module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 248,
"column": 6
} | {
"line": 255,
"column": 79
} | {
"line": 257,
"column": 0
} | [
{
"pp": "n✝ : Type u_1\ninst✝⁵ : Fintype n✝\nn : Type u_1\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : LinearOrder S\ninst✝ : IsStrictOrderedRing S\nA : Matrix n n S\nh1 : Pairwise fun i j ↦ A i j < 0\nh✝ : Nonempty n\nh2 : A.det = 0\nv : n → S\nh_vnz : v ≠ 0\nh_vA : ... | [] | obtain ⟨j₀, -, h_j₀⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty v
refine ⟨j₀, ?_⟩
rw [← mul_le_mul_iff_right₀ (h_j₀ ▸ h_sup), Finset.mul_sum, mul_zero]
rw [show 0 = ∑ i, v i * A i j₀ from (congrFun h_vA j₀).symm]
refine Finset.sum_le_sum (fun i hi => ?_)
by_cases h : i = j₀
· ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 248,
"column": 6
} | {
"line": 255,
"column": 79
} | {
"line": 257,
"column": 0
} | [
{
"pp": "n✝ : Type u_1\ninst✝⁵ : Fintype n✝\nn : Type u_1\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : LinearOrder S\ninst✝ : IsStrictOrderedRing S\nA : Matrix n n S\nh1 : Pairwise fun i j ↦ A i j < 0\nh✝ : Nonempty n\nh2 : A.det = 0\nv : n → S\nh_vnz : v ≠ 0\nh_vA : ... | [] | obtain ⟨j₀, -, h_j₀⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty v
refine ⟨j₀, ?_⟩
rw [← mul_le_mul_iff_right₀ (h_j₀ ▸ h_sup), Finset.mul_sum, mul_zero]
rw [show 0 = ∑ i, v i * A i j₀ from (congrFun h_vA j₀).symm]
refine Finset.sum_le_sum (fun i hi => ?_)
by_cases h : i = j₀
· ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | {
"line": 201,
"column": 4
} | {
"line": 204,
"column": 8
} | {
"line": 206,
"column": 0
} | [
{
"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 [weightedSumSquares_apply, Pi.smul_apply']
refine Finset.sum_congr rfl fun j hj => ?_
rw [smul_mul_smul, Units.smul_def, smul_smul, ← pow_two, ← h]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | {
"line": 201,
"column": 4
} | {
"line": 204,
"column": 8
} | {
"line": 206,
"column": 0
} | [
{
"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 [weightedSumSquares_apply, Pi.smul_apply']
refine Finset.sum_congr rfl fun j hj => ?_
rw [smul_mul_smul, Units.smul_def, smul_smul, ← pow_two, ← h]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 66
} | {
"line": 397,
"column": 2
} | [
{
"pp": "case h\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"MulZeroClass.toMul",
"AddGroupWit... | [
"case h\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"
] | have had : a * d = 1 := by rwa [hg, mul_zero, sub_zero] at h_det | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 343,
"column": 45
} | {
"line": 343,
"column": 77
} | {
"line": 343,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM₁ : Type u_4\nM₂ : Type u_5\nM₃ : Type u_6\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQ₃ : QuadraticForm R M₃\nf : Q... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nM₁ : Type u_4\nM₂ : Type u_5\nM₃ : Type u_6\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nQ₁ : QuadraticForm R M₁\nQ₂ : QuadraticForm R M₂\nQ₃ : QuadraticForm R M₃\nf : Q₂ →qᵢ Q₃\ng ... | QuadraticMap.Isometry.comp_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 608,
"column": 74
} | {
"line": 608,
"column": 94
} | {
"line": 610,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\n⊢ ↑(diag2 a ha) =\n diagonal fun i ↦\n match i with\n | 0 => a\n | 1 => a⁻¹",
"ppTerm": "?m.189",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Divi... | [] | by simp [diag2n_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GradedMonoid | {
"line": 683,
"column": 2
} | {
"line": 686,
"column": 9
} | {
"line": 688,
"column": 0
} | [
{
"pp": "case insert\nι : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : GradedMonoid A\nκ : Type u_5\ni : κ → ι\ng : κ → R\nF : Finset κ\na✝² : κ\ns✝ : Finset κ\na✝¹ : a✝² ∉ s✝\na✝ : (∀ k ∈ s✝, g k ∈ A (i k)) → ∏ k ∈ s✝, g k ∈ A (... | [] | · case insert j F' hF2 h3 =>
rw [Finset.prod_insert hF2, Finset.sum_insert hF2]
apply SetLike.mul_mem_graded (by grind)
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.DirectSum.Internal | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 50
} | {
"line": 175,
"column": 0
} | [
{
"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⊢ ∀ x ∈ antidiagonal n,\n x ∉ {ij... | [] | aesop (erase simp not_and) (add simp not_and_or) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 311,
"column": 55
} | {
"line": 311,
"column": 70
} | {
"line": 311,
"column": 70
} | [
{
"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 y' : M\n⊢ polar (⇑Q) x y + polar (⇑Q) x (-y') = polar (⇑Q) x y + -polar (⇑Q) x y'",
"ppTerm": "?m.57",
"assigned"... | [
"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 y' : M\n⊢ polar (⇑Q) x y + -polar (⇑Q) x y' = polar (⇑Q) x y + -polar (⇑Q) x y'"
] | polar_neg_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 366,
"column": 4
} | {
"line": 374,
"column": 49
} | {
"line": 376,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nι : Type u_8\ninst✝ : DecidableEq ι\nQ : QuadraticMap R M N\nf : ι → M\na : ι\ns : Finset ι\nha : a ∉ s\nih : Q (∑ i ∈ s, f i) = ∑ i ∈ s,... | [] | simp_rw [Finset.sum_cons, QuadraticMap.map_add, ih, add_assoc, Finset.sym2_cons,
Finset.sum_filter, Finset.sum_disjUnion, Finset.sum_map, Finset.sum_cons,
Sym2.mkEmbedding_apply, Sym2.mk_isDiag_iff, not_true, if_false, zero_add,
Sym2.map_mk, polarSym2_sym2Mk, ← polarBilin_apply_apply, _root_.map_sum,
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 366,
"column": 4
} | {
"line": 374,
"column": 49
} | {
"line": 376,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nι : Type u_8\ninst✝ : DecidableEq ι\nQ : QuadraticMap R M N\nf : ι → M\na : ι\ns : Finset ι\nha : a ∉ s\nih : Q (∑ i ∈ s, f i) = ∑ i ∈ s,... | [] | simp_rw [Finset.sum_cons, QuadraticMap.map_add, ih, add_assoc, Finset.sym2_cons,
Finset.sum_filter, Finset.sum_disjUnion, Finset.sum_map, Finset.sum_cons,
Sym2.mkEmbedding_apply, Sym2.mk_isDiag_iff, not_true, if_false, zero_add,
Sym2.map_mk, polarSym2_sym2Mk, ← polarBilin_apply_apply, _root_.map_sum,
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 802,
"column": 89
} | {
"line": 804,
"column": 71
} | {
"line": 806,
"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\nB : BilinMap R M N\nx y : M\n⊢ polar (⇑B.toQuadraticMap) x y = (B x) y + (B y) x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": ... | [] | by
simp only [polar, toQuadraticMap_apply, map_add, add_apply, add_assoc, add_comm (B y x) _,
add_sub_cancel_left, sub_eq_add_neg _ (B y y), add_neg_cancel_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 294,
"column": 10
} | {
"line": 294,
"column": 20
} | {
"line": 295,
"column": 4
} | [
{
