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.LinearAlgebra.Matrix.Basis | {
"line": 234,
"column": 2
} | {
"line": 236,
"column": 82
} | {
"line": 238,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_5\nM : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nb : Basis ι R M\nb' : Basis ι' R M\ninst✝¹ : Fintype ι\ninst✝ : Finite ι'\nm : M\n⊢ b'.toMatrix ⇑b *ᵥ ⇑(b.repr m) = ⇑(b'.repr m)",
"ppTerm": "?m.58",
"assigned": true... | [] | classical
cases nonempty_fintype ι'
simp [← LinearMap.toMatrix_id_eq_basis_toMatrix, LinearMap.toMatrix_mulVec_repr] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 234,
"column": 2
} | {
"line": 236,
"column": 82
} | {
"line": 238,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_5\nM : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nb : Basis ι R M\nb' : Basis ι' R M\ninst✝¹ : Fintype ι\ninst✝ : Finite ι'\nm : M\n⊢ b'.toMatrix ⇑b *ᵥ ⇑(b.repr m) = ⇑(b'.repr m)",
"ppTerm": "?m.58",
"assigned": true... | [] | classical
cases nonempty_fintype ι'
simp [← LinearMap.toMatrix_id_eq_basis_toMatrix, LinearMap.toMatrix_mulVec_repr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Basis | {
"line": 234,
"column": 2
} | {
"line": 236,
"column": 82
} | {
"line": 238,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_5\nM : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nb : Basis ι R M\nb' : Basis ι' R M\ninst✝¹ : Fintype ι\ninst✝ : Finite ι'\nm : M\n⊢ b'.toMatrix ⇑b *ᵥ ⇑(b.repr m) = ⇑(b'.repr m)",
"ppTerm": "?m.58",
"assigned": true... | [] | classical
cases nonempty_fintype ι'
simp [← LinearMap.toMatrix_id_eq_basis_toMatrix, LinearMap.toMatrix_mulVec_repr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 92
} | {
"line": 122,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nm n : ℕ\nh : m.Coprime n\n⊢ Disjoint (rootsOfUnity m M) (rootsOfUnity n M)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.Coprime",
"_private.Mathlib.RingTheory.RootsOfUnity.Basic.0.disjoint_rootsOfUnity_of_copr... | [] | simp [disjoint_iff_inf_le, rootsOfUnity_inf_rootsOfUnity, Nat.coprime_iff_gcd_eq_one.mp h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 92
} | {
"line": 122,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nm n : ℕ\nh : m.Coprime n\n⊢ Disjoint (rootsOfUnity m M) (rootsOfUnity n M)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.Coprime",
"_private.Mathlib.RingTheory.RootsOfUnity.Basic.0.disjoint_rootsOfUnity_of_copr... | [] | simp [disjoint_iff_inf_le, rootsOfUnity_inf_rootsOfUnity, Nat.coprime_iff_gcd_eq_one.mp h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 92
} | {
"line": 122,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nm n : ℕ\nh : m.Coprime n\n⊢ Disjoint (rootsOfUnity m M) (rootsOfUnity n M)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.Coprime",
"_private.Mathlib.RingTheory.RootsOfUnity.Basic.0.disjoint_rootsOfUnity_of_copr... | [] | simp [disjoint_iff_inf_le, rootsOfUnity_inf_rootsOfUnity, Nat.coprime_iff_gcd_eq_one.mp h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.Basic | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 39
} | {
"line": 267,
"column": 40
} | [
{
"pp": "R : Type u_4\nF : Type u_6\nk : ℕ\ninst✝⁴ : NeZero k\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : FunLike F R R\ninst✝ : MonoidHomClass F R R\nσ : F\nζ : ↥(rootsOfUnity k R)\nm : ℤ\nhm : ∀ (g : ↥(rootsOfUnity k R)), (restrictRootsOfUnity σ k) g = g ^ m\n⊢ ∃ m, σ ↑↑ζ = ↑↑ζ ^ m",
"ppTerm": "?m... | [
"R : Type u_4\nF : Type u_6\nk : ℕ\ninst✝⁴ : NeZero k\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : FunLike F R R\ninst✝ : MonoidHomClass F R R\nσ : F\nζ : ↥(rootsOfUnity k R)\nm : ℤ\nhm : ∀ (g : ↥(rootsOfUnity k R)), (restrictRootsOfUnity σ k) g = g ^ m\n⊢ ∃ m, ↑↑((restrictRootsOfUnity σ k) ζ) = ↑↑ζ ^ m"
] | ← restrictRootsOfUnity_coe_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 16
} | {
"line": 90,
"column": 17
} | [
{
"pp": "n : Type u_1\ninst✝⁴ : Fintype n\nR : Type u_2\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb : Basis n R M\ninst✝ : DecidableEq n\nA : Matrix n n R\nhA : IsUnit A.det\nx : M\n⊢ ((toLin b b) 1) x = x",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants"... | [
"n : Type u_1\ninst✝⁴ : Fintype n\nR : Type u_2\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb : Basis n R M\ninst✝ : DecidableEq n\nA : Matrix n n R\nhA : IsUnit A.det\nx : M\n⊢ LinearMap.id x = x"
] | toLin_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.ToLinearEquiv | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 16
} | {
"line": 93,
"column": 17
} | [
{
"pp": "n : Type u_1\ninst✝⁴ : Fintype n\nR : Type u_2\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb : Basis n R M\ninst✝ : DecidableEq n\nA : Matrix n n R\nhA : IsUnit A.det\nx : M\n⊢ ((toLin b b) 1) x = x",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants... | [
"n : Type u_1\ninst✝⁴ : Fintype n\nR : Type u_2\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nb : Basis n R M\ninst✝ : DecidableEq n\nA : Matrix n n R\nhA : IsUnit A.det\nx : M\n⊢ LinearMap.id x = x"
] | toLin_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 592,
"column": 45
} | {
"line": 592,
"column": 94
} | {
"line": 592,
"column": 94
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\nb c : F\nhc : c = b * (a ^ 2 - 1)\n⊢ diag2 a ha * SpecialLinearGroup.transvection ⋯ b * diag2 a⁻¹ ⋯ * (SpecialLinearGroup.transvection ⋯ b)⁻¹ =\n SpecialLinearGroup.transvection ⋯ c",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": ... | [
"F : Type u_1\ninst✝ : Field F\na : F\nha : a ≠ 0\nb c : F\nhc : c = b * (a ^ 2 - 1)\n⊢ diag2 a ha * SpecialLinearGroup.transvection ⋯ b * diag2 a⁻¹ ⋯ * SpecialLinearGroup.transvection ⋯ (-b) =\n SpecialLinearGroup.transvection ⋯ c"
] | SpecialLinearGroup.transvection_inv zero_ne_one b | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 665,
"column": 2
} | {
"line": 668,
"column": 34
} | {
"line": 670,
"column": 0
} | [
{
"pp": "⊢ ↑S * ↑S = -1",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Int.instAddCommMonoid",
"NegZeroClass.toNeg",
"neg_smul",
"instHSMul",
"Pi.instNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | simp only [S, Int.reduceNeg, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd,
