module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 427,
"column": 4
} | {
"line": 427,
"column": 27
} | {
"line": 428,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα : ↑(AddCommGrpCat.of (Ext X X✝ n))\n⊢ Ext.comp α (Ext.mk₀ (f ≫ f')) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯",
"ppTerm": "?m.61",
"assigned": true,
... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα : ↑(AddCommGrpCat.of (Ext X X✝ n))\n⊢ Ext.comp α ((Ext.mk₀ f).comp (Ext.mk₀ f') ⋯) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯"
] | rw [← Ext.mk₀_comp_mk₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 63
} | {
"line": 98,
"column": 2
} | [
{
"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 : A\ne₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\ne₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftF... | [
"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 : A\ne₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\ne₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a\n... | simp only [← cancel_mono (e₂.inv.app _ ≫ G.map (e₁.inv.app _)),
assoc, Iso.hom_inv_id_app_assoc, comp_id, ← Functor.map_comp,
Iso.hom_inv_id_app, Functor.comp_obj, Functor.map_id] at eq | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 37
} | {
"line": 73,
"column": 2
} | [
{
"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 ↔\n ∃ T',\n ∃ (_ : T' ∈ Pretriangulated.distinguish... | [
"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⊢ Opposite.unop ((triangleOpEquivalence C).inverse.obj T) ∈ Pretriangulated.distinguishedTriangles ↔\n... | rw [mem_distinguishedTriangles_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 612,
"column": 2
} | {
"line": 613,
"column": 84
} | {
"line": 615,
"column": 0
} | [
{
"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\nE : C ≌ D\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : E.functor.CommShift A\n⊢ E.CommShift A",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
... | [] | let := E.commShiftInverse A
exact CommShift.mk' _ _ (E.toAdjunction.commShift_of_leftAdjoint A).commShift_unit | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 612,
"column": 2
} | {
"line": 613,
"column": 84
} | {
"line": 615,
"column": 0
} | [
{
"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\nE : C ≌ D\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : E.functor.CommShift A\n⊢ E.CommShift A",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
... | [] | let := E.commShiftInverse A
exact CommShift.mk' _ _ (E.toAdjunction.commShift_of_leftAdjoint A).commShift_unit | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Ext.Basic | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 61
} | {
"line": 49,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM N : ModuleCat R\nr : R\ni : ℕ\n⊢ Mono (AddCommGrpCat.ofHom ((mk₀ (r • 𝟙 M)).postcomp N ⋯)) ↔ IsSMulRegular (Ext N M i) r",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.Category.ModuleCat.E... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM N : ModuleCat R\nr : R\ni : ℕ\n⊢ Function.Injective ⇑(ConcreteCategory.hom (AddCommGrpCat.ofHom ((mk₀ (r • 𝟙 M)).postcomp N ⋯))) ↔\n Function.Injective fun x ↦ r • x"
] | simp only [IsSMulRegular, AddCommGrpCat.mono_iff_injective] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 77
} | {
"line": 312,
"column": 0
} | [
{
"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\nh : solution.IsPresentation\n⊢ solution.π.ker = Submodule.span A (Set.range relations.relation)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstan... | [] | simpa only [← injective_fromQuotient_iff_ker_π_eq_span] using h.bijective.1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 77
} | {
"line": 312,
"column": 0
} | [
{
"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\nh : solution.IsPresentation\n⊢ solution.π.ker = Submodule.span A (Set.range relations.relation)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstan... | [] | simpa only [← injective_fromQuotient_iff_ker_π_eq_span] using h.bijective.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 77
} | {
"line": 312,
"column": 0
} | [
{
"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\nh : solution.IsPresentation\n⊢ solution.π.ker = Submodule.span A (Set.range relations.relation)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstan... | [] | simpa only [← injective_fromQuotient_iff_ker_π_eq_span] using h.bijective.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Nondegenerate | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 12
} | {
"line": 207,
"column": 2
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : ∀ (g : ι → R), ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis : Fintype κ\n⊢ ∀ (g : κ →... | [
"ι : Type u_1\nκ : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : Finite κ\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : ∀ (g : ι → R), ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0\nA : Matrix κ ι R\nhA : A.Nondegenerate\nthis : Fintype κ\nw : κ → R\nhw : ∑ i, w i... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 119,
"column": 54
} | {
"line": 120,
"column": 92
} | {
"line": 122,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nm n : ℕ\nh : m.Coprime n\n⊢ Disjoint (rootsOfUnity m M) (rootsOfUnity n M)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.Coprime",
"_private.Mathlib.RingTheory.RootsOfUnity.Basic.0.disjoint_rootsOfUnity_of_copr... | [] | by
simp [disjoint_iff_inf_le, rootsOfUnity_inf_rootsOfUnity, Nat.coprime_iff_gcd_eq_one.mp h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 74
} | {
"line": 208,
"column": 2
} | [
{
"pp": "ι✝ : Type u_1\nR✝ : Type u_2\nK : Type u_3\nM : Type u_4\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\nV : Type u_8\nN : Type u_9\ninst✝¹⁰ : Field K\ninst✝⁹ : Invertible 2\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module K V\ninst✝⁶ : FiniteDimensional K V\nι : Type u_10\nS : Type u_11\nR : Type u_12\ninst✝⁵ ... | [] | simp only [Pi.smul_apply', Pi.smul_apply, RingHom.id_apply, smul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 95
} | {
"line": 261,
"column": 4
} | [
{
