Type u) ⥤ _) :=
⟨fun n ↦ preservesLimitsOfShape_of_natIso (Final.colimIso F)⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 51
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nc x✝ : C\nx : Under c\ns s' : x✝ ⟶ (forget c).obj x\n⊢ ∃ t, s ≫ (forget c).map t = s' ≫ (forget c).map t",
"usedConstants": [
"CategoryTheory.instCategoryUnder... | use homMk (IsFiltered.coeqHom s s') (by simp) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 74
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nhF : F.Final\nd : D\n⊢ ∃ c, Nonempty (d ⟶ F.obj c)",
"usedConstants": [
"CategoryTheory.IsConnected.is_nonempty",
"CategoryTheory.Functor.Final.ou... | obtain ⟨f⟩ : Nonempty (StructuredArrow d F) := IsConnected.is_nonempty | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 341,
"column": 4
} | {
"line": 342,
"column": 38
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : IsFiltered C\nF : C ⥤ Discrete PUnit.{u_1 + 1}\nx✝ : Discrete PUnit.{u_1 + 1}\n⊢ ∃ c, Nonempty (x✝ ⟶ F.obj c)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Classical.choice",
"N... | use Classical.choice IsFiltered.nonempty
exact ⟨Discrete.eqToHom (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 341,
"column": 4
} | {
"line": 342,
"column": 38
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : IsFiltered C\nF : C ⥤ Discrete PUnit.{u_1 + 1}\nx✝ : Discrete PUnit.{u_1 + 1}\n⊢ ∃ c, Nonempty (x✝ ⟶ F.obj c)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Classical.choice",
"N... | use Classical.choice IsFiltered.nonempty
exact ⟨Discrete.eqToHom (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 166,
"column": 2
} | {
"line": 167,
"column": 6
} | [
{
"pp": "J₁ : Type u_1\nJ₂ : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J₁\ninst✝⁴ : Category.{v_2, u_2} J₂\nC₁ : Type u_3\nC₂ : Type u_4\nC : Type u_5\ninst✝³ : Category.{v_3, u_3} C₁\ninst✝² : Category.{v_4, u_4} C₂\ninst✝¹ : Category.{v_5, u_5} C\nK₁ : J₁ ⥤ C₁\nK₂ : J₂ ⥤ C₂\nG : C₁ ⥤ C₂ ⥤ C\ninst✝ : PreservesCol... | rw [← Category.assoc, ← Iso.eq_comp_inv]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 166,
"column": 2
} | {
"line": 167,
"column": 6
} | [
{
"pp": "J₁ : Type u_1\nJ₂ : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J₁\ninst✝⁴ : Category.{v_2, u_2} J₂\nC₁ : Type u_3\nC₂ : Type u_4\nC : Type u_5\ninst✝³ : Category.{v_3, u_3} C₁\ninst✝² : Category.{v_4, u_4} C₂\ninst✝¹ : Category.{v_5, u_5} C\nK₁ : J₁ ⥤ C₁\nK₂ : J₂ ⥤ C₂\nG : C₁ ⥤ C₂ ⥤ C\ninst✝ : PreservesCol... | rw [← Category.assoc, ← Iso.eq_comp_inv]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nc : Cofan E.X\nhc : IsColimit c\nd : Cofan E.Y'\nhd : IsColimit d\nF : Cᵒᵖ ⥤ A\ninst✝¹ : PreservesLimit (Discrete.functor fun i ↦ op (E.X i)) F\ninst✝ : PreservesLimit (Discrete.functor ... | letI c' : Fan (E.multicospanIndex F).left := Fan.mk _ fun i ↦ F.map (c.inj i).op | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 31
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nj : J\nk : K\n⊢ colimit.ι (F.flip.obj k) j ≫\n colimit.ι (F.flip ⋙ colim) k ≫\n ... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 31
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nk : K\nj : J\n⊢ colimit.ι (F.obj j) k ≫\n colimit.ι (F ⋙ colim) j ≫\n (((... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 309,
"column": 8
} | {
"line": 309,
"column": 57
} | [
{
"pp": "case h.refine_1.refine_2.refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E... | simp [this, pullback.condition_assoc, toPullback] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 312,
"column": 8
} | {
"line": 312,
"column": 57
} | [
{
"pp": "case h.refine_1.refine_2.refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E... | simp [this, pullback.condition_assoc, toPullback] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 31
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝³ : Category.{v_3, u_3} C\nG : J × K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nj : J\nk : K\n⊢ colimit.ι ((curry.obj (Pro... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 727,
"column": 2
} | {
"line": 727,
"column": 31
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝³ : Category.{v_3, u_3} C\nG : J × K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nj : J\nk : K\n⊢ colimit.ι ((curry.obj G).o... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 317,
"column": 6
} | {
"line": 318,
"column": 72
} | [
{
"pp": "case h.refine_2.refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = ... | simp only [Category.assoc] at u₁