vecMul_cons, head_cons, zero_smul, tail_cons, neg_smul, one_smul, neg_cons, neg_zero, neg_empty,
empty_vecMul, add_zero, zero_add, empty_mul, Equiv.symm_apply_apply]
exact Eq.symm (eta_fin_two (-1)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | {
"line": 665,
"column": 2
} | {
"line": 668,
"column": 34
} | {
"line": 670,
"column": 0
} | [
{
"pp": "⊢ ↑S * ↑S = -1",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Int.instAddCommMonoid",
"NegZeroClass.toNeg",
"neg_smul",
"instHSMul",
"Pi.instNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | simp only [S, Int.reduceNeg, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd,
vecMul_cons, head_cons, zero_smul, tail_cons, neg_smul, one_smul, neg_cons, neg_zero, neg_empty,
empty_vecMul, add_zero, zero_add, empty_mul, Equiv.symm_apply_apply]
exact Eq.symm (eta_fin_two (-1)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Internal | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 25
} | {
"line": 82,
"column": 4
} | [
{
"pp": "case negSucc\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddGroupWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubgroupClass σ R\nA : ι → σ\ninst✝ : GradedOne A\na✝ : ℕ\n⊢ ↑(Int.negSucc a✝) ∈ A 0",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"Add... | [
"case negSucc\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁴ : Zero ι\ninst✝³ : AddGroupWithOne R\ninst✝² : SetLike σ R\ninst✝¹ : AddSubgroupClass σ R\nA : ι → σ\ninst✝ : GradedOne A\na✝ : ℕ\n⊢ -↑(a✝ + 1) ∈ A 0"
] | rw [Int.cast_negSucc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 19
} | {
"line": 232,
"column": 2
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nQ : QuadraticMap R M N\nx y z : M\nB : BilinMap R M N\nh : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y\n⊢ Q (x + y + z) + (Q x + Q y + Q z) = ... | [
"R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nQ : QuadraticMap R M N\nx y z : M\nB : BilinMap R M N\nh : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y\n⊢ Q (x + y + z) + (Q x + Q y + Q z) = Q (x + y) + ... | rw [add_comm z x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 348,
"column": 26
} | {
"line": 348,
"column": 37
} | {
"line": 348,
"column": 38
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : Module S M\... | [
"S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\nQ : QuadraticMap R M N\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : Module S M\ninst✝² : Is... | polar_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 98
} | {
"line": 436,
"column": 0
} | [
{
"pp": "R : Type u_1\nN₂ : Type u_10\nn : Type u_11\nm : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid N₂\ninst✝⁴ : Module R N₂\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nB : (n → R) →ₗ[R] (m → R) →ₗ[R] N₂\ni✝ : n\nj✝ : m\n⊢ (toMatrix₂ (Pi.basisFun R n)... | [] | rw [LinearMap.toMatrix₂_apply, LinearMap.toMatrix₂'_apply, Pi.basisFun_apply, Pi.basisFun_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.DirectSum.Internal | {
"line": 459,
"column": 11
} | {
"line": 459,
"column": 21
} | {
"line": 460,
"column": 2
} | [
{
"pp": "case nil\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedA... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.DirectSum.Internal | {
"line": 459,
"column": 11
} | {
"line": 459,
"column": 21
} | {
"line": 460,
"column": 2
} | [
{
"pp": "case nil\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedA... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Internal | {
"line": 459,
"column": 11
} | {
"line": 459,
"column": 21
} | {
"line": 460,
"column": 2
} | [
{
"pp": "case nil\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedA... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Internal | {
"line": 470,
"column": 11
} | {
"line": 470,
"column": 21
} | {
"line": 471,
"column": 2
} | [
{
"pp": "case nil\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedA... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.DirectSum.Internal | {
"line": 470,
"column": 11
} | {
"line": 470,
"column": 21
} | {
"line": 471,
"column": 2
} | [
{
"pp": "case nil\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedA... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.DirectSum.Internal | {
"line": 470,
"column": 11
} | {
"line": 470,
"column": 21
} | {
"line": 471,
"column": 2
} | [
{
"pp": "case nil\nι : Type u_1\nσ : Type u_2\nR : Type u_4\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : LinearOrder ι\ninst✝⁶ : IsOrderedAddMonoid ι\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Semiring R\ninst✝³ : SetLike σ R\ninst✝² : AddSubmonoidClass σ R\nA : ι → σ\ninst✝¹ : SetLike.GradedMonoid A\ninst✝ : CanonicallyOrderedA... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 802,
"column": 2
} | {
"line": 803,
"column": 71
} | {
"line": 805,
"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": ... | [] | 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] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 802,
"column": 2
} | {
"line": 803,
"column": 71
} | {
"line": 805,
"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": ... | [] | 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] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 802,
"column": 2
} | {
"line": 803,
"column": 71
} | {
"line": 805,
"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": ... | [] | 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] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1393,
"column": 2
} | {
"line": 1393,
"column": 57
} | {
"line": 1394,
"column": 2
} | [
{
"pp": "case succ.inr\nK : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ ... | [
"case succ.inr\nK : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDime... | obtain ⟨x, hx⟩ := exists_bilinForm_self_ne_zero hB₁ hB₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1407,
"column": 8
} | {
"line": 1407,
"column": 95
} | {
"line": 1408,
"column": 8
} | [
{
"pp": "K : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensi... | [
