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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 317,
"column": 4
} | {
"line": 318,
"column": 65
} | {
"line": 319,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG✝ : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G✝\nG : Type u_7\ninst✝¹ : CommGroup G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nG' : Type u_8\ninst✝ : CommGroup G'\nζ : ↥(rootsOfUnity (... | [] | simpa only [orderOf_eq_card_of_forall_mem_zpowers hg, orderOf_dvd_iff_pow_eq_one,
← Units.val_pow_eq_pow_val, Units.val_eq_one] using! ζ.prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 139,
"column": 2
} | {
"line": 159,
"column": 73
} | {
"line": 161,
"column": 0
} | [
{
"pp": "case mpr\nn : Type u_1\ninst✝⁶ : Fintype n\nA : Type u_4\nK : Type u_5\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommRing A\ninst✝³ : Nontrivial A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nM : Matrix n n A\nthis : (∃ v, v ≠ 0 ∧ (algebraMap A K).mapMatrix M *ᵥ v = 0) ↔ M.det = 0\nv : ... | [] | · letI := Classical.decEq K
obtain ⟨⟨b, hb⟩, ba_eq⟩ :=
IsLocalization.exist_integer_multiples_of_finset (nonZeroDivisors A) (Finset.univ.image v)
choose f hf using ba_eq
refine
⟨fun i => f _ (Finset.mem_image.mpr ⟨i, Finset.mem_univ i, rfl⟩),
mt (fun h => funext fun i => ?_) hv, ?_⟩
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 28
} | {
"line": 284,
"column": 4
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type u_1\ninst✝ : CommRing S\ni : n\na : ↥(rootsOfUnity (Fintype.card n) R)\n⊢ (fun A ↦ rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯) ((fun a ↦ ⟨⟨a • 1, ⋯⟩, ⋯⟩) a) = a",
"ppTerm": "?m.147",
"assigned": true,
... | [
"n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type u_1\ninst✝ : CommRing S\ni : n\na inv✝ : R\nval_inv✝ : a * inv✝ = 1\ninv_val✝ : inv✝ * a = 1\nha : { val := a, inv := inv✝, val_inv := val_inv✝, inv_val := inv_val✝ } ∈ rootsOfUnity (Fintype.card n) R\n⊢ (fun A ↦ root... | obtain ⟨⟨a, _⟩, ha⟩ := a | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 858,
"column": 73
} | {
"line": 858,
"column": 84
} | {
"line": 859,
"column": 10
} | [
{
"pp": "S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹³ : CommSemiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra S R\ninst✝⁸ : Module S M\ninst✝⁷ : Module R M\ninst✝⁶ : Module Rᵐᵒᵖ M\ninst✝⁵ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁴ : IsScalarTower S R M\ninst✝³ : ... | [
"S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹³ : CommSemiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra S R\ninst✝⁸ : Module S M\ninst✝⁷ : Module R M\ninst✝⁶ : Module Rᵐᵒᵖ M\ninst✝⁵ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁴ : IsScalarTower S R M\ninst✝³ : IsScalarTowe... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 859,
"column": 20
} | {
"line": 859,
"column": 31
} | {
"line": 859,
"column": 32
} | [
{
"pp": "S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹³ : CommSemiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra S R\ninst✝⁸ : Module S M\ninst✝⁷ : Module R M\ninst✝⁶ : Module Rᵐᵒᵖ M\ninst✝⁵ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁴ : IsScalarTower S R M\ninst✝³ : ... | [
"S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹³ : CommSemiring S\ninst✝¹² : Semiring R\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra S R\ninst✝⁸ : Module S M\ninst✝⁷ : Module R M\ninst✝⁶ : Module Rᵐᵒᵖ M\ninst✝⁵ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁴ : IsScalarTower S R M\ninst✝³ : IsScalarTowe... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 396,
"column": 38
} | {
"line": 396,
"column": 47
} | {
"line": 396,
"column": 48
} | [
{
"pp": "R : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * 0 = 1\nhg : c = 0\n⊢ a * d = 1",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"MulZeroClass.toMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
... | [
"R : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - 0 = 1\nhg : c = 0\n⊢ a * d = 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 1051,
"column": 2
} | {
"line": 1051,
"column": 33
} | {
"line": 1052,
"column": 2
} | [
{
"pp": "S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹⁶ : CommSemiring S\ninst✝¹⁵ : Semiring R\ninst✝¹⁴ : CommSemiring R'\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Algebra S R\ninst✝¹¹ : Module S M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower S R M\ninst✝⁶... | [
"S : Type u_1\nR R' : Type u\nM : Type v\ninst✝¹⁶ : CommSemiring S\ninst✝¹⁵ : Semiring R\ninst✝¹⁴ : CommSemiring R'\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Algebra S R\ninst✝¹¹ : Module S M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower S R M\ninst✝⁶ : IsScalarT... | refine Equiv.trans ?_ liftEquiv | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.GradedMonoid | {
"line": 428,
"column": 89
} | {
"line": 429,
"column": 60
} | {
"line": 431,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_3\ninst✝¹ : AddMonoid ι\ninst✝ : GMonoid A\nn : ℕ\nf : Fin n → GradedMonoid A\n⊢ (List.ofFn f).prod =\n mk ((List.finRange n).dProdIndex fun i ↦ (f i).fst) ((List.finRange n).dProd (fun i ↦ (f i).fst) fun i ↦ (f i).snd)",
"ppTerm": "?m.24",
"assigned": true,
... | [] | by
rw [List.ofFn_eq_map, GradedMonoid.list_prod_map_eq_dProd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {
"line": 229,
"column": 21
} | {
"line": 229,
"column": 29
} | {
"line": 229,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_4\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nhf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)\na b : M\n⊢ f a * f b + f b * f a = (f a + f b) * (f a + f b) - f a... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nQ : QuadraticForm R M\nA : Type u_4\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nhf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)\na b : M\n⊢ f a * f b + f b * f a = (f a + f b) * f a + (f a + f b) * f b - f a... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.Ring | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 19
} | {