"pp": "case inr.refine_1\nn : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\ni : n\nh : A ∈ center (SpecialLinearGroup n R)\n⊢ ↑A i i ^ Fintype.card n = 1",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"Matri... | [
"case inr.refine_1\nn : Type u\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nA : SpecialLinearGroup n R\ni : n\nh : A ∈ center (SpecialLinearGroup n R)\nthis : ((scalar n) (↑A i i)).det = 1\n⊢ ↑A i i ^ Fintype.card n = 1"
] | have : det ((scalar n) (A i i)) = 1 := (scalar_eq_self_of_mem_center h i).symm ▸ A.property | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 438,
"column": 8
} | {
"line": 438,
"column": 30
} | {
"line": 438,
"column": 31
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\na b : R\nj : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ !![a, -v; b, u].det = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"Add... | [
"case refine_1\nR : Type u_1\ninst✝ : CommRing R\na b : R\nj : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ a * u - -v * b = 1"
] | Matrix.det_fin_two_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 438,
"column": 8
} | {
"line": 438,
"column": 30
} | {
"line": 438,
"column": 31
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\na b : R\nj : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ !![v, a; -u, b].det = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"Add... | [
"case refine_2\nR : Type u_1\ninst✝ : CommRing R\na b : R\nj : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ v * b - a * -u = 1"
] | Matrix.det_fin_two_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 449,
"column": 8
} | {
"line": 449,
"column": 30
} | {
"line": 449,
"column": 31
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\na b : R\ni : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ !![a, b; -v, u].det = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"Add... | [
"case refine_1\nR : Type u_1\ninst✝ : CommRing R\na b : R\ni : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ a * u - b * -v = 1"
] | Matrix.det_fin_two_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 449,
"column": 8
} | {
"line": 449,
"column": 30
} | {
"line": 449,
"column": 31
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\na b : R\ni : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ !![v, -u; a, b].det = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"Add... | [
"case refine_2\nR : Type u_1\ninst✝ : CommRing R\na b : R\ni : Fin 2\nu v : R\nh : u * a + v * b = 1\n⊢ v * b - -u * a = 1"
] | Matrix.det_fin_two_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.Algebra | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 88
} | {
"line": 66,
"column": 0
} | [
{
"pp": "ι : Type uι\nR : Type uR\nA : ι → Type uA\nB : Type uB\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (A i)\ninst✝⁵ : (i : ι) → Module R (A i)\ninst✝⁴ : AddMonoid ι\ninst✝³ : GSemiring A\ninst✝² : Semiring B\ninst✝¹ : GAlgebra R A\ninst✝ : Algebra R B\ns : R\nx y : GradedMonoid A\n⊢ s • (x ... | [] | rw [GAlgebra.smul_def, GAlgebra.smul_def, ← mul_assoc, GAlgebra.commutes, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 270,
"column": 66
} | {
"line": 270,
"column": 91
} | {
"line": 272,
"column": 0
} | [
{
"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 y : M\n⊢ Q (x - y) = Q (y - x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← neg_sub, Q.map_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 270,
"column": 66
} | {
"line": 270,
"column": 91
} | {
"line": 272,
"column": 0
} | [
{
"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 y : M\n⊢ Q (x - y) = Q (y - x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← neg_sub, Q.map_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 270,
"column": 66
} | {
"line": 270,
"column": 91
} | {
"line": 272,
"column": 0
} | [
{
"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 y : M\n⊢ Q (x - y) = Q (y - x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← neg_sub, Q.map_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 325,
"column": 75
} | {
"line": 326,
"column": 51
} | {
"line": 328,
"column": 0
} | [
{
"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\nι : Type u_8\nQ : QuadraticMap R M N\ng : ι → M\nl : ι → R\np : Sym2 ι\n⊢ polarSym2 (⇑Q) (Sym2.map (l • g) p) = (Sym2.map l p).mul • polarSym2 (⇑Q) (... | [] | by
obtain ⟨_, _⟩ := p; simp [← smul_assoc, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 572,
"column": 6
} | {
"line": 572,
"column": 38
} | {
"line": 572,
"column": 38
} | [
{
"pp": "R : Type u_1\nn : Type u_11\nn' : Type u_13\ninst✝⁴ : CommRing R\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nJ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ ((toLinearMap₂' R) J).IsAdjointPair ((toLinearMap₂' R) J') ⇑(toLin' ... | [
"R : Type u_1\nn : Type u_11\nn' : Type u_13\ninst✝⁴ : CommRing R\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nJ : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq n'\n⊢ (toLinearMap₂' R) J' ∘ₗ toLin' A = ((toLinearMap₂' R) J).compl₂ (toLin' A') ↔ J.... | isAdjointPair_iff_comp_eq_compl₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 488,
"column": 34
} | {
"line": 488,
"column": 80
} | {
"line": 489,
"column": 6
} | [
{
"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✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nQ : QuadraticMap R M N\na : R\nx : M\n⊢ (-⇑Q) (a • x) = (a * a) • (-⇑Q) x",
"ppTerm": "?m... | [] | simp only [Pi.neg_apply, Q.map_smul, smul_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 488,
"column": 34
} | {
"line": 488,
"column": 80
} | {
"line": 489,
"column": 6
} | [
{
"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✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nQ : QuadraticMap R M N\na : R\nx : M\n⊢ (-⇑Q) (a • x) = (a * a) • (-⇑Q) x",
"ppTerm": "?m... | [] | simp only [Pi.neg_apply, Q.map_smul, smul_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 488,
"column": 34
} | {
"line": 488,
"column": 80
} | {
"line": 489,
"column": 6
} | [
{
"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✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nQ : QuadraticMap R M N\na : R\nx : M\n⊢ (-⇑Q) (a • x) = (a * a) • (-⇑Q) x",