"pp": "R✝ : Type u1\ninst✝⁵ : CommRing R✝\nM✝ : Type u2\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nhn : ∀ (f : Fin n → M) (x y : Fin n), f x = f y → x < y → (List.ofFn fun i ↦ (ι R) (f i)).prod = 0\nf : F... | [] | exact hfxy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Localization | {
"line": 100,
"column": 79
} | {
"line": 102,
"column": 14
} | {
"line": 104,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\n⊢ Limits.PreservesFiniteColimits (localizedModuleFunctor S)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Limits.PreservesFiniteLimits",
... | [] | by
have := ((Functor.exact_tfae _).out 1 3).mp (ModuleCat.localizedModuleFunctor_map_exact S)
exact this.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 44
} | {
"line": 356,
"column": 2
} | [
{
"pp": "case h\nR : 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\nα_inj : Injective α\nα_card : (Finset.image α Finset.univ).card = n\n⊢ (ExteriorAlgebra.ιMulti R n)\n ((v ∘ α) ∘\n ⇑(((F... | [
"case h\nR : 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\nα_inj : Injective α\nα_card : (Finset.image α Finset.univ).card = n\n⊢ ExteriorAlgebra.ιMulti_family R n v ⟨Finset.image α Finset.univ, α_card⟩ ... | use ⟨(Finset.image α Finset.univ), α_card⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Products.Bifunctor | {
"line": 46,
"column": 37
} | {
"line": 46,
"column": 54
} | {
"line": 46,
"column": 55
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map ((𝟙 X ×ₘ g).1 ≫ (f ×ₘ 𝟙 Y').1 ×ₘ (𝟙 X ×ₘ g).2 ≫ (f ×ₘ 𝟙 Y').2) = F.map (f ×ₘ g)",
"ppTerm": "?m.83",... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map ((f ×ₘ 𝟙 Y').1 ×ₘ (𝟙 X ×ₘ g).2 ≫ (f ×ₘ 𝟙 Y').2) = F.map (f ×ₘ g)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Products.Bifunctor | {
"line": 51,
"column": 37
} | {
"line": 51,
"column": 54
} | {
"line": 51,
"column": 55
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map ((f ×ₘ 𝟙 Y).1 ≫ (𝟙 X' ×ₘ g).1 ×ₘ (f ×ₘ 𝟙 Y).2 ≫ (𝟙 X' ×ₘ g).2) = F.map (f ×ₘ g)",
"ppTerm": "?m.83",... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : Category.{v₃, u₃} E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map ((f ×ₘ 𝟙 Y).1 ≫ (𝟙 X' ×ₘ g).1 ×ₘ (𝟙 X' ×ₘ g).2) = F.map (f ×ₘ g)"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Generator | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 17
} | {
"line": 87,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nR : Cᵒᵖ ⥤ RingCat\nM N : PresheafOfModules R\nf₁ : M ⟶ N\n⊢ ∀ (g : M ⟶ N), (∀ (G : PresheafOfModules R), freeYoneda R G → ∀ (h : G ⟶ M), h ≫ f₁ = h ≫ g) → f₁ = g",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategorySt... | [
"C : Type u\ninst✝ : Category.{v, u} C\nR : Cᵒᵖ ⥤ RingCat\nM N : PresheafOfModules R\nf₁ f₂ : M ⟶ N\n⊢ (∀ (G : PresheafOfModules R), freeYoneda R G → ∀ (h : G ⟶ M), h ≫ f₁ = h ≫ f₂) → f₁ = f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Adjunction.PartialAdjoint | {
"line": 141,
"column": 22
} | {
"line": 141,
"column": 62
} | {
"line": 141,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nh : ∀ (X : C), (F ⋙ coyoneda.obj (op X)).IsCorepresentable\nX : C\nY Y' : D\ng : Y ⟶ Y'\nf : (F ⋙ coyoneda.obj (op X)).coreprX ⟶ Y\n⊢ (F ⋙ coyoneda.obj (op X)).corepresentableBy.homEquiv (f ≫ g) =\n (F ⋙ ... | [] | by apply CorepresentableBy.homEquiv_comp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal | {
"line": 52,
"column": 35
} | {
"line": 52,
"column": 46
} | {
"line": 52,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nR : Cᵒᵖ ⥤ CommRingCat\nM₁ M₂ M₃ M₄ : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nX Y : Cᵒᵖ\nf : X ⟶ Y\na : ↑((R ⋙ forget₂ CommRingCat RingCat).obj X)\nm₁ : ↑(M₁.obj X)\nm₂ : ↑(M₂.obj X)\n⊢ (ConcreteCategory.hom (M₁.map f)) (a • m₁) ⊗ₜ[↑(R.obj Y)] (C... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nR : Cᵒᵖ ⥤ CommRingCat\nM₁ M₂ M₃ M₄ : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nX Y : Cᵒᵖ\nf : X ⟶ Y\na : ↑((R ⋙ forget₂ CommRingCat RingCat).obj X)\nm₁ : ↑(M₁.obj X)\nm₂ : ↑(M₂.obj X)\n⊢ ((ConcreteCategory.hom ((R ⋙ forget₂ CommRingCat RingCat).map f)) a •\n ... | M₁.map_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.CompositionIso | {
"line": 180,
"column": 2
} | {
"line": 182,
"column": 40
} | {
"line": 184,
"column": 0
} | [
{
"pp": "C₀ : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₀\ninst✝² : Category.{v_2, u_2} C₁\ninst✝¹ : Category.{v_3, u_3} C₂\ninst✝ : Category.{v_4, u_4} C₃\nF₀₁ : C₀ ⥤ C₁\nF₁₂ : C₁ ⥤ C₂\nF₂₃ : C₂ ⥤ C₃\nF₀₂ : C₀ ⥤ C₂\nF₁₃ : C₁ ⥤ C₃\nF₀₃ : C₀ ⥤ C₃\nG₁₀ : C₁ ⥤ C₀\nG₂₁ : C... | [] | ext : 1
exact leftAdjointCompNatTrans_assoc _ _ _ _ _ _ _ _ _ _
(by simpa using congr_arg Iso.inv h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.CompositionIso | {
"line": 180,
"column": 2
} | {
"line": 182,
"column": 40
} | {
"line": 184,
"column": 0
} | [
{
"pp": "C₀ : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₀\ninst✝² : Category.{v_2, u_2} C₁\ninst✝¹ : Category.{v_3, u_3} C₂\ninst✝ : Category.{v_4, u_4} C₃\nF₀₁ : C₀ ⥤ C₁\nF₁₂ : C₁ ⥤ C₂\nF₂₃ : C₂ ⥤ C₃\nF₀₂ : C₀ ⥤ C₂\nF₁₃ : C₁ ⥤ C₃\nF₀₃ : C₀ ⥤ C₃\nG₁₀ : C₁ ⥤ C₀\nG₂₁ : C... | [] | ext : 1
exact leftAdjointCompNatTrans_assoc _ _ _ _ _ _ _ _ _ _
(by simpa using congr_arg Iso.inv h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Precoverage | {
"line": 175,
"column": 6
} | {
"line": 177,
"column": 69
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : Precoverage C\ninst✝ : J.IsStableUnderComposition\nι : Type w\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nhf : Presieve.ofArrows X f ∈ J.coverings S\nσ : ι → Type w'\nY : (i : ι) → σ i → C\ng : (i : ι) → (j : σ i) → Y i j ⟶ X i\nhg : ∀ (i : ι), P... | [] | convert! hf
refine le_antisymm (fun T u ⟨p⟩ ↦ .mk _) fun T u hu ↦ ?_
exact .mk' (Sum.inl ⟨⟨_, _⟩, hu⟩) (by cat_disch) (by cat_disch) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Precoverage | {
"line": 175,
"column": 6
} | {
"line": 177,
"column": 69
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : Precoverage C\ninst✝ : J.IsStableUnderComposition\nι : Type w\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nhf : Presieve.ofArrows X f ∈ J.coverings S\nσ : ι → Type w'\nY : (i : ι) → σ i → C\ng : (i : ι) → (j : σ i) → Y i j ⟶ X i\nhg : ∀ (i : ι), P... | [] | convert! hf
refine le_antisymm (fun T u ⟨p⟩ ↦ .mk _) fun T u hu ↦ ?_
exact .mk' (Sum.inl ⟨⟨_, _⟩, hu⟩) (by cat_disch) (by cat_disch) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 69
} | {
"line": 321,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ pushforward g (singleton f) = singleton (f ≫ g)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Presieve",
"congrArg",
"id",
"PUnit",
"Categor... | [] | rw [← ofArrows_pUnit.{0}, pushforward_ofArrows, ofArrows_pUnit.{0}] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 69
} | {
"line": 321,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ pushforward g (singleton f) = singleton (f ≫ g)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Presieve",