"K : Type v\ninst✝³ : Field K\nhK : Invertible 2\nd : ℕ\nih :\n ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] {B : BilinForm K V},\n IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v\nV : Type u\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nB ... | have := this (c • x) (Submodule.smul_mem _ _ <| Submodule.mem_span_singleton_self _) hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 16
} | {
"line": 88,
"column": 0
} | [
{
"pp": "R : 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\n⊢ (((foldl 0 (LinearMap.mk₂ R (fun m f i ↦ (f (i + 1)).curryLeft m) ⋯ ⋯ ⋯ ⋯) ⋯) f) 1 0) 0 = (f 0) 0",
"ppTerm":... | [] | rw [foldl_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {
"line": 280,
"column": 8
} | {
"line": 280,
"column": 43
} | {
"line": 281,
"column": 8
} | [
{
"pp": "case succ\nR✝ : 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... | [
"case succ\nR✝ : 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 : Fi... | rw [List.ofFn_succ, List.prod_cons] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Localization | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 46
} | {
"line": 112,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nS : Submonoid R\nM₁ M₂ M₃ : ModuleCat R\nf₁ : M₁ ⟶ M₂\nf₂ : M₂ ⟶ M₃\nh₁ : IsLocalizedModule S (Hom.hom f₁)\nh₂ : IsLocalizedModule S (Hom.hom (f₁ ≫ f₂))\nthis : Function.Bijective ⇑(Hom.hom f₂)\n⊢ IsIso f₂",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants... | [] | simpa [ConcreteCategory.isIso_iff_bijective] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Limits.IsConnected | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 48
} | {
"line": 122,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nx✝ : Nonempty (IsColimit (pUnitCocone C))\nh : IsColimit (pUnitCocone C)\ncolimitCocone : ColimitCocone (constPUnitFunctor C) := { cocone := pUnitCocone C, isColimit := h }\nthis : HasColimit (constPUnitFunctor C)\n⊢ Nonempty (colimit (constPUnitFunctor C) ≅ PUnit... | [] | exact ⟨colimit.isoColimitCocone colimitCocone⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 302,
"column": 2
} | {
"line": 304,
"column": 91
} | {
"line": 306,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsCofilteredOrEmpty C\n⊢ F.Initial ↔ ∀ (d : D), IsCofiltered (CostructuredArrow F d)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Categ... | [] | refine ⟨?_, fun h => initial_of_isCofiltered_costructuredArrow F⟩
rw [initial_iff_of_isCofiltered]
exact fun h d => isCofiltered_costructuredArrow_of_isCofiltered_of_exists F d (h.1 d) h.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Final | {
"line": 302,
"column": 2
} | {
"line": 304,
"column": 91
} | {
"line": 306,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : IsCofilteredOrEmpty C\n⊢ F.Initial ↔ ∀ (d : D), IsCofiltered (CostructuredArrow F d)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Categ... | [] | refine ⟨?_, fun h => initial_of_isCofiltered_costructuredArrow F⟩
rw [initial_iff_of_isCofiltered]
exact fun h d => isCofiltered_costructuredArrow_of_isCofiltered_of_exists F d (h.1 d) h.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 313,
"column": 20
} | {
"line": 313,
"column": 78
} | {
"line": 313,
"column": 79
} | [
{
"pp": "J : Type u_1\nK : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} K\nC : Type u_3\ninst✝ : Category.{v_3, u_3} C\nF : J ⥤ K ⥤ C\nG : J × K ⥤ C\nD : DiagramOfCocones F\nQ : (j : J) → IsColimit (D.obj j)\nc : Cocone (uncurry.obj F)\nP : IsColimit (coconeOfCoconeUncurry Q c)\nE : (j... | [] | by simpa [E] using s.ι.naturality ((Prod.sectL J k).map f) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 525,
"column": 2
} | {
"line": 527,
"column": 6
} | {
"line": 529,
"column": 0
} | [
{
"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\nF : J ⥤ K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimit (uncurry.obj F)\ninst✝ : HasColimit (F ⋙ colim)\nj : J\nk : K\n⊢ colimit.ι (F.obj j) k ≫ colimit.... | [] | dsimp [colimitUncurryIsoColimitCompColim, IsColimit.coconePointUniqueUpToIso,
IsColimit.uniqueUpToIso]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Fubini | {
"line": 525,
"column": 2
} | {
"line": 527,
"column": 6
} | {
"line": 529,
"column": 0
} | [
{
"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\nF : J ⥤ K ⥤ C\ninst✝² : HasColimitsOfShape K C\ninst✝¹ : HasColimit (uncurry.obj F)\ninst✝ : HasColimit (F ⋙ colim)\nj : J\nk : K\n⊢ colimit.ι (F.obj j) k ≫ colimit.... | [] | dsimp [colimitUncurryIsoColimitCompColim, IsColimit.coconePointUniqueUpToIso,
IsColimit.uniqueUpToIso]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 342,
"column": 4
} | {
"line": 343,
"column": 35
} | {
"line": 343,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [] | obtain ⟨g, rfl⟩ := (homEquiv hcR hcM').surjective g
simp [homEquiv_naturality_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 342,
"column": 4
} | {
"line": 343,
"column": 35
} | {
"line": 343,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM' : PresheafOfModules R\ncM' : Cocone M'.presheaf\nhc... | [] | obtain ⟨g, rfl⟩ := (homEquiv hcR hcM').surjective g
simp [homEquiv_naturality_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subfunctor.Basic | {
"line": 183,
"column": 4
} | {
"line": 187,
"column": 33
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nG : Subfunctor F\n⊢ IsIso G.ι → G = ⊤",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.NatIso.isIso_app_of_isIso",... | [] | intro H
ext U x
apply (iff_of_eq (iff_true _)).mpr
rw [← IsIso.inv_hom_id_apply (G.ι.app U) x]
exact ((inv (G.ι.app U)) x).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subfunctor.Basic | {
"line": 183,
"column": 4
} | {
"line": 187,
"column": 33
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nG : Subfunctor F\n⊢ IsIso G.ι → G = ⊤",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.NatIso.isIso_app_of_isIso",... | [] | intro H
ext U x
apply (iff_of_eq (iff_true _)).mpr
rw [← IsIso.inv_hom_id_apply (G.ι.app U) x]
exact ((inv (G.ι.app U)) x).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 10
} | {