"ppTerm": "?m... | [] | simp only [Pi.neg_apply, Q.map_smul, smul_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 679,
"column": 60
} | {
"line": 679,
"column": 93
} | {
"line": 679,
"column": 93
} | [
{
"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✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nN' : Type u_8\ninst✝¹ : AddCommMonoid N'\ninst✝ : Module R N'\nB : BilinMap R M N\nx y... | [] | simp [add_add_add_comm, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 679,
"column": 60
} | {
"line": 679,
"column": 93
} | {
"line": 679,
"column": 93
} | [
{
"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✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nN' : Type u_8\ninst✝¹ : AddCommMonoid N'\ninst✝ : Module R N'\nB : BilinMap R M N\nx y... | [] | simp [add_add_add_comm, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 679,
"column": 60
} | {
"line": 679,
"column": 93
} | {
"line": 679,
"column": 93
} | [
{
"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✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nN' : Type u_8\ninst✝¹ : AddCommMonoid N'\ninst✝ : Module R N'\nB : BilinMap R M N\nx y... | [] | simp [add_add_add_comm, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.GradedAlgebra.Basic | {
"line": 281,
"column": 8
} | {
"line": 281,
"column": 23
} | {
"line": 282,
"column": 8
} | [
{
"pp": "case refine_2.refine_3\nι : Type u_1\nR : Type u_2\nA : Type u_3\nσ : Type u_4\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ni : ... | [
"case refine_2.refine_3\nι : Type u_1\nR : Type u_2\nA : Type u_3\nσ : Type u_4\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ni : ι\nc : A\nhc... | intro _ _ hd he | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 589,
"column": 6
} | {
"line": 589,
"column": 38
} | {
"line": 589,
"column": 38
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_6\nM₂ : Type u_7\nn : Type u_11\nn' : Type u_13\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ : Matrix n n R\nJ' : Matrix... | [
"R : Type u_1\nM₁ : Type u_6\nM₂ : Type u_7\nn : Type u_11\nn' : Type u_13\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : Fintype n\ninst✝² : Fintype n'\nb₁ : Basis n R M₁\nb₂ : Basis n' R M₂\nJ : Matrix n n R\nJ' : Matrix n' n' R\nA ... | isAdjointPair_iff_comp_eq_compl₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 88,
"column": 4
} | {
"line": 102,
"column": 9
} | {
"line": 103,
"column": 2
} | [
{
"pp": "case mem\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\ni' : ZMod 2\nx' : CliffordAlgebra Q\ni : { n // ↑n = i' }\nx : CliffordAlgebra Q\nhx : x ∈ (ι Q).range ^ ↑i\n⊢ ((lift Q) ⟨GradedAlgebra.ι Q, ⋯⟩) x = (DirectSum.of (fun i ↦ ↥(ev... | [] | obtain ⟨i, rfl⟩ := i
dsimp only [Subtype.coe_mk] at hx
induction hx using Submodule.pow_induction_on_left' with
| algebraMap r =>
rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl
| add x y i hx hy ihx ihy =>
rw [map_add, ihx, ihy, ← map_add]
rfl
| mem_mul m hm i x hx ih =>
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Grading | {
"line": 88,
"column": 4
} | {
"line": 102,
"column": 9
} | {
"line": 103,
"column": 2
} | [
{
"pp": "case mem\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\ni' : ZMod 2\nx' : CliffordAlgebra Q\ni : { n // ↑n = i' }\nx : CliffordAlgebra Q\nhx : x ∈ (ι Q).range ^ ↑i\n⊢ ((lift Q) ⟨GradedAlgebra.ι Q, ⋯⟩) x = (DirectSum.of (fun i ↦ ↥(ev... | [] | obtain ⟨i, rfl⟩ := i
dsimp only [Subtype.coe_mk] at hx
induction hx using Submodule.pow_induction_on_left' with
| algebraMap r =>
rw [AlgHom.commutes, DirectSum.algebraMap_apply]; rfl
| add x y i hx hy ihx ihy =>
rw [map_add, ihx, ihy, ← map_add]
rfl
| mem_mul m hm i x hx ih =>
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.PowersetCard | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 17
} | {
"line": 215,
"column": 2
} | [
{
"pp": "α : Type u_1\nn : ℕ\ninst✝ : Fintype α\n⊢ Fintype ↑(powersetCard α n)",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"Set.Elem",
"id",
"Set.powersetCard.coe_finset",
"Fintype",
"Se... | [
"α : Type u_1\nn : ℕ\ninst✝ : Fintype α\n⊢ Fintype ↑↑(Finset.powersetCard n Finset.univ)"
] | rw [coe_finset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.PowersetCard | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 39
} | {
"line": 261,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\nn : ℕ\nh1 : 0 < n\nh2 : ↑n < ENat.card α\nval✝ : Fintype α\nthis : (Nat.card α).choose n = 0 ∨ (Nat.card α).choose n = 1\n⊢ False",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Nat.choose",
"ENat.instNatCast",
"congrArg",
"Eq.mp",
... | [
"case inl\nα : Type u_1\nn : ℕ\nh1 : 0 < n\nh2 : ↑n < ↑(Nat.card α)\nval✝ : Fintype α\nthis : (Nat.card α).choose n = 0 ∨ (Nat.card α).choose n = 1\n⊢ False"
] | rw [ENat.card_eq_coe_natCard] at h2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 17
} | {
"line": 213,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nhS : S.ShortExact\nn : ℕ\nh₂ : HasInjectiveDimensionLT S.X₂ n\nh₃ : HasInjectiveDimensionLT S.X₁ (n + 1)\nthis : HasExt C := HasExt.standard C\ni : ℕ\nhi : n ≤ i\nY : C\n⊢ ∀ (e : Ext Y S.X₃ i), e = 0",
"ppTerm": "?m.48",... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nhS : S.ShortExact\nn : ℕ\nh₂ : HasInjectiveDimensionLT S.X₂ n\nh₃ : HasInjectiveDimensionLT S.X₁ (n + 1)\nthis : HasExt C := HasExt.standard C\ni : ℕ\nhi : n ≤ i\nY : C\nx₁ : Ext Y S.X₃ i\n⊢ x₁ = 0"
] | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 338,
"column": 97
} | {
"line": 361,
"column": 5
} | {
"line": 363,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nn : ℕ\nM : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Type u_4\ninst✝ : LinearOrder I\nv : I → M\nα : Fin n → I\n⊢ (ExteriorAlgebra.ιMulti R n) (v ∘ α) ∈ span R (range (ExteriorAlgebra.ιMulti_family R n v))",
"ppTerm": "?m.30",
"assigned": true,... | [] | by
by_cases α_inj : Injective α; swap
· suffices ExteriorAlgebra.ιMulti R n (v ∘ α) = 0 by simp [this]
exact AlternatingMap.map_eq_zero_of_not_injective _ _ <| fun h ↦ α_inj (Injective.of_comp h)
suffices ∃ σ : Equiv.Perm (Fin n), (ExteriorAlgebra.ιMulti R n ((v ∘ α) ∘ σ)) ∈
Submodule.span R (Set.range ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits | {
"line": 110,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 111,
"column": 53
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝¹ :\n ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),\n PreservesColimit (F ⋙ evaluation R Y) (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))\ninst✝ : ∀ (X : Cᵒᵖ), HasColimi... | [] | ext1 X
simpa using colimit.w (F ⋙ evaluation R X) f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits | {
"line": 110,
"column": 8
} | {
"line": 111,
"column": 52
} | {
"line": 111,
"column": 53
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\nF : J ⥤ PresheafOfModules R\ninst✝¹ :\n ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),\n PreservesColimit (F ⋙ evaluation R Y) (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f)))\ninst✝ : ∀ (X : Cᵒᵖ), HasColimi... | [] | ext1 X
simpa using colimit.w (F ⋙ evaluation R X) f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 156,
"column": 83
} | {
"line": 158,
"column": 11
} | {
"line": 160,
"column": 0
} | [
{
"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... | [] | by
dsimp [isoObjCoconePointsOfIsColimit, Functor.mapCocone₂]
cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Filtered.FinallySmall | {
"line": 70,
"column": 7
} | {
"line": 70,
"column": 47
} | {
"line": 70,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : IsFiltered C\ninst✝⁴ : LocallySmall.{w, v, u} C\ninst✝³ : FinallySmall C\nC₀ : Type u\ninst✝² : Category.{w, u} C₀\ninst✝¹ : IsFiltered C₀\ninst✝ : FinallySmall C₀\nP : ObjectProperty C₀ := ⊤.strictMap (fromFinalModel C₀)\nhP : ∀ (X : C₀), ∃ Y, ∃ (_ : P ... | [] | by rintro ⟨_, Y, _, rfl⟩; exact ⟨Y, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 238,
"column": 10
} | {
"line": 238,
"column": 82
} | {
"line": 239,
"column": 10
} | [
{
"pp": "case refine_2\nJ₁ : Type u_1\nJ₂ : Type u_2\ninst✝⁷ : Category.{v_1, u_1} J₁\ninst✝⁶ : Category.{v_2, u_2} J₂\nC₁ : Type u_3\nC₂ : Type u_4\nC : Type u_5\ninst✝⁵ : Category.{v_3, u_3} C₁\ninst✝⁴ : Category.{v_4, u_4} C₂\ninst✝³ : Category.{v_5, u_5} C\nK₁ : J₁ ⥤ C₁\nK₂ : J₂ ⥤ C₂\nG : C₁ ⥤ C₂ ⥤ C\ninst✝... | [
"case refine_2\nJ₁ : Type u_1\nJ₂ : Type u_2\ninst✝⁷ : Category.{v_1, u_1} J₁\ninst✝⁶ : Category.{v_2, u_2} J₂\nC₁ : Type u_3\nC₂ : Type u_4\nC : Type u_5\ninst✝⁵ : Category.{v_3, u_3} C₁\ninst✝⁴ : Category.{v_4, u_4} C₂\ninst✝³ : Category.{v_5, u_5} C\nK₁ : J₁ ⥤ C₁\nK₂ : J₂ ⥤ C₂\nG : C₁ ⥤ C₂ ⥤ C\ninst✝² : Preserve... | simp only [Functor.mapCocone_pt, Functor.mapCocone_ι_app, Q₀, s] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 29
} | {
"line": 439,
"column": 4
} | [
{
"pp": "case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsFiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : d ⟶ (Pi.eval I s).obj c\nc't : (s : α) → (c' ... | [
"case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsFiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : d ⟶ (Pi.eval I s).obj c\nc't : (s : α) → (c' : I s) × (c ... | rw [Function.update_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 11
} | {
"line": 295,
"column": 0
} | [
{
"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✝ : PreservesLim... | [] | dsimp [isoObjConePointsOfIsLimit, Functor.mapCocone₂]
cat_disch | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 11
} | {
"line": 295,
"column": 0
} | [
{
"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✝ : PreservesLim... | [] | dsimp [isoObjConePointsOfIsLimit, Functor.mapCocone₂]
cat_disch | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 29
} | {
"line": 453,
"column": 4
} | [
{
"pp": "case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsCofiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : (Pi.eval I s).obj c ⟶ d\nc't : (s : α) → (c... | [
"case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\nα : Type u₁\nI : α → Type u₂\ninst✝¹ : (s : α) → Category.{v₂, u₂} (I s)\ninst✝ : ∀ (s : α), IsCofiltered (I s)\ns : α\nd : I s\nc : (i : α) → I i\nf g : (Pi.eval I s).obj c ⟶ d\nc't : (s : α) → (c' : I s) × (... | rw [Function.update_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Adjunction.PartialAdjoint | {
"line": 276,
"column": 14
} | {
"line": 278,
"column": 42
} | {
"line": 279,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX : F.PartialRightAdjointSource\n⊢ F.partialRightAdjointMap (𝟙 X) = 𝟙 (F.partialRightAdjointObj X)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"... | [] | by
apply F.partialRightAdjointHomEquiv.injective
simp [partialRightAdjointHomEquiv_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Pretopology | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 51
} | {
"line": 184,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nf : Y ⟶ X\nZ : C\ng : Z ⟶ X\ni : IsIso g\n⊢ pullbackArrows f (Presieve.singleton g) ∈ {S | ∃ Y_1 f, ∃ (_ : IsIso f), S = Presieve.singleton f}",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"CategoryTh... | [
"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nf : Y ⟶ X\nZ : C\ng : Z ⟶ X\ni : IsIso g\n⊢ IsIso (pullback.snd g f)",
"case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nf : Y ⟶ X\nZ : C\ng : Z ⟶ X\ni : IsIso g\n⊢ pullbackArrows f (Pre... | refine ⟨pullback g f, pullback.snd _ _, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 104,
"column": 4
} | {
"line": 105,
"column": 50
} | {
"line": 105,
"column": 50
} | [
{
"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 : Presieve X\nR : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → Presieve Y\nZ : C\nh : Z ⟶ X\nH : S.bind R h\n⊢ Nonempty (S.BindStruct R h)",
"ppTerm": "?m.25",
"assigned": true,
"usedCo... | [] | obtain ⟨Y, g, f, hf, hg, fac⟩ := H
exact ⟨{ hf := hf, hg := hg, fac := fac, .. }⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 104,
"column": 4
} | {
"line": 105,
"column": 50
} | {
"line": 105,
"column": 50
} | [
{
"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 : Presieve X\nR : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → Presieve Y\nZ : C\nh : Z ⟶ X\nH : S.bind R h\n⊢ Nonempty (S.BindStruct R h)",
"ppTerm": "?m.25",
"assigned": true,
"usedCo... | [] | obtain ⟨Y, g, f, hf, hg, fac⟩ := H