"line": 228,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ((mulHom A) ((of A 0) GradedMonoid.GOne.one)) ((of A i) xi) = (AddMonoidHom.id (⨁ (i : ι), A i)) ((of A i) xi)",
"ppTerm"... | [
"ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ (of A (0 + i)) (GradedMonoid.GMul.mul GradedMonoid.GOne.one xi) = (AddMonoidHom.id (⨁ (i : ι), A i)) ((of A i) xi)"
] | rw [mulHom_of_of] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.DirectSum.Ring | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 21
} | {
"line": 247,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\na b c : ⨁ (i : ι), A i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : A ci\n⊢ (of A ai) ax * (of A bi) bx * (of A ci) cx = (of A ai) ax * ((of A bi) bx * (of A... | [
"ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\na b c : ⨁ (i : ι), A i\nai : ι\nax : A ai\nbi : ι\nbx : A bi\nci : ι\ncx : A ci\n⊢ (of A (ai + bi + ci)) (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul ax bx) cx) =\n (of A (... | simp_rw [of_mul_of] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.DirectSum.Internal | {
"line": 160,
"column": 2
} | {
"line": 162,
"column": 7
} | {
"line": 163,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : (i : ι) → (x : ↥(A i)) → Decidable (x ≠ 0)\nr r' : ⨁ (i : ι), ↥(A i)\nn x✝¹ x✝... | [
"case neg\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Semiring R\ninst✝⁴ : SetLike σ R\ninst✝³ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝² : AddMonoid ι\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : (i : ι) → (x : ↥(A i)) → Decidable (x ≠ 0)\nr r' : ⨁ (i : ι), ↥(A i)\nn x✝¹ x✝ : ι\nh : ¬x... | · subst h
rw [of_eq_same]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.DirectSum.Internal | {
"line": 169,
"column": 66
} | {
"line": 173,
"column": 50
} | {
"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⊢ ↑((r * r') n) = ∑ ij ∈ antidiagona... | [] | by
classical
rw [coe_mul_apply]
apply Finset.sum_subset (fun _ ↦ by simp)
aesop (erase simp not_and) (add simp not_and_or) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.DirectSum.Internal | {
"line": 196,
"column": 38
} | {
"line": 196,
"column": 47
} | {
"line": 196,
"column": 48
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (DFinsupp... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : AddMonoid ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nj n : ι\nH : ∀ (x : ι), x + i = n ↔ x = j\n⊢ (DFinsupp.sum r fun i... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 290,
"column": 54
} | {
"line": 290,
"column": 68
} | {
"line": 290,
"column": 68
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nx x' y : M\n⊢ polar (⇑Q) x y + polar (⇑Q) (-x') y = polar (⇑Q) x y + -polar (⇑Q) x' y",
"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 x' y : M\n⊢ polar (⇑Q) x y + -polar (⇑Q) x' y = polar (⇑Q) x y + -polar (⇑Q) x' y"
] | polar_neg_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.DirectSum.Internal | {
"line": 243,
"column": 38
} | {
"line": 243,
"column": 47
} | {
"line": 243,
"column": 48
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Semiring R\ninst✝⁵ : SetLike σ R\ninst✝⁴ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : PartialOrder ι\ninst✝¹ : CanonicallyOrderedAdd ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' :... | [
"ι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Semiring R\ninst✝⁵ : SetLike σ R\ninst✝⁴ : AddSubmonoidClass σ R\nA : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : PartialOrder ι\ninst✝¹ : CanonicallyOrderedAdd ι\ninst✝ : SetLike.GradedMonoid A\nr : ⨁ (i : ι), ↥(A i)\ni : ι\nr' : ↥(A i)\nn :... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1108,
"column": 45
} | {
"line": 1109,
"column": 60
} | {
"line": 1111,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\n⊢ ¬Q.Anisotropic ↔ ∃ x, x ≠ 0 ∧ Q x = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"co... | [] | by
simp only [Anisotropic, not_forall, exists_prop, and_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 44
} | {
"line": 837,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
... | [] | simpa using separatingLeft_iff_det_ne_zero | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 44
} | {
"line": 837,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
... | [] | simpa using separatingLeft_iff_det_ne_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 44
} | {
"line": 837,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_11\ninst✝³ : CommRing R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : IsDomain R\nM : Matrix n n R\n⊢ ((toLinearMap₂' R) M).SeparatingLeft ↔ M.det ≠ 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
... | [] | simpa using separatingLeft_iff_det_ne_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 63
} | {
"line": 385,
"column": 4
} | [
{
"pp": "case e_6\nR : Type uR\nM₂ : Type v₂\nM' : Type v'\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nk l n : ℕ\ns : Finset (Fin n)\nhk : #s = k\nhl : #sᶜ = l\nf : MultilinearMap R (fun x ↦ M') (MultilinearMap R (fun x ↦ M') M₂)\nx ... | [
"case e_6.hi\nR : Type uR\nM₂ : Type v₂\nM' : Type v'\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nk l n : ℕ\ns : Finset (Fin n)\nhk : #s = k\nhl : #sᶜ = l\nf : MultilinearMap R (fun x ↦ M') (MultilinearMap R (fun x ↦ M') M₂)\nx y : M'\nx... | rw [finSumEquivOfFinset_inr, Finset.piecewise_eq_of_notMem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating | {
"line": 99,
"column": 16
} | {
"line": 99,
"column": 31
} | {
"line": 99,
"column": 32
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : (i : ℕ) → M [⋀^Fin i]→ₗ[R] N\nv : Fin 0 → M\n⊢ (liftAlternating f) (List.ofFn fun i ↦ (ι R) (v i)).prod = (f 0) v",
"ppTerm": "?ze... | [
"case zero\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : (i : ℕ) → M [⋀^Fin i]→ₗ[R] N\nv : Fin 0 → M\n⊢ (liftAlternating f) [].prod = (f 0) v"
] | List.ofFn_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.FilteredColimits | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 19
} | {
"line": 118,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\nJ : Type v\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ ModuleCat R\nr s : R\nj : J\nx : ↑(F.obj j)\n⊢ (r * s) • M.mk F ⟨j, x⟩ = r • s • M.mk F ⟨j, x⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"instHSMul... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 74