"line": 212,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\nhx : x.Compatible\nY₁✝ Y₂✝ Z✝ : C\ng₁✝ : Z✝ ⟶ Y₁✝\ng₂✝ : Z✝ ⟶ Y₂✝\nf₁✝ : Y₁✝ ⟶ X\nf₂✝ : Y₂✝ ⟶ X\nh₁ : (generate R).arrows f₁✝\nh₂ : (generate R).arrows f₂✝\ncomm : g₁✝ ≫ f₁✝ = g₂✝ ≫ f₂✝\n⊢ (Conc... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX : C\nR : Presieve X\nx : FamilyOfElements P R\nhx : x.Compatible\nY₁✝ Y₂✝ Z✝ : C\ng₁✝ : Z✝ ⟶ Y₁✝\ng₂✝ : Z✝ ⟶ Y₂✝\nf₁✝ : Y₁✝ ⟶ X\nf₂✝ : Y₂✝ ⟶ X\nh₁ : (generate R).arrows f₁✝\nh₂ : (generate R).arrows f₂✝\ncomm : g₁✝ ≫ f₁✝ = g₂✝ ≫ f₂✝\n⊢ (g₁✝ ≫ ⋯.choose) ... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 870,
"column": 22
} | {
"line": 870,
"column": 30
} | {
"line": 870,
"column": 30
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nB : C\nI : Type u_1\nX : I → C\nπ : (i : I) → X i ⟶ B\nh :\n ∀ (x : (i : I) → P.obj (op (X i))),\n Arrows.Compatible P π x → ∃! t, ∀ (i : I), (ConcreteCategory.hom (P.map (π i).op)) t = x i\nx✝¹ : Subtype (Arrows.Compatible P π)\ny : (i : ... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.IsSheafFor | {
"line": 986,
"column": 6
} | {
"line": 986,
"column": 17
} | {
"line": 986,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : X ⟶ Y\n⊢ IsSheafFor P (singleton f) ↔\n ∀ (x : P.obj (op X)),\n (∀ {Z : C} (p₁ p₂ : Z ⟶ X),\n p₁ ≫ f = p₂ ≫ f → (ConcreteCategory.hom (P.map p₁.op)) x = (ConcreteCategory.hom (P.map p₂.op)) x) →\n ∃! y, (Co... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP : Cᵒᵖ ⥤ Type w\nX Y : C\nf : X ⟶ Y\n⊢ (∀ (x : FamilyOfElements P (singleton f)), x.Compatible → ∃! t, x.IsAmalgamation t) ↔\n ∀ (x : P.obj (op X)),\n (∀ {Z : C} (p₁ p₂ : Z ⟶ X),\n p₁ ≫ f = p₂ ≫ f → (ConcreteCategory.hom (P.map p₁.op)) x = (ConcreteCatego... | IsSheafFor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.ConcreteSheafification | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 36
} | {
"line": 158,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝² : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMu... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁴ : Category.{w', w} D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)\ninstCC : ConcreteCategory D FD\ninst✝² : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.... | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Whiskering | {
"line": 166,
"column": 54
} | {
"line": 169,
"column": 89
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nA : Type u₂\ninst✝³ : Category.{v₂, u₂} A\nJ : GrothendieckTopology C\nFA : A → A → Type u_1\nCA : A → Type u_2\ninst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝¹ : ConcreteCategory A FA\ninst✝ : J.HasSheafCompose (forget A)\nF : Sheaf J A\n⊢ Preshea... | [] | by
rintro X S hS x y h
exact (((isSheaf_iff_isSheaf_of_type _ _).1
((sheafCompose J (forget A)).obj F).2).isSeparated S hS).ext (fun _ _ hf => h _ _ hf) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 256,
"column": 32
} | {
"line": 258,
"column": 35
} | {
"line": 260,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' : Sheaf J (Type w)\nf : F ⟶ F'\n⊢ toImage f ≫ imageι f = f",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Functor",
"CategoryTheory.Sheaf.ima... | [] | by
ext1
simp [Subfunctor.toRangeSheafify] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 15
} | {
"line": 51,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nι : Type v\nα : ι → TopCat\n⊢ ∀ (s : Cone (Discrete.functor α)) (m : s.pt ⟶ (piFan α).pt),\n (∀ (j : Discrete ι), m ≫ (piFan α).π.app j = s.π.app j) →\n m = ofHom { toFun := fun s_1 i ↦ (ConcreteCategory.hom (s.π.app { as := i })) s_1, continuous_toFun := ... | [
"J : Type v\ninst✝ : Category.{w, v} J\nι : Type v\nα : ι → TopCat\nS : Cone (Discrete.functor α)\nm : S.pt ⟶ (piFan α).pt\nh : ∀ (j : Discrete ι), m ≫ (piFan α).π.app j = S.π.app j\n⊢ m = ofHom { toFun := fun s i ↦ (ConcreteCategory.hom (S.π.app { as := i })) s, continuous_toFun := ⋯ }"
] | intro S m h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 15
} | {
"line": 92,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nι : Type v\nβ : ι → TopCat\n⊢ ∀ (s : Cocone (Discrete.functor β)) (m : (sigmaCofan β).pt ⟶ s.pt),\n (∀ (j : Discrete ι), (sigmaCofan β).ι.app j ≫ m = s.ι.app j) →\n m = ofHom { toFun := fun s_1 ↦ (ConcreteCategory.hom (s.ι.app { as := s_1.fst })) s_1.snd, ... | [
"J : Type v\ninst✝ : Category.{w, v} J\nι : Type v\nβ : ι → TopCat\nS : Cocone (Discrete.functor β)\nm : (sigmaCofan β).pt ⟶ S.pt\nh : ∀ (j : Discrete ι), (sigmaCofan β).ι.app j ≫ m = S.ι.app j\n⊢ m = ofHom { toFun := fun s ↦ (ConcreteCategory.hom (S.ι.app { as := s.fst })) s.snd, continuous_toFun := ⋯ }"
] | intro S m h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 15
} | {
"line": 139,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nX Y : TopCat\n⊢ ∀ (s : Cone (pair X Y)) (m : s.pt ⟶ (X.prodBinaryFan Y).pt),\n (∀ (j : Discrete WalkingPair), m ≫ (X.prodBinaryFan Y).π.app j = s.π.app j) →\n m =\n ofHom\n {\n toFun := fun s_1 ↦\n ((ConcreteCategory... | [
"J : Type v\ninst✝ : Category.{w, v} J\nX Y : TopCat\nS : Cone (pair X Y)\nm : S.pt ⟶ (X.prodBinaryFan Y).pt\nh : ∀ (j : Discrete WalkingPair), m ≫ (X.prodBinaryFan Y).π.app j = S.π.app j\n⊢ m =\n ofHom\n { toFun := fun s ↦ ((ConcreteCategory.hom (BinaryFan.fst S)) s, (ConcreteCategory.hom (BinaryFan.snd S)... | intro S m h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 271,
"column": 2
} | {
"line": 272,
"column": 83
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case toIsEmbedding\nW 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⊢ IsEmbedding ⇑... | [
"case isOpen_range\nW 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⊢ IsOpen (Set.range ⇑(Concre... | · apply
pullback_map_isEmbedding f₁ f₂ g₁ g₂ H₁.isEmbedding H₂.isEmbedding i₃ eq₁ eq₂ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 47
} | {
"line": 92,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nh : ∀ (A B : C), ∃ c x, Mono c.inl\nA B : C\nc' : BinaryCofan A B\nhc' : IsColimit c'\n⊢ Mono c'.inl",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Mono",
"Cat... | [] | obtain ⟨c, hc₁, hc₂⟩ := h A B
simpa only [mono_inl_iff hc' hc₁] using hc₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 91,
"column": 4
} | {
"line": 92,
"column": 47
} | {