exact ⟨{ hf := hf, hg := hg, fac := fac, .. }⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.SheafOfTypes | {
"line": 208,
"column": 83
} | {
"line": 221,
"column": 11
} | {
"line": 223,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\ns : Cocone S.arrows.diagram\n⊢ (S.arrows.yonedaFamilyOfElements_fromCocone s).Compatible",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Presieve.yonedaFamilyOfElements_fromCocone",
"CategoryThe... | [] | by
intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ hgf
have Hs := s.ι.naturality
simp only [yoneda_obj_obj, Opposite.unop_op, yoneda_obj_map, Quiver.Hom.unop_op]
dsimp [yonedaFamilyOfElements_fromCocone]
have hgf₁ : S.arrows (g₁ ≫ f₁) := by exact Sieve.downward_closed S hf₁ g₁
have hgf₂ : S.arrows (g₂ ≫ f₂) := by exact S... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Terminal | {
"line": 31,
"column": 2
} | {
"line": 33,
"column": 29
} | {
"line": 35,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nF : C ⥤ D\nhF : (X : C) → IsTerminal (F.obj X)\n⊢ IsTerminal F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.evaluationJointlyReflectsLimits",
"CategoryT... | [] | refine evaluationJointlyReflectsLimits _
fun X ↦ IsLimit.equivOfNatIsoOfIso (Functor.emptyExt _ _) _ _ ?_ (hF X)
exact Cone.ext (Iso.refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Terminal | {
"line": 31,
"column": 2
} | {
"line": 33,
"column": 29
} | {
"line": 35,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nF : C ⥤ D\nhF : (X : C) → IsTerminal (F.obj X)\n⊢ IsTerminal F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.evaluationJointlyReflectsLimits",
"CategoryT... | [] | refine evaluationJointlyReflectsLimits _
fun X ↦ IsLimit.equivOfNatIsoOfIso (Functor.emptyExt _ _) _ _ ?_ (hF X)
exact Cone.ext (Iso.refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1150,
"column": 4
} | {
"line": 1150,
"column": 37
} | {
"line": 1151,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ R ≤ functorPullback F S → functorPushforward F R ≤ S",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategorySt... | [
"case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : R ≤ functorPullback F S\nY : D\nX : C\ng : X ⟶ X✝\nh : Y ⟶ F.obj X\nhg : R.arrows g\n⊢ S.arrows (h ≫ F.map g)"
] | rintro hle Y f ⟨X, g, h, hg, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 53
} | {
"line": 123,
"column": 4
} | [
{
"pp": "case e'_2\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\... | [
"case e'_2\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : (unop S... | refine Multiequalizer.hom_ext _ _ _ (fun I => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 53
} | {
"line": 140,
"column": 4
} | [
{
"pp": "case e_a\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX Y Z : Cᵒᵖ\nf : X ⟶ Y\ng : Y ⟶ Z\n... | [
"case e_a\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX Y Z : Cᵒᵖ\nf : X ⟶ Y\ng : Y ⟶ Z\nS : (J.Cover... | refine Multiequalizer.hom_ext _ _ _ (fun I => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 53
} | {
"line": 207,
"column": 4
} | [
{
"pp": "case e_a\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX Y : Cᵒᵖ\nf : X ⟶ Y\ne : (J.pullba... | [
"case e_a\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX Y : Cᵒᵖ\nf : X ⟶ Y\ne : (J.pullback f.unop).o... | refine Multiequalizer.hom_ext _ _ _ (fun I => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 51
} | {
"line": 242,
"column": 2
} | [
{
"pp": "case e_a\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\n... | [
"case e_a\nC : 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)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶... | refine Multiequalizer.hom_ext _ _ _ (fun I => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 707,
"column": 2
} | {
"line": 708,
"column": 63
} | {
"line": 710,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nB : Type u₃\ninst✝¹ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ns : A ⥤ B\ninst✝ : ReflectsLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s\nh : IsSheaf J (P ⋙ s)\n⊢ IsSheaf J P",
"ppTerm": "?m.2... | [] | rw [isSheaf_iff_isLimit] at h ⊢
exact fun X S hS ↦ (h S hS).map fun t ↦ isLimitOfReflects s t | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 707,
"column": 2
} | {
"line": 708,
"column": 63
} | {
"line": 710,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nB : Type u₃\ninst✝¹ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ A\ns : A ⥤ B\ninst✝ : ReflectsLimitsOfSize.{v₁, max v₁ u₁, v₂, v₃, u₂, u₃} s\nh : IsSheaf J (P ⋙ s)\n⊢ IsSheaf J P",
"ppTerm": "?m.2... | [] | rw [isSheaf_iff_isLimit] at h ⊢
exact fun X S hS ↦ (h S hS).map fun t ↦ isLimitOfReflects s t | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 981,
"column": 4
} | {
"line": 981,
"column": 52
} | {
"line": 981,
"column": 53
} | [
{
"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)) (t₁ t₂ : P.obj (op Y)),\n (ConcreteCategory.hom (P.map f.op)) t₁ = (FamilyOfElements.singletonEquiv P f).symm b f ⋯ →\n (ConcreteCategory.hom (P.map f.op)) t₂ = (FamilyOfElements.singleto... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : X ⟶ Y\n⊢ (∀ (b : P.obj (op X)) (t₁ t₂ : P.obj (op Y)),\n (ConcreteCategory.hom (P.map f.op)) t₁ = b → (ConcreteCategory.hom (P.map f.op)) t₂ = b → t₁ = t₂) ↔\n Function.Injective ⇑(ConcreteCategory.hom (P.map f.op))"
] | FamilyOfElements.singletonEquiv_symm_apply_self, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Sites.Whiskering | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 24
} | {
"line": 150,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝⁵ : Category.{v₂, u₂} A\nB : Type u₃\ninst✝⁴ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF✝ G H : A ⥤ B\nη : F✝ ⟶ G\nγ : G ⟶ H\ninst✝³ : J.HasSheafCompose F✝\ninst✝² : J.HasSheafCompose G\ninst✝¹ : J.HasSheafComp... | [
"C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝⁵ : Category.{v₂, u₂} A\nB : Type u₃\ninst✝⁴ : Category.{v₃, u₃} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nF✝ G H : A ⥤ B\nη : F✝ ⟶ G\nγ : G ⟶ H\ninst✝³ : J.HasSheafCompose F✝\ninst✝² : J.HasSheafCompose G\ninst✝¹ : J.HasSheafCompose H\nF : A... | obtain ⟨h⟩ := hP X S | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 31
} | {
"line": 409,