} | {
"line": 190,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nn : ℕ\nM : Type u_1\nN✝ : Type u_2\nN' : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N✝\ninst✝² : Module R N✝\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nN : Type ?u.42\nx✝¹ : AddCommGroup N\nx✝ : Module R N\nf f' : ↥(⋀[R]^n M) →ₗ[R... | [
"R : Type u\ninst✝⁶ : CommRing R\nn : ℕ\nM : Type u_1\nN✝ : Type u_2\nN' : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N✝\ninst✝² : Module R N✝\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nN : Type ?u.42\nx✝¹ : AddCommGroup N\nx✝ : Module R N\nf f' : ↥(⋀[R]^n M) →ₗ[R] N\nh : (re... | rw [Submodule.linearMap_eq_iff_of_span_eq_top _ _ (ιMulti_span R n M)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.LeftResolution.Basic | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"pp": "A : Type u_1\nC : Type u_2\ninst✝⁵ : Category.{v_1, u_2} C\ninst✝⁴ : Category.{v_2, u_1} A\nι : C ⥤ A\nΛ : LeftResolution ι\nX : A\ninst✝³ : ι.Full\ninst✝² : ι.Faithful\ninst✝¹ : HasZeroMorphisms C\ninst✝ : Abelian A\nn : ℕ\n⊢ (Λ.chainComplexMap (𝟙 X)).f n = (𝟙 (Λ.chainComplex X)).f n",
"ppTerm":... | [] | induction n with
| zero => simp
| succ n hn => obtain _ | n := n <;> simp [hn] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Homology.LeftResolution.Basic | {
"line": 171,
"column": 2
} | {
"line": 173,
"column": 48
} | {
"line": 175,
"column": 0
} | [
{
"pp": "A : Type u_1\nC : Type u_2\ninst✝⁶ : Category.{v_1, u_2} C\ninst✝⁵ : Category.{v_2, u_1} A\nι : C ⥤ A\nΛ : LeftResolution ι\nX Y : A\ninst✝⁴ : ι.Full\ninst✝³ : ι.Faithful\ninst✝² : HasZeroMorphisms C\ninst✝¹ : Abelian A\ninst✝ : Λ.F.PreservesZeroMorphisms\nn : ℕ\n⊢ (Λ.chainComplexMap 0).f n = Homologic... | [] | induction n with
| zero => simp
| succ n hn => obtain _ | n := n <;> simp [hn] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 452,
"column": 41
} | {
"line": 453,
"column": 17
} | {
"line": 455,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : Fin 1 → M\n⊢ (oneEquiv R M) ((ιMulti R 1) f) = f 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"exteriorPower.alternatingMapLinearEquiv",
"LinearMap.id",
"Alternat... | [] | by
simp [oneEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.IsConnected | {
"line": 82,
"column": 15
} | {
"line": 86,
"column": 42
} | {
"line": 88,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : IsPreconnected C\ninst✝ : HasColimit (constPUnitFunctor C)\na b : colimit (constPUnitFunctor C)\n⊢ a = b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Cate... | [] | by
obtain ⟨c, ⟨⟩, rfl⟩ := jointly_surjective' a
obtain ⟨d, ⟨⟩, rfl⟩ := jointly_surjective' b
apply constant_of_preserves_morphisms (colimit.ι (constPUnitFunctor C) · PUnit.unit)
exact fun c d f => colimit_sound f rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 53
} | {
"line": 180,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : SmallCategory C\nX Y : C ⥤ Type u\nj : C\n⊢ colimit.ι ((tensor (C ⥤ Type u)).obj (X, Y)) j ≫\n (HasColimit.isoOfNatIso ((externalProductCompDiagIso C (Type u)).app (X, Y)).symm).hom ≫\n colimit.pre (X ⊠ Y) (Functor.diag C) ≫\n (PreservesColimit₂.isoColimitUncurr... | [
"C : Type u\ninst✝ : SmallCategory C\nX Y : C ⥤ Type u\nj : C\n⊢ colimit.ι (X ⊗ Y) j ≫\n (HasColimit.isoOfNatIso ((externalProductCompDiagIso C (Type u)).app (X, Y)).symm).hom ≫\n colimit.pre (uncurry.obj ((((whiskeringLeft₂ (Type u)).obj X).obj Y).obj (curriedTensor (Type u))))\n (Functor.di... | dsimp [externalProductBifunctor, CartesianMonoidalCategory.prodComparison,
externalProductBifunctorCurried, externalProduct] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Limits.Sifted | {
"line": 280,
"column": 2
} | {
"line": 281,
"column": 65
} | {
"line": 282,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : SmallCategory C\nD : Type u\ninst✝² : SmallCategory D\ninst✝¹ : IsSifted C\nF : C ⥤ D\ninst✝ : F.Final\n⊢ IsSifted D",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfShape",
"CategoryTheory.Functor",
"C... | [
"C : Type u\ninst✝³ : SmallCategory C\nD : Type u\ninst✝² : SmallCategory D\ninst✝¹ : IsSifted C\nF : C ⥤ D\ninst✝ : F.Final\nthis : PreservesFiniteProducts colim\n⊢ IsSifted D"
] | have : PreservesFiniteProducts (colim : (D ⥤ Type u) ⥤ _) :=
⟨fun n ↦ preservesLimitsOfShape_of_natIso (Final.colimIso F)⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 51
} | {
"line": 215,
"column": 6
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nc x✝ : C\nx : Under c\ns s' : x✝ ⟶ (forget c).obj x\n⊢ ∃ t, s ≫ (forget c).map t = s' ≫ (forget c).map t",
"ppTerm": "?h",
"assigned": true,
"usedConstants":... | [
"case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nc x✝ : C\nx : Under c\ns s' : x✝ ⟶ (forget c).obj x\n⊢ s ≫ (forget c).map (homMk (IsFiltered.coeqHom s s') ⋯) = s' ≫ (forget c).map (homMk (IsFiltered.coeqHom s s') ⋯)"
] | use homMk (IsFiltered.coeqHom s s') (by simp) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 257,
"column": 4
} | {
"line": 257,
"column": 74
} | {
"line": 258,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nhF : F.Final\nd : D\n⊢ ∃ c, Nonempty (d ⟶ F.obj c)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.... | [
"case refine_1\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsFilteredOrEmpty C\nhF : F.Final\nd : D\nf : StructuredArrow d F\n⊢ ∃ c, Nonempty (d ⟶ F.obj c)"
] | obtain ⟨f⟩ : Nonempty (StructuredArrow d F) := IsConnected.is_nonempty | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 6
} | {
"line": 170,
"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... | [] | rw [← Category.assoc, ← Iso.eq_comp_inv]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Bifunctor | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 6
} | {
"line": 170,
"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... | [] | rw [← Category.assoc, ← Iso.eq_comp_inv]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 342,
"column": 4
} | {
"line": 343,
"column": 38
} | {