"line": 92,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nh : ∀ (A B : C), ∃ c x, Mono c.inl\nA B : C\nc' : BinaryCofan A B\nhc' : IsColimit c'\n⊢ Mono c'.inl",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Mono",
"Cat... | [] | obtain ⟨c, hc₁, hc₂⟩ := h A B
simpa only [mono_inl_iff hc' hc₁] using hc₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 30
} | {
"line": 140,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nι : Type u_1\nX : ι → C\ninst✝¹ : CoproductDisjoint X\nc : Cofan X\nhc : IsColimit c\nY : C\nhY : IsInitial Y\ni j : ι\ninst✝ : HasPullback (c.inj i) (c.inj j)\nhij : i ≠ j\n⊢ Y ≅ pullback (c.inj i) (c.inj j)",
"ppTerm": "?refine_1",
"assig... | [
"case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nι : Type u_1\nX : ι → C\ninst✝¹ : CoproductDisjoint X\nc : Cofan X\nhc : IsColimit c\nY : C\nhY : IsInitial Y\ni j : ι\ninst✝ : HasPullback (c.inj i) (c.inj j)\nhij : i ≠ j\n⊢ IsInitial (pullback (c.inj i) (c.inj j))"
] | refine hY.uniqueUpToIso ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 43
} | {
"line": 251,
"column": 4
} | [
{
"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\nF : C ⥤ D\ninst✝² : MonoCoprod D\ninst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) F\ninst✝ : F.ReflectsMonomorphisms\nA B : C\nc : BinaryCofan A B\nh : IsColimit c\nc' : BinaryCofan (F.obj ((pair A B).o... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝² : MonoCoprod D\ninst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) F\ninst✝ : F.ReflectsMonomorphisms\nA B : C\nc : BinaryCofan A B\nh : IsColimit c\nc' : BinaryCofan (F.obj ((pair A B).obj { as := W... | refine Cocone.ext (φ := eqToIso rfl) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 116,
"column": 15
} | {
"line": 116,
"column": 31
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case mp.refine_5.mpr\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh✝ : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h✝ i\nH : H✝.IsVanKampen\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h✝ i ⋯).pt\neα : α ≫ (PushoutCocone.mk h✝ i ⋯).ι = ... | [] | exact ⟨h _, h _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 246,
"column": 10
} | {
"line": 246,
"column": 72
} | {
"line": 247,
"column": 10
} | [
{
"pp": "case right\nC : Type u\ninst✝² : Category.{v, u} C\nW E X Z : C\nc : BinaryCofan W E\ninst✝¹ : FinitaryExtensive C\ninst✝ : HasPullbacks C\nhc : IsColimit c\nf : W ⟶ X\nh : X ⟶ Z\ni : c.pt ⟶ Z\nH : IsPushout f c.inl h i\nhc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' ... | [
"C : Type u\ninst✝² : Category.{v, u} C\nW E X Z : C\nc : BinaryCofan W E\ninst✝¹ : FinitaryExtensive C\ninst✝ : HasPullbacks C\nhc : IsColimit c\nf : W ⟶ X\nh : X ⟶ Z\ni : c.pt ⟶ Z\nH : IsPushout f c.inl h i\nhc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\... | apply IsPullback.of_right _ e₂ (IsPullback.of_hasPullback _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 312,
"column": 2
} | {
"line": 326,
"column": 45
} | {
"line": 327,
"column": 2
} | [
{
"pp": "J : Type v'\ninst✝⁶ : Category.{u', v'} J\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝³ : Gr.Full\ninst✝² : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsUniversalColimit c\ninst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.... | [
"J : Type v'\ninst✝⁶ : Category.{u', v'} J\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝³ : Gr.Full\ninst✝² : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsUniversalColimit c\ninst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPull... | let cf : (Cocone.precompose β.hom).obj c' ⟶ Gl.mapCocone c'' := by
refine { hom := pullback.lift ?_ f ?_ ≫ (PreservesPullback.iso _ _ _).inv, w := ?_ }
· exact inv <| adj.counit.app c'.pt
· simp [← cancel_mono (adj.counit.app <| Gl.obj c.pt)]
· intro j
rw [← Category.assoc, Iso.comp_inv_eq]
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 74,
"column": 2
} | {
"line": 76,
"column": 58
} | {
"line": 78,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} 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 G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : ToType (F.obj (op U))\n⊢ imageSieve f ((ConcreteCategory.h... | [] | ext V i
simp only [Sieve.top_apply, iff_true, imageSieve_apply]
exact ⟨F.map i.op s, NatTrans.naturality_apply f i.op s⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 74,
"column": 2
} | {
"line": 76,
"column": 58
} | {
"line": 78,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} 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 G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : ToType (F.obj (op U))\n⊢ imageSieve f ((ConcreteCategory.h... | [] | ext V i
simp only [Sieve.top_apply, iff_true, imageSieve_apply]
exact ⟨F.map i.op s, NatTrans.naturality_apply f i.op s⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.LocallyBijective | {
"line": 150,
"column": 44
} | {
"line": 150,
"column": 81
} | {
"line": 151,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁷ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝⁶ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝⁵ : ConcreteCategory A FA\ninst✝⁴ : HasWeakSheafify J A\ninst✝³ : (forget A).ReflectsIsomorph... | [
"C : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁷ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝⁶ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝⁵ : ConcreteCategory A FA\ninst✝⁴ : HasWeakSheafify J A\ninst✝³ : (forget A).ReflectsIsomorphisms\ninst✝²... | Presheaf.comp_isLocallyInjective_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced | {
"line": 42,
"column": 32
} | {
"line": 44,
"column": 32
} | {
"line": 46,
"column": 0
} | [
{
"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\nF : C ⥤ D\ninst✝³ : F.ReflectsIsomorphisms\ninst✝² : F.PreservesEpimorphisms\ninst✝¹ : F.PreservesMonomorphisms\ninst✝ : Balanced D\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝¹ : Mono f\nx✝ : Epi f\n⊢ IsIso f",
"ppTerm": "?... | [] | by
rw [← isIso_iff_of_reflects_iso (F := F)]
exact isIso_of_mono_of_epi _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 710,
"column": 6
} | {
"line": 710,
"column": 18
} | {
"line": 711,
"column": 6
} | [
{