"congrArg",
"id",
"PUnit",
"Categor... | [] | rw [← ofArrows_pUnit.{0}, pushforward_ofArrows, ofArrows_pUnit.{0}] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 69
} | {
"line": 321,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ pushforward g (singleton f) = singleton (f ≫ g)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Presieve",
"congrArg",
"id",
"PUnit",
"Categor... | [] | rw [← ofArrows_pUnit.{0}, pushforward_ofArrows, ofArrows_pUnit.{0}] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.SheafOfTypes | {
"line": 208,
"column": 25
} | {
"line": 208,
"column": 27
} | {
"line": 208,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\ns : Cocone S.arrows.diagram\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\n⊢ ∀ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : S.arrows f₁) (h₂ : S.arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n (ConcreteCategory.hom ((yoneda.obj s.pt).map g₁.op)) (S.arrows.yonedaFamil... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\ns : Cocone S.arrows.diagram\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\n⊢ ∀ (h₁ : S.arrows f₁) (h₂ : S.arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n (ConcreteCategory.hom ((yoneda.obj s.pt).map g₁.op)) (S.arrows.yonedaFamilyOfElements_f... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 185,
"column": 21
} | {
"line": 185,
"column": 23
} | {
"line": 185,
"column": 24
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\ninst✝ : R.HasPairwisePullbacks\nt : x.Compatible\nY₁ Y₂ : C\nf₁ : Y₁ ⟶ X\n⊢ ∀ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂),\n (ConcreteCategory.hom (P.map (pullback.fst f₁ f₂).op)) (x f₁ h... | [
"case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\ninst✝ : R.HasPairwisePullbacks\nt : x.Compatible\nY₁ Y₂ : C\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\n⊢ ∀ (h₁ : R f₁) (h₂ : R f₂),\n (ConcreteCategory.hom (P.map (pullback.fst f₁ f₂).op)) (x f₁ h₁) =\n (... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 189,
"column": 29
} | {
"line": 189,
"column": 31
} | {
"line": 189,
"column": 32
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\ninst✝ : R.HasPairwisePullbacks\nt : x.PullbackCompatible\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\n⊢ ∀ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (Conc... | [
"case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\ninst✝ : R.HasPairwisePullbacks\nt : x.PullbackCompatible\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\n⊢ ∀ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 234,
"column": 23
} | {
"line": 234,
"column": 25
} | {
"line": 234,
"column": 26
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP Q : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nf : P ⟶ Q\ninst✝ : Mono f\nx : FamilyOfElements P R\nhx : (x.map f).Compatible\nY Z W : C\ng₁ : W ⟶ Y\ng₂ : W ⟶ Z\nf₁ : Y ⟶ X\n⊢ ∀ ⦃f₂ : Z ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.hom (... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP Q : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nf : P ⟶ Q\ninst✝ : Mono f\nx : FamilyOfElements P R\nhx : (x.map f).Compatible\nY Z W : C\ng₁ : W ⟶ Y\ng₂ : W ⟶ Z\nf₁ : Y ⟶ X\nf₂ : Z ⟶ X\n⊢ ∀ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.hom (P.map g₁.op))... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 255,
"column": 29
} | {
"line": 255,
"column": 31
} | {
"line": 255,
"column": 32
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nS : Sieve X\nx : FamilyOfElements P S.arrows\nh : x.SieveCompatible\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\n⊢ ∀ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : S.arrows f₁) (h₂ : S.arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.hom (... | [
"case mpr\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nS : Sieve X\nx : FamilyOfElements P S.arrows\nh : x.SieveCompatible\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\n⊢ ∀ (h₁ : S.arrows f₁) (h₂ : S.arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.hom (P.map g₁.op))... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 325,
"column": 25
} | {
"line": 325,
"column": 27
} | {
"line": 325,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nZ : D\nT : Presieve (F.obj Z)\nx : FamilyOfElements P T\nh : x.Compatible\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₁ : Z₁ ⟶ Z\n⊢ ∀ ⦃f₂ : Z₂ ⟶ Z⦄ (h₁ : Presieve.functorPullback F T f₁) (h₂ :... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nZ : D\nT : Presieve (F.obj Z)\nx : FamilyOfElements P T\nh : x.Compatible\nZ₁ Z₂ W : D\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₁ : Z₁ ⟶ Z\nf₂ : Z₂ ⟶ Z\n⊢ ∀ (h₁ : Presieve.functorPullback F T f₁) (h₂ : Presieve.fun... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.EqualizerSheafCondition | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 37
} | {
"line": 134,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type (max v u)\nX : C\nS : Sieve X\n⊢ forkMap P S.arrows ≫ firstMap P S = forkMap P S.arrows ≫ secondMap P S",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Opposite",
"CategoryThe... | [] | ext
simp [firstMap, secondMap, forkMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.EqualizerSheafCondition | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 37
} | {
"line": 134,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type (max v u)\nX : C\nS : Sieve X\n⊢ forkMap P S.arrows ≫ firstMap P S = forkMap P S.arrows ≫ secondMap P S",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Opposite",
"CategoryThe... | [] | ext
simp [firstMap, secondMap, forkMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 351,
"column": 15
} | {
"line": 351,
"column": 17
} | {
"line": 351,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nf : Y ⟶ X\nx : FamilyOfElements P S.arrows\nh : x.SieveCompatible\nW Z : C\nf₁ : W ⟶ Y\n⊢ ∀ (g : Z ⟶ W) (hf : (Sieve.pullback f S).arrows f₁),\n FamilyOfElements.pullback f x (g ≫ f₁) ⋯ = (ConcreteCategory.hom (P.map g... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nf : Y ⟶ X\nx : FamilyOfElements P S.arrows\nh : x.SieveCompatible\nW Z : C\nf₁ : W ⟶ Y\nf₂ : Z ⟶ W\n⊢ ∀ (hf : (Sieve.pullback f S).arrows f₁),\n FamilyOfElements.pullback f x (f₂ ≫ f₁) ⋯ =\n (ConcreteCategory.hom (P.map f₂.op... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 371,
"column": 25
} | {
"line": 371,
"column": 27
} | {