"line": 344,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : IsFiltered C\nF : C ⥤ Discrete PUnit.{u_1 + 1}\nx✝ : Discrete PUnit.{u_1 + 1}\n⊢ ∃ c, Nonempty (x✝ ⟶ F.obj c)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | use Classical.choice IsFiltered.nonempty
exact ⟨Discrete.eqToHom (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 342,
"column": 4
} | {
"line": 343,
"column": 38
} | {
"line": 344,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : IsFiltered C\nF : C ⥤ Discrete PUnit.{u_1 + 1}\nx✝ : Discrete PUnit.{u_1 + 1}\n⊢ ∃ c, Nonempty (x✝ ⟶ F.obj c)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | use Classical.choice IsFiltered.nonempty
exact ⟨Discrete.eqToHom (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.FinallySmall | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 50
} | {
"line": 74,
"column": 2
} | [
{
"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 ... | [] | exact ⟨P.FullSubcategory, small_of_surjective (f := G.obj)
(by rintro ⟨_, Y, _, rfl⟩; exact ⟨Y, rfl⟩), inferInstance, inferInstance, P.ι,
Functor.final_of_comp_full_faithful' G P.ι ⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 31
} | {
"line": 594,
"column": 2
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nj : J\nk : K\n⊢ colimit.ι (F.flip.obj k) j ≫\n colimit.ι (F.flip ⋙ colim) k ≫\n ... | [
"J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nj : J\nk : K\n⊢ (((colimit.ι (F.flip.obj k) j ≫ colimit.ι (F.flip ⋙ colim) k ≫ (colimitUncurryIso... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 604,
"column": 2
} | {
"line": 604,
"column": 31
} | {
"line": 605,
"column": 2
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nk : K\nj : J\n⊢ colimit.ι (F.obj j) k ≫\n colimit.ι (F ⋙ colim) j ≫\n (((... | [
"J : Type u_1\nK : Type u_2\ninst✝⁴ : Category.{v_1, u_1} J\ninst✝³ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝² : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nk : K\nj : J\n⊢ (((colimit.ι (F.obj j) k ≫ colimit.ι (F ⋙ colim) j ≫ (colimitUncurryIsoColimitCom... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 31
} | {
"line": 740,
"column": 2
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝³ : Category.{v_3, u_3} C\nG : J × K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nj : J\nk : K\n⊢ colimit.ι ((curry.obj (Pro... | [
"J : Type u_1\nK : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝³ : Category.{v_3, u_3} C\nG : J × K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nj : J\nk : K\n⊢ ((colimit.ι ((curry.obj (Prod.swap K J... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 31
} | {
"line": 753,
"column": 2
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝³ : Category.{v_3, u_3} C\nG : J × K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nj : J\nk : K\n⊢ colimit.ι ((curry.obj G).o... | [
"J : Type u_1\nK : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝³ : Category.{v_3, u_3} C\nG : J × K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasColimit (curry.obj G ⋙ colim)\nj : J\nk : K\n⊢ ((colimit.ι ((curry.obj G).obj j) k ≫ ... | slice_lhs 1 3 => simp only [] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal | {
"line": 139,
"column": 39
} | {
"line": 139,
"column": 67
} | {
"line": 140,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nR : Cᵒᵖ ⥤ CommRingCat\nX✝¹ Y✝ : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)\nx✝ : X✝¹ ⟶ Y✝\nX✝ : Cᵒᵖ\n⊢ (x✝ ▷ 𝟙_ (PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)) ≫ (ρ_ Y✝).hom).app X✝ = ((ρ_ X✝¹).hom ≫ x✝).app X✝",
"ppTerm": "?m.224",
"... | [] | apply rightUnitor_naturality | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 547,
"column": 24
} | {
"line": 547,
"column": 59
} | {
"line": 549,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ S R : Sieve X\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S.arrows f ∨ R.arrows f → ∀ (g : Z ⟶ Y), S.arrows (g ≫ f) ∨ R.arrows (g ≫ f)",
"ppTerm": "?m.25",
"assigned": true,
"usedCons... | [] | rintro _ _ _ (h | h) g <;> simp [h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 547,
"column": 24
} | {
"line": 547,
"column": 59
} | {
"line": 549,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ S R : Sieve X\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S.arrows f ∨ R.arrows f → ∀ (g : Z ⟶ Y), S.arrows (g ≫ f) ∨ R.arrows (g ≫ f)",
"ppTerm": "?m.25",
"assigned": true,
"usedCons... | [] | rintro _ _ _ (h | h) g <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 547,
"column": 24
} | {
"line": 547,
"column": 59
} | {
"line": 549,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS✝ R✝ S R : Sieve X\n⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S.arrows f ∨ R.arrows f → ∀ (g : Z ⟶ Y), S.arrows (g ≫ f) ∨ R.arrows (g ≫ f)",
"ppTerm": "?m.25",
"assigned": true,
"usedCons... | [] | rintro _ _ _ (h | h) g <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 943,
"column": 2
} | {
"line": 945,
"column": 34
} | {
"line": 947,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R : Sieve X\nf : Y ⟶ X\ninst✝ : IsSplitEpi f\n⊢ GaloisInsertion (pushforward f) (pullback f)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryThe... | [] | apply (galoisConnection f).toGaloisInsertion
intro S Z g hg
exact ⟨g ≫ section_ f, by simpa⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 943,
"column": 2
} | {
"line": 945,
"column": 34
} | {
"line": 947,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R : Sieve X\nf : Y ⟶ X\ninst✝ : IsSplitEpi f\n⊢ GaloisInsertion (pushforward f) (pullback f)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryThe... | [] | apply (galoisConnection f).toGaloisInsertion
intro S Z g hg
exact ⟨g ≫ section_ f, by simpa⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1125,
"column": 2
} | {
"line": 1125,
"column": 16
} | {
"line": 1127,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y✝ Z : C\nf✝ : Y✝ ⟶ X✝\nS✝ R : Sieve X✝\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\nG : D ⥤ E\ninst✝¹ : F.EssSurj\ninst✝ : F.Full\nX : C\nS : Sieve (F.obj X)\nY : D\nf : Y ⟶ F.obj X\nhf : S.arrows f\n⊢ (... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 21
} | {
"line": 425,
"column": 0
} | [
{