"pp": "case refine_2.refine_2\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\n⊢ ∀ (t : Cofan fun k ↦ if k = i then X i else ⊥_ C)\n (m : (Cofan.mk (X i) fun k ↦ if h : k = i then eqToHom ⋯ else ... | [
"case refine_2.refine_2\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\nm : (Cofan.mk (X i) fun k ↦ if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to ... | intro t m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification | {
"line": 130,
"column": 8
} | {
"line": 130,
"column": 54
} | {
"line": 131,
"column": 8
} | [
{
"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\nP₀ Q₀ : Pre... | apply (SheafOfModules.toSheaf _).map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 114,
"column": 2
} | {
"line": 117,
"column": 56
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nX : C\nn m : ℕ\nh : n ≤ m\ninst✝ : HasProjectiveDimensionLT X n\n⊢ HasProjectiveDimensionLT X m",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.Abelian.Ext.instAddCo... | [] | letI := HasExt.standard C
rw [hasProjectiveDimensionLT_iff]
intro i hi Y e
exact e.eq_zero_of_hasProjectiveDimensionLT n (by lia) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Projective.Dimension | {
"line": 114,
"column": 2
} | {
"line": 117,
"column": 56
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nX : C\nn m : ℕ\nh : n ≤ m\ninst✝ : HasProjectiveDimensionLT X n\n⊢ HasProjectiveDimensionLT X m",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.Abelian.Ext.instAddCo... | [] | letI := HasExt.standard C
rw [hasProjectiveDimensionLT_iff]
intro i hi Y e
exact e.eq_zero_of_hasProjectiveDimensionLT n (by lia) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 80
} | {
"line": 61,
"column": 4
} | [
{
"pp": "case succ\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\nR' : Type u'\ninst✝² : CommRing R'\ninst✝¹ : Small.{v', u'} R'\ne : R ≃+* R'\nn : ℕ\nih :\n ∀ {M : ModuleCat R} {N : ModuleCat R'} (e' : ↑M ≃ₛₗ[↑e] ↑N) [HasProjectiveDimensionLE M n],\n HasProjectiveDimensionLE N n\nM : ModuleCat ... | [
"case succ\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\nR' : Type u'\ninst✝² : CommRing R'\ninst✝¹ : Small.{v', u'} R'\ne : R ≃+* R'\nn : ℕ\nih :\n ∀ {M : ModuleCat R} {N : ModuleCat R'} (e' : ↑M ≃ₛₗ[↑e] ↑N) [HasProjectiveDimensionLE M n],\n HasProjectiveDimensionLE N n\nM : ModuleCat R\nN : Modul... | let eker : S.X₁ ≃ₛₗ[RingHomClass.toRingHom e] S'.X₁ :=
(LinearEquiv.ofEq _ _ this).trans (e2.symm.submoduleMap S'.g.hom.ker).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 105,
"column": 4
} | {
"line": 110,
"column": 19
} | {
"line": 111,
"column": 2
} | [
{
"pp": "J : Type v₁\ninst✝¹ : SmallCategory J\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF : J ⥤ C\nc₁ c₂ : Cone F\nX✝ Y✝ : Bicone J\nf : X✝ ⟶ Y✝\n⊢ (Bicone.casesOn X✝ c₁.pt c₂.pt fun j ↦ F.obj j) ⟶ Bicone.casesOn Y✝ c₁.pt c₂.pt fun j ↦ F.obj j",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants"... | [] | rcases f with (_ | _ | _ | _ | f)
· exact 𝟙 _
· exact 𝟙 _
· exact c₁.π.app _
· exact c₂.π.app _
· exact F.map f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 105,
"column": 4
} | {
"line": 110,
"column": 19
} | {
"line": 111,
"column": 2
} | [
{
"pp": "J : Type v₁\ninst✝¹ : SmallCategory J\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF : J ⥤ C\nc₁ c₂ : Cone F\nX✝ Y✝ : Bicone J\nf : X✝ ⟶ Y✝\n⊢ (Bicone.casesOn X✝ c₁.pt c₂.pt fun j ↦ F.obj j) ⟶ Bicone.casesOn Y✝ c₁.pt c₂.pt fun j ↦ F.obj j",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants"... | [] | rcases f with (_ | _ | _ | _ | f)
· exact 𝟙 _
· exact 𝟙 _
· exact c₁.π.app _
· exact c₂.π.app _
· exact F.map f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 79
} | {
"line": 182,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows\nM : Sieve X\nhM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)\nN : Sieve X\nhN : N ∈ (Functor.closedSie... | [
"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows\nM : Sieve X\nhM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)\nN : Sieve X\nhN : N ∈ (Functor.closedSieves J₁).obj ... | rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 125,
"column": 14
} | {
"line": 125,
"column": 21
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case left\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nk : Bicone J\n⊢ Fintype (Bicone.left ⟶ k)",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.diagram",
"CategoryTheory.CategoryStruct.toQui... | [
"case left.left\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\n⊢ Fintype (Bicone.left ⟶ Bicone.left)",
"case left.right\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\n⊢ Fintype (Bicone.left ⟶ Bicone.right)",
"case left.diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCa... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 125,
"column": 14
} | {
"line": 125,
"column": 21
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case right\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nk : Bicone J\n⊢ Fintype (Bicone.right ⟶ k)",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.diagram",
"CategoryTheory.CategoryStruct.to... | [
"case right.left\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\n⊢ Fintype (Bicone.right ⟶ Bicone.left)",
"case right.right\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\n⊢ Fintype (Bicone.right ⟶ Bicone.right)",
"case right.diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : ... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 125,
"column": 14
} | {
"line": 125,
"column": 21
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nk : Bicone J\nval✝ : J\n⊢ Fintype (Bicone.diagram val✝ ⟶ k)",
"ppTerm": "?diagram",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.right",
"CategoryTheory.Bicone.diagram",
"CategoryThe... | [
"case diagram.left\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝ : J\n⊢ Fintype (Bicone.diagram val✝ ⟶ Bicone.left)",
"case diagram.right\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝ : J\n⊢ Fintype (Bicone.diagram val✝ ⟶ Bicone.right)",
"case diagram.diagram\nJ : Typ... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.Limits.Bicones | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 45
} | {
"line": 135,
"column": 2
} | [
{