simp [← reassoc_of% h₁, w, ← reassoc_of% u₁, ← pullback.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 317,
"column": 6
} | {
"line": 318,
"column": 72
} | [
{
"pp": "case h.refine_2.refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = ... | simp only [Category.assoc] at u₁
simp [← reassoc_of% h₁, w, ← reassoc_of% u₁, ← pullback.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 463,
"column": 34
} | {
"line": 463,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ni✝ i' j✝ j' : E.I₀\nhii' : i✝ = i'\nhjj' : j✝ = j'\nu₀ v₀ : E.I₀ → F.I₀\nu... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 463,
"column": 49
} | {
"line": 463,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ni✝ i' j✝ j' : E.I₀\nhii' : i✝ = i'\nhjj' : j✝ = j'\nu₀ v₀ : E.I₀ → F.I₀\nu... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 652,
"column": 6
} | {
"line": 652,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE F : PreOneHypercover S\ns₀ : E.I₀ ≃ F.I₀\nh₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)\ns₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ ... | congrIndexOneOfEqIso_inv_p₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 35
} | [
{
"pp": "case h\nX : TopCat\nU : Opens ↑X\n⊢ ↑((map U.inclusion').obj U) = ↑⊤",
"usedConstants": [
"Subtype.coe_preimage_self",
"TopologicalSpace.Opens",
"TopologicalSpace.Opens.instSetLike",
"TopCat.str",
"TopCat.carrier",
"SetLike.coe"
]
}
] | exact Subtype.coe_preimage_self _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 408,
"column": 74
} | {
"line": 413,
"column": 16
} | [
{
"pp": "X : TopCat\n⊢ ⋯.functor = map (inclusionTopIso X).inv",
"usedConstants": [
"Set.ext",
"Lattice.toSemilatticeSup",
"trivial",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CompleteLattice.toLattice",
"CategoryTheory.Functor.ext",
"Topologica... | by
refine CategoryTheory.Functor.ext ?_ ?_
· intro U
ext x
exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩
· subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 28
} | [
{
"pp": "case h.h.mpr\nX Y : TopCat\nf : X ⟶ Y\nhf : IsOpenMap ⇑(ConcreteCategory.hom f)\nU : Opens ↑Y\nx✝ : ↑Y\n⊢ x✝ ∈ ↑(hf.functor.obj ⊤ ⊓ U) → x✝ ∈ ↑(hf.functor.obj ((map f).obj U))",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"TopologicalSpace.Opens.ins... | rintro ⟨⟨x, -, rfl⟩, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 1059,
"column": 2
} | {
"line": 1060,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\nS : C\nE : PreZeroHypercover S\ninst✝ : E.HasPullbacks\n⊢ E.toPreOneHypercover.HasPullbacks",
"usedConstants": [
"CategoryTheory.PreOneHypercover.toPreZeroHypercover",
"inferInstance",
"id",
... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 1059,
"column": 2
} | {
"line": 1060,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\nS : C\nE : PreZeroHypercover S\ninst✝ : E.HasPullbacks\n⊢ E.toPreOneHypercover.HasPullbacks",
"usedConstants": [
"CategoryTheory.PreOneHypercover.toPreZeroHypercover",
"inferInstance",
"id",
... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Lattice | {
"line": 101,
"column": 2
} | {
"line": 104,
"column": 42
} | [
{
"pp": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\n⊢ HasBinaryProducts α",
"usedConstants": [
"CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop",
"PartialOrder.toPreorder",
"Categ... | have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
infer_instance
apply hasBinaryProducts_of_hasLimit_pair | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Lattice | {
"line": 101,
"column": 2
} | {
"line": 104,
"column": 42
} | [
{
"pp": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\n⊢ HasBinaryProducts α",
"usedConstants": [
"CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop",
"PartialOrder.toPreorder",
"Categ... | have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
infer_instance
apply hasBinaryProducts_of_hasLimit_pair | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 64
} | [
{
"pp": "case h.h.hf.h\nC✝ : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁷ : Category.{v₂, u₂} D✝\nD' : Type u₃\ninst✝⁶ : Category.{v₃, u₃} D'\nD'' : Type u₄\ninst✝⁵ : Category.{v₄, u₄} D''\nJ✝ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nF✝ : C✝ ⥤ D✝\nS✝ : Sheaf J✝ RingCat\nR✝ : Shea... | change (X.val.presheaf.map (G.map (adj.counit.app U.unop)).op ≫
X.val.presheaf.map (adj.unit.app (G.obj U.unop)).op) _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
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