"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\nt : P.obj (op X)\nht : (x.map f).IsAmalgamation ((ConcreteCategory.hom (f.app (op X))) t)\nY : C\nu : Y ⟶ X\nhu : R u\n⊢ (ConcreteCategory.hom (f.app (op Y))) ((Con... | [] | simpa using ht _ hu | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.EqualizerSheafCondition | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 91
} | {
"line": 324,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type w\nB : C\nI : Type t\ninst✝¹ : Small.{w, t} I\nX : I → C\nπ : (i : I) → X i ⟶ B\ninst✝ : (Presieve.ofArrows X π).HasPairwisePullbacks\nx : I × I\nij : P.obj (op B)\n⊢ (ConcreteCategory.hom (Pi.π (fun ij ↦ P.obj (op (Limits.pullback (π ij.1) (π ij.2... | [] | simp [← comp_apply, -types_comp_apply, ← Functor.map_comp, ← op_comp, pullback.condition] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 407,
"column": 6
} | {
"line": 409,
"column": 19
} | {
"line": 410,
"column": 6
} | [
{
"pp": "case mpr.refine_2\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\nZ : C\nf : Z ⟶ Y\nhf : S.arrows f\nx :... | [
"case mpr.refine_2\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\nZ : C\nf : Z ⟶ Y\nhf : S.arrows f\nx : unop (op X)... | simp only [Functor.comp_obj, Functor.flip_obj_obj, yoneda_obj_obj, Functor.comp_map,
Functor.flip_obj_map, yoneda_map_app, ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk,
comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Plus | {
"line": 214,
"column": 49
} | {
"line": 221,
"column": 55
} | {
"line": 223,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{w', w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\n⊢ η ≫ J.toPlus Q = J.toPlus P ≫ ... | [] | by
ext
dsimp [toPlus, plusMap]
delta Cover.toMultiequalizer
simp only [ι_colimMap, Category.assoc]
simp_rw [← Category.assoc]
congr 1
exact Multiequalizer.hom_ext _ _ _ (fun I => by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 99
} | {
"line": 990,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : X ⟶ Y\n⊢ (∀ (b : P.obj (op X)),\n ((FamilyOfElements.singletonEquiv P f).symm b).Compatible →\n ∃! t, ((FamilyOfElements.singletonEquiv P f).symm b).IsAmalgamation t) ↔\n ∀ (x : P.obj (op X)),\n (∀ {Z : C} (p₁ ... | [] | simp_rw [FamilyOfElements.compatible_singleton_iff,
FamilyOfElements.isAmalgamation_singleton_iff, FamilyOfElements.singletonEquiv_symm_apply_self] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.CategoryTheory.Sites.Sheafification | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 27
} | {
"line": 206,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP : Cᵒᵖ ⥤ D\nhP : Presheaf.IsSheaf J P\n⊢ (isoSheafify J hP).inv = sheafifyLift J (𝟙 P) hP",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP : Cᵒᵖ ⥤ D\nhP : Presheaf.IsSheaf J P\n⊢ toSheafify J P ≫ (isoSheafify J hP).inv = 𝟙 P"
] | apply sheafifyLift_unique | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Sheafification | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 27
} | {
"line": 218,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nhR : Presheaf.IsSheaf J R\n⊢ sheafifyMap J η ≫ sheafifyLift J γ hR = sheafifyLift J (η ≫ γ) hR",
"ppTerm": "?m.51"... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\ninst✝ : HasWeakSheafify J D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nhR : Presheaf.IsSheaf J R\n⊢ toSheafify J P ≫ sheafifyMap J η ≫ sheafifyLift J γ hR = η ≫ γ"
] | apply sheafifyLift_unique | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Limits | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 44
} | {
"line": 128,
"column": 6
} | [
{
"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✝ : HasLimitsOfShape K D\nF : K ⥤ Sheaf J D\nE : Cone (F ⋙ sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : J.Cover X\nS : Multifork (W.index E.pt)\n... | [
"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✝ : HasLimitsOfShape K D\nF : K ⥤ Sheaf J D\nE : Cone (F ⋙ sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : J.Cover X\nS : Multifork (W.index E.pt)\nm : S.pt ⟶ (... | rw [(F.obj k).property.amalgamate_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 368,
"column": 8
} | {
"line": 369,
"column": 18
} | {
"line": 369,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝³ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMu... | [] | · cases I
exact hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 72,
"column": 6
} | {
"line": 78,
"column": 54
} | {
"line": 78,
"column": 54
} | [
{
"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\n⊢ ∀ {m : S.pt ⟶ (pullbackCone f g).pt},\n m ≫ (pullbackCone f g).fst = S.fst →\n m ≫ (pullbackCone f g).snd = S.snd →\n m =\n ofHom\n { toFun := ... | [] | · intro m h₁ h₂
ext x
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): used to be `ext x`.
apply Subtype.ext
apply Prod.ext
· simpa using! ConcreteCategory.congr_hom h₁ x
· simpa using! ConcreteCategory.congr_hom h₂ x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 270,
"column": 2
} | {
"line": 278,
"column": 27
} | {
"line": 281,
"column": 0
} | [
{
"pp": "W X Y Z S T : TopCat\nf₁ : W ⟶ S\nf₂ : X ⟶ S\ng₁ : Y ⟶ T\ng₂ : Z ⟶ T\ni₁ : W ⟶ Y\ni₂ : X ⟶ Z\nH₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₁)\nH₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₂)\ni₃ : S ⟶ T\nH₃ : Mono i₃\neq₁ : f₁ ≫ i₃ = i₁ ≫ g₁\neq₂ : f₂ ≫ i₃ = i₂ ≫ g₂\n⊢ IsOpenEmbedding ⇑(ConcreteCategor... | [] | constructor
· apply
pullback_map_isEmbedding f₁ f₂ g₁ g₂ H₁.isEmbedding H₂.isEmbedding i₃ eq₁ eq₂
· rw [range_pullback_map]
apply IsOpen.inter <;> apply Continuous.isOpen_preimage
· apply ContinuousMap.continuous_toFun
· exact H₁.isOpen_range
· apply ContinuousMap.continuous_toFun
· exact ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 270,
"column": 2
} | {
"line": 278,
"column": 27
} | {
"line": 281,
"column": 0
} | [
{
"pp": "W X Y Z S T : TopCat\nf₁ : W ⟶ S\nf₂ : X ⟶ S\ng₁ : Y ⟶ T\ng₂ : Z ⟶ T\ni₁ : W ⟶ Y\ni₂ : X ⟶ Z\nH₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₁)\nH₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₂)\ni₃ : S ⟶ T\nH₃ : Mono i₃\neq₁ : f₁ ≫ i₃ = i₁ ≫ g₁\neq₂ : f₂ ≫ i₃ = i₂ ≫ g₂\n⊢ IsOpenEmbedding ⇑(ConcreteCategor... | [] | constructor
· apply
pullback_map_isEmbedding f₁ f₂ g₁ g₂ H₁.isEmbedding H₂.isEmbedding i₃ eq₁ eq₂