"pp": "case left.diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nval✝ : J\n⊢ Fintype (Bicone.left ⟶ Bicone.diagram val✝)",
"ppTerm": "?left.diagram",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicone.diagram",
"CategoryTheory.CategoryStruct.toQuiver",
... | [
"case right.left\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\n⊢ Fintype (Bicone.right ⟶ Bicone.left)",
"case right.right\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\n⊢ Fintype (Bicone.right ⟶ Bicone.right)",
"case right.diagram\nJ : Type v₁\ninst✝¹ : SmallCategory J\ninst✝ : ... | · exact
{ elems := {BiconeHom.left _}
complete := fun f => by cases f; simp } | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 283,
"column": 58
} | {
"line": 283,
"column": 62
} | {
"line": 283,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX : C\nS : Sieve X\nhS : (c X) S = ⊤\nR : Sieve X\nhR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nJ₁ J₂ : GrothendieckTopology C\nc : (X : C) → ClosureOperator (Sieve X)\nhc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)\nX : C\nS : Sieve X\nhS : (c X) S = ⊤\nR : Sieve X\nhR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pull... | ← hS | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 123,
"column": 15
} | {
"line": 123,
"column": 17
} | {
"line": 123,
"column": 18
} | [
{
"pp": "case refine_1\nC : 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 : ... | [
"case refine_1\nC : 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... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 12
} | {
"line": 133,
"column": 4
} | [
{
"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 :... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 10
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case refine_2\nC : 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 : ... | [
"case refine_2\nC : 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... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 14
} | {
"line": 164,
"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 ∈... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 187,
"column": 6
} | {
"line": 192,
"column": 13
} | {
"line": 193,
"column": 6
} | [
{
"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... | [
"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.arrows f → ... | replace hleT (i : E.I₀) : ∃ (F : J.ZeroHypercover (E.X i)),
F.presieve₀ ≤ (Sieve.pullback (E.f i) T).arrows := by
obtain ⟨R', hR', hle'⟩ := hleT (hle _ _ ⟨i⟩)
rw [mem_iff_exists_zeroHypercover] at hR'
obtain ⟨F, rfl⟩ := hR'
use F | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.CategoryTheory.Sites.Continuous | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 18
} | {
"line": 251,
"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\nG : Sheaf K (Type t)\n⊢ Presheaf.IsSheaf K G.obj",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
... | [] | exact G.property | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.CoverPreserving | {
"line": 168,
"column": 15
} | {
"line": 168,
"column": 17
} | {
"line": 168,
"column": 18
} | [
{
"pp": "case hunique\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nhF₁ : CompatiblePreserving K F\nhF₂ : CoverPreserving J K F\nG : Sheaf K (Type (max u₁ v₁ u₂ v₂))\nX : C\nS : Sieve X\nhS : S ∈ J X\nx : ... | [
"case hunique\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nhF₁ : CompatiblePreserving K F\nhF₂ : CoverPreserving J K F\nG : Sheaf K (Type (max u₁ v₁ u₂ v₂))\nX : C\nS : Sieve X\nhS : S ∈ J X\nx : FamilyOfElem... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 10
} | {
"line": 500,
"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\nG : C ⥤ D\ninst✝² : G.IsCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nℱ : Sheaf K (Type u_5)\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.obj) T\n... | [
"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\nG : C ⥤ D\ninst✝² : G.IsCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nℱ : Sheaf K (Type u_5)\nZ : C\nT : Presieve Z\nx : FamilyOfElements (G.op ⋙ ℱ.obj) T\nhx : x.Compa... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 23
} | {
"line": 147,
"column": 4
} | [
{
"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\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝³ : Category.{u, v} A\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nX : C\nY : Ov... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝³ : Category.{u, v} A\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.IsLocallyFull K\ninst✝ : G.IsLocallyFaithful K\nX : C\nY : Over X\nT : ↑(... | convert! T.property | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 10
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX Z : C\nf : Z ⟶ X\ni : I\nφ : Z ⟶ Y i\n⊢ x.familyOfElements X f ⋯ = (ConcreteCategory.hom (F.map φ.op)) (x i)",
"ppTerm": "?m.37",
"assigned": true,
"us... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 10
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX Z : C\nf : Z ⟶ X\ni : I\nφ : Z ⟶ Y i\n⊢ x.familyOfElements X f ⋯ = (ConcreteCategory.hom (F.map φ.op)) (x i)",
"ppTerm": "?m.37",
"assigned": true,
"us... | [] | apply hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 10
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX Z : C\nf : Z ⟶ X\ni : I\nφ : Z ⟶ Y i\n⊢ x.familyOfElements X f ⋯ = (ConcreteCategory.hom (F.map φ.op)) (x i)",
"ppTerm": "?m.37",
"assigned": true,
"us... | [] | apply hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 150,
"column": 13
} | {
"line": 150,
"column": 15
} | {
"line": 150,
"column": 16
} | [
{
"pp": "case hunique\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nhY : J.CoversTop Y\nhF : IsSheaf J F\nH : Presieve.IsSheaf J F\ny₁ : ↑F.sections\n⊢ ∀ (y₂ : ↑F.sections),\n (∀ (i : I), ... | [
"case hunique\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nhY : J.CoversTop Y\nhF : IsSheaf J F\nH : Presieve.IsSheaf J F\ny₁ y₂ : ↑F.sections\n⊢ (∀ (i : I), ↑y₁ (Opposite.op (Y i)) = x i) → (∀ ... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.Over | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 63
} | {