"column": 4
} | [
{
"pp": "case left\nC : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape... | [
"case left\nC : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁶ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝⁴ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMul... | apply_fun fun e => e I at h | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 49,
"column": 10
} | {
"line": 53,
"column": 40
} | {
"line": 54,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nι : Type v\nα : ι → TopCat\n⊢ ∀ (s : Cone (Discrete.functor α)) (m : s.pt ⟶ (piFan α).pt),\n (∀ (j : Discrete ι), m ≫ (piFan α).π.app j = s.π.app j) →\n m = ofHom { toFun := fun s_1 i ↦ (ConcreteCategory.hom (s.π.app { as := i })) s_1, continuous_toFun := ... | [] | by
intro S m h
ext x
funext i
simp [ContinuousMap.coe_mk, ← h ⟨i⟩] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 22
} | {
"line": 77,
"column": 8
} | [
{
"pp": "case property.refine_3\nJ : Type v\ninst✝ : Category.{w, v} J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nS : PullbackCone f g\nm : S.pt ⟶ (pullbackCone f g).pt\nh₁ : m ≫ (pullbackCone f g).fst = S.fst\nh₂ : m ≫ (pullbackCone f g).snd = S.snd\nx : ↑S.pt\n⊢ ↑((ConcreteCategory.hom m) x) =\n ↑((ConcreteCat... | [
"case property.refine_3.fst\nJ : Type v\ninst✝ : Category.{w, v} J\nX Y Z : TopCat\nf : X ⟶ Z\ng : Y ⟶ Z\nS : PullbackCone f g\nm : S.pt ⟶ (pullbackCone f g).pt\nh₁ : m ≫ (pullbackCone f g).fst = S.fst\nh₂ : m ≫ (pullbackCone f g).snd = S.snd\nx : ↑S.pt\n⊢ (↑((ConcreteCategory.hom m) x)).1 =\n (↑((ConcreteCatego... | apply Prod.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 163,
"column": 23
} | {
"line": 168,
"column": 32
} | {
"line": 170,
"column": 0
} | [
{
"pp": "J : Type v\ninst✝³ : Category.{w, v} J\nX✝ Y✝ Z✝ : TopCat\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\nhf : Continuous[inst✝², inst✝] f\ng : Y → Z\nhg : IsEmbedding g\n⊢ Continuous[instTopologicalSpaceSubtype... | [] | by
apply Continuous.subtype_mk
refine continuous_subtype_val.prodMk <| hg.isInducing.continuous_iff.mpr ?_
convert! hf.comp continuous_subtype_val
ext x
exact Exists.choose_spec x.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 400,
"column": 2
} | {
"line": 401,
"column": 16
} | {
"line": 402,
"column": 2
} | [
{
"pp": "case mp\nX Y : TopCat\nf g : X ⟶ Y\nc : Cofork f g\nhc : IsColimit c\nU : Set ↑c.pt\n⊢ (∀ (j : WalkingParallelPair), IsOpen (⇑(ConcreteCategory.hom (c.ι.app j)) ⁻¹' U)) →\n IsOpen (⇑(ConcreteCategory.hom c.π) ⁻¹' U)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"CategoryT... | [
"case mpr\nX Y : TopCat\nf g : X ⟶ Y\nc : Cofork f g\nhc : IsColimit c\nU : Set ↑c.pt\n⊢ IsOpen (⇑(ConcreteCategory.hom c.π) ⁻¹' U) →\n ∀ (j : WalkingParallelPair), IsOpen (⇑(ConcreteCategory.hom (c.ι.app j)) ⁻¹' U)"
] | · intro h
exact h .one | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 49
} | {
"line": 111,
"column": 49
} | [
{
"pp": "case inr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B T✝ : Type u\nf₁ : A ⟶ T✝\nf₂ : B ⟶ T✝\nm : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).pt ⟶ T✝\nh₁ : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl ≫ m = f₁\nh₂ : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inr ≫ m = f₂\nx : B\n⊢ (ConcreteCategory.hom m).toF... | [] | exact ConcreteCategory.congr_hom h₂ x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 49
} | {
"line": 111,
"column": 49
} | [
{
"pp": "case inr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B T✝ : Type u\nf₁ : A ⟶ T✝\nf₂ : B ⟶ T✝\nm : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).pt ⟶ T✝\nh₁ : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl ≫ m = f₁\nh₂ : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inr ≫ m = f₂\nx : B\n⊢ (ConcreteCategory.hom m).toF... | [] | exact ConcreteCategory.congr_hom h₂ x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 49
} | {
"line": 111,
"column": 49
} | [
{
"pp": "case inr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B T✝ : Type u\nf₁ : A ⟶ T✝\nf₂ : B ⟶ T✝\nm : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).pt ⟶ T✝\nh₁ : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl ≫ m = f₁\nh₂ : (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inr ≫ m = f₂\nx : B\n⊢ (ConcreteCategory.hom m).toF... | [] | exact ConcreteCategory.congr_hom h₂ x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct | {
"line": 74,
"column": 23
} | {
"line": 76,
"column": 18
} | {
"line": 78,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nι : Type u_1\nX : ι → C\nc : Cofan X\nhc : IsColimit c\ninst✝ : ∀ (i : ι), Mono (c.inj i)\ns : {i j : ι} → i ≠ j → PullbackCone (c.inj i) (c.inj j)\nhs : {i j : ι} → (hij : i ≠ j) → IsLimit (s hij)\nH : {i j : ι} → (hij : i ≠ j) → IsInitial (s hij).pt\nd : Cofan ... | [] | by
rw [show d.inj i = c.inj i ≫ (hd.uniqueUpToIso hc).inv.hom by simp]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Extensive | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 32
} | {
"line": 208,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteCoproducts C\ninst✝ : HasPullbacksOfInclusions C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ c₀.inl = c'.inl ≫ f →\... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteCoproducts C\ninst✝ : HasPullbacksOfInclusions C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ c₀.inl = c'.inl ≫ f →\n αY ≫ ... | hl.paste_vert_iff hX.symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 397,
"column": 8
} | {
"line": 397,
"column": 93
} | {
"line": 398,
"column": 8
} | [
{
"pp": "case h'_w.h'_w\nJ : 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 ≫ p... | [
"case h'_w.h'_w\nJ : 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 ... | rw [pullback.condition_assoc, sq_f_u.w, sq_g_u.w, ← Category.assoc, ← Category.assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.LeftExact | {
"line": 257,
"column": 52
} | {
"line": 273,
"column": 50
} | {
"line": 275,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹² : Category.{t, w} D\ninst✝¹¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\ninst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁹ : (X Y ... | [] | by