· rw [range_pullback_map]
apply IsOpen.inter <;> apply Continuous.isOpen_preimage
· apply ContinuousMap.continuous_toFun
· exact H₁.isOpen_range
· apply ContinuousMap.continuous_toFun
· exact ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 83
} | {
"line": 157,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nI₁ : Type u_2\nI₂ : Type u_3\nX : I₁ ⊕ I₂ → C\nc : Cofan X\nc₁ : Cofan (X ∘ Sum.inl)\nc₂ : Cofan (X ∘ Sum.inr)\nhc : IsColimit c\nhc₁ : IsColimit c₁\nhc₂ : IsColimit c₂\ninst✝ : MonoCoprod C\ninl : c₁.pt ⟶ c.pt\nhinl : ∀ (i₁ : I₁), c₁.inj i₁ ≫ inl = c.inj (... | [] | rw [this]
exact MonoCoprod.binaryCofan_inl _ (isColimitBinaryCofanSum c c₁ c₂ hc hc₁ hc₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 83
} | {
"line": 157,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nI₁ : Type u_2\nI₂ : Type u_3\nX : I₁ ⊕ I₂ → C\nc : Cofan X\nc₁ : Cofan (X ∘ Sum.inl)\nc₂ : Cofan (X ∘ Sum.inr)\nhc : IsColimit c\nhc₁ : IsColimit c₁\nhc₂ : IsColimit c₂\ninst✝ : MonoCoprod C\ninl : c₁.pt ⟶ c.pt\nhinl : ∀ (i₁ : I₁), c₁.inj i₁ ≫ inl = c.inj (... | [] | rw [this]
exact MonoCoprod.binaryCofan_inl _ (isColimitBinaryCofanSum c c₁ c₂ hc hc₁ hc₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Extensive | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 79
} | {
"line": 231,
"column": 6
} | [
{
"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... | [
"case h₁.left.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\nX'✝ Y'✝ : Type u\nαX✝ : X'✝ ⟶ PUnit.{u + 1}\nαY✝ : Y'✝ ⟶ PUnit.{u + 1}\nf : (Types.binaryCoproductCocone X'✝ Y'✝).pt ⟶ (Types.binaryCoproductCocone PUnit.{... | refine ⟨↾(l), ?_, Types.isTerminalPUnit.hom_ext _ _, fun {l'} h₁ _ => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 485,
"column": 2
} | {
"line": 486,
"column": 51
} | {
"line": 487,
"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\nthis :... | [
"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✝ : ReflectsLi... | haveI : ReflectsColimitsOfShape WalkingSpan F :=
reflectsColimitsOfShape_of_reflectsIsomorphisms | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 11
} | {
"line": 149,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\n⊢ P.isoClosure.isColocal = P.isColocal",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.ObjectProperty.isoClosure",
"Cat... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nY Z : C\ng : Y ⟶ Z\n⊢ P.isoClosure.isColocal g ↔ P.isColocal g"
] | ext Y Z g | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 230,
"column": 10
} | {
"line": 230,
"column": 87
} | {
"line": 231,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : G ⊣ F\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nX Y : D\nf : X ⟶ Y\nthis : f ≫ adj.unit.app Y = adj.unit.app X ≫ F.map (G.map f)\n⊢ isLocal (fun x ↦ x ∈ Set.range F.obj) (adj.unit.app X ≫... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : G ⊣ F\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nX Y : D\nf : X ⟶ Y\nthis : f ≫ adj.unit.app Y = adj.unit.app X ≫ F.map (G.map f)\n⊢ isLocal (fun x ↦ x ∈ Set.range F.obj) (F.map (G.map f)) ↔ IsIso (G.... | (isLocal (· ∈ Set.range F.obj)).precomp_iff _ _ (isLocal_adj_unit_app adj X), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 99,
"column": 89
} | {
"line": 104,
"column": 18
} | {
"line": 106,
"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 : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\n⊢ IsLocallyInjective J (Functor.whiske... | [] | by
constructor
· intro
exact ⟨fun x y h => equalizerSieve_mem J (Functor.whiskerRight φ (forget D)) x y h⟩
· intro
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 144,
"column": 58
} | {
"line": 146,
"column": 54
} | {
"line": 148,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nFD : D → D → Type u_1\nCD : D → Type w\ninst✝² : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninst✝¹ : ConcreteCategory D FD\nJ : GrothendieckTopology C\nF₁ F₂ F₃ : Cᵒᵖ ⥤ D\nφ : F₁ ⟶ F₂\nψ : F₂ ⟶ F₃\nφψ : F₁ ⟶ F₃\nfac : φ... | [] | by
subst fac
exact isLocallyInjective_of_isLocallyInjective J φ ψ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.LocallyInjective | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 42
} | {
"line": 193,
"column": 2
} | [
{
"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 ... | [
"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 v)\n⊢ IsLoca... | rw [GrothendieckTopology.plusMap_toPlus] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 42
} | {
"line": 306,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝ : ConcreteCategory A FA\nP : Cᵒᵖ ⥤ Type (max u v)\n⊢ IsLocallySurjective J (J.toPlus P ≫ J.p... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝ : ConcreteCategory A FA\nP : Cᵒᵖ ⥤ Type (max u v)\n⊢ IsLocallySurjective J (J.toPlus P ≫ J.toPlus (J.plu... | rw [GrothendieckTopology.plusMap_toPlus] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 75
} | {
"line": 46,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nR : Cᵒᵖ ⥤ RingCat\nF : D ⥤ PresheafOfModules R\ninst✝¹ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections\nc : Cone F\ninst✝ : HasLimitsOfSh... | [] | exact Sheaf.isSheaf_of_isLimit G _ (isLimitOfPreserves (toPresheaf R) hc) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 707,
"column": 8
} | {
"line": 707,
"column": 21
} | {
"line": 707,
"column": 21
} | [
{
"pp": "case refine_2.refine_1.isTrue\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasInitial C\nι : Type u_3\nX : ι → C\nc : Cofan X\nhc : IsVanKampenColimit c\ni j✝ : ι\ninst✝ : DecidableEq ι\nt : Cofan fun k ↦ if k = i then X i else ⊥_ C\nj : ι\nh✝ : j = i\n⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj i = t.inj j",... | [
"case refine_2.refine_1.isTrue\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasInitial C\nι : Type u_3\nX : ι → C\nc : Cofan X\nhc : IsVanKampenColimit c\nj✝ : ι\ninst✝ : DecidableEq ι\nj : ι\nt : Cofan fun k ↦ if k = j then X j else ⊥_ C\n⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj j = t.inj j"
] | subst ‹j = i› | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 66
} | {
"line": 164,
"column": 66
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝ : HasWeakSheafify J AddCommGrpCat\... | [