"line": 157,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\nR : Presieve Y\nZ : C\nW : Over X\nu : W ⟶ Y\nv : Z ⟶ (Over.forget X).obj W\nhu : R u\n⊢ ((overEquiv Y) (generate R)).arrows (v ≫ (Over.forget X).map u)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | exact (overEquiv_iff _ _).mpr ⟨W, Over.homMk v, u, hu, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 18
} | {
"line": 72,
"column": 0
} | [
{
"pp": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝³ : HasWeakSheafify J AddCommGrpCat\ninst✝² : J.WEqualsLocallyBijective AddCommGrpCat\ninst✝¹ : J.HasSheafCompose (forget₂ RingCat AddCommGrpCat)\nM N P : SheafOfModules R\nσ : M.GeneratingSections\np : M ⟶... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Over | {
"line": 472,
"column": 2
} | {
"line": 478,
"column": 11
} | {
"line": 480,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX Y Z T : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Z ⟶ T\n⊢ (J.overMapPullbackComp A f (g ≫ h)).inv ≫\n Functor.whiskerRight (J.overMapPullbackComp A g h).inv (J.overMapPullback A f) ≫\n ((J.ov... | [] | ext
dsimp
simp only [overMapPullbackComp_inv_app_hom_app,
overMapPullbackComp_hom_app_hom_app, Functor.sheafPushforwardContinuous_obj_obj_map,
Quiver.Hom.unop_op, ← Functor.map_comp, ← op_comp, id_comp, assoc]
congr
cat_disch | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Over | {
"line": 472,
"column": 2
} | {
"line": 478,
"column": 11
} | {
"line": 480,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX Y Z T : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Z ⟶ T\n⊢ (J.overMapPullbackComp A f (g ≫ h)).inv ≫\n Functor.whiskerRight (J.overMapPullbackComp A g h).inv (J.overMapPullback A f) ≫\n ((J.ov... | [] | ext
dsimp
simp only [overMapPullbackComp_inv_app_hom_app,
overMapPullbackComp_hom_app_hom_app, Functor.sheafPushforwardContinuous_obj_obj_map,
Quiver.Hom.unop_op, ← Functor.map_comp, ← op_comp, id_comp, assoc]
congr
cat_disch | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Over | {
"line": 545,
"column": 4
} | {
"line": 545,
"column": 40
} | {
"line": 547,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\nK : Precoverage C\ninst✝¹ : K.HasPullbacks\ninst✝ : K.IsStableUnderBaseChange\nX : C\nY : Over X\nR : Presieve Y\nhR : Presieve.map (Over.forget X) R ∈ K.coverings ((Over.forget X).obj Y)\n⊢ Sieve.generate (Presieve.map (Over.forget X) R) ∈ K.toGro... | [] | exact Precoverage.Saturate.of _ _ hR | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Simple | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 16
} | {
"line": 98,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\nthis : IsIso (image.ι f)\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 357,
"column": 4
} | {
"line": 360,
"column": 47
} | {
"line": 362,
"column": 0
} | [
{
"pp": "case a\nX Y : TopCat\nf : X ⟶ Y\nhf : IsInducing ⇑(ConcreteCategory.hom f)\nU : Opens ↑X\n⊢ U ≤ (Opens.map f).obj (hf.functorObj U)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Topology.IsInducing.isOpen_iff",
"TopologicalSpace.Opens.instCompleteLatt... | [] | intro x hx
obtain ⟨U, hU⟩ := U
obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU
exact Opens.mem_sSup.mpr ⟨⟨_, ht⟩, rfl, hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.Opens | {
"line": 357,
"column": 4
} | {
"line": 360,
"column": 47
} | {
"line": 362,
"column": 0
} | [
{
"pp": "case a\nX Y : TopCat\nf : X ⟶ Y\nhf : IsInducing ⇑(ConcreteCategory.hom f)\nU : Opens ↑X\n⊢ U ≤ (Opens.map f).obj (hf.functorObj U)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Topology.IsInducing.isOpen_iff",
"TopologicalSpace.Opens.instCompleteLatt... | [] | intro x hx
obtain ⟨U, hU⟩ := U
obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU
exact Opens.mem_sSup.mpr ⟨⟨_, ht⟩, rfl, hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 122,
"column": 37
} | {
"line": 212,
"column": 47
} | {
"line": 212,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nι : Type u_2\nU : ι → Opens ↑X\nV : OpensLeCover U\nA B : StructuredArrow V (pairwiseToOpensLeCover U)\n⊢ ∃ l, List.IsChain Zag (A :: l) ∧ (A :: l).getLast ⋯ = B",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"List.ge... | [] | by
rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩ <;> rcases B with ⟨⟨⟨⟩⟩, ⟨i'⟩ | ⟨i', j'⟩, b⟩
· refine
⟨[{ left := ⟨⟨⟩⟩
right := pair i i'
hom := ObjectProperty.homMk (homOfLE
(by simpa using le_inf a.hom.le b.hom.le)) }, _], ?_, rfl⟩
exact
Lis... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 347,
"column": 4
} | {
"line": 348,
"column": 33
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X := ⋯\nx✝ : ↑X\n⊢ (∃ i, x✝ ∈ ↑(WalkingPair.casesOn i.down U V)) → x✝ ∈ ↑(U ⊔ V)",
"ppTerm"... | [] | · rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or.inr h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 15
} | {
"line": 362,
"column": 2
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V\nhι : U ⊔ V = iSup ι\ni j : CategoryTheory.Pai... | [
"case refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V\nhι : U ⊔ V = iSup ι\ni j : CategoryTheory.Pairwise (ULift... | clear_value g | Lean.Elab.Tactic.evalClearValue | Lean.Parser.Tactic.clearValue |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 399,
"column": 4
} | {
"line": 400,
"column": 33
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X := ⋯\nx✝ : ↑X\n⊢ (∃ i,\n x✝ ∈\n ↑(match i with\n | { down := j } => Walk... | [] | · rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or.inr h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.Semigrp.Basic | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 44
} | {
"line": 439,
"column": 4
} | [
{
"pp": "X✝ Y✝ : Type u\nX Y : MagmaCat\nf : X ⟶ Y\nx✝ : IsIso ((forget MagmaCat).map f)\n⊢ IsIso f",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MulHom",
"CategoryTheory.Iso",
"MagmaCat.instCategory",
"CategoryTheory.Functor.map",
"MulHom.funLike",
"... | [
"X✝ Y✝ : Type u\nX Y : MagmaCat\nf : X ⟶ Y\nx✝ : IsIso ((forget MagmaCat).map f)\ni : (forget MagmaCat).obj X ≅ (forget MagmaCat).obj Y := asIso ((forget MagmaCat).map f)\n⊢ IsIso f"
] | let i := asIso ((forget MagmaCat).map f) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
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