"line": 371,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP Q : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nf : P ⟶ Q\nx : FamilyOfElements P R\nh : x.Compatible\nZ₁ Z₂ W : C\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₁ : Z₁ ⟶ X\n⊢ ∀ ⦃f₂ : Z₂ ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n (ConcreteCategory.hom (Q.map g₁.op))... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP Q : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nf : P ⟶ Q\nx : FamilyOfElements P R\nh : x.Compatible\nZ₁ Z₂ W : C\ng₁ : W ⟶ Z₁\ng₂ : W ⟶ Z₂\nf₁ : Z₁ ⟶ X\nf₂ : Z₂ ⟶ X\n⊢ ∀ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n (ConcreteCategory.hom (Q.map g₁.op)) (x.map f f₁ ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 396,
"column": 25
} | {
"line": 396,
"column": 27
} | {
"line": 396,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : x.IsAmalgamation t\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\n⊢ ∀ ⦃f₂ : Y₂ ⟶ X⦄ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.hom (P.map g₁... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\nt : P.obj (op X)\nht : x.IsAmalgamation t\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\n⊢ ∀ (h₁ : R f₁) (h₂ : R f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ → (ConcreteCategory.hom (P.map g₁.op)) (x f₁ h... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 174,
"column": 13
} | {
"line": 174,
"column": 15
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝ : Category.{v₂, u₂} A\nP : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nh : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows\nE : A\nπ : (Functor.const S.arrows.categoryᵒᵖ).obj E ⟶ S.arrows.diagram.op ⋙ P\neqv : (S.arrows.diagram.op ⋙ P).... | [
"case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝ : Category.{v₂, u₂} A\nP : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nh : ∀ (E : Aᵒᵖ), IsSeparatedFor (P ⋙ coyoneda.obj E) S.arrows\nE : A\nπ : (Functor.const S.arrows.categoryᵒᵖ).obj E ⟶ S.arrows.diagram.op ⋙ P\neqv : (S.arrows.diagram.op ⋙ P).cones.obj (o... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 402,
"column": 29
} | {
"line": 402,
"column": 31
} | {
"line": 402,
"column": 32
} | [
{
"pp": "case mpr.refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S ∈ J Y\nz : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op X)) S.arrows\nhz : z.Compatible\nx : unop (op X)\nY₁ Y₂ Z : C\ng₁ : Z ⟶... | [
"case mpr.refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S ∈ J Y\nz : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op X)) S.arrows\nhz : z.Compatible\nx : unop (op X)\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 58,
"column": 61
} | {
"line": 58,
"column": 73
} | {
"line": 58,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category.{w', w} D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\nX Y : C\nf✝ : X ⟶ Y\nS T : (J.Cover Y)ᵒᵖ\nf : S ⟶ T\nI : (unop ((J.pullback f✝).op.obj T)).shape.L\n⊢... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 919,
"column": 8
} | {
"line": 919,
"column": 51
} | {
"line": 920,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nI : Type u_1\nB B' : C\np : B ⟶ B'\nP : (Over B')ᵒᵖ ⥤ Type w\nX : Over B\nY : I → Over B\nf : (i : I) → Y i ⟶ X\ns : Subtype (Arrows.Compatible ((Over.map p).op ⋙ P) f)\ni₁ i₂ : I\nZ : Over B'\ng₁ : Z ⟶ (fun i ↦ (Over.map p).obj (Y i)) i₁\ng₂ : Z ⟶ (fun i ↦ (Ov... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nI : Type u_1\nB B' : C\np : B ⟶ B'\nP : (Over B')ᵒᵖ ⥤ Type w\nX : Over B\nY : I → Over B\nf : (i : I) → Y i ⟶ X\ns : Subtype (Arrows.Compatible ((Over.map p).op ⋙ P) f)\ni₁ i₂ : I\nZ : Over B'\ng₁ : Z ⟶ (fun i ↦ (Over.map p).obj (Y i)) i₁\ng₂ : Z ⟶ (fun i ↦ (Over.map p).ob... | replace this := congr_arg (P.map φ.op) this | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.CategoryTheory.Subfunctor.Sieves | {
"line": 45,
"column": 25
} | {
"line": 45,
"column": 27
} | {
"line": 45,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nU : Cᵒᵖ\ns : F.obj U\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ unop U\n⊢ ∀ ⦃f₂ : Y₂ ⟶ unop U⦄ (h₁ : (G.sieveOfSection s).arrows f₁) (h₂ : (G.sieveOfSection s).arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n (ConcreteCategory... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nU : Cᵒᵖ\ns : F.obj U\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ unop U\nf₂ : Y₂ ⟶ unop U\n⊢ ∀ (h₁ : (G.sieveOfSection s).arrows f₁) (h₂ : (G.sieveOfSection s).arrows f₂),\n g₁ ≫ f₁ = g₂ ≫ f₂ →\n (ConcreteCategory.hom (G.toFun... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 1010,
"column": 33
} | {
"line": 1010,
"column": 35
} | {
"line": 1010,
"column": 36
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type u_1\nU : Sieve X\nB : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y\nhU : IsSheafFor P U.arrows\nhB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows\nhB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsS... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX : C\nP : Cᵒᵖ ⥤ Type u_1\nU : Sieve X\nB : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y\nhU : IsSheafFor P U.arrows\nhB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), IsSheafFor P (B hf).arrows\nhB' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), IsSeparatedFor ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Whiskering | {
"line": 143,
"column": 18
} | {
"line": 149,
"column": 74
} | {
"line": 151,
"column": 0
} | [
{
"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... | [] | by
rw [Presheaf.isSheaf_iff_multifork] at hP ⊢
intro X S
obtain ⟨h⟩ := hP X S
replace h := isLimitOfPreserves F h
replace h := Limits.IsLimit.ofIsoLimit h (S.mapMultifork F P)
exact ⟨Limits.IsLimit.postcomposeHomEquiv (S.multicospanComp F P) _ h⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Sheafification | {
"line": 232,
"column": 16
} | {
"line": 234,
"column": 40
} | {
"line": 235,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP : Sheaf J D\n⊢ { hom := (isoSheafify J ⋯).hom } ≫ { hom := (isoSheafify J ⋯).inv } = 𝟙 P",
"ppTerm": "?m.65... | [] | by
ext1
apply (isoSheafify J P.2).hom_inv_id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Limits | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 23
} | {
"line": 207,
"column": 24
} | [
{
"pp": "C : 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✝ : HasWeakSheafify J D\nF : K ⥤ Sheaf J D\nE : Cocone (F ⋙ sheafToPresheaf J D)\nk : K\n⊢ 𝟙 ((presheafToSheaf J D).obj (F.obj k).obj).obj ≫ ((presh... | [
"C : 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✝ : HasWeakSheafify J D\nF : K ⥤ Sheaf J D\nE : Cocone (F ⋙ sheafToPresheaf J D)\nk : K\n⊢ ((presheafToSheaf J D).map (E.ι.app k)).hom =\n ((sheafificationAdj... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Limits | {
"line": 248,
"column": 31
} | {
"line": 248,
"column": 60
} | {
"line": 249,
"column": 10
} | [
{