"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\nF : SheafOf... | ← isIso_iff_of_reflects_iso _ (SheafOfModules.toSheaf.{v} R) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 164,
"column": 4
} | {
"line": 178,
"column": 9
} | {
"line": 180,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrpCat\nφ : M₀.presheaf ⟶ A.obj\ninst✝¹ : Pr... | [] | apply A.isSeparated _ _ (J.pullback_stable f.unop hS)
rintro Z g hg
dsimp at hg
rw [← ConcreteCategory.comp_apply, ← A.obj.map_comp, ← NatTrans.naturality_apply (D := Ab)]
erw [M₀.map_smul] -- Mismatch between `M₀.map` and `M₀.presheaf.map`
refine (ha _ hg).trans (app_eq_of_isLocallyInjective α φ A.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 164,
"column": 4
} | {
"line": 178,
"column": 9
} | {
"line": 180,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrpCat\nφ : M₀.presheaf ⟶ A.obj\ninst✝¹ : Pr... | [] | apply A.isSeparated _ _ (J.pullback_stable f.unop hS)
rintro Z g hg
dsimp at hg
rw [← ConcreteCategory.comp_apply, ← A.obj.map_comp, ← NatTrans.naturality_apply (D := Ab)]
erw [M₀.map_smul] -- Mismatch between `M₀.map` and `M₀.presheaf.map`
refine (ha _ hg).trans (app_eq_of_isLocallyInjective α φ A.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 36
} | {
"line": 155,
"column": 8
} | [
{
"pp": "case diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝¹ val✝ : J\nf : val✝¹ ⟶ val✝\n⊢ BiconeHom.diagram f ∈ Finset.image BiconeHom.diagram Fintype.elems",
"ppTerm": "?diagram",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.elems",
"Categ... | [
"case diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝¹ val✝ : J\nf : val✝¹ ⟶ val✝\n⊢ ∃ a ∈ Fintype.elems, BiconeHom.diagram a = BiconeHom.diagram f"
] | simp only [Finset.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Continuous | {
"line": 111,
"column": 2
} | {
"line": 119,
"column": 54
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nX : C\nE : PreOneHypercover X\nF : C ⥤ D\ni₁ i₂ : E.I₀\ninst✝¹ : HasPullback (E.f i₁) (E.f i₂)\ninst✝ : PreservesLimit (cospan (E.f i₁) (E.f i₂)) F\nthis : HasPullback ((E.map F).f i₁) ((E.map F).f i₂)\... | [] | · rw [PreOneHypercover.sieve₁_eq_pullback_sieve₁' _ _ _
(by simp [← Functor.map_comp, pullback.condition])]
rintro W f ⟨Z, u, v, ⟨k⟩, h⟩
refine ⟨E.Y k, pullback.lift (E.p₁ k) (E.p₂ k) (E.w _), u, ?_, ?_⟩
· use E.Y k, 𝟙 _, pullback.lift (E.p₁ k) (E.p₂ k) (E.w _), ⟨k⟩
simp
· simp only [pullba... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.CoverPreserving | {
"line": 178,
"column": 4
} | {
"line": 187,
"column": 35
} | {
"line": 189,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : F.IsContinuous J K\nX : C\nS : Sieve X\nhS : S ∈ J X\n⊢ Sieve.functorPushforward F S ∈ K (F.obj X)",
"ppTerm": "?m.24",
"assigned": tr... | [] | rw [K.mem_iff_isSheafFor_closedSieves]
obtain ⟨ι, Y, f, rfl⟩ := S.exists_eq_ofArrows
rw [Sieve.ofArrows, ← Sieve.generate_map_eq_functorPushforward,
← Presieve.isSheafFor_iff_generate, Presieve.map_ofArrows]
have := Functor.op_comp_isSheaf_of_isSheaf_type F J (classifier_isSheaf K) _ hS
rw [Sieve.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.CoverPreserving | {
"line": 178,
"column": 4
} | {
"line": 187,
"column": 35
} | {
"line": 189,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝ : F.IsContinuous J K\nX : C\nS : Sieve X\nhS : S ∈ J X\n⊢ Sieve.functorPushforward F S ∈ K (F.obj X)",
"ppTerm": "?m.24",
"assigned": tr... | [] | rw [K.mem_iff_isSheafFor_closedSieves]
obtain ⟨ι, Y, f, rfl⟩ := S.exists_eq_ofArrows
rw [Sieve.ofArrows, ← Sieve.generate_map_eq_functorPushforward,
← Presieve.isSheafFor_iff_generate, Presieve.map_ofArrows]
have := Functor.op_comp_isSheaf_of_isSheaf_type F J (classifier_isSheaf K) _ hS
rw [Sieve.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 82
} | {
"line": 193,
"column": 2
} | [
{
"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 ... | [
"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 fun i ↦ op (... | letI c' : Fan (E.multicospanIndex F).left := Fan.mk _ fun i ↦ F.map (c.inj i).op | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 65
} | {
"line": 401,
"column": 2
} | [
{
"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\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\nX : Cᵒᵖ\nU : ... | [
"case hx\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\nX : Cᵒᵖ\nU : Aᵒᵖ... | apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 65
} | {
"line": 431,
"column": 2
} | [
{
"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\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.obj\nX : Dᵒᵖ\nU : Aᵒᵖ\nx✝ : (yon... | [
"case hx\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nα : ℱ ⟶ ℱ'.obj\nX : Dᵒᵖ\nU : Aᵒᵖ\nx✝ : (yoneda... | apply sheaf_eq_amalgamation ℱ' (G.is_cover_of_isCoverDense _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.CoverLifting | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 27
} | {
"line": 411,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝⁵ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝⁴ : G.IsCocontinuous J K\ninst✝³ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\ninst✝² :... | [
"C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝⁵ : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝⁴ : G.IsCocontinuous J K\ninst✝³ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\ninst✝² : G.IsContinu... | apply sheafifyLift_unique | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 331,
"column": 8
} | {
"line": 331,
"column": 57
} | {
"line": 332,
"column": 6
} | [
{
"pp": "case refine_1.refine_2.refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f... | [] | simp [this, pullback.condition_assoc, toPullback] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 57
} | {
"line": 335,
"column": 4
} | [
{
"pp": "case refine_1.refine_2.refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f... | [] | simp [this, pullback.condition_assoc, toPullback] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 339,
"column": 6
} | {
"line": 340,
"column": 72
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂... | [] | simp only [Category.assoc] at u₁