let e := (FinCategory.equivAsType K).symm.trans (AsSmall.equiv.{0, 0, t})
have : HasLimitsOfShape (AsSmall.{t} (FinCategory.AsType K)) D :=
Limits.hasLimitsOfShape_of_equivalence e
have : FinCategory (AsSmall.{t} (FinCategory.AsType K)) := by
constructor
· change Fintype (ULift _)
infer_insta... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 483,
"column": 2
} | {
"line": 484,
"column": 49
} | {
"line": 485,
"column": 2
} | [
{
"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\n⊢ Adhe... | [
"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 : ReflectsLim... | have : ReflectsLimitsOfShape WalkingCospan F :=
reflectsLimitsOfShape_of_reflectsIsomorphisms | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 191,
"column": 2
} | {
"line": 193,
"column": 16
} | {
"line": 195,
"column": 0
} | [
{
"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 ... | [] | dsimp [GrothendieckTopology.toSheafify]
rw [GrothendieckTopology.plusMap_toPlus]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 191,
"column": 2
} | {
"line": 193,
"column": 16
} | {
"line": 195,
"column": 0
} | [
{
"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 ... | [] | dsimp [GrothendieckTopology.toSheafify]
rw [GrothendieckTopology.plusMap_toPlus]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 194,
"column": 4
} | {
"line": 196,
"column": 36
} | {
"line": 197,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁴ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝² : ConcreteCategory A FA\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocally... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁴ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝² : ConcreteCategory A FA\nF₁ F₂ F₃ : Cᵒᵖ ⥤ A\nf₁ : F₁ ⟶ F₂\nf₂ : F₂ ⟶ F₃\ninst✝¹ : IsLocallyInjective J ... | have hS : S ∈ J X.unop := by
apply J.intersection_covering
all_goals apply imageSieve_mem | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 304,
"column": 2
} | {
"line": 306,
"column": 16
} | {
"line": 308,
"column": 0
} | [
{
"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.toSheafify P)"... | [] | dsimp [GrothendieckTopology.toSheafify]
rw [GrothendieckTopology.plusMap_toPlus]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 304,
"column": 2
} | {
"line": 306,
"column": 16
} | {
"line": 308,
"column": 0
} | [
{
"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.toSheafify P)"... | [] | dsimp [GrothendieckTopology.toSheafify]
rw [GrothendieckTopology.plusMap_toPlus]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.ChangeOfRings | {
"line": 59,
"column": 57
} | {
"line": 69,
"column": 7
} | {
"line": 69,
"column": 7
} | [
{
"pp": "C : Type u'\ninst✝¹ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR R' : Cᵒᵖ ⥤ RingCat\nα : R ⟶ R'\nM₁ M₂ : PresheafOfModules R'\nhM₂ : Presheaf.IsSheaf J M₂.presheaf\ninst✝ : Presheaf.IsLocallySurjective J α\ng : (restrictScalars α).obj M₁ ⟶ (restrictScalars α).obj M₂\nX : Cᵒᵖ\nr' : ↑(R'.obj X)\n... | [] | by
apply hM₂.isSeparated _ _ (Presheaf.imageSieve_mem J α r')
rintro Y p ⟨r : R.obj _, hr⟩
have hg : ∀ (z : M₁.obj X), g.app _ (M₁.map p.op z) = M₂.map p.op (g.app X z) :=
fun z ↦ CategoryTheory.congr_fun (g.naturality p.op) z
change M₂.map p.op (g.app X (r' • m)) = M₂.map p.op (r' • show M₂.obj X... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 424,
"column": 2
} | {
"line": 434,
"column": 75
} | {
"line": 435,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nS : C\nι : Type u_2\ninst✝¹ : Small.{w, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\ninst✝ : LocallySmall.{w, v, u} C\nc : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\nU : C\ng : U ⟶ S\nV : C\nv : V ⟶ U\nhv :\n (imageSieve (Cofan... | [
"case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\nS : C\nι : Type u_2\ninst✝¹ : Small.{w, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\ninst✝ : LocallySmall.{w, v, u} C\nc : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\nU : C\ng : U ⟶ S\nV : C\nv : V ⟶ U\n⊢ (∃ Y h g_1, Presieve.ofArrows X f g_... | · obtain ⟨w, hw⟩ := hv
obtain ⟨⟨i⟩, a, rfl⟩ := Types.jointly_surjective_of_isColimit
(isColimitOfPreserves ((evaluation _ _).obj (op V)) hc) w
obtain ⟨a : V ⟶ X i, rfl⟩ := shrinkYonedaObjObjEquiv.symm.surjective a
refine ⟨_, a, _, ⟨i⟩, shrinkYonedaObjObjEquiv.symm.injective ?_⟩
rw [← shrinkYoneda_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 218,
"column": 15
} | {
"line": 218,
"column": 17
} | {
"line": 219,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nhS : S.ShortExact\nn : ℕ\nh₂ : HasProjectiveDimensionLT S.X₂ n\nh₃ : HasProjectiveDimensionLT S.X₃ (n + 1)\nthis : HasExt C := HasExt.standard C\ni : ℕ\nhi : n ≤ i\nY : C\n⊢ ∀ (e : Ext S.X₁ Y i), e = 0",
"ppTerm": "?m.48... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nhS : S.ShortExact\nn : ℕ\nh₂ : HasProjectiveDimensionLT S.X₂ n\nh₃ : HasProjectiveDimensionLT S.X₃ (n + 1)\nthis : HasExt C := HasExt.standard C\ni : ℕ\nhi : n ≤ i\nY : C\nx₁ : Ext S.X₁ Y i\n⊢ x₁ = 0"
] | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 353,
"column": 2
} | {
"line": 354,
"column": 18
} | {
"line": 356,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreZeroHypercover S\ne : E ≅ F\ni : F.I₀\n⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.Hom.h₀",
"CategoryTheory.PreZeroHypercover.i... | [] | obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
simpa using hh i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 353,
"column": 2
} | {
"line": 354,
"column": 18
} | {
"line": 356,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreZeroHypercover S\ne : E ≅ F\ni : F.I₀\n⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.Hom.h₀",
"CategoryTheory.PreZeroHypercover.i... | [] | obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
simpa using hh i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 41
} | {
"line": 392,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\nX : E\ninst✝¹ : RepresentablyFlat F\ninst✝ : IsCofiltered (StructuredArrow X G)\nT : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow X G := Structu... | [