"pp": "C : 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✝ : HasWeakSheafify J D\nF : K ⥤ Sheaf J D\nh : ∀ (c : Cocone (F ⋙ sheafToPresheaf J D)) (x : IsColimit c), Presheaf.IsSheaf J c.pt\nE : Cocone (F ⋙ ... | [] | ext1; dsimp; rw [hE.fac]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Limits | {
"line": 248,
"column": 31
} | {
"line": 248,
"column": 60
} | {
"line": 249,
"column": 10
} | [
{
"pp": "C : 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✝ : HasWeakSheafify J D\nF : K ⥤ Sheaf J D\nh : ∀ (c : Cocone (F ⋙ sheafToPresheaf J D)) (x : IsColimit c), Presheaf.IsSheaf J c.pt\nE : Cocone (F ⋙ ... | [] | ext1; dsimp; rw [hE.fac]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 51,
"column": 8
} | {
"line": 52,
"column": 26
} | {
"line": 52,
"column": 27
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\nX Y : J\nf : X ⟶ Y\n⊢ ((Functor.const J).obj (of ↑{u | ∀ {i j : J} (f : i ⟶ j), (ConcreteCategory.hom (F.map f)) (u i) = u j})).map f ≫\n ofHom { toFun := fun u ↦ ↑u Y, continuous_toFun := ⋯ } =\n ofHom { toFun := fun u ↦ ↑u X, continuous_t... | [] | ext a
exact (a.2 f).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 51,
"column": 8
} | {
"line": 52,
"column": 26
} | {
"line": 52,
"column": 27
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\nX Y : J\nf : X ⟶ Y\n⊢ ((Functor.const J).obj (of ↑{u | ∀ {i j : J} (f : i ⟶ j), (ConcreteCategory.hom (F.map f)) (u i) = u j})).map f ≫\n ofHom { toFun := fun u ↦ ↑u Y, continuous_toFun := ⋯ } =\n ofHom { toFun := fun u ↦ ↑u X, continuous_t... | [] | ext a
exact (a.2 f).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 56
} | {
"line": 166,
"column": 0
} | [
{
"pp": "case mpr\nJ : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\na✝ : HasLimit (F ⋙ forget)\n⊢ HasLimit F",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"TopCat.instCategory",
"ContinuousMap",
"CategoryTheory.Functor.comp",
"CategoryTheory.Limits.LimitCone"... | [] | exact ⟨⟨_, isLimitConeOfForget _ (limit.isLimit _)⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 56
} | {
"line": 166,
"column": 0
} | [
{
"pp": "case mpr\nJ : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\na✝ : HasLimit (F ⋙ forget)\n⊢ HasLimit F",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"TopCat.instCategory",
"ContinuousMap",
"CategoryTheory.Functor.comp",
"CategoryTheory.Limits.LimitCone"... | [] | exact ⟨⟨_, isLimitConeOfForget _ (limit.isLimit _)⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 56
} | {
"line": 166,
"column": 0
} | [
{
"pp": "case mpr\nJ : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\na✝ : HasLimit (F ⋙ forget)\n⊢ HasLimit F",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"TopCat.instCategory",
"ContinuousMap",
"CategoryTheory.Functor.comp",
"CategoryTheory.Limits.LimitCone"... | [] | exact ⟨⟨_, isLimitConeOfForget _ (limit.isLimit _)⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 39
} | {
"line": 160,
"column": 2
} | [
{
"pp": "X Y : TopCat\n⊢ (X.prodIsoProd Y).hom ≫ prodFst = prod.fst",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [
"X Y : TopCat\n⊢ prodFst = (X.prodBinaryFan Y).π.app { as := WalkingPair.left }"
] | simp [← Iso.eq_inv_comp, prodIsoProd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 39
} | {
"line": 167,
"column": 2
} | [
{
"pp": "X Y : TopCat\n⊢ (X.prodIsoProd Y).hom ≫ prodSnd = prod.snd",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | [
"X Y : TopCat\n⊢ prodSnd = (X.prodBinaryFan Y).π.app { as := WalkingPair.right }"
] | simp [← Iso.eq_inv_comp, prodIsoProd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered | {
"line": 78,
"column": 42
} | {
"line": 78,
"column": 59
} | {
"line": 78,
"column": 60
} | [
{
"pp": "C : Type u_3\nι : Type u_5\ninst✝ : Category.{v_2, u_3} C\nF G : Discrete ι ⥤ C\nα : F ⟶ G\ni : ι\n⊢ 𝟙 (F.obj { as := i }) ≫ α.app { as := i } = α.app { as := i } ≫ 𝟙 (G.obj { as := i })",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Catego... | [
"C : Type u_3\nι : Type u_5\ninst✝ : Category.{v_2, u_3} C\nF G : Discrete ι ⥤ C\nα : F ⟶ G\ni : ι\n⊢ α.app { as := i } = α.app { as := i } ≫ 𝟙 (G.obj { as := i })"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered | {
"line": 125,
"column": 42
} | {
"line": 125,
"column": 59
} | {
"line": 125,
"column": 60
} | [
{
"pp": "C : Type u_3\ninst✝ : Category.{v_2, u_3} C\nι : Type u_6\nF G : Discrete ι ⥤ C\nα : F ⟶ G\ni : ι\n⊢ 𝟙 (F.obj { as := i }) ≫ α.app { as := i } = α.app { as := i } ≫ 𝟙 (G.obj { as := i })",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Catego... | [
"C : Type u_3\ninst✝ : Category.{v_2, u_3} C\nι : Type u_6\nF G : Discrete ι ⥤ C\nα : F ⟶ G\ni : ι\n⊢ α.app { as := i } = α.app { as := i } ≫ 𝟙 (G.obj { as := i })"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 257,
"column": 10
} | {
"line": 258,
"column": 92
} | {
"line": 259,
"column": 8
} | [
{
"pp": "case mp\nX Y : TopCat\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl) ∧\n IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr) ∧\n IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))",
"ppTerm": "?mp",
"assigned": true,
... | [
"case mp\nX Y : TopCat\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ IsOpenEmbedding\n ⇑(ConcreteCategory.hom\n ((X.binaryCofan Y).ι.app { as := WalkingPair.left } ≫\n (h.coconePointUniqueUpToIso (X.binaryCofanIsColimit Y)).inv)) ∧\n IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr) ∧\n IsCo... | ← show _ = c.inl from
h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.left⟩, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Extensive | {
"line": 219,
"column": 6
} | {
"line": 234,
"column": 65
} | {
"line": 235,
"column": 4
} | [
{
"pp": "case h₁.left\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\nX'✝ Y'✝ : Type u\nαX✝ : X'✝ ⟶ PUnit.{u + 1}\nαY✝ : Y'✝ ⟶ PUnit.{u + 1}\nf : (Types.binaryCoproductCocone X'✝ Y'✝).pt ⟶ (Types.binaryCoproductCocone PUni... | [] | refine ⟨⟨hαX.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
intro s
have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
intro x
rcases h : s.fst x with val | val
· simp
· apply_fun f at h
cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
(ConcreteCa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Extensive | {
"line": 219,
"column": 6
} | {
"line": 234,
"column": 65
} | {
"line": 235,
"column": 4
} | [
{
"pp": "case h₁.left\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\nX'✝ Y'✝ : Type u\nαX✝ : X'✝ ⟶ PUnit.{u + 1}\nαY✝ : Y'✝ ⟶ PUnit.{u + 1}\nf : (Types.binaryCoproductCocone X'✝ Y'✝).pt ⟶ (Types.binaryCoproductCocone PUni... | [] | refine ⟨⟨hαX.symm⟩, ⟨PullbackCone.isLimitAux' _ ?_⟩⟩
intro s
have : ∀ x, ∃! y, s.fst x = Sum.inl y := by
intro x
rcases h : s.fst x with val | val
· simp
· apply_fun f at h
cases ((ConcreteCategory.congr_hom s.condition x).symm.trans h).trans
(ConcreteCa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Extensive | {