simp [← reassoc_of% h₁, w, ← reassoc_of% u₁, ← pullback.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 339,
"column": 6
} | {
"line": 340,
"column": 72
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂... | [] | simp only [Category.assoc] at u₁
simp [← reassoc_of% h₁, w, ← reassoc_of% u₁, ← pullback.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Canonical | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 27
} | {
"line": 63,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP✝ : Cᵒᵖ ⥤ Type w\nX✝ : C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : Y ⟶ X\n⊢ Presieve.IsSheafFor P (Sieve.pullback f ⊤).arrows",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",... | [
"C : Type u\ninst✝ : Category.{v, u} C\nP✝ : Cᵒᵖ ⥤ Type w\nX✝ : C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : Y ⟶ X\n⊢ Presieve.IsSheafFor P ⊤.arrows"
] | rw [Sieve.pullback_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 19
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nX : C\nS : Sieve X\n⊢ (∃ T, Sieve.functorPullback G ↑T = S) → Sieve.functorPushforward G S ∈ ... | [
"case mpr\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nX : C\nT : ↑(K (G.obj X))\n⊢ Sieve.functorPushforward G (Sieve.functorPullback G ↑T) ∈ K (G.obj X)"
] | rintro ⟨T, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 491,
"column": 34
} | {
"line": 491,
"column": 46
} | {
"line": 491,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ni✝ i' j✝ j' : E.I₀\nhii' : i✝ = i'\nhjj' : j✝ = j'\nu₀ v₀ : E.I₀ → F.I₀\nu... | [] | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 491,
"column": 49
} | {
"line": 491,
"column": 61
} | {
"line": 491,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ni✝ i' j✝ j' : E.I₀\nhii' : i✝ = i'\nhjj' : j✝ = j'\nu₀ v₀ : E.I₀ → F.I₀\nu... | [] | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 681,
"column": 6
} | {
"line": 681,
"column": 33
} | {
"line": 681,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE F : PreOneHypercover S\ns₀ : E.I₀ ≃ F.I₀\nh₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)\ns₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS✝ : C\nE✝¹ E✝ : PreOneHypercover S✝\nF✝ : PreOneHypercover S✝\nG : PreOneHypercover S✝\nS : C\nE F : PreOneHypercover S\ns₀ : E.I₀ ≃ F.I₀\nh₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)\ns₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)\nh... | congrIndexOneOfEqIso_inv_p₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators | {
"line": 177,
"column": 9
} | {
"line": 179,
"column": 18
} | {
"line": 181,
"column": 0
} | [
{
"pp": "C : Type u'\ninst✝⁸ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝⁷ : HasWeakSheafify J AddCommGrpCat\ninst✝⁶ : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝⁵ : J.HasSheafCompose (forget₂ RingCat AddCommGrpCat)\nM✝ N P : SheafOfModules R\nC' : Type u₁\ninst✝⁴ : Category.... | [] | by
simp only [mapFreeHom, Equiv.symm_apply_apply, epi_comp_iff_of_epi]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 1099,
"column": 2
} | {
"line": 1100,
"column": 16
} | {
"line": 1102,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\nS : C\nE : PreZeroHypercover S\ninst✝ : E.HasPullbacks\n⊢ E.toPreOneHypercover.HasPullbacks",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreOneHypercover.toPreZeroHyperco... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 1099,
"column": 2
} | {
"line": 1100,
"column": 16
} | {
"line": 1102,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\nS : C\nE : PreZeroHypercover S\ninst✝ : E.HasPullbacks\n⊢ E.toPreOneHypercover.HasPullbacks",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreOneHypercover.toPreZeroHyperco... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 35
} | {
"line": 413,
"column": 0
} | [
{
"pp": "X : TopCat\nU : Opens ↑X\n⊢ ↑((map U.inclusion').obj U) = ↑⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Subtype.coe_preimage_self",
"TopologicalSpace.Opens",
"TopologicalSpace.Opens.instSetLike",
"TopCat.str",
"TopCat.carrier",
"SetLike.co... | [] | exact Subtype.coe_preimage_self _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 418,
"column": 74
} | {
"line": 423,
"column": 16
} | {
"line": 425,
"column": 0
} | [
{
"pp": "X : TopCat\n⊢ ⋯.functor = map (inclusionTopIso X).inv",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.ext",
"Lattice.toSemilatticeSup",
"trivial",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CompleteLattice.toLattice",
"C... | [] | by
refine CategoryTheory.Functor.ext ?_ ?_
· intro U
ext x
exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩
· subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 28
} | {
"line": 432,
"column": 4
} | [
{
"pp": "case mpr\nX Y : TopCat\nf : X ⟶ Y\nhf : IsOpenMap ⇑(ConcreteCategory.hom f)\nU : Opens ↑Y\nx✝ : ↑Y\n⊢ x✝ ∈ ↑(hf.functor.obj ⊤ ⊓ U) → x✝ ∈ ↑(hf.functor.obj ((map f).obj U))",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLat... | [
"case mpr\nX Y : TopCat\nf : X ⟶ Y\nhf : IsOpenMap ⇑(ConcreteCategory.hom f)\nU : Opens ↑Y\nx : ↑X\nhx : (ConcreteCategory.hom f) x ∈ ↑U\n⊢ (ConcreteCategory.hom f) x ∈ ↑(hf.functor.obj ((map f).obj U))"
] | rintro ⟨⟨x, -, rfl⟩, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Limits.Lattice | {
"line": 101,
"column": 2
} | {
"line": 104,
"column": 42
} | {
"line": 106,
"column": 0
} | [
{
"pp": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\n⊢ HasBinaryProducts α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop",
... | [] | have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
infer_instance
apply hasBinaryProducts_of_hasLimit_pair | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Lattice | {
"line": 101,
"column": 2
} | {
"line": 104,
"column": 42
} | {
"line": 106,
"column": 0
} | [
{
"pp": "α : Type u\nJ : Type w\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\n⊢ HasBinaryProducts α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop",
... | [] | have : ∀ x y : α, HasLimit (pair x y) := by
letI := hasFiniteLimits_of_hasFiniteLimits_of_size.{u} α
infer_instance
apply hasBinaryProducts_of_hasLimit_pair | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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