"case refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\nX : E\ninst✝¹ : RepresentablyFlat F\ninst✝ : IsCofiltered (StructuredArrow X G)\nT : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow X G := Stru... | refine ⟨fun A B ↦ ?_, fun A B f g ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 581,
"column": 4
} | {
"line": 582,
"column": 13
} | {
"line": 583,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nS : C\nR : Presieve S\n⊢ R.preZeroHypercover.presieve₀ ≤ R",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rintro - - ⟨i⟩
exact i.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 581,
"column": 4
} | {
"line": 582,
"column": 13
} | {
"line": 583,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nS : C\nR : Presieve S\n⊢ R.preZeroHypercover.presieve₀ ≤ R",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | rintro - - ⟨i⟩
exact i.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 771,
"column": 4
} | {
"line": 771,
"column": 43
} | {
"line": 773,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Precoverage C\nS T : C\ninst✝ : J.IsStableUnderSup\nE : J.ZeroHypercover S\nF : J.ZeroHypercover S\n⊢ E.presieve₀ ⊔ F.presieve₀ ∈ J.coverings S",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.presi... | [] | exact J.sup_mem_coverings E.mem₀ F.mem₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 923,
"column": 2
} | {
"line": 923,
"column": 24
} | {
"line": 925,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ K : Precoverage C\nh : ∀ ⦃X : C⦄ ⦃E : J.ZeroHypercover X⦄, E.presieve₀ ∈ K.coverings X\nX : C\nE : PreZeroHypercover X\nhR : E.presieve₀ ∈ J.coverings X\n⊢ E.presieve₀ ∈ K.coverings X",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"C... | [] | exact h (E := ⟨E, hR⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Adjunction.Whiskering | {
"line": 55,
"column": 35
} | {
"line": 55,
"column": 59
} | {
"line": 56,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX : D ⥤ C\n⊢ ((whiskeringLeft E D C).obj G).map\n ({ app := fun X ↦ ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.u... | [] | ext; simp [← X.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Whiskering | {
"line": 55,
"column": 35
} | {
"line": 55,
"column": 59
} | {
"line": 56,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX : D ⥤ C\n⊢ ((whiskeringLeft E D C).obj G).map\n ({ app := fun X ↦ ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.u... | [] | ext; simp [← X.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Whiskering | {
"line": 55,
"column": 32
} | {
"line": 55,
"column": 59
} | {
"line": 56,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX : D ⥤ C\n⊢ ((whiskeringLeft E D C).obj G).map\n ({ app := fun X ↦ ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.u... | [] | by ext; simp [← X.map_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adjunction.Whiskering | {
"line": 56,
"column": 36
} | {
"line": 56,
"column": 60
} | {
"line": 58,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX : E ⥤ C\n⊢ { app := fun X ↦ ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n ... | [] | ext; simp [← X.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Whiskering | {
"line": 56,
"column": 36
} | {
"line": 56,
"column": 60
} | {
"line": 58,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX : E ⥤ C\n⊢ { app := fun X ↦ ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n ... | [] | ext; simp [← X.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Whiskering | {
"line": 56,
"column": 33
} | {
"line": 56,
"column": 60
} | {
"line": 58,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : D ⥤ E\nG : E ⥤ D\nadj : F ⊣ G\nX : E ⥤ C\n⊢ { app := fun X ↦ ((𝟭 (D ⥤ C)).obj X).leftUnitor.inv ≫ whiskerRight adj.unit X ≫ (F.associator G X).hom,\n ... | [] | by ext; simp [← X.map_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 53
} | {
"line": 97,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nD : Type u_2\ninst✝¹ : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF G : C ⥤ D\ne : F ≅ G\ninst✝ : F.IsCocontinuous J K\nU : C\nS : Sieve (G.obj U)\nhS : S ∈ K (G.obj U)\nY : C\nf : Y ⟶ U\n⊢ (Sieve.functorPullback F (Sieve... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nD : Type u_2\ninst✝¹ : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF G : C ⥤ D\ne : F ≅ G\ninst✝ : F.IsCocontinuous J K\nU : C\nS : Sieve (G.obj U)\nhS : S ∈ K (G.obj U)\nY : C\nf : Y ⟶ U\nhf : S.arrows (F.map f ≫ e.hom.app U)\n⊢ (Sie... | (hf : S.arrows (F.map f ≫ e.hom.app U)) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 69
} | {
"line": 216,
"column": 2
} | [
{
"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\nG : C ⥤ D\nA : Type w\ninst✝¹ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : G.IsCocontinuous J K\nF : Cᵒᵖ ⥤ A\nhF : Presheaf.IsSheaf J F\nR : Dᵒᵖ ⥤ A\nα : G.op ⋙ R ⟶ F\nX ... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝¹ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : G.IsCocontinuous J K\nF : Cᵒᵖ ⥤ A\nhF : Presheaf.IsSheaf J F\nR : Dᵒᵖ ⥤ A\nα : G.op ⋙ R ⟶ F\nX : D\nS : K.C... | have eq₁ := liftAux_map hF α s f (g ≫ a) ⟨_, _, hg⟩ (𝟙 _) (by simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 617,
"column": 46
} | {
"line": 617,
"column": 61
} | {
"line": 617,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝² : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nG : C ⥤ D\nA : Type u_4\ninst✝¹ : Category.{v_4, u_4} A\ninst✝ : IsDenseSubsite J... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝² : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nG : C ⥤ D\nA : Type u_4\ninst✝¹ : Category.{v_4, u_4} A\ninst✝ : IsDenseSubsite J K G\nF : Sh... | ha.choose_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 632,
"column": 10
} | {
"line": 632,
"column": 25
} | {
"line": 632,
"column": 26
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝¹ : Category.{v_4, u_4} A\ninst✝ : IsDenseSubsite J K G\nF : Sheaf J A\nX Y Z : C\nf : G.obj X ⟶ G.obj Y\np : Z ⟶ X\ng : Z ⟶ ... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝¹ : Category.{v_4, u_4} A\ninst✝ : IsDenseSubsite J K G\nF : Sheaf J A\nX Y Z : C\nf : G.obj X ⟶ G.obj Y\np : Z ⟶ X\ng : Z ⟶ Y\nfac : G.m... | ha.choose_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
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