"line": 305,
"column": 4
} | {
"line": 306,
"column": 41
} | {
"line": 307,
"column": 2
} | [
{
"pp": "case refine_1\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 : TopCat\nf : Z ⟶ TopCat.of (PUnit.{u + 1} ⊕ PUnit.{u + 1})\nh₁ :\n Set.range\n ⇑(ConcreteCategory.hom\n (TopCat.pullbackFst f ((TopCa... | [] | refine ⟨(Homeomorph.prodPUnit Z).isEmbedding.comp .subtypeVal, ?_⟩
convert! f.hom.2.1 _ isOpen_range_inl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Extensive | {
"line": 305,
"column": 4
} | {
"line": 306,
"column": 41
} | {
"line": 307,
"column": 2
} | [
{
"pp": "case refine_1\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 : TopCat\nf : Z ⟶ TopCat.of (PUnit.{u + 1} ⊕ PUnit.{u + 1})\nh₁ :\n Set.range\n ⇑(ConcreteCategory.hom\n (TopCat.pullbackFst f ((TopCa... | [] | refine ⟨(Homeomorph.prodPUnit Z).isEmbedding.comp .subtypeVal, ?_⟩
convert! f.hom.2.1 _ isOpen_range_inl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Extensive | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 77
} | {
"line": 552,
"column": 2
} | [
{
"pp": "J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nX Z : C\nα : Type u_1\nf : X ⟶ Z\nY : α → C\ni : (a : α) → Y a ⟶ Z\ninst✝ : Finite α\nhi : IsIso (Sigma.desc i)\na : α\n⊢ HasPullb... | [
"J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nX Z : C\nα : Type u_1\nf : X ⟶ Z\nY : α → C\ni : (a : α) → Y a ⟶ Z\ninst✝ : Finite α\nhi : IsIso (Sigma.desc i)\na : α\n⊢ IsColimit (Cofan.mk ... | apply FinitaryPreExtensive.hasPullbacks_of_is_coproduct (c := Cofan.mk Z i) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.LeftExact | {
"line": 50,
"column": 12
} | {
"line": 50,
"column": 29
} | {
"line": 50,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{t, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nX : C\nK : Type s\ninst✝ : SmallCategory K\nF : K ⥤ Cᵒᵖ ⥤ D\nW : J.Cover X\ni : W.Arrow\nE : Cone (F ⋙ J.diagramFu... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{t, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nX : C\nK : Type s\ninst✝ : SmallCategory K\nF : K ⥤ Cᵒᵖ ⥤ D\nW : J.Cover X\ni : W.Arrow\nE : Cone (F ⋙ J.diagramFunctor D X ⋙ ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 407,
"column": 2
} | {
"line": 409,
"column": 47
} | {
"line": 410,
"column": 2
} | [
{
"pp": "J : Type v'\ninst✝⁸ : Category.{u', v'} J\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u_2\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasColimitsOfShape J C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsVanKampenColimit c\ninst... | [
"case refine_2\nJ : Type v'\ninst✝⁸ : Category.{u', v'} J\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u_2\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasColimitsOfShape J C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsVanKampenColimit c\ni... | have := ((H (colimit.cocone <| F' ⋙ Gr) (whiskerRight α' Gr)
(colimit.desc _ ⟨_, whiskerRight α' Gr ≫ c.2⟩) ?_ (hα'.whiskerRight Gr)).mp
⟨(getColimitCocone <| F' ⋙ Gr).2⟩ j).map Gl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 154,
"column": 15
} | {
"line": 154,
"column": 17
} | {
"line": 154,
"column": 18
} | [
{
"pp": "case mpr.left\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nY Z : C\ng : Y ⟶ Z\nhg : P.isColocal g\nX X' : C\nhX' : P X'\ne : X ≅ X'\nf₁ : X ⟶ Y\n⊢ ∀ ⦃a₂ : X ⟶ Y⦄, (fun f ↦ f ≫ g) f₁ = (fun f ↦ f ≫ g) a₂ → f₁ = a₂",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstan... | [
"case mpr.left\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nY Z : C\ng : Y ⟶ Z\nhg : P.isColocal g\nX X' : C\nhX' : P X'\ne : X ≅ X'\nf₁ f₂ : X ⟶ Y\n⊢ (fun f ↦ f ≫ g) f₁ = (fun f ↦ f ≫ g) f₂ → f₁ = f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.LocallyBijective | {
"line": 59,
"column": 29
} | {
"line": 59,
"column": 31
} | {
"line": 59,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type w)\nf : F ⟶ G\nh₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.hom.app X))\nright✝ : IsLocallySurjective f\nX : Cᵒᵖ\ns : G.obj.obj X\nH : Presieve.IsSheafFor F.obj (Presheaf.imageSieve f.hom s).arrows\... | [
"C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Sheaf J (Type w)\nf : F ⟶ G\nh₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.hom.app X))\nright✝ : IsLocallySurjective f\nX : Cᵒᵖ\ns : G.obj.obj X\nH : Presieve.IsSheafFor F.obj (Presheaf.imageSieve f.hom s).arrows\nt : Presiev... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 281,
"column": 40
} | {
"line": 281,
"column": 57
} | {
"line": 281,
"column": 58
} | [
{
"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\nF : Cᵒᵖ ⥤ Type (max u v)\n⊢ ((Subfunctor.sheafify J (Subfunctor.ran... | [
"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\nF : Cᵒᵖ ⥤ Type (max u v)\n⊢ ((Subfunctor.sheafify J (Subfunctor.range (J.toShea... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf | {
"line": 240,
"column": 20
} | {
"line": 240,
"column": 31
} | {
"line": 240,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR✝ : Sheaf J RingCat\nR : Cᵒᵖ ⥤ RingCat\nM₁ M₂ : PresheafOfModules R\nf : M₁ ⟶ M₂\nN : PresheafOfModules R\nhN : Presheaf.IsSheaf J N.presheaf\ninst✝² : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝¹ : IsLocallySurjective J f\ninst... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR✝ : Sheaf J RingCat\nR : Cᵒᵖ ⥤ RingCat\nM₁ M₂ : PresheafOfModules R\nf : M₁ ⟶ M₂\nN : PresheafOfModules R\nhN : Presheaf.IsSheaf J N.presheaf\ninst✝² : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝¹ : IsLocallySurjective J f\ninst✝ : IsLocall... | N.map_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 459,
"column": 6
} | {
"line": 459,
"column": 74
} | {
"line": 459,
"column": 74
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : LocallySmall.{w, v, u} C\nS : C\nι : Type u_2\ninst✝ : Small.{w, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\nc : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\nhf : Presheaf.IsLocallySurjective ... | [
"case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : LocallySmall.{w, v, u} C\nS : C\nι : Type u_2\ninst✝ : Small.{w, u_2} ι\nX : ι → C\nf : (i : ι) → X i ⟶ S\nc : Cofan fun i ↦ shrinkYoneda.{w, v, u}.obj (X i)\nhc : IsColimit c\nhf : Presheaf.IsLocallySurjective J (Cofan.IsC... | ← Presheaf.imageSieve_cofanIsColimitDesc_shrinkYoneda_map f hc (𝟙 S) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 647,
"column": 17
} | {
"line": 647,
"column": 19
} | {
"line": 648,
"column": 6
} | [
{
"pp": "case refine_2.hd₁\nC : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nj : Fin (n +... | [
"case refine_2.hd₁\nC : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nj : Fin (n + 1)\nF' : Fi... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 651,
"column": 17
} | {
"line": 651,
"column": 19
} | {
"line": 652,
"column": 6
} | [
{
"pp": "case refine_2.hd₂\nC : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nj : Fin (n +... | [
"case refine_2.hd₂\nC : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nj : Fin (n + 1)\nF' : Fi... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 657,
"column": 17
} | {
"line": 657,
"column": 19
} | {
"line": 657,
"column": 20
} | [
{
"pp": "case refine_2.uniq\nC : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nj : Fin (n ... | [
"case refine_2.uniq\nC : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nj : Fin (n + 1)\nF' : F... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 723,
"column": 2
} | {
"line": 723,
"column": 55
} | {
"line": 724,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nι : Type u_3\ni j : Discrete ι\nhi : i ≠ j\nf : ι → C\nc : Cocone (Discrete.functor f)\nhc : IsVanKampenColimit c\nthis : ∀ (i : Discrete ι), Subsingleton (⊥_ C ⟶ (Discrete.functor f).obj i)\n⊢ IsPullback (initial.to ((Discrete.functor f).ob... | [
"case e'_3\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nι : Type u_3\ni j : Discrete ι\nhi : i ≠ j\nf : ι → C\nc : Cocone (Discrete.functor f)\nhc : IsVanKampenColimit c\nthis : ∀ (i : Discrete ι), Subsingleton (⊥_ C ⟶ (Discrete.functor f).obj i)\n⊢ (⊥_ C) = if j.as = i.as then f i.as else ⊥_ C"
] | convert! isPullback_of_cofan_isVanKampen hc i.as j.as | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 95,
"column": 25
} | {
"line": 95,
"column": 27
} | {
"line": 95,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrpCat\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opp... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 155,
"column": 12
} | {
"line": 155,
"column": 41
} | {
"line": 155,
"column": 41
} | [
{
"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\... | [
"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\nM : SheafOf... | ← Adjunction.homEquiv_symm_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.LocallyDiscrete | {
"line": 156,
"column": 81
} | {
"line": 157,
"column": 14
} | {
"line": 159,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\na b : C\nh : a = b\n⊢ (eqToHom h).toLoc = eqToHom ⋯",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.LocallyDiscrete.categoryStruct",
"CategoryTheory.LocallyDiscrete.mk",
"CategoryTheory.CategoryStruct.toQuive... | [] | by
subst h; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 306,
"column": 19
} | {
"line": 306,
"column": 40
} | {
"line": 306,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ projectiveDimension X = ⊥ ↔ projectiveDimension X < 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
"Wi... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\n⊢ projectiveDimension X < ↑⊥ ↔ projectiveDimension X < 0"
] | ← WithBot.lt_coe_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 321,
"column": 61
} | {
"line": 321,
"column": 71
} | {
"line": 321,
"column": 71
} | [
{
"pp": "case coe.top\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : projectiveDimension X = ↑⊤\nn : ℕ\nhn : ⊤ ≤ ↑n\n⊢ False",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"WithBot",
"instCompleteLinearOrderENat",
"ChainCompletePartialOrder.in... | [
"case coe.top\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : projectiveDimension X = ↑⊤\nn : ℕ\nhn : ↑n = ⊤\n⊢ False"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Free | {
"line": 104,
"column": 35
} | {
"line": 104,
"column": 47
} | {
"line": 104,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ✝ : GrothendieckTopology C\nR : Sheaf J✝ RingCat\ninst✝¹ : HasWeakSheafify J✝ AddCommGrpCat\ninst✝ : J✝.WEqualsLocallyBijective AddCommGrpCat\nI J : Type u\nf : I → J\n⊢ (free J).freeHomEquiv.symm ((free J).freeHomEquiv (freeMap f)) = (free J).freeHomEquiv.sy... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 120,
"column": 4
} | {
"line": 123,
"column": 12
} | {
"line": 125,
"column": 0
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX Y : C\nf : Y ⟶ X\nS : Sieve X\n⊢ Sieve.pullback f (J₁.close S) ≤ J₁.close (Sieve.pullback f S)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver"... | [] | intro Z g hg
change _ ∈ J₁ _
rw [← Sieve.pullback_comp]
apply hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 120,
"column": 4
} | {
"line": 123,
"column": 12
} | {
"line": 125,
"column": 0
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX Y : C\nf : Y ⟶ X\nS : Sieve X\n⊢ Sieve.pullback f (J₁.close S) ≤ J₁.close (Sieve.pullback f S)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver"... | [] | intro Z g hg
change _ ∈ J₁ _
rw [← Sieve.pullback_comp]
apply hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 18
} | {
"line": 344,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreZeroHypercover S\ne : E ≅ F\ni : E.I₀\nhs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀\nhh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯\n⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯",
"ppTerm": "?m.81",
"assigned": true,
"used... | [] | simpa using hh i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 18
} | {
"line": 351,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreZeroHypercover S\ne : E ≅ F\ni : F.I₀\nhs : (e.inv ≫ e.hom).s₀ = (𝟙 F).s₀\nhh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (𝟙 F).h₀ i ≫ eqToHom ⋯\n⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯",
"ppTerm": "?m.81",
"assigned": true,
"used... | [] | simpa using hh i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 128,
"column": 27
} | {
"line": 128,
"column": 29
} | {
"line": 128,
"column": 30
} | [
{
"pp": "C : Type u_2\ninst✝ : Category.{v_1, u_2} C\nX : C\nS T : Presieve X\nP : Cᵒᵖ ⥤ Type u_1\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong S f\nhS : IsSeparatedFor P S ∧ ∀ (x : FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t\nW : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → ... | [
"C : Type u_2\ninst✝ : Category.{v_1, u_2} C\nX : C\nS T : Presieve X\nP : Cᵒᵖ ⥤ Type u_1\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, T f → ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong S f\nhS : IsSeparatedFor P S ∧ ∀ (x : FamilyOfElements P S), x.Compatible → ∃ t, x.IsAmalgamation t\nW : ⦃Z : C⦄ → ⦃g : Z ⟶ X⦄ → S g → C\ni : ⦃Z : ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 138,
"column": 36
} | {
"line": 138,
"column": 38
} | {
"line": 138,
"column": 39
} | [
{
"pp": "C : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)\nY : C\nf : Y ⟶ X\nH :\n ∀ {X Y : C} {f : Y ⟶ X},\n ∀ R ∈ J.coverings X,\n ... | [
"C : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)\nY : C\nf : Y ⟶ X\nH :\n ∀ {X Y : C} {f : Y ⟶ X},\n ∀ R ∈ J.coverings X,\n Presiev... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 304,
"column": 4
} | {
"line": 307,
"column": 62
} | {
"line": 308,
"column": 2
} | [
{
"pp": "case a\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nK : Coverage C\nJ : GrothendieckTopology C\nhJ : J ∈ {J | K ≤ J.toCoverage}\nX : C\nS : Sieve X\nhS : S ∈ K.toGrothendieck X\n⊢ S ∈ J X",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.GrothendieckTopology.tra... | [] | induction hS with
| of X S hS => apply hJ; assumption
| top => apply J.top_mem
| transitive X R S _ _ H1 H2 => exact J.transitive H1 _ H2 | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 145,
"column": 31
} | {
"line": 145,
"column": 33
} | {
"line": 145,
"column": 34
} | [
{
"pp": "C : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)\nY : C\nf : Y ⟶ X\nH :\n ∀ {X Y : C} {f : Y ⟶ X},\n ∀ R ∈ J.coverings X,\n ... | [
"C : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)\nY : C\nf : Y ⟶ X\nH :\n ∀ {X Y : C} {f : Y ⟶ X},\n ∀ R ∈ J.coverings X,\n Presiev... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 467,
"column": 78
} | {
"line": 470,
"column": 7
} | {
"line": 472,
"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\nK : Coverage C\nP : Cᵒᵖ ⥤ D\n⊢ IsSheaf K.toGrothendieck P ↔\n ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Nonempty (IsLimit (P.mapCone (Sieve.generate R).arrows.cocone.op))",
"ppTerm": "?m.42",
"assigned": true,
... | [] | by
simp only [Presheaf.IsSheaf, Presieve.isSheaf_coverage, isLimit_iff_isSheafFor,
← Presieve.isSheafFor_iff_generate]
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 53
} | {
"line": 162,
"column": 6
} | [
{
"pp": "case mpr.transitive.refine_2\nC : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)\nY : C\nf : Y ⟶ X\nH :\n ∀ {X Y : C} {f : Y ⟶ X}... | [
"case mpr.transitive.refine_2\nC : Type u_3\ninst✝ : Category.{u_2, u_3} C\nJ : Precoverage C\nP : Cᵒᵖ ⥤ Type u_1\nX✝ : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\nhS : J.Saturate X R\nh : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)\nY : C\nf : Y ⟶ X\nH :\n ∀ {X Y : C} {f : Y ⟶ X},\n ∀ R ∈... | refine (H2 hgg ggg).ext (fun ZZZZ gggg _ => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 30
} | {
"line": 198,
"column": 2
} | [
{
"pp": "case refine_1.transitive\nC : Type u_2\ninst✝⁴ : Category.{u_1, u_2} C\nJ : Precoverage C\ninst✝³ : J.IsStableUnderComposition\ninst✝² : J.IsStableUnderBaseChange\ninst✝¹ : J.HasPullbacks\ninst✝ : J.HasIsos\nX✝ : C\nS✝ : Sieve X✝\nX : C\nS T : Sieve X\nhS : J.Saturate X S\nhT : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S... | [] | exact hle' _ _ _ ⟨i.snd⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.CoverPreserving | {
"line": 119,
"column": 30
} | {
"line": 119,
"column": 32
} | {
"line": 119,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : Sheaf K (Type u_1)\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.obj) T\nhx : x.Compatible\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\n⊢ ∀ (f₂... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : RepresentablyFlat G\nℱ : Sheaf K (Type u_1)\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.obj) T\nhx : x.Compatible\nY₁ Y₂ : C\nX : D\nf₁ : X ⟶ G.obj Y₁\nf₂ : X ⟶ G.obj Y₂\n... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 371,
"column": 47
} | {
"line": 371,
"column": 64
} | {
"line": 372,
"column": 10
} | [
{
"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\nG : C ⥤ D\nA : Type w\ninst✝³ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝² : G.IsCocontinuous J K\ninst✝¹ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\ninst✝ : ... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝³ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝² : G.IsCocontinuous J K\ninst✝¹ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\ninst✝ : G.IsContinuo... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.LocallyFullyFaithful | {
"line": 49,
"column": 21
} | {
"line": 51,
"column": 32
} | {
"line": 53,
"column": 0
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{vC, uC} C\nD : Type uD\ninst✝ : Category.{vD, uD} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nU V : C\nf : G.obj U ⟶ G.obj V\n⊢ ∀ {Y Z : C} {f_1 : Y ⟶ U}, (∃ l, G.map l = G.map f_1 ≫ f) → ∀ (g : Z ⟶ Y), ∃ l, G.map l = G.map (g ≫ f_1) ≫ f",
"... | [] | by
rintro Y₁ Y₂ i₁ ⟨l, hl⟩ i₂
exact ⟨i₂ ≫ l, by simp [hl]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocallyFullyFaithful | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 20
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type uC\ninst✝² : Category.{vC, uC} C\nD : Type uD\ninst✝¹ : Category.{vD, uD} D\nK : GrothendieckTopology D\nG : C ⥤ D\ninst✝ : G.IsLocallyFull K\nℱ : Sheaf K (Type u_2)\nX Y : C\ni : G.obj X ⟶ G.obj Y\ns t : ℱ.obj.obj (op (G.obj X))\nh :\n ∀ ⦃Z : C⦄ (j : Z ⟶ X) (f : Z ⟶ Y),\n G.map f = G.map ... | [] | simp [h iWX iWY e] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 209,
"column": 2
} | {
"line": 218,
"column": 63
} | {
"line": 219,
"column": 2
} | [
{
"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\nK : GrothendieckTopology D\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : Sheaf K (Type v)\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nX : D\nx : ℱ.obj (op X)\n⊢ (pushforwardFamily α... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : Sheaf K (Type v)\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nX : D\nx : ℱ.obj (op X)\n⊢ ∀ {Z : D} {W₁ W₂ : C} (iWX₁ : G.... | suffices ∀ {Z W₁ W₂} (iWX₁ : G.obj W₁ ⟶ X) (iWX₂ : G.obj W₂ ⟶ X) (iZW₁ : Z ⟶ G.obj W₁)
(iZW₂ : Z ⟶ G.obj W₂), iZW₁ ≫ iWX₁ = iZW₂ ≫ iWX₂ →
ℱ'.1.map iZW₁.op (α.app _ (ℱ.map iWX₁.op x)) = ℱ'.1.map iZW₂.op (α.app _ (ℱ.map iWX₂.op x)) by
rintro Y₁ Y₂ Z iZY₁ iZY₂ f₁ f₂ h₁ h₂ e
simp only [pushforwardFamily... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 191,
"column": 69
} | {
"line": 211,
"column": 13
} | {
"line": 213,
"column": 0
} | [
{
"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 ... | [] | by
letI c' : Fan (E.multicospanIndex F).left := Fan.mk _ fun i ↦ F.map (c.inj i).op
letI hc' : IsLimit c' := isLimitFanMkObjOfIsLimit _ _ (fun i : E.I₀ ↦ _) (Cofan.IsColimit.op hc)
letI d' : Fan (E.multicospanIndex F).right := Fan.mk _ fun i ↦ F.map (d.inj i).op
letI hd' : IsLimit d' := isLimitFanMkObjOfIsLimit... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 280,
"column": 8
} | {
"line": 280,
"column": 24
} | {
"line": 280,
"column": 25
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝³ : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : S... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝³ : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : Sheaf K (Type... | appHom_restrict, | Lean.Elab.Tactic.evalRewriteSeq | null |
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